Electric and magnetic field of a plane wave through a slab?

[J_Physics]

Hello.

My question/problem is this:

I have a plane wave traveling in the positive z-direction through 3 different mediums. Also, medium 2 is comprised of a slab with thickness a. I want to express the electric and magnetic fields of the wave as it travels through these mediums.
More specifically, I want to know how to express the electric and magnetic field inside the slab as it is influenced by the slab. I specifically want to know this because I am not sure how to do it. I want to express these equations because I would like to know the reflection and transmission of the wave as it passes the 3 different mediums. Because the incident wave will be reflected back and transmitted through the 3 mediums.

If anybody could help me. Specially how to express the electric and magnetic field in medium 2 (slab), I would be extremely grateful!

 

[RUber]

Enforce your boundary conditions at each interface. Is there a reflective backing on the slabs, or will there be some transmitted field on the back side as well?

In each region you will have $E_{in} + E_{ref}$ If you know the material parameters, it is tedious but straightforward.

 

[J_Physics]

Thank you.

As you say, it is tedious but straightforward. I have no problem doing it in the case of it simply having a plane wave from one dielectric material to another. But when I have a slab I, for some reason, get stuck.
There is no reflective backing I think, but I'm not sure. If there is, does something change?

The plane wave will be incident on the first region and will also be reflected there. It will transmit into region 2, which is the slab. The plane wave will be affected due to the thickness of the slab as well. Finally, it will transmit through region 3. I really can't understand why I cannot express the electric and magnetic field in the region that contains the slab.

 

[RUber]

The only change with the reflective backing is that you end up with a full reflection at some point.
Are you saying normal incidence when you say the wave is moving in the +z direction? Without slab orientation, it is difficult to tell.

What progress have you made so far? From what I can tell, you should end up with an infinite series of reflections, but usually one bounce-back is sufficient depending on your error tolerance.

I assume you are having no problems calculating your reflection coefficients between the media.
I am working out of Belanis (Ch. 5), the biggest trick is that when you match coefficients, for a boundary at z = 0, this is easy. For a boundary at z=d, you end up with a coefficient that looks like exp(-j k c d) where c is some constant to make the boundary conditions match. You have to enforce both the continuity and transverse conditions to find all the coefficients, or you can look for a formulaic expression--which is often more confusing that working it out.
Find the transmitted field from region 1 into region 2, the reflected part of that from the boundary with region 3, and ... if necessary ... the re-reflected part of that field at the boundary from region 2 back to region 1. 3 terms should be enough to get a good approximation since the coefficients will be decreasing quickly.
To be really precise, you can make an infinite sum, but most people truncate after just a few terms.

 

[J_Physics]

Yes. I am talking about a plane wave at normal incidence moving in the +z-direction. The simplest case.

As you say, I do not have to worry about an infinite series of reflection nor do I have to make an infinite sum. One bounce back is enough. I am not attempting an experiment, I'm just looking at this case from a theoretical point view.

The transmitted wave from region 1 to region 2 at z = 0 should just be a straightforward transmitted wave E_tr right?

It is at the boundary z = d that I get stuck. The wave that transmits into region 3 and reflects back into region 2. Also can you say that the incident wave in region 2 will look like:

E_inc= E_0I * e^{i(k_2z - wt)} *e^{-i k_2 d} ?

Ps: I am sorry, but I do not know how to write a LaTex code into these threads..

 

[RUber]

Let's work in the time-harmonic system, for now. And to type LaTeX, just type a double hash mark $\#\#$.
Given an incident plane wave $E_{inc} = E_0 e^{-i k_1 z }$, incident on the slab at z = 0, $\Gamma_{1,2} = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1 } = \frac{E_{r}}{E_{inc}}$
So the transmitted wave will look like $E_{t} = (1+\Gamma_{1,2} ) E_0e^{-i k_2 z }$
When that wave gets to z= d, it will look like $(1+\Gamma_{1,2} )E_0 e^{-i k_2 d }$
Let $E_1 = (1+\Gamma_{1,2} )E_0$ and $\Gamma_{2,3} = \frac{\eta_2 - \eta_2}{\eta_3 + \eta_2 }$
You can define a reflection coefficient away from z = zero by the wave form:
$\Gamma(z = d) = \frac{E_{r}(d)}{E_{inc}(d) }= \frac{\Gamma_{2,3} E_1 e^{ i k_2 d }}{E_1 e^{-i k_2 d }} =\Gamma_{2,3}e^{i k_2 2d }$
And the transmitted field then will be:
$(1+\Gamma_{2,3}e^{i k_2 2d }) E_1 e^{ -i k_3 z }$

 

[J_Physics]

Thank you. I am with you that E_t will have that look at z=0. Just to be crystal clear though:
Why ##(1+\Gamma_{1,2})?

 

[RUber]

That's the basic formula for the transmission coefficient, right?
$T = 1+ \Gamma$

 

[J_Physics]

Yeah, you're correct. I was just confused because of the fact that the transmission and reflection coefficient equal 1. As in: R+T = 1

Thanks for the push. One final question (hopefully): How do I best express the electric and magnetic field, in your opinion, in the 3 different regions? As exponentials or in a sine-cosine notation? Maybe I should switch to sine-cosine notation in the slab region because of how the waves behave there..what do you think?

 

[RUber]

Right. Normally you will see "standing waves" expressed as sin/cos notation since they are only present in certain modes.
For these problems, I normally like to keep them in exponential form because i find it easier to keep track that way.

 

[J_Physics]

Thanks. I agree...it does become easier to just keep them in the exponential form..

May I ask a bit more of an intricate question? What if we have a slab with a complex refractive index? As in: n = n_1+in_2
How then would you go about expressing the electric and magnetic field?