diff --git "a/6NE2T4oBgHgl3EQfkgd1/content/tmp_files/load_file.txt" "b/6NE2T4oBgHgl3EQfkgd1/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/6NE2T4oBgHgl3EQfkgd1/content/tmp_files/load_file.txt" @@ -0,0 +1,606 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf,len=605 +page_content='Apodized photonic crystals: A non-dissipative system hosting multiple exceptional points Abhishek Mondal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Shailja Sharma and Ritwick Das∗ School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' National Institute of Science Education and Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' An OCC of Homi Bhabha National Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Jatni - 752050,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Odisha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' India (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2023) Optical systems obeying non-Hermitian dynamics have been the subject of intense and concerted investigation over the last two decades owing to their broad implications in photonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' acoustics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' electronics as well as atomic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A vast majority of such investigations rely on a dissipative, balanced loss-gain system which introduces unavoidable noise and consequently, this limits the coherent control of propagation dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, we show that an all-dielectric, non-dissipative photonic crystal (PC) could host, at least two exceptional points in its eigenvalue spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' By introducing optimum apodization in the PC architecture, namely 1D-APC, we show that such a configuration supports a spectrum of exceptional points which distinctly demarcates the PT - symmetric region from the region where PT -symmetry is broken in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The analytical framework allows us to estimate the geometric phase of the reflected beam and derive the constraint that governs the excitation of topologically-protected optical Tamm-plasmon modes in 1D-APCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' INTRODUCTION Optical systems which are governed by non-Hermitian Hamiltonian dynamics through an engineered gain and dissipation mechanism, provide a route to overcome the limitations imposed by closed optical systems that obey the Hermitian-Hamiltonian led dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such non- Hermitian systems give rise to a real eigenvalue spec- trum when the Hamiltonian commutes with the parity- time (PT ) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A continuous change in the pa- rameter governing the Hermiticity (of the Hamiltonian) breaks the PT symmetry which manifests in the form of complex eigenvalues for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In the phase space, such points where the real and complex eigenval- ues coalesce are termed as exceptional points (EPs) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This spontaneous PT -symmetry breaking has catalyzed a plethora of non-intuitive outcomes such as directional invisibility [3, 4], coherent perfect lasing and absorption [5–9], negative refraction [10], single-particle based sens- ing [11–13], distortion-free wireless optical power trans- fer [14] and a few more [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is, however, worth noting that the incommensurate gain and loss distribu- tion in non-Hermitian systems impose the primary limi- tation on the practical applications due to unpredictable signal-to-noise ratio near EP [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to cir- cumvent such bottlenecks, a few possibilities have been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' One such promising route is to create an asym- metric loss in the system (without gain) whose dynamics could be explored using a non-Hermitian Hamiltonian with a uniform background loss [20, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such a con- figuration would exhibit PT -symmetry which could be broken through scaling up the loss asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In a dif- ferent scheme, a pseudo-Hermitian system was explored which allowed strong coupling between a large number ∗ ritwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='das@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='in of modes via manipulation of the parameters governing the Hamiltonian [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This led to the existence of EPs of multiple order and the interaction of eigenvalues around each EP provides a robust control on the propagation dynamics [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In spite of the aforementioned devel- opments, a useful and practical proposition would be to devise a configuration hosting a multitude of EPs with the constraint that the electromagnetic (EM) energy lost due to the non-Hermitian dynamics is stored in a reser- voir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This essentially implies that the dissipative channel associated with a non-Hermitian system drives a separate Hermitian system which could allow reverse flow of EM energy by virtue of cyclical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such systems have been explored in the area of parametric frequency con- version processes where the EM energy lost in one of the parametric processes (obeying non-Hermitian dynamics) is coherently added to the other parametric process that follows a Hermitian dynamics [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A plausible transla- tion of such an idea in the non-absorptive linear systems would be to introduce a virtual loss in an intermodal interaction process thereby generating multiple EPs in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' One of the simplest configurations imitating such a process is a multimodal interaction in an all-dielectric one-dimensional (1D) photonic-crystal (PC) with a gradually varying duty cycle (for each unit cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In such an apodized 1D-PC, the forward (source) to backward (sink) mode-coupling dynamics is essentially governed by a pseudo-Hermitian Hamiltonian whose Her- miticity is determined by the apodization along the prop- agation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In the present work, we show the exis- tence of multiple EPs in an apodized 1D-PC and develop an analytical framework for ascertaining the possibility of exciting topologically-protected optical edge modes in such aperiodically stratified configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='03979v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='optics] 10 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' THEORETICAL FRAMEWORK AND COUPLED-MODE FORMALISM We consider a 1D-PC comprised of periodic bilayers with refractive indices n1 and n2 with thicknesses d1 and d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such conventional 1D-PCs or alternatively, dis- tributed Bragg reflectors (DBRs) are usually character- ized by photonic bandgaps (PBGs) which are separated from each other by high transmission (or pass) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to appreciate the EM wave propagation dynamics, we consider the coupling between pth-mode (|p⟩) with qth-mode (|q⟩) which could be represented employing the coupled-amplitude equations given by [29] dAq dz = −i βq |βq| � p � m ˜κ(m) qp Ape−i(βq−βp−m 2π Λ )z (1) where βp and βq are the longitudinal (z) components of wavevector kp and kq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' ˜κ(m) qp defines the strength of coupling (or coupling coefficient) between the pth and qth mode that is coupled through the mth Fourier component of the periodic dielectric distribution ( Λ = d1 + d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The factor ∆β = βq − βp − m 2π Λ (known as the phase-mismatch) is one of the dynamical variables (along with κqp) which dictate the measure of optical power transferred from one mode to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For the present work, we consider a contra-directional coupling set-up where a forward (along +z) propagating mode (|p⟩ ≡ |f⟩) is coupled to a backward (along −z) prop- agating mode (|q⟩ ≡ |b⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Accordingly, it could asserted that βb = −βf or alternatively ∆β = 2βf − 2π Λ and there- fore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (1) could be simplified to [29] dAb dz = i˜κAfe−i∆βz (2) dAf dz = −i˜κ∗Abei∆βz (3) where ˜κ = i(1−cos 2πζ) 2λ (n1 2−n2 2) ¯n = iκ and ζ is the di- electric filling fraction of layer with refractive index n1 in the unit cell i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' d1 Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The mean refractive index for an unit cell of thickness Λ is ¯n = � d1n12+d2n22 Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' By using a Gauge transformation given by [Af, Ab] → [ ˜Af, ˜Ab]ei/2[∆β0z− � 0 zq(z′)dz′], we obtain [30] i d dz � ˜Ab ˜Af � = � −∆k −˜κ ˜κ∗ ∆k � � ˜Ab ˜Af � (4) Equation (4) is analogous to time-dependent Schr¨odinger’s equation with t-coordinate being replaced by z-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, ∆k (= ∆β 2 ) and q(z) = 0 remains constant (for a given frequency) across the 1D-PC which has a fixed duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The autonomous Hamiltonian ˆH = −⃗σ · ⃗B with ⃗σ ≡ [σx, σy, σz] are the Pauli’s spin matrices and ⃗B ≡ [0, κ, ∆k] (magnetic field analog) rep- resents a pseudo-Hermitian evolution dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to appreciate this point, we note that the eigenvalues of ˆH which are given by e1,2 = ± � ∆k2 − κ2 whereas the eigenfunctions are |ψ1⟩ = � −i (∆k+√ ∆k2−κ2) κ 1 � and |ψ2⟩ = �� +i (−∆k+√ ∆k2−κ2) κ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, ˜κ = iκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A closer look into the eigenvectors reveals that the equality κ = ± ∆k manifests as coalescing of eigenvectors accompanied by vanishing eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Such points in parameter space where κ equals ±∆k are termed as exceptional points (EPs) and they distinctly demarcate the regions exhibiting Hermitian (PT -symmetric phase) and non-Hermitian (PT -broken phase) dynamical evolution of states (or modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to appreciate the aforementioned idea, we consider a practical 1D-PC with n1 ≡ TiO2 layer and n2 ≡ SiO2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The layer thicknesses are d1 = d2 = 150 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The reflection spectrum for N = 20 unit cells is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1(a) which exhibits a high reflection band (or PBG) spreading over a 75 THz band- width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to obtain the reflection spectrum, finite element method (FEM) based simulations were carried out using the commercially available computational tool (COMSOL Multiphysics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In the simulations, the peri- odic boundary condition is imposed along the transverse direction and a mesh size of 5 nm is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' We ignore the material dispersion for the simulations and assume n1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 (≡ TiO2) and n2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 (≡ SiO2) across the entire spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For this 1D-PC, we also plotted the eigenvalues e1 and e2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1(b)) as a function of the frequency of the incident electromagnetic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is ap- parent that the eigenvalues vanish at ν1 ≈ 210 THz and ν2 ≈ 285 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' These two frequencies (ν1 and ν2) define the EPs (κ = + ∆k and κ = − ∆k) for the periodic 1D-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A closer look would also reveal that the eigenval- ues are purely imaginary within the PBG and the band edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1 (a)) coincide with ν1 and ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The mode fields for frequencies lying inside the PBG (240 THz) and outside the PBG (310 THz) are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1(c) and (d) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is worth noting that the investigations on systems exhibiting PT -symmetry (or PT -broken symmetry) led dynamics in photonics essen- tially involve optimally balanced gain-loss architectures such as segmented waveguides and photonic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In such systems, a complex relative permittivity in different sections depicting actual gain or loss for the propagat- ing light beam gives rise to the PT -symmetry (or PT - broken symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The present configuration involving 1D-PC does not include an actual dissipative component for achieving the PT -symmetric to PT -symmetry bro- ken phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Alternatively, the coupling of opti- cal power to the backscattered mode |b⟩ is analogous to a virtual loss for a forward propagating |f⟩ mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' When this coupling is relatively weak i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' ∆k > ˜κ, |f⟩ and |b⟩ exhibits cyclic exchange of optical power (as a function of z) which is a primitive outcome for a PT -symmetric dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' On the other hand, a strong coupling regime 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' a) Shows the reflection spectrum of a conventional (periodic) 1D-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' b) Shows the variation in Re(e1) (dotted black curve), Im(e1) (dotted maroon curve), Re(e2) (solid black curve) and Im(e2) (solid maroon curve) as a function of frequency (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' c) and d) Shows the mode-field intensity for frequencies within the PBG 240 THz and that outside the PBG 310 THz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The solid red arrow represents the direction of incidence of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' where ∆k < ˜κ manifests through a monotonic growth of backscattered mode (|b⟩) that is a signature of PT - symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is worthwhile to reiterate the point that the two regimes depicted by the inequality of ∆k and ˜κ (in the parameter space) could be mapped onto the PBG and pass or transmission band (s) in the reflected spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Subsequently, each PBG is necessar- ily bounded by two EPs in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Additionally, these two EPs are fixed and could not be tailored for a given 1D-PC with a fixed duty cycle and fixed period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Also, the conventional 1D-PC geometry excludes the pos- sibility of realizing higher-order exceptional points [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Taking a cue from this critical viewpoint, we note that a small apodization or gradual change in dielectric fill- ing fraction (ζ) of each unit cell of the 1D-PC would allow us to realize discretely spaced (multiple) EPs at different optical frequencies (or wavelengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to elucidate this point, we recall that ∆k as well as ˜κ is a function of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' An optimum spatial variation in ζ could essentially give rise to the possibility of EPs at different physical locations (along z) in a 1D-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' As an example, we show below that an optimally apodized 1D-PC (1D- APC) which satisfies the adiabatic constraints enables us to observe EPs at discreetly separated points along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Design of an 1D apodized PC and intermodal coupling We consider a 1D-PC configuration that exhibits varying dielectric filling fraction (ζ) in each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This variation is essentially dictated through the relation d1M = d1 −Mδ and d2M = Λ−d1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, d1M and d2M are the thickness of TiO2 and SiO2 layers respectively in M th unit cell (M = 0, 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=', (N − 1) for N number of unit cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The unit cell period, however remains un- changed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Λ = d1M +d2M = d1 +d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This apodization FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' a) Shows the reflection spectrum for designed 1D- APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (b) and (c) Shows the mode-field intensities for two different frequencies νa = 250 THz and νb = 300 THz which are within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (d) and (e) Shows the vari- ation in Re(e1) (dotted black curve), Im(e1) (dotted maroon curve), Re(e2) (solid black curve) and Re(e2) (solid maroon curve) as a function of TiO2 layer thickness for each unit cell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' d1M) at frequencies νa = 250 THz and νb = 300 THz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' in 1D-PC could be visualized through a longitudinal variation in ∆k as well as ˜κ by virtue of a monotonic change in average refractive index (¯n) for an unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This variation in ∆k and ˜κ in a 1D-APC geometry leads to an adiabatic evolution of the Stokes vector along the propagation direction and manifests through a broader PBG (≈ 140 THz) in comparison with a conventional (periodic) 1D-PC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(a) which shows a broader reflection spectrum for the 1D-APC in comparison with the conventional 1D-PC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In addition, a flat transmission band and the absence of sharp transmission resonances is a distinct feature of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The mode-propagation characteristics for the frequencies within the PBG (of 1D-APC) is explored by drawing a comparison with the mode-field distributions for the equivalent modes within the PBG of a conventional 1D-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Figures 2(b) and (c) shows the mode-field distribution for two frequencies νa = 250 THz and νb = 300 THz which are within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In comparison with the mode-field distribution shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1(c), it could be observed that different modes are reflected from spatially separated z values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The smaller frequency (νa = 250 THz) is reflected from the regions which are closer to z = 0 edge of the 1D-APC in comparison to that for νb = 300 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This variation is indicative of the fact that the field is localized and exhibits instantaneous localization in different 1D-APC sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' From a different perspective, it is apparent that the variation in dielectric filling fraction (ζ) would result in different eigenvalues (and corresponding eigenvectors) for each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Accord- ingly, we plot the eigenvalues e1 and e2 as a function of d1M for two frequencies νa = 250 THz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(d)) and νb = 300 THz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(e)) which are within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Each one of the figures shows that the eigen- 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='15 a) b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 R 1,2′ 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 Re( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0 175 200 225 250 175 200 225 250 275 300 325 275 300 325 v (THz) v (THz) C 310THz 240THza) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 250THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 c) 300THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 0 160 200 250 300 350 380 V (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3 Re(e,).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Ree,)Im(e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='m(e,) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='04 d) e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 (V/) ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 (V /z) (V /Z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='02 (V /) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0 0 0 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 1 Re( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 250THz 300THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='04 0 50 100 150 200 250 300 0 50 100 150 200 250 300 d (nm) d (nm) 1M 1M4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (a) Shows the variation of ⃗B in parameter space (spanned by κ and ∆k) at different operating frequencies (ν1 = 400 THz, ν2 = 250 THz, ν3 = 160 THz) for the designed 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The blue and green solid lines repre- sent the ∆k = κ and ∆k = −κ curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (b) Shows the location of EPs in different unit cells (with different filling fraction ζ) as a function of frequency (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' values (e1 and e2) vanish at two different values of d1M i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' at the location of two different unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Therefore, the 1D-APC geometry hosts two EPs for every d1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Consequently, for a multitude of ζ, there would be multiple EPs in the 1D-APC for a forward-propagating mode to a backscattered mode-coupling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' As discussed before, the regions where ℜe1 and ℜe2 are non-zero in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(d) and 2(e) exhibit a PT -symmetric coupling dynamics between the forward-propagating and backscattered modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' On the other hand, in the regions where e1 and e2 are purely imaginary, the mode-coupling process exhibits PT -symmetry broken manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The illustrations presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(d) and 2(e) show that for each frequency within the PBG, the 1D-APC hosts two EPs at two different d1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This essentially implies that there exists one or more than one EPs hosted by each unit cell of the 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Therefore, an 1D-APC is expected to host multiple EPs which are spectrally as well as spatially separated from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order ascertain the spectral location of EPs in the 1D-APC, we plot the evolution of ⃗B in the parameter space for three different frequencies ν1 = 400 THz, ν2 = 250 THz, and ν3 = 160 THz as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It could be noted at ν1 and ν3 are situated outside PBG of 1D-APC (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Since, the EPs are depicted by the condition ∆k = |κ|, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3(a) also contains the curve ∆k = ±κ (solid blue and green curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is apparent that ∆k = ±κ curve intersects ⃗Bν2 at two points and it does not intersect the ⃗Bν1 curve as well as the ⃗Bν3 curve in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For frequencies close to the band-edge of 1D-APC (say 200 THz or 350 THz), it could be ascertained that there exists only one EP in the eigenvalue spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This is primarily due to the adiabatic constraints followed by the 1D-APC design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In other words, for the band-edge frequencies, the forward and backward propagating modes are decoupled (˜κ) at entry (z = 0) and exit (z = L) face of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Additionally, d1M = Λ for m = 0 (or d2M = Λ for m = N) in case of band-edge frequencies that leads to ∆k = 0 for ζ = (or ζ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Therefore, ˜κ = ∆k = 0 depicts the only EP for the band-edge frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to elucidate the aforementioned point, we present the spectral location of EPs as a function of di- electric filling fraction (ζ) or propagation direction (z) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It could be observed that there exists two (2) EPs (at different ζ or z) for all the frequencies well within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, for the band-edge frequencies (νl = 200 THz and νh = 330 THz), the 1D- APC hosts one EP only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Nevertheless, the area enclosed by the EPs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 3(b) represents the region of PT - symmetry broken phase for the 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is interesting to note that the separation between the two EPs for fre- quencies closer to the band-edges (say ν ≤ 210 THz or ν ≥ 310 THz) very less and they tend to overlap at the same filling fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is important to note that these EPs are physically positioned close to the entry (z = 0) and exit (z = L) face of the 1D-APC where ˜κ is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' By virtue of this, the PBG corresponding to that unit cell of 1D-APC is relatively smaller in comparison with the PBG for a unit cell close to the center (z ≈ L 2 ) of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Due to the fact that the EPs exist at the band-edges of PBG for each unit cell of APC, a smaller PBG would essentially imply closely spaced EPs near the band-edges (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Geometric phase estimation of reflection band It is well known that the geometric phase of a pass- band (or transmission band) for a one-dimensional con- ventional photonic crystal is quantized (0 or π) and it is known as the ‘Zak’ phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, the geomet- ric interpretation of backscattered (or reflection) phase from a 1D-PC remains irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, in case of 1D-APC, the reflection of different spectral components (within the PBG) takes place from different unit cells (or z) along the propagation direction [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For exam- ple, the adiabatic following constraint leads to conver- sion of optical power from the forward-propagating to the backscattered mode predominantly towards the exit face of 1D-APC for frequency ν = 250 THz which could be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Through a similar route, it could be shown that different spectral components within the PBG are reflected strongly from different unit cells of 1D-APC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The primary underlying reason could be traced to the variation in ˜κ and ∆k for each spectral component in the PBG which are non-identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Conse- quently, the estimation of geometric phase acquired by different backscattered modes is expected to be differ- ent and must play a crucial role in establishing the bulk- boundary correspondence in case of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to obtain the geometric phase γ, we consider a triad defining the state vector ⃗S (≡ [u, v, w]) where u = ˜Ai ˜A∗ r + ˜Ar ˜A∗ i , v = −i[ ˜Ai ˜A∗ r − ˜Ar ˜A∗ i ] and w = | ˜Ar| 2 − | ˜Ai| 2 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The z-component of the state-vector (w) represents the conversion efficiency of optical power from a forward- propagating to a backscattered mode [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is also worth 10 1 B(v/) a) b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 5 △k (μm" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 米米 △k= S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 米 B(v) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 B(v) Ak= - k 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 190 210 230 250 270 290 310 330 0 1 350 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 k(um v (THz)5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' a) Shows the variation in conversion efficiency ( w+1 2 ) for optical power transfer between a forward-propagating mode to a backscattered mode as a function of 1D-APC length (z) for a frequency ν2 = 250 THz which is within the PBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' (b) Presents the state-vector (⃗S = [u, v, w]) trajectory on the Bloch sphere for ν2 = 250 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' noting that the trajectory of state-vector (⃗S) correspond- ing to the frequencies within the PBG is non-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Al- ternatively, the geometric phase is not a conserved quan- tity during the dynamical evolution of states owing to the PT -symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In general, the solid angle subtended by the state-vector trajectory at the center of the Bloch sphere is used for computing the geomet- ric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, in case of an adiabatic evolution, the state-vector trajectory could be very complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 4(b), we have plotted such a state-vector trajec- tory (on the Bloch sphere) corresponding to a frequency ν = 250 THz (which is within the PBG of 1D-APC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is important to note that ⃗S = [0, 0, −1] and ⃗S = [0, 0, 1] represent states in which all the optical power (∝ | ˜Af,b|2) is present in the forward-propagating and backscattered mode respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Although, the adiabatic evolution of state-vector results in complete optical power trans- fer from the forward to backward-propagating mode i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' w = −1 to w = 1, the estimation of acquired geometric phase is quite complicated owing to the spiralling trajec- tory of ⃗S on the Bloch-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' However, it is interest- ing to note that ⃗S goes from [0, 0, −1] to [0, 0, 1] for all the frequencies within the PBG of 1D-APC by virtue of satisfying the adiabatic following constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The most important point is to note that the conversion efficiency (or reflectivity) is ‘unity’ for all the frequencies within the PBG of 1D-APC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In other words, ⃗B goes from [0, 0, −∆k] to [0, 0, ∆k] in the parameter space for all the PBG frequencies (through any trajectory) when the adi- abatic following constraints are satisfied [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' By virtue of the fact that the state-vector ⃗S adiabat- ically follows ⃗B (as per the Bloch equation), the initial and the final value of ⃗B could also yield the geometric phase (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is known that γ is estimated from angle φ (subtended by ⃗B at the origin ∆k = ˜κ = 0) through the relation γ = φ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In that case, the geometric phase for each spectral component within the PBG is π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In order to elucidate this point, we plot ⃗B at different z of 1D-APC in the parameter space for ν = 250 THz as FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Represents the evolution of ⃗B as a function of length (L) of 1D-APC in parameter (∆k − κ) space for a) ν2 = 250 THz and b) ν4 = 180 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' φ represents the angle subtended by curve ⃗B at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' At the entry face of 1D-APC (z = 0), ⃗B(z = 0) = [0, 0, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='7 µm−1] (black arrow) and gradu- ally goes to ⃗B(z = L) = [0, 0, +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='7 µm−1] (red arrow) at z = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' At z = L 2 , ∆k = 0 and ˜κ is maximum (green arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5(a)) The evolution of ⃗B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5(a) yields φ = π and consequently, γ = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In a similar manner, γ for all the frequencies within the PBG would be π 2 by virtue of adhering to the constraints imposed by adia- batic following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Hence, it could be asserted that a geo- metric phase of π 2 is acquired by a reflected beam in a 1D- APC for the values of parameters which results in PT - symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' On the contrary, the variation in ⃗B is plotted as a function of z for ν = 180 THz which is outside the PBG of 1D-APC (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' ⃗B(z = 0) (black arrow) and ⃗B(z = L) (red dashed arrow) are both negative as well as co-parallel in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Consequently, the geometric phase γ = φ 2 = 0 for �� = 180 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In addition, it is apparent that ∆k ̸= 0 at any point (or any z) in the 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Tamm-plasmon excitations in 1D-APC and topological connection The presence of a plasmon-active layer adjacent to the all-dielectric 1D-APC results in excitation of mul- tiple Tamm-plasmon modes which are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' As an example, we consider a thin (dAu = 30 nm) layer of gold placed in contact with high index layer (TiO2) of 1D-APC (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The simulated reflection spec- trum (using transfer matrix method) exhibits a sharp res- onance within the PBG as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' These res- onances are essentially due to Tamm-plasmon mode exci- tations which are highly localized electromagnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Figure 6(b) depicts the existence of 10 Tamm-plasmon modes within the PBG of 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Although there are a few sharp resonances outside the PBG, their mode-field signatures do not resemble that for a Tamm-plasmon mode [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In general, the existence of Tamm-plasmon modes is governed by the condition φAP C + φAu = 2sπ where s = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='. is an integer [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Here, φAP C is the total phase acquired by the reflected beam from the 1D-APC (light incident from Au side), and φAu a) b) (0,0,1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 (0,0,-1) 0 0 1 2 3 4 5 6 7 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='3 z (μm)3 B(z = L) 0 B (z = 0) a) b) 2 1 B (z = L/2) ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 0 B(z = L/2) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 2 250 THz B (z = L) 180 THz B(z = 0) 3 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content='2 k (μm"1) k (um=1)6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' a) Shows the schematic of the Au-1D-APC het- erostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The Au-layer is placed adjacent to the high- index TiO2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' The thick brown arrow depicts the direction of light incidence on the Au-1D-APC b) Shows the simulated reflection spectrum of 1D-APC without Au (black solid curve) and that of Au-1D-APC (maroon solid curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' is the phase acquired by reflected beam at the Au−TiO2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' It is worthwhile to reiterate that the dielec- tric layer (of 1D-APC) adjacent to the Au-film is TiO2 which is the high index layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In the present context φAP C = γ + α, where α is the dynamic phase acquired by the reflected beam [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This could be estimated by noting the fact that the EPs (for a given frequency) are situated in different unit cells (or ζ) of the 1D-APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' For a frequency ν, if the nearest EP (with respect to z = 0) is present in the pth-unit cell of 1D-APC, then α could be determined using α = 2πν c p � M=0 [n1d1M + n2d2M] (5) The knowledge of location for EPs in the 1D-APC (ob- tained from the eigenvalue spectrum of ˆH) would accu- rately yield the dynamic phase (α) for any frequency of operation (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In conjunction with the estimate of γ, this information would allow us to determine the Tamm- plasmon mode resonance frequencies (νr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This recipe provides a flexibility in terms of designing an 1D-APC which would facilitate excitation of Tamm-plasmon mode at a target (desirable) frequency (or wavelength) of op- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' One such application could be the generation of higher harmonics or frequency downconversion using optical surface states [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' In this case, the 1D-APC could be designed such that the Tamm-plasmon modes (localized modes) have resonance frequencies that are governed by the energy conservation and phase-matching constraints imposed by the frequency conversion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' CONCLUSIONS In conclusion, we presented an all-dielectric 1D-APC design which hosts multiple exceptional points in its eigenvalue spectrum by virtue of exhibiting a non- Hermitian dynamics for a mode-coupling process between a forward-propagating mode to its backscattered coun- terpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Although, the 1D-APC does not include any dissipative component, the intermodal coupling mecha- nism could be classified in terms of PT -symmetric and PT -broken phases which are connected through a quan- tum phase-transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' We also showed that the reflected beam (within the PBG) acquires a geometric phase of π 2 in the PT -symmetry broken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' As a consequence of this outcome, the 1D-APC could be designed for excit- ing the optical Tamm-plasmon modes at any desirable frequency within the PBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' This design flexibility allows us to employ such architectures for quite a few appli- cations such as efficiently carrying out optical frequency conversion using surface states [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' DISCLOSURES The authors declare that there are no conflicts of in- terest related to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Berry, Physics of nonhermitian degeneracies, Czechoslovak Journal of Physics 54, 1039 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfkgd1/content/2301.03979v1.pdf'} +page_content=' Heiss, 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