Abstract

We derive and test a generalization of the steady-state ab initio laser theory (SALT) to treat complex gain media. The generalized theory (C-SALT) is able to treat atomic and molecular gain media with diffusion and multiple lasing transitions, and semiconductor gain media in the free carrier approximation including fully the effect of Pauli blocking. The key assumption of the theory is stationarity of the level populations, which leads to coupled self-consistent equations for the populations and the lasing modes that fully include the effects of openness and non-linear spatial hole-burning. These equations can be solved efficiently for the steady-state lasing properties by a similar iteration procedure as in SALT, where a static gain medium with a single transition is assumed. The theory is tested by comparison to much less efficient finite difference time domain (FDTD) methods and excellent agreement is found. Using C-SALT to analyze the effects of varying gain diffusion constant we demonstrate a cross-over between the regime of strong spatial hole burning with multimode lasing to a regime of negligible spatial hole burning, leading to gain-clamping, and single mode lasing. The effect of spatially inhomogeneous pumping combined with diffusion is also studied and a relevant length scale for spatial inhomogeneity to persist under these conditions is determined. For the semiconductor gain model, we demonstrate the frequency shift due to Pauli blocking as the pumping strength changes.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
  5. S. Chua, Y. D. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by fano resonances,” Opt. Express 19, 1539–1562 (2011).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  27. Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
    [Crossref] [PubMed]
  28. J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
    [Crossref]
  29. A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation
  30. L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16,895–16,902 (2008).
    [Crossref]
  31. A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  34. H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
    [Crossref] [PubMed]
  35. F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
    [Crossref]
  36. A. Cerjan and A. D. Stone, “Steady-state ab initio theory of lasers with injected signals,” Phys. Rev. A 90, 013840 (2014).
    [Crossref]
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    [Crossref]
  39. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
    [Crossref]
  40. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
    [Crossref] [PubMed]
  41. M. Lindberg and S. W. Koch, “Effective bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
    [Crossref]

2014 (3)

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

A. Cerjan and A. D. Stone, “Steady-state ab initio theory of lasers with injected signals,” Phys. Rev. A 90, 013840 (2014).
[Crossref]

2013 (2)

T. Hisch, M. Liertzer, D. Pogany, F. Mintert, and S. Rotter, “Pump-controlled directional light emission from random lasers,” Phys. Rev. Lett. 111, 023902 (2013).
[Crossref] [PubMed]

X. Huang, J. L. Zhang, V. Tokranov, S. Oktyabrsky, and C. F. Gmachl, “Same-wavelength cascaded-transition quantum cascade laser,” Appl. Phys. Lett. 103, 051113 (2013).
[Crossref]

2012 (4)

Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
[Crossref] [PubMed]

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[Crossref] [PubMed]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

2011 (2)

S. Chua, Y. D. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by fano resonances,” Opt. Express 19, 1539–1562 (2011).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tuereci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

2010 (2)

J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A 81, 043818 (2010).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

2009 (1)

2008 (5)

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full-time-domain approach to spatio-temporal dynamics of semiconductor lasers. i. theoretical formulation,” Prog. Quantum Electron. 32, 159–246 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[Crossref]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16,895–16,902 (2008).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

2007 (2)

W. H. P. Pernice, F. P. Payne, and D. F. G. Gallagher, “A finite-difference time-domain method for the simulation of gain materials with carrier diffusion in photonic crystals,” J. Lightw. Technol. 25, 2306–2314 (2007).
[Crossref]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

2006 (2)

2004 (1)

M. Jungo, D. Erni, and W. Baechthold, “Alternative formulation of carrier transport in spatially-dependent laser rate equations,” Opt. Quantum Electron. 36, 881–891 (2004).
[Crossref]

2003 (1)

B. Bidégaray, “Time discretizations for maxwell-bloch equations,” Numer. Meth. Partial Differential Equations 19, 284–300 (2003).
[Crossref]

1998 (2)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

C. Sirtori, A. Tredicucci, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Dual-wavelength emission from optically cascaded intersubband transitions,” Opt. Lett. 23, 463–465 (1998).
[Crossref]

1997 (1)

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]

1996 (1)

M. Homar, J. V. Moloney, and M. S. Miguel, “Traveling wave model of a multimode fabry-perot laser in free running and external cavity configurations,” IEEE J. Quantum Electron. 32, 553–566 (1996).
[Crossref]

1995 (1)

P. Khandokhin, I. Koryukin, Y. Khanin, and P. Mandel, “Influence of carrier diffusion on the dynamics of a 2-mode laser,” IEEE J. Quantum Electron. 31, 647–652 (1995).
[Crossref]

1994 (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[Crossref] [PubMed]

1993 (1)

P. Mandel, C. Etrich, and K. Otsuka, “Laser rate-equations with phase-sensitive interactions,” IEEE J. Quantum Electron. 29, 836–843 (1993).
[Crossref]

1991 (1)

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref] [PubMed]

1989 (1)

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[Crossref] [PubMed]

1988 (1)

M. Lindberg and S. W. Koch, “Effective bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
[Crossref]

1987 (1)

H. Fu and H. Haken, “Semiclassical dye-laser equations and the unidirectional single frequency operation,” Phys. Rev. A 36, 4802–4816 (1987).
[Crossref] [PubMed]

1984 (1)

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Andreasen, J.

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[Crossref] [PubMed]

J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A 81, 043818 (2010).
[Crossref]

J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. 34, 3586 (2009).
[Crossref] [PubMed]

Arecchi, F. T.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Bachelard, N.

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[Crossref] [PubMed]

Baechthold, W.

M. Jungo, D. Erni, and W. Baechthold, “Alternative formulation of carrier transport in spatially-dependent laser rate equations,” Opt. Quantum Electron. 36, 881–891 (2004).
[Crossref]

Belyanin, A.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Bidégaray, B.

B. Bidégaray, “Time discretizations for maxwell-bloch equations,” Numer. Meth. Partial Differential Equations 19, 284–300 (2003).
[Crossref]

Böhringer, K.

K. Böhringer and O. Hess, “A full-time-domain approach to spatio-temporal dynamics of semiconductor lasers. i. theoretical formulation,” Prog. Quantum Electron. 32, 159–246 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[Crossref]

Bour, D.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Bravo-Abad, J.

Cao, H.

J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A 81, 043818 (2010).
[Crossref]

J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. 34, 3586 (2009).
[Crossref] [PubMed]

Capasso, F.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

C. Sirtori, A. Tredicucci, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Dual-wavelength emission from optically cascaded intersubband transitions,” Opt. Lett. 23, 463–465 (1998).
[Crossref]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[Crossref] [PubMed]

Cerjan, A.

A. Cerjan and A. D. Stone, “Steady-state ab initio theory of lasers with injected signals,” Phys. Rev. A 90, 013840 (2014).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

C. Sirtori, A. Tredicucci, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Dual-wavelength emission from optically cascaded intersubband transitions,” Opt. Lett. 23, 463–465 (1998).
[Crossref]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[Crossref] [PubMed]

Chong, Y. D.

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tuereci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

S. Chua, Y. D. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by fano resonances,” Opt. Express 19, 1539–1562 (2011).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation

Chow, W. W.

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer, 1999).
[Crossref]

Chua, S.

Collier, B.

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

Corzine, S.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Diehl, L.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Erni, D.

M. Jungo, D. Erni, and W. Baechthold, “Alternative formulation of carrier transport in spatially-dependent laser rate equations,” Opt. Quantum Electron. 36, 881–891 (2004).
[Crossref]

Esterhazy, S.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Etrich, C.

P. Mandel, C. Etrich, and K. Otsuka, “Laser rate-equations with phase-sensitive interactions,” IEEE J. Quantum Electron. 29, 836–843 (1993).
[Crossref]

Faist, J.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

C. Sirtori, A. Tredicucci, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Dual-wavelength emission from optically cascaded intersubband transitions,” Opt. Lett. 23, 463–465 (1998).
[Crossref]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[Crossref] [PubMed]

Fu, H.

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref] [PubMed]

H. Fu and H. Haken, “Semiclassical dye-laser equations and the unidirectional single frequency operation,” Phys. Rev. A 36, 4802–4816 (1987).
[Crossref] [PubMed]

Gallagher, D. F. G.

W. H. P. Pernice, F. P. Payne, and D. F. G. Gallagher, “A finite-difference time-domain method for the simulation of gain materials with carrier diffusion in photonic crystals,” J. Lightw. Technol. 25, 2306–2314 (2007).
[Crossref]

Ge, L.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tuereci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A 81, 043818 (2010).
[Crossref]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16,895–16,902 (2008).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

L. Ge, “Steady-state ab initio laser theory and its applications in random and complex media,” Ph.D. thesis, Yale University (2010).

Gigan, S.

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[Crossref] [PubMed]

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

Gmachl, C. F.

X. Huang, J. L. Zhang, V. Tokranov, S. Oktyabrsky, and C. F. Gmachl, “Same-wavelength cascaded-transition quantum cascade laser,” Appl. Phys. Lett. 103, 051113 (2013).
[Crossref]

Gordon, A.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Haken, H.

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991).
[Crossref] [PubMed]

H. Fu and H. Haken, “Semiclassical dye-laser equations and the unidirectional single frequency operation,” Phys. Rev. A 36, 4802–4816 (1987).
[Crossref] [PubMed]

H. Haken, Light: Laser Dynamics (North-Holland Phys. Publishing, 1985, vol. II).

Haug, H.

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[Crossref] [PubMed]

Hess, O.

K. Böhringer and O. Hess, “A full-time-domain approach to spatio-temporal dynamics of semiconductor lasers. i. theoretical formulation,” Prog. Quantum Electron. 32, 159–246 (2008).
[Crossref]

K. Böhringer and O. Hess, “A full time-domain approach to spatio-temporal dynamics of semiconductor lasers. II. spatio-temporal dynamics,” Prog. Quantum Electron. 32, 247–307 (2008).
[Crossref]

Hisch, T.

T. Hisch, M. Liertzer, D. Pogany, F. Mintert, and S. Rotter, “Pump-controlled directional light emission from random lasers,” Phys. Rev. Lett. 111, 023902 (2013).
[Crossref] [PubMed]

Ho, S.

Höfler, G.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Homar, M.

M. Homar, J. V. Moloney, and M. S. Miguel, “Traveling wave model of a multimode fabry-perot laser in free running and external cavity configurations,” IEEE J. Quantum Electron. 32, 553–566 (1996).
[Crossref]

Huang, X.

X. Huang, J. L. Zhang, V. Tokranov, S. Oktyabrsky, and C. F. Gmachl, “Same-wavelength cascaded-transition quantum cascade laser,” Appl. Phys. Lett. 103, 051113 (2013).
[Crossref]

Huang, Y.

Hutchinson, A. L.

Johnson, S.

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation

Johnson, S. G.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Jungo, M.

M. Jungo, D. Erni, and W. Baechthold, “Alternative formulation of carrier transport in spatially-dependent laser rate equations,” Opt. Quantum Electron. 36, 881–891 (2004).
[Crossref]

Kärtner, F. X.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Khandokhin, P.

P. Khandokhin, I. Koryukin, Y. Khanin, and P. Mandel, “Influence of carrier diffusion on the dynamics of a 2-mode laser,” IEEE J. Quantum Electron. 31, 647–652 (1995).
[Crossref]

Khanin, Y.

P. Khandokhin, I. Koryukin, Y. Khanin, and P. Mandel, “Influence of carrier diffusion on the dynamics of a 2-mode laser,” IEEE J. Quantum Electron. 31, 647–652 (1995).
[Crossref]

Koch, S. W.

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[Crossref] [PubMed]

M. Lindberg and S. W. Koch, “Effective bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
[Crossref]

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer, 1999).
[Crossref]

Koryukin, I.

P. Khandokhin, I. Koryukin, Y. Khanin, and P. Mandel, “Influence of carrier diffusion on the dynamics of a 2-mode laser,” IEEE J. Quantum Electron. 31, 647–652 (1995).
[Crossref]

Liertzer, M.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

T. Hisch, M. Liertzer, D. Pogany, F. Mintert, and S. Rotter, “Pump-controlled directional light emission from random lasers,” Phys. Rev. Lett. 111, 023902 (2013).
[Crossref] [PubMed]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

Lindberg, M.

M. Lindberg and S. W. Koch, “Effective bloch equations for semiconductors,” Phys. Rev. B 38, 3342–3350 (1988).
[Crossref]

Lippi, G. L.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Liu, D.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation

Liu, H. C.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Maier, T.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Makris, K. G.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Mandel, P.

P. Khandokhin, I. Koryukin, Y. Khanin, and P. Mandel, “Influence of carrier diffusion on the dynamics of a 2-mode laser,” IEEE J. Quantum Electron. 31, 647–652 (1995).
[Crossref]

P. Mandel, C. Etrich, and K. Otsuka, “Laser rate-equations with phase-sensitive interactions,” IEEE J. Quantum Electron. 29, 836–843 (1993).
[Crossref]

Melenk, J. M.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

Miguel, M. S.

M. Homar, J. V. Moloney, and M. S. Miguel, “Traveling wave model of a multimode fabry-perot laser in free running and external cavity configurations,” IEEE J. Quantum Electron. 32, 553–566 (1996).
[Crossref]

Mintert, F.

T. Hisch, M. Liertzer, D. Pogany, F. Mintert, and S. Rotter, “Pump-controlled directional light emission from random lasers,” Phys. Rev. Lett. 111, 023902 (2013).
[Crossref] [PubMed]

Moloney, J. V.

M. Homar, J. V. Moloney, and M. S. Miguel, “Traveling wave model of a multimode fabry-perot laser in free running and external cavity configurations,” IEEE J. Quantum Electron. 32, 553–566 (1996).
[Crossref]

Narimanov, E. E.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

Natsume, Y.

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

Nöckel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]

Oktyabrsky, S.

X. Huang, J. L. Zhang, V. Tokranov, S. Oktyabrsky, and C. F. Gmachl, “Same-wavelength cascaded-transition quantum cascade laser,” Appl. Phys. Lett. 103, 051113 (2013).
[Crossref]

Otsuka, K.

P. Mandel, C. Etrich, and K. Otsuka, “Laser rate-equations with phase-sensitive interactions,” IEEE J. Quantum Electron. 29, 836–843 (1993).
[Crossref]

Payne, F. P.

W. H. P. Pernice, F. P. Payne, and D. F. G. Gallagher, “A finite-difference time-domain method for the simulation of gain materials with carrier diffusion in photonic crystals,” J. Lightw. Technol. 25, 2306–2314 (2007).
[Crossref]

Pernice, W. H. P.

W. H. P. Pernice, F. P. Payne, and D. F. G. Gallagher, “A finite-difference time-domain method for the simulation of gain materials with carrier diffusion in photonic crystals,” J. Lightw. Technol. 25, 2306–2314 (2007).
[Crossref]

Pick, A.

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation

Pillay, J. C.

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

Pogany, D.

T. Hisch, M. Liertzer, D. Pogany, F. Mintert, and S. Rotter, “Pump-controlled directional light emission from random lasers,” Phys. Rev. Lett. 111, 023902 (2013).
[Crossref] [PubMed]

Puccioni, G. P.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Rodriguez, A.

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation

Rotter, S.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

T. Hisch, M. Liertzer, D. Pogany, F. Mintert, and S. Rotter, “Pump-controlled directional light emission from random lasers,” Phys. Rev. Lett. 111, 023902 (2013).
[Crossref] [PubMed]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tuereci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

Schneider, H.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Sebbah, P.

N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Taming random lasers through active spatial control of the pump,” Phys. Rev. Lett. 109, 033903 (2012).
[Crossref] [PubMed]

Sirtori, C.

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

C. Sirtori, A. Tredicucci, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Dual-wavelength emission from optically cascaded intersubband transitions,” Opt. Lett. 23, 463–465 (1998).
[Crossref]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[Crossref] [PubMed]

Soljacic, M.

Stone, A. D.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
[Crossref]

J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, “Generalized subSchawlow-Townes laser linewidths via material dispersion,” Phys. Rev. A 89, 033840 (2014).
[Crossref]

A. Cerjan and A. D. Stone, “Steady-state ab initio theory of lasers with injected signals,” Phys. Rev. A 90, 013840 (2014).
[Crossref]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

A. Cerjan, Y. D. Chong, L. Ge, and A. D. Stone, “Steady-state ab initio laser theory for n-level lasers,” Opt. Express 20, 474–488 (2012).
[Crossref] [PubMed]

Y. D. Chong and A. D. Stone, “General linewidth formula for steady-state multimode lasing in arbitrary cavities,” Phys. Rev. Lett. 109, 063902 (2012).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tuereci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

S. Chua, Y. D. Chong, A. D. Stone, M. Soljačić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by fano resonances,” Opt. Express 19, 1539–1562 (2011).
[Crossref] [PubMed]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16,895–16,902 (2008).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]

A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation

Tandy, R. J.

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16,895–16,902 (2008).
[Crossref]

Tokranov, V.

X. Huang, J. L. Zhang, V. Tokranov, S. Oktyabrsky, and C. F. Gmachl, “Same-wavelength cascaded-transition quantum cascade laser,” Appl. Phys. Lett. 103, 051113 (2013).
[Crossref]

Tredicce, J. R.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, and J. R. Tredicce, “Deterministic chaos in laser with injected signal,” Opt. Commun. 51, 308–314 (1984).
[Crossref]

Tredicucci, A.

Troccoli, M.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Tuereci, H. E.

L. Ge, Y. D. Chong, S. Rotter, H. E. Tuereci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[Crossref]

Türeci, H. E.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref] [PubMed]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref] [PubMed]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16,895–16,902 (2008).
[Crossref]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
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H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
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A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascase lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
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H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[Crossref]

J. Andreasen, C. Vanneste, L. Ge, and H. Cao, “Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems,” Phys. Rev. A 81, 043818 (2010).
[Crossref]

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A. Pick, A. Cerjan, D. Liu, A. Rodriguez, A. D. Stone, Y. D. Chong, and S. Johnson, In preparation

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[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Plot of modal intensities as a function of pump strength for a cavity with n = 1.5 and a gain medium consisting of atoms with two different atomic transitions, ωa,1 = 40, γ⊥,1 = 4, θ1 = .1, ωa,2 = 38, γ⊥,2 = 3, θ2 = .1, and 6 atomic levels in total, with decay rates as indicated in the schematic. Results from C-SALT using the stationary population approximation are shown as straight lines, results from FDTD simulations are shown as triangles. The different colors indicate different lasing modes. Inset shows the modal frequencies and their intensities at P = .0035. All values are reported in units of c/L.

Fig. 2
Fig. 2

(Left panel) Plot of the modal intensities calculated using C-SALT as a function of pump strength for a dielectric slab cavity, n = 1.5, and a two level, single transition atomic gain medium, with ωa = 40 and γ = 4, values again in units of c/L. Simulations for three different diffusion strengths are shown in solid, dashed, and dot-dashed lines. Different colors correspond to different lasing modes within each simulation. (Right panel) Plot of the inversion in the cavity as a function of position in the cavity at a pump strength of d0 = 0.345. Darker colors indicate increasing values of the diffusion coefficient. Schematic shows a one-sided dielectric slab cavity containing a two level atomic medium subject to uniform pumping.

Fig. 3
Fig. 3

(Left panel) Plot of the modal intensities calculated using SALT as a function of pump strength for a partially pumped dielectric slab cavity, n = 1.5, containing a four level, single transition atomic gain medium. Simulations of four different values of diffusion are shown as solid, dotted, dot-dashed, and dashed lines. The first lasing mode to turn on in all of the simulations is shown in red, and the second lasing mode, which only turns on in two of the simulations, in blue. (Right panel) Plot of the inversion in the cavity as a function of position in the cavity at a pump strength of d0 = 0.37. Darker colors indicate increasing values of the diffusion coefficient. Schematic shows a partially pumped one-sided dielectric slab cavity containing a four level atomic medium with a single lasing transition.

Fig. 4
Fig. 4

Plot of the non-interacting modal thresholds as a function of the diffusion coefficient in a quadrupole cavity with ε = 0.16, r0 = 3.45μm, ka = 6.27μm−1, γ = .19μm−1, n = 3 + .004i. Only the three modes that become the threshold lasing mode for different values of the diffusion are plotted. Left schematic within the plot shows the boundary of the simulated region, black circle, boundary of the cavity, blue quadrupole, and applied pump profile, red circle. The right schematic within the plot depicts the four level gain medium that fills the entire quadrupole cavity. Markers in the plot correspond to the threshold lasing mode and corresponding inversion profile accounting for the effects of diffusion at their locations. The threshold lasing mode profile and inversion profile are shown as false color plots, with red corresponding to large values, and blue corresponding to small values, with the specific values corresponding to each color indicated next to each plot. The white boundary in the plots denotes the cavity boundary.

Fig. 5
Fig. 5

(Left panel) Plot of the lasing thresholds for each of the non-interacting modes in the cavity as a function of the applied electric potential, ϕ, for a single sided, slab semiconductor laser. The energy gap at q = 0 has been set to Eg = 40, and the chemical potential set at half that, μ = 20. The non-interacting thresholds of the two modes studied in the right panel are given the same colors as appear in the right panel. (Right panel) Plot of the frequencies of the two lasing modes as a function of the applied electric potential. The frequency for the second lasing mode (red) is only shown after it reaches threshold.

Equations (71)

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[ × × ε c c 2 ( x ) t 2 ] E ( x , t ) = 4 π c 2 t 2 P g ( x , t )
P g ( x , t ) = α δ ( x x ( α ) ) Tr [ ρ ^ ( α ) e x ^ α ] ,
θ n m ( α ) = e n | x ^ ( α ) | m .
t P g ( x , t ) = N ( x ) n M m M t ( ρ n m ) θ m n ,
t ρ n m = i h ¯ n | [ H 0 + H I , ρ ^ ] | m ,
t ρ n m = i ω n m ρ n m i h ¯ k M ( θ n k ρ k m ρ k m θ k m ) E ( x , t ) ,
t ρ n m = ( γ , n m + i ω n m ) ρ n m + i h ¯ ( ρ n m ρ m m ) θ n m E ( x , t ) , ( m n )
P g ( x , t ) = j N T p j ( x , t )
p j + ( x , t ) = N ( x ) ρ n m θ m n
t p j + ( x , t ) = ( γ , j + i ω a , j ) p j + i d j h ¯ ( θ j E ) θ j * ,
t ρ n m = i h ¯ k M ( θ n k ρ k n ρ n k θ k n ) E ( x , t ) .
t ρ n = m n M γ n m ρ m m n M γ m n ρ n 1 i h ¯ j N T ξ n , j ( p j p j + ) E .
t P g + ( x , t ) = ( γ + i ω a ) P g + i N ( x ) d h ¯ ( θ E ) θ *
t d ( x , t ) = γ ( d d 0 ( x ) ) 2 i h ¯ ( P g P g + ) E
d 0 ( x ) = ( γ 21 γ 12 γ 21 + γ 12 ) N ( x )
E + ( x , t ) = μ N L Ψ μ ( x ) e i ω μ t
P g + ( x , t ) = μ N L p μ ( x ) e i ω μ t
p μ = | θ | 2 h ¯ d ω μ ω a + i γ Ψ μ .
0 = [ 2 + ( ε c ( x ) + 4 π χ g ( x , ω μ ) ) k μ 2 ] Ψ μ ( x ) ,
χ g ( x , ω ) = | θ | 2 h ¯ d 0 ( x ) ω ω a + i γ ( 1 1 + 4 | θ | 2 h ¯ 2 γ γ v N L Γ v | Ψ v | 2 ) ,
Ψ μ ( x ) = n a n ( μ ) f n ( x ; ω μ )
p j + ( x , t ) = μ N L p j , μ ( x ) e i ω μ t
p j , μ = | θ j | 2 h ¯ d j ω μ ω a , j + i γ , j Ψ μ .
t ρ n = m n M γ n m ρ m m n M γ m n ρ n j N T 2 | θ j | 2 ξ n , j d j h ¯ 2 γ , j ( v , μ Γ v , j Ψ v * Ψ μ e i ( ω μ ω v ) t ) .
0 = R ρ + j N T 2 | θ j | 2 h ¯ 2 γ , j ( v Γ v , j | Ψ v | 2 ) Ξ j ρ ,
n M ρ n ( x ) = N ( x ) ,
B ρ = N ( x ) ,
ρ = [ R + B + j N T 2 | θ j | 2 h ¯ 2 γ , j ( v N L Γ v , j | Ψ v | 2 ) Ξ j ] 1 N ( x ) .
χ g ( x , ω ) = 1 h ¯ j N T | θ j | 2 d j ω ω a , j + i γ , j ,
t ρ n = m n M γ n m ρ m m n M γ m n ρ n + D n 2 ρ n j N T 2 | θ j | 2 ξ n , j d j h ¯ 2 γ , j ( v , μ Γ v , j Ψ v * Ψ μ e i ( ω μ ω v ) t ) ,
0 = ( R + D 2 ) ρ + j N T 2 | θ j | 2 h ¯ 2 γ , j ( v N L Γ v , j | Ψ v | 2 ) Ξ j ρ ,
0 = 2 ( n M D n ρ n ) .
x ρ n | x = 0 , L = 0.
n M ρ n ( x ) = N .
B ρ = N ,
ρ = [ R + D 2 + B + j N T 2 | θ j | 2 h ¯ 2 γ , j ( v N L Γ v , j | Ψ v | 2 ) Ξ j ] 1 N ( x ) .
0 = t d = γ ( d d 0 ( x ) ) + D 2 d 4 | θ | 2 h ¯ γ v N L Γ v | E v | 2 d ,
γ SE ( I ) = 4 | θ | 2 h ¯ γ v N L Γ v | E v | 2 ,
d ( x ) = [ 1 + γ SE ( I ) γ D γ 2 ] 1 d 0 ( x ) .
k D 2 D γ 1 + γ SE ( I ) γ ,
( h ¯ ω Δ ε q + i h ¯ γ q ) ρ c v , q ( r , ω ) = ( f c , q ( r ) f v , q ( r ) ) [ θ q E ( r , ω ) + 1 V q V s ( q q ) ρ c v , q ( r , ω ) ] ,
P ( r , ω ) = 1 V q θ q ρ c v , q ( r , ω ) .
χ ( r , ω ) = d 3 q 2 ( 2 π ) 3 | θ | q 2 f c , q ( r ; ϕ , | E | ) f v , q ( r ; ϕ , | E | ) h ¯ ω Δ ε q + i h ¯ γ q ,
t D q ( r ) = γ , q ( D q ( r ) D q ( 0 ) ) 2 i h ¯ ( ( E θ q * ρ c v , q ) * c . c . )
D q ( r ) = f c , q ( r ) f v , q ( r )
D q ( 0 ) = f c , q ( 0 ) f v , q ( 0 ) = 1 e β ( ε c , q μ e ϕ ) + 1 1 e β ( ε v , q μ e ϕ ) + 1 ,
[ 2 + ( ε c + 4 π χ g ( r , ω ) ) k μ 2 ] Ψ μ = 0
χ g ( r , ω ) = d 3 q 2 ( 2 π ) 3 | θ q | 2 f c , q ( 0 ) f v , q ( 0 ) h ¯ ω Δ ε q + i h ¯ γ q ( 1 1 + 4 | θ q | 2 γ h ¯ v Γ v , q | Ψ v ( r ) | 2 )
Γ v , q = h ¯ γ q ( h ¯ ω v Δ ε q ) 2 + h ¯ 2 γ q 2
Δ ε q = h ¯ 2 q 2 2 m r + E g
θ q = θ 0 1 + h ¯ 2 q 2 2 m r 1 E g
θ 0 = λ | p ^ | λ
R n n ( x ) = m γ m n ( x )
R n m ( x ) = γ n m ( x ) , ( n m )
Ξ u u ( j ) = Ξ l l ( j ) = 1
Ξ u l ( j ) = Ξ l u ( j ) = 1
ρ ( x , t ) = ρ s ( x ) + ( ρ b ( x , t ) + c . c . ) ,
0 = D 2 ρ s ( x ) + R ρ s ( x ) + j = 1 1 i h ¯ ( Ψ 1 p 1 , j * + Ψ 2 p 2 , j * c . c . ) ξ j ,
t ρ b ( x , t ) = D 2 ρ b ( x , t ) + R ρ b ( x , t ) + j = 1 1 i h ¯ ( Ψ 1 p 2 , j * Ψ 2 * p 1 , j ) e i ( ω 1 ω 2 ) t ξ j ,
p 1 , j = | θ j | 2 γ 1 , j h ¯ ( Ψ 1 d j , s + Ψ 2 d j , Δ ) ,
γ μ , j = 1 ω μ ω a , j + i γ , j ,
p μ , j ( 0 ) = | θ j | 2 γ μ , j h ¯ Ψ μ d j , s ,
ρ Δ = 1 i ( ω 1 ω 2 ) ( D 2 + R ) ρ Δ + j | θ j | 2 h ¯ 2 ( γ 2 , j * γ 1 , j ω 1 ω 2 ) Ψ 1 Ψ 2 * Ξ j ρ s ,
ρ Δ = M ( ω 1 , ω 2 ) Ψ 1 Ψ 2 * ρ s ,
p 1 , j ( 1 ) = | θ j | 2 γ μ , j h ¯ Ψ 1 ( 1 + M ( ω 1 , ω 2 ) | Ψ 2 | 2 ) d j , s .
η n ( ω μ ) = 4 π 2 ( 2 π ) 3 d 3 q | θ q | 2 ( D q ( 0 ) h ¯ ω μ Δ ε q + i h ¯ γ q ) ,
0 = [ 2 + ( ε c ( x ) + η n F ( x ) ) ω 2 ] u n ( x )
x u n ( L ) = i k u n ( L ) ,
δ = Im [ η ( ω ) ] 4 π Im [ χ ( ω , ϕ t h r ) ] .
χ ( ω N + 1 , a N + 1 ) = χ ( ω N , a N ) + χ a | N ( a N + 1 a N ) + v χ ω v | N ( ω v , N + 1 ω v , N )
Ψ μ ( x ) = a ( μ ) u ( x ; ω μ ) ,

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