out-of-focus holography

IntroductionThe methodSimulations

Results at the GBT

Out-of-focus Holography

B. Nikolic1, R. E. Hills1, J. S. Richer1,R. M. Prestage2, D. S. Balser2, C. J. Chandler2

1Cavendish Lab, University of Cambridge, UK2National Radio Astronomy Observatory, USA

MPIfR Bonn, September 2007

B. Nikolic OOF Holography


Results at the GBT

Outline

1 Introduction

2 The method

3 Simulations

4 Results at the GBT



Results at the GBT

Motivation

Accuracy of the surface and collimation is one of the mainlimits on size/performance of large antennas.

ALMA Antennas: 12 m diameter, 20 µm accuracy=⇒ 1.6 : 106 accuracyGreen Bank Telescope: 100 m diameter, 200 µm accuracy=⇒ 2 : 106 accuracy

Sources of inaccuracy:Setting error: staticGravitation deformation: repeatableThermal deformation: ≈ 30 minute timescaleWind: short timescaleAgeing effects



Results at the GBT

Approaches

Conventional surveyingPhotogrammetryInterferometric holographyTransmitter with-phase holographyTransmitter phase-retrieval holographyOut-Of-Focus (OOF) holography



Results at the GBT

Introduction

Aim: measure errors in the telescope opticsSurface errors + mis-collimationRapidlyAs a function of elevation, time of day, etcWithout any extra equipment

How: Use beam power mapsAstronomical receiversAstronomical sources

Trick I: Obtain the beam-maps relatively far out-of-focusBreaks degeneraciesReduces the required signal to noise

Trick II: Appropriate parametrisation of errorsWe use Zernike PolynomialsTrades required signal to noise with resolution



Results at the GBT

Outline

1 Introduction

2 The method

3 Simulations




Results at the GBT

Simulated Out-Of-Focus Beams, Perfect Telescope

In-Focus -ve De-Focus +ve De-Focus

≈ −12 dB of taperDe-focus: ≈ λ of path across the aperture



Results at the GBT

Simulated Out-Of-Focus Beams


≈ −12 dB of taperRandom large-scale surface error added to the surface



Results at the GBT

Simulated Out-Of-Focus Beams, with noise


≈ −12 dB of taperSignal-To-Noise: 100:1 per pixel



Results at the GBT

The OOF Holography Algorithm Requirements

A classic non-linear inverse problem:Forward model

Transforms a description of the optics (including anyerrors), receiver properties and observing strategy to amodel for observed data

Parametrisation of surface errorsNeeds to describe the relevant error modes but also mustbe well constrained by observation.

Goodness-of-fit measureNoise-weighted difference between model and observation

Solver algorithmLevenberg-Marquardt minimisation



Results at the GBT

The Basics

ApertureFFT

Far fieldB. Nikolic OOF Holography


Results at the GBT

The Basics

ApertureFFT + ||2

Power onlyB. Nikolic OOF Holography


Results at the GBT

The OOF Holography Algorithm

Surface Errors Defocus

Aperture phase Aperture Amplitude

Telescope Beam

FFT

Observing Strategy

Model Observation−

Residual

MinimiseParametrisation



Results at the GBT

The OOF Holography Algorithm: Forward Model

Surface Errors Defocus

Aperture phase Aperture Amplitude

Telescope Beam

FFT

Observing Strategy

Model Observation−

Residual

MinimiseParametrisation



Results at the GBT

The Forward Model

Simple Fourier Relationship between aperture plane andbeams:

P(θ, φ) = |E(θ, φ)|2 = |FT [A(x , y)]|2

ComplicationsNon-regular sampling of beams: on-the-fly,under-sampled, missing dataBeam differencing or choppingOff-axis receiversElliptical or poorly centred receiver responseThe source used is extended



Results at the GBT

The Forward Model: Illustration of data

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Time (h) x1e-3

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Tb



Results at the GBT


1.60 1.65 1.70 1.75 1.80 1.85

Time (h) x1e-3

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Tb



Results at the GBT


1.0 1.5 2.0 2.5 3.0 3.5

Time (h) x1e-4

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

Tb



Results at the GBT

Parametrisation

Wavefront errors (aperture phase)Use Zernike polynomialsOrthonormal on the unit circle (not quite with the taperedastronomical receivers!)Low order polynomials correspond to classical aberrationsMaximum order used controls the resolution of retrievedsurface

Receiver Response (aperture amplitude)Model as GaussianCan fit for centre, taper, ellipticity



Results at the GBT

Zernike Polynomials: n = 1

Vertical Pointing Horizontal Pointing



Results at the GBT

Zernike Polynomials: n = 2

X astigmatism Focus + Astigmatism



Results at the GBT

Zernike Polynomials: n = 3Trefoil Coma



Results at the GBT

Zernike Polynomials: n = 4Spherical



Results at the GBT

Zernike Polynomials: n = 52nd Order Coma



Results at the GBT

Zernike Polynomials: orthogonality

1st Order Coma 2nd Order Coma



Results at the GBT

Sources

At longer millimetre wavelengths quasars usually idealtargetsAt short millimetre and sub-mm wavelengths planets maybe used:

Extended sources not a problem, sharp edges mostimportantNeed to model the extended source and any substructure(limb darkening; rings!)

Spectral line sources also a possibility



Results at the GBT

Outline

1 Introduction

2 The method

3 Simulations




Results at the GBT

Simulations

Necessary to work out optimum observing strategyAreas investigated:

Variation of error of the retrieved surface with the signal tonoise ratio of the input beamsEffect of the size of de-focusEffect of the extent of sourceEffect of tracking/pointing errors

Other areas remaining to be investigated:Differenced /chopped observationsUnder-sampled beam maps



Results at the GBT

Simulations: Error on retrieved surface Vs Peak S/N

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

ǫ(r

ad)

0.001 0.01 0.1

Noise/Signal



Results at the GBT

Simulations: Error on retrieved surface Vs De-focus

0.01

0.02

0.05

0.1

0.2

0.5

ǫ

0.2 0.5 1 2 5

dZ (mm)



Results at the GBT

Outline

1 Introduction

2 The method

3 Simulations




Results at the GBT

The Green Bank Telescope



Results at the GBT

Application at the Green Bank Telescope

The GBT has a fully active primary surface – instantadjustmentThe GBT is not exactly homologous:

The active surface can fully correct for non-homologousdeformationInitially used a Finite-Element model for non-homologousdeformationGain-elevation curve is curved at high frequenciesUse OOF holography to compute more accurate correctionfor non-homologous deformation

High signal to noise



Results at the GBT

Experiments at the GBT

Retrieval of known deformation (bump)Retrieval and correction of surface errors during bothnight-time and day-time conditionsClosure – repeated measure-correct-measure cycles tomeasure consistency and random error of techniqueDerivation of a refinement for the gravitational deformationmodel



Results at the GBT

GBT forward model



Results at the GBT

Sample GBT Observation



Results at the GBT

Sample GBT Observation: Retrieved surface



Results at the GBT

Closure

Measurement Analysis

Pointing, Focus

In-Focus Map

+ve De-Focus Map

-ve De-Focus Map

Set of OOF Maps

OOF Analsyis

Map of wavefront errors

Translate

Actuator ajustments

Active surfaceadjustment

Finite ElementModel



Results at the GBT

Closure: benign conditions

WRMS ≈ 150 µm WRMS ≈ 100 µm



Results at the GBT

Closure: Daytime

WRMS ≈ 340 µm WRMS ≈ 210 µm



Results at the GBT

Modelling Gravitational Deformation

Measurement

Pointing, Focus

In-Focus Map

+ve De-Focus Map

-ve De-Focus Map

Set of OOF Maps

Finite ElementModel

Active surfaceadjustment



Results at the GBT

Modelling Gravitational Deformation

Obtained 37 measurements over three sessions covering arange of elevationsFit a sin(θ) + b cos(θ) + c to each Zernike coefficientindividually

0

2

4

6

8

NN

0 20 40 60 80

θ (deg)θ (deg)



Results at the GBT

Gravitational Model: Vertical Coma

n = 3, l = −1−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80

Elevation (deg)Elevation (deg)



Results at the GBT

Gravitational Model: Horizontal Coma

n = 3, l = 1−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80




Results at the GBT

Gravitational Model: Trefoil

n = 3, l = −3−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80




Results at the GBT

Gravitational Model: Trefoil

n = 3, l = 3−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80




Results at the GBT

Gravitational Model: Astigmatism

n = 2, l = −2−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80




Results at the GBT

Gravitational Model: Astigmatism

n = 2, l = 2−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80




Results at the GBT

Gravitational Model

n = 4, l = −4−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80


n = 4, l = −2−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80


n = 4, l = 0−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80


n = 4, l = 2−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80


n = 4, l = 4−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80


n = 5, l = −5−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80


n = 5, l = −3−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80


n = 5, l = −1−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80


n = 5, l = 1−1

−0.5

0

0.5

1

1.5

Phase

(rad)

Phase

(rad)

20 40 60 80




Results at the GBT

Gravitational Model: Efficiency

FEM Only FEM andOOF gravitational model

0.25

0.3

0.35

0.4

0.45

0.5

0.55

ηa

ηa

0 20 40 60 80

E (degrees)E (degrees)

0.25

0.3

0.35

0.4

0.45

0.5

0.55

ηa

ηa

0 20 40 60 80

E (degrees)E (degrees)



Results at the GBT

Conclusions

Low-resolution measurement of the surfaceMinimal interruption to astronomical observingNo extra equipmentDemonstrated random error of about λ/100.Measure and model gravitational deformation

The model is in routine use for observing at GBT

Measure thermal deformationsGreat potential in the era of array receivers


out-of-focus holography

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