Abstract

In this paper, a theoretical model is developed to demonstrate that fluctuations in the Stokes signal and occurrence position contribute to the final compression ratio in stimulated Brillouin scattering (SBS). This theoretical analysis can be applied to the investigation of the temporal characteristics of SBS pulse compression. This model agrees well with the experimental results in a two-stage SBS compressor.

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References

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  1. C. B. Dane, W. A. Neuman, and L. A. Hackel, “High-energy SBS pulse compression,” IEEE J. Quantum Electron. 30(8), 1907–1915 (1994).
    [Crossref]
  2. V. Kmetik, H. Fiedorowicz, A. A. Andreev, K. J. Witte, H. Daido, H. Fujita, M. Nakatsuka, and T. Yamanaka, “Reliable stimulated Brillouin scattering compression of Nd:YAG laser pulses with liquid fluorocarbon for long-time operation at 10 Hz,” Appl. Opt. 37(30), 7085–7090 (1998).
    [Crossref] [PubMed]
  3. D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68(4), 671–675 (1999).
    [Crossref]
  4. X. Xu, C. Feng, and J.-C. Diels, “Optimizing sub-ns pulse compression for high energy application,” Opt. Express 22(11), 13904–13915 (2014).
    [Crossref] [PubMed]
  5. Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
    [Crossref]
  6. S. Schiemann, W. Ubachs, and W. Hogervorst, “Efficient temporal compression of coherent nanosecond pulses in a compact SBS generator-amplifier setup,” IEEE J. Quantum Electron. 33(3), 358–366 (1997).
    [Crossref]
  7. C. Feng, X. Xu, and J. C. Diels, “Generation of 300 ps laser pulse with 1.2 J energy by stimulated Brillouin scattering in water at 532 nm,” Opt. Lett. 39(12), 3367–3370 (2014).
    [Crossref] [PubMed]
  8. H. Yoshida, T. Hatae, H. Fujita, M. Nakatsuka, and S. Kitamura, “A high-energy 160-ps pulse generation by stimulated Brillouin scattering from heavy fluorocarbon liquid at 1064 nm wavelength,” Opt. Express 17(16), 13654–13662 (2009).
    [Crossref] [PubMed]
  9. M. S. Mangir, J. J. Ottusch, D. C. Jones, and D. A. Rockwell, “Time-resolved measurements of stimulated-Brillouin-scattering phase jumps,” Phys. Rev. Lett. 68(11), 1702–1705 (1992).
    [Crossref] [PubMed]
  10. A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44(5), 3205–3209 (1991).
    [Crossref] [PubMed]
  11. J. Munch, R. F. Wuerker, and M. J. Lefebvre, “Interaction length for optical phase conjugation by stimulated Brillouin scattering: an experimental investigation,” Appl. Opt. 28(15), 3099–3105 (1989).
    [Crossref] [PubMed]
  12. I. Velchev and W. Ubachs, “Statistical properties of the Stokes signal in stimulated Brillouin scattering pulse compressors,” Phys. Rev. A. 71(4), 043810 (2005).
  13. A. Zadok, A. Eyal, and M. Tur, “Stimulated Brillouin scattering slow light in optical fibers [Invited],” Appl. Opt. 50(25), E38–E49 (2011).
    [Crossref]
  14. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  15. I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35(12), 1812–1816 (1999).
    [Crossref]
  16. W. E. Kock, Laser Fundamentals (Springer US, 1975).
  17. M. Damzen and H. Hutchinson, “Laser pulse compression by stimulated Brillouin scattering in tapered waveguides,” IEEE J. Quantum Electron. 19(1), 7–14 (1983).
    [Crossref]
  18. R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990).
    [Crossref] [PubMed]
  19. K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
    [Crossref]

2014 (3)

2011 (1)

2009 (1)

2005 (1)

I. Velchev and W. Ubachs, “Statistical properties of the Stokes signal in stimulated Brillouin scattering pulse compressors,” Phys. Rev. A. 71(4), 043810 (2005).

2001 (1)

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

1999 (2)

D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68(4), 671–675 (1999).
[Crossref]

I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35(12), 1812–1816 (1999).
[Crossref]

1998 (1)

1997 (1)

S. Schiemann, W. Ubachs, and W. Hogervorst, “Efficient temporal compression of coherent nanosecond pulses in a compact SBS generator-amplifier setup,” IEEE J. Quantum Electron. 33(3), 358–366 (1997).
[Crossref]

1994 (1)

C. B. Dane, W. A. Neuman, and L. A. Hackel, “High-energy SBS pulse compression,” IEEE J. Quantum Electron. 30(8), 1907–1915 (1994).
[Crossref]

1992 (1)

M. S. Mangir, J. J. Ottusch, D. C. Jones, and D. A. Rockwell, “Time-resolved measurements of stimulated-Brillouin-scattering phase jumps,” Phys. Rev. Lett. 68(11), 1702–1705 (1992).
[Crossref] [PubMed]

1991 (1)

A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44(5), 3205–3209 (1991).
[Crossref] [PubMed]

1990 (1)

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990).
[Crossref] [PubMed]

1989 (1)

1983 (1)

M. Damzen and H. Hutchinson, “Laser pulse compression by stimulated Brillouin scattering in tapered waveguides,” IEEE J. Quantum Electron. 19(1), 7–14 (1983).
[Crossref]

Andreev, A. A.

Boyd, R. W.

A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44(5), 3205–3209 (1991).
[Crossref] [PubMed]

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990).
[Crossref] [PubMed]

Cheng, S. X.

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

Daido, H.

Damzen, M.

M. Damzen and H. Hutchinson, “Laser pulse compression by stimulated Brillouin scattering in tapered waveguides,” IEEE J. Quantum Electron. 19(1), 7–14 (1983).
[Crossref]

Dane, C. B.

C. B. Dane, W. A. Neuman, and L. A. Hackel, “High-energy SBS pulse compression,” IEEE J. Quantum Electron. 30(8), 1907–1915 (1994).
[Crossref]

Diels, J. C.

Diels, J.-C.

Eyal, A.

Feng, C.

Fiedorowicz, H.

Fujita, H.

Gaeta, A. L.

A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44(5), 3205–3209 (1991).
[Crossref] [PubMed]

Hackel, L. A.

C. B. Dane, W. A. Neuman, and L. A. Hackel, “High-energy SBS pulse compression,” IEEE J. Quantum Electron. 30(8), 1907–1915 (1994).
[Crossref]

Hasi, W. L. J.

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

Hatae, T.

He, W. M.

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

Hogervorst, W.

D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68(4), 671–675 (1999).
[Crossref]

I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35(12), 1812–1816 (1999).
[Crossref]

S. Schiemann, W. Ubachs, and W. Hogervorst, “Efficient temporal compression of coherent nanosecond pulses in a compact SBS generator-amplifier setup,” IEEE J. Quantum Electron. 33(3), 358–366 (1997).
[Crossref]

Hutchinson, H.

M. Damzen and H. Hutchinson, “Laser pulse compression by stimulated Brillouin scattering in tapered waveguides,” IEEE J. Quantum Electron. 19(1), 7–14 (1983).
[Crossref]

Jones, D. C.

M. S. Mangir, J. J. Ottusch, D. C. Jones, and D. A. Rockwell, “Time-resolved measurements of stimulated-Brillouin-scattering phase jumps,” Phys. Rev. Lett. 68(11), 1702–1705 (1992).
[Crossref] [PubMed]

Kato, S.

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

Kitamura, S.

Kmetik, V.

Kuwahara, K.

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

Lefebvre, M. J.

Lin, D. Y.

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

Lü, Z. W.

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

Majewski, W. A.

D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68(4), 671–675 (1999).
[Crossref]

Mangir, M. S.

M. S. Mangir, J. J. Ottusch, D. C. Jones, and D. A. Rockwell, “Time-resolved measurements of stimulated-Brillouin-scattering phase jumps,” Phys. Rev. Lett. 68(11), 1702–1705 (1992).
[Crossref] [PubMed]

Matsumoto, Y.

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

Matsushima, I.

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

Munch, J.

Nakatsuka, M.

Narum, P.

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990).
[Crossref] [PubMed]

Neshev, D.

I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35(12), 1812–1816 (1999).
[Crossref]

D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68(4), 671–675 (1999).
[Crossref]

Neuman, W. A.

C. B. Dane, W. A. Neuman, and L. A. Hackel, “High-energy SBS pulse compression,” IEEE J. Quantum Electron. 30(8), 1907–1915 (1994).
[Crossref]

Okuda, I.

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

Ottusch, J. J.

M. S. Mangir, J. J. Ottusch, D. C. Jones, and D. A. Rockwell, “Time-resolved measurements of stimulated-Brillouin-scattering phase jumps,” Phys. Rev. Lett. 68(11), 1702–1705 (1992).
[Crossref] [PubMed]

Owadano, Y.

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

Rockwell, D. A.

M. S. Mangir, J. J. Ottusch, D. C. Jones, and D. A. Rockwell, “Time-resolved measurements of stimulated-Brillouin-scattering phase jumps,” Phys. Rev. Lett. 68(11), 1702–1705 (1992).
[Crossref] [PubMed]

Rzaewski, K.

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990).
[Crossref] [PubMed]

Schiemann, S.

S. Schiemann, W. Ubachs, and W. Hogervorst, “Efficient temporal compression of coherent nanosecond pulses in a compact SBS generator-amplifier setup,” IEEE J. Quantum Electron. 33(3), 358–366 (1997).
[Crossref]

Takahashi, E.

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

Tur, M.

Ubachs, W.

I. Velchev and W. Ubachs, “Statistical properties of the Stokes signal in stimulated Brillouin scattering pulse compressors,” Phys. Rev. A. 71(4), 043810 (2005).

I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35(12), 1812–1816 (1999).
[Crossref]

D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68(4), 671–675 (1999).
[Crossref]

S. Schiemann, W. Ubachs, and W. Hogervorst, “Efficient temporal compression of coherent nanosecond pulses in a compact SBS generator-amplifier setup,” IEEE J. Quantum Electron. 33(3), 358–366 (1997).
[Crossref]

Velchev, I.

I. Velchev and W. Ubachs, “Statistical properties of the Stokes signal in stimulated Brillouin scattering pulse compressors,” Phys. Rev. A. 71(4), 043810 (2005).

I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35(12), 1812–1816 (1999).
[Crossref]

D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68(4), 671–675 (1999).
[Crossref]

Wang, X. Y.

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

Witte, K. J.

Wuerker, R. F.

Xu, X.

Yamanaka, T.

Yoshida, H.

Zadok, A.

Zhao, H.

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

Zheng, Z. X.

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

Appl. Opt. (3)

Appl. Phys. B (2)

Z. X. Zheng, W. L. J. Hasi, H. Zhao, S. X. Cheng, X. Y. Wang, D. Y. Lin, W. M. He, and Z. W. Lü, “Compression characteristics of two new SBS mediums to generate 100-ps pulse for shock ignition,” Appl. Phys. B 116(3), 659–663 (2014).
[Crossref]

D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68(4), 671–675 (1999).
[Crossref]

IEEE J. Quantum Electron. (4)

C. B. Dane, W. A. Neuman, and L. A. Hackel, “High-energy SBS pulse compression,” IEEE J. Quantum Electron. 30(8), 1907–1915 (1994).
[Crossref]

S. Schiemann, W. Ubachs, and W. Hogervorst, “Efficient temporal compression of coherent nanosecond pulses in a compact SBS generator-amplifier setup,” IEEE J. Quantum Electron. 33(3), 358–366 (1997).
[Crossref]

I. Velchev, D. Neshev, W. Hogervorst, and W. Ubachs, “Pulse compression to the subphonon lifetime region by half-cycle gain in transient stimulated Brillouin scattering,” IEEE J. Quantum Electron. 35(12), 1812–1816 (1999).
[Crossref]

M. Damzen and H. Hutchinson, “Laser pulse compression by stimulated Brillouin scattering in tapered waveguides,” IEEE J. Quantum Electron. 19(1), 7–14 (1983).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (2)

R. W. Boyd, K. Rzaewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42(9), 5514–5521 (1990).
[Crossref] [PubMed]

A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44(5), 3205–3209 (1991).
[Crossref] [PubMed]

Phys. Rev. A. (1)

I. Velchev and W. Ubachs, “Statistical properties of the Stokes signal in stimulated Brillouin scattering pulse compressors,” Phys. Rev. A. 71(4), 043810 (2005).

Phys. Rev. Lett. (1)

M. S. Mangir, J. J. Ottusch, D. C. Jones, and D. A. Rockwell, “Time-resolved measurements of stimulated-Brillouin-scattering phase jumps,” Phys. Rev. Lett. 68(11), 1702–1705 (1992).
[Crossref] [PubMed]

Proc. SPIE (1)

K. Kuwahara, E. Takahashi, Y. Matsumoto, I. Matsushima, I. Okuda, S. Kato, and Y. Owadano, “High-intensity pulse generation by saturated amplification of Stokes pulse with steep leading edge,” Proc. SPIE 4424, 155–158 (2001).
[Crossref]

Other (2)

W. E. Kock, Laser Fundamentals (Springer US, 1975).

R. W. Boyd, Nonlinear Optics (Academic, 2003).

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Figures (8)

Fig. 1
Fig. 1 Experimental optical path; green line: the Stokes signal in the generator cell that recorded by photon detector 3; indigo line: the final compressed pulse recorded by photon detector 2.
Fig. 2
Fig. 2 Variation of Stokes extraction efficiency with pump intensity.
Fig. 3
Fig. 3 Variation of output Stokes pulse width with pump intensity.
Fig. 4
Fig. 4 Variation of rising time of Stokes pulse with pump intensity.
Fig. 5
Fig. 5 Comparison between simulated pulse width obtained varies with input pump intensity and the experimental results; solid red squares: experimental results; cyan: simulation results.
Fig. 6
Fig. 6 Description of Stokes pulse generation and amplification in a two-stage SBS compressor. The injected pump intensity is 15 MW/cm2. (a) Final output Stokes pulse calculated by theoretical analysis. (b) Stokes signal in generation cell. (c) Output Stokes pulse obtained in experiment. (d) Experimental output Stokes pulse from generation cell.
Fig. 7
Fig. 7 (a) Final output Stokes pulse calculated by theoretical analysis. (b) Stokes signal in generation cell. (c) Output Stokes pulse obtained in experiment. (d) Experimental output Stokes pulse from generation cell. The injected pump intensity is 30 MW/cm2.
Fig. 8
Fig. 8 (a) Final output Stokes pulse calculated by theoretical analysis. (b) Stokes signal in generation cell. (c) Output Stokes pulse obtained in experiment. (d) Experimental output Stokes pulse from generation cell. The injected pump intensity is 140 MW/cm2.

Tables (1)

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Table 1 Parameters of FC-770

Equations (10)

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E L (z,t)= 1 2 E L (z,t) e i( k z ω L t) +c.c.
E s (z,t)= 1 2 E s (z,t) e i( k z ω s t) +c.c.
ρ (z,t)= 1 2 ρ(z,t) e i( q B z Ω B t) +c.c.
E L z + n c E L t = i ω L γ e 2nc ρ 0 ρ E S
E S z + n c E S t = i ω S γ e 2nc ρ 0 ρ * E L
2 ρ t 2 (2i Ω B Γ B ) ρ t (i Ω B Γ B )ρ= γ e 4π q B 2 E L E S *
E L z + n c E L t =igρ E S
E S z + n c E S t =ig ρ * E L
ρ(z,t)= 1 2π t f(tτ) E L (z,τ) E S * (z,τ)dτ
f(t)= 2π Ω B exp(( Γ B /2)t) Ω B Γ B 2 4 t exp(i Ω B t)sin( Ω B Γ B 2 4 t ),( t0 )

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