Abstract

The characteristics of a barium tungstate (BaWO4) anti-Stokes Raman laser at 968 nm are studied theoretically and experimentally. The BaWO4 Raman resonator is pumped by a Q-switched Nd:YAG laser at 1064 nm with its axis tilted from the pumping laser axis. The non-collinear phase matching for the generation of the first anti-Stokes wave in the same BaWO4 crystal is achieved. The output energy, temporal and spectral informations are investigated. At a pumping laser energy of 128 mJ, the anti-Stokes laser energy obtained is 2.2 mJ. The second Stokes radiation at 1324 nm as well as the first and the third Stokes waves at 1180 nm and 1509 nm is also generated at the same time. The maximum total Stokes energy output is 42.5 mJ. In the theory, the anti-Stokes laser intensity expression as a function of the pumping and the first Stokes laser intensities for the extracavity anti-Stokes Raman laser is deduced. The properties of the anti-Stokes Raman laser are simulated theoretically by solving the rate equations of the extracavity Raman laser and using the derived expression. The theoretical results are in good agreement with the experimental results.

© 2013 Optical Society of America

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2010 (2)

2009 (3)

2008 (1)

2006 (1)

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

2005 (4)

W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

H. Jelinkova, J. Sulc, T. T. Basiev, P. G. Zverev, and S. V. Kravtsov, “Stimulated Raman scattering in Nd:SrWO4,” Laser Phys. Lett. 2(1), 4–11 (2005).
[Crossref]

Y. F. Chen, K. W. Su, H. J. Zhang, J. Y. Wang, and M. H. Jiang, “Efficient diode-pumped actively Q-switched Nd:YAG/BaWO4 intracavity Raman laser,” Opt. Lett. 30(24), 3335–3337 (2005).
[Crossref] [PubMed]

A. Major, J. S. Aitchison, P. W. E. Smith, N. Langford, and A. I. Ferguson, “Efficient Raman shifting of high-energy picosecond pulses into the eye-safe 1.5-microm spectral region by use of a KGd(WO4)2 crystal,” Opt. Lett. 30(4), 421–423 (2005).
[Crossref] [PubMed]

2004 (1)

A. Z. Grasiuk, S. V. Kurbasov, and L. L. Losev, “Picosecond parametric Raman laser based on KGd(WO4)2 crystal,” Opt. Commun. 240(4-6), 239–244 (2004).
[Crossref]

2003 (2)

2002 (1)

2000 (1)

P. G. Zverev, T. T. Basiev, A. A. Sobol, V. V. Skornyakov, L. I. Ivleva, N. M. Polozkov, and V. V. Osiko, “Stimulated Raman scattering in alkaline-earth tungstate crystals,” Quantum Electron. 30(1), 55–59 (2000).
[Crossref]

1999 (2)

T. T. Basiev, A. A. Sobol, P. G. Zverev, L. I. Ivleva, V. V. Osiko, and R. C. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11(4), 307–314 (1999).
[Crossref]

A. S. Grabtchikov, A. N. Kuzmin, V. A. Lisinetskii, V. A. Orlovich, G. I. Ryabtsev, and A. A. Demidovich, “All solid-state diode-pumped Raman laser with self-frequency conversion,” Appl. Phys. Lett. 75(24), 3742–3744 (1999).
[Crossref]

1997 (1)

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-stokes stimulated Raman scattering,” Phys. Rev. Lett. 79(2), 209–212 (1997).
[Crossref]

1977 (1)

G. F. Bakhshieva and A. M. Morozov, “Refractive-Indexes of molybdate and tungstate single-crystals having scheelite structure,” Sov. J. Opt. Technol. 44, 542–543 (1977).

1970 (1)

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman Emission in Diamond: Spectrum, Gain, and Angular Distribution of Intensity,” Phys. Rev. A 1(3), 628–635 (1970).
[Crossref]

1967 (1)

S. P. S. Porto and J. F. Scott, “Raman spectra of CaWO4, SrWO4, CaMoO4, and SrMoO4,” Phys. Rev. 157(3), 716–719 (1967).
[Crossref]

Aitchison, J. S.

Bakhshieva, G. F.

G. F. Bakhshieva and A. M. Morozov, “Refractive-Indexes of molybdate and tungstate single-crystals having scheelite structure,” Sov. J. Opt. Technol. 44, 542–543 (1977).

Basiev, T. T.

H. Jelinkova, J. Sulc, T. T. Basiev, P. G. Zverev, and S. V. Kravtsov, “Stimulated Raman scattering in Nd:SrWO4,” Laser Phys. Lett. 2(1), 4–11 (2005).
[Crossref]

P. G. Zverev, T. T. Basiev, A. A. Sobol, V. V. Skornyakov, L. I. Ivleva, N. M. Polozkov, and V. V. Osiko, “Stimulated Raman scattering in alkaline-earth tungstate crystals,” Quantum Electron. 30(1), 55–59 (2000).
[Crossref]

T. T. Basiev, A. A. Sobol, P. G. Zverev, L. I. Ivleva, V. V. Osiko, and R. C. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11(4), 307–314 (1999).
[Crossref]

Chang, H. L.

Chang, J.

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Chang, Y. T.

Chen, X. H.

Chen, Y. F.

Cheng, X. F.

W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

Clements, W. R. L.

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman Emission in Diamond: Spectrum, Gain, and Angular Distribution of Intensity,” Phys. Rev. A 1(3), 628–635 (1970).
[Crossref]

Cong, Z. H.

Coutts, D. W.

Dekker, P.

Demidovich, A. A.

A. S. Grabtchikov, A. N. Kuzmin, V. A. Lisinetskii, V. A. Orlovich, G. I. Ryabtsev, and A. A. Demidovich, “All solid-state diode-pumped Raman laser with self-frequency conversion,” Appl. Phys. Lett. 75(24), 3742–3744 (1999).
[Crossref]

Ding, S.

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Eichler, H. J.

Erný, P.

Fan, L.

Fan, S.

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Fan, S. Z.

Fan, Y. X.

Ferguson, A. I.

Gad, G. M. A.

Ge, W. W.

W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

Grabtchikov, A. S.

A. S. Grabtchikov, A. N. Kuzmin, V. A. Lisinetskii, V. A. Orlovich, G. I. Ryabtsev, and A. A. Demidovich, “All solid-state diode-pumped Raman laser with self-frequency conversion,” Appl. Phys. Lett. 75(24), 3742–3744 (1999).
[Crossref]

Grasiuk, A. Z.

A. Z. Grasiuk, S. V. Kurbasov, and L. L. Losev, “Picosecond parametric Raman laser based on KGd(WO4)2 crystal,” Opt. Commun. 240(4-6), 239–244 (2004).
[Crossref]

Hakuta, K.

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-stokes stimulated Raman scattering,” Phys. Rev. Lett. 79(2), 209–212 (1997).
[Crossref]

Hu, X. B.

W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

Ivleva, L. I.

P. G. Zverev, T. T. Basiev, A. A. Sobol, V. V. Skornyakov, L. I. Ivleva, N. M. Polozkov, and V. V. Osiko, “Stimulated Raman scattering in alkaline-earth tungstate crystals,” Quantum Electron. 30(1), 55–59 (2000).
[Crossref]

T. T. Basiev, A. A. Sobol, P. G. Zverev, L. I. Ivleva, V. V. Osiko, and R. C. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11(4), 307–314 (1999).
[Crossref]

Jelinkova, H.

H. Jelinkova, J. Sulc, T. T. Basiev, P. G. Zverev, and S. V. Kravtsov, “Stimulated Raman scattering in Nd:SrWO4,” Laser Phys. Lett. 2(1), 4–11 (2005).
[Crossref]

Jelínková, H.

Jiang, M. H.

W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

Y. F. Chen, K. W. Su, H. J. Zhang, J. Y. Wang, and M. H. Jiang, “Efficient diode-pumped actively Q-switched Nd:YAG/BaWO4 intracavity Raman laser,” Opt. Lett. 30(24), 3335–3337 (2005).
[Crossref] [PubMed]

Kaminskii, A. A.

Katsuragawa, M.

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-stokes stimulated Raman scattering,” Phys. Rev. Lett. 79(2), 209–212 (1997).
[Crossref]

Kravtsov, S. V.

H. Jelinkova, J. Sulc, T. T. Basiev, P. G. Zverev, and S. V. Kravtsov, “Stimulated Raman scattering in Nd:SrWO4,” Laser Phys. Lett. 2(1), 4–11 (2005).
[Crossref]

Kurbasov, S. V.

A. Z. Grasiuk, S. V. Kurbasov, and L. L. Losev, “Picosecond parametric Raman laser based on KGd(WO4)2 crystal,” Opt. Commun. 240(4-6), 239–244 (2004).
[Crossref]

Kuzmin, A. N.

A. S. Grabtchikov, A. N. Kuzmin, V. A. Lisinetskii, V. A. Orlovich, G. I. Ryabtsev, and A. A. Demidovich, “All solid-state diode-pumped Raman laser with self-frequency conversion,” Appl. Phys. Lett. 75(24), 3742–3744 (1999).
[Crossref]

Langford, N.

Lee, A. J.

Li, H. X.

W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

Li, J. Z.

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-stokes stimulated Raman scattering,” Phys. Rev. Lett. 79(2), 209–212 (1997).
[Crossref]

Li, S.

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Li, S. T.

Li, Y. Q.

Lisinetskii, V. A.

A. S. Grabtchikov, A. N. Kuzmin, V. A. Lisinetskii, V. A. Orlovich, G. I. Ryabtsev, and A. A. Demidovich, “All solid-state diode-pumped Raman laser with self-frequency conversion,” Appl. Phys. Lett. 75(24), 3742–3744 (1999).
[Crossref]

Liu, J. H.

W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

Liu, Y.

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Liu, Z. J.

Losev, L. L.

A. Z. Grasiuk, S. V. Kurbasov, and L. L. Losev, “Picosecond parametric Raman laser based on KGd(WO4)2 crystal,” Opt. Commun. 240(4-6), 239–244 (2004).
[Crossref]

Major, A.

McQuillan, A. K.

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman Emission in Diamond: Spectrum, Gain, and Angular Distribution of Intensity,” Phys. Rev. A 1(3), 628–635 (1970).
[Crossref]

Mildren, R. P.

Morozov, A. M.

G. F. Bakhshieva and A. M. Morozov, “Refractive-Indexes of molybdate and tungstate single-crystals having scheelite structure,” Sov. J. Opt. Technol. 44, 542–543 (1977).

Orlovich, V. A.

A. S. Grabtchikov, A. N. Kuzmin, V. A. Lisinetskii, V. A. Orlovich, G. I. Ryabtsev, and A. A. Demidovich, “All solid-state diode-pumped Raman laser with self-frequency conversion,” Appl. Phys. Lett. 75(24), 3742–3744 (1999).
[Crossref]

Osiko, V. V.

P. G. Zverev, T. T. Basiev, A. A. Sobol, V. V. Skornyakov, L. I. Ivleva, N. M. Polozkov, and V. V. Osiko, “Stimulated Raman scattering in alkaline-earth tungstate crystals,” Quantum Electron. 30(1), 55–59 (2000).
[Crossref]

T. T. Basiev, A. A. Sobol, P. G. Zverev, L. I. Ivleva, V. V. Osiko, and R. C. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11(4), 307–314 (1999).
[Crossref]

Pask, H. M.

Piper, J. A.

Polozkov, N. M.

P. G. Zverev, T. T. Basiev, A. A. Sobol, V. V. Skornyakov, L. I. Ivleva, N. M. Polozkov, and V. V. Osiko, “Stimulated Raman scattering in alkaline-earth tungstate crystals,” Quantum Electron. 30(1), 55–59 (2000).
[Crossref]

Porto, S. P. S.

S. P. S. Porto and J. F. Scott, “Raman spectra of CaWO4, SrWO4, CaMoO4, and SrMoO4,” Phys. Rev. 157(3), 716–719 (1967).
[Crossref]

Powell, R. C.

T. T. Basiev, A. A. Sobol, P. G. Zverev, L. I. Ivleva, V. V. Osiko, and R. C. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11(4), 307–314 (1999).
[Crossref]

Ryabtsev, G. I.

A. S. Grabtchikov, A. N. Kuzmin, V. A. Lisinetskii, V. A. Orlovich, G. I. Ryabtsev, and A. A. Demidovich, “All solid-state diode-pumped Raman laser with self-frequency conversion,” Appl. Phys. Lett. 75(24), 3742–3744 (1999).
[Crossref]

Scott, J. F.

S. P. S. Porto and J. F. Scott, “Raman spectra of CaWO4, SrWO4, CaMoO4, and SrMoO4,” Phys. Rev. 157(3), 716–719 (1967).
[Crossref]

Skornyakov, V. V.

P. G. Zverev, T. T. Basiev, A. A. Sobol, V. V. Skornyakov, L. I. Ivleva, N. M. Polozkov, and V. V. Osiko, “Stimulated Raman scattering in alkaline-earth tungstate crystals,” Quantum Electron. 30(1), 55–59 (2000).
[Crossref]

Smith, P. W. E.

Sobol, A. A.

P. G. Zverev, T. T. Basiev, A. A. Sobol, V. V. Skornyakov, L. I. Ivleva, N. M. Polozkov, and V. V. Osiko, “Stimulated Raman scattering in alkaline-earth tungstate crystals,” Quantum Electron. 30(1), 55–59 (2000).
[Crossref]

T. T. Basiev, A. A. Sobol, P. G. Zverev, L. I. Ivleva, V. V. Osiko, and R. C. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11(4), 307–314 (1999).
[Crossref]

Spence, D. J.

Stoicheff, B. P.

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman Emission in Diamond: Spectrum, Gain, and Angular Distribution of Intensity,” Phys. Rev. A 1(3), 628–635 (1970).
[Crossref]

Su, F.

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Su, K. W.

Sulc, J.

H. Jelinkova, J. Sulc, T. T. Basiev, P. G. Zverev, and S. V. Kravtsov, “Stimulated Raman scattering in Nd:SrWO4,” Laser Phys. Lett. 2(1), 4–11 (2005).
[Crossref]

Suzuki, M.

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-stokes stimulated Raman scattering,” Phys. Rev. Lett. 79(2), 209–212 (1997).
[Crossref]

Tao, X. T.

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[Crossref] [PubMed]

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Wang, Q. P.

Wang, S.

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

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W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

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W. W. Ge, H. J. Zhang, J. Y. Wang, J. H. Liu, H. X. Li, X. F. Cheng, H. Y. Xu, X. G. Xu, X. B. Hu, and M. H. Jiang, “The thermal and optical properties of BaWO4 single crystal,” J. Cryst. Growth 276(1-2), 208–214 (2005).
[Crossref]

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S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Zhang, X.

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Zhang, X. L.

Zhang, X. Y.

Zverev, P. G.

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[Crossref]

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[Crossref]

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[Crossref]

Appl. Phys. B (1)

S. Ding, X. Zhang, Q. Wang, F. Su, S. Li, S. Fan, S. Zhang, J. Chang, S. Wang, and Y. Liu, “Theoretical models for the extracavity Raman laser with crystalline Raman medium,” Appl. Phys. B 85(1), 89–95 (2006).
[Crossref]

Appl. Phys. Lett. (1)

A. S. Grabtchikov, A. N. Kuzmin, V. A. Lisinetskii, V. A. Orlovich, G. I. Ryabtsev, and A. A. Demidovich, “All solid-state diode-pumped Raman laser with self-frequency conversion,” Appl. Phys. Lett. 75(24), 3742–3744 (1999).
[Crossref]

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[Crossref]

Laser Phys. Lett. (1)

H. Jelinkova, J. Sulc, T. T. Basiev, P. G. Zverev, and S. V. Kravtsov, “Stimulated Raman scattering in Nd:SrWO4,” Laser Phys. Lett. 2(1), 4–11 (2005).
[Crossref]

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T. T. Basiev, A. A. Sobol, P. G. Zverev, L. I. Ivleva, V. V. Osiko, and R. C. Powell, “Raman spectroscopy of crystals for stimulated Raman scattering,” Opt. Mater. 11(4), 307–314 (1999).
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Figures (8)

Fig. 1
Fig. 1

Experimental setup of the pulsed BaWO4 anti-Stokes Raman laser.

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Fig. 2
Fig. 2

Diagram of phase-matching condition.

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Fig. 3
Fig. 3

Transmission curves of the cavity mirrors.

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Fig. 4
Fig. 4

Output laser spectral information.

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Fig. 5
Fig. 5

Anti-Stokes output energies as functions of the pumping laser energy.

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Fig. 6
Fig. 6

Stokes output energies as functions of the pumping laser energy.

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Fig. 7
Fig. 7

Oscilloscope traces of the pumping, the depleted pumping, the first Stokes and the first anti-Stokes laser pulses.

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Fig. 8
Fig. 8

Numerically calculated traces of the pumping, the depleted pumping, the first Stokes and the first anti-Stokes laser pulses.

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Tables (3)

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Table 1 Parameter comparisons of the output beams between the calculated and the experimental values.

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Table 2 The parameters for the theoretical calculation.

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Table 3 The ratio of the anti-Stokes decrease due to the Raman conversion to the anti-Stokes increase due to the FWM process for different pumping energies.

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Equations (23)

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d A AS (z) dz = i ω AS 2 n AS c ε 0 P( ω AS ) e −i k AS z ,
P( ω AS )=6 ε 0 χ R ( ω AS )| A L (z) | 2 A AS (z) e i k AS z +3 ε 0 χ F ( ω AS ) A L 2 (z) A S1 ∗ e i(2 k L − k S1 )z ,
2 χ R ( ω S1 )= χ F ( ω AS ) ∗ , χ R ( ω S1 )= χ R ( ω AS ) ∗ .
d A AS (z) dz =i 3 ω AS n AS c χ R ( ω S1 ) ∗ | A L (z) | 2 A AS (z)+i 3 ω AS n AS c χ R ( ω S1 ) ∗ A L 2 (z) A S1 ∗ exp(iΔkz).
g R =− 12 ω S1 ε 0 c 2 n L n S1 χ R ′​′ ( ω S1 ),
χ R ( ω S1 )=i χ R ′​′ ( ω S1 ).
A AS (z)= ∫ 0 z i 3 ω AS c n AS χ R ( ω S1 ) ∗ A L 2 (z) A S1 ∗ dz,
A AS (z) ∗ = ∫ 0 z −i 3 ω AS c n AS χ R ( ω S1 ) [ A L (z) ∗ ] 2 A S1 dz.
I AS (z)= 1 2 n AS ε 0 μ 0 A AS (z) A AS (z) ∗                           = 1 2 n AS ε 0 μ 0 ∫ 0 z i 3 ω AS c n AS χ R ( ω S1 ) ∗ A L 2 (z) A S1 ∗ dz ∫ 0 z −i 3 ω AS c n AS χ R ( ω S1 ) [ A L (z) ∗ ] 2 A S1 dz                           = 9 ω AS 2 | χ R ( ω S1 ) | 2 n AS n S1 c 2 I S1 | FWM ∫ 0 z A L 2 (z) dz ∫ 0 z [ A L (z) ∗ ] 2 dz                                                                                                           ,
d A L (z) dz =i 3 ω L n L c [ χ R ( ω S1 ) ∗ | A S1 | 2 A L (z)+ χ R ( ω AS ) ∗ | A AS (z) | 2 A L (z) ]                               +i 3 ω L 2 n L c [ χ F ( ω AS )+ χ F ( ω AS ) ∗ ] A AS (z) A S1 A L (z) ∗ e iΔkz                                 .
d A L (z) dz =i 3 ω L n L c χ R ( ω S1 ) ∗ | A S1 | 2 A L (z).
A L (z)= A L0 exp( i 3 ω L n L c χ R ( ω S1 ) ∗ | A S1 | 2 z )= A L0 exp( − 1 2 ω L ω S1 g R I S1 z ),
A L (z) ∗ = A L0 ∗ exp( −i 3 ω L n L c χ R ( ω S1 )| A S1 | 2 z )= A L0 ∗ exp( − 1 2 ω L ω S1 g R I S1 z ),
I AS (z)= 72 π 2 | χ R ( ω S1 ) ∗ | 2 n L 2 n AS n S1 c 2 ε 0 2 λ AS 2 I L0 2 I S1 [ 1−exp( − g 0 I S1 z ) g 0 I S1 ] 2 ,
η= 72 π 2 | χ R ( ω S1 ) ∗ | 2 n L 2 n AS n S1 c 2 ε 0 2 λ AS 2 .
η= g 0 2 n S1 λ L 2 8 n AS λ AS 2 .
I AS (t)=η I L0 2 (t) I S1 (t) { 1−exp[ − g 0 I S1 (t) l R ] g 0 I S1 (t) } 2 .
d I S1 (t) dt = 2 υ S1 υ L t r I L0 (t)[ 1− e − g 0 I S1 (t) l R ]− I S1 (t) t r [ 2 g 1 l R I S2 (t)+ln( 1 R 11 R 12 )+L ]           + k sp I L0 (t)                                                                              ,
d I S2 (t) dt = I S2 (t) t r { 2 l R [ g 2 I S1 (t)− g 3 I S3 (t) ]−ln( 1 R 21 R 22 )−L }+ k sp I S1 (t),
d I S3 (t) dt = I S3 (t) t r [ 2 g 3 l R I S2 (t)−ln( 1 R 31 R 32 )−L ]+ k sp I S2 (t),
E AS =(1− R AS ) ∫ 0 ∞ ∬ I AS (t) dSdt                  =(1− R AS )η S o ∫ 0 ∞ I L0 2 (t) I S1 (t) { 1−exp[ − g 0 I S1 (t) l R ] g 0 I S1 (t) } 2 dt,
E Sj = 1 2 ln( 1 R j1 ) ∫ 0 ∞ ∬ I Sj (t) dSdt= 1 2 ln( 1 R j1 ) S S ∫ 0 ∞ I Sj (t) dt,j=1, 2, 3
r = ∫ 0 l R | | A L ( z ) | 2 A A S ( z ) | d z ∫ 0 l R | A L 2 ( z ) A S 1 ∗ | d z ≈ ∫ 0 l R I L ( z ) I A S ( z ) d z ∫ 0 l R I L ( z ) I S 1 d z ,

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