Unified Optical Scanning Technology

# Chapter 2 - Scanning Theory and Processes

 2.1 THE POINT SPREAD FUNCTION AND ITS CONVOLUTIONAs Introduced in Chapter 1, scanning transforms a multidimensional function of spatial coordinates and time into a signal that is uniquely a function of time. Or it may perform the reverse transformation from a signal to an output spatial pattern. The first operation represents "reading" of a pattern to derive a signal for a data channel, and the reciprocal operation represents 'writing,' to form the output pattern from its corresponding signal. The electrical signal theory and the optical image theory are fundamentally analogous, motivating the worker who is trained in one field to appreciate the common relationships that unify both fields [John].Establishing first the basic distinction between amplitude and intensity functions, Figure 2.1 illustrates one pair of functions of the variable x. The dashed curve forms the amplitude function A(x) and the solid curve, the intensity function I(x) such thatin which * denotes the complex conjugate and the angular brackets represent the time average of the extremely high frequency (1014-1015Hz) of the electric vector of an optical electromagnetic field. The analogous equivalent to A(x) is electrical voltage or current, which when squared to I(x) is proportional to power and, over time, to energy. In optics, theFig. 2.1 Amplitude A(x), dashed curve, and intensity I(x), solid curve, of variable x, where I(x) = A(x)2. Illustrating a typical PSF distribution (not to scale) derived from a uniformly illuminated rectangular aperture, where the nulls (1,2,3) are in units of Fλ. A(x) represents the coherent PSF and I(x) the incoherent PSF. I(x) represents the optical lines of flux, which when incident upon an area represent flux density or power density. The response of all photosensitive materials and optical detectors, the eye included, is to the intensity function.Figure 2.1 is recognized in the electrical domain as the 'impulse response' of a linear system to a Dirac delta function (unit impulse) and in the optical field as the 'point spread function' (PSF) image response to an ideal point object in a linear optical system. Because the PSF actually extends in two spatial dimensions (x and y), a more direct analogy to the one-dimensional signal impulse response is the one-dimensional 'line spread function' image of an ideal line object [D&S].

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