Turbulence bends the wavefront of a star; a deformable mirror cancels the part it can follow. Left to right: the distorted wavefront, the mirror shape, what remains, and the resulting star image. Real Zernike statistics (Noll) and an FFT point-spread function.
Runs 40 random turbulence screens for each mirror size and compares the mean with Marechal and the Noll prediction.
The science
Turbulence. Kolmogorov phase fluctuations expand in Zernike modes phi = sum a_j Z_j. The mean-square residual after removing the first J modes is Noll's Delta_J (D/r0)^(5/3), with Delta_1 = 1.0299, Delta_3 = 0.134 (tilt removed) and Delta_J ~ 0.2944 J^(-sqrt(3)/2) for large J. Fried's parameter r0 sets how coarse the seeing is.
Mirror. An ideal modal deformable mirror applies the opposite of modes 2..J. The residual phase variance is sigma^2.
Strehl ratio. Peak of the real image over the peak of a perfect one. For small errors the Marechal approximation is S = exp(-sigma^2); here S is measured from the FFT of the pupil field P e^(i phi).
Loop latency. Coefficients evolve as an AR(1) process, a(t+1) = rho a(t) + sqrt(1-rho^2) sigma xi. The mirror applies what was measured one frame ago times the gain, so a fast atmosphere (low coherence time) leaves a servo-lag error.
Honest limits. Diagonal Noll covariance (the small cross-correlations between modes of equal azimuthal order are ignored); no wavefront-sensor noise, no aliasing, no actuator influence functions (a perfect modal mirror); monochromatic light, near-field only (no scintillation); frame time and coherence time are illustrative, not a calibrated telescope. This is a teaching model, not the AETHERIS instrument design.