Download 6671-09_stacie_hvisc - LOFT, Large Optics Fabrication and

Structure Function Analysis
of Annular Zernike
Polynomials
Anastacia M. Hvisc*, James H. Burge
College of Optical Sciences/The University
of Arizona
* [email protected]
Outline


My work describes a method of converting
annular Zernike polynomials into structure
functions
This presentation will cover
Why we want to do this (background)?
 What is a structure function?
 Describe a tool for people to use to convert
commonly known information (Zernikes) into
knowledge useful for telescopes (structure functions)

Telescope Image Quality

The atmosphere, telescope and instrumentation
combined determine the observed image quality
of a distant star


Performance is usually limited by atmospheric
turbulence
The goal is to fabricate and support the
telescope mirrors so that the performance
degradation due to the optics matches the best
atmosphere you are statistically likely to see
Atmospheric Turbulence

Atmospheric turbulence
causes wavefront phase
errors on different spatial
scales

Due to variations in the
refractive index of the
atmosphere
Telescope
Images from Wikipedia
Structure Function Definition

The structure function for the phase fluctuations is
defined as
 
 2
D (r )  [ ( x  r )   ( x )]


x
It statistically describes the average variance in phase
between all pairs of points separated by a distance r
in the aperture
Units are waves2
Structure Function for Kolmogorov
Turbulence
The structure function for
Kolmogorov turbulence is
5/3

~20cm for a good atmosphere
r0 = 0.1m
6

r
  
D (r )  
 6.88 
 2 
 r0 
 (Equation was developed by
Tatarski for long exposures)
 r0 is the atmospheric
correlation length introduced
by Fried
2
8
10
D (r) [nm2]

10
r0 = 0.2m
4
10
2
10
-2
10
-1
10
r [m]
0
10
Structure Functions Mirror
Specifications



The goal is to fabricate and support the telescope
mirrors so that the performance degradation due to the
optics matches the best atmosphere you are statistically
likely to see on all spatial scales
Structure functions were first used as manufacturing
specifications for the William Herschel Telescope
(WHT) polished by Grubb-Parsons (1980s)
Other mirrors polished to specifications on several
spatial scales include the


Large Binocular Telescope (LBT)
Discovery Channel Telescope (DCT)
Telescope Figuring and Support
Errors

Zernike polynomials are a convenient set of
orthonormal basis functions frequently used to
describe wavefront errors on a unit circle or
annulus


Annular Zernikes describe telescope mirrors with a
central obscuration
Zernike polynomials describe the errors in
telescope mirrors due to figuring and support

Commonly output from interferometer data
Annular Zernike Polynomials

The first 10 Zernike annular
polynomials with a central
obscuration of
rinner

 0.4
router

Zernikes here are standard Zernikes


Terms are orthonormal such that the
magnitude of the coefficient of each
term is the RMS contribution of the
term
Annular Zernike polynomials were
developed by Mahajan
“Zernike annular polynomials for imaging systems
with annular pupils,” J. Opt. Soc. Am., Vol. 71,
No. 1 (1981).
Telescopes figuring and support
errors


Some of the errors polished into the telescope
surface or caused by gravity deflections can be
corrected using active supports
Zernike polynomial coefficients efficiently
describe the errors due to
Figuring/polishing
 Bending modes
 Residual after active support corrections

Zernike polynomials and Structure
Functions


Convert the Zernike polynomials describing the
surface into a structure function
Technique:
Find the structure function for each individual
Zernike term
 Add the individuals structure functions from each of
the Zernike coefficients linearly


Just as you add the Zernike terms linearly to find the total
surface
Finding the Structure Functions

The individual structure functions were found
numerically using MATLAB

For Zernike polynomials


For different annular pupils



Z = 1…28
Obscuration ratios of 0, 0.2, 0.4, 0.6
All these results listed in the paper
Available online
Finding the structure functions




For each obscuration ratio and for each Zernike polynomial
Create the surface using a matrix of points (>100 x 100)
For each point on the surface find a second point a distance r
away for a number of different angles (>15)
Then figure out if the second position also lies in the pupil




If the second point is also in the pupil, then add the squared difference
of the two values to the structure function
If the second point is not in the pupil, then ignore it
Sum the squared difference of the phase for all pairs of points a
distance r apart. Then divide by the total number of pairs to get
the average.
Save the resulting structure function
Example




As a example, point #1 is
chosen (in yellow) and seven
point #2’s are shown a
distance of s ~ 0.2 x
diameter away.
Two of the points fall off of
the pupil and so they are
ignored
The other 5 points are
included in the calculation
SF at r is from the variance
of all points separated by a
distance r that lie in the pupil
Sample results for astigmatism



For small separations,
the difference
approaches zero
For small distances, the
structure function is
linear on the log plot
For large distances, the
value of the structure
function is not
statistically significant

PSD has similar problem
solved by windowing
Giant Magellan Telescope Example

Structure Function for aberrations on a circle
1000

Difference in nm
100

10
1
0.01
0.1
1
Analysis came up with
Zernike coefficients
8.4m segment of Giant
Magellan Telescope
This is actually the 2x the
square root of the structure
function

10
Point Separation (in m )

Multiply by two since twice the
phase error after reflection
Units are difference in nm –
a physically significant
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0
0
0
0.05
0.12
0.79
7.03
0.93
0.33
0.28
5.57
5.63
0.84
0.19
0.07
3.65
25.45
4.04
4.70
0.06
0.03
10.48
2.04
13.62
25
26
27
28
29
30
31
32
33
34
35
36
37
0.44
0.92
0.45
0.26
4.67
0.68
0.59
0.56
0.15
0.17
0.09
0.15
2.23
Structure function for GMT optical
test
Excel example

Available online
Conclusion
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
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Zernike polynomials can be used to describe the residual
surface errors in telescope mirror due to polishing and support
errors
Using the conversions to structure functions from Zernike
polynomials developed here, a structure function for the mirror
can be found
Structure functions are useful specifications for manufacturing
mirrors because the errors compared to the atmosphere can be
specified at all spatial scales
R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am., Vol. 66, No. 3, (1976).
G.-m. Dai and V. N. Mahajan, “Zernike annular polynomials and atmospheric turbulence,” J. Opt. Soc. Am.
A, Vol. 24, No. 1 (2007).
D. L. Fried, “Statistics of a Geometric Representation of Wavefront Distortion,” J. Opt. Soc. Am., Vol. 55,
No. 11, (1965).
V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am.,
Vol. 71, No. 1 (1981).