Computing eigenfrequency sensitivities near exceptional points

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Computing eigenfrequency sensitivities near exceptional points

Felix Binkowski, Julius Kullig, Fridtjof Betz, Lin Zschiedrich, Andrea Walther, Jan Wiersig, and Sven Burger

Phys. Rev. Research 6, 023148 – Published 9 May 2024

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 for an optical microdisk cavity and derive a semianalytical 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.

Received 27 February 2024
Revised 12 April 2024
Accepted 17 April 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.023148

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Photonics

Non-Hermitian systems

Electromagnetic wave theory Maxwell's equations

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Felix Binkowski ¹, Julius Kullig ², Fridtjof Betz ¹, Lin Zschiedrich ³, Andrea Walther ^1,4, Jan Wiersig ², and Sven Burger ^1,3

¹Zuse Institute Berlin, 14195 Berlin, Germany
²Institut für Physik, Otto-von-Guericke-Universität Magdeburg, 39106 Magdeburg, Germany
³JCMwave GmbH, 14050 Berlin, Germany
⁴Department of Mathematics, Humboldt-Universität zu Berlin, 10099 Berlin, Germany

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Issue

Vol. 6, Iss. 2 — May - July 2024

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Images

Figure 1
Eigenfrequencies $ω_{l}$ and their sensitivities $\partial ω_{l} / \partial p$ in the complex frequency plane with respect to a parameter $p$ of an optical microdisk cavity. A specific change in the parameter causes the eigenfrequencies and the associated eigenmodes to coalesce at the EP. The sensitivities of the eigenfrequencies diverge near the EP.
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Figure 2
Optical microdisk cavity with an EP of second order. (a) Sketch of the two-dimensional system. Two concentric layers of different refractive indices form the cavity. (b) Electric field intensity of the eigenmode $E_{8, 1}^{z}$ corresponding to the EP. The associated eigenfrequency is given by $ω_{8, 1} \approx ω_{8, 2} = (6.96185 - 0.089761 i) c / R$ . The underlying parameters of the microdisk are $n_{1} = 3.1239791, n_{2} = 1.5$ , and $R_{1} = 0.4970147 R$ .
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Figure 3
Variation of the parameter $R_{1}$ near the EP with the parameters $n_{1} = 3.1239791, n_{2} = 1.5$ , and $R_{1} = 0.4970147 R$ . For each $R_{1}$ , two eigenfrequencies, $ω_{8, 1}$ and $ω_{8, 2}$ , are obtained. The values for $R_{1}$ result from an equidistant spacing in the interval $[0.999 R_{1}, 1.001 R_{1}]$ with 50 points, where the first point leads to $ω_{8, 1}^{A}$ and $ω_{8, 2}^{A}$ and the $25 th$ point leads to $ω_{8, 1}^{B}$ and $ω_{8, 2}^{B}$ . The numerical results can be compared with the semianalytical solution. (a) Real parts of the eigenfrequencies $ω_{8, l}$ . (b) Imaginary parts of $ω_{8, l}$ . (c) Real parts of the eigenfrequency sensitivities $\partial ω_{8, l} / \partial R_{1}$ . (d) Imaginary parts of $\partial ω_{8, l} / \partial R_{1}$ .
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Figure 4
Computing eigenfrequencies and their sensitivities near the EP. (a) Complex frequency plane with integration contour $C$ . The center of the contour is at $ω^{EP}$ , and the radius is given by $0.01 Re (ω^{EP})$ . Four eigenfrequencies from Fig. 3 are shown, where $ω_{8, 1}^{A}$ and $ω_{8, 2}^{A}$ or $ω_{8, 1}^{B}$ and $ω_{8, 2}^{B}$ are the result of a contour integration with a different $R_{1}$ in each case. (b) Relative error $| ω_{8, 1} - ω_{qex} | / | ω_{qex} |$ with respect to the number of integration points on the contour $C$ , where $ω_{qex}$ is the quasiexact solution computed with 64 integration points. The maximum side length of the FEM mesh is $0.05 R$ . (c) Relative error with respect to maximum side length of the FEM mesh, where $ω_{qex}$ is the quasiexact solution computed with a maximum side length of $0.0125 R$ . The number of integration points is 16. The legend is the same as in (b).
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Figure 5
Tracking the EP in the parameter space. (a) Numerical and semianalytical results for the trajectory of the eigenfrequency $ω^{EP}$ . The arrow indicates the eigenfrequency $ω^{EP}$ from Ref. [60]. (b) Corresponding parameter combinations. For each point in the parameter space, the parameter $n_{2}$ is shifted by 0.0025.
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