# Reflection transfer function of a Fabry-Perot cavity

I am trying to calculate the transfer function of a Fabry Perot optical cavity. All the resources I have looked up, do the following. They start with an incident light field $$e^{i\omega t}$$ and write down the reflected light field. The reflected light field will have contributions from :

1. The field that gets reflected from the first mirror.

2. The field that gets transmitted into the cavity, undergoes one roundtrip inside the cavity and then gets transmitted

3. Field that undergoes two roundtrips before getting out and so on.

Now when they write down the phase shifts(wrt to the incident field) undergone by these different components, they consider a phase shift of $$\pi$$ for the first case because it gets reflected from the mirror. However, for the later cases they only consider the phase shift picked up by the light due to the extra distance it covers(corresponding to different number of round trips). ie a phase shift of $$\frac{2\pi}{\lambda}*{\Delta x}$$, where $$\Delta x$$ is $$2L$$ for one round trip and $$4L$$ for two round trips etc, L being the length of the cavity. They do not take into account the phase picked up by reflection from the mirros. For a single round trip, the phase due to reflection from a mirror would be $$\pi$$. For multiple round trips, it would be $$3\pi$$, $$5\pi$$, etc, but they would all amount to $$\pi$$.

To calculate the transfer function you do $$F = \frac{I_{reflected}}{I_{incident}}$$. If I take the extra phase of $$\pi$$ for all the contributions, I get a different answer.

I am confused why would they not consider the phase shift of $$\pi$$ for all the terms in the reflected field.

Good question. Mirror phase shifts are tricky business. But here is what you need to know. The mirrors we are working with here are two sided mirrors. That is, light can be send on it from either side and will mostly be reflected (but a little bit transmitted).

The trick with this $$\pi$$-phase shift is that it only occurs for light incident from one side of the mirror. If light is incident from the other side it will actually collect no $$\pi$$-phase shift. The authors have chosen a convention where reflections inside the cavity cause no phase shift and reflections outside the cavity cause the phase shift. This makes their equations consistent.

See Lasers by Siegman (pages 401-406) for more information about mirror reflection phases and possible convention choices which might affect those phases. The upshot is that, if you choose a convention for your spatial reference planes (from which you reference the spatial phase) such that the transmitted light has zero phase shift then reflection from one side may also have no phase shift but the reflection from the second side will have a $$\pi$$-phase shift relative to the first.

This is a somewhat old question... nevertheless some hopefully useful notes:

1. I think it is important to remember what one is actually doing when calculating reflection functions in Fabry-Perot cavities... solving Maxwell's equations for a (usually purely) dielectric medium that constitutes the mirror/cavity material.
2. In particular, one solves the scattering problem, that is the steady state for a given monochromamtic plane wave incident from one side of the cavity. This input can be seen as an asymptotic boundary condition for the problem.
3. The method the OP describes, that is summing up all the reflection paths between the interfaces and the phase shifts they accumulate, is a method for solving precisely the above problem. It originally comes from the thin-film community and is known as Parratt's formalism. One can also rewrite the recursive summation formula as a matrix multiplication, which is then known as Abele's transfer matrix formalism.
4. After the above general comments, let's get a bit more concrete... The question about phase shifts in the OP depends on how the mirrors are realized. For the simplest case of having a cavity that is a slab of dielectric material the "mirrors" are just given by the interfaces. Their individual transmission and reflection coefficients are given by the Fresnel coefficients. The latter satisfy $$r_\mathrm{left} = -r_\mathrm{right},$$ where the minus sign explains the phase shift of $$\pi$$ the OP is asking about. This sign essentially follows from interface boundary conditions in combination with Maxwells equations.
5. More complex mirrors: In practice, mirrors are often multi-layer material stacks. In the limit of thin mirrors (i.e. away from any resonances inside the mirror itself), one can again treat this element as a set of reflection and transmission coefficients. However, these coefficients do not need to satisfy the left-right condition above.
6. If you are interested in such more general structures, there is a python software package to calculate such reflection/transmission coefficients.

A diagram would help. Lets say light is incident from the top, so the relected light escapes back up.

Ray 0: Reflected at top mirror (1 reflection) Ray 1: Goes into cavity, reflected at the bottom mirror, back to the top, transmitted through top mirror, then escapes (1 reflection)

Ray 2: Into cavity, reflected at the bottom, reflected at the top, reflected at the bottom, escapes at the top. (3 reflections)

Ray 3: (5 reflections)

You should now see that there is always an odd number of reflections, but if your etxra phase from each reflection is $$\pi$$, then the phase you pick up from 1 reflection is the same as from 3 ($$3\pi=full\,period+\pi$$), or from any odd number, so there is no need to account for the extras.

Note also, that you do not have to mess with all these rays. An alternative way of solving this problem is to solve the coupled boundary value problem with scattering boundaries. Personally, I prefer the second method.

• This suggests that the directly reflected light and light leaking out of the cavity are in phase. They aren't; and that is what the question is asking about. Commented Apr 29, 2021 at 5:49