# Calculating reflection of light when the refractive index changes continuously

Suppose you have 2 materials, one with refractive index $n_1$ and the other with refractive index $n_2$, and a plane-wave coming from the first material hits the interface with an incident angle of $0^\circ$.

Fresnel tells us that the reflected power will be $$r=\frac{n_1-n_2}{n_1+n_2} \Rightarrow R=(\frac{n_1-n_2}{n_1+n_2})^2$$

Now, if you have a set of materials each with thickness $d_i$ and refractive index $n_i$, you can use the simple formula above and interference to calculate the net reflection and net transmission (for example, one can multiply the matrices associated with the transmission and reflection of each interface and the matrices of propagating in each material).

I'm having trouble with a similar but different kind of problems. I have this optical fiber with refractive index $n_0$, and at some point along the fiber the refractive index changes periodically and continuously: $$n(x)=n_0+\epsilon \cdot \sin(\frac{2\pi}{d}x) ,\ \ \epsilon<<1$$

Where the wavelength $\lambda$ is not negligible compared to the period $d$. After N periods the refractive index returns to be $n_0$.

The question is: how do I calculate the net reflection and transmission of such grating? The refractive index varies continuously, not in discrete steps for which I can use Fresnel's equations.

Thank's a lot!

• You might consider using variational calculus. This would certainly allow you to solve the ray tracing problem for continuous index of refraction; whether or not it is useful for the problem you have in mind is not clear, but it seems like an obvious thing to look up. – ZeroTheHero Jan 21 '17 at 15:48
• You may find section 3 of this paper a good place to start. They are trying to solve the inverse problem - but start by solving a problem similar to the one you are asking about. – Floris Jan 21 '17 at 15:56
• @ZeroTheHero that's not my problem :/ . My problem is completely one dimensional. – Ofek Gillon Jan 21 '17 at 18:58
• @Floris thank you but unfortunately it didn't have the answer, do you have any tips for searching relative info? – Ofek Gillon Jan 21 '17 at 19:02
• May be this would be useful : "Principles of Optics" Max Born-Emil Wolf, 7th Edition 1999, first published 1959. '$\:\S\:$ 1.6 Wave propagation in a stratified medium. Theory of dielectric films' and especially '$\:\S\:$1.6.5 Periodically stratified media'. – Frobenius Jan 21 '17 at 20:45