Zernike polynomials

1. Zernike polynomials

2. What will Zernikes do for me? Widely used in industry outside of lens design Easy to estimate image quality from coefficients Continuous & orthogonal on unit circle, Seidels are not Can fit one at a time, discrete data not necessarily orthogonal ZP’s will give misleading, erroneous results if not circular aperture Balance aberrations as a user of an optical device would Formalism makes calculations easy for many problems Good cross check on lens design programs Applicable to slope and curvature measurement as well as wavefront or phase measurement

3. History of Zernikes Frits Zernike wrote paper in 1934 defining them Used to explain phase contrast microscopy He got a Nobel Prize in Physics in 1953 for above E. Wolf, et. al., got interested in 1956 & in his book Noll (1976) used them to describe turbulent air My interest started about 1975 at Itek with a report Shannon brought to OSC, John Loomis wrote FRINGE J. Schwiegerling used in corneal shape research Incorporated in ISO 24157 with double subscript

4. Practical historical note In 1934 there were no computers – stuff hard to calculate In 1965 computers starting to be used in lens design Still using mainframe computers in 1974 Personal calculators just becoming available at $5-10K each People needed quick way to get answers 36 coefficients described surface of hundreds of fringe centers Could manipulate surfaces without need to interpolate Same sort of reason for use of FFT, computationally fast Early 1980’s CNC grinder has 32K of memory Less computational need for ZP’s these days but they give insight into operations with surfaces and wavefronts

5. What are Zernike polynomials? Set of basis shapes or topographies of a surface Similar to modes of a circular drum head Real surface is constructed of linear combination of basis shapes or modes Polynomials are a product of a radial and azimuthal part Radial orders are positive, integers (n), 0, 1,2, 3, 4, …… Azimuthal indices (m) go from –n to +n with m – n even

6. Some Zernike details

7. Zernike Triangle

8. Rigid body or alignment terms

9. Third order aberrations

10. Zernike nomenclature Originally, Zernike polynomials defined by double indices More easily handled serially in computer code FRINGE order, standard order, Zygo order (confusing) Also, peak to valley and normalized PV, if coefficient is 1 unit, PV contour map is 2 units Normalized, coefficient equals rms departure from a plane Units, initially waves, but what wavelength? Now, generally, micrometers. Still in transition For class, use double indices, upper case coeff for PV lower case coefficient for normalized or rms

11. Examples of the problem

12. Zernike coefficients

13. Addition (subtraction) of wavefronts

14. Rotation of wavefronts

15. Rotation matrix in code

16. Aperture scaling

17. Aperture scaling matrix

18. Aperture shifting

19. Useful example of shift and scalingZernike coefficients over an off-axis aperture

20. Symmetry properties

21. Determining arbitrary symmetry

22. Symmetry of arbitrary surface

23. Symmetry properties of Zernikes

24. Symmetry applied to images

25. Same idea applied to slopes

26. References