How Do You Solve EM Wave Transmission Through Multiple Media?

[Drarp]

Hi there, I'm stuck with this problem, I've already worked out some algebra but I can't seem to get what I am asked, when trying to solve the equations I simply can't get the answer, so I was hoping you could help me. Here we go

Homework Statement

Calculate the transmission coefficient T if you have a plane EM wave that travels from right to left through the three media shown, each one with its respective refraction index shown in the figure. Assume you know the incident wave E field.

The wave incides on the frontier I-II, part of it gets refflected and part of it gets transmitted. The transmitted wave, when getting to the frontier II-III, gets split in 2, part of it gets refflected and the rest transmitted. Is this treatment okay?

Here goes the figure and my vectors (the problem doesn't ask you to complicate more than this according to my teacher). Sorry if it's not neat, I'm very sloppy.

https://www.physicsforums.com/attachment.php?attachmentid=52545&stc=1&d=1351833245

Homework Equations

After writing down the 10 equations for the E's and H's, we use the boundary conditions:
E (tangential in medium i) = E (tangential in medium j)
(that is, the continuity of the tangential component of the E field)
H (tangential in medium i) = H (tangential in medium j)
(continuity of the tangential component of H)

The Attempt at a Solution

We then use these equations in the boundaries at x=0 and x=d, which gives the next four equations (in which our unknowns are E1r, E2r, E2t and E3t) (remember, E1i is a given):

E1i - E1r = E2t - E2r

n1(E1i + E1r) = n2(E2t + E2r)

E2t*e^(i*k2*d) - E2r*e^-(i*k2*d) = E3t*e^(i*k3*d)

n2*(E2t*e^(i*k2*d) + E2r*e^-(i*k2*d) ) = n3*E3t*e^(i*k3*d)

The thing is, I'm not sure how to get through these equations. I tried two different ways: expressing the last 2 eqns by expanding the imaginary exponential, then trying to compare coefficients but something goes wrong and I wind up with contradictions; then I tried dividing the two last equations by the term that multiplies E3t on the RHS of the eqns but I also winded up with some ugly stuff: it would either make me impose an apparently unnecessary constraint that doesn't get me anywhere or it gives me inconsisten results. I hope I'm doing something wrong, any help would be appreciated. Also, sorry for not posting my whole procedure but it gets pretty ugly after a few steps.


I'm sorry if I'm not following the template correctly or if I am not typing correctly or doing something wrong, this is my first post in this forum. I'm just looking for some help :) Also, sorry for not using LaTeX, I know some but I'm not sure how to use it on this forum, yet.

 

[mark726]

Hi there,

First of all, don't worry about not using LaTeX, it's not necessary for this forum. Now, let's take a look at your problem. It seems like you have the right idea in terms of using boundary conditions to solve for the unknowns. However, I think you may have made a mistake in your equations.

In your second equation, you have n1(E1i + E1r) = n2(E2t + E2r). However, this should actually be n1(E1i - E1r) = n2(E2t + E2r). This is because when the wave is incident on the boundary, it is traveling in the opposite direction of the reflected wave, so the sign of E1r should be negative.

Similarly, in your third equation, you have E2t*e^(i*k2*d) - E2r*e^-(i*k2*d) = E3t*e^(i*k3*d). However, this should actually be E2t*e^(i*k2*d) - E2r*e^-(i*k2*d) = E3t*e^-(i*k3*d). Again, this is because when the wave is transmitted from medium 2 to medium 3, the transmitted wave is traveling in the opposite direction of the reflected wave, so the sign of E3t should be negative.

I would recommend checking your other equations as well to make sure they are correct. Once you have the correct equations, you should be able to solve for the unknowns using the boundary conditions. I hope this helps! Let me know if you have any other questions.