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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
IntroductionThe methodSimulations
Results at the GBT
Outline
1 Introduction
2 The method
3 Simulations
4 Results at the GBT
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Outline
1 Introduction
2 The method
3 Simulations
4 Results at the GBT
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Approaches
Conventional surveyingPhotogrammetryInterferometric holographyTransmitter with-phase holographyTransmitter phase-retrieval holographyOut-Of-Focus (OOF) holography
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Outline
1 Introduction
2 The method
3 Simulations
4 Results at the GBT
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Simulated Out-Of-Focus Beams
In-Focus -ve De-Focus +ve De-Focus
≈ −12 dB of taperRandom large-scale surface error added to the surface
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Simulated Out-Of-Focus Beams, with noise
In-Focus -ve De-Focus +ve De-Focus
≈ −12 dB of taperSignal-To-Noise: 100:1 per pixel
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
The Basics
ApertureFFT
Far fieldB. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
The Basics
ApertureFFT + ||2
Power onlyB. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
The OOF Holography Algorithm
Surface Errors Defocus
Aperture phase Aperture Amplitude
Telescope Beam
FFT
Observing Strategy
Model Observation−
Residual
MinimiseParametrisation
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
The Forward Model: Illustration of data
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
The Forward Model: Illustration of data
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Zernike Polynomials: n = 1
Vertical Pointing Horizontal Pointing
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Zernike Polynomials: n = 2
X astigmatism Focus + Astigmatism
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Zernike Polynomials: n = 3Trefoil Coma
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Zernike Polynomials: n = 4Spherical
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Zernike Polynomials: n = 52nd Order Coma
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Zernike Polynomials: orthogonality
1st Order Coma 2nd Order Coma
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Outline
1 Introduction
2 The method
3 Simulations
4 Results at the GBT
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Outline
1 Introduction
2 The method
3 Simulations
4 Results at the GBT
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
The Green Bank Telescope
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
GBT forward model
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Sample GBT Observation
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Sample GBT Observation
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Sample GBT Observation
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Sample GBT Observation: Retrieved surface
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Closure: benign conditions
WRMS ≈ 150 µm WRMS ≈ 100 µm
B. Nikolic OOF Holography
IntroductionThe methodSimulations
Results at the GBT
Closure: Daytime
WRMS ≈ 340 µm WRMS ≈ 210 µm
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
Elevation (deg)Elevation (deg)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
Elevation (deg)Elevation (deg)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
Elevation (deg)Elevation (deg)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
Elevation (deg)Elevation (deg)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
Elevation (deg)Elevation (deg)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
Elevation (deg)Elevation (deg)
n = 4, l = −2−1
−0.5
0
0.5
1
1.5
Phase
(rad)
Phase
(rad)
20 40 60 80
Elevation (deg)Elevation (deg)
n = 4, l = 0−1
−0.5
0
0.5
1
1.5
Phase
(rad)
Phase
(rad)
20 40 60 80
Elevation (deg)Elevation (deg)
n = 4, l = 2−1
−0.5
0
0.5
1
1.5
Phase
(rad)
Phase
(rad)
20 40 60 80
Elevation (deg)Elevation (deg)
n = 4, l = 4−1
−0.5
0
0.5
1
1.5
Phase
(rad)
Phase
(rad)
20 40 60 80
Elevation (deg)Elevation (deg)
n = 5, l = −5−1
−0.5
0
0.5
1
1.5
Phase
(rad)
Phase
(rad)
20 40 60 80
Elevation (deg)Elevation (deg)
n = 5, l = −3−1
−0.5
0
0.5
1
1.5
Phase
(rad)
Phase
(rad)
20 40 60 80
Elevation (deg)Elevation (deg)
n = 5, l = −1−1
−0.5
0
0.5
1
1.5
Phase
(rad)
Phase
(rad)
20 40 60 80
Elevation (deg)Elevation (deg)
n = 5, l = 1−1
−0.5
0
0.5
1
1.5
Phase
(rad)
Phase
(rad)
20 40 60 80
Elevation (deg)Elevation (deg)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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)
B. Nikolic OOF Holography
IntroductionThe methodSimulations
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
B. Nikolic OOF Holography