A first-principles theory (V. G. Bordo, JOSA B 39, 2407 (2022)) showed that an optically thin layer of active centers on a metal surface can lase either conventionally or without population inversion depending on layer thickness, because the surface-reflected field provides a sign-controllable feedback loop. The theory is linearized and stops at threshold. This problem asks for its completion under one objective: determine whether the predicted laser, in both branches, reaches a physically realizable steady state. Concretely: derive the nonlinear saturated steady state (amplitude, depleted inversion, output intensity, frequency pulling, linewidth) and its stability — in particular whether the inversionless branch, where saturation drives the inversion toward zero from below, sustains a stable steady state; establish the theory's domain of validity near grazing propagation, where the feedback eigenvalues diverge and the computed thresholds are lowest; and determine the achievable gain parameter for Er-doped glass on gold at non-cryogenic temperatures.

Background

Nanoscale coherent light sources (nanolasers, including plasmonic 'spasers' that exploit the subwavelength-confined field of surface plasmon polaritons — electromagnetic waves bound to a metal–dielectric interface) are pursued for on-chip optics, sensing, and spectroscopy. When the gain medium is confined within a wavelength-scale distance of a metal surface, an effect known from the classical problem of a dipole antenna above a conducting plane reappears: the field radiated by each emitting dipole is reflected by the surface and acts back on the dipole ensemble, so the local field acting on each active center is not the bulk Lorentz field (the standard 4π/(3ϵ2)4\pi/(3\epsilon_2) local-field correction of a dense medium of permittivity ϵ2\epsilon_2) but includes a reflected contribution expressible through the field-susceptibility tensor, the kernel that maps a point dipole source at one position to the field it produces at another in the presence of the interface.

In the source theory (Bordo, JOSA B 39, 2407 (2022)) a layer of active centers of number density NN and thickness hh occupies a dielectric on a metal described by a Drude dielectric function — the standard free-electron model of the metal's conduction electrons. The slowly varying polarization amplitude and the population inversion DD (the population difference between the excited and ground states of the active centers, times NN) obey the optical Bloch equations — the standard coupled equations of semiclassical laser theory — with transition dipole moment μ\mu, transverse (phase) relaxation rate γ⊥\gamma_\perp, transition frequency ω0\omega_0, and equilibrium inversion D0D_0 set by pumping through other levels. Linearizing in a weak seed field and decomposing into spatial Fourier components parallel to the interface, k∥k_\parallel, the poles of the polarization response give the growth rate

σj(k∥)=−γ⊥+2π3ℏμ2D0 Re Fj(k∥),\sigma_j(k_\parallel)=-\gamma_\perp+\frac{2\pi}{3\hbar}\mu^{2}D_0\,\mathrm{Re}\,\mathcal{F}_j(k_\parallel),

where Fj=S(k∥)ϕ~j(k∥)\mathcal{F}_j=S(k_\parallel)\tilde{\phi}_j(k_\parallel) encodes the surface feedback loop: SS is an averaged phase factor acquired by the dipole field propagating to the surface and back, and ϕ~j(k∥)\tilde{\phi}_j(k_\parallel) (for ss or pp polarization, i.e. with the electric field transverse to, or lying in, the plane of incidence) is fixed by the Fresnel reflection coefficients of the metal. Lasing requires σj>0\sigma_j>0, i.e.

2π3ℏμ2D0 Re Fj(k∥)>γ⊥.\frac{2\pi}{3\hbar}\mu^{2}D_0\,\mathrm{Re}\,\mathcal{F}_j(k_\parallel)>\gamma_\perp.

Problem Statement

For the model system of the source paper — an optically thin layer (thickness hh) of active centers in a dielectric of permittivity ϵ2\epsilon_2 on a Drude metal, described by the optical Bloch equations coupled self-consistently to the local field (Lorentz field plus the reflected-field integral contribution) — complete the theory of laser operation beyond the linearized threshold, with the single overarching objective of deciding whether the thickness-controlled lasing/lasing-without-inversion switching predicted at threshold survives as a physically realizable steady state. The source leaves three linked components open, and a solution must establish all of them:

(1) Nonlinear steady state. Retaining the saturation term in the Bloch equation for the inversion DD, derive the saturated steady state of the system — steady polarization amplitude, depleted inversion, output intensity, generation frequency (frequency pulling, i.e. the shift of the oscillation frequency away from the transition frequency ω0\omega_0 of the active centers), and spectral linewidth — together with a stability analysis, for both the conventional branch (D0>0D_0>0, Re Fj>0\mathrm{Re}\,\mathcal{F}_j>0) and the inversionless branch (D0<0D_0<0, Re Fj<0\mathrm{Re}\,\mathcal{F}_j<0), in particular whether the inversionless steady state exists and remains stable under realistic pumping.

(2) Validity near grazing propagation. The linearized theory and its frequency-pulling formula require ∣2π3ℏμ2D0 Im Fj(k∥)∣≪ω0\left|\frac{2\pi}{3\hbar}\mu^{2}D_0\,\mathrm{Im}\,\mathcal{F}_j(k_\parallel)\right|\ll\omega_0 and break down near grazing propagation in the gain medium (k∥→(ω/c)ϵ2k_\parallel\to(\omega/c)\sqrt{\epsilon_2}), where the feedback eigenvalues ϕ~j(k∥)\tilde{\phi}_j(k_\parallel) diverge — precisely the regime toward which the computed thresholds decrease. Determine the domain of validity of the theory there, or provide an approximation or regularization that covers this regime.

(3) Realistic gain parameter. Compute the achievable dimensionless gain parameter η=2πμ2D03ℏγ⊥\eta=\frac{2\pi\mu^{2}D_0}{3\hbar\gamma_\perp} for Er³⁺-doped glass on gold at the 1532 nm transition at practical (non-cryogenic) temperatures under realistic pumping constraints, and thereby assess the experimental feasibility of accessing both lasing regimes in this single geometry.

References

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