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Mathematically, I'm having trouble understanding where we can use what with light. I read somewhere on this site that Huygen's Principle is effectively just taking an expansion of a wave onto the spherical harmonics, (Is Huygen's Principle Axiomatic? comment by gatsu) and on another on of my question, I was told that the image produced from a lens is some how related to a fourier transform of the incoming light. That answer is here: Why can't we perfectly focus light-abberations aside So my question is, why are these two ideas appropriate for light? What is the general statement for either of these? Does the fourier transform always have some relation with lenses, and why does an expansion on the spherical harmonics have a name, is it a fundamental quality of light?

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It is not a fundamental quality of light so much as it is a fundamental quality of anything that can be modelled as a wave. The easiest illustration of this is diffraction by a single slit (this could be diffraction of light sound or any other wave). So if we have plane waves arriving at the slit and we look at any point, P, beyond the slit we can think of the waves arriving at P as a sum of waves arriving from every point within the slit. Let us indicate each point "along" the slit by its distance from one side of the slit, $x$. The width of the slit is $a$ so $x$ runs from 0 to $a$. Let $r(x)$ be the optical path length from a point at $x$ to the point P. So, the contribution by a "piece" of slit at x to the total amplitude at P is

$$dE_P = \frac{E_L dx}{r(x)} \exp \left[ i (kr-\omega t) \right]$$

where $E_L$ is just the amplitude of the plane wave. Now we can make various approximations to simplify this (many optics texts have a good presentation) but the upshot is that after simplifying how $r$ depends on $x$ and on the location of the point P we get something like

$$E_P \sim \int_0^a e^{ik x} dx$$

where $k$ is an "angular spatial frequency" which emerges from the analysis of the geometry. The punch line is that we can think of this as

$$E_P \sim \int_{-\infty}^\infty E_A(x) e^{ik x} dx$$

where $E_A(x)$ is the amplitude at the slit as a function of $x$. For a plain old ordinary slit and waves of uniform intensity falling on it this means $E_A(x)$ is just $E_L$ for $x$ from 0 to $a$ and zero elsewhere. But look at the structure of the above. $E_P$ is just the Fourier transform of the spatial dependence of the amplitude at the slit! The wave "takes the Fourier transform of the shape of the slit" and this is what we see projected on the wall as an interference pattern. Notice that I never assumed this was light. It is true of any wave as long as the wavelength is small compared to the slit size.

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