Zernike Polynomial Representation of Wavefront Shape

Application of Zernike Polynomials in Wavefront Analysis

In optical systems and ophthalmic medicine, describing wavefront distortions (such as lens aberrations or corneal irregularities) requires a mathematically orthogonal and intuitive decomposition method. Zernike polynomials serve as an ideal tool for wavefront shape representation due to their orthogonality over the unit circle.

Core Logic Orthogonal Basis Construction Zernike polynomials are composed of radial and angular function combinations, with each term corresponding to specific aberration types (e.g., defocus, astigmatism, coma). Through orthogonality, coefficients of different orders remain independent, facilitating separation and quantification of aberration contributions.

MATLAB Implementation Key Points Data Input: Wavefront data typically originates from interferometers or corneal topographers, stored as discrete phase values in matrix format. Fitting Process: Using least squares method to solve for Zernike coefficients requires constructing a matrix of polynomial values at sample points. Coefficient fitting is achieved through pseudo-inverse operations to avoid directly solving underdetermined equations. Result Visualization: Residual plots comparing reconstructed wavefronts with original data verify fitting accuracy, while higher-order coefficients reflect detailed distortions.

Extended Applications Dynamic Aberration Correction: Real-time calculation of lower-order coefficients (e.g., Z2-Z4) enables rapid adjustments in adaptive optical systems. Medical Diagnosis: Corneal Zernike analysis can identify early-stage asymmetry in conditions like keratoconus, supporting clinical decision-making.

This method decomposes complex wavefronts into interpretable physical quantities, providing quantitative basis for optical design and defect detection.