Fraunhoffer diffraction
Function 
Fourier transform 
Rectangular (h) 
sinc(uh/2) 
Triangle (2hbase) 
sinc2(uh/2) 
Dirac 
unity 
Gauss (s) 
Gauss(s^{1}) 
Series of Dirac (b spacing) 
Series of Dirac (2pi/b spacing) 
A way to describe how light behaves when it goes through the slits of a
grating is offered by the Fraunhofer diffraction according to which a
wave is diffracted into several outgoing waves when passed through an
aperture, slit or opening. Fraunhofer diffraction occurs when both
incident and diffracted waves are plane. To create such a situation,
one must make the distance from the light source to the diffracting
obstacle to the observation point large enough to neglect the curvature
of incident and diffracted light. In other words, when the light
reaches the diffracting aperture, the spherical wave front should be
large enough that it is virtually a planar wave front (basically a
flat, vertical line). The rays must then be parallel, or close to
parallel, as they reach the diffracting object. The point at which the
diffraction pattern is observed becomes the Fraunhofer plane. The
Fourier Transform, as it turns out, proves to be a powerful tool when
it comes to describing and analysing diffraction patterns in the
Fraunhofer plane. The Fourier transform decomposes a function into a
continuous spectrum of its frequency components, and the inverse
transform synthesises a function from its spectrum of frequency
components. A useful analogy is the relationship between a set of notes
in musical notation (the frequency components) and the sound of the
musical chord represented by these notes (the function/signal itself).
Using physical terminology, the Fourier transform of a signal x(t)
can be thought of as a representation of a signal in the "frequency
domain"; i.e. how much each frequency contributes to the signal. This
is similar to the basic idea of the various other Fourier transforms
including the Fourier series of a periodic function. Here are examples
of some functions and their Fourier transforms.
Convolution: The convolution, is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. The Fourier transform translates between convolution and multiplication of functions. If f(t) and h(t) are integrable functions with Fourier transforms F(ω) and H(ω) respectively, and if the convolution of f and h
exists and is absolutely integrable, then the Fourier transform of the
convolution is given by the product of the Fourier transforms F(ω) H(ω)
(possibly multiplied by a constant factor depending on the Fourier
normalisation convention). In the normalisation convention we have used
here, this means that if
where the cross denotes convolution operation, then

Concerning
a grating, it can be seen as n repetitions of a single feature
(Rect(a)) over a total width of B (Rect(B)), which can be represented
through the notation of the Fourier Transform and the convolution as
follows:
We then have a diffraction image, which means repetitions of the
diffraction image of B with intensity modulation by diffraction image
of a single feature.
First published on the web: 8 Decembre 2007
Authors: Aranka Derzsi and Giovanni Lotito. The text is based on a lecture given by Thomas Nelis at the first Gladnet training course in Antwerp Sept. 2007
