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## Zernike polynomials

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**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