# Is the plane wave model always valid in reflection and transmission?

my question is related to another one I asked, but I foolishly made that question about several things (experiment, computation, theory) at once so it was confused.

I was talking to my boss about theoretically modeling the reflection and transmission of incident light on a thin film, like this (warning: PDF). Essentially they use a plane wave model with propagation matrices through "slab" (thin film) and "matching matrices" to match the fields across boundaries. In each region (except the final transmitted region) there are waves going in both directions because both are solutions to MW's equations.

That seems to be pretty standard and straightforward to me. There's no inherent length scales in it, so intuitively it would seem to me like it'll work at all dimensions. However, he took one look at it and said that it's wrong to use plane waves here. I didn't entirely understand it at the time (hence, why I'm asking here), but I think he was saying:

• A plane wave is just one part of the solution of dipole radiation, the far field part (true, but why is this important here? Is it because the material is really just dipoles being oscillated and re-emitting?)
• He also mentioned something about the "amplitude not being well defined" I think, if the wave is decaying very quickly. This confused me because I thought if you have a notable decay index, you just find $n = n_r + i n_i \rightarrow k = k_r + i k_i \rightarrow E \propto e^{ikz} = e^{-k_i z}e^{ik_r z}$ and then your amplitude is determined by the exponential decay term at any point.

Can anyone shed some light on this?

• It is true that any arrangement of fields can be locally modeled as a sum over plane waves, so the question reduces to either "Am I satisfied with local correctness?" or "Is the plane wave description correct over a sufficiently extended area in this case?" depending on your problem. – dmckee --- ex-moderator kitten Mar 12 '14 at 15:14
• @dmckee Right, so that was my first thought in response: plane waves form a complete set, so anything else is just some linear combination of them. What do you mean by 'locally' though? Just within the film? – YungHummmma Mar 12 '14 at 15:24
• @dmckee Also, a question related to that: You need the full spectrum of frequencies to create arbitrary functions. So what happens if you shine monochromatic light on it? – YungHummmma Mar 12 '14 at 15:40
• Your boss might have some unspoken reason for thinking that this procedure won't work. Are you modeling an experiment where the light is not well collimated? – garyp Mar 12 '14 at 18:05

## 2 Answers

I realize this is an old question and you have probably moved on...

Flicking through the paper, the thing that strikes me is that the wave is normally incident. Unless that is the geometry you are using, the equations are not going to work for you. While an arbitrary wave front can be thought of as being composed of a smarties of plane waves, those wave will not all be normal to the surface.

A few lifetimes ago I did exactly this sort of calculation as part of my PhD work. See my answer to Make a semi transparent mirror with copper for some details. The procedure is explained in detail in Optical Properties of Thin Solid Films by O. S. Heavens. The paper you link considerably postdates my PhD (and the expiry of my memories of it) but it looks as if it's basically the same calculation, though I seem to remember the procedure in the Heavens book as being a lot simpler - matrices weren't involved.

The calculation assumes light is incident normally, which implies it must be a plane wave otherwise some portion of the wavefront wouldn't be normal.

My recollection is that the calculation is basically very simple - you just match up the electric and magnetic fields at the interface in the usual way. The trick is to start from the far side and work backwards i.e. opposite to the propagation of the light. As I recall the method is trivially extendable to any number of layers, including metal layers, and it correctly describes interference effects e.g. the film acting as an etalon.

I'm afraid I don't remember any more details, but if you can get a copy of Heavens' book from your library it would be worth a look.

• A detail: Suppose the incident light is a superposition of plane waves, each having a wave vector normal to the surface but different frequencies. Is that wave not a plane wave? The phase fronts are planar, I think. Or is "plane wave" a synonym for "monochromatic plane wave"? – garyp Mar 12 '14 at 18:09