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Introduction The Method Simulations

Out-of-focus Holography The Basics
B. Nikolic
The Cavendish Lab, University of Cambridge, UK

Green Bank, September 2007

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Outline

1

Introduction

2

The Method

3

Simulations

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Introduction The Method Simulations

Simulated Out-Of-Focus Beams, Perfect Telescope
In-Focus -ve De-Focus +ve De-Focus

-12 dB of taper De-focus: of path across the aper ture

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Introduction The Method Simulations

Surface with random large-scale errors
Receiver Response Surface Errors

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OOF Workshop­Basics


Introduction The Method Simulations

Simulated Out-Of-Focus Beams
In-Focus -ve De-Focus +ve De-Focus

-12 dB of taper Random large-scale surface error added to the surface

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OOF Workshop­Basics


Introduction The Method Simulations

Simulated Out-Of-Focus Beams, with noise
In-Focus -ve De-Focus +ve De-Focus

-12 dB of taper Signal-To-Noise: 100:1 per pixel

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OOF Workshop­Basics


Introduction The Method Simulations

Motivation
Accuracy of the surface and collimation is one of the main limits on size/performance of large antennas.
ALMA Antennas: 12 m diameter, 20 µm accuracy = 1.6 : 106 accuracy Green Bank Telescope: 100 m diameter, 200 µm accuracy = 2 : 106 accuracy

Sources of inaccuracy:
Setting error : static Gravitation deformation: repeatable Thermal deformation: 30 minute timescale Wind: shor t timescale Ageing effects

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Motivation
Accuracy of the surface and collimation is one of the main limits on size/performance of large antennas.
ALMA Antennas: 12 m diameter, 20 µm accuracy = 1.6 : 106 accuracy Green Bank Telescope: 100 m diameter, 200 µm accuracy = 2 : 106 accuracy

Sources of inaccuracy:
Setting error : static Gravitation deformation: repeatable Thermal deformation: 30 minute timescale Wind: shor t timescale Ageing effects

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Approaches

Conventional sur veying Photogrammetr y Interferometric holography Transmitter with-phase holography Transmitter phase-retrieval holography Out-Of-Focus (OOF) holography

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OOF Workshop­Basics


Introduction The Method Simulations

Introduction
Aim: measure errors in the telescope optics
Surface errors + mis-collimation Rapidly As a function of elevation, time of day, etc Without any extra equipment

How: Use beam power maps
Astronomical receivers Astronomical sources

Trick I: Obtain the beam-maps relatively far out-of-focus
Breaks degeneracies Reduces the required signal to noise

Trick II: Appropriate parametrisation of errors
We use Zernike Polynomials Trades required signal to noise with resolution
B. Nikolic OOF Workshop­Basics


Introduction The Method Simulations

Outline

1

Introduction

2

The Method

3

Simulations

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

The Basics

Aper ture FFT

Far field
B. Nikolic OOF Workshop­Basics


Introduction The Method Simulations

The Basics

Aper ture FFT + ||2

Power only
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Introduction The Method Simulations

Terminology

Aper ture function Aper ture fn A(x , y ) is a complex valued function representing the (scalar) electric field distribution at some imaginary plane very just above the telescope Transmitter/Receiver equivalence We interchangeably consider the telescope as a receiving or transmitting system

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

The OOF Holography Algorithm Requirements
A classic non-linear inverse problem: Forward model
Transforms a description of the optics (including any errors), receiver proper ties and obser ving strategy to a model for obser ved data

Parametrisation of surface errors
Needs to describe the relevant error modes but also must be well constrained by obser vation.

Goodness-of-fit measure
Noise-weighted difference between model and obser vation

Solver algorithm
Levenberg-Marquardt minimisation

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OOF Workshop­Basics


Introduction The Method Simulations

The OOF Holography Algorithm

Parametrisation

Minimise

Surface Errors

Defocus

Aperture phase

Aperture Amplitude

Residual

FFT

Telescope Beam

Observing Strategy

Model

-

Observation

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OOF Workshop­Basics


Introduction The Method Simulations

The OOF Holography Algorithm: Forward Model

Parametrisation

Minimise

Surface Errors

Defocus

Aperture phase

Aperture Amplitude

Residual

FFT

Telescope Beam

Observing Strategy

Model

-

Observation

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OOF Workshop­Basics


Introduction The Method Simulations

The Forward Model
Simple Fourier Relationship between aper ture plane and beams: P (, ) = |E (, )|2 = |F T [A(x , y )]|2 Complications Non-regular sampling of beams: on-the-fly, under-sampled, missing data Beam differencing or chopping Off-axis receivers Elliptical or poorly centred receiver response The source used is extended
B. Nikolic OOF Workshop­Basics


Introduction The Method Simulations

The Forward Model
Simple Fourier Relationship between aper ture plane and beams: P (, ) = |E (, )|2 = |F T [A(x , y )]|2 Complications Non-regular sampling of beams: on-the-fly, under-sampled, missing data Beam differencing or chopping Off-axis receivers Elliptical or poorly centred receiver response The source used is extended
B. Nikolic OOF Workshop­Basics


Introduction The Method Simulations

Parametrisation
Wavefront errors (aper ture phase) Use Zernike polynomials Or thonormal on the unit circle (not quite with the tapered astronomical receivers!) Low order polynomials correspond to classical aberrations Maximum order used controls the resolution of retrieved surface Receiver Response (aper ture amplitude) Model as Gaussian Can fit for centre, taper, ellipticity

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Parametrisation
Wavefront errors (aper ture phase) Use Zernike polynomials Or thonormal on the unit circle (not quite with the tapered astronomical receivers!) Low order polynomials correspond to classical aberrations Maximum order used controls the resolution of retrieved surface Receiver Response (aper ture amplitude) Model as Gaussian Can fit for centre, taper, ellipticity

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Zernike Polynomials: n = 1

Ver tical Pointing

Horizontal Pointing

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Introduction The Method Simulations

Zernike Polynomials: n = 2

X astigmatism

Focus

+ Astigmatism

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Introduction The Method Simulations

Zernike Polynomials: n = 3
Trefoil Coma

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Introduction The Method Simulations

Zernike Polynomials: n = 4
Spherical

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Introduction The Method Simulations

Zernike Polynomials: n = 5
2nd Order Coma

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Introduction The Method Simulations

Zernike Polynomials: or thogonality

1st Order Coma

2nd Order Coma

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Introduction The Method Simulations

Zernike Polynomials: or thogonality

1st Order Coma

2nd Order Coma

Illumination

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Introduction The Method Simulations

Correlations in retrieved parameters

Order of columns amp, z1, z2, z3, z4, z5, z6, z7, z8, z9, z10, z11, z12, z13, z14, z15, z16, z17, z18, z19, z20

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Introduction The Method Simulations

The minimisation algorithm

We use the Levenberg-Marquardt algorithm Evaluate derivatives by finite differences (i.e., numerically) Consider the entire residual vector (not just 2 ) Ensure good convergence: sequentially increase maximum order of Zernikes used in fit
Use the solution of each step as the initial condition of the next step

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Introduction The Method Simulations

Choosing sources
Ideal sources are strong and point-like = at longer mm-wavelengths quasars usually ideal targets At shor t millimetre and sub-mm wavelengths quasars may be weak = can use planets:
Extended sources not a problem, shar p edges most impor tant Need to model the extended source and any substructure (limb darkening; rings!)

Spectral line observations also possible:
High S/N with masers Excellent atmospheric rejection Can be problems reading out fast enough

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Outline

1

Introduction

2

The Method

3

Simulations

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Simulations

Necessar y to work out optimum observing strategy Areas investigated:
Variation of error of the retrieved surface with the signal to noise ratio of the input beams Effect of the size of de-focus Effect of tracking/pointing errors Effect of the extent of source

Other areas remaining to be investigated:
Differenced /chopped obser vations Under-sampled beam maps

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Simulations

Necessar y to work out optimum observing strategy Areas investigated:
Variation of error of the retrieved surface with the signal to noise ratio of the input beams Effect of the size of de-focus Effect of tracking/pointing errors Effect of the extent of source

Other areas remaining to be investigated:
Differenced /chopped obser vations Under-sampled beam maps

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Simulation strategy

Star t with perfect optics Simulate beams, add required noise Retrieve a surface = WRMS of surface is the error Results valid for telescope in fairly good shape
Large errors present in the wavefront may make retrieval more difficult

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Simulations: Error on the retrieved surface Vs S/N
1 0.5 0.2 (rad) 0.1 0.05 0.02 0.01 0.005 0.001 0.01 Noise/Signal
B. Nikolic OOF Workshop­Basics

0.1


Introduction The Method Simulations

Signal To Noise 200:1
In-Focus -ve De-Focus +ve De-Focus

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OOF Workshop­Basics


Introduction The Method Simulations

Signal To Noise 200:1 (Perfect beams shown)
In-Focus -ve De-Focus +ve De-Focus

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OOF Workshop­Basics


Introduction The Method Simulations

Signal To Noise 40:1
In-Focus -ve De-Focus +ve De-Focus

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Introduction The Method Simulations

Signal To Noise 20:1
In-Focus -ve De-Focus +ve De-Focus

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Introduction The Method Simulations

Signal To Noise: Retrieved Surfaces

200:1

40:1

20:1

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OOF Workshop­Basics


Introduction The Method Simulations

Simulations: Error on the retrieved surface Vs De-focus
0.5

0.2 0.1 0.05 0.02 0.01 0.2 0.5
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1 dZOOF Wor (mm)

2
kshop­Basics

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Introduction The Method Simulations

Magnitude of de-focus (dz=1)
In-Focus -ve De-Focus +ve De-Focus

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OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus (dz=2)
In-Focus -ve De-Focus +ve De-Focus

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Introduction The Method Simulations

Magnitude of de-focus (dz=5)
In-Focus -ve De-Focus +ve De-Focus

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Introduction The Method Simulations

Magnitude of de-focus (dz=7)
In-Focus -ve De-Focus +ve De-Focus

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OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus (dz=10)
In-Focus -ve De-Focus +ve De-Focus

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OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 100:1 (dz=1)
In-Focus -ve De-Focus +ve De-Focus

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 100:1 (dz=2)
In-Focus -ve De-Focus +ve De-Focus

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 100:1 (dz=5)
In-Focus -ve De-Focus +ve De-Focus

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 100:1 (dz=7)
In-Focus -ve De-Focus +ve De-Focus

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 100:1 (dz=10)
In-Focus -ve De-Focus +ve De-Focus

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 50:1 (dz=1)
In-Focus -ve De-Focus +ve De-Focus

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OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 50:1 (dz=2)
In-Focus -ve De-Focus +ve De-Focus

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 50:1 (dz=5)
In-Focus -ve De-Focus +ve De-Focus

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OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 50:1 (dz=7)
In-Focus -ve De-Focus +ve De-Focus

B. Nikolic

OOF Workshop­Basics


Introduction The Method Simulations

Magnitude of de-focus and S/N 50:1 (dz=10)
In-Focus -ve De-Focus +ve De-Focus

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OOF Workshop­Basics


Introduction The Method Simulations

Effect of pointing jitter
2 1 0.5 (rad) 0.2 0.1 0.05 0.02 0.01 0 0.25 0.5 0.75 1 (beam-width)
B. Nikolic OOF Workshop­Basics


Introduction The Method Simulations

Summary

Two features set OOF holography apar t from conventional phase-retrieval holography:
Large de-focus Parametrisation in terms of Zernike's

This allows reliable retrieval from beam maps with S/N as low as 100:1 Hence can use astronomical sources and receivers
Elevation dependence Minimum interruption to astronomical obser ving Measurement of full optical path Potential of array receivers

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OOF Workshop­Basics