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.

Introduction to laser guide star theory versus experiment

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.

The diffractive limits of atmospheric compensation and laser guide star return initial conditions

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.

The role of diffraction in an extended beacon laser guide star

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.

L. Whiting

Papers

The diffractive limits of atmospheric compensation and laser guide star return initial conditions

The role of diffraction in an extended beacon laser guide star