Byron M. Welsh

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.

Analysis of multiconjugate adaptive optics

We investigate a method for widening the compensated field of view of an adaptive-optical telescope with multiple deformable mirrors and an array of artificial guide stars. An ensemble of wavefront sensor measurements, made with the individual guide stars in the array, is used to estimate the contribution of a region of the atmosphere to the cumulative phase distortion. Our analysis includes the effects of measurement noise, wavefront-sensor sampling, and reconstruction of the wave front from slope measurements. We performed numerical computations for an atmosphere consisting of two turbulent layers: one at 1% of the guide-star altitude with 90% of the total turbulence strength and one at 10% of the guide-star altitude with the remaining 10% of the total turbulence strength. If we assume that r0 = 0.15 m and that a photon-limited wave-front sensor detecting 50 photons/r0-sized subaperture is used, the results indicate that a 0.9-m-square telescope with a diagonal field of view of ~92 μrad ≈ 19 arcsec can use two deformable mirrors, four laser guide stars, and a natural tilt reference star to achieve an rms residual phase error that is

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

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

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.

Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results

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.

Image-spectrum signal-to-noise-ratio improvements by statistical frame selection for adaptive-optics imaging through atmospheric turbulence

Micro-electro-mechanical deformable mirrors (MEM-DMs) are solid-state electronic devices with small, movable reflective surface elements that can be used to manipulate the phase of optical wavefronts. MEM-DMs differ from more conventional continuous-facesheet deformable mirrors in that the movable surface of a MEM-DM consists of a set of segmented moving surfaces. The segmented, reflective surfaces of a MEM-DM give rise to larger diffraction effects than those provided by continuous-facesheet deformable mirrors. However, MEM-DMs are still attractive due to their low cost and the low drive voltages. We explore the theoretical limits of performance of MEM-DMs for controlling fixed aberrations in optical systems, and we present laboratory results demonstrating reduction of a fixed aberration using a MEM-DM device. Results presented here show that while a MEM-DM does provide some degree of aberration control, diffraction effects arising from the static support structures of the MEM-DM surface are significant. An alternative design that uses a lenslet array in conjunction with the MEM-DM is shown through theoretical studies to provide superior aberration correction with lower residual effects due to diffraction.

Characterization of atmospheric turbulence phase statistics using wave-front slope measurements

Adaptive-optics systems have been used to overcome some of the effects of atmospheric turbulence on large-aperture astronomical telescopes. However, the correction provided by adaptive optics cannot restore diffraction-limited performance, due to discretized spatial sampling of the wavefront, limited degrees of freedom in the adaptive-optics system, and wavefront sensor measurement noise. Field experience with adaptive-optics imaging systems making short-exposure image measurements has shown that some of the images are better than others in the sense that the better images have higher resolution. This is a natural consequence of the statistical nature of the compensated optical transfer function in an adaptive-optics telescope. Hybrid imaging techniques have been proposed that combine adaptive optics and postdetection image processing to improve the high-spatial-frequency information of images. Performance analyses of hybrid methods have been based on prior knowledge of the ensemble statistics of the underlying random process. Improved image-spectrum SNRs have been predicted, and in some cases experimentally demonstrated. In this paper we address the issue of selecting and processing the best images from a finite data set of compensated short-exposure images. Image sharpness measures are used to select the data subset to be processed. Comparison of the image-spectrum SNRs for the cases of processing the entire data set and processing only the selected subset of the data shows a broad range of practical cases where processing the selected subset results in superior SNR.

Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution

Turbulence has long been recognized as one of the most significant factors limiting the performance of optical systems operating in the presence of atmosphere. Atmospheric turbulence over vertical paths has been well characterized, both theoretically and experimentally. Much less is known about turbulence over long, horizontal paths. Perturbations of the wave-front phase can be measured with a Hartmann wave-front sensor (H-WFS). One can use these measurements to characterize atmospheric turbulence directly. Theoretical expressions for the slope structure function of the H-WFS measurements are derived and evaluated with the use of numerical quadrature. By concentrating on the slope structure function, we avoid the phase reconstruction step and use the slope measurements in a more direct fashion. The theoretical slope structure function is compared with estimated slope structure functions computed from H-WFS measurements collected in a series of experiments conducted by researchers at the U.S. Air Force’s Phillips Laboratory. These experiments involved H-WFS measurements over high-altitude (airborne) horizontal paths 20–200 km in length.

Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions

Anew Fourier Series atmospheric phase screen generator is introduced. A Fourier Series (FS) is used to represent the wavefront phase as a two dimensional periodic function. The period of the function is chosen to be much larger than the outer scale of the turbulence and thus the FS accurately represents the power in the low spatial frequencies of the wavefront. The accuracy of the representation of the high spatial frequencies is determined by the number of terms used in the FS expansion. The FS based screen generator is capable of simulating atmospheric-induced wavefront phase distortions arising from temporal and/or anisoplanatic conditions. Both the spatial and temporal correlations between wavefront phases screens separated by time and/or angle are properly modeled. The conventional approach of simulating temporal evolution by making a large phase screen and then shifting is avoided. The phase screen generator is presented in an extremely compact and simple vector notation that lends itself for almost immediate implementation on modern mathematical analysis software packages.