May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS
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Spatio-temporal light springs: extended encoding of orbital angular momentum in ultrashort pulses G. Pariente and F. Quéré* Commissariat à l’Energie Atomique, Lasers, Interactions and Dynamics Laboratory, DSM/IRAMIS, CEN Saclay, Gif sur Yvette 91191, France *Corresponding author: [email protected] Received February 16, 2015; revised March 25, 2015; accepted March 26, 2015; posted March 31, 2015 (Doc. ID 234628); published April 27, 2015 We introduce a new class of spatio-temporally coupled ultrashort laser beams, which are obtained by superimposing Laguerre–Gauss beams whose azimuthal mode index is correlated to their frequency. These beams are characterized by helical structures for their phase and intensity profiles, which both encode the orbital angular momentum carried by the light. They can easily be engineered in the optical range, and are naturally produced at shorter wavelengths when attosecond pulses are generated by intense femtosecond Laguerre–Gauss laser beams. These spatio-temporal “light springs” will allow for the transfer of the orbital angular momentum to matter by stimulated Raman scattering. © 2015 Optical Society of America OCIS codes: (320.0320) Ultrafast optics; (320.5540) Pulse shaping; (080.4865) Optical vortices; (260.7120) Ultrafast phenomena. http://dx.doi.org/10.1364/OL.40.002037
In the last decades, ultrafast optics has become an important branch of optics, now producing femtosecond light pulses that can consist of a single optical period, or that can reach peak power in the petawatt range. The nonlinear interaction of such laser beams with matter can, for instance, produce beams of relativistic particles, or light pulses of attosecond durations that are already being used to temporally resolve electron dynamics. A common key feature of all these light sources is their large spectral width, with a well-defined phase relationship between the different frequency components. A current frontier of this research field is the control and metrology of ultrashort light pulses simultaneously in space and time. In ultrafast optics, a beam whose temporal and spatial properties cannot be separated, i.e., such that its electric field Ex; y; t ≠ E 0 f tgx; y, is said to be spatio-temporally coupled [1]. Spatio-temporal couplings (STCs) have long been considered to be detrimental, because they imply a loss of spatial coherence and thus systematically decrease the peak intensity at focus. However, recent results (e.g., [2,3]) indicate that moderate and controlled STCs could pave the way to a whole new range of experimental capabilities that mostly remain to be explored. In this Letter, we introduce a new class of spatiotemporally coupled light pulses of broad relevance, which we call spatio-temporal light springs (LS) for convenience, as a simple reference to the shape of their spatio-temporal intensity profile (Figs. 1 and 2). These beams are directly related to Laguerre–Gauss (LG) beams, which have also become a major tool in different fields of optics in the last decades, partly due to the fact that they carry orbital angular momentum (OAM) [4]. This property is related to the peculiar shape of their wavefronts, which have an helical structure. In the case of a LS, it is not only the phase, but also the intensity profile that has an helical structure, with different pitches for both. We formally describe the properties of these beams, and distinguish different subclasses from single light spring coils (Fig. 1) to intertwined light springs of 0146-9592/15/092037-04$15.00/0
different orders (Fig. 2). We explain how such beams can be produced, and discuss their possible applications. Let us start by recalling the equation for the spatial phase φl θ; z of a monochromatic LG beam E l jE l jeiφl of azimuthal mode index l and frequency ω, with cylindrical coordinates [4] (θ being the azimuthal angle, and z the propagation axis): φl θ; z lθ − jlj 1χz kz;
(1)
where k ω∕c, and χz arctanz∕zr is the Gouy phase with zr as the Rayleigh beam length. In such a beam, the average OAM per photon is lℏ. To simplify the calculations, we only consider LG beams with a radial mode index p 0 here, and restrict the study around the beam waist, so that the phase curvature can be neglected. In previous works on the generation of broadband beams carrying OAM (see, e.g., [5–8]), most of the efforts have been focused on ensuring that the mode index l is the same for all frequencies ω within the beam spectral width. But correlating l and ω can provide new degrees of freedom to shape ultrashort beams in space and time, and thus to tailor and control laser-matter interactions. Light springs are the simplest cases of this type of spacetime coupled beams. They consist of ultrashort pulses, for which each frequency component is associated with an LG mode with a different value of l, with the following linear relationship [Figs. 1(a) and 2(a)]: lω l0
Δl ω − ω0 ; Δω
(2)
where ω0 is the central frequency of the pulse, l0 is the mode index for this frequency, Δω is the spectral width of the pulse, and Δl is the variation of l over this width. This relationship results in a very peculiar spatiotemporal intensity profile for LS, which is easily understood intuitively. As is well known [4,9], when several LG modes of different l with a fixed frequency ω0 are © 2015 Optical Society of America
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Fig. 1. Example of a single coil of a light spring. (a) Pulse spectral intensity (black line, corresponding to τ 25 fs) and the variation of the azimuthal mode index l as a function of frequency (red line, corresponding to Δτ0 4τ 100 fs). (b) Spatio-temporal intensity profile of the pulsed beam obtained by superimposing LG modes with a continuously varying index l correlated to their frequency, as in panel (a). (c) Spatio-temporal intensity profile obtained after propagating this beam up to a focus. The main graphs of (b) and (c) show the isosurface corresponding to 20% of the peak intensity, while the side panels show the marginals of the intensity profile in the different subspaces.
superimposed with an appropriate phase relationship, their interference results in an azimuthal wavepacket: light is then localized around a given angle θ0 , independent of time. In the case of the LS, each LG mode is now associated with a different frequency ω [Eq. (2)], so that their relative phases vary linearly in time. This relative phase variation leads to a rotation of the azimuthal wavepacket in time, i.e., θ0 ∝ t. We can therefore expect the intensity profile of an LS to form an helix in space–time [Figs. 1(b) and 1(c) and 2(b) and 2(c)]. We now study the structure of this helix. The E-field of a LS in space and time is determined by the following Fourier transform: Z Eθ; z; t ∝ Z ∝
dωElω θ; z; ωe−iωt ;
(3)
dωjE lω je−iωtiφlω ;
(4)
with φl given by Eq. (1) and lω given by Eq. (2). Intensity maxima occur whenever the phase −ωt φlω of the integrand is stationary with respect to the integration variable ω, leading to the condition t ∂φlω ∕∂ω. A straightforward calculation provides the following locations θ0 t; z of the pulse maxima in space and time: θ0 t; z
Δω z t− χz; Δl c
(5)
where the sign of the last term is opposite to the sign of Δl. The intensity profile is thus indeed an helix, with a temporal pitch of Δτ0 2πΔl∕Δω Δlτ at any given z, where τ 2π∕Δω is the Fourier transform limited duration of the pulse. Neglecting the Gouy phase in Eq. (5), the spatial pitch at any given t is Δz ≈ cΔτ0 . This profile thus forms a “spring” in space, that essentially translates as it propagates, simply performing half a turn around its axis as it goes through focus, due to the π variation of the Gouy phase χz [10]. We now derive the exact temporal structure of LS at a given position z, assuming, for simplicity, a flat spectral phase. We can rewrite Eq. (4) as
Eθ; t ∝ eil0 θ−ω0 t
Z
dωjE lω je−iω−ω0 t−Δωθ ; Δl
(6)
(7) ∝ eil0 θ−ω0 t At − τ0 θ; R where At dωjE lω je−iω−ω0 t is the local temporal envelope of the ultrashort pulse, and τ0 θ Δl∕Δωθ is its local arrival time. This shows that the local temporal structure of the pulse is unaffected by the introduction of the STC. It still forms an ultrashort pulse of duration τ anywhere in space, but the arrival time τ0 of this pulse now varies azimuthally, leading to the formation of an intensity helix. According to Eq. (7), this helix is intertwined with the usual helical structure of the wavefronts of a LG beam with mode index l0 (see Media 1). We now discuss the practical issue of the generation of the LS, which will naturally lead us to separate them into different subclasses. The easiest way to generate a LS is to pass a normal ultrashort beam through a spiral waveplate, similar to those used to produce the usual LG beams [4], but now with a larger azimuthal variation of the plate thickness. For a phase velocity vφ ω and a thickness variation eθ Δeθ∕2π, this induces helical wavefronts corresponding to a mode index l0 c∕vφ ω0 − 1Δe∕cT 0 at the central frequency ω0 2π∕T 0 . But due to its group velocity vg ≠ c, it also induces an azimuthal group delay τ0 θ c∕vg − 1eθ∕c. According to Eq. (7), such an azimuthal group delay is equivalent to Eq. (2) when expressed in the frequency domain, with Δl given by Δl c∕vg − 1Δe∕cτ. This shows that the usual spiral waveplates, when used with ultrashort light sources, produce good approximations of a simple LG beam of a given l0 as long as Δe ∼ cT 0 ≪ cτ, but that they generate LS as soon as Δe ∼ cτ. Note that the independent control of Δl and l0 is not possible with such a single waveplate, but should be possible with additional optical elements, such as adaptive helical mirrors [11]. Passing an ultrashort beam through such a waveplate produces only a single coil of light spring, such as the one displayed in Fig. 1(b). This beam has a continuous frequency spectrum, across which lω also varies continuously [Fig. 1(a)]. For most frequencies, lω therefore takes noninteger values that correspond to spatial modes
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Fig. 2. Examples of ultrashort light springs. (a) Pulse spectrum (corresponding to a train of 25 fs pulses), and two different variations of l with ω (corresponding to Δτ0 2τ 50 fs in red, and Δτ0 4τ 100 fs in blue). In both cases, the l index takes integer values for all frequency modes. (b) Spatio-temporal intensity profiles of a first-order LS, corresponding to the red variation of l in (a), and (c) of a second-order LS, corresponding to the blue variation of l in (a) (same displays as in Fig. 1).
with fractional OAM. These beams are not solutions of the paraxial wave equation, and are thus known to distort as they propagate [12]. When considering the beam in space–time [Fig. 1(b)], the origin of this distortion upon propagation becomes obvious: it arises from the spatial singularities that occur at the front and rear temporal edges of the beam, which lead to diffraction. This, however, only affects these temporal edges, and diffraction smooths these singularities at focus, while the overall helical intensity structure is preserved, as illustrated in Fig. 1(c). Once a single coil has been produced, a complete LS [Figs. 2(b) and 2(c)] can be obtained simply by piling up such coils. This is most naturally achieved by passing this coil through a Fabry–Perot etalon, with a free spectral range given by ΔωFP pΔω∕Δl, with p as an integer. This etalon produces a train of replicas of the initial coil, with a time period ΔT FP 2π∕ΔωFP Δτ0 ∕p, where Δτ0 2πΔl∕Δω is the pitch of the coil. If p 1, the rear end of the ith replica perfectly matches with the front end of the (i 1)th replica, and a multiple-coil LS is formed, such as the one displayed in Fig. 2(b). If p > 1, the pitch of the initial coil is an integer multiple of ΔT FP . In this case, the optical system produces what we call a LS of pth-order, consisting of p intertwined intensity helices of pitch Δτ0 , such as the one displayed in Fig. 2(c) for p 2. The total length of these LS is determined by the finesse of the etalon. It is highly instructive to also analyze this configuration in the frequency domain. Starting from a pulse with a continuous spectrum, an etalon produces a spectrum consisting of a set of discrete spectral lines [Fig. 2(a)], separated by the free spectral range ΔωFP . The condition ΔωFP pΔω∕Δl implies that the azimuthal mode index lω varies by the integer p between two consecutive spectral lines. If the optical system is adjusted in such a way that lω is an integer for one of these spectral lines, then it will also be an integer for all of the others. The laser field Eθ; t then takes a very simple form: Eθ; t ∝
X
einpθΔωFP t ;
(8)
n
where the index n accounts for the multiple spectral lines produced by the etalon inside the bandwidth of the initial ultrashort pulse. It is easily shown that Eq. (8)
corresponds to an intensity profile formed of p intertwined helices. Compared to the single coil, this configuration eliminates all modes with fractional OAMs, and thus guarantees that the complete LS propagates without any distortion. There are other possible ways of generating light springs, which illustrates different circumstances where this general class of beams is relevant. One example are the optical ferris wheels described in [13] for applications in ultracold atoms physics. These are particular cases of light springs with a very large pitch (>100 ns), which is obtained by superimposing only two LG beams of slightly different frequencies with different mode indices l1 and l2 , leading to jl1 − l2 j intertwined helices. In a totally different context, intertwined LS are naturally produced when some nonlinear optical processes are driven with the usual LG beams. This is the case of high-order harmonic generation (HHG) at high laser intensities, associated in the time domain with the generation of trains of attosecond pulses. Indeed, when a LG laser beam of frequency ωi and index li is used for HHG in gases, for instance, the nth harmonic forms a LG beam of index lω nωi nli [14,15], as expected from conservation rules. This harmonic spectral comb then exactly corresponds to the case sketched in Fig. 2(a), i.e., a set of discrete spectral lines, with an integer variation δnli of lω between two successive lines, where δn is the order difference between the successive harmonics in the spectrum. In the time domain, the associated train of attosecond pulses therefore has the structure of a LS of order δnli . The pitch of any of the individual intensity helices is the slope of lω multiplied by 2π, and thus equals li T 0 (T 0 being the laser optical period). The number of intertwined helices corresponds to the variation of lω between the successive harmonics. Thus, there are δnli helices, the combination of which results in attosecond pulse trains of period T 0 ∕δn at any given position in the beam [see Fig. 2(c)]. It is precisely such LS that are observed in Fig. 4 of [14], that present numerical simulations of HHG in gases (for which δn 2) with a li 1 LG laser beam. For a many-cycle driving pulse, an attosecond light spring of order δnli 2 is produced (Fig. 4(a) of [14]), while two separate single coils of LSs are generated for a few-cycle laser pulse (Fig. 4(b) of [14]). Such attosecond light springs remain to be observed experimentally.
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We now turn to possible applications of LS. One of the great prospects of these beams is their use to transfer OAM to Raman-active media. Ultrashort pulses are ideal to drive stimulated Raman scattering, since their spectrum is broad enough to include both frequencies ωi and ωf of the absorbed and emitted photons. When a standard ultrashort LG beam (i.e., lω l0 ) is used, the OAM of each of these two spectral components is the same, l0 ℏ, and there is no net transfer of OAM to the medium. In contrast, in the case of LSs, the OAMs of the two frequencies differ by Δlℏωi − ωf ∕Δω, and this difference is necessarily imprinted to the scattering medium; note that this OAM will be imprinted to the medium as a whole, rather than to its individual atoms or molecule [16]. A simple example is the excitation of plasma waves in the wake of ultraintense femtosecond laser pulses in underdense plasmas, which is nowadays exploited to produce GeV electron beams over cm distances. As the basic underlying process is stimulated Raman scattering, one expects no transfer of OAM to the plasma waves when a standard LG beam is used. This has indeed been confirmed by recent theoretical studies [17], and has a simple intuitive explanation: the force driving the plasma dynamics is entirely determined by the laser intensity profile, while for a LG beam, OAM is encoded into the beam phase. In contrast, if a LS or a single coil of LS is used, the helical structure of the intensity profile will directly affect the plasma dynamics. This should allow for a transfer of OAM to the plasma, which might significantly affect the plasma dynamics, and possibly the relativistic electrons accelerated in the wake of the laser pulse. This example clearly shows that in LS, information on the light’s OAM is not only encoded in the beam phase profile, but also in its spatio-temporal intensity profile. For the sake of simplicity, we have so far neglected all effects related to the possible variations of the spatial field amplitude with frequency ω, an issue known as spatial chirp. In the present case, spatial chirp will unavoidably arise because the spatial amplitude profiles p of LG modes have a ring shape, with a radius r l ∝ l that depends on l, and hence also on frequency ω in the case of LS. This effect has already been taken into account in the calculations leading to the results of Figs. 1 and 2, and has a weak influence as long as Δl ≪ l0 . Even when it becomes significant, it will not alter the main conclusions of this Letter, which are the general consequences of the spatio-spectral phase structure of the beam. Equation (2) always results in beams described by an equation of the form of Eq. (7). In the case of significant spatial chirp, the
local temporal profile of the pulse [At in Eq. (7)] can now vary in space, leading to a more complex structure of the helix (see Figs. 3(b)–3(d) in [13] for an example). Spatial chirp can also lead to additional effects, such as the ultrafast radial rotation of the laser wavefront in time [3], which open further interesting possibilities to manipulate the field in space and time. In conclusion, spatio-temporal light springs arise in many different circumstances in linear and nonlinear optics. They allow for a new encoding of the orbital angular momentum in the intensity profile of a light beam, rather than in its phase profile. These beams are likely to find different applications in laser-matter interactions, especially when stimulated Raman scattering is involved. We would like to acknowledge useful discussions with R. Geneaux, T. Ruchon, J. Vieira, and J. T. Mendonça and IDEX Paris Saclay for G. Pariente’s Ph.D. grant. References 1. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, J. Opt. 12, 093001 (2010). 2. G. Zhu, J. Van Howe, M. Durst, W. Zipfel, and C. Xu, Opt. Express 13, 2153 (2005). 3. H. Vincenti and F. Quéré, Phys. Rev. Lett. 108, 113904 (2012). 4. A. Yao and M. Padgett, Adv. Opt. Photon. 3, 161 (2011). 5. I. Zeylikovitch, H. I. Sztul, V. Kartazaev, T. Le, and R. R. Alfano, Opt. Lett. 32, 2025 (2007). 6. M. Bock, J. Jahns, and R. Grunwald, Opt. Express 37, 3804 (2012). 7. J. Atencia, M. V. Collados, M. Quintanilla, J. Marn-Sez, and I. J. Sola, Opt. Lett. 21, 21056 (2013). 8. R. Grunwald, T. Elsaesser, and M. Bock, Sci. Rep. 4, 7148 (2014). 9. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, New J. Phys. 6, 103 (2004). 10. M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995). 11. D. P. Ghai, Appl. Opt. 50, 1374 (2011). 12. M. V. Berry, J. Opt. A 6, 259 (2004). 13. S. Franke-Arnold, J. Leach, M. Padgett, V. E. Lembessis, D. Elinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, Opt. Express 15, 8619 (2007). 14. C. Hernández-Garcia, A. Picón, J. San Román, and L. Plaja, Phys. Rev. Lett. 111, 083602 (2013). 15. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, Phys. Rev. Lett. 113, 153901 (2014). 16. M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Davila Romero, Phys. Rev. Lett. 89, 143161 (2002). 17. J. T. Mendonça and J. Vieira, Phys. Plasmas 21, 033107 (2014).
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Spatio-temporal light springs: extended encoding of orbital angular momentum in ultrashort pulses G. Pariente and F. Quéré* Commissariat à l’Energie Atomique, Lasers, Interactions and Dynamics Laboratory, DSM/IRAMIS, CEN Saclay, Gif sur Yvette 91191, France *Corresponding author: [email protected] Received February 16, 2015; revised March 25, 2015; accepted March 26, 2015; posted March 31, 2015 (Doc. ID 234628); published April 27, 2015 We introduce a new class of spatio-temporally coupled ultrashort laser beams, which are obtained by superimposing Laguerre–Gauss beams whose azimuthal mode index is correlated to their frequency. These beams are characterized by helical structures for their phase and intensity profiles, which both encode the orbital angular momentum carried by the light. They can easily be engineered in the optical range, and are naturally produced at shorter wavelengths when attosecond pulses are generated by intense femtosecond Laguerre–Gauss laser beams. These spatio-temporal “light springs” will allow for the transfer of the orbital angular momentum to matter by stimulated Raman scattering. © 2015 Optical Society of America OCIS codes: (320.0320) Ultrafast optics; (320.5540) Pulse shaping; (080.4865) Optical vortices; (260.7120) Ultrafast phenomena. http://dx.doi.org/10.1364/OL.40.002037
In the last decades, ultrafast optics has become an important branch of optics, now producing femtosecond light pulses that can consist of a single optical period, or that can reach peak power in the petawatt range. The nonlinear interaction of such laser beams with matter can, for instance, produce beams of relativistic particles, or light pulses of attosecond durations that are already being used to temporally resolve electron dynamics. A common key feature of all these light sources is their large spectral width, with a well-defined phase relationship between the different frequency components. A current frontier of this research field is the control and metrology of ultrashort light pulses simultaneously in space and time. In ultrafast optics, a beam whose temporal and spatial properties cannot be separated, i.e., such that its electric field Ex; y; t ≠ E 0 f tgx; y, is said to be spatio-temporally coupled [1]. Spatio-temporal couplings (STCs) have long been considered to be detrimental, because they imply a loss of spatial coherence and thus systematically decrease the peak intensity at focus. However, recent results (e.g., [2,3]) indicate that moderate and controlled STCs could pave the way to a whole new range of experimental capabilities that mostly remain to be explored. In this Letter, we introduce a new class of spatiotemporally coupled light pulses of broad relevance, which we call spatio-temporal light springs (LS) for convenience, as a simple reference to the shape of their spatio-temporal intensity profile (Figs. 1 and 2). These beams are directly related to Laguerre–Gauss (LG) beams, which have also become a major tool in different fields of optics in the last decades, partly due to the fact that they carry orbital angular momentum (OAM) [4]. This property is related to the peculiar shape of their wavefronts, which have an helical structure. In the case of a LS, it is not only the phase, but also the intensity profile that has an helical structure, with different pitches for both. We formally describe the properties of these beams, and distinguish different subclasses from single light spring coils (Fig. 1) to intertwined light springs of 0146-9592/15/092037-04$15.00/0
different orders (Fig. 2). We explain how such beams can be produced, and discuss their possible applications. Let us start by recalling the equation for the spatial phase φl θ; z of a monochromatic LG beam E l jE l jeiφl of azimuthal mode index l and frequency ω, with cylindrical coordinates [4] (θ being the azimuthal angle, and z the propagation axis): φl θ; z lθ − jlj 1χz kz;
(1)
where k ω∕c, and χz arctanz∕zr is the Gouy phase with zr as the Rayleigh beam length. In such a beam, the average OAM per photon is lℏ. To simplify the calculations, we only consider LG beams with a radial mode index p 0 here, and restrict the study around the beam waist, so that the phase curvature can be neglected. In previous works on the generation of broadband beams carrying OAM (see, e.g., [5–8]), most of the efforts have been focused on ensuring that the mode index l is the same for all frequencies ω within the beam spectral width. But correlating l and ω can provide new degrees of freedom to shape ultrashort beams in space and time, and thus to tailor and control laser-matter interactions. Light springs are the simplest cases of this type of spacetime coupled beams. They consist of ultrashort pulses, for which each frequency component is associated with an LG mode with a different value of l, with the following linear relationship [Figs. 1(a) and 2(a)]: lω l0
Δl ω − ω0 ; Δω
(2)
where ω0 is the central frequency of the pulse, l0 is the mode index for this frequency, Δω is the spectral width of the pulse, and Δl is the variation of l over this width. This relationship results in a very peculiar spatiotemporal intensity profile for LS, which is easily understood intuitively. As is well known [4,9], when several LG modes of different l with a fixed frequency ω0 are © 2015 Optical Society of America
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OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015
Fig. 1. Example of a single coil of a light spring. (a) Pulse spectral intensity (black line, corresponding to τ 25 fs) and the variation of the azimuthal mode index l as a function of frequency (red line, corresponding to Δτ0 4τ 100 fs). (b) Spatio-temporal intensity profile of the pulsed beam obtained by superimposing LG modes with a continuously varying index l correlated to their frequency, as in panel (a). (c) Spatio-temporal intensity profile obtained after propagating this beam up to a focus. The main graphs of (b) and (c) show the isosurface corresponding to 20% of the peak intensity, while the side panels show the marginals of the intensity profile in the different subspaces.
superimposed with an appropriate phase relationship, their interference results in an azimuthal wavepacket: light is then localized around a given angle θ0 , independent of time. In the case of the LS, each LG mode is now associated with a different frequency ω [Eq. (2)], so that their relative phases vary linearly in time. This relative phase variation leads to a rotation of the azimuthal wavepacket in time, i.e., θ0 ∝ t. We can therefore expect the intensity profile of an LS to form an helix in space–time [Figs. 1(b) and 1(c) and 2(b) and 2(c)]. We now study the structure of this helix. The E-field of a LS in space and time is determined by the following Fourier transform: Z Eθ; z; t ∝ Z ∝
dωElω θ; z; ωe−iωt ;
(3)
dωjE lω je−iωtiφlω ;
(4)
with φl given by Eq. (1) and lω given by Eq. (2). Intensity maxima occur whenever the phase −ωt φlω of the integrand is stationary with respect to the integration variable ω, leading to the condition t ∂φlω ∕∂ω. A straightforward calculation provides the following locations θ0 t; z of the pulse maxima in space and time: θ0 t; z
Δω z t− χz; Δl c
(5)
where the sign of the last term is opposite to the sign of Δl. The intensity profile is thus indeed an helix, with a temporal pitch of Δτ0 2πΔl∕Δω Δlτ at any given z, where τ 2π∕Δω is the Fourier transform limited duration of the pulse. Neglecting the Gouy phase in Eq. (5), the spatial pitch at any given t is Δz ≈ cΔτ0 . This profile thus forms a “spring” in space, that essentially translates as it propagates, simply performing half a turn around its axis as it goes through focus, due to the π variation of the Gouy phase χz [10]. We now derive the exact temporal structure of LS at a given position z, assuming, for simplicity, a flat spectral phase. We can rewrite Eq. (4) as
Eθ; t ∝ eil0 θ−ω0 t
Z
dωjE lω je−iω−ω0 t−Δωθ ; Δl
(6)
(7) ∝ eil0 θ−ω0 t At − τ0 θ; R where At dωjE lω je−iω−ω0 t is the local temporal envelope of the ultrashort pulse, and τ0 θ Δl∕Δωθ is its local arrival time. This shows that the local temporal structure of the pulse is unaffected by the introduction of the STC. It still forms an ultrashort pulse of duration τ anywhere in space, but the arrival time τ0 of this pulse now varies azimuthally, leading to the formation of an intensity helix. According to Eq. (7), this helix is intertwined with the usual helical structure of the wavefronts of a LG beam with mode index l0 (see Media 1). We now discuss the practical issue of the generation of the LS, which will naturally lead us to separate them into different subclasses. The easiest way to generate a LS is to pass a normal ultrashort beam through a spiral waveplate, similar to those used to produce the usual LG beams [4], but now with a larger azimuthal variation of the plate thickness. For a phase velocity vφ ω and a thickness variation eθ Δeθ∕2π, this induces helical wavefronts corresponding to a mode index l0 c∕vφ ω0 − 1Δe∕cT 0 at the central frequency ω0 2π∕T 0 . But due to its group velocity vg ≠ c, it also induces an azimuthal group delay τ0 θ c∕vg − 1eθ∕c. According to Eq. (7), such an azimuthal group delay is equivalent to Eq. (2) when expressed in the frequency domain, with Δl given by Δl c∕vg − 1Δe∕cτ. This shows that the usual spiral waveplates, when used with ultrashort light sources, produce good approximations of a simple LG beam of a given l0 as long as Δe ∼ cT 0 ≪ cτ, but that they generate LS as soon as Δe ∼ cτ. Note that the independent control of Δl and l0 is not possible with such a single waveplate, but should be possible with additional optical elements, such as adaptive helical mirrors [11]. Passing an ultrashort beam through such a waveplate produces only a single coil of light spring, such as the one displayed in Fig. 1(b). This beam has a continuous frequency spectrum, across which lω also varies continuously [Fig. 1(a)]. For most frequencies, lω therefore takes noninteger values that correspond to spatial modes
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Fig. 2. Examples of ultrashort light springs. (a) Pulse spectrum (corresponding to a train of 25 fs pulses), and two different variations of l with ω (corresponding to Δτ0 2τ 50 fs in red, and Δτ0 4τ 100 fs in blue). In both cases, the l index takes integer values for all frequency modes. (b) Spatio-temporal intensity profiles of a first-order LS, corresponding to the red variation of l in (a), and (c) of a second-order LS, corresponding to the blue variation of l in (a) (same displays as in Fig. 1).
with fractional OAM. These beams are not solutions of the paraxial wave equation, and are thus known to distort as they propagate [12]. When considering the beam in space–time [Fig. 1(b)], the origin of this distortion upon propagation becomes obvious: it arises from the spatial singularities that occur at the front and rear temporal edges of the beam, which lead to diffraction. This, however, only affects these temporal edges, and diffraction smooths these singularities at focus, while the overall helical intensity structure is preserved, as illustrated in Fig. 1(c). Once a single coil has been produced, a complete LS [Figs. 2(b) and 2(c)] can be obtained simply by piling up such coils. This is most naturally achieved by passing this coil through a Fabry–Perot etalon, with a free spectral range given by ΔωFP pΔω∕Δl, with p as an integer. This etalon produces a train of replicas of the initial coil, with a time period ΔT FP 2π∕ΔωFP Δτ0 ∕p, where Δτ0 2πΔl∕Δω is the pitch of the coil. If p 1, the rear end of the ith replica perfectly matches with the front end of the (i 1)th replica, and a multiple-coil LS is formed, such as the one displayed in Fig. 2(b). If p > 1, the pitch of the initial coil is an integer multiple of ΔT FP . In this case, the optical system produces what we call a LS of pth-order, consisting of p intertwined intensity helices of pitch Δτ0 , such as the one displayed in Fig. 2(c) for p 2. The total length of these LS is determined by the finesse of the etalon. It is highly instructive to also analyze this configuration in the frequency domain. Starting from a pulse with a continuous spectrum, an etalon produces a spectrum consisting of a set of discrete spectral lines [Fig. 2(a)], separated by the free spectral range ΔωFP . The condition ΔωFP pΔω∕Δl implies that the azimuthal mode index lω varies by the integer p between two consecutive spectral lines. If the optical system is adjusted in such a way that lω is an integer for one of these spectral lines, then it will also be an integer for all of the others. The laser field Eθ; t then takes a very simple form: Eθ; t ∝
X
einpθΔωFP t ;
(8)
n
where the index n accounts for the multiple spectral lines produced by the etalon inside the bandwidth of the initial ultrashort pulse. It is easily shown that Eq. (8)
corresponds to an intensity profile formed of p intertwined helices. Compared to the single coil, this configuration eliminates all modes with fractional OAMs, and thus guarantees that the complete LS propagates without any distortion. There are other possible ways of generating light springs, which illustrates different circumstances where this general class of beams is relevant. One example are the optical ferris wheels described in [13] for applications in ultracold atoms physics. These are particular cases of light springs with a very large pitch (>100 ns), which is obtained by superimposing only two LG beams of slightly different frequencies with different mode indices l1 and l2 , leading to jl1 − l2 j intertwined helices. In a totally different context, intertwined LS are naturally produced when some nonlinear optical processes are driven with the usual LG beams. This is the case of high-order harmonic generation (HHG) at high laser intensities, associated in the time domain with the generation of trains of attosecond pulses. Indeed, when a LG laser beam of frequency ωi and index li is used for HHG in gases, for instance, the nth harmonic forms a LG beam of index lω nωi nli [14,15], as expected from conservation rules. This harmonic spectral comb then exactly corresponds to the case sketched in Fig. 2(a), i.e., a set of discrete spectral lines, with an integer variation δnli of lω between two successive lines, where δn is the order difference between the successive harmonics in the spectrum. In the time domain, the associated train of attosecond pulses therefore has the structure of a LS of order δnli . The pitch of any of the individual intensity helices is the slope of lω multiplied by 2π, and thus equals li T 0 (T 0 being the laser optical period). The number of intertwined helices corresponds to the variation of lω between the successive harmonics. Thus, there are δnli helices, the combination of which results in attosecond pulse trains of period T 0 ∕δn at any given position in the beam [see Fig. 2(c)]. It is precisely such LS that are observed in Fig. 4 of [14], that present numerical simulations of HHG in gases (for which δn 2) with a li 1 LG laser beam. For a many-cycle driving pulse, an attosecond light spring of order δnli 2 is produced (Fig. 4(a) of [14]), while two separate single coils of LSs are generated for a few-cycle laser pulse (Fig. 4(b) of [14]). Such attosecond light springs remain to be observed experimentally.
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OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015
We now turn to possible applications of LS. One of the great prospects of these beams is their use to transfer OAM to Raman-active media. Ultrashort pulses are ideal to drive stimulated Raman scattering, since their spectrum is broad enough to include both frequencies ωi and ωf of the absorbed and emitted photons. When a standard ultrashort LG beam (i.e., lω l0 ) is used, the OAM of each of these two spectral components is the same, l0 ℏ, and there is no net transfer of OAM to the medium. In contrast, in the case of LSs, the OAMs of the two frequencies differ by Δlℏωi − ωf ∕Δω, and this difference is necessarily imprinted to the scattering medium; note that this OAM will be imprinted to the medium as a whole, rather than to its individual atoms or molecule [16]. A simple example is the excitation of plasma waves in the wake of ultraintense femtosecond laser pulses in underdense plasmas, which is nowadays exploited to produce GeV electron beams over cm distances. As the basic underlying process is stimulated Raman scattering, one expects no transfer of OAM to the plasma waves when a standard LG beam is used. This has indeed been confirmed by recent theoretical studies [17], and has a simple intuitive explanation: the force driving the plasma dynamics is entirely determined by the laser intensity profile, while for a LG beam, OAM is encoded into the beam phase. In contrast, if a LS or a single coil of LS is used, the helical structure of the intensity profile will directly affect the plasma dynamics. This should allow for a transfer of OAM to the plasma, which might significantly affect the plasma dynamics, and possibly the relativistic electrons accelerated in the wake of the laser pulse. This example clearly shows that in LS, information on the light’s OAM is not only encoded in the beam phase profile, but also in its spatio-temporal intensity profile. For the sake of simplicity, we have so far neglected all effects related to the possible variations of the spatial field amplitude with frequency ω, an issue known as spatial chirp. In the present case, spatial chirp will unavoidably arise because the spatial amplitude profiles p of LG modes have a ring shape, with a radius r l ∝ l that depends on l, and hence also on frequency ω in the case of LS. This effect has already been taken into account in the calculations leading to the results of Figs. 1 and 2, and has a weak influence as long as Δl ≪ l0 . Even when it becomes significant, it will not alter the main conclusions of this Letter, which are the general consequences of the spatio-spectral phase structure of the beam. Equation (2) always results in beams described by an equation of the form of Eq. (7). In the case of significant spatial chirp, the
local temporal profile of the pulse [At in Eq. (7)] can now vary in space, leading to a more complex structure of the helix (see Figs. 3(b)–3(d) in [13] for an example). Spatial chirp can also lead to additional effects, such as the ultrafast radial rotation of the laser wavefront in time [3], which open further interesting possibilities to manipulate the field in space and time. In conclusion, spatio-temporal light springs arise in many different circumstances in linear and nonlinear optics. They allow for a new encoding of the orbital angular momentum in the intensity profile of a light beam, rather than in its phase profile. These beams are likely to find different applications in laser-matter interactions, especially when stimulated Raman scattering is involved. We would like to acknowledge useful discussions with R. Geneaux, T. Ruchon, J. Vieira, and J. T. Mendonça and IDEX Paris Saclay for G. Pariente’s Ph.D. grant. References 1. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, J. Opt. 12, 093001 (2010). 2. G. Zhu, J. Van Howe, M. Durst, W. Zipfel, and C. Xu, Opt. Express 13, 2153 (2005). 3. H. Vincenti and F. Quéré, Phys. Rev. Lett. 108, 113904 (2012). 4. A. Yao and M. Padgett, Adv. Opt. Photon. 3, 161 (2011). 5. I. Zeylikovitch, H. I. Sztul, V. Kartazaev, T. Le, and R. R. Alfano, Opt. Lett. 32, 2025 (2007). 6. M. Bock, J. Jahns, and R. Grunwald, Opt. Express 37, 3804 (2012). 7. J. Atencia, M. V. Collados, M. Quintanilla, J. Marn-Sez, and I. J. Sola, Opt. Lett. 21, 21056 (2013). 8. R. Grunwald, T. Elsaesser, and M. Bock, Sci. Rep. 4, 7148 (2014). 9. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, New J. Phys. 6, 103 (2004). 10. M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995). 11. D. P. Ghai, Appl. Opt. 50, 1374 (2011). 12. M. V. Berry, J. Opt. A 6, 259 (2004). 13. S. Franke-Arnold, J. Leach, M. Padgett, V. E. Lembessis, D. Elinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, Opt. Express 15, 8619 (2007). 14. C. Hernández-Garcia, A. Picón, J. San Román, and L. Plaja, Phys. Rev. Lett. 111, 083602 (2013). 15. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, Phys. Rev. Lett. 113, 153901 (2014). 16. M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Davila Romero, Phys. Rev. Lett. 89, 143161 (2002). 17. J. T. Mendonça and J. Vieira, Phys. Plasmas 21, 033107 (2014).
