"Hidden phase" in two-wavelength adaptive optics

Two-wavelength adaptive optics (AO), where sensing and correcting (from a beacon) is performed at one wavelength $\lambda_\text{B}$ and compensation and observation (after transmission through the atmosphere) is performed at another $\lambda_\text{T}$, has historically been analyzed and practiced assuming negligible irradiance fluctuations (i.e., weak scintillation). Under these conditions, the phase corrections measured at $\lambda_\text{B}$ 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 $\lambda_\text{B}$ relate to the compensation (or observation) wavelength $\lambda_\text{T}$? If the hidden phase is highly correlated across $\lambda_\text{B}$ and $\lambda_\text{T}$, 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 $\lambda_\text{B}$, $\lambda_\text{T}$, 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 $\lambda_{\text{T}}$. Over the range $\lambda_\text{B},\lambda_\text{T} \in \left[1,3\right] \text{ } \mu\text{m}$ and in weak-to-moderate scintillation conditions (spherical-wave log-amplitude variance $\sigma_\chi^2 \in \left[0.1,0.5\right]$), we find the average hidden-phase correlation linewidth to be approximately $\text{0.35} \text{ } \mu\text{m}$. Consequently, for $\left|\lambda_\text{B}-\lambda_\text{T}\right|$ greater than this linewidth, including the hidden phase does not significantly improve AO performance over least-squares phase compensation.