# Snell's law for an interface with variable refractive index

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1. Oct 18, 2015

### Vaibhav Sahu

Consider an interface along x axis which seperates two media. The medium below y = 0 is air or vacuum and light is incident from this medium onto the surface. The refractive index of the medium above y = 0 varies with x as some function of x : μ = f(x). Does the Snell's law still hold good ??
If so please prove it using Huygen's principle. Also specify the path of the refracted ray.

Last edited: Oct 18, 2015
2. Oct 18, 2015

### Philip Wood

Set up not totally clear to me. Can we say that the plane z=0 (which of course includes the x-axis and y-axis) separates the media? Do you really mean that mu = f(x) rather than f(z) ?

What happens will depend on whether mu varies substantially over a distance of one wavelength, or whether the variation is small over a wavelength. The first case will be difficult to deal with, and, I'd guess, beyond the capabilities of Huygen's principle and calling for Maxwell's equations. The second case might be easier…

Now let wiser heads respond ...

3. Oct 18, 2015

### Vaibhav Sahu

Considering the setup in a plane might not be different. Still I'll clear it a bit. Here's an image.https://scontent-sin1-1.xx.fbcdn.net/hphotos-xaf1/v/t34.0-12/12167417_906070652794805_1640112228_n.jpg?oh=c670e33f48bae9df45946b176414dfe5&oe=5626B2E8

4. Oct 18, 2015

### Philip Wood

I note that you've now changed x=0 to y=0 on your original post! Your post now makes more sense!

5. Oct 18, 2015

### lightarrow

If the refraction index varies, which value do you take to verify Snell's law?

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lightarrow

6. Oct 18, 2015

### Vaibhav Sahu

I'm not quite sure about it. But if the ray strikes the surface at some point x = a, shouldn't the refractive index to be considered be the value μ = f(a) while applying Snell's law, if it is applicable.

7. Oct 19, 2015

### DrDu

8. Oct 19, 2015

### lightarrow

Ok, but if you do this you cannot use the initial (when the beam enters the material) and final (when the beam exit the material) values of the angles (*), you have to use the (variable) angles at every point of it; then Snell's law should (maybe :-) ) written as: $n(\theta) sin(\theta) = n(\theta+d\theta) sin(\theta+d\theta)$.

(*) Edit: actually it seems possible.

$n(\theta) sin(\theta) = n(\theta+d\theta) sin(\theta+d\theta)$

means, developing at first order $n(\theta+d\theta), sin(\theta+d\theta)$, making the product and neglecting the second order differential:

$n(\theta) sin(\theta) = n(\theta) sin(\theta) + [n'(\theta) sin(\theta) + n(\theta) cos(\theta)]d\theta$

where $n'(\theta) = dn(\theta)/d\theta$

and simplifying:

$n'(\theta) sin(\theta) + n(\theta) cos(\theta) = 0$

Solving the differential equation:

$n(\theta) sin(\theta) = n(\theta_0) sin(\theta_0)$

which is really amazing, at least for me!

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lightarrow

Last edited: Oct 19, 2015
9. Oct 19, 2015

### Philip Wood

You've retrieved the well known result $$n_1\ sin \theta_1 = n_2\ sin \theta_2$$

10. Oct 19, 2015

### Vaibhav Sahu

Even after we consider that the Snell's law holds, I still face problems drawing the path of the ray inside the medium. I am not sure of what has to be done inside the medium. It appears that the next refraction will be from a surface perpendicular to the plane we considered i.e. x = 0 if I consider the medium to be divided into planes with the same refractive index.

11. Oct 19, 2015

### DrDu

Yes, that's the way to go. You can then also let the thickness of sheets got to 0 with the difference in refractive index also scaling to 0 to obtain the continuous distribution.

12. Oct 19, 2015

### lightarrow

Yes, it's what I said. Isn't amazing that Snell's law is valid even considering the initial and the final values of angles and refractive indexes for a medium with variable index?

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lightarrow

13. Oct 19, 2015

### Philip Wood

Yes, it's a nice result. Can be thought of as :

n
sin θ = constant for a given light path

in a medium through which n varies in one direction only (θ being the angle between that direction and the direction of the light path). But you knew that!

14. Oct 19, 2015

### Vaibhav Sahu

We can indeed take μ sin θ = constant for the whole path. But as I was saying, the next refraction that takes place inside the medium is through a surface that is perpendicular to the surface we had considered earlier (y = 0). This makes the angle to be considered different.