Abstract

A basic analysis of complex-coupled distributed-feedback semiconductor lasers with stepwise constant coupling coefficients κ is derived. Solving coupled-wave equations at threshold reveals that the longitudinal distribution of κ as well as the relative amount of index and gain coupling plays a decisive role in the modal and spatial (internal fields) properties of complex-coupled structures. The standing-wave effect, extended to multisection devices, and the concept of apparent absorption induced by spatially dependent κ can explain the discrepancies between uniformly and nonuniformly complex-coupled structures. The complex-coupling profile is also discussed with respect to its influence on spatial hole burning and threshold gain margin, the usual criteria for optimizing sources in optical fiber telecommunication systems.

© 1998 Optical Society of America

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  1. H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
    [CrossRef]
  2. J. E. Whiteaway, G. H. B. Thompson, A. J. Collar, and C. J. Armistead, “The design and assessment of λ/4 shifted DFB structures,” IEEE J. Quantum Electron. 25, 1261–1279 (1989).
    [CrossRef]
  3. T. Fessant and J. Le Bihan, “Analysis of the spectral properties of multisection shifted DFB lasers with inhomogeneous coupling coefficient,” presented at the Semiconductor and Integrated Opto-Electronics Conference, SIOE’95, Cardiff, Wales, April 1995.
  4. B. S. K. Lo and H. Ghafouri-Shiraz, “Spectral characteristics of DFB laser diodes with distributed coupling coefficient,” J. Lightwave Technol. 13, 200–212 (1995).
    [CrossRef]
  5. T. Kimura and A. Sugimura, “Coupled phase-shift DFB semiconductor lasers for narrow linewidth operation,” IEEE J. Quantum Electron. 25, 678–683 (1989).
    [CrossRef]
  6. S. Ogita, Y. Kotaki, H. Ishikawa, and H. Imai, “Optimum design for multiple phase shift distributed feedback laser,” Electron. Lett. 24, 731–732 (1988).
    [CrossRef]
  7. J. I. Kinoshita and K. Matsumoto, “Yield analysis of SLM DFB lasers with axially-flattened internal field,” IEEE J. Quantum Electron. 25, 1324–1332 (1989).
    [CrossRef]
  8. H. Olesen, J. Salzman, B. Jonsson, and B. Tromborg, “Single mode stability of DFB lasers with longitudinal Bragg detuning,” IEEE Photonics Technol. Lett. 7, 461–463 (1995).
    [CrossRef]
  9. J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
    [CrossRef]
  10. H. Soda, K. Wakao, H. Sudo, T. Tanahashi, and H. Imai, “GaInAsP/InP phase-adjusted distributed feedback lasers with a step-like nonuniform stripe width structure,” Electron. Lett. 20, 1016–1018 (1984).
    [CrossRef]
  11. E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. QE-18, 66–71 (1982).
    [CrossRef]
  12. G. Morthier, P. Vankwikelberge, K. David, and R. Baets, “Improved performance of AR-coated DFB lasers by the introduction of gain-coupling,” IEEE Photonics Technol. Lett. 2, 170–172 (1990).
    [CrossRef]
  13. D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
    [CrossRef]
  14. K. Y. Kwon, “Effect of grating phase difference on single-mode yield in complex-coupled DFB lasers with gain and index gratings,” IEEE J. Quantum Electron. 32, 1937–1949 (1996).
    [CrossRef]
  15. Y. Boucher, “Non-reciprocal effects of complex-coupled distributed-feedback structures resulting from the phase difference between the coupling constants,” Opt. Commun. 136, 410–414 (1997).
    [CrossRef]
  16. K. David, J. Buus, and R. G. Baets, “Basic analysis of AR-coated, partly gain-coupled DFB lasers: the standing wave effect,” IEEE J. Quantum Electron. 28, 427–433 (1992).
    [CrossRef]
  17. F. Randone and I. Montrosset, “Analysis and simulation of gain-coupled distributed feedback semiconductor lasers,” IEEE J. Quantum Electron. 31, 1964–1973 (1995).
    [CrossRef]
  18. B. Jonsson, A. J. Lowery, H. Olesen, and B. Tromborg, “Instabilities and non-linear L-I characteristics in complex-coupled DFB lasers with antiphase gain and index gratings,” IEEE J. Quantum Electron. 32, 839–850 (1996).
    [CrossRef]
  19. J. Chen, A. Champagne, R. Maciejko, and T. Makino, “Improvement of single-mode gain margin in gain-coupled DFB lasers,” IEEE J. Quantum Electron. 33, 33–40 (1997).
    [CrossRef]
  20. W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
    [CrossRef]
  21. L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
    [CrossRef]
  22. R. Bonello and I. Montrosset, “Analysis of multisection and multielectrode semiconductor lasers,” J. Lightwave Technol. 10, 1890–1900 (1992).
    [CrossRef]
  23. G. P. Agrawal and N. K. Dutta, Long Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
  24. Y. Boucher, O. Dellea, and J. Le Bihan, “Quasi-periodic complex-coupled distributed feedback structures with an exponential-like gradient of coupling,” IEEE J. Quantum Electron. 33, 2137–2145 (1997).
    [CrossRef]
  25. H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  26. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
    [CrossRef]

1997 (3)

Y. Boucher, “Non-reciprocal effects of complex-coupled distributed-feedback structures resulting from the phase difference between the coupling constants,” Opt. Commun. 136, 410–414 (1997).
[CrossRef]

J. Chen, A. Champagne, R. Maciejko, and T. Makino, “Improvement of single-mode gain margin in gain-coupled DFB lasers,” IEEE J. Quantum Electron. 33, 33–40 (1997).
[CrossRef]

Y. Boucher, O. Dellea, and J. Le Bihan, “Quasi-periodic complex-coupled distributed feedback structures with an exponential-like gradient of coupling,” IEEE J. Quantum Electron. 33, 2137–2145 (1997).
[CrossRef]

1996 (3)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[CrossRef]

B. Jonsson, A. J. Lowery, H. Olesen, and B. Tromborg, “Instabilities and non-linear L-I characteristics in complex-coupled DFB lasers with antiphase gain and index gratings,” IEEE J. Quantum Electron. 32, 839–850 (1996).
[CrossRef]

K. Y. Kwon, “Effect of grating phase difference on single-mode yield in complex-coupled DFB lasers with gain and index gratings,” IEEE J. Quantum Electron. 32, 1937–1949 (1996).
[CrossRef]

1995 (5)

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[CrossRef]

F. Randone and I. Montrosset, “Analysis and simulation of gain-coupled distributed feedback semiconductor lasers,” IEEE J. Quantum Electron. 31, 1964–1973 (1995).
[CrossRef]

B. S. K. Lo and H. Ghafouri-Shiraz, “Spectral characteristics of DFB laser diodes with distributed coupling coefficient,” J. Lightwave Technol. 13, 200–212 (1995).
[CrossRef]

H. Olesen, J. Salzman, B. Jonsson, and B. Tromborg, “Single mode stability of DFB lasers with longitudinal Bragg detuning,” IEEE Photonics Technol. Lett. 7, 461–463 (1995).
[CrossRef]

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

1994 (1)

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
[CrossRef]

1992 (2)

R. Bonello and I. Montrosset, “Analysis of multisection and multielectrode semiconductor lasers,” J. Lightwave Technol. 10, 1890–1900 (1992).
[CrossRef]

K. David, J. Buus, and R. G. Baets, “Basic analysis of AR-coated, partly gain-coupled DFB lasers: the standing wave effect,” IEEE J. Quantum Electron. 28, 427–433 (1992).
[CrossRef]

1990 (1)

G. Morthier, P. Vankwikelberge, K. David, and R. Baets, “Improved performance of AR-coated DFB lasers by the introduction of gain-coupling,” IEEE Photonics Technol. Lett. 2, 170–172 (1990).
[CrossRef]

1989 (3)

J. I. Kinoshita and K. Matsumoto, “Yield analysis of SLM DFB lasers with axially-flattened internal field,” IEEE J. Quantum Electron. 25, 1324–1332 (1989).
[CrossRef]

T. Kimura and A. Sugimura, “Coupled phase-shift DFB semiconductor lasers for narrow linewidth operation,” IEEE J. Quantum Electron. 25, 678–683 (1989).
[CrossRef]

J. E. Whiteaway, G. H. B. Thompson, A. J. Collar, and C. J. Armistead, “The design and assessment of λ/4 shifted DFB structures,” IEEE J. Quantum Electron. 25, 1261–1279 (1989).
[CrossRef]

1988 (1)

S. Ogita, Y. Kotaki, H. Ishikawa, and H. Imai, “Optimum design for multiple phase shift distributed feedback laser,” Electron. Lett. 24, 731–732 (1988).
[CrossRef]

1987 (1)

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

1984 (1)

H. Soda, K. Wakao, H. Sudo, T. Tanahashi, and H. Imai, “GaInAsP/InP phase-adjusted distributed feedback lasers with a step-like nonuniform stripe width structure,” Electron. Lett. 20, 1016–1018 (1984).
[CrossRef]

1982 (1)

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. QE-18, 66–71 (1982).
[CrossRef]

1977 (1)

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Albrektsen, O.

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

Armistead, C. J.

J. E. Whiteaway, G. H. B. Thompson, A. J. Collar, and C. J. Armistead, “The design and assessment of λ/4 shifted DFB structures,” IEEE J. Quantum Electron. 25, 1261–1279 (1989).
[CrossRef]

Baets, R.

G. Morthier, P. Vankwikelberge, K. David, and R. Baets, “Improved performance of AR-coated DFB lasers by the introduction of gain-coupling,” IEEE Photonics Technol. Lett. 2, 170–172 (1990).
[CrossRef]

Baets, R. G.

K. David, J. Buus, and R. G. Baets, “Basic analysis of AR-coated, partly gain-coupled DFB lasers: the standing wave effect,” IEEE J. Quantum Electron. 28, 427–433 (1992).
[CrossRef]

Bonello, R.

R. Bonello and I. Montrosset, “Analysis of multisection and multielectrode semiconductor lasers,” J. Lightwave Technol. 10, 1890–1900 (1992).
[CrossRef]

Boucher, Y.

Y. Boucher, “Non-reciprocal effects of complex-coupled distributed-feedback structures resulting from the phase difference between the coupling constants,” Opt. Commun. 136, 410–414 (1997).
[CrossRef]

Y. Boucher, O. Dellea, and J. Le Bihan, “Quasi-periodic complex-coupled distributed feedback structures with an exponential-like gradient of coupling,” IEEE J. Quantum Electron. 33, 2137–2145 (1997).
[CrossRef]

Burnham, R. D.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Buus, J.

K. David, J. Buus, and R. G. Baets, “Basic analysis of AR-coated, partly gain-coupled DFB lasers: the standing wave effect,” IEEE J. Quantum Electron. 28, 427–433 (1992).
[CrossRef]

Cardimona, D. A.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[CrossRef]

Carroll, J. E.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
[CrossRef]

Champagne, A.

J. Chen, A. Champagne, R. Maciejko, and T. Makino, “Improvement of single-mode gain margin in gain-coupled DFB lasers,” IEEE J. Quantum Electron. 33, 33–40 (1997).
[CrossRef]

Chen, J.

J. Chen, A. Champagne, R. Maciejko, and T. Makino, “Improvement of single-mode gain margin in gain-coupled DFB lasers,” IEEE J. Quantum Electron. 33, 33–40 (1997).
[CrossRef]

Collar, A. J.

J. E. Whiteaway, G. H. B. Thompson, A. J. Collar, and C. J. Armistead, “The design and assessment of λ/4 shifted DFB structures,” IEEE J. Quantum Electron. 25, 1261–1279 (1989).
[CrossRef]

David, K.

K. David, J. Buus, and R. G. Baets, “Basic analysis of AR-coated, partly gain-coupled DFB lasers: the standing wave effect,” IEEE J. Quantum Electron. 28, 427–433 (1992).
[CrossRef]

G. Morthier, P. Vankwikelberge, K. David, and R. Baets, “Improved performance of AR-coated DFB lasers by the introduction of gain-coupling,” IEEE Photonics Technol. Lett. 2, 170–172 (1990).
[CrossRef]

Dellea, O.

Y. Boucher, O. Dellea, and J. Le Bihan, “Quasi-periodic complex-coupled distributed feedback structures with an exponential-like gradient of coupling,” IEEE J. Quantum Electron. 33, 2137–2145 (1997).
[CrossRef]

Gavrielides, A.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[CrossRef]

Ghafouri-Shiraz, H.

B. S. K. Lo and H. Ghafouri-Shiraz, “Spectral characteristics of DFB laser diodes with distributed coupling coefficient,” J. Lightwave Technol. 13, 200–212 (1995).
[CrossRef]

Hanberg, J.

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

Hardy, A.

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. QE-18, 66–71 (1982).
[CrossRef]

Imai, H.

S. Ogita, Y. Kotaki, H. Ishikawa, and H. Imai, “Optimum design for multiple phase shift distributed feedback laser,” Electron. Lett. 24, 731–732 (1988).
[CrossRef]

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

H. Soda, K. Wakao, H. Sudo, T. Tanahashi, and H. Imai, “GaInAsP/InP phase-adjusted distributed feedback lasers with a step-like nonuniform stripe width structure,” Electron. Lett. 20, 1016–1018 (1984).
[CrossRef]

Ishikawa, H.

S. Ogita, Y. Kotaki, H. Ishikawa, and H. Imai, “Optimum design for multiple phase shift distributed feedback laser,” Electron. Lett. 24, 731–732 (1988).
[CrossRef]

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Jonsson, B.

B. Jonsson, A. J. Lowery, H. Olesen, and B. Tromborg, “Instabilities and non-linear L-I characteristics in complex-coupled DFB lasers with antiphase gain and index gratings,” IEEE J. Quantum Electron. 32, 839–850 (1996).
[CrossRef]

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

H. Olesen, J. Salzman, B. Jonsson, and B. Tromborg, “Single mode stability of DFB lasers with longitudinal Bragg detuning,” IEEE Photonics Technol. Lett. 7, 461–463 (1995).
[CrossRef]

Kapon, E.

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. QE-18, 66–71 (1982).
[CrossRef]

Katzir, A.

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. QE-18, 66–71 (1982).
[CrossRef]

Kimura, T.

T. Kimura and A. Sugimura, “Coupled phase-shift DFB semiconductor lasers for narrow linewidth operation,” IEEE J. Quantum Electron. 25, 678–683 (1989).
[CrossRef]

Kinoshita, J. I.

J. I. Kinoshita and K. Matsumoto, “Yield analysis of SLM DFB lasers with axially-flattened internal field,” IEEE J. Quantum Electron. 25, 1324–1332 (1989).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Kotaki, Y.

S. Ogita, Y. Kotaki, H. Ishikawa, and H. Imai, “Optimum design for multiple phase shift distributed feedback laser,” Electron. Lett. 24, 731–732 (1988).
[CrossRef]

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Kovanis, V.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[CrossRef]

Kwon, K. Y.

K. Y. Kwon, “Effect of grating phase difference on single-mode yield in complex-coupled DFB lasers with gain and index gratings,” IEEE J. Quantum Electron. 32, 1937–1949 (1996).
[CrossRef]

Le Bihan, J.

Y. Boucher, O. Dellea, and J. Le Bihan, “Quasi-periodic complex-coupled distributed feedback structures with an exponential-like gradient of coupling,” IEEE J. Quantum Electron. 33, 2137–2145 (1997).
[CrossRef]

Lo, B. S. K.

B. S. K. Lo and H. Ghafouri-Shiraz, “Spectral characteristics of DFB laser diodes with distributed coupling coefficient,” J. Lightwave Technol. 13, 200–212 (1995).
[CrossRef]

Lowery, A. J.

B. Jonsson, A. J. Lowery, H. Olesen, and B. Tromborg, “Instabilities and non-linear L-I characteristics in complex-coupled DFB lasers with antiphase gain and index gratings,” IEEE J. Quantum Electron. 32, 839–850 (1996).
[CrossRef]

Maciejko, R.

J. Chen, A. Champagne, R. Maciejko, and T. Makino, “Improvement of single-mode gain margin in gain-coupled DFB lasers,” IEEE J. Quantum Electron. 33, 33–40 (1997).
[CrossRef]

Makino, T.

J. Chen, A. Champagne, R. Maciejko, and T. Makino, “Improvement of single-mode gain margin in gain-coupled DFB lasers,” IEEE J. Quantum Electron. 33, 33–40 (1997).
[CrossRef]

Marcenac, D. D.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
[CrossRef]

Matsumoto, K.

J. I. Kinoshita and K. Matsumoto, “Yield analysis of SLM DFB lasers with axially-flattened internal field,” IEEE J. Quantum Electron. 25, 1324–1332 (1989).
[CrossRef]

Moller-Larsen, A.

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

Montrosset, I.

F. Randone and I. Montrosset, “Analysis and simulation of gain-coupled distributed feedback semiconductor lasers,” IEEE J. Quantum Electron. 31, 1964–1973 (1995).
[CrossRef]

R. Bonello and I. Montrosset, “Analysis of multisection and multielectrode semiconductor lasers,” J. Lightwave Technol. 10, 1890–1900 (1992).
[CrossRef]

Morthier, G.

G. Morthier, P. Vankwikelberge, K. David, and R. Baets, “Improved performance of AR-coated DFB lasers by the introduction of gain-coupling,” IEEE Photonics Technol. Lett. 2, 170–172 (1990).
[CrossRef]

Norregaard, J.

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

Nowell, M. C.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
[CrossRef]

Ogita, S.

S. Ogita, Y. Kotaki, H. Ishikawa, and H. Imai, “Optimum design for multiple phase shift distributed feedback laser,” Electron. Lett. 24, 731–732 (1988).
[CrossRef]

Olesen, H.

B. Jonsson, A. J. Lowery, H. Olesen, and B. Tromborg, “Instabilities and non-linear L-I characteristics in complex-coupled DFB lasers with antiphase gain and index gratings,” IEEE J. Quantum Electron. 32, 839–850 (1996).
[CrossRef]

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

H. Olesen, J. Salzman, B. Jonsson, and B. Tromborg, “Single mode stability of DFB lasers with longitudinal Bragg detuning,” IEEE Photonics Technol. Lett. 7, 461–463 (1995).
[CrossRef]

Plumb, R. G. S.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
[CrossRef]

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[CrossRef]

Randone, F.

F. Randone and I. Montrosset, “Analysis and simulation of gain-coupled distributed feedback semiconductor lasers,” IEEE J. Quantum Electron. 31, 1964–1973 (1995).
[CrossRef]

Salzman, J.

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

H. Olesen, J. Salzman, B. Jonsson, and B. Tromborg, “Single mode stability of DFB lasers with longitudinal Bragg detuning,” IEEE Photonics Technol. Lett. 7, 461–463 (1995).
[CrossRef]

Scifres, D. R.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Sharma, M. P.

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[CrossRef]

Soda, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

H. Soda, K. Wakao, H. Sudo, T. Tanahashi, and H. Imai, “GaInAsP/InP phase-adjusted distributed feedback lasers with a step-like nonuniform stripe width structure,” Electron. Lett. 20, 1016–1018 (1984).
[CrossRef]

Streifer, W.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

Sudo, H.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

H. Soda, K. Wakao, H. Sudo, T. Tanahashi, and H. Imai, “GaInAsP/InP phase-adjusted distributed feedback lasers with a step-like nonuniform stripe width structure,” Electron. Lett. 20, 1016–1018 (1984).
[CrossRef]

Sugimura, A.

T. Kimura and A. Sugimura, “Coupled phase-shift DFB semiconductor lasers for narrow linewidth operation,” IEEE J. Quantum Electron. 25, 678–683 (1989).
[CrossRef]

Tanahashi, T.

H. Soda, K. Wakao, H. Sudo, T. Tanahashi, and H. Imai, “GaInAsP/InP phase-adjusted distributed feedback lasers with a step-like nonuniform stripe width structure,” Electron. Lett. 20, 1016–1018 (1984).
[CrossRef]

Thompson, G. H. B.

J. E. Whiteaway, G. H. B. Thompson, A. J. Collar, and C. J. Armistead, “The design and assessment of λ/4 shifted DFB structures,” IEEE J. Quantum Electron. 25, 1261–1279 (1989).
[CrossRef]

Tromborg, B.

B. Jonsson, A. J. Lowery, H. Olesen, and B. Tromborg, “Instabilities and non-linear L-I characteristics in complex-coupled DFB lasers with antiphase gain and index gratings,” IEEE J. Quantum Electron. 32, 839–850 (1996).
[CrossRef]

H. Olesen, J. Salzman, B. Jonsson, and B. Tromborg, “Single mode stability of DFB lasers with longitudinal Bragg detuning,” IEEE Photonics Technol. Lett. 7, 461–463 (1995).
[CrossRef]

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

Vankwikelberge, P.

G. Morthier, P. Vankwikelberge, K. David, and R. Baets, “Improved performance of AR-coated DFB lasers by the introduction of gain-coupling,” IEEE Photonics Technol. Lett. 2, 170–172 (1990).
[CrossRef]

Wakao, K.

H. Soda, K. Wakao, H. Sudo, T. Tanahashi, and H. Imai, “GaInAsP/InP phase-adjusted distributed feedback lasers with a step-like nonuniform stripe width structure,” Electron. Lett. 20, 1016–1018 (1984).
[CrossRef]

Whiteaway, J. E.

J. E. Whiteaway, G. H. B. Thompson, A. J. Collar, and C. J. Armistead, “The design and assessment of λ/4 shifted DFB structures,” IEEE J. Quantum Electron. 25, 1261–1279 (1989).
[CrossRef]

Yamakoshi, S.

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

Yu, S. F.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
[CrossRef]

Zhang, L. M.

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
[CrossRef]

Electron. Lett. (2)

S. Ogita, Y. Kotaki, H. Ishikawa, and H. Imai, “Optimum design for multiple phase shift distributed feedback laser,” Electron. Lett. 24, 731–732 (1988).
[CrossRef]

H. Soda, K. Wakao, H. Sudo, T. Tanahashi, and H. Imai, “GaInAsP/InP phase-adjusted distributed feedback lasers with a step-like nonuniform stripe width structure,” Electron. Lett. 20, 1016–1018 (1984).
[CrossRef]

IEEE J. Quantum Electron. (14)

E. Kapon, A. Hardy, and A. Katzir, “The effect of complex coupling coefficients on distributed feedback lasers,” IEEE J. Quantum Electron. QE-18, 66–71 (1982).
[CrossRef]

J. I. Kinoshita and K. Matsumoto, “Yield analysis of SLM DFB lasers with axially-flattened internal field,” IEEE J. Quantum Electron. 25, 1324–1332 (1989).
[CrossRef]

H. Soda, Y. Kotaki, H. Sudo, H. Ishikawa, S. Yamakoshi, and H. Imai, “Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers,” IEEE J. Quantum Electron. QE-23, 804–814 (1987).
[CrossRef]

J. E. Whiteaway, G. H. B. Thompson, A. J. Collar, and C. J. Armistead, “The design and assessment of λ/4 shifted DFB structures,” IEEE J. Quantum Electron. 25, 1261–1279 (1989).
[CrossRef]

T. Kimura and A. Sugimura, “Coupled phase-shift DFB semiconductor lasers for narrow linewidth operation,” IEEE J. Quantum Electron. 25, 678–683 (1989).
[CrossRef]

K. David, J. Buus, and R. G. Baets, “Basic analysis of AR-coated, partly gain-coupled DFB lasers: the standing wave effect,” IEEE J. Quantum Electron. 28, 427–433 (1992).
[CrossRef]

F. Randone and I. Montrosset, “Analysis and simulation of gain-coupled distributed feedback semiconductor lasers,” IEEE J. Quantum Electron. 31, 1964–1973 (1995).
[CrossRef]

B. Jonsson, A. J. Lowery, H. Olesen, and B. Tromborg, “Instabilities and non-linear L-I characteristics in complex-coupled DFB lasers with antiphase gain and index gratings,” IEEE J. Quantum Electron. 32, 839–850 (1996).
[CrossRef]

J. Chen, A. Champagne, R. Maciejko, and T. Makino, “Improvement of single-mode gain margin in gain-coupled DFB lasers,” IEEE J. Quantum Electron. 33, 33–40 (1997).
[CrossRef]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. QE-13, 134–141 (1977).
[CrossRef]

L. M. Zhang, S. F. Yu, M. C. Nowell, D. D. Marcenac, J. E. Carroll, and R. G. S. Plumb, “Dynamic analysis of radiation side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model,” IEEE J. Quantum Electron. 30, 1389–1395 (1994).
[CrossRef]

D. A. Cardimona, M. P. Sharma, V. Kovanis, and A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[CrossRef]

K. Y. Kwon, “Effect of grating phase difference on single-mode yield in complex-coupled DFB lasers with gain and index gratings,” IEEE J. Quantum Electron. 32, 1937–1949 (1996).
[CrossRef]

Y. Boucher, O. Dellea, and J. Le Bihan, “Quasi-periodic complex-coupled distributed feedback structures with an exponential-like gradient of coupling,” IEEE J. Quantum Electron. 33, 2137–2145 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Salzman, H. Olesen, A. Moller-Larsen, O. Albrektsen, J. Hanberg, J. Norregaard, B. Jonsson, and B. Tromborg, “Distributed feedback lasers with an S-bent waveguide for high-power single-mode operation,” IEEE J. Sel. Top. Quantum Electron. 1, 346–355 (1995).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

H. Olesen, J. Salzman, B. Jonsson, and B. Tromborg, “Single mode stability of DFB lasers with longitudinal Bragg detuning,” IEEE Photonics Technol. Lett. 7, 461–463 (1995).
[CrossRef]

G. Morthier, P. Vankwikelberge, K. David, and R. Baets, “Improved performance of AR-coated DFB lasers by the introduction of gain-coupling,” IEEE Photonics Technol. Lett. 2, 170–172 (1990).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

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B. S. K. Lo and H. Ghafouri-Shiraz, “Spectral characteristics of DFB laser diodes with distributed coupling coefficient,” J. Lightwave Technol. 13, 200–212 (1995).
[CrossRef]

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[CrossRef]

Opt. Commun. (1)

Y. Boucher, “Non-reciprocal effects of complex-coupled distributed-feedback structures resulting from the phase difference between the coupling constants,” Opt. Commun. 136, 410–414 (1997).
[CrossRef]

Phys. Rev. E (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E 54, 2963–2975 (1996).
[CrossRef]

Other (2)

G. P. Agrawal and N. K. Dutta, Long Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).

T. Fessant and J. Le Bihan, “Analysis of the spectral properties of multisection shifted DFB lasers with inhomogeneous coupling coefficient,” presented at the Semiconductor and Integrated Opto-Electronics Conference, SIOE’95, Cardiff, Wales, April 1995.

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Figures (10)

Fig. 1
Fig. 1

(a) Schematic view of in-phase index and gain gratings with (b) the possible longitudinal dependence of the resultant complex-coupling coefficient. In this paper three-section structures with stepwise constant complex-coupling coefficients are studied. (c) Uniform (conventional) coupling κ(1)=κ(2), (d) stronger outer coupling κ(1)>κ(2), (e) stronger center coupling κ(2)>κ(1).

Fig. 2
Fig. 2

Normalized net threshold gain αL of the main mode versus average normalized coupling-coefficient modulus |κLav| with Φ as a parameter and the three coupling configurations. Φ represents the relative amount of index and gain contribution. (a) (pure index coupling), (e) (pure gain coupling).

Fig. 3
Fig. 3

Normalized net threshold gain αL of the first side modes versus |κLav| with Φ as a parameter and the three coupling configurations.

Fig. 4
Fig. 4

Corresponding normalized threshold gain margin ΔαL versus |κLav| with Φ as a parameter and the three coupling configurations.

Fig. 5
Fig. 5

Longitudinal profile of the normalized standing-wave pattern of the main and first side modes. The upper curve in each case represents a schematic view of the grating shape. |κLav|=2 and Φ=π/2: (a) uniform coupling, (b) stronger outer coupling, (c) stronger center coupling.

Fig. 6
Fig. 6

FST coefficient as defined by Ref. 24 (solid curves) and normalized total loss per facet αendL (dashed curves) versus |κLav| with Φ as a parameter.

Fig. 7
Fig. 7

Longitudinal profile of the main-mode normalized photon density envelope in the case of a stronger center coupling κ(2)>κ(1), with |κLav| as a parameter.  

Fig. 8
Fig. 8

Longitudinal profile of the main-mode normalized photon density envelope in the case of stronger outer coupling κ(1)>κ(2), with |κLav| as a parameter.

Fig. 9
Fig. 9

Longitudinal profile of the normalized standing-wave pattern of the main mode in the specific case |κLav|=π/2 and Φ=π/2. The three coupling configurations exhibit the same photon density envelope (d).

Fig. 10
Fig. 10

Longitudinal profile of the main-mode normalized photon density envelope with Φ as a parameter for uniform coupling, stronger outer coupling, and stronger center coupling. |κLav|=2.

Equations (37)

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n(z)=n+Δn(z)cos[2πz/Λ+Ω(z)],
α(z)=α+Δα(z)cos[2πz/Λ+Ω(z)],
R(z)=(α-iδ)R(z)-iκ(z)S(z)exp[-iΩ(z)],
S(z)=-(α-iδ)S(z)+iκ˜(z)R(z)exp[+iΩ(z)],
E(m)+(z)=R(m)(z)exp[-iβ(z-zm)],
E(m)-(z)=S(m)(z)exp[+iβ(z-zm)],
E(m)(x, y, z)=Φ(x, y)[E(m)+(z)+E(m)-(z)],
E(m)+(zm+1)E(m)-(zm+1)=T11(m)T12(m)T21(m)T22(m) E(m)+(zm)E(m)-(zm).
T11(m)=cosh[γ(m)L(m)]+(α-iδ)γ(m) sinh[γ(m)L(m)]×exp[-iβL(m)],
T12(m)=-iκ(m)γ(m) sinh[γ(m)L(m)]exp(-iΩ(m))exp[-iβL(m)],
T21(m)=+iκ˜(m)γ(m) sinh[γ(m)L(m)]exp(+iΩ(m))exp[+iβL(m)],
T22(m)=cosh[γ(m)L(m)]-(α-iδ)γ(m) sinh[γ(m)L(m)]×exp[+iβL(m)].
γ(m)=[(α-iδ)2+κ(m)κ˜(m)]1/2,
κ(m)=κ˜(m)=κind(m)+iκgain(m),
E(m)+(z)=E(m)+(zm+1)cosh[γ(m)(zm+1-z)]-(α-iδ)γ(m) sinh[γ(m)(zm+1-z)]×exp[+iβ(zm+1-z)]+E(m)-(zm+1)×iκ(m)γ(m) sinh[γ(m)(zm+1-z)]×exp[-iΩ(m)]×exp[-iβ(zm+1+z)]exp(2iβzm),
E(m)-(z)=E(m)+(zm+1)-iκ˜(m)γ(m) sinh[γ(m)(zm+1-z)]×exp[+iΩ(m)]exp[+iβ(zm+1+z)]×exp(-2iβzm)+E(m)-(zm+1)×cosh[γ(m)(zm+1-z)]+(α-iδ)γ(m) sinh[γ(m)(zm+1-z)]×exp[-iβ(zm+1-z)].
P(m)(z)=|E(m)+(z)+E(m)-(z)|2,
P(m)(z)= |E(m)+(z)|2+|E(m)-(z)|2.
ddz (RR*-SS*)=2α(RR*+SS*)+2 Im{SR* exp[-iΩ(z)]×[κ(z)-κ˜*(z)]},
m=1N[R(m)R*(m)-S(m)S*(m)]zmzm+1
=[|R(L)|2-|S(L)|2]+[|S(0)|2-|R(0)|2],
E+(m)(zm+1)=E+(m+1)(zm+1),
E-(m)(zm+1)=E-(m+1)(zm+1),
PTOTAL=m=1NPtot(m)=m=1Nzmzm+1[R(m)R*(m)+S(m)S*(m)]dz,
αend, right=[|R(L)|2-|S(L)|2]/PTOTAL,
αend, left=[|S(0)|2-|R(0)|2]/PTOTAL.
 αend, right=αend, left=αend.
2αm=1Nzmzm+1[R(m)R*(m)+S(m)S*(m)]dz
=2αm=1NPtot(m)=2αPTOTAL.
fst(m)=Re2zmzm+1R(m)S*(m) exp[iΩ(m)]dzPTOTAL.
m=1Nzmzm+12 Im{2iκgain(m)S(m)R*(m) exp[-iΩ(m)]}dz
=m=1N2κgain(m) Re2zmzm+1R(m)S*(m) exp[iΩ(m)]dz
=PTOTALm=1N2κgain(m) fst(m).
αL=αend L-m=1Nfst(m)κgain(m) L.
κLav=m=13κ(m)L(m)=m=13|κ(m)|exp[iΦ(m)]L(m),
κLav= |κ(1)L(1)+κ(2)L(2)+κ(3)L(3)|exp(iΦ)= |κLav|exp(iΦ).
FST=m=13fst(m)κgain(m) L.

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