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Why does anyone care about Zernike polynomials? A little history about their development. Definitions and math - what are they? How do they make certain questions easy to answer? A couple of practical applications. Zernike polynomials. What will Zernikes do for me?.

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## Zernike polynomials

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**Why does anyone care about Zernike polynomials?**A little history about their development. Definitions and math - what are they? How do they make certain questions easy to answer? A couple of practical applications Zernike polynomials**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**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**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**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 The only proper way to refer to the polynomials is with two indices**Zernike Triangle**n = 0 1 2 3 4 m = -4 -3 -2 -1 0 1 2 3 4**Rigid body or alignment terms**Tilt y and x Focus z For these terms n + m = 2 Location of a point has 3 degrees of freedom, x, y and z Alignment refers to object under test relative to test instrument**Third order aberrations**Astigmatism n = 2, m = +/- 2 Coma n = 3, m = +/- 1 Spherical aberration n = 4, m = 0 For 3rd order aberrations, n + m = 4 These are dominant errors due to mis-alignment and mounting**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**Examples of the problem**Z 1 1 Z 2 4^(1/2) (p) * COS (A) Z 3 4^(1/2) (p) * SIN (A) Z 4 3^(1/2) (2p^2 - 1) Z 5 6^(1/2) (p^2) * SIN (2A) Z 6 6^(1/2) (p^2) * COS (2A) Z 7 8^(1/2) (3p^3 - 2p) * SIN (A) Z 8 8^(1/2) (3p^3 - 2p) * COS (A) Z 9 8^(1/2) (p^3) * SIN (3A) Z 1 1 Z 2 (p) * COS (A) Z 3 (p) * SIN (A) Z 4 (2p^2 - 1) Z 5 (p^2) * COS (2A) Z 6 (p^2) * SIN (2A) Z 7 (3p^2 - 2) p * COS (A) Z 8 (3p^2 - 2) p * SIN (A) Z 9 (6p^4 - 6p^2 + 1) FRINGE order, P-V Standard order, normalized Normalization coefficient is the ratio between P-V and normalized One unit of P-V coefficient will give an rms equal normalization factor**Rotation of wavefronts**These equations look familiar Derived from multi-angle formulas Work in pairs like coord. rotation**Useful example of shift and scalingZernike coefficients over**an off-axis aperture**Determining arbitrary symmetry**Flip by changing sign of appropriate coefficients**Symmetry of arbitrary surface**For alignment situations, symmetry may be all you need This is a simple way of finding the components**Symmetry properties of Zernikes**e-e even-even o-o odd-odd e-o even-odd o-e odd-even n = 1 2 3 4 o-o e-o o-o e-o rot o-e e-e o-e e-e If radial order is odd, then e-o or o-e, if even the e-e or o-o**References**Born & Wolf, Principles of Optics – but notation is dense Malacara, Optical Shop Testing, Ch 13, V. Mahajan, “Zernike Polynomials and Wavefront Fitting” – includes annular pupils Zemax and CodeV manuals have relevant information for their applications http://www.gb.nrao.edu/~bnikolic/oof/zernikes.html http://wyant.optics.arizona.edu/zernikes/zernikes.htm http://en.wikipedia.org/wiki/Zernike_polynomials

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