Second harmonic generation of spatiotemporal optical vortices and conservation of orbital angular momentum

Spatiotemporal optical vortices (STOVs) are a new type of optical orbital angular momentum (OAM) structure in which the OAM vector is orthogonal to the propagation direction [Optica 6, 1547, (2019)] and the optical phase circulates in space-time. Here, we experimentally and theoretically demonstrate the generation of the second harmonic of a STOV-carrying pulse along with the conservation of STOV-based OAM.


Introduction
A spatiotemporal optical vortex (STOV) [1,2] is an electromagnetic structure with orbital angular momentum (OAM) and optical phase circulation defined in space-time, and is supported by a polychromatic pulse [3]. For a STOV-carrying pulse propagating in free space [2], the OAM vector is perpendicular to the direction of propagation. This is in contrast to a conventional space-defined optical vortex, which can be supported by a monochromatic beam, and where the OAM vector is parallel/anti-parallel to the direction of propagation and the optical phase winding is in the plane transverse to propagation [4]. Examples of the latter include Bessel-Gauss (BG ) or Laguerre-Gaussian (LG ) modes with nonzero azimuthal index [4]. STOVs are naturally emergent from filamentation processes [1] and can be constructed using a 4 pulse shaper, as originally proposed in [5] and experimentally demonstrated in [2,6], with free-space STOV propagation examined in [2] and later confirmed by [7].
In second harmonic generation ( → 2 ) of conventional OAM beams [8][9][10][11][12], the second harmonic photons carry twice the OAM of fundamental beam photons ( ℏ → 2 ℏ), where is the OAM quantum number or beam topological charge. Similarly, in sum or difference frequency generation, the OAM of two fundamental modes add [13]. In the case of ℎ order high harmonic generation with a mode of charge , the resulting photons have OAM ℏ [14][15][16][17]. The conservation of conventional OAM under these wide conditions has prompted our assessment of harmonic generation and OAM conservation in nonlinear interactions of STOVcarrying pulses, as presented in [18,19]. In this Letter, we demonstrate second harmonic generation (SHG) of STOVs and conservation of STOV orbital angular momentum. We use a single shot measurement technique [2,20] that captures the fundamental and SHG STOV amplitude and phase structure in midflight. Accompanying the measurements are simulations exploring the conversion process and the propagation of STOVs in material media.

Experimental Setup
In order to observe the spatiotemporal phase and amplitude of the fundamental and SHG STOVs and measure their OAM, we used transient grating single-shot supercontinuum spectral interferometry (TG-SSSI) [2,20]. TG-SSSI is an extension of single-shot supercontinuum spectral interferometry (SSSI) [21,22], which is a single-shot pump-probe technique for measuring ultrafast pump pulse-induced refractive index shifts in time and one or two spatial dimensions [23]. In prior work, we used SSSI to measure the nonlinear refractive index of air constituents from the visible (=400nm) through the long wave infrared (=11.0 m) [24,25], rates of ionization of noble gases [23] and the rovibrational response of diatomic gases [26]. While SSSI can also be used to characterize the spatiotemporal amplitude of ultrashort pulses [25], it fails to capture the spatiotemporal phase. As shown in Fig. 1(a), TG-SSSI recovers the spatiotemporal phase of a pump pulse by adding a spatial interferometric reference pulse ℇ (of the same central frequency, narrowed in bandwidth, and with a planar phasefront), which crosses the pump pulse at a small angle in an instantaneous nonlinear "witness plate", here a thin fused silica wafer. The interaction in the plate is probed by a supercontinuum (SC) probe pulse [21], yielding an output probe pulse ( , ) ∝ (3) (| | 2 + |ℇ | 2 + ℇ * + * ℇ ) that is relayed by an achromatic lens L2 to an imaging spectrometer and interfered with SC reference pulse , forming a broadband spectral interferogram. Here, is a coordinate transverse to the beam in the witness plate and parallel to the spectrometer entrance slit, and (3) is the third order susceptibility of the witness plate. The first two terms of express the cross-Kerr effect on the probe of the pump and reference ℇ . The second two terms are induced by the nonlinear transient grating from interference of and ℇ . The contribution of the term (3) |ℇ | 2 is subtracted as background by placing a chopper in the beam path. The spatiospectral nonlinear phase shift ∆ ( , ) encoded on in the witness plate is extracted from the spectral interferogram, after which Fourier analysis yields the spatio- Here, ( , ) = cos(2 ( /2) + ΔΦ( , )) is the grating function, = 0 0 is the central wavenumber of and ℇ in the witness plate (of refractive index 0 ) and ΔΦ( , ) is the spatiotemporal phase of . Low pass filtering in of ∆ ( , ) yields the pump pulse spatiotemporal intensity envelope ( , ) ∝ | ( , )| 2 , and high pass filtering yields the spatiotemporal phase ΔΦ( , ) of . Here, the pump is a STOV-carrying pulse, either the fundamental or the output 2 of a frequency doubling crystal. STOV pulses, , at the fundamental central frequency 0 are generated by the 4 pulse shaper shown in Fig. 1(b), to which is input = 800 nm, 50 fs pulses of energy up to 700 μJ from a Ti:Sapphire laser. Split off from the main beam upstream of the shaper were the interferometric reference pulse ℇ and a separate beam focused at /150 into a 2 atm Xe filamentation cell to generate a 400nm- Figure 1. TG-SSSI geometry and single-shot measurement of a STOV pulse. (a) Spatial interferometric reference pulse ℇ and STOV-carrying pump pulse 2 imaged from near field of pulse shaper by = 20cm lens L1 to the 200µm thick fused silica witness plate. Supercontinuum reference and probe pulses and are combined collinearly with the pump pulse through a dichroic mirror, with leading by 1.6ps. The pump and interferometric reference are rejected by another dichroic mirror and the SC is imaged onto the slit of an imaging spectrometer. The first lens in the relay imaging, L2, is an 18cm achromatic lens. (b) 4 pulse shaper with 100µm BBO crystal located 20cm from output grating. A fused silica phase plate with a 882 step (corresponding to phase shift) oriented at 40° to the grating dispersion direction was used to generate a = +1 STOV at 800nm. 750nm SC pulse which was first stretched to ~1.5 ps (from dispersion in the cell exit window and achromatic relay lenses) and then split by a Michelson interferometer into the replica probe and reference pulses and pulses (each ~1 μJ), with leading by 1.6 ps. The pulse energies of and ℇ were controlled using a /2 waveplate followed by a thin-film polarizer. The polarization orientation of and was controlled by a /2 plate before the filamentation cell so that it could be rotated to match the polarization of either or 2 . The probe is passed through a dichroic mirror to spatially overfill and temporally overlap with the transient grating formed in the witness plate by the interference of and ℇ , while precedes this interaction by ~1.6 ps.
The 4 pulse shaper depicted in Fig. 1(b) consisted of two 1200 groove/mm gratings, two 10 cm focal length cylindrical lenses and a transmissive phase plate at the common Fourier plane of the lenses. The fused silica phase plate has a step (~882 nm for 0 = 800 nm) across its diameter to generate fundamental STOVs with = 1 or = −1 at central frequency 0 in the near-field of the pulse shaper [2]. For STOVs of = ±1, the step is oriented at an angle = ±40 ∘ to the dispersion direction of the input diffraction grating. This angle is determined by the diameter of the beam and the shaper grating dispersion, and is tuned experimentally by real time TG-SSSI extraction [2]. SHG of was accomplished by placing a 100 m thick, type I BBO (beta barium borate) crystal at the immediate output of the 4 pulse shaper (in the near field), with its SHG output 2 imaged by MgF 2 lens L1 (focal length 20 cm) from the BBO crystal into the witness plate, as shown in Fig. 1(a).
The second harmonic of ℇ was generated using a separate, 1 mm thick BBO crystal. Here, the reduced phase matching bandwidth ensured that a spectrally narrower (temporally longer) ℇ 2 interfered with 2 in the witness plate, producing higher contrast interference modulations. ℇ 2 was directed through L1 with vertical offset so that it crossed 2 at an angle ~2.5° in the witness plate.
The process of SHG involving monochromatic and polychromatic beams is well known [27], where, given perfect phase matching, the nonlinear polarization and second harmonic field output is proportional to the square of the input field. Applying the same process to the STOV pulse of Eq. (1) gives 2 ( ⊥ , , ) ∝ 2 ( ± sgn( ) ) 2| | 2 ( ⊥ , , ) = 2 ( , ) 2 Φ s−t 2 ( ⊥ , , ). (2) Here, Eq. (2) predicts that the frequency doubled pulse will have twice the vorticity, topological charge, and angular momentum as the fundamental STOV-carrying pulse. This result is plotted in Fig. 2, which shows the field intensity and phase of the (a) fundamental and (b) second harmonic fields, where we see that the 2 phase winding of is transformed into a 4 phase winding of 2 , accompanied by a narrowing of the intensity ring by a factor √2. The measurements of the fundamental and SHG STOVs are shown in Fig. 3, where the red colormap panels of (a) show the spatiotemporal intensity ( , ) and phase ∆Φ( , ) of the fundamental = +1 STOV ( , ) at the near-field output of the 4 pulse shaper, which is imaged to the witness plate, interfered with ℇ , and probed by TG-SSSI, as shown in Fig. 1(a). The intensity shows the characteristic edge-first "flying donut" profile, with the pulse propagating right-to-left, while the phase profile is a single 2 winding centred at ( , ) = (0,0). As discussed, 2 is generated by passing the shaper's near field output through the BBO crystal, whose output is imaged to the witness plate, interfered with ℇ 2 , and then probed by TG-SSSI. Figure 3(b), in blue colormap, shows the frequency-doubled spatiotemporal intensity 2 ( , ) and phase ∆Φ 2 ( , ) of 2 ( , ). Instead of a single = +2 STOV , for which 2 ( , ) would have a single donut hole and ∆Φ 2 ( , ) would have a 4 phase winding (as in Fig. 2), we see that 2 ( , ) and ∆Φ 2 ( , ) show two nearby, spatiotemporally offest flying donuts around whose centers are two 2 phase windings. This constitutes two = +1 STOVs, and thus energy conservation dictates that the 2 pulse carries twice the OAM per photon of the fundamental .
⁄ ) ≈ 19 fs for the SHG crystal length = 100 . The integral yields two spatially offset = +1 STOVs, as depicted in Fig 4(e). This is essentially the STOV equivalent to the splitting observed due to spatial walk-off of LG beams in nonlinear crystals [32]. We note that the addition of nonzero GDD L1 leads to the diagonal (spatiotemporal) offset of Fig. 4(d).
The conservation of photon number implied by the Manley-Rowe relations for SHG, 2 ⁄ ( ( ) ℏ ⁄ ) = ⁄ ( (2 ) ℏ ⁄ ) [27], implies that photons at the second harmonic carry twice the OAM of photons at the fundamental. Recognizing from Fig. 4 that the two spatiotemporally offset = +1 STOVs represents a superposition of time-shifted = +2 STOV pulses, we find that OAM conservation in second harmonic generation also applies to STOVs.

Summary
We have experimentally and theoretically demonstrated the conservation of STOV-based OAM in second harmonic generation. Group velocity mismatch between the fundamental and second harmonic STOVs is the primary cause for = +2 STOVs to quickly separate into two = +1 STOVs after only a short propagation distance in the SHG crystal. In general, once a higher order STOV with | | > 1 is generated, spatiotemporal decomposition of vorticity is driven by diffraction and dispersion in propagation media. In particular, the space-time separation of topological charge during SHG could be mitigated via group velocity matching by using noncollinear SHG geometry.