Coherent theory of lasing in a multiple-scattering medium: emission coherence and the approach to the photon-localization threshold

In 1996 John and Pang developed a microscopic theory of lasing in the disordered 'paint-on' laser systems (rhodamine 640 dye molecules with TiO2 scatterers) that had recently shown laser-like emission without a cavity. Their model combined dye singlet–triplet rate equations with a nonlinear diffusion equation for the light intensity, and reproduced the observed threshold, linewidth collapse, bichromatic emission, and mean-free-path dependence for a transport mean free path $l^{*}$ much larger than the emission wavelength $\lambda_0$. The paper itself flagged what its intensity-diffusion framework could not deliver: the coherence properties of the emitted light (phase-coherent laser action versus amplified spontaneous emission) were 'not yet elucidated', and the predicted threshold lowering and intensity enhancement as $l^{*}\to\lambda_0$ near the photon-localization threshold was a suggestive extrapolation rather than a controlled result. In the subsequent literature, full-wave self-consistent lasing theories (steady-state ab initio laser theory, Maxwell–Bloch and Euclidean-matrix computations), a diffusion-level master-equation theory of the photon statistics and second-order coherence, and measurements of spatial coherence, photon statistics, and fluctuations now exist for diffusive and lower-dimensional localized systems, and lasing into localized modes has been demonstrated in layered and transversely localized media. What survives as open is the coherent core of the original question: a controlled three-dimensional theory of gain, interference, and localization coupling to a realistic molecular gain medium, a field-level, validated account of emission coherence (phase coherence versus amplified spontaneous emission) across the diffusive-to-localized crossover, and the experimental confirmation or correction of the strong-scattering-limit predictions, together with the microscopic singlet–triplet input parameters that such a theory must be validated against.

Background

A random laser is a laser in which the mirrors of a conventional cavity are replaced by multiple light scattering: photons are confined inside the gain medium by disorder instead of by cavity resonances. The archetype is the 'paint-on' or laser-paint system studied in the mid-1990s: rhodamine 640 (R640) dye molecules in methanol, providing gain, mixed with TiO2 nanoparticles that scatter light. Pumping such a suspension produced isotropic, laser-like emission with a sharp threshold and an abrupt collapse of the emission linewidth from tens of nanometers to a few nanometers. The possibility of laser action in a scattering medium with negative (inverted) absorption had been predicted as early as 1968 (Letokhov's diffusive amplification mechanism, in which multiple scattering lengthens the photon path and thus the amplification); independent diffusion-with-gain theories were developed in parallel with the John–Pang work (Wiersma and Lagendijk, 1996).

John and Pang (Phys. Rev. A 54, 3642, 1996) gave this phenomenon a microscopic theory. They solved the laser rate equations for a dye with excited singlet and triplet electronic manifolds — including absorption and stimulated-emission cross sections $\sigma_{a}^{(\alpha)}(\omega)$ and $\sigma_{e}^{(\alpha)}(\omega)$ for each manifold $\alpha$, nonradiative decay rates, and singlet–triplet intersystem crossing parameterized by an effective triplet-population fraction $\eta$ — and coupled the resulting nonlinear, intensity- and position-dependent gain to photon transport in the diffusion approximation. The diffusion approximation treats light propagation in a scattering medium as a random walk with diffusion constant $D=(1/3)\,c\,l^{*}$, where $l^{*}$ is the transport mean free path (the length over which the photon direction is randomized) and $c$ is the light speed in the medium; it is an intensity-level, incoherent description that discards all wave interference. With this model they accounted for nearly all experimentally reported properties of laser paint in the regime $l^{*}\gg\lambda_0$, and attributed the observed bichromatic emission (a secondary peak near 650 nm) to stimulated emission involving the triplet manifold.

Two limitations of that framework define the problem. First, coherence: a laser is normally distinguished from amplified spontaneous emission (ASE, the noise-amplified output of a gain medium without coherent feedback) by the phase coherence of its field, measurable through the first-order (electric-field) autocorrelation and the photon-counting statistics. An intensity-diffusion equation cannot say whether the emission is phase-coherent laser action or partially coherent ASE. Second, strong scattering: interference between multiple-scattering paths modifies the diffusion constant and, at the photon-localization threshold (the optical analog of Anderson localization of electrons, expected when $l^{*}$ becomes comparable to $\lambda_0$), light transport is suppressed and modes become spatially localized — the diffusion approximation, and any extrapolation of its predictions to $l^{*}\to\lambda_0$, then fails. John and Pang extrapolated their solutions to this limit and predicted a dramatic increase of the peak output intensity (a power law in $l^{*}/\lambda_0$) and a further lowering of the lasing threshold near the photon-localization threshold, while stating that this extrapolation was suggestive rather than rigorous.

Later work supplied the missing wave-level machinery. Finite-difference time-domain and related Maxwell–Bloch simulations solve the full electromagnetic field with gain in a fixed disorder realization. Steady-state ab initio laser theory (SALT; Türeci, Stone, and co-workers) finds stationary multimode lasing solutions of the semiclassical Maxwell–Bloch equations self-consistently, including nonlinear gain saturation and spatial hole burning (the local depletion of the gain where the lasing mode is intense), and has been applied to random lasers whose passive systems have no detectable resonances. On the coherence side, master-equation treatments of the photon statistics of diffusive random lasers (Florescu and John, 2004) derived the photon-number distribution and the second-order coherence of the emission, predicting a transition from chaotic to coherent emission above a pumping threshold that itself decreases with stronger scattering — a coherence answer, but one still built on a diffusion-level description with simplified, spatially uncorrelated isotropic scatterers, not on a field-level, localization-resolving theory. These theories resolve the wave-interference structure that the diffusion model omits; the open question is whether they, or a successor theory, can be carried into the regime of incipient localization in three dimensions with a realistic gain medium, and what they predict there for thresholds, output, and coherence.

Problem Statement

Construct and validate a theory of laser action in a multiple-scattering gain medium that goes beyond the intensity-diffusion treatment of John and Pang (Phys. Rev. A 54, 3642, 1996) and settles the questions that treatment explicitly left open. The single scientific objective is a self-consistent active-medium theory of the electromagnetic field — not only of the intensity — in a disordered amplifying medium, together with the experimental discrimination it calls for. Concretely, the problem comprises: (1) incorporating wave interference, and in the strong-scattering limit approaching the photon-localization threshold ($l^{*}\to\lambda_0$) localization itself, into a self-consistent theory of lasing, including the modification of the diffusion constant $D$ and of the escape probabilities (the probabilities with which photons reaching the sample boundary leave the sample rather than scatter back into it) in that regime; (2) within such a theory, computing the coherence properties of the emitted light — the electric-field autocorrelation and the photon statistics, going beyond the existing diffusion-level master-equation treatments of the photon statistics — so as to determine whether, and under what conditions (scattering strength, feedback character, pump intensity and geometry), the observed emission is genuinely phase-coherent laser action rather than amplified spontaneous emission with partial coherence; (3) determining rigorously whether the predicted lowering of the lasing threshold and the dramatic enhancement of peak output intensity as $l^{*}\to\lambda_0$ (extrapolated in the source as a power law in $l^{*}/\lambda_0$) survive a controlled treatment of gain and localization, and specifying the experimental test that would confirm or refute it; and (4) anchoring the theory in the microscopic input of the experimental dye/scatterer systems (frequency-dependent singlet and triplet absorption and emission cross sections, nonradiative rates, and the singlet–triplet intersystem-crossing dynamics, including the concentration- and pulse-duration dependence of the effective triplet population), including the suggested discriminating experiments such as triplet quenching, systematic coherence measurements, and varied pump and detection geometry. The theory of (1)–(3), together with the microscopic input and the discriminating experiments specified in (4), is the scientific content of the problem; those experiments are to be specified rather than performed. The theory should remain valid from the diffusive regime $l^{*}\gg\lambda_0$, where its intensity-level limit must reproduce the threshold, linewidth-collapse, spectral, and bichromatic-emission behavior already accounted for by the diffusion model, through the approach to the photon-localization threshold.

References

1. Sajeev John and Gendi Pang, 'Theory of lasing in a multiple-scattering medium', Phys. Rev. A 54, 3642 (1996), DOI 10.1103/PhysRevA.54.3642, https://doi.org/10.1103/PhysRevA.54.3642
2. Lucia Florescu and Sajeev John, 'Photon Statistics and Coherence in Light Emission from a Random Laser', Phys. Rev. Lett. 93, 013602 (2004), DOI 10.1103/PhysRevLett.93.013602, https://doi.org/10.1103/PhysRevLett.93.013602
3. Lucia Florescu and Sajeev John, 'Theory of photon statistics and optical coherence in a multiple-scattering random-laser medium', Phys. Rev. E 69, 046603 (2004), DOI 10.1103/PhysRevE.69.046603, https://doi.org/10.1103/PhysRevE.69.046603
4. Lucia Florescu and Sajeev John, 'Lasing in a random amplifying medium: Spatiotemporal characteristics and nonadiabatic atomic dynamics', Phys. Rev. E 70, 036607 (2004), DOI 10.1103/PhysRevE.70.036607, https://doi.org/10.1103/PhysRevE.70.036607
5. V. S. Letokhov, 'Generation of light by a scattering medium with negative resonance absorption', Sov. Phys. JETP 26, 835–840 (1968) (no DOI registered for the 1968 English translation)
6. Diederik S. Wiersma and Ad Lagendijk, 'Light diffusion with gain and random lasers', Phys. Rev. E 54, 4256 (1996), DOI 10.1103/PhysRevE.54.4256, https://doi.org/10.1103/PhysRevE.54.4256
7. R. Pierrat and R. Carminati, 'Threshold of random lasers in the incoherent transport regime', Phys. Rev. A 76, 023821 (2007), DOI 10.1103/PhysRevA.76.023821, https://doi.org/10.1103/PhysRevA.76.023821
8. Hakan E. Türeci, A. Douglas Stone, B. Collier, 'Self-consistent multimode lasing theory for complex or random lasing media', Phys. Rev. A 74, 043822 (2006), DOI 10.1103/PhysRevA.74.043822, https://doi.org/10.1103/PhysRevA.74.043822
9. Hakan E. Türeci, A. Douglas Stone, Li Ge, Stefan Rotter, Robert J. Tandy, 'Ab initio self-consistent laser theory and random lasers', Nonlinearity 22, C1 (2009), arXiv:0811.3542, DOI 10.1088/0951-7715/22/1/C01, https://arxiv.org/abs/0811.3542
10. J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, C. Vanneste, 'Modes of random lasers', Advances in Optics and Photonics 3, 88 (2011), arXiv:1001.4671, DOI 10.1364/aop.3.000088, https://doi.org/10.1364/aop.3.000088
11. A. Goetschy and S. E. Skipetrov, 'Euclidean matrix theory of random lasing in a cloud of cold atoms', EPL 96, 34005 (2011), arXiv:1104.2711, DOI 10.1209/0295-5075/96/34005, https://arxiv.org/abs/1104.2711
12. C. Conti and A. Fratalocchi, 'Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals', Nature Physics 4, 794–798 (2008), arXiv:0802.3775 (preprint posted under the title 'Dynamic light diffusion, Anderson localization and lasing in disordered inverted opals: 3D ab-initio Maxwell-Bloch computation'), DOI 10.1038/nphys1035, https://doi.org/10.1038/nphys1035
13. W. Guerin, Y. D. Chong, Q. Baudouin, M. Liertzer, S. Rotter, R. Kaiser, 'Diffusive to quasi-ballistic random laser: incoherent and coherent models', J. Opt. Soc. Am. B 33, 1888 (2016), arXiv:1606.03679, DOI 10.1364/JOSAB.33.001888, https://arxiv.org/abs/1606.03679
14. C. W. J. Beenakker, 'Photon statistics of a random laser', in J.-P. Fouque (ed.), 'Diffuse Waves in Complex Media', NATO Science Series C vol. 531 (Kluwer, Dordrecht, 1999), pp. 137–164, arXiv:quant-ph/9808066, DOI 10.1007/978-94-011-4572-5_6, https://arxiv.org/abs/quant-ph/9808066
15. M. Patra, 'Theory for photon statistics of random lasers', Phys. Rev. A 65, 043809 (2002), arXiv:quant-ph/0105069, DOI 10.1103/PhysRevA.65.043809, https://arxiv.org/abs/quant-ph/0105069
16. Brandon Redding, Michael A. Choma, Hui Cao, 'Spatial coherence of random laser emission', Optics Letters 36, 3404 (2011), arXiv:1107.5586, DOI 10.1364/OL.36.003404, https://arxiv.org/abs/1107.5586
17. Jason W. Merrill, Hui Cao, Eric R. Dufresne, 'Fluctuations and Correlations of Emission from Random Lasers', Phys. Rev. A 93, 021801 (2016), arXiv:1508.07339, DOI 10.1103/PhysRevA.93.021801, https://arxiv.org/abs/1508.07339
18. Anderson S. L. Gomes, Ernesto P. Raposo, André L. Moura, Serge I. Fewo, Pablo I. R. Pincheira, Vladimir Jerez, Lauro J. Q. Maia, Cid B. de Araújo, 'Observation of Lévy distribution and replica symmetry breaking in random lasers from a single set of measurements', Scientific Reports 6, 27987 (2016), arXiv:1509.00276 (preprint posted under the title 'Random lasers, Lévy statistics and spin glasses: Synergy between photonics and complex systems'), DOI 10.1038/srep27987, https://doi.org/10.1038/srep27987
19. R. Frank, A. Lubatsch, J. Kroha, 'Light transport and localization in diffusive random lasers', Journal of Optics A: Pure and Applied Optics 11, 114012 (2009), DOI 10.1088/1464-4258/11/11/114012, https://doi.org/10.1088/1464-4258/11/11/114012
20. Regine Frank, Andreas Lubatsch, Johann Kroha, 'Coherent feedback from dissipation: the lasing mode volume of random lasers', arXiv:1111.1906 (2011), https://arxiv.org/abs/1111.1906
21. Andreas Lubatsch and Regine Frank, 'A Self-Consistent Quantum Field Theory for Random Lasing', Applied Sciences 9, 2477 (2019), arXiv:1509.01873, DOI 10.3390/app9122477, https://arxiv.org/abs/1509.01873
22. Valery Milner and Azriel Z. Genack, 'Photon Localization Laser: Low-Threshold Lasing in a Random Amplifying Layered Medium via Wave Localization', Phys. Rev. Lett. 94, 073901 (2005), arXiv:physics/0409044, DOI 10.1103/PhysRevLett.94.073901, https://arxiv.org/abs/physics/0409044
23. Behnam Abaie, Esmaeil Mobini, Salman Karbasi, Thomas Hawkins, John Ballato, Arash Mafi, 'Random lasing in an Anderson localizing optical fiber', arXiv:1612.03835 (2016), https://arxiv.org/abs/1612.03835
24. Bhupesh Kumar, Ran Homri, Priyanka Priyanka, Santosh Maurya, Melanie Lebental, Patrick Sebbah, 'Localized modes revealed in Random Lasers', arXiv:2106.12172 (2021), https://arxiv.org/abs/2106.12172
25. Q. Baudouin, N. Mercadier, V. Guarrera, W. Guerin, R. Kaiser, 'A cold-atom random laser', Nature Physics 9, 357 (2013), arXiv:1301.0522, DOI 10.1038/nphys2614, https://arxiv.org/abs/1301.0522
26. Messaoud Bahoura and M. A. Noginov, 'Determination of the transport mean free path in a solid-state random laser', J. Opt. Soc. Am. B 20, 2389 (2003), DOI 10.1364/josab.20.002389, https://doi.org/10.1364/JOSAB.20.002389
27. Hai Liu, Xiangjun Zhang, Shengyu Jing, Zihui Ren, 'Dispersive effect on dual-color laser action from one-dimensional scattering gain media', Journal of Optics 14, 055203 (2012), DOI 10.1088/2040-8978/14/5/055203, https://doi.org/10.1088/2040-8978/14/5/055203