# 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*.

*ω _{m}* and

*ω*arecomplex roots of unity defined by the following equations.

_{n}*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).').'`

.

### 2-D Diffraction Pattern

### 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 [1]. 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

[1] Fowles, G. R. *Introductionto Modern Optics*. New York: Dover, 1989.

## See Also

`fft`

`fft2`

`fftn`

`fftshift`

`ifft2`