Development and verification of the signal to noise ratio for a layer of turbulence in a multi-layer atmosphere

Wide-field image correction in systems that look through the atmosphere generally requires a tomographic reconstruction of the turbulence volume to compensate for anisoplanatism. The reconstruction is conditioned by estimating the turbulence volume as a profile of thin homogeneous layers. We present the signal to noise ratio (SNR) of a layer, which quantifies how difficult a single layer of homogeneous turbulence is to detect with wavefront slope measurements. The signal is the sum of wavefront tip and tilt variances at the signal layer, and the noise is the sum of wavefront tip and tilt auto-correlations given the aperture shape and projected aperture separations at all non-signal layers. An analytic expression for layer SNR is found for Kolmogorov and von Kármán turbulence models, then verified with a Monte Carlo simulation. We show that the Kolmogorov layer SNR is a function of only layer Fried length, the spatio-angular sampling of the system, and normalized aperture separation at the layer. In addition to these parameters, the von Kármán layer SNR also depends on aperture size, and layer inner and outer scales. Due to the infinite outer scale, layers of Kolmogorov turbulence tend to have lower SNR than von Kármán layers. We conclude that the layer SNR is a statistically valid performance metric to be used when designing, simulating, operating, and quantifying the performance of any system that measures properties of layers of turbulence in the atmosphere from slope data.