Influence of the aberration structure on the accuracy of thin-diffuser wavefront sensing

We present a theoretical and numerical analysis on how the aberration structure governs speckle decorrelation and, consequently, thin-diffuser wavefront sensor (TDWFS) performance. A field decomposition is developed that separates quadratic and higher-order wavefront contributions, distinguishing purely geometric speckle deformation from additional structural changes arising from higher-order diffraction effects. We show that wavefronts producing affine displacement fields are fully described by the quadratic field contribution and are therefore recoverable by image registration, whereas non-affine wavefronts additionally contain higher-order field components that introduce irreducible speckle decorrelation. Using a Sobolev space decomposition approach, we isolate the affine-removable components of the wavefront and show that irreducible speckle decorrelation is controlled by the Hessian of the non-affine component. We derive critical correlation- and deformation-based propagation distance metrics governed by the pupil radius, aberration amplitude, and the radial and azimuthal Zernike indices. We further analyze the Fisher information associated with individual Zernike modes and show that modes with low azimuthal order exhibit greater measurement sensitivity due to stronger speckle intensity variation. Numerical simulations are performed to confirm theoretical predictions and show that purely affine modes, such as defocus and astigmatism, remain robust across propagation distances beyond a sensitivity threshold, whereas non-affine modes exhibit degradation beyond a critical distance that strongly depends on their spatial curvature distribution. These results establish that wavefront curvature, rather than global metrics such as Zernike order or RMS amplitude, is the primary determinant of TDWFS reconstruction limits and provides a framework for predicting performance across aberration types.