Can an optical pulse traverse a non-dispersive material at the speed of light in vacuum? Because traditional approaches for controlling the group velocity of light manipulate either the material or structural resonances, an absence of dispersion altogether appears to exclude such a prospect. Here we demonstrate theoretically and experimentally that “space–time” wave packets—pulsed beams in which the spatial and temporal degrees of freedom are tightly intertwined—can indeed traverse a non-dispersive transparent optical material at the speed of light in vacuum. We synthesize wave packets whose spatio-temporal spectra lie along the intersection of the material’s light-cone with a spectral hyperplane tilted to coincide with the vacuum light-line. By measuring the group delay interferometrically with respect to a generic reference pulse, we confirm that the wave packet group velocity in a variety of materials (including water, glass, and sapphire) is the speed of light in vacuum.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
It has long been known that the group velocity of an optical pulse can vary substantially when traversing an optical material if the pulse spectrum overlaps with a resonance . To date, systems that realize so-called slow light and fast light  exercise control over the group velocity through the effects of a carefully crafted wavelength-dependent refractive index in diverse physical embodiments, via material [3–5] or structural resonances [6,7]. Such control facilitates optical buffering, enables the implementation of true optical delays, and offers advantages with respect to interferometric precision . However, in transparent materials at wavelengths far from resonance, the wavelength dependence of the refractive index may be ignored, especially for narrow bandwidths, in which case the group velocity is , where is the speed of light in vacuum, and is the refractive index. Under these assumptions, can a pulse travel in such a medium at ? In absence of dispersion, the large required change in group velocity from to does not seem possible.
Here, we demonstrate that the key to versatile control over the group velocity of a pulsed beam traveling in an optical material is to manipulate the spatial and temporal degrees of freedom jointly. Propagation-invariant space–time (ST) wave packets [8,9] endowed with tight spatio-temporal spectral correlations [10–14] can indeed achieve this apparently impossible goal: to travel diffraction-free and dispersion-free for extended distances at a group velocity of in linear isotropic non-dispersive materials (and also in dispersive media [15,16]). Because we do not rely on an optical resonance, this behavior is achieved independently of the refractive index or the spectral band, without amplification or attenuation, and potentially over large bandwidths. Indeed, spatio-temporal structuring produces arbitrary subluminal or superluminal shifts that can readily change the group velocity from to —or to any other velocity desired in the material [17–20]. The question concerning the various definitions of group velocities has recently received careful consideration in Ref. . Because we rely on measurements that are resolved in space and time, the group velocity we refer to henceforth is the average group velocity in the direction of the optical axis .
Prior efforts for spatio-temporal pulse shaping have yielded only limited control over the speed of light. In free space, typical velocities realized to date have been , , and  (but larger changes can be achieved by the different methodology of “tilted-pulse fronts” [25–27]). This falls short of the required variation needed to change the group velocity in a material from to . Previous approaches to endow optical fields with spatio-temporal correlations have utilized techniques for producing Bessel beams (e.g., annular apertures  or axicons [22,23,29]), nonlinear processes such as second-harmonic generation  and laser filamentation [31,32], or spatio-temporal spectral filtering [33,34]. In contrast, we make use here of a spatio-temporal synthesis approach that relies on a phase-only spectral modulation scheme implemented by a spatial light modulator (SLM) [35,36]. We have exploited this energy-efficient technique in synthesizing ST wave packets that propagate in free space with controllable group velocity . The precision achievable with this generalized pulse-shaping approach has been verified by the high quality of the fit between measurements and theory in a variety of experiments, including the observation of self-healing after opaque obstructions , propagation for extended distances in free space , and synthesis of broadband ST pulses . However, to the best of our knowledge, there have been no demonstrations of the propagation of ST wave packets synthesized via this novel approach in optical materials, whereupon the coupling from free space to the material is expected to result in substantial variation in the spatio-temporal spectral correlations initially introduced in free space.
In general, the spatio-temporal spectra of propagation-invariant ST wave packets lie along the intersection of the material light-cone with a tilted spectral hyperplane [35,41] whose spectral tilt angle solely determines its group velocity, as described theoretically by Donnelly and Ziolkowski  and by Sheppard  (Chapter 18), and experimentally in Ref. . Crossing the interface between two materials at normal incidence results in a change in this tilt angle. By judiciously selecting the spectral tilt angle in free space, we arrange for the spectral hyperplane in the material to coincide with the light-line in vacuum, leading to the group velocity in the material becoming exactly . The requisite spatio-temporal spectral correlations are introduced into a pulsed plane wave via a SLM, and the group delay resulting from traversing an optical material is measured with respect to a generic short reference pulse via an interferometric arrangement. We confirm that the synthesized ST wave packets travel at in a variety of solid and liquid optical materials having refractive indices in the range from 1.3 to 1.76. Furthermore, we demonstrate smooth and continuous control of the ST wave packet group velocity in these materials below and above .
2. THEORY OF SPACE–TIME WAVE PACKETS IN NON-DISPERSIVE MEDIA
We start our theoretical analysis from the decomposition of pulsed beams in a non-dispersive medium of refractive index into monochromatic plane waves , each represented by a point on the surface of the material’s light-cone [Fig. 1(a)]. Here and are the components of the wave vector along the transverse - and longitudinal -axes, respectively, and is the temporal frequency. The spatio-temporal spectra of propagation-invariant ST wave packets lie along conic sections at the intersection of the light-cone with the spectral hyperplane given by the equation [35,41], where the spectral tilt angle is with respect to the -axis, is an optical carrier frequency, and . This implies that each temporal frequency is associated with a spatial frequency pair . The field has a slowly varying envelope whose plane-wave expansion is
The key to synthesizing the associated ST wave packet in free space, which produces the targeted wave packet once coupled into the material, is exploiting the invariance of and across planar interfaces between different media. Given a target plane defining the ST wave packet in the material, the projection of the hyperbola at its intersection with the light cone [Fig. 1(a)] onto the -plane is also a hyperbola. This projected spatio-temporal spectrum is invariant across the planar interface between the material and free space [Fig. 1(b)]. Whatever form is taken by the spatio-temporal spectrum on the free-space light-cone, its projection onto the -plane must be identical with this hyperbola [Fig. 1(c)]. It can be shown (Appendix A) that this projection is consistent with the intersection of the free-space light-cone with a non-planar surface that is parallel to the -axis. The projection onto the -plane is hence not a straight line, thereby indicating dispersive behavior. Nevertheless, for narrow temporal bandwidths , the projection onto the -plane of the free-space ST wave packet can be approximated to first order as a straight line making a spectral tilt angle with respect to the -axis (Appendix A); [Fig. 1(c)].
This simplification amounts to approximating the exact hyperbola in the vicinity of with a parabola. Within this approximation, the transition of the ST wave packet from free space to the material changes the spectral tilt angle from to (Appendix A) such that
The experimental synthesis of ST wave packets utilizes a two-dimensional pulse shaper that simultaneously modulates the transverse spatial profile and temporal linewidth of a pulsed plane wave from a Ti:Sapphire femtosecond laser ; (Fig. 2). This process endows the field with the requisite spatio-temporal spectral correlations and realizes the prescribed spectral tilt angle . A reflective phase-only SLM assigns the appropriate spatial-frequency pair to each wavelength after spatially resolving the spectrum via a diffraction grating (Fig. 2, inset). The retro-reflected wave front returns to the grating that reconstitutes the pulse and produces the ST wave packet (Appendix A). We make use of a temporal bandwidth of to avoid dispersive effects in the materials investigated. The spatio-temporal spectrum is recorded after a double Fourier transform in space and time [Fig. 3(a)] from which we can confirm that the targeted value of has been realized [Fig. 3(b)]. A charge coupled device (CCD) camera is scanned axially along to capture the diffraction-free propagation of the time-averaged intensity of the ST wave packet [Fig. 3(c)].
Measurements of the group delay of ST wave packets are carried out interferometrically with respect to a short reference pulse  (Fig. 2). We first reconstruct the spatio-temporal intensity profile of the wave packet (pulse width, 9 ps) by superposing it with a shorter plane-wave reference pulse (100 fs). Observing high-visibility spatially resolved fringes indicates that the two wave packets overlap in space and time, whereupon the fringe visibility is proportional to the spatio-temporal intensity distribution . The spatio-temporal intensity is traced by varying the delay in the path of the reference pulse [Fig. 3(d)]. Of course, using this linear interferometric technique, only the amplitude of the spatio-temporal envelope is reconstructed and not the phase, but that suffices for our purpose here of estimating . Previous measurements of the spatio-temporal profiles of propagation invariant wave packets have made use of spatially resolved ultrafast pulse measurement techniques  or self-referenced interferometry [33,35,36].
We measure the group velocity of the ST wave packet in free space by introducing an extra propagation distance in the common path of the ST wave packet and reference pulse, whose group delays are and , respectively. When is larger than the ST wave packet pulse width, the fringe visibility is lost, but it can be restored by introducing a delay in the reference path. Measuring yields , and thus further confirmation of the value of .
4. MEASUREMENT RESULTS
We exploit a similar strategy to estimate the group velocity of the ST wave packet in a material. Placing a layer of length and index in the common path results in a loss of interference, which is restored by inserting a delay length in the reference path. The group delay of the ST wave packet traversing the material normalized by the group delay of a traditional pulse traversing a length of free space isA). For a traditional pulse, , but this expectation is thwarted when utilizing a ST wave packet. A wave packet for which traverses the material at a group velocity . Furthermore, it can be shown (Appendix A) that this condition implies that the delay length introduced into the reference path normalized to the material layer thickness depends only on the material refractive index,
An example of the delayed pulse profile after a sample (N-BK7) is plotted in Fig. 3(e) along with the initial pulse. In Fig. 4(a), we report the measured values of the normalized group delay for glass ( and ) and distilled water ( and ). Making use of Eq. (3), the required spectral tilt angle for in these materials is for glass and for distilled water. The results confirm that we have achieved the condition in both samples at the expected spectral tilt angles , in addition to demonstrating precise control over above and below . We also confirm the expected linear increase in the group delay with length of the optical material via measurements carried out on the BK7 glass samples of increasing thickness from to 60 mm, as shown in Fig. 4(b), while holding at 77.1° to maintain in the sample. Finally, we confirm the relation in Eq. (6) that applies to any optical material when the group velocity of the ST wave packet traversing is . The measured values of for all the materials investigated are plotted in Fig. 5, and the theoretical expectation of provides an excellent fit.
5. DISCUSSION AND CONCLUSIONS
The effect described here does not exploit material or structural resonances to modify the propagation of light. Only spatio-temporal correlations that can be readily encoded into the wave packet spectrum are required. Spatio-temporal structuring of light is a powerful tool enabling unprecedented control over optical propagation, effectively rendering the group velocity in a material independent of the refractive index or wavelength range. Careful spectral engineering coupled with the use of high-resolution phase plates in lieu of a SLM  yields ST wave packets that can propagate for extended distances , thus making them candidates for remote sensing and standoff detection. Moreover, our approach may potentially be extended to accelerating or decelerating wave packets .
The versatile control over the group velocity in optical materials enabled by the proposed wave-shaping approach can help solve many critical problems. Controlling the group velocity of optical pulses is essential for efficient frequency conversion [17,25,27,43]. Our versatile approach paves the way to perform group-velocity matching to boost the efficiency of a larger set of processes, such as non-degenerate two-photon absorption, crucial for the detection of infrared radiation and two-photon lasers [44,45].
Moreover, the spatio-temporal spectral correlations that we have studied here in the context of non-dispersive optical materials can be modified to ensure propagation invariance in dispersive materials at arbitrary group velocities. Finally, the bandwidth that can be exploited in synthesizing ST wave packets that travel in optical materials at a prescribed group velocity is—in principle—extended (enabling the utilization of ultrashort pulses). This is in contradistinction to the narrow linewidths associated with material and structural resonances exploited in traditional approaches to slow- and fast-light, which consequently enforce the utilization of only long pulses in such scenarios.
1. Space–Time Wave Packets in a Non-Dispersive Material
In a non-dispersive material of index , the light-cone is . Here we assume that the field is uniform along , and thus is negligible. The spatio-temporal spectrum of a propagation-invariant ST wave packet in such a material lies at the intersection of this light-cone with the spectral hyperplane given by . The projection of this intersection onto the -plane is a conic section given by
Both and are invariant across planar interfaces between different materials. Therefore, the spatio-temporal spectrum projected onto the -plane in Eq. (A1) remains invariant when going from the material to air. However, such a projected spatio-temporal spectrum is not consistent with the intersection of the free-space light-cone with a spectral hyperplane. Instead, it corresponds to the intersection of the free-space light-cone with a non-planar surface, whose projection onto the -plane is not a straight line, but is instead a curve given by the following equation:1(a)], although the equation refers to the spatio-temporal spectrum in air [Fig. 1(c)].
By limiting the bandwidth of the ST wave packet , the curved trajectory in the -plane given in Eq. (A2) becomes approximately a straight line, . To the first order, the projection onto the -plane given in Eq. (A1) can be approximated as a parabola in free space and in the material:A2)], ignoring the term quadratic in , and approximating for , we obtain, A4).
2. Measuring the Group Velocity of theSpace–Time Wave Packet
The ST wave packet and the reference pulse have group velocities and in air, respectively. Maximal interference occurs when the delays encountered by both wave packets, when they reach the detector, are equal. If the ST wave packet propagates a distance , and the reference pulse propagates a distance , then . Inserting a distance of free space in the common path results in a loss of interference. The spatially resolved high-visibility fringes are regained by introducing a distance in the reference path, to reach a new temporal balance , such that , from which and hence in air can be obtained, and which can be rewritten as , where , , and .
Introducing a material of length and index in the common path diminishes the observed interference fringes. The group velocities of the ST wave packet and the reference pulse in the material are and , respectively. However, adding an extra free-space length in the path of the reference pulse can restore the temporal balance such that . Substituting for in terms of from Eq. (A4), we obtain—within the parabolic approximation valid for narrow bandwidths—the following relationship:5). In the special case of , this relationship simplifies to .
3. Synthesis of Space–Time Wave Packets
The experimental setup combines spatial beam modulation with ultrafast pulse shaping. We start with a horizontally polarized pulsed laser beam from a Ti:Sapphire femtosecond laser (Tsunami, Spectra Physics) having a bandwidth of 8.5 nm and a central wavelength of 800 nm (pulse width, 100 fs). The beam is expanded spatially and collimated to produce a 25-mm-diameter plane-wave front. A reflective diffraction grating G (1200 lines/mm, area ; Newport 10HG1200-800-1) disperses the spectrum in space, and the second diffraction order is directed to a reflective spatial light modulator (Hamamatsu X10468-02) through a cylindrical lens in a arrangement (), which results in an estimated spectral uncertainty of [Fig. 3(a)].
The SLM introduces controllable spatio-temporal spectral correlations. We implement on the SLM at the position associated with each wavelength the linear phase distribution that corresponds to the required spatial frequency after introducing a reduction in scale by a factor of (i.e., we replace with ). We then subsequently compensate for this reduction at the output via a telescope system that introduces a demagnification by a factor of along the -direction (cylindrical lenses and of focal lengths 40 cm and 10 cm, respectively). This system implements the transformation , which compensates for the transformation implemented at the SLM. An aperture A is introduced to reduce the temporal bandwidth to 0.2 nm. The modulated wave front is retro-reflected back to the grating through , where the pulse is reconstituted.
4. Measurement Procedure for the Space–Time Wave Packets
To measure the spatio-temporal spectrum , we sample a portion of the retro-reflected field from the SLM after passing through the lens via a beam splitter . The field is directed through a spherical lens of focal length to a CCD camera (, the Imaging Source, DMK 72AUC02) in a configuration to collimate the spectrum along and carry out a Fourier transform spatially along .
The reference pulse is obtained from the initial laser pulses via beam splitter . A neutral density filter adjusts the power, and the beam is expanded and spatially filtered using two spherical lenses (focal lengths 50 cm and 10 cm) and a pinhole (diameter 30 μm). The reference pulses traverse an optical delay line and are then combined with the ST wave packet via beam splitter , and the resulting interferogram is recorded with a CCD camera (, the Imaging Source, DMK 33UX178). The optical-material samples are placed in the common path after . The solid samples (UV fused silica, BK7, and sapphire) are in the form of flat windows, whereas the liquid samples (distilled water, sea water, tetrahydrofuran, DMSO, benzene, and ) are placed in a cylindrical cell (Starna Cells 34-Q-50, interior length 50 mm, and diameter 19 mm). To accommodate the samples, was kept at a fixed distance of 60 mm from .
5. Measurement of Refractive Index of the Samples
For the liquid samples, interference is first recorded with an empty cylindrical cell placed in the reference path, and a ST wave packet of the spectral tilt angle is synthesized (any other value of can be used). The cylindrical cell is next filled with a liquid, resulting in a loss of the interference. The delay line in the reference arm is reduced to regain the interference. Using the measured delay and the inner length of the cylindrical cell (50 mm), we estimate the refractive index. The sea water was initially filtered using an Acrodisc PTFE CR syringe filter (25 mm diameter, 0.45 μm pore size) to remove any impurities. A similar approach was implemented for the solid samples. In Table 1, we list all the materials used and their relevant parameters.
Office of Naval Research (ONR) (N00014-17-1-2458).
We thank Scott Webster and H. Esat Kondakci for assistance.
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