# Sinc function in nature?

• Bassalisk

#### Bassalisk

Its my favourite function.

$\Large{\frac{sin(x)}{x}}$

I first saw it 1 year ago, when we studied limits. I don't know why, but I really like this function.

Can anybody tell me an example, in nature where we have behaviour that has sinc function characteristics.

Fourier transform of aperiodic square wave is this function(give or take few constants).
And this function would be impulse response of lowpass filter.

So a natural process, that can be described with sinc function. Or approximated with it.

Hmm, off top of the head if you shine light through a slit it should diffract like sinc function (the amplitude being sinc function squared) for exact same reason why Fourier transform of aperiodic square is sinc.

Yes yes! How could have I missed that?! The slit experiment. So the slit would be a lowpass filter here, and my light would be a impulse?

Yes yes! How could have I missed that?! The slit experiment. So the slit would be a lowpass filter here, and my light would be a impulse?
Sort of, though I wouldn't describe it this way.
The intensity is given as integral of the phase from the slit to the screen over the slit width. When you are calculating intensity of the light in direction a (in radians off the centre) you're summing the contributions from every point on the slit multiplied by sines of their phases. The factor varies over the slit as sin(x*sin(a)*2pi/wavelength) (edit: better way to put it is to use complex numbers and e^(i*x*sin(a)*2pi/wavelength) for the multiplier). For a slit that is big relatively to wavelength, the relevant a is small and sin(a)~=a . The summing is same as in Fourier transform, essentially.

You can have some pattern of slits, and then shine laser light through and on the screen see Fourier transform* of that slit pattern (provided that the screen is far enough away). Isn't that cool. Doing the Fourier transform with laser.
*the amplitudes, if you want phase you'll need a beam-splitter to add the beam here so that you can see just the imaginary or just the real part. That's actually the principle behind holography. A hologram is sort of frequency-domain image of the object (not exactly so though).

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Sort of, though I wouldn't describe it this way.
The intensity is given as integral of the phase from the slit to the screen over the slit width. When you are calculating intensity of the light in direction a (in radians off the centre) you're summing the contributions from every point on the slit multiplied by their phases. The phase varies over the slit as sin(x*sin(a)*2pi/wavelength). For a slit that is big relatively to wavelength, the relevant a is small and sin(a)~=a . The summing is same as in Fourier transform, essentially.

You can have some pattern of slits, and then shine laser light through and on the screen see Fourier transform of that slit pattern (provided that the screen is far enough away). Isn't that cool. Doing the Fourier transform with laser.

Interesting. I will research this through. Thanks! Gave me a lot to think about.