Subluminal and superluminal velocities of free-space photons
DOI (Low Temperature Physics):
https://doi.org/10.1063/10.0044812Ключові слова:
free-space photons, wave packets, group velocity, phase velocityАнотація
Розглянуто прямолінійне поширення електромагнітних хвильових пакетів у вільному просторі, використовуючи теорію електромагнітного поля, скалярне поширення хвильових пакетів та квантово-механічний формалізм. Показано, що просторово локалізовані хвильові пакети за своєю природою характеризуються субсвітловою груповою швидкістю та надсвітловою фазовою швидкістю, добуток яких дорівнює с2. Ці швидкості також відомі як «енергетична» та «імпульсна» швидкості, запроваджені К. Мілтоном та Дж. Швінгером. Наведено загальні висновки за допомогою явних обчислень для гаусових та більш високих порядків пучків та хвильових пакетів, а також підкреслено тонкощі квантово-механічного опису, заснованого на «хвильовій функції фотона».
Посилання
1. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, London, 1960).
2. R. Y. Chiao and A. M. Steinberg, “Tunneling times and superluminality,” Prog. Opt. 37, 345 (1997). https://doi.org/10.1016/S0079-6638(08)70341-X
3. H. G. Winful, “Tunneling time, the hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. 436, 1 (2006). https://doi.org/10.1016/j.physrep.2006.09.002
4. R. W. Boyd and D. J. Gauthier, “Slow and fast light,” Prog. Opt. 43, 497 (2002). https://doi.org/10.1016/S0079-6638(02)80030-0
5. J. B. Khurgin, “Slow light in various media: A tutorial,” Adv. Opt. Photonics 2, 287 (2010). https://doi.org/10.1364/AOP.2.000287
6. M. Asano, K. Y. Bliokh, Y. P. Bliokh, A. G. Kofman, R. Ikuta, T. Yamamoto, Y. S. Kivshar, L. Yang, N. Imoto, S. K. Özdemir, and F. Nori, “Anomalous time delays and quantum weak measurements in optical micro-resonators,” Nat. Commun. 7, 13488 (2016). https://doi.org/10.1038/ncomms13488
7. J. I. Latorre, P. Pascual, and R. Tarrach, “Speed of light in non-trivial vacua,” Nucl. Phys. B 437, 60 (1995). https://doi.org/10.1016/0550-3213(94)00490-6
8. M. V. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A: Math. Theor. 43, 415302 (2010). https://doi.org/10.1088/1751-8113/43/41/415302
9. M. V. Berry, “Superluminal speeds for relativistic random waves,” J. Phys. A: Math. Theor. 45, 185308 (2012). https://doi.org/10.1088/1751-8113/45/18/185308
10. K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: A classical interpretation,” New J. Phys. 15, 073022 (2013). https://doi.org/10.1088/1367-2630/15/7/073022
11. K. Y. Bliokh and F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86, 033824 (2012). https://doi.org/10.1103/PhysRevA.86.033824
12. H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets having arbitrary group velocities in free space,” Nat. Commun. 10, 929 (2019). https://doi.org/10.1038/s41467-019-08735-8
13. M. Yessenov, L. A. Hall, K. L. Schepler, and A. F. Abouraddy, “Space-time wave packets,” Adv. Opt. Photonics 14, 455 (2022). https://doi.org/10.1364/AOP.450016
14. Z. L. Horváth, J. Vinkó, and Z. Bor, and D. von der Linde, “Acceleration of femtosecond pulses to superluminal velocities by Gouy phase shift,” Appl. Phys. B 63, 481 (1996). https://doi.org/10.1007/BF01828944
15. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002). https://doi.org/10.1103/PhysRevE.65.026606
16. M. A. Porras, I. Gonzalo, and A. Mondello, “Pulsed light beams in vacuum with superluminal and negative group velocities,” Phys. Rev. E 67, 066604 (2003). https://doi.org/10.1103/PhysRevE.67.066604
17. D. Giovannini, J. Romero, V. Potocek, G. Ferenczi, F. Speirits, S. M. Barnett, D. Faccio, and M. J. Padgett, “Spatially structured photons that travel in free space slower than the speed of light,” Science 347, 857 (2015). https://doi.org/10.1126/science.aaa3035
18. F. Bouchard, J. Harris, H. Mand, R. W. Boyd, and E. Karimi, “Observation of subluminal twisted light in vacuum,” Optica 3, 351 (2016). https://doi.org/10.1364/OPTICA.3.000351
19. G. Gouesbet and J. A. Lock, “Consequences of the angular spectrum decomposition of a focused beam, including slower than c beam propagation,” J. Quant. Spectrosc. Radiat. Transfer 178, 142 (2016). https://doi.org/10.1016/j.jqsrt.2015.10.012
20. N. D. Bareza and N. Hermosa, “Subluminal group velocity and dispersion of laguerre gauss beams in free space,” Sci. Rep. 6, 26842 (2016). https://doi.org/10.1038/srep26842
21. R. R. Alfano and D. A. Nolan, “Slowing of bessel light beam group velocity,” Opt. Commun. 361, 25 (2016). https://doi.org/10.1016/j.optcom.2015.10.016
22. P. Saari, “Reexamination of group velocities of structured light pulses,” Phys. Rev. A 97, 063824 (2018). https://doi.org/10.1103/PhysRevA.97.063824
23. N. I. Petrov, “Speed of structured light pulses in free space,” Sci. Rep. 9, 18332 (2019). https://doi.org/10.1038/s41598-019-54921-5
24. K. Y. Bliokh, “Photon centroids and their subluminal propagation,” J. Phys. A: Math. Theor. 56, 375701 (2023). https://doi.org/10.1088/1751-8121/acef7f
25. G. Gouesbet and L. A. Ambrosio, “On subluminal laser light propagation in free space: The case of Bessel–Gauss beams,” J. Quant. Spectrosc. Radiat. Transfer 354, 109858 (2026). https://doi.org/10.1016/j.jqsrt.2026.109858
26. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
27. D. E. Soper, Classical Field Theory (Wiley, New York, 1976).
28. K. Milton and J. Schwinger, Classical Electrodynamics, 2nd ed. (CRC Press, 2024).
29. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Butterworth-Heinemann, Oxford, 1980).
30. B. Ghosh, A. Daniel, B. Gorzkowski, A. Y. Bekshaev, R. Lapkiewicz, and K. Y. Bliokh, “Canonical and poynting currents in propagation and diffraction of structured light: Tutorial,” J. Opt. Soc. Am. B 41, 1276 (2024). https://doi.org/10.1364/JOSAB.522393
31. D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15 (2005). https://doi.org/10.1080/0010751042000275259
32. S. Vo, R. Gutiérrez-Cuevas, and M. A. Alonso, “Closed forms for spatiotemporal optical vortices and sagittal skyrmionic pulses,” J. Opt. 26, 095607 (2024). https://doi.org/10.1088/2040-8986/ad6a26
33. C. F. R. Caron and R. M. Potvliege, “Free-space propagation of ultrashort pulses: Space-time couplings in Gaussian pulse beams,” J. Mod. Opt. 46, 1881 (1999). https://doi.org/10.1080/09500349908231378
34. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
35. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485 (2001). https://doi.org/10.1364/OL.26.000485
36. R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643 (1983). https://doi.org/10.1364/AO.22.000643
37. I. Bialynicki-Birula, “Photon wave function,” Prog. Opt. 36, 245 (1996). https://doi.org/10.1016/S0079-6638(08)70316-0
38. I. Bialynicki-Birula and Z. Bialynicka-Birula, “The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism,” J. Phys. A: Math. Theor. 46, 053001 (2013). https://doi.org/10.1088/1751-8113/46/5/053001
39. K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1 (2015). https://doi.org/10.1016/j.physrep.2015.06.003
40. B. Thaller, The Dirac Equation (Springer, Berlin, 1992).
41. H. Bacry, Localizability and Space in Quantum Physics (Springer-Verlag, 1988).
42. 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). https://doi.org/10.1103/PhysRevA.82.063825
43. K. Y. Bliokh, M. R. Dennis, and F. Nori, “Position, spin, and orbital angular momentum of a relativistic electron,” Phys. Rev. A 96, 023622 (2017). https://doi.org/10.1103/PhysRevA.96.023622
44. S. Weinberg and E. Witten, “Limits on masless particles,” Phys. Lett. B 96, 59 (1980). https://doi.org/10.1016/0370-2693(80)90212-9
45. K. Y. Bliokh and F. Nori, “Relativistic Hall effect,” Phys. Rev. Lett. 108, 120403 (2012). https://doi.org/10.1103/PhysRevLett.108.120403
46. K. Y. Bliokh, “Orbital angular momentum of optical, acoustic, and quantum-mechanical spatiotemporal vortex pulses,” Phys. Rev. A 107, L031501 (2023). https://doi.org/10.1103/PhysRevA.107.L031501
47. K. Y. Bliokh, “On the Ehrenfest theorem and centroids of relativistic particles,” Phys. Lett. A 542, 130425 (2025). https://doi.org/10.1016/j.physleta.2025.130425
48. K. Y. Bliokh, “Expectation Pauli-Lubanski vector and intrinsic angular momentum of relativistic wavepackets,” New J. Phys. (in press) (2026). https://doi.org/10.1088/1367-2630/ae98a5