- 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

Conservation law for angular momentum based on optical field derivatives: Analysis of optical spin-orbit conversion

Shun Hashiyada* and Yoshito Y. Tanaka†

  • Laboratory of Nanosystem Optical Manipulation, Department of Photonics and Optical Science, Research Institute for Electronic Science, Hokkaido University, Nishi 10, Kita 21, Kita-ku, Sapporo, Hokkaido 001-0021, Japan

  • *Contact author: shun.hashiyada@es.hokudai.ac.jp
  • †Contact author: ytanaka@es.hokudai.ac.jp
PDF

Phys. Rev. Research 7, L022052 – Published 4 June, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L022052

Abstract

We present a theoretical framework for analyzing the loss of optical angular momentum (AM), including spin AM (SAM) and orbital AM (OAM) components, in light-matter interactions. Conventional SAM and OAM conservation laws rely on transverse field components, neglecting longitudinal fields and limiting applicability to a vacuum. Our approach defines optical AM using time derivatives of the electric and magnetic fields, yielding a gauge-invariant formulation that includes both transverse and longitudinal components and explicitly incorporates charge and current densities into SAM and OAM conservation laws. This enables a more complete description of AM dissipation in materials. We apply this framework to analyze spin-orbit conversion (SOC) in two scenarios: the scattering of circularly polarized (CP) Gaussian beams by a gold nanoparticle and focusing of CP Gaussian beams and linearly polarized optical vortex beams by a lens. The results show that SOC depends on particle size and polarization, with notable OAM loss in larger particles and CP Gaussian beam focusing. This framework enables the evaluation of previously overlooked SAM and OAM losses, providing a powerful tool for studying systems in which the analysis of AM losses is intrinsically important, such as chiral materials, as well as for designing photonic devices and exploring light-matter interactions at the nanoscale.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (58)

  1. J. H. Poynting, On the transfer of energy in the electromagnetic field, Philos. Trans. R. Soc. Lond. 175, 343 (1884).
  2. D. M. Lipkin, Existence of a new conservation law in electromagnetic theory, J. Math. Phys. 5, 696 (1964).
  3. Y. Tang and A. E. Cohen, Optical chirality and its interaction with matter, Phys. Rev. Lett. 104, 163901 (2010).
  4. C. G. Darwin, Notes on the theory of radiation, Proc. R. Soc. Lond. Ser. A 136, 36 (1932).
  5. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1951).
  6. J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, New York, 1962).
  7. P. H. Jones, O. M. Maragò, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge University Press, Cambridge, England, 2015).
  8. J. H. Poynting, The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light, Proc. R. Soc. Lond. Ser. A 82, 560 (1909).
  9. R. A. Beth, Mechanical detection and measurement of the angular momentum of light, Phys. Rev. 50, 115 (1936).
  10. P. L. Marston and J. H. Crichton, Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave, Phys. Rev. A 30, 2508 (1984).
  11. E. Santamato, B. Daino, M. Romagnoli, M. Settembre, and Y. R. Shen, Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave, Phys. Rev. Lett. 57, 2423 (1986).
  12. T. Sugiura, S. Kawata, and S. Minami, Optical rotation of small particles by a circularly-polarized laser beam in an optical microscope, J. Spectrosc. Soc. Jpn. 39, 342 (1990).
  13. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Optical rotation of small particles by a circularly-polarized laser beam in an optical microscope, Opt. Lett. 23, 1 (1998).
  14. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms, J. Mod. Opt. 42, 217 (1995).
  15. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity, Phys. Rev. Lett. 75, 826 (1995).
  16. N. B. Simpson, L. Allen, and M. J. Padgett, Optical tweezers and optical spanners with Laguerre–Gaussian modes, J. Mod. Opt. 43, 2485 (1996).
  17. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner, Opt. Lett. 22, 52 (1997).
  18. V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle, Phys. Rev. Lett. 91, 093602 (2003).
  19. Y. Zhao, D. Shapiro, D. Mcgloin, D. Chiu, and S. Marchesini, Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam, Opt. Express 17, 23316 (2009).
  20. H. Fujiwara, K. Sudo, Y. Sunaba, C. Pin, S. Ishida, and K. Sasaki, Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized gaussian beam, Nano Lett. 21, 6268 (2021).
  21. P. Rivera, K. Seyler, H. Yu, J. Schaibley, J. Yan, D. Mandrus, W. Yao, and X. Xu, Valley-polarized exciton dynamics in a 2D semiconductor heterostructure, Science 351, 688 (2016).
  22. C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. Poschinger, and F. Schmidt-Kaler, Transfer of optical orbital angular momentum to a bound electron, Nat. Commun. 7, 12998 (2016).
  23. T. Arikawa, T. Hiraoka, S. Morimoto, F. Blanchard, S. Tani, T. Tanaka, K. Sakai, H. Kitajima, K. Sasaki, and K. Tanaka, Transfer of orbital angular momentum of light to plasmonic excitations in metamaterials, Sci. Adv. 6, eaay1977 (2020).
  24. A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, Transfer of orbital angular momentum of light to plasmonic excitations in metamaterials, Phys. Rev. Lett. 90, 107404 (2003).
  25. T. Vallius, K. Jefimovs, J. Turunen, P. Vahimaa, and Y. Svirko, Optical activity in subwavelength-period arrays of chiral metallic particles, Appl. Phys. Lett. 83, 234 (2003).
  26. S. Zhang, H. Wei, K. Bao, U. Håkanson, N. J. Halas, P. Nordlander, and H. Xu, Chiral surface plasmon polaritons on metallic nanowires, Phys. Rev. Lett. 107, 096801 (2011).
  27. L. Zhang and Q. Niu, Angular momentum of phonons and the Einstein–de Haas effect, Phys. Rev. Lett. 112, 085503 (2014).
  28. H. Chen, W. Zhang, Q. Niu, and L. Zhang, Chiral phonons in two-dimensional materials, 2D Mater. 6, 012002 (2019).
  29. K. Ishito, H. Mao, Y. Kousaka, Y. Togawa, S. Iwasaki, T. Zhang, S. Murakami, J. Kishine, and T. Satoh, Truly chiral phonons in α-HgS, Nat. Phys. 19, 35 (2023).
  30. H. Ueda, M. García-Fernández, S. Agrestini, C. P. Romao, J. van den Brink, N. A. Spaldin, K.-J. Zhou, and U. Staub, Chiral phonons in quartz probed by x-rays, Nature (London) 618, 946 (2023).
  31. E. Oishi, Y. Fujii, and A. Koreeda, Selective observation of enantiomeric chiral phonons in α-quartz, Phys. Rev. B 109, 104306 (2024).
  32. A. Rodger and B. Norden, Circular Dichroism and Linear Dichroism: A Textbook on Polarized-Light Spectroscop (Oxford University Press, Oxford, 1997).
  33. N. Berova, K. Nakanishi, and R. W. Woody, Circular Dichroism: Principles and Applications (Wiley-VCH, New York, 2000).
  34. L. D. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge University Press, Cambridge, England, 2004).
  35. W. Brullot, M. K. Vanbel, T. Swusten, and T. Verbiest, Resolving enantiomers using the optical angular momentum of twisted light, Sci. Adv. 2, e1501349 (2016).
  36. K. A. Forbes and D. L. Andrews, Orbital angular momentum of twisted light: Chirality and optical activity, J. Phys. Photon. 3, 022007 (2021).
  37. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 45, 8185 (1992).
  38. L. Marrucci, C. Manzo, and D. Paparo, Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media, Phys. Rev. Lett. 96, 163905 (2006).
  39. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Spin-to-orbital angular momentum conversion in a strongly focused optical beam, Phys. Rev. Lett. 99, 073901 (2007).
  40. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, Angular momenta and spin-orbit interaction of nonparaxial light in free space, Phys. Rev. A 82, 063825 (2010).
  41. K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodriguez-Herrera, D. Lara, and C. Dainty, Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems, Opt. Express 19, 26132 (2011).
  42. S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, Orbital-to-spin angular momentum conversion employing local helicity, Phys. Rev. B 99, 075155 (2019).
  43. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, Photonic spin Hall effect at metasurfaces, Science 339, 1405 (2013).
  44. F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O'Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, Near-field interference for the unidirectional excitation of electromagnetic guided modes, Science 340, 328 (2013).
  45. K. Y. Bliokh, D. Smirnova, and F. Nori, Quantum spin Hall effect of light, Science 348, 1448 (2015).
  46. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, New York, 1989).
  47. F. Crimin, N. MacKinnon, J. Götte, and S. M. Barnett, Continuous symmetries and conservation laws in chiral media, in Proceedings of SPIE 11297 Complex Light and Optical Forces XIV (SPIE, San Francisco, CA, 2020), p. 112970J.
  48. M. Nieto-Vesperinas, Optical torque: Electromagnetic spin and orbital-angular-momentum conservation laws and their significance, Phys. Rev. A 92, 043843 (2015).
  49. G. Nienhuis, Conservation laws and symmetry transformations of the electromagnetic field with sources, Phys. Rev. A 93, 023840 (2016).
  50. K. Y. Bliokh, J. Dressel, and F. Nori, Conservation of the spin and orbital angular momenta in electromagnetism, New J. Phys. 16, 093037 (2014).
  51. R. P. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Boston, 1964).
  52. T. A. Morgan, Two classes of new conservation laws for the electromagnetic field and for other massless fields, J. Math. Phys. 5, 1659 (1964).
  53. T. G. Philbin, Lipkin's conservation law, Noether's theorem, and the relation to optical helicity, Phys. Rev. A 87, 043843 (2013).
  54. I. Proskurin, A. S. Ovchinnikov, P. Nosov, and J. Kishine, Optical chirality in gyrotropic media: symmetry approach, New J. Phys. 19, 063021 (2017).
  55. A. A. Wu, Y. Y. Tanaka, R. Fukuhara, and T. Shimura, Continuity equation for spin angular momentum in relation to optical chirality, Phys. Rev. A 102, 023531 (2020).
  56. K. Cho, Reconstruction of Macroscopic Maxwell Equations: A Single Susceptibility Theory (Springer, Berlin, 2010).
  57. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, New York, 2004).
  58. S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, Optical skyrmion lattice in evanescent electromagnetic fields, Science 361, 993 (2018).

Outline

Information

Phys. Rev. Research 7, L022052– Published 4 June, 2025
  • Received 18 October 2024
  • Revised 4 April 2025
  • Accepted 16 May 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