Abstract

Considering the amplified spontaneous emission, the saturation effect and the energy distributions of the incident pump and seed lasers, a physical model is established to describe the kinetic process and the output performance of a four sided diode pumped alkali vapor laser amplifier. According to the experimental parameters of a single-side pumped configuration with a diffuse type hollow cylinder cavity, energy distributions in the cell and influences of several important factors are simulated and analyzed. The model is validated by comparing the simulation result with the experimental data, which shows the model can provide an effective way for designing an efficient diode four-side symmetrically pumped alkali vapor laser amplifier.

© 2015 Optical Society of America

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References

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  1. J. Han, Y. Wang, H. Cai, W. Zhang, L. Xue, and H. Wang, “Algorithm for evaluation of temperature distribution of a vapor cell in a diode-pumped alkali laser system: part I,” Opt. Express 22(11), 13988–14003 (2014).
    [Crossref] [PubMed]
  2. W. F. Krupke, R. J. Beach, V. K. Kanz, and S. A. Payne, “Resonance transition 795-nm rubidium laser,” Opt. Lett. 28(23), 2336–2338 (2003).
    [Crossref] [PubMed]
  3. B. V. Zhdanov, A. Stooke, G. Boyadjian, A. Voci, and R. J. Knize, “Rubidium vapor laser pumped by two laser diode arrays,” Opt. Lett. 33(5), 414–415 (2008).
    [Crossref] [PubMed]
  4. T. Ehrenreich, B. Zhdanov, T. Takekoshi, S. P. Phipps, and R. J. Knize, “Diode pumped caesium laser,” Electron. Lett. 41(7), 415–416 (2005).
    [Crossref]
  5. B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
    [Crossref]
  6. A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
    [Crossref]
  7. G. A. Pitz, G. D. Hager, T. B. Tafoya, J. W. Young, G. P. Perram, and D. A. Hostutler, “An experimental high pressure line shape study of the rubidium D1 and D2 transitions with the noble gases, methane, and ethane,” In SPIE LASE. International Society for Optics and Photonics (2014), pp. 896208.
  8. D. A. Hostutler and W. L. Klennert, “Power enhancement of a Rubidium vapor laser with a master oscillator power amplifier,” Opt. Express 16(11), 8050–8053 (2008).
    [Crossref] [PubMed]
  9. B. V. Zhdanov and R. J. Knize, “Efficient diode pumped cesium vapor amplifier,” Opt. Commun. 281(15), 4068–4070 (2008).
    [Crossref]
  10. B. L. Pan, Y. J. Wang, Q. Zhu, and J. Yang, “Modeling of an alkali vapor laser MOPA system,” Opt. Commun. 284(7), 1963–1966 (2011).
    [Crossref]
  11. B. V. Zhdanov, M. K. Shaffer, and R. J. Knize, “Scaling of diode-pumped Cs laser: transverse pump, unstable cavity, MOPA,” In SPIE LASE. International Society for Optics and Photonics (2010), pp. 75810F.
  12. A. I. Parkhomenko and A. M. Shalagin, “An alkali metal vapor laser amplifier,” J. Exp. Theor. Phys. 119(1), 24–35 (2014).
    [Crossref]
  13. Z. Yang, H. Wang, Q. Lu, W. Hua, and X. Xu, “Modeling of an optically side-pumped alkali vapor amplifier with consideration of amplified spontaneous emission,” Opt. Express 19(23), 23118–23131 (2011).
    [Crossref] [PubMed]
  14. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004).
    [Crossref]
  15. A. E. Siegman, Lasers (University science books, 1986), Chapter 7.
  16. G. A. Pitz and G. P. Perram, “Pressure broadening of the D1 and D2 lines in diode pumped alkali lasers,” In High-Power Laser Ablation 2008. International Society for Optics and Photonics (2008), pp. 700526–700526.
  17. W. Xie, S. C. Tam, Y. L. Lam, J. Liu, H. Yang, J. Gu, and W. Tan, “Influence of the thermal effect on the TEM00 mode output power of a laser-diode side-pumped solid-state laser,” Appl. Opt. 39(30), 5482–5487 (2000).
    [Crossref] [PubMed]
  18. M. K. Shaffer, T. C. Lilly, B. V. Zhdanov, and R. J. Knize, “In situ non-perturbative temperature measurement in a Cs alkali laser,” Opt. Lett. 40(1), 119–122 (2015).
    [Crossref] [PubMed]
  19. Z. Li, R. Tan, W. Huang, and D. Zhang, “Quasicontinuous wave linearly polarized rubidium vapor laser pumped by a 5-bar laser diode stack,” Opt. Eng. 53(11), 116113 (2014).
    [Crossref]

2015 (1)

2014 (3)

J. Han, Y. Wang, H. Cai, W. Zhang, L. Xue, and H. Wang, “Algorithm for evaluation of temperature distribution of a vapor cell in a diode-pumped alkali laser system: part I,” Opt. Express 22(11), 13988–14003 (2014).
[Crossref] [PubMed]

A. I. Parkhomenko and A. M. Shalagin, “An alkali metal vapor laser amplifier,” J. Exp. Theor. Phys. 119(1), 24–35 (2014).
[Crossref]

Z. Li, R. Tan, W. Huang, and D. Zhang, “Quasicontinuous wave linearly polarized rubidium vapor laser pumped by a 5-bar laser diode stack,” Opt. Eng. 53(11), 116113 (2014).
[Crossref]

2012 (1)

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

2011 (2)

2008 (3)

2007 (1)

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

2005 (1)

T. Ehrenreich, B. Zhdanov, T. Takekoshi, S. P. Phipps, and R. J. Knize, “Diode pumped caesium laser,” Electron. Lett. 41(7), 415–416 (2005).
[Crossref]

2004 (1)

2003 (1)

2000 (1)

Beach, R. J.

Bogachev, A. V.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Boyadjian, G.

Cai, H.

Dubinskii, M. A.

Dudov, A. M.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Ehrenreich, T.

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

T. Ehrenreich, B. Zhdanov, T. Takekoshi, S. P. Phipps, and R. J. Knize, “Diode pumped caesium laser,” Electron. Lett. 41(7), 415–416 (2005).
[Crossref]

Eroshenko, V. A.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Flusche, B.

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

Garanin, S. G.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Gu, J.

Han, J.

Havko, A.

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

Hostutler, D. A.

Hua, W.

Huang, W.

Z. Li, R. Tan, W. Huang, and D. Zhang, “Quasicontinuous wave linearly polarized rubidium vapor laser pumped by a 5-bar laser diode stack,” Opt. Eng. 53(11), 116113 (2014).
[Crossref]

Kanz, V. K.

Klennert, W. L.

Knize, R. J.

M. K. Shaffer, T. C. Lilly, B. V. Zhdanov, and R. J. Knize, “In situ non-perturbative temperature measurement in a Cs alkali laser,” Opt. Lett. 40(1), 119–122 (2015).
[Crossref] [PubMed]

B. V. Zhdanov, A. Stooke, G. Boyadjian, A. Voci, and R. J. Knize, “Rubidium vapor laser pumped by two laser diode arrays,” Opt. Lett. 33(5), 414–415 (2008).
[Crossref] [PubMed]

B. V. Zhdanov and R. J. Knize, “Efficient diode pumped cesium vapor amplifier,” Opt. Commun. 281(15), 4068–4070 (2008).
[Crossref]

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

T. Ehrenreich, B. Zhdanov, T. Takekoshi, S. P. Phipps, and R. J. Knize, “Diode pumped caesium laser,” Electron. Lett. 41(7), 415–416 (2005).
[Crossref]

Koval, N.

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

Krupke, W. F.

Kulikov, S. M.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Lam, Y. L.

Li, Z.

Z. Li, R. Tan, W. Huang, and D. Zhang, “Quasicontinuous wave linearly polarized rubidium vapor laser pumped by a 5-bar laser diode stack,” Opt. Eng. 53(11), 116113 (2014).
[Crossref]

Lilly, T. C.

Liu, J.

Lu, Q.

Maes, C.

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

Meeker, T.

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

Merkle, L. D.

Mikaelian, G. T.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Pan, B. L.

B. L. Pan, Y. J. Wang, Q. Zhu, and J. Yang, “Modeling of an alkali vapor laser MOPA system,” Opt. Commun. 284(7), 1963–1966 (2011).
[Crossref]

Panarin, V. A.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Parkhomenko, A. I.

A. I. Parkhomenko and A. M. Shalagin, “An alkali metal vapor laser amplifier,” J. Exp. Theor. Phys. 119(1), 24–35 (2014).
[Crossref]

Pautov, V. O.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Payne, S. A.

Phipps, S. P.

T. Ehrenreich, B. Zhdanov, T. Takekoshi, S. P. Phipps, and R. J. Knize, “Diode pumped caesium laser,” Electron. Lett. 41(7), 415–416 (2005).
[Crossref]

Rus, A. V.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Shaffer, M. K.

Shalagin, A. M.

A. I. Parkhomenko and A. M. Shalagin, “An alkali metal vapor laser amplifier,” J. Exp. Theor. Phys. 119(1), 24–35 (2014).
[Crossref]

Stooke, A.

Sukharev, S. A.

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Takekoshi, T.

T. Ehrenreich, B. Zhdanov, T. Takekoshi, S. P. Phipps, and R. J. Knize, “Diode pumped caesium laser,” Electron. Lett. 41(7), 415–416 (2005).
[Crossref]

Tam, S. C.

Tan, R.

Z. Li, R. Tan, W. Huang, and D. Zhang, “Quasicontinuous wave linearly polarized rubidium vapor laser pumped by a 5-bar laser diode stack,” Opt. Eng. 53(11), 116113 (2014).
[Crossref]

Tan, W.

Voci, A.

Wang, H.

Wang, Y.

Wang, Y. J.

B. L. Pan, Y. J. Wang, Q. Zhu, and J. Yang, “Modeling of an alkali vapor laser MOPA system,” Opt. Commun. 284(7), 1963–1966 (2011).
[Crossref]

Worker, B.

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

Xie, W.

Xu, X.

Xue, L.

Yang, H.

Yang, J.

B. L. Pan, Y. J. Wang, Q. Zhu, and J. Yang, “Modeling of an alkali vapor laser MOPA system,” Opt. Commun. 284(7), 1963–1966 (2011).
[Crossref]

Yang, Z.

Zhang, D.

Z. Li, R. Tan, W. Huang, and D. Zhang, “Quasicontinuous wave linearly polarized rubidium vapor laser pumped by a 5-bar laser diode stack,” Opt. Eng. 53(11), 116113 (2014).
[Crossref]

Zhang, W.

Zhdanov, B.

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

T. Ehrenreich, B. Zhdanov, T. Takekoshi, S. P. Phipps, and R. J. Knize, “Diode pumped caesium laser,” Electron. Lett. 41(7), 415–416 (2005).
[Crossref]

Zhdanov, B. V.

Zhu, Q.

B. L. Pan, Y. J. Wang, Q. Zhu, and J. Yang, “Modeling of an alkali vapor laser MOPA system,” Opt. Commun. 284(7), 1963–1966 (2011).
[Crossref]

Appl. Opt. (1)

Electron. Lett. (1)

T. Ehrenreich, B. Zhdanov, T. Takekoshi, S. P. Phipps, and R. J. Knize, “Diode pumped caesium laser,” Electron. Lett. 41(7), 415–416 (2005).
[Crossref]

J. Exp. Theor. Phys. (1)

A. I. Parkhomenko and A. M. Shalagin, “An alkali metal vapor laser amplifier,” J. Exp. Theor. Phys. 119(1), 24–35 (2014).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

B. V. Zhdanov and R. J. Knize, “Efficient diode pumped cesium vapor amplifier,” Opt. Commun. 281(15), 4068–4070 (2008).
[Crossref]

B. L. Pan, Y. J. Wang, Q. Zhu, and J. Yang, “Modeling of an alkali vapor laser MOPA system,” Opt. Commun. 284(7), 1963–1966 (2011).
[Crossref]

B. Zhdanov, C. Maes, T. Ehrenreich, A. Havko, N. Koval, T. Meeker, B. Worker, B. Flusche, and R. J. Knize, “Optically pumped potassium laser,” Opt. Commun. 270(2), 353–355 (2007).
[Crossref]

Opt. Eng. (1)

Z. Li, R. Tan, W. Huang, and D. Zhang, “Quasicontinuous wave linearly polarized rubidium vapor laser pumped by a 5-bar laser diode stack,” Opt. Eng. 53(11), 116113 (2014).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Quantum Electron. (1)

A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Eroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, “Diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation,” Quantum Electron. 42(2), 95–98 (2012).
[Crossref]

Other (4)

G. A. Pitz, G. D. Hager, T. B. Tafoya, J. W. Young, G. P. Perram, and D. A. Hostutler, “An experimental high pressure line shape study of the rubidium D1 and D2 transitions with the noble gases, methane, and ethane,” In SPIE LASE. International Society for Optics and Photonics (2014), pp. 896208.

B. V. Zhdanov, M. K. Shaffer, and R. J. Knize, “Scaling of diode-pumped Cs laser: transverse pump, unstable cavity, MOPA,” In SPIE LASE. International Society for Optics and Photonics (2010), pp. 75810F.

A. E. Siegman, Lasers (University science books, 1986), Chapter 7.

G. A. Pitz and G. P. Perram, “Pressure broadening of the D1 and D2 lines in diode pumped alkali lasers,” In High-Power Laser Ablation 2008. International Society for Optics and Photonics (2008), pp. 700526–700526.

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Figures (13)

Fig. 1
Fig. 1 Schematic diagram of an alkali vapor laser MOPA system in symmetrically four-side pumped configuration.
Fig. 2
Fig. 2 The cross-sectional geometry for four-side pumped amplifier cell.
Fig. 3
Fig. 3 Quarter cross-sectional geometry of the amplifier cell. Where the black bold lines are the present positions of the pump lights that being calculated, while the gray bold lines are the former positions of them. (a) shows the pump lights propagate along the two vertical directions and (b) shows pump lights come from four symmetrical sides.
Fig. 4
Fig. 4 Flowchart of the process of iterative algorithm for four-side pumped configuration.
Fig. 5
Fig. 5 Pump energy distribution for a four-side pumped alkali vapor amplifier with ω p =1mm and ω s =2.5mm .
Fig. 6
Fig. 6 Pump energy distribution for a four-side pumped alkali vapor amplifier with ω p = ω s =2.5mm .
Fig. 7
Fig. 7 Laser energy distribution in the x-y plane at the output end for a four-side pumped alkali vapor amplifier with ω p =1mm and ω s =2.5mm , where the broken circle represents the waist of the seed laser. The peak value of the output laser energy is 600W/cm2.
Fig. 8
Fig. 8 Laser energy distribution in the x-y plane at the output end for a four-side pumped alkali vapor amplifier with ω p = ω s =2.5mm , where the broken circle represents the waist of the seed laser. The peak value of the output laser energy is 180W/cm2.
Fig. 9
Fig. 9 Output power of the power amplifier vs. cell temperature when Psl = 5W.
Fig. 10
Fig. 10 Output power of the transversely pumped amplifier as function of the input seed power from the master oscillator when T = 103°C.
Fig. 11
Fig. 11 Output power of the transversely pumped amplifier as function of the seed power with different pump power and ω p =1mm and ω s =2.5mm .
Fig. 12
Fig. 12 Length of the cell influence on output power of the transversely pumped amplifier.
Fig. 13
Fig. 13 Dependences of the amplifier output power on η mode with ω s =R and different pump intensity.

Equations (20)

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

d n 1 ( x,y ) dt = Γ p ( x,y )+ Γ l ( x,y )+ Γ ASE ( x,y )+ n 2 ( x,y ) τ D1 + n 3 ( x,y ) τ D2
   d n 2 ( x,y ) dt = Γ l ( x,y ) Γ ASE ( x,y )+ γ 32 [ n 3 ( x,y )2 n 2 ( x,y )exp( ΔE KT ) ] n 2 ( x,y ) τ D1
d n 3 ( x,y ) dt = Γ p ( x,y ) γ 32 [ n 3 ( x,y )2 n 2 ( x,y )exp( ΔE KT ) ] n 3 ( x,y ) τ D2
P x ( x,y,λ )=0 P y ( x,y,λ )=0
ω p = ω p 0 ( sλ π ω p 0 2 ) 2 +1
P p + ( x,s,λ )= P x + ( x,s,λ )+ P y + ( x,s,λ ) P p + ( s,y,λ )= P y + ( s,y,λ )+ P x + ( s,y,λ )
P p + ( x,s,λ )= P y + ( x,s,λ ) P p + ( s,y,λ )= P y + ( s,y,λ )
P p + ( x,s,λ )=2× P x + ( x,s,λ )
P p + ( s,y,λ )=2× P y + ( s,y,λ )
P y ± ( x 0 ,s,λ )=Pp 2 π Δx ω p exp( 2 x 0 2 ω p 2 )× ln2 π 2 Δ λ p exp[ 4ln2 Δ v p 2 ( c λ c λ p ) 2 ]
P p + ( x,s,λ )= P x + ( x,s,λ )+ P y + ( x,s,λ ) P p ( x,s,λ )= P x ( x,s,λ )+ P y ( x,s,λ )
Γ p ( x,s )= 1 h v p V p P p + ( x,s,λ )×{ 1exp[ ( n 1 ( x,s ) 1 2 n 3 ( x,s ) ) σ 13 ( λ )Δy ] }dλ + 1 h v p V p P p ( x,s,λ )×{ exp[ ( n 1 ( x,s ) 1 2 n 3 ( x,s ) ) σ 13 ( λ )Δy ]1 }dλ
Γ l ( x,s )= P l ( x,s ) P s ( x,s ) h v l V l
η mode = V o V s
P l ( x,s )= η mode P s ( x,s )exp[ ( n 2 ( x,s ) n 1 ( x,s ) ) σ 21 L ] ×exp[ ( P l ( x,s ) P s ( x,s ) )/ P sat ]
P s ( x,s )= P sl 2 π ΔxΔy ω s 2 exp[ 2( x 2 + s 2 ) ω s 2 ]
Γ ASE ( x,s )= 2 V l π τ D1 L 4 n 2 ( x,s ) n 2 ( x,s ) n 1 ( x,s ) l( λ ) × exp[ ( n 2 ( x,s ) n 1 ( x,s ) ) σ 21 ( λ )L ]1 σ 21 ( λ ) dλ
P y + ( x,s+Δy,λ )= P y + ( x,s,λ )exp[ ( n 1 ( x,s ) 1 2 n 3 ( x,s ) ) σ 13 ( λ )Δy ] P y ( x,s+Δy,λ )= P y ( x,s,λ )exp[ ( n 1 ( x,s ) 1 2 n 3 ( x,s ) ) σ 13 ( λ )Δy ]
n i ( 0 ) * ( x,y )={ n i ( 0 ) ( x,y ) x[ 0,R ],y[ 0,R ] n i ( 0 ) ( x,y ) x[ 0,R ],y[ 0,R ] n i ( 0 ) ( x,y ) x[ 0,R ],y[ 0,R ] n i ( 0 ) ( x,y ) x[ 0,R ],y[ 0,R ] ( i=1,2,3 )
P p tot( 1 ) ( x,y,λ )= P x + ( x,y,λ )+ P x ( x,y,λ )+ P y + ( x,y,λ )+ P y ( x,y,λ )

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