- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Extending completeness of the eigenmodes of an open system beyond its boundary, for Green's function and scattering-matrix calculations

Z. Sztranyovszky1,2, W. Langbein1, and E. A. Muljarov1

PDF

Phys. Rev. Research 7, L012035 – Published 18 February, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L012035

Abstract

The asymptotic completeness of a set of the eigenmodes of an open system with increasing number of modes enables an accurate calculation of the system response in terms of these modes. Using the exact eigenmodes, such completeness is limited to the interior of the system. Here we show that when the eigenmodes of a target system are obtained by the resonant-state expansion, using the modes of a basis system embedding the target system, the completeness extends beyond the boundary of the target system. We illustrate this by using the Mittag-Leffler series of the Green's function expressed in terms of the eigenmodes, which converges to the correct solution anywhere within the basis system, including the space outside the target system. Importantly, this property allows one to treat perturbations outside the target system and to calculate the scattering cross-section using the boundary conditions for the basis system. Choosing a basis system of spherical geometry, these boundary conditions have simple analytical expressions, allowing for an efficient calculation of the response of the target system, as we demonstrate for a resonator in a form of a finite dielectric cylinder.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (40)

  1. G. Mie, Beiträge zur optik trüber medien, speziell kolloidaler metallösungen, Ann. Phys. 330, 377 (1908).
  2. C. F. Bohren and D. F. Huffman, Absorption and scattering of light by small particles (John Wiley and Sons, New York, 1998).
  3. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications Inc., New York, 1998).
  4. F. Michael Kahnert, Numerical methods in electromagnetic scattering theory, J. Quant. Spectrosc. Radiat. Transfer 79-80, 775 (2003).
  5. Y. Wang, A. Zilli, Z. Sztranyovszky, W. Langbein, and P. Borri, Quantitative optical microspectroscopy, electron microscopy, and modelling of individual silver nanocubes reveal surface compositional changes at the nanoscale, Nanoscale Adv. 2, 2485 (2020).
  6. Y. Wang, Z. Sztranyovszky, A. Zilli, W. Albrecht, S. Bals, P. Borri, and W. Langbein, Quantitatively linking morphology and optical response of individual silver nanohedra, Nanoscale 14, 11028 (2022).
  7. M. B. Doost, W. Langbein, and E. A. Muljarov, Resonant-state expansion applied to planar open optical systems, Phys. Rev. A 85, 023835 (2012).
  8. T. Weiss and E. A. Muljarov, How to calculate the pole expansion of the optical scattering matrix from the resonant states, Phys. Rev. B 98, 085433 (2018).
  9. Q. Bai, M. Perrin, C. Sauvan, J.-P. Hugonin, and P. Lalanne, Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure, Opt. Express 21, 27371 (2013).
  10. C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators, Phys. Rev. Lett. 110, 237401 (2013).
  11. S. V. Lobanov, W. Langbein, and E. A. Muljarov, Resonant-state expansion of three-dimensional open optical systems: Light scattering, Phys. Rev. A 98, 033820 (2018).
  12. E. A. Muljarov and W. Langbein, Exact mode volume and Purcell factor of open optical systems, Phys. Rev. B 94, 235438 (2016).
  13. P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, Light interaction with photonic and plasmonic resonances, Laser Photonics Rev. 12, 1700113 (2018).
  14. S. Both and T. Weiss, Resonant states and their role in nanophotonics, Semicond. Sci. Technol. 37, 013002 (2022).
  15. P. T. Leung, S. Y. Liu, and K. Young, Completeness and orthogonality of quasinormal modes in leaky optical cavities, Phys. Rev. A 49, 3057 (1994).
  16. P. T. Leung and K. M. Pang, Completeness and time-independent perturbation of morphology-dependent resonances in dielectric spheres, J. Opt. Soc. Am. B 13, 805 (1996).
  17. P. T. Leung, S. S. Tong, and K. Young, Two-component eigenfunction expansion for open systems described by the wave equation I: Completeness of expansion, J. Phys. A: Math. Gen. 30, 2139 (1997).
  18. K. M. Lee, P. T. Leung, and K. M. Pang, Dyadic formulation of morphology-dependent resonances. I. Completeness relation, J. Opt. Soc. Am. B 16, 1409 (1999).
  19. C. Sauvan, T. Wu, R. Zarouf, E. A. Muljarov, and P. Lalanne, Normalization, orthogonality, and completeness of quasinormal modes of open systems: The case of electromagnetism [Invited], Opt. Express, 30, 6846 (2022).
  20. E. A. Muljarov, Full electromagnetic Green's dyadic of spherically symmetric open optical systems and elimination of static modes from the resonant-state expansion, Phys. Rev. A 101, 053854 (2020).
  21. W. Yan, R. Faggiani, and P. Lalanne, Rigorous modal analysis of plasmonic nanoresonators, Phys. Rev. B 97, 205422 (2018).
  22. R. Colom, R. McPhedran, B. Stout, and N. Bonod, Modal expansion of the scattered field: Causality, nondivergence, and nonresonant contribution, Phys. Rev. B 98, 085418 (2018).
  23. M. Ismail Abdelrahman and B. Gralak, Completeness and divergence-free behavior of the quasi-normal modes using causality principle, OSA Continuum 1, 340 (2018).
  24. P. T. Kristensen, K. Herrmann, F. Intravaia, and K. Busch, Modeling electromagnetic resonators using quasinormal modes, Adv. Opt. Photon. 12, 612 (2020).
  25. M. Kamandar Dezfouli and S. Hughes, Regularized quasinormal modes for plasmonic resonators and open cavities, Phys. Rev. B 97, 115302 (2018).
  26. S. Franke, J. Ren, and S. Hughes, Impact of mode regularization for quasinormal-mode perturbation theories, Phys. Rev. A 108, 043502 (2023).
  27. E. A. Muljarov, W. Langbein, and R. Zimmermann, Brillouin-Wigner perturbation theory in open electromagnetic systems, Europhys. Lett. 92, 50010 (2010).
  28. M. B. Doost, W. Langbein, and E. A. Muljarov, Resonant-state expansion applied to three-dimensional open optical systems, Phys. Rev. A 90, 013834 (2014).
  29. S. V. Lobanov, W. Langbein, and E. A. Muljarov, Resonant-state expansion applied to three-dimensional open optical systems: Complete set of static modes, Phys. Rev. A 100, 063811 (2019).
  30. S. F. Almousa and E. A. Muljarov, Exact theory and approximations for optical resonators in a changing external medium, Phys. Rev. B 107, L081401 (2023).
  31. Z. Sztranyovszky, W. Langbein, and E. A. Muljarov, Optical resonances in graded index spheres: A resonant-state-expansion study and analytic approximations, Phys. Rev. A 105, 033522 (2022).
  32. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.7.L012035 for details of the method, derivations, and additional data supporting the claims of this letter, containing: Secs. S.I. review of resonant-state expansion; S.II. modes for l=10; S.III. details on VG modes; S.IV. Mittag-Leffler expansion of the Green's dyadic; S.V. scattering matrix in terms of the Green's dyadic; S.VI. Green's dyadic of the Maxwell's equation and of the wave equation; S.VII. effect of modes excluded from the spectra; S.VIII convergence study of the error.
  33. C. P. Dettmann, G. V. Morozov, M. Sieber, and H. Waalkens, Internal and external resonances of dielectric disks, Europhys. Lett. 87, 34003 (2009).
  34. K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide, 3rd ed. (Cambridge University Press, Cambridge, 2006).
  35. G. Demésy, T. Wu, Yoann Brûlé, F. Zolla, A. Nicolet, P. Lalanne, and B. Gralak, Dispersive perfectly matched layers and high-order absorbing boundary conditions for electromagnetic quasinormal modes, J. Opt. Soc. Am. A 40, 1947 (2023).
  36. Z. Ruan and S. Fan, Temporal coupled-mode theory for light scattering by an arbitrarily shaped object supporting a single resonance, Phys. Rev. A 85, 043828 (2012).
  37. C. Sauvan, Quasinormal modes expansions for nanoresonators made of absorbing dielectric materials: Study of the role of static modes, Opt. Express 29, 8268 (2021).
  38. M. Besbes and C. Sauvan, Role of static modes in quasinormal modes expansions: When and how to take them into account? Mathematics 10, 3542 (2022).
  39. COMSOL Multiphysics®, www.comsol.com, COMSOL AB, Stockholm, Sweden.
  40. https://doi.org/10.17035/cardiff.27108967.

Outline

Information

Phys. Rev. Research 7, L012035– Published 18 February, 2025
  • Received 27 September 2024
  • Accepted 31 January 2025
Crossmark - Check for updates button
Creative Commons Logo

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

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation