Can we explain diffraction without using Huygens principle?
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3$\begingroup$ I could do the detailed math for a particular wave equation, but why? Huygens may have found it by intuition, but it is a consequence of the mathematical structure of linear wave equations. Avoiding it just makes everyone's life harder. $\endgroup$– dmckee --- ex-moderator kittenCommented Oct 18, 2014 at 17:05
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Yes. There are more complete developments based on electromagnetic theory. Exact solutions are known for a few cases. Almost all applications rely on approximations, many of which are very good approximations under the right conditions.
But the more complete explanations end up looking like Huygens Principle, and serve to justify it. Huygens' original statement had a few details not quite right, mainly some multiplicative factors. But the essential part, the wavelet construction, was there. Fresnel and Kirchoff used electromagnetic theory to fix the few missing details, and put the principle on firmer theoretical grounding... but still not completely rigorous. Full rigorous solutions are available only for a few special cases.
