As an addition to textbooks, it may present some. We wish to describe the grating in terms of simpler objects with diffraction patterns that are easy to calculate. These can be regarded as the respective far-field diffraction patterns. 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.įor example, consider a one-dimensional diffraction grating with a finite number of slits of width w. In the analysis of diffraction experiments, the convolution theorem is very useful. The Fourier transform of the product of two functions is equal to the convolution of their separate Fourier transforms: The Fourier transform of the convolution of two functions is equal to the product of their separate Fourier transforms: The convolution operation involves 'mapping' the function g onto the function f, illustrated graphically below: The convolution of two functions f and g is denoted f g. We will use the notation F( f) to mean 'the Fourier transform of the function f'. The diffraction pattern arising from an 'object function' f is the Fourier transform of f. The formal treatment of convolution and Fourier transforms is beyond the scope of this package, but the concept is still very useful. The convolution theorem can be used to greatly simplify the analysis of diffraction experiments. Far-field is proportional to the 2D Fourier transform of the shape of the aperture L z L x Or: sin() ' ' 2 2 2 ' 2 2 E e e ' dz e dx r j k E r x x x z z z L L jk x L L jkr jk z ff a r r ECE 303 Fall 2005 Farhan Rana Cornell University Fourier Transforms and the Rectangular Aperture Far-Field x 2. Previous Next Diffraction pattern from a grating: the convolution theorem
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