Abstract

The group velocity of ‘space-time’ wave packets – propagation-invariant pulsed beams endowed with tight spatio-temporal spectral correlations – can take on arbitrary values in free space. Here we investigate theoretically and experimentally the maximum achievable group delay that realistic finite-energy space-time wave packets can achieve with respect to a reference pulse traveling at the speed of light. We find that this delay is determined solely by the spectral uncertainty in the association between the spatial frequencies and wavelengths underlying the wave packet spatio-temporal spectrum – and not by the beam size, bandwidth, or pulse width. We show experimentally that the propagation of space-time wave packets is delimited by a spectral-uncertainty-induced ‘pilot envelope’ that travels at a group velocity equal to the speed of light in vacuum. Temporal walk-off between the space-time wave packet and the pilot envelope limits the maximum achievable differential group delay to the width of the pilot envelope. Within this pilot envelope the space-time wave packet can locally travel at an arbitrary group velocity and yet not violate relativistic causality because the leading or trailing edge of superluminal and subluminal space-time wave packets, respectively, are suppressed once they reach the envelope edge. Using pulses of width ∼ 4 ps and a spectral uncertainty of ∼ 20 pm, we measure maximum differential group delays of approximately ±150 ps, which exceed previously reported measurements by at least three orders of magnitude.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (4)

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” Nat. Commun. 10, 929 (2019).
[Crossref]

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-pulse-front space-time wave packets,” ACS Photon. 6, 475–481 (2019).
[Crossref]

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time light-sheets in free space: Theory and experiments,” Phys. Rev. A 99, 023856 (2019).
[Crossref]

B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Space-time wave packets that travel in optical materials at the speed of light in vacuum,” Optica 6, 139–146 (2019).
[Crossref]

2018 (7)

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26, 13628–13638 (2018).
[Crossref] [PubMed]

B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Meters-long propagation of diffraction-free space-time light sheets,” Opt. Express 26, 20111–20121 (2018).
[Crossref] [PubMed]

H. E. Kondakci and A. F. Abouraddy, “Self-healing of space-time light sheets,” Opt. Lett. 43, 3830–3833 (2018).
[Crossref] [PubMed]

D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, “Spatiotemporal control of laser intensity,” Nat. Photon. 12, 262–265 (2018).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Airy wavepackets accelerating in space-time,” Phys. Rev. Lett. 120, 163901 (2018).
[Crossref]

M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A 97, 063803 (2018).
[Crossref]

P. Saari, “Reexamination of group velocities of structured light pulses,” Phys. Rev. A 97, 063824 (2018).
[Crossref]

2017 (7)

2016 (3)

2015 (1)

D. Giovannini, J. Romero, V. Potoč, 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, 857–860 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (2)

2010 (1)

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2010).
[Crossref]

2009 (6)

2008 (2)

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

C. J. Zapata-Rodríguez, M. A. Porras, and J. J. Miret, “Free-space delay lines and resonances with ultraslow pulsed Bessel beams,” J. Opt. Soc. Am. A 25, 2758–2763 (2008).
[Crossref]

2007 (3)

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13077–13095 (2007).
[Crossref] [PubMed]

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

2006 (3)

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[Crossref] [PubMed]

M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166–2176 (2006).
[Crossref]

C. J. Zapata-Rodríguez and M. A. Porras, “X-wave bullets with negative group velocity in vacuum,” Opt. Lett. 31, 3532–3534 (2006).
[Crossref] [PubMed]

2004 (3)

2003 (6)

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).
[Crossref]

M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, and C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

2002 (2)

I. Alexeev, K. Y. Kim, and H. M. Milchberg, “Measurement of the superluminal group velocity of an ultrashort Bessel beam pulse,” Phys. Rev. Lett. 88, 073901 (2002).
[Crossref] [PubMed]

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

2000 (3)

K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
[Crossref]

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
[Crossref]

A. M. Shaarawi and I. M. Besieris, “Relativistic causality and superluminal signalling using X-shaped localized waves,” J. Phys. A 33, 7255–7263 (2000).
[Crossref]

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[Crossref]

1996 (1)

1995 (1)

1993 (2)

1992 (1)

J. E. Hernandez, R. W. Ziolkowski, and S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,” J. Acoust. Soc. Am. 92, 550–562 (1992).
[Crossref]

1989 (2)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[Crossref]

R. W. Ziolkowski, D. K. Lewis, and B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[Crossref] [PubMed]

1985 (2)

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[Crossref]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
[Crossref]

1983 (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

1970 (1)

R. L. Smith, “The velocities of light,” Am. J. Phys. 38, 978–983 (1970).
[Crossref]

1952 (1)

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev. 85, 166–179 (1952).
[Crossref]

Abouraddy, A. F.

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” Nat. Commun. 10, 929 (2019).
[Crossref]

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-pulse-front space-time wave packets,” ACS Photon. 6, 475–481 (2019).
[Crossref]

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time light-sheets in free space: Theory and experiments,” Phys. Rev. A 99, 023856 (2019).
[Crossref]

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D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, “Spatiotemporal control of laser intensity,” Nat. Photon. 12, 262–265 (2018).
[Crossref]

Parker, K. J.

Parker, S. R.

J. E. Hernandez, R. W. Ziolkowski, and S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,” J. Acoust. Soc. Am. 92, 550–562 (1992).
[Crossref]

Pessina, E. M.

K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, “Spatial and temporal characterization of a bessel beam produced using a conical mirror,” Phys. Rev. A 79, 043802 (2009).
[Crossref]

Piché, M.

M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal bessel beams: theory and experiments,” Opt. Express 17, 18148–18164 (2009).
[Crossref] [PubMed]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Piksarv, P.

Piskarskas, A.

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13077–13095 (2007).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Polesana, P.

Porras, M. A.

M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A 97, 063803 (2018).
[Crossref]

M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42, 4679–4682 (2017).
[Crossref] [PubMed]

C. J. Zapata-Rodríguez, M. A. Porras, and J. J. Miret, “Free-space delay lines and resonances with ultraslow pulsed Bessel beams,” J. Opt. Soc. Am. A 25, 2758–2763 (2008).
[Crossref]

C. J. Zapata-Rodríguez and M. A. Porras, “X-wave bullets with negative group velocity in vacuum,” Opt. Lett. 31, 3532–3534 (2006).
[Crossref] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[Crossref] [PubMed]

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).
[Crossref]

M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[Crossref]

Potoc, V.

D. Giovannini, J. Romero, V. Potoč, 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, 857–860 (2015).
[Crossref] [PubMed]

Quéré, F.

Recami, E.

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, and C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

Reivelt, K.

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[Crossref]

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[Crossref]

Reyes, D.

Richardson, M.

Rini, M.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Romero, J.

D. Giovannini, J. Romero, V. Potoč, 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, 857–860 (2015).
[Crossref] [PubMed]

Rousseau, G.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Saari, P.

P. Saari, “Reexamination of group velocities of structured light pulses,” Phys. Rev. A 97, 063824 (2018).
[Crossref]

M. Lõhmus, P. Bowlan, P. Piksarv, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort optical pulses from circularly symmetric binary phase gratings,” Opt. Lett. 37, 1238–1240 (2012).
[Crossref] [PubMed]

P. Piksarv, H. Valtna-Lukner, A. Valdmann, M. Lõhmus, R. Matt, and P. Saari, “Temporal focusing of ultrashort pulsed Bessel beams into Airy-Bessel light bullets,” Opt. Express 20, 17220–17229 (2012).
[Crossref]

P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009).
[Crossref] [PubMed]

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[Crossref]

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[Crossref]

Sainte-Marie, A.

Sedky, S. M.

Sezginer, A.

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
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Shaarawi, A.

Shaarawi, A. M.

Shaarawi, M.

Shaw, J. L.

D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, “Spatiotemporal control of laser intensity,” Nat. Photon. 12, 262–265 (2018).
[Crossref]

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Smith, R. L.

R. L. Smith, “The velocities of light,” Am. J. Phys. 38, 978–983 (1970).
[Crossref]

Speirits, F.

D. Giovannini, J. Romero, V. Potoč, 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, 857–860 (2015).
[Crossref] [PubMed]

Taiel, F. M. M.

Tamosauskas, G.

Thul, D.

Trebino, R.

Treffer, A.

Trillo, S.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Trull, J.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

Turnbull, D.

D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, “Spatiotemporal control of laser intensity,” Nat. Photon. 12, 262–265 (2018).
[Crossref]

Turunen, J.

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2010).
[Crossref]

Valdmann, A.

Valiulis, G.

O. Jedrkiewicz, Y.-D. Wang, G. Valiulis, and P. Di Trapani, “One dimensional spatial localization of polychromatic stationary wave-packets in normally dispersive media,” Opt. Express 21, 25000–25009 (2013).
[Crossref] [PubMed]

M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

Valtna, H.

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

Valtna-Lukner, H.

Wang, Y.-D.

Weiner, A. M.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
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Wong, L. J.

L. J. Wong and I. Kaminer, “Ultrashort tilted-pulsefront pulses and nonparaxial tilted-phase-front beams,” ACS Photon. 4, 2257–2264 (2017).
[Crossref]

L. J. Wong and I. Kaminer, “Abruptly focusing and defocusing needles of light and closed-form electromagnetic wavepackets,” ACS Photon. 4, 1131–1137 (2017).
[Crossref]

Yessenov, M.

Youn, S. H.

K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, “Spatial and temporal characterization of a bessel beam produced using a conical mirror,” Phys. Rev. A 79, 043802 (2009).
[Crossref]

Zamboni-Rached, M.

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
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M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166–2176 (2006).
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E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, and C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

Zapata-Rodríguez, C. J.

Ziolkowski, R.

R. Donnelly and R. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

Ziolkowski, R. W.

A. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and F. M. M. Taiel, “Effect of the switching pattern of the illumination of dynamic apertures on the ranges of the generated localized pulses,” J. Opt. Soc. Am. A 13, 1712–1718 (1996).
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M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, and S. M. Sedky, “The generation of approximate focus wave mode pulses from wide-band dynamic gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
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R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
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J. E. Hernandez, R. W. Ziolkowski, and S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,” J. Acoust. Soc. Am. 92, 550–562 (1992).
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R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
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R. W. Ziolkowski, D. K. Lewis, and B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
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R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
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ACS Photon. (3)

L. J. Wong and I. Kaminer, “Abruptly focusing and defocusing needles of light and closed-form electromagnetic wavepackets,” ACS Photon. 4, 1131–1137 (2017).
[Crossref]

L. J. Wong and I. Kaminer, “Ultrashort tilted-pulsefront pulses and nonparaxial tilted-phase-front beams,” ACS Photon. 4, 2257–2264 (2017).
[Crossref]

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-pulse-front space-time wave packets,” ACS Photon. 6, 475–481 (2019).
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Am. J. Phys. (1)

R. L. Smith, “The velocities of light,” Am. J. Phys. 38, 978–983 (1970).
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IEEE J. Sel. Top. Quantum Electron. (1)

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, and C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

J. Acoust. Soc. Am. (1)

J. E. Hernandez, R. W. Ziolkowski, and S. R. Parker, “Synthesis of the driving functions of an array for propagating localized wave energy,” J. Acoust. Soc. Am. 92, 550–562 (1992).
[Crossref]

J. Appl. Phys. (2)

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
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J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
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J. Math. Phys. (1)

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
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J. Opt. Soc. Am. A (6)

J. Phys. A (1)

A. M. Shaarawi and I. M. Besieris, “Relativistic causality and superluminal signalling using X-shaped localized waves,” J. Phys. A 33, 7255–7263 (2000).
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Nat. Commun. (1)

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” Nat. Commun. 10, 929 (2019).
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Nat. Photon. (2)

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photon. 11, 733–740 (2017).
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D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, “Spatiotemporal control of laser intensity,” Nat. Photon. 12, 262–265 (2018).
[Crossref]

Opt. Commun. (2)

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

R. R. Alfano and D. A. Nolan, “Slowing of Bessel light beam group velocity,” Opt. Commun. 361, 25–27 (2016).
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Opt. Express (12)

B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Meters-long propagation of diffraction-free space-time light sheets,” Opt. Express 26, 20111–20121 (2018).
[Crossref] [PubMed]

P. Piksarv, H. Valtna-Lukner, A. Valdmann, M. Lõhmus, R. Matt, and P. Saari, “Temporal focusing of ultrashort pulsed Bessel beams into Airy-Bessel light bullets,” Opt. Express 20, 17220–17229 (2012).
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S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12, 935–940 (2004).
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H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
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K. J. Parker and M. A. Alonso, “The longitudinal iso-phase condition and needle pulses,” Opt. Express 24, 28669–28677 (2016).
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F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” Opt. Express 17, 9804–9809 (2009).
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H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26, 13628–13638 (2018).
[Crossref] [PubMed]

D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, “Spatio-temporal reshaping and X wave dynamics in optical filaments,” Opt. Express 15, 13077–13095 (2007).
[Crossref] [PubMed]

M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal bessel beams: theory and experiments,” Opt. Express 17, 18148–18164 (2009).
[Crossref] [PubMed]

O. Jedrkiewicz, Y.-D. Wang, G. Valiulis, and P. Di Trapani, “One dimensional spatial localization of polychromatic stationary wave-packets in normally dispersive media,” Opt. Express 21, 25000–25009 (2013).
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M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express 17, 7465–7478 (2009).
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I. M. Besieris and A. M. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12, 3848–3864 (2004).
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Opt. Lett. (7)

Optica (2)

Phys. Rev. (1)

D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev. 85, 166–179 (1952).
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Phys. Rev. A (7)

P. Saari, “Reexamination of group velocities of structured light pulses,” Phys. Rev. A 97, 063824 (2018).
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R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[Crossref]

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
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M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time light-sheets in free space: Theory and experiments,” Phys. Rev. A 99, 023856 (2019).
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M. A. Porras, “Nature, diffraction-free propagation via space-time correlations, and nonlinear generation of time-diffracting light beams,” Phys. Rev. A 97, 063803 (2018).
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K. B. Kuntz, B. Braverman, S. H. Youn, M. Lobino, E. M. Pessina, and A. I. Lvovsky, “Spatial and temporal characterization of a bessel beam produced using a conical mirror,” Phys. Rev. A 79, 043802 (2009).
[Crossref]

Phys. Rev. E (4)

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).
[Crossref]

M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
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K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

Phys. Rev. Lett. (8)

I. Alexeev, K. Y. Kim, and H. M. Milchberg, “Measurement of the superluminal group velocity of an ultrashort Bessel beam pulse,” Phys. Rev. Lett. 88, 073901 (2002).
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C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[Crossref] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[Crossref] [PubMed]

R. W. Ziolkowski, D. K. Lewis, and B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[Crossref] [PubMed]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
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H. E. Kondakci and A. F. Abouraddy, “Airy wavepackets accelerating in space-time,” Phys. Rev. Lett. 120, 163901 (2018).
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Proc. R. Soc. Lond. A (1)

R. Donnelly and R. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

Prog. Opt. (1)

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2010).
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Rev. Sci. Instrum. (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
[Crossref]

Science (2)

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[Crossref] [PubMed]

Other (6)

B. Bhaduri, M. Yessenov, and A. F. Abouraddy, “Dynamical refraction of optical space-time wave packets,” unpublished (2019).

H. E. Kondakci, M. A. Alonso, and A. F. Abouraddy, “Classical entanglement underpins the propagation invariance of space-time wave packets,” arXiv:1812.10566 (2018).

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[Crossref]

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[Crossref]

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Figures (6)

Fig. 1
Fig. 1 (a)–(c) Schematic illustration of finite-energy ST wave packets as the product of an ideal ST wave packet (narrow pulse, solid blue) traveling at vg and a pilot envelope (wide pulse, dashed red) traveling at c: (a) superluminal vg > c; (b) negative vg <0; and (c) subluminal vg < c. In all cases, the properties of the ideal ST wave packet are maintained only over the length of the pilot envelope. (d) Snapshots of the spatio-temporal intensity profile I(x, z, t) at different z showing the evolution from a symmetric to an asymmetric wave packet (along the time axis) when the ST wave packet approaches the edge of the pilot envelope. The amplitudes are adjusted by the factors given in the bottom right corner for clarity. (e) The time-averaged intensity profile I(x, z), showing the axial locations where the snapshots in (d) are calculated. In (d) and (e), X is normalized to the inverse of the spatial bandwidth 1/Δkx, T to 1/ΔΩ, and Z to the axial distance where the on-axis intensity drops to 1/e. See also [10].
Fig. 2
Fig. 2 Schematic of the optical setup for synthesizing and characterizing ST wave packets. DL: Delay line; G: diffraction grating; SLM: spatial light modulator; A: aperture.
Fig. 3
Fig. 3 (a) Measured spatio-temporal spectral intensity |ψ̃(kx, λ)|2 for a subluminal (θ =35°; an ellipse) and superluminal (θ =70°; a hyperbola) ST wave packets. Both spectra appear approximately as parabolas because of the limited bandwidth (Δλ≈0.4 nm), and they are shifted vertically with respect to each other for clarity. (b) Measured spatio-temporal spectra in (a) projected onto the ( k z , ω c ) -plane are tilted straight lines. The insets illustrate the intersection of the tilted spectral planes planes with the light-cone.
Fig. 4
Fig. 4 (a) Time-averaged intensity I(x, z) of the subluminal ST wave packet θ = 35°. (b) Time-resolved intensity I(x, z, τ) at z =0, 45, 100 mm. The white curves are the intensity profiles of the pilot envelope. (c) Normalized intensity at the center of the spatial profile I(x =0, z, τ) for the axial locations in (b). (d) Same as (c) at x =50 μm.
Fig. 5
Fig. 5 Same as Fig. 4 for a superluminal ST wave packet θ =70°.
Fig. 6
Fig. 6 (a) Measured Lm for different θ at fixed δλ. Theoretical curve is Lmax = Lp/|1−cot θ|, with Lp =45 mm. (b) Maximum DGD as a function of θ at fixed δλ.

Equations (9)

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ψ ( x , z , t ) = d k x d Ω ψ ˜ ( k x , Ω ) e i ( k x x + ( k z k o ) z Ω t ) ;
ψ ( x , z , t ) = d k x ψ ˜ ( k x ) e i k x x e i Ω ( t z / v g ) = ψ ( x , 0 , t z / v g ) ,
Ω ( k x ) ω o = f ( θ ) k x 2 2 k o 2
I ( x , z , t ) = | ψ ( x , z , t ) | 2 = I ST ( x , z , t ) I p ( x , z , t ) ,
I ST ( x , z , t ) = 2 π Δ k x 1 + [ Δ Ω ( t z / v g ) ] 2 exp { x 2 ( Δ k x ) 2 1 + [ Δ Ω ( t z / v g ) ] 2 } = I ST ( x , 0 , t z / v g ) ;
I p ( x , z , t ) = 2 π δ Ω exp { ( t z c ) 2 ( δ Ω ) 2 } = I p ( x , 0 , t z c ) .
τ max = L max ( 1 v g 1 c ) ~ ± 1 δ Ω ~ ± τ p ,
L max = c δ Ω 1 | 1 c / v g | = L p | 1 cot θ | ;
I ( x , z ) = Δ k x 0 d s e s s ( 1 + s / κ 2 ) exp { ( x Δ k x s z / z R ) 2 1 + s / κ 2 } ,

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