Abstract

We report on experimental tests of the trend of random laser linewidth versus pumping power as predicted by a Haus master equation that is formally identical to the one-dimensional Gross–Pitaevskii equation in a harmonic potential. Experiments are done by employing picosecond pumped dispersions of titanium dioxide particles in dye-doped methanol. The derivation of the master equation is also detailed and shown to be in agreement with experiments analytically predicting the value of the threshold linewidth.

© 2010 Optical Society of America

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  1. V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Zh. Eksp. Teor. Fiz. 53, 1442–1452 (1967) V. S. Letokhov, “[Sov. Phys. JETP 26, 835–840 (1968)].
  2. N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994).
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  3. L. S. Froufe-Pérez, W. Guerin, R. Carminati, and R. Kaiser, “Threshold of a random laser with cold atoms,” Phys. Rev. Lett. 102, 173903 (2009).
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  4. D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008).
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  7. H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
    [CrossRef] [PubMed]
  8. A. Einstein, “Über die von molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen,” Ann. Phys. 322, 549–560 (1905).
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  9. D. S. Wiersma and A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
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    [CrossRef]
  11. K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, “Spatial extent of random laser modes,” Phys. Rev. Lett. 98, 143901 (2007).
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  12. L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, “Glassy behavior of light,” Phys. Rev. Lett. 96, 065702 (2006).
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  13. H. E. Tureci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
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  14. O. Zaitsev and L. Deych, “Recent developments in the theory of multimode random lasers,” J. Opt. 12, 024001 (2010).
    [CrossRef]
  15. 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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    [CrossRef]
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    [CrossRef]
  18. Y. S. Kivshar, T. J. Alexander, and S. K. Turitsyn, “Nonlinear modes of a macroscopic quantum oscillator,” Phys. Lett. A 278, 225–230 (2001).
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  20. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
    [CrossRef]
  21. R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95, 013903 (2005).
    [CrossRef] [PubMed]
  22. L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: From order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
    [CrossRef] [PubMed]
  23. D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
    [CrossRef] [PubMed]
  24. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
    [CrossRef] [PubMed]
  25. S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett. 34, 130–132 (2009).
    [CrossRef] [PubMed]
  26. J. Fallert, R. J. B. Dietz, J. Sartor, D. Schneider, C. Klingshirn, and H. Kalt, “Co-existence of strongly and weakly localized random laser modes,” Nat. Photonics 3, 279–282 (2009).
    [CrossRef]
  27. W. E. Lamb, Jr., “Theory of an optical maser,” Phys. Rev. 134, A1429–A1450 (1964).
    [CrossRef]
  28. H. A. Haus and W. H. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
    [CrossRef]
  29. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
    [CrossRef]
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    [CrossRef]

2010 (1)

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

2009 (4)

L. S. Froufe-Pérez, W. Guerin, R. Carminati, and R. Kaiser, “Threshold of a random laser with cold atoms,” Phys. Rev. Lett. 102, 173903 (2009).
[CrossRef] [PubMed]

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: From order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef] [PubMed]

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett. 34, 130–132 (2009).
[CrossRef] [PubMed]

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

2008 (3)

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

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]

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

2007 (1)

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

2006 (3)

L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, “Glassy behavior of light,” Phys. Rev. Lett. 96, 065702 (2006).
[CrossRef] [PubMed]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

2005 (2)

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

2004 (1)

L. Florescu and S. John, “Photon statistics and coherence in light emission from a random laser,” Phys. Rev. Lett. 93, 013602 (2004).
[CrossRef]

2001 (2)

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

Y. S. Kivshar, T. J. Alexander, and S. K. Turitsyn, “Nonlinear modes of a macroscopic quantum oscillator,” Phys. Lett. A 278, 225–230 (2001).
[CrossRef]

2000 (2)

1999 (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

1996 (1)

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

1995 (1)

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef] [PubMed]

1994 (1)

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

1993 (1)

1991 (1)

H. A. Haus and W. H. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

1967 (1)

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Zh. Eksp. Teor. Fiz. 53, 1442–1452 (1967) V. S. Letokhov, “[Sov. Phys. JETP 26, 835–840 (1968)].

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Zh. Eksp. Teor. Fiz. 53, 1442–1452 (1967) V. S. Letokhov, “[Sov. Phys. JETP 26, 835–840 (1968)].

1964 (1)

W. E. Lamb, Jr., “Theory of an optical maser,” Phys. Rev. 134, A1429–A1450 (1964).
[CrossRef]

1958 (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

1905 (1)

A. Einstein, “Über die von molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen,” Ann. Phys. 322, 549–560 (1905).
[CrossRef]

Aegerter, C. M.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

Alexander, T. J.

Y. S. Kivshar, T. J. Alexander, and S. K. Turitsyn, “Nonlinear modes of a macroscopic quantum oscillator,” Phys. Lett. A 278, 225–230 (2001).
[CrossRef]

Angelani, L.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: From order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef] [PubMed]

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett. 34, 130–132 (2009).
[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. Angelani, C. Conti, G. Ruocco, and F. Zamponi, “Glassy behavior of light,” Phys. Rev. Lett. 96, 065702 (2006).
[CrossRef] [PubMed]

Auzel, F.

Balachandran, R. M.

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

Cao, C. Q.

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

Cao, H.

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

Carminati, R.

L. S. Froufe-Pérez, W. Guerin, R. Carminati, and R. Kaiser, “Threshold of a random laser with cold atoms,” Phys. Rev. Lett. 102, 173903 (2009).
[CrossRef] [PubMed]

Connaughton, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Conti, C.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: From order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef] [PubMed]

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett. 34, 130–132 (2009).
[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. Angelani, C. Conti, G. Ruocco, and F. Zamponi, “Glassy behavior of light,” Phys. Rev. Lett. 96, 065702 (2006).
[CrossRef] [PubMed]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Deych, L.

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

Dietz, R. J. B.

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

Einstein, A.

A. Einstein, “Über die von molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen,” Ann. Phys. 322, 549–560 (1905).
[CrossRef]

Fallert, J.

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

Filippidis, G.

Fischer, B.

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

Florescu, L.

L. Florescu and S. John, “Photon statistics and coherence in light emission from a random laser,” Phys. Rev. Lett. 93, 013602 (2004).
[CrossRef]

Folli, V.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: From order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef] [PubMed]

Fratalocchi, A.

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett. 34, 130–132 (2009).
[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]

Froufe-Pérez, L. S.

L. S. Froufe-Pérez, W. Guerin, R. Carminati, and R. Kaiser, “Threshold of a random laser with cold atoms,” Phys. Rev. Lett. 102, 173903 (2009).
[CrossRef] [PubMed]

Gat, O.

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

Ge, L.

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

Gentilini, S.

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Gomes, A. S. L.

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

Gordon, A.

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

Gouedard, C.

Gross, P.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

Guerin, W.

L. S. Froufe-Pérez, W. Guerin, R. Carminati, and R. Kaiser, “Threshold of a random laser with cold atoms,” Phys. Rev. Lett. 102, 173903 (2009).
[CrossRef] [PubMed]

Hakne, H.

H. Hakne, Synergetics (Springer-Verlag, 1978).
[CrossRef]

Haus, H. A.

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[CrossRef]

H. A. Haus and W. H. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

Huang, W. H.

H. A. Haus and W. H. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

Husson, D.

John, S.

L. Florescu and S. John, “Photon statistics and coherence in light emission from a random laser,” Phys. Rev. Lett. 93, 013602 (2004).
[CrossRef]

Josserand, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Kaiser, R.

L. S. Froufe-Pérez, W. Guerin, R. Carminati, and R. Kaiser, “Threshold of a random laser with cold atoms,” Phys. Rev. Lett. 102, 173903 (2009).
[CrossRef] [PubMed]

Kalt, H.

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

Kivshar, Y. S.

Y. S. Kivshar, T. J. Alexander, and S. K. Turitsyn, “Nonlinear modes of a macroscopic quantum oscillator,” Phys. Lett. A 278, 225–230 (2001).
[CrossRef]

Klingshirn, C.

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

Kumar, P.

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

Kutz, J. N.

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

Lagendijk, A.

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

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

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef] [PubMed]

Lamb, W. E.

W. E. Lamb, Jr., “Theory of an optical maser,” Phys. Rev. 134, A1429–A1450 (1964).
[CrossRef]

Lawandy, N. M.

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

Leonetti, M.

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]

Letokhov, V. S.

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Zh. Eksp. Teor. Fiz. 53, 1442–1452 (1967) V. S. Letokhov, “[Sov. Phys. JETP 26, 835–840 (1968)].

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Zh. Eksp. Teor. Fiz. 53, 1442–1452 (1967) V. S. Letokhov, “[Sov. Phys. JETP 26, 835–840 (1968)].

Leuzzi, L.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: From order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef] [PubMed]

Ling, Y.

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

Maret, G.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

Migus, A.

Mosk, A. P.

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

Papadogiannis, N. A.

Papazoglou, T. G.

Picozzi, A.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Pitaevskii, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose–Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Pomeau, Y.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Rica, S.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Rosen, A.

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical behavior of light in mode-locked lasers,” Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

Rotter, S.

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

Ruocco, G.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, “Phase diagram and complexity of mode-locked lasers: From order to disorder,” Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef] [PubMed]

S. Gentilini, A. Fratalocchi, L. Angelani, G. Ruocco, and C. Conti, “Ultrashort pulse propagation and the Anderson localization,” Opt. Lett. 34, 130–132 (2009).
[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]

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Figures (7)

Fig. 1
Fig. 1

Scheme of losses and gain profile as modeled by our theory.

Fig. 2
Fig. 2

Shape of the intensity spectrum for different values of the nonlinear eigenvalue E.

Fig. 3
Fig. 3

Waist (standard deviation) theoretically predicted for the RL spectrum in the high scattering regime. The curves are shown for different values of the κ parameter.

Fig. 4
Fig. 4

Peak intensity theoretically predicted for RL in the high scattering regime. The curves are shown for different values of the κ parameter.

Fig. 5
Fig. 5

Enhanced backscattering cone from disorderly arranged titanium dioxide particles (≈300 nm diameter, 0.2 packing fraction) in methanol.

Fig. 6
Fig. 6

Measured spectral linewidth versus energy (dots, left scale); the thick continuous line (left scale) is the best fit from the theory. The right scale shows the trend of the adimensional nonlinear eigenvalue (thin line) versus the input energy, as obtained from the fit.

Fig. 7
Fig. 7

As in Fig. 6 for the measured peak spectrum.

Equations (28)

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α ( w ) = α 0 j = 1 N α j ( ω ω j ) ,
g [ ω , A ( ω ) ] = α ( w ) A ( ω ) ,
g [ ω , A ( ω ) ] = g 0 { ( 1 t g 2 ω 2 ) A ( ω ) + d ω 1 d ω 2 d ω 3 δ ( ω + ω 1 ω 2 ω 3 ) χ ( ω 1 ; ω 2 ω 3 ) A ( ω 1 ) A ( ω 2 ) A ( ω 3 ) } ,
g [ ω , A ( ω ) ] = α 0 A ( ω ) j = 1 N α j ( ω ω j ) A ( ω j ) ,
g [ ω , A ( ω ) ] = α 0 A ( ω ) α avg ( ω Ω ) A ( Ω ) d Ω .
g [ A i ] i = j K i j ( ω i ω j ) A j ,
F [ a ( ω ) ] = 1 2 π a ( t ) exp ( i ω t ) d t ,
g [ t , a ( t ) ] = [ α 0 ϕ ( t ) ] a ( t ) ,
ϕ L ( t ) ( α 0 α L ) [ 1 ( t / t L ) 2 ] ,
g [ t , a ( t ) ] = g 0 [ a ( t ) + t g 2 d 2 a ( t ) d t 2 γ s | a ( t ) | 2 a ( t ) ] ,
g 0 [ a ( t ) + t g 2 d 2 a ( t ) d t 2 γ s | a ( t ) | 2 a ( t ) ] = [ α 0 ( α α L ) ] [ 1 ( t / t L ) 2 ] a ( t )
a 0 2 = t g α 0 α L ,
t 0 2 = t g t L g 0 α 0 α L .
d 2 φ d τ 2 + τ 2 φ + | φ | 2 φ = E φ ,
E = t L t g g 0 α L ( α 0 α L ) g 0 = p 1 κ p .
κ t g t L ( α 0 α L 1 ) .
p t h = 1 + κ 2 2 + κ 4 + κ 2 2 .
S ( ω ) = | A ( ω 2 ) | = t g 2 γ s | φ ̃ ( ω t 0 ) | 2 .
g [ t , a ( t ) ] = g 0 [ t g 2 d 2 a ( t ) d t 2 + a ( t ) 1 + γ s | a ( t ) | 2 ] ,
d 2 φ d τ 2 + τ 2 φ + 1 ϵ ( 1 1 1 + ϵ | φ | 2 ) φ = E φ ,
φ ( τ ) 2 1 / 4 E 1   exp ( τ 2 / 2 ) ,
S ( ω ) = t g 2 2 π γ S ( E 1 ) exp [ ω 2 8 π 2 W t h 2 ] ,
2 π t g W t h = κ 2 = t g 2 t L α 0 α L 1 .
κ 2 π t g λ D t L ,
W t h 1 4 π t g λ D t L .
I ( ω ) = I 0 [ 1 ( ω ω 0 ) 2 t g 2 ]
1 α 0 = c n a v 3   fs ,
p = C E ¯

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