Structure Function Structure Function Analysis of Annular Analysis of Annular Zernike PolynomialsZernike Polynomials

Anastacia M. Hvisc*, James H. Anastacia M. Hvisc*, James H. BurgeBurge

College of Optical Sciences/The College of Optical Sciences/The University of ArizonaUniversity of Arizona

* amh21@email.arizona.edu* amh21@email.arizona.edu

OutlineOutline

My work describes a method of My work describes a method of converting annular Zernike converting annular Zernike polynomials into structure functionspolynomials into structure functions

This presentation will coverThis presentation will cover Why we want to do this (background)?Why we want to do this (background)? What is a structure function?What is a structure function? Describe a tool for people to use to Describe a tool for people to use to

convert commonly known information convert commonly known information (Zernikes) into knowledge useful for (Zernikes) into knowledge useful for telescopes (structure functions)telescopes (structure functions)

Telescope Image QualityTelescope Image Quality

The atmosphere, telescope and The atmosphere, telescope and instrumentation combined determine instrumentation combined determine the observed image quality of a distant the observed image quality of a distant starstar Performance is usually limited by Performance is usually limited by

atmospheric turbulence atmospheric turbulence The goal is to fabricate and support the The goal is to fabricate and support the

telescope mirrors so that the telescope mirrors so that the performance degradation due to the performance degradation due to the optics matches the best atmosphere you optics matches the best atmosphere you are statistically likely to seeare statistically likely to see

Atmospheric TurbulenceAtmospheric Turbulence

Atmospheric Atmospheric turbulence causes turbulence causes wavefront phase wavefront phase errors on different errors on different spatial scales spatial scales Due to variations in Due to variations in

the refractive index the refractive index of the atmosphereof the atmosphere

Telescope

Images from Wikipedia

Structure Function Structure Function DefinitionDefinition

The structure function for the phase The structure function for the phase fluctuations is defined asfluctuations is defined as

It statistically describes the average It statistically describes the average variance in phase between all pairs of variance in phase between all pairs of points separated by a distance points separated by a distance rr in the in the apertureaperture

Units are wavesUnits are waves22

xxrxrD 2)]()([)(

Structure Function for Structure Function for Kolmogorov TurbulenceKolmogorov Turbulence

The structure function The structure function for Kolmogorov for Kolmogorov turbulence isturbulence is

(Equation was developed (Equation was developed by Tatarski for long by Tatarski for long exposures)exposures)

rr00 is the atmospheric is the atmospheric correlation length correlation length introduced by Friedintroduced by Fried

~20cm for a good ~20cm for a good atmosphereatmosphere

3/5

0

2

88.62

)(

r

rrD

10-2

10-1

100

102

104

106

108

r [m]

D(

r) [

nm2 ]

r0 = 0.1m

r0 = 0.2m

Structure Functions Mirror Structure Functions Mirror SpecificationsSpecifications

The goal is to fabricate and support the The goal is to fabricate and support the telescope mirrors so that the performance telescope mirrors so that the performance degradation due to the optics matches the best degradation due to the optics matches the best atmosphere you are statistically likely to see atmosphere you are statistically likely to see on on all spatial scalesall spatial scales

Structure functions were first used as Structure functions were first used as manufacturing specifications for the William manufacturing specifications for the William Herschel Telescope (WHT) polished by Grubb-Herschel Telescope (WHT) polished by Grubb-Parsons (1980s)Parsons (1980s)

Other mirrors polished to specifications on Other mirrors polished to specifications on several spatial scales include theseveral spatial scales include the Large Binocular Telescope (LBT)Large Binocular Telescope (LBT) Discovery Channel Telescope (DCT)Discovery Channel Telescope (DCT)

Telescope Figuring and Telescope Figuring and Support ErrorsSupport Errors

Zernike polynomials are a convenient Zernike polynomials are a convenient set of orthonormal basis functions set of orthonormal basis functions frequently used to describe wavefront frequently used to describe wavefront errors on a unit circle or annuluserrors on a unit circle or annulus Annular Zernikes describe telescope Annular Zernikes describe telescope

mirrors with a central obscurationmirrors with a central obscuration Zernike polynomials describe the errors Zernike polynomials describe the errors

in telescope mirrors due to figuring and in telescope mirrors due to figuring and supportsupport Commonly output from interferometer dataCommonly output from interferometer data

Annular Zernike Annular Zernike PolynomialsPolynomials

The first 10 Zernike annular The first 10 Zernike annular polynomials with a central polynomials with a central obscuration ofobscuration of

Zernikes here are standard Zernikes here are standard ZernikesZernikes Terms are orthonormal such Terms are orthonormal such

that the magnitude of the that the magnitude of the coefficient of each term is the coefficient of each term is the RMS contribution of the termRMS contribution of the term

Annular Zernike polynomials Annular Zernike polynomials were developed by Mahajanwere developed by Mahajan

““Zernike annular polynomials for imaging Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. systems with annular pupils,” J. Opt. Soc. Am., Vol. 71, No. 1 (1981).Soc. Am., Vol. 71, No. 1 (1981).

4.0outer

inner

r

r

Telescopes figuring and Telescopes figuring and support errorssupport errors

Some of the errors polished into the Some of the errors polished into the telescope surface or caused by gravity telescope surface or caused by gravity deflections can be corrected using deflections can be corrected using active supports active supports

Zernike polynomial coefficients Zernike polynomial coefficients efficiently describe the errors due toefficiently describe the errors due to Figuring/polishingFiguring/polishing Bending modesBending modes Residual after active support correctionsResidual after active support corrections

Zernike polynomials and Zernike polynomials and Structure FunctionsStructure Functions

Convert the Zernike polynomials Convert the Zernike polynomials describing the surface into a structure describing the surface into a structure functionfunction

Technique:Technique: Find the structure function for each Find the structure function for each

individual Zernike term individual Zernike term Add the individuals structure functions from Add the individuals structure functions from

each of the Zernike coefficients linearly each of the Zernike coefficients linearly Just as you add the Zernike terms linearly to find Just as you add the Zernike terms linearly to find

the total surfacethe total surface

Finding the Structure Finding the Structure FunctionsFunctions

The individual structure functions The individual structure functions were found numerically using were found numerically using MATLABMATLAB For Zernike polynomials For Zernike polynomials

Z = 1…28Z = 1…28 For different annular pupilsFor different annular pupils

Obscuration ratios of 0, 0.2, 0.4, 0.6Obscuration ratios of 0, 0.2, 0.4, 0.6

All these results listed in the All these results listed in the paperpaper Available onlineAvailable online

Finding the structure Finding the structure functionsfunctions

For each obscuration ratio and for each Zernike For each obscuration ratio and for each Zernike polynomialpolynomial

Create the surface using a matrix of points (>100 x 100)Create the surface using a matrix of points (>100 x 100) For each point on the surface find a second point a For each point on the surface find a second point a

distance distance rr away for a number of different angles (>15) away for a number of different angles (>15) Then figure out if the second position also lies in the Then figure out if the second position also lies in the

pupilpupil If the second point is also in the pupil, then add the squared If the second point is also in the pupil, then add the squared

difference of the two values to the structure functiondifference of the two values to the structure function If the second point is not in the pupil, then ignore itIf the second point is not in the pupil, then ignore it

Sum the squared difference of the phase for all pairs of Sum the squared difference of the phase for all pairs of points a distance points a distance rr apart. Then divide by the total apart. Then divide by the total number of pairs to get the average.number of pairs to get the average.

Save the resulting structure functionSave the resulting structure function

ExampleExample As a example, point #1 As a example, point #1

is chosen (in yellow) and is chosen (in yellow) and seven point #2’s are seven point #2’s are shown a distance of s ~ shown a distance of s ~ 0.2 x diameter away.0.2 x diameter away.

Two of the points fall off Two of the points fall off of the pupil and so they of the pupil and so they are ignoredare ignored

The other 5 points are The other 5 points are included in the included in the calculationcalculation

SF at r is from the SF at r is from the variance of all points variance of all points separated by a distance separated by a distance r that lie in the pupilr that lie in the pupil

Sample results for Sample results for astigmatismastigmatism

For small separations, For small separations, the difference the difference approaches zeroapproaches zero

For small distances, For small distances, the structure function the structure function is linear on the log is linear on the log plotplot

For large distances, For large distances, the value of the the value of the structure function is structure function is not statistically not statistically significantsignificant PSD has similar PSD has similar

problem solved by problem solved by windowingwindowing

Giant Magellan Telescope Giant Magellan Telescope ExampleExample

Structure Function for aberrations on a circle

1

10

100

1000

0.01 0.1 1 10

Point Separation (in m )

Dif

fere

nce

in n

m

Analysis came up with Analysis came up with Zernike coefficientsZernike coefficients

8.4m segment of Giant 8.4m segment of Giant Magellan TelescopeMagellan Telescope

This is actually the 2x This is actually the 2x the square root of the the square root of the structure functionstructure function Multiply by two since Multiply by two since

twice the phase error twice the phase error after reflectionafter reflection

Units are difference in Units are difference in nm – a physically nm – a physically significant numbersignificant number1 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 Structure function for GMT optical testoptical test

Excel exampleExcel example

Available onlineAvailable online

ConclusionConclusion Zernike polynomials can be used to describe the Zernike polynomials can be used to describe the

residual surface errors in telescope mirror due to residual surface errors in telescope mirror due to polishing and support errorspolishing and support errors

Using the conversions to structure functions from Using the conversions to structure functions from Zernike polynomials developed here, a structure Zernike polynomials developed here, a structure function for the mirror can be foundfunction for the mirror can be found

Structure functions are useful specifications for Structure functions are useful specifications for manufacturing mirrors because the errors manufacturing mirrors because the errors compared to the atmosphere can be specified at compared to the atmosphere can be specified at all spatial scalesall spatial scales

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am., Vol. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am., Vol. 66, No. 3, (1976).66, No. 3, (1976).

G.-m. Dai and V. N. Mahajan, “Zernike annular polynomials and atmospheric G.-m. Dai and V. N. Mahajan, “Zernike annular polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A, Vol. 24, No. 1 (2007).turbulence,” J. Opt. Soc. Am. A, Vol. 24, No. 1 (2007).

D. L. Fried, “Statistics of a Geometric Representation of Wavefront Distortion,” J. Opt. D. L. Fried, “Statistics of a Geometric Representation of Wavefront Distortion,” J. Opt. Soc. Am., Vol. 55, No. 11, (1965).Soc. Am., Vol. 55, No. 11, (1965).

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am., Vol. 71, No. 1 (1981).pupils,” J. Opt. Soc. Am., Vol. 71, No. 1 (1981).