Abstract

Considering field-free molecular alignment enhancement by multiple pulses in the hollow fiber, we theoretically analyze how long-distance propagation influences the final alignment degree. Based on this, we propose a way to enhance frequency tuning of few-cycle femtosecond laser pulse by multi-pump-induced N2 index modulation. It is found that two pumps are optimized with parameters of 100 fs and 3×1013Wcm2. The tuning range of the initial 5 fs 800 nm probe pulse by double aligning pumps is about twice that by a single pump.

© 2012 Optical Society of America

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  1. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
    [CrossRef]
  2. X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
    [CrossRef]
  3. Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong, “Selective steering of molecular multiple dissociative channels with strong few-cycle laser pulses,” Phys. Rev. Lett. 106, 073004 (2011).
    [CrossRef]
  4. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
    [CrossRef]
  5. I. N. Ross and P. Matousek, “Analysis and optimization of optical parametric chirped pulse amplification,” J. Opt. Soc. Am. B 19, 2945–2956 (2002).
    [CrossRef]
  6. N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
    [CrossRef]
  7. S.-J. Im, A. Husakou, and J. Herrmann, “High-power soliton-induced supercontinuum generation and tunable sub-10 fs VUV pulses from kagome-lattice HC-PCFs,” Opt. Express 18, 5367–5374 (2010).
    [CrossRef]
  8. I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
    [CrossRef]
  9. W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
    [CrossRef]
  10. M. Y. Emelin, M. Y. Ryabikin, and A. M. Sergeev, “Frequency tunable single attosecond pulse production from aligned diatomic molecules ionized by intense laser field,” Opt. Express 18, 2269–2278 (2010).
    [CrossRef]
  11. M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
    [CrossRef]
  12. 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, 183–187 (2009).
    [CrossRef]
  13. S. Pabst and R. Santra, “Alignment of asymmetric-top molecules using multiple-pulse trains,” Phys. Rev. A 81, 065401 (2010).
    [CrossRef]
  14. 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, 173004 (2004).
    [CrossRef]
  15. H. Jiang, C. Wu, H. Zhang, H. Jiang, H. Yang, and Q. Gong, “Alignment structures of rotational wavepacket created by two strong femtosecond laser pulses,” Opt. Express 18, 8990–8997(2010).
    [CrossRef]
  16. J. P. Cryan, P. H. Bucksbaum, and R. N. Coffee, “Field-free alignment in repetitively kicked nitrogen gas,” Phys. Rev. A 80, 063412 (2009).
    [CrossRef]
  17. F. Zhong, H. Jiang, and Q. Gong, “Tuning the frequency of few-cycle femtosecond laser pulses by molecular phase modulation,” Opt. Express 17, 1472–1477 (2009).
    [CrossRef]
  18. 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, 013903 (2001).
    [CrossRef]
  19. F. Zhong, H. Jiang, and Q. Gong, “Generation of intense frequency-tunable few-cycle femtosecond pulses in hollow fiber,” J. Opt. Soc. Am. B 27, 128–131 (2010).
    [CrossRef]
  20. N. Milosevic, G. Tempea, and T. Brabec, “Optical pulse compression: bulk media versus hollow waveguides,” Opt. Lett. 25, 672–674 (2000).
    [CrossRef]
  21. M. Spanner and M. Yu. Ivanov, “Optimal generation of single-dispersion precompensated 1-fs pulses by molecular phase modulation,” Opt. Lett. 28, 576–578 (2003).
    [CrossRef]
  22. A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
    [CrossRef]
  23. J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
    [CrossRef]
  24. V. Kalosha, M. Spanner, J. Herrmann, and M. Ivanov, “Generation of single dispersion precompensated 1 fs pulses by shaped-pulse optimized high-order stimulated Raman scattering,” Phys. Rev. Lett. 88, 103901 (2002).
    [CrossRef]
  25. D. W. Broege, R. N. Coffee, and P. H. Bucksbaum, “Strong-field impulsive alignment in the presence of high temperatures and large centrifugal distortion,” Phys. Rev. A 78, 035401 (2008).
    [CrossRef]

2011 (1)

Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong, “Selective steering of molecular multiple dissociative channels with strong few-cycle laser pulses,” Phys. Rev. Lett. 106, 073004 (2011).
[CrossRef]

2010 (5)

2009 (4)

F. Zhong, H. Jiang, and Q. Gong, “Tuning the frequency of few-cycle femtosecond laser pulses by molecular phase modulation,” Opt. Express 17, 1472–1477 (2009).
[CrossRef]

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, 183–187 (2009).
[CrossRef]

J. P. Cryan, P. H. Bucksbaum, and R. N. Coffee, “Field-free alignment in repetitively kicked nitrogen gas,” Phys. Rev. A 80, 063412 (2009).
[CrossRef]

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

2008 (2)

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

D. W. Broege, R. N. Coffee, and P. H. Bucksbaum, “Strong-field impulsive alignment in the presence of high temperatures and large centrifugal distortion,” Phys. Rev. A 78, 035401 (2008).
[CrossRef]

2006 (1)

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

2004 (1)

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, 173004 (2004).
[CrossRef]

2003 (4)

I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
[CrossRef]

M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
[CrossRef]

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

M. Spanner and M. Yu. Ivanov, “Optimal generation of single-dispersion precompensated 1-fs pulses by molecular phase modulation,” Opt. Lett. 28, 576–578 (2003).
[CrossRef]

2002 (2)

I. N. Ross and P. Matousek, “Analysis and optimization of optical parametric chirped pulse amplification,” J. Opt. Soc. Am. B 19, 2945–2956 (2002).
[CrossRef]

V. Kalosha, M. Spanner, J. Herrmann, and M. Ivanov, “Generation of single dispersion precompensated 1 fs pulses by shaped-pulse optimized high-order stimulated Raman scattering,” Phys. Rev. Lett. 88, 103901 (2002).
[CrossRef]

2001 (3)

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, 013903 (2001).
[CrossRef]

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
[CrossRef]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

2000 (2)

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
[CrossRef]

N. Milosevic, G. Tempea, and T. Brabec, “Optical pulse compression: bulk media versus hollow waveguides,” Opt. Lett. 25, 672–674 (2000).
[CrossRef]

Agostini, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

Akozbek, N.

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
[CrossRef]

Auge, F.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

Averbukh, I. Sh.

M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
[CrossRef]

Baertschy, M.

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, 013903 (2001).
[CrossRef]

Balcou, Ph.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

Baltuška, A.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Bartels, R. A.

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, 013903 (2001).
[CrossRef]

Becker, A.

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
[CrossRef]

Bisgaard, C. Z.

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, 173004 (2004).
[CrossRef]

Boutu, W.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Bowden, C. M.

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
[CrossRef]

Brabec, T.

Breger, P.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

Broege, D. W.

D. W. Broege, R. N. Coffee, and P. H. Bucksbaum, “Strong-field impulsive alignment in the presence of high temperatures and large centrifugal distortion,” Phys. Rev. A 78, 035401 (2008).
[CrossRef]

Bucksbaum, P. H.

J. P. Cryan, P. H. Bucksbaum, and R. N. Coffee, “Field-free alignment in repetitively kicked nitrogen gas,” Phys. Rev. A 80, 063412 (2009).
[CrossRef]

D. W. Broege, R. N. Coffee, and P. H. Bucksbaum, “Strong-field impulsive alignment in the presence of high temperatures and large centrifugal distortion,” Phys. Rev. A 78, 035401 (2008).
[CrossRef]

Caillat, J.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Carre, B.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Cerullo, G.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

Chang, Z.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Chin, S. L.

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
[CrossRef]

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
[CrossRef]

Chini, M.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Coffee, R. N.

J. P. Cryan, P. H. Bucksbaum, and R. N. Coffee, “Field-free alignment in repetitively kicked nitrogen gas,” Phys. Rev. A 80, 063412 (2009).
[CrossRef]

D. W. Broege, R. N. Coffee, and P. H. Bucksbaum, “Strong-field impulsive alignment in the presence of high temperatures and large centrifugal distortion,” Phys. Rev. A 78, 035401 (2008).
[CrossRef]

Corkum, P. B.

I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
[CrossRef]

Cryan, J. P.

J. P. Cryan, P. H. Bucksbaum, and R. N. Coffee, “Field-free alignment in repetitively kicked nitrogen gas,” Phys. Rev. A 80, 063412 (2009).
[CrossRef]

De Silvestri, S.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

Deng, Y.

Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong, “Selective steering of molecular multiple dissociative channels with strong few-cycle laser pulses,” Phys. Rev. Lett. 106, 073004 (2011).
[CrossRef]

Dooley, P. W.

I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
[CrossRef]

Emelin, M. Y.

Feng, X.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Frasinski, L. J.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Fuji, T.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Gilbertson, S.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Gong, Q.

Greene, C. H.

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, 013903 (2001).
[CrossRef]

Haessler, S.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Herrmann, J.

S.-J. Im, A. Husakou, and J. Herrmann, “High-power soliton-induced supercontinuum generation and tunable sub-10 fs VUV pulses from kagome-lattice HC-PCFs,” Opt. Express 18, 5367–5374 (2010).
[CrossRef]

V. Kalosha, M. Spanner, J. Herrmann, and M. Ivanov, “Generation of single dispersion precompensated 1 fs pulses by shaped-pulse optimized high-order stimulated Raman scattering,” Phys. Rev. Lett. 88, 103901 (2002).
[CrossRef]

Husakou, A.

Im, S.-J.

Ishii, N.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Ivanov, M.

V. Kalosha, M. Spanner, J. Herrmann, and M. Ivanov, “Generation of single dispersion precompensated 1 fs pulses by shaped-pulse optimized high-order stimulated Raman scattering,” Phys. Rev. Lett. 88, 103901 (2002).
[CrossRef]

Ivanov, M. Yu.

Jiang, H.

Kalosha, V.

V. Kalosha, M. Spanner, J. Herrmann, and M. Ivanov, “Generation of single dispersion precompensated 1 fs pulses by shaped-pulse optimized high-order stimulated Raman scattering,” Phys. Rev. Lett. 88, 103901 (2002).
[CrossRef]

Kapteyn, H. C.

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, 013903 (2001).
[CrossRef]

Kasparian, J.

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
[CrossRef]

Khan, S. D.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Köhler, S.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Krausz, F.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Lee, K. F.

I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
[CrossRef]

Leibscher, M.

M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
[CrossRef]

Li, R.

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, 183–187 (2009).
[CrossRef]

Li, Y.

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, 183–187 (2009).
[CrossRef]

Litvinyuk, I. V.

I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
[CrossRef]

Liu, P.

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, 183–187 (2009).
[CrossRef]

Liu, X.

Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong, “Selective steering of molecular multiple dissociative channels with strong few-cycle laser pulses,” Phys. Rev. Lett. 106, 073004 (2011).
[CrossRef]

Liu, Y.

Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong, “Selective steering of molecular multiple dissociative channels with strong few-cycle laser pulses,” Phys. Rev. Lett. 106, 073004 (2011).
[CrossRef]

Maquet, A.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Mashiko, H.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Matousek, P.

Merdji, H.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Metzger, T.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Milosevic, N.

Monchicourt, P.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Muller, H. G.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

Mullot, G.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

Murnane, M. M.

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, 013903 (2001).
[CrossRef]

Oral, E.

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
[CrossRef]

Pabst, S.

S. Pabst and R. Santra, “Alignment of asymmetric-top molecules using multiple-pulse trains,” Phys. Rev. A 81, 065401 (2010).
[CrossRef]

Paul, P. M.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

Peronne, E.

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, 173004 (2004).
[CrossRef]

Poulsen, M. D.

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, 173004 (2004).
[CrossRef]

Rabitz, H.

M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
[CrossRef]

Rayner, D. M.

I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
[CrossRef]

Ross, I. N.

Ryabikin, M. Y.

Salieres, P.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Santra, R.

S. Pabst and R. Santra, “Alignment of asymmetric-top molecules using multiple-pulse trains,” Phys. Rev. A 81, 065401 (2010).
[CrossRef]

Sauerbrey, R.

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
[CrossRef]

Serebryannikov, E. E.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Sergeev, A. M.

Spanner, M.

M. Spanner and M. Yu. Ivanov, “Optimal generation of single-dispersion precompensated 1-fs pulses by molecular phase modulation,” Opt. Lett. 28, 576–578 (2003).
[CrossRef]

V. Kalosha, M. Spanner, J. Herrmann, and M. Ivanov, “Generation of single dispersion precompensated 1 fs pulses by shaped-pulse optimized high-order stimulated Raman scattering,” Phys. Rev. Lett. 88, 103901 (2002).
[CrossRef]

Stankiewicz, M.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Stapelfeldt, H.

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, 173004 (2004).
[CrossRef]

Taieb, R.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Teisset, C. Y.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Tempea, G.

Toma, E. S.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

Viftrup, S. S.

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, 173004 (2004).
[CrossRef]

Vijayalakshmi, K.

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
[CrossRef]

Villeneuve, D. M.

I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
[CrossRef]

Wagner, N.

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, 013903 (2001).
[CrossRef]

Wang, H.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Waters, G.

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Weinacht, T. C.

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, 013903 (2001).
[CrossRef]

Wu, C.

Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong, “Selective steering of molecular multiple dissociative channels with strong few-cycle laser pulses,” Phys. Rev. Lett. 106, 073004 (2011).
[CrossRef]

H. Jiang, C. Wu, H. Zhang, H. Jiang, H. Yang, and Q. Gong, “Alignment structures of rotational wavepacket created by two strong femtosecond laser pulses,” Opt. Express 18, 8990–8997(2010).
[CrossRef]

Wu, Y.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Xu, Z.

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, 183–187 (2009).
[CrossRef]

Yang, H.

Zeng, Z.

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, 183–187 (2009).
[CrossRef]

Zhang, H.

Zhao, K.

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Zhao, S.

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, 183–187 (2009).
[CrossRef]

Zheltikov, A. M.

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Zhong, F.

Appl. Phys. B (2)

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001).
[CrossRef]

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
[CrossRef]

Chem. Phys. Lett. (1)

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, 183–187 (2009).
[CrossRef]

Field-free alignment in repetitively kicked nitrogen gas (1)

J. P. Cryan, P. H. Bucksbaum, and R. N. Coffee, “Field-free alignment in repetitively kicked nitrogen gas,” Phys. Rev. A 80, 063412 (2009).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nat. Phys. (1)

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taieb, J. Caillat, A. Maquet, P. Monchicourt, B. Carre, and P. Salieres, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (2)

D. W. Broege, R. N. Coffee, and P. H. Bucksbaum, “Strong-field impulsive alignment in the presence of high temperatures and large centrifugal distortion,” Phys. Rev. A 78, 035401 (2008).
[CrossRef]

S. Pabst and R. Santra, “Alignment of asymmetric-top molecules using multiple-pulse trains,” Phys. Rev. A 81, 065401 (2010).
[CrossRef]

Phys. Rev. E (1)

N. Ishii, C. Y. Teisset, S. Köhler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuška, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Phys. Rev. Lett. (7)

I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90, 233003 (2003).
[CrossRef]

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. 103, 183901 (2009).
[CrossRef]

Y. Liu, X. Liu, Y. Deng, C. Wu, H. Jiang, and Q. Gong, “Selective steering of molecular multiple dissociative channels with strong few-cycle laser pulses,” Phys. Rev. Lett. 106, 073004 (2011).
[CrossRef]

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, 173004 (2004).
[CrossRef]

M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
[CrossRef]

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, 013903 (2001).
[CrossRef]

V. Kalosha, M. Spanner, J. Herrmann, and M. Ivanov, “Generation of single dispersion precompensated 1 fs pulses by shaped-pulse optimized high-order stimulated Raman scattering,” Phys. Rev. Lett. 88, 103901 (2002).
[CrossRef]

Rev. Sci. Instrum. (1)

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

Science (1)

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Initial N2 alignment signals by a different number of pulses, all with the same duration of 50 fs and intensity of 3×1013Wcm2. One single (red dashed curve), double (black dashed–dotted curve), and triple (blue solid curve) aligning pulses. The temperature of N2 is 90 K, and the density is 2.6875×1025m3.

Fig. 2.
Fig. 2.

N2 alignment signals after 105 cm by a different number of pulses, all with the same initial duration of 50 fs and intensity of 3×1013Wcm2. One single (red dashed curve), double (black dashed–dotted curve), and triple (blue solid curve) aligning pulses. The N2 system has a density of 2.6875×1025m3, and the temperature is 90 K.

Fig. 3.
Fig. 3.

Spectrum of the third aligning pulse after 0 (black solid curve), 30 (red dashed curve), and 105 cm (blue dashed–dotted curve) propagation. Here the initial duration of all three aligning pumps is 50 fs, and it shares the same intensity of 3×1013Wcm2. The N2 system has a density of 2.6875×1025m3, and the temperature is 90 K.

Fig. 4.
Fig. 4.

(a) Time distribution of the alignment induced by the first two pumps and the normalized intensity of the third pulse shown together and (b) corresponding alignments contrast, all at a distance of 30 cm propagation. In Fig. 3(a), the normalized intensity is shown by the black solid curve, and the alignment is shown by the red dashed curve. While in Fig. 4(b), the red dashed curve shows the alignment by the first pulse, the black dashed–dotted curve shows that by the first two pulses, and the blue solid curve shows that by all three pulses. Here the duration of all three aligning pumps is 50 fs, and it shares the same intensity of 3×1013Wcm2. The N2 system has a density of 2.6875×1025m3, and the temperature is 90 K.

Fig. 5.
Fig. 5.

N2 initial alignment signals by different number of pulses, all at the same duration of 100 fs and intensity of 3×1013Wcm2. One single (red dashed curve) and double (black solid curve) aligning pulses. The N2 system has a density of 2.6875×1025m3, and the temperature is 90 K.

Fig. 6.
Fig. 6.

N2 alignment signals after 105 cm by different number of pulses, all at the same duration of 100 fs and intensity of 3×1013Wcm2. One single (red dashed curve) and double (black solid curve) aligning pulses. The N2 system has a density of 2.6875×1025m3, and the temperature is 90 K.

Fig. 7.
Fig. 7.

Normalized intensity of the second pump (black solid), alignment signal by single pump (red dashed–dotted curve) and double pumps (blue dashed curve) shown together. Both pulse parameters are 100 fs and 3×1013Wcm2. The N2 system has a density of 2.6875×1025m3, and the temperature is 90 K.

Fig. 8.
Fig. 8.

Frequency blueshift contrast of 5 fs weak probe pumped by a single or double laser pulses. One single (red dashed curve) and double (blue dashed–dotted curve) aligning pulses. The black solid curve stands for its initial spectrum. Both pulse parameters are 50 fs and 3×1013Wcm2. The N2 system has a density of 2.6875×1025m3, and the temperature is 90 K.

Fig. 9.
Fig. 9.

Frequency redshift contrast of 5 fs weak probe pumped by a single or double laser pulses. One single (red dashed curve) and double (blue dashed–dotted curve) aligning pulses. The black solid curve stands for its initial spectrum. Both pulse parameters are 50 fs and 3×1013Wcm2. The N2 system has a density of 2.6875×1025m3, and the temperature is 90 K.

Tables (1)

Tables Icon

Table 1. Frequency Tuning δν=(νν0) Contrast for Different Alignment Cases, with the Temperature of 90 k and All Pulse Intensity at 3×1013Wcm2 and N2 System Density of 2.6875×1025m3a

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

n(t)1+2πN(α+Δαcos2θ(t)),
itψ(θ,t)=[J22IΔαπ|Epump(t)|2cos2θ]ψ(θ,t).
E(z,t)z+1cE(z,t)t=12cP(z,t)t,

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