Interferometer Circuits
Professor David H. Staelin
Massachusetts Institute of Technology
Lec18.5 - 1
2/6/01
Basic Aperture Synthesis Equation
Recall:
( )
E r,t
( )
↓↓
( )
E ⎡RE τ λ ⎤ = φE τ λ
⎣
⎦
Lec18.5 - 2
2/6/01
↔
↔
( )
E ψ,t
{
↓↓
( )
E E Ψ,t
2
}
( ) ( )
∝ Ι Ψ ,TA Ψ
E1
Simple Adding Interferometers
( )
v o ψ = a2 + b2 + 2ab
G1
G2
(overbars mean
"time average" here)
Point
source
response
a(t)
+
( )2
Lec18.5 - 3
2/6/01
vo ( ψ x )
b(t)
θB
(single
antenna)
ψx
v o (t )
First null
λ /L
T
E2
Simple Adding Interferometers
ψ n u ll = sin − 1 ( λ / 2L )
≅ λ / 2L if λ < < L
S p re a d so u rce re sp o n se
v max
vo ( ψ x )
θB
λ /2
ψ n u ll
ψx
v min
DC: a2 + b2
AC: 2ab
L
=∆" T1 "
v max − v min j2πT1 T
∝ RE ( τλ ) ↔ TA ( ψ )
Complex fringe visibility v =
e
v max + v min
∆
Lec18.5 - 4
2/6/01
E3
Interferometry as Fourier Analysis
( )
1
2
−D x 0
( )
( )
RE τλ
R e c a ll: φ A τ λ = R E τ λ
1
Dλ
τ λx
Example: 2 small
duplicate antennas
separated by Dλ in
the x direction
( )
∼
( )
TA ψ = G ψ
T A ( fψ
)
∼
= G ( fψ
∼
∗
∼
•
( )
TB ψ
∼
TB ( f ψ
)
1 + co s D λ ψ x
G sin ( ψ x ) ∝
0 1 + s in D ψ
λ
x
0
Lec18.5 - 5
2/6/01
∼
observed = antenna times signal
G co s ( ψ x ) ∝
2-element adding
interferometer
)
( )
φE τλ
•
ψx
ψx
E4
Interferometry as Fourier Analysis
1 + cos D λ ψ x
G co s ( ψ x ) ∝
2-element adding
interferometer
ψx
0 1 + s in D ψ
λ
x
G sin ( ψ x ) ∝
ψx
0
Interferometer directly measures Fourier components of source
Fringe patterns for all frequencies
Effect of finite bandwidth:
(colors) add in phase to create
"w h ite fr in g e "
ψx
fo + ∆ (H z )
Lec18.5 - 6
2/6/01
ψx
+
fo (H z )
∆ψ x ∝
ψx
⇒
+
fo − ∆ (H z )
1
B
B = 2∆
A delay line in one interferometer arm can redirect
the strong white fringe in other directions.
E5
Broad-Bandwidth Effects in Interferometers
E ( x, y,t )( the "slowly varying" part )
may vary rapidly enough that the offset
time L sin ϕx c is significant
ϕx
v2 ( t)
L
L sin φx
seconds
c
x
v1 ( t )
{
L sin ϕ x ⎞
⎡
⎛
E [ v1 ( t ) v 2 ( t ) ] =
G12 ( φ x ) E R e E ⎜ x, y, t +
⎟
⎢
2
2c ⎠
⎝
⎣
L sin ϕ x ⎞
jωL sin ϕ x / c − jγ ⎤
∗⎛
E ⎜ x − τx , y − τy , t −
e
e
•
⎟
⎥⎦
2c ⎠
⎝
⎡ e jω( t + L sin ϕx / 2c )e − jω( t −L sin ϕx / 2c ) = e jωL sin ϕx / c ⎤
⎣
⎦
k1k 2
E [ v1 ( t ) v 2 ( t ) ] =
k1k 2
2
( )
G12 ϕs R e
cross-gain
Lec18.5 - 7
2/6/01
{
}
}
⎛ L sin ϕ x ⎞
j2πL sin ϕx λ
e e
• φE( t ) ⎜
⎟
c
⎝
⎠
monochromatic
fringe envelope
fringe
−j
γ
E6
Bandwidth-Limited Angular Response
φ ( ) ( τ)
E ⎡⎣ v ( t ) v ( t ) ⎤⎦
E(f)
2
E t
E.G.
−B 0
null at
1/2B sec
↔
B
f
0 sinφ
x
Null @ L
c
τ sec.
null
( )
G12 ϕ s
1
sec
=
2B
1
2
white fringe
ϕx
Fringe envelope
φE( t ) (L sin ϕ x / c )
In broadband optical interferometer all colors contribute to
central "white" fringe; sidelobe fringes appear colored.
Therefore ϕxnull = sin−1 ( c / 2LB ) ≈ c / 2LB for ϕx ≅ 1
3 × 108
e.g. ϕ xnull ≅
= 1.5 radians for L = 10 m, B = 10 MHz
7
2 × 10 × 10
↑ 10 MHz
10-m
If B = 1 GHz, and L = 100 m, then ϕ xnull = 1.5 mrad ≅ 5 arc min
B = 3 × 1014 Hz and L = 100 m, then ϕ xnull = 10 −3 arc sec.
Lec18.5 - 8
2/6/01
E7
Dicke Adding Interferometer
y ( ψ ) point source response
+1 Dicke
−1
a2 + b2 + 2ab
ψ
(Also called "lobe-switching"
interferometer)
"Adding" interferometer
Dicke switch
oscillator
±1
∫
τ
×
×
a
b
+
⇒
( )2
y ( ϕ ) = ( ±a + b )
This circuit cancels D.C.
term, leaving only 4ab
as source traverses
beam ⇒
ψ
Lec18.5 - 9
2/6/01
2
⎡a2 + b2 + 2ab⎤ − ⎡a2 − 2ab + b2 ⎤ = 4ab
⎣
⎦ ⎣
⎦
Can add second adder and
square-law device operating
on a and –jb to yield sine
terms in Fourier expansion
G1
Dicke Adding Interferometer
±1
Yields φ ( τ λ ) for
even a stationary
source
"Multiplying"
interferometer
×
×
±1
90
×
×
Note: If no Dicke
switch, have gain
fluctuation
vulnerability
Lec18.5 - 10
2/6/01
∫
×
±1
τ
{ ( )}
Re φ τ λ
∫
τ
{
Ιm φ ( τ λ )
}
G2
“Lobe-Scanning” Interferometer
Lobes are scanned at
ωm , demodulated, and
averaged to yield φ ( τ λ )
Because all bias and
large-scale sources yield
no fine-scale response,
can integrate long times
seeking fine structure,
e.g. 10-3 Jansky point
sources like stars (can
measure stellar diameters
at ~10-3 arc sec)
τλ
ωs
L.O.
×
×
×
USB
Filter
×
ωm
Filter
ωm
90
×
×
∫
∫
τ
{
∝ G12 Re φ ( τ λ )
Lec18.5 - 11
2/6/01
ωs
τ
}
{
∝ G12Ιm φ ( τ λ )
}
G3
Cross-Correlation Interferometer Spectrometer
Let a ( t ) =∆ a1
a ( t − τ ) =∆ a2
×
a
b
+
Delay line or shift register
±1
±1
(
)
×
∫
∫
×
×
c + = a1a2 + b1b2 + a1b2 + a2b1
c − = −a1a2 − b1b2 + a1b2 + a2b1
φ τλ , f
×
T
T
(
φˆ τ λ , τ
) ∫
T
Computer
FFT
( )
if a ⊥ b, φ ( τ , f ) → 0
Note: if a = b, φ τ λ , f → φ ( 0, f ) = S ( f )
λ
Lec18.5 - 12
2/6/01
G4
Alternate Cross-Correlation Interferometer Spectrometer
L.O.
a
b
L.O.
Delay line
×
×
×
∫
∫
∫
Computer
(
φ τλ , f
)
Note: Figures omit down-converters and bandlimiting filters
Lec18.5 - 13
2/6/01
G5
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
Lec18.5 - 14
2/6/01
Feedback
"Line stretcher" varies path so that ∆φ ≅ 0
Communications and telemetry systems can
similarly be phase synchronized
J1
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- long baseline" interferometry, "VLBI"
×
Hydrogen maser
Cesium beam
Global Positioning System (GPS)
Loran − C
Temperature-stabilized crystal oscillators
Crystal oscillators
Tape
⇒
⇒
⇒
⇒
⇒
⇒
∼ 10 −14 , 10 −15
∼ 10 −12
∼ 10 −8 sec
∼ 10 −6 sec
∼ 10 −5 − 10 −8
∼ 10 −4 − 10 −5
If clocks perfect: cross-correlate to find time offset, correct for it, then
correlate the signals, albeit with an unknown fixed phase offset φ0
[unless reference source (in the sky) or phase is available].
If clocks imperfect and delays each way are identical: 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.
Lec18.5 - 15
2/6/01
J2
Synchronizing with Remote Atomic Clocks
Alternative approaches if clocks and transmissions are imperfect:
a) Track and correct phase shifts:
then average φˆ ( τ λ , f )
+π
Phase β
t
0
−π
Example: A 10 −12 cesium clock drifts 2π in ~ 100 sec, so φˆ ( τλ , f ) might be
computed for 5 − 10 sec blocks before averaging; then only φˆ is known.
b) Same, but set phase using strong resonant line point source,
ψy
Φ(f)
monochromatic
Strong point source
point source
ψx
f
c) or separable point source in space, modulation, etc.
To source
To reference
Lec18.5 - 16
2/6/01
J3
Multiband Synchronization of Clocks
Use multiple frequencies for wideband sources:
Which fringe F is source on?
ψ
0
Switch across all f's within coherence time of clock.
2πF
Source 1
Source 2
Observed bands
(collectively adequate)
0
Lec18.5 - 17
2/6/01
f
J4