Research Article Vol. 6, No. 12 / December 2019 / Optica 1547

Free-space propagation of spatiotemporal opticalvorticesS. W. Hancock, S. Zahedpour, A. Goffin, AND H. M. Milchberg*Institute for Research in Electronics and Applied Physics, University ofMaryland, College Park, Maryland 20742, USA*Corresponding author: milch@umd.edu

Received 26 September 2019; revised 3 November 2019; accepted 17 November 2019 (Doc. ID 378855); published 18 December 2019

Spatiotemporal optical vortices (STOVs) are a new type of optical orbital angular momentum (OAM) with optical phasecirculation in space–time. In prior work [Phys. Rev. X. 6, 031037 (2016)], we demonstrated that a STOV is a universalstructure emerging from the arrest of self-focusing collapse leading to nonlinear self-guiding in material media. Here, wedemonstrate linear generation and propagation in free space of STOV-carrying pulses. Our measurements and simula-tions demonstrate STOV mediation of space–time energy flow within the pulse and conservation of OAM in space–time.Single-shot amplitude and phase images of STOVs are taken using a new diagnostic, transient grating single-shot super-continuum spectral interferometry. © 2019 Optical Society of America under the terms of the OSA Open Access PublishingAgreement

https://doi.org/10.1364/OPTICA.6.001547

1. INTRODUCTION

Optical vortices are electromagnetic structures characterized bya rotational flow of energy density around a phase singularity,comprising a null in the field amplitude and a discontinuity inthe azimuthal phase. In the most common type of optical vortexin a laser beam, the azimuthal phase circulation resides in spatialdimensions transverse to the propagation direction. An exampleis the well-known orbital angular momentum (OAM) modes [1],typified by Bessel–Gauss (BGl ) or Laguerre–Gaussian (LGpl )modes with nonzero azimuthal index l . OAM beams have beenused in optical trapping [2] and superresolution microscopy [3],with proposed applications such as turbulence-resilient free-spacecommunications [4,5] and quantum key distribution [6]. Opticalvortex formation is ubiquitously observed in the speckle pat-tern of randomly scattered coherent light [7]. We note that all ofthese standard OAM vortices can, in principle, be supported bymonochromatic beams and hence are fundamentally CW phe-nomena. Standard OAM vortices embedded in short pulse beams[8–10], necessarily polychromatic, have also been experimentallyand theoretically studied, while theoretical work has exploredpolychromatic vortices that can exist in space–time [11].

Recently [12], we reported on the experimental discoveryand analysis of the spatiotemporal optical vortex (STOV), whosephase winding resides in the spatiotemporal domain. ToroidalSTOVs were found to be a universal electromagnetic structurethat naturally emerges from arrested self-focusing collapse of shortpulses, which occurs, for example, in femtosecond filamentationin air [13] or relativistic self-guiding in laser wakefield accelerators[14]. As this vortex is supported on the envelope of a short pulse,

its description is necessarily polychromatic. For femtosecondfilamentation in air, a pulse with no initial vorticity collapses andgenerates plasma at beam center. The ultrafast onset of plasmaprovides sufficient transient phase shear to spawn two toroidalspatiotemporal vortex rings of charge l =−1 and l =+1 thatwrap around the pulse. In air, the delayed rotational response ofN2 and O2 [15] provides additional transient phase shear, gener-ating additional l =±1 ring STOVs on the trailing edge of thepulse [12]. After some propagation distance and STOV–STOVdynamics, the self-guided pulse is accompanied by the l =+1vortex, which governs the intrapulse energy flow supportingself-guiding [12].

The requirement of transient phase shear for such nonlinearlygenerated STOVs suggested that phase shear linearly applied in thespatiospectral domain could also lead to STOVs, and use of a zerodispersion (4 f ) pulse shaper and phase masks have been proposed[16] and demonstrated [17] for this purpose. In this paper, we usesuch a 4 f pulse shaper to impose STOVs on Gaussian pulses andrecord single-shot in-flight phase and amplitude images of thesestructures using a new diagnostic developed for this purpose. Thestructures generated are “line-STOVs” as described in [12,16]; thephase circulates around a straight axis normal to the spatiotem-poral plane. An electric field component of a simple |l |th orderline-STOV-carrying pulse at position z along the propagation axiscan be described as

E (r⊥, z, τ )= a(τ/τs + i sgn(l)x/xs )|l |E0(r⊥, z, τ )

= A(x , τ )e il8s−t E0(r⊥, z, τ ), (1)

2334-2536/19/121547-07 Journal © 2019Optical Society of America

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Research Article Vol. 6, No. 12 / December 2019 / Optica 1548

Fig. 1. Top, setup for TG-SSSI. The STOV-carrying pump pulse (center wavelength λ0 = 800 nm) at the output of a 4 f pulse shaper is focused(∼1.5 µJ) or imaged (∼20 µJ) into a 500 µm thick fused silica witness plate. The pump pulse energy is kept sufficiently low so that the STOV pulse prop-agates nearly linearly in the plate. A probe pulse εi (λ0 = 795 nm, 2 nm bandwidth) crosses the STOV pulse direction at angle θ = 6◦, forming a transientgrating with modulations ∝ cos(kx sin θ +18(x , τ )), where the symbols are defined in the main text and reference coordinates are shown next to thewitness plate. The transient grating is probed by SSSI [19,20], which uses∼1.5 ps long chirped SC reference and probe pulses E ref and Epr (λ0 ∼ 575 nm).The result is single-shot time- and space-resolved images of amplitude and phase of STOV-carrying pulses. Bottom left, cylindrical lens-based 4 f pulseshaper [16,17] for imposing a line-STOV on a 45 fs, λ= 800 nm input pulse. A phase mask is inserted in the Fourier plane at the common focus of thecylindrical lenses. For the current experiment, we use spiral phase masks (l = 1, l =−1, and l = 8) and a π -step mask, all etched on fused silica, wherethe π -step angle α and the spiral orientation (for l =−1) are also shown. Both the l = 1 and l = 8 plates have 16 levels (steps) every 2 π . Shaper gratings:1200 line/mm; cylindrical lenses: focal length 20 cm.

(a)

(b)

(c)

Fig. 2. (a) Output of pulse shaper with no phase plate. The 50 fs input pulse, with a weakly parabolic temporal phase, is recovered. (b, c) Intensity andphase of pulse in far field of pulse shaper with l = 1 and l =−1 spiral phase plates; white-bordered insets, pulse shaper near-field intensity images. The redarrows show the direction of phase circulation. Headings of each column are described in the text. In all panels, the temporal leading edge of the pulse is onthe left (τ < 0), so propagation is right to left. The pulse energy for the three far-field cases above is∼1 µJ. For the near field cases (insets), the pulse energy isincreased to∼20 µJ to offset the reduced signal due to magnification.

where r⊥ = (x , y ), τ = t − z/vg is a time coordinate local tothe pulse, vg is the group velocity, τs and xs are temporal andspatial scale widths of the STOV, 8s−t(x , τ ) is the space–time

phase circulation in x − τ space, l =±1,±2, . . ., A(x , τ )=a((τ/τs )2 + (x/xs )2)|l |/2, a =

√2((x0/xs )2 + (τ0/τs )2)−1/2 for

l =±1, and E0 is the STOV-free near-Gaussian pulse input to the

Research Article Vol. 6, No. 12 / December 2019 / Optica 1549

shaper, where x0 and τ0 are its spatial and temporal widths. Herea is a normalization factor ensuring that pulse energy is conservedthrough the shaper:

∫d2r⊥dτ |E |2 =

∫d2r⊥dτ |E0|2.

2. EXPERIMENTAL SETUP

In order to confirm the presence of an ultrafast STOV-carryingpulse, both the phase and the amplitude of the electric field enve-lope must be measured, preferably in a single shot. In work byanother group, amplitude and phase have been retrieved fromfemtosecond pulses undergoing filamentation using a multishotscanning technique in combination with an iterative algorithm[18]. In our group, we have used single-shot supercontinuumspectral interferometry (SSSI) [19,20] to measure the space- andtime-resolved envelope (but not phase) of ultrafast laser pulsesfrom the near-UV to the long wave infrared [21,22]. It is worth firstbriefly reviewing SSSI. Figure 1 shows the three beams employed inSSSI: a pump pulse E S (the optical signal of interest, here a STOV)and twin supercontinuum (SC) reference and probe pulses E ref andEpr. The reference and probe SC pulses are generated upstream ofFig. 1 in a 2 atm SF6 cell followed by a Michelson interferometer(not shown). The transient amplitude of E S is measured via thephase modulation it induces in a spatially and temporally over-lapped chirped SC probe pulse Epr in a thin instantaneous Kerr“witness plate,” such as the thin fused silica window used here.The resulting spatiospectral phase shift 1ϕ(x , ω) imposed onthe probe is extracted from interfering E outpr ∼ χ

(3)E S E ∗S Einpr with

E ref in an imaging spectrometer. Here, E inpr and Eoutpr are the probe

fields entering and exiting the fused silica witness plate, χ (3) is thefused silica nonlinear susceptibility, and x is the position within aone-dimensional (1D) transverse spatial slice through the pumppulse at the witness plate (axes shown in Fig. 1). Fourier analysis ofthe extracted 1ϕ(x , ω) [19] then determines the spatiotemporalphase shift 1φ(x , τ )∝ IS(x , τ ), where IS(x , τ ) is the 1D space+ time pump intensity envelope.

For measurements of STOVs, in which space–time phase circu-lation is the key feature, ordinary SSSI is insufficient. To measurespace–time-resolved amplitude and phase in a single shot, we havedeveloped a new diagnostic, transient grating SSSI (TG-SSSI).In TG-SSSI, a weak auxiliary probe pulse εi (same central wave-length of the pump pulse and spectrally filtered by a 2 nm bandpassfilter; see Fig. 1) is interfered with the pump to form a transientspatial interference grating in the witness plate. Here εi is crossedwith E S at an angle θ = 6◦. The transient grating is now thestructure probed by SSSI, with the output probe pulse becomingE outpr ∼ χ

(3)E Sε∗i Einpr. As before, 1ϕ(x , ω) is extracted from the

interference of E outpr and E ref in the imaging spectrometer, leadingto1φ(x , τ ). Now, however,1φ(x , τ ) is encoded with the pumpenvelope modulated by the time-dependent spatial interferencepattern (transient grating):1φ(x , τ )∝ IS(x , τ ) f (x , τ ), where kis the pump wavenumber, f (x , τ )= cos(kx sin θ +18(x , τ )) isthe transient grating, and18(x , τ ) is the spatiotemporal phase ofE S . In the analysis of the two-dimensional (2D)1φ(x , τ ) images,18(x , τ ) is extracted using standard interferogram analysis tech-niques [19,20], and IS(x , τ ) is extracted using a low-pass imagefilter (suppressing the sideband imposed by the transient grating).While we can extract 2D amplitude and phase maps from a singleshot, averaging shots yields a better signal-to-noise ratio. However,due to mechanical vibrations in the lab, the fringe positions arenot stable shot-to-shot. To prepare frames for averaging, the phase

of the fringes in each frame is shifted to enforce alignment. Theframes are then averaged, after which the peak of the Fouriersideband is windowed and shifted to zero in the frequency domain.

STOVs were generated by a cylindrical lens-based 4 f pulseshaper [16], depicted in the lower left of Fig. 1. The pulseshaper imposes a line-STOV on an input Gaussian pulse (50 fs,1.5–20 µJ) using a 2π l spiral transmissive phase plate (withl =+1,−1, or 8) or a π -step plate at the shaper’s Fourier plane(common focus of the cylindrical lenses). The vertical and hori-zontal axes on the phase masks lie in the spatial (x ) and spectral(ω) domains. The phase plate orientations are shown in the figure,where for the step plate, the adjustable angle α is with respect tothe spectral (dispersion) direction. While the shaper imposes a

( )

()

(a)

(b)

(c)

(d)

(e)

= 25°

= −25°

( , ) ΔΦ ( , )

Fig. 3. (a) Flying donut near-field intensity and phase from π -steppulse shaper (αstep =+25◦, l = 1), obtained from imaging shaper out-put into witness plate. (b, c) Offset lobe far-field intensity and phase,obtained by focusing shaper output into witness plate for step orienta-tions αstep =±25◦(l =±1); (d) simulation for αstep =−25◦ of far-fieldintensity and phase for (d) no dispersion; (e) group dispersion delay(GDD= 100 fs2). The addition of parabolic temporal phase to the spa-tiotemporal phase step of (d) explains the phase pattern in (b, c). Headingsof each column are described in the text. The pulse energy in panel (a) is∼20 µJ and∼1 µJ in panels (b) and (c). Propagation is right to left.

Research Article Vol. 6, No. 12 / December 2019 / Optica 1550

spatiospectral (x , ω) phase at the phase plate, leading to a spa-tiotemporal (x , τ ) pulse immediately at its output at the exitgrating (near field), our desired spatial effects appear in the far fieldof the shaper, where the desired STOV-carrying pulse emerges.Here, we project to the far field by focusing the shaper output withlens L1 into the 500 µm fused silica witness plate, whereupon it ismeasured using TG-SSSI. In this context, the subscript on E S cannow be read as referring to a STOV-carrying pulse.

3. EXPERIMENTAL RESULTS AND DISCUSSION

In Fig. 2, row (a) shows the pulse with no phase plate in the pulseshaper. This is the far-field output of the shaper as measured byTG-SSSI in the witness plate. The temporal leading edge is atτ < 0. The left column shows1φ(x , τ ). The fringes are removedwith a low-pass filter, yielding IS(x , τ ) in the next column, while inthe third column, a high-pass filter leaves the fringe image f (x , τ ).The far-right column shows the extracted spatiotemporal phase18(x , τ ). It is seen that the pulse envelope IS closely agrees withthe 50 fs pulse input to the shaper, and that 18(x , τ ) is weaklyparabolic in time (small chirp) and relatively flat in space. Theslight curvature of the fringes of f (x , τ ) seen in Fig. 2 is attributedto a spectral phase mismatch between E S and εi .

One form of line-STOV-carrying pulse can be generated witha spiral phase plate in the pulse shaper. For a l = 1 plate, row (b)of Fig. 2 shows, as in (a), the various extractions from TG-SSSI.The presence of a spatiotemporal phase singularity is evident fromthe characteristic forked pattern in f (x , τ ). The spatiotemporalenvelope IS(x , τ ) and phase 18(x , τ ) of the STOV are shownin the second and fourth columns, where the pulse appears as anedge-first flying donut with a 2π phase circulation around thephase singularity at the donut null. Using an l =−1 plate (flip-ping the l = 1 plate) generates the opposite spatiotemporal phasecirculation, as seen in row (c). The small insets in (b) and (c) showthe corresponding near-field intensity envelopes from the shaper(obtained by imaging the shaper output at the witness plate), con-sisting of two lobes separated by a space–time diagonal. Owing toconservation of angular momentum, the associated spatiotemporalphase windings (not shown) are the same as in the far field.

Line-STOVs of charge l =±1 can also be generatedwith a π -step phase plate in the shaper’s Fourier plane,

rotated to an angle αstep with respect to the grating disper-sion direction, so that the step lies along the spatiospectral lined x̄/d ω̄=∓ 12 (xs /x0)(τs /τ0)

−1 (see discussion below), where x0and τ0 are the width and duration of the shaper input pulse, withx̄ = x/x0 and ω̄=ωτ0. In practice, αstep is finely adjusted to geta line-STOV output as measured by TG-SSSI. As seen in Fig. 3,for αstep = 25◦, the near-field output of the shaper is a flying donut[row (a)] with l = 1, while the lens-focused, far-field envelope[row (b)] is two lobes separated by a space–time diagonal. Going toαstep =−25◦ [row (c)] gives a STOV-carrying pulse envelope thatis the space reflection of (b). In (b), it is seen that the vortex chargeadds to+1 [consistent with (a)] and in (c) the charge adds to−1.

We simulate the near-field output of the pulse shaper byFourier-transforming an input spatiotemporal pulse E0(x , τ )to the spatiospectral domain Ẽ0(x , ω), applying the spatiospec-tral phase shift represented by the phase mask, along with anydispersion, and then Fourier-transforming the field back to the spa-tiotemporal domain as E (x , τ ). Here we ignore the y dependence,which is near-Gaussian throughout. To simulate the far-field out-put of the shaper, we transform E (x , τ )→ Ẽ (kx , τ )= E ′(x ′, τ ),where x ′ ∝ kx is the local transverse coordinate in the far field.Simulations of the far field of the π -step shaper are shown inpanels (d) with no dispersion and (e) with group dispersion delayGDD= 100 fs2, corresponding to the measured 18(x , τ ) inFig. 2(a). The result of (d) is in agreement with the expression forẼ (kx , y , τ ), while (e) resembles the experimental result (b). Theorigin of this effect is that optimizing the SC pulse for TG-SSSIleaves the pump pulse with a very small chirp [parabolic phase intime, as seen in Fig. 2(a)]. Adding this phase to the diagonalπ -stepphase of 3(d) gives 3(e). Comparing Figs. 2 and 3, we note that theπ -step and l =±1 spiral phase shaper outputs appear to be com-plementary: the near field of one these “quasi-modes” correspondsto the far -field of the other. As discussed, going from the Fourierplane in the shaper to the shaper output (near field) and then tothe far field requires two transforms: (x , ω)→ (x , τ )→ (kx , τ ).If we start with Eq. (1) (for l =±1) and ignore z, x→ kx yieldsẼ (kx , y , τ )= a(τ/τs ± 12 kx x

20/xs )Ẽ0(kx , y , τ ) and then τ→

ω yields Ẽ (kx , y , ω)= 12 a(iωτ20 /τs ± kx x

20/xs )Ẽ0(kx , y , ω),

where we have assumed a pulse shaper input E0(r⊥, τ )=�(y )e−(x/x0)

2e−(τ/τ0)

2, with spatial and temporal widths x0

and τ0, and where �(y ) in our experiment is near-Gaussian

Fig. 4. 3D+time UPPE simulation of STOV-carrying pulse launched from a pulse shaper with a l =+1 spatiospectral spiral phase factor e i1ϕ(x ,ω) =Z/|Z| corresponding to our experiment (see text). Near field is right after a 3 m lens at the pulse shaper output; far field is at the lens focus. Rayleigh range iszR = 2.3 m. The propagation direction within each panel is right to left. Top row, intensity profiles IS(x , τ ); bottom row, phase profiles18(x , τ )with redarrows showing phase increase direction; bottom row white-bordered insets, intensity profiles simulated using l =+1 spatiospectral spiral phase factorZ.

Research Article Vol. 6, No. 12 / December 2019 / Optica 1551

(∝ e−(y/y0)2) but can be arbitrary but bounded. However, we can

swap kx ↔ 2i x/x 20 and ω↔−2iτ/τ20 in any of these expressions

to calculate the field at any location given either of the other two.Therefore, a flying donut STOV with an l =±1 spiral phase in(x , τ ) in the far field requires an l =±1 spiral phase plate in (x , ω)in the shaper. A flying donut in (x , τ ) in the near field requires aπ -step plate in (x , ω) in the shaper, which yields spatiotemporallyoffset lobes in the far field separated by aπ -step in phase.

To estimate the optimum angle αstep for the π -step plateto produce a near-field l =±1 STOV at the shaper out-put, making the appropriate swap in the above expressionsgives Ẽ (x , y , ω) = ia

(12ωτ

20 /τs ± x/xs

)Ẽ0(x , y , ω) at

the phase plate, where a π phase shift occurs across the line12ωτ

20 /τs ± x/xs = 0. The spatiospectral orientation of the

plate’s π step is therefore d x̄/d ω̄=∓ 12 (xs /x0)(τs /τ0)−1, as cited

earlier, and clearly enables control of the STOV space–time aspectratio. For example, we have observed that for αstep→ 0, the STOVappears as two lobes reflected across the time axis. This is consis-tent with d x̄/d ω̄→ 0 and τ0/τs → 0, corresponding to extremetime-axis-stretching of the donut hole.

As most experiments with STOVs will take place in the far fieldof a pulse shaper, selecting among a flying donut, spatiotemporallyoffset lobes, or other possible space–time structures will depend onthe far-field STOV profile desired for applications. In any case, theelectromagnetic angular momentum is conserved through the spa-tiotemporal/spatiospectral domains.

To visualize how a STOV-carrying pulse evolves from thenear field at the pulse shaper to the far field, we have performed3D+time unidirectional pulse propagation equation (UPPE)propagation simulations [23,24], as shown in Fig. 4. The inputto the shaper is E0(r⊥, τ )= �0e−(y/y0)

2e−(x/x0)

2e−(τ/τ0)

2, to

which is applied the l =+1 spiral spatiospectral phase factore i1ϕ(x ,ω) =Z/|Z| (phase-only mask corresponding to our exper-iment), where Z= a( 12 ω̄(τ0/τs )+ i x̄ (x0/xs )), the prefactor ofẼ0(x , y , ω) as calculated using the theoretical treatment above.The pulse was then propagated to the far field through a 3 m lensat Rayleigh range zR = 2.3 m (our finite graphical processingunit (GPU)-based computer memory limited the simulations tolower spatial resolution, necessitating use of a long focal lengthlens). The top and bottom row of panels in Fig. 4 show amplitudeIS(x , τ ) and phase 18(x , τ ) of the STOV. The y dependencemaintains its Gaussian envelope. The white-bordered insets inthe bottom row show simulations with the phase and amplitudemaskZ applied, corresponding to our theoretical treatment above,which is based on the form of STOV assumed in Eq. (1). Theresults for both masks are very similar, and either works to generateSTOVs. The simulation clearly shows the continuous evolutionof the STOV pulse from space–time diagonally-separated lobes todonut, with the STOV angular momentum conserved through-out. It is important to reiterate that while the form of STOVassumed in Eq. (1) necessitates a spatiospectral phase and ampli-tude mask of form Z, we actually use a pure phase mask Z/|Z| inour experiment—that is, our (x , ω) pulse profile is mismatched tothe phase plate profile—but as shown by our 3D+time propagationsimulations, this leads to very similar results.

The transformation of one quasi-mode into the other canbe viewed as STOV mediation of the energy flow within thepulse. In a frame moving at the pulse group velocity, as shownin [25] and more recently applied to STOVs [12], the localPoynting flux consistent with the paraxial wave equation is

Fig. 5. Results from pulse shaper with l = 8 spiral phase plate. Pulsepropagation is right to left. Left column, intensity profiles IS(x , τ ); rightcolumn, phase profiles 18(x , τ ). All profiles are spatially rescaled forcomparison. (a) Near-field intensity and phase of shaper output, obtainedfrom imaging exit grating onto witness plate. The l = 8 vortex appears aseight π -step phase jumps (only six visible owing to underfilling of imageby probe SC pulse). Laser pulse energy ∼20 µJ; (b) far-field intensityand phase obtained by focusing shaper output onto witness plate. Here,eight l = 1 STOVs are seen in the phase plot (here, the SC reference pulseprofile overfills the smaller spot). Laser pulse energy∼2 µJ; (c, d) Fouriertransform simulation of near-field and far-field intensity and phase wherethe (x , ω) spatial profile of the pulse in the shaper is not matched to thephase mask; (e, f ) Fourier transform simulation of near-field and far-fieldintensity and phase where the (x , ω) pulse profile in the shaper is matchedto the phase plate.

S = (c/8πk0) | E S |2(∇⊥8s−t − β2(∂8s−t/∂ξ) ξ̂), whereξ = vg τ ,8s−t is the spatiotemporal phase [see Eq. (1)], ξ̂ is a unitvector along ξ , and β2 = c 2k0(∂2k/∂ω2)0 is the normalized groupvelocity dispersion, where βair2 ∼ 10

−5 and βglass2 ∼ 2× 10−2.

Because the first term in S is dominant for both air and glass, theweakly saddle-shaped energy flow [12] is mostly along±x , provid-ing the necessary transformation from donut to spatiotemporallyoffset lobes or back again. This is a remarkable effect: we note thatin a STOV-free beam, the term ∇⊥8 would act on a local spatialphase curvature to transversely direct energy (diffract) to both sidesof the beam propagation direction (here ±x and ±y ). However,in an l = 1 linear STOV whose axis is along y [see Fig. 2(b)],∇⊥8

Research Article Vol. 6, No. 12 / December 2019 / Optica 1552

points along (−x ) in front of the pulse and along (+x ) in the back,directing energy density to one side in front of the pulse and theopposite side in the back. This is seen in the transformation of thespatiotemporally offset lobes from the near field [Fig. 2(b) insetimage] to the far-field flying donut. Similar dynamics apply to thel =−1 STOV of Fig. 2(c), and to the l = 1 STOV of Figs. 3(a)and 3(b).

To explore higher-order STOVs, we use an l = 8(16π) spiralphase plate in the pulse shaper. Figure 5(a) shows the near-fieldintensity envelope and phase, while 5(b) shows the intensity andphase in the far field. In the near field, as shown in (a), six π -stepphase jumps appear, corresponding to nulls in the intensity enve-lope (rather than eight because the SC probe pulse underfilledthe larger image of the exit grating in the witness plate). In the farfield, where the SC probe pulse overfilled the pump pulse, enablingcoverage of all the vortices, it is seen that the pulse has formedeight l =+1 STOVs. Such splitting of high-charge vortices intomultiple single-charge vortices has been explained for standardmonochromatic OAM as originating from interference with acoherent background, or with a coherent probe beam used tomeasure the presence of vortices [26].

In our case, the splitting has a different origin: a mismatch ofour (x , ω) beam profile in the pulse shaper to the profile of thel = 8 spiral phase plate. While this mismatch has only minoreffects for generating l = 1 STOVs, as discussed in the context ofFig. 4, it reveals itself for higher-order STOVs. Fourier transformsimulations, including glass dispersion, are shown in Figs. 5(c)and 5(d) for the near and far fields, reproducing the main featuresof the measurements, including the “splitting” into eight l =+1vortices. The (x , ω) beam profile–phase plate mismatch leads toslightly different orientations of adjacent pairs of near-field lobes inFig. 5(a) [5(c)]; these form slightly displaced l = 1 windings in thefar field in Fig. 5(b) [5(d)]. So in our case, it appears that the l = 8STOV never forms and eight l = 1 STOVs are formed directly.We expect that careful dispersion management and a better matchof our spatiospectral beam profile with the phase plate will enablegeneration of high-order STOVs that can propagate into the farfield. This is shown in the simulations of Figs. 5(e) and 5(f ) forthe case where the spatiospectral profile and the phase plate arematched: an l = 8 flying donut is formed, accompanied by a singlevortex of the same charge.

4. CONCLUSION

In conclusion, we have demonstrated the linear generation andpropagation in free space of pulses that carry a new type of opticalOAM whose associated vortex phase circulation exists in space–time: the STOV. Our measurements show that freely propagatingSTOVs conserve angular momentum in space–time and mediatespace–time energy flow within the pulse. We have introduceda new ultrafast diagnostic, TG-SSSI to measure the space- andtime-resolved amplitude and phase of a STOV in a single shot. Weexpect that nonlinear propagation of STOV-carrying pulses orpropagation of STOVs through fluctuating media will provide arich area of study, and in such experiments sensitive to shot-to-shotfluctuations, TG-SSSI will be an important tool.

Funding. Air Force Office of Scientific Research (FA9550-16-10121, FA9550-16-10284); Office of Naval Research

(N00014-17-1-2705, N00014-17-12778); National ScienceFoundation (PHY1619582).

Acknowledgment. The authors thank N. Jhajj andJ. Wahlstrand for early work on the pulse shaper, andJ. Griff-McMahon and I. Larkin for help with the simulations.

Disclosures. The authors declare no conflicts of interest.

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