We give an analytical basis for the theory of optimal beam splitting by one-dimensional gratings. In particular, we use methods from the calculus of variations to derive analytical expressions for the optimal phase function.
Industrial, military, medical, and research and development applications of lasers frequently require a beam with a specified irradiance distribution in some plane. A common requirement is a laser profile that is uniform over some cross-section. Such applications include laser/material processing, laser material interaction studies, fiber injection systems, optical data/image processing, lithography, medical applications, and military applications. Laser beam shaping techniques can be divided in to three areas: apertured beams, field mappers, and multi-aperture beam integrators. An uncertainty relation exists for laser beam shaping that puts constraints on system design. In this paper we review the basics of laser beam shaping and present applications and limitations of various techniques.
Edge detection and enhancement are widely used in image processing applications. In this paper we consider the problem of optimizing spatial frequency domain filters for detecting edges in digital pictures. The filter is optimum in that it produces maximum energy within a resolution interval of specified width in the vicinity of the edge.We show that, in the continuous case, the filter transfer function is specified in terms of the prolate spheroidal wave function. In the discrete case, the filter transfer function is specified in terms of the sampled values of the first-order prolate spheroidal wave function or in terms of the sampled values of an asymptotic approximation of the wave function. Both versions can be implemented via the fast Fourier transform (FFT). We show that the optimum filter is very effective for detecting blufred and noisy edges. Finally, we compare the perfor-
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