Single-shot femtosecond x-ray holography using extended references.

In the context of x-ray lensless imaging, we present a recent approach for Fourier transform holography based on the use of extended references. Major advances shown here rely on a high signal efficiency and on the direct image reconstruction of the object performed by a simple linear derivative. Moreover, the extended holographic reference is easy to manufacture and can be applied to a variety of imaging experiments. Here we demonstrate single-shot imaging with a table-top, laser-based coherent soft x-ray source. A spatial resolution of 110 nm was obtained with an integration time of 20 fs.

Aberration correction by maximizing generalized sharpness metrics.

The technique of maximizing sharpness metrics has been used to estimate and compensate for aberrations with adaptive optics, to correct phase errors in synthetic-aperture radar, and to restore images. The largest class of sharpness metrics is the sum over a nonlinear point transformation of the image intensity. How the second derivative of the point nonlinearity varies with image intensity determines the effects of various metrics on the imagery. Some metrics emphasize making shadows darker, and other emphasize making bright points brighter. One can determine the image content needed to pick the best metric by computing the statistics of the image autocorrelation or of the Fourier magnitude, either of which is independent of the phase error. Computationally efficient, closed-form expressions for the gradient make possible efficient search algorithms to maximize sharpness.

Synthetic-aperture radar autofocus by maximizing sharpness.

To focus a synthetic-aperture radar image that is suffering from phase errors, a phase-error estimate is found that, when it is applied, maximizes the sharpness of the image. Closed-form expressions are derived for the gradients of a sharpness metric with respect to phase-error parameters, including both a point-by-point (nonparametric) phase function and coefficients of a polynomial expansion. Use of these expressions allows for a highly efficient gradient-search algorithm for high-order phase errors. The effectiveness of the algorithm is demonstrated with an example.

Invariant error metrics for image reconstruction.

Expressions are derived for the normalized root-mean-square error of an image relative to a reference image. Different versions of the error metric are invariant to different combinations of effects, including the image's (a) being multiplied by a real or complex-valued constant, (b) having a constant added to its phase, (c) being translated, or (d) being complex conjugated and rotated 180 degrees . Invariance to these effects is particularly important for the phase-retrieval problem. One can also estimate the parameters of those effects. Similarly, two wave fronts can be compared, allowing for arbitrary constant (piston) and linear (tilt) phase terms. One can also include a weighting function. The relation between the error metric and other quality measures is derived.

Phase-retrieval algorithms for a complicated optical system.

Phase-retrieval algorithms have been developed that handle a complicated optical system that requires multiple Fresnellike transforms to propagate from one end of the system to the other including the absorption by apertures in more than one plane and allowance for bad detector pixels. Gradientsearch algorithms and generalizations of the iterative-transform phase-retrieval algorithms are derived. Analytic expressions for the gradient of an error metric, with respect to polynomial coefficients and with respect to point-by-point phase descriptions, are given. The entire gradient can be computed with the number of transforms required to propagate a wave front from one end of the optical system to the other and back again, independent of the number of coefficients or phase points. This greatly speeds the computation. The reconstruction of pupil amplitude is also given. A convergence proof of the generalized iterative transform algorithm is given. These improved algorithms permit a more accurate characterizationof complicated optical systems from their point spread functions.

Hubble Space Telescope characterized by using phase-retrieval algorithms.

We describe several results characterizing the Hubble Space Telescope from measured point spread functions by using phase-retrieval algorithms. The Cramer-Rao lower bounds show that point spread functions taken well out of focus result in smaller errors when aberrations are estimated and that, for those images, photon noise is not a limiting factor. Reconstruction experiments with both simulated and real data show that the calculation of wave-front propagation by the retrieval algorithms must be performed with a multiple-plane propagation rather than a simple fast Fourier transform to ensure the high accuracy required. Pupil reconstruction was performed and indicates a misalignment of the optical axis of a camera relay telescope relative to the main telescope. After we accounted for measured spherical aberration in the relay telescope, our estimate of the conic constant of the primary mirror of the HST was - 1.0144.

Image synthesis from nonimaged laser-speckle patterns.

We demonstrate that unspeckled images of coherently illuminated, diffuse objects can be formed from measurements of backscattered laser-speckle intensity. The theoretical basis for this imaging technique is outlined, and results of computer experiments that successfully construct images from digitally simulated laser-speckle measurements are presented.

Phase retrieval algorithms: a comparison.

Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input-output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.

Holographic optics for a matched-filter optical processor.

The requirements on and the performance of holographic Fourier transform lenses for a matched-filter optical processor are discussed. The holographic aberrations were analyzed using a ray-tracing computer program, and holographic lenses were fabricated and assembled into an optical processor. Analysis and experimental results show that holographic optics are capable of processing images of large space-bandwidth product and can be an attractive alternative to conventional optics.

Optical residue arithmetic computer with programmable computation modules.

The concept of a programmable multipurpose computation module for residue arithmetic is introduced. A possible design for the module is also presented. The application of the computation module in mathematical computation, coding, decoding, and scaling is demonstrated. Very high throughput rate is achieved by the use of parallel computation structures and pipelining of sequential operations.

Reconstruction of an object from the modulus of its Fourier transform.

We present a digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform. This technique should be useful for obtaining high-resolution imagery from interferometer data.