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The Airy disk intensity formula is given in the mathematical details section of the Wikipedia Airy disk article in terms of the Bessel function of the first kind.

I am interested in the asymptotic behavior of the tail of this function. Is it exponential or powerlaw? If powerlaw, what power? Can you give a Taylor series expansion? TIA.

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migrated from Dec 9 '12 at 23:52

This question came from our site for active researchers, academics and students of physics.

(Moderators: This is a very mathematical question. Feel free to migrate it to Math SE if you think it will get better answers there.) – Jim Graber Dec 9 '12 at 21:52
up vote 3 down vote accepted

Answering one wiki entry with another: The asymptotic forms of the Bessel functions are described here. From the article, $$ J_\alpha(x) \sim \sqrt{\frac{2}{\pi x}} \cos\left(x - \frac{\alpha\pi}{2} - \frac{\pi}{4}\right) $$ for large $x$, so $J_1(x)/x$ asymptotically approaches a sinusoid with power-law envelope proportional to $x^{-3/2}$. Thus the Airy pattern in intensity, $$ I(\theta) = I_0 \left(\frac{2J_1(x)}{x}\right)^2, \quad x = ka\sin(\theta), $$ goes as the square of a sinusoid with an $x^{-3}$ envelope. Here $k$ is the wavenumber of the light, $a$ is the radius of the aperture, and $\theta$ is the angle away from the normal to the aperture.

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Thanks for the very useful answer. All those sines and cosines confused me. – Jim Graber Dec 10 '12 at 14:50

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