A Reflectivity with gradient in refractive index

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Determining reflectance with a gradual change in refractive index can be complex, as traditional formulas for sharp discontinuities do not apply. The discussion suggests using Hamiltonian optics or the Eikonal equation, although specific references are scarce. When the refractive index increases, the power reflection coefficient can be calculated using the formula Rpower = [(Z2 - Z1) / (Z2 + Z1)]^2, yielding a maximum of 25% power reflection in this scenario. If the index transition is gradual, particularly if exponential, the reflected power could approach zero. Gradient index optical fibers are highlighted as relevant applications, with additional resources available for further exploration.
thepolishman
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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.
 
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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.
 
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thepolishman said:
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.
 
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.
 
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Thanks!
 
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.
 
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