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Interferometry 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

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Two Slit Interference constructive interference destructive interference light sources Interference pattern λ/2d

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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

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Adding Interferometer diagram courtesy of 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

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

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

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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 Image courtesy of

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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 contributes no signal contributes strong signal contributes weak signal

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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 = E o e i(kd-wt) θ θ xsinθ x 0- a- P r dx

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Fourier Analysis: Single Slit Diffraction at P, the electric field due to a small dx is dE= E o e i(k(r+xsinθ)-wt) dx at P, the electric field is E(θ)= o a E o e i(k(r+xsinθ)-wt) dx generally, for any aperture = E o aperture e ikxsinθ dx let A(x) describe the aperture = E o all space e ikxsinθ A(x)dx Generalizing this to two dimensions, and any aperture, the E field at P is E(θ,Ф)= E o all space e ikxp+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

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θ θ xsinθ x 0- a- P r dx Fourier Analysis: Generalized Diffraction detector sky The intensity (proportional to E 2 ) on the sky is therefore I(θ,Ф)= C*(E(θ,Ф)) 2 = C*( all space e ikxp+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 space e ikxp+ikyq A(x,y)dxdy) 2 *S(θ,Ф)dθdФ

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Fourier Analysis: Image Reconstruction P = C* sky ( all space e ikxp+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., which is beyond the scope of this presentation. 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

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