# Reflectivity with gradient in refractive index

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• thepolishman
In summary, the question is how to determine the amount of reflectance that occurs when there is a gradual change in refractive index, and if there is a formula similar to R=[(1-sqrt(e_r))/(1+sqrt(e_r))]^2 for sharp discontinuities. The answer is that when the index of refraction is increased, it corresponds to a reduction in the wave impedance of the medium, and the power reflection coefficient can be calculated using [(Z2 - Z1) / (Z2 + Z1)]^2. The worst case scenario is 25% power reflection, but it could be near zero for certain thicknesses and if the transition is gradual or exponential. There is also a lot of information available
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.

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.

vanhees71
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.

thepolishman
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.

## 1. What is reflectivity with gradient in refractive index?

Reflectivity with gradient in refractive index refers to the phenomenon where the refractive index of a material changes gradually over a distance, causing a gradual change in the amount of light reflected at the interface between two materials with different refractive indices.

## 2. How does reflectivity with gradient in refractive index occur?

Reflectivity with gradient in refractive index occurs due to the gradual change in the refractive index of a material, causing a gradual change in the angle of light as it passes through the material. This results in a gradual change in the amount of light reflected at the interface between two materials with different refractive indices.

## 3. What is the importance of reflectivity with gradient in refractive index?

Reflectivity with gradient in refractive index is important in many applications, such as in optical coatings and anti-reflective coatings. It can also be used to control the amount of light reflected in optical devices, such as lenses and mirrors.

## 4. How is reflectivity with gradient in refractive index measured?

Reflectivity with gradient in refractive index can be measured using various techniques, such as ellipsometry, interferometry, and spectroscopy. These methods involve measuring the change in light intensity or phase as it passes through a material with a gradient in refractive index.

## 5. Can reflectivity with gradient in refractive index be controlled?

Yes, reflectivity with gradient in refractive index can be controlled by adjusting the refractive index of a material or by changing the gradient of the refractive index. This can be achieved through various methods, such as doping, ion implantation, and chemical vapor deposition.

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