Solution to the shearing problem

Lateral shearing interferometry is a promising reference-free measurement technique for optical wave-front reconstruction. The wave front under study is coherently superposed by a laterally sheared copy of itself, and from the interferogram difference measurements of the wave front are obtained. From these difference measurements the wave front is then reconstructed. Recently, several new and efficient algorithms for evaluating lateral shearing interferograms have been suggested. So far, however, all evaluation methods are somewhat restricted, e.g., assume a priori knowledge of the wave front under study, or assume small shears, and so on. Here a new, to our knowledge, approach for the evaluation of lateral shearing interferograms is presented, which is based on an extension of the difference measurements. This so-called natural extension allows for reconstruction of that part of the underlying wave front whose information is contained in the given difference measurements. The method is not restricted to small shears and allows for high lateral resolution to be achieved. Since the method uses discrete Fourier analysis, the reconstructions can be efficiently calculated. Furthermore, it is shown that, by application of the method to the analysis of two shearing interferograms with suitably chosen shears, exact reconstruction of the underlying wave front at all evaluation points is obtained up to an arbitrary constant. The influence of noise on the results obtained by this reconstruction procedure is investigated in detail, and its stability is shown. Finally, applications to simulated measurements are presented. The results demonstrate high-quality reconstructions for single shearing interferograms and exact reconstructions for two shearing interferograms.     

Improved Saunders method for the analysis of lateral shearing interferograms

An interferogram obtained by use of ordinary interferometers, such as Fizeau and Twyman-Green interferometers, will show a contour map of the wave front under test. A lateral-shearing interferogram, however, will show a contour map of the difference between the wave front under test and a sheared wave front, that is, a contour map of the derivative of the wave front under test. Therefore one can reconstruct the shape of the wave front under test by analyzing that difference. Many methods for reconstructing a wave front have been proposed. The Saunders method reconstructs a wave front; rapidly however the wave-front data are reconstructed only at intervals of the amount of shear along the direction of the shear. Therefore the method has low spatial resolution. A method for reconstructing a wave front that is based on the Saunders method and has high spatial resolution is proposed. The method analyzes the differences that are produced by shearing of the wave front under test in many directions. This method requires a large number of interferograms for reconstructing the wave front. Here the method is described, and its validity is confirmed by simulation.     

Analysis of interferograms with a spatial radial carrier or closed fringes and its holographic analogy

It is well known that an interferogram can be demodulated to find the wave-front shape if a linear carrier is introduced. We show that it can also be demodulated if it has many closed fringes or a circular carrier appears. A basic assumption is that the carrier fringes are of a bandwidth adequate to contain the wave-front distortion. This phase determination, called here demodulation, is made in the space domain, as opposed to demodulation in Fourier space, but the low-pass filter characteristics must be properly chosen. For academic purposes a holographic analogy of this demodulation process is also presented, which shows that the common technique of multiplying by a sine function and a cosine function is equivalent to holographically reconstructing with a tilted-flat wave front. Alternatively, a defocused (spherical) wave front can be used as a reference to perform the reconstruction or demodulation of some closed-fringe interferograms.     

Risley prisms to control wave-front tilt and displacement in a vectorial shearing interferometer

A pair of thin prisms is used to deviate a light beam without changing the image orientation in a vectorial shearing interferometer. The relative angle between prisms determines the displacement of the wave front and its tilt. The direction of the beam displacement is controlled by means of changing the relative angle between prisms. This system is employed to control the displacement of a sheared wave front as a vector quantity and to introduce a controlled amount of tilt in what we believe is a novel interferometric shearing system. The predicted performance of this wave-front director is confirmed experimentally.     

High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials

Interferograms obtained with ordinary interferometers, such as the Fizeau interferometer or the Twyman-Green interferometer, show the contour maps of a wave front under test. On the other hand, lateral shearing interferograms show the difference between a wave front under test and a sheared wave front, that is, the inclination of the wave front. Therefore the shape of the wave front under test is reconstructed by means of analyzing the difference. To reconstruct the wave front, many methods have been proposed. An integration method is usually used to reconstruct the wave front under test rapidly. However, this method has two disadvantages: The analysis accuracy of the method is low, and part of the wave front cannot be measured. To overcome these two problems, a new, to our knowledge, integration method, improved by use of polynomials, is proposed. The validity of the proposed method is evaluated by computer simulations. In the simulations the analysis accuracy achieved by the proposed method is compared with the analysis accuracy of the ordinary integration method and that of the method proposed by Rimmer and Wyant. The results of the simulations show that the analysis accuracy of the newly proposed method is better than that of the integration method and that of the Rimmer-Wyant method.     

Liquid-crystal Hartmann wave-front scanner

The liquid-crystal wave-front scanner (LCWS) is a highly sensitive wave-front sensor suited to the measurement of aberrations in optical systems and, more generally, of static wave fronts, and it is based on the Hartmann test. In the LCWS an incoming wave front is scanned sequentially by a programmable moving aperture that is implemented by use of a liquid-crystal display. The position of the diffraction spot is recorded behind an observation lens with a CCD detector and provides an estimation of the local slopes in two orthogonal directions at the aperture position. The wave front is then reconstructed from slope data by use of a least-squares method. Experiments are reported for nearly planar wave fronts as well as for strongly aberrated wave fronts, demonstrating both the large dynamic range and the great sensitivity of the LCWS. The LCWS is compared with the Shack-Hartmann wave-front sensor in terms of dynamic range and sensitivity.     

Tip/tilt estimation from defocused images

In astronomical imaging, the errors in the wave-front slope are a significant cause of aberrations in the detected image. We investigate how the slope can be estimated optimally using an intensity measurement of the propagated wave front. We show that the optimal location for detection of wave-front tilt is the focal plane, and we quantify the error in using defocused images, such as would be obtained from a curvature sensor, for estimating the wave-front tilt. The effect of using broadband light is also quantified.     

Real-time optical aberration correction with a ferroelectric liquid-crystal spatial light modulator

Real-time correction of an optically aberrated wave front by use of a 10 x 10 ferroelectric liquid-crystal spatial light modulator as the correction device and a point-diffraction interferometer as the wave-front sensor is demonstrated. This type of interferometer requires no reference arm and so can be used, in theory, in an astronomical adaptive-optics system. We discuss some of the unusual features of the point-diffraction interferometer for wave-front sensing.     

Wave-front measurement errors from restricted concentric subdomains

In interferometry and optical testing, system wave-front measurements that are analyzed on a restricted subdomain of the full pupil can include predictable systematic errors. In nearly all cases, the measured rms wave-front error and the magnitudes of the individual aberration polynomial coefficients underestimate the wave-front error magnitudes present in the full-pupil domain. We present an analytic method to determine the relationships between the coefficients of aberration polynomials defined on the full-pupil domain and those defined on a restricted concentric subdomain. In this way, systematic wave-front measurement errors introduced by subregion selection are investigated. Using vector and matrix representations for the wave-front aberration coefficients, we generalize the method to the study of arbitrary input wave fronts and subdomain sizes. While wave-front measurements on a restricted subdomain are insufficient for predicting the wave front of the full-pupil domain, studying the relationship between known full-pupil wave fronts and subdomain wave fronts allows us to set subdomain size limits for arbitrary measurement fidelity.     

Residual phase variance in partial correction: application to the estimate of the light intensity statistics

Although the wave-front correction provided by an adaptive optics system should be as complete as possible, only a partial compensation is attainable in the visible. An estimate of the residual phase variance in the compensated wave front can be used to calibrate system performance, but it is not a simple task when errors affect the compensation process. We propose a simple method for estimation of the residual phase variance that requires only the measurement of the Strehl ratio value. It provides good results over the whole range of compensation degrees. The estimate of the effective residual phase variance is useful not only for system calibration but also for determining the light intensity statistics to be expected in the image as a function of the degree of compensation introduced.     

Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils

An iterative zonal wave-front estimation algorithm for slope or gradient-type data in optical testing acquired with regular or irregular pupil shapes is presented. In the mathematical model proposed, the optical surface, or wave-front shape estimation, which may have any pupil shape or size, shares a predefined wave-front estimation matrix that we establish. Owing to the finite pupil of the instrument, the challenge of wave front shape estimation in optical testing lies in large part in how to properly handle boundary conditions. The solution we propose is an efficient iterative process based on Gerchberg-type iterations. The proposed method is validated with data collected from a 15 x 15-grid Shack-Hartmann sensor built at the Nanjing Astronomical Instruments Research Center in China. Results show that the rms deviation error of the estimated wave front from the original wave front is less than lambda/130-lambda/150 after approximately 12 iterations and less than lambda/100 (both for lambda = 632.8 nm) after as few as four iterations. Also, a theoretical analysis of algorithm complexity and error propagation is presented.     

Variational solution for modal wave-front projection functions of minimum-error norm.

Common wave-front sensors such as the Hartmann or curvature sensor provide measurements of the local gradient or Laplacian of the wave front. The expression of wave fronts in terms of a set of orthogonal basis functions thus generally leads to a linear wave-front-estimation problem in which modal cross coupling occurs. Auxiliary vector functions may be derived that effectively restore the orthogonality of the problem and enable the modes of a wave front to be independently and directly projected from slope measurements. By using variational methods, we derive the necessary and sufficient condition for these auxiliary vector functions to have minimum-error norm. For the specific case of a slope-based sensor and a basis set comprising the Zernike circular polynomials, these functions are precisely the Gavrielides functions.     

Jamin double-shearing interferometer for diffraction-limited wave-front test

A Jamin double-shearing interferometer with three changeable schemes is proposed for the measurement of diffraction-limited laser wave front. A concept of detectable wave-front height is thus defined, and on this basis the limits of detectable wave-front height from the suggested schemes of interferometer are analyzed. The design is detailed, the simulation for wave aberrations is given, and the experiment is demonstrated. One of the major features of this interferometer is that it is capable of visually testing a diffraction-limited wave front immediately by the fringes with the matched accuracy and minimum detectable wave-front height on the order of 0.1lambda.     

Wave-front sensing from subdivision of the focal plane with a lenslet array

A wave-front sensing scheme based on placing a lenslet array at the focal plane of the telescope with each lenslet reimaging the aperture is analyzed. This wave-front sensing arrangement is the dual of the Shack-Hartmann sensor, with the wave front partitioned in the focal plane rather than in the aperture plane. This arrangement can be viewed as the generalization of the pyramid sensor and allows direct comparisons of this sensor with the Shack-Hartmann sensor. We show that, as with the Shack-Hartmann sensor, when subdividing in the focal plane, the quality of the wave-front estimate is a trade-off between the quality of the slope measurements over each region in the aperture and the resolution to which the slope measurements are obtained. Open-loop simulation results demonstrate that the performance of the lenslet array at the focal plane is equivalent to that of the Shack-Hartmann sensor when no modulation is applied to the lenslet array. However, when the array is modulated in a manner akin to that of the pyramid sensor, subdivision at the focal plane provides advantages when compared with the Shack-Hartmann sensor.     

Measurement of telescope aberrations through atmospheric turbulence by use of phase diversity

We conduct computer simulations of the reconstruction of a wave front at a telescope pupil with the phase-diversity method. An instantaneous wave front is reconstructed from focused and defocused specklegrams of a point star. In the wave-front reconstruction we do not fit the wave front to Zernike polynomials but retrieve the phase with a phase-unwrapping procedure. Averaging over many atmospherically perturbed wave fronts leads to the residual phase error, namely, the aberration of the telescope. The scintillation effect, nonuniformity of amplitude on a telescope pupil, is also discussed.     

Calibration of a deformable mirror and Strehl ratio measurements by use of phase diversity

Calibration experiments with a bimorph mirror are presented. Phase-diversity wave-front sensing is used for measuring the control matrix, nulling wave-front errors in the optical setup, including the mirror, and measuring Strehl ratios and residual higher-order aberrations. The Strehl ratio of the calibrated system is measured to be 0.975, corresponding to 1/40 wave rms in the residual wave front. The conclusion is that a phase-diversity wave-front sensor is easier to install and use than interferometers and can replace them in optical setups for testing adaptive optics systems.     

Design of lenses to project the image of a pupil in optical testing interferometers

When an optical surface or lens in an interferometer (Twyman-Green or Fizeau interferometer) is tested, the wave front at the pupil of the element being tested does not have the same shape as at the observation plane, because this shape changes along its propagation trajectory if the wave front is not flat or spherical. An imaging lens must then be used, as reported many times in the literature, to project the image of the pupil of the system being tested over the observation plane. This lens is especially necessary if the deviation of the wave front from sphericity is large, as in the case of testing paraboloidal or hyperboloidal surfaces. We show that the wave front at both positions does not need to have the same shape. The only condition is that the interferograms at both places be identical, which is a different condition. This leads to some considerations that should be taken into account in the optical design of such lenses.     

Experimental measurements of estimator bias and the signal-to-noise ratio for deconvolution from wave-front sensing

Deconvolution from wave-front sensing (DWFS) has been proposed as a method for achieving high-resolution images of astronomical objects from ground-based telescopes. The technique consists of the simultaneous measurement of a short-exposure focal-plane speckled image, as well as the wave front, by use of a Shack-Hartmann sensor placed at the pupil plane. In early studies it was suspected that some problems would occur in poor seeing conditions; however, it was usually assumed that the technique would work well as long as the wave-front sensor subaperture spacing was less than r(0) (L/r(0) < 1). Atmosphere-induced phase errors in the pupil of a telescope imaging system produce both phase errors and magnitude errors in the effective short-exposure optical transfer function (OTF) of the system. Recently it has been shown that the commonly used estimator for this technique produces biased estimates of the magnitude errors. The significance of this bias problem is that one cannot properly estimate or correct for the frame-to-frame fluctuations in the magnitude of the OTF but can do so only for fluctuations in the phase. An auxiliary estimate must also be used to correct for the mean value of the magnitude error. The inability to compensate for the magnitude fluctuations results in a signal-to-noise ratio (SNR) that is less favorable for the technique than was previously thought. In some situations simpler techniques, such as the Knox-Thompson and bispectrum methods, which require only speckle gram data from the focal plane of the imaging system, can produce better results. We present experimental measurements based on observations of bright stars and the Jovian moon Ganymede that confirm previous theoretical predictions.     

Adaptive control of a micromachined continuous-membrane deformable mirror for aberration compensation

The nonlinear response and strong coupling of control channels in micromachined membrane deformable mirror (MMDM) devices make it difficult for one to control the MMDM to obtain the desired mirror surface shapes. A closed-loop adaptive control algorithm is developed for a continuous-surface MMDM used for aberration compensation. The algorithm iteratively adjusts the control voltages of all electrodes to reduce the variance of the optical wave front measured with a Hartmann-Shack wave-front sensor. Zernike polynomials are used to represent the mirror surface shape as well as the optical wave front. An adaptive experimental system to compensate for the wave-front aberrations of a model eye has been built in which the developed adaptive mirror-control algorithm is used to control a deformable mirror with 19 active channels. The experimental results show that the algorithm can adaptively update control voltages to generate an optimum continuous mirror surface profile, compensating for the aberrations within the operating range of the deformable mirror.