Coherent multiple scattering of light in ($2+1$) dimensions

We formulate a multiple scattering theory of light in media spatially disordered along two directions and homogeneous along the third one without making any paraxial approximation on the wave equation and fully treating the vector character of light. With this formalism, we calculate the distribution of transverse momenta of a beam as it evolves along the optical axis and unveil a phenomenon not captured by the paraxial equation: a crossover from a scalar to a vector regime, visible in the coherent backscattering peak as polarization gets randomized.

Figure 1

We consider the propagation of a quasi-plane-wave beam of transverse wave vector ${\mathit{k}}_0$ and polarization $\mathbf{\epsilon }$ through a medium spatially disordered along $x$ and $y$ and homogeneous along the optical axis $z$. Light is detected on a polarization channel ${\mathbf{\epsilon }}^{\prime }$ belonging to the plane $(x,y)$.

Figure 2

(top) Diagrammatic representation of the Bethe–Salpeter equation (12) for the ladder series ${\mathrm{\Gamma }}^{\text{(L)}}$. Upper solid lines symbolize the Green tensor $\mathbf{G}$, and lower dashed lines symbolize its complex conjugate. Vertical dotted lines refer to the correlation function of the disorder potential, Eq. (2). (bottom) Reciprocity relation (32) between ladder and crossed series.

Figure 3

Factors $1-{\lambda }_n(k_z,q_z=0)$ of the mode decomposition (17). The mode $n=6$ fulfills $1-{\lambda }_6(k_z,0)=0$ for all ${\widehat{k}}_z$. It thus always controls the large-$z$ (small-$q_z$) limit of the structure factor (18). In the experimentally relevant limit where ${\widehat{k}}_z$ is close to 1, the two modes $n=1$ and 2 obey $1-{\lambda }_{1,2}(k_z,0)\ll 1$, so they also contribute in general. The modes $n=3$, 4, and 5 are, on the other hand, strongly attenuated in this regime.

Figure 4

(top): Diffusive contribution ${\mathcal{F}}_{\text{L}}$ to the momentum distribution as a function of $z$, Eq. (26), in the four polarization channels. (middle) CBS contribution ${\mathcal{F}}_{\text{C}}$ at ${\mathit{k}}_{\perp }=-{\mathit{k}}_0$, Eq. (27), in the four channels. (bottom) Contrast of the CBS peak, ${\mathcal{F}}_{\text{C}}/{\mathcal{F}}_{\text{L}}$ (curves $\sigma \perp \sigma$ and $l\parallel l$ overlap). The insets display the shape of the momentum distribution in the plane $(k_x,k_y)$, in three configurations where the CBS peak is fully contrasted, partially contrasted and not present. Note that, in the channels $\sigma \perp \sigma$ and $l\perp l$, the CBS peak and the diffusive ring are both very small when $z\ll z_{1,2}$, but their ratio is finite.