The many faces of electromagnetic waves In my waves and optics class, we have learned several ways to treat electromagnetic waves: light rays (geometric optics), electromagnetic plane waves, spherical waves, cylindrical waves (2D). One thing still confuses me: How can one determine which method to use when approaching a problem? In other words, when can I treat the wave as light rays? When is the electromagnetic wave a plane wave? When is it a spherical wave?
 A: I will answer this question in two parts:


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*first comparing plane waves, spherical waves and cylindrical waves (which are really the same thing as I will explain).

*how this relates to ray optics (which is completely different).


Plane waves, spherical waves and cylindrical waves are 3 different examples of doing a decomposition of a wavefield. The method is that you represent a general wavefield in terms of an infinite sum over a complete set of basis functions that have some property that is convenient. Plane waves are the most widely used since they are the same as the Fourier decompositions. Plane and cylindrical waves would be more convenient if you have some symmetry in your system, for example for a point source you would use spherical waves, for a line source most likely cylindrical. All of these 3 decompositions contain the same physics (namely light as an electromagnetic wavefield), but represent it in a different mathematical way. Which one you choose depends on the situation and you choose the most convenient one. You can solve most problems with either of these, it just gets messier.
Ray optics is different, since this represents a different approximation. Instead of as a wavefield you represent light as a ray. This for example does not allow you to describe interference effects and is less general than the wavepicture, can still be useful for systems like lenses etc.
