1. Structure Function Analysis of Annular Zernike Polynomials
Anastacia M. Hvisc*, James H. Burge
College of Optical Sciences/The University of Arizona *

2. 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)

3. 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

4. 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

5. Structure Function Definition
The structure function for the phase fluctuations is defined as
It statistically describes the average variance in phase between all pairs of points separated by a distance r in the aperture
Units are waves2

6. Structure Function for Kolmogorov Turbulence
The structure function for Kolmogorov turbulence is
(Equation was developed by Tatarski for long exposures)
r0 is the atmospheric correlation length introduced by Fried
~20cm for a good atmosphere

7. 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)

8. 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

9. Annular Zernike Polynomials
The first 10 Zernike annular polynomials with a central obscuration of
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).

10. 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

11. 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

12. Finding the Structure Functions
The individual structure functions were found numerically using MATLAB
For Zernike polynomials
Z = 1…28
For different annular pupils
Obscuration ratios of 0, 0.2, 0.4, 0.6
All these results listed in the paper
Available online

13. 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

14. 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

15. 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

16. Giant Magellan Telescope Example
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
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
0.05
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
2.23

17. Structure function for GMT optical test

18. Excel example
Available online

19. Conclusion
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).