Optical path difference finiteness

However, it is to be noted that the Fourier transform integrals have infinite limits while the optical path differences are finite so modifications or approximations must be made. We will use the approximation of the limits between —L and + L where L is the maximum distance of the mirror drive. So... 

The second factor is related to the fact that in real life the interferogram is truncated at finite optical path difference. In addition, in the fast Fourier transform (FFT) algorithm, according to Cooley and TTikey [30], which is used to perform the Fourier transform faster than the classical method, certain assumptions and simplifications are made. The result is that the FFT of a monochromatic source is not an infinitely narrow line. 

Next, let us consider that the finite dimensions of the source impose a limit to the resolving power of the Michdson interferometer in the same way as the finite slit width does for the grating spectrometer. The finite size of the source means that also rays enter the interferometer which are inclined by an angle d to the optical axis (see Fig. 34) hence, a displacement s/2 of the movable mirror produces the path difference s cos d instead of s. The total interferogram is the sum of the contributions from all points of the (circular ) source... 

Equation (5.9) shows that in order to measure the complete spectrum, we would have to scan the moving mirror of the interferometer an infinitely long distance, with (5 varying between -oo and +cx) centimeters. In practice, the optical path length difference is finite. By restricting the maximum retardation to /, we are effectively multiplying the complete interferogram by the boxcar truncation function (see Fig. 5.3a left)... 

---