Origin of light localization from orientational disorder in one- and two-dimensional random media with uniaxial scatterers

An origin of lightwave localization is revealed by analyzing the spatial distribution and the spectral intensity of lightwave as well as lasing threshold in one- and two-dimensional random media made of uniaxial scatterers arranged to be ordered in the spatial location but disordered in the spatial orientation of their optical axes. The time dependent theory of random lasers is developed from isotropic scatterers to anisotropic ones in order to simulate the behavior of random lasing. Results show that, as the strength of the orientational disorder increases, the spatial distribution of lightwave in the medium changes from the diffusion state dominant to the localized state dominant, accompanied by the decrease of the lasing threshold. At the given system size there is a characteristic value for the orientational disorder, above which lightwave can be localized obviously.

Figure 1

Schematic illustration of 2D random media in which the uniaxial scatterers are arranged in the spatial location but disordered in the spatial orientation. (a) The configuration for spatial orientation in which $x_1$ and $x_2$ denote the principal axes, and (b) the schematic pattern of a random realization in which the ellipse denotes the refractive index ellipse in the medium plan.

Figure 2

The spatial distribution of the field amplitude in the medium for $\kappa =n_o∕n_e=1$ with $n_o=1.5$ and $n_e=1.5$ at $W_p=3\times {10}^{10}{\mathrm{s}}^{-1}$. The other system parameters are $l=4\mathrm{\mu }\mathrm{m}$, $r=60\mathrm{nm}$, $\varphi =60\%$, $h=v=140\mathrm{nm}$, and $n_1=1$.

Figure 3

(a) The spatial distribution of the field amplitude, (b) the corresponding emission spectrum, and (c) the spatial map of the mode with its peak wavelength denoted by ${\lambda }_a$ in the spectrum. $\kappa =n_o∕n_e=1.27$ with $n_o=1.9$ and $n_e=1.5$ at $W_p=3\times {10}^{10}{\mathrm{s}}^{-1}$. The other system parameters are $l=4\mathrm{\mu }\mathrm{m}$, $r=60\mathrm{nm}$, $\varphi =60\%$, $h=v=140\mathrm{nm}$, and $n_1=1$.

Figure 4

(a) The spatial distribution of the field amplitude, (b) the corresponding emission spectrum, and (c) the spatial map of the mode with its peak wavelength denoted by ${\lambda }_b$ in the spectrum. $\kappa =n_o∕n_e=1.87$ with $n_o=2.8$ and $n_e=1.5$. The other parameters and the random realization are the same as that in Fig. 3.

Figure 5

(a) Schematic for 1D random media in which the uniaxial dielectric layers (black ones) are ordered in spatial location but disordered in spatial orientation. (b) The configuration for spatial orientation. The plane of the principal axes (denoted by $x_1$ and $x_2$) of the uniaxial layer is within the $x\text{−}y$ plan. The axis $x_2$ takes an arbitrary angle $\theta$ with the $x$ direction.

Figure 6

The spatial distribution of the electric field amplitude and the corresponding emission spectrum for a 50 cell 1D sample with $L=20\mathrm{\mu }\mathrm{m}$, $w_g=250\mathrm{nm}$, $w_s=150\mathrm{nm}$, and $n_1=1$ at $W_p=3\times {10}^{10}{\mathrm{s}}^{-1}$ (a) $\kappa =1$ for $n_o=1.5$ and $n_e=1.5$; (b) $\kappa =1.2$ for $n_o=1.8$ and $n_o=1.5$; (c) $\kappa =1.67$ for $n_o=2.5$ and $n_e=1.5$; and (d) $\kappa =2.3$ for $n_o=3.5$ and $n_e=1.5$.

Figure 7

The spectral intensity in arbitrary units vs the wavelength for a 75 cell 1D sample with $L=30\mathrm{\mu }\mathrm{m}$, $w_g=250\mathrm{nm}$, $w_s=150\mathrm{nm}$, and $n_1=1$ for different pump rates $W_p$ and disorder degrees $\kappa$. (a) $\kappa =1.2$ for $n_o=1.8$ and $n_o=1.5$; (a) $W_p={10}^8{\mathrm{s}}^{-1}$; (b) $W_p=5.3\times {10}^9{\mathrm{s}}^{-1}$; (c) $W_p=2\times {10}^{10}{\mathrm{s}}^{-1}$; (d) $\kappa =1.67$ for $n_o=2.5$ and $n_e=1.5$; (d) $W_p=3\times {10}^9{\mathrm{s}}^{-1}$; (e) $W_p=4\times {10}^9{\mathrm{s}}^{-1}$; (f) $W_p=6.2\times {10}^9{\mathrm{s}}^{-1}$; (g) $\kappa =2.3$ for $n_o=3.5$ and $n_e=1.5$; (g) $W_p=2\times {10}^9{\mathrm{s}}^{-1}$; (h) $W_p=4\times {10}^9{\mathrm{s}}^{-1}$; (i) $W_p={10}^{10}{\mathrm{s}}^{-1}$.

Figure 8

The pump rate dependence of the peak intensity of the mode that has the minimal lasing threshold corresponding to the spectral intensity in Fig. 7a $\kappa =1.2$ and $W_{pth1}\approx 5.1\times {10}^9{\mathrm{s}}^{-1}$; (b) $\kappa =1.67$ and $W_{pth2}\approx 3.6\times {10}^9{\mathrm{s}}^{-1}$; (c) $\kappa =2.3$ and $W_{pth3}\approx 2.9\times {10}^9{\mathrm{s}}^{-1}$.

Figure 9

Plotting of the lasing threshold $W_{pth}$ varying with the disorder degrees $\kappa$ for the 1D sample as the same as that in Fig. 7.