Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

## Diffraction pattern from a grating: the convolution theorem

The convolution theorem can be used to greatly simplify the analysis of diffraction experiments. The formal treatment of convolution and Fourier transforms is beyond the scope of this package, but the concept is still very useful.

The diffraction pattern arising from an 'object function' f is the Fourier transform of f. We will use the notation F(f) to mean 'the Fourier transform of the function f'.

The convolution of two functions f and g is denoted f g. The convolution operation involves 'mapping' the function g onto the function f, illustrated graphically below: The convolution theorem is as follows.

The Fourier transform of the convolution of two functions is equal to the product of their separate Fourier transforms:

F(f g) = F(f)F(g)

The Fourier transform of the product of two functions is equal to the convolution of their separate Fourier transforms:

F(fg) = F(f) F(g)

In the analysis of diffraction experiments, the convolution theorem is very useful. It allows diffraction patterns from complex objects to be explained in terms of the convolution and multiplication of simple object functions with simple diffraction patterns.

For example, consider a one-dimensional diffraction grating with a finite number of slits of width w. We wish to describe the grating in terms of simpler objects with diffraction patterns that are easy to calculate. These objects are:

 OBJECT INTENSITY OF DIFFRACTION PATTERN A  an infinite array of infinitesimally thin slits an infinite 'comb' function with infinitely sharp lines B  an aperture with the same dimensions as the grating a narrow 'sinc' function C  a single slit of width w a wide 'sinc' function

The grating and its corresponding diffraction pattern are constructed as follows:

First, construct a finite array of infinitely narrow slits, taking the PRODUCT of objects A and B: The diffraction pattern associated with this is the CONVOLUTION of the diffraction patterns from objects A and B: A diffraction grating of finite size with slits of width w can be constructed by CONVOLUTING the finite array of infinitely narrow slits (labelled D) with the single slit of finite width (object C): The diffraction pattern from the grating is the PRODUCT of the diffraction pattern from object D and the diffraction pattern from object C: 