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I have a GRIN lens with the refractive index varying linearly with y, and supposedly this lens tilts the wavefront. enter image description here

Since the rays are travelling normally in both refractive index materials, they stay normal to the plane of the lens at all times. However the wavefront travelling with them tilts because of different speed of light for different indices, and gives a wavefront like this. Is this correct? Or do the rays themselves tilt? I don't think this is a possibility since the rays are always perpendicular to the plane.

If my reasoning is correct, does this satisfy Huygens' Law? It states that the wavefront must be perpendicular to the path of the waves at all times, but this doesn't seem to be the case here, which leads me to thinking its the waves themselves that are tilting.

Diagrams like this can be found on the internet, but I do not understand how are the waves tilting when they enter normally, enter image description here

What am I missing? I found this situation while solving a problem in a book (Pathfinder for Olympiad Physics) but upon thinking a bit more was fell into this confusing situation.

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If by Huygens' principle, properly modified by the obliquity factor, you mean that each point of the wavefront is a source of spherical wavelets then it is not true for a variable index material.

If by Huygens' principle, properly modified by the obliquity factor, you mean that each point of the wavefront is a local point-source of wavelets then it is also true for a variable index material.

The reason for the difference is that a spherical wavefront in a homogeneous medium stays spherical while in a variable index medium the wavefront gets distorted as it propagates. But Huygens' principle taken as to mean that every point of a propagating wavefront is a source of a new wave still holds in this sense. Unfortunately, in an inhomogeneous medium the principle is then less useful because the type of elementary wavelets that are emitted from each point on a wavefront varies form wavefront to another wavefront and it also varies from one point to another point on the same wavefront.

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  • $\begingroup$ Does this means that the wavelets emitted get distorted? In that case does it imply that perpendicularity of travelling waves to the wavefront is not a necessary criterion for Huygens to be followed? $\endgroup$
    – Shivansh Jaiswal
    Sep 19 at 15:41
  • $\begingroup$ The usual definition of the ray vector that it is parallel with the direction of the local propagation of the Poynting vector in an isotropic medium, and it is the gradient of the wavefront function, the latter being the constant phase surfaces. The local gradient is, of course, perpendicular to the local surface tangent. This is true for an inhomogeneous medium as well. Huygens' principle is true still but the constant phase surfaces emitted from any point source are not spherical. $\endgroup$
    – hyportnex
    Sep 19 at 15:42
  • $\begingroup$ Alright, makes sense. The wavelets are no more spherical, so this implies that the diagram in question is right since it follows from Snell's laws and the wavelets distort in such a way that the wavefront is as such. Is this correct? $\endgroup$
    – Shivansh Jaiswal
    Sep 19 at 15:45
  • $\begingroup$ yes, that is right, but do not forget that even Snell's law must be understood in a local sense if the medium is continuously varying. You can only use the conventional form of Snell's law if the medium is piece-wise homogeneous as in a layered material. $\endgroup$
    – hyportnex
    Sep 19 at 15:56
  • $\begingroup$ for a layered material, because of the wavefront distortion, it is usually easier to calculate first the rays via Snell's law and then if still interested calculate the surface to which they are orthogonal, the wavefront itself. The Malus-Dupin theorem assures that those surfaces to which the rays are orthogonal always exist and unique. There are methods to calculate image distortion using rays and others using the wavefront itself. By distortion is meant how closely the rays meet at a point or how closely a wavefront is shrinking to a sphere, etc. $\endgroup$
    – hyportnex
    Sep 19 at 16:09

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