Wavefront errors in two-wavelength adaptive optics systems

Two-wavelength adaptive optics (AO) systems sense turbulence-induced wavefront distortions using an artificial beacon or natural guidestar at one wavelength, while correcting and possibly transmitting at another. Although most existing AO systems employ this methodology, the literature on atmospheric turbulence correction and AO system design generally focuses on performance at a single wavelength, neglecting the two-wavelength nature of the problem. In this paper, we undertake a rigorous study of the relevant wavefront errors necessary to quantify two-wavelength AO system performance. Since most AO systems employ separate tilt and higher-order correcting subsystems, our analysis mirrors this division, and we begin with higher-order wavefront errors. Utilizing Mellin transform techniques, we derive closed-form relations for the piston-removed and piston- and tilt-removed variances. The former is a measure of the total, residual wavefront error that a two-wavelength AO systems experiences; while the latter, quantifies the residual wavefront error due to higher-order aberrations. We then proceed to tilt or tracking errors and derive the two-wavelength Zernike- and gradient-tilt variances. Zernike tilt is the actual amount of tilt in the turbulent atmosphere; yet, most AO tracking subsystems measure gradient tilt. Consequently, we also derive the two-wavelength gradient-tilt, Zernike-tilt variance -- also known as centroid anisoplanatism -- to quantify this error. Lastly, we validate our analysis by performing two-wavelength wave-optics simulations and comparing the results to theory. We observe excellent agreement among the simulated results and our theoretical predictions. The analysis and findings presented in this paper will be useful in the characterization of existing, and the design of new, two-wavelength AO systems.