The wave-particle duality is vividly demonstrated by experimental 'Young's fringes' using coherent electron beams under conditions in which the flight time is less than the time between particle emission. Phase plays a similarly fundamental role in matter-wave interferometry, for which the field-emission electron microscope provides ideal instrumentation. We have found that 16 levels is adequate for most systems, giving greater than 99% of efficiency.Thomas Young's quantitative analysis of interference effects provided the confidence needed to revive the wave theory of light, and firmly established the concept of phase in optics. They are binary in the sense that they use discrete phase levels, not in the sense of using only two levels (they might more properly be called digital optics). We have applied binary optics technology to more » construct various wavefront sensing using four mask processes to create 16 level optics. While these optics have a large number of applications, they are extremely useful for systems that require arrays of small optics or aperture multiplexing, since these are fabricated using computer controlled photo-lithography and etching processes. In fact, optics with no symmetry, no smooth surfaces, and that perform multiple functions can be readily fabricated. Optical fabrication is no longer limited by surfaces that can be made by grinding and polishing, or even diamond turning. The advent of micro- or binary optics technology has made possible the fabrication of a variety of new optical devices. Furthermore, this is confirmed by experimental results obtained with various pixel distributions and induced fabrication errors. Analysis shows that the optimized pixel distribution starting from a high-noise distribution defined by a random-draw algorithm should be more resilient to fabrication errors than the optimized pixel distributions starting from a low-noise, error-diffusion algorithm, while leading to similar beamshaping performance. The optimization process preserves the pixel distribution statistical properties. Simulations using a design transmission chosen in the context of high-energy lasers show that the beam-fluence modulation at an image plane can be reduced by a factor of 2, leading to performance similar to using a non-optimized spatial-dithering algorithm with pixels of size reduced by a more » factor of 2 without the additional fabrication complexity or cost. The binary-pixel distribution can be iteratively optimized to lower an error function that takes into account the design transmission and the characteristics of the required far-field filter. The optimization of components that rely on spatially dithered distributions of transparent or opaque pixels and an imaging system with far-field filtering for transmission control is demonstrated. Experimental results obtained with Cr-on-glass devices for amplitude modulation and liquid crystal devices operating in the Mauguin condition for polarization modulation are in very good agreement with simulations. Polarization modulation improves the efficiency obtained by amplitude-only modulation, with a gain that depends on the aberration. Wavefront aberrations are corrected by modulation of the field in the pupil more » plane to prevent destructive interference in the focal plane of an ideal focusing element. Binary pixelated devices that approximate the sinusoidal transmission profile of a Gabor zone plate by spatial dithering are also investigated with amplitude and polarization modulation. The conventional Soret zone plate (binary amplitude profile) is expanded to a polarization Soret zone plate with twice the focusing efficiency. Focusing is performed by selectively modulating the field in different zones of the pupil to obtain on-axis constructive interference at a given distance. Here, we investigate the focusing and correcting wavefront aberration of an optical wave using binary amplitude and polarization modulation.
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