Theory of the spatial structure of nonlinear lasing modes

A self-consistent integral equation is formulated and solved iteratively which determines the steady-state lasing modes of open multimode lasers. These modes are naturally decomposed in terms of frequency dependent biorthogonal modes of a linear wave equation and not in terms of resonances of the cold cavity. A one-dimensional cavity laser is analyzed and the lasing mode is found to have nontrivial spatial structure even in the single-mode limit. In the multimode regime spatial hole-burning and mode competition is treated exactly. The formalism generalizes to complex, chaotic, and random laser media.

Figure 1

(Color) Convergence and solution of the multimode lasing map for a 1D resonator with $n_0=1.5$, $k_a=19.89$, and ${\gamma }_{\perp }=4.0$ vs pump $D_0$. Three modes lase in this range. At threshold they correspond to CF states $m=8,9$, and 10 with threshold lasing frequencies $k_t^{\left(8\right)}=18.08$, $k_t^{\left(9\right)}=19.91$, and $k_t^{\left(10\right)}=21.76$, and noninteracting thresholds $D_0^{\left(9\right)}=1.204$, $D_0^{\left(10\right)}=1.445$, and $D_0^{\left(8\right)}=1.482$ (green dots). $k_t^{\left(9\right)}\approx k_a$ and $m=9$ has the lowest threshold. Due to mode competition, modes 2 and 3 do not lase until much higher values $(D_0=2.25$ and 2.53). Each mode is represented by an 11 component vector of CF states; we plot the sum of ${\mid a_m\mid }^2$ vs pump $D_0$. Below threshold the vectors flows to zero (blue dots). For $D_0⩾1.204$ the sum flows (red dots) to a non-zero value (black dashed line), and above $D_0=2.25$ and 2.53, two additional nonzero vector fixed points (modes) are found (convergence only shown for modes 1 and 2).

Figure 2

(Color) The spatial structure of three consecutive CF modes calculated at the external frequency $k=19.90$ and $n_0=1.5$. The CF frequencies are $k_m=19.90-0.536i$ (black), $21.99-0.466i$ (red), and $17.81-0.640i$ (blue). The upper inset compares the CF mode frequencies $(+)$ with the quasibound mode frequencies $\left(엯\right)$ in the range $m=[6,12]$. Lower inset: schematic of the edge-emitting laser cavity.

Figure 3

(Color) Nonlinear electric field intensity for a single-mode edge-emitting laser with $n_0=1.5$, $k_a=19.89$, ${\gamma }_{\perp }=0.5$, and $D_0=9$. The full field (red line) has an appreciably larger amplitude at the output $x=a$ than the “single-pole” approximation (blue) which neglects the sideband CF components. Inset: The ratio of the two largest CF sideband components to that of the central pole for $n_0=1.5$ $(◻\times )$, and $n_0=3$ $(◻,+)$ vs pump strength $D_0$.