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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