Abstract
Refraction at the interface between two materials is fundamental to the interaction of light with photonic devices and to the propagation of light through the atmosphere at large1. Underpinning the traditional rules for the refraction of an optical field is the tacit presumption of the separability of its spatial and temporal degrees of freedom. We show here that endowing a pulsed beam with precise spatiotemporal spectral correlations2,3,4 unveils remarkable refractory phenomena, such as group-velocity invariance with respect to the refractive index, group-delay cancellation, anomalous group-velocity increase in higher-index materials, and tunable group velocity by varying the angle of incidence. A law of refraction for ‘spacetime’ (ST) wave packets5,6,7,8,9,10 encompassing these effects is verified experimentally in a variety of optical materials. Spacetime refraction defies our expectations derived from Fermat’s principle and offers new opportunities for moulding the flow of light and other wave phenomena.
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Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
References
Saleh, B. E. A. & Teich, M. C. Principles of Photonics (Wiley, 2007).
Donnelly, R. & Ziolkowski, R. Designing localized waves. Proc. R. Soc. Lond. A 440, 541–565 (1993).
Longhi, S. Gaussian pulsed beams with arbitrary speed. Opt. Express 12, 935–940 (2004).
Saari, P. & Reivelt, K. Generation and classification of localized waves by Lorentz transformations in Fourier space. Phys. Rev. E 69, 036612 (2004).
Kondakci, H. E. & Abouraddy, A. F. Diffraction-free pulsed optical beams via space-time correlations. Opt. Express 24, 28659–28668 (2016).
Parker, K. J. & Alonso, M. A. The longitudinal iso-phase condition and needle pulses. Opt. Express 24, 28669–28677 (2016).
Wong, L. J. & Kaminer, I. Ultrashort tilted-pulsefront pulses and nonparaxial tilted-phase-front beams. ACS Photon. 4, 2257–2264 (2017).
Porras, M. A. Gaussian beams diffracting in time. Opt. Lett. 42, 4679–4682 (2017).
Kondakci, H. E. & Abouraddy, A. F. Diffraction-free space-time beams. Nat. Photon. 11, 733–740 (2017).
Kondakci, H. E. & Abouraddy, A. F. Optical space-time wave packets of arbitrary group velocity in free space. Nat. Commun. 10, 929 (2019).
Sabra, A. I. Theories of Light from Descartes to Newton (Cambridge University Press, 1981).
Koenderink, A. F., Alú, A. & Polman, A. Nanophotonics: shrinking light-based technology. Science 348, 516–521 (2015).
Yessenov, M., Bhaduri, B., Kondakci, H. E. & Abouraddy, A. F. Classification of propagation-invariant space-time light-sheets in free space: theory and experiments. Phys. Rev. A 99, 023856 (2019).
Saari, P. Reexamination of group velocities of structured light pulses. Phys. Rev. A 97, 063824 (2018).
Besieris, I. M., Shaarawi, A. M. & Ziolkowski, R. W. A bidirectional travelling plane representation of exact solutions of the scalar wave equation. J. Math. Phys. 30, 1254–1269 (1989).
Saari, P. & Reivelt, K. Evidence of X-shaped propagation-invariant localized light waves. Phys. Rev. Lett. 79, 4135–4138 (1997).
Salo, J. & Salomaa, M. M. Diffraction-free pulses at arbitrary speeds. J. Opt. A 3, 366–373 (2001).
Turunen, J. & Friberg, A. T. Propagation-invariant optical fields. Prog. Opt. 54, 1–88 (2010).
Hernández-Figueroa, H. E., Recami, E. & Zamboni-Rached, M. (eds) Non-diffracting Waves (Wiley-VCH, 2014).
Efremidis, N. K. Spatiotemporal diffraction-free pulsed beams in free-space of the Airy and Bessel type. Opt. Lett. 42, 5038–5041 (2017).
Bhaduri, B., Yessenov, M. & Abouraddy, A. F. Space–time wave packets that travel in optical materials at the speed of light in vacuum. Optica 6, 139–146 (2019).
Faccio, D. et al. Spatio-temporal reshaping and X wave dynamics in optical filaments. Opt. Express 15, 13077–13095 (2007).
Hillion, P. How do focus wave modes propagate across a discontinuity in a medium? Optik 93, 67–72 (1993).
Donnelly, R. & Power, D. The behavior of electromagnetic localized waves at a planar interface. IEEE Trans. Antennas Propag. 45, 580–591 (1997).
Attiya, A. M., El-Diwany, E., Shaarawi, A. M. & Besieris, I. M. Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation. Prog. Electromagn. Res. 30, 191–211 (2001).
Salem, M. A. & Bağcí, H. Reflection and transmission of normally incident full-vector X waves on planar interfaces. J. Opt. Soc. Am. A 29, 139–152 (2012).
Bhaduri, B. et al. Broadband space-time wave packets propagating 70 m. Opt. Lett. 44, 2073–2076 (2019).
Yessenov, M. et al. What is the maximum differential group delay achievable by a space-time wave packet in free space? Opt. Express 27, 12443–12457 (2019).
Liberal, I. & Engheta, N. Near-zero refractive index photonics. Nat. Photon. 11, 149–158 (2017).
Yu, N. et al. Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science 334, 333–337 (2011).
Acknowledgements
We thank D. N. Christodoulides, A. Dogariu and K. L. Schepler for useful discussions. This work was supported by the US Office of Naval Research (ONR) under contracts N00014-17-1-2458 and N00014-19-1-2192.
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A.F.A. developed the concept and supervised the research. B.B. designed the experiments, carried out the measurements, and analysed the data, with assistance from M.Y. The Supplementary information was prepared by M.Y. with assistance from B.B. All the authors contributed to writing the paper.
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Supplementary Figs. 1–15 and discussion (including derivations, details of the experimental set-up and measurement procedure).
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Bhaduri, B., Yessenov, M. & Abouraddy, A.F. Anomalous refraction of optical spacetime wave packets. Nat. Photonics 14, 416–421 (2020). https://doi.org/10.1038/s41566-020-0645-6
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DOI: https://doi.org/10.1038/s41566-020-0645-6
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