In class, we have derived the formula for a single slit diffraction (slit length $a$) and it seems that the same equations are used when talking about diffraction around an obstacle of width $a$ (like a human hair). Why's that?
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1$\begingroup$ Because they are complementary functions - if you add the slit function plus the block (obstacle) function you will get the original source function back again. $\endgroup$– Jon CusterCommented Jan 5, 2016 at 17:59
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$\begingroup$ This is known as Babinet's principle (en.wikipedia.org/wiki/Babinet's_principle) $\endgroup$– Punk_PhysicistCommented Jan 5, 2016 at 22:15
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1 Answer
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The Fraunhofer diffraction pattern you see is the Fourier Transform of the aperture function.
The Fourier Transform of a completely open aperture is a delta function (the light continues in a straight line). Because the Fourier Transform is a linear operation, if you have $\mathcal{F}(a)$ and $\mathcal{F}(1-a)$, their sum must be $\mathcal{F}(1)$ - that is, zero at every point except "straight through".
So the amplitude of the diffraction pattern has the same magnitude but opposite sign at any point in the pattern. But since we only observe intensity, we can't see the opposite sign. And so the two patterns look exactly the same (except for the "forward" direction: obviously a thin slit will let much less light through than the light that diffracts past a hair).
