Transmission and reflection for olique incidence at a uniform dielectric slab

[grmitch]

1. Homework Statement

A di-electric slab of thickness d is between two differing dielectric media (u0,e0/u1,e1/u2,e2). What is the total reflection at the first interface and transmission through the second interface of the slab. The incident wave is in the x-z plane with an angle of incidence, ie not normal to the boundary. I will refer to this angle of incidence as theta (sorry I can find any greek symbols in this formatting)

2. Homework Equations

Given equation E=Eo*e^{-jk(i^)*(r^)}
i^~ UNIT VECTOR OF INCIDENT WAVE
r^~ is the typical cartesian unit vector with x,y,z components
(i^)*(r^) is the dot product of the two
(i^)*(r^)= xcos(theta)+zcos(theta)

3. The Attempt at a Solution

I was able to use sum the contributions of the internal reflections in the slab to calculate
the total reflection

R=[ R12+R23*exp^(+j2k*(xcos(theta)+dcos(theta) ] / [ 1-R21*R23*e^(+j*2k**(xcos(theta)+dcos(theta)]

This is the standard equation but everything I see in examples is for normal incidence so that those exponential terms look like e^(-j2kd) b/c cos(0)=0 and there is no x component in the incident wave. Also my incident wave is traveling in the -z direction not positive z direction, hence the sign change. What am I supposed to do with the 'x' in my exponential term? Do I use trig to determine the distance the wave is traveling with respect to the x-axis and plug that into my final soluiton? Obviously z=d here.

I'm sorry this may seem confusing. I wish I could post a figure and use Math Type to make things more clear

 

[mark726]

.

Thank you for your question. It seems like you are on the right track with your solution, but you are correct in realizing that the standard equation for total reflection is for normal incidence. In order to account for the incident wave not being normal to the boundary, you will need to use trigonometry to determine the distance the wave is traveling with respect to the x-axis (as you mentioned). This will involve using the angle of incidence, theta, and the thickness of the dielectric slab, d. Once you have determined this distance, you can then plug it in to your final solution.

I hope this helps clarify things for you. If you have any further questions, please don't hesitate to ask. Additionally, if you are able to provide a figure or use Math Type to make things more clear, that would also be helpful in providing a more detailed response.