Abstract

Recent numerical and theoretical studies have demonstrated that the modes at threshold of a random laser are in direct correspondence with the resonances of the same system without gain, a feature which is well known in conventional lasers but not known until recently for random lasers. This paper presents numerical results of the multimode regime that takes place when the pumping rate is progressively increased above threshold. Behavior that is already known in standard lasers, such as mode competition and nonlinear wave mixing, are shown to also take place in random lasers thus reinforcing their recent modal description. However, due to the complexity of the laser modes and to the openness of such lasers, which require large external pumping to compensate for strong loss, one observes that these effects are systematic and can be more pronounced than in a conventional laser.

© 2011 Optical Society of America

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2011 (5)

2010 (5)

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, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A 82, 063835 (2010).
[CrossRef]

A. Tulek, R. C. Polson, and Z. V. Vardeny, “Naturally occurring resonators in random lasing of π-conjugated polymer films,” Nat. Phys. 6, 303–310 (2010).
[CrossRef]

O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. 12, 024001 (2010).
[CrossRef]

O. Zaitsev and L. Deych, “Diagrammatic semiclassical laser theory,” Phys. Rev. A 81, 023822 (2010).
[CrossRef]

2009 (5)

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

O. Zaitsev, L. Deych, and V. Shuvayev, “Statistical properties of one-dimensional random lasers,” Phys. Rev. Lett. 102, 043906(2009).
[CrossRef] [PubMed]

J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. 34, 3586–3588 (2009).
[CrossRef] [PubMed]

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009).
[CrossRef]

K. L. van der Molen, A. P. Mosk, and A. Lagendijk, “Relaxation oscillations in long-pulsed random lasers,” Phys. Rev. A 80, 055803 (2009).
[CrossRef]

2008 (3)

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

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: Theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (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, 16895–16902 (2008).
[CrossRef] [PubMed]

2007 (1)

C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef] [PubMed]

2006 (2)

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]

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
[CrossRef]

2004 (7)

E. Y. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Quantum Electron. 34, 227–232 (2004).
[CrossRef]

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
[CrossRef]

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
[CrossRef] [PubMed]

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[CrossRef]

M. A. Noginov, G. Zhu, A. A. Frantz, J. Novak, S. N. Williams, and I. Fowlkes, “Dependence of NdSc3(BO3)4 random laser parameters on particle size,” J. Opt. Soc. Am. B 21, 191–200 (2004) and references cited therein.
[CrossRef]

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903(2004).
[CrossRef] [PubMed]

2003 (3)

H. Cao, “Lasing in random media,” Waves Random Media 13, R1–R39 (2003) and references therein.
[CrossRef]

H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101(R) (2003).
[CrossRef]

B. Liu, A. Yamilov, Y. Ling, J. Y. Xu, and H. Cao, “Dynamic nonlinear effect on lasing in a random medium,” Phys. Rev. Lett. 91, 063903 (2003). The surprising drift of mode 1 across the maximum of the gain curve, toward mode 2 [Fig. ], is attributed to the nonlinear Kerr effect.
[CrossRef] [PubMed]

2002 (4)

C. M. Soukoulis, X. Jiang, J. Y. Xu, and H. Cao, “Dynamic response and relaxation oscillations in random lasers,” Phys. Rev. B 65, 041103(R) (2002).
[CrossRef]

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

R. C. Polson, M. E. Raikh, and Z. V. Vardeny, “Universality in unintentional laser resonators in π-conjugated polymer films,” C. R. Acad. Sci. Ser. IV A, 509–521 (2002).

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
[CrossRef] [PubMed]

2001 (1)

C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903(2001).
[CrossRef]

2000 (3)

H. Cao, J. Y. Xu, S.-H. Chang, and S. T. Ho, “Transition from amplified spontaneous emission to laser action in strongly scattering media,” Phys. Rev. E 61, 1985–1989 (2000).
[CrossRef]

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

S. M. Dutra and G. Nienhuis, “Quantized modes of a leaky cavity,” Phys. Rev. B 62, 063805 (2000).

1998 (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[CrossRef]

1996 (2)

S. John and G. Pang, “Theory of lasing in a multiple-scattering medium,” Phys. Rev. A 54, 3642–3652 (1996).
[CrossRef] [PubMed]

D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[CrossRef]

1975 (1)

C. F. Dewey Jr. and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[CrossRef]

1968 (1)

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP 26, 835–840(1968).

1964 (1)

R. C. Miller, “Optical harmonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964).
[CrossRef]

Andreasen, J.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

J. Andreasen, P. Sebbah, and C. Vanneste, “Coherent instabilities in random lasers,” Phys. Rev. A 84, 023826 (2011).
[CrossRef]

J. Andreasen and H. Cao, “Spectral behavior of partially pumped weakly scattering random lasers,” Opt. Express 19, 3418–3433(2011).
[CrossRef] [PubMed]

J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A 82, 063835 (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]

J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. 34, 3586–3588 (2009).
[CrossRef] [PubMed]

Angelani, L.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: Theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef] [PubMed]

Apalkov, V. M.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
[CrossRef] [PubMed]

Asatryan, A.

Asatryan, A. A.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Bardoux, R.

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
[CrossRef] [PubMed]

Botten, L.

Botten, L. C.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Byrne, M.

Cao, H.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

J. Andreasen and H. Cao, “Spectral behavior of partially pumped weakly scattering random lasers,” Opt. Express 19, 3418–3433(2011).
[CrossRef] [PubMed]

J. Andreasen and H. Cao, “Numerical study of amplified spontaneous emission and lasing in random media,” Phys. Rev. A 82, 063835 (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]

J. Andreasen and H. Cao, “Creation of new lasing modes with spatially nonuniform gain,” Opt. Lett. 34, 3586–3588 (2009).
[CrossRef] [PubMed]

C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef] [PubMed]

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
[CrossRef]

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[CrossRef]

B. Liu, A. Yamilov, Y. Ling, J. Y. Xu, and H. Cao, “Dynamic nonlinear effect on lasing in a random medium,” Phys. Rev. Lett. 91, 063903 (2003). The surprising drift of mode 1 across the maximum of the gain curve, toward mode 2 [Fig. ], is attributed to the nonlinear Kerr effect.
[CrossRef] [PubMed]

H. Cao, “Lasing in random media,” Waves Random Media 13, R1–R39 (2003) and references therein.
[CrossRef]

H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101(R) (2003).
[CrossRef]

C. M. Soukoulis, X. Jiang, J. Y. Xu, and H. Cao, “Dynamic response and relaxation oscillations in random lasers,” Phys. Rev. B 65, 041103(R) (2002).
[CrossRef]

H. Cao, J. Y. Xu, S.-H. Chang, and S. T. Ho, “Transition from amplified spontaneous emission to laser action in strongly scattering media,” Phys. Rev. E 61, 1985–1989 (2000).
[CrossRef]

Chabanov, A. A.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Chang, S.-H.

H. Cao, J. Y. Xu, S.-H. Chang, and S. T. Ho, “Transition from amplified spontaneous emission to laser action in strongly scattering media,” Phys. Rev. E 61, 1985–1989 (2000).
[CrossRef]

Chirkin, A. S.

E. Y. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Quantum Electron. 34, 227–232 (2004).
[CrossRef]

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]

Conti, C.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: Theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef] [PubMed]

de Valcárcel, G. J.

E. Roldán, G. J. de Valcárcel, F. Prati, F. Mitschke, and T. Voigt, “Multilongitudinal mode emission in ring cavity class B lasers,” in “Trends in Spatiotemporal Dynamics in Lasers. Instabilities, Polarization Dynamics, and Spatial Structures,” O.Gomez-Calderon and J.M.Guerra, eds. (Research Signpost, 2005), pp. 1–80.

Dewey, C. F.

C. F. Dewey Jr. and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[CrossRef]

Deych, L.

O. Zaitsev and L. Deych, “Diagrammatic semiclassical laser theory,” Phys. Rev. A 81, 023822 (2010).
[CrossRef]

O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. 12, 024001 (2010).
[CrossRef]

O. Zaitsev, L. Deych, and V. Shuvayev, “Statistical properties of one-dimensional random lasers,” Phys. Rev. Lett. 102, 043906(2009).
[CrossRef] [PubMed]

Dutra, S. M.

S. M. Dutra and G. Nienhuis, “Quantized modes of a leaky cavity,” Phys. Rev. B 62, 063805 (2000).

Fang, W.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Fedotov, A. B.

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
[CrossRef]

Feng, S.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[CrossRef]

Fowlkes, I.

Frantz, A. A.

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C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: Theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
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Ge, L.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[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]

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[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]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[CrossRef] [PubMed]

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V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
[CrossRef]

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A. Taflove and S. Hagness, Computational Electrodynamics (Artech House, 2005), 3rd ed.

Haïdar, R.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
[CrossRef] [PubMed]

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H. Cao, J. Y. Xu, S.-H. Chang, and S. T. Ho, “Transition from amplified spontaneous emission to laser action in strongly scattering media,” Phys. Rev. E 61, 1985–1989 (2000).
[CrossRef]

Hocker, L. O.

C. F. Dewey Jr. and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[CrossRef]

Jiang, X.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[CrossRef]

H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101(R) (2003).
[CrossRef]

C. M. Soukoulis, X. Jiang, J. Y. Xu, and H. Cao, “Dynamic response and relaxation oscillations in random lasers,” Phys. Rev. B 65, 041103(R) (2002).
[CrossRef]

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

Joannopoulos, J. D.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
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S. John and G. Pang, “Theory of lasing in a multiple-scattering medium,” Phys. Rev. A 54, 3642–3652 (1996).
[CrossRef] [PubMed]

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Kashkarov, P. K.

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
[CrossRef]

Kawakami, Y.

Kikuchi, A.

Kishino, K.

Konorov, S. O.

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
[CrossRef]

Kupecek, P.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
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Lagendijk, A.

K. L. van der Molen, A. P. Mosk, and A. Lagendijk, “Relaxation oscillations in long-pulsed random lasers,” Phys. Rev. A 80, 055803 (2009).
[CrossRef]

D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[CrossRef]

Lemasson, P.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
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C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: Theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
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Ling, Y.

H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101(R) (2003).
[CrossRef]

B. Liu, A. Yamilov, Y. Ling, J. Y. Xu, and H. Cao, “Dynamic nonlinear effect on lasing in a random medium,” Phys. Rev. Lett. 91, 063903 (2003). The surprising drift of mode 1 across the maximum of the gain curve, toward mode 2 [Fig. ], is attributed to the nonlinear Kerr effect.
[CrossRef] [PubMed]

Liu, B.

B. Liu, A. Yamilov, Y. Ling, J. Y. Xu, and H. Cao, “Dynamic nonlinear effect on lasing in a random medium,” Phys. Rev. Lett. 91, 063903 (2003). The surprising drift of mode 1 across the maximum of the gain curve, toward mode 2 [Fig. ], is attributed to the nonlinear Kerr effect.
[CrossRef] [PubMed]

Ma, X.

Mel’nikov, V. A.

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
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E. Y. Morozov and A. S. Chirkin, “Stochastic quasi-phase matching in nonlinear-optical crystals with an irregular domain structure,” Quantum Electron. 34, 227–232 (2004).
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K. L. van der Molen, A. P. Mosk, and A. Lagendijk, “Relaxation oscillations in long-pulsed random lasers,” Phys. Rev. A 80, 055803 (2009).
[CrossRef]

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S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903(2004).
[CrossRef] [PubMed]

Muzychenko, D. A.

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
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A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
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S. M. Dutra and G. Nienhuis, “Quantized modes of a leaky cavity,” Phys. Rev. B 62, 063805 (2000).

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Novak, J.

Okamoto, K.

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S. John and G. Pang, “Theory of lasing in a multiple-scattering medium,” Phys. Rev. A 54, 3642–3652 (1996).
[CrossRef] [PubMed]

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A. Tulek, R. C. Polson, and Z. V. Vardeny, “Naturally occurring resonators in random lasing of π-conjugated polymer films,” Nat. Phys. 6, 303–310 (2010).
[CrossRef]

R. C. Polson, M. E. Raikh, and Z. V. Vardeny, “Universality in unintentional laser resonators in π-conjugated polymer films,” C. R. Acad. Sci. Ser. IV A, 509–521 (2002).

Prati, F.

E. Roldán, G. J. de Valcárcel, F. Prati, F. Mitschke, and T. Voigt, “Multilongitudinal mode emission in ring cavity class B lasers,” in “Trends in Spatiotemporal Dynamics in Lasers. Instabilities, Polarization Dynamics, and Spatial Structures,” O.Gomez-Calderon and J.M.Guerra, eds. (Research Signpost, 2005), pp. 1–80.

Raikh, M. E.

R. C. Polson, M. E. Raikh, and Z. V. Vardeny, “Universality in unintentional laser resonators in π-conjugated polymer films,” C. R. Acad. Sci. Ser. IV A, 509–521 (2002).

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
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Ricci, M.

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903(2004).
[CrossRef] [PubMed]

Roldán, E.

E. Roldán, G. J. de Valcárcel, F. Prati, F. Mitschke, and T. Voigt, “Multilongitudinal mode emission in ring cavity class B lasers,” in “Trends in Spatiotemporal Dynamics in Lasers. Instabilities, Polarization Dynamics, and Spatial Structures,” O.Gomez-Calderon and J.M.Guerra, eds. (Research Signpost, 2005), pp. 1–80.

Rosencher, E.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
[CrossRef] [PubMed]

Rotter, S.

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
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H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef] [PubMed]

Ruocco, G.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, “Condensation in disordered lasers: Theory, 3d+1 simulations, and experiments,” Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef] [PubMed]

Sebbah, P.

J. Andreasen, P. Sebbah, and C. Vanneste, “Coherent instabilities in random lasers,” Phys. Rev. A 84, 023826 (2011).
[CrossRef]

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009).
[CrossRef]

C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef] [PubMed]

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903(2001).
[CrossRef]

Shapiro, B.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
[CrossRef] [PubMed]

Shuvayev, V.

O. Zaitsev, L. Deych, and V. Shuvayev, “Statistical properties of one-dimensional random lasers,” Phys. Rev. Lett. 102, 043906(2009).
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Siegman, A. E.

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

Skipetrov, S. E.

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[CrossRef]

H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101(R) (2003).
[CrossRef]

C. M. Soukoulis, X. Jiang, J. Y. Xu, and H. Cao, “Dynamic response and relaxation oscillations in random lasers,” Phys. Rev. B 65, 041103(R) (2002).
[CrossRef]

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

Stone, A. D.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[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]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[CrossRef] [PubMed]

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]

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics (Artech House, 2005), 3rd ed.

Tandy, R. J.

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[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, 16895–16902 (2008).
[CrossRef] [PubMed]

Timoshenko, V. Y.

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
[CrossRef]

Torre, R.

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903(2004).
[CrossRef] [PubMed]

Tulek, A.

A. Tulek, R. C. Polson, and Z. V. Vardeny, “Naturally occurring resonators in random lasing of π-conjugated polymer films,” Nat. Phys. 6, 303–310 (2010).
[CrossRef]

Türeci, H. E.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[CrossRef]

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[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]

L. Ge, R. J. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895–16902 (2008).
[CrossRef] [PubMed]

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]

van der Molen, K. L.

K. L. van der Molen, A. P. Mosk, and A. Lagendijk, “Relaxation oscillations in long-pulsed random lasers,” Phys. Rev. A 80, 055803 (2009).
[CrossRef]

Vanneste, C.

J. Andreasen, P. Sebbah, and C. Vanneste, “Coherent instabilities in random lasers,” Phys. Rev. A 84, 023826 (2011).
[CrossRef]

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. 3, 88–127 (2011).
[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]

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009).
[CrossRef]

C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007).
[CrossRef] [PubMed]

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903(2001).
[CrossRef]

Vardeny, Z. V.

A. Tulek, R. C. Polson, and Z. V. Vardeny, “Naturally occurring resonators in random lasing of π-conjugated polymer films,” Nat. Phys. 6, 303–310 (2010).
[CrossRef]

R. C. Polson, M. E. Raikh, and Z. V. Vardeny, “Universality in unintentional laser resonators in π-conjugated polymer films,” C. R. Acad. Sci. Ser. IV A, 509–521 (2002).

Voigt, T.

E. Roldán, G. J. de Valcárcel, F. Prati, F. Mitschke, and T. Voigt, “Multilongitudinal mode emission in ring cavity class B lasers,” in “Trends in Spatiotemporal Dynamics in Lasers. Instabilities, Polarization Dynamics, and Spatial Structures,” O.Gomez-Calderon and J.M.Guerra, eds. (Research Signpost, 2005), pp. 1–80.

Wiersma, D. S.

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903(2004).
[CrossRef] [PubMed]

D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[CrossRef]

Williams, S. N.

Wu, X.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Xiang, L.

Xu, J. Y.

H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101(R) (2003).
[CrossRef]

B. Liu, A. Yamilov, Y. Ling, J. Y. Xu, and H. Cao, “Dynamic nonlinear effect on lasing in a random medium,” Phys. Rev. Lett. 91, 063903 (2003). The surprising drift of mode 1 across the maximum of the gain curve, toward mode 2 [Fig. ], is attributed to the nonlinear Kerr effect.
[CrossRef] [PubMed]

C. M. Soukoulis, X. Jiang, J. Y. Xu, and H. Cao, “Dynamic response and relaxation oscillations in random lasers,” Phys. Rev. B 65, 041103(R) (2002).
[CrossRef]

H. Cao, J. Y. Xu, S.-H. Chang, and S. T. Ho, “Transition from amplified spontaneous emission to laser action in strongly scattering media,” Phys. Rev. E 61, 1985–1989 (2000).
[CrossRef]

Xu, M.

Yamilov, A.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
[CrossRef]

B. Liu, A. Yamilov, Y. Ling, J. Y. Xu, and H. Cao, “Dynamic nonlinear effect on lasing in a random medium,” Phys. Rev. Lett. 91, 063903 (2003). The surprising drift of mode 1 across the maximum of the gain curve, toward mode 2 [Fig. ], is attributed to the nonlinear Kerr effect.
[CrossRef] [PubMed]

Yang, D.

York, R. A.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[CrossRef]

Zaitsev, O.

O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. 12, 024001 (2010).
[CrossRef]

O. Zaitsev and L. Deych, “Diagrammatic semiclassical laser theory,” Phys. Rev. A 81, 023822 (2010).
[CrossRef]

O. Zaitsev, L. Deych, and V. Shuvayev, “Statistical properties of one-dimensional random lasers,” Phys. Rev. Lett. 102, 043906(2009).
[CrossRef] [PubMed]

Zheltikov, A. M.

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
[CrossRef]

Zhu, G.

Zi, J.

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Phys. B (1)

V. A. Mel’nikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Y. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225–228 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

C. F. Dewey Jr. and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[CrossRef]

C. R. Acad. Sci. Ser. IV (1)

R. C. Polson, M. E. Raikh, and Z. V. Vardeny, “Universality in unintentional laser resonators in π-conjugated polymer films,” C. R. Acad. Sci. Ser. IV A, 509–521 (2002).

IEEE Trans. Antennas Propag. (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[CrossRef]

J. Opt. (1)

O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. 12, 024001 (2010).
[CrossRef]

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

Nat. Phys. (1)

A. Tulek, R. C. Polson, and Z. V. Vardeny, “Naturally occurring resonators in random lasing of π-conjugated polymer films,” Nat. Phys. 6, 303–310 (2010).
[CrossRef]

Nature (2)

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
[CrossRef] [PubMed]

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

Nonlinearity (1)

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. (1)

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Phys. Rev. A (9)

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[CrossRef]

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[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]

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
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C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009).
[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]

J. Andreasen, P. Sebbah, and C. Vanneste, “Coherent instabilities in random lasers,” Phys. Rev. A 84, 023826 (2011).
[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

Phys. Rev. B (5)

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H. Cao, X. Jiang, Y. Ling, J. Y. Xu, and C. M. Soukoulis, “Mode repulsion and mode coupling in random lasers,” Phys. Rev. B 67, 161101(R) (2003).
[CrossRef]

X. Jiang, S. Feng, C. M. Soukoulis, J. Zi, J. D. Joannopoulos, and H. Cao, “Coupling, competition, and stability of modes in random lasers,” Phys. Rev. B 69, 104202 (2004).
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P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

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[CrossRef]

Phys. Rev. E (2)

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[CrossRef]

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Phys. Rev. Lett. (8)

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C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903(2001).
[CrossRef]

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C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007).
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B. Liu, A. Yamilov, Y. Ling, J. Y. Xu, and H. Cao, “Dynamic nonlinear effect on lasing in a random medium,” Phys. Rev. Lett. 91, 063903 (2003). The surprising drift of mode 1 across the maximum of the gain curve, toward mode 2 [Fig. ], is attributed to the nonlinear Kerr effect.
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Quantum Electron. (1)

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Science (1)

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Sov. Phys. JETP (1)

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Other (3)

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Figures (11)

Fig. 1
Fig. 1

Spatially dependent index of refraction. (a)  n ( x ) of a 1D random structure. (b)  n ( x , y ) of a 2D random structure, 5 × 5 μm 2 . The outer black border represents the absorbing boundary.

Fig. 2
Fig. 2

Relaxation oscillations in (a) population inversion and (b) intensity for P r = 0.25 ns 1 (solid black line), P r = 0.26 ns 1 (solid gray line), P r = 0.27 ns 1 (dashed dark red line), and P r = 0.28 ns 1 (dotted red line) where single-mode lasing occurs. The intensity and oscillation frequency increases with P r .

Fig. 3
Fig. 3

(a) Differences between the intensity distributions of the threshold lasing mode with uniform and flat gain and that with successively including (PP) partial gain, (FD) frequency- dependent gain, and (GS) gain saturation. (Triangles) first lasing mode, (squares) second lasing mode. (Open and closed squares) two different realizations for second lasing modes to verify the effect of gain saturation. (b) Intensity distributions of second threshold lasing mode with (solid red line) uniform gain and (dashed black line) gain saturation with gain only in the air gaps.

Fig. 4
Fig. 4

Lasing thresholds for 10 realizations of 1D random lasers. (a) First (filled symbols) and second (open symbols) lasing thresholds without mode interaction (circles) and with mode interaction (triangles) included. (b) Ratio of the second lasing threshold over the first lasing threshold without and with mode interaction included. The single-mode regime persisted for realizations 7 and 10, even for the largest ratio checked (9000).

Fig. 5
Fig. 5

(a) Intensity and (b) wavelength versus pumping rate P r of the (solid black line) first and (dotted gray line) second lasing mode. Their respective thresholds are at P r = 0.24 ns 1 and P r = 0.30 ns 1 . Mode 1 is suppressed for P r 0.44 ns 1 .

Fig. 6
Fig. 6

Spectrograms of output intensity (color on a log scale ranging from 10 10 8 ) of a 1D random laser. (a)  P r = 0.24 ns 1 , single-mode lasing. (b)  P r = 0.29 ns 1 , just below the lasing threshold of the second mode. The second mode appears in the transient regime. (c–e)  P r = 0.30 , 0.31, 0.43 ns 1 , multimode lasing. (f)  P r = 0.47 ns 1 , the first lasing mode is suppressed, though it appears in the transient regime.

Fig. 7
Fig. 7

Differences D between (circles) lasing mode 1 and lasing mode 1 at threshold, (triangles) lasing mode 2 and lasing mode 2 at threshold, and (squares) lasing mode 1 and 2 at the same pumping rate P r .

Fig. 8
Fig. 8

Intensity versus pumping rate P r of the (solid black line) first, (dotted gray line) second, and (dashed dark-gray line) third lasing mode of a 2D random laser, size 1 × 1 μm 2 .

Fig. 9
Fig. 9

Emission spectrum at the threshold of the first lasing mode and (inset) peak from third-harmonic generation for a (a) 1D random laser at P r = 0.24 ns 1 and (b) 2D random laser at P r = 2.50 ns 1 .

Fig. 10
Fig. 10

Emission spectrum for a (a–b) 1D random laser at P r = 0.30 ns 1 and (c–d) 2D random laser at P r = 3.00 ns 1 . (a,c) Two lasing modes (labeled 1 and 2) and the peaks resulting from four-wave mixing (labeled 3 and 4). (b,d) Peaks resulting from third-harmonic generation (labeled 1t, 2t) and sum-frequency generation (labeled 3t, 4t).

Fig. 11
Fig. 11

Quasimode Q-factors and spectral distance from the gain center wavelength λ a for 10 realizations of passive random structures in 1D. Quasimodes correspond to the first lasing mode (red circles) and second lasing mode (blue triangles) found without mode interaction (TM); the first and second modes for the same realization of disorder are connected with a line. With mode interaction (FDTD), the two cases in which the first lasing mode strongly suppresses the second lasing mode are marked as 7 and 10.

Equations (10)

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D i j = 0 L E i 2 ( x ) E j 2 ( x ) d x ,
1 / λ 3 1 / λ 2 = 1 / λ 2 1 / λ 1 ,
1 / λ 4 1 / λ 1 = 1 / λ 1 1 / λ 2 .
k 1 + k 2 k 3 + k 4 .
1 / λ 3 t = 2 / λ 2 + 1 / λ 1 ,
1 / λ 4 t = 2 / λ 1 + 1 / λ 2 .
μ 0 H x / t = E z / y μ 0 H y / t = E z / x ϵ i ϵ 0 E z / t + P / t = H y / x H x / y ,
d N 1 / d t = N 2 / τ 21 P r N 1 d N 2 / d t = N 3 / τ 32 N 2 / τ 21 ( E z / ω a ) d P / d t d N 3 / d t = N 4 / τ 43 N 3 / τ 32 + ( E z / ω a ) d P / d t d N 4 / d t = N 4 / τ 43 + P r N 1 ,
d 2 P / d t 2 + Δ ω a d P / d t + ω a 2 P = κ Δ N E z ,
κ = 6 π ϵ 0 c 3 / ω a 2 T 1 .

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