The symmetry operation associated with propagation reciprocity is complex conjugation and adaptive optics is used to physically carry out this symmetry operation. We use a plane-to-plane framework to describe the fundamental limits placed on implementing propagation reciprocity that arise due to diffraction. Compensation system performance is often analyzed using the ray optics limit (e.g. defining the isoplanatic angle). This limits the applicability of such results by ignoring the diffractive limits on the ability to sense the laser guide star phase and amplitude information. We describe how the diffractive limits of phase-only and full-field compensation arise in terms of this flow of information. The plane-to-plane framework also shows the role of the beacon initial conditions as the end result of complete spatial reciprocity.
We discuss the reliance on backscatter by a laser guide star to generate a propagative probe and we show that spatial reciprocity can be accomplished by compensating phase alone. An analytic plane to plane propagation framework is introduced in which the spatial reciprocity of Maxwell's equations Is utilized to demonstrate that the adaptive optics compensation can do no better than the beacon initial conditions (I.e. cannot correct for the beacon too.) It is shown analytically that use of point to point reciprocity reasoning fails. While the laser guide star itself may be compensated to optimize uplink spatial coherence at altitude, the backscattering process is completely incoherent and the backscattering volume constitutes a very bad mirror or diffuse source. While diffraction restores some coherency as described by the van Cittert-Zernike theorem[2,5], the consequences of the incoherency of the beacon, lead to problems for the adaptive optics system which do not affect natural guide stars. The main consequence for the laser guide star system is that the wave sensor of the adaptive optics cannot distinguish between the
phase aberrations from the backscattering process and those phase aberrations induced by turbulence. The question of the beacon and propagative path being different is weighed within the context of correlated versus uncorrelated ensemble members of turbulence.
The limits to the ability of adaptive optics to achieve spatial propagation reciprocity are determined by diffraction. The beacon is a prominent component in defining the diffractive limit, so diffraction plays a role in the optimal choice of beacon parameters. We show with an explicit example that a point-source beacon is not the optimal choice, and that a point-source beacon cannot be used to measure the diffractive limit of phase-only compensation. At the single scattering level, diffraction dictates the use of an extended coherent beacon. We also show with an explicit example that optical vortices are not branch points, thus a well-defined phase reconstruction from an initially coherent beacon propagated through strong or extensive turbulence will not be hindered by the presence of optical vortices.
We investigate a fundamental limit of atmospheric compensation for adaptive optical systems that use uncooperative beacons or laser guide stars. Laser guide stars are generated by backscattering processes that are naturally incoherent. The limit arises because the wavefront sensing component of the adaptive optics cannot differentiate between the incoherency introduced by atmospheric turbulence from that of the laser guide star generation process. The limit is significant and restrictive. Under absolutely ideal conditions with perfect compensation, the compensated Strehl ratio must be less than 0.3. Realistic conditions will reduce this by more than a factor of 2.
We present the analytic theory for the interaction of thermal blooming and its interaction with optical turbulence in the form of an exact solution  which gives the asymtotics of the MCF and yields a Strehl. From that functional scaling is derived and a reconstruction algorithm is presented in terms of the eigenfunctions.
The resulting systems model, AMPERES, which runs in real time on a PC matches the results of full non linear wave optics simulations. It is shown that thermal blooming is an asymtotic instability that requires
a stable heating pattern to be imparted into the medium. Any kind of of wind shear disrupts the heating pattern and destroys the instability growth. Targets are not static. They are moving rapidly in cases of interest for HEL and the beam slew required to track the target naturally introduces a wind shear.
The SOPHI (Segmented Optics PHase Integration) Phase 1 project has demonstrated the original goals for the SOPHI breadboard to enable it to meet the challenge of the Phase II full 1 meter optical quality segmented mirror project: one of these goals is that the mirror segments can be phased without edge sensors and without referencing each other (no 2 π ambiguities). To do this, an optical metrology is utilized (the subject of a separate paper and reference 1). The stiffness and lightweight criteria are met by utilizing CSIC and Cesic ceramics.
The additional degrees of freedom that arise from segmenting the optical surfaces naturally allow for the development of active control of orbital thermal fluctuations and mechanical stresses as well as atmospheric compensation. The successful performance of segmented optical systems depends on ability to position and control the segments. We briefly review the goal of segment control and our exact analytical solution to adaptive optics segmented mirror control1. Our completely deterministic control solution eliminates the need to do iterations and eliminates the need to develop iterative control algorithms. We also present diffractive concepts for segment position metrology that exploit and transfer diffractive measurement technology from crystallography and condensed matter physics.
Segmentation of the primary mirror offers an inexpensive method to produce large, active telescopes or beam directors. We discuss the goal of segment control and our analytic solution configuration functional. We show that the nature of the segmentation, control, and solution generates an additional source of diffractive errors that must be accounted for in the design of these systems. These diffractive errors and not r0 ultimately set the segment size.
We define the diffractive limit of phase-only compensation and compute this limit in the form of the strehl ratio of a perfectly phase-only compensated scoring laser beam on target. A comparison of uplink, downlink including horizontal compensated propagation, is made.
From the solution for the linear theory of thermal blooming, the propagator is a 2 X 2 matrix that satisfies an integral equation of Fredholm type. We develop a generalized Fredholm series solution to this integral equation. Since the Kernel is a matrix, the usual determinants in the Fredholm series contain ordering ambiguities. We resolve all ordering ambiguities using the standard diagrammatic representation of the series. The Fredholm denominator is computed for the case of uncompensated and compensated propagation in a uniform atmosphere with uniform wind. When the Fredholm denominator vanishes, the propagator contains poles. In the compensated case, the denominator does develop zeros. The single mode phase compensation instability gains computed from the zeros agrees with results obtained from other methods.
We present the path integral representation of the Green's function for the linearized equations of isobaric thermal blooming for uniform wind and atmosphere. We first describe the path integrals for the Fresnel and Helmholtz propagators as an introduction to the path integral, and then use these results to define a perturbative expansion of the thermal blooming path integral. Each term in the perturbation series may be computed in closed form. Finally, a Feynman diagrammatic representation of the series is given.
We give an overview of functional reconstruction theory for predicting whole beam Strehl ratios as applied to thermal blooming. We begin by writing down the reconstruction formula that relates a finite sized beam Strehl ratio to a sum involving the Strehl ratio of an infinite beam or periodic patch. We then define what a dynamically equivalent patch is and follow by describing the functional approach to scaling patch Strehl curves using a metric on the space of absorption profiles. We end by comparing results from our systems model based on functional reconstruction, SWATM2, which takes only seconds on any machine, with results from large nonlinear 4D wave optics simulations.
A substantial amount of progress has been made in the past three years in understanding the small scale physics of thermal blooming, and in particular, in understanding the interaction of thermal blooming with atmospheric turbulence. The authors present a brief review of some of the theoretical aspects of small scale thermal blooming.
The linearized equations of thermal blooming for an infinite beam in a uniform atmosphere and wind are analytically solved as a perturbation series in blooming for the case of compensated and uncompensated propagation. A Feynman diagram representation of the series is presented. The propagators are used to compute the mutual coherence function (MCF) and Strehl also as a perturbation series in blooming. The dependence of the results on the actuator Fresnel number is discussed along with the relative roles of the phase compensation instability and stimulated thermal Rayleigh scattering. A brief comparison is made with nonlinear numerical simulations to show that the nonlinearities may be neglected.