Abstract

We study experimentally as well as numerically the transport and generation of light in multiple scattering media with optical gain. By imaging the spatial distribution of light escaping from the side of the sample, the propagation depth is analyzed. Far below and far above random laser threshold, the spatial profile of emission light is independent of pump intensity, while around threshold, the spatial distribution of emission light changes profoundly. The experimental results are explained by interpreting the numerical solutions to a set of coupled time-dependent diffusion equations on a nonuniform spatial grid. Our studies provide a new and easily accessible method for observing the random laser threshold.

© 2010 Optical Society of America

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  1. A. Yodh and B. Chance, Phys. Today 48, 34 (1995).
    [CrossRef]
  2. A. Lagendijk, B. A. van Tiggelen, and D. S. Wiersma, Phys. Today 62, 24 (2009).
    [CrossRef]
  3. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
    [CrossRef] [PubMed]
  4. B. Shapiro, Phys. Rev. Lett. 83, 4733 (1999).
    [CrossRef]
  5. D. S. Wiersma, Nature Phys. 4, 359 (2008).
    [CrossRef]
  6. M. A. Noginov, Solid-State Random Lasers (Springer, 2005).
  7. H. Cao, J. Phys. A 38, 10497 (2005).
    [CrossRef]
  8. P. M. Johnson, S. Faez, and A. Lagendijk, Opt. Express 16, 7435 (2008).
    [CrossRef] [PubMed]
  9. X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
    [CrossRef]
  10. O. L. Muskens and A. Lagendijk, Opt. Express 16, 1222 (2008).
    [CrossRef] [PubMed]
  11. G. van Soest, F. P. Poelwijk, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 86, 1522 (2001).
    [CrossRef] [PubMed]
  12. W. E. Schiesser and G. W. Griffiths, A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab (Cambridge U. Press, 2009).
    [CrossRef]
  13. A. E. P. Veldman and K. Rinzema, J. Eng. Math. 26, 119 (1992).
    [CrossRef]
  14. A. E. Siegman, Lasers (University Science, 1986).
  15. G. A. Berger, M. Kempe, and A. Z. Genack, Phys. Rev. E 56, 6118 (1997).
    [CrossRef]
  16. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
    [CrossRef]

2010

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef] [PubMed]

2009

A. Lagendijk, B. A. van Tiggelen, and D. S. Wiersma, Phys. Today 62, 24 (2009).
[CrossRef]

W. E. Schiesser and G. W. Griffiths, A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab (Cambridge U. Press, 2009).
[CrossRef]

2008

2006

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

2005

M. A. Noginov, Solid-State Random Lasers (Springer, 2005).

H. Cao, J. Phys. A 38, 10497 (2005).
[CrossRef]

2001

G. van Soest, F. P. Poelwijk, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 86, 1522 (2001).
[CrossRef] [PubMed]

1999

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
[CrossRef]

B. Shapiro, Phys. Rev. Lett. 83, 4733 (1999).
[CrossRef]

1997

G. A. Berger, M. Kempe, and A. Z. Genack, Phys. Rev. E 56, 6118 (1997).
[CrossRef]

1995

A. Yodh and B. Chance, Phys. Today 48, 34 (1995).
[CrossRef]

1992

A. E. P. Veldman and K. Rinzema, J. Eng. Math. 26, 119 (1992).
[CrossRef]

1986

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

Asatryan, A. A.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Berger, G. A.

G. A. Berger, M. Kempe, and A. Z. Genack, Phys. Rev. E 56, 6118 (1997).
[CrossRef]

Botten, L. C.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Cao, H.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

H. Cao, J. Phys. A 38, 10497 (2005).
[CrossRef]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
[CrossRef]

Chabanov, A. A.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Chance, B.

A. Yodh and B. Chance, Phys. Today 48, 34 (1995).
[CrossRef]

Chang, R. P. H.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
[CrossRef]

Faez, S.

Fang, W.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Garcia, P. D.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef] [PubMed]

Genack, A. Z.

G. A. Berger, M. Kempe, and A. Z. Genack, Phys. Rev. E 56, 6118 (1997).
[CrossRef]

Griffiths, G. W.

W. E. Schiesser and G. W. Griffiths, A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab (Cambridge U. Press, 2009).
[CrossRef]

Ho, S. T.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
[CrossRef]

Johnson, P. M.

Kempe, M.

G. A. Berger, M. Kempe, and A. Z. Genack, Phys. Rev. E 56, 6118 (1997).
[CrossRef]

Lagendijk, A.

A. Lagendijk, B. A. van Tiggelen, and D. S. Wiersma, Phys. Today 62, 24 (2009).
[CrossRef]

P. M. Johnson, S. Faez, and A. Lagendijk, Opt. Express 16, 7435 (2008).
[CrossRef] [PubMed]

O. L. Muskens and A. Lagendijk, Opt. Express 16, 1222 (2008).
[CrossRef] [PubMed]

G. van Soest, F. P. Poelwijk, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 86, 1522 (2001).
[CrossRef] [PubMed]

Lodahl, P.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef] [PubMed]

Muskens, O. L.

Noginov, M. A.

M. A. Noginov, Solid-State Random Lasers (Springer, 2005).

Poelwijk, F. P.

G. van Soest, F. P. Poelwijk, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 86, 1522 (2001).
[CrossRef] [PubMed]

Rinzema, K.

A. E. P. Veldman and K. Rinzema, J. Eng. Math. 26, 119 (1992).
[CrossRef]

Sapienza, L.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef] [PubMed]

Schiesser, W. E.

W. E. Schiesser and G. W. Griffiths, A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab (Cambridge U. Press, 2009).
[CrossRef]

Seelig, E. W.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
[CrossRef]

Shapiro, B.

B. Shapiro, Phys. Rev. Lett. 83, 4733 (1999).
[CrossRef]

Siegman, A. E.

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

Smolka, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef] [PubMed]

Sprik, R.

G. van Soest, F. P. Poelwijk, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 86, 1522 (2001).
[CrossRef] [PubMed]

Stobbe, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef] [PubMed]

Thyrrestrup, H.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef] [PubMed]

van Soest, G.

G. van Soest, F. P. Poelwijk, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 86, 1522 (2001).
[CrossRef] [PubMed]

van Tiggelen, B. A.

A. Lagendijk, B. A. van Tiggelen, and D. S. Wiersma, Phys. Today 62, 24 (2009).
[CrossRef]

Veldman, A. E. P.

A. E. P. Veldman and K. Rinzema, J. Eng. Math. 26, 119 (1992).
[CrossRef]

Wang, Q. H.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
[CrossRef]

Wiersma, D. S.

A. Lagendijk, B. A. van Tiggelen, and D. S. Wiersma, Phys. Today 62, 24 (2009).
[CrossRef]

D. S. Wiersma, Nature Phys. 4, 359 (2008).
[CrossRef]

Wu, X.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Yamilov, A.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Yodh, A.

A. Yodh and B. Chance, Phys. Today 48, 34 (1995).
[CrossRef]

Zhao, Y. G.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
[CrossRef]

J. Eng. Math.

A. E. P. Veldman and K. Rinzema, J. Eng. Math. 26, 119 (1992).
[CrossRef]

J. Phys. A

H. Cao, J. Phys. A 38, 10497 (2005).
[CrossRef]

Nature Phys.

D. S. Wiersma, Nature Phys. 4, 359 (2008).
[CrossRef]

Opt. Express

Phys. Rev. A

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, Phys. Rev. A 74, 053812 (2006).
[CrossRef]

Phys. Rev. E

G. A. Berger, M. Kempe, and A. Z. Genack, Phys. Rev. E 56, 6118 (1997).
[CrossRef]

Phys. Rev. Lett.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999).
[CrossRef]

B. Shapiro, Phys. Rev. Lett. 83, 4733 (1999).
[CrossRef]

G. van Soest, F. P. Poelwijk, R. Sprik, and A. Lagendijk, Phys. Rev. Lett. 86, 1522 (2001).
[CrossRef] [PubMed]

Phys. Today

A. Yodh and B. Chance, Phys. Today 48, 34 (1995).
[CrossRef]

A. Lagendijk, B. A. van Tiggelen, and D. S. Wiersma, Phys. Today 62, 24 (2009).
[CrossRef]

Science

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
[CrossRef] [PubMed]

Other

M. A. Noginov, Solid-State Random Lasers (Springer, 2005).

W. E. Schiesser and G. W. Griffiths, A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab (Cambridge U. Press, 2009).
[CrossRef]

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

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Figures (4)

Fig. 1
Fig. 1

Schematic top view of the experimental apparatus. Green, pump beam; red, emission light. The orange bars represent color filters.

Fig. 2
Fig. 2

(a) Peak spectral radiance versus the pump power. (b) Spectral width versus the pump power on a semilog scale. The red solid curve is a sigmoidal fit and serves as a guide to the eye.

Fig. 3
Fig. 3

(a) Measured 50% intensity contour of emission light exiting the sample from the side below (red) and above threshold (black) in the y z plane. Gray line, boundary of the sample; crosses, the maxima of the profiles. (b) Time-integrated calculated 50% intensity contour of emission light in the r z plane below (red) and above threshold (black); crosses, maxima of the profiles. (c) Theoretical and experimental characteristics of the spatial profiles along the dashed curves in (a) and (b) versus pump power. Black open squares ( □ ), numerically calculated values of the FWHM of the cut of the spatial distribution of laser light; black solid squares (▪), experimental values from the profile cut; red open circles ( ○ ), numerically determined position of the laser light maximum in depth; red solid circles (•), experimental values.

Fig. 4
Fig. 4

Calculated 50% contours of the excited state profile in the z t plane for different pump powers far below and far above threshold.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

W t = D 2 W + ( σ e c n 1 σ r c n 0 ) W + β τ n 1 ,
W p t = D 2 W p σ a c n 0 W p + 1 I in ,
n 1 t = σ a c n 0 W p ( σ e c n 1 σ r c n 0 ) W 1 τ n 1 .
f x f ( x + h + ) f ( x h ) h + + h ,
2 f x 2 h f ( x + h + ) ( h + + h ) f ( x ) + h + f ( x h ) 0.5 h + h ( h + + h ) .

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