Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

The `fft2`

function transforms
2-D data into frequency space. For example, you can transform a 2-D
optical mask to reveal its diffraction pattern.

The following formula defines the discrete Fourier transform *Y* of
an *m*-by-*n* matrix *X*.

$${Y}_{p+1,q+1}={\displaystyle \sum _{j=0}^{m-1}{\displaystyle \sum _{k=0}^{n-1}{\omega}_{m}^{jp}{\omega}_{n}^{kq}{X}_{j+1,k+1}}}$$

*ω _{m}* and

$$\begin{array}{l}{\omega}_{m}={e}^{-2\pi i/m}\\ {\omega}_{n}={e}^{-2\pi i/n}\end{array}$$

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

Computing the 2-D Fourier transform of *X* is
equivalent 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 other
words, the command `fft2(X)`

is equivalent to ```
Y
= fft(fft(X).').'
```

.

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.

n = 2^10; % size of mask M = zeros(n); I = 1:n; x = I-n/2; % mask x-coordinates y = n/2-I; % mask y-coordinates [X,Y] = meshgrid(x,y); % create 2-D mask grid R = 10; % aperture radius A = (X.^2 + Y.^2 <= R^2); % circular aperture of radius R M(A) = 1; % set mask elements inside aperture to 1 imagesc(M) % plot mask axis image

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.

```
DP = fftshift(fft2(M));
imagesc(abs(DP))
axis image
```

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.

```
imagesc(abs(log2(DP)))
axis image
```

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

`fft`

| `fft2`

| `fftn`

| `fftshift`

| `ifft2`