Abstract

I theoretically investigate the simultaneous generation of cascaded stimulated Brillouin scattering (CSBS) and four-wave mixing (FWM) in a fiber Fabry–Perot resonator with reflection mirrors. The coupled-wave equations for a pump, Stokes, and anti-Stokes waves are derived. To discuss the stationary interplay between the two nonlinear processes, I further deduce the coupled-power equations by introducing some approximations into them. I investigate the effect of FWM on CSBS for asymmetric and symmetric resonators by numerically solving the coupled-power equations under the boundary conditions at the two mirrors. Compared with the case without FWM, the FWM process leads to the gradual generation of higher-order Stokes waves, splitting of the intersection of two transmitted or reflected optical powers, and the generation of anti-Stokes waves. The present analysis can well explain an earlier experiment [Opt. Commun. 32, 385 (1980)].

© 2003 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  2. D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10–19 (1983).
    [Crossref]
  3. K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “cw Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
    [Crossref]
  4. K. O. Hill, D. C. Johnson, and B. S. Kawasaki, “cw generation of multiple Stokes and anti-Stokes Brillouin-shifted frequencies,” Appl. Phys. Lett. 29, 185–187 (1976).
    [Crossref]
  5. L. F. Stokes, M. Chodorow, and H. J. Show, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509–511 (1982).
    [Crossref] [PubMed]
  6. P. Labudde, P. Anliker, and H. P. Weber, “Transmission of narrow band high power laser radiation through optical fibers,” Opt. Commun. 32, 385–390 (1980).
    [Crossref]
  7. C. Montes, D. Bahloul, I. Bongrand, J. Botineau, G. Cheval, A. Mamhould, E. Picholle, and A. Picozzi, “Self-pulsing and dynamic bistability in cw-pumped Brillouin fiber ring lasers,” J. Opt. Soc. Am. B 16, 932–951 (1999).
    [Crossref]
  8. V. Lecoeuche, B. Ségard, and J. Zemmouri, “On route of chaos in stimulated Brillouin scattering with feedback,” Opt. Commun. 172, 335–345 (1999).
    [Crossref]
  9. H. Li and K. Ogusu, “Transient stimulated Brillouin scattering in a fiber ring resonator and its effect on optical Kerr bistability,” J. Opt. Soc. Am. B 18, 93–100 (2001).
    [Crossref]
  10. K. Ogusu, “Dynamics of stimulated Brillouin scattering in a short and high-finesse fiber Fabry-Perot resonator,” Opt. Rev. 8, 358–363 (2001).
    [Crossref]
  11. S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
    [Crossref] [PubMed]
  12. K. Ogusu and A. Sakai, “Generation and dynamics of cascaded stimulated Brillouin scattering in a high-finesse fiber Fabry-Perot resonator,” Jpn. J. Appl. Phys. 41, 609–616 (2002).
    [Crossref]
  13. K. Ogusu, “Analysis of steady-state cascaded stimulated Brillouin scattering in a fiber Fabry-Pérot resonator,” IEEE Photonics Technol. Lett. 14, 947–949 (2002).
    [Crossref]
  14. K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
    [Crossref]
  15. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
    [Crossref]
  16. A. Vatarescu, “Light conversion in nonlinear monomode optical fibers,” J. Lightwave Technol. LT-5, 1652–1659 (1987).
    [Crossref]
  17. N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
    [Crossref]
  18. G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
    [Crossref]
  19. P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Fiber four-wave mixing demultiplexing with inherent parametric amplification,” J. Lightwave Technol. 15, 2051–2058 (1997).
    [Crossref]
  20. S. Song, C. T. Allen, K. R. Demarest, and R. Hui, “Intensity-dependent phase-matching effects on four-wave mixing in optical fibers,” J. Lightwave Technol. 17, 2285–2290 (1999).
    [Crossref]
  21. M. Salhi, A. Hideur, T. Chartier, M. Brunel, G. Martel, C. Ozkul, and F. Sanchez, “Evidence of Brillouin scattering in an ytterbium-doped double-clad fiber laser,” Opt. Lett. 27, 1294–1296 (2002).
    [Crossref]
  22. T. H. Russell and W. B. Roh, “Threshold of second-order stimulated Brillouin scattering in optical fiber,” J. Opt. Soc. Am. B 19, 2341–2345 (2002).
    [Crossref]
  23. R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1160 (1979).
    [Crossref]

2002 (4)

K. Ogusu and A. Sakai, “Generation and dynamics of cascaded stimulated Brillouin scattering in a high-finesse fiber Fabry-Perot resonator,” Jpn. J. Appl. Phys. 41, 609–616 (2002).
[Crossref]

K. Ogusu, “Analysis of steady-state cascaded stimulated Brillouin scattering in a fiber Fabry-Pérot resonator,” IEEE Photonics Technol. Lett. 14, 947–949 (2002).
[Crossref]

M. Salhi, A. Hideur, T. Chartier, M. Brunel, G. Martel, C. Ozkul, and F. Sanchez, “Evidence of Brillouin scattering in an ytterbium-doped double-clad fiber laser,” Opt. Lett. 27, 1294–1296 (2002).
[Crossref]

T. H. Russell and W. B. Roh, “Threshold of second-order stimulated Brillouin scattering in optical fiber,” J. Opt. Soc. Am. B 19, 2341–2345 (2002).
[Crossref]

2001 (2)

H. Li and K. Ogusu, “Transient stimulated Brillouin scattering in a fiber ring resonator and its effect on optical Kerr bistability,” J. Opt. Soc. Am. B 18, 93–100 (2001).
[Crossref]

K. Ogusu, “Dynamics of stimulated Brillouin scattering in a short and high-finesse fiber Fabry-Perot resonator,” Opt. Rev. 8, 358–363 (2001).
[Crossref]

1999 (3)

1997 (1)

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Fiber four-wave mixing demultiplexing with inherent parametric amplification,” J. Lightwave Technol. 15, 2051–2058 (1997).
[Crossref]

1995 (1)

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[Crossref] [PubMed]

1991 (1)

1987 (2)

A. Vatarescu, “Light conversion in nonlinear monomode optical fibers,” J. Lightwave Technol. LT-5, 1652–1659 (1987).
[Crossref]

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

1983 (1)

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10–19 (1983).
[Crossref]

1982 (2)

L. F. Stokes, M. Chodorow, and H. J. Show, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509–511 (1982).
[Crossref] [PubMed]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[Crossref]

1980 (1)

P. Labudde, P. Anliker, and H. P. Weber, “Transmission of narrow band high power laser radiation through optical fibers,” Opt. Commun. 32, 385–390 (1980).
[Crossref]

1979 (1)

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1160 (1979).
[Crossref]

1978 (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

1976 (2)

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “cw Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

K. O. Hill, D. C. Johnson, and B. S. Kawasaki, “cw generation of multiple Stokes and anti-Stokes Brillouin-shifted frequencies,” Appl. Phys. Lett. 29, 185–187 (1976).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

Allen, C. T.

Andrekson, P. A.

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Fiber four-wave mixing demultiplexing with inherent parametric amplification,” J. Lightwave Technol. 15, 2051–2058 (1997).
[Crossref]

Anliker, P.

P. Labudde, P. Anliker, and H. P. Weber, “Transmission of narrow band high power laser radiation through optical fibers,” Opt. Commun. 32, 385–390 (1980).
[Crossref]

Bahloul, D.

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[Crossref]

Bongrand, I.

Botineau, J.

Braun, R. P.

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

Brunel, M.

Cappellini, G.

Chartier, T.

Cheval, G.

Chodorow, M.

Cotter, D.

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10–19 (1983).
[Crossref]

Demarest, K. R.

Hedekvist, P. O.

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Fiber four-wave mixing demultiplexing with inherent parametric amplification,” J. Lightwave Technol. 15, 2051–2058 (1997).
[Crossref]

Hideur, A.

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “cw Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

K. O. Hill, D. C. Johnson, and B. S. Kawasaki, “cw generation of multiple Stokes and anti-Stokes Brillouin-shifted frequencies,” Appl. Phys. Lett. 29, 185–187 (1976).
[Crossref]

Hui, R.

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “cw Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

K. O. Hill, D. C. Johnson, and B. S. Kawasaki, “cw generation of multiple Stokes and anti-Stokes Brillouin-shifted frequencies,” Appl. Phys. Lett. 29, 185–187 (1976).
[Crossref]

Karlsson, M.

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Fiber four-wave mixing demultiplexing with inherent parametric amplification,” J. Lightwave Technol. 15, 2051–2058 (1997).
[Crossref]

Kawasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

K. O. Hill, D. C. Johnson, and B. S. Kawasaki, “cw generation of multiple Stokes and anti-Stokes Brillouin-shifted frequencies,” Appl. Phys. Lett. 29, 185–187 (1976).
[Crossref]

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “cw Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

Labudde, P.

P. Labudde, P. Anliker, and H. P. Weber, “Transmission of narrow band high power laser radiation through optical fibers,” Opt. Commun. 32, 385–390 (1980).
[Crossref]

Lecoeuche, V.

V. Lecoeuche, B. Ségard, and J. Zemmouri, “On route of chaos in stimulated Brillouin scattering with feedback,” Opt. Commun. 172, 335–345 (1999).
[Crossref]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[Crossref] [PubMed]

Li, H.

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Mamhould, A.

Martel, G.

Montes, C.

Ogusu, K.

K. Ogusu, “Analysis of steady-state cascaded stimulated Brillouin scattering in a fiber Fabry-Pérot resonator,” IEEE Photonics Technol. Lett. 14, 947–949 (2002).
[Crossref]

K. Ogusu and A. Sakai, “Generation and dynamics of cascaded stimulated Brillouin scattering in a high-finesse fiber Fabry-Perot resonator,” Jpn. J. Appl. Phys. 41, 609–616 (2002).
[Crossref]

K. Ogusu, “Dynamics of stimulated Brillouin scattering in a short and high-finesse fiber Fabry-Perot resonator,” Opt. Rev. 8, 358–363 (2001).
[Crossref]

H. Li and K. Ogusu, “Transient stimulated Brillouin scattering in a fiber ring resonator and its effect on optical Kerr bistability,” J. Opt. Soc. Am. B 18, 93–100 (2001).
[Crossref]

Ozkul, C.

Picholle, E.

Picozzi, A.

Randoux, S.

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[Crossref] [PubMed]

Roh, W. B.

Russell, T. H.

Sakai, A.

K. Ogusu and A. Sakai, “Generation and dynamics of cascaded stimulated Brillouin scattering in a high-finesse fiber Fabry-Perot resonator,” Jpn. J. Appl. Phys. 41, 609–616 (2002).
[Crossref]

Salhi, M.

Sanchez, F.

Ségard, B.

V. Lecoeuche, B. Ségard, and J. Zemmouri, “On route of chaos in stimulated Brillouin scattering with feedback,” Opt. Commun. 172, 335–345 (1999).
[Crossref]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[Crossref] [PubMed]

Shibata, N.

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

Show, H. J.

Song, S.

Stokes, L. F.

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[Crossref]

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1160 (1979).
[Crossref]

Trillo, S.

Vatarescu, A.

A. Vatarescu, “Light conversion in nonlinear monomode optical fibers,” J. Lightwave Technol. LT-5, 1652–1659 (1987).
[Crossref]

Waarts, R. G.

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

Weber, H. P.

P. Labudde, P. Anliker, and H. P. Weber, “Transmission of narrow band high power laser radiation through optical fibers,” Opt. Commun. 32, 385–390 (1980).
[Crossref]

Zemmouri, J.

V. Lecoeuche, B. Ségard, and J. Zemmouri, “On route of chaos in stimulated Brillouin scattering with feedback,” Opt. Commun. 172, 335–345 (1999).
[Crossref]

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, “cw Brillouin laser,” Appl. Phys. Lett. 28, 608–609 (1976).
[Crossref]

K. O. Hill, D. C. Johnson, and B. S. Kawasaki, “cw generation of multiple Stokes and anti-Stokes Brillouin-shifted frequencies,” Appl. Phys. Lett. 29, 185–187 (1976).
[Crossref]

IEEE J. Quantum Electron. (3)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[Crossref]

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1160 (1979).
[Crossref]

IEEE Photonics Technol. Lett. (1)

K. Ogusu, “Analysis of steady-state cascaded stimulated Brillouin scattering in a fiber Fabry-Pérot resonator,” IEEE Photonics Technol. Lett. 14, 947–949 (2002).
[Crossref]

J. Appl. Phys. (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

J. Lightwave Technol. (3)

A. Vatarescu, “Light conversion in nonlinear monomode optical fibers,” J. Lightwave Technol. LT-5, 1652–1659 (1987).
[Crossref]

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Fiber four-wave mixing demultiplexing with inherent parametric amplification,” J. Lightwave Technol. 15, 2051–2058 (1997).
[Crossref]

S. Song, C. T. Allen, K. R. Demarest, and R. Hui, “Intensity-dependent phase-matching effects on four-wave mixing in optical fibers,” J. Lightwave Technol. 17, 2285–2290 (1999).
[Crossref]

J. Opt. Commun. (1)

D. Cotter, “Stimulated Brillouin scattering in monomode optical fiber,” J. Opt. Commun. 4, 10–19 (1983).
[Crossref]

J. Opt. Soc. Am. B (4)

Jpn. J. Appl. Phys. (1)

K. Ogusu and A. Sakai, “Generation and dynamics of cascaded stimulated Brillouin scattering in a high-finesse fiber Fabry-Perot resonator,” Jpn. J. Appl. Phys. 41, 609–616 (2002).
[Crossref]

Opt. Commun. (2)

P. Labudde, P. Anliker, and H. P. Weber, “Transmission of narrow band high power laser radiation through optical fibers,” Opt. Commun. 32, 385–390 (1980).
[Crossref]

V. Lecoeuche, B. Ségard, and J. Zemmouri, “On route of chaos in stimulated Brillouin scattering with feedback,” Opt. Commun. 172, 335–345 (1999).
[Crossref]

Opt. Lett. (2)

Opt. Rev. (1)

K. Ogusu, “Dynamics of stimulated Brillouin scattering in a short and high-finesse fiber Fabry-Perot resonator,” Opt. Rev. 8, 358–363 (2001).
[Crossref]

Phys. Rev. A (1)

S. Randoux, V. Lecoeuche, B. Ségard, and J. Zemmouri, “Dynamical behavior of a Brillouin fiber ring laser emitting two Stokes components,” Phys. Rev. A 52, 2327–2334 (1995).
[Crossref] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

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Figures (6)

Fig. 1
Fig. 1

Schematic of cascaded stimulated Brillouin scattering in a fiber Fabry–Perot resonator. Ep, Es, and ρ represent the pump, Stokes, and acoustic waves, respectively.

Fig. 2
Fig. 2

Output powers of the pump, Stokes, and anti-Stokes waves as a function of the input power for an asymmetric fiber resonator with L=10 m, Aeff=30 μm2, R1=0.1, and R2=0.04. (a) Without FWM, (b) with FWM, (c) experimental results by Labudde et al.6

Fig. 3
Fig. 3

Output powers of the pump, Stokes, and anti-Stokes waves as a function of the input power for an asymmetric fiber resonator with L=10 m, Aeff=30 μm2, R1=0.3, and R2=0.04. (a) Without FWM and (b) with FWM.

Fig. 4
Fig. 4

Output powers of the pump, Stokes, and anti-Stokes waves as a function of the input power for a symmetric fiber resonator with L=10 m, Aeff=30 μm2, R1=R2=0.16. (a) Without FWM and (b) with FWM.

Fig. 5
Fig. 5

Output powers of the pump, Stokes, and anti-Stokes waves as a function of the input power for a symmetric fiber resonator with L=10 m, Aeff=30 μm2, R1=R2=0.49. The numerical results were calculated with the model for FWM.

Fig. 6
Fig. 6

Dependence of threshold power Pth for the first two Stokes waves on the mirror amplitude reflectance r for symmetric and asymmetric resonators with no FWM. The first- and second-order Stokes waves are generated for Pin>Ps1th, Ps2th, respectively.  

Tables (1)

Tables Icon

Table 1 Combinations of FWM When Two Pump Waves p+(ωp, kp), p-(ωp, -kp) and Two First-Order Stokes Waves s1+(ωp-ωA, kp), s1-(ωp-ωA, -kp) Exist in a Fiber Resonator

Equations (54)

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p±(z, t)=12 Ep±(z)exp[i(ωptkpz)]+c.c.,
sj±(z, t)=12 Esj±(z)exp[i(ωsjtksjz)]+c.c.(j=1, 2,),
aj±(z, t)=12 Eaj±(z)exp[i(ωajtkajz)]+c.c.(j=1, 2,),
dEp+dz=-α2 Ep+-ngB4η0 |Es1-|2Ep+-in2k0[(2|ET|2-|Ep+|2)Ep++2(Es1-Ea1+Ep-*+Es1+Ea1-Ep-*+Es1+Es1-Es2-*)],
dEp-dz=α2 Ep-+ngB4η0 |Es1+|2Ep-+in2k0[(2|ET|2-|Ep-|2)Ep-+2(Es1-Ea1+Ep+*+Es1+Ea1-Ep+*+Es1+Es1-Es2+*)],
dEs1+dz=-α2 Es1+-ngB4η0 (|Es2-|2-|Ep-|2)Es1+-in2k0[(2|ET|2-|Es1+|2)Es1++2(Ep+Ep-Ea1-*+Ep+Es2-Es1-*+Ep-Es2+Es1-*)],
dEs1-dz=α2 Es1-+ngB4η0 (|Es2+|2-|Ep+|2)Es1-+in2k0[(2|ET|2-|Es1-|2)Es1-+2(Ep+Ep-Ea1+*+Ep+Es2-Es1+*+Ep-Es2+Es1+*)],
dEs2+dz=-α2 Es2+-ngB4η0 (|Es3-|2-|Es1-|2)Es2+-in2k0[(2|ET|2-|Es2+|2)Es2++Es1+Es1+Ep+*+2Es1+Es1-Ep-*],
dEs2-dz=α2 Es2-+ngB4η0 (|Es3+|2-|Es1+|2)Es2-+in2k0[(2|ET|2-|Es2-|2)Es2-+Es1-Es1-Ep-*+2Es1+Es1-Ep+*],
dEa1+dz=-α2 Ea1+-in2k0[(2|ET|2-|Ea1+|2)Ea1++2Ep+Ep-Es1-*+Ep+Ep+Es1+*],
dEa1-dz=α2 Ea1-+in2k0[(2|ET|2-|Ea1-|2)Ea1-+2Ep+Ep-Es1+*+Ep-Ep-Es1-*],
|ET|2=j(|Ej+|2+|Ej-|2)
(j=p, s1, s2, s3,anda1),
dEp+dz=-i2gPEs1-Ea1+Ep-*,
dEp-dz=i2gPEs1-Ea1+Ep+*,
dEs1-dz=i2gPEp+Ep-Ea1+*,
dEa1+dz=-i2gPEp+Ep-Es1-*,
dEs2-dz=igPEs1-Es1-Ep-*,
dAp+dz=2gPAp-As1-Aa1+sin ϕ,
dAp-dz=-2gPAp+As1-Aa1+sin ϕ,
dAs1-dz=2gPAp+Ap-Aa1+sin ϕ,
dAa1+dz=-2gPAp+Ap-As1-sin ϕ,
Ap+dϕp+dz=-2gPAp-As1-Aa1+cos ϕ,
Ap-dϕp-dz=2gPAp+As1-Aa1+cos ϕ,
As1-dϕs1-dz=2gPAp+Ap-Aa1+cos ϕ,
Aa1+dϕa1+dz=-2gPAp+Ap-As1-cos ϕ,
ϕ=ϕa1++ϕs1--ϕp+-ϕp-.
Ap+Ap-As1-Aa1+cos ϕ=0,
dAp+dz=-ngB4η0 |As1-|2Ap+-2gP(As1-Aa1+Ap-+As1+Aa1-Ap--As1+As1-As2-),
dAp-dz=ngB4η0 |As1+|2AP-+2gP(As1-Aa1+Ap++As1+Aa1-Ap+-As1+As1-As2+),
dAs1+dz=ngB4η0 (|Ap-|2-|As2-|2)As1++2gP(Ap+Ap-Aa1--Ap+As2-As1--Ap-As2+As1-),
dAs1-dz=-ngB4η0 (|Ap+|2-|As2+|2)As1--2gP(Ap+Ap-Aa1+-Ap+As2-As1+-Ap-As2+As1+),
dAs2+dz=ngB4η0 (|As1-|2-|As3-|2)As2++gP(|As1+|2Ap++2As1+As1-Ap-),
dAs2-dz=-ngB4η0 (|As1+|2-|As3+|2)As2--gP(|As1-|2Ap-+2As1+As1-Ap+),
dAa1+dz=gP(|Ap+|2As1++2Ap+Ap-As1-),
dAa1-dz=-gP(|Ap-|2As1-+2Ap+Ap-As1+).
Pp+(0)=(1-R1)Pin+R1Pp-(0),
Psj+(0)=R1Psj-(0),Paj+(0)=R1Paj-(0),
Psj-(L)=R2Psj+(L),Paj-(L)=R2Paj+(L),
dPp+(z)dz=-gBAeff Ps1-Pp+(z)-gFAeff [(Ps1-Pa1+Pp-)1/2+(Ps1+Pa1-Pp-)1/2-(Ps1+Ps1-Ps2-)1/2+(Ps2-Pa2+Pp-)1/2+(Ps2+Pa2-Pp-)1/2-(Ps2+Pa2-Ps4-)1/2-(1/2)Ps1+(Ps2+)1/2][Pp+(z)]1/2,
dPp-(z)dz=gBAeff Ps1+Pp-(z)+gFAeff [(Ps1-Pa1+Pp+)1/2+(Ps1+Pa1-Pp+)1/2-(Ps1+Ps1-Ps2+)1/2+(Ps2-Pa2+Pp+)1/2+(Ps2+Pa2-Pp+)1/2-(Ps2+Pa2-Ps4+)1/2-(1/2)Ps1-(Ps2-)]1/2[Pp-(z)]1/2,
dPs1+(z)dz=gBAeff (Pp--Ps2-)Ps1+(z)+gFAeff [(Pp+Pp-Pa1-)1/2-(Pp+Ps2-Ps1-)1/2-(Pp-Ps2+Ps1-)1/2+(Ps2+Ps2-Ps3-)1/2-(Ps3-Pa1+Ps1-)1/2-(Ps3+Pa1-Ps1-)1/2][Ps1+(z)]1/2,
dPs1-(z)dz=-gBAeff (Pp+-Ps2+)Ps1-(z)-gFAeff [(Pp+Pp-Pa1+)1/2-(Pp+Ps2-Ps1+)1/2-(Pp-Ps2+Ps1+)1/2+(Ps2+Ps2-Ps3+)1/2-(Ps3-Pa1+Ps1+)1/2-(Ps3+Pa1-Ps1+)1/2][Ps1-(z)]1/2,
dPs2+(z)dz=gBAeff (Ps1--Ps3-)Ps2+(z)+gFAeff [(1/2)Ps1+(Pp+)1/2+(Ps1+Ps1-Pp-)1/2+(Pp+Pp-Pa2-)1/2-(Pp+Ps4-Ps2-)1/2-(Pp-Ps4+Ps2-)1/2-(Ps1+Ps3-Ps2-)1/2-(Ps1-Ps3+Ps2-)1/2]×[Ps2+(z)]1/2,
dPs2-(z)dz=-gBAeff (Ps1+-Ps3+)Ps2-(z)-gFAeff [(1/2)Ps1-(Pp-)1/2+(Ps1+Ps1-Pp+)1/2+(Pp+Pp-Pa2+)1/2-(Pp+Ps4-Ps2+)1/2-(Pp-Ps4+Ps2+)1/2-(Ps1+Ps3-Ps2+)1/2-(Ps1-Ps3+Ps2+)1/2]×[Ps2-(z)]1/2,
dPs3+(z)dz=gBAeff (Ps2--Ps4-)Ps3+(z)+gFAeff [(1/2)Ps2+(Ps1+)1/2+(Ps2+Ps2-Ps1-)1/2+(Ps1+Ps1-Pa1-)1/2]×[Ps3+(z)]1/2,
dPs3-(z)dz=-gBAeff (Ps2+-Ps4+)Ps3-(z)-gFAeff [(1/2)Ps2-(Ps1-)1/2+(Ps2+Ps2-Ps1+)1/2+(Ps1+Ps1-Pa1+)1/2]×[Ps3-(z)]1/2,
dPs4+(z)dz=gBAeff Ps3-Ps4+(z)+gFAeff [(1/2)Ps2+(Pp+)1/2+(Ps2+Ps2-Pp-)1/2][Ps4+(z)]1/2,
dPs4-(z)dz=-gBAeff Ps3+Ps4+(z)-gFAeff [(1/2)Ps2-(Pp-)1/2+(Ps2+Ps2-Pp+)1/2][Ps4-(z)]1/2,
dPa1+(z)dz=gFAeff [(1/2)Pp+(Ps1+)1/2+(Pp+Pp-Ps1-)1/2+(1/2)Ps1+(Ps3+)1/2+(Ps1+Ps1-Ps3-)1/2]×[Pa1+(z)]1/2,
dPa1-(z)dz=-gFAeff [(1/2)Pp-(Ps1-)1/2+(Pp+Pp-Ps1+)1/2+(1/2)Ps1-(Ps3-)1/2+(Ps1+Ps1-Ps3+)1/2]×[Pa1-(z)]1/2,
dPa2+(z)dz=gFAeff [(1/2)Pp+(Ps2+)1/2+(Pp+Pp-Ps2-)1/2]×[Pa2+(z)]1/2,
dPa2-(z)dz=-gFAeff [(1/2)Pp-(Ps2-)1/2+(Pp+Pp-Ps2+)1/2]×[Pa2-(z)]1/2,
gF=8gPη0n=8n2k0η0n=4n2Ik0,

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