1. Introduction

The interaction of high intensity ultrashort pulses and gaseous media is rich in fundamental phenomena and applications including optical field ionization [1–3], high harmonic generation [4, 5], relativistic self-channeling [6, 7], and laser-induced wake-fields in plasmas [8]. In particular, optical field ionization is a fundamental and universal process that occurs in a wide range of media under high intensity femtosecond laser irradiation.

One of the earliest works on optical field ionization was the measurement of the peak intensity dependence of ion yields for various gases [9], over a high dynamic range of laser intensity. Use of similar ion or electron spectroscopy techniques has triggered the discovery of nonsequential ionization in helium (He → He⁺² directly, compared to the sequential process He → He⁺ → He⁺²) [10]. However, these types of measurements, which involve charged particle detection long after the ionization event, do not provide information on the time-resolved ionization dynamics.

Optical field ionization of a gas sample of sufficient density can significantly alter its macroscopic refractive index. This transient index change can then encode itself on the propagation phase ϕ(z, t)=k ₀ nz-ω ₀ t of laser beams that either probe or drive the ionization process. Here k ₀ is the vacuum wave number of the laser beam, ω ₀ is its angular frequency, z is distance along the propagation axis, and n is the refractive index. The non-relativistic refractive index experienced by a laser beam in an ionizing gas is n(r,t) = (1 + 4πN ₀(r,t)γ-N_e (r,t)/N_cr )^1/2 where N ₀(r, t) is the neutral atom (molecule) density, γ is the atomic (molecular) polarizability, N_e (r, t) is the electron density generated during the ionization process (N ₀ is N _e), and N_cr =mω ²/4πe ² is the critical electron density. Here e and m are the electron charge and mass, and ω is the laser angular frequency. In this expression for n, the optical field-ionized electrons are considered to enter the continuum where they are freely driven by the laser field. One example of a phase sensitive measurement of ionization dynamics is the monitoring of spectral blue shifts. A rapid increase in electron density owing to ionization results in a rapid reduction of n, which in turn causes a spectral blue shift in either the pump pulse inducing the ionization [11] or in a probe beam co-propagating with the pump [12]. The time-dependent spectral shift is given by Δω(t,τ)=ω(t,τ)-ω ₀ where ω(t,τ)=-∂ϕ(t,τ)/∂t=ω ₀-k ₀ z∂n(t,τ)/∂t, where τ is the time delay between the pump and probe pulses (τ= 0 for pump alone). The blue shift method for extraction of ionization dynamics depends on the time derivative of the refractive index ∂n/∂t, which in many practical cases can be small. As a result, the spectral change Δω is too small to measure from the ionization of low-density and low-Z gases.

A much more sensitive technique, spectral interferometry (SI), has been widely used for the measurement of refractive index transients. Examples of its use are measurements of induced phase modulation in solids [13], time evolution of femtosecond laser-induced plasmas [14], and plasma oscillations [15–18]. Here, a pump pulse induces an index transient in some medium, and a probe pulse co-propagates with the pump at some delay. The phase-shifted probe is then interfered with an unperturbed reference pulse inside a spectrometer. The interference allows even small phase shifts to be detected. Early versions of this method used probe pulses of much shorter duration than the index transient to be measured. Scanning of the probe pulse delay with respect to the pump pulse then allowed shot-by-shot reconstruction of the phase transient over some useful temporal window. This version of SI is sensitive to shot-to-shot variations in the laser pulses and in the sample response, which can result in degradation of the acquired phase information. Single-shot SI, or SSI, avoids this problem. SSI has been demonstrated with chirped [19–21] or transform-limited probe pulses [22]. However, the shortest temporal resolution has been limited to ~ 100 fs by the pulse spectral bandwidth [21]. Furthermore, chirp in the probe pulse can limit the temporal resolution even further [19, 21, 23]. A clear route to improved temporal resolution is the development of SSI probe pulses with very large spectral bandwidths. Recently, we have developed the technique of single-shot supercontinuum spectral interferometry (SSSI) [23]. Here, a probe pulse of ~100 nm bandwidth is generated by the self-focusing of an 80 fs, 1 mJ Ti: Sapphire laser pulse in atmospheric pressure air. The large bandwidth allows temporal resolution of ~10 fs. We have used this SSSI diagnostic to measure the transient nonlinear Kerr effect induced by a pump pulse in glass [23] and the dynamics of exploding laser-heated Ar clusters [24].

In this paper, our SSSI diagnostic is used for the spatio-temporal measurement of laser-induced optical field ionization of a helium gas jet. We observe the stepwise transition He → He⁺ followed by He⁺ → He²⁺. In addition, we show that probe beam propagation effects in the finite laser-gas interaction length must be carefully considered in order for proper interpretation of extracted refractive index transients.

2. Theoretical background

First we simulate the spatio-temporal evolution of the electron density for the case of a low-pressure helium gas jet irradiated by a femtosecond laser pulse. For sub-atmospheric pressure gases exposed to intense femtosecond laser pulses, the dominant ionization mechanism is optical field ionization or tunneling ionization for cases where the Keldysh parameter γ < 1 [1]. For our experimental pump pulses with λ=800 nm and peak intensity I_peak = 3.8 × 10¹⁶ W/cm², γ ~ 0.07. We use the tunneling ionization rate calculated by Ammosov, Delone, and Krainov (ADK) [25]. The ADK ionization rate averaged over a laser period is

where ω_a = 4.134 × 10¹⁶ s^-1 is the atomic frequency, E _H the laser field normalized to the atomic field seen by the ground state electron in hydrogen, χ _H the ionization potential normalized to that of hydrogen, n_eff = Z${{\chi }_{\mathrm{H}}}^{-1/2}$ the effective principal quantum number, and Z is the resulting ion charge. The full simulation code includes optical field ionization, collisional ionization, thermal transport, and hydrodynamics [26].

Figure 1 (a) shows results of a simulation for a spatial and temporal Gaussian pump pulse

where τ_FWHM = 240 fs the pump pulse full width at half maximum (FWHM), t_peak = 288.3 fs is the time at which peak intensity of I_peak ~3.8×10¹⁶ W/cm² is achieved, and r_FWHM = 10.3μm the pump spot radius. The neutral helium gas density was taken to be N _He = 1.7×10¹⁷ cm^-3 to correspond with the experimental value (see below). The two-step ionization (He → He⁺ → He²⁺) in space and time is clearly seen in Fig. 1(a). The simulation confirms that optical field ionization is by far the dominant effect. Collisional ionization plays no role here: increasing the initial neutral He density by more than a factor of 50 increases the spatio-temporal electron density profile only proportionally. The effects of thermal transport and hydrodynamics are similarly negligible at these densities and time scales. The distinctiveness of the field ionization steps in space and time results from the large ionization potential (I.P.) gaps for He → He⁺ (I.P. = 24.58741 eV) and He⁺ → He²⁺ (I.P. = 54.41778 eV) [27]. Figure 1(b) shows the on-axis (r = 0) transient electron density evolution and the pump pulse envelope. The temporal step is clearly seen. The time evolution of the spatial steps in electron density is seen in Fig. 1(c), which shows a sequence of electron density profiles at 20 fs increments.

 

Fig. 1. (a) Theoretical spatio-temporal electron density profile at the laser focus using ADK theory with τ_FWHM = 240 fs, r_FWHM = 10.3μm, and I_peak = 3.8 × 10¹⁶ W/cm². (b) On-axis electron density evolution (blue line) and pump pulse envelope (red line). (c) Electron density profiles at Δt = 20 fs time increments.

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3. Experiment

The experimental setup is shown in Fig. 2(a). A 20 mJ, λ = 800nm, 240 fs FWHM pump pulse from a Ti:Sapphire laser system is focused at f/4 by a BK7 lens into a helium gas jet. The focal spot is elliptical with FWHM dimensions 10.3μm × 16.1μm. This corresponds to a peak vacuum intensity I_peak = 3.8 × 10¹⁶ W/cm². The pump beam confocal parameter is 2z₀ ~ 0.6 mm. A thin sheet of helium gas was produced by a nozzle with a 10 mm × 0.4 mm exit orifice. The pump laser beam was incident normal to the gas sheet and 0.5mm above the nozzle mouth, with a resulting laser-gas interaction length of 0.5 mm. The setup of our SSSI diagnostic is described as follows. A synchronous ~1 mJ pulse was split off from the main laser beam and was focused in 1 atm of air to generate a supercontinuum (SC) pulse with ~100 nm bandwidth centered at 700 nm. [23]. The SC pulse was collimated and then split into τ-delayed reference and probe pulses in a Michelson splitter. The reference and probe pulses were then sent through a 1” thick SF4 glass window and positively chirped to ~1.5 ps. This sets the maximum temporal window for the single-shot observation of ionization dynamics. The reference and probe pulses were then collinearly focused by the same lens as the pump pulse into the helium gas jet, but to a larger FWHM spot size of ~100 μm, thus overfilling the pump region. The SC beam confocal parameter is 2z₀ ~ 65mm. The timing of the pulses was adjusted so that the reference SC pulse preceded the pump, while the probe SC pulse was temporally overlapped on the pump. While the reference pulse is unaffected, the probe acquires a transient phase shift due to the increasing population of free electrons generated in the pump pulse envelope. The pump pulse is then removed from the beam by reflection from an 800 nm mirror. The spatial mode of the reference and probe SC pulses emerging from the gas jet is imaged (with 15× magnification) onto the entrance slit of a visible spectrometer. A charge coupled device (CCD) camera in the focal plane of the spectrometer then records 1D space-resolved spectral interferograms of the reference and probe SC pulses. Sample spectral interferograms are shown in Fig. 2 with the gas jet off (b) and on (c), where a wavelength-dependent fringe shift is seen only in (c).

 

Fig. 2. (a) Experimental setup with pump beam and chirped supercontinuum (SC) reference and probe pulses combined at a beam splitter and focused into a helium gas jet. The pump is dumped and the reference and probe SC pulses are relayed to the imaging spectrometer. Sample spectral interferograms are shown with the helium gas jet (b) off and (c) on .

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Using a simple Fourier transformation technique [28], the spectral phase variation Δϕ(x,ω) can be extracted from the spectral interferograms. Here, the x coordinate represents the 1D spatial section of the beam measured by our SSSI diagnostic. The corresponding temporal phase ΔΦ(x,t) can then be reconstructed from the spectral phase shift Δϕ(x,ω) in two ways: direct mapping and Fourier transform methods [23]. Since the reference and probe pulses are linearly chirped in time, the transient phase can be directly obtained from the spectral phase ΔΦ(x, t(ω))=Δϕ(x, a(ω-ω ₀)) where the instantaneous frequency of the probe is ω(t)=ω ₀+at. This is the direct mapping method. However, this method has a strong chirp-dependent temporal distortion, which worsens with increased pulse stretching [19, 21, 23]. To take advantage of the excellent temporal resolution promised by our broadband SC pulses, we perform a full Fourier transform extraction of the temporal phase transient as follows [23]:

Here, Ẽ_{r0, pr0} and ϕ_{r, pr} are the real spectral amplitudes and phases of the reference and probe pulses, Δϕ(ω)=ϕ _pr(ω)-ϕ _r(ω) is the spectral phase difference between the probe and reference pulses, and τ is the reference-probe temporal separation introduced by the Michelson splitter. The spectral amplitudes are known from the measured SC spectra, and the SC pulse phase is obtained from cross phase modulation (XPM) with a pump pulse in a thin glass slide. More details of the SSSI diagnostic and data extraction are described in references [23, 29].

 

Fig. 3. (a) Experimental spatio-temporal phase profile ΔΦ(x, t) from optical field ionization of helium at 15 psi jet backing pressure. (b) Central line-outs for jet backing pressures of 5psi (line with solid triangles) and 15psi (line with squares). The pump pulse envelope obtained from XPM in glass is also shown (line with circles). The inset shows spatial phase profiles from the 5 psi case at 20 fs increments.

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An experimental spatio-temporal phase variation profile ΔΦ(x,t) at 15 psi jet backing pressure is shown in Fig 3(a). This profile was extracted using the procedure embodied by Eq. (3). ΔΦ(x, t) is proportional to the transient electron density variation across the pump beam profile at the jet. The transient phase along the pump beam axis, ΔΦ(x=0, t), is plotted in Fig. 3(b) for 5 psi jet backing pressure (line with triangles) and 15 psi (line with squares). For the laser-gas interaction length of 0.5 mm, the maximum phase shift in the 5 psi case implies an electron density of 3.4 × 10¹⁷ cm^-3, which implies a neutral helium density of N _He = 1.7 × 10¹⁷ cm^-3. This number was used in the simulations of Fig. 1. The on-axis pump temporal envelope is also shown (line with circles) in Fig. 3(b). The pump pulse duration τ_FWHM = 240 fs is measured from XPM between the pump and probe pulses in a thin glass slide. When the probe propagates along with the pump inside the glass, the probe phase is modulated by the pump-induced nonlinear refractive index Δn(x, t)=n ₂ I_pump(x, t) where n ₂ is the nonlinear refractive index coefficient of the glass and I_pump(x, t) is a 1D transverse spatial slice of the pump pulse envelope. Using Eg. (3), the on-axis pump envelope I_pump(x=0, t) is then extracted.

The plots in Fig. 3(b) show that the helium gas is ionized in two sequential steps. The phase rises rapidly, and then slightly jogs near -250 fs for both the 5 psi and 15 psi cases. The phase then rises again, saturating near -200 fs for both cases. The coincidence of these temporal features implies that the ionization dynamics are density independent. This in turn implies that collisional ionization is negligible, as predicted in Sec. 2. The phase jogs near -200 fs are also seen in the inset to Fig. 3(b), which shows the radial phase profile at a sequence of 20 fs incremental delays. Initially the phase profile grows at a constant rate, but it lingers near its half maximum value, after which it grows again. We interpret the jogs, or steps, as the temporal pause that occurs between the pump achieving ionization saturation to He⁺ and the early stages of ionization to He²⁺. The measured steps, however, are less distinct than predicted in the simulation of Fig. 1(b). In addition, the step in the 15 psi result is less distinct than the 5 psi step. These observations are explained by considering the finite interaction length of the laser pulses in the gas jet, as will be discussed in the following section.

4. Simulation

To examine how the finite laser-gas interaction length affects the probe phase profiles observed at the exit of the helium gas sheet, we simulated probe pulse propagation through refractive index profiles generated by the co-propagating pump pulse. There are three main effects that can result in sufficient probe pulse phase distortion to mask the index transient we wish to uncover. The simulation covers all three. The first is caused by mismatch between the natural divergences of the pump and probe beams. In our experimental geometry, the probe beam overfills the pump at the focus, so the beam divergences are different. Under such conditions, a probe ray would sample a range of radial electron densities along its propagation path. Thus, the transverse phase dependence of the probe beam at the jet exit would differ from the desired transverse phase profile ΔΦ(x, t), which should be proportional to the purely axially integrated electron density. This phase distortion effect is mitigated by making the pump and probe beams as planar as possible with respect to the gas jet. This was achieved in practice by ensuring that the confocal parameters of the pump and probe beams (2z₀ ~ 0.6 mm and 65 mm, respectively) were greater than the gas sheet thickness (here 0.5 mm).

The second, and more significant phase distortion effect is described as follows. The pump intensity varies along the propagation axis z, so the degree of ionization can change along that axis if the gas jet is sufficiently extended along z. For example, for an extended enough gas jet, helium can be found completely doubly ionized around the pump focus but singly ionized axially away from it. Hence, when the probe pulse propagates along z with the pump beam, it acquires a phase shift through the He⁺ plasma well before it reaches the pump focal region near z=0, whereupon it picks up an additional phase shift due to the He²⁺ plasma there. Hence, the accumulated probe phase may not reveal a sharp transition from He⁺ to He²⁺. In general, for thick jets, or even worse, for backfill gas targets [15–17], pump intensity variation along the propagation axis within the gas volume can significantly degrade transient phase shift measurements. A diagram showing how the chirped probe samples the transient refractive index profile generated by the co-propagating pump pulse is shown in Fig. 4(a). Here, the reference pulse is not shown. The chirped probe pulse can be decomposed into a series of temporal slices δ_pr(t-τ) with delays τ with respect to the pump. Each slice of the probe δ_pr(t-τ) propagates in an electron density disturbance N _e(x, z, t’-τ) generated by the co-propagating pump pulse E _p(t) where N _e(x, z, t) is the electron density in the lab frame and t’=t-z/v_g is a time coordinate local to the pump pulse. The electron density disturbance and the probe pulse are assumed to move at the pump group velocity v_g. Figure 4(b) shows electron density profiles N _e(x, z, t’-τ) left in a helium gas volume at 50 fs intervals by a Gaussian pump pulse propagating along the z-axis. The pulse was modeled to correspond to the experimental one: peak intensity I_peak ~ 3.8 × 10¹⁶ W/cm², λ _pump = 800 nm, FWHM pulse width τ_FWHM = 240 fs, and focal radius x_FWHM = 10.3μm, corresponding to a confocal parameter of 2z₀ ~ 0.6 mm. The neutral helium target was taken to be of density N _He = 1.7 × 10¹⁷ cm^-3, corresponding to the 5 psi experiment, but with a jet thickness of 2 mm. A boundary box in the figure shows the actual extent of our helium gas sheet. The calculation of pump pulse propagation ignored ionization-induced refraction [30, 31]. For τ= 50 and 100 fs delays between the pump E _p(t) and the probe slice δ_pr(t-τ), the probe slice samples only a singly ionized helium plasma, but with τ= 150, 200, 250, and 300 fs, the probe samples both He⁺ plasma away from the pump focus and He²⁺ plasma around the focus. This results in smearing out of the temporal step from He⁺ to He²⁺. However, with the thin 0.5 mm gas jet actually used, which is less than the pump beam confocal parameter 2z₀ = 0.6 mm, distortion of the measured is minimized.

 

Fig. 4. (a) Schematic pump-probe diagram. A slice of the probe pulse samples the volume of the helium plasma. (b) Theoretical electron density profiles N _e(x, z, t’-τ) that a slice of the probe pulse δ_pr(t-τ) samples with τ= 50 fs time increments.

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The third distortion effect is the refraction of the probe beam owing to the radial distribution of electron density induced by the pump. To simulate the spatio-temporal phase profiles ΔΦ(x, t-τ) of the probe pulse at the exit of the helium gas sheet, we first compute the electron density profiles N _e(x, z, t’-τ) that a series of probe slices δ_pr(t-τ) propagate through (See Fig. 4. (b)). Once a time-invariant refractive index profile is given for a probe slice δ_pr(t-τ), the simple Beam Propagation Method (BPM) [32] is used to achieve the spatio-temporal phase profiles ΔΦ(x, z, t-τ). Each τ-delayed temporal slice of the probe δ_pr(r, z, t-τ)=δ_pr0(r, z, t-τ) exp(iΦ_pr(r, z, t-τ)) with an initial Gaussian transverse spatial profile propagates through the helium jet through the time-invariant refractive index n(r, z, t’-τ)=(1-N _e(x, z, t’-τ)/N _cr)^1/2 along the z-axis. Each reference pulse δ_r(r, z, t-τ)=β_r0(r, z, t-τ) exp (iΦ_r(r, z, t-τ)) propagates in neutral helium gas. At each grid point along z, the electric fields of the reference and probe slices are decomposed into a superposition of plane waves via a discrete Fourier transform (DFT), and the plane waves propagate a distance δz through the refractive index profile < n > locally averaged along δz. At z + δz, a phase correction term exp(ik₀ (n(r, z, t’-τ)-< n >)δz) is added to take into account the local space variation of the refractive index profile. Finally, an inverse DFT converts the superposition of the plane waves into the electric fields of the reference and probe δ_{r, pr}(r, z+δz, t-τ) at z + δz. The transverse fields give the reference and probe phases Φ_{r, pr}(r, z, t-τ) at z + δz. This process is repeated until the waves reach the end of the plasma Δz. At each z, the phase difference between the reference and probe beams ΔΦ(r, z, t-τ)=Φ_pr(r, z, t-τ)-Φ_r(r, z, t-τ) can be obtained. At the end of the helium plasma, the phase difference ΔΦ(r, z=Δz, t-τ)=ΔΦ(r, t-τ) simulates the experimentally measured probe phase shift owing to the ionization-induced transient refractive index. In the BPM calculation, the propagation step length is taken to Δz = 10 μm for the reference and probe beams, which is much less than z ₀ for both. The reference and probe beams have λ _probe = 700 nm and focal spot radii x_FWHM = 100 μm, corresponding to the experimental values. The simulation implicitly accounts for distortions owing to the second effect (axial pump variation) and the third effect (ionization-induced refraction).

Figure 5 shows a sequence of calculated probe phase profiles ΔΦ(x, t-τ) at the jet exit for laser-gas interaction lengths Δz of (a) 0.1 mm, (b) 0.25 mm, (c) 0.5 mm, (d) 0.75 mm, (e) 1 mm, and (f) 2 mm. With the shortest interaction length, Δz = 0.1 mm, the double-step ionization feature is prominent. However, as the interaction length Δz increases, the step gradually smears out. Also, radial phase oscillations develop due to increasing probe refraction with interaction length. With Δz = 0.5 mm, which corresponds to the gas sheet of our experiment, the temporal double-step in the phase is still discernable. However, both the radial flat-top near the beam center and the step on the beam edge predicted by Fig. 1 are substantially smoothed owing to the probe refraction. This agrees with the experimentally measured phase shown in the inset of Fig. 3(b). By Δz = 1 mm (Fig. 5(e)), the temporal step has washed out as well. The washing out of the temporal step is mostly caused by the second distortion effect: axial pump intensity variation within the laser-gas interaction volume. Consequently, it is vitally important to keep the laser-gas interaction length as small as possible. In our case, it appears that a gas sheet not too much thicker than the 0.5 mm of our experiment would have ended up washing out the temporal effects we were trying to uncover.

To summarize, the results of our simulations show that a short laser-gas interaction length must be used to localize the electron density measurement in space and time. Otherwise, the measured spatio-temporal phase at the jet exit is significantly affected by (a) the divergence mismatch of the pump and probe pulses, (b) axial variation of the pump intensity in the interaction volume, and (c) probe beam refraction. A short interaction length mitigates all three effects.

In recent SI-based measurements of laser-driven electron density wakefields [15–17] in a static-filled helium gas volume (with laser-gas interaction length L ≫ z ₀ at densities N _e ~ 10¹⁷ cm^-3 similar to those of our experiment), both temporal [16] and spatio-temporal [15, 17] phases ΔΦ were extracted, from which were inferred axial and axial/radial plasma oscillations, respectively. Our calculation cautions that all such measurements where L > z ₀ must be accompanied by a careful analysis of probe beam propagation.

 

Fig. 5. Theoretical probe phase profiles ΔΦ(x, t) at the end of the helium plasma with the laser-gas interaction length Δz (a) 0.1 mm, (b) 0.25 mm, (c) 0.5 mm, (d) 0.75 mm, (e) 1 mm, and (f) 2 mm.

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5. Conclusion

We have observed laser-driven double step helium optical field ionization (He → He⁺ → He²⁺) with single-shot supercontinuum spectral interferometry (SSSI). SSSI provides ~10 fs resolution and a ~1 ps observation window to observe the ionization dynamics. The experimental measurements of electron density evolution N _e(x, t) are in good agreement with the tunneling ionization model. The large spectral bandwidth of the supercontinuum probe pulse, the single-shot pump-probe operation, and the minimal laser-gas interaction length of ~0.5 mm made it possible to observe the double-step transient.

It is worth emphasizing again that a minimal laser-target interaction length is essential for clear measurements of refractive index profile transients. This applies not only to the various versions of spectral interferometry, but to FROG-related measurements as well [33]. Otherwise, three effects can act to mask the true index profile transient. First, any difference in divergence between the pump and probe beams can result in transverse spatial mixing of temporal information encoded on the probe beam. Second, the axial pump intensity variation within the interaction volume can act to smear out any temporal phase structure picked up by the probe. Finally, probe beam refraction from the pump-induced index profile can smear out temporal and spatial phase structure. These three effects can be greatly suppressed with a short laser-target interaction length. Otherwise, all such measurements must be accompanied by a careful analysis of probe beam propagation.