Orthogonal mode decomposition of aspheric surface form errors by principal components analysis

Abstract. Surface form errors generated by process variation in optical lens fabrication are optimally described using polynomial representations that are orthogonal, physically relevant, and result in small model fitting residuals. Similarly, discrete vector space methods are appropriate for the analysis of surface form data to the extent they satisfy the above criteria. A statistical approach to the analysis of aspheric surface form errors using principal components analysis (PCA) is presented. The method returns a set of independent data vectors that (1) satisfies the orthogonality requirement, (2) minimizes the number of fitting terms, and (3) represents the most significant modes of spatial variation. Following a description of the computational methods, modeling results are provided with an emphasis on surface reconstruction fidelity and statistical validation for sample sizes n  >  25. Examples of the PCA surface modes are then discussed in terms of relevant physical interpretations. A PCA-based bootstrap resampling method is also developed from which an empirical Monte Carlo distribution of the RMS surface form error is generated.