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Review of Zernike polynomials and
their use in describing the impact of
misalignment in optical systemsJim Schwiegerling, PhD
Ophthalmology & Vision Science
Optical Sciences
The University of Arizona
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Background
• The mathematical functions were
originally described by Frits
Zernike in 1934.
• They were developed to describe
the diffracted wavefront in phase
contrast imaging.
• Zernike won the 1953 Nobel
Prize in Physics for developing
Phase Contrast Microscopy.
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Phase Contrast Microscopy
Transparent specimens leave the amplitude of the illumination
virtually unchanged, but introduces a change in phase.
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Applications
• Optical Design – describing complex
shapes such as freeform surfaces and
fabrication errors.
• Optical Testing - fitting reflected and
transmitted wavefront data.
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Surface Fitting
• Fitting a complex, non-rotationally symmetric surfaces (phase
fronts) over a circular domain.
• Possible goals of fitting a surface:
– Exact fit to measured data points?
– Minimize “Error” between fit and data points?
– Extract Features from the data?
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1D Curve Fitting
0
5
10
15
20
25
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
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Low-order Polynomial Fit
y = 9.9146x + 2.3839
R2 = 0.9383
0
5
10
15
20
25
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
In this case, the error is the vertical distance between the line and
the data point. The sum of the squares of the error is minimized.
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High-order Polynomial Fit
0
5
10
15
20
25
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5-400
-200
0
200
400
600
800
1000
1200
1400
1600
1800
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
y = a0 + a1x + a2x2 + … a15x
15
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Fitting Issues
• Know your data. Too many terms in the fit can be numerically
unstable and/or fit noise in the data. Too few terms may miss real
trends in the surface.
• Typically want “nice” properties for the fitting function such as
smooth surfaces with continuous derivatives.
• Typically want to represent many data points with just a few terms
of a fit. This gives compression of the data, but leaves some
residual error. For example, the line fit represents 16 data points
with two numbers: a slope and an intercept.
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Why Zernikes?
• Zernike polynomials have nice mathematical properties.
– They are orthogonal over the continuous unit circle.
– All their derivatives are continuous.
– They efficiently represent common errors (e.g. coma,
spherical aberration) seen in optics.
– They form a complete set, meaning that they can represent
arbitrarily complex continuous surfaces given enough terms.
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Orthogonality - Zernike
Orthogonality means we have an easy means of calculating
expansion coefficients.
𝑊 𝜌, 𝜃 =
𝑛,𝑚
𝑎𝑛𝑚𝑍𝑛𝑚 𝜌, 𝜃
𝑎𝑛𝑚 =1
𝜋 0
2𝜋
0
1
𝑊 𝜌, 𝜃 𝑍𝑛𝑚 𝜌, 𝜃 𝜌𝑑𝜌𝑑𝜃
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Discrete Data
• Typically, we do not have a continuous description of 𝑊 ,
instead the data is discretely sampled (e.g. pixels on the digital
sensor of an interferometer.
• In this case, we use matrix methods to find the expansion
coefficients. This method is the same technique that we would use
for fitting non-orthogonal functions.
• So why use orthogonal function?
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XY Polynomials
XY Polynomials
Most of the variation for the higher order terms
occurs at the edges.
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Zernike Polynomials
The variation oscillates in the radial and azimuthal
direction.
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Even Asphere
When the fitting functions
have most of their change at
the edge, then we need huge
values of high order terms to
represent small total sag.
Numerical precision becomes
an issue here as small
changes to coefficients can
cause large changes in total
sag.
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Zernike Equivalent
When the fitting functions
have most of their change at
the edge, then we need huge
values of high order terms to
represent small total sag.
Numerical precision becomes
an issue here as small
changes to coefficients can
cause large changes in total
sag.
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StandardsANSI Z80.28-2010
ISO 14999-2:2005
ISO 24157:2008 Normalized
Unnormalized
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Unit Circle
x
y
r
q
1
Divide the real
radial coordinate
by the maximum radius
to get a normalized
coordinate r
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ANSI Z80.28/ISO 24157 Zernikes
qr
qrqr
0 mfor ; msin)(RN
0mfor ; mcos)(RN),(Z
m
n
m
n
m
n
m
nm
n
Double Index
n is radial order
m is azimuthal
frequency
Normalization
Radial
Component
Azimuthal
Component
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ANSI Z80.28/ISO 24157 Zernikes
only depends
on |m| (i.e. same
for both sine &
cosine terms)
Powers of r
s2n
2/)mn(
0s
sm
n! s)mn(5.0! s)mn(5.0!s
)!sn()1()(R
r
r
Constant that depends
on n and m
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ANSI Z80.28/ISO 24157 Zernikes
constant that
depends on n & m
0m
m
n1
2n2N
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Wavefront Variance
• Wavefront variance and its square root RMS wavefront error are
metrics of image quality.
• For the normalized Zernikes, the wavefront variance is trivial to
calculate.
• Basically, the squared magnitude of each term describes its
contribution to the variance.
• RMS Error is just square root of the variance.
𝜎𝑊2 =
𝑛≥1
𝑎𝑛𝑚2 − 𝑎00
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Wavefront Fitting
=
-0.003 x
+ 0.002 x
+ 0.001 x
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Different Zernike Sets
“Standard” or Noll Zernike Fringe Zernike
The Fringe Zernike set is a subset of the Zernike
polynomials.
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Zernike Polynomials
Azimuthal Frequency, q
Ra
dia
lP
oly
nom
ial,
r
Z00
Z11Z1
1
Z20
Z31 Z3
1
Z40 Z4
2
Z22
Z42
Z33 Z3
3
Z44Z4
4
Z22
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Caveats to the Definition of
Zernike Polynomials• At least six different schemes exist for the Zernike polynomials.
• Some schemes only use a single index number instead of n and m.
With the single number, there is no unique ordering or definition
for the polynomials, so different orderings are used.
• Some schemes set the normalization to unity for all polynomials.
• Some schemes measure the polar angle in the clockwise direction
from the y axis.
• The expansion coefficients depend on pupil size, so the maximum
radius used must be given.
• Make sure which set is being given for a specific application.
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Zernike Polynomials - Single Index
Azimuthal Frequency, q
Ra
dia
lP
oly
nom
ial,
r
Z0
Z1
Z4 Z5Z3
Z9Z8Z7Z6
Z10 Z11 Z12 Z13 Z14
Z2
ANSI Z80.28/ISO
24157 STANDARDStarts at 0
Left-to-Right
Top-to-Bottom
Normalized
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Other Single Index Schemes
Z0
Z2
Z3 Z4 Z5
Z9Z6 Z7 Z10
Z17Z12Z8 Z11 Z16
Z1
ISO 14999-2 STANDARDStarts at 0
increases along diagonal
cosine terms first
No Normalization
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Other Single Index Schemes
Z1
Z3
Z4 Z6Z5
Z10Z8Z7 Z9
Z15Z13Z11 Z12 Z14
Z2
NON-STANDARDStarts at 1
cosines are even terms
sines are odd terms
Normalized
Noll, RJ. Zernike polynomials and atmospheric turbulence. J Opt Soc Am 66; 207-211 (1976).
Also Zemax “Standard Zernike Coefficients”
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Other Single Index Schemes
Z1
Z3
Z4 Z5 Z6
Z10Z7 Z8 Z11
Z18Z13Z9 Z12 Z17
Z2
NON-STANDARDStarts at 1
increases along diagonal
cosine terms first
35 terms plus two extra
spherical aberration terms.
No Normalization!!!
Zemax “Zernike Fringe Coefficients”
Code V Zernikes
Also, Air Force or University of Arizona
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Other Single Index Schemes
• Born & Wolf
• Malacara
• Others??? Plus mixtures of non-normalized, coordinate systems.
NON-STANDARD
Use two indices n, m to unambiguously define polynomials.
Use a single standard index only if needed to avoid confusion.
Noll or Zemax “Standard” is closest to ANSI Z80.28/ISO 24157
Fringe set is closest to ISO 14999-2, but has limited terms.
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Summary
• Zernike polynomials are a useful set of functions for representing
surface form and wavefronts on circular domains.
• The normalized version of the Zernikes gives a direct quality
metric in the form of variance.
• Many different schemes and definitions exists, so be careful when
comparing results from different sources.
• Two-index scheme is always unambiguous.
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Aligned
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Y-Decenter
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General Decenter
𝑡𝑎𝑛−1𝑎1,−1𝑎1,1
Direction of decentration