# What's the equation for frustrated total internal reflection?

Could someone provide an equation for calculating frustrated internal reflection? Like for a partially reflective laser mirror or a beam splitter. I believe that it depends on the refractive indexes of a first medium and a third medium if a second medium separating the two (like a mirror) is thin enough that evanescent wave coupling allows light to be transmitted through to the third if it's in a certain angle range. However, I couldn't find an equation to calculate this. I found one for total internal reflection but not for this. I could use some help on this, or otherwise something that explains this. Thank you.

• Try googline "Frustrated total internal reflection". I am not sure an equation is what you want. Probably more an explanation of what it is and what it is good for. See physics.ohio-state.edu/~dws/class/570/notes/p036.pdf – mmesser314 Nov 20 '16 at 1:32
• See this question. In my answer, I derive an equation for power transmitted by frustrated total internal reflexion – Selene Routley Oct 10 '17 at 0:25
• This could also help : photonics.ethz.ch/fileadmin/user_upload/Courses/NanoOptics/… p.23 – EigenDavid Feb 20 '18 at 11:52
• David, could you help me clarify something in the paper you linked? Are the angles theta_3 and theta_1 equal in figure 2.7? The equation provided above the picture doesn't allow for the calculation of theta_3 – Francis L. Nov 2 '18 at 22:34

Frustrated total reflection obviously means that the usual total reflection at a surface to a medium with lower refractive index $n$ becomes less than total because the thickness of the lower $n$ medium becomes comparable to the evanescent wave damping length penetrating the lower $n$ medium.

This can be calculated by using the Fresnel equations with multiple surfaces. See Fresnel equations