Interferometry Jack Replinger Observational Cosmology Lab Professor Peter

Interferometry Ø Ø Ø Jack 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 pattern λ/2 d constructive interference destructive interference light sources

Two Slit Interference Reversed “Interference pattern” λ/2 d detected in phase detected out of phase The detectors are therefore sensitive to the interference pattern detectors

Adding Interferometer 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 diagram courtesy of http: //www. geocities. com/Cape. Canaveral/2309/page 3. html

Antenna Optics Ø Ø Antenna is designed such that parallel rays converge at focus Use reversibility: Analogous to single slit diffraction

Diffraction Reversed diffraction pattern detectors sources λ/D total destructive undetectable interference maximum response side lobe The single antenna is therefore sensitive to the diffraction pattern antenna light source

Two Slit Diffraction Ø Ø Ø Envelope due to antenna sensitivity (Diffraction) Peaks due to baseline (Interference) Angular Resolution (Rayleigh Criterion) is λ/2 d from baseline, instead of λ/D from diffraction limit Image courtesy of http: //img. sparknotes. com/figures/C/c 33 e 2 bffc 162212 e 1 d 9 aa 769 ad 3 ae 54 f/envelope. gif Image courtesy of http: //www. ece. utexas. edu/~becker/diffract. pdf

Interferometer Sensitivity Ø Interferometer is like diffraction in reverse, in 2 D Ø Ø Ø 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 Ø Ø Sources in light regions are detected, signal strength varies with intensity Sources in dark regions are undetectable contributes weak signal contributes strong signal contributes no signal courtesy of http: //www. ee. surrey. ac. uk/Personal/D. Jefferies/aperture. html

Fourier Analysis: Background r Ø In order to understand how to reconstruct an image it is important to understand the mathematics of diffraction and interference θ 0θ x dx a- P xsin Ø θ Notation: Ø x (and y) describe the plane of the aperture Ø 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)

Fourier Analysis: Single Slit Diffraction r θ 0θ x dx a- xsin θ at P, the electric field due to a small dx is at P, the electric field is d. E = Eoei(k(r+xsinθ)-wt)dx E(θ) = ∫oa Eoei(k(r+xsinθ)-wt)dx generally, for any aperture = Eo ∫aperture eikxsinθdx let A(x) describe the aperture = Eo ∫all space eikxsinθA(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 P

Fourier Analysis: Generalized Diffraction P r detector θ 0 - sky θ x dx a- xsin θ The intensity (proportional to E 2) 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 )dθd. Ф = C*∫sky(∫all spaceeikxp+ikyq. A(x, y)dxdy)2*S(θ, Ф)dθd. Ф

Fourier Analysis: P Image Reconstruction r detector θ 0 - sky θ x dx a- xsin θ P = C*∫sky(∫all spaceeikxp+ikyq A(x, y)dxdy)2*S(θ, Ф) dθd. Ф 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.