Richard J. Sasiela

LASER-GUIDE-STAR SYSTEMS FOR ASTRONOMICAL APPLICATIONS

With computational techniques developed in the investigation of high-energy laser-beam-control systems, a set of concise analytic models describing the essential properties of a laser-guide-star phase-conjugation system has been assembled. With the aid of these models an optimization strategy for mating adaptive optics to a 4-m-class optical telescope is evolved, and it is shown that such a system might be expected to improve the effective atmospheric seeing conditions by nearly a factor of 10 within the isoplanatic patch of the turbulence probe. For operation at visible wavelengths, a compensation system having ~300 actuators and a closed-loop bandwidth of 20 Hz is recommended. All the key hardware components have already been built and tested, with the exception of a suitable laser source for high-repetition-rate illumination of the Earth’s sodium layer.

Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects

First-order weak-fluctuation Rytov theory predicts that the longitudinal (on-axis) component of the scintillation index of an uplink collimated beam will become significantly smaller as the size of the transmitter aperture increases up to around 100 cm. However, the results of recent computer simulations are at odds with this behavior, and we believe that this discrepancy is due to the fact that the conventional Rytov theory does not correctly account for the effects of beam wander on the scintillation index. We present a theoretical structure that accurately describes far-field irradiance fluctuations caused by uncorrected beam wander. This new theory is validated by demonstrating excellent agreement between the predicted scintillation index and computer code results for both tracked and untracked beams. For many applications of practical interest, such as free-space optical communications, a good understanding of the time-average Strehl ratio is also essential simulation results for this parameter are presented and shown to be in good agreement with the theory.

Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism

The process of setting up problems of wave propagation through turbulence and reducing the expressions to integrals is typically lengthy. Furthermore, to yield useful results the integrals must be evaluated numerically, except for the simplest problems. Here procedures are given for quickly writing an integral expression and easily evaluating it analytically, yielding a series solution that requires only a few terms to yield accurate results. The solution can also be expressed as a finite sum of generalized hypergeometric functions. The approach uses the Rytov approximation and filter functions in the spatial domain to express quantities of interest such as Zernike modes and effects of anisoplanatism for single or counterpropagating or copropagating plane or spherical waves in integral form. The integrals are readily evaluated with Mellin transforms. We illustrate the technique by deriving the tilt jitter of a single wave and the jitter between two waves with outer-scale effects present. It is shown that outer scale has a significant effect on tilt even for large outer-scale sizes. The effect of outer scale on tilt anisoplanatism is less pronounced.

Beam wander effects on the scintillation index of a focused beam

First-order weak-fluctuation Rytov theory predicts that the longitudinal (on-axis) scintillation component of an untracked focused beam projected along a horizontal path will become significantly smaller as the size of the transmitter aperture increases. At the same time, the radial component near the diffractive edge of the beam is predicted to increase without bound. The results of recent computer simulations are at odds with this behavior, and we believe that this discrepancy is due to the fact that Rytov theory does not correctly account for the effects of beam wander. We present a theoretical structure that accurately describes far-field irradiance fluctuations caused by uncorrected tilt jitter. This new theory is validated by demonstrating excellent agreement between the predicted scintillation index and computer code results for both tracked and untracked beams. For many applications of practical interest, such as free-space optical communications, a good understanding of the time-average Strehl ratio is also essential; simulation results for this parameter are presented and shown to be in good agreement with theory.

WAVE-FRONT CORRECTION BY ONE OR MORE SYNTHETIC BEACONS

Adaptive optics has been used in a cooperative mode to measure the phase distortion of the light from a star and to correct its image with a deformable mirror. A wave-front sensor in the adaptive-optics system that measures the phase aberrations requires that an object be fairly bright for accurate performance of the measurement. The use of synthetic beacons provides a means of correcting the images of objects that are too dim to allow one to use their light to provide correction in a cooperative mode. Synthetic beacons at a finite distance do not provide a perfect correction in imaging an object at a greater distance. The error in making a correction with one or more beacons is analyzed. Analytical expressions that can be used to determine performance in a variety of geometries, with various beacon altitudes and numbers, are derived. This analysis is applied to 60-cm and 4-m systems.

Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations

Mellin transform methods have been used to obtain Taylor series or asymptotic approximations to definite integrals whose integrands consist of the product of two generalized hypergeometric functions. This approach has proved extremely useful in many applied physics problems including the analysis of electromagnetic propagation in a turbulent medium. The main result in this article is a generalization of the method to integrands containing the product of an arbitrary number of hypergeometric functions providing a very powerful and general technique for integral evaluation with many applications in physics.

Strehl ratios with various types of anisoplanatism

There are many ways in which the paths of two waves through turbulence can become separated, thereby leading to anisoplanatic effects. Among these are a parallel path separation, an angular separation, one caused by a time delay, and one that is due to differential refraction at two wavelengths. All these effects can be treated in the same manner. Gegenbauer polynomials are used to obtain an approximation for the Strehl ratio for these anisoplanatic effects, yielding a greater range of applicability than the Marechal approximation.

An Optical Phase Reconstructor Based On Using A Multiplier-Accumulator Approach

Phase sensors that are most commonly used in the adaptive-optics area typically measure the gradient of the phase. A phase reconstructor is necessary to obtain the phase at the actuator positions of the deformable mirror. In the past reconstructors to obtain the optical phase from gradient measurements have been built using resistive nets. These nets simulate a least-squares reconstruction algorithm. There are other algorithms which can be used to mate wavefront sensors and deformable mirrors with different geometries or which can improve the noise performance by using the spatial correlation of the phase. These types of algorithms are difficult to implement and change using analog techniques. In addition, since the movement of an actuator can influence the position of adjacent actuators it is desirable to include this effect in the reconstructor. One may also want to remove the piston and the tip and tilt from the signal applied to the deformable mirror, and determine the values of the focus and tip and tilt terms in order to provide signals to auxiliary mirrors. A digital reconstructor can provide this capability. An approach to a digital reconstructor which can calculate an optical phase which is any linear function of the gradient measurements is described. This reconstructor is based on using a multiplier-accumulator circuit in each channel. A single phase value is calculated in each channel by summing the result of multiplying each gradient measurement by a stored matrix coefficient. Several sets of matrix coefficients are stored in memory to allow one to change the reconstruction algorithm quickly. The circuitry used and the time taken to perform the reconstruction will be described.

PDF models for uplink to space in the presence of beam wander

In this paper we discuss several models for the probability density function (pdf) of the irradiance of a Gaussian-beam wave from ground to space. We consider cases of tracked beams and untracked beams, both of which involve a certain amount of beam wander. The various pdf models that we introduce are all compared with simulation data over a broad range of beam diameters. We find that certain well-known models fit the simulation data in one of the regimes defined by the ratio of beam radius W0 to Frieds parameter r0 (W0/r0 <<1, W0/r0 ~ 1, W0/r0 >> 1), but not generally in the other regimes. This is true for tracked beams as well as untracked beams. Two new pdf models, developed here as a modulation of either the gamma-gamma pdf or the gamma pdf, are shown to provide excellent fits to the simulation data over all three regimes defined above.

Modeling the PDF for the irradiance of an uplink beam in the presence of beam wander

Recent studies of ground-to-space beam propagation have revealed significant departures from Rytov theory when the beam diameter w0 is on the order of the atmospheric coherence width r0. It has been shown that such departures from Rytov theory are primarily a consequence of beam wander and other low-order aberrations. In this paper we discuss modeling of the probability density function (PDF) for uplink beams. In particular, we show how the PDF transitions from lognormal statistics when w0/r0 << 1 to the negative exponential distribution when w0/r0 >> 1. The most interesting regime is the transition region near w0/r0 = 1, where the statistical behavior of tracked and untracked beams differs significantly.