Laboratorio di Astrofisica - IRA Homeddallaca/Lab_1.pdf · Daniele Dallacasa Laboratorio di...

Daniele Dallacasa

Laboratorio di Astrofisica (laboratorio radio)

Basic Theory:1. Fraunhofer diffraction & Fourier Transforms why radio telescopes are diffraction limited

2. Antenna concepts (as specific to radio astronomy) 3. Radio interferometres (i.e. the quest for angular resolution)

4. Present day and future instruments (Single dish, interferometres,...)

5. How to handle interferometric data (practical stuff) From a set of complex measurements to a sky radio image

[email protected] ricevimento: Martedi' & Giovedi' 15:3017:00

mailto:[email protected]

Why bother radio waves

LOFAR EVLA ALMA

Fraunhofer diffraction

aperture

screen a O

r

Q

Pa

R

H


aperture

a

O

x sin

x

dx

dr

parallel wave


Point source at infinite distancefrom aperture

to the (far away) screen


The electric field of a randomly located source in a given point of the aperture is

F(x), (x) and = c/

are amplitude, phase and frequency of e.m. radiation.This is also known as “grading”.

Aim:compute the electric field in any

point of the screen at a given time to .

g x =F x e−i x e−i t


Each element dx contributes at a given point Pa with:

In each point of the screen we must:1. evaluate the contribution of each point which has a distance r from the screen2. integrate dx over the whole aperture.

1. r = R + x sin ~ R + x(small angle approximation) therefore

E Pa ~ g x e−2 i r /dx = F x e−i x e−i t e−2 i r /dx

E P a ~ g x e−2 i R /e−2 i x /sindxE P a ~ g x e−2 i R /e−2 i x /dx


2. Let's integrate over dx to describe the whole aperture. It is possible to write the electric field in a given point P at a distance from the axis of the screen.

Let's use sin ~ and then u = x/ :

Formalization of Huygens – Fresnel principle (each point in the aperture is a source of spherical waves with the same v of the incomingradiation; in the image plane at a distance from the axis each contributes with a vectorwith amplitude g(u) and phase 2u sin ~ 2u ))

The electric field in a generic point P in the screen is the FT of the gradingIt does NOT depend on either x or , but from their ratio

it is characteristic of the optical system

E ~ e−2 i R /∫−a /2

a /2g x e−2 i x /sindx

E ~ e−2 i R /∫−∞

∞

g x e−2 i x /sindx

E ~ ∫−∞

∞

g u e−2 i u du


The approximation holds when:

1. Monochromatic (unpolarized) radiation (u does make sense)

2. Far field

3. Plane aperture

has various names: Field radiation pattern, Angular spectrum of the radiation field, Far Field.

The intensity of the signal on the screen to is the image of the aperture. It varies between positive and negative amplitudes.

In order to have a signal that can be “accumulated” we must consider

R ≫a2

E

I = ∣E ∣2

and now..... FT festival!


is known as Power Pattern, Beam, PSF.

From FT theory

The Beam represents the FT of the autocorrelation function of the grading. is known as transfer function of the instrument.

I = ∣E ∣2

I = ∣E ∣2= g u ∨ g u

g u ∗g u

Fourier Transforms

Any periodic function with period = T, can be represented by means of a trigonometric (Fourier) seriesIt can be considered a representation of a periodic function in the frequency domain.

Also nonperiodic functions can be arranged with FT:

f(t) 0 over a given interval, =0 elsewheref(t) non periodic T The Fourier Series can be seen as the limiting case of the periodic function.

F t =1T∑k =−∞

∞

e ik t∫0

Tf x e−ik x dx

Fourierspectrum

if k = 2u F t = ∫−∞

∞

e2 iut dt∫−∞

∞

f x e−2 iux dx

Fourier Transforms

For a given function f(t) defined over ℝ we have:

F(u) is the FT of f(t) & f(t) is the antitransform of F(u).

If t is a time then u is a frequency length (space) spatial frequency

Radio astronomy: F(u) is complex function of a real f(t) distribution.

Maths: F(u) FT of f(t) exists in case:

1. exists2. Discontinuities in f(t) are finite (in amplitude and number)3. In a given (finite) interval, the number of maxima and minima are finite.

In physics these mathematical requirements are much relaxed

F u = ∫−∞

∞

f t e−2 iut dt

f t = ∫−∞

∞

F u e 2 iut du

∫a

b∣f x ∣dx

Fourier Transforms: properties

FT of a FT function:

F(u) is FT of f(t). Let's compute the FT of F(u):

FT and FT1 are identical except that reverse the function (inversion of t)

furthermore...

Equivalent width (figure)

∫−∞

∞

F u e−2 iut du = ∫−∞

∞

F u e2 iu −t du = f −t

Le =∫−∞

∞

f t dt

f 0=

F 0

∫−∞

∞

F u du=

1LE

if u=0 F 0 = ∫−∞

∞

f x dx

if x=0 f 0 = ∫−∞

∞

F u du


Linearity:

this is known as superposition theorem.

Parity:a function is even when f(t) = f(t); it is odd when f(t) = – f (t)Any function can be uniquely divided into two complex functions: one even and one odd

a f 1 t b f 2t ⇒a F 1u b F 2 u

∫−∞

∞

[af 1t bf 2 t ]e−2ut dt = a∫−∞

∞

f 1x e−2ut dtb∫−∞

∞

bf 2t e−2ut dt

i.e. = aF 1u b F 2u

p t =f t f −t

2even

d t =f t −f −t

2odd

then f t = p t d t ⇒ F u = P u D u


Simmetry

Similarity

N.B. f and F change in opposite ways. In case one shrinks/broadens, the other broadens/shrinks. Furthermore f (0) is independent from a, therefore the area of the FT remains constant.

Shifting theorem:

f .∗.t ⇒ F .∗.

−u f .∗.−t ⇒ F .∗.

u

f at ⇒1∣a∣

F ua

∫−∞

∞

f at e−2i u t dt =1∣a∣∫−∞

∞

f at e−2 i u

aat

d at =1∣a∣

F ua

f ts ⇒ F u e2i us

∫−∞

∞

f ts e−2i u t dt = ∫−∞

∞

f ts e−2i u tse2 i u s dt = F u e2 i u s


Derivation

N.B. A derivation in the t domain corresponds to a multiplication in the u domain (& viceversa) Real & Imaginary parts are swapped, 0 is suppressed, high frequencies are enhanced, while low frequencies are damped

Parseval Theorem

is known as Spectral Energy of the function f(t).

f ' t ⇒ 2 i u F u

f [n ]t ⇒ 2 i u n F u

∫−∞

∞

∣F u ∣2du = ∫

−∞

∞

∣f t ∣2dt

∣F u ∣2

Fourier Transforms: CONVOLUTION

Many operations are gathered by this operation (smearing, blurring, scanningsmoothing, crosscorrelation, running mean, etc.) and can be interpreted asthe way and instrument measures a physical quantity.

definition:

f may represent the physical quantityg may represent the observing tool (e.g. telescope)h may represent the observation

The function g is inverted and displaced by a given amount s prior to execute the multiplication.Convolution does not change the peak value of f, but may change its distribution

h t = ∫−∞

∞

f s g s−t ds = f t ∗ g t

Fourier Transforms: CONVOLUTION 2

Properties of h :

The convolution product becomes simple multiplication in the Fourier space.It is also true that:

Convolution is commutative:

associative

distributive

h t = f t ∗ g t ⇒ F u ⋅ G u = H u

∫−∞

∞

∫−∞

∞

f s g s−t e−2 i u s dt ds = ∫−∞

∞

∫−∞

∞

f s g s−t e−2 i u s−t e−2 i u t dt d s−t = .

F u ⋅ G u = H u

f t ⋅ g t ⇒ F u ∗ G u

f t ∗ g t = g t ∗ f t

f t ∗ [ g t ∗ h t ] = [ f t ∗ g t ] ∗ h t

f t ∗ [ g t h t ] = [ f t ∗ g t ] f t ∗ h t

Fourier Transforms: CROSS – CORRELATION

Similar to convolution but the cross–correlating function is not swapped

definition:

In particular, in case the functions are real, then g*(s+t) = g(s+t)and the autocorrelation becomes

It is noteworthy to mention the AUTOCORRELATION of a given function

The autocorrelation peaks at t=0.

f t ∨f t = ∫−∞

∞

f s f .∗.st ds ≤ ∫

−∞

∞

∣f s ∣2ds

h t = f t ∨g t = ∫−∞

∞

f s g st ds

h t = f t ∨g t = ∫−∞

∞

f s g .∗.st ds

Fourier Transforms: CROSS – CORRELATION 2

Properties:

distributive wrt addition

It is NOT commutative

In case of real functions, cross correlation and convolution are coincident.

h t = f t ∨ g t ≠ g t ∨ f t = h −t g t ∨ f t = ∫

−∞

∞

g s f st ds if st=x

∫−∞

∞

g x−t f x dx = ∫−∞

∞

g [x−t ] f x dx = g −t ∨ f t

f t ∨ [ g t h t ] = f t ∨ g t f t ∨ h t [ f t g t ] ∨ h t = f t ∨ h t f t ∨ h t

Fourier Transforms

Some examples....

1. The pulse: (x – xo )

Fourier Transforms

Some examples....

2. The rectangle: (x)={1 ∀ −0.5x0.50 elsewhere

FT sinc u =sinu

u

Fourier Transforms

Some examples....

3. The Gaussian: f x = A e−a2 x 2

FT F u =1∣a∣

e−

u 2

a2

Fourier Transforms

Some examples....

4. The double pulse: g(u) = (x – xo ) + (x + xo)

Fourier Transforms

Elements of a Radio Telescope 1

Detection of lowenergy radiation:

use the wave formalism (not enough energy for photoelectric effect)

Hardware:A (main) mirror/collector, a secondary mirror (subreflector) and possibly more,drive the radiation on a given place (focus).

A detector must be sensible to an incoming electromagnetic wave.In particular, the easiest component to detect is the electric field of the wave.


The mirror(s) must collect most of the radiation (E field).

Shape and surface are related to the science driving the construction of the RT

Westerbork Synthesis Radio TelescopeCaltech Submillimeter Observatory


F is small !

Most detectors at secondary focus

Effelsberg


Surface accuracy: solid panels, mesh, wires.

Parkes has a parabolic dish antenna, 64 m in diameter with a collecting area of 3,216 m2. The dish is made up of aluminium panels supported by a latticework of supporting struts. To incoming radio waves from space, the dish surface acts in the same manner as a smooth mirror.The waves are reflected and focused into a feedhorn in the base of the telescope's focus cabin.The dish has a mass of 300 tonnes and distorts under its own weight as it points to differentparts of the sky. Due to clever engineering design, however, this distortion is accounted for so that the radio waves are always reflected to the focus cabin..

The same applies to any radio telescope.

The surface type and telescope characteristics are critical to the main scientificdrivers for which a given instrument has been built.

Antenna basics surface accuracy

Deviations from the ideal shape can be measured and corrected either witholographic techniques or with laser ranging, as a function of elevation (andweather conditions). Actuators are supposed to provide closeloop real timecorrection to each mirror panel.

Irregularities in the reflecting surface cannot be corrected for and lead to anunrecoverable signal loss

the radiation reflected from the hatched area gets into the focus at a later time(with some phase delay, 2) leading to a reduction of the coherence.


the signal reflected from the bowl acquires a phase

which turns into an additional phase at the focus. The contribution of the hatched portion of the mirror becomes

reducing the amplitude of the total signal. If (u) is small, then

and now let's consider the reduction of the effective area originated by an irregular reflecting surface

= 2×2

g u = go u e i u

e i u 1i −

2/2...

Ae

Aeo

=∣∫ g u du ∣

2

∣∫ go u du ∣2


which becomes

for a given irregularity , the effect in reducing the effective area depends on :the efficiency of a mirror becomes smaller and smaller with since increases.

The other way round: for a given surface, we can define a minimum operating wavelength min

therefore , conservatively, one can chose min = 20 .

Ae

Aeo =

∣∫ go u [1i −2/2]du ∣2

∣∫ go u du ∣2 =

= 1−⟨2⟩⟨⟩

2

Ae min

Aeo

=1e

= 2×2

min

= 1rad


Westerbork Synthesis Radio TelescopeNorthern Cross, Medicina

Effellsberg

Alma


Detection of lowenergy radiation:

use the wave formalism (not enough energy for photoelectric effect)

Hardware:A (main) mirror/collector, a secondary mirror (subreflector) and possibly more,drive the radiation on a given place (focus).



Mixer; down conversion to IF

Elements of a Radio Telescope

Each detector can reveal radiation within a given bandwidth wrt the referencefrequency.

In general, the total available bandwidth is a small fraction of the referencefrequency

Modern receivers can have larger bandwidths.

Future radio telescopes aim at a continuous (but not simultaneous)frequency coverage


SEFD =T sys

G[Jy ]

Plan for the remaining theory....

Elements of a Radio Telescope (RT) – collecting area & the PSF (resolution) – the detector (receiver) – how the data look like – frequency coverage (&.... the proper receiver for each frequency)

Why RT have little resolution

Where to go then (interferometer)

Adding interferometer

Simple (crosscorrelation) interferometer – geometric delay – how the data look like – many element interferometer – Earth rotation synthesis and the uvcoverage – Modern (and future) interferometers (WSRT, VLA, ATCA, GMRT, LOFAR, ALMA, SKA)

Plan for the remaining theory 2

Fourier Inversion – the effect of the uvcoverage & the need of cleaning – the clean & restore (various methods) – the final image

Image analysis

Antenna basics

(normalized) PSF / beam / power pattern

beam solid angle

Pn , =P ,

P max

where P max=P 0,0

A = ∫4P n ,d =∫0

2

∫0

P n ,sind d

if Pn ,∀ , A=4 isotropic antenna

for a dipole A dipole = ∫0

2

∫0

sin2sind d =

83

Various features limit the antenna efficiency: surface deformation of the mirror(s) [gravity, wind, temperature, etc.], surface accuracy, blocking, etc.

Antenna basics

Main beam solid angle

Directivity

Effective aperture

Efficiency

M = ∫main−lobeP n ,d

D =4

A

Ae = AgA

A =Ae

Ag

≤ 1

Antenna basics

Aea = 2

Soss = S trueA

W = m Ae∫∫B , ,P n o− ,o−d = kT A

T A =∫T B , ,P n o− ,o−d

∫P n o− ,o−d ≈

≈ T B

∫Pn o− ,o−d

∫Pn o− ,o−d

T A =m Ae S

k= G S G =

mAe

kAntenna GAIN .o K /Jy

Antenna basics

Antenna gain is both a measure of the collecting area and of the efficiency ofthe mirror, namely of the effective area. It is function of frequency

Gains for a small sample of (old) radio telescopes at = 6 cm

Often related to SEFD (system equivalent flux density) to take into accountalso the contribution of electronics to the signal on the detector (see further)

Antenna Diam (m)

25 0.09 VLA (1) 25 0.11

32 0.20 OVRO 43 0.40

100 1.0300 8.0

Gain (°K/Jy)

Onsala

Medicina

Effelsberg Arecibo

Antenna basics

What the antenna measures is

and the system temperatures is largely dominated by the “receiver” temperature(tens of K, depending on the observing frequency).

This also determines the noise level of an observation.

The detector reveals N wavepackets in a time t and over a bandwidth and integrated for .

The system temperature represents the average of many measures N, and it holds

TSYS = TA TR

N =

t= 2 being t=

12

T rms =T SYS

Antenna basics

and the signaltonoise ratio is

– weak sources TA≪ TR to increase the SNR, larger TA = GS() = S()mAe /kareas are needed, implying larger mirrors.

– strong sources (high TA ) the SNR does not depend on the size of the mirror.

SN

=T A

T rms

=T A

T SYS

≈T A

T AT R


SEFD =T sys

G[Jy ]

Antenna basics Resolving power

Assuming a 1D telescope of size D, its grading is represented by a rectangle function

whose FT is the far field

and the modulus of the far field provides the antenna pattern (beam, PSF)

whose first nulls occur when

gu = ∏ u = {1 ∀ −D/2≤u≤D /20 elsewhere }

E = sinc D =sinD

D

I = sinc2D =

sin2D

D2

D = =1

D

=

Drad


For a 2D (3D) telescope, the grading is more complex and its FT is a modifiedBessel function, which goes to zero when

and the same happens to the PSF.It is a very bad case, since lambdas are large in the radio domain.

Exercise:Which is the PSF size (HPBW, FWFN, etc) of a telescope like Effellsberg (100m)operating at = 20 cm?

How it is possible to increase the resolution of a RT?

D =1.22D

= 1.22

Drad


For a 2D (3D) telescope, the grading is more complex and its FT is a modifiedBessel function, which goes to zero when

D =1.22D

= 1.22

Drad

Confusion noise

A contribution to the noise arises from unresolved sources falling within the beam of a RT (A )

It is related to the distributionP(S) of the sources as a functionof the flux density (and of thefrequency )

The confusion limit is defined as

namely, to reveal a given source witha SNR=5, it is necessary that withinthe A fall less than 75 sources brighter than Slim.

It is a function of v, slope in the source counts and A

Nmin A1

75

Confusion noise

From single dish to interferometry

From the Fraunhofer Fourier theory, the simpliest interferometer can berepresented by two pointlike slits in 1D (e.g. just the detectors!).

The grading is a double pulse

The far field is a cosine

The antenna pattern is a square cosine

The resolution is given by the separation between the two.

+ as the separation increases, the PSF gets narrower and narrower – ...but it is not possible to know where the radio source is! – the collecting area remains small

Interferometry

Interferometry the “geometric delay”

Interferometry

Signal on the elements of an interferometer

adding interferometer

rapidly varying, average is 0 depends on the geometric delay, slowly varying its the only term surviving time averaging, measures the phase differenceon the two antennas.

A B

A = E o sin2 t

B = E o sin[2t g ] where g =D sin

c=

D

c

A B 2= A2

B 22AB

2AB = 2E o2 sin2t sin [2tg ] =

= 2E o2⋅12{cos [22tg ] cos2g }

cos2g = cos 2 D sin/c = cos 2D

sin

Interferometry

(the first) null[s] occur(s) when

P = ⟨A2B 2

2AB⟩2= E o

2 [1cosD

sin]Pn =

1cos2

where = D

sin

cercare altra figura

cos = −1 namely D

sin= HPBW ≈1

Dsin=

1B

Interferometry

However, the continuous term is often disturbed by RFI, while the product is not, and therefore the switching interferometer has been introduced. It can be considered the progenitor of modern correlation interferometers

the first nulls occur when

Pn = cos where = D

sin

cos = 0 namely D

sin=

2 HPBW ≈

12Dsin

=1

2B

A B 2− A − B

2=

= A2 B2

2AB − A2− B 2

2AB = 4AB

= 4E o2⋅12{cos[22tg ] cos2g }


Real interferometers are made of small apertures (compared to the maximumbaseline)

a new expression of the grading

the far field becomes

and the beam (antenna pattern) is

g u = d ∗ [x−D /2xD /2]

E = sinc 2dcos2

D = sinc 2Dcos 2a

P = sinc 22d cos2

2D

D

d


The cosine term is modulated (and quenched) by the sinc2 and now maximaare not identical anymore.

P = sinc 22d cos2

2D

“From infinity..... to here!” (adapted from Buzz Lightyear, 1995)

Each antenna (ith single dish) measures a Voltage, i.e. the E field of the incoming monochromaticradiation is converted into V, to be sampled

V i= V o sin t

Everybody knows.... life is more complicated!2 hands / linears for polarization (together they fully sample incoming radiation)

.... well there is the noise as well!!!!

We are not interested in most of the signal collected at each antenna [ n(t) ]

V Li= V L sin t L

V Ri

= V R sin t R

V Li= V L sin t L nL t

V Ri

= V R sin t R nR t

radio Interferometry: the signals from two antennas _i and _ j are (cross)CORRELATED

V i= V sin t

i n i

t V j

= V sin t j n j

t wavefront gets in j with some delay

The signal arrives at one of the antennas first and then, after g , gets to the other. In case it gets into i first, then in j the signal is

after some (boring) algebra and approximations we obtain:

vary with earth rotation

baseline length ( units!!!)

V i V j = V 2 sin ti sintg j

V i V j≈V 2cosg =V 2cos2

Dc

sin=V 2 cos2D

sin

Indeed the CORRELATOR performs a more complicated operation (i.e. thetrue crosscorrelation) to deliver VISIBILITIES:

In the (2D) uvplane each visibility samples the FT of the (2D) B()

Modern correlators:Are special computing devicesHandle and deliver a HUGE amount of data

Visibilities are: Complex numbers (amplitude & phase), with ancillary informationComputed over the integration time = T

V ij g = V i V j = limT ∞∫−T /2

T /2V i t V j∗ tg dt

Correlator throughput

X N (N1)/2 baselines

X 4 (RR LL RL LR if full polarization mode)

X subbands (sensitivity requires large bandwidths, which are arranged into a number of subbands)

X channels (each subband is divided into a number of spectral channels, i.e. small widows in frequency, over which the data are averaged)

X total observing time / integration time

Nowadays N is a few tens, the number of subbands is a few tens, the numberof channels can reach a few thousands, and the integration time is of the orderof 1 sec. This makes A LOT of measurements!!!

Deep radio images implies sensitive continuum observations with:Large bandwidths, often organized in subbands, in turn sliced into channels

Necessary to prevent (minimize) bandwidth smearing [radial]Allow an accurate RFI removal

Long onsource times [(repeated) full tracks of the target(s)]Short integration times prevent (minimize) time smearing [tangential]Effective to remove time variable (intermittent) RFI

All this also allow to image wide fields [surveys!](large primary beams at low frequencies)

(a huge amount of significant pixels on the sky!)

“external” RFI @ a single antenna

Warnings:

The Visibility Function is not sampled in a uvpoint, but it is rather averaged over an area (depending on integration time and bandwidth/frequency)⇒ this leads to image distortions known as smearing

Oldtype continuum datasets are nowadays out of date. Nonetheless stilluseful to understand how CALIBRATION works in practice

Modern datasets perform a sort of frequency synthesis: the field of view may appear rather different at the edges of the observing bandwidths

Many effects are (strongly) frequency dependente.g. the FoV is small / large at high / low frequenciesand the data handling must take this into account

More appropriate presentations will be given by other lecturers!

Real interferometry:

An interferometer samples the Visibility Function as transmitted by theatmosphere and the instrumentation (antenna, receiver, electronics, cables,correlator, etc.)

With a number of fair assumptions, CALIBRATION is the process to determine Gi

aiming at transforming the observed quantities to the proper scale.

All the quantities are COMPLEX, and therefore we need to find two values, AMPLITUDEand PHASE, for each antenna, polarization, subband, channel, ....possibly as a function of time

V obsij

= V trueij G i G j


We need to know the true visibility in order to determine the complex gains

Simplest (ideal) case:Pointlike source of known flux density S, observed at the centre of the field of view.

⇒ ALL amplitudes are identical to S⇒ ALL phases are 0 (zero)

warnings: the number of such ideal sources is ridiculously small (0) sources (and FoV) are different as a function of – frequency [high (10s GHz)/ low (a few 100s MHz)] – interferometer (baseline length, also depends on frequency) sources are often variable in both flux density and polarization

Let's try the simplest approach

V obsij

= V trueij G i G j


The complex gain G can be generally split into two terms:

Amplitude aPhase and the new relationship can be written as:

Calibration means to find appropriate a and for the raw data.

V obsij

= V trueij G i G j

Aobsij e i obs

ij

= A trueij a i a j e i true

ij i− j

We can term the modification of the true signal into the observed measure asa corruption of the information.

Basic assumption (1st order)all the signal corruption can be determined and

corrected solving an element/antenna based system

Each interferometric element will have a correction forAMPLITUDE (t)

PHASE (t)to be applied (in combination) to ALL the measurements delivered by

the correlator. This operation MUST be done prior of Fourier inversion.

The complex gain Gi contains many components (along the signal path):

F = ionospheric Faraday rotation T = tropospheric effectsP = parallactic angle (altazmounts)E = antenna voltage patternD = polarisation leakage J = electronic gainB = bandpass response

K = geometric compensation

They are either additive (phases) or multiplicative (amplitudes). In most cases, when performing calibration we can forget the origin of the contribution to be removed. Some of them are specific to each type ofobservation (VLBI, Spectral line, wide field) and of the observing frequency.

G i = K i B i J i D i E i P i T i F i

The complex gain Gi contains many components (along the signal path):

F = ionospheric Faraday rotation T = tropospheric effectsP = parallactic angle (altazmounts)E = antenna voltage patternD = polarisation leakage J = electronic gainB = bandpass response

K = geometric compensation

Each term on the right has matrix form.

The full matrix equation Gi is very complex, but usually only need to consider the

terms individually or in pairs, and rarely in open form Existing software does the job (... more or less) but it is software....!

G i = K i B i J i D i E i P i T i F i

Ionospheric Faraday Rotation F i

ionosphere is inhomogeneous (t,):

Ionospheric Faraday Rotation F i

ionosphere is inhomogeneous: various directions have different refraction indices:It is birefringent:one hand of circular polarisation is delayed w.r.t.the other, introducing a phase shift:

• progressively relevant at long wavelengths (2) [ @ 20cm could be tens of deg.)

• + at solar maximum and at sunrise/sunset (high and variable TEC)

• Distant antennas have very different signal paths across the ionosphere (coherence length)

• Direction dependent within fieldofview

≈ 812 2∫l

B ∥ ne dl rad ; =ne

FRL=ei e

i 00 e−i ; F XY

=ei cos −sin

sin cos

The Tropospheric contribution Ti

the troposphere causes polarizationindependent amplitude and phase variationsdue to emission/opacity and refraction

• + relevant above ~15 GHz where water vapour (& oxygen, ...) absorbs/emits

• elevation dependent (path length across the troposphere) (gain curve!)

• Distant antennas are have very different signal paths across the troposphere

• May be critical on bad weather (Tsys and noise contribution, very short coherence time)

T RL= t 1 0

0 1

The polarisation leakage Di

Polarisers are not ideal:orthogonal polarisations are not perfectly isolated and mix.

• A geometric property of the feed design & frequency dependent

• Vital for linear polarization imaging (RL & LR).

• Plays a role in very high dynamic range Stokes' I

• Good receivers may have Dterms of a few percent or less

V Ri

= V R sin t R DR V L sin t L

V Li= V L sin t L D LV R sin t R

The Parallactic angle χ = Pi

• Alt – az antennas rotate during tracking (equatorial have a fixed orientation)

• Imply a rotation of the FoV and of the polarization response (intrinsic + leakage terms)

• Plays a role in very high dynamic range Stokes' I

• Good receivers may have Dterms of a few percent or less

t = atan coslat sin[HAt ]sinlat cos Dec − coslat sinDec cos[HAt ]

PRL = ei t 0

0 e−i t ; P XY = cos t −sin t sint cos t

The antenna voltage pattern Ei

• Individual antennas have direction dependent gains (non uniform illumination)

• Relevant when the full FoV (Primary Beam) is imaged

• (Fields are wider and much more populated at low frequencies)

• Rotates with azimuth

• Related to elevation (gain curve)

N.B. Changes with AzEl

The electronic antenna GAIN Ji

It accounts for most of amplitude and phase effects introduced by antenna electronics (amplifiers, mixers, digitizers, samplers, ...) and characteristics (collecting area, efficiency,....). It is the dominant term

• In practice in this term, many other effects (mentioned earlier) can be included

• No frequency dependence is deliberately considered (B i)

• Can be considered the tribute to engineers: implies the need to convert to physical units.

J RL = jR 00 j L ; J XY = j X 0

0 j Y

The bandpass response Bi

It represents the frequencydependence of the performance of the whole system (mainly electronics)

•It is dominated by the filter design and performance•Disturbances introduced by spurious electronic behaviour (e.g. with temperature)

A strong (a good SNR per channel is necessary) calibrator, possiblypointlike, observed by all the antennas

BRL = bR 0

0 bL ; B XY = b

X 00 bY

The geometric compensation Ki

The geometric model (antenna i, antenna j, source position) must be (ideally)perfect so that Synthesis Fourier Transform relation can work in real time; strong dependence on baseline length

• arises from uncertainties in antenna and source position (geodesy/astrometry)• independent clocks and LO are a problem!• specific of VLBI, gets worse with frequency

Specific correlation and data handling techniques are necessary (Fringe Fitting)to recover residual errors. In general it is not relevant for “conventional interferometers” (EVLA, WSRT, GMRT, ATCA, MERLIN, ....)

K RL = kR 0

0 k L ; K XY = kX 0

0 k Y

Summary of arguments for the Lab test

Things to remember:

1. FT of particular functions and its relation to Fraunhofer diffraction2. Interferometers: 2.1 simple interferometer (2elements) 2.2 multi element interferometer3. The UV plane4. Visibility function 4.1 Visibility for a pointlike source 4.2 Visibility for an extended source5. Fourier Inversion6. Clean & Restore7. Modern interferometers: WSRT, (E)VLA, ATCA......LOFAR, ALMA8. The VLBI9. The VLA and the lab test

Interferometry

Uvplane

Source visibility (point, extended)

Primary beam

87

11B

11C

Laboratorio di Astrofisica - IRA Homeddallaca/Lab_1.pdf · Daniele Dallacasa Laboratorio di...

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