Abstract

In conventional lasers, the optical cavity that confines the photons also determines essential characteristics of the lasing modes such as wavelength, emission pattern, directivity, and polarization. In random lasers, which do not have mirrors or a well-defined cavity, light is confined within the gain medium by means of multiple scattering. The sharp peaks in the emission spectra of semiconductor powders, first observed in 1999, has therefore lead to an intense debate about the nature of the lasing modes in these so-called lasers with resonant feedback. We review numerical and theoretical studies aimed at clarifying the nature of the lasing modes in disordered scattering systems with gain. The past decade has witnessed the emergence of the idea that even the low-Q resonances of such open systems could play a role similar to the cavity modes of a conventional laser and produce sharp lasing peaks. We focus here on the near-threshold single-mode lasing regime where nonlinear effects associated with gain saturation and mode competition can be neglected. We discuss in particular the link between random laser modes near threshold and the resonances or quasi-bound (QB) states of the passive system without gain. For random lasers in the localized (strong scattering) regime, QB states and threshold lasing modes were found to be nearly identical within the scattering medium. These studies were later extended to the case of more lossy systems such as random systems in the diffusive regime, where it was observed that increasing the openness of such systems eventually resulted in measurable and increasing differences between quasi-bound states and lasing modes. Very recently, a theory able to treat lasers with arbitrarily complex and open cavities such as random lasers established that the threshold lasing modes are in fact distinct from QB states of the passive system and are better described in terms of a new class of states, the so-called constant-flux states. The correspondence between QB states and lasing modes is found to improve in the strong scattering limit, confirming the validity of initial work in the strong scattering limit.

© 2010 Optical Society of America

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2009 (4)

A. Lagendijk, B. van Tiggelen, D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2009).
[CrossRef]

J. Fallert, R. J. B. Dietz, J. Sartor, D. Schneider, C. Klingshirn, H. Kalt, “Co-existence of strongly and weakly localized random laser modes,” Nat. Photonics 3, 279–282 (2009).
[CrossRef]

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

2008 (3)

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

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

D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008).
[CrossRef]

2007 (6)

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, A. Lagendijk, “Spatial extent of random laser modes,” Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef] [PubMed]

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

S. Mujumdar, V. Türck, R. Torre, D. S. Wiersma, “Chaotic behavior of a random laser with static disorder,” Phys. Rev. A 76, 033807 (2007).
[CrossRef]

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

X. Wu, J. Andreasen, H. Cao, A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B 24, A26–A33 (2007).
[CrossRef]

Ü. Pekel, R. Mittra, “A finite-element-method frequency-domain application of the perfectly matched layer (PML) concept,” Microwave Opt. Technol. Lett. 9(3) 117–122 (2007).
[CrossRef]

2006 (3)

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, “Frequency shift of sources embedded in finite two-dimensional photonic clusters,” Waves Random Complex Media 16, 151–165 (2006).
[CrossRef]

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

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

2005 (1)

V. Milner, A. Z. Genack, “Photon localization laser: low-threshold lasing in a random amplifying layered medium via wave localization,” Phys. Rev. Lett. 94, 073901 (2005).
[CrossRef] [PubMed]

2004 (3)

V. M. Apalkov, M. E. Raikh, B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004).
[CrossRef]

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

F. A. Pinheiro, M. Rusek, A. Orlowski, B. A. van Tiggelen, “Probing Anderson localization of light via decay rate statistic,” Phys. Rev. E 69, 026605 (2004)..
[CrossRef]

2003 (5)

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Complex Media 13, 9–25 (2003).
[CrossRef]

D. P. Fussell, R. C. McPhedran, C. Martijn de Sterke, A. A. Asatryan, “Three-dimensional local density of states in a finite two-dimensional photonic crystal composed of cylinders,” Phys. Rev. E 67, 045601(R) (2003).
[CrossRef]

A. A. Chabanov, Z. Q. Zhang, A. Z. Genack, “Breakdown of diffusion in dynamics of extended waves in mesoscopic media,” Phys. Rev. Lett. 90, 203903 (2003).
[CrossRef] [PubMed]

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

H. Cao, J. Y. Xu, Y. Ling, A. L. Burin, E. W. Seeling, X. Liu, R. P. H. Chang, “Random lasers with coherent feedback,” IEEE J. Sel. Top. Quantum Electron. 9, 111–118 (2003).
[CrossRef]

2002 (5)

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

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

T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B 10, 2322–2330 (2002).
[CrossRef]

B. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, R. C. McPhedran, “Multipole method for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 10, 2331–2340 (2002).
[CrossRef]

X. Jiang, C. M. Soukoulis, “Localized random lasing modes and a path for observing localization,” Phys. Rev. E 65, 025601(R) (2002).
[CrossRef]

2001 (3)

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
[CrossRef]

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, C. Q. Cao, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

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

2000 (4)

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

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000).
[CrossRef] [PubMed]

D. S. Wiersma, “The smallest random laser,” Nature 406, 132–135 (2000).
[CrossRef] [PubMed]

S. M. Dutra, G. Nienhuis, “Quantized mode of a leaky cavity,” Phys. Rev. A 62, 063805 (2000).
[CrossRef]

1999 (5)

X. Jiang, Q. Li, C. M. Soukoulis, “Symmetry between absorption and amplification in disordered media,” Phys. Rev. B 59, R9007–R9010 (1999).
[CrossRef]

E. Centeno, D. Felbacq, “Characterization of defect modes in finite bidimensional photonic crystals,” J. Opt. Soc. Am. A 16, 2705–2712 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, H. C. Ong, R. P. H. Chang, “Far-field characteristics of random lasers,” Phys. Rev. B 59, 15107–15111 (1999).
[CrossRef]

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[CrossRef]

1998 (2)

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycristalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
[CrossRef]

A. A. Asatryan, N. A. Nicorovici, L. C. Botten, M. C. de Sterke, P. A. Robinson, R. C. McPhedran, “Electromagnetic localization in dispersive stratified media with random loss and gain,” Phys. Rev. B 57, 13535–13549 (1998).
[CrossRef]

1996 (1)

J. C. J. Paasschens, T. Sh. Misirpashaev, C. W. J. Beenakker, “Localization of light: dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
[CrossRef]

1995 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1995).
[CrossRef]

1994 (4)

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of optical wave-guides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

N. M. Lawandy, R. M. Balachandra, A. S. L. Gomes, E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994).
[CrossRef]

W. L. Sha, C. H. Liu, R. R. Alfano, “Spectral and temporal measurements of laser action of Rhodamine 640 dye in strongly scattering media,” Opt. Lett. 19, 1922–1924 (1994).
[CrossRef] [PubMed]

1993 (2)

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

1960 (1)

A. G. Fox, T. Li, “Resonant modes in an optical maser,” Proc. IRE 48, 1904–1905 (1960).

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[CrossRef]

Alfano, R. R.

W. L. Sha, C. H. Liu, R. R. Alfano, “Spectral and temporal measurements of laser action of Rhodamine 640 dye in strongly scattering media,” Opt. Lett. 19, 1922–1924 (1994).
[CrossRef] [PubMed]

Ambre, P.

P. Ambre, “Modélisation et caractérisation des fibres microstructurées air/silice pour application aux télécommunications optiques,” Ph.D. thèses (Université de Limoges, 2003).

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[CrossRef]

Andreasen, J.

X. Wu, J. Andreasen, H. Cao, A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B 24, A26–A33 (2007).
[CrossRef]

Apalkov, V. M.

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Recently it was shown that one can define a variant of the CF states, termed the threshold constant flux (TCF) states, for which one TCF state is the TLM. Above threshold additional TCF states are needed to describe the lasing modes, but in general fewer than for the CF states defined here. The above threshold theory using these TCF states is almost identical to the theory described in [33, 35]. Details are in L. Ge, Y. Chong, A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” arXiv.org, arXiv:1008.0628v1 (submitted to Phys. Rev. A).

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J. Fallert, R. J. B. Dietz, J. Sartor, D. Schneider, C. Klingshirn, H. Kalt, “Co-existence of strongly and weakly localized random laser modes,” Nat. Photonics 3, 279–282 (2009).
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X. Jiang, Q. Li, C. M. Soukoulis, “Symmetry between absorption and amplification in disordered media,” Phys. Rev. B 59, R9007–R9010 (1999).
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A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
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[CrossRef]

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000).
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[CrossRef]

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[CrossRef]

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T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B 10, 2322–2330 (2002).
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B. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, R. C. McPhedran, “Multipole method for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 10, 2331–2340 (2002).
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A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Complex Media 13, 9–25 (2003).
[CrossRef]

B. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, R. C. McPhedran, “Multipole method for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 10, 2331–2340 (2002).
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T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B 10, 2322–2330 (2002).
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A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
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Milner, V.

V. Milner, A. Z. Genack, “Photon localization laser: low-threshold lasing in a random amplifying layered medium via wave localization,” Phys. Rev. Lett. 94, 073901 (2005).
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K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of optical wave-guides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Misirpashaev, T. Sh.

J. C. J. Paasschens, T. Sh. Misirpashaev, C. W. J. Beenakker, “Localization of light: dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
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Ü. Pekel, R. Mittra, “A finite-element-method frequency-domain application of the perfectly matched layer (PML) concept,” Microwave Opt. Technol. Lett. 9(3) 117–122 (2007).
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K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, A. Lagendijk, “Spatial extent of random laser modes,” Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef] [PubMed]

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S. Mujumdar, V. Türck, R. Torre, D. S. Wiersma, “Chaotic behavior of a random laser with static disorder,” Phys. Rev. A 76, 033807 (2007).
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S. Mujumdar, M. Ricci, R. Torre, D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903 (2004).
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A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, “Frequency shift of sources embedded in finite two-dimensional photonic clusters,” Waves Random Complex Media 16, 151–165 (2006).
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A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Complex Media 13, 9–25 (2003).
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A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001).
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A. A. Asatryan, N. A. Nicorovici, L. C. Botten, M. C. de Sterke, P. A. Robinson, R. C. McPhedran, “Electromagnetic localization in dispersive stratified media with random loss and gain,” Phys. Rev. B 57, 13535–13549 (1998).
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H. Cao, Y. G. Zhao, H. C. Ong, R. P. H. Chang, “Far-field characteristics of random lasers,” Phys. Rev. B 59, 15107–15111 (1999).
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H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycristalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
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F. A. Pinheiro, M. Rusek, A. Orlowski, B. A. van Tiggelen, “Probing Anderson localization of light via decay rate statistic,” Phys. Rev. E 69, 026605 (2004)..
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J. C. J. Paasschens, T. Sh. Misirpashaev, C. W. J. Beenakker, “Localization of light: dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
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Ü. Pekel, R. Mittra, “A finite-element-method frequency-domain application of the perfectly matched layer (PML) concept,” Microwave Opt. Technol. Lett. 9(3) 117–122 (2007).
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F. A. Pinheiro, M. Rusek, A. Orlowski, B. A. van Tiggelen, “Probing Anderson localization of light via decay rate statistic,” Phys. Rev. E 69, 026605 (2004)..
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V. M. Apalkov, M. E. Raikh, B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004).
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V. M. Apalkov, M. E. Raikh, B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
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B. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, R. C. McPhedran, “Multipole method for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 10, 2331–2340 (2002).
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T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B 10, 2322–2330 (2002).
[CrossRef]

Ricci, M.

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

Robinson, P. A.

A. A. Asatryan, N. A. Nicorovici, L. C. Botten, M. C. de Sterke, P. A. Robinson, R. C. McPhedran, “Electromagnetic localization in dispersive stratified media with random loss and gain,” Phys. Rev. B 57, 13535–13549 (1998).
[CrossRef]

Rotter, S.

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, 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, A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643 (2008).
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F. A. Pinheiro, M. Rusek, A. Orlowski, B. A. van Tiggelen, “Probing Anderson localization of light via decay rate statistic,” Phys. Rev. E 69, 026605 (2004)..
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M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, 1982).

Sartor, J.

J. Fallert, R. J. B. Dietz, J. Sartor, D. Schneider, C. Klingshirn, H. Kalt, “Co-existence of strongly and weakly localized random laser modes,” Nat. Photonics 3, 279–282 (2009).
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J. Fallert, R. J. B. Dietz, J. Sartor, D. Schneider, C. Klingshirn, H. Kalt, “Co-existence of strongly and weakly localized random laser modes,” Nat. Photonics 3, 279–282 (2009).
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C. Vanneste, P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009).
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C. Vanneste, P. Sebbah, H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007).
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P. Sebbah, C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
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C. Vanneste, P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001).
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P. Sebbah, D. Sornette, C. Vanneste, “Anomalous diffusion in two-dimensional Anderson-localization dynamics,” Phys. Rev. B 48, 12506–12510 (1993).
[CrossRef]

The idea of considering localized modes for random lasing can already be found in P. Sebbah, D. Sornette, C. Vanneste, “Wave automaton for wave propagation in the time domain,” Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, 1994), p. 68.

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H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000).
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H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Seeling, E. W.

H. Cao, J. Y. Xu, Y. Ling, A. L. Burin, E. W. Seeling, X. Liu, R. P. H. Chang, “Random lasers with coherent feedback,” IEEE J. Sel. Top. Quantum Electron. 9, 111–118 (2003).
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V. M. Apalkov, M. E. Raikh, B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004).
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V. M. Apalkov, M. E. Raikh, B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
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P. Sebbah, D. Sornette, C. Vanneste, “Anomalous diffusion in two-dimensional Anderson-localization dynamics,” Phys. Rev. B 48, 12506–12510 (1993).
[CrossRef]

The idea of considering localized modes for random lasing can already be found in P. Sebbah, D. Sornette, C. Vanneste, “Wave automaton for wave propagation in the time domain,” Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, 1994), p. 68.

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H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, 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, A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643 (2008).
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L. Ge, R. Tandy, A. D. Stone, H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895 (2008).
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H. E. Türeci, A. D. Stone, 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, B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

Recently it was shown that one can define a variant of the CF states, termed the threshold constant flux (TCF) states, for which one TCF state is the TLM. Above threshold additional TCF states are needed to describe the lasing modes, but in general fewer than for the CF states defined here. The above threshold theory using these TCF states is almost identical to the theory described in [33, 35]. Details are in L. Ge, Y. Chong, A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” arXiv.org, arXiv:1008.0628v1 (submitted to Phys. Rev. A).

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

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

Tayeb, G.

Tjerkstra, R. W.

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, A. Lagendijk, “Spatial extent of random laser modes,” Phys. Rev. Lett. 98, 143901 (2007).
[CrossRef] [PubMed]

Torre, R.

S. Mujumdar, V. Türck, R. Torre, D. S. Wiersma, “Chaotic behavior of a random laser with static disorder,” Phys. Rev. A 76, 033807 (2007).
[CrossRef]

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

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S. Mujumdar, V. Türck, R. Torre, D. S. Wiersma, “Chaotic behavior of a random laser with static disorder,” Phys. Rev. A 76, 033807 (2007).
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H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
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L. Ge, R. Tandy, A. D. Stone, H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895 (2008).
[CrossRef] [PubMed]

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

H. E. Türeci, A. D. Stone, 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, 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, R. W. Tjerkstra, A. P. Mosk, A. Lagendijk, “Spatial extent of random laser modes,” Phys. Rev. Lett. 98, 143901 (2007).
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A. Lagendijk, B. van Tiggelen, D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2009).
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van Tiggelen, B. A.

F. A. Pinheiro, M. Rusek, A. Orlowski, B. A. van Tiggelen, “Probing Anderson localization of light via decay rate statistic,” Phys. Rev. E 69, 026605 (2004)..
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C. Vanneste, P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009).
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C. Vanneste, P. Sebbah, H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007).
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P. Sebbah, C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
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C. Vanneste, P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001).
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P. Sebbah, D. Sornette, C. Vanneste, “Anomalous diffusion in two-dimensional Anderson-localization dynamics,” Phys. Rev. B 48, 12506–12510 (1993).
[CrossRef]

The idea of considering localized modes for random lasing can already be found in P. Sebbah, D. Sornette, C. Vanneste, “Wave automaton for wave propagation in the time domain,” Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, 1994), p. 68.

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S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[CrossRef]

Wang, Q. H.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
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White, T. P.

B. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, R. C. McPhedran, “Multipole method for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 10, 2331–2340 (2002).
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T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B 10, 2322–2330 (2002).
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Wiersma, D. S.

A. Lagendijk, B. van Tiggelen, D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2009).
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D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008).
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S. Mujumdar, V. Türck, R. Torre, D. S. Wiersma, “Chaotic behavior of a random laser with static disorder,” Phys. Rev. A 76, 033807 (2007).
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S. Mujumdar, M. Ricci, R. Torre, D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903 (2004).
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D. S. Wiersma, “The smallest random laser,” Nature 406, 132–135 (2000).
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Wu, J. Y.

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycristalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
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Wu, X.

X. Wu, J. Andreasen, H. Cao, A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B 24, A26–A33 (2007).
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X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
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Xu, J. Y.

H. Cao, J. Y. Xu, Y. Ling, A. L. Burin, E. W. Seeling, X. Liu, R. P. H. Chang, “Random lasers with coherent feedback,” IEEE J. Sel. Top. Quantum Electron. 9, 111–118 (2003).
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H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, C. Q. Cao, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
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H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000).
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Yamilov, A.

X. Wu, J. Andreasen, H. Cao, A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B 24, A26–A33 (2007).
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X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006).
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Yoshino, K.

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
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Zakhidov, A.

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
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H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000).
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A. A. Chabanov, Z. Q. Zhang, A. Z. Genack, “Breakdown of diffusion in dynamics of extended waves in mesoscopic media,” Phys. Rev. Lett. 90, 203903 (2003).
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H. Cao, Y. G. Zhao, H. C. Ong, R. P. H. Chang, “Far-field characteristics of random lasers,” Phys. Rev. B 59, 15107–15111 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycristalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
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Appl. Phys. Lett. (1)

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycristalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998).
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X. Wu, J. Andreasen, H. Cao, A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B 24, A26–A33 (2007).
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T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B 10, 2322–2330 (2002).
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Ü. Pekel, R. Mittra, “A finite-element-method frequency-domain application of the perfectly matched layer (PML) concept,” Microwave Opt. Technol. Lett. 9(3) 117–122 (2007).
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Nat. Photonics (1)

J. Fallert, R. J. B. Dietz, J. Sartor, D. Schneider, C. Klingshirn, H. Kalt, “Co-existence of strongly and weakly localized random laser modes,” Nat. Photonics 3, 279–282 (2009).
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Phys. Rev. A (1)

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Phys. Rev. E (1)

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Phys. Rev. Lett. (3)

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Phys. Rev. B (3)

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Phys. Rev. Lett. (8)

X. Jiang, C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
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C. Vanneste, P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001).
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H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000).
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Proc. IRE (1)

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Waves Random Complex Media (1)

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, “Frequency shift of sources embedded in finite two-dimensional photonic clusters,” Waves Random Complex Media 16, 151–165 (2006).
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A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Complex Media 13, 9–25 (2003).
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H. Cao, “Lasing in random media,” Waves Random Complex Media 13, R1–R39 (2003).
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Other (10)

H. Cao, “Random lasers with coherent feedback,” in Optical Properties of Nanostructured Random MediaV. M. Shalaev, ed., Vol. 82 of Topics in Applied Physics (Springer-Verlag, 2002), pp. 303–330
[CrossRef]

The idea of considering localized modes for random lasing can already be found in P. Sebbah, D. Sornette, C. Vanneste, “Wave automaton for wave propagation in the time domain,” Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, 1994), p. 68.

P. Sebbah, “A new approach for the study of wave propagation and localization,” Ph.D. thesis (Université de Nice—Sophia Antipolis, 1993).

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

A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 1995).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, 1982).

H. Haken, Light: Laser Dynamics (North-Holland, 1985), vol. 2.

Recently it was shown that one can define a variant of the CF states, termed the threshold constant flux (TCF) states, for which one TCF state is the TLM. Above threshold additional TCF states are needed to describe the lasing modes, but in general fewer than for the CF states defined here. The above threshold theory using these TCF states is almost identical to the theory described in [33, 35]. Details are in L. Ge, Y. Chong, A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” arXiv.org, arXiv:1008.0628v1 (submitted to Phys. Rev. A).

P. Ambre, “Modélisation et caractérisation des fibres microstructurées air/silice pour application aux télécommunications optiques,” Ph.D. thèses (Université de Limoges, 2003).

Supplementary Material (4)

» Media 1: MOV (495 KB)     
» Media 2: MOV (602 KB)     
» Media 3: MOV (2073 KB)     
» Media 4: MOV (2620 KB)     

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Figures (20)

Fig. 1
Fig. 1

Example of a random realization of 896 circular scatterers contained in a square box of size L = 5 μm and optical index n = 1 . The radius and the optical index of the scatterers are, respectively, r = 60 nm and n = 2 . The total system of size 9 μ m is bounded by perfectly matched layers (not shown) to simulate an open system.

Fig. 2
Fig. 2

(a) Spatial distribution of the amplitude of a lasing mode in the localized regime ( n = 2 ) and (b) that of the corresponding QB states of the same random system without gain. The squares delimit the scattering medium. The amplitude rather than the intensity is represented for a better display of the small values of the field.

Fig. 3
Fig. 3

(a) Spatial distribution of the amplitude of a lasing mode in the diffusive regime. (b) Spatial distribution of the field amplitude after the pump has been stopped and the polarization term has been set to zero. The spatial distribution of scatterers is the collection shown in Fig. 1, but here the optical index of the scatterers is n = 1.25 instead of n = 2 in Fig. 2

Fig. 4
Fig. 4

Short-time behavior over a few cycles of the correlation function, C E ( t 0 , t ) , for (a) a localized lasing mode as in Fig. 2 and for (b) a diffusive lasing mode as in Fig. 3. The periodic square function in (a) is typical of a standing wave, while the sine-like function in (b) is characteristic of a traveling wave [28].

Fig. 5
Fig. 5

Correlation function (solid curves) and energy decay (dashed curves) versus time of (a) the lasing mode when the pump is turned off and (b) an arbitrary field distribution at the frequency of the lasing mode.

Fig. 6
Fig. 6

The frequencies k of QB states (crosses) and lasing modes with linear gain (open diamonds) together with the decay rates k 0 of QB states and the lasing thresholds k of lasing modes. The horizontal dashed lines separate three different regions of behavior: (a) lasing modes are easily matched to QB states, (b) clear differences appear but matching lasing modes to QB states is still possible, (c) lasing modes have shifted so much it is difficult to match them to QB states. The QB state with the largest decay rate and the lasing mode with the largest threshold are circled, though they may not be a matching pair.

Fig. 7
Fig. 7

Spatial intensity distributions of QB states I QB ( x ) (red solid lines) and lasing modes I LG ( x ) (black dashed lines) from each of the three regions in Fig. 6. Representative samples were chosen for each case. (a) The lasing mode intensity is nearly identical to the QB state intensity with σ d = 1.7 % . (b) A clear difference appears between the lasing mode and the QB state, with σ d = 21.8 % , but they are still similar. (c) The lasing mode with the largest threshold and QB state with the largest decay rate are compared, with σ d = 198 % . Though these two modes are fairly close to each other [circled in Fig. 6 region (c)], their intensity distributions are quite different.

Fig. 8
Fig. 8

(a) Intensity | E | 2 of the localized QB state (Media 1) and (b) corresponding lasing mode (Media 2) calculated by using a multipole method for a 2D disordered scattering system of the kind shown in Fig. 1 with the refractive index of the cylinders n l = 2.0 .

Fig. 9
Fig. 9

(a) Intensity | E | 2 of the diffusive QB state (Media 3) and (b) the lasing mode (Media 4) calculated by using the multipole method for the same random configuration as in Fig. 8 but with the refractive index of the cylinders of n l = 1.25 .

Fig. 10
Fig. 10

Intensity | E | 2 of the diffusive QB state (blue dashed curve) and lasing mode (red solid curve) for x = 2.75 and n l = 1.25 .

Fig. 11
Fig. 11

(a) Intensity | E | 2 of a QB state and (b) a lasing mode calculated by using multipole method for the same random configuration as above but with the refractive index of the cylinders n l = 1.75 .

Fig. 12
Fig. 12

Same as in Fig. 11 but for n l = 1.5 .

Fig. 13
Fig. 13

Intensity | E | 2 of QB state (blue dashed curves) and lasing mode (red solid curves) at x = 2.75 for (a) n l = 1.75 , (b) n = 1.5 , (c) n = 1.25 .

Fig. 14
Fig. 14

Shift of the poles of the S matrix in the complex plane onto the real axis to form TLMs when the imaginary part of the dielectric function ϵ ϵ c + ϵ g varies for a simple 1D edge-emitting cavity laser [34]. The cavity is a region of length L and uniform index (a) n c = 1.5 , (b) n = 1.05 ( ϵ c = 2.25 , 1.0025 ) terminated in vacuum at both ends. The calculations are based on the MB model discussed in Section 5, with parameters k a L = 39 and γ = 2 . (a) n c = 1.5 ; squares of different colors represent Im [ ϵ g ] = 0 , 0.032 , 0.064 , 0.096 , 0.128 ; (b) n c = 1.05 ; squares of different colors represent Im [ ϵ g ] = 0 , 0.04 , 0.08 , 0.12 , 0.16 . Note the increase in the frequency shift in the complex plane for the leakier cavity. The center of the gain curve is at k L = 39 , which determines the visible line-pulling effect.

Fig. 15
Fig. 15

Typical values of the threshold matrix elements T ( 0 ) in a 2D random laser schematized in the inset of Fig. 18 below, using sixteen CF states. The off-diagonal elements are one to two orders of magnitude smaller than the diagonal ones.

Fig. 16
Fig. 16

(a) False color plot of one TLM in a 2D random laser modeled as an aggregate of subwavelength particles of index of refraction n = 1.2 and radius r = R 30 against a background index n = 1 imbedded in a uniform disk of gain material of radius R [see inset, panel (d)]. The frequency of the lasing mode is k R = 59.9432 , which is pulled from (b) the real part of the dominating CF state k m R = 59.8766 0.8593 i towards the transition frequency k a R = 60 . The spatial profile of the TLM and CF state agree very well, whereas (c) the corresponding QB state k ̃ m R = 59.8602 0.8660 i differs from that of the TLM and the CF state noticeably, as can be seen in (d), where we plot the internal intensity along the θ = 200 ° direction [white line in (a)].

Fig. 17
Fig. 17

(a) False color plot of one TLM in a 2D random laser similar to that in Fig. 16 but with particles of radius r = R 60 , corresponding to weaker scattering [see inset, panel (d)]. The frequency of the lasing mode is k R = 29.9959 , which is very close to (b) the CF state k m R = 30.0058 1.3219 i but shifted from (c) the corresponding QB state k ̃ m R = 29.8813 1.3790 i . (d) Internal intensity of the three states in the θ = π direction [white line in (a)]; because of weaker scattering the QB state now differs substantially from the CF and TLM, which still agree quite well with each other.

Fig. 18
Fig. 18

(a) CF (dots) and QB (crosses) frequencies in a 2D random laser modeled as an aggregate of subwavelength particles of index of refraction n = 1.2 against a background index n = 1 imbedded in a uniform disk of gain material (see inset). The two sets of complex frequencies are statistically similar but differ substantially. The solid curve shows the gain curve Γ ( k ) with γ = 1 . (b) Lasing frequencies of the same random system well above threshold (colored lines). Colored circles denote the CF state dominating the correspondingly colored modes at threshold.

Fig. 19
Fig. 19

Geometry and local coordinate systems.

Fig. 20
Fig. 20

Close up of a typical mesh created by Comsol to describe the 2D random system of Fig. 2.

Tables (1)

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Table 1 QB State Values a    for Four Index Values n of Scatterers

Equations (45)

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d N 1 d t = N 2 τ 21 W p N 1 ,
d N 2 d t = N 3 τ 32 N 2 τ 21 ( E ω a ) d P d t ,
d N 3 d t = N 4 τ 43 N 3 τ 32 + ( E ω a ) d P d t ,
d N 4 d t = N 4 τ 43 + W p N 1 ,
d 2 P d t 2 + Δ ω a d P d t + ω a 2 P = κ Δ N E ,
H t = c × E ,
ϵ ( r ) E / t = c × H 4 π P t .
C E ( t 0 , t ) = D d 2 r E ( r , t 0 ) E ( r , t ) ,
σ d = | I QB I LG | d A I LG d A × 100 % .
E ̈ + = 1 ϵ c ( x ) 2 E + 4 π ϵ c ( x ) P ̈ + ,
P ̇ + = ( i ω a + γ ) P + + g 2 i E + D ,
D ̇ = γ ( D 0 D ) 2 i ( E + ( P + ) * P + ( E + ) * ) .
E + ( x , t ) = μ = 1 N Ψ μ ( x ) e i k μ t , P + ( x , t ) = μ = 1 N P μ ( x ) e i k μ t .
P μ ( x ) = i D 0 g 2 Ψ μ ( x ) ( γ i ( k μ k a ) ) .
[ 2 + ϵ c ( x ) k μ 2 ] Ψ μ ( x ) = i D 0 4 π g 2 k μ 2 Ψ μ ( x ) ( γ i ( k μ k a ) ) ,
[ 2 + ( ϵ c ( x ) + ϵ g ( x ) ) k μ 2 ] Ψ μ ( x ) = 0 ,
ϵ g ( x ) = D 0 k a 2 [ γ ( k μ k a ) γ 2 + ( k μ k a ) 2 + i γ 2 γ 2 + ( k μ k a ) 2 ] .
ϵ g i D 0 k a 2 ,
n ( x ) = ϵ c ( x ) + ϵ g ( D 0 , k μ k a , γ )
[ ϵ c ( x ) 1 2 + k μ 2 ] Ψ μ ( x ) = ϵ g k μ 2 ϵ c ( x ) Ψ μ ( x ) ,
Ψ μ ( x ) = i D 0 γ γ i ( k μ k a ) k μ 2 k a 2 D d x G ( x , x ; k μ ) Ψ μ ( x ) ϵ c ( x ) .
[ ϵ c ( x ) 1 2 + k 2 ] G ( x , x | k ) = δ d ( x x )
G ( x , x | k ) = m φ m ( x , k ) φ ¯ m * ( x , k ) ( k 2 k m 2 ) .
[ ϵ c ( x ) 1 2 + k m 2 ] φ m ( x , k ) = 0
D d x φ m ( x , k ) φ ¯ n * ( x , k ) = δ m n
Ψ μ ( x ) = m = 1 a m μ φ m μ ( x ) .
a m μ = D 0 Λ m ( k μ ) D d x φ ¯ m μ * ( x ) p N a p μ φ p μ ( x ) ϵ c ( x ) D 0 p N T m p ( 0 ) a p μ ,
D 0 D 0 1 + ν Γ ( k ν ) ( | Ψ ν ( x ) | 2 ) ,
Ψ μ ( x ) = i D 0 γ γ i ( k μ k a ) k μ 2 k a 2 D d x G ( x , x ; k μ ) Ψ μ ( x ) ϵ c ( x ) ( 1 + ν Γ ν | Ψ ν ( x ) | 2 ) .
2 V ( r ) + k 2 n 2 ( r ) V ( r ) = 0 .
V ( r ) = m = [ A m l J m ( k n b r l ) + B m l H m ( 1 ) ( k n b r l ) ] e i m θ l .
V ( r ) = q = 1 N c m = B m q H m ( 1 ) ( k | r c q | ) e i m arg ( r c q ) .
V ( r ) = m = C m l J m ( k n l | r c l | ) e i m arg ( r c l ) .
A m l = q = 1 , q l N c p = H m p l q B p q ,
H m p l q = H m p ( 1 ) ( k c l q ) e i ( m p ) θ l q .
B m l = R m l A m l ,
C m l = T m l A m l ,
R m l = ξ n l J m ( k n l a l ) J m ( k n b a l ) n b J m ( k n l a l ) J m ( k n b a l ) ξ n l J m ( k n l a l ) H m ( 1 ) ( k n b a l ) n b J m ( k n l a l ) H m ( 1 ) ( k n b a l ) ,
T m l = 2 i ( π k a L ) ξ n l J m ( n l k a l ) H m ( 1 ) ( k n b a l ) n b J m ( k n l a l ) H m ( 1 ) ( k n b a l ) ,
A l = q H l q B q ,
B = R A ,
C = T A ,
( I R H ) B = 0 .
D = 0 , where D = det ( S 1 )
S 1 ( λ ) = ( I R H ) .

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