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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 9 — Apr. 26, 2010
  • pp: 8990–8997
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Alignment structures of rotational wavepacket created by two strong femtosecond laser pulses

Hongyan Jiang, Chengyin Wu, He Zhang, Hongbing Jiang, Hong Yang, and Qihuang Gong  »View Author Affiliations


Optics Express, Vol. 18, Issue 9, pp. 8990-8997 (2010)
http://dx.doi.org/10.1364/OE.18.008990


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Abstract

We experimentally and theoretically study the alignment structures of the rotational wavepacket created by linear molecules and two strong femtosecond laser pulses. In the experiment, we observe that the alignment structures depend on the time delay between the two laser pulses. In the theory, we find that the alignment structures are composed of the self-coupling term and the cross-coupling term. The contributions of these two terms are separately calculated. Their coherent superposition reproduces the alignment structures observed in the experiment.

© 2010 OSA

1. Introduction

2. Theory

Fig. 1
Fig. 1 Rotational wavepacket created by linear molecules and two strong femtosecond laser pulses.
shows the generation process of molecular rotational wavepacket created by linear molecules and two strong femtosecond laser pulses. A molecule initially in a rotational state |J0,M0 is irradiated by a strong femtosecond laser pulse. The laser-molecule interaction generates a rotational wavepacket via a series of Raman processes. The wavepacket can be expanded as:
Ψ1(t)=JiPJ0,Jiexp(iEJit)|Ji ,
(1)
whereEJi and PJ0,Ji are respectively the energy eigenvalue and the coefficient of rotational state |Ji. By numerically solving the time-dependant Schrödinger equation for the linear molecule irradiated by strong femtosecond laser pulses, PJ0,Jiis obtained at the end of the laser pulse. The formulas are written in atomic units. In the expression, we omit the quantum number M0 because the cylindrical symmetry of the interaction keeps the azimuthal quantum number unchanged. By a kind of phase matching between different rotational states, the evolution of the wavepacket results in transient molecular alignment.

During the evolution of the wavepacket, a second laser pulse is applied. Each Ji component of the wavepacket Ψ1(t) produces a sub-wavepacket ΦJi(t), which can be written as:
ΦJi(t)=PJ0,Jiexp(iEJiΔt)JPJi,Jexp[iEJ(tΔt)]|J ,
(2)
where Δt is the delay time between the two laser pulses. Coherent superpositions of these sub-wavepackets generate the final wavepacket Ψ2(t), which can be written as:

Ψ2(t)=JiΦJi(t)=JiPJ0,Jiexp(iEJiΔt)JPJi,Jexp[iEJ(tΔt)]|J  .
(3)

For the wavepacket created by two laser pulses with a time delay Δt, the alignment <cos2θ> can be written as:
cos2θ(t)=J0,M0gJ0,M0<cos2θ>(t)J0,M0=J0,M0gJ0,M0Ψ2(t)|cos2θ|Ψ2(t)           =J0,M0gJ0,M0JiΦJi(t)|cos2θ|ΦJi(t)+J0,M0gJ0,M0JiJi'JiΦJi(t)|cos2θ|ΦJi'(t) ,
(4)
where gJ0,M0 is the Boltzmann averaging factor of different initial state |J0,M0. This expression demonstrates that, for the rotational wavepacket created by two laser pulses, the alignment structures are composed of the self-coupling term J0,M0gJ0,M0JiΦJi(t)|cos2θ|ΦJi(t) and the cross-coupling term J0,M0gJ0,M0JiJi'JiΦJi(t)|cos2θ|ΦJi'(t). Substituting Eq. (2) into the self-coupling term, we find that the phase factor exp(iEJiΔt) is totally cancelled. The self-coupling term can therefore be expanded into a superposition of a series of AJcos[ωJ(tΔt)+φJ] withωJ the Raman frequency. The expansion means that the self-coupling term produces one series of structures. These structures are called basal alignment structures. They locate at nTr/2+Δt with Tr the molecular rotational period. Unlike the self-coupling term, the phase factor exp(iEJiΔt) cannot be cancelled in the cross-coupling term. Additional phase factor exp[i(EJi'EJi)Δt] is left. This phase factor has a modulation effect on the alignment structures. Depending on the sign of the additional phase, the cross-coupling term can be expanded into a series of AJcos[ωJt+φJ]and a series of AJcos[ωJ(t2Δt)+φJ]. Based on these expansions, we know that the cross-coupling term generate two series of structures. These structures are called modulation structures and locate at t=nTr/2 and nTr/2+2Δt, respectively. For different time delays between the two laser pulses, the superposition of the basal alignment structures and the modulation structures generates different alignment structures, which have been observed in the experiment.

3. Experiment

Fig. 2
Fig. 2 Experimentally measured alignment signal of N2O created by (a) a single laser pulse and (b)-(e) double laser pulses with different time delays. The arrow marks the time the second laser pulse is applied.
shows the alignment signal for the rotational wavepacket of N2O created by a single or double laser pulses. The initial N2O molecules are at room temperature. Nonadiabatic rotational excitation generates a coherent rotational wavepacket after the irradiation of one strong femtosecond laser pulse. The laser intensity is estimated to be 6.0 × 1012 W/cm2 through measuring its pulse duration and focusing size. Fig. 2(a) shows the alignment structure for the rotational wavepacket of N2O created by a single laser pulse. The classical rotational period Tr of N2O is 39.9 ps. At the half or full rotational period, all the components of the wavepacket evolve in phase and the wavepacket exhibits transient alignment or antialignment. The alignment structure fully revives every rotational period. Fig. 2(b)-2(e) show the alignment structures for the rotational wavepacket of N2O created by two laser pulses with different time delays. These observations demonstrate that the alignment structure created by the first aligning laser pulse can be annihilated, enhanced or split by the second laser pulse depending on the time delay between the two laser pulses. In the following section, we will analyze these different alignment structures using the model we propose in the theory section.

4. Results and discussion

Fig. 3(a)
Fig. 3 (a) Evolution of rotational wavepacket of N2O created by two laser pulse with ∆t = 8.21 ps. Black line represents the measured alignment signal, red line represents the theoretical superposition of the self-coupling term and the cross-coupling term, (b) basal alignment structures described by the self-coupling term and (c) modulation structures described by the cross-coupling term.
shows the experimentally measured alignment structures for the rotational wavepacket of N2O created by two laser pulses with ∆t = 8.21 ps. There are three series of alignment structures. One locates at nTr/2+Δt. The other two locate respectively at nTr/2 and nTr/2+2Δt. Fig. 3(b) exhibits the basal alignment structures described by the self-coupling term. These structures located at nTr/2+Δt. Fig. 3(c) exhibits the modulation structures described by the cross-coupling term. There are two series of structures, respectively located at nTr/2and nTr/2+2Δt. The coherent superposition of the self-coupling term and the cross-coupling term is shown with the red line in Fig. 3(a). It reproduces the three series of alignment structures observed in the experiment for the rotational wavepacket of N2O created by two laser pulses with ∆t = 8.21 ps.Fig. 4(a)
Fig. 4 The same as Fig. 3 but ∆t = 9.99 ps.
shows the experimentally measured alignment structures for the rotational wavepacket of N2O created by two laser pulses with ∆t = 9.99 ps. There are only two series of alignment structures when the time delay is approximately a quarter of a rotational period. One locates at nTr/2+Δt. The other locates at nTr/2. Fig. 4(b) exhibits the basal alignment structures described by the self-coupling term. The structures locate at nTr/2+Δtand their shapes are similar to those shown in Fig. 3(b). Fig. 4(c) exhibits the modulation structures described by the cross-coupling term. Different from the two series of modulation structures shown in Fig. 3(c), there is only one series of modulation structures when the time delay is around a quarter rotational period. These structures locate at nTr/2. The superposition of the self-coupling term and the cross-coupling term also reproduces the two series of alignment structures observed in the experiment for the rotational wavepacket of N2O created by two laser pulses with ∆t = 9.99 ps.The calculations above show that the self-coupling term generates one series of basal alignment structures, whose shapes have no relationship with the time delay between the two laser pulses. However, the modulation structures produced by the cross-coupling term sensitively depend on the time delay. Two series of modulation structures are generated for a general time delay. The coherent superposition of the basal alignment structures and the modulation structures generates three series of alignment structures. However, when the time delay is around a quarter rotational period, the two series of modulation structures produced by the cross-coupling term will overlap in time and merge into one series of structures. With this time delay, the coherent superposition of the self-coupling term and the cross-coupling term generates two series of alignment structures. These calculations are consistent with our experimental observations.

Fig. 5(a)
Fig. 5 The same as Fig. 3 but ∆t = 19.93 ps.
shows the experimentally measured alignment structures for the rotational wavepacket of N2O created by two laser pulses with ∆t = 19.93 ps. The alignment structures are annihilated when the time delay is around a half rotational period. Fig. 5(b) and 5(c) show the basal alignment structures and the modulation structures described by the self-coupling term and the cross-coupling term, respectively. Based on these calculations, we know that the basal alignment structures and the modulation structures have opposite phases in addition to the temporal overlap. The superposition cancels each other out and leads to the annihilation of the alignment structures.

Fig. 6(a)
Fig. 6 The same as Fig. 3 but ∆t = 39.82 ps.
shows the experimentally measured alignment structures for the rotational wavepacket of N2O created by two laser pulses with ∆t = 39.82 ps. There are only one series of alignment structures when the time delay is approximately a full rotational period. Moreover, the alignment structures are enhanced after the irradiation of the two laser pulses. Fig. 6(b) and 6(c) exhibit the basal alignment structures and the modulation structures described by the self-coupling term and the cross-coupling term, respectively. These calculations show that the basal alignment structures and the modulation structures are in phase in addition to the temporal overlap. Their superposition therefore enhances the alignment structures and is consistent with the experimental observation.

5. Conclusion

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant Nos. 10974005, 10634020, and 10821062 and the National Basic Research Program of China under grant No. 2006CB921601.

References and links

1.

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

2.

P. W. Dooley, I. V. Litvinyuk, K. 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(2), 023406 (2003). [CrossRef]

3.

V. Renard, M. Renard, S. Guérin, Y. T. Pashayan, B. Lavorel, O. Faucher, and H. R. Jauslin, “Postpulse molecular alignment measured by a weak field polarization technique,” Phys. Rev. Lett. 90(15), 153601 (2003). [CrossRef] [PubMed]

4.

N. Xu, C. Wu, Y. Gao, H. Jiang, H. Yang, and Q. Gong, “Measurement of the field-free alignment of diatomic molecules,” J. Phys. Chem. A 112(4), 612–617 (2008). [CrossRef] [PubMed]

5.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432(7019), 867–871 (2004). [CrossRef] [PubMed]

6.

D. Pavicić, K. F. Lee, D. M. Rayner, P. B. Corkum, and D. M. Villeneuve, “Direct measurement of the angular dependence of ionization for N2, O2, and CO2 in intense laser fields,” Phys. Rev. Lett. 98(24), 243001 (2007). [CrossRef] [PubMed]

7.

B. K. McFarland, J. P. Farrell, P. H. Bucksbaum, and M. Gühr, “High harmonic generation from multiple orbitals in N2.,” Science 322(5905), 1232–1235 (2008). [CrossRef] [PubMed]

8.

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460(7258), 972–977 (2009). [CrossRef] [PubMed]

9.

M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavicic, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, and P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320(5882), 1478–1482 (2008). [CrossRef] [PubMed]

10.

K. Yoshii, G. Miyaji, and K. Miyazaki, “Measurement of molecular rotational temperature in a supersonic gas jet with high-order harmonic generation,” Opt. Lett. 34(11), 1651–1653 (2009). [CrossRef] [PubMed]

11.

N. Xu, C. Wu, J. Huang, Z. Wu, Q. Liang, H. Yang, and Q. Gong, “Field-free alignment of molecules at room temperature,” Opt. Express 14(12), 4992–4997 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-12-4992. [CrossRef] [PubMed]

12.

I. S. Averbukh and R. Arvieu, “Angular focusing, squeezing, and rainbow formation in a strongly driven quantum rotor,” Phys. Rev. Lett. 87(16), 163601 (2001). [CrossRef] [PubMed]

13.

M. Leibscher, I. S. Averbukh, and H. Rabitz, “Enhanced molecular alignment by short laser pulses,” Phys. Rev. A 69(1), 013402 (2004). [CrossRef]

14.

C. Z. Bisgaard, M. D. Poulsen, E. Péronne, S. S. Viftrup, and H. Stapelfeldt, “Observation of enhanced field-free molecular alignment by two laser pulses,” Phys. Rev. Lett. 92(17), 173004 (2004). [CrossRef] [PubMed]

15.

C. Z. Bisgaard, S. S. Viftrup, and H. Stapelfeldt, “Alignment enhancement of a symmetric top molecule by two short laser pulses,” Phys. Rev. A 73(5), 053410 (2006). [CrossRef]

16.

C. Horn, M. Wollenhaupt, M. Krug, T. Baumert, R. de Nalda, and L. Banares, “Adaptive control of molecular alignment,” Phys. Rev. A 73(3), 031401 (2006). [CrossRef]

17.

E. Hertz, A. Rouzee, S. Guerin, B. Lavorel, and O. Faucher, “Optimization of field-free molecular alignment by phase-shaped laser pulses,” Phys. Rev. A 75(3), 031403 (2007). [CrossRef]

18.

A. Rouzée, E. Hertz, B. Lavorel, and O. Faucher, “Towards the adaptive optimization of field-free molecular alignment,” J. Phys. B 41(7), 074002 (2008). [CrossRef]

19.

K. F. Lee, I. V. Litvinyuk, P. W. Dooley, M. Spanner, D. M. Villeneuve, and P. B. Corkum, “Two-pulse alignment of molecules,” J. Phys. B 37(3), L43–L48 (2004). [CrossRef]

20.

K. F. Lee, E. A. Shapiro, D. M. Villeneuve, and P. B. Corkum, “Coherent creation and annihilation of rotational wave packets in incoherent ensembles,” Phys. Rev. A 73(3), 033403 (2006). [CrossRef]

21.

Y. Li, P. Liu, S. Zhao, Z. Zeng, R. Li, and Z. Xu, “Active control of the molecular rotational wave packet using two laser pulses,” Chem. Phys. Lett. 475(4-6), 183–187 (2009). [CrossRef]

22.

S. Fleischer, I. Sh. Averbukh, and Y. Prior, “Selective alignment of molecular spin isomers,” Phys. Rev. Lett. 99(9), 093002 (2007). [CrossRef] [PubMed]

23.

Y. Gao, C. Wu, N. Xu, G. Zeng, H. Jiang, H. Yang, and Q. Gong, “Manipulating molecular rotational wave packets with strong femtosecond laser pulses,” Phys. Rev. A 77(4), 043404 (2008). [CrossRef]

24.

G. Zeng, C. Wu, H. Jiang, Y. Gao, N. Xu, and Q. Gong, “Rotational wave packet of N2O created by two strong femtosecond laser pulses,” J. Phys. B 42(16), 165508 (2009). [CrossRef]

25.

C. Wu, G. Zeng, Y. Gao, N. Xu, L. Y. Peng, H. Jiang, and Q. Gong, “Controlling molecular rotational population by wave-packet interference,” J. Chem. Phys. 130(23), 231102 (2009). [CrossRef] [PubMed]

26.

C. Wu, G. Zeng, H. Jiang, Y. Gao, N. Xu, and Q. Gong, “Molecular rotational excitation by strong femtosecond laser pulses,” J. Phys. Chem. A 113(40), 10610–10618 (2009). [CrossRef] [PubMed]

27.

A. S. Meijer, Y. Zhang, D. H. Parker, W. J. van der Zande, A. Gijsbertsen, and M. J. J. Vrakking, “Controlling rotational state distributions using two-pulse stimulated Raman excitation,” Phys. Rev. A 76(2), 023411 (2007). [CrossRef]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(270.6620) Quantum optics : Strong-field processes
(320.7100) Ultrafast optics : Ultrafast measurements

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: March 8, 2010
Revised Manuscript: April 7, 2010
Manuscript Accepted: April 10, 2010
Published: April 14, 2010

Citation
Hongyan Jiang, Chengyin Wu, He Zhang, Hongbing Jiang, Hong Yang, and Qihuang Gong, "Alignment structures of rotational wavepacket created by two strong femtosecond laser pulses," Opt. Express 18, 8990-8997 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-8990


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References

  1. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]
  2. P. W. Dooley, I. V. Litvinyuk, K. 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(2), 023406 (2003). [CrossRef]
  3. V. Renard, M. Renard, S. Guérin, Y. T. Pashayan, B. Lavorel, O. Faucher, and H. R. Jauslin, “Postpulse molecular alignment measured by a weak field polarization technique,” Phys. Rev. Lett. 90(15), 153601 (2003). [CrossRef] [PubMed]
  4. N. Xu, C. Wu, Y. Gao, H. Jiang, H. Yang, and Q. Gong, “Measurement of the field-free alignment of diatomic molecules,” J. Phys. Chem. A 112(4), 612–617 (2008). [CrossRef] [PubMed]
  5. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432(7019), 867–871 (2004). [CrossRef] [PubMed]
  6. D. Pavicić, K. F. Lee, D. M. Rayner, P. B. Corkum, and D. M. Villeneuve, “Direct measurement of the angular dependence of ionization for N2, O2, and CO2 in intense laser fields,” Phys. Rev. Lett. 98(24), 243001 (2007). [CrossRef] [PubMed]
  7. B. K. McFarland, J. P. Farrell, P. H. Bucksbaum, and M. Gühr, “High harmonic generation from multiple orbitals in N2.,” Science 322(5905), 1232–1235 (2008). [CrossRef] [PubMed]
  8. O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460(7258), 972–977 (2009). [CrossRef] [PubMed]
  9. M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavicic, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve, and P. B. Corkum, “Laser-induced electron tunneling and diffraction,” Science 320(5882), 1478–1482 (2008). [CrossRef] [PubMed]
  10. K. Yoshii, G. Miyaji, and K. Miyazaki, “Measurement of molecular rotational temperature in a supersonic gas jet with high-order harmonic generation,” Opt. Lett. 34(11), 1651–1653 (2009). [CrossRef] [PubMed]
  11. N. Xu, C. Wu, J. Huang, Z. Wu, Q. Liang, H. Yang, and Q. Gong, “Field-free alignment of molecules at room temperature,” Opt. Express 14(12), 4992–4997 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-12-4992 . [CrossRef] [PubMed]
  12. I. S. Averbukh and R. Arvieu, “Angular focusing, squeezing, and rainbow formation in a strongly driven quantum rotor,” Phys. Rev. Lett. 87(16), 163601 (2001). [CrossRef] [PubMed]
  13. M. Leibscher, I. S. Averbukh, and H. Rabitz, “Enhanced molecular alignment by short laser pulses,” Phys. Rev. A 69(1), 013402 (2004). [CrossRef]
  14. C. Z. Bisgaard, M. D. Poulsen, E. Péronne, S. S. Viftrup, and H. Stapelfeldt, “Observation of enhanced field-free molecular alignment by two laser pulses,” Phys. Rev. Lett. 92(17), 173004 (2004). [CrossRef] [PubMed]
  15. C. Z. Bisgaard, S. S. Viftrup, and H. Stapelfeldt, “Alignment enhancement of a symmetric top molecule by two short laser pulses,” Phys. Rev. A 73(5), 053410 (2006). [CrossRef]
  16. C. Horn, M. Wollenhaupt, M. Krug, T. Baumert, R. de Nalda, and L. Banares, “Adaptive control of molecular alignment,” Phys. Rev. A 73(3), 031401 (2006). [CrossRef]
  17. E. Hertz, A. Rouzee, S. Guerin, B. Lavorel, and O. Faucher, “Optimization of field-free molecular alignment by phase-shaped laser pulses,” Phys. Rev. A 75(3), 031403 (2007). [CrossRef]
  18. A. Rouzée, E. Hertz, B. Lavorel, and O. Faucher, “Towards the adaptive optimization of field-free molecular alignment,” J. Phys. B 41(7), 074002 (2008). [CrossRef]
  19. K. F. Lee, I. V. Litvinyuk, P. W. Dooley, M. Spanner, D. M. Villeneuve, and P. B. Corkum, “Two-pulse alignment of molecules,” J. Phys. B 37(3), L43–L48 (2004). [CrossRef]
  20. K. F. Lee, E. A. Shapiro, D. M. Villeneuve, and P. B. Corkum, “Coherent creation and annihilation of rotational wave packets in incoherent ensembles,” Phys. Rev. A 73(3), 033403 (2006). [CrossRef]
  21. Y. Li, P. Liu, S. Zhao, Z. Zeng, R. Li, and Z. Xu, “Active control of the molecular rotational wave packet using two laser pulses,” Chem. Phys. Lett. 475(4-6), 183–187 (2009). [CrossRef]
  22. S. Fleischer, I. Sh. Averbukh, and Y. Prior, “Selective alignment of molecular spin isomers,” Phys. Rev. Lett. 99(9), 093002 (2007). [CrossRef] [PubMed]
  23. Y. Gao, C. Wu, N. Xu, G. Zeng, H. Jiang, H. Yang, and Q. Gong, “Manipulating molecular rotational wave packets with strong femtosecond laser pulses,” Phys. Rev. A 77(4), 043404 (2008). [CrossRef]
  24. G. Zeng, C. Wu, H. Jiang, Y. Gao, N. Xu, and Q. Gong, “Rotational wave packet of N2O created by two strong femtosecond laser pulses,” J. Phys. B 42(16), 165508 (2009). [CrossRef]
  25. C. Wu, G. Zeng, Y. Gao, N. Xu, L. Y. Peng, H. Jiang, and Q. Gong, “Controlling molecular rotational population by wave-packet interference,” J. Chem. Phys. 130(23), 231102 (2009). [CrossRef] [PubMed]
  26. C. Wu, G. Zeng, H. Jiang, Y. Gao, N. Xu, and Q. Gong, “Molecular rotational excitation by strong femtosecond laser pulses,” J. Phys. Chem. A 113(40), 10610–10618 (2009). [CrossRef] [PubMed]
  27. A. S. Meijer, Y. Zhang, D. H. Parker, W. J. van der Zande, A. Gijsbertsen, and M. J. J. Vrakking, “Controlling rotational state distributions using two-pulse stimulated Raman excitation,” Phys. Rev. A 76(2), 023411 (2007). [CrossRef]

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