Snell's law for an interface with variable refractive index

[Vaibhav Sahu]

Consider an interface along x-axis which separates 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.

 

[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 ...

 

[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

 

[Philip Wood]

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

 

[lightarrow]

Consider an interface along x-axis which separates 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.

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

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lightarrow

 

[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.

 

[DrDu]

As pointed out by phillip wood, this becomes problematic if the variation is on the order of the wavelength of light. For variations on larger distances, you may have a look at GRIN optics: https://en.wikipedia.org/wiki/Gradient-index_optics

 

[lightarrow]

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.

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

 

[Philip Wood]

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

 

[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.

 

[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.

 

[lightarrow]

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

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

 

[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!

 

[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.