New PDF release: Applied optics and optical engineering,Vol.III

By R et al Kingslake

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Extra info for Applied optics and optical engineering,Vol.III

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When the zero is moved to infinity, the situation becomes physical; the simplest possible signal is introduced within the aperture. The Dirac delta functions locating the aperture edges have now disappeared. The aperture becomes encoded not by nonphysical markers, but by the signal itself, confined to the interval (a,b). Any dc level introduced in (a,b) renders the edge markers immediately redundant: it automatically defines the aperture in the primary space and translates the zero at the origin along the y a xis to infinity.

To be more precise, the function will be zero on a (n-l)-dimensional line. 28,29] but it is far more complicated to interpret. Of course, the description of the encoding of information in the r space by the complex zeros of a function of class E applies to all complex planes associated with values of d for which the Fresnel approximation is valid. In each complex plane, a 40 suitably perturbed lattice of complex zeros encodes all the information about the scattered field at the distance d, and hence about the object wave U(r).

The consequence of this on the x axis is a smoothing process. For the particular case in which the multiplying function goes to zero at the ends of the interval (a,b), Theorem 2 is satisfied, the function defined in the z space E L1(_00 , 00) and thus converges as a simple limit towards the infinite Fourier transform, the requirement w(a)=w(b)=O being maintained while a .... - oo , b ... oo . The density of zeros increases as this limit is taken ; hence they cannot stay in the vicinity of the real axis, since then the convergence would be possible only as a limit in the mean and not as a simple limit.

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