Fraction of light reflected inside a diamond

[Davidllerenav]

Homework Statement

Diamonds have an index of refraction of n = 2.42 which allows total internal
reflection to occur at relatively shallow angles of incidence. What fraction of the light reflects for internal angles ##\theta_i = 40.5^°## and ##\theta_i = 50.6^°##?

Relevant Equations

Fresnel equations
Snell's law

So i do now that it is a case of total internal reflection, but i didn't get R=1 for ##\theta_i=40.5^°##. I used the Fresnel equations for both s and p-polarized light and for s I got ##r_s=\frac{n_i\cos\theta_i-n_t\cos\theta_t}{n_i\cos\theta_i+
n_t\cos\theta_t}=0.296## using ##n_i=2.42## and ##n_t=1##. For p I got ##r_p=\frac{n_i\cos\theta_t-n_t\cos\theta_i}{n_i\cos\theta_t+
n_t\cos\theta_i}=0.522##. What am I doing wrong?

 

[nasu]

What values are you using for ##\theta_t##?

 

[Davidllerenav]

What values are you using for ##\theta_t##?

Well since both incident angles are greater than the critical angle ##\theta_c=24.4^°##, then ##\theta_t=0##.

 

[haruspex]

Are Fresnel's equations still valid beyond the critical angle?

 

[Davidllerenav]

Are Fresnel's equations still valid beyond the critical angle?

I thought so, but reading the chapter again I think not because the transmited angle isn't in facr zero, but complex.

 

[nasu]

Well since both incident angles are greater than the critical angle ##\theta_c=24.4^°##, then ##\theta_t=0##.

No, the angle is not zero. For angles larger than the critical angle there is no real ##\theta_c ##. The Fresnel reflection coefficient becomes a complex number with a magnitude of 1 for any angle larger than the critical angle. The phase of the complex coefficient still changes with the angle but the magnitude doesn't. Fresnel's equations are still valid. They can be written in terms of just incident angle and index of refraction so there is no problem with the transmission angle.