Reflection from multiple thin films: accounting for lost light due to small surface area

I have a problem similar to reflection of multiple thin films. I have light coming in from medium 1 and I want to find the total reflected intensity after being reflected inside 2 layers. However, I want to account for the fact that the surface area of medium 4 is smaller than the light's spot size and so some of the light is lost.

I already derived the total reflection for the regular 2 layer case: (I am assuming a zero incident angle) $$R =\left| r + \frac{tt'r_{34}e^{i\delta}}{1-r'r_{34}e^{i\delta}} \right|^2$$ $r$ is the total reflected electric field amplitude from the first layer only, $t$ total transmitted amplitude through the first layer( $r'$ and $t'$ are in the opposite direction), $r_{34}$ is the reflection Fresnel coefficient for the n3-n4 boundary and $\delta$ the phase corresponding to the n3 layer.

Now I want to take into account that not all of the light transmitted through the first layer hits the last boundary. I thought about just multiplying the second term in my expression by some factor, say 0.5, which would make the transmitted amplitude smaller. However, since this will effectively multiply the complex electric field amplitude I am not sure if that make sense.

You already have the expression for the 2-layer case, and if I observe correctly, you are only concerned with the reflected part, not the transmitted part. So, a smaller reflecting area would mean lesser reflected intensity and larger transmitted intensity, but we are not going to bother about the latter. So, as long as the concerned length parameter is not comparable with the wavelength ($\lambda$) of light, one can construct an effective r for the third layer. Since a larger amount of light gets reflected from a larger area in such macroscopic circumstances, I can safely assume, as a zeroth order approximation, that this $$r_{\rm eff} = r_{\rm original} \times \left(\frac{A_{\rm layer \ area}}{A_{\rm spot \ size}} \right)$$ Of course, then $t_{\rm eff} = 1 - r_{\rm eff}$, neglecting losses etc., but again, you probably don't want to dive into these.
Now, using your earlier derivation, use this $r_{\rm eff}$ to calculate $R$. I feel this is better than just using a number, $0.5$ or anything, and should work as a zeroth level approximation. And certainly don't insert factors into the final answer.
• So would it be wrong to do $t'_{eff} = t'\times loss factor$? Aug 6, 2014 at 23:50
• @user1830663 - This may be another way of doing the same thing only. My main point was, whichever way you estimate the loss, use a $t_{\rm eff}$, or $r_{\rm eff}$, for the last layer, in the calculation, instead of multiplying only the final answer by some factor. Anyways, how are you planning to go about finding this 'loss factor'? Aug 7, 2014 at 5:34