Fresnel amplitude coefficients for unpolarized light

[Nono713]

Hi,
I have an issue with the Fresnel amplitude coefficients. I know they are given in two versions, for s-polarization and p-polarization. A version for unpolarized (randomly polarized) light is available for the power coefficients - it's just an average - but I could not find such an expression for the amplitude coefficients. From the power coefficient average expression I get:

$$r_\text{avg} = \sqrt{\frac{|r_s|^2 + |r_p|^2}{2}}$$

Which is obviously wrong, since this value is always real and positive which doesn't make sense for amplitude coefficients (I think).

So is there a simple expression for the average fractional amplitude of the reflected light assuming light is randomly polarized (equal amount of s-polarized and p-polarized waves)? And can the result be extended to an arbitrary proportion of s-polarized and p-polarized waves (say 25%, 30%, etc..)

Or does it simply not make sense to try and find an average of amplitudes? I need this to analyze interference effects, I could go with two versions dependent on the polarization of light, and average the final reflectance at the end, but it seems cleaner and easier to have a single expression. Thanks...

I should mention this is for a computer graphics project and I do not know that much about electromagnetism, please go easy on me :)

 

[Valerie Park]

Thanks for your help!Yes, it is possible to find an average fractional amplitude of the reflected light assuming light is randomly polarized. The average reflection coefficient can be written as:$$r_{avg} = \frac{1}{2}\left[|r_s|e^{i\phi_s} + |r_p|e^{i\phi_p}\right]$$ where $r_s$ and $r_p$ are the reflection coefficients for s- and p-polarized light, respectively, and $\phi_s$ and $\phi_p$ are the corresponding phase shifts.This expression can also be extended to an arbitrary proportion of s-polarized and p-polarized waves, simply by multiplying each coefficient by the appropriate fraction. For example, if the light is 25% s-polarized and 75% p-polarized, the expression becomes:$$r_{avg} = \frac{1}{2}\left[0.25|r_s|e^{i\phi_s} + 0.75|r_p|e^{i\phi_p}\right]$$ Hope this helps!