Abstract

In this paper, an approximate explicit solution for the first-order Raman fiber laser is obtained by using Lambert W function. Good agreement between the explicit solution and numerical simulation is demonstrated. Furthermore, the optimal design of Raman fiber laser is discussed using the proposed solution. The optimal values of fiber length, reflectivity of output fiber Bragg grating and power transfer efficiency are obtained under different pump power. There exists a certain tolerance of the optimal parameters, in which the output power decreases only slightly. The optimal fiber length and reflectivity of output FBG decrease with increasing pump power.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
    [CrossRef]
  2. N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000).
    [CrossRef]
  3. M. Rini, I. Cristiani, and V. Degiorgio, "Numerical modeling and optimization of cascaded CW Raman fiber lasers," IEEE J. Quantum Elect. 36, 1117-1122 (2000).
    [CrossRef]
  4. S. Cierullies, H. Renner, and E. Brinkmeyer, "Numerical optimization of multi-wavelength and cascaded Raman fiber lasers," Opt. Commun. 217, 233-238 (2003).
    [CrossRef]
  5. I. A. Bufetov and E. M. Dianov, "A simple analytic model of a cw multicascade fibre Raman laser," Quantum Elect. 30,873-877 (2000).
    [CrossRef]
  6. S. A. Babin, D. V. Churkin, and E. V. Podivilov, "Intensity interactions in cascades of a two-stage Raman fiber laser," Opt. Commun. 226, 329-335 (2003).
    [CrossRef]
  7. B. Burgoyne, N. Godbout, and S. Lacroix, "Theoretical analysis of nth-order cascaded continuous-wave Raman fiber lasers. I. Model and resolution," J. Opt. Soc. Am. B 22, 764-771 (2005).
    [CrossRef]
  8. B. Burgoyne, N. Godbout, and S. Lacroix, "Theoretical analysis of nth-order cascaded continuous-wave Raman fiber lasers. II. Optimization and design rules," J. Opt. Soc. Am. B 22, 772-776 (2005).
    [CrossRef]
  9. J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006).
    [CrossRef]
  10. C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007).
    [CrossRef]
  11. S. D. Jackson and P. H. Muir, "Theory and numerical simulation of nth-order cascaded Raman fiber lasers," J. Opt. Soc. Am. B 18, 1297-1306 (2001).
    [CrossRef]
  12. J. AuYeung and A. Yariv, "Theory of cw Raman oscillation in optical fibers," J. Opt. Soc. Am. B 69, 803-807(1979).
    [CrossRef]
  13. E. M. Wright, "Solution of the Equation zez=a," Bull. Am. Math. Soc. 65, 89-93(1959).
    [CrossRef]
  14. J. C. Bouteiller, "Spectral modeling of Raman fiber lasers," IEEE Photon. Technol. Lett. 15, 1698-1700(2003).
    [CrossRef]

2007

C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007).
[CrossRef]

2006

J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006).
[CrossRef]

2005

2003

S. A. Babin, D. V. Churkin, and E. V. Podivilov, "Intensity interactions in cascades of a two-stage Raman fiber laser," Opt. Commun. 226, 329-335 (2003).
[CrossRef]

S. Cierullies, H. Renner, and E. Brinkmeyer, "Numerical optimization of multi-wavelength and cascaded Raman fiber lasers," Opt. Commun. 217, 233-238 (2003).
[CrossRef]

J. C. Bouteiller, "Spectral modeling of Raman fiber lasers," IEEE Photon. Technol. Lett. 15, 1698-1700(2003).
[CrossRef]

2001

2000

I. A. Bufetov and E. M. Dianov, "A simple analytic model of a cw multicascade fibre Raman laser," Quantum Elect. 30,873-877 (2000).
[CrossRef]

N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000).
[CrossRef]

M. Rini, I. Cristiani, and V. Degiorgio, "Numerical modeling and optimization of cascaded CW Raman fiber lasers," IEEE J. Quantum Elect. 36, 1117-1122 (2000).
[CrossRef]

1997

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

1979

J. AuYeung and A. Yariv, "Theory of cw Raman oscillation in optical fibers," J. Opt. Soc. Am. B 69, 803-807(1979).
[CrossRef]

1959

E. M. Wright, "Solution of the Equation zez=a," Bull. Am. Math. Soc. 65, 89-93(1959).
[CrossRef]

Babin, S. A.

S. A. Babin, D. V. Churkin, and E. V. Podivilov, "Intensity interactions in cascades of a two-stage Raman fiber laser," Opt. Commun. 226, 329-335 (2003).
[CrossRef]

Belov, A. V.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Bouteiller, J. C.

J. C. Bouteiller, "Spectral modeling of Raman fiber lasers," IEEE Photon. Technol. Lett. 15, 1698-1700(2003).
[CrossRef]

Brinkmeyer, E.

S. Cierullies, H. Renner, and E. Brinkmeyer, "Numerical optimization of multi-wavelength and cascaded Raman fiber lasers," Opt. Commun. 217, 233-238 (2003).
[CrossRef]

Bubnov, M. M.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Bufetov, I. A.

I. A. Bufetov and E. M. Dianov, "A simple analytic model of a cw multicascade fibre Raman laser," Quantum Elect. 30,873-877 (2000).
[CrossRef]

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Burgoyne, B.

Cai, Z. P.

C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007).
[CrossRef]

Chen, J. P.

J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006).
[CrossRef]

Churkin, D. V.

S. A. Babin, D. V. Churkin, and E. V. Podivilov, "Intensity interactions in cascades of a two-stage Raman fiber laser," Opt. Commun. 226, 329-335 (2003).
[CrossRef]

Cierullies, S.

S. Cierullies, H. Renner, and E. Brinkmeyer, "Numerical optimization of multi-wavelength and cascaded Raman fiber lasers," Opt. Commun. 217, 233-238 (2003).
[CrossRef]

Cristiani, I.

M. Rini, I. Cristiani, and V. Degiorgio, "Numerical modeling and optimization of cascaded CW Raman fiber lasers," IEEE J. Quantum Elect. 36, 1117-1122 (2000).
[CrossRef]

Degiorgio, V.

M. Rini, I. Cristiani, and V. Degiorgio, "Numerical modeling and optimization of cascaded CW Raman fiber lasers," IEEE J. Quantum Elect. 36, 1117-1122 (2000).
[CrossRef]

Dianov, E. M.

I. A. Bufetov and E. M. Dianov, "A simple analytic model of a cw multicascade fibre Raman laser," Quantum Elect. 30,873-877 (2000).
[CrossRef]

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Fursa, D. G.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Godbout, N.

Huang, C. H.

C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007).
[CrossRef]

Jackson, S. D.

Kim, N. S.

N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000).
[CrossRef]

Koltashev, V. V.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Lacroix, S.

Li, C.

N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000).
[CrossRef]

Li, X. W.

J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006).
[CrossRef]

Luo, Z. Q.

C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007).
[CrossRef]

Medvedkov, O. I.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Muir, P. H.

Plotnichenko, V. G.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Podivilov, E. V.

S. A. Babin, D. V. Churkin, and E. V. Podivilov, "Intensity interactions in cascades of a two-stage Raman fiber laser," Opt. Commun. 226, 329-335 (2003).
[CrossRef]

Prabhu, M.

N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000).
[CrossRef]

Prokhorov, A. M.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Renner, H.

S. Cierullies, H. Renner, and E. Brinkmeyer, "Numerical optimization of multi-wavelength and cascaded Raman fiber lasers," Opt. Commun. 217, 233-238 (2003).
[CrossRef]

Rini, M.

M. Rini, I. Cristiani, and V. Degiorgio, "Numerical modeling and optimization of cascaded CW Raman fiber lasers," IEEE J. Quantum Elect. 36, 1117-1122 (2000).
[CrossRef]

Semjonov, S. L.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Song, J.

N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000).
[CrossRef]

Ueda, K.

N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000).
[CrossRef]

Vasiliev, S. A.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

Wang, Y. P.

J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006).
[CrossRef]

Wright, E. M.

E. M. Wright, "Solution of the Equation zez=a," Bull. Am. Math. Soc. 65, 89-93(1959).
[CrossRef]

Wu, G. L.

J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006).
[CrossRef]

Xu, H.Y.

C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007).
[CrossRef]

Ye, C. C.

C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007).
[CrossRef]

Zhou, J. H.

J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006).
[CrossRef]

Bull. Am. Math. Soc.

E. M. Wright, "Solution of the Equation zez=a," Bull. Am. Math. Soc. 65, 89-93(1959).
[CrossRef]

Electron. Lett.

E. M. Dianov, D. G. Fursa, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov, V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov, and A. M. Prokhorov, "CW high power 1.24μm and 1.48μm Raman lasers based on low loss phosphosilicate fibre," Electron. Lett. 33, 1542-1544 (1997).
[CrossRef]

IEEE J. Quantum Elect.

M. Rini, I. Cristiani, and V. Degiorgio, "Numerical modeling and optimization of cascaded CW Raman fiber lasers," IEEE J. Quantum Elect. 36, 1117-1122 (2000).
[CrossRef]

IEEE Photon. Technol. Lett.

J. H. Zhou, J. P. Chen, X. W. Li, G. L. Wu, Y. P. Wang, "Exact analytical solution for Raman fiber laser," IEEE Photon. Technol. Lett. 18,1097-1099 (2006).
[CrossRef]

J. C. Bouteiller, "Spectral modeling of Raman fiber lasers," IEEE Photon. Technol. Lett. 15, 1698-1700(2003).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun

N. S. Kim, M. Prabhu, C. Li, J. Song, and K. Ueda, "1239/1484 nm cascaded phosphosilicate Raman fiber laser with CW output power of 1.36 W at 1484 nm pumped by CW Yb-doped double-clad fiber laser at 1064 nm and spectral continuum generation," Opt. Commun 176, 219-222 (2000).
[CrossRef]

Opt. Commun.

S. Cierullies, H. Renner, and E. Brinkmeyer, "Numerical optimization of multi-wavelength and cascaded Raman fiber lasers," Opt. Commun. 217, 233-238 (2003).
[CrossRef]

S. A. Babin, D. V. Churkin, and E. V. Podivilov, "Intensity interactions in cascades of a two-stage Raman fiber laser," Opt. Commun. 226, 329-335 (2003).
[CrossRef]

Opt. Fiber Technol.

C. H. Huang, Z. P. Cai, C. C. Ye, H.Y. Xu, and Z. Q. Luo, "Analytic modeling of the P-doped cascaded Raman fiber lasers," Opt. Fiber Technol. 13, 22-26 (2007).
[CrossRef]

Quantum Elect.

I. A. Bufetov and E. M. Dianov, "A simple analytic model of a cw multicascade fibre Raman laser," Quantum Elect. 30,873-877 (2000).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram of a double-pass pumping Raman fiber laser

Fig. 2.
Fig. 2.

The comparison between analytic and numerical solution (L=250m, R1=30%). (squares: numerical; lines: analytic). (a) Output power of pump and Stokes radiations versus input pump power; (b) Power distributions of pump and Stokes radiations in Raman fiber when Pin =5W

Fig. 3.
Fig. 3.

Design optimization of RFL when Pin =5W (squares: numerical; lines: analytical). (a) Optimal fiber length and power transfer efficiency versus R 1; (b) Power transfer efficiency versus R 1 and L

Fig. 4.
Fig. 4.

The optimal fiber length, reflectivity of output FBG and power transfer efficiency vesus input pump power

Equations (21)

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

± 1 P 0 ± dP 0 ± dz = α 0 g λ 1 λ 0 ( P 1 + + P 1 )
± 1 P 1 ± dP 1 ± dz = α 1 + g ( P 0 + + P 0 )
P 0 + ( 0 ) = P in ´ , P 0 ( L ) = R 0 L P 0 + ( L )
P 0 + ( 0 ) = R 1 0 P 1 ( 0 ) , P 1 ( L ) = R 1 L P 1 + ( L )
du 0 dz = α 0 2 g λ 1 λ 0 c 1 cosh ( u 1 )
du 1 dz = α 1 + 2 g c 0 cosh ( u 0 )
{ u 0 ( 0 ) = ln ( P ́ in c 0 ) , u 0 ( L ) = ln ( R 0 L ) 2 u 1 ( 0 ) = ln ( R 0 L ) 2 , u 1 ( L ) = −ln ( R 1 L ) 2
2 g λ 1 λ 0 c 1 Ll 1 eff = ln P ´ in c 0 δ 0
2 g c 0 Ll 0 eff = δ 1
c 1 = λ 0 λ 1 . α 1 [ u 0 ( L ) u 0 ( 0 ) ] 2 g c 0 { sinh [ u 0 ( L ) ] sinh [ u 0 ( 0 ) ] } α 0 [ u 1 ( L ) u 1 ( 0 ) ] 2 g { sinh [ u 1 ( L ) ] sinh [ u 1 ( 0 ) ] }
P th = 1 η in α 0 δ 1 g ( 1 e α 0 L ) ( 1 + R 0 L e α 0 L )
l 0 eff = { sinh [ u 0 ( L ) ] sinh [ u 0 ( 0 ) ] } [ u 0 ( L ) u 0 ( 0 ) ]
u 0 ( 0 ) = ln P in ´ c 0 = P in ´ gL δ 1 + 1 2 W 0 [ 2 P in ´ gL δ 1 exp ( 2 P in ´ gL δ 1 ) ]
l 1 eff = { sinh [ u 1 ( L ) ] sinh [ u 1 ( 0 ) ] } [ u 1 ( L ) u 1 ( 0 ) ]
c 1 = λ 0 λ 1 u 0 ( 0 ) δ 0 2 g Ll 1 eff
P i out = T i c i e u i ( L ) ( i = 0,1 )
u 1 ( z ) = α 1 z + u 1 ( 0 ) + 2 gL c 0 { sinh [ u 0 ( z ) ] sinh [ u 0 ( 0 ) ] } [ u 0 ( L ) u 0 ( 0 ) ]
P 0 ( 0 ) = P in ´ exp [ 2 u 0 ( 0 ) ]
P 0 + ( L ) = P in ´ exp [ u 0 ( 0 ) ] R 0 L
η s λ 0 λ 1 T 1 R 1 L 1 2 δ 1 1 l 1 eff
L = 1 2 ln 1 R 1 0 R 1 L ln 2 P in ´ g α 1 ( 2 P in ´ g α 1 α 1 ln 2 P in ´ g α 1 )

Metrics