InterferometryJack Replinger

Observational Cosmology Lab

Professor Peter Timbie

University of Wisconsin-Madison

Interferometry uses an array of small dishes to gain the resolution of a much larger single dish

The purpose of this tutorial is to give a viewer a basic understanding of the physics of interferometry

We proceed by discussing interference and diffraction.

picture courtesy of http://aries.phys.yorku.ca/~bartel/GPBmovie/Vla.jpg

Two Slit Interference

constructive interference destructive interference

light sources

Interference pattern

λ/2d

Two Slit Interference

detected in phase detected out of phase

detectors

“Interference pattern”

Reversed

λ/2d

The detectors are therefore sensitive to the

interference pattern

Adding Interferometer

diagram courtesy of http://www.geocities.com/CapeCanaveral/2309/page3.html

signal emitted from source reaches right

antenna δt sooner than left antenna

signal detected by right antenna delayed by

δt such that at the tee, two corresponding

signals are interfered

Antenna is designed such that parallel rays converge at focus

Use reversibility: Analogous to single slit diffraction

Antenna Optics

Diffraction

diffraction pattern

total destructive

interference

maximum side lobe

detectors

light source

sources

undetectable maximum response side lobe

antenna

Reversed

λ/D

The single antenna is therefore sensitive to

the diffraction pattern

Two Slit Diffraction Envelope due to

antenna sensitivity (Diffraction)

Peaks due to baseline (Interference)

Angular Resolution (Rayleigh Criterion) is λ/2d from baseline, instead of λ/D from diffraction limit

Image courtesy of http://img.sparknotes.com/figures/C/c33e2bffc162212e1d9aa769ad3ae54f/envelope.gif

Image courtesy of http://www.ece.utexas.edu/~becker/diffract.pdf

Interferometer Sensitivity

Interferometer is like diffraction in reverse, in 2D

Each antenna is analogous to a circular aperture

Example at left is the Diffraction pattern from two circular apertures (shown at upper left)

The interferometer is sensitive to projection of the diffraction pattern on the sky

Sources in light regions are detected, signal strength varies with intensity

Sources in dark regions are undetectable

courtesy of http://www.ee.surrey.ac.uk/Personal/D.Jefferies/aperture.html

contributes no signal

contributes strong signal

contributes weak signal

Fourier Analysis:Background

In order to understand how to reconstruct an image it is important to understand the mathematics of diffraction and interference

Notation:

x (and y) describe the plane of the aperture

θ (and Ф) describe the “plane” of the diffraction pattern

Equation describing the E-field from a point source of the wave at any distance d

E = Eoei(kd-wt)

θ

θ

xsinθ

x

0-

a-

P

r

dx

Fourier Analysis:Single Slit Diffraction

at P, the electric field due to a small dx is

dE = Eoei(k(r+xsinθ)-wt)

dx

at P, the electric field is

E(θ) = ∫oa Eoe

i(k(r+xsinθ)-wt)dx

generally, for any aperture = Eo ∫aperture eikxsinθ

dx

let A(x) describe the aperture = Eo ∫all space eikxsinθ

A(x)dx

Generalizing this to two dimensions, and any aperture, the E field at P is

E(θ,Ф) = Eo ∫all space eikxp+ikyq

A(x,y)dxdy

p and q are functions of θ and Ф, describing the phase shifts

θ

θ

xsinθ

x

0-

a-

P

r

dx

θ

θ

xsinθ

x

0-

a-

P

r

dx

Fourier Analysis:Generalized Diffraction

detector

sky

The intensity (proportional to E2

) on the sky is therefore

I(θ,Ф) = C*(E(θ,Ф))2

= C*(∫all space eikxp+ikyq

A(x,y)dxdy)2

If the aperture is the detector, then the diffraction pattern describes the sensitivity of the instrument G(θ,Ф) = I(θ,Ф)

The power P recorded by the detector is the product of the sensitivity function and the intensity of the sky S( θ,Ф) integrated over the sky

P = ∫sky G(θ,Ф)*S(θ,Ф )dθdФ

= C*∫sky(∫all spaceeikxp+ikyq

A(x,y)dxdy)2

*S(θ,Ф)dθdФ

Fourier Analysis:Image Reconstruction

P = C*∫sky(∫all spaceeikxp+ikyq

A(x,y)dxdy)2

*S(θ,Ф) dθdФ

P is recorded each time the detector is pointed

A(x) is determined by the aperture, p and q are known functions that describe the phase shifts

By recording P and varying A, by changing the baselines or using multiple baselines if there are enough detectors, we obtain enough information to solve

for S(θ,Ф) for that patch of the sky, fortunately this can be done on computers with existing software. For MBI the program will be written by Siddharth

Malu.

It is important to note that this is an oversimplification of the situation, when applying this is a diffuse source the coherence of the light from different

regions of the source must be addressed with a coherence function. This is addressed by the Van Cittert-Zernike theorem, which is beyond the scope of this

presentation.

detector

sky

θ

θ

xsinθ

x

0-

a-

P

r

dx