1. Separable functions University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo A function of more than one variable is separable with respect to a certain coordinate system if it can be written as a product of functions, where each function depends on only one variable. Cartesian coordinates: Polar coordinates: Separable functions are easier to deal with and, in particular, the Fourier transform may be written as: (1.22)

2. University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo In polar coordinates, the expression of the Fourier transform of a separable function is not as straightforward as it is for rectangular cartesian coordinates. However, it is possible to show that: where: and is the Hankel transform operator of order k, defined by: (1.25) (1.23) (1.24)

3. Functions with circular simmetry University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo These functions represent a special class of separable functions in polar coordinates that depend only one the variable r, i.e. Many problems in optics have this kind of simmetry, hence we consider more in detail the expression of the Fourier transform of functions with circular simmetry. We start from: then we consider the polar representation of both the (x,y) and the (f x,f y ) planes: (1.28) (1.27) (1.26)

4. University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo Introducing the change of variables into our original transform, we obtain: Bessel function identity: And: The Fourier transform itself is circularly simmetric. It is also possible to show that the inverse Fourier transform is: Fourier-Bessel transform OR Hankel transform of zero order (1.32) (1.31) (1.30) (1.29) (1.33)

5. University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo We observe that: 1)There is no difference between direct and inverse Fourier-Bessel transform operation for circularly symmetric functions. 2)If is the symbol that denotes the transform operation, then: 3)The similarity theorem becomes: (1.34) (1.35)

6. Useful functions University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo We will frequently use the following functions: The Fourier transform of two-dimensional functions that are separable and include some of the functions listed above are given in table 2.1 of the textbook.

7. Spatial frequency University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo One interpretation of the inverse Fourier transform considers the superposition of the elementary waves of the form: that are defined over all the (x,y) plane. However, in some practical cases we observe images that contain parallel lines and certain fixed spacing. Recalling our previous interpretation of the direction of an elementary wave and the spacing between wavefront lines, we are tempted to associate the values of (f x,f y ) to a certain region of an image. This association occurs with the idea of local spatial frequencies. Consider a complex-valued function of the form: where a(x,y) is a real non-negative amplitude and  x,y) is a real phase distribution. (1.36)

8. University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo We assume that a(x,y) is a slowly varying function and define local spatial frequency the pair (f lx,f ly ) given by For example, if we apply this definition to an elementary wave we obtain As a second example, we consider a space-limited quadratic-phase exponential function defined all over the (x,y) plane Rectangle of dimensions (1.37) (1.38) (1.39)

9. University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo In this case, the local spatial frequencies depend on the location. In fact, they are different from zero only inside a rectangle of dimensions and correspond to: If we computed the exact spectrum of g(x,y), we would find that the spectrum is almost flat inside the rectangle and almost zero outside. So, in this case, the local spatial frequencies provide a good approximation of where significant values of the spectrum are. In general: 1)good agreement is found if f(x,y) is sufficiently slow-varying, i.e. it can be well approximated with only the first three terms of its Taylor series (constant value plus the two first partial derivatives). 2) local spatial frequencies of a coherent optical wavefront correspond to the ray directions of the geometrical optics description of that wavefront. (1.40)