Abstract

It is a generally accepted fact of laser physics that in a homogeneously broadened gain medium, above threshold the output power of the laser grows linearly with the pump power. The derivation requires only a few simple lines in laser textbooks, and the linear growth is a direct result of the fact that above threshold, the intracavity optical intensity will increase to the point that the gain is saturated to the level of the net loss—so-called gain pinning or clamping. Such a derivation, however, assumes that the mirror loss is distributed (the approximation of uniform gain saturation) which is only a good assumption for cavities whose end mirrors have reflectivities close to one. Furthermore, in gain media with a distributed loss there is a maximum achievable intracavity intensity that in turn limits the output power. We show that the approximation of uniform gain saturation leads to output powers that violate this limit. More generally, for lasers with low mirror reflectivities that also have distributed loss, we prove that the output power grows sublinearly with the pump power close to threshold. Furthermore, after threshold the output grows linearly, but with a slope efficiency that can be substantially smaller than predicted by the uniform gain saturation theory, with the largest deviation occurring for traveling-wave lasers and asymmetric Fabry–Perot lasers. These results are particularly applicable to semiconductor lasers, and specific applications to quantum cascade lasers are discussed.

© 2015 Optical Society of America

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References

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  1. A. Siegman, Lasers (University Science Books, 1986).
  2. O. Svelto, Principles of Lasers, 5th Ed. (Springer, 2010).
  3. W. T. Silfvast, Laser Fundamentals, 2nd Ed. (Cambridge University, 2004).
  4. A. Yariv, P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th Ed. (Oxford University, 2007).
  5. B. E. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd Ed. (Wiley, 2007).
  6. S. L. Chuang, Physics of Photonic Devices, 2nd Ed. (Wiley, 2009).
  7. L. A. Coldren, S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).
  8. E. Schulz-DuBois, “Pulse sharpening and gain saturation in traveling-wave masers,” Bell Syst. Tech. J. 43, 625–658 (1964).
    [Crossref]
  9. W. Rigrod, “Gain saturation and output power of optical masers,” J. Appl. Phys. 34, 2602–2609 (1963).
    [Crossref]
  10. N. Karlov, Lectures on Quantum Electronics, 2nd Ed. (CRC, 1993).
  11. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
    [Crossref]
  12. W. Rigrod, “Homogeneously broadened CW lasers with uniform distributed loss,” IEEE J. Quantum Electron. 14, 377–381 (1978).
    [Crossref]
  13. G. M. Schindler, “Optimum output efficiency of homogeneously broadened lasers with constant loss,” IEEE J. Quantum Electron. 16, 546–549 (1980).
    [Crossref]
  14. T. R. Ferguson, “Lasers with saturable gain and distributed loss,” Appl. Opt. 26, 2522–2527 (1987).
    [Crossref]
  15. T. R. Ferguson, W. P. Latham, “Efficiency and equivalence of homogeneously broadened lossy lasers,” Appl. Opt. 31, 4113–4121 (1992).
    [Crossref]
  16. S. Hasuo, T. Ohmi, “Spatial distribution of the light intensity in the injection lasers,” Jpn. J. Appl. Phys. 13, 1429–1434 (1974).
    [Crossref]
  17. W. W. Fang, C. Bethea, Y. Chen, S. L. Chuang, “Longitudinal spatial inhomogeneities in high-power semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 117–128 (1995).
    [Crossref]
  18. F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
    [Crossref]
  19. B. Ryvkin, E. Avrutin, “Spatial hole burning in high-power edge-emitting lasers: A simple analytical model and the effect on laser performance,” J. Appl. Phys. 109, 043101 (2011).
    [Crossref]
  20. Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
    [Crossref]
  21. L. W. Casperson, “Laser power calculations: sources of error,” Appl. Opt. 19, 422–434 (1980).
    [Crossref]
  22. A. White, E. Gordon, J. Rigden, “Output power of the 6328-Å gas maser,” Appl. Phys. Lett. 2, 91–93 (1963).
    [Crossref]
  23. L. W. Casperson, “Threshold characteristics of multimode laser oscillators,” J. Appl. Phys. 46, 5194–5201 (1975).
    [Crossref]
  24. L. W. Casperson, M. Khoshnevissan, “Threshold characteristics of multimode semiconductor lasers,” J. Appl. Phys. 75, 737–747 (1994).
    [Crossref]
  25. C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. 64, 1533–1601 (2001).
    [Crossref]
  26. A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
    [Crossref]

2012 (1)

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

2011 (1)

B. Ryvkin, E. Avrutin, “Spatial hole burning in high-power edge-emitting lasers: A simple analytical model and the effect on laser performance,” J. Appl. Phys. 109, 043101 (2011).
[Crossref]

2008 (1)

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

2002 (1)

F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
[Crossref]

2001 (1)

C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. 64, 1533–1601 (2001).
[Crossref]

1995 (1)

W. W. Fang, C. Bethea, Y. Chen, S. L. Chuang, “Longitudinal spatial inhomogeneities in high-power semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 117–128 (1995).
[Crossref]

1994 (1)

L. W. Casperson, M. Khoshnevissan, “Threshold characteristics of multimode semiconductor lasers,” J. Appl. Phys. 75, 737–747 (1994).
[Crossref]

1992 (1)

1987 (1)

1980 (2)

G. M. Schindler, “Optimum output efficiency of homogeneously broadened lasers with constant loss,” IEEE J. Quantum Electron. 16, 546–549 (1980).
[Crossref]

L. W. Casperson, “Laser power calculations: sources of error,” Appl. Opt. 19, 422–434 (1980).
[Crossref]

1978 (1)

W. Rigrod, “Homogeneously broadened CW lasers with uniform distributed loss,” IEEE J. Quantum Electron. 14, 377–381 (1978).
[Crossref]

1975 (1)

L. W. Casperson, “Threshold characteristics of multimode laser oscillators,” J. Appl. Phys. 46, 5194–5201 (1975).
[Crossref]

1974 (1)

S. Hasuo, T. Ohmi, “Spatial distribution of the light intensity in the injection lasers,” Jpn. J. Appl. Phys. 13, 1429–1434 (1974).
[Crossref]

1965 (1)

W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

1964 (1)

E. Schulz-DuBois, “Pulse sharpening and gain saturation in traveling-wave masers,” Bell Syst. Tech. J. 43, 625–658 (1964).
[Crossref]

1963 (2)

W. Rigrod, “Gain saturation and output power of optical masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[Crossref]

A. White, E. Gordon, J. Rigden, “Output power of the 6328-Å gas maser,” Appl. Phys. Lett. 2, 91–93 (1963).
[Crossref]

Avrutin, E.

B. Ryvkin, E. Avrutin, “Spatial hole burning in high-power edge-emitting lasers: A simple analytical model and the effect on laser performance,” J. Appl. Phys. 109, 043101 (2011).
[Crossref]

Bai, J.

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Bao, L.

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Bethea, C.

W. W. Fang, C. Bethea, Y. Chen, S. L. Chuang, “Longitudinal spatial inhomogeneities in high-power semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 117–128 (1995).
[Crossref]

Capasso, F.

C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. 64, 1533–1601 (2001).
[Crossref]

Casperson, L. W.

L. W. Casperson, M. Khoshnevissan, “Threshold characteristics of multimode semiconductor lasers,” J. Appl. Phys. 75, 737–747 (1994).
[Crossref]

L. W. Casperson, “Laser power calculations: sources of error,” Appl. Opt. 19, 422–434 (1980).
[Crossref]

L. W. Casperson, “Threshold characteristics of multimode laser oscillators,” J. Appl. Phys. 46, 5194–5201 (1975).
[Crossref]

Chen, Y.

W. W. Fang, C. Bethea, Y. Chen, S. L. Chuang, “Longitudinal spatial inhomogeneities in high-power semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 117–128 (1995).
[Crossref]

Chen, Z.

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Cho, A. Y.

C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. 64, 1533–1601 (2001).
[Crossref]

Chuang, S. L.

W. W. Fang, C. Bethea, Y. Chen, S. L. Chuang, “Longitudinal spatial inhomogeneities in high-power semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 117–128 (1995).
[Crossref]

S. L. Chuang, Physics of Photonic Devices, 2nd Ed. (Wiley, 2009).

Coldren, L. A.

L. A. Coldren, S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Corzine, S. W.

L. A. Coldren, S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

DeVito, M.

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Faist, J.

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

Fang, W. W.

W. W. Fang, C. Bethea, Y. Chen, S. L. Chuang, “Longitudinal spatial inhomogeneities in high-power semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 117–128 (1995).
[Crossref]

Ferguson, T. R.

Friedmann, P.

F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
[Crossref]

Gini, E.

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

Giovannini, M.

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

Gmachl, C.

C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. 64, 1533–1601 (2001).
[Crossref]

Gordon, E.

A. White, E. Gordon, J. Rigden, “Output power of the 6328-Å gas maser,” Appl. Phys. Lett. 2, 91–93 (1963).
[Crossref]

Gresch, T.

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

Grimshaw, M.

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Haden, J.

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Hasuo, S.

S. Hasuo, T. Ohmi, “Spatial distribution of the light intensity in the injection lasers,” Jpn. J. Appl. Phys. 13, 1429–1434 (1974).
[Crossref]

Hoyler, N.

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

Hvozdara, L.

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

Karlov, N.

N. Karlov, Lectures on Quantum Electronics, 2nd Ed. (CRC, 1993).

Khoshnevissan, M.

L. W. Casperson, M. Khoshnevissan, “Threshold characteristics of multimode semiconductor lasers,” J. Appl. Phys. 75, 737–747 (1994).
[Crossref]

Latham, W. P.

Leisher, P.

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Martinsen, R.

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Mikulla, M.

F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
[Crossref]

Ohmi, T.

S. Hasuo, T. Ohmi, “Spatial distribution of the light intensity in the injection lasers,” Jpn. J. Appl. Phys. 13, 1429–1434 (1974).
[Crossref]

Poprawe, R.

F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
[Crossref]

Rigden, J.

A. White, E. Gordon, J. Rigden, “Output power of the 6328-Å gas maser,” Appl. Phys. Lett. 2, 91–93 (1963).
[Crossref]

Rigrod, W.

W. Rigrod, “Homogeneously broadened CW lasers with uniform distributed loss,” IEEE J. Quantum Electron. 14, 377–381 (1978).
[Crossref]

W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

W. Rigrod, “Gain saturation and output power of optical masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[Crossref]

Rinner, F.

F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
[Crossref]

Rogg, J.

F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
[Crossref]

Ryvkin, B.

B. Ryvkin, E. Avrutin, “Spatial hole burning in high-power edge-emitting lasers: A simple analytical model and the effect on laser performance,” J. Appl. Phys. 109, 043101 (2011).
[Crossref]

Saleh, B. E.

B. E. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd Ed. (Wiley, 2007).

Schindler, G. M.

G. M. Schindler, “Optimum output efficiency of homogeneously broadened lasers with constant loss,” IEEE J. Quantum Electron. 16, 546–549 (1980).
[Crossref]

Schulz-DuBois, E.

E. Schulz-DuBois, “Pulse sharpening and gain saturation in traveling-wave masers,” Bell Syst. Tech. J. 43, 625–658 (1964).
[Crossref]

Siegman, A.

A. Siegman, Lasers (University Science Books, 1986).

Silfvast, W. T.

W. T. Silfvast, Laser Fundamentals, 2nd Ed. (Cambridge University, 2004).

Sivco, D. L.

C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. 64, 1533–1601 (2001).
[Crossref]

Svelto, O.

O. Svelto, Principles of Lasers, 5th Ed. (Springer, 2010).

Teich, M. C.

B. E. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd Ed. (Wiley, 2007).

Weimann, G.

F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
[Crossref]

White, A.

A. White, E. Gordon, J. Rigden, “Output power of the 6328-Å gas maser,” Appl. Phys. Lett. 2, 91–93 (1963).
[Crossref]

Wittmann, A.

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

Yariv, A.

A. Yariv, P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th Ed. (Oxford University, 2007).

Yeh, P.

A. Yariv, P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th Ed. (Oxford University, 2007).

Appl. Opt. (3)

Appl. Phys. Lett. (2)

F. Rinner, J. Rogg, P. Friedmann, M. Mikulla, G. Weimann, R. Poprawe, “Longitudinal carrier density measurement of high power broad area laser diodes,” Appl. Phys. Lett. 80, 19–21 (2002).
[Crossref]

A. White, E. Gordon, J. Rigden, “Output power of the 6328-Å gas maser,” Appl. Phys. Lett. 2, 91–93 (1963).
[Crossref]

Bell Syst. Tech. J. (1)

E. Schulz-DuBois, “Pulse sharpening and gain saturation in traveling-wave masers,” Bell Syst. Tech. J. 43, 625–658 (1964).
[Crossref]

IEEE J. Quantum Electron. (3)

W. Rigrod, “Homogeneously broadened CW lasers with uniform distributed loss,” IEEE J. Quantum Electron. 14, 377–381 (1978).
[Crossref]

G. M. Schindler, “Optimum output efficiency of homogeneously broadened lasers with constant loss,” IEEE J. Quantum Electron. 16, 546–549 (1980).
[Crossref]

A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, “High-performance bound-to-continuum quantum-cascade lasers for broad-gain applications,” IEEE J. Quantum Electron. 44, 36–40 (2008).
[Crossref]

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

W. W. Fang, C. Bethea, Y. Chen, S. L. Chuang, “Longitudinal spatial inhomogeneities in high-power semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 117–128 (1995).
[Crossref]

J. Appl. Phys. (5)

W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

B. Ryvkin, E. Avrutin, “Spatial hole burning in high-power edge-emitting lasers: A simple analytical model and the effect on laser performance,” J. Appl. Phys. 109, 043101 (2011).
[Crossref]

L. W. Casperson, “Threshold characteristics of multimode laser oscillators,” J. Appl. Phys. 46, 5194–5201 (1975).
[Crossref]

L. W. Casperson, M. Khoshnevissan, “Threshold characteristics of multimode semiconductor lasers,” J. Appl. Phys. 75, 737–747 (1994).
[Crossref]

W. Rigrod, “Gain saturation and output power of optical masers,” J. Appl. Phys. 34, 2602–2609 (1963).
[Crossref]

Jpn. J. Appl. Phys. (1)

S. Hasuo, T. Ohmi, “Spatial distribution of the light intensity in the injection lasers,” Jpn. J. Appl. Phys. 13, 1429–1434 (1974).
[Crossref]

Proc. SPIE (1)

Z. Chen, L. Bao, J. Bai, M. Grimshaw, R. Martinsen, M. DeVito, J. Haden, P. Leisher, “Performance limitation and mitigation of longitudinal spatial hole burning in high-power diode lasers,” Proc. SPIE 8277, 82771J (2012).
[Crossref]

Rep. Prog. Phys. (1)

C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, “Recent progress in quantum cascade lasers and applications,” Rep. Prog. Phys. 64, 1533–1601 (2001).
[Crossref]

Other (8)

N. Karlov, Lectures on Quantum Electronics, 2nd Ed. (CRC, 1993).

A. Siegman, Lasers (University Science Books, 1986).

O. Svelto, Principles of Lasers, 5th Ed. (Springer, 2010).

W. T. Silfvast, Laser Fundamentals, 2nd Ed. (Cambridge University, 2004).

A. Yariv, P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th Ed. (Oxford University, 2007).

B. E. Saleh, M. C. Teich, Fundamentals of Photonics, 2nd Ed. (Wiley, 2007).

S. L. Chuang, Physics of Photonic Devices, 2nd Ed. (Wiley, 2009).

L. A. Coldren, S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1. Laser cavity geometry and relevant parameters: (a) FP laser and (b) unidirectional traveling-wave ring laser.
Fig. 2.
Fig. 2. Sketch illustrates that if the slope of β DLA out (dashed) exceeds the slope of β ring out , max (solid), the DLA prediction exceeds the allowable maximum output of the ring laser above a certain gain. Therefore, we can expect the actual output β ring out to qualitatively follow the dotted curve. Inset, parameter ranges of α 0 L and R that result in such a violation, namely, those which satisfy Inequality (7), are shaded in light blue.
Fig. 3.
Fig. 3. Ring laser: ratio of the asymptotic to the threshold slope efficiency as a function of R for various values of α 0 L .
Fig. 4.
Fig. 4. (a)  β out versus g 0 for α 0 = 20 cm 1 , L = 5 mm , and R = 0.25 , shown for the DLA (black), symmetric FP with R 1 = R 2 = 0.25 (red), maximally asymmetric FP with R 1 = 1 and R 2 = 0.25 2 = 0.0625 (blue), and ring laser with R = 0.25 (green). The maximum output intensity is shown in correspondingly colored dashed lines. Inset, slope efficiencies of the curves (i.e., first derivatives), together with the corresponding slopes of β out , max for each laser; (b) intracavity intensity at the right facet, β 2 + + β 2 , is normalized to β max and plotted against g 0 over a larger range than in (a).
Fig. 5.
Fig. 5. For the same parameters as those used in Fig. 4 ( α 0 = 20 cm 1 , L = 5 mm , and R = 0.25 ), the intensity envelopes β + and β of the right- and left-propagating waves are plotted for 1.2 g 0 , th to 3 g 0 , th (in steps of 0.3 g 0 , th ); (a) for a symmetric FP with R 1 = R 2 = 0.25 ; (b) for a maximally asymmetric FP with R 1 = 1 and R 2 = 0.25 2 = 0.0625 ; (c) for a ring laser with R = 0.25 . Total intracavity intensity, β + + β , is normalized to β max and plotted in (d), (e), and (f) for the same three lasers, respectively. See Supplement 1 for a mathematical discussion of the intensity variation.
Fig. 6.
Fig. 6. Output power β aFP out of the maximally asymmetric FP laser with α 0 = 15 cm 1 , R 1 = 1 , and R 2 = 0.01 , for cavities of length 4 mm (red) and 5 mm (blue), as computed numerically accounting for nonuniform gain saturation (solid lines) and as predicted by the DLA (dashed lines). Maximum output power β aFP out , max is also plotted (black).

Equations (18)

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1 β + d β + d z = 1 β d β d z = g 0 1 + β + + β α 0 .
g 0 , th = α 0 + 1 L ln ( 1 R ) .
1 β + d β + d z = g 0 1 + β ( α 0 + α m ) = 0 ,
β DLA out = α m L α 0 + α m ( g 0 g 0 , th ) .
β max = 1 α 0 ( g 0 α 0 ) ,
β ring out , max = ( 1 R ) β max = 1 R α 0 ( g 0 α 0 ) .
α m L α 0 + α m > 1 R α 0 ,
α 0 L ln ( β 1 + / β 2 + ) = g 0 ln [ F ( β 1 + ) / F ( β 2 + ) ] ( g 0 α 0 ) 2 ( 2 α 0 β 0 ) 2 ,
F ( β i + ) ( g 0 α 0 ) 2 ( 2 α 0 β 0 ) 2 + ( g 0 α 0 2 α 0 β i + ) ( g 0 α 0 ) 2 ( 2 α 0 β 0 ) 2 ( g 0 α 0 2 α 0 β i + ) .
β 1 + = R 1 β 1 , β 2 = R 2 β 2 + ,
β 0 2 = β 1 + β 1 = β 2 + β 2 ,
β ring out = ( 1 R ) ( g 0 / α 0 1 ) × R exp [ ( 1 α 0 / g 0 ) ( α 0 + α m ) L ] 1 R exp [ ( 1 α 0 / g 0 ) ( α 0 + α m ) L ] R .
d β ring out d g 0 = α m L α 0 + α m α 0 L ( α 0 + α m ) 2 ( 1 + R 1 R ln ( 1 / R ) 2 ) ( g 0 g 0 , th ) .
β ring out | g g 0 , th = ( 1 R ) ( 1 e α 0 L ) α 0 ( 1 R e α 0 L ) ( g 0 g 0 , th eff ) ,
g 0 , th eff ( α 0 + α m ) ( α 0 L ) ( 1 1 e α 0 L 1 1 R e α 0 L ) + α 0 ,
β ring out | g g 0 , th & exp ( α 0 L ) 1 = 1 R α 0 ( g 0 α 0 ) ,
β sFP out , max = 2 ( 1 R 1 + R ) β max ,
β aFP out , max = ( 1 R 2 1 + R 2 ) β max .

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