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.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Communications Physics Open Access 21 November 2022
Scientific Reports Open Access 27 September 2022
Nature Communications Open Access 05 August 2022
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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).
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.
The authors declare no competing interests.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
This article is cited by
Nature Physics (2023)
Nature Photonics (2023)
Nature Communications (2022)
Communications Physics (2022)
Nature Photonics (2022)