Transcript Document 7875610

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 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
Some Zernike details
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
Z
Z
Z
Z
Z
Z
Z
Z
1
2
3
4
5
6
7
8
9
1
(p) * COS (A)
(p) * SIN (A)
(2p^2 - 1)
(p^2) * COS (2A)
(p^2) * SIN (2A)
(3p^2 - 2) p * COS (A)
(3p^2 - 2) p * SIN (A)
(6p^4 - 6p^2 + 1)
FRINGE order, P-V
Z
Z
Z
Z
Z
Z
Z
Z
Z
1
2
3
4
5
6
7
8
9
1
4^(1/2)
4^(1/2)
3^(1/2)
6^(1/2)
6^(1/2)
8^(1/2)
8^(1/2)
8^(1/2)
(p) * COS (A)
(p) * SIN (A)
(2p^2 - 1)
(p^2) * SIN (2A)
(p^2) * COS (2A)
(3p^3 - 2p) * SIN (A)
(3p^3 - 2p) * COS (A)
(p^3) * SIN (3A)
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
Zernike coefficients
Addition (subtraction) of wavefronts
Rotation of wavefronts
These equations look familiar
Derived from multi-angle formulas
Work in pairs like coord. rotation
a'lk  .14  cos 2   .20  sin  2   (.14  .5)  (.20  .866)  .243
a'2 2  .2  cos( 2 )  .14  sin( 2 )  ( .2  .5)  (.14  .866)  .021
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
 
 Rot      a  b
Aperture scaling
Z 20 ( r , )  2r 2  1  2r' 2 c 2 1  ( 2r' 2 1 )c 2  c 2  1
Aperture scaling matrix
1
c^2-1
c
2c^2(c^2-1)
c
2c^2(c^2-1)
c^2
c^2
c^2
c^3
c^3
c^3
B  Mc A
Aperture shifting
1
h
1
2h^2
h^2
2h
1
4h
3h^2
3h^2
3h
3h^2
2h
1
1
1
1
B  MxA
1
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
e-e even-even
o-o odd-odd
n=
e-o even-odd
1
o-e odd-even
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
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
http://www.gb.nrao.edu/~bnikolic/oof/zernikes.html
http://wyant.optics.arizona.edu/zernikes/zernikes.htm
http://en.wikipedia.org/wiki/Zernike_polynomials