A highly stable version of spectral interferometry is demonstrated, allowing single shot measurement of ultrafast high field processes using modest energy lasers, with pump and probe pulses totaling less than 1 mJ. The technique makes possible reconstruction of ultrafast refractive index transients with one-dimensional spatial resolution, limited only by the bandwidth of the supercontinuum pulse (~100 nm) and instrument resolution. The ultrafast nonlinear Kerr effect in glass, and in Ar, N2, and N2O gases is measured, along with plasma generation in Ar. The inertial contribution to the nonlinear index from N2 and N2O molecular rotation is also observed.
© 2007 Optical Society of America
Spectral or frequency domain interferometry (SI)  was developed to study the ultrafast transient refractive index change induced by the interaction of intense short duration pump laser pulse with a medium. This technique has been applied, for example, to measure the pump-induced phase modulation in absorptive materials  and optical fibers [3, 4]. Because of its sensitivity and also its simplicity as a linear method, SI has also proven useful in the temporal characterization of intense laser-plasma interactions, including density evolution of femtosecond laser produced plasma [5, 6], plasma shock waves , laser-cluster interactions , and laser wakefields [9, 10, 11].
In SI, a weak reference pulse first passes through the interaction zone, followed collinearly by the pump pulse, with no temporal overlap. A weak probe pulse, which is a replica of the reference pulse, is then sent collinearly through the interaction zone with adjustable time delay. The pump is removed from the beam path and the reference and probe are sent to a spectrometer, where they interfere in the spectral domain, recording a wavelength (frequency) dependent series of fringes in the spectrometer focal plane with frequency spacing 2π/τ, where τ is the time delay between reference and probe pulses. If the probe pulse experiences a phase shift caused by the pump-induced perturbation of refractive index (in other words, cross-phase modulation), the interference fringes shift accordingly. The standard version of SI (for example, ref. ) requires the probe pulse duration to be shorter than the fastest index modulation time scale; the temporal evolution of the refractive index is then retrieved via step-by-step scanning of the delay between the pump and probe pulses.
Thus there are two stringent requirements for obtaining a complete, high temporal resolution trace of refractive index evolution: an ultrashort probe pulse, and a high degree of shot-to-shot reproducibility. However, both shot-to-shot fluctuations in the pump pulse and the typically highly nonlinear response of the excited medium make good reproducibility difficult. Usually necessary is multi-shot averaging for each probe delay, which is time-consuming and may also result in the averaging to zero of real but small phase effects swamped by the shot-to-shot fluctuations. Standard SI is thus almost impractical for measuring the effects of high intensity laser pulses with repetition rate much lower than 1 Hz. For higher repetition rate systems, for example, the widely used 10 Hz Ti: Sapphire tabletop terawatt laser, in addition to fluctuations there may be long term drift of output energy, frequency and pulse shape, which can introduce systematic errors with probe delay.
To overcome these difficulties, various versions of single-shot spectral interferometry (SSI) have been developed. These utilize chirped reference and probe pulses [12, 13, 14], or an unchirped reference pulse and a chirped probe pulse . For a linearly chirped pulse with a Gaussian spectrum of full width at half maximum (FWHM) Δω centered at ω = ω 0 and group delay dispersion (GDD) β2 = 1/2 (∂ 2ϕ/∂ω 2)ω0 , where ϕ(ω) is the pulse phase in the frequency domain, the frequency sweep is given by ω = ω 0 + bt with chirp parameter . The chirped probe pulse is temporally overlapped onto the full transient index evolution, so that the varying phase shift is encoded onto the chirped pulse’s frequency components.
The frequency-dependent fringe shift Δϕ(ω) recorded by the spectrometer allows extraction of the index-induced phase transient Φ (t = (ω - ω 0)/b) from a single interferogram. However, this frequency-to-time ”direct mapping” approach is resolution-limited to , indicating that bandwidth-limited resolution of Δtres ~ (Δω)-1 is achievable only for β2(Δω)2 ≪ 1. Thus for fixed bandwidth pulses that are stretched longer (smaller b and larger β2) in order to capture longer duration events, Δtres increases and time resolution is degraded .
In order to take advantage of a potentially large Δω and to achieve the best time resolution, a different approach is needed for analyzing the spectral interferogram. For chirped probe and reference pulses Ẽpr(ω) exp(iϕpr(ω) and Ẽr(ω) exp(iϕr(ω)), where Ẽpr(ω) and Ẽr(ω) are real and ϕpr(ω) = ϕr + Δϕ(ω), the spectral interferogram allows extraction of Δϕ(ω). Knowledge of , and ϕr(ω), where Ipr(ω) and Ir(ω) are probe and reference spectra measured by the spectrometer and ϕr(ω) is the reference phase measured by cross-phase modulation , allows Fourier transformation to find the time-domain probe signal Epr(x,t) exp(iΔΦ(x,t)), from which the refractive index transient n(x,t) is obtained from kn(x,t)L = ΔΦ(x,t), where k is the central probe vacuum wavenumber and L is the effective interaction length in the medium. Here we consider transverse (to the probe beam) spatial variation in x of the measured phase, as discussed below.
Essential to SSI is a broadband probe pulse. Specialized Ti:Sapphire oscillators and optical parametric amplifiers (OPA) might fulfill this requirement for output bandwidths exceeding 100 nm. However, this would dramatically increase the system cost and complexity. A convenient way to obtain broad bandwidth is through supercontinuum (SC) generation. By focusing a 80 fs, 1 mJ, 800 nm Ti: Sapphire laser pulse in atmospheric air, ~100 nm bandwidth SC probe and reference pulses centered at ~690 nm were generated (with total SC energy 10–100 μJ), making single-shot supercontinuum spectral interferometry (SSSI) feasible . In SSSI, temporal resolution of ~10 fs was achieved , with probe bandwidth and spectrometer resolution as the only limiting factors. An added feature is that for experiments using 800 nm pump pulses, the probe wavelength of ~700nm introduces negligible pump-probe walkoff during propagation through the interaction region. Pump-probe walkoff can be a problem for schemes using second harmonic probe pulses . SSSI has been used to measure the transient Kerr nonlinearity in a solid , laser-induced double step ionization of helium , laser-heated cluster explosion [8, 18], and intense laser coupling into plasma waveguides . Recently, SSI with a broadband chirped second harmonic (SHG) probe pulse at ~400 nm was used to measure laser wakefields induced by 800 nm pump pulses , but with less temporal resolution and more walkoff than with SSSI.
We note that a recent and popular method to generate a broadband supercontinuum is to guide a femtosecond laser pulse through a photonic crystal fiber . However, fiber damage limits the pump laser pulse energy to the nanojoule level . For a single-shot measurement where there may be significant background light, such as in laser-plasma experiments , nanojoule supercontinuum pulse energy is too low for practical application.
In this paper we report an improved SSSI setup employing a commercial kilohertz regenerative amplifier system producing 1 mJ, 110 fs pulses. SC pulses are generated with much lower pulse energy in a sealed Xe gas cell, leaving sufficient pulse energy to use as a pump in a wide range of experiments. The SC pulses also have excellent shot-to-shot stability, making possible the averaging of results over many thousands of shots if desired. We discuss in detail this new configuration and report results for the transient Kerr nonlinearity in BK7 glass, and samples of monatomic gas (argon) and gases of linear molecules N2 and N2O.
2. Experimental setup
Two laser pulses were split at beamsplitter BS1 from the output of a commercial Ti:Sapphire regenerative amplifier (RGA) (Spectra-Physics Spitfire) with 1 kHz repetition rate (see Fig. 1). We note that in our previous work [8, 16, 17, 18, 19] SC was generated by focusing in 1 atm air a ~1 mJ, 70 fs laser pulse split from a 10 Hz, 2 TW Ti: Sapphire laser system based on a 10 Hz regenerative amplifier followed by two power amplifiers. Pulse-to-pulse output energy fluctuations of ~10–15% were determined by fluctuations of the 10 Hz, 532 nm pump laser pulses. Here, the 1 kHz RGA is pumped by a CW arc lamp-pumped, intra-cavity doubled Q-switched Nd:YLF laser (Spectra Physics Merlin), with pulse-to-pulse energy fluctuation less than 2%. The result is very stable SC on a shot-to-shot basis.
For SC generation, one of the pulses (~300 μJ) was focused at f/6 into an 11-cm long xenon-filled (0–2 atm) gas cell (XGC) with 1-mm thick fused silica entrance and exit windows. Xenon gas has previously been observed to generate very broad supercontinuum spectra under femtosecond laser pulse illumination . The SC pulse (along with the fundamental) emerges from the propagation filament induced by χ (3) self-focusing. The cell windows were sufficiently far from the beam waist/filament that they provided no contribution to the SC generation. The conical emission was transversely spatially chirped, with frequency increasing with radial position. This emission, with approximately 10 μJ/pulse in the SC component and the rest at the fundamental frequency, was collected by a lens at f/3 and converted into weakly converging beam, from which the fundamental component was removed by passing the beam through a high reflection dielectric mirror (M) centered at λ = 800 nm. The slightly converging SC pulse was then passed through a Michelson interferometer (MI) to generate a pair of co-propagating, identical pulses with variable delay (the reference and probe pulses). Beyond the Michelson, the converging beam spot was now small enough to efficiently reduce its spatial chirp and shape its transverse profile by placing a 500-μm diameter pinhole (P) in its path. By fine tuning the transverse position of the pinhole, a SC beam with high brightness, broad bandwidth, good spatial coherence, and uniform beam profile was obtained. The SC beam was then collimated by a telescope with 2× magnification, and the pulse duration and chirp parameter were tuned by adding appropriate lengths of dispersive material in the beam path. In the results shown here, we used a 2.5-cm thick optical grade SF4 glass window. This stretched the reference/probe pulses to ~2 ps, providing a 2 ps window for single-shot measurements of refractive index transients.
The other beam from BS1 was passed through an adjustable delay line and served as the pump. A half waveplate (HWP in Fig. 1) in this beam allowed independent pump polarization adjustment with respect to the SC beam. The SC and the pump beam paths were collinearly recombined at BS2, and focused by a f=41 cm lens into the sample to be measured. In the work presented here, the sample was either 200 μm thick BK7 window or a 45 cm long high pressure gas cell with 1 cm thick broadband anti-reflection coated fused silica windows. In the case of the gas cell, the windows were far enough from the pump focus so as to not contribute to any pump-induced phase shifts (cross phase modulation) to the probe. To keep the pump intensity low at the cell windows, the pump beam was expanded with 2 × magnification before the focusing lens. The pump Rayleigh range in the cell was z 0, p = 4.5 mm with a full width at half-maximum (FWHM) focal spot size of 36 μm by 27 μm. Pump peak intensities were determined by the known pulse energy, pulse shape (from SSSI (see below) and independently from a Grenouille measurement ), and relay images of the pump spot recorded on a 14-bit CCD camera. The SC beam Rayleigh range was z 0, sc = 24.6 cm with a FWHM spot size of 270 μm. In the interaction region, the probe beam therefore significantly overfilled the pump in the transverse plane, allowing observation of the pump-induced phase shift across the full pump profile. The ”exit” plane of the pump interaction region was imaged beyond the sample onto the spectrometer slit at 6.9× magnification. Along this beam path, the combined pump/SC beam exiting the sample was passed through a zero degree dielectric Ti:Sapphire mirror (M) to reject the pump beam. The f/2 imaging spectrometer consisted of a diffraction grating with 1200 mm-1 groove density and a 10-bit CCD camera (SONY XCD-SX910), which captures full frame images of 1280 × 960 pixels at 7.5 frames per second. The ~72 nm spectral window projected on the CCD sensor chip ranged from 651.7 nm to 723.2 nm, and the one-dimensional source spatial resolution was 0.67 μm/pixel along the entrance slit direction.
As discussed earlier, extraction of the probe temporal phase shift ΔΦ(x,t), where x is the coordinate along the spectrometer slit axis in the image plane, can be achieved by either direct frequency-to-time mapping or through Fourier transforms. For extraction by Fourier transform, the full spectral phase ϕpr(ω) = ϕr(ω) + Δϕ(ω) of the probe pulse is required, necessitating knowledge of the reference phase ϕr(ω). Determining this through the second order dispersion ϕr(ω) ≅ β2 (ω - ω 0)2 and neglecting higher order terms has been found to be sufficient for pump pulses > 20 fs . To obtain β 2, a calibration procedure using cross-phase modulation, similar to the method in Ref. , was applied: interferograms were recorded under varying delay τ between pump and probe pulses in 100 psi argon, giving a sequence of identical Δϕ(ω) cal Δϕ(ω) traces, but shifted in frequency. For each trace the frequency ω′ of maximum Δϕ(ω) was identified and plotted against τ. A linear fit to this plot gave for the linear chirp parameter 1/b = a = 2β 2 (1 + (2ln(2))2/β -2 2 (Δω)-4) = 7820 fs2. This agrees well with the calculated total dispersion introduced by total lengths of 1.1 cm of fused silica, 3.5 cm of BK7, and 2.5 cm of SF4 in the SC beam path. Our SC probe spectral width of ~100 nm gives β -2 2 (Δω)-4 ≪ 1, and therefore β 2 ≈ a/2.
Figure 2 shows sample spectral interferograms and extracted transient refractive index shifts Δn(x, t) using the gas cell filled with argon at room temperature. The CCD shutter speed was set to ~ 1 ms to ensure that only one shot was recorded per image. Argon is a monatomic gas where the lowest order nonvanishing nonlinearity (χ (3)) at 700-800 nm is electronic, nonresonant, and nearly instantaneous, so below the ionization threshold the time- and 1D-space- dependent nonlinear phase shift is given by ΔΦAr(x,t) = k ∫ Δn(x,z,t)dz = kn 2,Ar ∫I(x,z,t)dz, where n 2,Ar is the nonlinear refractive index for argon . We can define an effective interaction length L by ΔΦAr(x,t) = kΔn(x,t)L = kn 2,Ar I(x,t)L. Thus the phase shift follows the time and one-dimensional transverse envelope I(x, t) of the pump pulse intensity.
In Fig. 2(a) the pump pulse intensity was intentionally kept far below the argon field ionization threshold (~ 1014 W/cm2 ). In Fig. 2(b) the intensity was increased so that plasma was generated. The wavelength-dependent interference fringe shifts in Fig. 2(a) and Fig. 2(b) represent the transient modification of refractive index in the argon gas and gas/plasma. Figure 2(c) shows the 1D space and time variation of the argon nonlinear refractive index shift Δn(x,t) extracted from Fig. 2(a), using an effective nonlinear interaction length LAr = 5.7 mm. We note that for well-defined gas interaction lengths, such as provided by a thin (≪ 2z0, p) gas jet , L could be considered a known quantity and n 2 could be extracted. Here, however, for the longer gas cell, where the effective nonlinear interaction length is less well-defined, we wish to extract L. The procedure was to compare the nonlinear Kerr effect phase shift in the gas cell, ΔΦAr(x, t), to that in a thin BK7 window ΔΦBK7(x,t) = kn 2,BK7 L BK7 I(x,t), where the window thickness is L BK7 - 200 μm≪2z0,p. Thus using values of n 2,BK7 - 3.5 × 10-16 cm2/W obtained from SSSI measurement described later, and n 2,Ar = 9.8 × 10-20 cm2 W-1 atm-1 from Ref. . The measured n 2,BK7 value is in good agreement with various values (3.43 × 10-16, 3.63 × 10-16, 3 × 10-16 cm2/W) given by, respectively, Refs. , , and .
Figure 2(d) shows Δn(x,t) extracted from Fig. 2(b), including the generation of plasma. The initial profile of Δn is similar to Fig. 2(c), then the onset of plasma generation drives Δn to a value Δn plasma ~ -0.4 × 10-5, corresponding to an on-axis electron density of 1.9 × 1016 cm-3, which stays constant for the remainder of the 2 ps probe window. The gas density is 1.2 × 1020 cm , which means only ~ 0.02% of argon atoms are ionized. Plasma recombination occurs on a longer, nanosecond time scale. As an example of the good shot-to-shot stability made possible through use of a kHz regenerative amplifier system, Fig. 3 shows a 250 shot average and a single shot sample of the phase and refractive index transient from an unionized 5.1 atm nitrogen sample. The results agree well. Evidence of shot-to-shot stability of the SC generation in both spectrum and transverse spatial distribution is further demonstrated by Fig. 4, which shows a comparison of a single shot spectral interferogram to an interferogram averaged over 300 shots, of pump interaction with 5.1 atm of N2O.
Figure 5 shows the nonlinear phase shift Δn BK7(x,t) for the 200 μm thick BK7 (borosilicate glass) window, which compares quite well to ΔnAr(x,t) in Fig. 2(c), as it should: in BK7 glass, the dominant low order nonlinearity (χ (3)) is also electronic, non-resonant, and nearly instantaneous. Thus both phase shifts are proportional to I(x,t), justifying our method above for finding LAr.
Figure 6 shows a comparison of the pump-induced nonlinear index change in Ar, N2, and N2O samples for times near the pump laser pulse. Unlike Ar, the other species are linear molecules with an inertial contribution to their nonlinearity, which corresponds to delayed molecular axis alignment along the laser polarization resulting from the torque experienced by the induced molecular dipole in the laser field . The prompt and delayed refractive index response can be expressed as Δn(t) = n 2 I(t) + ∫ ∞ 0 R(τ)I(t - τ)dτ, where R is the molecular response function . As discussed earlier, the response of Ar is near instantaneous, as expressed by the first term only. The curves in Fig. 5 show the nonlinear response of Ar peaking first, followed by N2 and then N2O, with increasing broadening of the peaks. This is consistent with the increasing moment of inertia of N2 and N2O (decreasing ground state rotational constants B of 2.01 cm-1  and 0.42 cm-1 , respectively). The decay in Δn of the molecular response shown here can be viewed as resulting from the dephasing of the superposition of rotational quantum states excited by the pump pulse , where the timescale for such dephasing increases with molecular moment of inertia or decreasing B.
Note that the 2 ps measurement window of our SSSI diagnostic can be moved with an optical delay line to times well past the pump pulse. In this manner, we can measure, in a single shot, the refractive index effect of the quantum rotational recurrences  induced by pump pulses in molecular gases. This will be the subject of a future paper.
In conclusion, we have developed a spectral interferometer capable of recording single shot records of refractive index transients with ~10 fs time resolution and 1D space resolution in a 2 ps window. It uses chirped supercontinuum probe pulses generated from the self-focusing of few hundred microjoule, femtosecond pulses in a Xe gas cell. This diagnostic is suitable for use with modest energy femtosecond laser systems such as those based on kHz or multi-kilohertz Ti:Sapphire regenerative amplifiers, which are the workhorse system in many ultrafast optics and molecular physics laboratories.
The authors thank A. York for help with the interferogram analysis. This work is supported by the U.S. Department of Energy and the National Science Foundation.
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