Michael C. Roggemann

Optical propagation in non-Kolmogorov atmospheric turbulence

Several observations of atmospheric turbulence statistics have been reported which do not obey Kolmogorovs power spectral density model. These observations have prompted the study of optical propagation through turbulence described by non-classical power spectra. This paper presents an analysis of optical propagation through turbulence which causes fluctuations in the index of refraction. The index fluctuations are assumed to have spatial power spectra that obey arbitrary power laws. The spherical and plane wave structure functions are derived using Mellin transform techniques. The wave structure function is used to compute the Strehl ratio of a focused plane wave propagating in turbulence as the power law for the spectrum of the index of refraction fluctuations is varied from -3 to -4. The relative contributions of the log amplitude and phase structure functions to the wave structure function are computed. At power laws close to -3, the magnitude of the log amplitude and phase perturbations are determined by the system Fresnel ratio. At power laws approaching -4, phase effects dominate in the form of random tilts.

Two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere.

A two-deformable-mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere is presented. This system uses a deformable mirror and a Fourier-transforming mirror to adjust the amplitude of the wave front in the telescope pupil, similar to kinoforms used in laser beam shaping. A second deformable mirror is used to correct the phase of the wave front before it leaves the aperture. The phase applied to the deformable mirror used for controlling the beam amplitude is obtained with a technique based on the Fienup phase-retrieval algorithm. Simulations of propagation through a single turbulent layer sufficiently distant from the beacon observation and laser beam transmission aperture to cause scintillation shows that, for an ideal deformable-mirror system, this field-conjugation approach improves the on-axis field amplitude by a factor of approximately 1.4 to 1.5 compared with a conventional phase-only correction system.

Limited degree-of-freedom adaptive optics and image reconstruction

The use of limited degree-of-freedom adaptive optics in conjunction with statistical averaging and a linear image reconstruction algorithm is addressed. Image reconstruction is traded for full predetection compensation. It is shown through analytic calculations that the average optical transfer function (OTF) is significant for high spatial frequencies in the case of imaging through atmospheric turbulence with an adaptive optics system composed of a Hartmann-type wave-front sensor and a deformable mirror possessing far fewer actuators than one per atmospheric coherence diameter (r(0)). Statistical averaging is used to overcome the effects of measurement noise and randomness in individual realizations of the OTF. The imaging concept and signal-to-noise considerations are presented.

Digital simulation of scalar optical diffraction: revisiting chirp function sampling criteria and consequences

Accurate simulation of scalar optical diffraction requires consideration of the sampling requirement for the phase chirp function that appears in the Fresnel diffraction expression. We describe three sampling regimes for FFT-based propagation approaches: ideally sampled, oversampled, and undersampled. Ideal sampling, where the chirp and its FFT both have values that match analytic chirp expressions, usually provides the most accurate results but can be difficult to realize in practical simulations. Under- or oversampling leads to a reduction in the available source plane support size, the available source bandwidth, or the available observation support size, depending on the approach and simulation scenario. We discuss three Fresnel propagation approaches: the impulse response/transfer function (angular spectrum) method, the single FFT (direct) method, and the two-step method. With illustrations and simulation examples we show the form of the sampled chirp functions and their discrete transforms, common relationships between the three methods under ideal sampling conditions, and define conditions and consequences to be considered when using nonideal sampling. The analysis is extended to describe the sampling limitations for the more exact Rayleigh-Sommerfeld diffraction solution.

Cramér–Rao analysis of phase-diverse wave-front sensing

Phase-diverse wave-front sensing (PDWFS) is a methodology for estimating aberration coefficients from multiple incoherent images whose pupil phases differ from one another in a known manner. With the use of previous work by other authors, the Cramer–Rao lower-bound (CRLB) expression for the phase diversity aberration estimation problem is developed and is generalized slightly to allow for multiple phase-diverse images, various beam-splitting configurations, and imaging of known extended objects. The CRLB for a given problem depends implicitly on the true underlying value of the aberration being estimated. Therefore we use numerical evaluation and Monte Carlo analysis of the PDWFS CRLB expressions. The numerical evaluation is performed on an ensemble of aberration phase screens while simulating a number of different imaging configurations. We demonstrate the use of average CRLB values as figures of merit in comparing these various PDWFS configurations. For simulated point-source imaging we quantify the effects of varying the amounts and the types of diversity phase and briefly address the issue of the number of diversity images. Our results show that there is a diversity defocus configuration that is optimal in a Cramer–Rao sense for estimating certain aberrations. We also show that PDWFS Cramer–Rao squared-error values can be orders of magnitude higher for imaging of an extended target object than those from a point-source target.

Surface micromachined segmented mirrors for adaptive optics

This paper presents recent results for aberration correction and beam steering experiments using polysilicon surface micromachined piston micromirror arrays. Microfabricated deformable mirrors offer a substantial cost reduction for adaptive optic systems. In addition to the reduced mirror cost, microfabricated mirrors typically require low control voltages (less than 30 V), thus eliminating high-voltage amplifiers. The greatly reduced cost per channel of adaptive optic systems employing microfabricated deformable mirrors promise high-order aberration correction at low cost. Arrays of piston micromirrors with 128 active elements were tested. Mirror elements are on a 203-/spl mu/m 12/spl times/12 square grid (with 16 inactive elements, four in each corner of the array). The overall array size is 2.4 mm square. The arrays were fabricated in a commercially available surface micromachining process. The cost per mirror array in this prototyping process is less than

Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive optics and bispectrum postprocessing

200. Experimental results are presented for a hybrid correcting element comprised of a lenslet array and piston micromirror array, and for a piston micromirror array only. Also presented is a novel digital deflection micromirror that requires no digital to analog converters, further reducing the cost of adaptive optics systems.

Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor

The use of predetection compensation for the effects of atmospheric turbulence combined with postdetection image processing for imaging applications with large telescopes is addressed. Full and partial predetection compensation with adaptive optics is implemented by varying the number of actuators in the deformable mirror. The theoretical expression for the single-frame power spectrum signal-to-noise ratio (SNR) is reevaluated for the compensated case to include the statistics of the compensated optical transfer function. Critical to this analysis is the observation that the compensated optical transfer function does not behave as a circularly complex Gaussian random variable except at high spatial frequencies. Results from a parametric study of performance are presented to demonstrate improvements in power spectrum estimation for both point sources and an extended object and improvements in the Fourier phase spectrum estimation for an extended object. Full compensation is shown to provide a large improvement in the power spectrum SNR over the uncompensated case, while successively smaller amounts of predetection compensation provide smaller improvements, until a low degree of compensation gives results essentially identical to those of the uncompensated case. Three regions of performance were found with respect to the object Fourier phase spectrum estimate obtained from bispectrum postprocessing: (1) the fully compensated case in which bispectrum postprocessing provides no improvement in the phase estimate over that obtained from a fully compensated long-exposure image, (2) a partially compensated regime in which applying bispectrum postprocessing to the compensated images provides a phase spectrum estimation superior to that of the uncompensated bispectrum case, and (3) a poorly compensated regime in which the results are essentially indistinguishable from those of the uncompensated case. Accurate simulations were used to obtain some parameters for the power spectrum SNR analysis and to obtain the Fourier phase spectrum results.

Algorithm to increase the largest aberration that can be reconstructed from Hartmann sensor measurements

The performance of ground-based optical imaging systems is severely degraded from the diffraction limit by the random effects of the atmosphere. Adaptive-optics techniques have been used to compensate for atmospheric-turbulence effects. A critical component in the adaptive-optics system is the wave-front sensor. At present, two types of sensors are common: the Hartmann-Shack wave-front sensor and the shearing interferometer wave-front sensor. In this paper we make a direct performance comparison of these two sensors. The performance calculations are restricted to common configurations of these two sensors and the fundamental limits imposed by shot noise and atmospheric effects. These two effects encompass the effects of extended reference beacons and sensor subaperture spacings larger than the Fried parameter r(0). Our results indicate comparable performance for good seeing conditions and small beacons. However, for poor seeing conditions and extended beacons, the Hartmann sensor has lower error levels than the shearing interferometer.

Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors

Conventional Hartmann sensor processing relies on locating the centroid of the image that is formed behind each element of a lenslet array. These centroid locations are used for computing the local gradient of the incident aberration, from which the phase of the incident wave front is calculated. The largest aberration that can reliably be sensed in a conventional Hartmann sensor must have a local gradient small enough that the spot formed by each lenslet is confined to the area behind the lenslet: If the local gradient is larger, spots form under nearby lenslets, causing a form of cross talk between the wave-front sensor channels. We describe a wave-front reconstruction algorithm that processes the whole image measured by a Hartmann sensor and a conventional image that is formed by use of the incident aberration. We show that this algorithm can accurately estimate aberrations for cases in which the aberration is strong enough to cause many of the images formed by individual lenslets to fall outside the local region of the Hartmann sensor detector plane defined by the edges of a lenslet.