Abstract

Because of their broad spectral width, ultrashort lasers provide unique possibilities to shape light beams and control their properties, in particular through the use of spatio-temporal couplings. In this context, we present a theoretical investigation of the linear propagation of ultrashort laser beams that combine temporal chirp and a standard aberration known as longitudinal chromatism. When such beams are focused in a vacuum, or in a linear medium, the interplay of these two effects can be exploited to set the velocity of the resulting intensity peak to arbitrary values within the Rayleigh length, i.e., precisely where laser pulses are generally used. Such beams could find groundbreaking applications in the control of laser–matter interactions, in particular for laser-driven particle acceleration.

© 2017 Optical Society of America

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References

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    [Crossref]

2016 (3)

2015 (5)

C. G. Durfee and J. A. Squier, “Breakthroughs in photonics 2014: spatiotemporal focusing: advances and applications,” IEEE Photon. J. 7, 1–6 (2015).
[Crossref]

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3, e3 (2015).

B. Sun, P. S. Salter, and M. J. Booth, “Pulse front adaptive optics: a new method for control of ultrashort laser pulses,” Opt. Express 23, 19348–19357 (2015).
[Crossref]

D. Giovannini, J. Romero, V. Potoček, 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]

A. Depresseux, E. Oliva, J. Gautier, F. Tissandier, J. Nejdl, M. Kozlova, G. Maynard, J. P. Goddet, A. Tafzi, A. Lifschitz, H. T. Kim, S. Jacquemot, V. Malka, K. Ta Phuoc, C. Thaury, P. Rousseau, G. Iaquaniello, T. Lefrou, A. Flacco, B. Vodungbo, G. Lambert, A. Rousse, P. Zeitoun, and S. Sebban, “Table-top femtosecond soft x-ray laser by collisional ionization gating,” Nat. Photonics 9, 817–821 (2015).
[Crossref]

2014 (3)

2013 (2)

K. T. Kim, C. Zhang, T. Ruchon, J.-F. Hergott, T. Auguste, D. M. Villeneuve, P. B. Corkum, and F. Quéré, “Photonic streaking of attosecond pulse trains,” Nat. Photonics 7, 651–656 (2013).
[Crossref]

A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751–793 (2013).
[Crossref]

2012 (2)

H. Vincenti and F. Quéré, “Attosecond lighthouses: how to use spatiotemporally coupled light fields to generate isolated attosecond pulses,” Phys. Rev. Lett. 108, 113904 (2012).
[Crossref]

J. Wheeler, A. Borot, S. Monchocé, H. Vincenti, A. Ricci, A. Malvache, R. Lopez-Martens, and F. Quéré, “Attosecond lighthouse from plasma mirrors,” Nat. Photonics 6, 829–833 (2012).
[Crossref]

2010 (1)

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12, 093001 (2010).
[Crossref]

2009 (2)

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[Crossref]

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229–1285 (2009).
[Crossref]

2008 (1)

2007 (2)

2006 (2)

H.-M. Heuck, P. Neumayer, T. Kühl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (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]

2005 (3)

2003 (1)

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]

2002 (1)

P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B 35, R31–R56 (2002).
[Crossref]

2000 (2)

1997 (1)

P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. 140, 245–249 (1997).
[Crossref]

1993 (1)

Z. Horvath and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[Crossref]

1989 (2)

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464–2468 (1989).
[Crossref]

Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14, 119–121 (1989).
[Crossref]

1988 (1)

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[Crossref]

Abouraddy, A. F.

Akturk, S.

Alonso, M. A.

Auguste, T.

F. Quéré, H. Vincenti, A. Borot, S. Monchocé, T. J. Hammond, K. T. Kim, J. A. Wheeler, C. Zhang, T. Ruchon, T. Auguste, J. F. Hergott, D. M. Villeneuve, P. B. Corkum, and R. Lopez-Martens, “Applications of ultrafast wavefront rotation in highly nonlinear optics,” J. Phys. B 47, 124004 (2014).
[Crossref]

K. T. Kim, C. Zhang, T. Ruchon, J.-F. Hergott, T. Auguste, D. M. Villeneuve, P. B. Corkum, and F. Quéré, “Photonic streaking of attosecond pulse trains,” Nat. Photonics 7, 651–656 (2013).
[Crossref]

Bahk, S.-W.

Barnett, S. M.

D. Giovannini, J. Romero, V. Potoček, 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]

Biegert, J.

Blanchot, N.

Booth, M. J.

Bor, Z.

Z. Horvath and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[Crossref]

Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14, 119–121 (1989).
[Crossref]

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[Crossref]

Borghesi, M.

A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751–793 (2013).
[Crossref]

Borot, A.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10, 547–553 (2016).
[Crossref]

F. Quéré, H. Vincenti, A. Borot, S. Monchocé, T. J. Hammond, K. T. Kim, J. A. Wheeler, C. Zhang, T. Ruchon, T. Auguste, J. F. Hergott, D. M. Villeneuve, P. B. Corkum, and R. Lopez-Martens, “Applications of ultrafast wavefront rotation in highly nonlinear optics,” J. Phys. B 47, 124004 (2014).
[Crossref]

J. Wheeler, A. Borot, S. Monchocé, H. Vincenti, A. Ricci, A. Malvache, R. Lopez-Martens, and F. Quéré, “Attosecond lighthouse from plasma mirrors,” Nat. Photonics 6, 829–833 (2012).
[Crossref]

Bowlan, P.

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12, 093001 (2010).
[Crossref]

Boyd, R. W.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[Crossref]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Bromage, J.

Cheng, Z.

Clerici, M.

Comte, M.

Corkum, P. B.

F. Quéré, H. Vincenti, A. Borot, S. Monchocé, T. J. Hammond, K. T. Kim, J. A. Wheeler, C. Zhang, T. Ruchon, T. Auguste, J. F. Hergott, D. M. Villeneuve, P. B. Corkum, and R. Lopez-Martens, “Applications of ultrafast wavefront rotation in highly nonlinear optics,” J. Phys. B 47, 124004 (2014).
[Crossref]

K. T. Kim, C. Zhang, T. Ruchon, J.-F. Hergott, T. Auguste, D. M. Villeneuve, P. B. Corkum, and F. Quéré, “Photonic streaking of attosecond pulse trains,” Nat. Photonics 7, 651–656 (2013).
[Crossref]

Danson, C.

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3, e3 (2015).

Dausinger, F.

S. Nolte, F. Schrempel, and F. Dausinger, Ultrashort Pulse Laser Technology: Laser Sources and Applications, Springer Series in Optical Sciences (Springer, 2016).

Depresseux, A.

A. Depresseux, E. Oliva, J. Gautier, F. Tissandier, J. Nejdl, M. Kozlova, G. Maynard, J. P. Goddet, A. Tafzi, A. Lifschitz, H. T. Kim, S. Jacquemot, V. Malka, K. Ta Phuoc, C. Thaury, P. Rousseau, G. Iaquaniello, T. Lefrou, A. Flacco, B. Vodungbo, G. Lambert, A. Rousse, P. Zeitoun, and S. Sebban, “Table-top femtosecond soft x-ray laser by collisional ionization gating,” Nat. Photonics 9, 817–821 (2015).
[Crossref]

Di Trapani, P.

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]

Durfee, C. G.

C. G. Durfee and J. A. Squier, “Breakthroughs in photonics 2014: spatiotemporal focusing: advances and applications,” IEEE Photon. J. 7, 1–6 (2015).
[Crossref]

Durst, M.

Efron, U.

U. Efron, Spatial Light Modulator Technology (CRC Press, 1994).

Esarey, E.

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229–1285 (2009).
[Crossref]

Faccio, D.

D. Giovannini, J. Romero, V. Potoček, 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]

M. Clerici, D. Faccio, A. Lotti, E. Rubino, O. Jedrkiewicz, J. Biegert, and P. D. Trapani, “Finite-energy, accelerating Bessel pulses,” Opt. Express 16, 19807–19811 (2008).
[Crossref]

Fedorov, N.

Ferenczi, G.

D. Giovannini, J. Romero, V. Potoček, 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]

Flacco, A.

A. Depresseux, E. Oliva, J. Gautier, F. Tissandier, J. Nejdl, M. Kozlova, G. Maynard, J. P. Goddet, A. Tafzi, A. Lifschitz, H. T. Kim, S. Jacquemot, V. Malka, K. Ta Phuoc, C. Thaury, P. Rousseau, G. Iaquaniello, T. Lefrou, A. Flacco, B. Vodungbo, G. Lambert, A. Rousse, P. Zeitoun, and S. Sebban, “Table-top femtosecond soft x-ray laser by collisional ionization gating,” Nat. Photonics 9, 817–821 (2015).
[Crossref]

Gabolde, P.

Gallet, V.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10, 547–553 (2016).
[Crossref]

Gauthier, D. J.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[Crossref]

Gautier, J.

A. Depresseux, E. Oliva, J. Gautier, F. Tissandier, J. Nejdl, M. Kozlova, G. Maynard, J. P. Goddet, A. Tafzi, A. Lifschitz, H. T. Kim, S. Jacquemot, V. Malka, K. Ta Phuoc, C. Thaury, P. Rousseau, G. Iaquaniello, T. Lefrou, A. Flacco, B. Vodungbo, G. Lambert, A. Rousse, P. Zeitoun, and S. Sebban, “Table-top femtosecond soft x-ray laser by collisional ionization gating,” Nat. Photonics 9, 817–821 (2015).
[Crossref]

Giovannini, D.

D. Giovannini, J. Romero, V. Potoček, 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]

Gobert, O.

Goddet, J. P.

A. Depresseux, E. Oliva, J. Gautier, F. Tissandier, J. Nejdl, M. Kozlova, G. Maynard, J. P. Goddet, A. Tafzi, A. Lifschitz, H. T. Kim, S. Jacquemot, V. Malka, K. Ta Phuoc, C. Thaury, P. Rousseau, G. Iaquaniello, T. Lefrou, A. Flacco, B. Vodungbo, G. Lambert, A. Rousse, P. Zeitoun, and S. Sebban, “Table-top femtosecond soft x-ray laser by collisional ionization gating,” Nat. Photonics 9, 817–821 (2015).
[Crossref]

Gu, X.

Guillaumet, D.

Habib, J.

Hammond, T. J.

F. Quéré, H. Vincenti, A. Borot, S. Monchocé, T. J. Hammond, K. T. Kim, J. A. Wheeler, C. Zhang, T. Ruchon, T. Auguste, J. F. Hergott, D. M. Villeneuve, P. B. Corkum, and R. Lopez-Martens, “Applications of ultrafast wavefront rotation in highly nonlinear optics,” J. Phys. B 47, 124004 (2014).
[Crossref]

Hergott, J. F.

F. Quéré, H. Vincenti, A. Borot, S. Monchocé, T. J. Hammond, K. T. Kim, J. A. Wheeler, C. Zhang, T. Ruchon, T. Auguste, J. F. Hergott, D. M. Villeneuve, P. B. Corkum, and R. Lopez-Martens, “Applications of ultrafast wavefront rotation in highly nonlinear optics,” J. Phys. B 47, 124004 (2014).
[Crossref]

Hergott, J.-F.

K. T. Kim, C. Zhang, T. Ruchon, J.-F. Hergott, T. Auguste, D. M. Villeneuve, P. B. Corkum, and F. Quéré, “Photonic streaking of attosecond pulse trains,” Nat. Photonics 7, 651–656 (2013).
[Crossref]

Heuck, H.-M.

H.-M. Heuck, P. Neumayer, T. Kühl, and U. Wittrock, “Chromatic aberration in petawatt-class lasers,” Appl. Phys. B 84, 421–428 (2006).
[Crossref]

Hillier, D.

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3, e3 (2015).

Hopps, N.

C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3, e3 (2015).

Horvath, Z.

Z. Horvath and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[Crossref]

Iaquaniello, G.

A. Depresseux, E. Oliva, J. Gautier, F. Tissandier, J. Nejdl, M. Kozlova, G. Maynard, J. P. Goddet, A. Tafzi, A. Lifschitz, H. T. Kim, S. Jacquemot, V. Malka, K. Ta Phuoc, C. Thaury, P. Rousseau, G. Iaquaniello, T. Lefrou, A. Flacco, B. Vodungbo, G. Lambert, A. Rousse, P. Zeitoun, and S. Sebban, “Table-top femtosecond soft x-ray laser by collisional ionization gating,” Nat. Photonics 9, 817–821 (2015).
[Crossref]

Jacquemot, S.

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Supplementary Material (3)

NameDescription
» Supplement 1       Supplemental document
» Visualization 1       Left: Temporal evolution of the beam intensity profile, I(r,z,t), using common color scale. Right: Corresponding plots of the on-axis intensity profile, I(r=0,z,t).
» Visualization 2       Temporal evolution of the beam intensity profile, I(r,z,t), with color scales adjusted for each image.

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Figures (5)

Fig. 1.
Fig. 1. Generation and properties of CPLC beams. (a) A chromatic linear optical system induces a combination of temporal chirp and frequency-dependent wavefront curvature on an ultrashort beam. When this beam is subsequently focused, its different frequency components have their best focus at different z (longitudinal chromatism). (b) Spatio-temporal field E(r,t) of the beam prior to focusing in a saturated color scale (negative field in blue, positive in red), in the presence of PFC/LC (α=3  fs/cm2 and beam waist wi=5  cm before focusing, corresponding to τP=αwi2=75  fs) and in the absence of chirp, for a local pulse duration τF=25  fs. The carrier frequency has been reduced for the sake of readability. Pulse front curvature is clearly observed. (c) Same plot, now in the presence of chirp (β=6380  fs2), i.e., for a CPLC beam. In (b) and (c), the lower panels show the local temporal intensity profile |E(r=0,t)|2 at the center of the beam (red curve), and the spatially integrated temporal intensity profile d2r|E(r,t)|2 (blue curve). These profiles, respectively, have characteristic temporal widths of τF1/Δω and τP in the absence of chirp, and τC|β|Δω for strong chirps. (d) On-axis spectrum |E(z,ω)|2 of the focused beam, as a function of the longitudinal coordinate z along the extended Rayleigh length. The right panel compares the local spectrum at z=0 (blue curve, of characteristic width 1/τP) with the spatially integrated spectrum dz|E(z,ω)|2 (red curve, of characteristic width 1/τFΔω).
Fig. 2.
Fig. 2. How CPLC enable adjustable light pulse velocities. Because of PFC/LC, different frequencies of the pulse (here ω1<ω0<ω2) are focused at different longitudinal positions z0(ω), indicated by the horizontal color lines. In (a), no chirp is applied, and the pulse intensity peak (black line with arrow) propagates at c. In (b), the frequencies that have their best focus at large z are retarded in time, leading to an effective velocity smaller than c. In (c), these same frequencies are now advanced in time, in such a way that the intensity peak first appears at large z, and then moves in a direction opposite to the beam propagation direction—i.e., an apparent backward propagation within the extended Rayleigh length. In (d), a higher-order spectral phase is applied to the beam, leading to a longitudinal acceleration of the pulse peak as it propagates.
Fig. 3.
Fig. 3. Simulations of CPLC of different propagation velocities. Simulated on-axis spatio-temporal intensity profile I(z,t) of different CPLC beams along the extended Rayleigh length, for a fixed PFC/LC parameter α=3  fs/cm2 and different spectral phases. The beam parameters are λ0=800  nm, τF=25  fs, and wi=5  cm, focused by an optic of focal length f=1  m (i.e., f/20 focusing). Panel (a) corresponds to a reference case without chirp, where the pulse envelope propagates at c. In (b) and (c), a linear chirp has been applied (β=12230  fs2 and β=53420  fs2), that respectively lead to the propagation regimes of Figs. 2(b) and 2(c), with v=0.7c in (b) and v=c in (c). In (d), a third-order spectral phase is applied to the beam, leading to a longitudinal acceleration of the pulse as it propagates, as illustrated in Fig. 2(d). In each case, the lower plots show the pulse temporal intensity profile at z=0, I(0,t). By convention, I=1 corresponds to peak intensity obtained at best focus for the STC-free unchirped beam. For comparison with this reference case, all curves have been multiplied by the numerical factors indicated next to the curves, which thus correspond to the inverse of the intensity reduction factor ϵ resulting from the combination of PFC/LC and chirp.
Fig. 4.
Fig. 4. Quantitative properties of CPLC beams, for α=3  fs/cm2, λ0=800  nm, τF=25  fs, wi=5  cm, and f=1  m as in Fig. 3. (a) Propagation velocity v of the intensity peak of a CPLC beam around focus, as a function of chirp β. The black line corresponds to the prediction of Eq. (3), while the red one shows the velocity deduced from numerical simulations. (b) Corresponding peak intensity reduction factor ϵ=IM/I0 as a function of β. In both panels, the insets show zooms on these curves for small values of β. (c) Peak intensity reduction factor ϵ (full lines) as a function of α, for fixed propagation velocities v=c, v=1.1c or 0.91c, and v=1.3c or 0.81c, corresponding to β/α=0, ±7.71×102  fs·cm2 and ±1.95×103  fs·cm2. The dotted parts of the curves correspond to a range where the calculation no longer makes sense, because PFC/LC becomes too weak. The red dashed line shows the evolution of the extended Rayleigh length zre with α, obtained from numerical simulations.
Fig. 5.
Fig. 5. Pulse with an oscillating propagation velocity produced using CPLC with a high-order spectral phase. Simulated on-axis spatio-temporal intensity profile I(z,t), along the extended Rayleigh length, of a CPLC beam with an oscillating spectral phase, leading to a velocity v oscillating between c and 0.3c. Compared to Fig. 3, the PFC/LC parameter has also been increased to α=30  fs/cm2 (τP=750  fs), while all other beam parameters are identical.

Equations (12)

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z0(ω)=2f2cαδω/ω0.
t(ω)ϕ(z,ω)ω=z/c+βδω.
vc=11+(ω0/2f2)×(β/α).
E(z,ω)=|A0(δω)|[1+(zτPzrδω)2/zr2]1/2eiϕ(z,ω),
ϕ(z,ω)=arctan(zτPzrδωzr)β2δω2ωcz.
|E(z,ω)|2|A0(Δωz/zre)|2[1+τP2(δωΔωz/zre)2].
E(z,t)|A0(t/β)|[1+(t/τe)2]1/2eiϕ(z,t),
t=tβτpzrz
=tzv,
ϕ(z,t)=(ω0+t2β)tarctan(t/τe).
I(z,t)|E(z,t)|2|A0(Δωz/zre)|2[1+(t/τe)2].
2ϕω2=τPzr(1v(z)1c),

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