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  • Open Access

Exceptional-point-controlled mode interaction in three-dimensional microcavities represented by generalized Husimi functions

Tom Simon Rodemund

Shilong Li

Síle Nic Chormaic

Martina Hentschel*

  • *Contact author: martina.hentschel@physik.tu-chemnitz.de
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Phys. Rev. A 112, 033528 – Published 17 September, 2025

DOI: https://doi.org/10.1103/2wrn-z5fg

Abstract

Non-Hermitian photonics has attracted significant interest and influences several key areas such as optical metamaterials, laser physics, and nonlinear optics. While non-Hermitian effects have been widely addressed in two-dimensional (2D) systems, we focus on realistic three-dimensional devices. To this end we generalize established phase-space methods from mesoscopic optics and introduce Husimi functions for three-dimensional systems that deepen the insight and access to the mode morphology and their dynamics. We illustrate that four-dimensional Husimi functions can be represented using a specific projection on two dimensions and illustrate it for (conical) cylindrical cavities. The non-Hermitian character of the intrinsically open photonic systems is in particular revealed when examining the transverse electric and transverse magnetic polarization character of the resonance modes. Unlike the 2D case, polarization is not conserved in three-dimensional cavities, and we use generalized Husimi function to represent the interaction of polarization modes. We find their dynamics to be ruled by a network of exceptional points in the parameter space spanned by the refractive index and the cavity geometry tilt angle. This approach not only enhances our understanding of cavity modes but also aids in the design of more efficient photonic devices and systems.

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References (78)

  1. K. J. Vahala, Optical microcavities, Nature (London) 424, 839 (2003).
  2. S. Yang, Y. Wang, and H. Sun, Advances and prospects for whispering gallery mode microcavities, Adv. Opt. Mater. 3, 1136 (2015).
  3. H. Cao and J. Wiersig, Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics, Rev. Mod. Phys. 87, 61 (2015).
  4. H.-J. Stöckmann, Quantum Chaos (Cambridge University Press, Cambridge, England, 1999).
  5. J. U. Nöckel and A. D. Stone, Ray and wave chaos in asymmetric resonant optical cavities, Nature (London) 385, 45 (1997).
  6. M. Hentschel and K. Richter, Quantum chaos in optical systems: The annular billiard, Phys. Rev. E 66, 056207 (2002).
  7. R. Ketzmerick, K. Clauß, F. Fritzsch, and A. Bäcker, Chaotic resonance modes in dielectric cavities: Product of conditionally invariant measure and universal fluctuations, Phys. Rev. Lett. 129, 193901 (2022).
  8. M. Hentschel and T.-Y. Kwon, Designing and understanding directional emission from spiral microlasers, Opt. Lett. 34, 163 (2009).
  9. C. Yan, Q. J. Wang, L. Diehl, M. Hentschel, J. Wiersig, N. Yu, C. Pflügl, F. Capasso, M. A. Belkin, T. Edamura et al., Directional emission and universal far-field behavior from semiconductor lasers with limaçon-shaped microcavity, Appl. Phys. Lett. 94, 251101 (2009).
  10. Q. Jie Wang, C. Yan, L. Diehl, M. Hentschel, J. Wiersig, N. Yu, C. Pflügl, M. A. Belkin, T. Edamura, M. Yamanishi et al., Deformed microcavity quantum cascade lasers with directional emission, New J. Phys. 11, 125018 (2009).
  11. Y. Wu, J. M. Ward, and S. Nic Chormaic, Ultralow threshold green lasing and optical bistability in ZBNA (ZrF4−BaF2−NaF−AlF3⁠⁠) microspheres, J. Appl. Phys. 107, 033103 (2010).
  12. T. Harayama and S. Shinohara, Two-dimensional microcavity lasers, Laser Photonics Rev. 5, 247 (2011).
  13. H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, Cavity quantum electrodynamics, Rep. Prog. Phys. 69, 1325 (2006).
  14. J. C. Owens, M. G. Panetta, B. Saxberg, G. Roberts, S. Chakram, R. Ma, A. Vrajitoarea, J. Simon, and D. I. Schuster, Chiral cavity quantum electrodynamics, Nat. Phys. 18, 1048 (2022).
  15. M. A. D. Taylor, A. Mandal, and P. Huo, Light–matter interaction Hamiltonians in cavity quantum electrodynamics, Chem. Phys. Rev. 6, 011305 (2025).
  16. M. D. Baaske, M. R. Foreman, and F. Vollmer, Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform, Nat. Nanotechnol. 9, 933 (2014).
  17. Y. Zhi, X.-C. Yu, Q. Gong, L. Yang, and Y.-F. Xiao, Single nanoparticle detection using optical microcavities, Adv. Mater. 29, 1604920 (2017).
  18. M. R. Foreman, D. Keng, E. Treasurer, J. R. Lopez, and S. Arnold, Whispering gallery mode single nanoparticle detection and sizing: the validity of the dipole approximation, Opt. Lett. 42, 963 (2017).
  19. T. Kato, Perturbation Theory for Linear Operators (Springer Science & Business Media, New York, 1976), Vol. 132.
  20. I. Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems, J. Phys. A: Math. Theor. 42, 153001 (2009).
  21. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  22. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, Parity-time–symmetric microring lasers, Science 346, 975 (2014).
  23. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Science 346, 972 (2014).
  24. P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, Orbital angular momentum microlaser, Science 353, 464 (2016).
  25. B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yılmaz, J. Wiersig, S. Rotter, and L. Yang, Chiral modes and directional lasing at exceptional points, Proc. Natl. Acad. Sci. USA 113, 6845 (2016).
  26. S. Longhi, Non-Hermitian gauged topological laser arrays, Ann. Phys. 530, 1800023 (2018).
  27. B. Zhu, Q. Wang, D. Leykam, H. Xue, Q. J. Wang, and Y. D. Chong, Anomalous single-mode lasing induced by nonlinearity and the non-Hermitian skin effect, Phys. Rev. Lett. 129, 013903 (2022).
  28. J. Wiersig, Sensors operating at exceptional points: General theory, Phys. Rev. A 93, 033809 (2016).
  29. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature (London) 548, 192 (2017).
  30. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature (London) 548, 187 (2017).
  31. J. Wiersig, Review of exceptional point-based sensors, Photonics Res. 8, 1457 (2020).
  32. R. Kononchuk, J. Cai, F. Ellis, R. Thevamaran, and T. Kottos, Exceptional-point-based accelerometers with enhanced signal-to-noise ratio, Nature (London) 607, 697 (2022).
  33. W. Mao, Z. Fu, Y. Li, F. Li, and L. Yang, Exceptional–point–enhanced phase sensing, Sci. Adv. 10, eadl5037 (2024).
  34. M. Parto, C. Leefmans, J. Williams, R. M. Gray, and A. Marandi, Enhanced sensitivity via non-Hermitian topology, Light Sci. Appl. 14, 6 (2025).
  35. M. Yang, Z. Ye, H. Pan, M. Farhat, A. E. Cetin, and P.-Y. Chen, Electromagnetically unclonable functions generated by non-Hermitian absorber-emitter, Sci. Adv. 9, eadg7481 (2023).
  36. H. Zhao, X. Qiao, T. Wu, B. Midya, S. Longhi, and L. Feng, Non-Hermitian topological light steering, Science 365, 1163 (2019).
  37. T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, and F. Nori, Second-order topological phases in non-Hermitian systems, Phys. Rev. Lett. 122, 076801 (2019).
  38. S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Topological funneling of light, Science 368, 311 (2020).
  39. S. Xia, D. Kaltsas, D. Song, I. Komis, J. Xu, A. Szameit, H. Buljan, K. G. Makris, and Z. Chen, Nonlinear tuning of PT symmetry and non-Hermitian topological states, Science 372, 72 (2021).
  40. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  41. S. Sunada and T. Harayama, Design of resonant microcavities: application to optical gyroscopes, Opt. Express 15, 16245 (2007).
  42. W. Liang, V. S. Ilchenko, A. A. Savchenkov, E. Dale, D. Eliyahu, A. B. Matsko, and L. Maleki, Resonant microphotonic gyroscope, Optica 4, 114 (2017).
  43. C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, Accessing the exceptional points of parity-time symmetric acoustics, Nat. Commun. 7, 11110 (2016).
  44. Z. Gu, H. Gao, H. Xue, J. Li, Z. Su, and J. Zhu, Transient non-Hermitian skin effect, Nat. Commun. 13, 7668 (2022).
  45. H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topological energy transfer in an optomechanical system with exceptional points, Nature (London) 537, 80 (2016).
  46. H. Yang, C. Wang, T. Yu, Y. Cao, and P. Yan, Antiferromagnetism emerging in a ferromagnet with gain, Phys. Rev. Lett. 121, 197201 (2018).
  47. Y. Xu, L. Li, H. Jeong, S. Kim, I. Kim, J. Rho, and Y. Liu, Subwavelength control of light transport at the exceptional point by non-Hermitian metagratings, Sci. Adv. 9, eadf3510 (2023).
  48. W. Deng, W. Zhu, T. Chen, H. Sun, and X. Zhang, Ultrasensitive integrated circuit sensors based on high-order non-Hermitian topological physics, Sci. Adv. 10, eadp6905 (2024).
  49. K. Husimi, Some formal properties of the density matrix, Proc. Phys. Soc. Japan. 22, 264 (1940).
  50. W. P. Schleich, Quantum Optics in Phase Space (John Wiley & Sons, New York, 2011).
  51. S. Nonnenmacher and A. Voros, Chaotic eigenfunctions in phase space, J. Stat. Phys. 92, 431 (1998).
  52. A. Bäcker, S. Fürstberger, and R. Schubert, Poincaré Husimi representation of eigenstates in quantum billiards, Phys. Rev. E 70, 036204 (2004).
  53. B. Crespi, G. Perez, and S.-J. Chang, Quantum Poincaré sections for two-dimensional billiards, Phys. Rev. E 47, 986 (1993).
  54. R. Hofferbert, H. Alt, C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, H. Rehfeld, and A. Richter, Experimental investigations of chaos-assisted tunneling in a microwave annular billiard, Phys. Rev. E 71, 046201 (2005).
  55. M. Hentschel, H. Schomerus, and R. Schubert, Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond, Europhys. Lett. 62, 636 (2003).
  56. S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, Scarred resonances and steady probability distribution in a chaotic microcavity, Phys. Rev. A 72, 061801(R) (2005).
  57. H. Schomerus and M. Hentschel, Correcting ray optics at curved dielectric microresonator interfaces: Phase-space unification of Fresnel filtering and the Goos-Hänchen shift, Phys. Rev. Lett. 96, 243903 (2006).
  58. J. Unterhinninghofen, J. Wiersig, and M. Hentschel, Goos-Hänchen shift and localization of optical modes in deformed microcavities, Phys. Rev. E 78, 016201 (2008).
  59. S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C.-M. Kim, Ray and wave dynamical properties of a spiral-shaped dielectric microcavity, J. Phys. A: Math. Theor. 41, 275102 (2008).
  60. J. Kullig and J. Wiersig, Frobenius–Perron eigenstates in deformed microdisk cavities: Non-Hermitian physics and asymmetric backscattering in ray dynamics, New J. Phys. 18, 015005 (2016).
  61. T. S. Rodemund, S. Nic Chormaic, and M. Hentschel, Coupled deformed microdisk cavities featuring non-Hermitian properties, Appl. Phys. Lett. 124, 051107 (2024).
  62. J. Wiersig and M. Hentschel, Combining directional light output and ultralow loss in deformed microdisks, Phys. Rev. Lett. 100, 033901 (2008).
  63. Y.-J. Qian, H. Liu, Q.-T. Cao, J. Kullig, K. Rong, C.-W. Qiu, J. Wiersig, Q. Gong, J. Chen, and Y.-F. Xiao, Regulated photon transport in chaotic microcavities by tailoring phase space, Phys. Rev. Lett. 127, 273902 (2021).
  64. X. Jiang, S. Yin, H. Li, J. Quan, H. Goh, M. Cotrufo, J. Kullig, J. Wiersig, and A. Alù, Coherent control of chaotic optical microcavity with reflectionless scattering modes, Nat. Phys. 20, 109 (2024).
  65. M. Bosch, A. Behrens, S. Sinzinger, and M. Hentschel, Husimi functions for coupled optical resonators, J. Opt. Soc. Am. A 38, 573 (2021).
  66. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, New York, 1999).
  67. S. Bittner, B. Dietz, M. Miski-Oglu, P. O. Iriarte, A. Richter, and F. Schäfer, Experimental test of a two-dimensional approximation for dielectric microcavities, Phys. Rev. A 80, 023825 (2009).
  68. M. Firmbach, S. Lange, R. Ketzmerick, and A. Bäcker, Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories, Phys. Rev. E 98, 022214 (2018).
  69. J.-W. Ryu, S.-Y. Lee, and S. W. Kim, Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission, Phys. Rev. A 79, 053858 (2009).
  70. I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, Hexagonal microlasers based on organic dyes in nanoporous crystals, Appl. Phys. B 70, 335 (2000).
  71. J. Wiersig, Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities, Phys. Rev. Lett. 97, 253901 (2006).
  72. W. D. Heiss, Repulsion of resonance states and exceptional points, Phys. Rev. E 61, 929 (2000).
  73. W. D. Heiss, Exceptional points of non-Hermitian operators, J. Phys. A: Math. Gen. 37, 2455 (2004).
  74. J. Wiersig, Response strengths of open systems at exceptional points, Phys. Rev. Res. 4, 023121 (2022).
  75. J. Kullig, D. Grom, S. Klembt, and J. Wiersig, Higher-order exceptional points in waveguide-coupled microcavities: Perturbation induced frequency splitting and mode patterns, Photonics Res. 11, A54 (2023).
  76. C.-H. Yi, J. Kullig, M. Hentschel, and J. Wiersig, Non-Hermitian degeneracies of internal–external mode pairs in dielectric microdisks, Photonics Res. 7, 464 (2019).
  77. V. Igoshin, M. Tsimokha, A. Nikitina, M. Petrov, I. Toftul, and K. Frizyuk, Exceptional points in single open acoustic resonator due to symmetry breaking, Phys. Rev. B 109, 144102 (2024).
  78. T.-Y. Kwon, S.-Y. Lee, J.-W. Ryu, and M. Hentschel, Phase-space analysis of lasing modes in a chaotic microcavity, Phys. Rev. A 88, 023855 (2013).

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Phys. Rev. A 112, 033528– Published 17 September, 2025
  • Received 28 March 2025
  • Accepted 5 August 2025
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