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Fraunhoffer diffraction

Function

Fourier transform

Rectangular (h)

sinc(uh/2)        

Triangle (2h-base)

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

Convolution definitionwhere the cross denotes convolution operation, then

Fourier Transform of Convoltion


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: Grating in Space


Grating in angles
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.

Marie Curie ActionsFirst 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