Interferometry Basics
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Transcript of Interferometry Basics
Interferometry Basics
Andrea IsellaCaltech
Caltech ALMA Community DayPasadena, March 16, 2011
References
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
• Thompson, A.R., Moran, J.M., & Swensen, G.W. 2004, “Interferometry and Synthesis in Radio Astronomy” 2nd edition (WILEY-VCH)
• NRAO Summer School proceedings– http://www.aoc.nrao.edu/events/synthesis/2010/lectures10.html
(in particular D.Wilner lecture)– Perley, R.A., Schwab, F.R. & Bridle, A.H., eds. 1989, ASP Conf. Series 6, Synthesis
Imaging in Radio Astronomy (San Francisco: ASP)• Chapter 6: Imaging (Sramek & Schwab), Chapter 8: Deconvolution (Cornwell)
• IRAM Summer School proceedings– http://www.iram.fr/IRAMFR/IS/archive.html– Guilloteau, S., ed. 2000, “IRAM Millimeter Interferometry Summer School”
• Chapter 13: Imaging Principles, Chapter 16: Imaging in Practice (Guilloteau)– J. Pety 2004, 2006, 2008 Imaging and Deconvolution lectures
• CARMA Summer School proceedings- http://carma.astro.umd.edu/wiki/index.php/School2010- CARMA SUMMER school, July 2011, ask J. Carpenter for information
1. An Interferometer measures the interference pattern produced by two apertures.
2. The interference pattern is directly related to the source brightness. In particular, for small fields of view the complex visibility, V(u,v), is the 2D Fourier transform of the brightness on the sky, T(x,y)
From Sky Brightness to Visibility
(van Cittert-Zernike theorem) T(x,y)x
y
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
uv plane
Fourier space/domain
Image space/domain
image plane
| V|
b (meters)0
0.5
1
b1
=/bb1b2
=/b
b2
phase
Visibility and Sky Brightness
The visibility is a complex quantity:- amplitude tells “how much” of a certain frequency component- phase tells “where” this component is located€
V =Imax − IminImax+Imin
=Fringe Amplitude
Average Intensity
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Vb (meters)
0
0.5
1
b1
b1
Visibility and Sky Brightness
b2b3
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
The visibility is a complex quantity:- amplitude tells “how much” of a certain frequency component- phase tells “where” this component is located€
V =Imax − IminImax+Imin
=Fringe Amplitude
Average Intensity
2D Fourier Transform Pairs
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
T(x,y) |V(u,v)|
Gaussian
Function Constant
Gaussian
2D Fourier Transform Pairs
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
T(x,y) |V(u,v)|
ellipticalGaussian
sharp edges result in many high spatial frequencies
ellipticalGaussian
Disk Bessel
CARMA A-array configuration
θ = 0.15’’ at 1.3 mm
2 Km1.2 miles
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Need for high angular resolution
Single telescope Array of N telescopes
Angular resolution, θ ~ λ/D ~λ/Dmax
Collecting area ~ D2 ~ ND2
10-3
10
-2
10-1
1
00 1
01 1
02
EVLA
ALMA
θ (‘’
)
λ (μm)10-1 100 101 102 103 104
HST
Spitzer
VLTI
Herschel
CARMA
Keck
SMA PdBI
CCATD=50 m
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Aperture Synthesis
V(u,v) can be measured on a discrete number of points. A good image quality requires a good coverage of the uv plane. We can use the earth rotation to increase the uv coverage
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Synthesized beam
€
′ T (x,y) = W (u,v)V (u,v)e2πi(uv+vy )dudv∫∫Discrete sampling:
The weighting function W(u,v) is 0 where V is not sampled
T’(x,y) is FT of the product of W and V, which is the convolution of the FT of V and W:
B(x,y) is the synthesized beam, analogous of the point-spread function in an optical telescope.
€
′ T (x,y) = B(x,y)⊗T(x,y)
€
B(x,y) = W (u,v)e2πi(uv+vy )dudv∫∫
11
Synthesized Beam (i.e.,PSF) for 2 Antennas
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
12
3 Antennas
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
13
4 Antennas
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
14
5 Antennas
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
15
6 Antennas
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
16
7 Antennas
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
17
8 Antennas
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
18
8 Antennas x 6 Samples
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
19
8 Antennas x 30 Samples
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
20
8 Antennas x 60 Samples
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
21
8 Antennas x 120 Samples
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
22
8 Antennas x 240 Samples
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
23
8 Antennas x 480 Samples
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Effects of a sparse uv coverage
The synthesized beam approaches a 2D gaussian function and can be described in terms of its Full Width at Half
Maximum (FWHM) and Position Angle (PA)
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Characteristic angular scales1. An interferometer has at least THREE important characteristic angular scales:
- angular resolution: ~ λ/Dmax , where Dmax is the maximum separation between the apertures.- shortest spacing problem: the source is resolved if θ>λ/Dmin, where Dmin is the minimum separation between apertures.
An interferometer is sensitive to a range of angular sizes, λ/Dmax - λ/Dmin
andsince Dmin> Aperture diameter, an interferometer is not sensitive to the large angular scales and cannot recover the total flux of resolved sources (you need a single dish, e.g., CSO, APEX, IRAM 30 m, ALMA total power array, CCAT).
2. Field of view of the single aperture ~ λ/D, where D is the diameter of the telescope. Source more extended than the field of view can be observed using multiple pointing centers in a mosaic.
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Primary beam and Field of View
A telescope does not have uniform response across the entire sky- main lobe approximately Gaussian, fwhm ~1.2/D = “primary beam”
- limited field of view ~ /D
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Characteristic angular scales during ALMA early science
Band λ λ/Dmax λ/Dmin FOV
3 3 mm 2.5’’ 30’’ 50’’
6 1.3 mm 1’’ 13’’ 22’’
7 0.9 mm 0.75’’ 9’’ 15’’
9 0.45 mm 0.37’’ 4.5’’ 8’’
Dmax = 250 m, Dmin = 20 m, D = 12 m
1. How complex is the source? Is good (u,v) coverage needed? This may set additional constraints on the integration time if the source has a complex morphology.
2. How large is the source? If it is comparable to the primary beam (l/D), you should mosaic several fields (NOTE that during early science ALMA will have limited mosaicing capabilities)
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
From visibilities to images
visibilities dirty image sky brightness
deconvo
lve
Fourier tr
ansform
• uv plane analysis– best for “simple” sources, e.g. point sources, disks
• image plane analysis– Fourier transform V(u,v) samples to image plane, get T’(x,y)– but difficult to do science on dirty image– deconvolve b(x,y) from T’(x,y) to determine (model of) T(x,y)
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Weighting function
€
′ T (x,y) = B(x,y)⊗T(x,y)
€
B(x,y) = W (u,v)e2πi(uv+vy )dudv∫∫
Measured flux:
Synthesized beam:
You can change the angular resolution and sensitivity of the final image by changing the
weighting function W(u,v)
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Weighting function
• “Natural” weighting: W(u,v) = 1/σ2(u,v), where σ2(u,v) is the noise variance of the (u,v) sample:
Advantage: gives the lowest noise in the final image, highlight extended structures.
Disadvantage: generally gives more weights to the short baseline (where there are more measurements of V) degrading the resolution
• “Uniform” weighting: W(u,v) is inversely proportional to the local density of (u,v) points. It generally gives more weights to the long baseline therefore leading to higher angular resolution. Advantage: better resolution and lower sidelobes
Disadvantage: higher noise in the final map
• “Robust” (Briggs) weighting: W(u,v) depends on a given threshold value S, so that a large S gives natural weighting and a small S gives uniform weighting.
Advantage: continuous variation of the angular resolution.
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Weighting functionN
ATU
RAL
WEI
GH
TIN
G
UN
IFRO
M W
EIG
HTI
NG
FWHM beam size = 1.7’’ x 1.2’’ 0.21’’ x 0.19’’
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Deconvolution: Dirty Vs Clean imageA non complete coverage of the uv plane gives a synthesized beam with a lot of sidelobes, i.e. a ‘dirty’ beam. And since
,
T’(x,y) is also characterized by sidelobes, i.e., ‘dirty’ image.
€
′ T (x,y) = B(x,y)⊗T(x,y)
The deconvolution process consists in giving reasonable values to the visibility in the unmeasured (u,v) areas in order to get a nice gaussian beam without sidelobes. The most successful deconvolution procedure is the algorithm CLEAN (Hogbom 1974).
Deconvolution or CleaningDirty image Clean image
Andrea Isella :: ALMA community day :: Caltech, March 16, 2011
Concluding remarks
• Interferometry samples visibilities that are related to a sky brightness image by the Fourier transform
• In planning your observations keep in mind these three important scales: λ/Dmax , λ/Dmin, λ/D
• remember… there are usually an infinite number of images compatible with the sampled visibilities. Play around with the weighting functions while using the ALMA simulator
• Deconvolution (e.g. cleaning) try to correct for the incomplete sampling of the uv-plane. Play around with ‘clean’ while using the ALMA simulator