A design method is presented for computing the phase functions of an energy efficient system using two holographic elements for converting a Gaussian beam into a uniform beam with rectangular support in the far field of the source. The method is based on a modification of the Gerchberg-Saxton algorithm which includes an x-y separability constraint on the phase of one of the holographic elements. A beamforming system was fabricated using this method, and experimental results were obtained which support the design approach.

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A novel wave-front sensor that estimates phase from Fourier intensity measurements is described, and an explicit expression is found and numerically evaluated for the Cramtr-Rao lower bound on integrated rms wave-front phase estimation error. For comparison, turbulence-aberrated wave-front phases and corresponding noisy Fourier intensity measurements were computer simulated. An iterative phase-retrieval algorithm was then used to estimate the phase from the Fourier intensity measurements and knowledge of the shape of an aperture through which the wave front passed. The simulation error approaches the lower bound asymptotically as the noise is reduced.

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Wavefront sensors that can operate at low light levels, be built from present technology components, and provide accurate wavefront phase estimates in real -time are required for use with adaptive optics systems. The use of estimation theory makes possible the evaluation of wavefront sensors without specification of the wavefront phase estimation algorithms.The Cramer -Rao method was used to find a lower bound on integrated rms wavefront sensor estimation error.In addition to an analysis of the general case, the error lower bound was numerically evaluated for the shearing interferometer wavefront sensor. Computer simulations of the atmosphere and wavefront sensor measurements including noise were performed.Using an appropriate algorithm, the phase was estimated and the resulting phase error was compared with the lower bound.The results support the validity of using the Cramer -Rao lower bound to evaluate wavefront sensor performance.