Transient-grating single-shot supercontinuum spectral interferometry (TG-SSSI)
TTransient-grating single-shot supercontinuum spectral interferometry (TG-SSSI)
S. W. Hancock, S. Zahedpour, H. M. Milchberg
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland, 20742, USA Abstract:
We present a technique for the single-shot measurement of the space- and time-resolved spatiotemporal amplitude and phase of an ultrashort laser pulse. The method, transient-grating single-shot supercontinuum spectral interferometry (TG-SSSI), is demonstrated by the space-time imaging of short pulses carrying spatiotemporal optical vortices (STOVs). TG-SSSI is well-suited for characterizing ultrashort laser pulses that contain singularities associated with spin/orbital angular momentum or polarization.
I. Introduction
The need to characterize ultrashort laser pulses has spawned a large and increasing number of single-shot techniques including autocorrelation [1], multiple versions of frequency resolved optical gating (FROG) [2-6], spectral phase interferometry for direct electric-field reconstruction (SPIDER) and related methods [7-12], STRIPED FISH [13], d-scan [14], plus single-shot supercontinuum spectral interferometry (SSSI) [15-17] and other spectral interferometry methods [18,19]. While the basic FROG and SPIDER techniques extract only the space-independent temporal amplitude and phase, more complicated techniques [12-14] have recovered the spatiotemporal phase and amplitude of a laser pulse in a single-shot, albeit only with simple features such as pulsefront tilt. STRIPED-FISH [13] and d-scan [14] methods use iterative algorithms which, to our knowledge, have not been shown to converge for complicated structured light containing singularities, and SEA-SPIDER requires ancillary assumptions in determining the timing of spatial slices [12]. While SSSI does not recover the spatiotemporal phase of a pulse, it does recover the spatiotemporal pulse envelope, which has enabled measurement of ionization rates and ultrafast plasma evolution [20], electronic, vibrational and rotational nonlinearities [21,22], as well as nonlinear refractive indices and pulse front tilt [23]. In this paper, we present a new method that can measure, in a single-shot, the spatiotemporal phase and amplitude of an ultrafast laser pulse. It was developed for recent measurements [24] of pulses embedded with spatiotemporal optical vortices (STOVs) [25] and is well-suited for characterizing ultrashort laser pulses that contain singularities associated with spin/orbital angular momentum (SAM/OAM) [26-28] or polarization [29].
II. Experimental Setup
We first briefly review SSSI by examining three of the beams in Fig. 1: the βstructured pulseβ πΈ π which we want to measure, the reference pulse πΈ πππ , and the probe pulse πΈ ππ . Here, the structured pulse has spatiotemporal phase and amplitude imposed by the zero dispersion pulse shaper [30-32] in the lower left of the figure. The reference and probe supercontinuum (SC) pulses are generated upstream of Fig. 1 by filamentation in a 2 atm SF cell followed by a Michelson interferometer (not shown), with πΈ πππ leading ππ by ~ 2 ps. The transient amplitude of πΈ π is measured via the phase modulation it induces in a spatially and temporally overlapped SC probe pulse πΈ ππ in an instantaneous Kerr βwitness plateβ, here a thin (100-500 Β΅m) fused silica window. The resulting spatio-spectral phase shift βπ(π₯, π) imposed on the probe is extracted from interfering πΈ ππππ’π‘ ~π (3) πΈ π πΈ πβ πΈ ππππ with πΈ πππ in an imaging spectrometer. Here, πΈ ππππ and πΈ ππππ’π‘ are the probe fields entering and exiting the witness plate, π (3) is the fused silica nonlinear susceptibility, and π₯ is position within a 1D transverse spatial slice through the pump pulse at the witness plate (axes shown in Fig. 1). Fourier analysis of the extracted βπ(π₯, π) [20] then determines the spatio-temporal phase shift βπ(π₯, π) β |πΈ π (π₯, π)| β πΌ π (π₯, π) , yielding the 1D space + time spatio-temporal intensity envelope πΌ π . Measurement of the spatiotemporal phase of πΈ π is enabled by the addition of an interferometric reference pulse β π , which is crossed with πΈ π at a small angle π π ( π π€ = 3.15Β° in the witness plate). This forms a nonlinear transient refractive index grating, where β π has the same center wavelength as πΈ π but is bandpassed to be temporally longer . The transient grating (TG) is now the signal probed by SSSI (yielding the new method we call TG-SSSI), where the output probe pulse from the witness plate becomes πΈ ππππ’π‘ β π (3) πΈ π β πβ πΈ ππππ . The interference of πΈ ππππ’π‘ and πΈ πππ in the imaging spectrometer then enables extraction of βπ(π₯, π) , yielding βπ(π₯, π) as before. We note that βπ(π₯, π) is the envelope of πΈ π modulated by the transient grating: βπ(π₯, π) β πΌ π (π₯, π)π(π₯, π) , where π(π₯, π) = cos(2π π€ π₯ π ππ(π π€ /2) + ΞΞ¦(π₯, π)) is the transient grating, π π€ = π π is the pump central wavenumber in the witness plate, π =1.45, and ΞΞ¦(π₯, π) is the spatiotemporal phase of πΈ π . Fig. 1.
Setup for transient grating single-shot spectral interferometry (TG-SSSI). The structured pulse πΈ π and the interferometric reference pulse β π cross at angle π π€ at the focus of lens L1 in a fused silica witness plate ( π π in air), where a transient grating is formed. The grating is probed by a supercontinuum (SC) probe πΈ ππππ , preceded ~2ps earlier by a reference SC pulse πΈ πππ . Imaged by L2, πΈ πππ and πΈ ππππ’π‘ interfere at an imaging spectrometer and the interferogram is analyzed in the Fourier domain, yielding a single-shot time and space resolved image of amplitude and phase of πΈ π . Bottom left : pulse shaper [30-32] for generating spatiotemporally structured pulses πΈ π , here STOVs [24,33,34] imposed on a 50 fs, Ξ»=800nm input pulse. The STOV phase windings are imposed by π = Β± 1 , and π = β2 spiral phase plates in the Fourier plane of the pulse shaper. The phase plates are etched on fused silica and have 16 levels (steps) every 2Ο. n the analysis of the 2D βπ(π₯, π) images, ΞΞ¦(π₯, π) is extracted using standard interferogram analysis techniques [13,14], and πΌ π (π₯, π) is extracted using a low pass image filter (suppressing the sidebands imposed by the transient grating). Due to group velocity mismatch in the witness plate between πΈ π (centre wavelength π = 800 nm ) and the SC probe ( π ππ = 600nm ), the extracted phase shift is smeared slightly in time by ~4 fs per 100ΞΌm of fused silica. The laser used in the experiments is a 4 mJ/pulse, 50 fs FWHM, π = 800 nm , 1 kHz Ti:Sapphire system. The beam is split 3 ways, with ~100 ΞΌJ directed to SC generation (400-700nm) for πΈ ππ and πΈ πππ , and a portion of the rest for πΈ π and β π , whose energies were controlled using ο¬ /2 plates and thin-film polarizers. The structured pulse πΈ π was embedded with spatiotemporal phase windings by placing π = Β±1 or π = β2 spiral phase plates at the Fourier plan of the pulse shaper [24]. As depicted in Fig. 1, the SC reference and probe pulses, πΈ πππ and πΈ ππππ , are combined collinearly with the pulse πΈ π using dichroic mirror DM , with πΈ π , β π , and πΈ ππππ overlapping temporally in the witness plate, while πΈ πππ precedes them (by 2 ps). From the output face of the witness plate, πΈ πππ and πΈ ππππ’π‘ were magnified and relay imaged onto the spectrometer slit using high numerical aperture (NA) telescope with achromatic lenses. The large NA is necessary to collect the first order diffraction ( π = Β±1 ) of πΈ ππππ’π‘ from the transient grating. It is important for the imaging lenses to be achromatic for the image at the spectrometer slit to be in focus for all SC wavelengths and to minimize spherical aberration, which could spatially offset the diffracted orders of πΈ ππππ’π‘ from the zero order. Background and signal data were collected at 40 Hz by placing Fig. 2 . Simulated spectrally-resolved scattering of supercontinuum probe pulse πΈ ππ from a transient nonlinear grating in a 500 Β΅m thick fused silica plate, generated by the interference of pulses πΈ π and β π , where π§ is distance from the output face of the plate. Plotted as |βπΈ ππ | = |πΈ ππππ’π‘ β πΈ ππππ | .(a) Bragg regime transient grating: π π€ = 3.15 β , grating period Ξ = 10ππ , and
π = 13.0 . Here, the scattering is captured at π§ = 1 cm owing to the rapid escape of the single diffracted order ( π = β1 ) from the simulation window. (b) Raman-Nath regime transient grating: π π€ = 0.31Β° , Ξ = 100ππ , and
π = 0.13 , showing π = Β±1 diffraction orders. π₯ ππ π ππ π§ = 1 π π§ = 8 π -0.500.51 (a) π = +1π = β1 (b) chopper in the path of πΈ π , which allowed for the subtraction of the phase shift induced by β π in the witness plate. In principle, achromatic imaging of all diffracted orders precludes the need for detailed analysis of the diffraction. However, it is interesting to note that in our experiment, we observe only the zero order ( π = 0 ) and the π = β1 order diffraction of the probe. To understand this, we assess whether probe diffraction is in the Bragg regime (one dominant diffraction peak) or in the Raman-Nath regime (multiple positive and negative diffraction orders) [35] by considering the dimensionless parameter π = 2ππ π πΏ Ξ πΜ β , where π π is the vacuum wavelength of incident light, Ξ is the interference grating period, πΜ is the mean refractive index, and πΏ is the grating thickness. From ref. [35], diffraction is in the Raman-Nath regime for π β€ 1 and in the Bragg regime for
π β« 1 . Our TG-SSSI configuration (with a π = 500 ΞΌm fused silica witness plate,
Ξ(π π€ =3.15Β°) = 10 ΞΌm , π = 2.5 Γ 10 cm /W , and πΜ = π + π π πΌ = π + Ξπ ππΊ /ππΏ , where Ξπ ππΊ is the modulated phase shift amplitude of the transient grating) gives π β 13.0 , which is in the Bragg regime. (Both Ξπ ππΊ and Ξ are from our measurements.) This explains the observation of only one diffracted order. This result is confirmed by simulations of scattering of πΈ ππ from the nonlinear grating formed by the interference of πΈ π and β π . The simulation uses our implementation [36] of the unidirectional pulse propagation equation (UPPE) method [37], where all 3 beams intersect in the 500Β΅m thick fused silica plate (with πΈ π and β π crossing at angle π π€ and πΈ ππππ normal to the surface). The beam parameters are πΈ π ( ο¬ =800nm, 50 fs FWHM, π€ = 100ππ , πΌ π,ππππ = 28 GW/cm , π = +1 STOV), β π ( ο¬ =800nm, 300 fs FWHM, π€ = 300ππ , πΌ π,ππππ = 28 GW/cm , GDD = 0 fs ), and πΈ ππππ ( ο¬ = 600nm, οο¬ = 350nm, 2.4 ps FWHM, π€ = 500ππ , GDD = 1200 fs , TOD =200 fs ). The output electric field is numerically propagated 4 cm beyond the witness plate in air and then πΈ π and β π are spectrally filtered out, leaving the field πΈ ππππ’π‘ . Figure 2(a) shows simulation results of probe diffraction for conditions similar to our experimental parameters ( π π€ = 3.15Β° and π = 13.0 ), where only the π = β1 diffraction order is present (Bragg regime), agreeing with our experiments. The crossing angle for Fig. 2(b) was chosen to be π π€ = 0.31Β° , giving π = 0.13 , in the Raman-Nath regime, and the π = Β±1 orders are present. We note that our current TG-SSSI setup could be adjusted to operate in the Raman- Nath regime by increasing the grating period Ξ (to Ξ β₯ β2ππ π πΏ/πΜ ), but increasing the intensity of πΈ π or β π to increase π could result in non-negligible plasma formation in the witness plate and refractive distortion of πΈ ππ . Figure 3(a) shows an example of a raw TG-SSSI interferogram frame recorded on the imaging spectrometer camera. Here, the pulse shaper generates πΈ π as a π = β2 spatiotemporal optical vortex (STOV) pulse [24,25]. The vertical spectral fringe spacing is set by the Michelson-imposed time delay between the πΈ πππ and πΈ ππππ’π‘ pulses. The 2D phase shift βπ(π₯, π) is extracted in the same way as with all other SSSI interferograms [14,15], yielding βπ(π₯, π) β πΌ π (π₯, π)π(π₯, π) , which is plotted in Fig. 3(b). Here, the horizontal fringes imposed by π(π₯, π) show the time-dependent interference between πΈ π and β π . The spatiotemporal pulse envelope is recovered by low pass image filtering of βπ(π₯, π) to remove π(π₯, π) , yielding πΌ π (π₯, π) in Fig. 3(c). Extraction of the spatiotemporal phase ΞΞ¦(π₯, π) is performed by Fourier analysis along π₯ [38], ΞΞ¦(π₯, π) = arg(β± π {β± π₯ {ΞΟ(π₯, π)}Ξ(π)}), (1) here β± π₯ {Ξπ(π₯, π)} = ΞπΜ(π, π) is the Fourier transform along π₯ , β± π {β} is the inverse Fourier transform along π , Ξ(π) is a sideband windowing and shifting ( π β π β 2π Ξβ ) function, and π is the π₯ -component of the spatial frequency. This is shown in Fig. 3(d) and (e). If the sideband is too close to the π -spectrum of the pulse envelope (which is centered at π = 0 ), Ξ(π) cannot separate the transient grating from the pulse envelope. This necessitates a larger spatial sample and/or finer grating period, considerations that have informed our pump-probe beam geometry. Even though TG-SSSI is a single-shot method, averaging many shots of a reproducible transient process enables significant enhancement of the signal-to-noise ratio. Before averaging, however, the shot-to-shot shifting of the spatial interference fringes (from mechanical vibrations in the optical setup) must be compensated. The fringes are effectively forced into common alignment by adding a constant phase
ΞπΜ π (π π , π π ) to each frame, giving ΞΞ¦Μ (π₯, π) = arg ( 1π β β± π {[π₯πΜ π (π, π) Γ exp(πarg(βπΜ π (π π , π π )))]Ξ(π) ππ= }), (2) where
ΞπΜ π (π, π) = β± π₯ {Ξπ π (π₯, π)} , ΞπΜ π (π π , π π ) is the constant phase added to frame π to align the fringes, (π π , π π ) is a common point across all π frames, and ΞΞ¦Μ (π₯, π) is the mean spatiotemporal phase. The point (π π , π π ) is chosen at a location in the sideband where the signal is sufficiently larger than the phase noise, otherwise each frame would be offset by a random phase factor. Fig. 3.
Measurement of a π = β2
STOV-carrying pulse (interferometric reference β π at 800nm and 10nm FWHM bandwidth); (a) Raw 1D space-resolved spectral interferogram; (b) extracted
Ξπ(π₯, π) ; (c) pulse envelope πΌ π (π₯, π) from low pass filtering of Ξπ(π₯, π) ; (d) log(|β± π₯ {Ξπ(π₯, π)}| + 1) . The red lines show the region to be spectrally windowed and the green circle identifies (π π , π π ) for frame averaging; (e) extracted spatiotemporal phase of the pulse, ΞΞ¦(π₯, π) . π ππ π ππ π ππ π₯ π π π₯ π π π₯ π π π π π (a) (b) (c) (d) (e) II. Results and Discussion
To demonstrate TG-SSSI, we used the pulse-shaper to generate (a) a Gaussian pulse ( π =0 , no phase plate), and STOV-carrying pulses with topological charge (b) π = +1 , (c) π = β1 , and (d) π = β2 , using corresponding spiral phase plates in the Fourier plane of the shaper. The columns of Fig. 4 show Ξπ(π₯, π) , πΌ π (π₯, π) , π(π₯, π) , and ΞΞ¦(π₯, π) for pulses carrying π = 0, Β±1, and β2 . For π = 0 (row (a)), we see a slight fringe curvature in the transient grating π(π₯, π) , indicating a dispersion mismatch between πΈ π and β° π . For the π = Β±1 STOVs in rows (b) and (c), π(π₯, π) clearly shows the transient fringe fusing or splitting identifying the opposite phase windings shown in the
ΞΞ¦(π₯, π) column. For π = β2 (row (d)), one fringe in π(π₯, π) splits into three at the center of the pulse. Upon phase extraction,
ΞΞ¦(π₯, π) has two nearby π = β1 phase windings rather than a single π = β2 winding. We attribute this to a mismatch between the transverse beam dimensions at the Fourier plane of the pulse shaper and the radially independent phase winding of the π = β2 phase plate. Since the profile of the beam in the Fourier plane of the shaper (itself dictated by the grating periodicity and cylindrical lens focal length) is slightly elliptical, one of the axes of the phase plate should ideally be scaled to match the ellipticity of the beam. Utilizing a programmable
Fig. 4.
Experimental results from TG-SSSI after fringe alignment. Columns show the extracted full TG-SSSI signal
Ξπ(π₯, π) , which is low-pass filtered to yield the pulse intensity envelope πΌ π (π₯, π) , or high-pass filtered to give the transient grating π(π₯, π) , from which the spatiotemporal phase ΞΞ¦(π₯, π) is extracted. The rows show results for (a) a Gaussian pulse ( π = 0) , (b) a π = +1
STOV, (b) a π = β1
STOV, and (c) results from an π = β2 phase plate. The red arrows denote the direction of increasing spatiotemporal phase. -100 0 100-30030 -100 0 100 fs -100 0 100 -0.05 0 0.05 -100 0 100 - 0 -100 0 100-30030 -100 0 100 -100 0 100 -100 0 100 -30030 -100 0 100 -100 0 100 -100 0 100 -100 0 100 -100 0 100-100 0 100 -30030 -100 0 100 -100 0 100 ( , ) ( , ) ( , ) ( , ) π ππ π₯ π π π = 0 π = +1 π = β1π = β2 (a)(b)(c)(d) patial light modulator rather than a fixed phase plate in the pulse shaper would enable scaling of the phase mask to match the beam profile, making possible the generation of π = Β±2 and even higher order STOVs. IV. Conclusion
In summary, we have presented a new single-shot diagnostic of ultrashort spatio-temporally structured laser pulses, transient grating single-shot supercontinuum spectral interferometry (TG-SSSI) and have used it to measure simple Gaussian and STOV-carrying pulses generated by a pulse-shaper. Among multiple possible applications, TG-SSSI should prove useful in the study of nonlinear propagation, collapse and collapse arrest of intense laser pulses in transparent media, where spatio-temporal optical pulse structures naturally emerge [25]. Finally, we note for experiments where the structured pulse πΈ π is highly repetitive and reproducible, TG-SSSI could be extended to two spatial dimensions ( π₯ and π¦ ) by transversely scanning πΈ ππππ’π‘ across the spectrometer entrance slit in the π¦ direction, as is done in 2D+1 SSSI [20], to obtain πΌ π (π₯, π¦, π) and ΞΞ¦(π₯, π¦, π) . Acknowledgements
The authors thank I. Larkin for discussions and technical assistance. This work is supported by Air Force Office of Scientific Research (FA9550-16-10121, FA9550-16-10284); Office of Naval Research (N00014-17-1-2705, N00014-20-1-2233); National Science Foundation (PHY2010511).
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