Abstract

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

1. INTRODUCTION

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 [1]. To date, systems that realize so-called slow light and fast light [2] exercise control over the group velocity through the effects of a carefully crafted wavelength-dependent refractive index in diverse physical embodiments, via material [35] 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 [2]. 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 vgc/n, where c is the speed of light in vacuum, and n is the refractive index. Under these assumptions, can a pulse travel in such a medium at c? In absence of dispersion, the large required change in group velocity from c/n to c 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 [1014] can indeed achieve this apparently impossible goal: to travel diffraction-free and dispersion-free for extended distances at a group velocity of c 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 c/n to c—or to any other velocity desired in the material [1720]. The question concerning the various definitions of group velocities has recently received careful consideration in Ref. [21]. 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 [21].

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 1.00022c [22], 1.00012c [23], and 1.00015c [24] (but larger changes can be achieved by the different methodology of “tilted-pulse fronts” [2527]). This falls short of the required variation needed to change the group velocity in a material from c/n to c. Previous approaches to endow optical fields with spatio-temporal correlations have utilized techniques for producing Bessel beams (e.g., annular apertures [28] or axicons [22,23,29]), nonlinear processes such as second-harmonic generation [30] 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 [37]. 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 [38], propagation for extended distances in free space [39], and synthesis of broadband ST pulses [40]. 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 [10] and by Sheppard [9] (Chapter 18), and experimentally in Ref. [37]. 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 c. 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 c 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 c.

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 n into monochromatic plane waves ei(kxx+kzzωt), each represented by a point on the surface of the material’s light-cone kx2+kz2=n2(ωc)2 [Fig. 1(a)]. Here kx and kz are the components of the wave vector along the transverse x- and longitudinal z-axes, respectively, and ω 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 P(θ) given by the equation ωc=ko+(kznko)tanθ [35,41], where the spectral tilt angle θ is with respect to the kz-axis, ωo is an optical carrier frequency, and ko=ωo/c. This implies that each temporal frequency ω is associated with a spatial frequency pair ±kx. The field E(x,z,t)=ei(nkozωot)ψ(x,z,t) has a slowly varying envelope whose plane-wave expansion is

ψ(x,z,t)=dkxψ˜(kx)eikxxei(ωωo)(tzccotθ)=ψ(x,0,tzvg),
corresponding to a wave packet of phase velocity of c/n and group velocity vg=ctanθ, and ψ˜(kx) is the Fourier transform of ψ(x,0,0). The intersection of the plane P(θm) with the material light-cone is a hyperbola when |tanθm|>1/n, which is the range of interest here. By setting tanθm=1, the ST wave packet propagates in the material at the speed of light in vacuum vg=c.

 figure: Fig. 1.

Fig. 1. Controlling the group velocity of a ST wave packet in a non-dispersive optical material. (a) The spatio-temporal spectrum of the ST wave packet in a material of refractive index n lies at the intersection of the light-cone kx2+kz2=(nωc)2, having an apex angle of tan1n, with a plane P(θm); we assume n>1 throughout. The intersection here (black curve) is a hyperbola on the surface of the light-cone. The projection of this spatio-temporal spectrum onto the (kz,ωc)-plane is a straight line that makes an angle θm with respect to the kz-axis. The group velocity in the medium is vg=ctanθm, and is independent of the refractive index. Our goal is to synthesize ST wave packets in any given material with θm=45° such that vg=c in the material. (b) The projection of the spatio-temporal spectrum in (a) onto the (kx,ωc)-plane, which is invariant upon traversing the planar boundary between different materials at normal incidence. (c) The projection in (b) is consistent with the intersection of the free-space light-cone kx2+kz2=(ωc)2 with a non-planar surface. The projection of this surface on the (kz,ωc)-plane is a conic section (solid curve), but it can be approximated for narrow bandwidths as a straight line (dotted line) making an angle θa with the kz-axis.

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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 kx and ω across planar interfaces between different media. Given a target plane P(θm) 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 (kx,ωc)-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 (kx,ωc)-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 kx-axis. The projection onto the (kz,ωc)-plane is hence not a straight line, thereby indicating dispersive behavior. Nevertheless, for narrow temporal bandwidths Δωωo, the projection onto the (kz,ωc)-plane of the free-space ST wave packet can be approximated to first order as a straight line making a spectral tilt angle θa with respect to the kz-axis (Appendix A); [Fig. 1(c)].

This simplification amounts to approximating the exact hyperbola in the vicinity of kx=0 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 θa to θm (Appendix A) such that

tanθa=tanθmn(n21)tanθm.
Therefore, to produce a ST wave packet that travels in the medium at the speed of light in vacuum (vg=c or tanθm=1), one must first synthesize a ST wave packet in free space for which
tanθa=11+nn2.
Because tanθa>1 for n>1, the ST wave packet in air must be superluminal. For example, a ST wave packet in air with θa77.1° will propagate in BK7 glass (n1.51) at the speed of light in vacuum instead of c/n. For a ST wave packet to travel at c in materials with higher indices, it may require first synthesizing a ST wave packet in free space with a negative group velocity (θa>90°). Interestingly, the transition from positive to negative values of the required group velocity when incident from air—to propagate in the material at c—occurs at a refractive index equal to the golden mean:
ϕ=1+521.618.

3. EXPERIMENT

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 [35]; (Fig. 2). This process endows the field with the requisite spatio-temporal spectral correlations and realizes the prescribed spectral tilt angle θa. A reflective phase-only SLM assigns the appropriate spatial-frequency pair ±kx to each wavelength λ 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 Δλ0.2nm 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 θa has been realized [Fig. 3(b)]. A charge coupled device (CCD) camera is scanned axially along z to capture the diffraction-free propagation of the time-averaged intensity I(x,z)=dtI(x,z,t) of the ST wave packet [Fig. 3(c)].

 figure: Fig. 2.

Fig. 2. Schematic of the setup used to synthesize and characterize ST wave packets traveling in a medium at the speed of light in vacuum. BS, beam splitter; Lx, Ly, cylindrical lenses along the x- and y-axes, respectively; Ls, spherical lens; A, aperture; SLM, spatial light modulator; G, diffraction grating; CCD, charge coupled device camera. Inset is the two-dimensional phase pattern imparted by the SLM to the impinging spatially resolved input-pulse spectrum.

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 figure: Fig. 3.

Fig. 3. (a) Measured spatio-temporal spectrum |E˜(kx,λ)|2 for a ST wave packet having θa=77.1° (the value corresponding to propagation at c in BK7, n1.51). (b) The spatio-temporal spectrum projected onto the (kz,ωc)-plane after normalizing both axes with respect to ko=2π/λo, where λo=797nm. The dotted red line corresponds to the light-line in vacuum θ=45°. The ST wave packet synthesized in free space is confirmed to have a spectral tilt angle of θ=77.1°, as illustrated in the inset, which is required for propagation at c in BK7 glass. (c) Spatial evolution of the time-averaged intensity I(x,z) along the optical axis, showing a diffraction-free length of 70mm. Here θa=60.6° (the value corresponding to propagation at c in distilled water, n1.33). Also shown on the same axial scale is the Rayleigh range zR=0.7mm of a traditional beam of the same transverse width of the ST wave packet. (d) Measured spatio-temporal intensity profile I(x,0,τ) of the ST wave packet after traversing a BK7 glass sample of length L=48mm, which is almost identical to the profile in absence of the sample. (e) Normalized pulse profile I(x=0,0,t) at the center of the transverse profile in (d) before and after inserting the BK7 sample (the pulses have been smoothed using median filtering). Here, z=0 corresponds to the location of the beam splitter BS4 in Fig. 2. The plotted delay is τd=41.4ps, which corresponds to a group delay of τm=160.11ps for the ST wave packet traversing a BK7 sample of length L=48mm.

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Measurements of the group delay of ST wave packets are carried out interferometrically with respect to a short reference pulse [37] (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 [37]. The spatio-temporal intensity I(x,z=0,τ)=|E(x,0,τ)|2 is traced by varying the delay τ 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 vg. Previous measurements of the spatio-temporal profiles of propagation invariant wave packets have made use of spatially resolved ultrafast pulse measurement techniques [23] 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 L in the common path of the ST wave packet and reference pulse, whose group delays are τa=L/(ctanθa) and τo=L/c, respectively. When τoτa is larger than the ST wave packet pulse width, the fringe visibility is lost, but it can be restored by introducing a delay τd=τaτo in the reference path. Measuring τd yields τa, and thus further confirmation of the value of θa.

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 L and index n in the common path results in a loss of interference, which is restored by inserting a delay length Δ in the reference path. The group delay of the ST wave packet traversing the material τm normalized by the group delay τo of a traditional pulse traversing a length L of free space is

τmτo=n+τa+τdτoτo,
where τm=L/(ctanθm), τo=L/c, and τa=L/(ctanθa) is the group delay of the ST wave packet traversing a distance L in free space, and τd=Δ/c is the delay inserted in the reference arm (Appendix A). For a traditional pulse, τm/τo=n, but this expectation is thwarted when utilizing a ST wave packet. A wave packet for which τm=τo traverses the material at a group velocity vg=c. Furthermore, it can be shown (Appendix A) that this condition implies that the delay length Δ introduced into the reference path normalized to the material layer thickness L depends only on the material refractive index,
ΔL=τdτo=(n1)2,
which provides a simple test to confirm that the ST wave packet propagates in the material at the speed of light in vacuum.

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 τm/τo for glass (n1.51 and L=48mm) and distilled water (n1.33 and L=50mm). Making use of Eq. (3), the required spectral tilt angle for vg=c in these materials is θa=77.14° for glass and θa=60.64° for distilled water. The results confirm that we have achieved the condition vg=c in both samples at the expected spectral tilt angles θa, in addition to demonstrating precise control over vg above and below c. 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 L=12 to 60 mm, as shown in Fig. 4(b), while holding θa at 77.1° to maintain vg=c 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 c. The measured values of ΔL for all the materials investigated are plotted in Fig. 5, and the theoretical expectation of (n1)2 provides an excellent fit.

 figure: Fig. 4.

Fig. 4. (a) Measured normalized group delay τm/τo (and normalized group velocity vg/c in the material) after traversing a length L=50mm in distilled water (n1.33) and L=48mm in BK7 glass (n1.51) versus the spectral tilt angle θa of ST wave packets synthesized in air. The points are experimental data and the continuous curves are the theoretical expectation, Eq. (5). (b) Measured delay length Δ and normalized group delay for BK7 glass while increasing the sample length L. The points are experimental results, and the continuous line is the theoretical expectation. Inset shows the group delay in the material τm normalized with respect to the delay in free space τo.

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 figure: Fig. 5.

Fig. 5. Measured delay length Δ normalized with respect to the sample length L for ST wave packets traversing a variety of optical materials at c. The points are data, and the continuous curve is the theoretical expectation [Eq. (6)]. Errors in estimating the group delays used to obtain the group velocity correspond to 40 μm of delay in free space (133 fs, the minimum detectable shift in the pulse peak). This represents an accuracy of 1.43% error with respect to a pulse width of 9.3 ps.

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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 [40] yields ST wave packets that can propagate for extended distances [39], thus making them candidates for remote sensing and standoff detection. Moreover, our approach may potentially be extended to accelerating or decelerating wave packets [42].

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.

APPENDIX A

1. Space–Time Wave Packets in a Non-Dispersive Material

In a non-dispersive material of index n, the light-cone is kx2+kz2=n2(ωc)2. Here we assume that the field is uniform along y, and thus ky 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 P(θ) given by ωc=ko+(kznko)tanθ. The projection of this intersection onto the (kx,ωc)-plane is a conic section given by

(1+ntanθ)2n2ko2tan2θ(ωcko1+ntanθ)2kx2n2ko2ntanθ+1ntanθ1=1,
which may be an ellipse, a hyperbola, or a parabola, according to the value of θ; in the range |tanθ>1n|, it is a hyperbola.

Both ω and kx are invariant across planar interfaces between different materials. Therefore, the spatio-temporal spectrum projected onto the (kx,ωc)-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 (kz,ωc)-plane is not a straight line, but is instead a curve given by the following equation:

(kzko)2=12Ω{n21ntanθm}Ω2{n211tan2θm},
where Ω=(ωωo)/ωo. Note that θm is the spectral tilt angle in the material [Fig. 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 Δωωo, the curved trajectory in the (kz,ωc)-plane given in Eq. (A2) becomes approximately a straight line, ωc=ko+(kzko)tanθa. To the first order, the projection onto the (kx,ωc)-plane given in Eq. (A1) can be approximated as a parabola in free space and in the material:

ωωo=1+kx22ko2tanθatanθa1,ωωo=1+kx22n2ko2ntanθmntanθm1.
Because the projection onto the (kx,ωc)-plane is invariant upon transition from the material to air, the following transformations between θa and θm are obtained:
tanθa=tanθmn(n21)tanθm,tanθm=ntanθa(n21)tanθa+1.
Alternatively, starting from the conic section projected onto the (kz,ωc)-plane in air [Eq. (A2)], ignoring the term quadratic in Ω, and approximating 1+x1+12x for x1, we obtain,
kzko+n(n21)tanθmtanθm(ωcko),
which can be rearranged as
ωc=ko+tanθmn(n21)tanθm(kzko)=ko+(kzko)tanθa,
from which we obtain the same relationship in Eq. (A4).

2. Measuring the Group Velocity of theSpace–Time Wave Packet

The ST wave packet and the reference pulse have group velocities ctanθa and c 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 L1, and the reference pulse propagates a distance L2, then L1/(ctanθa)=L2/c. Inserting a distance L 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 Δ in the reference path, to reach a new temporal balance (L1+L)/(ctanθa)=(L2+L+Δ)/c, such that Δ/L=cotθa1, from which θa and hence vg=ctanθa in air can be obtained, and which can be rewritten as τa=τo+τd, where τa=L/(ctanθa), τo=L/c, and τd=Δ/c.

Introducing a material of length L and index n 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 ctanθm and cn, respectively. However, adding an extra free-space length Δ in the path of the reference pulse can restore the temporal balance such that Δ/L=cotθmcotθa(n1). Substituting for tanθm in terms of tanθa from Eq. (A4), we obtain—within the parabolic approximation valid for narrow bandwidths—the following relationship:

ΔL=(11n)(11tanθa).
This relationship can then be recast into group delays as provided in Eq. (5). In the special case of tanθm=1, this relationship simplifies to Δ/L=(n1)2.

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 25×25mm2; 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 L1y in a 2f arrangement (f=50cm), which results in an estimated spectral uncertainty of δλ30pm [Fig. 3(a)].

The SLM introduces controllable spatio-temporal spectral correlations. We implement on the SLM at the position associated with each wavelength λ the linear phase distribution that corresponds to the required spatial frequency kx after introducing a reduction in scale by a factor of 4× (i.e., we replace kx with kx/4). We then subsequently compensate for this reduction at the output via a 4f telescope system that introduces a demagnification by a factor of 4× along the x-direction (cylindrical lenses L2x and L3x of focal lengths 40 cm and 10 cm, respectively). This system implements the transformation kx4kx, which compensates for the transformation kxkx/4 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 L1y, where the pulse is reconstituted.

4. Measurement Procedure for the Space–Time Wave Packets

To measure the spatio-temporal spectrum |E˜(kx,λ)|2, we sample a portion of the retro-reflected field from the SLM after passing through the lens L1y via a beam splitter BS3. The field is directed through a spherical lens L4s of focal length f=7.5cm to a CCD camera (CCD2, the Imaging Source, DMK 72AUC02) in a 2f configuration to collimate the spectrum along y and carry out a Fourier transform spatially along x.

The reference pulse is obtained from the initial laser pulses via beam splitter BS1. 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 BS4, and the resulting interferogram is recorded with a CCD camera (CCD1, the Imaging Source, DMK 33UX178). The optical-material samples are placed in the common path after BS4. 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 CS2) are placed in a cylindrical cell (Starna Cells 34-Q-50, interior length 50 mm, and diameter 19 mm). To accommodate the samples, CCD1 was kept at a fixed distance of 60 mm from BS4.

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 θ=60° is synthesized (any other value of θ 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.

Tables Icon

Table 1. List of the Materials Utilized and Their Relevant Parameters: The Measured Refractive Index n, the Corresponding Spectral Tilt Angle θa of the ST Wave Packet in Air, and Length of the Sample La

Funding

Office of Naval Research (ONR) (N00014-17-1-2458).

Acknowledgment

We thank Scott Webster and H. Esat Kondakci for assistance.

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13. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016). [CrossRef]  

14. K. J. Parker and M. A. Alonso, “The longitudinal iso-phase condition and needle pulses,” Opt. Express 24, 28669–28677 (2016). [CrossRef]  

15. S. Szatmári, P. Simon, and M. Feuerhake, “Group-velocity-dispersion-compensated propagation of short pulses in dispersive media,” Opt. Lett. 21, 1156–1158 (1996). [CrossRef]  

16. H. Sõnajalg and P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996). [CrossRef]  

17. R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996). [CrossRef]  

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

19. 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]  

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

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

22. 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). [CrossRef]  

23. 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]  

24. 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]  

25. J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express 10, 1161–1166 (2002). [CrossRef]  

26. J. Hebling, K.-L. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B 25, B6–B19 (2008). [CrossRef]  

27. J. A. Fülöp and J. Hebling, “Applications of tilted-pulse-front excitation,” in Recent Optical and Photonic Technologies, K. Y. Kim, ed. (InTech, 2010).

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

29. 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]  

30. 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]  

31. 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]  

32. 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]  

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

34. 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]  

35. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photonics 11, 733–740 (2017). [CrossRef]  

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

37. H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” arXiv:1810.08893 (2018).

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

39. 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]  

40. 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]  

41. 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,” arXiv:1809.08375 (2018).

42. 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]  

43. J. P. Torres, M. Hendrych, and A. Valencia, “Angular dispersion: an enabling tool in nonlinear and quantum optics,” Adv. Opt. Photon. 2, 319–369 (2010). [CrossRef]  

44. D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics 5, 561–565 (2011). [CrossRef]  

45. M. Reichert, A. Smirl, G. Salamo, D. Hagan, and E. Van Stryland, “Observation of nondegenerate two-photon gain in GaAs,” Phys. Rev. Lett. 117, 073602 (2016). [CrossRef]  

46. G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973). [CrossRef]  

47. M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. 23, 1980–1985 (1984). [CrossRef]  

48. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965). [CrossRef]  

49. I. Z. Kozma, P. Krok, and E. Riedle, “Direct measurement of the group-velocity mismatch and derivation of the refractive-index dispersion for a variety of solvents in the ultraviolet,” J. Opt. Soc. Am. B 22, 1479–1485 (2005). [CrossRef]  

50. K. Moutzouris, M. Papamichael, S. C. Betsis, I. Stavrakas, G. Hloupis, and D. Triantis, “Refractive, dispersive and thermo-optic properties of twelve organic solvents in the visible and near-infrared,” Appl. Phys. B 116, 617–622 (2014). [CrossRef]  

51. R. Jedamzik, S. Reichel, and P. Hartmann, “Optical glass with tightest refractive index and dispersion tolerances for high-end optical designs,” Proc. SPIE 8982, 89821F (2014). [CrossRef]  

52. S. Kedenburg, M. Vieweg, T. Gissibl, and H. Giessen, “Linear refractive index and absorption measurements of nonlinear optical liquids in the visible and near-infrared spectral region,” Opt. Mater. Express 2, 1588–1611 (2012). [CrossRef]  

53. I. H. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 1377–1379 (1962). [CrossRef]  

References

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  1. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
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  3. L. V. Hau, S. E. Harris, Z. Dutton, and C. Behroozi, “Light speed reduction to 17 m per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
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  4. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
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  5. K. Y. Song, M. G. Herráez, and L. Thévenaz, “Gain-assisted pulse advancement using single and double Brillouin gain peaks in optical fibers,” Opt. Express 13, 9758–9765 (2005).
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  6. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
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  7. K. L. Tsakmakidis, O. Hess, R. W. Boyd, and X. Zhang, “Ultraslow waves on the nanoscale,” Science 358, eaan5196 (2017).
    [Crossref]
  8. J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2010).
    [Crossref]
  9. H. E. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds., Non-diffracting Waves (Wiley-VCH, 2014).
  10. R. Donnelly and R. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
    [Crossref]
  11. S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12, 935–940 (2004).
    [Crossref]
  12. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
    [Crossref]
  13. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
    [Crossref]
  14. K. J. Parker and M. A. Alonso, “The longitudinal iso-phase condition and needle pulses,” Opt. Express 24, 28669–28677 (2016).
    [Crossref]
  15. S. Szatmári, P. Simon, and M. Feuerhake, “Group-velocity-dispersion-compensated propagation of short pulses in dispersive media,” Opt. Lett. 21, 1156–1158 (1996).
    [Crossref]
  16. H. Sõnajalg and P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
    [Crossref]
  17. R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
    [Crossref]
  18. K. Reivelt and P. Saari, “Optical generation of focus wave modes,” J. Opt. Soc. Am. A 17, 1785–1790 (2000).
    [Crossref]
  19. 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]
  20. C. J. Zapata-Rodríguez and M. A. Porras, “X-wave bullets with negative group velocity in vacuum,” Opt. Lett. 31, 3532–3534 (2006).
    [Crossref]
  21. P. Saari, “Reexamination of group velocities of structured light pulses,” Phys. Rev. A 97, 063824 (2018).
    [Crossref]
  22. 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).
    [Crossref]
  23. 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]
  24. 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]
  25. J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express 10, 1161–1166 (2002).
    [Crossref]
  26. J. Hebling, K.-L. Yeh, M. C. Hoffmann, B. Bartal, and K. A. Nelson, “Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities,” J. Opt. Soc. Am. B 25, B6–B19 (2008).
    [Crossref]
  27. J. A. Fülöp and J. Hebling, “Applications of tilted-pulse-front excitation,” in Recent Optical and Photonic Technologies, K. Y. Kim, ed. (InTech, 2010).
  28. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [Crossref]
  29. 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]
  30. 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]
  31. 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]
  32. 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]
  33. M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal Bessel beams: theory and experiments,” Opt. Express 17, 18148–18164 (2009).
    [Crossref]
  34. 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]
  35. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photonics 11, 733–740 (2017).
    [Crossref]
  36. H. E. Kondakci and A. F. Abouraddy, “Airy wavepackets accelerating in space-time,” Phys. Rev. Lett. 120, 163901 (2018).
    [Crossref]
  37. H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” arXiv:1810.08893 (2018).
  38. H. E. Kondakci and A. F. Abouraddy, “Self-healing of space-time light sheets,” Opt. Lett. 43, 3830–3833 (2018).
    [Crossref]
  39. 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]
  40. 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]
  41. 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,” arXiv:1809.08375 (2018).
  42. 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]
  43. J. P. Torres, M. Hendrych, and A. Valencia, “Angular dispersion: an enabling tool in nonlinear and quantum optics,” Adv. Opt. Photon. 2, 319–369 (2010).
    [Crossref]
  44. D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics 5, 561–565 (2011).
    [Crossref]
  45. M. Reichert, A. Smirl, G. Salamo, D. Hagan, and E. Van Stryland, “Observation of nondegenerate two-photon gain in GaAs,” Phys. Rev. Lett. 117, 073602 (2016).
    [Crossref]
  46. G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [Crossref]
  47. M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. 23, 1980–1985 (1984).
    [Crossref]
  48. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965).
    [Crossref]
  49. I. Z. Kozma, P. Krok, and E. Riedle, “Direct measurement of the group-velocity mismatch and derivation of the refractive-index dispersion for a variety of solvents in the ultraviolet,” J. Opt. Soc. Am. B 22, 1479–1485 (2005).
    [Crossref]
  50. K. Moutzouris, M. Papamichael, S. C. Betsis, I. Stavrakas, G. Hloupis, and D. Triantis, “Refractive, dispersive and thermo-optic properties of twelve organic solvents in the visible and near-infrared,” Appl. Phys. B 116, 617–622 (2014).
    [Crossref]
  51. R. Jedamzik, S. Reichel, and P. Hartmann, “Optical glass with tightest refractive index and dispersion tolerances for high-end optical designs,” Proc. SPIE 8982, 89821F (2014).
    [Crossref]
  52. S. Kedenburg, M. Vieweg, T. Gissibl, and H. Giessen, “Linear refractive index and absorption measurements of nonlinear optical liquids in the visible and near-infrared spectral region,” Opt. Mater. Express 2, 1588–1611 (2012).
    [Crossref]
  53. I. H. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 1377–1379 (1962).
    [Crossref]

2018 (5)

2017 (2)

K. L. Tsakmakidis, O. Hess, R. W. Boyd, and X. Zhang, “Ultraslow waves on the nanoscale,” Science 358, eaan5196 (2017).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photonics 11, 733–740 (2017).
[Crossref]

2016 (3)

2014 (2)

K. Moutzouris, M. Papamichael, S. C. Betsis, I. Stavrakas, G. Hloupis, and D. Triantis, “Refractive, dispersive and thermo-optic properties of twelve organic solvents in the visible and near-infrared,” Appl. Phys. B 116, 617–622 (2014).
[Crossref]

R. Jedamzik, S. Reichel, and P. Hartmann, “Optical glass with tightest refractive index and dispersion tolerances for high-end optical designs,” Proc. SPIE 8982, 89821F (2014).
[Crossref]

2013 (1)

2012 (1)

2011 (1)

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics 5, 561–565 (2011).
[Crossref]

2010 (2)

2009 (5)

2008 (3)

2007 (1)

2006 (2)

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]

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 (2)

2004 (2)

S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12, 935–940 (2004).
[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]

2003 (2)

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. 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]

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]

J. Hebling, G. Almási, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area THz-pulse generation,” Opt. Express 10, 1161–1166 (2002).
[Crossref]

2000 (2)

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

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[Crossref]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. Behroozi, “Light speed reduction to 17 m per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[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 (3)

1993 (1)

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

1984 (1)

1973 (1)

1965 (1)

1962 (1)

Abouraddy, A. F.

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

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

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]

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]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photonics 11, 733–740 (2017).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” arXiv:1810.08893 (2018).

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,” arXiv:1809.08375 (2018).

Alexeev, I.

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]

Almási, G.

Alonso, M. A.

Andreoni, A.

Averchi, A.

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

Bartal, B.

Behroozi, C.

L. V. Hau, S. E. Harris, Z. Dutton, and C. Behroozi, “Light speed reduction to 17 m per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Betsis, S. C.

K. Moutzouris, M. Papamichael, S. C. Betsis, I. Stavrakas, G. Hloupis, and D. Triantis, “Refractive, dispersive and thermo-optic properties of twelve organic solvents in the visible and near-infrared,” Appl. Phys. B 116, 617–622 (2014).
[Crossref]

Bhaduri, B.

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]

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,” arXiv:1809.08375 (2018).

Biegert, J.

Bonaretti, F.

Bowlan, P.

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Adv. Opt. Photon. (1)

Appl. Opt. (2)

Appl. Phys. B (1)

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Figures (5)

Fig. 1.
Fig. 1. Controlling the group velocity of a ST wave packet in a non-dispersive optical material. (a) The spatio-temporal spectrum of the ST wave packet in a material of refractive index n lies at the intersection of the light-cone kx2+kz2=(nωc)2, having an apex angle of tan1n, with a plane P(θm); we assume n>1 throughout. The intersection here (black curve) is a hyperbola on the surface of the light-cone. The projection of this spatio-temporal spectrum onto the (kz,ωc)-plane is a straight line that makes an angle θm with respect to the kz-axis. The group velocity in the medium is vg=ctanθm, and is independent of the refractive index. Our goal is to synthesize ST wave packets in any given material with θm=45° such that vg=c in the material. (b) The projection of the spatio-temporal spectrum in (a) onto the (kx,ωc)-plane, which is invariant upon traversing the planar boundary between different materials at normal incidence. (c) The projection in (b) is consistent with the intersection of the free-space light-cone kx2+kz2=(ωc)2 with a non-planar surface. The projection of this surface on the (kz,ωc)-plane is a conic section (solid curve), but it can be approximated for narrow bandwidths as a straight line (dotted line) making an angle θa with the kz-axis.
Fig. 2.
Fig. 2. Schematic of the setup used to synthesize and characterize ST wave packets traveling in a medium at the speed of light in vacuum. BS, beam splitter; Lx, Ly, cylindrical lenses along the x- and y-axes, respectively; Ls, spherical lens; A, aperture; SLM, spatial light modulator; G, diffraction grating; CCD, charge coupled device camera. Inset is the two-dimensional phase pattern imparted by the SLM to the impinging spatially resolved input-pulse spectrum.
Fig. 3.
Fig. 3. (a) Measured spatio-temporal spectrum |E˜(kx,λ)|2 for a ST wave packet having θa=77.1° (the value corresponding to propagation at c in BK7, n1.51). (b) The spatio-temporal spectrum projected onto the (kz,ωc)-plane after normalizing both axes with respect to ko=2π/λo, where λo=797nm. The dotted red line corresponds to the light-line in vacuum θ=45°. The ST wave packet synthesized in free space is confirmed to have a spectral tilt angle of θ=77.1°, as illustrated in the inset, which is required for propagation at c in BK7 glass. (c) Spatial evolution of the time-averaged intensity I(x,z) along the optical axis, showing a diffraction-free length of 70mm. Here θa=60.6° (the value corresponding to propagation at c in distilled water, n1.33). Also shown on the same axial scale is the Rayleigh range zR=0.7mm of a traditional beam of the same transverse width of the ST wave packet. (d) Measured spatio-temporal intensity profile I(x,0,τ) of the ST wave packet after traversing a BK7 glass sample of length L=48mm, which is almost identical to the profile in absence of the sample. (e) Normalized pulse profile I(x=0,0,t) at the center of the transverse profile in (d) before and after inserting the BK7 sample (the pulses have been smoothed using median filtering). Here, z=0 corresponds to the location of the beam splitter BS4 in Fig. 2. The plotted delay is τd=41.4ps, which corresponds to a group delay of τm=160.11ps for the ST wave packet traversing a BK7 sample of length L=48mm.
Fig. 4.
Fig. 4. (a) Measured normalized group delay τm/τo (and normalized group velocity vg/c in the material) after traversing a length L=50mm in distilled water (n1.33) and L=48mm in BK7 glass (n1.51) versus the spectral tilt angle θa of ST wave packets synthesized in air. The points are experimental data and the continuous curves are the theoretical expectation, Eq. (5). (b) Measured delay length Δ and normalized group delay for BK7 glass while increasing the sample length L. The points are experimental results, and the continuous line is the theoretical expectation. Inset shows the group delay in the material τm normalized with respect to the delay in free space τo.
Fig. 5.
Fig. 5. Measured delay length Δ normalized with respect to the sample length L for ST wave packets traversing a variety of optical materials at c. The points are data, and the continuous curve is the theoretical expectation [Eq. (6)]. Errors in estimating the group delays used to obtain the group velocity correspond to 40 μm of delay in free space (133 fs, the minimum detectable shift in the pulse peak). This represents an accuracy of 1.43% error with respect to a pulse width of 9.3 ps.

Tables (1)

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Table 1. List of the Materials Utilized and Their Relevant Parameters: The Measured Refractive Index n, the Corresponding Spectral Tilt Angle θa of the ST Wave Packet in Air, and Length of the Sample La

Equations (13)

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ψ(x,z,t)=dkxψ˜(kx)eikxxei(ωωo)(tzccotθ)=ψ(x,0,tzvg),
tanθa=tanθmn(n21)tanθm.
tanθa=11+nn2.
ϕ=1+521.618.
τmτo=n+τa+τdτoτo,
ΔL=τdτo=(n1)2,
(1+ntanθ)2n2ko2tan2θ(ωcko1+ntanθ)2kx2n2ko2ntanθ+1ntanθ1=1,
(kzko)2=12Ω{n21ntanθm}Ω2{n211tan2θm},
ωωo=1+kx22ko2tanθatanθa1,ωωo=1+kx22n2ko2ntanθmntanθm1.
tanθa=tanθmn(n21)tanθm,tanθm=ntanθa(n21)tanθa+1.
kzko+n(n21)tanθmtanθm(ωcko),
ωc=ko+tanθmn(n21)tanθm(kzko)=ko+(kzko)tanθa,
ΔL=(11n)(11tanθa).

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