diff --git "a/2NAzT4oBgHgl3EQfRvtg/content/tmp_files/load_file.txt" "b/2NAzT4oBgHgl3EQfRvtg/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2NAzT4oBgHgl3EQfRvtg/content/tmp_files/load_file.txt" @@ -0,0 +1,1616 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf,len=1615 +page_content='Caustic spin wave beams in soft, thin films: properties and classification Alexis Wartelle,∗ Franz Vilsmeier, Takuya Taniguchi, and Christian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Back Fakult¨at fur Physik, Technische Universit¨at M¨unchen, Garching, Germany (Dated: January 4, 2023) In the context of wave propagation, caustics are usually defined as the envelope of a finite-extent wavefront;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' folds and cusps in a caustic result in enhanced wave amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Here, we tackle a related phenomenon, namely the existence of well-defined beams originating solely from the geometric properties of the corresponding dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This directional emission, termed caustic beam, is enabled by a stationary group velocity direction, and has been observed first in the case of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We propose an overview of this “focusing” effect in the context of spin waves excited in soft, thin ferromagnetic films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Based on an analytical dispersion relation, we provide tools for a systematic survey of caustic spin wave beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Our theoretical approach is validated by time- resolved microscopy experiments using the magneto-optical Kerr effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Then, we identify two cases of particular interest both from fundamental and applicative perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Indeed, both of them enable broadband excitations (in terms of wave vectors) to result in narrowband beams of low divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' INTRODUCTION The collective motion of magnetic moments in a ma- terials, referred to as spin waves, has shown remarkable properties from a fundamental perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Examples range from anisotropic dispersion in thin films [1], rel- evant for the field of magnonics, to Bose-Einstein con- densation of magnons [2], through restricted-relativity- like bounded domain wall velocities [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Applications of magnetization dynamics also abound, starting with the infinite-wavelength ferromagnetic resonance (FMR) [4] and going all the way towards sub-micrometer wave- lengths, which are currently viewed as promising alterna- tive information carriers in the fields of magnonics [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In addition to the absence of Joule heating and the potential device downscaling (using small wavelengths), spin wave interference is an appealing prospect [6] as it allows logic operations through the design of the propagation lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Several experimental techniques are readily available for the study of spin waves [1], especially in the case of thin films or patterned elements thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Among them, micro-/phase-resolved Brillouin Light Scattering (BLS) [7], Time-Resolved Magneto-Optical Kerr Effect (TR-MOKE) microscopy [8], and time-resolved Scan- ning Transmission X-ray Microscopy (TR-STXM) with magnetic sensitivity through X-ray Magnetic Circular Dichroism (XMCD [9]) [10] have demonstrated outstand- ing imaging capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Nevertheless, the usually very small amplitudes of magnetization precession associated to spin waves as well as their attenuation lengths (typi- cally on the micrometer scale) pose a significant challenge both for fundamental investigations and for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To be of practical use, spin waves must be harnessed via a power-efficient strategy: some approaches like Win- ter’s magnons rely on channeling along domain walls [11], ∗ alexis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='wartelle@ens-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='org;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Present address: Universit´e Greno- ble Alpes, CNRS, Grenoble INP, SIMaP, 38000 Grenoble, France others rely on careful control of spin wave scattering [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Another possibility would take advantage of caustic spin wave beams (CSWBs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' spin wave beams of well- defined propagation direction, narrow angular width and higher power compared to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Damon-Eshbach-type [13] spin waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Furthermore, caustics in soft, thin ferro- magnetic films can be very different from the well-known acoustical or optical caustics, which originate from inho- mogeneous media [14–16], : here, spin wave caustics can arise in perfectly homogeneous films in broad ranges of conditions solely because of sufficient anisotropies in their dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The latter indeed allows the direc- tion of the group velocity to be stationary around some wave vectors, leading to well-defined directions of wave propagation associated to significantly stronger emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In the context of phonon propagation, such phenomena have been referred to as “focussing” [17], and they have been observed and investigated since 1969 [17–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' By contrast, caustics in ferromagnetic films were re- ported for the first time ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 30 years later [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' There has been quite a few reports since then [23–31] but, to the best of our knowledge, there exists to date no sys- tematic survey of the properties of spin wave caustics, not even focusing on a certain type of systems e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' ul- trathin films with perpendicular anisotropy, or soft thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In this work, we restrict ourselves to the latter and give an overview of caustics in soft thin films, as well as tools to further investigate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Moreover, we high- light two special cases which seem particularly appealing notably for application in magnonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' General considerations Our starting point is the model derived by Kalinikos and Slavin [32] for spin waves in soft ferromagnetic thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' These excitations correspond to a time- and space-dependent magnetization −→ M(⃗r, t), yet its norm arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='01220v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='mes-hall] 3 Jan 2023 2 Ms = ||−→ M(⃗r, t)|| the spontaneous magnetization is uni- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As a result, it is simpler to consider the re- duced magnetization −→ m(⃗r, t) = −→ M(⃗r, t)/Ms with norm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We focus on the linear regime i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' the deviation δ−→ m(⃗r, t) = −→ m(⃗r, t) − −→ m0(⃗r, t) from the equilibrium mag- netization (when no excitation is applied) −→ m0 is such that ||δ−→ m|| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Under the assumption of negligible mode mixing and of a perfectly isotropic ferromagnetic mate- rial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' one may write the dispersion relation of a thin film as: ω2 = � γ0Ha + 2Aγ0 µ0Ms k2�� γ0 � Ms + Ha � + 2Aγ0 µ0Ms k2� −γ2 0M 2 s · ξ(kd) � 1 − ξ(kd) + Ha Ms + 2Aγ0 µ0M 2s k2� cos2 ϕ +γ2 0M 2 s · ξ(kd) · [1 − ξ(kd)] (1) where ω is the spin wave angular frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' γ0 = µ0|γ| with γ = qe/(2me) the electron’s gyromagnetic ratio (qe = −e and me being the electron’s charge and mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' respectively) and µ0 the permeability of vacuum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A is the micromagnetic exchange constant for the soft ferromag- netic material of interest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Ms its spontaneous magnetiza- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' k the spin wave’s wavenumber corresponding to its wave vector −→k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Ha = ||−→ Ha|| the strength of the externally applied magnetic field −→ Ha from which ϕ = angle �−→ Ha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' −→k � the wavefront angle is defined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' d the film thickness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' and ξ is the function whose values are defined as: ξ(u) = 1 − 1 − e−u |u| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' (2) As a consequence of the ferromagnetic material’s soft- ness, in the absence of excitation, the equilibrium mag- netization configuration in our thin film is the single- domain state, with a corresponding reduced magnetiza- tion −→ m0 exactly along the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The orientations of −�� m0, −→ Ha, and −→k are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 1, which also highlights the natural wavelength λ0 = 2π/||−→k || of the spin wave as well as the unit vectors −→ ex, −→ ey and −→ ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Here, we focus on spin waves with no amplitude node across the film thickness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' we do not consider per- pendicular standing spin waves (PSSWs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' However, we do note that the latter may play a role in experiments performed on sufficiently thick films where a realistic an- tenna for instance could excite them due to its inhomo- geneous magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We introduce the following quantities: the Larmor angular frequencies associated to magnetization ωM = γ0Ms and to the applied magnetic field ωH = γ0Ha, the material’s dipolar-exchange length lex = � 2A/(µ0M 2s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We then rewrite the equation as: − → ex − → k −→ Ha −→ m0 φ λ0= 2π ||− → k|| = 2π k 0 δmz − → ez − → ey FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Schematic representation of a spin plane wave prop- agating in a soft thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The grey scale codes the local perpendicular component of the dynamic component of mag- netization, δmz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' ω2 ω2 M = � ωH ωM + l2 exk2 � � 1 + ωH ωM + l2 exk2 � −ξ(kd) � 1 − ξ(kd) + ωH ωM + l2 exk2� cos2 ϕ +ξ(kd) � 1 − ξ(kd) � (3) Introducing the reduced frequency ν = ω/ωM and ap- plied field h = ωH/ωM = Ha/Ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' and normalizing both the dipolar-exchange length and wavenumber to the film thickness d using η = lex/d and ˜k = kd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' we arrive at: ν2 = � h + η2˜k2�� 1 + h + η2˜k2� −ξ(˜k) � 1 − ξ(˜k) + h + η2˜k2� cos2 ϕ +ξ(˜k) � 1 − ξ(˜k) � (4) With this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' it is clear that any given experiment of spin wave excitation corresponds to a specific value of the di- mensionless triplet (η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In other words: they are the only independent parameters within this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' For a value of (η, ν, h), the solution to (4) is the pos- sibly empty set of accessible dimensionless wave vectors −→k d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The existence and properties of spin wave caus- tics depend on the geometrical characteristics of this set, which is why we are first going to review several of its general properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Keeping in mind that we focus on applied fields below the ferromagnetic resonance field at the excitation fre- quency, we actually always have a non-empty solution, which is usually a closed curve winding around the ori- gin in wave-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This is the so-called slowness curve, in reference to the fact that at fixed frequency k ∝ 1/||−→ vp|| where −→ vp is the phase velocity [33], oriented of course along the wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Considering the parity of the cosine function and its antisymmetry for the re- flection ϕ → π − ϕ, we may restrict our analysis to only the quadrant ϕ ∈ [0, π/2] and deduce the others using mirror symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 3 One can also parametrize the slowness curve using a curvilinear abscissa: we define it to be zero for the lowest dimensionless wavenumber ˜kmin at ϕ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' One can in- deed show that the reduced wavenumber solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' (4) at ϕ = π/2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 0) is minimum (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' maximum) on the quadrant ϕ ∈ [0, π/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Thus, at the largest dimen- sionless wavenumber ˜kmax = ˜k(ϕ = 0), the correspond- ing curvilinear abscissa sM corresponds to the length of the slowness curve in the quadrant ϕ ∈ [0, π/2] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' one fourth of the whole length of this curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Another important geometrical aspect of the slowness curve that is central to the present work is the local normal to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Considering its definition as a constant- frequency intercept of the dispersion relation in wave- vector space, by nature, the frequency gradient −→ ∇− → k ω is perpendicular to the slowness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As a result, the di- rection of the group velocity of spin waves −→ vg = −→ ∇− → k ω can be directly read from the direction of the local normal to the slowness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In our notations, we point out that: −→ ∇− → k ω ≡ � β=x,y,z ∂ω ∂kβ −→ eβ where kβ = −→k · −→ eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In the following, we will use the angle θV = angle(−→ Ha, −→ vg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We point out that in the present case, phase and group velocities need not be collinear: on the contrary, there can be differences between θV and ϕ much larger than in cases of light propagation through anisotropic media [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 2 illustrates this on the ex- ample of a slowness curve reconstructed for a vanishing reduced applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Distinctive features of dispersion relation caustics Typically, caustics in inhomogeneous media occur when a wavefront folds onto itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' in this situation, there exists a surface (or a line in 2D wave propagation) such that across it the number of rays passing through a point in space changes by an even number [15, 16]: this is the caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Equivalently, it can be viewed as the set of the local extrema of positions on the ray bundle on the wave- front, for all the wavefronts along the wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' It is this extremal nature that grants these caustics large and localized intensities compared to other points on the ray bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In a geometrical optics approach, the in- tensity diverges as an initially finite-sized portion of the wavefront shrinks to a vanishing area [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A wave op- tics treatment however reveals that the intensity remains finite due to interferences: illumination profiles across caustics can in principle be determined by taking into account the variations of phase as a function of distance to the caustic [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Such an approach has been used by Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [26] for spin wave caustics excited by the scattering of a spin wave travelling in a waveguide terminating into a θV φ0 − → k0d a) b) c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a) Exemplary slowness curve for (ν, h, η) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2873, 10−20, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As can be clearly seen in the polar plot of kd = ˜k(ϕ), the direction (ϕ0 =32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00°) of the phase veloc- ity −→ vp and that (θV =108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9°) of the group velocity −→ vg at the point −→ k0d are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b) Radiation pattern (δmz is grey-coded) of a hypothetical source exciting only wavenum- bers very close to ||−→ k0||d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' c) Plane wave corresponding to the carrier wave vector −→ k0d (red lines are guides to the eye).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' full permalloy (Ni80Fe20) film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' However, this is a very different situation compared to the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Indeed, the wavefront does not fold onto itself due to spatial varia- tions of medium properties, rather, its extent is deter- mined almost exclusively (owing to the sub-wavelength source size) by the characteristics of spin wave propaga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The latter are determined by the anisotropic spin wave dispersion relation, which allows caustics to form thanks to the possibility of stationary group velocity di- rection i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a beam with a well-defined propagation direc- tion yet comprising a range of wave vectors in the vicinity of a carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' More precisely, caustics correspond to local extrema of the group velocity direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' in other words, a caustic spin wave beam implies the existence of a caustic point ˜kc on the slowness curve such that: dθV d˜k ����˜kc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' (5) The CSWB has then a carrier wavenumber ˜kc, corre- sponding to a central wavefront angle ϕc = ϕ(˜kc) and a 90 75° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='09 45° 30° Ug 15° kd 0° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 kd4 beam direction θV,c = θV(˜kc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Coming back to the wavefront extent, rays from wave vectors not close enough to the carrier cannot play a role in the caustic wave amplitude simply because of differ- ences in propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' More specifically, the experimental data presented by Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' sug- gests that beam divergences of 2° or less can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Thus, there seems to be a contradiction between the cu- bic dispersion which is assumed to define the beam pro- file and the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The question of the CSWB’s profile goes however beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Nev- ertheless, it is clear from the low beam divergences ob- served in many experimental reports [23, 27, 35] that only small, almost straight parts of the slowness curve must contribute to CSWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In fact, integrating the contribution of wave vectors all the way to infinity as done in [26] neglects the geometric impossibility for them to create waves travelling from the point source to a far-away point on the caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To put it differently: for geometrical reasons, caustics originating solely from anisotropies in the dispersion relation and excited by a point-like source naturally restrict the range of relevant wave vectors, in contrast to the case of caustics due to inhomogeneities in the propagation medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We wish to emphasize the above by reminding that in most cases [15, 36], caustics are treated on the basis of wave propagation in an isotropic or weakly anisotropic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' One consequence is the fact that the flow of power, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' the group velocity, is along the wave vector or close to parallel to it [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' While this remains a rea- sonable approximation for slightly anisotropic media (as in usual crystal optics), in the case of perfectly soft but fully polarized thin ferromagnetic films this collinearity may break down dramatically, as was illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Therefore, even small changes in wave vector may re- sult in drastic changes in group velocity direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' By contrast, large changes in wave vectors do not necessar- ily lead to strong variations in the apparent wavelength λ which we define as: λ = 2π · ||−→ vg|| −→k · −→ vg = 2π −→k · −→ eg = λ0(ϕ) cos (θV − ϕ), (6) where we have introduced −→ eg as a unit vector along the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The apparent wavelength is simply the spatial period measured along the beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Since large differences θV − ϕ can easily be obtained (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 2, where cos (θV − ϕ0) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='227), and more importantly since the projection ˜k(ϕ) cos (θV − ϕ) may remain almost constant over significant portions of the slowness curve, one should consider notions such as propagation-induced phase or spectral breadth [37] of a spin wave beam care- fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Limit of model applicability: thick films We start by providing an example of situation where the model we use cannot be fully trusted, so as to high- light its limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 3 we show a case where the reconstructed slowness curve splits into two separate con- nected components above a certain threshold frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' ˜k FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Slowness curves for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='015, h = 10−20, and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='331 (dashed blue line) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='333 (full red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Such a behaviour has been described by Kreisel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [38]: the model chosen for spin wave dispersion predicts a local maximum in the ω(k, ϕ = π/2) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' wavenumber curve, but this extremum is not reproduced by a formal approach not based on the thin-film approximation [39], and designed to tackle the dipole-exchange regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The maximum’s presence leads to an additional pair of so- lutions in terms of wavenumber in a certain frequency range, corresponding to a splitting of the slowness curve into two separate components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Clearly, results obtained within our approach about caustics deep in the dipole-exchange regime are not trust- worthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Empirically, we see the slowness curve splitting into separate components for values of η up to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='075;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' for the sake of comparison, the thinnest films investigated by Kreisel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' feature η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='035 according to literature data on yttrium iron garnet (YIG) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Nevertheless, the absence of this splitting is no proof that the reconstructed slowness curve is accurate, and we shall remain cautious in discussing results concerning CSWBs with wavenum- bers in the dipole-exchange regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Finally, we note that promising theoretical developments such as the dipole- exchange dispersion relations recently derived by Harms and Duine [39] could eventually allow a more accurate treatment of caustics in the dipole-exchange regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 90° 75° 60° 45° v =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='333 v =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='331 30° 15° 5 10 15 20 25 30 35 0 kd5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' General features Let us have a look at a first example of frequency and field map of caustic properties in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In the presented graphs, the red color means that either the corresponding (h, ν) point was not investigated because its reduced field is above the reduced FMR field hFMR, or because no caustic points were found there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' First of all, one can see that there is indeed a portion of the (h, ν) plane where no caustic points exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This occurs for frequencies above a certain νm(h, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Then, going down in reduced frequency, there appears to be an oblique boundary between two regions of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Above it, ˜kc quickly enters the dipole-exchange regime, which we will only present but not discuss quantitatively as it cor- responds to a situation where our model is less reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Below the boundary, the reduced caustic wavenumber is much smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Correspondingly, a boundary which we will label νb(h, η) appears at the same position on the plot of ϕc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' this angle also seems close to constant over much of the region below the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In both cases, its sharpness decreases towards low h, and at vanishing reduced field the transitions in ˜kc or ϕc are both smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' All these features are represented on a simplified repre- sentation of the map of ˜kc shown as inset on the ϕc map, including the point (hc, νc) at which the sharp boundary seems to end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A zoomed-in view on (hc,νc) is shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In the following, we will refer to the lowest reduced field at which this boundary is sharp as hc and denote νc = νb(hc, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As we shall see in more details, this abrupt boundary corresponds to a change in the num- ber of caustic points by two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The lowest point (hc, νc) is actually a cusp in the domain of existence of the two additional caustic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We point out that for all re- duced fields and frequencies, the maps shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 4 displays the lowest caustic wavenumber respectively the associated wavefront angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Before moving on to discussing the low-frequency pocket, its boundary and the existence of additional caus- tic points, and finally the threshold frequency for the ab- sence of caustic points, we stress that the behaviour of caustics strongly depends on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As an example, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 5 field and frequency maps for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='09, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' At the lowest value, the boundary νb extends all the way to h = 0, whereas the two other maps do not display such a sharp behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In addition to the expected changes in range of values for ˜kc, one can see that the overall shape of the domain of existence of CSWBs also changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' From here on, we will call this area D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' From η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='09 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3, we see that D has expanded in the vertical direction at low h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In even thinner films, for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6, the average slope of νm(h, η) has not changed much, yet νm(0, η) has decreased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' as a result, D shrinks vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' By contrast, even if the caustic group velocity direction displays a similar wealth of features as the caustic wave- front angle and reduced wavenumber, the jumps across the boundary νb are much less significant when they ex- ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' An example of this is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 6, which shows maps for θV,c at the same values of η as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In a certain range of reduced dipolar-exchange length, we find that there may actually be more than one caustic point on the slowness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Empirically, we observe that the additional caustic points may exist for ˜kc < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' When this inequality holds, the number of caustic points is ei- ther equal to one or to three;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' two being possible but only on a 1D curve in the field and frequency plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' this curve includes the aforementioned boundary νb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Qualitatively, this is due to the fact that in the corresponding range of field and frequency, when dθV/d˜k crosses 0, it does so with a local behaviour somewhat reminiscent of a poly- nomial of the type P(˜k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a, b) = (˜k − ˜kc)3 +a·(˜k − ˜kc)+b, where a and b are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' If a > 0, there ex- ists only one root, whereas if a < 0 and |b| is sufficiently small, there exists three distinct roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The domain in the field and frequency plane with these three roots will be referred to as D3 from now on, by contrast with D1 = D \\ D3 in which there is only one caustic point instead of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We will now describe D3 using the P(˜k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a, b) approximant to dθV/d˜k for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Let us start with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 7, which displays the same field and frequency map for ˜kc as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 4 along with the maps for the two other reduced caustic wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The two additional solutions can be shown to coincide on the rounded boundary of D3 to the lower left, which will be referred to as ∂D3,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Entering D3 through this bound- ary by increasing ν corresponds to the situation where |b| becomes small enough to allow the two additional caustic points (with respect to the one with lowest ˜kc), thanks to a being negative enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Increasing h on the other hand mostly decreases a: upon crossing ∂D3,l, a pair of caus- tic points with higher ˜kc’s appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Of course, exactly on ∂D3,l the two additional roots of dθ/d˜k are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Starting from inside D3, if one increases the reduced frequency, eventually the caustic point with the interme- diate value of ˜kc merges with the one featuring the small- est reduced wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This happens on the other boundary of D3, which we will call ∂D3,u from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This situation corresponds to ν = νb(h, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Just above this boundary, the value of b is low enough so that only one root of dθ/d˜k remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' That is the reason for the dis- continuity in ˜kc in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 4: the lowest caustic wavenumber jumps to what was the highest of the three ˜kc’s below νb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Experimentally, this could imply that SW excitation around this threshold wavenumber would have marked changes in intensity as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Based on the above, since the two boundaries other than ferromagnetic resonance each imply that a differ- ent pair of caustic points coincide, we can infer that on the cusped intersection of ∂D3,l and ∂D3,u, there exists a single caustic point corresponding to three of them coin- ciding on the slowness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This is precisely the point (hc, νc) from the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' It is important to note that while a purely math- 6 φc (◦) ˜kc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='98 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='40 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='60 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='66 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='19 0 h 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='48 ν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 Low- frequency pocket νm(h, η) h ν 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 νb(h, η) νc hc a) b) h 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Frequency and field maps for a value of η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' For high enough fields, a sharp upturn in both properties can be seen for reduced frequencies above ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We remind the reader that fields above ferromagnetic resonance are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Only few level curves are displayed for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a) Caustic wavefront angle ϕc, with a schematic representation of the map’s distinctive features as inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b) Normalized wavenumber ˜kc = kcd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a zoomed-in view on the area where the upturn’s sharpness drastically changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 ˜kc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='19 0 ˜kc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='69 0 ˜kc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='611 0 a) b) c) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='09 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Examples of field and frequency maps for a) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='09, b) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3, and c) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' only the reduced caustic wavenumber is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Caustic wavefront angles Φc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9000 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4001 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1547 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6190 hTNormalized wavenumber at Φc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='662 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='000 h000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1900 7Normalized wavenumber at Φc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9991 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7993 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6994 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4996 kc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6189 hNormalized wavenumber at Φc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4996 kc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6189 hNormalized wavenumber at Φc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9991 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7993 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5995 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4996 kc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2998 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6189 h7 ematical analysis yields well-defined, separate caustic points, experimentally the distinction between close caus- tic points may well be impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In fact, there ex- ists no straightforward experimental signature of dθV/d˜k crossing 0, and portions of the slowness curve where this derivative is small but non-zero can behave similarly to an actual caustic point, as was noted by Gallardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Nevertheless, the presence of more than one caustic point constrains a slowness curve to be almost straight in their vicinities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' this should then favour marked caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Low-frequency pocket The low-frequency regime is important as it corre- sponds to a well established domain of validity of our theoretical model as well as wavelengths which can still be excited and detected reasonably easily in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Analytics As could be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5, the shape or even the existence of the low-frequency pocket strongly depends on the chosen value of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Nevertheless, we can in- vestigate the behaviour of caustics there by taking the limit ν → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In order to remain below ferromagnetic resonance, we also take the limit h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Assum- ing h = 0 simplifies the computation of the quantity tan θV = tan ϕ · [1 + f(˜k, ν, η)], where f is a function given in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We can then dif- ferentiate this with respect to ˜k, take the limit ν → 0 and Taylor-expand the derivative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' the details are provided in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Eventually, we find that: ˜kc(ν → 0) = 3ν2 + O(ν4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' (7) It was expected that the caustic wavenumber goes to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' we can furthermore show that the lowest reduced wavenumber on the slowness curve (still in zero applied field) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' the Damon-Eshbach wavenumber goes to zero as: ˜kmin(ν → 0) = 2ν2 + O(ν4) (8) which proves that CSWBs exist down to vanishing re- duced frequencies, regardless of their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In this limit, the associated caustic wavefront angle is such that: cos ϕc = 1 √ 3 + O(ν2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' (9) From the latter, we also get the CSWB direction θV,c: tan θV (˜kc, h → 0, ν → 0) = −2 √ 2 + O(ν2) (10) The strength of this result lies with its independence on η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' this is not surprising as in the limit we are considering, the CSWB’s wavelength diverges which means it must be much larger than both the film thickness d and the dipolar-exchange length lex, however large they may be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The numerical values for the limits of ϕc and θV,c are ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='74° and 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5°, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Comparison with literature We present in Table I a comparison between experi- mental reports on caustics and predictions we make for the same conditions, focusing on the CSWB direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Whenever there are three caustic points, the indicated predicted value for θV,c is the closest found across all three caustic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We find a reasonable agreement in quite a few cases, generally for the larger values of η (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' for thinner films) with the notable exception of the report by Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' However, in this case, the theoretical dispersion relation that we use may not be accurate any more due to the strong lateral confinement of spin waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Furthermore, we find much larger discrepancies in sev- eral cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' For instance, if we consider the excitation of a caustic-like beam by Gieniusz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [43] at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 GHz and under an induction of 98 mT in a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 µm thick YIG film, our model predicts a caustic point at reduced wavenum- ber 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2, with a beam direction 169°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' However, the rele- vant reduced wavenumbers in this experiment are in the range of a few percents [43], and the measured beam di- rection is 128°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The origin of this strong disagreement is easily understood by observing the derivative dθV/d˜k in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 8 reveals, there exists a local minimum at ˜k ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0659 for dθV/d˜k deep in the dipolar-dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Moreover, the associated group velocity direc- tion is 128°, and past the next local maximum, similar values of dθV/d˜k are reached again only for ˜k ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This illustrates the impossibility to distinguish a close- to-straight slowness curve from a true caustic point from measurements alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Discrepancies may also arise due to the source’s non- ideal excitation efficiency, for instance if it is too direc- tional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This is illustrated by the excitation of caustic- like spin wave beams by K¨orner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' One of the reported TR-MOKE measurements deals with a 60 nm thin permalloy film driven at an excitation frequency of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='08 GHz, under 160 mT applied induction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' the authors observe twin beams with a wavefront angle of 65°, a beam direction 114°, and a reduced wavenumber of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Yet, the expected caustic spin wave beams in these condi- tions should feature a reduced wavenumber of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7063, a beam direction 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62°, not to mention a wavefront angle of 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='27°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In this case, it appears that the excited spin waves simply correspond to the rather narrow portion of the slowness curve that could be excited by the authors’ tapered coplanar waveguide segments [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Indeed, at the measured wavefront angle of 65°, in the authors’ experi- 8 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Comparison between reports on CSWBs and our predictions for the beam direction θV,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Excitation method Material (thickness in nm) Predicted θV,c Measured θV,c h ν η [28] Edge modes of a waveg- uide and nonlinearities Co2Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4Si (30) 113° 123° 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='81·10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='15 [42] Corners of slotline termi- nation and scattering off a defect YIG (235) 123° 124°, 122° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='427 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='36·10−2 [35] Corners of slotline termination YIG (245) 119° 118° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='427 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='06·10−2 [43] Spin wave scattering off antidots YIG (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5·103) 169° 128° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='939 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='84·10−3 [27] Collapsing spin-wave bullet YIG (5·103) 137° 137° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='442 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='46·10−3 [22] Spin wave scattering off a defect YIG (7·103) 139° 135° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='47·10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='616 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='47·10−3 mental conditions, the expected reduced wavenumber is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='28 (which falls rather far from zeroes in the an- tenna’s expected excitation efficiency [46]), and the beam direction 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We do not have an explanation for the remaining deviation in beam direction, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Experimental results We now present results from experiments we have carried out in order to validate our theoretical ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Our aim here is to measure CSWBs and compare their properties with our predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In order to ac- cess CSWBs experimentally, the reciprocal-space Fourier components of its magnetic field must span a broad range of wave vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The ideal situation where all wave vec- tors are accessible corresponds to an unrealistic point source, which can obviously not correspond to any high- frequency antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As a result, we choose a compromise between ease of fabrication, and broad-band excitation efficiency, namely a half-ring shaped stripline antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This design allows for a spin wave excitation of the slow- ness curve within ϕ ∈ [0, π], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' twice the quadrant previously investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Of course, this excitation is not uniform because of the microwave antenna dimensions on the order of a micrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Our experiments were carried out using Time-Resolved Magneto-Optical Kerr Effect (TR-MOKE) microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Here, the dynamic out-of plane component of the mag- netization δmz is spatially mapped in the xy-plane at a fixed phase between the microwave excitation frequency and the laser probing pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This enables direct imag- ing of the spin wave propagation in the magnetic film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The wavenumber resolution of the set-up lies within the dipolar-dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Indeed, our spatial resolution r is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='29 µm (see Supplementary Materials), so that for a film thickness t ∼100 nm, the largest accessible re- duced wavenumbers are 2π/(2r) · t ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' It shall be noted that the position of the microwave an- tenna in the resulting Kerr images is extracted from the topography image which is acquired simultaneously and is proportional to the reflectivity of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Further information on TR-MOKE can be found in the Supple- mentary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' These experiments were performed on a 200 nm thick YIG film grown on a gadolinium gal- lium garnet (GGG) substrate using liquid phase epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Considering this materials’ parameters [40], if not stated otherwise, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='087 for all measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' On top of the YIG film the 2 µm to 3 µm wide microwave antenna was patterned by optical lithography with subsequent Ar- presputtering and electron-beam-induced evaporation of Cr(5 nm)/Au(100 nm to 220 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' During the measure- ment the external bias field −→ Ha was always kept fixed such that it aligned with the legs of the antenna structure along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A sketch of the measurement ge- ometry can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' At this stage, we point out one complication resulting from this design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' When driv- ing the antenna with a microwave field, the legs them- selves excite spin waves in the Damon-Eshbach geometry [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' These modes are not of interest for the generation of CSWBs, but due to the relatively long attenuation length in YIG [35] they may propagate to the tip of the antenna and interfere with the spin waves excited by the half-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In order to suppress this effect, two different approaches where applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Either the length of the an- tenna was set to 50 µm and the YIG between the legs and tip was etched away, or the antenna was patterned to be 1 mm long in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The first Kerr image shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='a) was ob- tained at a constant microwave frequency f =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='44 GHz and an external field µ0Ha =5 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This corresponds to h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='028, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The width of the waveguide was 2 µm and the distance between the legs and the tip was 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In the spatial map, two spin wave beams with well-defined propagation directions are visible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' moreover, the phase and group velocities are clearly non-collinear to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Here, beam II stems from the waveguide excitation in the quadrant ϕ ∈ [π/2, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The beam angles 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 a) b) c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='09 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 θV,c (◦) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='25 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='25 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 θV,c (◦) θV,c (◦) θV,c (◦) h h h FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Examples of field and frequency maps for the CSWB direction θV,c, at a) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='09, b) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3, and c) η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 ˜kc a) b) c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='555 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='826 ˜kc ˜kc h h h FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Field and frequency maps for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='12, looking at the three reduced caustic wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Note the distinct grey scales for each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a) Lowest ˜kc in the presence of several caustic points, and single value for ˜kc otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b) Intermediate value for ˜kc if several caustic points exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' c) Largest reduced caustic wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Caustic beam directions Avc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9991 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7993 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5995 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3997 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1999 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1000 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6189 hCaustic beam directions Oyc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9991 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8992 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6994 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4996 0vc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3997 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1999 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1000 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6189 hCaustic beam directions Oyc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9991 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8992 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6994 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4996 0vc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3997 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1999 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1000 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6189 hNormalized wavenumber at Φc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6190 hNormalized wavenumber at Φc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5000 kc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6190 hNormalized wavenumber at Φc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' v and h 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='9000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4000 kc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6190 h10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 ˜k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7 dθV/d˜k FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Calculated derivative of the group velocity direction with respect to the reduced wavenumber in the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='62 GHz spin wave excitation described by Gieniusz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' YIG film 2 µm − → k x y z −→ Ha FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Schematic of the measurement geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The half- ring shaped antenna excites spin wave propagation within a broad angular spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' of beams I and II with respect to the positive x direction are found to be 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00◦ (beam I) and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='28◦ (beam II) which results in effective beam directions of θI = 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00◦ and θII = 180◦ − 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='28◦ = 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='72◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The discrepancy between θI and θII simply originates from a small misalignment of the external field with respect to the waveguide legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Since −→ Ha is not fully parallel to the x-axis, the slowness curve is rotated by a small an- gle αH = (θI − θII)/2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='64◦ in our frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Keeping this in mind, we extract an average beam direc- tion θV,e = 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='36◦, a wavefront angle ϕe = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='66◦ and a reduced wavenumber ˜ke = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' These experimental findings are in good agreement with our theoretical ap- proach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' indeed, values of θV,c = 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='05◦, ϕc = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='29◦ and ˜kc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='223 are predicted for a CSWB in our experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We can obtain further insight in reciprocal space with 30 20 10 10 20 30 x (µm) y (µm) 0 0 ˜kx 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0 ˜ky |FT(δmz)|2 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=') δmz (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=') 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 a) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Measurement data obtained for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='087, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='028 and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a) Kerr image acquired from TR-MOKE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Two spin wave beams highlighted in yellow and red propa- gate from the tip of the antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b) Squared modulus of the Fourier transform (FT) of the Kerr image and expected slowness curve (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The yellow and red points and ar- rows indicate the expected caustic points and their respective group velocity directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Caustic points I and II correspond to beams I and II in the Kerr image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' the Fourier-transformed (FT) data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Generally speaking, the FT data allows for a direct ob- servation of the slowness curve in ˜k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In order to re- duce spectral leakage, a Hanning windowing was applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' the latter provides a good trade-off between frequency and amplitude accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We see that the chosen antenna structure indeed excites a wide range of wave vector di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The gaps in the spectrum arise from the finite antenna dimensions, as previously mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We find a good agreement between the slowness curve (blue curve) derived from our model (and corrected by the external field angle αH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' More importantly, this graph confirms that the antenna structure grants access to the expected caustic points (yellow and red points) since the Fourier magnitude is still sufficiently large in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To con- clude, caustic points I and II can be assigned to beams I and II from the Kerr image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We may now turn to the additional caustic points pre- dicted by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The chosen triplet (η, h, ν) is an ele- ment of the D3 set, and we would expect two further caus- tic points θV,c,2 = 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='74◦, ϕc,2 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='00◦, ˜kc,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='662 and θV,c,3 = 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='02◦, ϕc,3 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='78◦, ˜kc,3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' These reduced wave vectors could actually be resolved by our experimental set-up where ˜kres ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The reciprocal space image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='b), however, displays a very low amplitude for ˜k ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='55 meaning that the microwave an- tenna cannot excite the other caustic points very effi- ciently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hence, only the low frequency pocket can be accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Further Kerr images were taken for the same ν, but I II11 for different h values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The h values were chosen such that they lie beneath the expected FMR field hFMR ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='078778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A selection of the resulting Kerr images is il- lustrated in the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In each of them, twin spin wave beams are apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' An overview of all the beam properties for the corresponding h values is plotted in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Here, the relevant pa- rameters from every individual beam are extracted with image processing and bootstrapping least squares regres- sion procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' An example on how one set of experi- mental data points is obtained can be found in the Sup- plementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The reasonable, sometimes even very good agreement between predicted and experimen- tal values of θV,c and ˜kc strongly suggests true CSWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The deviation of the beam directions is mostly within the range of the external field angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The larger discrepancy between predicted and measured wavefront angles ϕc is attributed to the narrowness of the CSWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b1) b2) b3) a1) a2) a3) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0341 h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0398 h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0511 Theory Experiment θV,c (◦) φc (◦) ˜kc 30 42 54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='26 110 113 116 119 122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='060 h 30 20 10 0 10 20 30 x (µm) y (µm) 30 20 10 0 x (µm) 30 20 10 0 x (µm) 0 0 0 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Measurement data obtained for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='087 and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Upper part: acquired Kerr images for reduced fields of a1) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0341, a2) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0398, and a3) h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0511,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b1-3) comparison between experiment and theoretical predictions of caustic point properties θV,c, ˜kc, ϕc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The error bars are the standard deviations from a bootstrapping fit procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Beam-like features which do not coincide with a caus- tic point were detected as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This time, the measure- ments were conducted with the 50 µm antennna struc- ture and partially etched film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The width of the antenna was 3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The resulting Kerr map for f =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='84 GHz (ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='372) and µ0Ha =5 mT (h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='028) is shown in the left upper half of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In this geometry, a Damon Eshbach-like mode propagating from the YIG edge could not be fully suppressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' it is visible as a plane wave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Our procedure to analyze spin wave beams yields θV,e = 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='33◦, ϕe = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='97◦ and ˜ke = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='522, whereas our model predicts a caustic point with θV,c = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='39◦, ϕc = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='84◦ and ˜kc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The origin of the experimentally observed beams may be twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Firstly, a close-to-straight slowness curve similar to the case of Gieniusz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [43] is predicted to exist within relatively close distance to ˜ke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The dθV/d˜k plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='b) displays almost a constant behaviour between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 ≲ ˜k ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 (marked with green dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The proximity of the experimental caustic point to a straight-to-close slowness curve is also illustrated in the FT data in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Here, the dashed green semicircle represents the lower bound of ˜k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 and the extracted beam points are highlighted in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' For this portion of the slowness curve, group velocity directions of up to 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='39◦ are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This beam di- rection, however, is still in stark contrast with the mea- surement result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Moreover, the calculated slowness curve (blue curve) deviates significantly from the FT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The difference between reciprocal space image and our model may show the limit of the model applicability, since a film with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='087 may not be considered a thin film any- more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This results in predictions which are less reliable at higher ν values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A second possible origin of the beams is the excitation efficiency of the microwave antenna as there are many gaps in the FFT spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The beams appear to be located close to some of them, and hence, may correspond to the excitation of only a small portion of the slowness curve within this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' |FT(δmz)|2 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=') c) Theory Fit data ˜kx 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0 ˜ky 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 30 20 10 10 20 30 y (µm) 0 0 δmz (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=') 0 a) b) dθV/d˜k 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 x (µm) ˜k FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a) Kerr image with twin beams obtained with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='087 h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='028 and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b) Calculated derivative of the group velocity direction with respect to the reduced wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Dashed green lines highlight close-to-straight slowness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' c) FT of Kerr image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The experimentally observed beam parameters are depicted in yellow, the calcu- lated slowness curve in blue and the calculated caustic points in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Dashed green semicircle illustrates lower limit of close- to-straight portion of slowness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Caustic point of higher order Based on the conclusions from section III B, we know that the intersection of ∂D3,l and ∂D3,u there exists a sin- gle caustic point on the slowness curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' in the schematic discussion from the above based on the approximant P(˜k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a, b), it corresponds to a = 0 and b = 0, which means that dθV/d˜k ∼ (˜k − ˜kc)3 around this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To put it differently: at this intersection, corresponding to the cusp seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 7, the caustic point is not a simple extremum for θV on the slowness curve but an undulation point, in the vicinity of which θV − θV,c ∼ (˜k − ˜kc)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The existence of such an undulation point is of par- ticular interest since the higher order in the dependence of θV on ˜k implies a flatter extremum in group veloc- ity direction and therefore the possibility of larger por- tions of the slowness curve contributing to the CSWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Moreover, as was discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='II B, this does not necessarily mean an increase in spectral breadth of the CSWB since the latter depends on the apparent wave- length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In order to evidence this, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 13 how the group velocity direction as well as the natural and apparent wavelengths vary around a caustic point very close to one of higher order, here the one such that its corresponding critical field hc is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The considered slowness curve corresponds to h = h1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='15 · 10−21, ν = ν1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='315279504, η = η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='10253664614147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Let us briefly outline how the coordinates νc,0 = νc(hc = 0) and ηc,0 = ηc(hc = 0) were found with a good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' More details can be found in the Sup- plementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The starting point was a rough, hand-performed search for a value of η bringing the cusp of D3 to lie on the ordinate axis in a field and fre- quency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This yielded a starting point of η(0) c,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='10 and ν(0) c,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In these conditions, a caustic point was found for ˜k(0) c,0 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We then began an itera- tive procedure using appropriate Taylor expansions of the dispersion relation and of an exact expression for θV(h = 0, η, ν, ˜k, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Updating these at each step with the new solutions found by looking for the undulation point allows to converge to numerical values which we assimilate to the intersection of ∂D3,l and ∂D3,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Over three iterations, the relative changes in the esti- mates steadily decrease in absolute value, from at most 5% in the first step to at most 5 · 10−6 in the last one, which provides the following guesses : ˜k(g) c,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='731717, η(g) c,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1025366, ν(g) c,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3152796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The latter can be compared with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' the hand-refined values used for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 13: ν = ν1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='315279504, η = η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='10253664614147, corresponding to ˜kc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='725904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' It must be noted that the somewhat larger relative difference in terms of ˜kc,0 is due to the very steep dependence of ˜kc(ν, η, h → 0) on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We do emphasize that the exact location (νc,0,ηc,0) is nec- essarily different from (ν1, η1) but close enough to high- light the qualitatively different behaviour of several char- acteristics of the slowness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Finally, we note that for ˜k ˜k ˜k φ ˜kc φc 0 1 2 3 4 0 ◦ 15 ◦ 30 ◦ 45 ◦ 60 ◦ 75 ◦ 90 ◦ b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='04 θV/θV,c−1 λ0/λ0,c−1 λ/λc−1 a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a) Plots of the relative deviations from the fol- lowing caustic point properties as a function of ˜k: its group velocity direction θV, its natural wavelength λ0 = 2π/˜k and its apparent wavelength λ = 2π/[˜k cos (θV − ϕ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Main graph: h = h1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='15 · 10−21, ν = ν1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='315279504, η = η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='10253664614147, which are extremely close to the values of νc and η for which hc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Inset: same h and η = η1, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='95 · ν1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2995155288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b) Slowness curve for ν1, η1 and h1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' ˜kc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The slowness curve at ν2 is not shown for clarity, as it is very similar to the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 90° 75° 60° 45° 30° 15° 0 2 3 1 4 kd13 the parameters from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 13, θV,c =118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='36°, ϕc ≃42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='75°, λ0,c = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='41lex = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='655d, and λc ≃ 339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7lex = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='83d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We now examine the properties of the caustic point of higher order in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 13, the depen- dence of θV,c and the apparent wavelength λ on ˜k (in blue and green, respectively) clearly appears to be quar- tic rather than quadratic around the caustic point, which is where the deviations in natural wavelength (in red) go through 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Its much steeper behaviour is easily under- stood by looking at the corresponding slowness curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='b): around ˜kc it is not only almost straight but the angle γ between −→˜k and d −→˜k /ds is low, γ ≃14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='38°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hence, since d(˜k2)/ds is large, λ0 ∝ 1/˜k varies fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' By contrast, one can show that in the Taylor expansion of λ in (s−sc)/˜kc around λc, the first coefficient is always exactly zero at a caustic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We stress again that this is caused by an unchanging projection of −→k on −→ eg across the caustic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' If it is of higher order, it may be shown (see Supplementary Materials) that in this term, the con- tributions due to the second- and third-order variations of ϕ and to those of ˜k cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To put it differently, the projection k · cos (θV − ϕ) is now constant up to fourth order in (s − sc)/˜kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' On the other hand, if the consid- ered caustic point is a regular extremum for θV, the term ∝ (s − sc)2 will be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To summarize the above paragraph: for geometrical reasons, the caustic point of higher order suppresses the quadratic and cubic variations of the apparent wave- length around λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hence, λ has then a markedly quartic behaviour at a caustic point of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Further- more, we point out that even a small offset in frequency makes it display a clearly quadratic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This is shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 13, showing the same relative variations for the slowness curve at h = h1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='15·10−21, η = η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='10253664614147, but ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='95 · ν1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2995155288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We have thus shown that in a sufficiently close vicin- ity of a higher-order caustic point, a broadband excita- tion in terms of wavenumber can result in a narrowband CSWB with a very well-defined direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As a result, this phenomenon is expected to be extremely favourable in experiments, since any realistic antenna cannot have an arbitrarily narrow excitation efficiency as a function of wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Provided that its design yields AC mag- netic fields with Fourier components in the (broad) range of interest and with phases in a given interval of width < π, all the corresponding spin waves will coherently add in a beam with very small spectral breadth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In other words: in such a situation, counter-intuitively, exciting additional wave vectors with different wavenumbers does not average out the carrier wave’s amplitude but rather increase it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This naturally prompts the question of how much stronger the emission from a caustic point of higher order would be with respect to that of a regular caustic point, and more generally, of the spin wave amplitude enhancement due to the caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This, however, goes beyond the scope of the present manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To conclude this section, we point out that the reduced field hc(η) corresponding to the caustic point of higher order decreases as a function of reduced dipolar-exchange length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Thus, this feature is expected to exist only for η < ηc,0 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1025366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Merged caustic spin wave beams We now move on to the topic of the threshold fre- quency νm(h, η) corresponding to the upper boundary of D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' above which there are no caustic points any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As was shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 6, the CSWB direction θV,c goes to π/2 as ν → νm(h, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 14, where we show a slowness curve for η1, h1, and ν2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='71836419052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We stress again that νm(h, η) is strictly speaking an infinitely narrow boundary and therefore ν2 ̸= νm(h1, η1), but in these conditions, we find a unique caustic point on the slowness curve, with π/2 − ϕc ≃0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='32 µrad, and θV,c is equal to π/2 (within numerical precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Moreover, at ν′ 2 = ν2 + δν, where δν = 1 · 10−11, we do not find any caustic point on the slowness curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As a result, we take the slowness curve at (ν2, h1, η1) to be assimilable to the one at (νm(h1, η1), h1, η1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Its very straight aspect around ϕ = π/2 is somewhat rem- iniscent of the one seen in the discussion of the caus- tic point of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To illustrate this in more de- tail, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='b) displays the relative deviations in group velocity direction θV, natural wavelength and apparent wavelength around the caustic point at ϕc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We point out that in the present case, the deviations are plotted against the curvilinear abscissa s normalized to the slow- ness curve’s length sM instead of ˜k as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This choice is motivated by (i) the fact that in this case, to lowest order ˜k − ˜kc = O(s2) instead of O(s − sc) as be- fore, and (ii) the much smaller relative difference between the smallest and largest normalized wavenumbers ˜km re- spectively ˜kM: ˜km ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='17 and ˜kM ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='91, compared to ˜km ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='240 and ˜kM ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='66 before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' (i) implies that for (ν2, h1, η1), ˜k cannot serve as a meaningful abscissa along the curve since d˜k/ds = 0, which was not the case for (ν1, h1, η1), while (ii) shows that the slowness curve for (ν2, h1, η1) is much closer to a fourth of a circle than that for (ν1, h1, η1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' as a matter of fact, for (ν2, h1, η1), we find that 1 − [π/2 · (˜km + ˜kM)/2]/sM = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' There- fore, s/sM provides a better feeling for how much of the slowness curve contributes to the CSWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' From the graph, it seems that the apparent wavelength has once more a quartic behaviour around the caustic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We show in the Supplementary Materials that this is indeed the case: in the conditions where ν = νm(h, η), to the lowest non-zero order, θV(s → 0) − π/2 varies with an s3 dependence around s = 0, and the lowest- order variations in ˜k and ϕ (around ˜km and π/2) cancel each other out in the projection ˜k · cos (θV − ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As a result, a caustic point at νm(h, η) is such that 14 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='14 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 s/sM φ 0 ◦ 15 ◦ 30 ◦ 45 ◦ 60 ◦ 75 ◦ 90 ◦ ˜k 0 2 4 8 6 a) b) θV/θV,c−1 λ0/λ0,c−1 λ/λc−1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a) Slowness curve at (ν2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='71836419052, h1, η1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b) Relative deviations in group velocity direction θV (blue), natural wavelength λ0 (red) and apparent wavelength λ (green), as a function of curvilinear abscissa along the slow- ness curve normalized by its total length sM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' an excitation from a suitable, moderately directional an- tenna would be effectively narrowband, and weakly di- vergent around the group velocity direction θV,c = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This orientation is itself also advantageous in practice: as long as the used antenna can excite sufficiently high wavenumbers, the CSWB direction becomes in this case simply perpendicular to the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Moreover, ow- ing to the symmetries of the dispersion relation, the CSWB benefits from the part of the slowness curve at ϕ ≳ π/2, which also feature θV ≃ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' That is why large spin wave amplitudes can be expected, as effectively two CSWB have merged at this particular frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We note that this merging phenomenon has already been observed in simulations by Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [29] in perpendicularly mag- netized ultrathin films and by Gallardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [41] in syn- thetic antiferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' For the sake of completeness, let us comment on what happens from an analytical point of view when the two caustic points just below and above ϕ = π/2 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' It must be kept in mind that they respectively correspond to a maximum and a minimum for θV, temporarily considered for ϕ ∈ [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Thus, when they do coincide at ϕc = π/2, strictly speaking there is no caustic point any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' To put it differently: be- low νm(h, η), over ϕ ∈ [0, π], θV increases up to the first caustic point where it reaches θV,c > π/2, decreases until the second one (for ϕ > π/2) where it reaches π − θV,c, then increases again to reach π when ϕ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Exactly at νm(h, ��), it is monotonously increasing with an inflex- ion point, and above νm(h, η), it is strictly monotonously increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In order to go beyond the particular case presented here, we now investigate the evolution of νm(h → 0, η) = νm,0(η) as a function of reduced dipolar-exchange length η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Similarly to the caustic point of higher order, the evolutions as a function of reduced field quickly become cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This is why we focus on the νm,0(η), which is both the lowest frequency at which CSWBs merge and a threshold frequency that is easier to reach in experi- ments owing to the vanishing applied field, provided that the studied film is soft enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We do keep in mind that below a certain limit in terms of reduced dipolar-exchange length, the model we use loses its validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' However, it has been shown that at sufficiently high frequency [38], the analytical dispersion relation derived by Kalinikos and Slavin describes spin waves once more with a good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 15 displays the numerically determined depen- dence of νm0 on η, as well as that of λm/lex the wave- length of the corresponding CSWB, normalized by the dipolar-exchange length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The procedure to find first a coarse estimate of this curve (before refining it with ac- tual field and frequency maps) is described in the Supple- mentary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We point out that in the case of the merged CSWBs, the apparent and natural wavelengths are equal since θV,c = ϕc = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The minimum value of η in these graphs corresponds to the smallest one we used such that the slowness curve (in vanishing fields) has only one connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' While we may not expect our findings to hold at the lowest η’s, we do expect their accuracy to improve as η increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' it should be suf- ficient at least for η > 1 since in this case the considered ferromagnetic film can truly be considered thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' If we think about searching for the merged CSWBs, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='b) indicates that for realistic values of η = lex/d, the CSWBs’ apparent wavelengths λ are only about one order of magnitude larger than the material’s dipolar- exchange length, typically λ ≲ 25lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This is in stark contrast with the case of the caustic point of higher order in vanishing field, where the natural wavelength was λ0,HO ≃ 84lex, and the apparent wavelength λHO ≃ 334lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As a result, it seems that while caustic points of higher order may readily be excited by antennas cre- ated with even conventional electron beam lithography, in the case of the merged CSWB achieving a sufficient excitation efficiency at the proper wavevectors should .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='15 a) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' a) νm,0(η) as a function of reduced dipolar-exchange length η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' b) The corresponding reduced wavelength ˜λm = (2π/˜k)/η = λ/lex = λ0/lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Here, natural and apparent wave- lengths coincide as phase and group velocities are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' prove quite challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' For instance, even the low- magnetization, low-damping and rather soft ferrimagnet YIG features lex =17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='3 nm [40], meaning that high-end antennas with a characteristic periodicity down to about 200 nm would be required in this easiest of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' CONCLUSIONS We have focused on some properties displayed by spin wave caustics in soft, thin ferromagnetic films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' On the theoretical side, our approach relied on the analytical dispersion relation established by Kalinikos and Slavin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We could show that many reports on CSWBs in the lit- erature can be interpreted within this frame, although the absence of characteristic signs of a true CSWB may still cause some ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Following up on most stud- ies, we have performed time-resolved magneto-optical Kerr-effect-based microscopy on samples designed for the study of CSWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Despite the large thickness of the fer- romagnetic material, our measurements are in very good agreement with our predictions, thus validating the ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Furthermore, we have specifically highlighted the large misalignment between phase and group velocities in this case, and succeeded in observing narrow CSWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Just at the boundary of the dipolar-dominated regime accessible in our experiments, we have predicted the ex- istence of a special caustic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We refer to it as caustic point of higher order because it corresponds to an undu- lation point for the group velocity direction rather than a quadratic extremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' This configuration was shown to be of particular interest because the apparent wave- length also featured a quartic behaviour, which implies a low spectral breadth for the CSWB even in the case of a broadband excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Although we focused on the special value ηc of reduced dipolar-exchange length such that the caustic point of higher order occurs at vanish- ing applied fields, we stress that this phenomenon would appear at non-zero fields for η < ηc, as long as the dis- persion relation we use is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Finally, we have investigated the merging of CSWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Once again, we have studied in detail the case of van- ishing applied fields, yet the merging may occur for any field value, provided that the excitation frequency is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' In terms of model validity, it must be recalled that while vanishing values of η are problematic for the chosen dispersion relation, the merging always occurs at frequencies close to the exchange-dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' The discrepancies between the actual spin wave disper- sion and the model by Kalinikos and Slavin decrease in this frequency range [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As a result, our claim is that the merging frequencies νm0 obtained for low η may be slightly inaccurate yet the phenomenology should remain the same as for larger η, where we expect our predictions to be more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' As the CSWBs merge, a very sig- nificant portion of the slowness curve contributes to spin wave emission around θV = ϕ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Therefore, this configuration appears promising in terms of channelling strong spin wave beams with short wavelengths, as low as ∼ 15lex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' One of the most important questions remaining un- addressed so far concerns the quantification and predic- tion of the enhancement of amplitude associated with CSWBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' More precisely, the crucial distinction between natural and apparent wavelength as well as the inad- equacy of the usual Huygens-Fresnel approach (due to the strong non-collinearity between phase and group ve- locities) in the construction of CSWBs calls for alterna- tive evaluations of their amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' We intend to clar- ify these points and to go beyond the usually described amplitude divergence so as to reconcile the theoretically vanishing curvature (on the slowness curve) and experi- mentally finite amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Xiao, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Schultheiss, Magnetic texture based magnonics, Physics Reports 905, 1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Divinskiy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Merbouche, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Demidov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Nikolaev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Soumah, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Gou´er´e, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Lebrun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Cros, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Ben Youssef, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bortolotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Anane, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Demokritov, Evidence for spin current driven bose- einstein condensation of magnons, Nature Communica- tions 12, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1038/s41467-021-26790-y (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bouzidi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Suhl, Motion of a bloch domain wall, Physical Review Letters 65, 2587 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Stancil and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Prabhakar, Spin waves - Theory and applications (Springer New York, NY, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Chumak, Fundamentals of magnon- based computing, ArXiv e-prints (2019), arXiv:http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='org/abs/1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='08934v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Petti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Tacchi, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Albisetti, Review on magnon- ics with engineered spin textures, Journal of Physics D: Applied Physics 55, 293003 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='8 : : : : : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=" '." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6 2 : 1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4 : : 1 : -- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=" '." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=" ', : 1 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' : : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='0 i -- 0 1 2 3 n80 70 60 50 40 uu 30 20 10 0 0 1 2 3 n16 [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Seo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hwang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Yoon, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Cho, Tunable asymmetric spin wave excitation and propagation in a magnetic system with two rectangular blocks, Scientific Reports 11, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1038/s41598-021-02967- 9 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Au, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Dvornik, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Davison, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Ahmad, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Keat- ley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Vansteenkiste, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Waeyenberge, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kruglyak, Direct excitation of propagating spin waves by focused ultrashort optical pulses, Physical Review Let- ters 110, 097201 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [9] St¨ohr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' and Siegmann, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=', Magnetism: From Fun- damentals to Nanoscale Dynamics (Springer Berlin Hei- delberg, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Sluka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Schneider, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Gallardo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' K´akay, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Weigand, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Warnatz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Mattheis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Rold´an- Molina, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Landeros, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Tiberkevich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Slavin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Sch¨utz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Erbe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Deac, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Lindner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Raabe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Fassbender, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Wintz, Emission and propaga- tion of 1d and 2d spin waves with nanoscale wavelengths in anisotropic spin textures, Nature Nanotechnology 14, 328 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Aliev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Awad, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Dieleman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Lara, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Metlushko, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Guslienko, Localized domain- wall excitations in patterned magnetic dots probed by broadband ferromagnetic resonance, Physical Review B 84, 144406 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Golebiewski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Gruszecki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Krawczyk, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Serebryannikov, Spin-wave talbot effect in thin ferromag- netic film, Physical Review B 102 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Damon and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Eshbach, Magnetostatic modes of a ferromagnet slab, Journal of Physics and Chemistry of Solids 19, 308 (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kravtsov and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Orlov, Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenom- ena, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 6 (Springer-Verlag Berlin Heidelberg, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kravtsov and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Orlov, Caustics, Catastrophes and Wave Fields, Springer Series on Wave Phenomena, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 15 (Springer, Berlin, Heidelberg, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Poston and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Stewart, Catastrophe Theory and Its Ap- plications, Surveys and reference works in mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 2 (Dover Publications, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Northrop and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Wolfe, Ballistic phonon imaging in germanium, Physical Review B 22, 6196 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Taylor, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Maris, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Elbaum, Phonon focusing in solids, Physical Review Letters 23, 416 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [19] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Maris, Enhancement of heat pulses in crystals due to elastic anisotropy, The Journal of the Acoustical So- ciety of America 50, 812 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Every, Effects of first-order spatial dispersion on phonon focusing: Application to quartz, Physical Review B 36, 1448 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [21] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Maris, Phonon focusing, in Nonequilibrium phonons in nonmetallic crystals, Modern Problems in Condensed Matter Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 16, edited by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Eisenmenger and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kaplyanskii (North Holland, 1986) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' 51–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [22] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' B¨uttner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Demokritov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hillebrands, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kivshar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Grimalsky, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Rapoport, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Slavin, Linear and nonlinear diffraction of dipolar spin waves in yttrium iron garnet films observed by space- and time-resolved brillouin light scattering, Physical Review B 61, 11576 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [23] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Veerakumar and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Camley, Magnon focusing in thin ferromagnetic films, Physical Review B 74, 214401 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Khomeriki, Self-focusing magnetostatic beams in thin magnetic films, The European Physical Journal B 41, 219 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Demidov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Demokritov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Birt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' O’Gorman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Tsoi, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Li, Radiation of spin waves from the open end of a microscopic magnetic-film waveguide, Physical Review B 80, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1103/physrevb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='014429 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Schneider, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Serga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Chumak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Sandweg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Trudel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Wolff, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kostylev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Tiberkevich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Slavin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hillebrands, Non- diffractive subwavelength wave beams in a medium with externally controlled anisotropy, Physical Review Letters 104, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1103/physrevlett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='197203 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kostylev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Serga, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hillebrands, Radi- ation of caustic beams from a collapsing bullet, Physi- cal Review Letters 106, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1103/physrevlett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='134101 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [28] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Sebastian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Br¨acher, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Pirro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Serga, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hille- brands, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kubota, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Naganuma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Oogane, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Ando, Nonlinear Emission of Spin-Wave Caustics from an Edge Mode of a Microstructured Co2Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='6Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='4Si Waveguide, Physical Review Letters 110, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1103/phys- revlett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='067201 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kim, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Stamps, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Camley, Spin wave power flow and caustics in ultrathin ferromagnets with the dzyaloshinskii-moriya interaction, Physical Review Letters 117, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1103/physrevlett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='197204 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [30] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Heussner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Serga, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Br¨acher, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hillebrands, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Pirro, A switchable spin-wave signal splitter for magnonic networks, Applied Physics Letters 111, 122401 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Muralidhar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Khymyn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Awad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Alem´an, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hanstorp, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' ˚Akerman, Femtosecond laser pulse driven caustic spin wave beams, Physical Review Letters 126, 037204 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [32] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kalinikos and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Slavin, Theory of dipole- exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions, Journal of Physics C: Solid State Physics 19, 7013 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Lax and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Narayanamurti, Phonon magnification and the gaussian curvature of the slowness surface in anisotropic media: Detector shape effects with applica- tion to GaAs, Physical Review B 22, 4876 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [34] Even for markedly anisotropic dielectrics the differences in refractive indices lead to maximal angles between phase and group velocities below 10° [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [35] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bertelli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Carmiggelt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Yu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Simon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Pothoven, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bauer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Blanter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Aarts, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' van der Sar, Magnetic resonance imaging of spin-wave transport and interference in a magnetic insulator, Sci- ence Advances 6, eabd3556 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [36] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Felsen and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Wave Theory (Wiley-IEEE Press, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [37] We define the spectral breadth of a non-monochromatic beam with well-defined propagation direction and negli- gible divergence as ∆λ/λcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=', where λcar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' is the carrier’s mean apparent wavelength, and ∆λ is the beam’s stan- dard deviation in apparent wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kreisel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Sauli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bartosch, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Kopietz, Mi- croscopic spin-wave theory for yttrium-iron garnet films, The European Physical Journal B 71, 59 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Harms and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Duine, Theory of the dipole- exchange spin wave spectrum in ferromagnetic films with 17 in-plane magnetization revisited, Journal of Magnetism and Magnetic Materials 557, 169426 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Klingler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Chumak, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Mewes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Khodadadi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Mewes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Dubs, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Surzhenko, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Hillebrands, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Conca, Measurements of the exchange stiffness of YIG films using broadband ferromagnetic resonance tech- niques, Journal of Physics D: Applied Physics 48, 015001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Gallardo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Alvarado-Seguel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' K´akay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Lind- ner, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Landeros, Spin-wave focusing induced by dipole-dipole interaction in synthetic antiferromagnets, Physical Review B 104, 174417 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [42] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bertelli, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Simon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Aarts, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bauer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Blanter, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' van der Sar, Imaging spin- wave damping underneath metals using electron spins in diamond, Advanced Quantum Technologies 4, 2100094 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [43] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Gieniusz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Ulrichs, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Bessonov, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Guzowska, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Stognii, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Maziewski, Single antidot as a pas- sive way to create caustic spin-wave beams in yttrium iron garnet films, Applied Physics Letters 102, 102409 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [44] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' K¨orner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Stigloher, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' Back, Excitation and tailoring of diffractive spin-wave beams in NiFe using nonuniform microwave antennas, Physical Review B 96, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='1103/physrevb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='100401 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [45] We point out that the segment perpendicular to the ta- pered waveguide segments was at an angle of about 60° with respect to the applied field in these experiments [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' [46] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' K¨orner, Time-resolved Kerr microscopy of spin waves propagating in magnetic nanostructures, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'} +page_content=' thesis, Universit¨at Regensburg (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfRvtg/content/2301.01220v1.pdf'}