Plane Wave Total Internal Reflection Problem

[marksambob]

Homework Statement

A plane wave is incident at 70˚ to surface normal traveling within a medium with relative permittivity = 4, striking the boundary with air. Where is the nearest magnetic field max. to the boundary in the initial medium? Find the 1/e penetration depth of the evanescent wave.

Homework Equations

No equations are given, but I've been using:
θC= sin-1(εr2/ εr1)^.5

The Attempt at a Solution

For the second part, i said that in the air, Et and Ht vary with the factor: exp(-α₂z)exp(-jβ_2xx), where
α₂ = β₂(εr1/ εr2*sin2θi-1)^.5 = 1.59β2 the 1/e penetration distance is then just
z = 1/(1.59β2) = .628β2

The first part, however, is where I am having my main difficulty. I think I know how to do it were this to be a plane wave incident on a conductor, but I am not sure if I can use the same logic for the air interface given that I don't think I can assume that E = 0 at the boundary. ( for a conductor, I've been able to solve for H for a TE wave being H₁=2*E_i0/Z₁*cos(β₁zcosθ_i)*exp(-jβ₁xsinθ_i)
from here I would just find where β₁zcosθ_i = 0 and that would give the max. Does this still work for an air incidence though? And is there any max for a TM wave?

Any help would be greatly appreciated! Thanks!

 

[BerryBoy]

The first part can be done by just considering that the field inside the medium can be represented as a sum of cosine/sine waves corresponding to the reflected and incident wave. The reflected ray will have experienced a phase delay upon reflection according to the Frensel equations. So the corresponding maximum (if we consider cosine waves and set the boundary at z = 0), will be the distance that the half of this phase delay corresponds to.

I agree with your expression for $\alpha_2$ except for the $\beta_2$ term. Unless the material in which the wave is traveling is lossy, why would you expect there to be attenuation in the x direction?

 

[BerryBoy]

Sorry, missed the j term in the exponential with $\beta_2$. But I still think it doesn't belong in the expression for $\alpha_2$. It corresponds to the propagation constant in the x direction.

 

[marksambob]

Thanks for the help!

Solving for the reflection coefficient, I get -.69-.73j which corresponds to an angle of -133.47˚. It then would make sense that the maximum would be at half of this away, ie 66.735˚. However, this should only correspond to .185λ. From adding up the equations for incident and reflected waves, however, I get a dependency on cos(β₁*z*cos(θ_i)). Setting this = 1 gives me a result of z = 1.462, quite a different answer. Am I using the reflection coefficient correctly?

Thanks again!

 

[DeeNos]

I would approach this problem by first understanding the concept of total internal reflection. Total internal reflection occurs when a light wave travels from a medium with a higher refractive index to a medium with a lower refractive index, and the angle of incidence is greater than the critical angle (θc). In this case, the incident angle (θi) is 70˚, which is greater than the critical angle given by the equation θc= sin^-1(εr2/ εr1)^.5. The critical angle is the angle at which the refracted ray would have an angle of 90˚, and beyond this angle, total internal reflection occurs.

Now, for the first part of the problem, we need to find the nearest magnetic field maximum to the boundary in the initial medium. Since the incident angle is greater than the critical angle, we know that total internal reflection will occur at the boundary. This means that the magnetic field maximum will be at the boundary itself. We can use the same logic as you did for a conductor, but in this case, we will use the boundary between the two media as the reference point. Therefore, the magnetic field maximum will be at z=0.

For the second part of the problem, we need to find the 1/e penetration depth of the evanescent wave. This refers to the distance at which the intensity of the evanescent wave decreases by a factor of 1/e. In the case of total internal reflection, the evanescent wave only exists within a small distance from the boundary and decays exponentially as it moves away from the boundary. The equation you used, z = 1/(1.59β2), is correct for finding the penetration depth of the evanescent wave. However, it is important to note that this equation is valid for both TE and TM waves.

In conclusion, for a plane wave incident at 70˚ to the surface normal traveling within a medium with relative permittivity of 4 and striking the boundary with air, the nearest magnetic field maximum will be at the boundary and the 1/e penetration depth of the evanescent wave is given by z = 1/(1.59β2).