Interferometer Circuits. Professor David H. Staelin

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1 Interferometer Circuits Professor David H. Staelin Massachusetts Institute of Technology Lec18.5-1

2 Basic Aperture Synthesis Equation Recall: E( ) E r,t,t E E λ = λ { } Ι ( Ψ ) E R E τ φ τ Ψ,t,T E A Ψ 2 Lec E1

3 G 1 a(t) Simple Adding Interferometers + 2 G 2 b(t) v(t o ) Point source response 2 2 v o = a + b + 2ab ( overbars mean "time average" here) v o θ B (single antenna) First null λ / L T Lec E2

4 λ /2 Simple Adding Interferometers 1 nu ll = λ / λ /2L sin null if ( 2L ) λ<< L S p read source re sponse v ma v min v o θ B L DC: a + b AC: 2ab 2 2 = 1 "T " v vmin j2 π T Comple fringe visibility v = ma e 1 T R E τ T A v v λ ma + min Lec E3

5 R Interferometry as Fourier Analysis E ( τ ) λ D 0 D λ Eample: 2 small duplicate antennas separated by D λ in the direction τ λ 2-element adding interferometer Re c a ll: φ A τ = R E τ φ E τ G G co s sin ( ) ( ) A λ λ λ ( ) = ( ) B T G T T f G f T f A ( ) = ( ) B ( ) observed = antenna times signal co s D λ 1 + s in D λ Lec E4

6 Interferometry as Fourier Analysis 2-element adding interferometer G G ( ) ( ) 0 0 λ 1 + s in D λ Interferometer directly measures Fourier components of source Effect of finite bandwidth: + f o + ( H z ) fo ( H z ) f o ( H z ) co s sin "w hite fringe" c o s D B = 2 Fringe patterns for all frequencies (colors) add in phase to create 1 B Lec A delay line in one interferometer arm can redirect the strong white fringe in other directions. E5

7 Lec Broad-Bandwidth Effects in Interferometers v1 ( t ) ϕ ( l ) E, y,t the "slow y varying" part v2 ( t ) may vary rapidly enough that the offset time L sin ϕ c is significant L Lsin φ seconds c kk [ ] { 1 2 L sinϕ E v 1 () t v 2 () t = G 12 φ E R + e E, y, t 2 2c } Lsinϕ jω L sin ϕ γ τ τ / c j E, y y, t e e 2c ( L s 2c ) ω( L s 2c ) = j ω t + in ϕ / j t in ϕ / j ω Lsin ϕ / c e e e { j2 π L sin ϕ λ L sin ϕ } kk [ ] γ 1 2 j E v 1 () t v 2 () t = G 12 ϕ s R e e e φ E () t 2 c cross-gain monochromatic fringe envelope fringe E6

8 E.G. Lec Bandwidth-Limited Angular Response Ef 2 φ () τ E v 1 ( t ) v 2 ( t) () Et null at 1/2B sec ( ϕ ) 12 s f τ sec. ϕ B 0 B 0 sin φ 1 Fringe envelope null L = sec c 2B φ Et ( Lsin ϕ / c) In broadband optical interferometer al l colors contribute to centra l "white" fringe; sidelobe fringes appear colored. Therefore sin c / 2LB c / 2LB for ϕ 1 1 ϕ = null 8 G white fringe 3 10 e.g. ϕ = 1.5 radians for L = 10 m, B = 10 MHz null MHz 10-m If B = 1 GHz, and L = 100 m, then ϕ = 1.5 mrad 5 arc min null 14 3 B = 3 10 Hz and L = 100 m, then ϕ = 10 arc sec. null E7

9 y ( ) This circuit cancels D.C. term, leaving only 4ab as source traverses beam Lec Dicke Adding Interferometer point source response + 1 Dicke 1 a 2 + b 2 + 2ab (Also called "lobe-switching" i nterferometer) "Adding" interferometer Dicke switch oscillator ± 1 τ y a + 2 b ( ϕ ) = ( ± a + b ) a + b + 2ab a 2ab + b = 4ab Can add second adder and square-law device operating on a and jb to yield sine terms in Fourier epansion G1

10 Dicke Adding Interferometer Yields φτ λ for even a stationary source Note: If no Dicke switch, have gain fluctuation vu nerabi ± 1 "Multiplying" interferometer ± 1 Re λ l lity φτ τ 90 { } ± 1 Ι m τ { φτ } λ Lec G2

11 m averaged to yield Lec Lobe-Scanning Interferometer Lobes are scanned at ω, demodulated, and φ τ Because all bias and large-scale sources yield no fine-scale response, can integrate long times seeking fine structure, e.g Jansky point sources like stars (can measure stellar diameters at ~10-3 arc sec) λ ω m Filter ω s L.O. τ G Re ω m { φτ } 12 λ τ λ 90 ω s USB Filter τ { φ τ } G12Ι m λ G3

12 Cross-Correlation Interferometer Spectrometer c c () Let a t a t + = a τ = a 2 1 ± 1 a ± 1 + = a 1 a 2 + b 1b 2 + a 1b 2 + a 2 b 1 = a a b b + a b + a b b Del ay line or shift register T T λ φˆ τ, τ T Lec λ φτ, f FFT Computer λ, f λ Note: i f a = b, φ τ φ 0, f = S f if a b, φ τ, f 0 G4

13 Alternate Cross-Correlation Interferometer Spectrometer L.O. a b L.O. Delay line Computer φτ ( λ, f) Note: Figures omit down-converters and bandlimiting filters Lec G5

14 Mechanical Long Distance Phase Synchronization For L >> λ, L.O. synchronization can be degraded by random phase variations in path length L between two (or more) sites (due principally to thermal and acoustic variations) One standard solution: Line Stretcher φ Lec Feedback "Line stretcher" varies path so that φ 0 Communications and telemetry systems can similarly be phase synchronized J1

15 Synchronizing with Remote Atomic Clocks If distance L is too great to synchronize L.O.'s, then we can use a remote clock, e.g. "very l ong baseli ne" interferometry, "VLBI" Tape Hydrogen maser 10, 10 Cesium beam Global Posi tioning System ( GPS) sec Loran C 6 10 sec Temperature-stabilized crystal oscillators Crystal oscillators If clocks perfect: cross-correlate to find time offset, correct for it, then correlate the signals, albeit with an unknown fied phase offset φ 0 [ unless reference source (i n the sky ) or phase is availab le]. If clocks imperfect and delays each way are identica l: at site A measure delay between clock B and A; do the same at B, and subtract results to yield twice the clock offsets. Use this offset to align A and B data streams. Lec J2

16 Synchronizing with Remote Atomic Clocks Alternative approaches if clocks and transmissions are imperfect: a) Track and correct phase shifts: then average φτ ˆ ( λ, f ) +π Phase β t π 0 12 Eample: A 10 cesium clock drifts 2 π in ~ 100 sec, so φˆ ( τ λ, f ) might be computed for 5 10 sec blocks before averaging; then only φ ˆ is known. b) Same, but set phase using strong resonant line point source, Φ(f) monochromatic y Strong point source point source f c) or separable point source in space, modulation, etc. To source To reference Lec J3

17 Multiband Synchronization of Clocks Use multiple frequencies for wideband sources: Which fringe F is source on? 0 Switch across al l f's within coherence time of clock. 2π F Source 1 Source 2 0 f Observed bands ( collectively adequate) Lec J4