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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 18 — Sep. 3, 2007
  • pp: 11341–11357
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Single-shot, space- and time-resolved measurement of rotational wavepacket revivals in H2, D2, N2, O2, and N2O

Y.-H. Chen, S. Varma, A. York, and H. M. Milchberg  »View Author Affiliations


Optics Express, Vol. 15, Issue 18, pp. 11341-11357 (2007)
http://dx.doi.org/10.1364/OE.15.011341


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Abstract

Femtosecond laser-induced alignment and periodic recurrences in hydrogen and deuterium are measured in a single shot for the first time, in a room temperature gas cell. Single-shot Supercontinuum Spectral Interferometry (SSSI) is employed, with measurements also performed in room temperature samples of nitrogen, oxygen, and nitrous oxide. Unlike previous optical techniques for probing molecular alignment in gases or liquids, SSSI quantitatively and directly measures the degree of molecular alignment without reliance on model fits, and it can do so with spatial resolution transverse to the pump beam. In addition, wavepacket collisional dephasing rates can be directly measured in gas samples at useful densities.

© 2007 Optical Society of America

1. Introduction

Alignment of molecules in intense laser fields has assumed a significant level of activity in recent years [1

1. H. Stapelfeldt and T. Seideman, “Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75, 543–557 (2003). [CrossRef]

]. Applications include enhancement of high harmonic generation [2

2. R. Velotta, N. Hay, M. B. Mason, M. Castillejo, and J. P. Marangos, “High-Order Harmonic Generation in Aligned Molecules,” Phys. Rev. Lett. 87, 183901 (2001); [CrossRef]

4

4. C. Vozzi, F. Calegari, E. Benedetti, J.-P. Caumes, G. Sansone, S. Stagira, M. Nisoli, R. Torres, E. Heesel, N. Kajumba, J. P. Marangos, C. Altucci, and R. Velotta, “Controlling Two-Center Interference in Molecular High Harmonic Generation,” Phys. Rev. Lett. 95, 153902 (2005). [CrossRef] [PubMed]

], orientation of molecules for field-driven wavepacket tomography [5

5. J. Itatani, J. Levesque, D. Zeidler, Hiromichi Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867–871 (2004) [CrossRef] [PubMed]

,6

6. S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing Proton Dynamics in Molecules on an Attosecond Time Scale,” Science 21, 424–427 (2006). [CrossRef]

], structural studies of molecules [7

7. P. M. Felker, J. S. Baskin, and A. H. Zewail, “Rephasing of collisionless molecular coherence in large molecules,” J. Phys. Chem. 90, 724–728 (1986) [CrossRef]

,8

8. L. L. Connell, T. C. Corcoran, P. W. Joireman, and P. M. Felker, “Observation and description of a new type of transient in rotational coherence spectroscopy,” J. Phys. Chem. 94, 1229–1232 (1990). [CrossRef]

], and effects on long-range atmospheric propagation of intense femtosecond laser pulses [9

9. J. R. Peñano, P. Sprangle, P. Serafim, B. Hafizi, and A. Ting, “Stimulated Raman scattering of intense laser pulses in air,” Phys. Rev. E 68, 056502 (2003). [CrossRef]

,10

10. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–660 (1997). [CrossRef]

].

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

11. P. W. Dooley, I. V. Litvinyuk, Kevin F. Lee, D. M. Rayner, M. Spanner, D. M. Villeneuve, and P. B. Corkum, “Direct imaging of rotational wave-packet dynamics of diatomic molecules,” Phys. Rev. A 68, 023406 (2003). [CrossRef]

, 12

12. F. Rosca-Pruna and M. J. J. Vrakking, “Experimental Observation of Revival Structures in Picosecond Laser-Induced Alignment of I2,” Phys. Rev. Lett. 87, 153902 (2001). [CrossRef] [PubMed]

]. 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

13. M. A. Duguay and J. W. Hansen, “An ultrafast light gate,” Appl. Phys. Lett. 15,192–194 (1969). [CrossRef]

, 14

14. E. P. Ippen and C. V. Shank, “Picosecond response of a high-repetition-rate CS2 optical Kerr gate,” Appl. Phys. Lett. 26, 92–93 (1975). [CrossRef]

]. 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

15. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of coherent transient birefringence in CS2 vapor,” Phys. Rev. Lett. 34, 1299–1302 (1975). [CrossRef]

]. A version of this technique, now called Optical Kerr Effect (OKE) spectroscopy [16

16. R. Righini, “Ultrafast optical Kerr effect in liquids and solids,” Science 262, 1386–1390 (1993). [CrossRef] [PubMed]

], 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

17. V. Renard, O. Faucher, and B. Lavorel, “Measurement of laser-induced alignment of molecules by cross defocusing,” Opt. Lett. 30, 70–72 (2005). [CrossRef] [PubMed]

] and ionization of these samples [18

18. V. Loriot, E. Hertz, A. Rouzée, B. Sinardet, B. Lavorel, and O. Faucher, “Strong-field molecular ionization: determination of ionization probabilities calibrated with field-free alignment,” Opt. Lett. 31, 2897–2899 (2006). [CrossRef] [PubMed]

]. 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

19. J.-F. Ripoche, G. Grillon, B. Prade, M. France, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of n2 in air,” Opt. Commun. 135, 310–314 (1997). [CrossRef]

, 20

20. I. V. Fedotov, A. D. Savvin, A. B. Fedotov, and A. M. Zheltikov, “Controlled rotational Raman echo recurrences and modulation of high-intensity ultrashort laser pulses by molecular rotations in the gas phase,” Opt. Lett. 32, 1275–1277 (2007). [CrossRef] [PubMed]

].

The only technique described above which is capable of direct quantitative measurement of alignment is Coulomb explosion imaging [11

11. P. W. Dooley, I. V. Litvinyuk, Kevin F. Lee, D. M. Rayner, M. Spanner, D. M. Villeneuve, and P. B. Corkum, “Direct imaging of rotational wave-packet dynamics of diatomic molecules,” Phys. Rev. A 68, 023406 (2003). [CrossRef]

, 12

12. F. Rosca-Pruna and M. J. J. Vrakking, “Experimental Observation of Revival Structures in Picosecond Laser-Induced Alignment of I2,” Phys. Rev. Lett. 87, 153902 (2001). [CrossRef] [PubMed]

], 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.

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

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

]. 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

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

].

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.

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

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

], 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

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

, 24

24. K. Y. Kim, I. Alexeev, and H. M. Milchberg, “Single-shot supercontinuum spectral interferometry,” Appl. Phys. Lett. 81, 4124–4126 (2002). [CrossRef]

]. 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

24. K. Y. Kim, I. Alexeev, and H. M. Milchberg, “Single-shot supercontinuum spectral interferometry,” Appl. Phys. Lett. 81, 4124–4126 (2002). [CrossRef]

]. 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

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

]. The time resolution of measured index transients is ~10 fs, limited by the SC bandwidth of ~100 nm [23

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

].

In anticipation of potentially tiny phase shifts compared to the experiments of ref. [23

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

], 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 δΦshot/N int 1/2, where δΦshot 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=<eiαijej>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 αzz=α and αxx=αyy=α . Owing to molecular symmetry about the z-axis, the laser electric field can be taken as E=x̂Ex+ Ez 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 αxx+<e 2 z>t αzz, or

n2(t)=1+4πN(Δα<cos2θ>t+α),
(1)

where Δα=α -α and ez=ê·;=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

Δn(t)=2πNn01Δα(<cos2θ>t13),
(2)

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 n2(t=)=n02=1+4πN(13Δα+α)..

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

n2(t)14πN=Δα<cos2θ>t+α
+E(t)2(<sin4θ>tαxxxx(3)+12<sin22θ>tαxxzz(3)+<cos4θ>tαzzzz(3)),
(3)

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 12[<sin4θ>t=αxxxx(3)+12<sin22θ>t=αxxzz(3)+<cos4θ>t=αzzzz(3)]E(t)2=n2I(t),, 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)=ρkl〈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/kBT≫ 1 and ΔE vib/kBT≫1, where ΔE elec and ΔE vib are the energies of the first excited electronic and vibrational states and kB 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

[ρ1(t)]kl=itdτ[𝓱(τ),ρ(0)]kle(iωkl+γkl)(τt),
(4)

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, ωkl=(Ek-El)/ħ corresponds to rotational states |k〉=|j,m〉 and |l〉=j′,m′ with energies Ek=E j,m=hcBj(j+1) and El=E j′,mhcBj′(j′+1) (B is the rotational constant), γkl 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=(ρl (0)-ρk (0)) 𝓱kl, where ρl (0)ρll (0)(no sum), ρk (0)ρkk (0) (no sum), and 𝓱kl=-Δα|E|2k|cos2θ|l〉-α δkl|E|2, where δkl is the unity matrix. As the rotational eigenstates are the spherical harmonics, |j,m〉=Yjm(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

ρj,j2,m(1)(t)=i2(ρj,m(0)ρj2,m(0))ΔαQj,j2me(iωi,j2+γi,j2)ttdτε2(τ)e(iωj,j2+γj,j2)τ,
(5)

where ω j,j-2=(Ej-E j-2)/ħ=4πcB(2j-1) and ρj,m(0)=DjehcBj(j+1)kBTZ, where Z=k=0Dk(2k+1)ehcBk(k+1)kBT is the rotational partition function and Dj is the statistical weighting factor for the number of nuclear spin states, |IN,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)(eiωt+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+ρkl Qlk=1/3+(ρ (1) j,j-2,m +(ρ (1) j,j-2,m)*)Q m j,j-2 or

<cos2θ>t=131(ρj,m(0)ρj2,m(0))Δα(Qj,j2m)2
×Im(e(iωj,j2γj,j2)ttdτε2(τ)e(iωj,j2+γj,j2)τ),
(6)

where we have used the fact that Q m j,j-2=(Q m j,j-2)*=Qm 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 (Qm 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

<cos2θ>t=13215j(j1)2j1(ρj(0)ρj2(0))Δα
×Im(e(iωj,j2γj,j2)ttdτε2(τ)e(iωj,j2+γj,j2)τ),
(7)

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)(τ) and Eq. (7) becomes

<cos2θ>t=13115j(j1)2j116πFc(ρj(0)ρj2(0))Δαeγj,j2tsin(ωj,j2t).
(8)

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

<cos2θ>t=13215j(j1)2j1(ρj(0)ρj2(0))ΔαE02(πln2)12τp
×e(ωj,j22γj,j22)τp216ln2eγj,j2tsin(ωj,j2tγj,j2ωj,j2τp28ln2).
(9)

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 ε2(τ)=E02cos2(π2ττ0) for |τ|≤τ 0 and ε 2(τ)=0 for |τ|>τ 0. For times |t|≤τ 0, Eq. (7) becomes

<cos2θ>t=13215j(j1)2j1(ρj(0)ρj2(0))ΔαE02[a1+(a2+a3)cos(πtτ0)
+(a4a5)sin(πtτ0)+(a4+a5a6)eγj,j2(t+τ0)sinωj,j2(t+τ0)
+(a1+a2+a3)eγj,j2(t+τ0)cosωj,j2(t+τ0)],
(10)

and for t>τ 0 it becomes

<cos2θ>t=13+215j(j1)2j1(ρj(0)ρj2(0))ΔαE02
×[(a6a4a5)eγj,j2(tτ0)sinωj,j2(tτ0)+(a6+a4+a5)eγj,j2(t+τ0)sinωj,j2(t+τ0)
+(a1a2a3)eγj,j2(tτ0)cosωj,j2(tτ0)+(a1+a2+a3)eγj,j2(t+τ0)cosωj,j2(t+τ0)],
(11)

where

a1=12ωj,j2(γj,j22+ωj,j22), a2=14(ωj,j2πτ0)(γj,j22+(ωj,j2πτ0)2), a3=14(ωj,j2+πτ0)(γj,j22+(ωj,j2+πτ0)2), a4=14γj,j2(γj,j22+(ωj,j2πτ0)2), a5=14γj,j2(γj,j22+(ωj,j2+πτ0)2), a6=12γj,j2(γj,j22+ωj,j22).

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

9. J. R. Peñano, P. Sprangle, P. Serafim, B. Hafizi, and A. Ting, “Stimulated Raman scattering of intense laser pulses in air,” Phys. Rev. E 68, 056502 (2003). [CrossRef]

, 10

10. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–660 (1997). [CrossRef]

, 19

19. J.-F. Ripoche, G. Grillon, B. Prade, M. France, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of n2 in air,” Opt. Commun. 135, 310–314 (1997). [CrossRef]

, 25

25. I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Penano, R. F. Hubbard, and P. Sprangle, “Longitudinal compression of short laser pulses in air,” Appl. Phys. Lett. 84, 4080–4082 (2004). [CrossRef]

]. 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

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

] 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 BN2=2.0cm1, ΔαN2=0.93×1024cm3, BN2O=0.41cm1, and ΔαN2O=2.79×1024 [26

26. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence arising from the reorientation of the polarizability anisotropy of molecules in collisionless gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

]. 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

10. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–660 (1997). [CrossRef]

] 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.
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.

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

27. H. Harde and D. Grischkowsky, “Coherent transients excited by subpicosecond pulses of terahertz radiation,” J. Opt. Soc. Am. B 8, 1642–1651 (1991). [CrossRef]

] or non-resonant Raman excitation (for example [11

11. P. W. Dooley, I. V. Litvinyuk, Kevin F. Lee, D. M. Rayner, M. Spanner, D. M. Villeneuve, and P. B. Corkum, “Direct imaging of rotational wave-packet dynamics of diatomic molecules,” Phys. Rev. A 68, 023406 (2003). [CrossRef]

, 12

12. F. Rosca-Pruna and M. J. J. Vrakking, “Experimental Observation of Revival Structures in Picosecond Laser-Induced Alignment of I2,” Phys. Rev. Lett. 87, 153902 (2001). [CrossRef] [PubMed]

]), 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 |ψ=j,maj,m|j,meiωjt, where ωj=Ej/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(-jt). However, when time passes through values t=q𝕋, where q is an integer and 𝕋 = (2cB)-1, the phases become ωjt=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, ωjt=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

28. W. Demtroder, Molecular Physics, Wiley-VCH (Weinheim, 2005). [CrossRef]

].

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(( 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 12Δωt=q2π, or t=qT. Peaks that are π out of phase occur at 12Δωt=qπ for odd q, or t=q𝕋/2; these are the half-revival peaks representing anti-alignment discussed above. The peak width is approximately ~𝕋/Nrot, where Nrot 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 BN2=2.0cm1 [26

26. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence arising from the reorientation of the polarizability anisotropy of molecules in collisionless gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

], 𝕋=(2cBN2)1=8.33ps. 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/Nrot 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=jmax34(1+(1+89kBTBhc)12). For kBT/Bhc≫1, which is the case for N2, j max ~(kBT/Bhc)1/2~10. For large j max, the frequency width of the thermal distribution of rotational states available for pumping is Δω rot~kBT/ħ~4×1013 s-1. By comparison, our pump laser frequency bandwidth corresponding to Δλlaser~10 nm is Δωlaser ~3×1013 s-1. As Δωlaserωrot, 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

23. Y.-H. Chen, S. Varma, I. Alexeev, and H. M. Milchberg, “Measurement of transient nonlinear refractive index in gases using xenon supercontinuum single-shot spectral interferometry,” Opt. Express 15, 7458–7467 (2007). [CrossRef] [PubMed]

].

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=(2cBO2)1=11.6ps [26

26. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence arising from the reorientation of the polarizability anisotropy of molecules in collisionless gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

]. Note that unlike in N2, the t=14𝕋 and t=34𝕋 revivals in O2 are comparable in amplitude to the full- and half-revivals. This derives from the IN=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 BO2=1.44cm1 and ΔαO2=1.14×1024cm3 [26

26. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence arising from the reorientation of the polarizability anisotropy of molecules in collisionless gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

]) 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.
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).
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).

Results for N2O, a linear molecule with a much greater moment of inertia (and lower rotational constant, BN2O=0.412cm1 [26

26. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence arising from the reorientation of the polarizability anisotropy of molecules in collisionless gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

]) 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 BN2O=0.412cm1 and ΔαN2O=2.79×1024cm3 [26

26. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence arising from the reorientation of the polarizability anisotropy of molecules in collisionless gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

]. 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 Δωrot~kBT/ħ applies even better, with Δωlaserωrot 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 ΔαD2=1.14×1024cm3 [26

26. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence arising from the reorientation of the polarizability anisotropy of molecules in collisionless gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

] and BD2=30.4cm1 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

29. K. F. Lee, F. Legare, D. M. Villeneuve, and P. B. Corkum, “Measured field-free alignment of deuterium by few-cycle pulses,” J. Phys. B: At. Mol. Opt. Phys. 39, 4081–4086 (2006). [CrossRef]

] 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

24. K. Y. Kim, I. Alexeev, and H. M. Milchberg, “Single-shot supercontinuum spectral interferometry,” Appl. Phys. Lett. 81, 4124–4126 (2002). [CrossRef]

, 30

30. K. Kim, I. Alexeev, and H. Milchberg, “Single-shot measurement of laser-induced double step ionization of helium,” Opt. Express 10, 1563–1572 (2002). [PubMed]

], we estimate an error of (δΦshot/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.

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.

For the H2 wavepacket, the most populated state is estimated to be j max~2. The spin of the H nucleus is IN=12, 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 ω20=12πcBH27×1013s1. 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 (δΦshot/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(- 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.

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.

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.

Acknowledgements

The authors thank I. Alexeev for discussions and technical assistance. This work was supported by the National Science Foundation and the U.S. Department of Energy.

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OCIS Codes
(020.0020) Atomic and molecular physics : Atomic and molecular physics
(020.1670) Atomic and molecular physics : Coherent optical effects
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(320.2250) Ultrafast optics : Femtosecond phenomena
(320.7100) Ultrafast optics : Ultrafast measurements

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: July 12, 2007
Revised Manuscript: August 19, 2007
Manuscript Accepted: August 20, 2007
Published: August 22, 2007

Citation
Y-H. Chen, S. Varma, A. York, and H. M. Milchberg, "Single-shot, space- and time-resolved measurement of rotational wavepacket revivals in H2, D2, N2, O2, and N2O," Opt. Express 15, 11341-11357 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-18-11341


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