## Single-shot, space- and time-resolved measurement of rotational wavepacket revivals in H_{2}, D_{2}, N_{2}, O_{2}, and N_{2}O

Optics Express, Vol. 15, Issue 18, pp. 11341-11357 (2007)

http://dx.doi.org/10.1364/OE.15.011341

Acrobat PDF (1538 KB)

### 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

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

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. 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]

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. 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]

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. 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]

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. 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, N_{2}, and O_{2} by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B **14**, 650–660 (1997). [CrossRef]

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. F. Rosca-Pruna and M. J. J. Vrakking, “Experimental Observation of Revival Structures in Picosecond Laser-Induced Alignment of I_{2},” Phys. Rev. Lett. **87**, 153902 (2001). [CrossRef] [PubMed]

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

14. E. P. Ippen and C. V. Shank, “Picosecond response of a high-repetition-rate CS_{2} optical Kerr gate,” Appl. Phys. Lett. **26**, 92–93 (1975). [CrossRef]

_{2}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 CS

_{2}vapour [15

15. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of coherent transient birefringence in CS_{2} vapor,” Phys. Rev. Lett. **34**, 1299–1302 (1975). [CrossRef]

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

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]

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]

19. J.-F. Ripoche, G. Grillon, B. Prade, M. France, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of n_{2} in air,” Opt. Commun. **135**, 310–314 (1997). [CrossRef]

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]

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. F. Rosca-Pruna and M. J. J. Vrakking, “Experimental Observation of Revival Structures in Picosecond Laser-Induced Alignment of I_{2},” Phys. Rev. Lett. **87**, 153902 (2001). [CrossRef] [PubMed]

21. S. Varma, Y.-H. Chen, H. M. Milchberg, and I. Alexeev, “Single-Shot Time Resolved Measurement of Molecular Alignment in Laser-Irradiated Gases,” in *Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies*, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JFC3

22. S. Varma, Y.-H. Chen, I. Alexeev, R. Moon, and H. M. Milchberg, “Single-shot time resolved measurement of molecular alignment in laser-irradiated gases: application to ‘self-channeled’ plasma columns,” presented at *48th Annual Meeting of the Division of Plasma Physics, American Physical Society*, Philadelphia, 30 Oct.- 3 Nov. 2006, abstract #NO3.015.

## 2. Experimental setup

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]

*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.

^{-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]

*x*,

*t*) induced on the probe pulse is extracted from Fourier analysis of the spectral interferograms Δ

*ϕ*(

*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]

*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*)+Δ

*ϕ*(

*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

**15**, 7458–7467 (2007). [CrossRef] [PubMed]

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

*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]

*n*is then determined by 1 Δ

*n*(

*x*,

*t*)=(

*k*

_{0}

*L*)

^{-1}ΔΦ(

*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

**15**, 7458–7467 (2007). [CrossRef] [PubMed]

**15**, 7458–7467 (2007). [CrossRef] [PubMed]

**15**, 7458–7467 (2007). [CrossRef] [PubMed]

*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

*δ*Φ

_{shot}/

*N*

_{int}

^{1/2}, where

*δ*Φ

_{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

*ε*=

*n*

^{2}=1+4

*πN*<

*α*>

*, where*

_{t}*N*is the number of molecules per cm

^{3}, <

*α*>

*is the time-dependent ensemble average molecular polarizability along the laser electric field, and*

_{t}*n*is the refractive index. Here, <

*α*>

*=<*

_{t}**ê**·

**α**̿·

**ê**>

*=<*

_{t}*e*>

*α*_{i}_{ij}e_{j}*, where*

_{t}**ê**is the electric field polarization,

**α**̿ 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

**α**̿ 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**=

**x**̂

*E*+

_{x}**ẑ**

*E*for a particular molecular orientation. Therefore, for an ensemble of molecular orientations in the space-fixed field,

_{z}*n*

^{2}=1+4

*πN*(<

*e*

^{2}

*>*

_{x}

_{t}*α*+<

_{xx}*e*

^{2}

*>*

_{z}

_{t}*α*, or

_{zz}*α*=

*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

^{2}

*θ*>

_{t=-∞}1/3 (early time ensemble averages are just averages over solid angle) and

*α*

^{(3)}

*is the fourth rank molecular polarizability tensor and the ‘prompt’ contribution is proportional to the square of the field envelope amplitude |*

_{ijkl}**E**(

*t*)|

^{2}. As will be seen, the angular ensemble averages < >

*consist of a constant term (from at*

_{t}*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

*n*

_{2}is the isotropic nonlinear index of refraction,

*I*(

*t*) is the laser intensity, and where <sin

^{4}

*θ*>=

_{t-∞}=8/15, <sin

^{2}2

*θ*>=

_{t-∞}=8/15, and <cos

^{4}

*θ*>

_{t=-∞}=1/5.

^{2}

*θ*; its time-dependent classical ensemble average is calculated as <cos

^{2}

*θ*>

*=Tr(*

_{t}**ρ**(

*t*)⊗cos

^{2}

*θ*)=

*ρ*〈l cos

_{kl}^{2}

*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*, Δ

*E*

_{elec}/

*k*≫ 1 and Δ

_{B}T*E*

_{vib}/

*k*≫1, where Δ

_{B}T*E*

_{elec}and Δ

*E*

_{vib}are the energies of the first excited electronic and vibrational states and

*k*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,

_{B}**ρ**(

*t*)=

**ρ**

^{(0)}+

**ρ**

^{(1)}(

*t*), where

*𝓱*=-

**p**·

**E**, where

**p**=

**α**̿·

**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*is the dephasing rate between states

_{kl}*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 <cos

^{2}

*θ*>

*deviates from the unperturbed <cos*

_{t}^{2}

*θ*>

_{t=-∞}=Tr(

**ρ**

^{(0)}(

*t*)⊗cos

^{2}

*θ*)=1/3 by small amounts.

*𝓱*,

*ρ*

^{(0)}]

*=(*

_{kl}*ρ*

_{l}^{(0)}-

*ρ*

_{k}^{(0)})

*𝓱*, where

_{kl}*ρ*

_{l}^{(0)}≡

*ρ*

_{ll}^{(0)}(no sum),

*ρ*

_{k}^{(0)}≡

*ρ*

_{kk}^{(0)}(no sum), and

*𝓱*=-Δ

_{kl}*α*|

**E**|

^{2}〈

*k*|cos

^{2}θ|

*l*〉-

*α*

_{⊥}

*δ*|

_{kl}**E**|

^{2}, where

*δ*is the unity matrix. As the rotational eigenstates are the spherical harmonics, |

_{kl}*j*,

*m*〉=

*Y*(

_{jm}*θ*,

*φ*), the matrix element 〈

*k*|cos

^{2}

*θ*|

*l*〉=〈

*j*,

*m*|cos

^{2}

*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.

*Q*

^{m}_{j,j}′=〈

*j*,

*m*|cos

^{2}

*θ*|

*j*′,

*m*〉 m and noting from above that [

*𝓱*,

*ρ*

^{(0)}]

*is nonvanishing only for the non-diagonal components*

_{kl}*j*′=

*j*+2 and

*j*′=

*j*-2, Eq. (4) becomes

*ω*

_{j,j-2}=(

*E*-

_{j}*E*

_{j-2})/

*ħ*=4

*πcB*(2

*j*-1) and

*D*is the statistical weighting factor for the number of nuclear spin states, |

_{j}*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 2

*π*/

*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.

^{2}

*θ*>

*=1/3+*

_{t}*ρ*

_{kl}*Q*=1/3+(

_{lk}*ρ*

^{(1)}

_{j,j-2,m}+(

*ρ*

^{(1)}

_{j,j-2,m})*)

*Q*

^{m}

_{j,j-2}or

*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})/[(2

*j*-1)

^{2}(2

*j*+1)(2

*j*-3)]. Summing Eq. (6) from

*m*=-

*j*through

*m*=

*j*eliminates

*m*to yield

*ρ*

^{(0)}

*=*

_{j}*ρ*

^{(0)}

_{j,m}.

*F*(in erg/cm

^{2}),

*ε*

^{2}(

*τ*)=(8

*π*/

*c*)

*Fδ*(

*τ*) and Eq. (7) becomes

*τ*and peak field amplitude

_{p}*E*

_{0}, and for times

*t*≫

*τ*, the upper limit of the integral in Eq. (7) can be taken to ∞ yielding

_{p}*τ*|≤

*τ*

_{0}and

*ε*

^{2}(

*τ*)=0 for |

*τ*|>

*τ*

_{0}. For times |

*t*|≤

*τ*

_{0}, Eq. (7) becomes

*t*>

*τ*

_{0}it becomes

## 4. Experimental results and discussion

*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. 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, N_{2}, and O_{2} by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B **14**, 650–660 (1997). [CrossRef]

19. J.-F. Ripoche, G. Grillon, B. Prade, M. France, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of n_{2} in air,” Opt. Commun. **135**, 310–314 (1997). [CrossRef]

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]

_{2}and N

_{2}O at 4.4 atm. The 110 fs pump pulse energy was 95 µJ (peak intensity I=6.7×10

^{13}W/cm

^{2}) for Ar, 60 µJ (peak intensity I=4.2×10

^{13}W/cm

^{2}) for N

_{2}, and 20 µJ (peak intensity I=1.4×10

^{13}W/cm

^{2}) for N

_{2}O. 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

**15**, 7458–7467 (2007). [CrossRef] [PubMed]

*t*=0. The later-peaking shifts for N

_{2}and N

_{2}O are the result of the delayed response of the molecular alignment. In Fig. 3, we plot calculations of Δ

*n*(

*t*) using Eqs. (2), (10) and (11), which consider alignment effects only, for (a) N

_{2}and (b) N

_{2}O, using

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]

_{2}and N

_{2}O there is good agreement between the calculations and measurements. The conclusion is that the orientational effects in N

_{2}O and N

_{2}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, N_{2}, and O_{2} by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B **14**, 650–660 (1997). [CrossRef]

_{2}, 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 H

_{2}and D

_{2}under our conditions, the prompt

*n*response is larger in amplitude than the alignment response.

_{2}I27. H. Harde and D. Grischkowsky, “Coherent transients excited by subpicosecond pulses of terahertz radiation,” J. Opt. Soc. Am. B **8**, 1642–1651 (1991). [CrossRef]

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. F. Rosca-Pruna and M. J. J. Vrakking, “Experimental Observation of Revival Structures in Picosecond Laser-Induced Alignment of I_{2},” Phys. Rev. Lett. **87**, 153902 (2001). [CrossRef] [PubMed]

*t*~0 the impulsive alignment torque administered by the short pump pulse locks the relative phases of the rotational states in the superposition wavepacket

*ω*=

_{j}*E*/

_{j}*ħ*=2

*πcBj*(

*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*). However, when time passes through values

_{j}t*t*=

*q*𝕋, where

*q*is an integer and 𝕋 = (2

*cB*)

^{-1}, the phases become

*ω*=

_{j}t*qπj*(

*j*+1), an integer multiple of 2

*π*. 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*=

_{j}t*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]

^{2}

*θ*>

*is described by a sum of terms with factors exp((*

_{t}*iω*

_{j,j-2}-

*γ*

_{j,j-2})

*t*). The frequency difference between 2 successive terms is Δ

*ω*=

*ω*

_{j+1,j-1}-

*ω*

_{j,j-2}=8

*πcB*, so that the full revival period of the alignment occurs at times

*t*=

*q*T. Peaks that are π out of phase occur at

*q*, or

*t*=

*q*𝕋/2; these are the half-revival peaks representing anti-alignment discussed above. The peak width is approximately ~𝕋/

*N*, where

_{rot}*N*is the number of rotational states contributing to the wavepacket, which is a function of temperature and pump pulse energy, duration and bandwidth.

_{rot}^{2}

*θ*>

*-1/3 for 6.4 atm of N*

_{t}_{2}for time windows centered at

*t*=0 through

*t*=1.25𝕋 in 0.25𝕋 steps, where for

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]

^{13}W/cm

^{2}. Note that preceding the half and full revivals by an interval

*δt*~𝕋/

*N*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 <cos

_{rot}^{2}θ>

*t*-1/3 for N

_{2}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 4.5×10

^{7}erg/cm

^{2}. 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

*k*/

_{B}T*Bhc*≫1, which is the case for N

_{2},

*j*

_{max}~(

*k*/

_{B}T*Bhc*)

^{1/2}~10. For large

*j*

_{max}, the frequency width of the thermal distribution of rotational states available for pumping is Δ

*ω*

*~*

_{rot}*k*/

_{B}T*ħ*~4×10

^{13}s

^{-1}. By comparison, our pump laser frequency bandwidth corresponding to Δ

*λ*~10 nm is Δ

_{laser}*ω*~3×10

_{laser}^{13}s

^{-1}. As Δ

*ω*~Δ

_{laser}*ω*, 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 N

_{rot}_{2}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

**15**, 7458–7467 (2007). [CrossRef] [PubMed]

_{2}the format of Fig. 4 for a cell pressure of 5.1 atm and pump pulse energy 40 µJ (peak intensity I=2.7×10

^{13}W/cm

^{2}). Figure 5(a) shows probe beam-centre lineouts of the measured O

_{2}alignment <cos

^{2}θ>

*-1/3 for time windows centered at*

_{t}*t*=0 through

*t*=1.25𝕋 in 0.25𝕋 steps, where

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]

_{2}, the

_{2}are comparable in amplitude to the full- and half-revivals. This derives from the I

_{N}=0 spin of the

^{16}

_{8}O 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 N

_{2}. Figure 5(b) shows the same revivals plotted versus space and time, and 5(c) shows calculations of alignment (using

**13**, 813–829 (1976). [CrossRef]

*γ*

_{j,j-2}=γ=4.31×

^{10}10 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 O

_{2}, as for N

_{2}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 N

_{2}, the rotational state wavepacket contribution is thermally limited rather than pump laser spectrum limited.

_{2}O, a linear molecule with a much greater moment of inertia (and lower rotational constant,

**13**, 813–829 (1976). [CrossRef]

^{13}W/cm

^{2}) 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 N

_{2}O 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

**13**, 813–829 (1976). [CrossRef]

*j*

_{max}~22, so the approximation Δ

*ω*~

_{rot}*k*/

_{B}T*ħ*applies even better, with Δ

*ω*>Δ

_{laser}*ω*as in the N

_{rot}_{2}and O

_{2}cases.

_{2}and H

_{2}, 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 D

_{2}for a pump energy of 65 µJ (peak intensity I=4.4×10

^{13}W/cm

^{2}), 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

**13**, 813–829 (1976). [CrossRef]

_{2}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]

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

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]

*δ*Φ

_{shot}/

*N*

_{int}

^{1/2})(ΔΦ)

^{-1}~2%.

_{2}, 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 D

_{2}

_{N}=1, so that for the D

_{2}molecule, even

*j*states are twice as populated as odd

*j*. Dominant coupled Δ

*j*=2 states near

*j*

_{max}are therefore

*j*=2 and

*j*=4, which have a frequency spacing

*j*=0 and

*j*=2, with

^{13}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*=3.44×10

^{13}s

^{-1}is immediately identified as

*ω*

_{2,0}, and allows us to extract the value of

*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 N

_{2}, O

_{2}, and N

_{2}O.

_{2}are shown in Figure 8, for a pump energy of 65µJ (peak intensity I=4.4×10

^{13}W/cm

^{2}) 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 D

_{2}. 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

**13**, 813–829 (1976). [CrossRef]

**13**, 813–829 (1976). [CrossRef]

_{2}, 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 D

_{2}, the prompt peak in Fig. 8(a) is dominated by the isotropic

*n*

_{2}

*I*contribution.

_{2}wavepacket, the most populated state is estimated to be

*j*

_{max}~2. The spin of the H nucleus is

*j*states are 3 times more populated than odd

*j*states. The likely coupled Δ

*j*=2 states are

*j*=2 and

*j*=0, with

_{2}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 H

_{2}revivals are generated by beating of the two low amplitude

*j*=0 and

*j*=2 rotational modes. For H

_{2}, the error in the extracted phase shift is estimated to be (

*δ*Φ

_{shot}/

*N*

_{int}

^{1/2})(ΔΦ)

^{-1}~15%.

*t*past the pump pulse, the integral portion can be written ∫

^{t}_{-∞}

*dτI*(

*τ*)exp(-

*iω*

_{j,j-2})≈

*I*̃(

*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 Δ

*j*=2 beat frequencies lying within the pump pulse spectrum.

*t*=0.5𝕋 revival in N

_{2}O 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 1.46×10

^{10}s

^{-1}atm

^{-1}.

## 5. Conclusions

## Acknowledgements

## References and Links

1. | H. Stapelfeldt and T. Seideman, “Aligning molecules with strong laser pulses,” Rev. Mod. Phys. |

2. | R. Velotta, N. Hay, M. B. Mason, M. Castillejo, and J. P. Marangos, “High-Order Harmonic Generation in Aligned Molecules,” Phys. Rev. Lett. |

3. | N. Hay, R. Velotta, M. Lein, R. de Nalda, E. Heesel, M. Castillejo, and J. P. Marangos, “High-order harmonic generation in laser-aligned molecules,” Phys. Rev. A |

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. |

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 |

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 |

7. | P. M. Felker, J. S. Baskin, and A. H. Zewail, “Rephasing of collisionless molecular coherence in large molecules,” J. Phys. Chem. |

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. |

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 |

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, N |

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 |

12. | F. Rosca-Pruna and M. J. J. Vrakking, “Experimental Observation of Revival Structures in Picosecond Laser-Induced Alignment of I |

13. | M. A. Duguay and J. W. Hansen, “An ultrafast light gate,” Appl. Phys. Lett. |

14. | E. P. Ippen and C. V. Shank, “Picosecond response of a high-repetition-rate CS |

15. | J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of coherent transient birefringence in CS |

16. | R. Righini, “Ultrafast optical Kerr effect in liquids and solids,” Science |

17. | V. Renard, O. Faucher, and B. Lavorel, “Measurement of laser-induced alignment of molecules by cross defocusing,” Opt. Lett. |

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. |

19. | J.-F. Ripoche, G. Grillon, B. Prade, M. France, E. Nibbering, R. Lange, and A. Mysyrowicz, “Determination of the time dependence of n |

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. |

21. | S. Varma, Y.-H. Chen, H. M. Milchberg, and I. Alexeev, “Single-Shot Time Resolved Measurement of Molecular Alignment in Laser-Irradiated Gases,” in |

22. | S. Varma, Y.-H. Chen, I. Alexeev, R. Moon, and H. M. Milchberg, “Single-shot time resolved measurement of molecular alignment in laser-irradiated gases: application to ‘self-channeled’ plasma columns,” presented at |

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 |

24. | K. Y. Kim, I. Alexeev, and H. M. Milchberg, “Single-shot supercontinuum spectral interferometry,” Appl. Phys. Lett. |

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. |

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 |

27. | H. Harde and D. Grischkowsky, “Coherent transients excited by subpicosecond pulses of terahertz radiation,” J. Opt. Soc. Am. B |

28. | W. Demtroder, |

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. |

30. | K. Kim, I. Alexeev, and H. Milchberg, “Single-shot measurement of laser-induced double step ionization of helium,” Opt. Express |

**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 H_{2}, D_{2}, N_{2}, O_{2}, and N_{2}O," Opt. Express **15**, 11341-11357 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-18-11341

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### References

- H. Stapelfeldt and T. Seideman, "Aligning molecules with strong laser pulses," Rev. Mod. Phys. 75, 543-557 (2003). [CrossRef]
- 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]
- N. Hay, R. Velotta, M. Lein, R. de Nalda, E. Heesel, M. Castillejo, and J. P. Marangos, "High-order harmonic generation in laser-aligned molecules," Phys. Rev. A 65, 053805 (2002) [CrossRef]
- 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]
- J. Itatani, J. Levesque, D. Zeidler, Hiromichi Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, D. M. Villeneuve, "Tomographic imaging of molecular orbitals," Nature 432, 867-871 (2004) [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, 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]
- P. W. Dooley, I. V. Litvinyuk, KevinF. 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]
- 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]
- M. A. Duguay and J. W. Hansen, "An ultrafast light gate," Appl. Phys. Lett. 15,192-194 (1969). [CrossRef]
- 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]
- 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]
- R. Righini, "Ultrafast optical Kerr effect in liquids and solids," Science 262, 1386-1390 (1993). [CrossRef] [PubMed]
- 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]
- 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]
- J.-F. Ripoche, G. Grillon, B. Prade, M. France, E. Nibbering, R. Lange, A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997). [CrossRef]
- 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]
- S. Varma, Y.-H. Chen, H. M. Milchberg, and I. Alexeev, "Single-Shot Time Resolved Measurement of Molecular Alignment in Laser-Irradiated Gases," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JFC3
- S. Varma, Y.-H. Chen, I. Alexeev, R. Moon, and H. M. Milchberg, "Single-shot time resolved measurement of molecular alignment in laser-irradiated gases: application to ‘self-channeled’ plasma columns," presented at 48th Annual Meeting of the Division of Plasma Physics, American Physical Society, Philadelphia, 30 Oct.-3 Nov. 2006, abstract #NO3.015.
- 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]
- K. Y. Kim, I. Alexeev, and H. M. Milchberg, "Single-shot supercontinuum spectral interferometry," Appl. Phys. Lett. 81, 4124-4126 (2002). [CrossRef]
- 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]
- 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]
- H. Harde and D. Grischkowsky, "Coherent transients excited by subpicosecond pulses of terahertz radiation," J. Opt. Soc. Am. B 8, 1642-1651 (1991). [CrossRef]
- W. Demtroder, Molecular Physics, Wiley-VCH (Weinheim, 2005). [CrossRef]
- 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]
- 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]

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