Abstract

A fiber loop mirror (FLM) filter with two-stage polarization-maintaining fibers (PMFs) and polarization controllers (PCs) is presented. The transmission function of this FLM is calculated in detail by Jones matrix. The wavelength interval depends on both the PMFs and the PCs. The side frequencies can be restrained by choosing appropriate length of the PMFs. Furthermore, an erbium-doped fiber ring laser based on this FLM filter is proposed and demonstrated. Stable single-, double- and triple-wavelength are achieved respectively. The 3dB line-width is less than 0.03nm, and the fluctuation of wavelength and peak power is less than 0.05nm and 0.1dB in 30 minutes.

© 2009 Optical Society of America

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  1. S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
    [CrossRef]
  2. F.-W. Sheu, C.-Y. Chiou, and S.-C. Yang, "Performance of a wavelength-tunable erbium-doped fiber laser using a Sagnac interferometer," Opt. Commun. 281, 4719-4722 (2008).
    [CrossRef]
  3. C.-H. Chang, H. Lin, Y.-S. Huang, and S.-H. Tong, "Multi-wavelength-switchable and uniform erbium-doped fiber laser using unbalanced in-line Sagnac interferometer," Opt. Express. 15, 12450-12456 (2007).
    [CrossRef] [PubMed]
  4. D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
    [CrossRef]
  5. M. Jinno and T. Matsumoto, "Nonlinear Sagnac interferometer switch and its applications," IEEE J. Quantum Electron. 28, 875-882 (1992).
    [CrossRef]
  6. M. A. Mirza and G. Stewart, "Theory and design of a simple tunable Sagnac loop filter for multiwavelength fiber lasers," Appl. Opt. 47, 5242-5252 (2008).
    [CrossRef] [PubMed]
  7. E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, "Polarization independent nonlinear fiber sagnac interferometer," Opt. Commun. 183, 389-393 (2000).
    [CrossRef]
  8. X. P. Dong, S. Li, K. S. Chiang, M. N. Ng, and B. C. B. Chu, "Multiwavelength erbium-doped fibre laser based on a high-birefringence fibre loop mirror," Electron. Lett. 36, 1609-1610 (2000).
    [CrossRef]
  9. S. Feng, Q. Mao, L. Shang, and J. W. Y. Lit, "Reflectivity characteristics of the fiber loop mirror with a polarization controller," Opt. Commun. 277, 322-328 (2007).
    [CrossRef]
  10. P. P. Yupapin., and P. Chunpang, "An experimental investigation of the optical switching characteristics using an optical Sagnac interferometer incorporating one and two ring resonators," Optics & Laser Technol. 40, 273-277 (2008).
    [CrossRef]
  11. F. Heismann, "Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems," J. Lightwave Technol. 12, 690-699 (1994).
    [CrossRef]
  12. E. Desurvire, J. L. Zyskind, and J. R. Simpson, "Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers," IEEE Photon. Technol. Letts. 2, 246-248 (1990).
    [CrossRef]
  13. J. Sun, J. Qiu, and D. Huang, "Multiwavelength erbium-doped fiber lasers exploiting polarization hole burning," Opt. Commun. 182, 193-197 (2000).
    [CrossRef]
  14. D. S. Lim, H. K. Lee, K. H. Kim, S. B. Kang, J. T. Ahn, and M.-Y. Jeon, "Generation of multiorder Stokes and anti-Stokes lines in a Brillouin erbium-fiber laser with a Sagnac loop mirror," Opt. Lett. 23, 1671-1673 (1998).
    [CrossRef]
  15. X. Xu, Y. Yao, and D. Chen, "Numerical analysis of multiwavelength erbium-doped fiber ring laser exploiting four-wave mixing," Opt. Express. 16, 12397-12402 (2008).
    [CrossRef] [PubMed]
  16. S. Yamashita and K. Hotate, "Multiwavelength erbium-doped fibre laser using intracavity etalon and cooled by liquid nitrogen," Electron. Lett. 32, 1298-1299 (1996).
    [CrossRef]

2008 (4)

F.-W. Sheu, C.-Y. Chiou, and S.-C. Yang, "Performance of a wavelength-tunable erbium-doped fiber laser using a Sagnac interferometer," Opt. Commun. 281, 4719-4722 (2008).
[CrossRef]

M. A. Mirza and G. Stewart, "Theory and design of a simple tunable Sagnac loop filter for multiwavelength fiber lasers," Appl. Opt. 47, 5242-5252 (2008).
[CrossRef] [PubMed]

P. P. Yupapin., and P. Chunpang, "An experimental investigation of the optical switching characteristics using an optical Sagnac interferometer incorporating one and two ring resonators," Optics & Laser Technol. 40, 273-277 (2008).
[CrossRef]

X. Xu, Y. Yao, and D. Chen, "Numerical analysis of multiwavelength erbium-doped fiber ring laser exploiting four-wave mixing," Opt. Express. 16, 12397-12402 (2008).
[CrossRef] [PubMed]

2007 (2)

S. Feng, Q. Mao, L. Shang, and J. W. Y. Lit, "Reflectivity characteristics of the fiber loop mirror with a polarization controller," Opt. Commun. 277, 322-328 (2007).
[CrossRef]

C.-H. Chang, H. Lin, Y.-S. Huang, and S.-H. Tong, "Multi-wavelength-switchable and uniform erbium-doped fiber laser using unbalanced in-line Sagnac interferometer," Opt. Express. 15, 12450-12456 (2007).
[CrossRef] [PubMed]

2005 (1)

S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
[CrossRef]

2000 (3)

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, "Polarization independent nonlinear fiber sagnac interferometer," Opt. Commun. 183, 389-393 (2000).
[CrossRef]

X. P. Dong, S. Li, K. S. Chiang, M. N. Ng, and B. C. B. Chu, "Multiwavelength erbium-doped fibre laser based on a high-birefringence fibre loop mirror," Electron. Lett. 36, 1609-1610 (2000).
[CrossRef]

J. Sun, J. Qiu, and D. Huang, "Multiwavelength erbium-doped fiber lasers exploiting polarization hole burning," Opt. Commun. 182, 193-197 (2000).
[CrossRef]

1998 (1)

1996 (1)

S. Yamashita and K. Hotate, "Multiwavelength erbium-doped fibre laser using intracavity etalon and cooled by liquid nitrogen," Electron. Lett. 32, 1298-1299 (1996).
[CrossRef]

1994 (1)

F. Heismann, "Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems," J. Lightwave Technol. 12, 690-699 (1994).
[CrossRef]

1992 (1)

M. Jinno and T. Matsumoto, "Nonlinear Sagnac interferometer switch and its applications," IEEE J. Quantum Electron. 28, 875-882 (1992).
[CrossRef]

1990 (1)

E. Desurvire, J. L. Zyskind, and J. R. Simpson, "Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers," IEEE Photon. Technol. Letts. 2, 246-248 (1990).
[CrossRef]

1988 (1)

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

Ahn, J. T.

Chang, C.-H.

C.-H. Chang, H. Lin, Y.-S. Huang, and S.-H. Tong, "Multi-wavelength-switchable and uniform erbium-doped fiber laser using unbalanced in-line Sagnac interferometer," Opt. Express. 15, 12450-12456 (2007).
[CrossRef] [PubMed]

Chen, D.

X. Xu, Y. Yao, and D. Chen, "Numerical analysis of multiwavelength erbium-doped fiber ring laser exploiting four-wave mixing," Opt. Express. 16, 12397-12402 (2008).
[CrossRef] [PubMed]

Chiang, K. S.

X. P. Dong, S. Li, K. S. Chiang, M. N. Ng, and B. C. B. Chu, "Multiwavelength erbium-doped fibre laser based on a high-birefringence fibre loop mirror," Electron. Lett. 36, 1609-1610 (2000).
[CrossRef]

Chiou, C.-Y.

F.-W. Sheu, C.-Y. Chiou, and S.-C. Yang, "Performance of a wavelength-tunable erbium-doped fiber laser using a Sagnac interferometer," Opt. Commun. 281, 4719-4722 (2008).
[CrossRef]

Chu, B. C. B.

X. P. Dong, S. Li, K. S. Chiang, M. N. Ng, and B. C. B. Chu, "Multiwavelength erbium-doped fibre laser based on a high-birefringence fibre loop mirror," Electron. Lett. 36, 1609-1610 (2000).
[CrossRef]

Chunpang, P.

P. P. Yupapin., and P. Chunpang, "An experimental investigation of the optical switching characteristics using an optical Sagnac interferometer incorporating one and two ring resonators," Optics & Laser Technol. 40, 273-277 (2008).
[CrossRef]

Desurvire, E.

E. Desurvire, J. L. Zyskind, and J. R. Simpson, "Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers," IEEE Photon. Technol. Letts. 2, 246-248 (1990).
[CrossRef]

Dong, X. P.

X. P. Dong, S. Li, K. S. Chiang, M. N. Ng, and B. C. B. Chu, "Multiwavelength erbium-doped fibre laser based on a high-birefringence fibre loop mirror," Electron. Lett. 36, 1609-1610 (2000).
[CrossRef]

Feng, S.

S. Feng, Q. Mao, L. Shang, and J. W. Y. Lit, "Reflectivity characteristics of the fiber loop mirror with a polarization controller," Opt. Commun. 277, 322-328 (2007).
[CrossRef]

Haus, J. W.

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, "Polarization independent nonlinear fiber sagnac interferometer," Opt. Commun. 183, 389-393 (2000).
[CrossRef]

Heismann, F.

F. Heismann, "Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems," J. Lightwave Technol. 12, 690-699 (1994).
[CrossRef]

Hotate, K.

S. Yamashita and K. Hotate, "Multiwavelength erbium-doped fibre laser using intracavity etalon and cooled by liquid nitrogen," Electron. Lett. 32, 1298-1299 (1996).
[CrossRef]

Hu, S.

S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
[CrossRef]

Huang, D.

J. Sun, J. Qiu, and D. Huang, "Multiwavelength erbium-doped fiber lasers exploiting polarization hole burning," Opt. Commun. 182, 193-197 (2000).
[CrossRef]

Huang, Y.-S.

C.-H. Chang, H. Lin, Y.-S. Huang, and S.-H. Tong, "Multi-wavelength-switchable and uniform erbium-doped fiber laser using unbalanced in-line Sagnac interferometer," Opt. Express. 15, 12450-12456 (2007).
[CrossRef] [PubMed]

Ibarra-Escamilla, B.

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, "Polarization independent nonlinear fiber sagnac interferometer," Opt. Commun. 183, 389-393 (2000).
[CrossRef]

Jeon, M.-Y.

Jinno, M.

M. Jinno and T. Matsumoto, "Nonlinear Sagnac interferometer switch and its applications," IEEE J. Quantum Electron. 28, 875-882 (1992).
[CrossRef]

Kang, S. B.

Kim, K. H.

Korneev, N.

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, "Polarization independent nonlinear fiber sagnac interferometer," Opt. Commun. 183, 389-393 (2000).
[CrossRef]

Kuzin, E. A.

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, "Polarization independent nonlinear fiber sagnac interferometer," Opt. Commun. 183, 389-393 (2000).
[CrossRef]

Lee, H. K.

Li, S.

X. P. Dong, S. Li, K. S. Chiang, M. N. Ng, and B. C. B. Chu, "Multiwavelength erbium-doped fibre laser based on a high-birefringence fibre loop mirror," Electron. Lett. 36, 1609-1610 (2000).
[CrossRef]

Li, W.

S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
[CrossRef]

Lim, D. S.

Lin, H.

C.-H. Chang, H. Lin, Y.-S. Huang, and S.-H. Tong, "Multi-wavelength-switchable and uniform erbium-doped fiber laser using unbalanced in-line Sagnac interferometer," Opt. Express. 15, 12450-12456 (2007).
[CrossRef] [PubMed]

Lit, J. W. Y.

S. Feng, Q. Mao, L. Shang, and J. W. Y. Lit, "Reflectivity characteristics of the fiber loop mirror with a polarization controller," Opt. Commun. 277, 322-328 (2007).
[CrossRef]

Luo, S. Y.

S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
[CrossRef]

Mao, Q.

S. Feng, Q. Mao, L. Shang, and J. W. Y. Lit, "Reflectivity characteristics of the fiber loop mirror with a polarization controller," Opt. Commun. 277, 322-328 (2007).
[CrossRef]

Matsumoto, T.

M. Jinno and T. Matsumoto, "Nonlinear Sagnac interferometer switch and its applications," IEEE J. Quantum Electron. 28, 875-882 (1992).
[CrossRef]

Mirza, M. A.

Mortimore, D. B.

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

Ng, M. N.

X. P. Dong, S. Li, K. S. Chiang, M. N. Ng, and B. C. B. Chu, "Multiwavelength erbium-doped fibre laser based on a high-birefringence fibre loop mirror," Electron. Lett. 36, 1609-1610 (2000).
[CrossRef]

Qiu, J.

J. Sun, J. Qiu, and D. Huang, "Multiwavelength erbium-doped fiber lasers exploiting polarization hole burning," Opt. Commun. 182, 193-197 (2000).
[CrossRef]

Shang, L.

S. Feng, Q. Mao, L. Shang, and J. W. Y. Lit, "Reflectivity characteristics of the fiber loop mirror with a polarization controller," Opt. Commun. 277, 322-328 (2007).
[CrossRef]

Sheu, F.-W.

F.-W. Sheu, C.-Y. Chiou, and S.-C. Yang, "Performance of a wavelength-tunable erbium-doped fiber laser using a Sagnac interferometer," Opt. Commun. 281, 4719-4722 (2008).
[CrossRef]

Simpson, J. R.

E. Desurvire, J. L. Zyskind, and J. R. Simpson, "Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers," IEEE Photon. Technol. Letts. 2, 246-248 (1990).
[CrossRef]

Song, Y. J.

S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
[CrossRef]

Stewart, G.

Sun, J.

J. Sun, J. Qiu, and D. Huang, "Multiwavelength erbium-doped fiber lasers exploiting polarization hole burning," Opt. Commun. 182, 193-197 (2000).
[CrossRef]

Tong, S.-H.

C.-H. Chang, H. Lin, Y.-S. Huang, and S.-H. Tong, "Multi-wavelength-switchable and uniform erbium-doped fiber laser using unbalanced in-line Sagnac interferometer," Opt. Express. 15, 12450-12456 (2007).
[CrossRef] [PubMed]

Xia, Y. X.

S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
[CrossRef]

Xu, X.

X. Xu, Y. Yao, and D. Chen, "Numerical analysis of multiwavelength erbium-doped fiber ring laser exploiting four-wave mixing," Opt. Express. 16, 12397-12402 (2008).
[CrossRef] [PubMed]

Yamashita, S.

S. Yamashita and K. Hotate, "Multiwavelength erbium-doped fibre laser using intracavity etalon and cooled by liquid nitrogen," Electron. Lett. 32, 1298-1299 (1996).
[CrossRef]

Yang, S.-C.

F.-W. Sheu, C.-Y. Chiou, and S.-C. Yang, "Performance of a wavelength-tunable erbium-doped fiber laser using a Sagnac interferometer," Opt. Commun. 281, 4719-4722 (2008).
[CrossRef]

Yao, Y.

X. Xu, Y. Yao, and D. Chen, "Numerical analysis of multiwavelength erbium-doped fiber ring laser exploiting four-wave mixing," Opt. Express. 16, 12397-12402 (2008).
[CrossRef] [PubMed]

Yupapin, P. P.

P. P. Yupapin., and P. Chunpang, "An experimental investigation of the optical switching characteristics using an optical Sagnac interferometer incorporating one and two ring resonators," Optics & Laser Technol. 40, 273-277 (2008).
[CrossRef]

Zhan, L.

S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
[CrossRef]

Zyskind, J. L.

E. Desurvire, J. L. Zyskind, and J. R. Simpson, "Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers," IEEE Photon. Technol. Letts. 2, 246-248 (1990).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (2)

X. P. Dong, S. Li, K. S. Chiang, M. N. Ng, and B. C. B. Chu, "Multiwavelength erbium-doped fibre laser based on a high-birefringence fibre loop mirror," Electron. Lett. 36, 1609-1610 (2000).
[CrossRef]

S. Yamashita and K. Hotate, "Multiwavelength erbium-doped fibre laser using intracavity etalon and cooled by liquid nitrogen," Electron. Lett. 32, 1298-1299 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Jinno and T. Matsumoto, "Nonlinear Sagnac interferometer switch and its applications," IEEE J. Quantum Electron. 28, 875-882 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. Hu, L. Zhan, Y. J. Song, W. Li, S. Y. Luo, and Y. X. Xia, "Switchable multiwavelength erbium-doped fiber ring laser with a multisection high-birefringence fiber loop mirror," IEEE Photon. Technol. Lett. 17, 1387-1389 (2005).
[CrossRef]

IEEE Photon. Technol. Letts. (1)

E. Desurvire, J. L. Zyskind, and J. R. Simpson, "Spectral gain hole-burning at 1.53 μm in erbium-doped fiber amplifiers," IEEE Photon. Technol. Letts. 2, 246-248 (1990).
[CrossRef]

J. Lightwave Technol. (2)

F. Heismann, "Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems," J. Lightwave Technol. 12, 690-699 (1994).
[CrossRef]

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

Opt. Commun. (4)

F.-W. Sheu, C.-Y. Chiou, and S.-C. Yang, "Performance of a wavelength-tunable erbium-doped fiber laser using a Sagnac interferometer," Opt. Commun. 281, 4719-4722 (2008).
[CrossRef]

S. Feng, Q. Mao, L. Shang, and J. W. Y. Lit, "Reflectivity characteristics of the fiber loop mirror with a polarization controller," Opt. Commun. 277, 322-328 (2007).
[CrossRef]

E. A. Kuzin, N. Korneev, J. W. Haus, and B. Ibarra-Escamilla, "Polarization independent nonlinear fiber sagnac interferometer," Opt. Commun. 183, 389-393 (2000).
[CrossRef]

J. Sun, J. Qiu, and D. Huang, "Multiwavelength erbium-doped fiber lasers exploiting polarization hole burning," Opt. Commun. 182, 193-197 (2000).
[CrossRef]

Opt. Express. (2)

X. Xu, Y. Yao, and D. Chen, "Numerical analysis of multiwavelength erbium-doped fiber ring laser exploiting four-wave mixing," Opt. Express. 16, 12397-12402 (2008).
[CrossRef] [PubMed]

C.-H. Chang, H. Lin, Y.-S. Huang, and S.-H. Tong, "Multi-wavelength-switchable and uniform erbium-doped fiber laser using unbalanced in-line Sagnac interferometer," Opt. Express. 15, 12450-12456 (2007).
[CrossRef] [PubMed]

Opt. Lett. (1)

Optics & Laser Technol. (1)

P. P. Yupapin., and P. Chunpang, "An experimental investigation of the optical switching characteristics using an optical Sagnac interferometer incorporating one and two ring resonators," Optics & Laser Technol. 40, 273-277 (2008).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Fiber loop mirror (FLM) of two-stage polarization-maintaining fibers (PMF) and polarization controllers (PC).

Fig. 2.
Fig. 2.

Polarization independent of (a) the PC-PMF FLM, and b) the PCs-PMFs FLM.

Fig. 3.
Fig. 3.

Interval dependence of PC-PMF FLM for (a) different length of PMF, and (b) different refractive index difference of PMF.

Fig. 4.
Fig. 4.

Interval dependence of the PCs-PMFs FLM for (a) different PMFs, and (b) different PCs.

Fig. 5.
Fig. 5.

Transmission dependence for different PMFs length with (a) L 2/L 1≠constant and (b) L 2/L 1=constant.

Fig. 6.
Fig. 6.

Transmission spectrum of the PC-PMF FLM in (a) PC (θ1 , π/4, π/6), (b) PC (π/2, θ2 , π/9) and (c) PC (π/6, π/4, θ3 ).

Fig. 7.
Fig. 7.

Transmission spectrum of the PCs-PMFs in (a) PCs (θ 11, π/3, π/6, π/3, π/4, π/9), and (b) PCs(π/9, θ 12, π/6, π/3, π/4, π/9).

Fig. 8.
Fig. 8.

Testing transmission spectra of FLM with two-stage PMFs.

Fig. 9.
Fig. 9.

Schematic diagram of erbium-doped fiber ring laser based on the PCs-PMFs FLM.

Fig. 10.
Fig. 10.

Output spectra of erbium-doped fiber ring laser.

Equations (10)

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

E=[ExEy]=[EcosθEsinθ]
[E3E4]=Mc[E1E2]
T(θj)=[cosφj2+isinφj2cos2θjisinφj2sin2θjisinφj2sin2θjcosφj2isinφj2cos2θj]
MPC(θ1,θ2,θ3)=T(θ3)T(θ2)T(θ1)=i2[A+iBC+iDCiDA+iB]
{A=cos(2θ2)cos(2θ32θ2+2θ1)B=cos(2θ32θ2)+cos(2θ22θ1)C=sin(2θ2)sin(2θ32θ2+2θ1)D=sin(2θ32θ2)+sin(2θ22θ1)
MPMF(φ)=[100eiΔφ]
M=MPMF·MPC
M=MPC·MPMF=T (θ1) T(θ2)T(θ3)·MPMF
[E1E2]=MC[E3E4]=12[i(M+M)E1(MM)E1]
T(λ)=E22E12

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