Pump-Controlled Directional Light Emission from Random Lasers

The angular emission pattern of a random laser is typically very irregular and difficult to tune. Here we show by detailed numerical calculations that one can overcome the lack of control over this emission pattern by actively shaping the spatial pump distribution. We demonstrate, in particular, how to obtain customized pump profiles to achieve highly directional emission. Going beyond the regime of strongly scattering media where localized modes with a given directionality can simply be selected by the pump, we present an optimization-based approach which shapes extended lasing modes in the weakly scattering regime according to any predetermined emission pattern.

Figure 1

Sketch of the envisioned setup for realizing pump-controlled emission from a random laser: Before being directed to the random medium, an applied pump beam is sent through a spatial light modulator which shapes the beam following an optimization procedure. The targeted emission pattern can be chosen at will. The geometry of the disk-shaped random laser (radius $R=1\mu \mathrm{m}$) consists of square shaped scatterers with index $n_s$ (red), embedded in a uniform background medium with index $n_b=1$ (gray).

Figure 2

First threshold laser mode ${\overline{\Psi }}_1$ at ${\overline{k}}_1=29.82\mu {\mathrm{m}}^{-1}$ of a random laser in the strongly scattering regime (see Fig. 1 with $n_s=3.0$). Shown are (a) the mode profile $|{\overline{\Psi }}_1(\mathbf{x})|$ inside the random laser, (b) the Gaussian-shaped pump profile that selects the mode (darker means more pump), and (c) the far-field profile (wiggly curve, blue) as well as the desired emission profile used as a reference (narrow peak, red). The inset in (c) depicts the near field directly outside the random laser disk and is complementary to (a). Note that the best directionality of a localized mode ($\mathcal{D}=48.19\%$) matches the desired profile only approximately. The employed gain curve has a center frequency at $k_a=30.0\mu {\mathrm{m}}^{-1}$ and a width of ${\gamma }_{\perp }=0.2\mu {\mathrm{m}}^{-1}$. For better visibility the linear color functions used in the plots differ in absolute scale.

Figure 3

First threshold laser mode ${\overline{\Psi }}_1$ at ${\overline{k}}_1=30.00\mu {\mathrm{m}}^{-1}$ of a random laser in the weakly scattering regime (see Fig. 1 with $n_s=1.2$). Shown are (a) the optimized mode profile $|{\overline{\Psi }}_1(\mathbf{x})|$ inside the random laser, (b) the corresponding pump profile resulting from the optimization algorithm, and (c) the far-field profile featuring a directionality of $\mathcal{D}=96.08\%$. The inset in (c) shows the near field of the laser mode, just outside the pumped disk. Note the very good agreement which we find in the main panel of (c) between the optimized emission pattern (wiggly curve, blue) and the targeted Gaussian emission profile $G(\phi )$ (narrow peak, red) centered at $\phi =180°$ and with a width of 8.3°. In (d) the directionality is shown with respect to the number of iterations used in the optimization procedure. The parameters of the employed gain curve are $k_a=30.0\mu {\mathrm{m}}^{-1}$ and ${\gamma }_{\perp }=0.01\mu {\mathrm{m}}^{-1}$.

Figure 4

Results from the optimization of the random laser for emitting into two prescribed directions (65° and 190°) simultaneously (the same disorder is used as in Fig. 3). Shown are (a) the optimized mode profile $|{\overline{\Psi }}_1(\mathbf{x})|$ inside the random laser, (b) the corresponding pump profile resulting from the optimization algorithm, and (c) the far-field profile featuring a convergence degree of $\mathcal{D}=96.12\%$. The inset in (c) shows the near field of the laser mode, just outside the pumped disk. Note the very good agreement which we find in the main panel of (c) between the optimized emission pattern (wiggly curve, blue) and the targeted double-peak emission profile $G(\phi )$ (narrow peaks, red).