Zernike polynomials 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

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 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

Some Zernike details

Zernike Triangle m = n =

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 3 rd 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 (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) 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) FRINGE order, P-VStandard 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

Zernike coefficients

Addition (subtraction) of wavefronts

Rotation of wavefronts These equations look familiar Derived from multi-angle formulas Work in pairs like coord. rotation

Rotation matrix in code a00a00 b00b00 0 cos  sin  a 1 -1 b sin  cos  a11a11 b11b cos2  0 sin2  00 a 2 -2 b a20a20 b20b sin2  0 cos2  00 a22a22 b22b cos3  0 a 3 -3 b cos  a 3 -1 b 3 -1

Aperture scaling

Aperture scaling matrix 1c^2-1 c2c^2(c^2-1) c c^2 c^3

Aperture shifting 1h2h^2h^2 12h3h^2 14h2h 13h3h^

Useful example of shift and scaling Zernike coefficients over an off-axis aperture

Symmetry properties

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 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 e-e even-even o-o odd-odd e-o even-odd o-e odd-even n =

Symmetry applied to images

Same idea applied to slopes

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