A Stochastic Differential Equation For Laser Propagation In Medias With Random Gaussian Absorption Coefficients: A Modified Beer's Law Solution Via A Van Kampen Cluster Expansion

Let $\mathbb{I}\text{\hspace{0.17em}⁣}\mathbb{D}=[0,\mathrm{L}]\subset {\mathbb{R}}^{+}$ be a slab geometry with boundaries $z=0$ and $z=\mathrm{L}$. A laser beam with a flat incident intensity ${\psi }_o$ enters the slab along the z-axis or unit vector ${\widehat{\mathbf{e}}}_3$ at $z=0$. The slab contains matter with an absorption coefficient of $\mathsf{A}$ with respect to the wavelength. If $\mathsf{A}$ is constant and homogenous then the beam decays as Beer’s law $\psi (z,{\widehat{\mathbf{e}}}_3)={\psi }_o\exp (-\mathsf{A}z)$. If the absorption coefficient is randomly fluctuating in space as $\mathbf{A}(z)=\mathsf{A}(1+\alpha \mathbf{G}(z))$–where $\alpha >0$ determines the magnitude of the fluctuations, and the Gaussian random function has expectation $\mathbb{E}\{\mathbf{G}(z)\}=0$ and a binary correlation $\mathbb{E}\{\mathbf{G}(z_1)\otimes \mathbf{G}(z_2)\}=\phi (z_1,z_2;\xi )=\mathsf{C}\exp (-\mathrm{\mid }{z}_1-z_2{\mathrm{\mid }}^2{\xi }^{-2})$ for all $(z_1,z_2)\in \mathbb{I}\text{\hspace{0.17em}⁣}\mathbb{D}$ with correlation length $\xi$–then the beam propagation and attentuation within the medium is described by the stochastic differential equation \begin{equation} d\widehat{ψ(z,\widehat{$\mathbf{e}$}{3})}=-$\mathsf{A}$\widehat{ψ(z,$\mathbf{e}${3})}dz-α$\mathsf{A}$\widehat{ψ(z,$\mathbf{e}${3})}$\mathbf{G}$(z)dz \end{equation} The stochastically averaged solution is derived via a Van Kampen-type cluster expansion, truncated at 2nd order for Gaussianality, giving a modified Beer’s law \begin{equation} $\mathbb{I}$(z,\widehat{$\mathbf{e}$}{3})=$\mathbb{E}$\big\lbrace\widehat{ψ(z,\widehat{$\mathbf{e}$}{3})}\big\rbrace=ψ{o}\exp(-$\mathsf{A}$z)\exp\bigg(\frac{1}{4}α^{2}$\mathsf{A}$^{2}{$\mathsf{C}$}ξ\bigg[\exp(-z^{2}/ξ^{2})\bigg(\sqrtπz Erf(\tfrac{z}ξ)\exp\bigg(\frac{z^{2}}{ξ^{2}}\bigg)+ξ\bigg)-ξ\bigg]\bigg) \end{equation} The deterministic Beer’s law is recovered as $\alpha \to 0$.