"Hidden phase" in two-wavelength adaptive optics

Two-wavelength adaptive optics (AO), where sensing and correcting (from a beacon) are performed at one wavelength λ and compensation and observation (after transmission through the atmosphere) are performed at another λ , has historically been analyzed and practiced assuming negligible irradiance fluctuations (i.e., weak scintillation). Under these conditions, the phase corrections measured at λ are robust over a relatively large range of wavelengths, resulting in a negligible decrease in AO performance. In weak-to-moderate scintillation conditions, which result from distributed-volume atmospheric aberrations, the pupil-phase function becomes discontinuous, producing what Fried called the "hidden phase" because it is not sensed by traditional least-squares phase reconstructors or unwrappers. Neglecting the hidden phase has a significant negative impact on AO performance even with perfect least-squares phase compensation. To the authors' knowledge, the hidden phase has not been studied in the context of two-wavelength AO. In particular, how does the hidden phase sensed at λ relate to the compensation (or observation) wavelength λ ? If the hidden phase is highly correlated across λ and λ , like the least-squares phase, it is worth sensing and correcting; otherwise, it is not. Through a series of wave optics simulations, we find an approximate expression for the hidden-phase correlation coefficient as a function of λ , λ , and the scintillation strength. In contrast to the least-squares phase, we determine that the hidden phase (when present) is correlated over a small band of wavelengths centered on λ . Over the range λ ,λ ∈[1,3]µm and in weak-to-moderate scintillation conditions (spherical-wave log-amplitude variance σ χ2∈[0.1,0.5]), we find the average hidden-phase correlation linewidth to be approximately 0.35 µm. Consequently, for |λ -λ | greater than this linewidth, including the hidden phase does not significantly improve AO performance over least-squares phase compensation.