Circular Diffraction

Diffraction from a Circular aperture The intensity distribution 15! 1 = 1.22 ' a has a ﬁrst minima at The region inside of this is called the “Airy disk”. The size of the airy disk is the minimum size a circular lens can focus light to. If this is larger than any other aberrations the lens or optical system is said to be “diffraction. Fraunhofer diffraction from a circular aperture The 2D Fourier transform of a circular aperture, radius = b, is given by a Bessel function of the first kind: 1, 11 Jkbz FT Circular aperture x y kbz where is the radial coordinate in the x 1-y 1 plane. 22 xy 11 0 7.5 15 A plot of J 1(r)/r first zero at r = 3.83. Fraunhofer diffraction by a circular aperture. Lecture aims to explain: 1. Diffraction problem for a circular aperture 2. Diffraction pattern produced by a circular aperture, Airy rings. Importance of diffraction for imaging: Rayleigh criterion.

2-D Fourier Transforms

The fft2 function transforms2-D data into frequency space. For example, you can transform a 2-Doptical mask to reveal its diffraction pattern.

Two-Dimensional Fourier Transform

The following formula defines the discrete Fourier transform Y ofan m-by-n matrix X. ωm and ωn arecomplex roots of unity defined by the following equations.

i is the imaginary unit, p and j areindices that run from 0 to m–1, and q and k areindices that run from 0 to n–1. The indicesfor X and Y are shifted by 1in this formula to reflect matrix indices in MATLAB®.

Circular Diffraction Grating

Computing the 2-D Fourier transform of X isequivalent to first computing the 1-D transform of each column of X,and then taking the 1-D transform of each row of the result. In otherwords, the command fft2(X) is equivalent to Y= fft(fft(X).').'. Circular Diffraction Patterns Mastering Physics

In optics, the Fourier transform can be used to describe the diffraction pattern produced by a plane wave incident on an optical mask with a small aperture . This example uses the fft2 function on an optical mask to compute its diffraction pattern.

Create a logical array that defines an optical mask with a small, circular aperture.

Circular Diffraction Equation

Use fft2 to compute the 2-D Fourier transform of the mask, and use the fftshift function to rearrange the output so that the zero-frequency component is at the center. Plot the resulting diffraction pattern frequencies. Blue indicates small amplitudes and yellow indicates large amplitudes.

Circular Diffraction Grating

To enhance the details of regions with small amplitudes, plot the 2-D logarithm of the diffraction pattern. Very small amplitudes are affected by numerical round-off error, and the rectangular grid causes radial asymmetry.

References Fowles, G. R. Introductionto Modern Optics. New York: Dover, 1989.