Applications of Zernike Polynomials

Overview

• This short course introduces the Zernike polynomials and their use in various areas of optics.  Topics covered include applications of Zernike polynomials in optics, general issues of fitting data, orthogonal functions, standard and non-standard forms of Zernike polynomials, root-mean-square error and least squares fitting to both phase and phase derivative data.

Intended Audience

• Optical engineers and scientists who need to fit surface metrology and wavefront data to Zernike polynomials. 

• Astronomers using adaptive optics systems.

• Lens designers who use Zernike polynomials to evaluate the aberration structure and performance of optical systems. 

• Vision scientists who measure corneal topography and ocular wavefront aberrations to determine visual performance.

• Image processing researchers who need rotationally invariant means for recognizing objects.

Learning Objectives

This course will enable participants to:

• Understand issues that arise when fitting data.

• Verify if sets of functions satisfy orthogonality and derive expansion coefficients for these functions.

• Fit 2D surfaces over a circular region.

• Determine which of the standard and non-standard forms of the Zernike polynomials is presented and convert between various sets.

• Calculate root-mean square error as a measure of the irregularity of the surface.

• Generate expansion coefficients for discrete data for both phase (sag) and phase derivative (surface slope).

Course Outline

History of Zernike polynomials

Applications of Zernike polynomials in optics

• Astronomy

• Diffraction theory

• Ophthalmic optics

• Optical testing

General Issues of Fitting Data

• Exact fit versus minimizing residual error

• Numerical stability

• Features versus reconstruction

Orthogonal Functions

• Definition of orthogonal functions

• Expansion coefficients

• Example: 1D Fourier series

Zernike Polynomials

• Fitting 2D surfaces to the unit circle

• Coordinate conventions

• ANSI standard definition of Zernike polynomials

• Common alternative definitions of the Zernike polynomials

• Example: fitting wavefronts for different pupil sizes

• rms error

Least Squares Fitting

• Least squares method for fitting discrete phase (sag) data

• Shack-Hartmann wavefront sensing

• Least squares method for fitting discrete slope data

Concluding Remarks

Contact:

Ms. Cindy Robertson

520-626-4719

cindyr@optics.arizona.edu

 

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