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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 6 — Mar. 20, 2006
  • pp: 2520–2532
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Optimal single-pulse excitation of rotational impulsive molecular phase modulation

Omid Masihzadeh, Mark Baertschy, and Randy A. Bartels  »View Author Affiliations


Optics Express, Vol. 14, Issue 6, pp. 2520-2532 (2006)
http://dx.doi.org/10.1364/OE.14.002520


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Abstract

The full transient macroscopic linear optical susceptibility tensor induced in a transiently aligned molecular gas by a single, linearly polarized intense alignment pulse is studied. We determine the optimal properties of the pulse that forms the rotational wave packet. Significantly, we demonstrate that the optimal pulse for phase modulation differs from the optimal alignment pulse. Finally, we show that the limited information about rotational wave packets obtained by measuring the linear optical susceptibility can be augmented by also measuring the time-varying nonlinear optical susceptibilities.

© 2006 Optical Society of America

1. Introduction

In recent years, molecular phase modulation of light has been vigorously investigated as a method for optical pulse manipulation [1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

, 2

2. S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022 (1997). [CrossRef]

, 3

3. S. E. Harris and A. V. Sokolov, “Subfemtosecond pulse generation by molecular modulation,” Phys. Rev. Lett. 81, 2894–2897 (1998). [CrossRef]

, 4

4. N. Zhavoronkov and G. Korn, “Generation of single intense short optical pulses by ultrafast molecular phase modulation,” Phys. Rev. Lett. 88, (2002). [CrossRef] [PubMed]

] where coherent Raman motion driven by strong laser fields produces an ultrafast, time-varying index of refraction in a molecular gas. The transient index of refraction directly modifies the temporal phase of an optical pulse, producing large changes in the optical spectrum. In one approach, adiabatic molecular modulation (AMM), two nanosecond laser pulses (slightly detuned from the vibrational frequency of a molecule) create a Raman coherence, present only during the interaction pulses, that generates very high order, phase-coherent sidebands on these pulses that can span from the IR to the UV [2

2. S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022 (1997). [CrossRef]

, 3

3. S. E. Harris and A. V. Sokolov, “Subfemtosecond pulse generation by molecular modulation,” Phys. Rev. Lett. 81, 2894–2897 (1998). [CrossRef]

] and which have been used to study sub-femtosecond synthesis of laser pulses. In the impulsive molecular modulation (IMM) limit a pump pulse with a duration shorter than a molecular rotational [1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

] or vibrational period [4

4. N. Zhavoronkov and G. Korn, “Generation of single intense short optical pulses by ultrafast molecular phase modulation,” Phys. Rev. Lett. 88, (2002). [CrossRef] [PubMed]

, 5

5. R. A. Bartels, T. C. Weinacht, S. R. Leone, H. C. Kapteyn, and M. M. Murnane, “Nonresonant control of multi-mode molecular wave packets at room temperature,” Phys. Rev. Lett. 88(3) (2002). [CrossRef] [PubMed]

, 6

6. R. A. Bartels, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Impulsive stimulated Raman scattering of molecular vibrations using nonlinear pulse shaping,” Chem. Phys. Lett. 374, 326–333 (2003). [CrossRef]

] creates a wave packet leading to ultrafast index transients that persist in the gas until collisional dephasing destroys the wave packet coherence. A separate, time-delayed probe pulse may then be phase-modulated by propagation in the time-varying optical susceptibility.

Impulsive molecular phase modulation of optical pulses was first demonstrated with ultrafast transients in the index of refraction of a molecular gas that occur during rotational revivals [1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

]. Rotational IMM relies on wave packets formed in a gas of linear molecules that produces a time dependent index of refraction [7

7. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of Coherent Transient Birefringence in Cs2 Vapor,” Phys. Rev. Lett. 34(21), 1299–1302 (1975). [CrossRef]

, 8

8. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence Arising from Reorientation of Polarizability Anisotropy of Molecules in Collisionless Gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

]. In order to produce an ultrafast index transient, a collection of anisotropic molecules is made to align along the polarization direction of an intense, linearly polarized laser pulse [7

7. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of Coherent Transient Birefringence in Cs2 Vapor,” Phys. Rev. Lett. 34(21), 1299–1302 (1975). [CrossRef]

, 8

8. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence Arising from Reorientation of Polarizability Anisotropy of Molecules in Collisionless Gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

, 9

9. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

, 1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

]. Although the molecules quickly go out of alignment, a periodic rephasing of the wave packet brings the molecules back into alignment at regular intervals. During these so-called rotational revivals the wave packet evolves through states where the molecules are both aligned and anti-aligned (i.e. perpendicular to) with the polarization of the laser pulse — producing the observed ultrafast temporal perturbation of the optical susceptibility.

The optimization of this so-called field-free molecular alignment has been the subject of intense study in recent years [9

9. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

]. In 2001, Averbukh et al. demonstrated that successive alignment of a rigid rotor model leads to angular squeezing [10

10. I. S. Averbukh and R. Arvieu, “Angular focusing, squeezing, and rainbow formation in a strongly driven quantum rotor,” Phys. Rev. Lett. 8716, 163,601 (2001).

] — with the promise of fully aligning linear molecules in a molecular gas [11

11. M. Leibscher, Averbukh I. S., and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, (2003). [CrossRef] [PubMed]

]. The predicted improvement in molecular alignment has been verified [12

12. C. Z. Bisgaard, M. D. Poulsen, E. Peronne, S. S. Viftrup, and H. Stapelfeldt, “Observation of enhanced field-free molecular alignment by two laser pulses,” Phys. Rev. Lett. 92,) (2004). [CrossRef] [PubMed]

]. Experimentally, rotational revivals are traditionally probed by observing the transient birefringence of the molecular gas [7

7. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of Coherent Transient Birefringence in Cs2 Vapor,” Phys. Rev. Lett. 34(21), 1299–1302 (1975). [CrossRef]

] forming the basis of a commonly used method of molecular spectroscopy [13

13. P. M. Felker, “Rotational Coherence Spectroscopy - Studies of the Geometries of Large Gas-Phase Species by Picosecond Time-Domain Methods,” J. Phys. Chem. 96, 7844–7857 (1992). [CrossRef]

, 14

14. M. Gruebele and A. H. Zewail, “Femtosecond Wave Packet Spectroscopy - Coherences, the Potential, and Structural Determination,” J. Chem. Phys. 98(2), 883–902 (1993). [CrossRef]

]. The transient birefringence induced in the molecular gas can also be used for phase matching nonlinear interactions [15

15. R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346–348 (2003). [CrossRef] [PubMed]

, 16

16. K. Hartinger, S. Nirmalgandhi, J. Wilson, and R. A. Bartels, “Efficient nonlinear frequency conversion with a dynamically structured nonlinearity,” Opt. Express 13, 6919–6930 (2005); http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-6919. [CrossRef] [PubMed]

].

In this paper, we study the optimal conditions for preparation of rotational IMM with a single linearly-polarized alignment pump pulse. Specifically, we consider a pump-probe experiment in which the pump (i.e., an alignment) pulse is used to form a rotational wave packet in a molecular gas. The time-delay between the pump and probe pulses is adjusted so that the probe pulse encounters a particular portion of a revival feature. As we show in this paper, if the direction of the linearly polarized probe pulse coincides with (or is perpendicular to) that of the alignment pulse, the polarization state of the probe pulse is not altered with propagation in the transiently aligned gas. Thus, the polarization directions parallel with and perpendicular to the alignment pulse polarization are eigenpolarizations of the transient optical medium. When the probe pulse is polarized along an eigenpolarization direction, the transient refractive index of the gas can substantially modulate the optical spectrum of the probe pulse [1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

].

Our goal is to find conditions on the pump pulse that maximize the potential for spectral changes in the probe pulse. This requires maximizing the rate at which the linear susceptibility of the gas varies in time. To do so, we isolate the component of the rotational wave packet that contributes to the linear susceptibility. Then in a series of numerical simulations, the pulse parameters that maximize the components of the rotational wave packet relevant to IMM are determined. Significantly, the simulations show that the optimal conditions required for IMM and molecular alignment do not coincide.

In this study we consider only a single, transform-limited alignment pulse and investigate the effects of varying the duration and peak intensity of that pulse as well as the initial temperature of the gas. The strength of the rotational wave packet is generally proportional to the peak intensity of the pulse. However, if the alignment pulse intensity becomes too large, there is a high probability of molecular dissociation. Thus, our optimization approach is to consider an intensity just below the dissociation threshold and to vary the pulse duration (or equivalently pulse energy for a fixed beam size) to determine the strength of the IMM produced at the fractional revivals.

2. Theory

Rotational wave packet excitation and dynamics has been well studied [17

17. B. Friedrich and D. Herschbach, “Polarization of molecules induced by intense nonresonant laser fields,” J. Phys. Chem. 99, 15,686–15,693 (1995). [CrossRef]

, 18

18. B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623–4626 (1995). [CrossRef] [PubMed]

, 7

7. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of Coherent Transient Birefringence in Cs2 Vapor,” Phys. Rev. Lett. 34(21), 1299–1302 (1975). [CrossRef]

] and recently reviewed by Seideman and Stapelfeldt [9

9. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

]. For clarity, we provide a concise overview of the subject. Interaction of a short, linearly polarized pump pulse with a gas of anisotropic, linear molecules forms a rotational wave packet that can be interpreted as the time-varying orientational probability of the gas

Gθϕt=JogJoP(Jo)Mo=JoJoψJoMoθϕt2,
(1)

where P(Jo ) gives the (thermal) distribution of molecules with total angular momentum quantum number Jo prior to interaction with the laser pulse, and gJo is a degeneracy factor determined by the spin statistics [19

19. G. Herzberg, Molecular spectra and molecular structure (Krieger, 1950).

]. For a CO2 molecule, gJo will be zero for odd and unity for even Jo values, respectively. Here

ψJoMoθϕt=JM=JJCJMJ0M0YJM(θ,ϕ)eiEJtħ
(2)

is the wave function that describes a molecule that was initially in the rotational state specified by quantum numbers Jo and Mo and the expansion coefficients, cJ M J0 Mo, are found through numerical integration of Schrödinger’s equation as described in [20

20. J. Ortigoso, M. Rodriguez, M. Gupta, and B. Friedrich, “Time evolution of pendular states created by the interaction of molecular polarizability with a pulsed nonresonant laser field,” J. Chem. Phys. 110(8), 3870–3875 (1999). [CrossRef]

].

When first formed, the wave packet corresponds to a partial molecular alignment of the gas along the direction of the polarization of the pulse. This alignment is transient, but as the wave packet evolves the molecules will periodically come back into alignment. Full revivals of the rotational wave packet occur at time intervals of TR=h2B, where h is Planck’s constant and B is the rotational constant of the molecule.

This time dependent orientational probability due to the evolving rotational wave packet leads to transients in the optical properties of the gas [8

8. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence Arising from Reorientation of Polarizability Anisotropy of Molecules in Collisionless Gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

]. These transients are given by the induced polarization density averaged over the molecular ensemble. The corresponding macroscopic polarization density is written as [21

21. J. D. Le Grange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567–605 (1987).

]

P=NμIε0χ̿E,
(3)

where χ̿ defines the macroscopic susceptibility tensor of the gas, N is the molecular density of the gas, μI = αIJEJ is the linear approximation of the dipole moment induced by the applied electric field EA , and αIJ is the linear polarizability tensor given in the principle coordinate frame (indices [I,J]) of the molecule.

The induced polarization density given in Eq. (3) defines the macroscopic linear susceptibility tensor that is attributable to the orientational average of the molecular polarizability and is given by

χij=Nε0αIJij,
(4)

where the orientational average is defined by

αIJij=αIJ4πriIrjJGθϕtdΩ,
(5)

and rqQ are the direction cosines between the molecular (indices I,J) and lab (indices i, j) frames [21

21. J. D. Le Grange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567–605 (1987).

].

We restrict our discussion to linear molecules in a dilute gas aligned by a linearly polarized femtosecond laser pulse. Under these conditions M (defined along the polarization direction) is conserved in the alignment pulse interaction and only transitions of the total angular momentum, J, meeting the selection rules ΔJ = ±2,0 are allowed [?]. Conservation of M guarantees an azimuthally symmetric orientational probability distribution, i.e., G(θ,ϕ,t) → G(θ,t). Therefore, we can expand the orientational average of the molecular polarizability tensor in terms of azimuthally independent (i.e. M = 0) spherical harmonics,

αIJij=2παIJLbL(t)0πriIrjJYL0(θ)sinθdθ.
(6)

Using the above expression, we can relate the molecular alignment created by the interaction of the molecules and the alignment pulse, as given by Eq. (1), to the macroscopic linear optical susceptibility tensor of the gas, χ̿. This definition separates the temporal and spatially varying components of the orientational probability, allowing the susceptibility of the molecular gas to be written in terms of the time-varying expansion coefficients bL (t) with the spatial integrals being performed analytically.

For alignment, as opposed to orientation, the bL for odd integer values of L are identically zero, which is indicative of the symmetry of molecular alignment [22

22. B. Friedrich, P. P.D., and D. Herschbach, “Alignment and orientation of rotationally cool molecules,” J. Phys. Chem. 95, 8118–8129 (1991). [CrossRef]

]. Moreover, only the L = 2 coefficients contribute to the linear susceptibility tensor. Higher-order terms appear in the ensemble averaged nonlinear susceptibility tensor as demonstrated with the third-harmonic macroscopic third-order susceptibility tensor for aligned linear molecules [16

16. K. Hartinger, S. Nirmalgandhi, J. Wilson, and R. A. Bartels, “Efficient nonlinear frequency conversion with a dynamically structured nonlinearity,” Opt. Express 13, 6919–6930 (2005); http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-6919. [CrossRef] [PubMed]

]. The details of the derivations are deferred to Appendices A and B. The non-zero linear susceptibility tensor components are given by

χxx=χyy=Nε0[α+13Δα]23ΔαNε0π5b2(t)
(7)

and

χzz=Nε0[α+13Δα]+43ΔαNε0π5b2(t),
(8)

where Δα = α - α is the polarizability anisotropy of the linear molecule.

At thermal equilibrium, the gas molecules are randomly oriented, i.e., G(θ,ϕ) = const., and the optical response of the gas is isotropic, indicating that bL = 0 for L > 0. The isotropic susceptibility tensor is given by the average polarizability along the principle directions of the molecule and can be expressed as

χ̿iso=N3ε0(α+2α)I,
(9)

where I is the 3 × 3 identity matrix.

Rotational wave packet revivals were first observed through a time-varying birefringence that was detected as an amplitude modulation of a probe pulse that was scanned through the revivals [7

7. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of Coherent Transient Birefringence in Cs2 Vapor,” Phys. Rev. Lett. 34(21), 1299–1302 (1975). [CrossRef]

]. A measurement of the birefringence will not fully resolve the transient susceptibility tensor. The susceptibility tensor for the ensemble of aligned molecules is diagonal and can be separated into isotropic, stationary anisotropic, and transient birefringent components as χ̿ = {χxx , χyy , χzz } = χ̿ iso + χ̿ st + χ̿ tr (t). The coordinates {x,y,z} are defined such that z is along the polarization direction of the alignment pulse. The structure of the transient susceptibility tensor is determined by the symmetry of G(θ,ϕ,t) where the azimuthal symmetry of G is reflected in the equality χ tr(t)xx = χ tr(t)yy.

For weak alignment pulses, the static birefringence is vanishingly small. However, we show that this term becomes significant for strong rotational wave packet excitation. Both the static and transient birefringent tensors have the form of a uniaxial crystal with a 2:1 ratio between the extraordinary (z) and ordinary (x and y) directions. The magnitude of the birefringence of the static and transient anisotropic susceptibilities is dependent on the alignment pulse that produces the molecular alignment. In the following section, we study the optimization of the strength of the transient birefringence for a single, linearly-polarized alignment pulse.

3. Results and discussion

We found optimal alignment pulses by first choosing an alignment pulse peak intensity, I 0, and then studying the field-free alignment and transient phase modulation that persisted in the molecular gas after the alignment pulse as a function of both temperature and pulse duration τp (FWHM of a Gaussian pulse). The model molecule used in the calculations was comparable to CO2, so only even values of J and J 0 were used. The peak intensity of the pulse was chosen to be below the dissociation threshold of linear molecules typically used in experiments [1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

].

Fig. 1. Expansion coefficients, bL (t), as defined in App. A for the transient molecular alignment induced in CO2 gas at 300 K for (a) τp = 20 fs, I 0 = 2 × 1013 W cm-2, (b) τp = 200 fs, I 0 = 2 × 1013 W cm-2, and (c) τp = 140 fs, I 0 = 6 × 1013 W cm-2. The trace colors correlate with the order of expansion for L = 2 (blue), L = 4 (green), L = 6 (red), and L = 8 (cyan).

Examples of expansion coefficients bL (t) for the time-varying molecular alignment are shown in Fig. 1. The time-varying molecular alignment for the alignment pulses of Fig. 1 is shown in Fig. 2. A case of weak molecular alignment is shown in Fig. 1(a) for alignment induced by a 20 fs laser pulse. The b 2(t) expansion coefficient exhibits the quarter, half, and full rotational revival structure (at intervals of TR /4) typically seen for symmetric, linear molecules [7

7. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of Coherent Transient Birefringence in Cs2 Vapor,” Phys. Rev. Lett. 34(21), 1299–1302 (1975). [CrossRef]

]. No significant deviations from zero are seen in the coefficients for higher L. Expansion coefficients resulting from the alignment pulse with the pulse duration that gives optimal frequency shifting (i.e. max{dn/dt}) for peak intensities of 2 × 1013 W cm-2 and 6 × 1013 W cm-2 are shown in Figs. 1(b) and (c), respectively. For these more energetic alignment pulses, the average value of b 2(t) is raised above zero, indicating partial molecular alignment between the fraction revivals as has been reported by Machholm [23

23. M. Machholm, “Postpulse alignment of molecules robust to thermal averaging,” J. Chem. Phys. 115, 10,724–10,730 (2001). [CrossRef]

]. The time-dependent linear susceptibility tensor is completely specified by the b 2(t) coefficient, as indicated by Eq. (7) and Eq. (8). Clearly, a linear optical measurement (e.g., with polarization rotation [7

7. J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of Coherent Transient Birefringence in Cs2 Vapor,” Phys. Rev. Lett. 34(21), 1299–1302 (1975). [CrossRef]

] or phase modulation [1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

]) will provide limited information about the rotational wave packet for energetic alignment pulses.

Fig. 2. Time-dependent orientational probability densities, G(θ,ϕ,t) generated by the expansion coefficients shown in Fig 1. The animations show the evolution of the orientational probability density with time. The moving circle in the right panels highlights the corresponding alignment cosine of the rotational wave packet. [Media 1]

Our interest here is to find the alignment pulse that optimally phase modulates a probe pulse that is timed to coincide with a revival feature. The temporal phase modulation accumulated by a probe pulse propagating in a molecular gas of length L is given by ϕ(t)=12k0Lχtr(t)aa, where the transient susceptibility is given in Appendix B and a corresponds to an eigenpolarization direction of the macroscopic aligned susceptibility tensor (i.e., either along or perpendicular to the alignment polarization direction). For optimal IMM, we need to specify what spectral changes are desired. To spectrally tune the probe pulse central frequency, we must maximize dϕ(t)dtddtcos2θ(t), whereas to optimize spectral broadening, we must maximize d2ϕ(t)dt2d2dt2cos2θ(t) [1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

].

We studied the optimal single-pulse alignment for a fixed peak intensity of I 0 = 2 × 1013 W cm-2 as a function of pulse duration for three different initial temperatures of the gas. The results of these calculations are shown in Fig. 3. The magnitude of the static alignment that persists between rotational revivals, shown in Fig. 3(a), rises with decreasing temperature, but has a weak dependence on the alignment pulse duration. The maximum transient alignment shows a similar trend. The sum of the static and transient alignments gives the total field-free alignment, which is known to improve rapidly with decreasing molecular gas temperature [9

9. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

]. We also see that for low-energy alignment pulses (i.e., τp < 100 fs), the maximum frequency shifting is independent of temperature and the spectral broadening improves with increasing temperature, which is consistent with predictions of perturbation theory [8

8. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence Arising from Reorientation of Polarizability Anisotropy of Molecules in Collisionless Gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

, 24

24. M. Comstock, V. V. Lozovoy, and M. Dantus, “Rotational wavepacket revivals for phase modulation of ultrafast pulses,” Chem. Phys. Lett. 372, 739–744 (2003). [CrossRef]

]. However, with more energetic pulses, the lower temperature gas is better at both frequency shifting and spectral broadening. A summary of the optimal pulse durations for the alignment conditions presented in Fig. 3 is presented in Table 1.

Table 1. Optimal pulse durations for alignment, static & transient birefringence, and spectral modifications as a function of temperature.

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The spectrum of the transient optical response given by the coefficients |aωJ|2, as defined in Appendix B, is plotted in Fig. 4 as a function of alignment pulse duration at 300 K for a peak intensity of 2 × 1013 W cm-2. We see that with increasing pulse energy (i.e., pulse duration) there is an increase in the high-frequency transient spectral content. The peak spectral width coincides with maximum spectral broadening as shown in Fig. 3(d) for a pulse duration of ~220 fs. When further increasing the pulse duration, the alignment pulse intensity profile lacks the proper spectral content to excite the high frequency Raman rotational transitions [6

6. R. A. Bartels, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Impulsive stimulated Raman scattering of molecular vibrations using nonlinear pulse shaping,” Chem. Phys. Lett. 374, 326–333 (2003). [CrossRef]

] — reducing the high-frequency spectral content and lowering the angular momentum of the wave packet. With pulse elongation, both the spectral width and centroid begin to decrease, reducing the ability of the transient susceptibility to phase modulate the probe pulse.

Fig. 3. Maximum static birefringence (a), transient birefringence (b), slope for frequency shifting (c), and chirp rate for spectral broadening (d) versus alignment pulse duration for a peak pulse intensity of Io = 2 × 1013W cm-2. Results for initial temperatures of 20 K(square), 100 K(circle), and 300 K(triangle) are shown.
Fig. 4. Alignment spectrum given by |aωJ|2 as defined in App. B. at room temperature (300 K) with variation of the alignment pulse duration for a peak intensity of Io = 2 × 1013Wcm-2.

The results of the temperature sweep indicate a weak dependence on the optimal alignment pulse durations as the temperature of the molecular gas is varied. Moreover, substantial modulation of the probe pulse spectrum requires an interaction length of 10’s of cm and near atmospheric pressures [1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

]. Typically, cooled gas experiments are performed in a supersonic expansion in vacuum, leading to extremely short interaction lengths and low gas densities. As a result of these experimental difficulties with gas cooling, we restrict the remainder of our discussion to the optimization of IMM alignment pulses at room temperature.

Results from our calculations when varying peak intensity and alignment pulse duration (all at room temperature) are shown in Fig. 5. The combination of Figs. 5 (a) & (b) indicate the pulse duration for optimal molecular alignment is extremely sensitive to the peak intensity of the pulse. Figs. 5 (c) & (d) also show a strong dependence on the optimal alignment pulse duration both for maximizing frequency shifting and for maximizing spectral broadening. A summary of the optimal pulse durations for the conditions shown in Fig. 5 are shown in Table 2. These results demonstrate that the optimal pulse duration for alignment is not the same as required for either maximizing frequency shifting or broadening of a probe pulse.

4. Conclusion

Through numerical propagation of Schrödinger’s equation, we have studied the preparation of rotational wave packets for the purposes of optimizing spectral changes in a probe pulse that propagates through the induced time-varying susceptibility. We find that optimizing IMM has a weak dependence on the initial gas temperature in contrast with the enhancement of molecular alignment that accompanies a reduction in the initial gas temperature. Furthermore, we demonstrate that the single-pulse pump pulse parameters that maximize IMM are different for the conditions of optimal alignment. In relating the rotational wave packet to the linear optical susceptibility tensor through a spherical harmonic expansion, we found that spherical harmonic orders correlate with optical nonlinearity order. Moreover, the fractional revivals present for order L can be shown to have a simple relationship to the spectral content of bL (t). Obtaining more detailed information about the wave packet requires measuring nonlinear susceptibilities.

Fig. 5. Maximum static birefringence (a), transient birefringence (b), slope for frequency shifting (c), and chirp rate for spectral broadening (d) versus alignment pulse duration at room temperature (300 K) for peak pulse intensities of I 0 = 6× (square), I 0 = 3× (triangle), I 0 = 2× (circle), and I 0 = 1 × 1013Wcm-2 (x).

Table 2. Optimal pulse durations for alignment, static & transient birefringence, and spectral modifications as a function of pulse peak intensity.

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Appendix A

In this appendix, we derive the coefficients for expanding the orientational probability density in terms of spherical harmonics. The probability of finding a molecule aligned in a given direction may be expressed in terms of a spherical harmonic expansion by defining

Gθϕt=L,mbLm(t)YLm(θ,ϕ),
(10)

where YLm (θ, ϕ) are the spherical harmonics. The coefficients bLm (t) for the expansion of the orientational probability are expressed as

bLm=04πY¯LmGθϕtdΩ,
(11)

where the overbar implies the complex conjugate. This simplifies to the form

bLm(t)=(1)L2L+14πJo,MoP(Jo)JMcJMJoMoJ,McJMJoMoeiωJ,JtL0J0J0LmJMJM,

where ωJ′,J = (EJ′ - EJ )/ħ. Conservation of M collapses the sums over M and M′ to just the M = M 0 and the M′ = M 0 terms. The final Clebsch-Gordan coefficient then becomes 〈L-mJ′M 0|JM 0〉 which is nonzero only for m = 0. Therefore only m = 0 terms appear in the spherical harmonic expansion giving the required azimuthal symmetry of G(θ,ϕ). With this simplification, the expression for the expansion coefficients is

bL(t)=(1)L2L+14πJo,MoP(Jo)JJcJMoJoMocJMoJoMoeiωJ,JtL0J0J0L0JMoJMo

The temporal evolution of the bL (t) expansion coefficients relate the rotational wave packet dynamics to the quantum beat frequencies of the excited rotational coherences. Writing J′ = J + ΔJ, we can express these beat frequencies as a set of frequency combs indexed by the total angular momentum quantum number

ħB1ωJ,ΔJ=ΔJ(ΔJ+1)+2ΔJJħB1(ωoff+ωbeatJ),
(12)

where ħB -1 ω off = ΔJJ + 1) defines the offset beat frequency of the comb and ħB -1 ω beat = 2ΔJ is the comb spacing. The ΔJ dependence of ω off prevents the combs for successive ΔJ’s from overlapping spectrally. As a result, the revival fraction that occurs at time TR /f scales with |ΔJ| as f = 2J|/2+1. The Clebsch-Gordan triangle rule requires that |ΔJ| ≤ L which leads to a maximum revival fraction, f, for a given L to be constrained to f = 2L/2+1.

Appendix B

The components of the linear susceptibility tensor of an aligned molecular gas (assuming gas densities low enough that molecule-molecule interactions and local field corrections are negligible) are given by

χij(1)=Nε0αIJij=Nε04παlab(θ,ϕ)ijGϕθtdΩ.
(13)

Here, α̿ lab = R-1(θ,ϕ)α̿R(θ,ϕ) = riIrjJαIJ , R(θ,ϕ) is the Euler rotation matrix between the molecular and laboratory frames, i, j are coordinates in lab frame, I, J are coordinates in the principle molecular frame, rqQ are the direction cosines between the molecular and laboratory coordinates, and summation over repeated indices is implied.

In general χxx , χyy , χzz , χxy = χyx , χxz = χzx, and χyz = χyz must be evaluated. For linear molecules, with the (diagonal) polarizability tensor components α̿ = {α , α , α } in the principle frame of the molecule, the nonzero susceptibility tensor components are given by

χxx=Nε0L,mbLm(t)4π{α+Δαsin2θsin2ϕ}YLm(θ,ϕ)dΩ,
(14)
χyy=Nε0L,mbLm(t)4π{α+Δαcos2θsin2ϕ}YLm(θ,ϕ)dΩ,and
(15)
χzz=Nε0L,mbLm(t)4π{α+Δαcos2(θ)}YLm(θ,ϕ)dΩ.
(16)

in the principle frame of the alignment.

Due to the azimuthal symmetry of G(θ), we consider only m = 0 and the transient linear susceptibility tensor components simplify to

χxx=χyy=Nε0[α+13Δα]23ΔαNε0π5b2(t),
χzz==Nε0[α+13Δα]+43ΔαNε0π5b2(t),

and

χxy=χyx=χxz=χzx=χyz=χzy=0.

The expansion coefficient can also be written as bL(t)=2L+14πPL(cosθ)(t), which allows one to express the tensor component in terms of the alignment cosine as

χxx=χyy=12Nε0[α+α]12Nε0Δαcos2θ(t)
χzz=Nε0α+Nε0Δαcos2θ(t),

where the transient molecular alignment cosine is given by [8

8. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence Arising from Reorientation of Polarizability Anisotropy of Molecules in Collisionless Gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

, 1

1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

, 9

9. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

]

cos2θ(t)=J0P(J0)M0=J0J0ψJ0M0cos2θψJ0M0=43π5b2(t)+13.
(17)

The linear susceptibility tensor components of transiently aligned linear molecules can be separated into a superposition of the isotropic (χ̿ iso), statically birefringent (χ̿ st), and transiently birefringent (χ̿ tr) components written as χ̿ = χ̿ iso + χ̿ st + χ̿ tr. The isotropic susceptibility tensor is identical to the susceptibility for un-aligned molecules at thermal equilibrium and is given by χ̿iso=N3ε0(α+2α)I where χiso=N3ε0(α+2α) is the isotropic susceptibility and I is the 3 × 3 identity matrix.

Both the static and transient susceptibility components are present only in an aligned molecular gas, i.e., when b 2 ≠ 0. We can rewrite the second-order expansion coefficient as b 2(t) = b2dc + 2(t), where b2dc and 2(t) represent the time independent and the time-varying components of the alignment, respectively. Note that the molecular alignment is conveniently represented by the discrete Fourier spectrum defined by b 2(t) = 2R{∑J JeJt}, where ωJħ = E J+2 - EJ ; clearly, b2dc = a 0. With this definition, the static birefringent susceptibility tensor produced by the molecular alignment is diagonal and can be written as χ̿st={χstxx,χstyy,χstzz}=13ΔχstM,, where M = {-1,-1,2} is a diagonal matrix and

Δχst=2π5ΔαNε0b2dc=3N2ε0Δα(cos2θ(t)¯13),

with cos2θ(t)¯ defining the time-averaged value of the alignment cosine. Note that we have defined the birefringence of the anisotropic medium as Δχ = χzz - χxx . Similarly, the time-varying portion of the linear susceptibility tensor is given as χ̿tr={χtr(t)xx,χtr(t)yy,χtr(t)zz}=13Δχtr(t)M with

Δχtr(t)=2π5ΔαNε0b˜2(t)=N2ε0Δα(cos2θ(t)cos2(t)¯).

Acknowledgments

References and links

1.

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, (2002). [PubMed]

2.

S. E. Harris and A. V. Sokolov, “Broadband spectral generation with refractive index control,” Phys. Rev. A 55, R4019–R4022 (1997). [CrossRef]

3.

S. E. Harris and A. V. Sokolov, “Subfemtosecond pulse generation by molecular modulation,” Phys. Rev. Lett. 81, 2894–2897 (1998). [CrossRef]

4.

N. Zhavoronkov and G. Korn, “Generation of single intense short optical pulses by ultrafast molecular phase modulation,” Phys. Rev. Lett. 88, (2002). [CrossRef] [PubMed]

5.

R. A. Bartels, T. C. Weinacht, S. R. Leone, H. C. Kapteyn, and M. M. Murnane, “Nonresonant control of multi-mode molecular wave packets at room temperature,” Phys. Rev. Lett. 88(3) (2002). [CrossRef] [PubMed]

6.

R. A. Bartels, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Impulsive stimulated Raman scattering of molecular vibrations using nonlinear pulse shaping,” Chem. Phys. Lett. 374, 326–333 (2003). [CrossRef]

7.

J. P. Heritage, T. K. Gustafson, and C. H. Lin, “Observation of Coherent Transient Birefringence in Cs2 Vapor,” Phys. Rev. Lett. 34(21), 1299–1302 (1975). [CrossRef]

8.

C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, “Birefringence Arising from Reorientation of Polarizability Anisotropy of Molecules in Collisionless Gases,” Phys. Rev. A 13, 813–829 (1976). [CrossRef]

9.

H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]

10.

I. S. Averbukh and R. Arvieu, “Angular focusing, squeezing, and rainbow formation in a strongly driven quantum rotor,” Phys. Rev. Lett. 8716, 163,601 (2001).

11.

M. Leibscher, Averbukh I. S., and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, (2003). [CrossRef] [PubMed]

12.

C. Z. Bisgaard, M. D. Poulsen, E. Peronne, S. S. Viftrup, and H. Stapelfeldt, “Observation of enhanced field-free molecular alignment by two laser pulses,” Phys. Rev. Lett. 92,) (2004). [CrossRef] [PubMed]

13.

P. M. Felker, “Rotational Coherence Spectroscopy - Studies of the Geometries of Large Gas-Phase Species by Picosecond Time-Domain Methods,” J. Phys. Chem. 96, 7844–7857 (1992). [CrossRef]

14.

M. Gruebele and A. H. Zewail, “Femtosecond Wave Packet Spectroscopy - Coherences, the Potential, and Structural Determination,” J. Chem. Phys. 98(2), 883–902 (1993). [CrossRef]

15.

R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346–348 (2003). [CrossRef] [PubMed]

16.

K. Hartinger, S. Nirmalgandhi, J. Wilson, and R. A. Bartels, “Efficient nonlinear frequency conversion with a dynamically structured nonlinearity,” Opt. Express 13, 6919–6930 (2005); http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-6919. [CrossRef] [PubMed]

17.

B. Friedrich and D. Herschbach, “Polarization of molecules induced by intense nonresonant laser fields,” J. Phys. Chem. 99, 15,686–15,693 (1995). [CrossRef]

18.

B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623–4626 (1995). [CrossRef] [PubMed]

19.

G. Herzberg, Molecular spectra and molecular structure (Krieger, 1950).

20.

J. Ortigoso, M. Rodriguez, M. Gupta, and B. Friedrich, “Time evolution of pendular states created by the interaction of molecular polarizability with a pulsed nonresonant laser field,” J. Chem. Phys. 110(8), 3870–3875 (1999). [CrossRef]

21.

J. D. Le Grange, M. G. Kuzyk, and K. D. Singer, “Effects of order on nonlinear optical processes in organic molecular materials,” Mol. Cryst. Liq. Cryst. 150b, 567–605 (1987).

22.

B. Friedrich, P. P.D., and D. Herschbach, “Alignment and orientation of rotationally cool molecules,” J. Phys. Chem. 95, 8118–8129 (1991). [CrossRef]

23.

M. Machholm, “Postpulse alignment of molecules robust to thermal averaging,” J. Chem. Phys. 115, 10,724–10,730 (2001). [CrossRef]

24.

M. Comstock, V. V. Lozovoy, and M. Dantus, “Rotational wavepacket revivals for phase modulation of ultrafast pulses,” Chem. Phys. Lett. 372, 739–744 (2003). [CrossRef]

OCIS Codes
(190.2640) Nonlinear optics : Stimulated scattering, modulation, etc.
(190.5650) Nonlinear optics : Raman effect
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(260.1440) Physical optics : Birefringence
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Ultrafast Optics

History
Original Manuscript: January 31, 2006
Revised Manuscript: March 10, 2006
Manuscript Accepted: March 11, 2006
Published: March 20, 2006

Citation
Omid Masihzadeh, Mark Baertschy, and Randy A. Bartels, "Optimal single-pulse excitation of rotational impulsive molecular phase modulation," Opt. Express 14, 2520-2532 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2520


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References

  1. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, "Phase modulation of ultrashort light pulses using molecular rotational wave packets," Phys. Rev. Lett. 88, (2002). [PubMed]
  2. S. E. Harris and A. V. Sokolov, "Broadband spectral generation with refractive index control," Phys. Rev. A 55, 4019-4022 (1997). [CrossRef]
  3. S. E. Harris and A. V. Sokolov, "Subfemtosecond pulse generation by molecular modulation," Phys. Rev. Lett. 81, 2894-2897 (1998). [CrossRef]
  4. N. Zhavoronkov and G. Korn, "Generation of single intense short optical pulses by ultrafast molecular phase modulation," Phys. Rev. Lett. 88, (2002). [CrossRef] [PubMed]
  5. R. A. Bartels, T. C. Weinacht, S. R. Leone, H. C. Kapteyn, and M. M. Murnane, "Nonresonant control of multimode molecular wave packets at room temperature," Phys. Rev. Lett. 88(2) (2002). [CrossRef] [PubMed]
  6. R. A. Bartels, S. Backus, M. M. Murnane, and H. C. Kapteyn, "Impulsive stimulated Raman scattering of molecular vibrations using nonlinear pulse shaping," Chem. Phys. Lett. 374, 326-333 (2003). [CrossRef]
  7. 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]
  8. C. H. Lin, J. P. Heritage, T. K. Gustafson, R. Y. Chiao, and J. P. McTague, "Birefringence Arising from Reorientation of Polarizability Anisotropy of Molecules in Collisionless Gases," Phys. Rev. A 13, 813-829 (1976). [CrossRef]
  9. H. Stapelfeldt and T. Seideman, "Colloquium: Aligning molecules with strong laser pulses," Rev. Mod. Phys. 75, 543-557 (2003). [CrossRef]
  10. I. S. Averbukh and R. Arvieu, "Angular focusing, squeezing, and rainbow formation in a strongly driven quantum rotor," Phys. Rev. Lett. 8716, 163,601 (2001).
  11. M. Leibscher, I. S. Averbukh, and H. Rabitz, "Molecular alignment by trains of short laser pulses," Phys. Rev. Lett. 90, (2003). [CrossRef] [PubMed]
  12. C. Z. Bisgaard, M. D. Poulsen, E. Peronne, S. S. Viftrup, and H. Stapelfeldt, "Observation of enhanced field-free molecular alignment by two laser pulses," Phys. Rev. Lett. 92, (2004). [CrossRef] [PubMed]
  13. P. M. Felker, "Rotational Coherence Spectroscopy - Studies of the Geometries of Large Gas-Phase Species by Picosecond Time-Domain Methods," J. Phys. Chem. 96, 7844-7857 (1992). [CrossRef]
  14. M. Gruebele and A. H. Zewail, "FemtosecondWave Packet Spectroscopy - Coherences, the Potential, and Structural Determination," J. Chem. Phys. 98, 883-902 (1993). [CrossRef]
  15. R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, "Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases," Opt. Lett. 28, 346-348 (2003). [CrossRef] [PubMed]
  16. K. Hartinger, S. Nirmalgandhi, J. Wilson, and R. A. Bartels, "Efficient nonlinear frequency conversion with a dynamically structured nonlinearity," Opt. Express 13, 6919-6930 (2005); http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-18-6919. [CrossRef] [PubMed]
  17. B. Friedrich and D. Herschbach, "Polarization of molecules induced by intense nonresonant laser fields," J. Phys. Chem. 99, 15,686-15,693 (1995). [CrossRef]
  18. B. Friedrich and D. Herschbach, "Alignment and Trapping of Molecules in Intense Laser Fields," Phys. Rev. Lett. 74, 4623-4626 (1995). [CrossRef] [PubMed]
  19. G. Herzberg, Molecular spectra and molecular structure (Krieger, 1950).
  20. J. Ortigoso, M. Rodriguez, M. Gupta, and B. Friedrich, "Time evolution of pendular states created by the interaction of molecular polarizability with a pulsed nonresonant laser field," J. Chem. Phys. 110, 3870-3875 (1999). [CrossRef]
  21. J. D. Le Grange, M. G. Kuzyk, and K. D. Singer, "Effects of order on nonlinear optical processes in organic molecular materials," Mol. Cryst. Liq. Cryst. 150, 567-605 (1987).
  22. B. Friedrich, P. P.D., and D. Herschbach, "Alignment and orientation of rotationally cool molecules," J. Phys. Chem. 95, 8118-8129 (1991). [CrossRef]
  23. M. Machholm, "Postpulse alignment of molecules robust to thermal averaging," J. Chem. Phys. 115, 10,724-10,730 (2001). [CrossRef]
  24. M. Comstock, V. V. Lozovoy, and M. Dantus, "Rotational wavepacket revivals for phase modulation of ultrafast pulses," Chem. Phys. Lett. 372, 739-744 (2003). [CrossRef]

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