Abstract

We interpret cooperative scattering by a collection of cold atoms as a multiple-scattering process. Starting from microscopic equations describing the response of N atoms to a probe light beam, we represent the total scattered field as an infinite series of multiple-scattering events. As an application of the method, we obtain analytical expressions of the coherent intensity in the double-scattering approximation for Gaussian density profiles. In particular, we quantify the contributions of coherent backward and forward scattering.

© 2014 Optical Society of America

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  3. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons, 1st ed. (Cambridge University, 2007).
  4. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption by nonsphereical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  5. A. Ishimaru, Wave Propagation and Scattering in Random Media and Rough Surfaces, Vol. 2 (Academic, 1978).
  6. A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 79, 1359–1366 (1991).
    [CrossRef]
  7. H. C. van de Hulst and R. Stark, “Accurate eigenvalues and exact extrapolation lengths in radiative transfer,” Astron. Astrophys. 235, 511–520 (1990).
  8. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
    [CrossRef]
  9. P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
    [CrossRef]
  10. M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
    [CrossRef]
  11. M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).
  12. G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
    [CrossRef]
  13. Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
    [CrossRef]
  14. G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by an inhomogeneous cloud of cold atoms,” Phys. Rev. A 67, 033814 (2003).
    [CrossRef]
  15. O. Morice, Y. Castin, and J. Dalibard, “Refractive index of a dilute bose gas,” Phys. Rev. A 51, 3896–3901 (1995).
    [CrossRef]
  16. A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Cooperative spontaneous emission of n atoms: many-body eigenstates, the effect of virtual lamb shift processes, and analogy with radiation of n classical oscillators,” Phys. Rev. A 81, 053821 (2010).
    [CrossRef]
  17. P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
    [CrossRef]
  18. T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, “Atom and photon measurement in cooperative scattering by cold atoms,” J. Mod. Opt. 58, 1942–1950 (2011).
    [CrossRef]
  19. R. Bachelard, N. Piovella, and P. W. Courteille, “Cooperative scattering and radiation pressure force in dense atomic clouds,” Phys. Rev. A 84, 013821 (2011).
    [CrossRef]
  20. T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
    [CrossRef]
  21. H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
    [CrossRef]
  22. T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
    [CrossRef]
  23. I. M. Sokolov, M. D. Kupriyanova, D. V. Kupriyanov, and M. D. Havey, “Light scattering from a dense and ultracold atomic gas,” Phys. Rev. A 79, 053405 (2009).
    [CrossRef]
  24. T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169, 191–194 (1992).
    [CrossRef]
  25. R. Bachelard, P. W. Courteille, R. Kaiser, and N. Piovella, “Resonances in Mie scattering by an inhomogeneous atomic cloud,” EPL 97, 14004 (2012).
  26. R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems ot two-level atoms,” Phys. Rep. 7, 101–179 (1973).
    [CrossRef]
  27. J. T. Manassah, “Cooperative radiation from atoms in different geometries: decay rate and frequency shift,” Adv. Opt. Photon. 4, 108–156 (2012).
  28. V. N. Fadeeva, Computational Methods of Linear Algebra (Dover, 1959).
  29. S. B. Singham and C. F. Bohren, “Light scattering by an arbitrary particle: the scattering-order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
    [CrossRef]
  30. A. Goetschy and S. E. Skipetrov, “Non-Hermitian Euclidean random matrix theory,” Phys. Rev. E 84, 011150 (2011).
    [CrossRef]
  31. E. Akkermans, A. Gero, and R. Kaiser, “Photon localization and dicke superradiance in atomic gases,” Phys. Rev. Lett. 101, 103602 (2008).
    [CrossRef]
  32. S. E. Skipetrov and A. Goetschy, “Eigenvalue distributions of large euclidean random matrices for waves in random media,” J. Phys. A 44, 065102 (2011).
    [CrossRef]
  33. M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of n atoms: timing is everything,” Phys. Rev. Lett. 96, 010501 (2006).
    [CrossRef]
  34. A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Dynamical evolution of correlated spontaneous emission of a single photon from a uniformly excited cloud of n atoms,” Phys. Rev. Lett. 100, 160504 (2008).
    [CrossRef]
  35. T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
    [CrossRef]
  36. B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Multiple-scattering effects for the propagation of light in 3D slabs,” J. Phys. Condens. Matter 2, 7653–7677 (1990).
    [CrossRef]

2014 (1)

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

2012 (3)

R. Bachelard, P. W. Courteille, R. Kaiser, and N. Piovella, “Resonances in Mie scattering by an inhomogeneous atomic cloud,” EPL 97, 14004 (2012).

T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
[CrossRef]

J. T. Manassah, “Cooperative radiation from atoms in different geometries: decay rate and frequency shift,” Adv. Opt. Photon. 4, 108–156 (2012).

2011 (4)

A. Goetschy and S. E. Skipetrov, “Non-Hermitian Euclidean random matrix theory,” Phys. Rev. E 84, 011150 (2011).
[CrossRef]

S. E. Skipetrov and A. Goetschy, “Eigenvalue distributions of large euclidean random matrices for waves in random media,” J. Phys. A 44, 065102 (2011).
[CrossRef]

T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, “Atom and photon measurement in cooperative scattering by cold atoms,” J. Mod. Opt. 58, 1942–1950 (2011).
[CrossRef]

R. Bachelard, N. Piovella, and P. W. Courteille, “Cooperative scattering and radiation pressure force in dense atomic clouds,” Phys. Rev. A 84, 013821 (2011).
[CrossRef]

2010 (4)

T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
[CrossRef]

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Cooperative spontaneous emission of n atoms: many-body eigenstates, the effect of virtual lamb shift processes, and analogy with radiation of n classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

2009 (1)

I. M. Sokolov, M. D. Kupriyanova, D. V. Kupriyanov, and M. D. Havey, “Light scattering from a dense and ultracold atomic gas,” Phys. Rev. A 79, 053405 (2009).
[CrossRef]

2008 (2)

E. Akkermans, A. Gero, and R. Kaiser, “Photon localization and dicke superradiance in atomic gases,” Phys. Rev. Lett. 101, 103602 (2008).
[CrossRef]

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Dynamical evolution of correlated spontaneous emission of a single photon from a uniformly excited cloud of n atoms,” Phys. Rev. Lett. 100, 160504 (2008).
[CrossRef]

2006 (1)

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of n atoms: timing is everything,” Phys. Rev. Lett. 96, 010501 (2006).
[CrossRef]

2003 (1)

G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by an inhomogeneous cloud of cold atoms,” Phys. Rev. A 67, 033814 (2003).
[CrossRef]

2002 (1)

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

1999 (2)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
[CrossRef]

1995 (1)

O. Morice, Y. Castin, and J. Dalibard, “Refractive index of a dilute bose gas,” Phys. Rev. A 51, 3896–3901 (1995).
[CrossRef]

1992 (1)

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169, 191–194 (1992).
[CrossRef]

1991 (1)

A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 79, 1359–1366 (1991).
[CrossRef]

1990 (2)

H. C. van de Hulst and R. Stark, “Accurate eigenvalues and exact extrapolation lengths in radiative transfer,” Astron. Astrophys. 235, 511–520 (1990).

B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Multiple-scattering effects for the propagation of light in 3D slabs,” J. Phys. Condens. Matter 2, 7653–7677 (1990).
[CrossRef]

1988 (2)

S. B. Singham and C. F. Bohren, “Light scattering by an arbitrary particle: the scattering-order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

1985 (2)

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef]

M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef]

1973 (2)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption by nonsphereical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems ot two-level atoms,” Phys. Rep. 7, 101–179 (1973).
[CrossRef]

Akkermans, E.

E. Akkermans, A. Gero, and R. Kaiser, “Photon localization and dicke superradiance in atomic gases,” Phys. Rev. Lett. 101, 103602 (2008).
[CrossRef]

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons, 1st ed. (Cambridge University, 2007).

Albada, M. P. V.

M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef]

Bachelard, R.

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

R. Bachelard, P. W. Courteille, R. Kaiser, and N. Piovella, “Resonances in Mie scattering by an inhomogeneous atomic cloud,” EPL 97, 14004 (2012).

R. Bachelard, N. Piovella, and P. W. Courteille, “Cooperative scattering and radiation pressure force in dense atomic clouds,” Phys. Rev. A 84, 013821 (2011).
[CrossRef]

Bellando, L.

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

Bender, H.

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

Bernard, J. C.

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

Bernard, J.-C.

G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
[CrossRef]

Bidel, Y.

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

Bienaimé, T.

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, “Atom and photon measurement in cooperative scattering by cold atoms,” J. Mod. Opt. 58, 1942–1950 (2011).
[CrossRef]

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
[CrossRef]

Bigerni, D.

T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, “Atom and photon measurement in cooperative scattering by cold atoms,” J. Mod. Opt. 58, 1942–1950 (2011).
[CrossRef]

Bohren, C. F.

Bux, S.

T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
[CrossRef]

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

Castin, Y.

O. Morice, Y. Castin, and J. Dalibard, “Refractive index of a dilute bose gas,” Phys. Rev. A 51, 3896–3901 (1995).
[CrossRef]

Chabé, J.

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

Chang, J.-T.

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Cooperative spontaneous emission of n atoms: many-body eigenstates, the effect of virtual lamb shift processes, and analogy with radiation of n classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Dynamical evolution of correlated spontaneous emission of a single photon from a uniformly excited cloud of n atoms,” Phys. Rev. Lett. 100, 160504 (2008).
[CrossRef]

Cherroret, N.

T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
[CrossRef]

Courteille, P. W.

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

R. Bachelard, P. W. Courteille, R. Kaiser, and N. Piovella, “Resonances in Mie scattering by an inhomogeneous atomic cloud,” EPL 97, 14004 (2012).

R. Bachelard, N. Piovella, and P. W. Courteille, “Cooperative scattering and radiation pressure force in dense atomic clouds,” Phys. Rev. A 84, 013821 (2011).
[CrossRef]

T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
[CrossRef]

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

Dalibard, J.

O. Morice, Y. Castin, and J. Dalibard, “Refractive index of a dilute bose gas,” Phys. Rev. A 51, 3896–3901 (1995).
[CrossRef]

de Tomasi, F.

G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
[CrossRef]

Delande, D.

G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by an inhomogeneous cloud of cold atoms,” Phys. Rev. A 67, 033814 (2003).
[CrossRef]

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

Fadeeva, V. N.

V. N. Fadeeva, Computational Methods of Linear Algebra (Dover, 1959).

Friedberg, R.

R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems ot two-level atoms,” Phys. Rep. 7, 101–179 (1973).
[CrossRef]

Fry, E. S.

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of n atoms: timing is everything,” Phys. Rev. Lett. 96, 010501 (2006).
[CrossRef]

Gero, A.

E. Akkermans, A. Gero, and R. Kaiser, “Photon localization and dicke superradiance in atomic gases,” Phys. Rev. Lett. 101, 103602 (2008).
[CrossRef]

Goetschy, A.

A. Goetschy and S. E. Skipetrov, “Non-Hermitian Euclidean random matrix theory,” Phys. Rev. E 84, 011150 (2011).
[CrossRef]

S. E. Skipetrov and A. Goetschy, “Eigenvalue distributions of large euclidean random matrices for waves in random media,” J. Phys. A 44, 065102 (2011).
[CrossRef]

Grémaud, B.

T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
[CrossRef]

Hartmann, S.

R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems ot two-level atoms,” Phys. Rep. 7, 101–179 (1973).
[CrossRef]

Havey, M. D.

I. M. Sokolov, M. D. Kupriyanova, D. V. Kupriyanov, and M. D. Havey, “Light scattering from a dense and ultracold atomic gas,” Phys. Rev. A 79, 053405 (2009).
[CrossRef]

Ishimaru, A.

A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 79, 1359–1366 (1991).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media and Rough Surfaces, Vol. 2 (Academic, 1978).

Kaiser, R.

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

R. Bachelard, P. W. Courteille, R. Kaiser, and N. Piovella, “Resonances in Mie scattering by an inhomogeneous atomic cloud,” EPL 97, 14004 (2012).

T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, “Atom and photon measurement in cooperative scattering by cold atoms,” J. Mod. Opt. 58, 1942–1950 (2011).
[CrossRef]

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
[CrossRef]

E. Akkermans, A. Gero, and R. Kaiser, “Photon localization and dicke superradiance in atomic gases,” Phys. Rev. Lett. 101, 103602 (2008).
[CrossRef]

G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by an inhomogeneous cloud of cold atoms,” Phys. Rev. A 67, 033814 (2003).
[CrossRef]

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
[CrossRef]

Karpiuk, T.

T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
[CrossRef]

Klappauf, B.

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

Kupriyanov, D. V.

I. M. Sokolov, M. D. Kupriyanova, D. V. Kupriyanov, and M. D. Havey, “Light scattering from a dense and ultracold atomic gas,” Phys. Rev. A 79, 053405 (2009).
[CrossRef]

Kupriyanova, M. D.

I. M. Sokolov, M. D. Kupriyanova, D. V. Kupriyanov, and M. D. Havey, “Light scattering from a dense and ultracold atomic gas,” Phys. Rev. A 79, 053405 (2009).
[CrossRef]

Labeyrie, G.

G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by an inhomogeneous cloud of cold atoms,” Phys. Rev. A 67, 033814 (2003).
[CrossRef]

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Lagendijk, A.

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169, 191–194 (1992).
[CrossRef]

B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Multiple-scattering effects for the propagation of light in 3D slabs,” J. Phys. Condens. Matter 2, 7653–7677 (1990).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).

M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef]

Lauber, K.

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

Lee, K. L.

T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
[CrossRef]

Lucioni, E.

T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
[CrossRef]

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

Manassah, J.

R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems ot two-level atoms,” Phys. Rep. 7, 101–179 (1973).
[CrossRef]

Manassah, J. T.

Maret, G.

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef]

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

Miniatura, C.

T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
[CrossRef]

G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by an inhomogeneous cloud of cold atoms,” Phys. Rev. A 67, 033814 (2003).
[CrossRef]

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons, 1st ed. (Cambridge University, 2007).

Morice, O.

O. Morice, Y. Castin, and J. Dalibard, “Refractive index of a dilute bose gas,” Phys. Rev. A 51, 3896–3901 (1995).
[CrossRef]

Müller, C. A.

T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
[CrossRef]

G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by an inhomogeneous cloud of cold atoms,” Phys. Rev. A 67, 033814 (2003).
[CrossRef]

G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
[CrossRef]

Nieuwenhuizen, T. M.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169, 191–194 (1992).
[CrossRef]

Ooi, C. H. R.

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of n atoms: timing is everything,” Phys. Rev. Lett. 96, 010501 (2006).
[CrossRef]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption by nonsphereical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Petruzzo, M.

T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, “Atom and photon measurement in cooperative scattering by cold atoms,” J. Mod. Opt. 58, 1942–1950 (2011).
[CrossRef]

Piovella, N.

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

R. Bachelard, P. W. Courteille, R. Kaiser, and N. Piovella, “Resonances in Mie scattering by an inhomogeneous atomic cloud,” EPL 97, 14004 (2012).

T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, “Atom and photon measurement in cooperative scattering by cold atoms,” J. Mod. Opt. 58, 1942–1950 (2011).
[CrossRef]

R. Bachelard, N. Piovella, and P. W. Courteille, “Cooperative scattering and radiation pressure force in dense atomic clouds,” Phys. Rev. A 84, 013821 (2011).
[CrossRef]

T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
[CrossRef]

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption by nonsphereical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Rouabah, M. T.

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

Scully, M. O.

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Cooperative spontaneous emission of n atoms: many-body eigenstates, the effect of virtual lamb shift processes, and analogy with radiation of n classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Dynamical evolution of correlated spontaneous emission of a single photon from a uniformly excited cloud of n atoms,” Phys. Rev. Lett. 100, 160504 (2008).
[CrossRef]

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of n atoms: timing is everything,” Phys. Rev. Lett. 96, 010501 (2006).
[CrossRef]

Singham, S. B.

Skipetrov, S. E.

S. E. Skipetrov and A. Goetschy, “Eigenvalue distributions of large euclidean random matrices for waves in random media,” J. Phys. A 44, 065102 (2011).
[CrossRef]

A. Goetschy and S. E. Skipetrov, “Non-Hermitian Euclidean random matrix theory,” Phys. Rev. E 84, 011150 (2011).
[CrossRef]

Slama, S.

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

Sokolov, I. M.

I. M. Sokolov, M. D. Kupriyanova, D. V. Kupriyanov, and M. D. Havey, “Light scattering from a dense and ultracold atomic gas,” Phys. Rev. A 79, 053405 (2009).
[CrossRef]

Stark, R.

H. C. van de Hulst and R. Stark, “Accurate eigenvalues and exact extrapolation lengths in radiative transfer,” Astron. Astrophys. 235, 511–520 (1990).

Stehle, C.

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

Svidzinsky, A. A.

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Cooperative spontaneous emission of n atoms: many-body eigenstates, the effect of virtual lamb shift processes, and analogy with radiation of n classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Dynamical evolution of correlated spontaneous emission of a single photon from a uniformly excited cloud of n atoms,” Phys. Rev. Lett. 100, 160504 (2008).
[CrossRef]

Tip, A.

B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Multiple-scattering effects for the propagation of light in 3D slabs,” J. Phys. Condens. Matter 2, 7653–7677 (1990).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

van Albada, M. P.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).

van de Hulst, H. C.

H. C. van de Hulst and R. Stark, “Accurate eigenvalues and exact extrapolation lengths in radiative transfer,” Astron. Astrophys. 235, 511–520 (1990).

van der Mark, M. B.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).

van Rossum, M. C. W.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

van Tiggelen, B. A.

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169, 191–194 (1992).
[CrossRef]

B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Multiple-scattering effects for the propagation of light in 3D slabs,” J. Phys. Condens. Matter 2, 7653–7677 (1990).
[CrossRef]

Wilkowski, D.

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

Wódkiewicz, K.

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of n atoms: timing is everything,” Phys. Rev. Lett. 96, 010501 (2006).
[CrossRef]

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

Wolf, P.-E.

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef]

Zimmermann, C.

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

Adv. Opt. Photon. (1)

Astron. Astrophys. (1)

H. C. van de Hulst and R. Stark, “Accurate eigenvalues and exact extrapolation lengths in radiative transfer,” Astron. Astrophys. 235, 511–520 (1990).

Astrophys. J. (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption by nonsphereical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

EPL (1)

R. Bachelard, P. W. Courteille, R. Kaiser, and N. Piovella, “Resonances in Mie scattering by an inhomogeneous atomic cloud,” EPL 97, 14004 (2012).

Eur. Phys. J. D (1)

P. W. Courteille, S. Bux, E. Lucioni, K. Lauber, T. Bienaimé, R. Kaiser, and N. Piovella, “Modification of radiation pressure due to cooperative scattering of light,” Eur. Phys. J. D 58, 69–73 (2010).
[CrossRef]

J. Mod. Opt. (2)

T. Bienaimé, M. Petruzzo, D. Bigerni, N. Piovella, and R. Kaiser, “Atom and photon measurement in cooperative scattering by cold atoms,” J. Mod. Opt. 58, 1942–1950 (2011).
[CrossRef]

T. Bienaimé, R. Bachelard, J. Chabé, M. T. Rouabah, L. Bellando, P. W. Courteille, N. Piovella, and R. Kaiser, “Interplay between radiation pressure force and scattered light intensity in the cooperative scattering by cold atoms,” J. Mod. Opt. 61, 18–24 (2014).
[CrossRef]

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

J. Phys. A (1)

S. E. Skipetrov and A. Goetschy, “Eigenvalue distributions of large euclidean random matrices for waves in random media,” J. Phys. A 44, 065102 (2011).
[CrossRef]

J. Phys. Condens. Matter (1)

B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Multiple-scattering effects for the propagation of light in 3D slabs,” J. Phys. Condens. Matter 2, 7653–7677 (1990).
[CrossRef]

Phys. Lett. A (1)

T. M. Nieuwenhuizen, A. Lagendijk, and B. A. van Tiggelen, “Resonant point scatterers in multiple scattering of classical waves,” Phys. Lett. A 169, 191–194 (1992).
[CrossRef]

Phys. Rep. (1)

R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems ot two-level atoms,” Phys. Rep. 7, 101–179 (1973).
[CrossRef]

Phys. Rev. A (6)

I. M. Sokolov, M. D. Kupriyanova, D. V. Kupriyanov, and M. D. Havey, “Light scattering from a dense and ultracold atomic gas,” Phys. Rev. A 79, 053405 (2009).
[CrossRef]

H. Bender, C. Stehle, S. Slama, R. Kaiser, N. Piovella, C. Zimmermann, and P. W. Courteille, “Observation of cooperative mie scattering from an ultracold atomic cloud,” Phys. Rev. A 82, 011404 (2010).
[CrossRef]

R. Bachelard, N. Piovella, and P. W. Courteille, “Cooperative scattering and radiation pressure force in dense atomic clouds,” Phys. Rev. A 84, 013821 (2011).
[CrossRef]

G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by an inhomogeneous cloud of cold atoms,” Phys. Rev. A 67, 033814 (2003).
[CrossRef]

O. Morice, Y. Castin, and J. Dalibard, “Refractive index of a dilute bose gas,” Phys. Rev. A 51, 3896–3901 (1995).
[CrossRef]

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Cooperative spontaneous emission of n atoms: many-body eigenstates, the effect of virtual lamb shift processes, and analogy with radiation of n classical oscillators,” Phys. Rev. A 81, 053821 (2010).
[CrossRef]

Phys. Rev. B (1)

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, “Light scattering in strongly scattering media: multiple scattering and weak localization,” Phys. Rev. B 37, 3575–3592 (1988).

Phys. Rev. E (1)

A. Goetschy and S. E. Skipetrov, “Non-Hermitian Euclidean random matrix theory,” Phys. Rev. E 84, 011150 (2011).
[CrossRef]

Phys. Rev. Lett. (10)

E. Akkermans, A. Gero, and R. Kaiser, “Photon localization and dicke superradiance in atomic gases,” Phys. Rev. Lett. 101, 103602 (2008).
[CrossRef]

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of n atoms: timing is everything,” Phys. Rev. Lett. 96, 010501 (2006).
[CrossRef]

A. A. Svidzinsky, J.-T. Chang, and M. O. Scully, “Dynamical evolution of correlated spontaneous emission of a single photon from a uniformly excited cloud of n atoms,” Phys. Rev. Lett. 100, 160504 (2008).
[CrossRef]

T. Karpiuk, N. Cherroret, K. L. Lee, B. Grémaud, C. A. Müller, and C. Miniatura, “Coherent forward scattering peak induced by anderson localization,” Phys. Rev. Lett. 109, 190601 (2012).
[CrossRef]

G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, “Coherent backscattering of light by cold atoms,” Phys. Rev. Lett. 83, 5266–5269 (1999).
[CrossRef]

Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, C. Miniatura, D. Wilkowski, and R. Kaiser, “Coherent light transport in a cold strontium cloud,” Phys. Rev. Lett. 88, 203902 (2002).
[CrossRef]

T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, and R. Kaiser, “Observation of a cooperative radiation force in the presence of disorder,” Phys. Rev. Lett. 104, 183602 (2010).
[CrossRef]

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[CrossRef]

P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef]

M. P. V. Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef]

Proc. IEEE (1)

A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 79, 1359–1366 (1991).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[CrossRef]

Other (4)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons, 1st ed. (Cambridge University, 2007).

A. Ishimaru, Wave Propagation and Scattering in Random Media and Rough Surfaces, Vol. 2 (Academic, 1978).

V. N. Fadeeva, Computational Methods of Linear Algebra (Dover, 1959).

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Figures (6)

Fig. 1.
Fig. 1.

Profile of the radiation field inside a Gaussian cloud, E(n)(r)=κ(δ)jG(rrj)E¯(n1)(rj), in the y=0 plane for different orders n (from left to right) and two different optical thicknesses (top and bottom). For small optical thickness (b(δ)=0.061, top line), the field decreases as the scattering order n increases, and the series (10) converges. For larger optical thickness (b(δ)=5, bottom line), the presence of eigenvalues of modulus larger than unity makes the multiple-scattering series diverge. In both cases, the presence of local fields much stronger than the incident one is due to the divergent field radiated in the vicinity of the atoms, which can be arbitrary close to the y=0 plane. Simulations realized for a Gaussian cloud of N=500 atoms with an on-resonant optical thickness b0=5 and standard deviation σR=17.32/k, where b0=3N/(kσR)2; top pictures correspond to δ=4.5 and b(δ)=0.061, bottom pictures to the resonant case δ=0 and b(δ)=5.

Fig. 2.
Fig. 2.

Far-field power P(n)=(ε0c/2)|E(n)|2dS radiated by the atomic cloud (measured on a spherical surface of radius r1/k) versus the scattering order n for the same parameters as in Fig. 1. For small optical thickness (b(δ)=0.061, red crosses), the scattered power decreases as the scattering order n increases, whereas for larger optical thickness (b(δ)=5, blue dots), the power diverges. P(n) is in units of the independent-atom power NP1, where P1=(4πI0/k2)/(1+4δ2) and I0 is the incident intensity.

Fig. 3.
Fig. 3.

Intensity diagram, I(n)=(ε0c/2)|j=1nE(j)|2, versus the polar angle θ, for n=1, 2, 3, derived from Eq. (13). The intensity scattered by the dipoles, Ib=(ε0c/2)|Eb|2, is derived from Eq. (4) and contains all the scattering orders. The modulation of the background is due to the vectorial nature of the light (linearly polarized light): the single-atom intensity I1(1+cos2θ) is plotted as a plain green line, where I1=I0/[k2r2(1+4δ2)]. The inset shows a zoom of the radiation (linear scale). Simulations have been realized for a Gaussian cloud of N=1000 atoms with on-resonance optical thickness b0=1, detuning δ=0, and rms size σR54.8/k given by b0=3N/(kσR)2. The intensity is averaged over the azimuthal angle ϕ and is in units of NI1.

Fig. 4.
Fig. 4.

Radiation profile |E(n)(θ)|2 in the far-field limit for scattering orders n=1, 2. The single-scattering order E(1) exhibits only a forward contribution (peaks pointing to the right) and a homogeneous background. The double-scattering contribution E(2) shows both forward and backward (CBS) patterns (peaks pointing to the left) in addition to the background. The theoretical curves (thr) are derived from Eqs. (30), (31), and (32). Simulations are realized for a Gaussian spherical cloud consisting of N=400 atoms with resonant optical thickness b0=2N/(kσR)2=1, standard deviation σR28.3/k, and detuning δ=1, and averaged over 1000 realizations. The incoming field is unity E0=1, and the radius of observation is 3·104/k. Scale is logarithmic.

Fig. 5.
Fig. 5.

Far-field scattered intensity versus θ up to the first, second, and third scattering orders, i.e., I(n)=(ε0c/2)|j=1nE(j)|2, in units of NI1. The inset is a zoom of the backscattering region (linear scale). Simulations realized for a Gaussian spherical cloud of on-resonant optical thickness b0=1, N=200 particles, scaled size σ=20, and laser detuning δ=0.5. The intensity has been averaged over 1000 realizations.

Fig. 6.
Fig. 6.

Experimental and theoretical CBS enhancement E(θ) for a Gaussian sphere of normalized standard deviation σ=8098 and δ=0. The circles correspond to the experimental values reported in Fig. 2 of Ref. [13], while the plain curve reproduces Eq. (33). It must be noted that the only free parameter is a 3% adjustment of the background.

Equations (60)

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dbjαdt=(iΔγ/2)bjαi(d/)Einα(rj)(γ/2)αmjGα,α(rjrm)bmα,
Gα,α(r)=32eikrikr{[δα,αn^αn^α]+[δα,α3n^αn^α][i/(kr)1/(kr)2]}
Gα,α(r)=32[δα,α+1k22αα]G(r).
bj=1Δ+iγ/2[dEin(rj)iγ2mjG(rjrm)·bm].
Esca(r)=idk36πϵ0m=1NG(rrm)·bm.
Esca(rj)=Eself(rj)idk36πϵ0mjG(rjrm)·bm.
bj=d(Δ+iγ/2)[Ein(rj)+E¯(rj)],
E¯(rj)=κ(δ)mjG(rjrm)·[Ein(rm)+E¯(rm)],
E¯tot=(IG)1Ein,
E¯(n)(rj)=κ(δ)mjG(rjrm)·E¯(n1)(rm),
E¯(rj)=n=1E¯(n)(rj),
E¯tot(rj)=Ein(rj)+κ(δ)mjG(rjrm)Ein(rm)+κ2(δ)mjG(rjrm)lmG(rmrl)Ein(rl)+
Esca(r)=κ(δ)j=1NG(rrj)·[Ein(rj)+E¯(rj)].
Gα,α(rrj)32eikrikr[δα,αn^αn^α]eik·rj,
Escafar(r)=E1far(r)+Emsfar(r),
E1far(r)=κ(δ)E0eikrikr[n^×(n^×e^0)]NSN(kk0)
Emsfar(r)=κ(δ)eikrikrNn^×[n^×F(k)],
F(k)=1Nj=1NE¯(rj)eik·rj.
Esca(r)=κ(δ)j=1NG(|rrj|)[Ein(rj)+E¯(rj)].
E¯(rj)κ(δ)mjG(|rjrm|)Ein(rm),
Etot(r)=Ein(r)+κ(δ)j=1NG(|rrj|)Ein(rj)+κ2(δ)m=1NjmG(|rrm|)G(|rmrj|)Ein(rj).
Esca(r)=κ(δ)eikrikrE0j=1Nei(k0k)·rj+κ2(δ)eikrikrE0m=1NjmG(|rmrj|)ei(k0·rjk·rm),
TN(k,k0)=1NmjmG(|rjrm|)ei(k0·rjk·rm),
Isca(r)=I1N2|SN(kk0)+κ(δ)TN(k,k0)|2,
S=1Ndrρ(r)exp[i(kk0)·r],
|TN(k,k0)|pair21N2mjm1+cos[(k+k0)·(rjrm)]k2|rjrm|2.
T(k,k0)=1Ndr1ρ(r1)dr2ρ(r2)exp(ik|r1r2|)ik|r1r2|×ei(k0·r1k·r2).
Isca=I1N{1+N1+4δ2|TN|pair2+N|F|2},
|TN|pair2=1N2jmj1k2rjm2×{1+sin[2krjmcos(θ/2)]2krjmcos(θ/2)},
|TN(θ)|pair2=12σ2{1+π2erf[2σcos(θ/2)]2σcos(θ/2)}=E(θ)2σ2
S(θ)=exp[2σ2sin2(θ/2)],
T(θ)N4σ2exp[4σ2sin2(θ/4)].
Isca(r)=I1N{1+b(δ)4E(θ)+N|F(θ)|2},
F(θ)=e2σ2sin2(θ/2)(1+2iδ)b(δ)8e4σ2sin2(θ/4),
×E=iωB,
×B=iωμ0(ϵ0E+P),
·E=(1/ϵ0)·P.
××E=(ω2/c2)[E+(1/ϵ0)P],
(2+k2)E=k2ϵ0[P+1k2(·P)].
E(r)=ik34πϵ0drG(|rr|)[P(r)+1k2(·P(r))]
Eα(r)=ik34πϵ0βdrG(|rr|)×[δα,β+1k22xαxβ]Pβ(r).
Eα(r)=ik36πϵ0βdrGα,β(rr)Pβ(r).
E(r)=idk36πϵ0m=1NG(rrm)·bm.
(k+k0)·rjm=krjm[sinθsinθjmcos(ϕjmϕ)+(1+cosθ)cosθjm].
|TN|pair2=1N2j,mj1k2rjm2×{1+14π02πdϕjm0πdθjmsinθjmcos{krjm[sinθsinθjmcos(ϕjmϕ)+(1+cosθ)cosθjm]}}.
|TN|pair2=1N2jmj1krjm2{1+120πdθjmsinθjmcos[krjm(1+cosθ)cosθjm]J0[krjmsinθsinθjm]}.
0πdθsinθcos(acosθ)J0(bsinθ)=2sina2+b2a2+b2,
|TN|pair2=1N2jmj1krjm2{1+sinc[2krjmcos(θ/2)]},
I=dr1ρ(r1)dr2ρ(r2)f(|r1r2|).
I=dRdsρ(Rs/2)ρ(R+s/2)f(|s|).
I=2N2πσR60dRR2eR2/σR20dss2es2/4σR2f(s)=4N2π0dxx2ex2f(2σRx).
f(s)=1k2s2{1+sin[2kscos(θ/2)]2skcos(θ/2)},
I=N2σ2π0dxex2[1+sin(ax)ax]=N22σ2[1+πaerf(a/2)],
T(k,k0)=1ikNdRdsρ(R+s/2)ρ(Rs/2)×s1ei(k0k)·R+i(k0+k)·s/2+iks.
T(k,k0)=ρ02ikNdReR2/σR2+i(k0k)·R×dss1es2/4σR2+i(k0+k)·s/2+iks.
I1=ρ0dReR2/σR2+i(k0k)·R=N22eσ2(1cosθ)/2,
I2=2Ni2πσ2cos(θ/2)0dxex2/4+iσxsin[σcos(θ/2)x].
0dxex2/4+iaxsin(bx)=π2{e4ab[erfi(ab)i]+erfi(a+b)i}e(a+b)2
T(θ)=Ne2σ2(1+cosθ/2)4iσ2cosθ/2{erfi[σ(1+cosθ/2)]e4σ2cosθ/2erfi[σ(1cosθ/2)]+i(e4σ2cosθ/21)}.
T(θ)N4σ2e4σ2sin2(θ/4)b08e4σ2sin2(θ/4).

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