Numerical study of amplified spontaneous emission and lasing in random media

We simulate the transition from amplified spontaneous emission (ASE) to lasing in random systems with varying degrees of mode overlap. This is accomplished by solving the stochastic Maxwell-Bloch equations with the finite-difference time-domain method. Below lasing threshold, the continuous emission spectra are narrowed by frequency-dependent amplification. Our simulations reproduce the stochastic emission spikes in the spectra. Well-defined peaks, corresponding to the system resonances, emerge at higher pumping and are narrowed by stimulated emission before lasing takes place. Noise tends to distribute pump energy over many modes, resulting in multimode operation. Well above the lasing threshold, the effects of noise lessen and results become similar to those without noise. By comparing systems of different scattering strength, we find that weaker scattering extends the transition region from ASE to lasing, where the effects of noise are most significant.

Figure 1

(a) Transmission spectra $\mathcal{T}(k)$ of passive random systems with $g=0.5$ (solid line) and $g=1.0$ (dashed line). The gain curve (dotted line) for the MB simulations is also shown. (b) Intensity distribution of a representative quasimode in a random system with $g=0.5$ (solid line) and $g=1.0$ (dashed line).

Figure 2

Frequencies $k$ and decay rates $k_i$ of quasimodes in two random systems with $g=1.0$ ($+$) and $g=0.5$ ($\times$). The vertical dashed gray line marks the atomic transition frequency $k_a$ in the MB simulations. The two strongest lasing modes found in the following section are bounded by (circles) for $g=1.0$ and (squares) for $g=0.5$. The circled modes have smaller decay rates than the modes nearest $k_a$.

Figure 3

Average steady-state emission intensity $I_o$ (black $\times$) vs pumping rate $P_r$ for a random system with $g=1.0$. Both $I_o$ and $P_r$ are plotted on log${}_{10}$ scales to clearly show the regions of spontaneous emission (SE), amplified spontaneous emission (ASE), and laser emission (LE). A red solid line and a blue dotted line are linear fits to the intensities of SE (with nearly constant reabsorption at $P_r<0.1$) and LE (well above the lasing threshold at $P_r>10$), respectively. The slopes, written next to the lines, are equal to one, reflecting linear increase of $I_o$ with $P_r$. The intensity of ASE increases superlinearly with $(P_r){}^p$.

Figure 4

Steady-state emission spectra $|E(k)|{}^2$ with noise (upper black line in each panel) compared to those without noise (lower red line and circles in the same panel) at the same pumping rate $P_r$ for a random system with $g=1.0$. The values of $P_r$ are written in each panel.

Figure 5

Spectral width $\delta k$ (black squares) and amplitude $A_l$ (red circles) of the broad emission peak vs pumping rate $P_r$.

Figure 6

(a) Steady-state emission spectra $|E(k)|{}^2$ with noise over a small frequency range illustrating stochastic emission spikes. Crosses mark the wavelengths at which the emission intensities are obtained by the Fourier transform. Arrows mark spikes identified by a three-point peak-finding method. (b) Statistical distribution of frequency spacing of spikes $P(\delta k_n)$ at $P_r=1.00$ ($+$) and $P_r=2.00$ ($\times$), plotted on the linear scale (main panel) and logarithmic scale (inset). An exponential fit is marked in the inset by a straight red line.

Figure 7

Steady-state emission spectra $|E(k)|{}^2$ with noise (upper black line in each panel) compared to those without noise (lower red line and circles in the same panel) at the same pumping rate $P_r$ for a random system with $g=1.0$. The values of $P_r$ are written in each panel.

Figure 8

Modal intensities without noise (solid lines) compared to those with noise (dashed lines) for a random system with $g=1.0$. (a) Intensity of mode 1 $I_1$ (red thick lines) and mode 2 $I_2$ (blue thin lines) vs $P_r$. (b) Intensity of mode 3 $I_3$ with (green thick lines) and mode 4 $I_4$ (black thin lines) vs $P_r$. Modes are enumerated in Fig. 7d.

Figure 9

Linewidths of lasing mode 1 $\delta k_1$ (red crosses) and lasing mode 2 $\delta k_2$ (blue diamonds) vs the corresponding mode intensity $I_1$ and $I_2$ for a random system with $g=1.0$. Linear fits to the data give the power by which the linewidths decrease. $\delta k_1\propto I_1^{-0.97}$. $\delta k_2\propto I_2^{-0.70}$.

Figure 10

Steady-state emission spectra $|E(k)|{}^2$ with noise (upper black line in each panel) compared to those without noise (lower red line and circles in the same panel) at the same pumping rate $P_r$ for a random system with $g=0.5$. The values of $P_r$ are written in each panel.

Figure 11

Modal intensities without noise (solid lines) compared to those with noise (dashed lines). Intensity of mode 1 (red thick lines) and mode 2 (blue thin lines) vs $P_r$ for a random system with $g=0.5$. Modes are enumerated in Fig. 10d.