Abstract

Using the classical treatment of the stimulated Raman-scattering process, we use a theoretical model to simulate the operation of an nth-order cascaded Raman fiber laser. We introduce the partial differential equations employed to describe the propagation and time dependence of the forward and reverse-propagating fields of an nth-order cascaded Raman fiber laser. Under steady-state conditions, these equations form the well-known system of first-order, nonlinear boundary-value ordinary differential equations, with separated boundary conditions. We solve this system of equations numerically with the use of mono-implicit Runge–Kutta methods within a defect-control framework. We consider cascaded Raman fiber lasers of orders 2 through 5 and examine the parameters that influence the operation of these devices. We also provide preliminary results on the investigation of a time-dependent model in which the pump power is assumed to vary periodically with time. The associated system of first-order, hyperbolic, partial differential equations is treated by employing a transverse method-of-lines algorithm; the time derivatives are discretized with a finite-difference scheme, yielding a large system of boundary-value ordinary differential equations. We establish that for sinusoidal modulation of the pump the Stokes cavity modes exhibit antiphase dynamics typical of a system of locally coupled nonlinear oscillators.

© 2001 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2000

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

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

M. Prabhu, N. S. Kim, L. Jianren, and K. Ueda, “Simultaneous two-color CW Raman fiber laser with maximum output power of 1.05 W/1239 nm and 0.95 W/1484 nm using phosphosilicate fiber,” Opt. Commun. 182, 305–309 (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 fibre laser at 1064 nm and spectral continuum generation,” Opt. Commun. 176, 219–222 (2000).
[CrossRef]

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Vasilev, and O. I. Medvedkov, “Three-cascaded 1407-nm Raman laser based on phosphorous-doped silica fiber,” Opt. Lett. 25, 402–404 (2000).
[CrossRef]

1999

V. I. Karpov, E. M. Dianov, V. M. Paramonov, O. I. Medvedkov, M. M. Bubnov, S. L. Semyonov, S. A. Vasiliev, V. N. Protopopov, O. N. Egorova, V. F. Hopin, A. N. Guryanov, M. P. Bachynski, and W. R. L. Clements, “Laser-diode-pumped phosphosilicate-fiber Raman laser with an output power of 1 W at 1.48 μm,” Opt. Lett. 24, 887–889 (1999).
[CrossRef]

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

P. H. Muir, “Optimal discrete and continuous mono-implicit Runge–Kutta schemes for BVODE’s,” Adv. Comput. Math. 10, 135–167 (1999).
[CrossRef]

1998

C. Szwaj, S. Bielawski, and D. Derozier, “Acoustical and optical branches in the spectral waves of a laser,” Phys. Rev. A 57, 3022–3027 (1998).
[CrossRef]

A. Bertoni and G. C. Reali, “1.24-μm cascaded Raman laser for 1.31-μm Raman fiber amplifiers,” Appl. Phys. B 67, 5–10 (1998).
[CrossRef]

G. Vareille, O. Audouin, and E. Desurive, “Numerical optimisation of power conversion efficiency in 1480 nm multi-Stokes Raman fibre lasers,” Electron. Lett. 34, 675–676 (1998).
[CrossRef]

S. V. Chernikov, N. S. Platonov, D. V. Gapontsev, D. I. Chang, M. J. Guy, and J. R. Taylor, “Raman fibre laser operating at 1.24 μm,” Electron. Lett. 34, 680–681 (1998).
[CrossRef]

1997

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

A. Bertoni, “Analysis of the efficiency of a third order cascaded Raman operating at the wavelength of 1.24 μm,” Opt. Quantum Electron. 29, 1047–1058 (1997).
[CrossRef]

1996

P. Persephonis, S. V. Chernikov, and J. R. Taylor, “Cascaded CW Raman laser source 1.6–1.9 μm,” Electron. Lett. 32, 1486–1487 (1996).
[CrossRef]

C. Szwaj, S. Bielawski, and D. Derozier, “Propagation of waves in the spectrum of a multimode laser,” Phys. Rev. Lett. 77, 4540–4543 (1996).
[CrossRef] [PubMed]

W. H. Enright and P. H. Muir, “Runge–Kutta software with defect control for boundary value ODE’s,” SIAM J. Sci. Stat. Comput. 17, 479–497 (1996).
[CrossRef]

1991

J. R. Cash and M. H. Wright, “A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation,” SIAM J. Sci. Stat. Comput. 12, 971–989 (1991).
[CrossRef]

1987

G. Bader and U. M. Ascher, “A new basis implementation for a mixed order boundary value ODE solver,” SIAM J. Sci. Stat. Comput. 8, 483–500 (1987).
[CrossRef]

1983

J. C. Diaz, G. Fairweather, and P. Keast, “Algorithm 603. COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 376–380 (1983).
[CrossRef]

J. C. Diaz, G. Fairweather, and P. Keast, “FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 358–375 (1983).
[CrossRef]

1981

U. M. Ascher, J. Christiansen, and R. D. Russell, “Collocation software for boundary value ODE’s,” ACM Trans. Math. Software 7, 209–222 (1981).
[CrossRef]

1979

1977

M. Lentini and V. Pereyra, “An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers,” SIAM J. Numer. Anal. 14, 94–111 (1977).
[CrossRef]

Amodio, P.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

Ascher, U. M.

G. Bader and U. M. Ascher, “A new basis implementation for a mixed order boundary value ODE solver,” SIAM J. Sci. Stat. Comput. 8, 483–500 (1987).
[CrossRef]

U. M. Ascher, J. Christiansen, and R. D. Russell, “Collocation software for boundary value ODE’s,” ACM Trans. Math. Software 7, 209–222 (1981).
[CrossRef]

Audouin, O.

G. Vareille, O. Audouin, and E. Desurive, “Numerical optimisation of power conversion efficiency in 1480 nm multi-Stokes Raman fibre lasers,” Electron. Lett. 34, 675–676 (1998).
[CrossRef]

AuYeung, J.

Bachynski, M. P.

Bader, G.

G. Bader and U. M. Ascher, “A new basis implementation for a mixed order boundary value ODE solver,” SIAM J. Sci. Stat. Comput. 8, 483–500 (1987).
[CrossRef]

Belov, A. V.

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Bertoni, A.

A. Bertoni and G. C. Reali, “1.24-μm cascaded Raman laser for 1.31-μm Raman fiber amplifiers,” Appl. Phys. B 67, 5–10 (1998).
[CrossRef]

A. Bertoni, “Analysis of the efficiency of a third order cascaded Raman operating at the wavelength of 1.24 μm,” Opt. Quantum Electron. 29, 1047–1058 (1997).
[CrossRef]

Bielawski, S.

C. Szwaj, S. Bielawski, and D. Derozier, “Acoustical and optical branches in the spectral waves of a laser,” Phys. Rev. A 57, 3022–3027 (1998).
[CrossRef]

C. Szwaj, S. Bielawski, and D. Derozier, “Propagation of waves in the spectrum of a multimode laser,” Phys. Rev. Lett. 77, 4540–4543 (1996).
[CrossRef] [PubMed]

Bubnov, M. M.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Vasilev, and O. I. Medvedkov, “Three-cascaded 1407-nm Raman laser based on phosphorous-doped silica fiber,” Opt. Lett. 25, 402–404 (2000).
[CrossRef]

V. I. Karpov, E. M. Dianov, V. M. Paramonov, O. I. Medvedkov, M. M. Bubnov, S. L. Semyonov, S. A. Vasiliev, V. N. Protopopov, O. N. Egorova, V. F. Hopin, A. N. Guryanov, M. P. Bachynski, and W. R. L. Clements, “Laser-diode-pumped phosphosilicate-fiber Raman laser with an output power of 1 W at 1.48 μm,” Opt. Lett. 24, 887–889 (1999).
[CrossRef]

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Bufetov, I. A.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Vasilev, and O. I. Medvedkov, “Three-cascaded 1407-nm Raman laser based on phosphorous-doped silica fiber,” Opt. Lett. 25, 402–404 (2000).
[CrossRef]

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Cash, J. R.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

J. R. Cash and M. H. Wright, “A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation,” SIAM J. Sci. Stat. Comput. 12, 971–989 (1991).
[CrossRef]

Chang, D. I.

S. V. Chernikov, N. S. Platonov, D. V. Gapontsev, D. I. Chang, M. J. Guy, and J. R. Taylor, “Raman fibre laser operating at 1.24 μm,” Electron. Lett. 34, 680–681 (1998).
[CrossRef]

Chernikov, S. V.

S. V. Chernikov, N. S. Platonov, D. V. Gapontsev, D. I. Chang, M. J. Guy, and J. R. Taylor, “Raman fibre laser operating at 1.24 μm,” Electron. Lett. 34, 680–681 (1998).
[CrossRef]

P. Persephonis, S. V. Chernikov, and J. R. Taylor, “Cascaded CW Raman laser source 1.6–1.9 μm,” Electron. Lett. 32, 1486–1487 (1996).
[CrossRef]

Christiansen, J.

U. M. Ascher, J. Christiansen, and R. D. Russell, “Collocation software for boundary value ODE’s,” ACM Trans. Math. Software 7, 209–222 (1981).
[CrossRef]

Clements, W. R. L.

Costantini, D.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Cristiani, I.

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

Derozier, D.

C. Szwaj, S. Bielawski, and D. Derozier, “Acoustical and optical branches in the spectral waves of a laser,” Phys. Rev. A 57, 3022–3027 (1998).
[CrossRef]

C. Szwaj, S. Bielawski, and D. Derozier, “Propagation of waves in the spectrum of a multimode laser,” Phys. Rev. Lett. 77, 4540–4543 (1996).
[CrossRef] [PubMed]

Desurive, E.

G. Vareille, O. Audouin, and E. Desurive, “Numerical optimisation of power conversion efficiency in 1480 nm multi-Stokes Raman fibre lasers,” Electron. Lett. 34, 675–676 (1998).
[CrossRef]

Dianov, E. M.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Vasilev, and O. I. Medvedkov, “Three-cascaded 1407-nm Raman laser based on phosphorous-doped silica fiber,” Opt. Lett. 25, 402–404 (2000).
[CrossRef]

V. I. Karpov, E. M. Dianov, V. M. Paramonov, O. I. Medvedkov, M. M. Bubnov, S. L. Semyonov, S. A. Vasiliev, V. N. Protopopov, O. N. Egorova, V. F. Hopin, A. N. Guryanov, M. P. Bachynski, and W. R. L. Clements, “Laser-diode-pumped phosphosilicate-fiber Raman laser with an output power of 1 W at 1.48 μm,” Opt. Lett. 24, 887–889 (1999).
[CrossRef]

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Diaz, J. C.

J. C. Diaz, G. Fairweather, and P. Keast, “Algorithm 603. COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 376–380 (1983).
[CrossRef]

J. C. Diaz, G. Fairweather, and P. Keast, “FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 358–375 (1983).
[CrossRef]

Dvarelas, D.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Egorova, O. N.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

V. I. Karpov, E. M. Dianov, V. M. Paramonov, O. I. Medvedkov, M. M. Bubnov, S. L. Semyonov, S. A. Vasiliev, V. N. Protopopov, O. N. Egorova, V. F. Hopin, A. N. Guryanov, M. P. Bachynski, and W. R. L. Clements, “Laser-diode-pumped phosphosilicate-fiber Raman laser with an output power of 1 W at 1.48 μm,” Opt. Lett. 24, 887–889 (1999).
[CrossRef]

Enright, W. H.

W. H. Enright and P. H. Muir, “Runge–Kutta software with defect control for boundary value ODE’s,” SIAM J. Sci. Stat. Comput. 17, 479–497 (1996).
[CrossRef]

Fairweather, G.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

J. C. Diaz, G. Fairweather, and P. Keast, “Algorithm 603. COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 376–380 (1983).
[CrossRef]

J. C. Diaz, G. Fairweather, and P. Keast, “FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 358–375 (1983).
[CrossRef]

Gapontsev, D. V.

S. V. Chernikov, N. S. Platonov, D. V. Gapontsev, D. I. Chang, M. J. Guy, and J. R. Taylor, “Raman fibre laser operating at 1.24 μm,” Electron. Lett. 34, 680–681 (1998).
[CrossRef]

Gladwell, I.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

Grekov, M. V.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Vasilev, and O. I. Medvedkov, “Three-cascaded 1407-nm Raman laser based on phosphorous-doped silica fiber,” Opt. Lett. 25, 402–404 (2000).
[CrossRef]

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Gur’yanov, A. N.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Guryanov, A. N.

Guy, M. J.

S. V. Chernikov, N. S. Platonov, D. V. Gapontsev, D. I. Chang, M. J. Guy, and J. R. Taylor, “Raman fibre laser operating at 1.24 μm,” Electron. Lett. 34, 680–681 (1998).
[CrossRef]

Hopin, V. F.

Iocco, A.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Jianren, L.

M. Prabhu, N. S. Kim, L. Jianren, and K. Ueda, “Simultaneous two-color CW Raman fiber laser with maximum output power of 1.05 W/1239 nm and 0.95 W/1484 nm using phosphosilicate fiber,” Opt. Commun. 182, 305–309 (2000).
[CrossRef]

Karpov, V. I.

Keast, P.

J. C. Diaz, G. Fairweather, and P. Keast, “FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 358–375 (1983).
[CrossRef]

J. C. Diaz, G. Fairweather, and P. Keast, “Algorithm 603. COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 376–380 (1983).
[CrossRef]

Khopin, V. F.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

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 fibre laser at 1064 nm and spectral continuum generation,” Opt. Commun. 176, 219–222 (2000).
[CrossRef]

M. Prabhu, N. S. Kim, L. Jianren, and K. Ueda, “Simultaneous two-color CW Raman fiber laser with maximum output power of 1.05 W/1239 nm and 0.95 W/1484 nm using phosphosilicate fiber,” Opt. Commun. 182, 305–309 (2000).
[CrossRef]

Koltashev, V. V.

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Kraut, G. L.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

Lentini, M.

M. Lentini and V. Pereyra, “An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers,” SIAM J. Numer. Anal. 14, 94–111 (1977).
[CrossRef]

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 fibre laser at 1064 nm and spectral continuum generation,” Opt. Commun. 176, 219–222 (2000).
[CrossRef]

Limberger, H. G.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Medvedkov, O. I.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, S. A. Vasilev, and O. I. Medvedkov, “Three-cascaded 1407-nm Raman laser based on phosphorous-doped silica fiber,” Opt. Lett. 25, 402–404 (2000).
[CrossRef]

V. I. Karpov, E. M. Dianov, V. M. Paramonov, O. I. Medvedkov, M. M. Bubnov, S. L. Semyonov, S. A. Vasiliev, V. N. Protopopov, O. N. Egorova, V. F. Hopin, A. N. Guryanov, M. P. Bachynski, and W. R. L. Clements, “Laser-diode-pumped phosphosilicate-fiber Raman laser with an output power of 1 W at 1.48 μm,” Opt. Lett. 24, 887–889 (1999).
[CrossRef]

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Muir, P. H.

P. H. Muir, “Optimal discrete and continuous mono-implicit Runge–Kutta schemes for BVODE’s,” Adv. Comput. Math. 10, 135–167 (1999).
[CrossRef]

W. H. Enright and P. H. Muir, “Runge–Kutta software with defect control for boundary value ODE’s,” SIAM J. Sci. Stat. Comput. 17, 479–497 (1996).
[CrossRef]

Paprzycki, M.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

Paramonov, V. M.

Pereyra, V.

M. Lentini and V. Pereyra, “An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers,” SIAM J. Numer. Anal. 14, 94–111 (1977).
[CrossRef]

Persephonis, P.

P. Persephonis, S. V. Chernikov, and J. R. Taylor, “Cascaded CW Raman laser source 1.6–1.9 μm,” Electron. Lett. 32, 1486–1487 (1996).
[CrossRef]

Platonov, N. S.

S. V. Chernikov, N. S. Platonov, D. V. Gapontsev, D. I. Chang, M. J. Guy, and J. R. Taylor, “Raman fibre laser operating at 1.24 μm,” Electron. Lett. 34, 680–681 (1998).
[CrossRef]

Plotnichenko, V. G.

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Prabhu, M.

M. Prabhu, N. S. Kim, L. Jianren, and K. Ueda, “Simultaneous two-color CW Raman fiber laser with maximum output power of 1.05 W/1239 nm and 0.95 W/1484 nm using phosphosilicate fiber,” Opt. Commun. 182, 305–309 (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 fibre laser at 1064 nm and spectral continuum generation,” Opt. Commun. 176, 219–222 (2000).
[CrossRef]

Prokhorov, A. M.

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Protopopov, V. N.

Reali, G. C.

A. Bertoni and G. C. Reali, “1.24-μm cascaded Raman laser for 1.31-μm Raman fiber amplifiers,” Appl. Phys. B 67, 5–10 (1998).
[CrossRef]

Rini, M.

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

Roussos, G.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

Russell, R. D.

U. M. Ascher, J. Christiansen, and R. D. Russell, “Collocation software for boundary value ODE’s,” ACM Trans. Math. Software 7, 209–222 (1981).
[CrossRef]

Salathe, R.-P.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Semenov, S. L.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Semjonov, S. L.

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Semyonov, S. L.

Shubin, A. V.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[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 fibre laser at 1064 nm and spectral continuum generation,” Opt. Commun. 176, 219–222 (2000).
[CrossRef]

Szwaj, C.

C. Szwaj, S. Bielawski, and D. Derozier, “Acoustical and optical branches in the spectral waves of a laser,” Phys. Rev. A 57, 3022–3027 (1998).
[CrossRef]

C. Szwaj, S. Bielawski, and D. Derozier, “Propagation of waves in the spectrum of a multimode laser,” Phys. Rev. Lett. 77, 4540–4543 (1996).
[CrossRef] [PubMed]

Taylor, J. R.

S. V. Chernikov, N. S. Platonov, D. V. Gapontsev, D. I. Chang, M. J. Guy, and J. R. Taylor, “Raman fibre laser operating at 1.24 μm,” Electron. Lett. 34, 680–681 (1998).
[CrossRef]

P. Persephonis, S. V. Chernikov, and J. R. Taylor, “Cascaded CW Raman laser source 1.6–1.9 μm,” Electron. Lett. 32, 1486–1487 (1996).
[CrossRef]

Ueda, K.

M. Prabhu, N. S. Kim, L. Jianren, and K. Ueda, “Simultaneous two-color CW Raman fiber laser with maximum output power of 1.05 W/1239 nm and 0.95 W/1484 nm using phosphosilicate fiber,” Opt. Commun. 182, 305–309 (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 fibre laser at 1064 nm and spectral continuum generation,” Opt. Commun. 176, 219–222 (2000).
[CrossRef]

Vareille, G.

G. Vareille, O. Audouin, and E. Desurive, “Numerical optimisation of power conversion efficiency in 1480 nm multi-Stokes Raman fibre lasers,” Electron. Lett. 34, 675–676 (1998).
[CrossRef]

Vasil’ev, S. A.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Vasilev, S. A.

Vasiliev, S. A.

V. I. Karpov, E. M. Dianov, V. M. Paramonov, O. I. Medvedkov, M. M. Bubnov, S. L. Semyonov, S. A. Vasiliev, V. N. Protopopov, O. N. Egorova, V. F. Hopin, A. N. Guryanov, M. P. Bachynski, and W. R. L. Clements, “Laser-diode-pumped phosphosilicate-fiber Raman laser with an output power of 1 W at 1.48 μm,” Opt. Lett. 24, 887–889 (1999).
[CrossRef]

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

Wright, M. H.

J. R. Cash and M. H. Wright, “A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation,” SIAM J. Sci. Stat. Comput. 12, 971–989 (1991).
[CrossRef]

Wright, R. W.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

Yariv, A.

Yashkov, M. V.

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

ACM Trans. Math. Software

U. M. Ascher, J. Christiansen, and R. D. Russell, “Collocation software for boundary value ODE’s,” ACM Trans. Math. Software 7, 209–222 (1981).
[CrossRef]

J. C. Diaz, G. Fairweather, and P. Keast, “Algorithm 603. COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 376–380 (1983).
[CrossRef]

J. C. Diaz, G. Fairweather, and P. Keast, “FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination,” ACM Trans. Math. Software 9, 358–375 (1983).
[CrossRef]

Adv. Comput. Math.

P. H. Muir, “Optimal discrete and continuous mono-implicit Runge–Kutta schemes for BVODE’s,” Adv. Comput. Math. 10, 135–167 (1999).
[CrossRef]

Appl. Phys. B

A. Bertoni and G. C. Reali, “1.24-μm cascaded Raman laser for 1.31-μm Raman fiber amplifiers,” Appl. Phys. B 67, 5–10 (1998).
[CrossRef]

Electron. Lett.

G. Vareille, O. Audouin, and E. Desurive, “Numerical optimisation of power conversion efficiency in 1480 nm multi-Stokes Raman fibre lasers,” Electron. Lett. 34, 675–676 (1998).
[CrossRef]

P. Persephonis, S. V. Chernikov, and J. R. Taylor, “Cascaded CW Raman laser source 1.6–1.9 μm,” Electron. Lett. 32, 1486–1487 (1996).
[CrossRef]

E. M. Dianov, M. V. Grekov, 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 1.24 μm and 1.48 μm Raman lasers based on low loss phosphosilicate fibre,” Electron. Lett. 33, 1542–1544 (1997).
[CrossRef]

S. V. Chernikov, N. S. Platonov, D. V. Gapontsev, D. I. Chang, M. J. Guy, and J. R. Taylor, “Raman fibre laser operating at 1.24 μm,” Electron. Lett. 34, 680–681 (1998).
[CrossRef]

IEEE J. Quantum Electron.

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

J. Opt. Soc. Am.

Kvant. Elektron. (Moscow)

E. M. Dianov, I. A. Bufetov, M. M. Bubnov, M. V. Grekov, A. V. Shubin, S. A. Vasil’ev, O. I. Medvedkov, S. L. Semenov, O. N. Egorova, A. N. Gur’yanov, V. F. Khopin, M. V. Yashkov, D. Dvarelas, A. Iocco, D. Costantini, H. G. Limberger, and R.-P. Salathe, “Continuous-wave highly efficient phosphosilicate fibre-based Raman laser (λ=1.24 μm),” Kvant. Elektron. (Moscow) 29, 97–100 (1999).
[CrossRef]

Numer. Linear Algebra Appl.

P. Amodio, J. R. Cash, G. Roussos, R. W. Wright, G. Fairweather, I. Gladwell, G. L. Kraut, and M. Paprzycki, “Almost block diagonal linear systems: sequential and parallel solution techniques, and applications,” Numer. Linear Algebra Appl. 7, 275–317 (2000).
[CrossRef]

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 fibre laser at 1064 nm and spectral continuum generation,” Opt. Commun. 176, 219–222 (2000).
[CrossRef]

M. Prabhu, N. S. Kim, L. Jianren, and K. Ueda, “Simultaneous two-color CW Raman fiber laser with maximum output power of 1.05 W/1239 nm and 0.95 W/1484 nm using phosphosilicate fiber,” Opt. Commun. 182, 305–309 (2000).
[CrossRef]

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Opt. Quantum Electron.

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of an nth-order cascaded Raman fiber laser. HR represents a highly reflecting Bragg grating, and R% indicates a Bragg grating with <100% reflectivity at the nth Stokes wavelength. λn represents the wavelength of the nth Stokes cavity mode.

Fig. 2
Fig. 2

Calculated values of (a) the pump, (b) the first Stokes, and (c) the second Stokes cavity modes as a function of length along the second-order cascaded Raman fiber laser. For this simulation the launched pump power was 3.3 W, fiber length was 1000 m, core diameter was 5 µm, and the reflectivity of the second Stokes radiation at the output end of the fiber was 10%. Potential excess losses as a result of splices, etc., were neglected, α0=1.7 dB/km, and α1=α2=1 dB/km.

Fig. 3
Fig. 3

Calculated values of (a) the pump, (b) the first Stokes, (c) the second Stokes, and (d) the third Stokes cavity modes as a function of length along the third-order cascaded Raman fiber laser. For this simulation the launched pump power was 3.3 W, fiber length was 1000 m, core diameter was 5 µm, the reflectivity of the third Stokes radiation at the output end of the fiber was 10%. Potential excess losses as a result of splices, etc., were neglected, α0=1.7 dB/km, and α1=α2=α3=1 dB/km.

Fig. 4
Fig. 4

Calculated values of (a) the pump, (b) the first Stokes, (c) the second Stokes, (d) the third Stokes, and (e) the fourth Stokes cavity modes as a function of length along the fourth-order cascaded Raman fiber laser. For this simulation the launched pump power was 3.3 W, fiber length was 1000 m, and the reflectivity of the fourth Stokes radiation at the output end of the fiber was 10%. Potential excess losses arising from splices, etc., were neglected, α0=1.7 dB/km, and α1=α2=α3=α4=1 dB/km.

Fig. 5
Fig. 5

Calculated values of (a) the pump, (b) the first Stokes, (c) the second Stokes, (d) the third Stokes, (e) the fourth Stokes, and (f) the fifth Stokes cavity modes as a function of length along the fifth-order cascaded Raman fiber laser. For this simulation the launched pump power was 3.3 W, fiber length was 1000 m, core diameter was 5 µm, and the reflectivity of the fifth Stokes radiation at the output end of the fiber was 10%. Potential excess losses arising from splices, etc., were neglected, α0=1.7 dB/km, and α1=α2=α3=α4=α5=1 dB/km.

Fig. 6
Fig. 6

Calculated values of (a) the slope efficiency and (b) the pump power at threshold for a second-order cascaded Raman fiber laser. For this simulation the reflectivity of the second Stokes radiation at the output end of the fiber was 15%, core diameter was 5 µm, α0=1.7 dB/km, and α1=α2=1 dB/km.

Fig. 7
Fig. 7

Calculated values of (a) the pump (taken at z=0), (b) the forward-propagating first Stokes cavity mode (taken at z=1000 m), and (c) the forward-propagating second Stokes cavity mode (taken at z=1000 m) as a function of time for sinusoidal modulation of the pump. The pump modulation had a 50% modulation depth, a 7 ms period, and 3.3-W time-averaged launched power. The fiber length was 1000 m, core diameter was 5 µm, and the second Stokes reflection coefficient was 95% (i.e., 5% output coupling).

Fig. 8
Fig. 8

Calculated values of (a) the pump (taken at z=0), (b) the forward-propagating first Stokes cavity mode (taken at z=1000 m), and (c) the forward-propagating second Stokes cavity mode (taken at z=1000 m) as a function of time for sinusoidal modulation of the pump. The pump modulation had a 100% modulation depth, a 7 ms period, and 3.3-W time-averaged launched power. The fiber length was 1000 m, the core diameter was 5 µm, and the second Stokes reflection coefficient was 95% (i.e., 5% output coupling).

Fig. 9
Fig. 9

Calculated values of (a) the forward-propagating first Stokes cavity mode (taken at z=L), and (b) the forward-propagating second Stokes cavity mode (taken at z=L) for sinusoidal modulation of the pump as a function of time and for various values of the fiber length. The pump modulation had a 100% modulation depth, a 7-ms period, and 3.3-W time-averaged launched power. The second Stokes reflection coefficient was 95% (i.e., 5% output coupling), and the core diameter was 5 µm.

Equations (15)

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1P0+(z, t) P0+(z, t)z+nc P0+(z, t)t
=1-P0-(z, t) P0-(z, t)z+nc P0-(z, t)t
=-α0-ω0ω1γ1[P1+(z, t)+P1-(z, t)],
1Pi+(z, t) Pi+(z, t)z+nc P0+(z, t)t
=1-Pi-(z, t) Pi-(z, t)z+nc Pi-(z, t)t
=-αi+γi[Pi-1+(z, t)+Pi-1-(z, t)]-γi+1 ωiωi+1[Pi+1+(z, t)+Pi+1-(z, t)]fori=1ton-1,
1Pn+(z, t) Pn+(z, t)z+nc Pn+(z, t)t
=1-Pn-(z, t) Pn-(z, t)z+nc Pn-(z, t)t
=-αn+γn[Pn-1+(z, t)+Pn-1-(z, t)].
P0+(0)=PLaunch+R0P0-(0),
Pi+(0)=Pi-(0)fori=1ton.
Pi-(L)=Pi+(L)fori=0to(n-1),
Pn-(L)=RnPn+(L).
ddz i=1n ω0ωiPi+(z)+P0+(z)
-ddz i=1n ω0ωiPi-(z)+P0-(z)=0.

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