GLOW Interferometry School, Bielefeld 2011 1.Fourier optics 2.Fourier theorems 3.Visibility function...

GLOW Interferometry School, Bielefeld 2011

1. Fourier optics

2. Fourier theorems

3. Visibility function

4. Aperture synthesis

5. Coordinates

6. Correlation interferometer

7. Imaging

8. Cleaning

Radio Interferometry Basics

U. Klein


Fundamental understanding of the functionality of radio interferometry


1. Fourier optics

Aperture with constant extent perpendicular to screen, with extent a in this y-direction, constant current Gy(x) in y-direction. Infinitesimal far-field element dE resulting from an E-field component E(x) along infinitesimal element dx

and integrating over y and x leads to

This is Huygens’ Principle for the far-field: superposition of field components along the aperture Far-field: R x

(… ties up to Jackson, Chapt. 14)

dydx eR

E(x) i E d rki

22

dxE(x) e R

a i dE rki

2

sin xR r


Normalising by wavelength

Lumping constants and R dependence into

leads to simple Fourier intergral (x = x / , direction cosine ξ = sin )

xx

ikR

deER

eai E

x sin2

2

xikR E eR

a i E

2

deE xE

dxexE E

xi

xi

2

2

)(

)(


slit or uniformly illuminated aperture: sinc function

otherwise xE

Dxforconst

0)(

2

)(sinc)sin(

)(

DD

DE

circular aperture: Bessel function

antenna diagram: P(,) = |E(, )|2

D

DJE

)(2)( 1


2. Fourier theorems

Fourier transform F(s) of function f(x)

with inverse transform

f(x)dxexf sF xsi F

2)(

)()( 2 sFdsesF xf xsi F


Convolution:

)()()()( xgxfduuxguf xh


Consider the two domains:

x s application

time t [s] frequency [Hz]

acoustics

angle [rad] spatial frequ. u [rad-1]

radio astronomy

Oboe: waves Oboe: spectrum

LMC: radio brightness LMC: spatial frequ. spectrum


Let F f(x) = F(s), F g(x) = G(s)

addition theorem

)()()]()([ sGsFxgxf F

a

sF

axaf

1)(F

example: an aperture has beam size

/D

similarity theorem


shift theorem

sFeaxf sai 2)( Fexample: electronic steering


convolution theorem

example: beam smearing

)()()]()([ sGsFxgxf F

convolution with broader beam

multiplication with narrower ‘filter‘


autocorrelation theorem

2* )()( sFduxufuf

F

example: antenna diagramme


convolution and auto-/cross-correlation theorem: reception pattern of an interferometer

dxuxEuExP

)()()( * )()()( * EEP

)2cos()()( DPP SD


sampling theorem: how to properly convert continuous into discretised functions

comb or „Sha“ function:

multiplication

f(x) · III(x)

converts continuous

into

discretised function

)( xnxxx

x

n

III


A function whose Fourier transform is zero for |s| > sc is fully specified by values spaced at equal intervals not exceeding ½ sc

-1 save for any harmonic term with zeros at the sampling points.

‘aliasing‘!


3. Visibility function

Correlated power received from source with brightness B

where (D = D/)

D is the baseline. Integration yields

dBAdP )(cos)(

sDss 2,0

Ωd)σ · Dπ() · σ) · B(σA() sDπ(- Δ

Ωd)σ · Dπ() · σ) · B(σA() sDπ( Δ

dΩ)σs(Dπ)σB()σA(Δν)s,DP(

Source

λ

Source

λ

Source

λλ

2sin2sin

2cos2cos

2cos

0

0

00


With

it is convenient to introduce complex visibility

such that

)()0()0(

)()0()(

NAAF

FAA

dΩe)σB()σ(AeVVSource

σDπiN

i λ 2

VdΩ)σDπ()σB()σ(AφV

VdΩ)σDπ()σB()σ(AφV

Source

λN

Source

λN

Im

Re

2sinsin

2coscos


Since AN 1 (the single-antenna diagramme is very broad), and writing

we obtain

This is the basic relation for aperture synthesis:

Comparing with 1st version of P we arrive at

dude)V(u)B(

dde)B(uV

uπi

uπi

)(2

)(2

,,

,),(

In order to retrieve the brightness B(ξ,η), we need to measure the visibility V(u,v) for as many baselines Dλ= (u,v) as possible and Fourier-transform it.

)2cos(),( 000 VsDVAsDP

uD


example:

double source (schematically):

apply convolution theorem

source brightness = convolutionof -function with two -functions

visibility = multiplication of cosinewith sinc-function

Now observe with different base-lines


astronomical telescope = spatial filter:

single dish: low-pass filter

interferometer: high-pass filter


4. Aperture synthesis

need many D = (u,v)

consider 4 4 apertures, producing n (n-1) 2 = 120 baselines, most of which are redundant, while we need only 24 of them to cover the (u,v)plane

aim: preferably little redundancy!

ddueuV)(B uπiobsobs )(2),(,


Earth-rotation synthesis

need good sampling of (u,v) plane with many (non-redundant) baselines

earth-rotation synthesis helps a lot

due to Sir Martin Ryle (Nobel Prize 1974 with A. Hewish)


144 m 144 m

9144 m

36 m

300 m 180 m9144 m

0 1 2 3 4 9 A B C D

east-west array: e.g. WSRT


earth rotation produces a goodtransfer function: (u,v) tracks

e.g. pure east-west interferometer

FT antenna diagramme („dirty beam“)


for the kth ring, the grating rings have functional form

where sinc1/2 implies the “half-order derivative“ of the sinc function

derívative theorem:

and from this:

(intrigued? Bracewell & Thompson, Ap.J. 188, 77, 1973)

kDN)F( 22

2

12sinc, 2

1

)(2)( sFsixfdxd F

)(2)(21

21sFsixf

dx

d F


actually need max. 12 hrs. to obtain full (u,v) coverage

VLA: 8 hrs. (Y-shape)

V(u,v) is Hermitean, since thebrightness is a real quantity.

… we get the other half for free!

),(),( * uVuV


5. Coordinates

Complex visibility reads

Define coordinate system (u,v,w), whichare related to unit vectors and baseline:

We may have to deal with non-zero w-component!

dΩe)σB()σ(AeVVSource

σDπiN

i λ 2

d

ddddd

wusλ

D

wsλ

D

22

22

0

1

1


, , are direction cosines

and the visibility becomes

cos

cos

cos

dde

)B()(AwuV

wuπi

N

22

)11(2

1,,),,(

22


Coordinate system (,) corresponds to projection of celestial sphere onto a tangential plane at the origin of the field centre.

possible projections:

• tan: optical astronomy• arc: Schmidt telescopes, single dishes• sin : aperture synthesis

(more: Calabretta & Greisen, A&A 395, 1077, 2002)


direction cosines and their relation to other coordinate systems:

spherical coordinates ,

equatorial coordinates ,

cossin

sinsin

arctan

arcsin 22

22

00

022

0

0

1sincos

sin1cosarctan

)cos(sincoscossin

)sin(cos

0000

0


Using polar coordinates , , the solid-angle element d reads

The releation between the infinitesimal elements is given by

where |J| is the determinant of the Jacobi matrix

which leads to

ddd sin

221

dd

d

ddJdd

d

d

d

dd

d

d

d

J ),(


Antenna coordinates and baseline components

defines a double ellipse:

Z

Y

X

L

L

L

hh

hh

hh

w

u

sinsincoscoscos

cossinsincossin

0cossin1

2

22

2

2

sin

cos

YX

Z

LLL

u


Coverage of (u,v) plane

e.g. VLA e.g. LOFAR

a bit weird: component LZ gives rise to non-coplanar baselines not simple Fourier integral anymore!

dde

)B()(AwuV

wuπi

N

22

)11(2

1,,),,(

22


6. Correlation interferometer

has power response

with phase lag

due to geometric time delay

cos)()( 0PP

Dg 22

sinc

Dg


heterodyne principle:

down-convert from high (RF or HF) to intermediate frequency (IF)

produces frequency spectrum including

IFLOs

IFLOi

signal frequ. (USB)

image frequ. (LSB)


Effect of finite bandwidth (1):

Delay compensation i indispensible in order to avoid ‚fringe washing‘.We measure correlation of two voltages V1, V2:

By definition, this is also the Fourier transform of the bandpass |H()|

Calculate power over the band, assuming rectangular bandpass and (roughly) constant power as a function of frequency:

T

T

g dttVtVT

P )()(2

1lim)( 12 T

deHP ti 22)()(

2

2

0

0

0

)2cos()(

dVAP Vgg


Hence finally:

So the correlated signal is modulated with a sinc function!

= 50 MHz

D = 1 km

tracking source over 1.6‘ withoutdelay compensation produces~1% of loss

needs g 1.6 ns

)2cos()sin(

)( 00 Vgg

gg VAP


the phases from telescopes 1 and 2 are

hence the correlated power is

where = g - i and ‘‘ refers to USB and LSB.

gIFLOg )(221

LOi 22

])(2cos[

)cos(

0

210

LOVIFgLO

V

VA

VAP

Measure complex visibility with quadrature network:

V

V

VVV

V ReIm

ImRe

arctan

22


Fringe rotation/stopping: earth‘s rotation modulates correlated power quasi-sinusoidally with natural fringe rate

where e = dhdt = angular rotation frequency of earth, = declination. At the VLA, the fringe frequency may exceed 150 Hz.

Not interested in fringe rate, but rather in change ofV; hence compensate fringe rotation by modulating LO phase. In the IF section, we have

So control LO phase such that

vanishes. This now requires a complex correlator in order to measure the amplitude and phase separately!

cosudt

dh

dh

dw

dt

dwe

LOVgLOLOVIFgLO 2)(221

dt

d

dt

d

dt

d LOgLO

2)( 21


Effect of finite bandwidth (2):

Fourier integral is precise only for monochromatic radiation! At the centre frequency 0 we have the precise relation

Away from the centre frequency we have

Generalised similarity theorem in n dimensions:

00)(2

0000),(, ddueuV)(B uπi

obsobs

0

00

00

0

00

00

1

1

uuu

a

sF

axaf n

n

1)(F


application to 2-D Fourier relation between visibility and brightness:

so that (“~“ indicating the influence of the bandpass)

Delay tracking is exact for 0 = 0 at = 0. For signals at frequeny arriving from direction (, ) there will be a lag error

so that the phase is imprecise by an amount

),(,,,2

000

00

2

000

uVuVB)B( oo

deHuV)(uVuπi

IF

)(22

200

00

000

000

0

)(,,~

0

00

u

)()(2

)(2 000

00

u


Consider point source at location 0, assume rectangular bandpass. Then

where we have assumed ( 0)2 1. Thence, with du = ( 0) du0 du0, we obtain the convolution

The Fourier transform of the sinc function is a box function

where

This effect called bandwidth smearing (chromatic aberration) is irreversible!

0

200

00

)(2 000~

dueudue)(B uπiuπi IFsinc

2

2

22

0

0

01~

dudee)(B uπiuπi

IF

V(u)Fourier kernel

00

1

IFIF

)S( otherwise0

2

1for1 0


Short version of this: radiation within range of frequencies produces u-v tracks of different size produces images of different scale if referred to same frequency when doing the Fourier transform.

13‘ to fie

ld centre

radial smearing 202

0

IF

0

u

v20

20


7. ImagingIn practice: convert Fourier integral

into a sum and apply an FFT. This requires a retabulation of the measured visibilities onto a 2N 2M grid. In what follows, we use the integral notation for the sake of better readability of the equations. The true visibilities are strongly modified owing to:

ddueuV)B( uπi

)(2),(,

1. sampling by transfer function: S(u,v)

2. regridding onto regular grid for FFT with grid function: G(u,v)

3. weighting to shape the synthesized beam and control sensitivity: R(u,v)

4. FFT


transfer function: S(u,v) gives rise to ‘dirty beam‘

weighting with W(u,v)

W = R T D

R: Aeff, Tsys, ,

T: taper (shapes the beam)

D: density of visibilities

brightness (‘dirty image‘) now is

ddueuVuS)(B uπiD

)(2),(),(,

ddueuVuSuW)(B uπiD

)(2),(),(),(,

Plateau de Bure Interferometer


now prepare for FFT by retabulation S(u,v); needs two steps:

(i) convolve with appropriatefunction C(u,v)

(ii) multiply with grid function G(u,v) (fakir‘s bed of nails)

this is a significant modi-fication of the data

GCVSWV D ])[(

VSWCGBD FFFFF

use convolution theorem to see modification in theimage domain:


can ‘undo‘ gridding by dividing VD by the inverse transform of G:, yielding the ‘grid-corrected‘ dirty image:

This is the true brightness distribution, convolved with the dirty beam

C

BB

-

DDC

1F

SWGF D FFF


8. Cleaning

Goal: deconvolution in order to get rid of sidelobes


basic clean algorithm (Högbom 1974):

(i) First compute the dirty map BD(,) via FT of VD(u,v) and the dirty beam FD(,) via the FT of S(u,v) · W(u,v).

(ii) Find position of maximum brightness and subtract ·FD(,) · Bmax; “loop gain” controls speed of the cleaning process (0 < < 1); store Bmax at the corresponding position in a work array (creates “fakir’s bed of nails”).

(iii) Go to (ii) until a user-defined level is reached (e.g. number of iterations or rms noise).

(iv) Convolve the so obtained “point source model” (“fakir’s bed of nails”) with the “clean beam”, or synthesized beam, e.g. a Gaussian of width ~ / Dmax.

(v) Add this convolved model to the residual map to yield the clean image BC(,).

There are variants with major (check in the u-v domain) and minor cycles (Clark 1980, Cotton-Schwab 1984)


Zero-spacing problem

Missing short spacings: interferometer measures zero total flux!

hence

since

which cannot be measured “negative bowl”

dde)B(uV uπi )(2,),(

!,)0,0( totSdd)B(V dBS


need to add in single-dish measurements (in the u,v domain!)


Don‘t forget that the brightness distribution is multiplied by the enveloping ‘primary beam‘, i.e. the beam of the individual antennae. Need to correct forthis degradation by dividing the clean image by the primary beam:

),(

),(),(

PCP

C F

BB

Naturally gives rise to increasingnoise at radial distances largerthan the primary HPBW.


3C 120


Thanks for your attention!

GLOW Interferometry School, Bielefeld 2011 1.Fourier optics 2.Fourier theorems 3.Visibility function...

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Transcript of GLOW Interferometry School, Bielefeld 2011 1.Fourier optics 2.Fourier theorems 3.Visibility function...