# statistical property of light reflected from a coarse mirror

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This is a problem comes up in my study in correlation reflectometry. I'm wondering if there is a straight-forward way to interpret the phase and amplitude of the reflected light from a perturbed reflecting surface.

A simplified description is as follows:

Consider an infinitely large plane mirror placed vertically in front of a wall at a distance $L$. On the wall, there is a circular hole with diameter $a$, $a\ll L$. A monochromatic light with wave length $\lambda$, $\lambda \ll a$, is passing through the hole horizontally towards the mirror, and then reflected back to the hole. The phase difference between the incidental and reflected waves at the center of the hole is measured(denoted as $\phi$), as well as the reflected amplitude comparing to the incidental amplitude (denoted as $E$). If the mirror is a perfect plane, the reflected wave can be calculated analytically, denote this as "unperturbed solution" $(E_0,\phi_0)$. The problem is to find the relationship between statistical property of the mirror deformation and statistical property of the reflected wave's amplitude and phase perturbation $(\tilde{E},\tilde{\phi)}$, where $\tilde{E}=E-E_0$ and $\tilde\phi = \phi-\phi_0$.

To be specific, let's use coordinates $(x,y,z)$, $x$ being horizontal direction pointing from the mirror to the wall, $y$ the other horizontal direction, and $z$ the vertical direction. The deformed mirror surface can be described by a function $x=f(y,z)$, with $f=0$ corresponds to the unperturbed mirror surface. $f$ is a random function specified by its statistical property, $\left<f\right>=0$ and $\left<f(y_1,z_1)f(y_2,z_2)\right> = \delta^2\cdot exp(-\frac{r^2}{l_{corr}})$, where $r \equiv \sqrt{(y_1-y_2)^2+(z_1-z_2)^2}$ is the distance between $(y_1,z_1)$ and $(y_2,z_2)$, $\left< ... \right>$ denotes ensemble average, $\delta$ and $l_{corr}$ are constants determining the magnitude and correlation length of the deformation.

Is there a easy way to calculate $\left<\tilde\phi\right>,\left<\tilde E\right>,\left<\tilde\phi^2\right>,\left<\tilde E^2\right>$, etc. ?

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