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

PreviousNext

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 Convolution symbol g. The convolution operation involves 'mapping' the function g onto the function f, illustrated graphically below:

Diagram illustrating the convolution operation

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 Convolution symbol 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) Convolution symbol 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.

Diagram of a one-dimensional difraction grating

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 Diagram of an infinite array of infinitesimally thin slits Diagram of an infinite ‘comb’ function with infinitely sharp lines
 

an infinite array of infinitesimally thin slits

an infinite 'comb' function with infinitely sharp lines

B Diagram of an aperture with the same dimensions as the grating Diagram of a narrow ‘sinc’ function
 

an aperture with the same dimensions as the grating

a narrow 'sinc' function

C Diagram of a single slit of width w Diagram of a wide ‘sinc’ function
 

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:

Diagram illustrating product of objects A and B

The diffraction pattern associated with this is the CONVOLUTION of the diffraction patterns from objects A and B:

Diagram illustrating diffraction pattern resulting from convolution of 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):

Diagram illustrating convolution of a finite array of infinitely narrow slits with a single slit of finite width

The diffraction pattern from the grating is the PRODUCT of the diffraction pattern from object D and the diffraction pattern from object C:

Diagram illustrating diffraction pattern resulting from convolution of objects D and C