Scattering phase delay and momentum transfer of light in disordered media

Disordered dielectric materials with short-range spatial correlations on length scales comparable to the wavelength of light display a rich variety of optical phenomena: photonic band gaps, structural coloring, strong scattering, and whiteness. However, a lack of reliable and straightforward analytical models complicates the rationale design of optical materials for specific applications. Here, we demonstrate how to accurately introduce collective scattering and define the effective medium in single scattering from heterogeneous dielectrics with a substantial refractive index contrast, starting from fundamental principles. Our model captures the effective medium's role in the momentum transfer definition for the particle form and structure factor in the forward scattering regime. We support our claims through transfer matrix calculations scattering from particle clusters.

Figure 1

Phase maps in light scattering from soft dielectrics. (a) Scattering geometry with an incoming plane wave ${\mathit{k}}_{\text{in}}$ propagating in $z$ direction and a scattered wave ${\mathit{k}}_{\text{out}}$. (b) Phase map calculated using Mie theory (equatorial cross section, incoming field perpendicular to the plane) inside a sphere illuminated with a plane wave, $kR=19.8, m=1.2$. (c) Phase map calculated using the T-matrix approach inside a spherical cluster containing $N=435$ dielectric spheres illuminated with a plane wave, $kR=2.2, m=1.2$. The coloured bars show the distance (wavelength) between the wave fronts having the same phase ($\varphi =0$) in different materials.

Figure 2

(a) Angular dependence of the differential scattering cross-section (SCS) of single dielectric spheres with $m=1.2, kR=6.5$. Open symbols show the prediction by Mie theory and the lines are predicted by single scattering approximations RGD (orange line), mRGD (dash-dotted blue line), and WKBA (solid-blue line). (b) Mie differential scattering cross-section (SCS) for $m=1.2$ and $kR\in [2,6]$ plotted as a function of $q_{\text{mod}}R$, see Eq. (5). Dash-dotted blue line: mRGD approximation. Inset: Mie differential scattering cross section at $\theta =0$ (open symbols) for different particle sizes (left) and refractive indices (right). Dash-dotted lines show the mRGD-prediction (WKBA-prediction for $\theta =0$ is identical). The horizontal lines show the RGD prediction.

Figure 3

Shift of the position of the minima of the scattering function ${\theta }_{\text{min}}$ with increasing $m$. Open squares: predictions by Mie theory for a dielectric sphere with $kR=6.5$. The plot shows sharp transitions, e.g., around $m=1.6$ where the first minimum disappears. Dashed lines: prediction for the minima position from mRGD for $kR=6.5$, solving $q_{\mathrm{mod}}R=x_{\min }$ for values where $j\left(x_{\min }\right)=0$. Solid lines: prediction from WKBA model.

Figure 4

Scattering from a cluster of $N=435$ particles with $kR=2.2$ and $m=1.2$, panel (a) and (b). Open squares: T-matrix calculations of a cluster shown in the inset of panel (a). The solid line in panel (a) shows the numerical WKBA (ray tracing) prediction. (b) Lines: predictions by the closed-form analytical model Eq. (10) using either $q^{\text{eff}}$ or $q_{\text{mod}}^{\text{eff}}$ for the structure factor. (c) Experimental differential scattering cross section of dilute dispersions of spherical particle clusters (mean diameter ${\overline{d}}_{\text{PB}}=3.3\mu \mathrm{m}$, polydispersity $\delta d_{\text{PB}}/{\overline{d}}_{\text{PB}}\sim 0.45$) suspended in water ($n_{\mathrm{h}}=1.33$) measured by static light scattering, $\lambda =660$ nm [36]. The clusters are made of closely packed $R=174$ nm ($kR\approx 2.2$) polystyrene spheres ($n_{\mathrm{p}}=1.59$) with a polydispersity of $\sim 5\%$ and a filling fraction of $\varphi \sim 0.6$ [36]. Lines: same color label as in panel (b), calculated using Eq. (9) with Mie scattering amplitudes and averaged over the PB-size distribution.