Abstract

An automatic step adjustment (ASA) method for average power analysis (APA) technique used in fiber amplifiers is proposed in this paper for the first time. In comparison with the traditional APA technique, the proposed method has suggested two unique merits such as a higher order accuracy and an ASA mechanism, so that it can significantly shorten the computing time and improve the solution accuracy. A test example demonstrates that, by comparing to the APA technique, the proposed method increases the computing speed by more than a hundredfold under the same errors. By computing the model equations of erbium-doped fiber amplifiers, the numerical results show that our method can improve the solution accuracy by over two orders of magnitude at the same amplifying section number. The proposed method has the capacity to rapidly and effectively compute the model equations of fiber Raman amplifiers and semiconductor lasers.

© 2006 Optical Society of America

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  1. C. Berkdemir, and S. Ozsoy "The temperature dependent performance analysis of EDFAs pumped at 1480 nm: A more accurate propagation equation," Opt. Express 13, 5179-5185 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-5179
    [CrossRef] [PubMed]
  2. M. A. Quintela, J. Lopez-Higuera, and C. Jauregui, "Polarization characteristics of a reflective erbium doped fiber amplifier with temperature changes at the Faraday rotator mirror," Opt. Express 13, 1368-1376 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1368
    [CrossRef] [PubMed]
  3. M. Bolshtyansky, I. Mandelbaum, and F. Pan, "Signal excited-state absorption in the L-band EDFA: Simulation and measurements," J. Lightwave Technol. 23, 2796-2799 (2005).
    [CrossRef]
  4. J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
    [CrossRef]
  5. P. S. Chan and H. K. Tsang, "Minimizing gain transient dynamics by optimizing the erbium concentration and cavity length of a gain clamped EDFA," Opt. Express 13, 7520-7526 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-19-7520
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    [CrossRef]
  18. C. R. Giles and E. Desurvire, ‘‘Modeling erbium-doped fiber amplifiers,’’J. Lightwave Technol. 9, 271-283 (1991).
    [CrossRef]
  19. B. Pedersen, K. Dybdal, C. D. Hansen,  et al., "Detailed theoretical and experimental investigation of high-gain erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 2, 863-865 (1990).
    [CrossRef]
  20. B. Pedersen, S. A. Zemon, and W. J. Miniscalco, "Analysis of erbium-doped fiber amplifiers pumped at 800nm," Fiber and Integr. Opt.,  10, 127-136 (1991).
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  26. B. K. Min, W. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photon. Technol. Lett. 12, 1486-1488 (2000).
    [CrossRef]
  27. Z. Tong, H. Wei, and S. Jian, "A novel algorithm to simulate DWDM transmission systems amplified by backward multipumped fiber Raman amplifiers," Microwave Opt. Technol. Lett. 35, 333-337 (2002).
    [CrossRef]
  28. M. Menif, M. Karasek, and L. Rusch, "Cross-gain modulation in Raman fiber amplifier: Experimentation and Modeling," IEEE Photonics Technol. Lett. 14, 1261-1263 (2002).
    [CrossRef]

2005 (7)

J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
[CrossRef]

M. A. Quintela, J. Lopez-Higuera, and C. Jauregui, "Polarization characteristics of a reflective erbium doped fiber amplifier with temperature changes at the Faraday rotator mirror," Opt. Express 13, 1368-1376 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1368
[CrossRef] [PubMed]

C. Berkdemir, and S. Ozsoy "The temperature dependent performance analysis of EDFAs pumped at 1480 nm: A more accurate propagation equation," Opt. Express 13, 5179-5185 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-5179
[CrossRef] [PubMed]

P. S. Chan and H. K. Tsang, "Minimizing gain transient dynamics by optimizing the erbium concentration and cavity length of a gain clamped EDFA," Opt. Express 13, 7520-7526 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-19-7520
[CrossRef] [PubMed]

M. Bolshtyansky, I. Mandelbaum, and F. Pan, "Signal excited-state absorption in the L-band EDFA: Simulation and measurements," J. Lightwave Technol. 23, 2796-2799 (2005).
[CrossRef]

Z. G. Lu, J. Liu, F. Sun,  et al., "A hybrid fiber amplifier with 36.9-dBm output power and 70-dB gain," Opt. Commun. 256, 352-357 (2005).
[CrossRef]

C. Cheng, and M. Xiao, "Optimization of an erbium-doped fiber amplifier with radial effects," Opt. Commun. 254, 215-222 (2005).
[CrossRef]

2004 (1)

2002 (5)

L. Zhu, Y. Ma, G. Wang,  et al., "General computer model for both erbium-doped fiber amplifier and fiber Raman amplifier," Opt. Eng. 41, 1805-1808 (2002).
[CrossRef]

M. M. Tiesler, J. Witkowski, and K. Abramski, "Quality criterion for numerical methods in EDFA modeling," Optica Applicata 32, 187-196 (2002).

S. K. Kim, S. Chang, J. Han,  et al., "Design of hybrid optical amplifiers for high capacity optical transmission," ETRI J. 24, 81-96 (2002).
[CrossRef]

Z. Tong, H. Wei, and S. Jian, "A novel algorithm to simulate DWDM transmission systems amplified by backward multipumped fiber Raman amplifiers," Microwave Opt. Technol. Lett. 35, 333-337 (2002).
[CrossRef]

M. Menif, M. Karasek, and L. Rusch, "Cross-gain modulation in Raman fiber amplifier: Experimentation and Modeling," IEEE Photonics Technol. Lett. 14, 1261-1263 (2002).
[CrossRef]

2001 (1)

M. Karasek, M. Menif, A. Bellemare, "Design of wideband hybrid amplifiers for local area networks," IEE Proc. J. Optoelectron. 148, 150-155 (2001).
[CrossRef]

2000 (2)

J. Bryce, Y. Zhao, and R. Minasian, "Modeling and optimization of add-drop dynamics in gain-clamped fiber amplifiers," Appl. Opt. 39, 4270-4277 (2000).
[CrossRef]

B. K. Min, W. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photon. Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

1999 (1)

I. Roudas, D. Richards, N. Antoniades,  et al., "An efficient simulation model of the erbium-doped fiber for the study of multiwavelength optical networks," Opt. Fiber Technol. 5, 363-389 (1999).
[CrossRef]

1998 (1)

1992 (2)

R. J. Mears, and S. R. Baker, "Erbium fiber amplifiers and lasers," Opt. Quantum Electron. 24, 517-538 (1992).
[CrossRef]

T. G. Hodgkinson, "Improved average power analysis technique for erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 4, 1273-1275 (1992).
[CrossRef]

1991 (3)

T. G. Hodgkinson, "Average power analysis technique for erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 3, 1082-1084 (1991).
[CrossRef]

C. R. Giles and E. Desurvire, ‘‘Modeling erbium-doped fiber amplifiers,’’J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

B. Pedersen, S. A. Zemon, and W. J. Miniscalco, "Analysis of erbium-doped fiber amplifiers pumped at 800nm," Fiber and Integr. Opt.,  10, 127-136 (1991).
[CrossRef]

1990 (1)

B. Pedersen, K. Dybdal, C. D. Hansen,  et al., "Detailed theoretical and experimental investigation of high-gain erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 2, 863-865 (1990).
[CrossRef]

1985 (1)

M. J. Adams, J. V. Collins, and I. D. Henning, "Analysis of semiconductor laser optical amplifiers", IEE. Proc.J Optoelectron. 132, 58-63 (1985).
[CrossRef]

Abramski, K.

M. M. Tiesler, J. Witkowski, and K. Abramski, "Quality criterion for numerical methods in EDFA modeling," Optica Applicata 32, 187-196 (2002).

Adams, M. J.

M. J. Adams, J. V. Collins, and I. D. Henning, "Analysis of semiconductor laser optical amplifiers", IEE. Proc.J Optoelectron. 132, 58-63 (1985).
[CrossRef]

Antoniades, N.

I. Roudas, D. Richards, N. Antoniades,  et al., "An efficient simulation model of the erbium-doped fiber for the study of multiwavelength optical networks," Opt. Fiber Technol. 5, 363-389 (1999).
[CrossRef]

Arradi, R.

J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
[CrossRef]

Baker, S. R.

R. J. Mears, and S. R. Baker, "Erbium fiber amplifiers and lasers," Opt. Quantum Electron. 24, 517-538 (1992).
[CrossRef]

Bellemare, A.

M. Karasek, M. Menif, A. Bellemare, "Design of wideband hybrid amplifiers for local area networks," IEE Proc. J. Optoelectron. 148, 150-155 (2001).
[CrossRef]

Berkdemir, C.

Bolshtyansky, M.

Bononi, A.

Bryce, J.

Chan, P. S.

Chang, S.

S. K. Kim, S. Chang, J. Han,  et al., "Design of hybrid optical amplifiers for high capacity optical transmission," ETRI J. 24, 81-96 (2002).
[CrossRef]

Cheng, C.

C. Cheng, and M. Xiao, "Optimization of an erbium-doped fiber amplifier with radial effects," Opt. Commun. 254, 215-222 (2005).
[CrossRef]

Collins, J. V.

M. J. Adams, J. V. Collins, and I. D. Henning, "Analysis of semiconductor laser optical amplifiers", IEE. Proc.J Optoelectron. 132, 58-63 (1985).
[CrossRef]

Coral, A. D.

J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
[CrossRef]

Desurvire, E.

C. R. Giles and E. Desurvire, ‘‘Modeling erbium-doped fiber amplifiers,’’J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

Dybdal, K.

B. Pedersen, K. Dybdal, C. D. Hansen,  et al., "Detailed theoretical and experimental investigation of high-gain erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 2, 863-865 (1990).
[CrossRef]

Fragnito, H.

Giles, C. R.

C. R. Giles and E. Desurvire, ‘‘Modeling erbium-doped fiber amplifiers,’’J. Lightwave Technol. 9, 271-283 (1991).
[CrossRef]

Han, J.

S. K. Kim, S. Chang, J. Han,  et al., "Design of hybrid optical amplifiers for high capacity optical transmission," ETRI J. 24, 81-96 (2002).
[CrossRef]

Hansen, C. D.

B. Pedersen, K. Dybdal, C. D. Hansen,  et al., "Detailed theoretical and experimental investigation of high-gain erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 2, 863-865 (1990).
[CrossRef]

Henning, I. D.

M. J. Adams, J. V. Collins, and I. D. Henning, "Analysis of semiconductor laser optical amplifiers", IEE. Proc.J Optoelectron. 132, 58-63 (1985).
[CrossRef]

Hodgkinson, T. G.

T. G. Hodgkinson, "Improved average power analysis technique for erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 4, 1273-1275 (1992).
[CrossRef]

T. G. Hodgkinson, "Average power analysis technique for erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 3, 1082-1084 (1991).
[CrossRef]

Jauregui, C.

Jian, S.

Z. Tong, H. Wei, and S. Jian, "A novel algorithm to simulate DWDM transmission systems amplified by backward multipumped fiber Raman amplifiers," Microwave Opt. Technol. Lett. 35, 333-337 (2002).
[CrossRef]

Juriollo, A.

J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
[CrossRef]

Karasek, M.

M. Menif, M. Karasek, and L. Rusch, "Cross-gain modulation in Raman fiber amplifier: Experimentation and Modeling," IEEE Photonics Technol. Lett. 14, 1261-1263 (2002).
[CrossRef]

M. Karasek, M. Menif, A. Bellemare, "Design of wideband hybrid amplifiers for local area networks," IEE Proc. J. Optoelectron. 148, 150-155 (2001).
[CrossRef]

Kim, S. K.

S. K. Kim, S. Chang, J. Han,  et al., "Design of hybrid optical amplifiers for high capacity optical transmission," ETRI J. 24, 81-96 (2002).
[CrossRef]

Lee, W.

B. K. Min, W. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photon. Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

Liu, J.

Z. G. Lu, J. Liu, F. Sun,  et al., "A hybrid fiber amplifier with 36.9-dBm output power and 70-dB gain," Opt. Commun. 256, 352-357 (2005).
[CrossRef]

Lopez-Higuera, J.

Lu, Z. G.

Z. G. Lu, J. Liu, F. Sun,  et al., "A hybrid fiber amplifier with 36.9-dBm output power and 70-dB gain," Opt. Commun. 256, 352-357 (2005).
[CrossRef]

Ma, Y.

L. Zhu, Y. Ma, G. Wang,  et al., "General computer model for both erbium-doped fiber amplifier and fiber Raman amplifier," Opt. Eng. 41, 1805-1808 (2002).
[CrossRef]

Mandelbaum, I.

Mears, R. J.

R. J. Mears, and S. R. Baker, "Erbium fiber amplifiers and lasers," Opt. Quantum Electron. 24, 517-538 (1992).
[CrossRef]

Menif, M.

M. Menif, M. Karasek, and L. Rusch, "Cross-gain modulation in Raman fiber amplifier: Experimentation and Modeling," IEEE Photonics Technol. Lett. 14, 1261-1263 (2002).
[CrossRef]

M. Karasek, M. Menif, A. Bellemare, "Design of wideband hybrid amplifiers for local area networks," IEE Proc. J. Optoelectron. 148, 150-155 (2001).
[CrossRef]

Min, B. K.

B. K. Min, W. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photon. Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

Minasian, R.

Miniscalco, W. J.

B. Pedersen, S. A. Zemon, and W. J. Miniscalco, "Analysis of erbium-doped fiber amplifiers pumped at 800nm," Fiber and Integr. Opt.,  10, 127-136 (1991).
[CrossRef]

Oliveira, J. C. R.

J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
[CrossRef]

Ozsoy, S.

Pan, F.

Park, N.

B. K. Min, W. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photon. Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

Pedersen, B.

B. Pedersen, S. A. Zemon, and W. J. Miniscalco, "Analysis of erbium-doped fiber amplifiers pumped at 800nm," Fiber and Integr. Opt.,  10, 127-136 (1991).
[CrossRef]

B. Pedersen, K. Dybdal, C. D. Hansen,  et al., "Detailed theoretical and experimental investigation of high-gain erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 2, 863-865 (1990).
[CrossRef]

Quintela, M. A.

Richards, D.

I. Roudas, D. Richards, N. Antoniades,  et al., "An efficient simulation model of the erbium-doped fiber for the study of multiwavelength optical networks," Opt. Fiber Technol. 5, 363-389 (1999).
[CrossRef]

Rieznik, A. A.

Romero, M. A.

J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
[CrossRef]

Rosolem, J. B.

J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
[CrossRef]

Roudas, I.

I. Roudas, D. Richards, N. Antoniades,  et al., "An efficient simulation model of the erbium-doped fiber for the study of multiwavelength optical networks," Opt. Fiber Technol. 5, 363-389 (1999).
[CrossRef]

Rusch, L.

M. Menif, M. Karasek, and L. Rusch, "Cross-gain modulation in Raman fiber amplifier: Experimentation and Modeling," IEEE Photonics Technol. Lett. 14, 1261-1263 (2002).
[CrossRef]

A. Bononi, and L. Rusch, "Doped-fiber amplifier dynamics: a system perspective," J. Lightwave Technol. 16, 945-956 (1998).
[CrossRef]

Sun, F.

Z. G. Lu, J. Liu, F. Sun,  et al., "A hybrid fiber amplifier with 36.9-dBm output power and 70-dB gain," Opt. Commun. 256, 352-357 (2005).
[CrossRef]

Tiesler, M. M.

M. M. Tiesler, J. Witkowski, and K. Abramski, "Quality criterion for numerical methods in EDFA modeling," Optica Applicata 32, 187-196 (2002).

Tong, Z.

Z. Tong, H. Wei, and S. Jian, "A novel algorithm to simulate DWDM transmission systems amplified by backward multipumped fiber Raman amplifiers," Microwave Opt. Technol. Lett. 35, 333-337 (2002).
[CrossRef]

Tsang, H. K.

Wang, G.

L. Zhu, Y. Ma, G. Wang,  et al., "General computer model for both erbium-doped fiber amplifier and fiber Raman amplifier," Opt. Eng. 41, 1805-1808 (2002).
[CrossRef]

Wei, H.

Z. Tong, H. Wei, and S. Jian, "A novel algorithm to simulate DWDM transmission systems amplified by backward multipumped fiber Raman amplifiers," Microwave Opt. Technol. Lett. 35, 333-337 (2002).
[CrossRef]

Witkowski, J.

M. M. Tiesler, J. Witkowski, and K. Abramski, "Quality criterion for numerical methods in EDFA modeling," Optica Applicata 32, 187-196 (2002).

Xiao, M.

C. Cheng, and M. Xiao, "Optimization of an erbium-doped fiber amplifier with radial effects," Opt. Commun. 254, 215-222 (2005).
[CrossRef]

Zemon, S. A.

B. Pedersen, S. A. Zemon, and W. J. Miniscalco, "Analysis of erbium-doped fiber amplifiers pumped at 800nm," Fiber and Integr. Opt.,  10, 127-136 (1991).
[CrossRef]

Zhao, Y.

Zhu, L.

L. Zhu, Y. Ma, G. Wang,  et al., "General computer model for both erbium-doped fiber amplifier and fiber Raman amplifier," Opt. Eng. 41, 1805-1808 (2002).
[CrossRef]

Appl. Opt. (1)

ETRI J. (1)

S. K. Kim, S. Chang, J. Han,  et al., "Design of hybrid optical amplifiers for high capacity optical transmission," ETRI J. 24, 81-96 (2002).
[CrossRef]

Fiber and Integr. Opt. (1)

B. Pedersen, S. A. Zemon, and W. J. Miniscalco, "Analysis of erbium-doped fiber amplifiers pumped at 800nm," Fiber and Integr. Opt.,  10, 127-136 (1991).
[CrossRef]

IEE Proc. J. Optoelectron. (1)

M. Karasek, M. Menif, A. Bellemare, "Design of wideband hybrid amplifiers for local area networks," IEE Proc. J. Optoelectron. 148, 150-155 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

B. K. Min, W. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photon. Technol. Lett. 12, 1486-1488 (2000).
[CrossRef]

IEEE Photonics Technol. Lett. (5)

B. Pedersen, K. Dybdal, C. D. Hansen,  et al., "Detailed theoretical and experimental investigation of high-gain erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 2, 863-865 (1990).
[CrossRef]

M. Menif, M. Karasek, and L. Rusch, "Cross-gain modulation in Raman fiber amplifier: Experimentation and Modeling," IEEE Photonics Technol. Lett. 14, 1261-1263 (2002).
[CrossRef]

J. B. Rosolem, A. Juriollo, R. Arradi, A. D. Coral, J. C. R. Oliveira and M. A. Romero, "All silica S-band double-pass erbium-doped fiber amplifier," IEEE Photonics Technol. Lett. 17, 1399-1401 (2005).
[CrossRef]

T. G. Hodgkinson, "Average power analysis technique for erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 3, 1082-1084 (1991).
[CrossRef]

T. G. Hodgkinson, "Improved average power analysis technique for erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 4, 1273-1275 (1992).
[CrossRef]

J Optoelectron. (1)

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J. Opt. Soc. Am. B (1)

Microwave Opt. Technol. Lett. (1)

Z. Tong, H. Wei, and S. Jian, "A novel algorithm to simulate DWDM transmission systems amplified by backward multipumped fiber Raman amplifiers," Microwave Opt. Technol. Lett. 35, 333-337 (2002).
[CrossRef]

Opt. Commun. (2)

Z. G. Lu, J. Liu, F. Sun,  et al., "A hybrid fiber amplifier with 36.9-dBm output power and 70-dB gain," Opt. Commun. 256, 352-357 (2005).
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C. Cheng, and M. Xiao, "Optimization of an erbium-doped fiber amplifier with radial effects," Opt. Commun. 254, 215-222 (2005).
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L. Zhu, Y. Ma, G. Wang,  et al., "General computer model for both erbium-doped fiber amplifier and fiber Raman amplifier," Opt. Eng. 41, 1805-1808 (2002).
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Opt. Express (3)

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Figures (2)

Fig. 1.
Fig. 1.

Relationship of relative error with t. The circular symbols in figure represent the points that are calculated in the computing procedure.

Fig. 2.
Fig. 2.

Relationship of the relative error Er with the elemental amplifying section number n. Each section length h is L/n in the APA technique, but the length h is adaptive in the proposed method.

Equations (21)

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d P k ± dz = ± { [ ( σ 21 k + σ 12 k ) N 2 ( z ) σ 12 k N t ] Γ k α k } P k ± ( z ) ,
d P i ± dz = ± ( { [ ( σ 21 i + σ 12 i ) N 2 ( z ) σ 12 i N t ] Γ i α i } + 2 σ 12 i N 2 ( z ) Γ i h v i Δ v P i ± ( z ) ) P i ± ( z ) .
P k ± out = P k ± in G k ( z ) , P i ± out = 2 h v i n sp Δ v ( G i ( z ) 1 ) ,
G k , i ( z ) = exp ( { [ ( σ 21 k , i + σ 12 k , i ) N 2 σ 12 k , i N t ] Γ k , i α k , i } z ) ,
n sp = N 2 [ ( 1 + σ 12 k , i σ 21 k , i ) N 2 N t σ 12 k , i σ 21 k , i α k , i Γ k , i σ 21 k , i ] .
P k = P k ± in ( G k 1 In ( G k ) ) , P i = 2 h v i n sp Δ v ( G i 1 In ( G i ) 1 ) .
d P dz = f z P P .
dy dz = f ( z ) .
y n + 1,1 = y n + h f z n y n ,
y n + 1,2 = y n + 0.5 h f z n y n + 0.5 h f ( z n + 0.5 h , y n + 0.5 h f z n y n ) ,
y n + 1,3 = 2 y n + 1,2 y n + 1,1 .
ε 1 = y ( z n + h ) y n + 1,1 = K h 2 + O 1 ( h 3 ) ,
ε 2 = y ( z n + h ) y n + 1,2 = 0.5 K h 2 + O 2 ( h 3 ) ,
ε 3 = y ( z n + h ) y n + 1,3 = 2 O 2 ( h 3 ) O 1 ( h 3 ) ,
ε r = y n + 1,2 y n + 1,1 h 0.5 K h .
h ' = ε 0 h ε r .
P n + 1,1 = P n . exp ( h f z n P n ) ,
P n + 1,2 = P n exp [ 0.5 h f z n P n ] exp { 0.5 h f ( z n + 0.5 h , P n exp [ 0.5 h f z n P n ] ) } ,
P n + 1,3 = P n + 1,2 2 P n + 1,1 .
ε r = In ( P n + 1,2 ) In ( P n + 1,1 ) h 0.5 K h .
dx dt = x t + 1 = ( 1 t x + 1 x ) x , and x ( 2 ) = exp ( 2 ) + 2 .

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