Abstract

We develop an approximate analytical solution to the steady-state equations that describe a nested fiber, Raman laser cavity that incorporates Bragg reflectors. From this solution the output power, the threshold power, and the efficiency are found. Numerical simulation of a six-step cascade shows that the approximate analytical solution is accurate within 1.5%. It appears that the laser behaves differently depending on whether it has an even or odd number of Stokes waves.

© 2005 Optical Society of America

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References

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  1. R. W. Boyd, "Stimulated Raman scattering and stimulated Rayleigh-Wing scattering," in Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 365-388.
  2. S. G. Grubb, T. Strasser, W. Y. Cheung, W. A. Reed, V. Mizrahi, T. Erdogan, P. J. Lemaire, A. M. Vengsarkar, and D. J. DiGiovanni, "High-power 1.48 µm cascaded Raman laser in germanosilicate fibers," in Digest of Topical Meeting on Optical Amplifiers and Their Applications , Vol. 18 of 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 197-199.
  3. W. A. Reed, W. C. Coughran, and S. G. Grubb, "Numerical modeling of cascaded, CW, Raman fiber amplifiers and lasers," in Digest of Optical Fiber Communications Conference , Vol. 8 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 107-108.
  4. M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
    [CrossRef]
  5. 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]
  6. S. D. Jackson and P. H. Muir, "Theory and numerical simulation of nth-order cascaded CW Raman fiber lasers," J. Opt. Soc. Am. B 18, 1297-1306 (2001).
    [CrossRef]
  7. G. Vareille, O. Audouin, and E. Desurvire, "Numerical optimisation of power conversion efficiency in 1480 nm multi-Stokes Raman fibre lasers," Electron. Lett. 34, 675-676 (1998).
    [CrossRef]
  8. I. A. Bufetov and E. M. Dianov, "A simple analytic model of a CW multicascade fibre Raman laser," Sov. J. Quantum Electron. 30, 873-877 (2000).
    [CrossRef]
  9. J. AuYeung and A. Yariv, "Theory of CW Raman oscillation in optical fibers," J. Opt. Soc. Am. 69, 803-807 (1979).
    [CrossRef]
  10. C. Headley III and G. P. Agrawal, "Noise Characteristics and statistics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering," IEEE J. Quantum Electron. 31, 2058-2067 (1995).
    [CrossRef]
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, "Two point boundary value problem," in Numerical Recipes in C: The Art of Scientific Computing , 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 757-760.
  12. 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]

2005 (1)

2001 (2)

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

S. D. Jackson and P. H. Muir, "Theory and numerical simulation of nth-order cascaded CW Raman fiber lasers," J. Opt. Soc. Am. B 18, 1297-1306 (2001).
[CrossRef]

2000 (2)

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]

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

1998 (1)

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

1995 (1)

C. Headley III and G. P. Agrawal, "Noise Characteristics and statistics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering," IEEE J. Quantum Electron. 31, 2058-2067 (1995).
[CrossRef]

1979 (1)

Agrawal, G. P.

C. Headley III and G. P. Agrawal, "Noise Characteristics and statistics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering," IEEE J. Quantum Electron. 31, 2058-2067 (1995).
[CrossRef]

Audouin, O.

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

AuYeung , J.

Bouteiller, J. C.

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

Bufetov , I. A.

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

Burgoyne, B.

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]

Desurvire, E.

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

Dianov, E. M.

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

Eggleton, B. J.

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

Feder, K.

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

Godbout, N.

Headley, C.

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

Headley , C.

C. Headley III and G. P. Agrawal, "Noise Characteristics and statistics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering," IEEE J. Quantum Electron. 31, 2058-2067 (1995).
[CrossRef]

Horn, C.

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

Jackson , S. D.

Lacroix, S.

Mermelstein, M. D.

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

Muir, P. H.

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]

Steinvurzel, P.

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

Vareille, G.

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

Yariv, A.

Electron. Lett. (1)

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

IEEE J. Quantum Electron. (2)

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]

C. Headley III and G. P. Agrawal, "Noise Characteristics and statistics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering," IEEE J. Quantum Electron. 31, 2058-2067 (1995).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

M. D. Mermelstein, C. Headley, J. C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, and B. J. Eggleton, "Configurable three-wavelength Raman fiber laser for Raman amplification and dynamic gain flattening," IEEE Photonics Technol. Lett. 13, 1286-1288 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Sov. J. Quantum Electron. (1)

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

Other (4)

R. W. Boyd, "Stimulated Raman scattering and stimulated Rayleigh-Wing scattering," in Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 365-388.

S. G. Grubb, T. Strasser, W. Y. Cheung, W. A. Reed, V. Mizrahi, T. Erdogan, P. J. Lemaire, A. M. Vengsarkar, and D. J. DiGiovanni, "High-power 1.48 µm cascaded Raman laser in germanosilicate fibers," in Digest of Topical Meeting on Optical Amplifiers and Their Applications , Vol. 18 of 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 197-199.

W. A. Reed, W. C. Coughran, and S. G. Grubb, "Numerical modeling of cascaded, CW, Raman fiber amplifiers and lasers," in Digest of Optical Fiber Communications Conference , Vol. 8 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 107-108.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, "Two point boundary value problem," in Numerical Recipes in C: The Art of Scientific Computing , 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 757-760.

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Figures (6)

Fig. 1
Fig. 1

Diagram of an nth-order cascaded Raman fiber laser. λj is the wavelength of the jth Stokes wave. The vertical lines represent the Bragg gratings used as reflectors with reflectivities Rj±. Only the output coupler (OC) is not highly reflective.

Fig. 2
Fig. 2

Comparison between simulated (symbols) and analytical (curves) Stokes powers inside the cavity by use of the model with the depleted-pump approximation. The forward-propagating waves are in solid curves and circles, whereas the backward-propagating waves are in dashed curves and squares. Pj± indicate the power of the jth Stokes wave (zero being the pump, ± referring to the propagation direction). We use 96.7%-reflectivity mirrors except for the output mirror, which is 10%. The cavity length is 150 m and the injected pump power 6 W.

Fig. 3
Fig. 3

Relative difference along the cavity between the simulated solutions and the analytical solution with the depleted-pump approximation for the case presented in Fig. 2. The forward-propagating waves are in solid curves, whereas the backward-propagating waves are in dashed curves. Pj± indicate the power of the jth Stokes wave (zero being the pump, ± referring to the propagation direction).

Fig. 4
Fig. 4

Comparison between simulated (symbols) and analytical (curves) Stokes powers inside the cavity by use of the model with the depleted-pump approximation. The convention and simulation parameters are the same as in Fig. 2 except for the output coupler reflectivity, which is 50%. The agreement is clearly not as good as in Fig. 1 (where a 10%-reflectivity output coupler was used), because the pump is not completely depleted.

Fig. 5
Fig. 5

Comparison between simulated (symbols) and analytical (curves) Stokes powers inside the cavity by use of the model without the depleted-pump approximation. The convention and simulation parameters are the same as in Fig. 4. The agreement is better than in Fig. 4.

Fig. 6
Fig. 6

Relative difference along the cavity between simulated results and the full analytical solution for the case presented in Fig. 5. The forward-propagating waves are in solid curves, whereas the backward-propagating waves are in dashed curves. Pj± indicate the power of the jth Stokes wave (zero being the pump, ± referring to the propagation direction). With the full solution the simulated and analytical results agree within 1.5%.

Tables (1)

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Table 1 Fiber Characteristics

Equations (53)

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1P0+ dP0+dz=-1P0- dP0-dz=-g0(P1++P1-)-α0,
1Pj+ dPj+dz=-1Pj- dPj-dz=-gj(Pj+1++Pj+1-)+gj(Pj-1++Pj-1-)-αj,
1Pn+ dPn+dz=-1Pn- dPn-dz=gn(Pn-1++Pn-1-)-αn.
P0+(0)=Pin,P0-(L)=R0+P0+(L),
Pj+(0)=Rj-Pj-(0),Pj-(L)=Rj+Pj+(L),
1Pj+ dPj+dz=-1Pj- dPj-dz,
Pj+Pj-=Cj2,
Pout=(1-Rn+)Pn+(L)=1-Rn+(Rn+)1/2Cn.
Pj±=Cj exp(±θj),
dθ0dz=-2g0C1 cosh θ1-α0,
dθjdz=-2gjCj+1 cosh θj+1+2gjCj-1 cosh θj-1-αj,
dθndz=2gnCn-1 cosh θn-1-αn,
θ0(0)=ln(Pin/C0),θ0(L)=-ln(R0+)1/2
θj(0)=ln(Rj-)1/2,θj(L)=-ln(Rj+)1/2,
cosh θj1for0<j<n.
dθ0dz-2g0C1-α0,
dθ1dz-2g1C2+2g1C0 cosh θ0-α1,
dθjdz-2gjCj+1+2gjCj-1-αj,
dθn-1dz-2gn-1Cn cosh θn+2gn-1Cn-2-αn-1,
dθndz2gnCn-1-αn,
θj=2gj-Cj+1+Cj-1-αj2gjz+K,
K|z=0=ln(Rj-)1/2=K|z=L=2gjCj+1-Cj-1+αj2gjL-ln(Rj+)1/2.
C1=-α02g0+12g0L lnPin(R0+)1/2C0,
Cj+1=Cj-1-αj2gj+ln(Rj-Rj+)4gjL
for1<j<n-1,
Cn-1=αn2gn-ln(Rn-Rn+)4gnL,
dθ1dz=-2g1C2+2g1C0 cosh-zL lnPin(R0+)1/2C0+lnPinC0-α1,
dθn-1dz=-2gn-1Cn cosh-z2L ln(Rn-Rn+)+ln(Rn-)1/2+2gn-1Cn-2-αn-1.
C2=-α12g1+ln(R1-R1+)4g1L+C0 sinh(ln R0+)+sinhlnPinC0lnPinR0+C0
Cn-2=αn-12gn-1-ln(Rn-1-Rn-1+)4gn-1L+Cn sinh(ln Rn+)+sinh(ln Rn-)ln Rn+Rn-
C0=PinR0+ exp-Pin-Pr2C2+α12g1-ln(R1-R1+)4g1L,
C0=PinR0+ exp-2g0Lj=2jevennρj+α02g0,
ρj=αj2gj-ln(Rj-Rj+)4gjL.
ρj=Cj-1-Cj+1=(Pj-1+Pj-1-)1/2-(Pj+1+Pj+1-)1/2.
Cn=Pin4g0Lj=2jevennρj+α02g0-j=1joddn-1ρj×-Rn+ ln(Rn+Rn-)(1-Rn+),
C0=PinR0+ exp-Pin2j=1joddnρj,
Cn=Pin4g0Lj=1joddnρj-j=2jevenn-1ρj-α02g0×-Rn+ ln(Rn+Rn-)(1-Rn+).
Pout=-ln(Rn+Rn-)Pin4g0Lj=2jevennρj+α02g0-j=1joddn-1ρj,
Pthres=4g0Lj=2jevennρj+α02g0j=1joddn-1ρj,
η=-ln(Rn+Rn-)4g0Lj=2jevennρj+α02g0;
Pout=-ln(Rn+Rn-)Pin4g0Lj=1joddnρj-j=2jevenn-1ρj-α02g0,
Pthres=4g0Li=1ioddnρji=2ievenn-1ρj+α02g0,
η=-ln(Rn+Rn-)4g0Lj=1joddnρj.
Pr=C02Pin=PinR0+ exp-4g0Lj=2jevennρj+α02g0,
Pr=C02Pin=PinR0+ exp-Pin-Prj=1joddnρj.
Pout=-ln(Rn+Rn-)1+Rn+Rn-(1-Rn-)(1-Rn+)×Pinfp4g0Lj=2jevennρj+α02g0-j=1joddn-1ρj,
Pthres=4g0Lj=2jevennρj+α02g0j=1joddn-1ρj 1fp,
η=-ln(Rn+Rn-)1+Rn+Rn-(1-Rn-)(1-Rn+) fp4g0Lj=2jevennρj+α02g0,
fp=1-R0+ exp-4g0Lj=2jevennρj+α02g0;
Pout=-ln(Rn+Rn-)1+Rn+Rn-(1-Rn-)(1-Rn+)×Pin-Pr4g0Lj=1joddnρj-j=2jevenn-1ρj-α02g0,
Pthres=4g0Li=1ioddnρji=2ievenn-1ρj+α02g0+Pr(Pthres),
η=-ln(Rn+Rn-)1+Rn+Rn-(1-Rn-)(1-Rn+)  1-dPrdPin4g0Lj=1joddnρj.
Rn+Rn-1-Rn-1-Rn+0.04746.

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