Reflectivity with gradient in refractive index

[thepolishman]

Hey all. Was wondering if anyone knew how I would go about determining the amount of reflectance that occurs when there is a gradual change in the refractive index. For example, if I have a material in air whose refractive index begins at e_r=1 (i.e. it matches the refractive index of the air) and slowly increases to e_r=3, how would I go about determining how much light is reflected?

Normally, I'd use the formula R=[(1-sqrt(e_r))/(1+sqrt(e_r))]^2 when there is a sharp discontinuity. But I'm not sure how to proceed when the change in refractive index is gradual.

 

[tech99]

A similar problem occurs when a radio wave passes through the Ionosphere. I think the method I have seen is to divide the medium into layers and plot the ray as it passes across the boundaries.

 

[Andy Resnick]

Hey all. Was wondering if anyone knew how I would go about determining the amount of reflectance that occurs when there is a gradual change in the refractive index. For example, if I have a material in air whose refractive index begins at e_r=1 (i.e. it matches the refractive index of the air) and slowly increases to e_r=3, how would I go about determining how much light is reflected?

Normally, I'd use the formula R=[(1-sqrt(e_r))/(1+sqrt(e_r))]^2 when there is a sharp discontinuity. But I'm not sure how to proceed when the change in refractive index is gradual.

Interesting question! My first thought is that Hamiltonian optics or the Eikonal equation would be used to solve this problem (with the result in terms of energy diffusion), but I couldn't easily find a reference.

 

[tech99]

I have just read the question more carefully and I see it refers to reflection not refraction.
When the index of refraction is increased, it corresponds to a reduction in the wave impedance of the medium, n1/n2 = Z2/Z1. If this takes place over a small fraction of the wavelength, power will be reflected. The power reflection coefficient will be [(Z2 - Z1) / (Z2 + Z1)]^2. So for the present case, Rpower = [(3 -1) / (3 +1)]^2 = 0.25.

The worst case is therefore 25% power reflection. If the transition is very gradual, and especially if it takes place exponentially, the reflected power could be near zero. It could also be near zero for certain thicknesses of material.

 

[thepolishman]

Thanks!

 

[DaveE]

Check out this book. No, not a recommendation. I haven't read it either. But all of the foundations are there. Particularly at page 48. Plus it's free.

There should be a lot of info available since gradient index optical fibers are a common application addressing exactly this sort of problem.

 

[DrDu]

This classic article by Berry and Mount contains quite some interesting discussion of the reflection in the semiclassical approximation:
https://iopscience.iop.org/article/...ayG5ptSR1zFvUmQUDGSEBuPWkEEul1QNfgj355YaAEt8o