Computing eigenfrequency sensitivities near exceptional points (2402.17648v1)
Abstract: Exceptional points are spectral degeneracies of non-Hermitian systems where both eigenfrequencies and eigenmodes coalesce. The eigenfrequency sensitivities near an exceptional point are significantly enhanced, whereby they diverge directly at the exceptional point. Capturing this enhanced sensitivity is crucial for the investigation and optimization of exceptional-point-based applications, such as optical sensors. We present a numerical framework, based on contour integration and algorithmic differentiation, to accurately and efficiently compute eigenfrequency sensitivities near exceptional points. We demonstrate the framework to an optical microdisk cavity and derive a semi-analytical solution to validate the numerical results. The computed eigenfrequency sensitivities are used to track the exceptional point along an exceptional surface in the parameter space. The presented framework can be applied to any kind of resonance problem, e.g., with arbitrary geometry or with exceptional points of arbitrary order.
- S. Nie and S. R. Emory, Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering, Science 275, 1102 (1997).
- P. Senellart, G. Solomon, and A. White, High-performance semiconductor quantum-dot single-photon sources, Nat. Nanotechnol. 12, 1026 (2017).
- S. Dyatlov and M. Zworski, Mathematical theory of scattering resonances (American Mathematical Society, Providence, Rhode Island, 2019).
- W. Yan, P. Lalanne, and M. Qiu, Shape Deformation of Nanoresonator: A Quasinormal-Mode Perturbation Theory, Phys. Rev. Lett. 125, 013901 (2020).
- J. Jensen and O. Sigmund, Topology optimization for nano-photonics, Laser Photonics Rev. 5, 308 (2011).
- M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
- T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer-Verlag Berlin Heidelberg, 1995).
- J. Wiersig, Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection, Phys. Rev. Lett. 112, 203901 (2014).
- J. Wiersig, Sensors operating at exceptional points: General theory, Phys. Rev. A 93, 033809 (2016).
- J. Wiersig, Review of exceptional point-based sensors, Photon. Res. 8, 1457 (2020a).
- J. Kullig and J. Wiersig, Microdisk cavities with a Brewster notch, Phys. Rev. Res. 3, 023202 (2021).
- P. Chamorro-Posada, F. Fraile-Pelaez, and F. J. Diaz-Otero, Micro-Ring Chains With High-Order Resonances, J. Lightwave Technol. 29, 1514 (2011).
- J. Wiersig, Prospects and fundamental limits in exceptional point-based sensing, Nat. Commun. 11, 2454 (2020b).
- J. Wiersig, Petermann factors and phase rigidities near exceptional points, Phys. Rev. Res. 5, 033042 (2023).
- L. M. Delves and J. N. Lyness, A numerical method for locating the zeros of an analytic function, Math. Comp. 21, 543 (1967).
- P. Kravanja and M. V. Barel, Computing the Zeros of Analytic Functions, Lect. Notes Math. 1727 (Springer, New York, 2000).
- A. P. Austin, P. Kravanja, and L. N. Trefethen, Numerical Algorithms Based on Analytic Function Values at Roots of Unity, SIAM J. Numer. Anal. 52, 1795 (2014).
- J. Wiersig, Revisiting the hierarchical construction of higher-order exceptional points, Phys. Rev. A 106, 063526 (2022).
- L. N. Trefethen and J. Weideman, The Exponentially Convergent Trapezoidal Rule, SIAM Rev. 56, 385 (2014).
- F. Betz, F. Binkowski, and S. Burger, RPExpand: Software for Riesz projection expansion of resonance phenomena, SoftwareX 15, 100763 (2021).
- N. Nikolova, J. Bandler, and M. Bakr, Adjoint techniques for sensitivity analysis in high-frequency structure CAD, IEEE Trans. Microw. Theory Techn. 52, 403 (2004).
- A. Griewank, K. Kulshreshtha, and A. Walther, On the numerical stability of algorithmic differentiation, Computing 94, 125 (2012).
- A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (SIAM, Philadelphia, 2008).
- U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation (SIAM, Philadelphia, 2012).
- A. Griewank and A. Walther, Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation, ACM Trans. Math. Softw. 26, 19 (2000).
- B. Bell, CppAD, https://github.com/coin-or/CppAD.
- A. Walther and A. Griewank, Getting started with ADOL-C, in Combinatorial Scientific Computing, edited by U. Naumann and O. Schenk (Chapman-Hall CRC Computational Science, 2012) Chap. 7, pp. 181–202.
- M. Hentschel and K. Richter, Quantum chaos in optical systems: The annular billiard, Phys. Rev. E 66, 056207 (2002).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.