1. Introduction

Alignment of molecules in intense laser fields has assumed a significant level of activity in recent years [1]. Applications include enhancement of high harmonic generation [2–4], orientation of molecules for field-driven wavepacket tomography [5,6], structural studies of molecules [7,8], and effects on long-range atmospheric propagation of intense femtosecond laser pulses [9,10].

Many methods have been developed for producing and measuring molecular alignment. Thus far, all methods use variations of the multi-shot pump and probe technique. For very low density molecular samples, an intense laser pulse (pump pulse) is used to align the molecule, and an intense ultrashort secondary pulse stepped through a variable delay (probe pulse) is used to remove electrons by field ionization and initiate coulomb explosion. The ionic fragment velocities are along the molecular axis direction at the time of electron removal, angularly resolved with respect to the pump polarization. This ‘Coulomb explosion imaging’ technique measures the time resolved molecular alignment [11, 12]. All optical multi-shot pump-probe methods have also been used. The earliest optical measurements of alignment were applied to picosecond optical Kerr gating in liquids [13, 14]. A linearly polarized pump pulse torqued CS₂ molecules into alignment, creating a transient birefringence sampled by the polarization rotation imposed on a probe pulse variably delayed in the temporal vicinity of the pump. Later, it was realized that probe pulse delays long after the pump could sample quantum echoes of the molecular alignment (also called rotational recurrences) if this measurement were performed in much less collisional CS₂ vapour [15]. A version of this technique, now called Optical Kerr Effect (OKE) spectroscopy [16], depends on the birefringence, or small anisotropic refractive index change in a gas or liquid induced by the presence of pump-aligned molecules, and has been used to measure a wide range of molecular alignment dynamics, including the prompt response near the pump pulse and quantum echoes. Other optical methods have recently been developed which depend on probe beam refraction from aligned molecular gas samples [17] and ionization of these samples [18]. Spectral modulations imposed on sequentially delayed short probe pulses have also been used to map out wavepacket recurrences, although the time resolution of these measurements is limited by the relatively long (> 50 fs) probe pulse duration, and quantitative molecular response is obtained only through propagation model-dependent fits to the shifted spectra [19, 20].

The only technique described above which is capable of direct quantitative measurement of alignment is Coulomb explosion imaging [11, 12], but this method is not capable of spatial resolution, and it is a time-consuming multi-shot pump-probe method. None of the above techniques combine the direct measurement of alignment, spatially resolved across the laser beam, with alignment evolution measured in a single shot.

In this paper we present single shot measurements of the prompt molecular alignment induced by intense ultrafast laser fields and the coherent alignment recurrences that appear long after the laser pulse has passed. Unlike all previous measurements of such molecular response, we observe the evolution of the alignment and periodic recurrences in response to a single pump laser pulse and spatially resolved [21, 22]. Single shot measurement is an improvement over previous techniques for several reasons. First, in laser systems having even small shot to shot variations in energy (typical of most femtosecond laser systems) or pulse width, the alignment response will vary from shot to shot and the result of composite measurements could obscure low levels of response. Second, in experiments where conditions such as pump laser pulse energy or gas pressure are varied, collecting the full temporal response for each set of conditions is far less time consuming than doing shot by shot temporal scans.

Our method for determining the transient molecular alignment is to measure the time- and space-dependent phase shift imposed on probe pulse by the pump pulse-induced molecular dynamics in a gas sample. In essence the pump pulse, mediated by the molecular response, induces cross-phase modulation of the probe pulse. The probe phase shift, which is equivalent to a time and space dependent refractive index shift, represents the ensemble average of the transient molecular response. There are two components to the molecular contribution to the transient refractive index: a fast component, nearly instantaneous on the ~3 fs (at 800 nm) time scale of a laser oscillation period, which is the bound valence electron response (electronic polarizability), and a slower component related to orientation of the molecular axis. This component is maximized when the molecular axis coincides with the laser E-field polarization, but molecular rotation induced by the torque of the field on the field-induced dipole occurs on a much longer timescale determined by the molecular moment of inertia.

2. Experimental setup

The experimental setup for SSSI has been improved recently for efficient and highly stable operation with a commercial 1 kHz femtosecond regenerative amplifier system [23]. Figure 1 shows the experimental setup. A broadband (~100 nm FWHM) supercontinuum (SC) pulse is generated using self-focusing of a ~200–300 µJ, 100 fs, 800 nm Ti:Sapphire laser pulse in a xenon gas cell (not shown). The SC pulse is split into collinear twin pulses separated by time τ using a Michelson interferometer (reference pulse followed by probe pulse), which were then passed through a 1” thick window of SF4 glass, dispersively stretching and linearly chirping them up to ~2 ps. For measuring pump-probe delays longer than the 2 ps probe pulse window, a delay line was implemented for placing the 2 ps probe window at delays up to 5 ns with respect to the pump. In this manner we were able to measure recurrences that occurred well after the pump pulse.

The probe and reference SC beams were combined at a beamsplitter with the pump, and the three pulses were collinearly focused into a 45 cm long gas cell. The combined pump/SC beam exiting the cell was passed through a zero degree dielectric Ti:Sapphire mirror to reject the pump beam. The SC beam was imaged from the exit plane of the nonlinear interaction zone in the cell onto the entrance slit of an f/2 imaging spectrometer with a 1200 groove mm^-1 grating and a 10-bit CCD camera at the focal plane, recording interference patterns in the spectral domain (spectral interferograms). These 2D patterns had a ~100 nm wide wavelength axis and perpendicular to that, a 0.67 µm/pixel spatial resolution along the entrance slit direction. Full details of the setup are given in reference [23].

 

Fig. 1. Experimental setup. HWP: half waveplate, BS: beamsplitter for combining pump and SC pulses, M: zero degree Ti:Sapphire dielectric mirror. The inset shows interferograms in 3 temporal windows, corresponding to laser field alignment (0 ps), quarter revival (~3 ps), and half revival (~6 ps) of the rotational wave packet in 5.1 atm O₂.

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The time and 1D-space-dependent phase shift ΔΦ(x,t) induced on the probe pulse is extracted from Fourier analysis of the spectral interferograms Δϕ(x,ω) recorded by the CCD sensor in the spectrometer image plane [23], where x is the coordinate along the spectrometer slit axis. Fourier extraction uses the full spectral phase of the probe pulse φ _pr(x,ω)=ϕ _r(ω)+Δϕ(x,ω), requiring determination of the reference pulse phase ϕ _r(ω). This is easily measured by cross-phase modulation induced in the probe or reference by a variably delayed pump in a thin glass window or a gas sample [23, 24]. Determining the reference phase through second order dispersion ϕ _r(ω)≅β ₂(ω-ω ₀)² and neglecting higher order terms has been found to be sufficient for pump pulses >20 fs [24]. The refractive index transient Δn is then determined by 1 Δn(x,t)=(k ₀ L)^-1ΔΦ(x,t), where k ₀ is the vacuum wavenumber of the probe pulse (at the peak of its spectrum) and L is the effective nonlinear interaction length of the pump in the sample [23]. The time resolution of measured index transients is ~10 fs, limited by the SC bandwidth of ~100 nm [23].

In anticipation of potentially tiny phase shifts compared to the experiments of ref. [23], in the current work we implemented a new method for interferogram analysis. Recognizing that the dominant contribution to noise in the extracted phase is dark current and readout noise of the CCD camera pixels, we averaged 300 consecutive interferograms before phase extraction. This procedure averaged out the interferogram noise and allowed observation of signals levels smaller than the noise of an individual shot. An estimate of the error in the extracted phase is δΦ_shot/N _int ^1/2, where δΦ_shot is the maximum noise amplitude in the extracted phase of an individual shot and N _int is the number of averaged interferograms. We note that this procedure depends on the excellent shot-to-shot stability of the interferogram fringe locations and fringe visibility made possible by the very stable kHz pump laser and SC generation technique. For experiments with more shot-to-shot variability (such as typical with 10 Hz pump lasers), a much lower noise CCD camera would be required to achieve in a single shot the low levels of phase shift extracted in the current experiment.

3. Effect of field-induced molecular alignment of linear molecules on the transient refractive index

As our SSSI diagnostic measures a dynamic phase shift that is expressed as a transient refractive index, it is worthwhile to understand the connection between the refractive index and the underlying laser-induced molecular response.

The dielectric response of a sample of gas molecules to an applied electric field is given by ε=n ²=1+4πN<α>_t, where N is the number of molecules per cm³, <α>_t is the time-dependent ensemble average molecular polarizability along the laser electric field, and n is the refractive index. Here, <α>_t=<ê·α̿·ê>_t=<e_iα_ije_j>_t, where ê is the electric field polarization, α̿ is the second rank molecular polarizability tensor, and the Einstein summation convention for repeated indices is assumed. Instances of repeated indices later in the paper where no summation is intended are clear from their context. For a linear molecule, where we choose the body-fixed axis z to be along the molecular axis, the only nonzero components of α̿ are α_zz=α _″ and α_xx=α_yy=α _⊥. Owing to molecular symmetry about the z-axis, the laser electric field can be taken as E=x̂E_x+ẑ E_z for a particular molecular orientation. Therefore, for an ensemble of molecular orientations in the space-fixed field, n ²=1+4πN(<e ² _x>_t α_xx+<e ² _z>_t α_zz, or

where Δα=α _″-α _⊥ and e_z=ê·;ẑ=cos θ is the cosine of the angle between the molecular (z) axis and the electric field. The index shift measured by SSSI is then given by

where we have used the results that well before the pump pulse, <cos² θ>_t=-∞1/3 (early time ensemble averages are just averages over solid angle) and $n^2\left(t=-\infty \right)=n_0^2=1+4\pi N\left(\frac{1}{3}\Delta \alpha +{\alpha }_{\perp }\right).$.

Equation (1) neglects the ‘prompt’ contribution of the nonlinear distortion of the molecular electron cloud by the laser electric field. This is important for consideration of refractive index transients during the pump pulse involving sufficiently large pump laser fields. In that case, it can be shown that the index is given by

where α ⁽³⁾ _ijkl is the fourth rank molecular polarizability tensor and the ‘prompt’ contribution is proportional to the square of the field envelope amplitude |E(t)|². As will be seen, the angular ensemble averages < >_t consist of a constant term (from at t=-∞) and a term which is second order in peak field amplitude E ₀. The prompt response therefore has an isotropic part proportional to |E(t)|², and an orientational part proportional to E ² ₀|E(t)|² which is significantly smaller. The contribution to the refractive index of the prompt isotropic part is written as $\frac{1}{2}\left[{<{\sin }^4\theta >}_{t=-\infty }{\alpha }_{\mathrm{xxxx}}^{\mathit{\left(}3\mathit{\right)}}+\frac{1}{2}{<{\sin }^{2}2\theta >}_{t=-\infty }{\alpha }_{\mathrm{xxzz}}^{\mathit{\left(}3\mathit{\right)}}+{<{\cos }^4\theta >}_{t=-\infty }{\alpha }_{\mathrm{zzzz}}^{\mathit{\left(}3\mathit{\right)}}\right]{\mid \mathbf{E}\left(t\right)\mid }^2=n_{2}I\left(t\right),$, where n ₂ is the isotropic nonlinear index of refraction, I(t) is the laser intensity, and where <sin⁴ θ>=_t-∞=8/15, <sin²2θ>=_t-∞=8/15, and <cos⁴ θ>_t=-∞=1/5.

For a particular molecule, we define the degree of alignment to be cos² θ; its time-dependent classical ensemble average is calculated as <cos² θ>_t=Tr(ρ(t)⊗cos² θ)=ρ_kl〈l cos² θ|k〉, where ρ(t) is the density matrix, ⊗ denotes operator multiplication, Tr is the trace operation, and where |l〉 and |k〉 are molecular rotational eigenstates of the field-free Hamiltonian. At room temperature T, ΔE _elec/k_BT≫ 1 and ΔE _vib/k_BT≫1, where ΔE _elec and ΔE _vib are the energies of the first excited electronic and vibrational states and k_B is the Boltzmann constant, so that the molecules dominantly occupy the ground electronic and vibrational states. The density matrix is calculated to first order in the optical perturbation, ρ(t)=ρ ⁽⁰⁾+ρ ⁽¹⁾(t), where

is the first order correction to the density matrix induced by the perturbation Hamiltonian 𝓱=-p·E, where p=α̿·E is the induced molecular dipole moment and E(τ) is the laser field, whose pulse envelope peak is located at time τ=0. In Eq. (4), [ ] denotes a commutator, ω_kl=(E_k-E_l)/ħ corresponds to rotational states |k〉=|j,m〉 and |l〉=j′,m′ with energies E_k=E _j,m=hcBj(j+1) and E_l=E _j′,m′hcBj′(j′+1) (B is the rotational constant), γ_kl is the dephasing rate between states k and l, ρ ⁽⁰⁾ is the zeroeth order density matrix describing a thermal equilibrium distribution of rotational states at t=-∞, j is the quantum number for total rotational angular momentum J, and m is the quantum number corresponding to the component of J along E. The perturbation Hamiltonian 𝓱 is the driving mechanism for molecular alignment. Use of first order perturbation theory is justified by our experimental results showing that <cos² θ>_t deviates from the unperturbed <cos² θ>_t=-∞=Tr(ρ ⁽⁰⁾(t)⊗cos² θ)=1/3 by small amounts.

The commutator matrix element in Eq. (4) is [𝓱,ρ ⁽⁰⁾]_kl=(ρ_l ⁽⁰⁾-ρ_k ⁽⁰⁾) 𝓱_kl, where ρ_l ⁽⁰⁾≡ρ_ll ⁽⁰⁾(no sum), ρ_k ⁽⁰⁾≡ρ_kk ⁽⁰⁾ (no sum), and 𝓱_kl=-Δα|E|²〈k|cos²θ|l〉-α _⊥ δ_kl|E|², where δ_kl is the unity matrix. As the rotational eigenstates are the spherical harmonics, |j,m〉=Y_jm(θ,φ), the matrix element 〈k|cos² θ|l〉=〈j,m|cos² θ|j′,m′〉 is nonvanishing only for m′=m and j′=j+2, j′=j, or j′=j-2. The non-coupling between different m states corresponds to the interaction symmetry about the molecular (z) axis (or conservation of angular momentum), while the j coupling corresponds to the two-photon non-resonant Raman excitation process which results in population of the spectrum of rotational states.

Defining Q ^m _j,j′=〈j,m|cos² θ|j′,m〉 m and noting from above that [𝓱,ρ ⁽⁰⁾]_kl is nonvanishing only for the non-diagonal components j′=j+2 and j′=j-2, Eq. (4) becomes

where ω _j,j-2=(E_j-E _j-2)/ħ=4πcB(2j-1) and ${\rho }_{j,m}^{\left(0\right)}=D_j{e}^{-hcBj\left(j+1\right)⁄k_{B}T}⁄Z$, where $Z=\sum _{k=0}^{\infty }{D}_k\left(2k+1\right)e^{-hcBk\left(k+1\right)⁄k_{B}T}$ is the rotational partition function and D_j is the statistical weighting factor for the number of nuclear spin states, |I_N,M〉 associated with each j. Note that the equation for ρ ⁽¹⁾ _j,j+2,m is implicitly included in Eq. (5) since ρ ⁽¹⁾ _j,j+2,m=(ρ ⁽¹⁾ _j+2,j,m)*, which is equivalent to ρ ⁽¹⁾ _j-2,j,m=(ρ ⁽¹⁾ _j,j-2,m)*. The laser field was taken to be E(t)=ï ε(t)(e^iωt+e ^-iωt)/2, where ω is the optical carrier frequency and ε (t) is the slowly varying field envelope whose temporal width is much greater than 2π/ω. Equation (5) is obtained after averaging over a laser optical cycle: the refractive index transients we seek have physical significance only on a pulse envelope timescale.

Finally, for the ensemble average alignment we get <cos² θ>_t=1/3+ρ_kl Q_lk=1/3+(ρ ⁽¹⁾ _j,j-2,m +(ρ ⁽¹⁾ _j,j-2,m)*)Q ^m _j,j-2 or

where we have used the fact that Q ^m _j,j-2=(Q ^m _j,j-2)*=Q^m _j-2,j is real, and summation over j and m is assumed. Performing the solid angle integration of the spherical harmonics, it can be shown that (Q^m _j,j-2)²=(j ²-m ²)((j-1)²-m ²)/[(2j-1)²(2j+1)(2j-3)]. Summing Eq. (6) from m=-j through m=j eliminates m to yield

where ρ ⁽⁰⁾ _j=ρ ⁽⁰⁾ _j,m.

Several special cases can be calculated. For a delta-function pump pulse having a fluence F (in erg/cm²), ε ²(τ)=(8π/c)Fδ(τ) and Eq. (7) becomes

This case models an extremely short pump pulse with wide bandwidth. For a Gaussian pump pulse with temporal full-width-at- half-maximum τ_p and peak field amplitude E ₀, and for times t≫τ_p, the upper limit of the integral in Eq. (7) can be taken to ∞ yielding

Both Eqs. (8) and (9) are appropriate for examining the alignment response for times well after the laser pulse. For the alignment response during and immediately after the pump pulse, a simple expression can be obtained by assuming a laser envelope of the form ${\epsilon }^2\left(\tau \right)=E_0^2{\cos }^2\left(\frac{\pi }{2}\frac{\tau }{{\tau }_0}\right)$ for |τ|≤τ ₀ and ε ²(τ)=0 for |τ|>τ ₀. For times |t|≤τ ₀, Eq. (7) becomes

and for t>τ ₀ it becomes

where

$a_1=\frac{1}{2}{\omega }_{j,j-2}⁄\left({\gamma }_{j,j-2}^2+{\omega }_{j,j-2}^2\right),$ $a_2=\frac{1}{4}({\omega }_{j,j-2}-\pi ⁄{\tau }_0)⁄\left({{\gamma }_{j,j-2}^2+({\omega }_{j,j-2}-\pi ⁄{\tau }_0)}^2\right),$ $a_3=\frac{1}{4}({\omega }_{j,j-2}+\pi ⁄{\tau }_0)⁄\left({\gamma }_{j,j-2}^2+{({\omega }_{j,j-2}+\pi ⁄{\tau }_0)}^2\right),$ $a_4=\frac{1}{4}{\gamma }_{j,j-2}⁄\left({\gamma }_{j,j-2}^2+{({\omega }_{j,j-2}-\pi ⁄{\tau }_0)}^2\right),$ $a_5=\frac{1}{4}{\gamma }_{j,j-2}⁄\left({\gamma }_{j,j-2}^2+{({\omega }_{j,j-2}+\pi ⁄{\tau }_0)}^2\right),$ $a_6=\frac{1}{2}{\gamma }_{j,j-2}⁄\left({\gamma }_{j,j-2}^2+{\omega }_{j,j-2}^2\right).$

4. Experimental results and discussion

The effect of molecular rotational inertia is immediately seen from a comparison of the response of various gases during a time window which includes the pump pulse. Molecular gas response at the time of the pulse is of special interest for studies of long range propagation of high power femtosecond pulses in the atmosphere [9, 10, 19, 25]. Figure 2 shows the transient refractive index shift experienced by the probe pulse for Ar, N₂ and N₂O at 4.4 atm. The 110 fs pump pulse energy was 95 µJ (peak intensity I=6.7×10¹³ W/cm²) for Ar, 60 µJ (peak intensity I=4.2×10¹³ W/cm²) for N₂, and 20 µJ (peak intensity I=1.4×10¹³ W/cm²) for N₂O. The shift for Ar, a monatomic gas, represents the purely prompt nonlinear response owing to electron cloud distortion, and as such its temporal shape directly follows the pump pulse envelope [23] and is centered at t=0. The later-peaking shifts for N₂ and N₂O are the result of the delayed response of the molecular alignment. In Fig. 3, we plot calculations of Δn(t) using Eqs. (2), (10) and (11), which consider alignment effects only, for (a) N₂ and (b) N₂O, using $B_{N_2}=2.0{\mathrm{cm}}^{-1}$, $\Delta {\alpha }_{N_2}=0.93\times {10}^{-24}{\mathrm{cm}}^3$, $B_{N_{2}O}=0.41{\mathrm{cm}}^{-1}$, and $\Delta {\alpha }_{N_{2}O}=2.79\times {10}^{-24}$ [26]. The data curves from Fig. 2 are overlaid on these plots. For both N₂ and N₂O there is good agreement between the calculations and measurements. The conclusion is that the orientational effects in N₂O and N₂ dominate the isotropic prompt response n ₂ I in the vicinity of the 110 fs pump pulse. This differs from the conclusion reached in reference [10] for N₂, where the index contributions are considered to be approximately the same size for prompt and orientational effects. Our determination may have important implications for studies of long range atmospheric propagation studies of femtosecond pulses. However, as seen later in this paper, for H₂ and D₂ under our conditions, the prompt n₂I response is larger in amplitude than the alignment response.

 

Fig. 2. Transient shift of refractive index induced by a 110-fs laser pulse in 4.4 atm Ar, N₂, and N₂O, with 95 µJ (6.7×10¹³ W/cm²), 60 µJ (4.2×10¹³ W/cm²), and 20 µJ (1.4×10¹³ W/cm²) pulse energy (peak intensity), respectively.

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Fig. 3. Normalized Δn profiles comparing experimental results from Fig. 2 (open circles) and calculations (solid red line) for (a) N₂ and (b) N₂O. The pump pulse envelope, with its peak centered at t=0, is shown in solid blue circles.

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As mentioned earlier, at times past the pump pulse, molecular gases can exhibit echoes in their refractive index response. As is well known, the excitation of a broad superposition of rotational states by a short pump pulse, either through absorption [27] or non-resonant Raman excitation (for example [11, 12]), gives rise to quantum echoes in the molecular alignment. The effect is qualitatively explained as follows. At t~0 the impulsive alignment torque administered by the short pump pulse locks the relative phases of the rotational states in the superposition wavepacket $|\psi 〉={\displaystyle {\sum }_{j,m}{a}_{j,m}|j,m〉e^{-i{\omega }_{j}t}}$, where ω_j=E_j/ħ=2πcBj(j+1). As there is net alignment along the E-field, an enhanced refractive index is observed. As t increases and the superposition evolves forward in time, its terms tend to cancel owing to differing factors exp(-iω_jt). However, when time passes through values t=q𝕋, where q is an integer and 𝕋 = (2cB)^-1, the phases become ω_jt=qπj(j+1), an integer multiple of 2π. The wavepacket’s constituent states are therefore rephased to their t=0 values and an alignment ‘full-revival’ occurs. When time passes through the ‘half-revival’ times t=q(𝕋/2), with q odd, ω_jt=qπj(j+1)/2, so that the set of states with j(j+1)/2 even are all in phase and the set of states with j(j+1)/2 odd are separately all in phase, with the two sets differing in phase by π. These sets interfere destructively, representing ‘anti-alignment’ perpendicular to the original t=0 alignment. Because j(j+1) is even, there are also partial revivals at times 𝕋/4, 𝕋/8, 𝕋/16, etc. In general these revivals are progressively weaker as they contain more subsets of in-phase states. Also, their peak amplitudes depend on nuclear spin statistics [28].

These qualitative considerations are borne out by examination of Eq. (7), where well after the pump pulse, the alignment <cos² θ>_t is described by a sum of terms with factors exp((iω _j,j-2-γ _j,j-2)t). The frequency difference between 2 successive terms is Δω=ω _j+1,j-1-ω _j,j-2=8πcB, so that the full revival period of the alignment occurs at times $\frac{1}{2}\Delta \omega t=q2\pi$, or t=qT. Peaks that are π out of phase occur at $\frac{1}{2}\Delta \omega t=q\pi$ for odd q, or t=q𝕋/2; these are the half-revival peaks representing anti-alignment discussed above. The peak width is approximately ~𝕋/N_rot, where N_rot is the number of rotational states contributing to the wavepacket, which is a function of temperature and pump pulse energy, duration and bandwidth.

Figure 4(a) shows probe beam-centre lineouts of the measured alignment <cos² θ>_t-1/3 for 6.4 atm of N₂ for time windows centered at t=0 through t=1.25𝕋 in 0.25𝕋 steps, where for $B_{N_2}=2.0{\mathrm{cm}}^{-1}$ [26], $𝕋={\left(2c{B}_{N_2}\right)}^{-1}=8.33\mathrm{ps}$. The pump pulse was 60 µJ, 110 fs, with peak intensity I=4.1×10¹³ W/cm². Note that preceding the half and full revivals by an interval δt ~𝕋/N_rot are positive and negative excursions corresponding to alignment and anti-alignment, respectively. Figure 4(b) shows the corresponding space-time images across the probe beam, clearly showing the radial intensity dependence of alignment. Figure 4(c) shows a calculation of <cos²θ>t-1/3 for N₂ comparing the finite pulse response (Eqs. (10) and (11)) with the delta function pump response (Eq. (8)), in which the laser fluence is matched to the experimental value of 4.5×10⁷ erg/cm². The delta function models an extremely short, highly broadband pulse. It is seen that the finite pulse result is an excellent match to the experimental curves, but the delta-function result is still quite reasonable. To explain this, we note that for a thermal rotational distribution, it can be shown that the most populous state contributing to the wavepacket has $j=j_{max}\sim \frac{3}{4}\left({1+\left(1+\frac{8}{9}\frac{k_{B}T}{Bhc}\right)}^{1⁄2}\right)$. For k_BT/Bhc≫1, which is the case for N₂, j _max ~(k_BT/Bhc)^1/2~10. For large j _max, the frequency width of the thermal distribution of rotational states available for pumping is Δω _rot~k_BT/ħ~4×10¹³ s^-1. By comparison, our pump laser frequency bandwidth corresponding to Δλ_laser~10 nm is Δω_laser ~3×10¹³ s^-1. As Δω_laser ~Δω_rot, the bandwidth of the laser pulse adequately overlaps the thermal distribution and therefore one would expect reasonable agreement in Fig. 4(c) with the delta function pump. We note that for N₂ and for the other molecules studied in this paper, the shapes of the calculated finite pulse alignment response curves are an excellent match to the experimental results except for a persistent overall amplitude mismatch of approximately a factor of 2, which we are unable to definitively explain at this time. One possible explanation is an error in the pump’s effective nonlinear interaction length L in the gas cell, which was determined to be 5.7 mm [23].

Figure 5 repeats for O₂ the format of Fig. 4 for a cell pressure of 5.1 atm and pump pulse energy 40 µJ (peak intensity I=2.7×10¹³ W/cm²). Figure 5(a) shows probe beam-centre lineouts of the measured O₂ alignment <cos²θ>_t-1/3 for time windows centered at t=0 through t=1.25𝕋 in 0.25𝕋 steps, where $𝕋={\left(2c{B}_{O_2}\right)}^{-1}=11.6\mathrm{ps}$ [26]. Note that unlike in N₂, the $t=\frac{1}{4}𝕋$ and $t=\frac{3}{4}𝕋$ revivals in O₂ are comparable in amplitude to the full- and half-revivals. This derives from the I_N=0 spin of the ¹⁶ ₈O nucleus, making it a boson, and requiring the total molecular wave function to be symmetric upon nuclear interchange. Thus only odd j states are populated, so that at the quarter revivals there is no cancellation of aligned and anti-aligned states as observed in N₂. Figure 5(b) shows the same revivals plotted versus space and time, and 5(c) shows calculations of alignment (using $B_{O_2}=1.44{\mathrm{cm}}^{-1}$ and $\Delta {\alpha }_{O_2}=1.14\times {10}^{-24}{\mathrm{cm}}^3$ [26]) using the delta function pulse and the finite duration (110 fs) pulse, where a damping rate of γ _j,j-2=γ=4.31×¹⁰10 s^-1 (or dephasing time 1/γ=23.2 ps) was used. The damping rate was obtained from a fit to the declining peak amplitudes in the time sequence data of Fig. 5(a). Since dephasing is dominated by elastic molecular collisions, there should be little j dependence of the rate; hence we put γ _j,j-2=γ. For O₂, as for N₂ in Fig. 4, the best agreement with the data is obtained with the finite pulse, with the delta function response fitting the finite pulse response reasonably well. For oxygen at room temperature, j _max ~12 and therefore, to a similar extent as in N₂, the rotational state wavepacket contribution is thermally limited rather than pump laser spectrum limited.

 

Fig. 4. Measured N₂ alignment <cos² θ>_t-1/3 up to t=1.25T for 6.4 atm gas pressure and 4.1×10¹³ W/cm² pump peak intensity: (a) probe beam central lineout, and (b) corresponding 2-D space- and time-resolved image across the probe beam. Simulated <cos² θ>_t-1/3 is shown in (c), assuming two different excitation pulse shapes: δ-function (blue line) and cos² function with 110 fs FWHM (red line). Dephasing is not included in this calculation.

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Fig. 5. Measured O₂ alignment <cos² θ>_t-1/3 up to t=1.25T for 5.1 atm gas pressure and 2.7×10¹³ W/cm² pump peak intensity: (a) probe beam central lineout, and (b) corresponding 2-D space- and time-resolved image across the probe beam. The dephasing time constant 1/γ=23.2 ps was obtained by fitting the peak amplitudes of (a) to an exponential. Simulated <cos² θ>_t-1/3 is shown in (c), including dephasing by using the extracted value of γ, and assuming two different excitation pulse shapes: δ-function (blue line) and cos² function with 110 fs FWHM (red line).

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Fig. 6. Measured N₂O alignment <cos² θ>_t-1/3 up to t=T for 2.4 atm gas pressure and 1.4×10¹³ W/cm² pump peak intensity: (a) probe beam central lineout, and (b) corresponding 2-D spaceand time-resolved image across the probe beam. Note that the 1/4 and 3/4 revivals are not present due to the axial asymmetry of the N₂O molecule. The dephasing time constant 1/γ=23.8 ps was obtained by fitting the peak amplitudes of (a) to an exponential. Simulated <cos² θ>_t-1/3 is shown in (c), using the extracted dephasing rate γ and assuming two different excitation pulse shapes: δ-function (blue line) and cos² function with 110 fs FWHM (red line).

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Results for N₂O, a linear molecule with a much greater moment of inertia (and lower rotational constant, $B_{N_{2}O}=0.412{\mathrm{cm}}^{-1}$ [26]) are shown in Fig. 6, for pump energy of 20 µJ (peak intensity I=1.4×10¹³ W/cm²) and cell pressure 2.4 atm. Beam centre lineouts of the alignment are shown in 6(a) for windows centered at t=0, t=0.5𝕋 and t=𝕋. The 1/4 and 3/4 revivals are not present owing to the axial asymmetry of the linear N₂O molecule, in which the atoms are ordered N-N-O. Thus even and odd j rotational states are populated with equal weight, causing the aligned and anti-aligned contributions to cancel. Figure 6(b) shows the same revivals in full space-time plots, and Fig. 6(c) again shows calculations for finite and delta function pulses using $B_{N_{2}O}=0.412{\mathrm{cm}}^{-1}$ and $\Delta {\alpha }_{N_{2}O}=2.79\times {10}^{-24}{\mathrm{cm}}^3$ [26]. Here the delta function result shows even better agreement with the finite pulse result and the experimental revival curves. Here j _max ~22, so the approximation Δω_rot~k_BT/ħ applies even better, with Δω_laser>Δω_rot as in the N₂ and O₂ cases.

We now show results for D₂ and H₂, which have the smallest moments of inertia and largest values of B and therefore rotate the fastest. In Fig. 7(a) is shown a beam centre lineout of the alignment recurrences in 7.8 atm D₂ for a pump energy of 65 µJ (peak intensity I=4.4×10¹³ W/cm²), with the corresponding time-space plot shown in Fig. 7(b). The delta function and finite pulse calculations are shown in Fig. 7(c), where we have used $\Delta {\alpha }_{D_2}=1.14\times {10}^{-24}{\mathrm{cm}}^3$ [26] and $B_{D_2}=30.4{\mathrm{cm}}^{-1}$ as determined below in connection with Fig. 7(d). This corresponds to 𝕋 = 548 fs, so that our single-shot temporal window is ~2.5𝕋 long. Earlier results for D₂ using Coulomb explosion imaging [29] showed revivals through ~0.5𝕋, with error bars comparable to the revival amplitudes. In our case, apart from any effects due to probe bandwidth limitation or phase front distortion [24, 30], we estimate an error of (δΦ_shot/N _int ^1/2)(ΔΦ)^-1~2%.

 

Fig. 7. Measured D₂ molecule alignment <cos² θ>_t-1/3 up to t=2.5T for 7.8 atm gas pressure, 110 fs pump pulse duration, and 4.4×10¹³ W/cm² pump peak intensity: (a) probe beam central lineout, and (b) corresponding 2-D space- and time-resolved image across the probe beam. Simulated <cos² θ>_t-1/3 is shown in (c), assuming two different excitation pulse shapes: δ-function (blue line) and cos² function with 110 fs FWHM (red line). (d) Fourier transform of measured D₂ alignment response of (a), with the first peak in (a) (partly contributed by instantaneous n₂ I) excluded. The peak in (d) is identified as the beat frequency between j=0 and j=2.

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For D₂, there are two significant differences with the simulations for the smaller B molecules discussed earlier. First, the delta-function and finite pulse results are strikingly different. The large value of B results in much lower j states dominating the wavepacket so that for D₂ $j_{max}\sim \frac{3}{4}\left(1+{\left(1+\frac{8}{9}\frac{k_{B}T}{Bhc}\right)}^{\frac{1}{2}}\right)\sim 2-3$ ~2–3. The spin of the D nucleus is I_N=1, so that for the D₂ molecule, even j states are twice as populated as odd j. Dominant coupled Δj=2 states near j _max are therefore j=2 and j=4, which have a frequency spacing ${\omega }_{4,2}={\omega }_4-{\omega }_2=28\pi c{B}_{D_2}\sim 8.0\times {10}^{13}{s}^{-1}$, or j=0 and j=2, with ${\omega }_{2,0}=12\pi c{B}_{D_2}\sim 3.4\times {10}^{13}{s}^{-1}$. Thus, the pump bandwidth of 3×10¹³ s^-1 is barely adequate to overlap the latter two rotation states. Fourier transforming the revivals of Fig. 7(a) shows explicitly, as seen in Fig. 7(d), the sparse modal content of the wavepacket: the peak at ω=3.44×10¹³ s^-1 is immediately identified as ω _2,0, and allows us to extract the value of $B_{D_2}=30.4{\mathrm{cm}}^{-1}$ used in the calculations as discussed above. There is negligible contribution from other states.

The other difference is between the experimental results (Fig. 7(a)) and the finite pulse theory (Fig. 7(c)) where it is seen that the experiment shows a much bigger initial prompt peak relative to the following revivals. We note that the calculation, based on Eqs. (10) and (11), accounts only for the rotational effect on the refractive index. In the experiment, the relative contribution of the isotropic n ₂ I to the prompt response (see Eq. (3) and discussion) is much greater than for the smaller B molecules, where the delayed rotational response dominates near the pump, as seen by the excellent match of calculations and experiment shown for N₂, O₂, and N₂O.

Results for H₂ are shown in Figure 8, for a pump energy of 65µJ (peak intensity I=4.4×10¹³ W/cm²) and gas cell pressure of 7.8 atm. A beam centre lineout of alignment is shown in Fig. 8(a), and it is seen that the effect is much smaller than in D₂. If not for the full space-time plot in Fig. 8(b), where the revivals are seen to follow the pulse like a wake, one might not have recognized the small amplitude revivals in the lineout. The revivals are surprisingly well-modeled by the finite pulse simulation, where the rotational constant that best fits the Fourier-transformed response is $B_{H_2}=61.8{\mathrm{cm}}^{-1}$, in good agreement with previous values [26], and using $\Delta {\alpha }_{H_2}=0.30\times {10}^{-24}{\mathrm{cm}}^3$ [26]. This corresponds to 𝕋=270 fs, so that our single-shot temporal window is ~3𝕋 long. As in the case for D₂, the delta function and finite pulse simulations differ substantially because of the insufficient pump bandwidth of the finite pulse to populate many rotational states. Also, as in the case of D₂, the prompt peak in Fig. 8(a) is dominated by the isotropic n ₂ I contribution.

For the H₂ wavepacket, the most populated state is estimated to be j _max~2. The spin of the H nucleus is ${\mathbf{I}}_N=\frac{1}{2}$, so that even j states are 3 times more populated than odd j states. The likely coupled Δj=2 states are j=2 and j=0, with ${\omega }_{20}=12\pi c{B}_{H_2}\sim 7\times {10}^{13}{s}^{-1}$. The pump bandwidth is even less adequate than in D₂ to populate these states, so their amplitudes would be expected to be weak. Figure 8(d) shows a Fourier transform of the revivals of Fig. 8(a), confirming that our H₂ revivals are generated by beating of the two low amplitude j=0 and j=2 rotational modes. For H₂, the error in the extracted phase shift is estimated to be (δΦ_shot/N _int ^1/2)(ΔΦ)^-1~15%.

Much of the above discussion on the pump bandwidth effect on wavepacket excitation is summarized explicitly by Eq. (7), where for negligible dephasing and time t past the pump pulse, the integral portion can be written ∫^t _-∞ dτI(τ)exp(-iω _j,j-2)≈Ĩ(ω _j,j-2), which is simply the pump intensity spectral amplitude at the beat frequency. Therefore, those rotational modes contributing to the wavepacket have Δj=2 beat frequencies lying within the pump pulse spectrum.

 

Fig. 8. Measured H₂ alignment <cos² θ>_t-1/3 up to t=3T, for 7.8 atm gas pressure, 110 fs pump pulse duration, and 4.4×10¹³ W/cm² pump peak intensity: (a) probe beam central lineout, and (b) corresponding 2-D space- and time-resolved image across the probe beam. Simulated <cos² θ>_t-1/3 is shown in (c), assuming δ-function (blue line) and 110 fs FWHM cos² (red line) excitation pulses. Experiment and simulation show that the revival amplitude is much smaller than for D₂. (d) Fourier transform of measured H₂ alignment response in (a) (neglecting the first peak) shows that the dominant contribution is beating between the j=0 and j=2 states.

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Finally, results are presented for the gas pressure dependence of collisional dephasing. The t=0.5𝕋 revival in N₂O is shown in Fig. 9(a) normalized to the t=0 peak for several cell pressures: 2.4, 3.7, 5.1, and 6.4 atm. The dephasing rate γ for each pressure was obtained from a fit to the decay of revival amplitudes from t=0 through t=𝕋. The dephasing rate is plotted versus pressure in Fig. 9(b), and it is seen that the dependence is linear, as expected from a collisional process. The dephasing rate per unit pressure is 1.46×10¹⁰ s^-1 atm^-1.

 

Fig. 9. (a) Measured N₂O revivals at t=1/2 T at pressures of 2.4, 3.7, 5.1, and 6.4 atm, normalized to the peak alignment amplitude near t=0. (b) Dephasing rate γ versus pressure (open circles) showing linear fit (solid line). The dephasing rate per unit pressure is 1.46×10¹⁰ s^-1 atm^-1. The other measurement conditions are the same as for Fig. 6.

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5. Conclusions

In conclusion, we have applied single-shot supercontinuum spectral interferometry (SSSI) to measure for the first time, in a single-shot, the space- and time-resolved quantum rotational echo response of a number of molecular gases to femtosecond pump pulse excitation. In particular, these measurements have been achieved for H₂ and D₂, for which the low level of the effect could easily have been hidden in the shot-to-shot fluctuations characteristic of multi-shot pump-probe techniques.