Completing the Polarium model: cooperative reflection and transmission by a resonant atomic slab beyond the ideal coherent limit

The Polarium analysis of Glauber and Prasad [Phys. Rev. A 61, 063815 (2000)] gave a closed-form modal theory of the reflection and transmission of coherent radiation by an optically thick slab of two-level atoms treated as a continuous linear resonant medium, under a rapid-transit approximation, with a single dominant resonance, a frequency-independent background index, and infinite transverse extent. This record states the completion problem: determine how the collective radiative eigenvalues, the closed-form interaction kernel, and the resonant reflection/transmission structures are modified (i) when incoherent processes (dephasing, inhomogeneous broadening, additional relaxation) are included and outside the rapid-transit limit, (ii) for media with multiple nearby resonances or dispersive backgrounds, and (iii) for TM-polarized oblique incidence, where the eigenvalue relations are implicit, the modes biorthogonal, and the spectrum was never computed; (iv) design mode-resolved experiments measuring the even/odd superposition amplitudes the authors proposed; and (v) treat finite transverse size and edge effects.

Background

An ensemble of atoms sharing a resonant transition and coupled only through their common radiation field does not fluoresce as independent atoms: the field radiated by one atom drives the others, and the medium possesses collective decay modes whose radiative rates can greatly exceed the single-atom rate (superradiance) or fall far below it (subradiance). The effect was identified by Dicke for samples small compared with the wavelength; in spatially extended samples the collective modes form a rich spectrum of spatially structured excitations, each with its own complex eigenvalue combining a cooperative frequency shift and a decay rate. Glauber and Prasad gave this medium the name 'polarium' and analyzed its simplest geometry, a slab: the companion Paper I (Prasad and Glauber, Phys. Rev. A 61, 063814 (2000)) constructed the collective radiative decay modes of an initially excited slab, and Paper II (Glauber and Prasad, Phys. Rev. A 61, 063815 (2000)) solved the complementary driven problem — reflection and transmission of an incident coherent plane wave — as a sum over the same modes.

Several conventions of Paper II are needed to state what remains open. (1) The treatment is explicitly carried out in the absence of incoherent relaxation processes: no dephasing, no inhomogeneous broadening, and no relaxation channels other than cooperative radiation. (2) The rapid-transit approximation neglects the radiation phase accumulated across the slab during a cooperative decay time, of order $2\pi L/(c\tau_R)$ for slab thickness $L$ and characteristic decay time $\tau_R$. (3) All atomic transitions except one dominant resonance are lumped into a frequency-independent background index $\sigma$. (4) The slab is of infinite transverse extent, so the fields carry a fixed transverse wave-vector component $k_y = k\sin\theta$ and the analysis is translationally invariant in the slab plane. Within these assumptions the theory is complete: at normal incidence the reflection and transmission amplitudes reduce to those of a Fabry–Perot cavity (two parallel mirrors forming a standing-wave resonator) filled with a resonant medium; the mode sums $\tilde t + \tilde r$ and $\tilde t - \tilde r$ isolate the interleaved even- and odd-parity mode families; and at oblique incidence the modes become chevron modes, radiating symmetrically but obliquely. For TE polarization (electric field transverse to the plane of incidence) the eigenvalue relations are explicit, but for TM polarization (magnetic field transverse to that plane) the relations contain the eigenvalue $\lambda$ on both sides and the TM mode family is non-self-adjoint: its members are biorthogonal to a dual set of modes radiating at the angle $-\theta$, a structure familiar from the non-Hermitian mode theory of lasers. The paper notes that when $|1 - 2i/(\beta\lambda)|$ is large ($\beta = kL$ being the paper's optical-thickness parameter, the ratio of the slab thickness $L$ to the reduced wavelength $1/k$ of the incident light of wave number $k$) the TM eigenvalue solutions are qualitatively different from the TE ones, but it never computes the TM spectrum, obtaining TM reflection and transmission instead in closed Fresnel-type form.

The subsequent landscape defines the stakes of completion. A parallel eigenmode program (Friedberg and Manassah) developed the same one-dimensional kernel for initially excited samples and computed TE and TM spectra for an infinite cylinder but not for the oblique slab; atom-by-atom electrodynamics simulations and nanometer-vapor-cell experiments probe neighboring regimes (dense cold gases, subwavelength layers) without mode-resolved slab spectroscopy. What no later work supplies is the polarium scattering theory with the idealizations (1)–(4) removed, or a measurement resolving its individual collective modes.

Problem Statement

Complete the cooperative modal theory of reflection and transmission of coherent radiation by a resonant atomic slab — the Polarium analysis of Glauber and Prasad (2000) — beyond the idealizations under which its closed-form results were derived, and test it at the level of individual collective modes. Concretely, for a slab of two-level atoms treated as in the source analysis, determine:

(i) how the collective eigenvalues, the closed-form interaction kernel, and the resonant structures of the reflection and transmission amplitudes are modified when incoherent processes (dephasing, inhomogeneous broadening of the transition frequencies across the atoms, and other relaxation channels) are included, and when the rapid-transit approximation fails (the phase $2\pi L/(c\tau_R)$ accumulated across the slab of thickness $L$ within the cooperative decay time $\tau_R$ no longer being small);

(ii) the accuracy of the single-resonance model with frequency-independent background index $\sigma$ for media with multiple nearby resonances, and the impact of dispersive (frequency-dependent) backgrounds on the cooperative eigenvalues;

(iii) a full numerical or analytic characterization of the TM-polarized oblique-incidence eigenvalue spectrum — the implicit relations whose eigenvalue $\lambda$ appears on both sides, with biorthogonal (non-self-adjoint) modes — including the parameter regimes, such as large $|1-2i/(\beta\lambda)|$ (with $\beta = kL$ the optical thickness and $\lambda$ the collective eigenvalue), in which the TM spectrum departs qualitatively from the TE spectrum;

(iv) the feasibility and design of experiments that selectively excite and measure single chevron modes or mode-superposition amplitudes, in particular the even/odd reduced amplitudes $\tilde t \pm \tilde r$ whose measurement the authors proposed;

(v) the effect of finite transverse extent — edge effects and transverse mode quantization — on the radiation pattern and the modal overlap integrals.

These components share one objective: a realistic, mode-resolved theory of cooperative reflection and transmission by a resonant slab. The problem is stated at the source's own level of generality — no restriction on slab thickness, density, or incidence angle is intended.

References

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