Effects of spatially nonuniform gain on lasing modes in weakly scattering random systems

A study on the effects of optical gain nonuniformly distributed in one-dimensional random systems is presented. It is demonstrated numerically that even without gain saturation and mode competition, the spatial nonuniformity of gain can cause dramatic and complicated changes to lasing modes. Lasing modes are decomposed in terms of the quasimodes of the passive system to monitor the changes. As the gain distribution changes gradually from uniform to nonuniform, the amount of mode mixing increases. Furthermore, we investigate new lasing modes created by nonuniform gain distributions. We find that new lasing modes may disappear together with existing lasing modes, thereby causing fluctuations in the local density of lasing states.

Figure 1

Effective potential (wavelet power spectrum) $|W(x)|{}^2$ of the dielectric function ${\epsilon }_r(x)$ as a function of position $x$ and wavelength $\lambda$. Regions of high power indicate potential barriers and regions of low power indicate potential wells where intensities are typically trapped. The black lines on the top represent the spatial distribution of dielectric constant ${\epsilon }_r(x)=n^2(x)$.

Figure 2

A mapping of the phase $\theta$ of $M_{22}$ for the passive 1D random system without gain. The topological charge of all quasimodes seen here is $-1$. Modes are enumerated from left to right. Quasimode 27 is encircled in black.

Figure 3

Normalized intensity $|\psi (x)|{}^2$ of a leaky mode—quasimode 27 ($\mathrm{k̃}=11.8-i1.03$ $\mu$m${}^{-1}$) of the passive random 1D structure. The intensity is peaked at the left boundary of the structure, similar to doorway states [40].

Figure 4

Frequencies and thresholds ($k$, $n_i$) of the lasing modes of the 1D random structure with gain. Lasing modes 1, 17, 18, and 33 are explicitly marked. The gain region length $l_G$ reduces from uniform gain ($l_G=L$) to nonuniform gain $l_G<L$. The color indicates the value of $l_G$ (units of nm) decremented along the layer interfaces. Darker shades represent larger $l_G$ values where $\left|n_i\right|$ values are smallest. The shades lighten as $l_G$ reduces and $\left|n_i\right|$ increases. Due to the random thicknesses of the layers, the $l_G$ decrements are unequal. Hence, the reason for the unequal spacing of the color code.

Figure 5

Normalized intensity of lasing mode 17 with uniform gain $l_G=24.1\times {10}^3$ nm (solid red lines) and nonuniform gain $l_G=14.284\times {10}^3$ nm (dashed black lines). Gain is only located in the region $0⩽x⩽l_G$. (a) Total intensity $|\Psi (x)|{}^2$, (b) traveling wave intensity $|{\Psi }^{(T)}(x)|{}^2$, and (c) standing wave intensity $|{\Psi }^{(S)}(x)|{}^2$. Nonuniform gain significantly changes the spatial intensity envelope as well as the standing wave and traveling wave components.

Figure 6

Standing to traveling wave ratio $A_{\mathrm{ST}}(x)$ (solid red lines) of lasing mode 17 with uniform gain $l_G=L$ (a) and nonuniform gain $l_G=14.284\times {10}^3$ nm (b). Gain is located in the regions left of the vertical black solid line. We term the location at which the ratio diverges ($A_{\mathrm{ST}}(x)\to \infty$), the standing wave center (SWC). The potential profile $|W(x)|{}^2$ (dashed black lines) of the dielectric function at the wavelength of mode 17 is overlaid in both (a) and (b).

Figure 7

Decomposition of lasing mode 17 in terms of passive quasimodes. (a) Decomposition with uniform gain (red crosses) and nonuniform gain (black circles). Leaky quasimodes, that is, modes with large $\left|k_i\right|$ such as modes 7, 14, and 23, contribute to lasing modes markedly different than the others. (b) Five largest coefficients from the decomposition of lasing mode 17. As $l_G$ is reduced, the amount of mode mixing increases dramatically. The reconstruction error $E_R$ for lasing mode 17 is close to ${10}^{-4}$ until $l_G=11\times {10}^3$ nm then rises to ${10}^{-2}$ at $l_G=3.2\times {10}^3$ nm. Some coefficients are greater than one, which is possible in open systems.

Figure 8

(Left) Red (dark gray) lines represent real zero lines of $M_{22}$ and green (light gray) lines represent imaginary zero lines of $M_{22}$. Their crossings indicate ($k$, $n_i$) values of lasing modes. (Right) Phase maps of $M_{22}$. All data are in the ranges ($10.3\hspace{0.5em}\mu {\mathrm{m}}^{-1}<k<10.8\hspace{0.5em}\mu {\mathrm{m}}^{-1}$) and ($0⩾n_i⩾-0.074$) covering lasing modes 17 and 18 for $l_G=14.961\times {10}^3$ nm (a–b), $14.553\times {10}^3$ nm (c–d), $14.523\times {10}^3$ nm (f–g), $14.472\times {10}^3$ nm (h–i), $14.284\times {10}^3$ nm (j–k), and $14.042\times {10}^3$ nm (l–m). The joining of zero lines in (c) results in the formation of a new lasing mode (new zero line crossing is encircled in black). The inset in (c) is an enlargement of the mode 17 and new mode solutions. In (d), the phase singularity at the new mode has a topological charge of $+1$, opposite to that of mode 17. The real and imaginary zero lines pull apart in (f) so that the mode 17 and new mode solutions are nearly identical. The phase map in (g) reveals the existence of the two phase singularities. The lines completely separate in (h) resulting in the disappearance of mode 17 and the new mode. The phase map in (i) confirms the disappearance of the two modes. This process reverses itself in (j–m) yielding behavioral symmetry around $l_G=14.472\times {10}^3$ nm.

Figure 9

(a) Standing to traveling wave ratio $A_{\mathrm{ST}}(x)$ of the new lasing mode in red (light gray) and lasing mode 17 in black for $l_G=14.553\times {10}^3$ nm (solid lines) and $l_G=14.523\times {10}^3$ nm (dotted lines). The potential profile $|W(x)|{}^2$ (dashed black line) of the dielectric function for this wavelength is overlaid in both (a) and (b) and major potential barriers are marked 1) through 4). The ratio $A_{\mathrm{ST}}(x)$ of the new lasing mode and lasing mode 17 become more similar and converge on each other as $l_G$ reduces. Reducing $l_G$ further causes these two lasing modes to first disappear then reappear as the process reverses itself. (b) Standing to traveling wave ratio $A_{\mathrm{ST}}(x)$ of the new lasing mode in red (light gray) and lasing mode 17 in black after they have reappeared for $l_G=14.284\times {10}^3$ nm. The ratios are similar to the ratios for $l_G=14.553\times {10}^3$ nm in (a). The vertical black solid line marks the gain edge. The ratios of the modes diverge now as $l_G$ is reduced.

Figure 10

Decomposition at $l_G=14.284\times {10}^3$ nm of lasing mode 17 (black circles), lasing mode 18 (blue open diamonds), and the new lasing mode (red crosses) in terms of the quasimodes of the passive system. Lasing modes 17 and 18 are mostly composed of their respective quasimodes while the new mode is dominated by a mixture of both quasimode 17 and 18. The inset shows the decomposition coefficients of outlying quasimodes for lasing mode 17 (dotted black line), lasing mode 18 (dashed blue line), and the new lasing mode (solid red line).