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What i know about snell's law: It is applied when a ray of light meets the interface of some other medium and we can find the fourth quantity if we know any of the three quantities in the following relation

$u_isin(i)=u_rsin(r)$

where $u_i$ & $u_r$ is refractive index of incident & refractive medium respectively and $sin(i) , sin(r)$ are angle the ray of light makes with the normal to the interface in incidence medium and refractive medium respectively

What I want to ask:Can the snell's law be applied (in the case where a ray of light is incident with some angle on the rectangular slab whose refractive index changes with depth of the rectangular slab and refractive index has a linear relation with depth of the slab),between the points,the initial point of contact of ray of light with the slab and the point at some depth say x in the slab considering we know the angle of refraction the light ray makes at depth x.

enter image description here

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  • $\begingroup$ It can be done but an integration would be needed. There is a whole branch called gradient index optics although in a lot of instances the variation of refractive index is not necessarily linear. en.m.wikipedia.org/wiki/Gradient-index_optics $\endgroup$
    – Farcher
    Commented Feb 8, 2016 at 0:26
  • $\begingroup$ @Farcher can't we use it directly without using integration in my given case. $\endgroup$
    – Kartik Watwani
    Commented Feb 8, 2016 at 0:29
  • $\begingroup$ I cannot immediately think how you could do it without integration. If there is a continuous change of refractive index with depth then the ray would be curved as it went though the medium. $\endgroup$
    – Farcher
    Commented Feb 8, 2016 at 0:36
  • $\begingroup$ @Farcher how can we do it with integration? $\endgroup$
    – Kartik Watwani
    Commented Feb 8, 2016 at 0:41
  • $\begingroup$ I think that it is no simple matter. Perhaps with the linear variation and for small angle so that sine theta can be approximated to theta it can be done? $\endgroup$
    – Farcher
    Commented Feb 8, 2016 at 1:23

3 Answers 3

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Yes, you can still use Snell's law if u=u(z), where u in the index of refraction and z is the axis parallel to the normal of the surface.

Depending upon the length scale of the variation you may have to treat it as a differential.

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The eikonal equation should be used when the index of refraction has varies with position; it is derived directly from Fermat's Princaple of Least Time.

OTOH, if you just have slabs of finite thickness, each with its own index of refraction, then just apply Snell's Law multiple times; matrix methods help. Multi-layer hin film coatings can be designed this way.

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This is just an example I created to help you understand how to solve such problems. Snell's law is applicable in such cases. Hope this helps. Don't mind my handwriting though. enter image description here

Sorry I missed the 2 in denominator at last.

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