Zernike polynomials

1
Zernike polynomials
Why does anyone care about Zernike polynomials? A
little history about their development. Definition
s and math - what are they? How do they make
certain questions easy to answer? A couple of
practical applications
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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
• ZPs 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

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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 1980s CNC grinder has 32K of memory
• Less computational need for ZPs these days but
  they give insight into operations with surfaces
  and wavefronts

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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
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Some Zernike details
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Zernike Triangle
n 0 1 2 3 4
m -4 -3 -2 -1 0 1
2 3 4
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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
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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
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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

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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) (2p2 - 1) Z 5 6(1/2) (p2)
SIN (2A) Z 6 6(1/2) (p2) COS (2A) Z
7 8(1/2) (3p3 - 2p) SIN (A) Z 8
8(1/2) (3p3 - 2p) COS (A) Z 9
8(1/2) (p3) SIN (3A)
Z 1 1 Z 2 (p) COS (A) Z 3
(p) SIN (A) Z 4 (2p2 - 1) Z 5
(p2) COS (2A) Z 6 (p2) SIN (2A) Z
7 (3p2 - 2) p COS (A) Z 8 (3p2
- 2) p SIN (A) Z 9 (6p4 - 6p2 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
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Zernike coefficients
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Addition (subtraction) of wavefronts
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Rotation of wavefronts
These equations look familiar Derived from
multi-angle formulas Work in pairs like coord.
rotation
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Rotation matrix in code
1 0 0 0 0 0 0 0 a00 b00
0 cos? sin? 0 0 0 0 0 a1-1 b1-1
0 -sin? cos? 0 0 0 0 0 a11 b11
0 0 0 cos2? 0 sin2? 0 0 a2-2 b2-2
0 0 0 0 1 0 0 0 a20 b20
0 0 0 -sin2? 0 cos2? 0 0 a22 b22
0 0 0 0 0 0 cos3? 0 a3-3 b3-3
0 0 0 0 0 0 0 cos? a3-1 b3-1

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Aperture scaling
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Aperture scaling matrix
1 c2-1 c2-1
c 2c2(c2-1)
c 2c2(c2-1)
c2
c2
c2
c3
c3
c3
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Aperture shifting
1 h 2h2 h2
1 2h 3h2 3h2
1 4h 2h
1 3h 3h2
1
1
1
1

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Useful example of shift and scalingZernike
coefficients over an off-axis aperture
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Symmetry properties

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Determining arbitrary symmetry
Flip by changing sign of appropriate coefficients
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Symmetry of arbitrary surface
For alignment situations, symmetry may be all you
need This is a simple way of finding the
components
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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
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Symmetry applied to images
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Same idea applied to slopes
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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/zern
ikes/zernikes.htm http//en.wikipedia.org/wiki/Ze
rnike_polynomials