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Structure Function Analysis of Annular Zernike Polynomials

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### Structure Function Analysis of Annular Zernike Polynomials

Anastacia M. Hvisc*, James H. Burge

College of Optical Sciences/The University of Arizona

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
- 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
- (Equation was developed by Tatarski for long exposures)
- r0 is the atmospheric correlation length introduced by Fried
- ~20cm for a good atmosphere

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
- 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
- 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

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

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

Giant Magellan Telescope ExampleExcel example

- Available online

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).

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