Effect of the Finite Record Length Leakage and Apodization

4 Effect of the Finite Record Length Leakage and Apodization 

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) 

According to the convolution theorem of Fourier analysis, the Fourier transform of a product of two functions is given by the convolution (here indicated by the symbol ) of their individual Fourier transforms. Hence, the effect of multiplying 1(8) by the boxcar function D(8) is to yield a spectrum that is the convolution of the Fourier transform of 1(8) measured with an infinitely long retardation and the Fourier transform of D(8). The Fourier transform of 1(8) is the true spectrum 5(v), while the Fourier transform of D(8), /(v), is given by 

Therefore, the spectrum 5/(v) measured with a finite length retardation is described by 

The result is that the Fourier transform of a monochromatic source is not an infinitely narrow line, but has the shape of the (sinx)/jc function. As shown in Fig. 5.3a right, this function is centred about v = 0 and intersects the v axis at V = n/2l, where n = 1, 2, 3. so that the first intersection occurs at a wave-number 1/2/. Obviously, the main maximum at v = 0 has a series of negative and positive side lobes or feet with diminishing amplitudes. These side lobes cause a leakage of the spectral intensity, i.e. the intensity is not strictly 

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Effect of the Finite Record Length Leakage and Apodization 47 Table 5.1. Coefficients of the Norton-Beer apodization functions. 

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