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C. Carilli, Synthesis Summer School, 24 June 2002
1
Noise and Interferometry Chris Carilli (NRAO)
VLA 5e9 Hz VLTI 5e14 Hz
C. Carilli, Synthesis Summer School, 24 June 2002
2
Radio vs. Optical Interferometry
Radio: heterodyne + correlator
Amplifier
Optical: mirrors + direct detector (CCD)
Correlator
Delay
Baseband
Mixer
C. Carilli, Synthesis Summer School, 24 June 2002
3
Stability = 1e8
C. Carilli, Synthesis Summer School, 24 June 2002
4
References‘Noise and interferometry ,’ V. Radhakrishnan 1999, SIRA II
‘Bolometers for infrared and millimeter waves,’ P. Richards, J. 1994, Appl. Phys., 76, 1
‘Letter on Brown and Twiss effect,’ E. Purcell 1956, Nature, 178, 1449
‘Thermal noise and correlations in photon detection,’ Zmuidzinas 2000, Caltech
‘The intensity interferometer,’ Brown and Twiss 1974, (Taylor-Francis)
‘Multiple telescope infrared interferometry ,’ Townes and Sutton 1981, (ESO conference, eds. Ulich and van de Stadt)
‘Coherent vs. incoherent detection for interferometry ,’ de Graauw1981, (ESO conference, eds Ulich and va de Stadt)
‘Measurement of power spectra,’ Blackman and Tuckey (Dover)
‘Celestial masers,’ Cook (Cambridge Univ. Press) chap 4
‘Fundamentals of statistical physics,’ Reif (McGraw-Hill) chap 9
C. Carilli, Synthesis Summer School, 24 June 2002
5
Photon statistics: Bose-Einstein statistics for gas without number conservation ( indistiguishable particles or wave function symmetric under particle exchange, spin 0; Reif Chap 9)
Thermal equilibrium => Planck distribution function
n_s = relative number of particles in state s =
number of photons in standing mode in box at temperature T =
number of photons/s/Hz/ster/pol in beam in free space (Richards 1994)
Photon noise: fluctuations in # of photons arriving each second in free space beam
( )
signal'classical'ainnoisenoiseWaven
statsBoltzmannMaxwell
statscountingnoiseshotStatsPoissonn
nnnnn
2s
s
2
ss
2
ss2s
==
−
====
+=−≡∆
( ) 1T/k?hs 1en s
−−=
C. Carilli, Synthesis Summer School, 24 June 2002
6
Electron statistics: Fermi-Dirac (indistiguishable particles, but number of particles in each state = 0 or 1, or antisymmetric wave function under particle exchange, spin ½)
0variancefilledarestatesall1nmaximumeg. s =∴⇒=
( ) 2ss
2ss
2s nnnnn −=−≡∆
( ) 1T/k?hs 1en s
−+=
2
C. Carilli, Synthesis Summer School, 24 June 2002
7
RJ
Wien
Photon Noise II
'regimeclassical'noise)(wavenrms1n:(radio)RJ
regime'quantum'stats)(countingnrms1n:(optical)Wien
ss
ss
=∝⇒>
=∝⇒<
sn
?B
( ) 2ss
2ss
2s nnnnn +=−≡∆
C. Carilli, Synthesis Summer School, 24 June 2002
8
Photon noise III: Black bodies, spatial modes, and temperatures (Richards 94)
C. Carilli, Synthesis Summer School, 24 June 2002
9
Disclaimer on Wave noise
Richards 1994: ‘The first term in equation 11 can be obtained more directly. For Poisson statistics the mean square fluctuatio n in the number of photons arriving in 1s is just equal to the number of photons arriving. This term has been verified experimentally in many experiments. The second term, by contrast, has not been measured unambiguously.’
Zmuidzinas 2000: ‘Richards has recently discussed the second term of Eq. 4.1, raising questions about the theoretical and experimental justification for this term. However, as discussed in section III, the second term is needed in order to recover the Dickey radiometer equation for single mode detectors in the highbackground limit; we therefore disagree that there is no empirical justification for this term.’
C. Carilli, Synthesis Summer School, 24 June 2002
10
Origin of wave noise I: Young’s 2 slit experiment
( )( ) 'photons4to'02EI:sourcescoherentTwo
photons''2E2I:sourcesincoherentTwo
photon''1EI:SourceSingle
2
2
2
=∝
=∝
=∝
C. Carilli, Synthesis Summer School, 24 June 2002
11
Origin of wave noise II: ‘Bunching of Bosons’ in phase space (time and frequency) allows for interference (ie. coherence).
Bosons can, and will, occupy the exact same phase space if allowed, such that interference (destructive or constructive) will occur. Restricting phase space (ie. narrowing the bandwidth and sampling time) leads to interference within the beam. This naturally leads to fluctuations that are proportional to intensity (= wave noise).
C. Carilli, Synthesis Summer School, 24 June 2002
12
“Think then, of a stream of wave packets each about c/∆ν long, in a random sequence. There is a certain probability that two such trains accidentally overlap. When this occurs they interfere and one may find four photons, or none, or something in between as a result. It is proper to speak of interference inthis situation because the conditions of the experiment are justsuch as will ensure that these photons are in the same quantum state. To such interference one may ascribe the ‘abnormal’ density fluctuations in any assemblage of bosons .
Were we to carry out a similar experiment with a beam of electrons we should find a suppression of the normal fluctuations instead of an enhancement. The accidental overlapping wave trains are precisely the configurations excluded by the Pauli principle.” Purcell 1959
Origin of wave noise III
3
C. Carilli, Synthesis Summer School, 24 June 2002
13
Photon arrival time: normalized probability of detecting a second photoelectron after interval t in a plane wave of linearly polarized light withGaussian spectral profile of width ∆ν (Mandel 1963). Exactly the same factor 2 as in Young’s slits!
Maxwell-Boltzmann
Fermi-Dirac
Origin of wave noise IV
Photons ( Bose-Einstein wo. # conservation)
ν∆≈1/timescaleRelevantC. Carilli, Synthesis Summer School, 24
June 200214
‘If we were to split a beam of electrons by anonpolarizing mirror, allowing the beams to fall on separate electron multipliers, the outputs of the latter would show a negative cross-correlation. A split beam of classical particles would, of course, show zero cross correlation. As usual in fluctuation phenomena, the behavior of fermions and bosons deviate in opposite directions from that of classical particles. The Brown-Twiss effect is thus, from a particle point of view, a characteristic quantum effect.’’ Purcell 1959
Origin of wave noise V
C. Carilli, Synthesis Summer School, 24 June 2002
15
Intensity Interferometry: rectifying signal with square-law detector (‘photon counter’) destroys phase information. Cross correlation of inten sities still results in a finite correlation, proportional to the square of the E-field correlation coefficient as measured by a ‘normal’ interferometer. Exact same phenomenon as increased co rrelation for t < 1/ ∆νin lag -space above, ie. correlation of the wave noise itself = ‘Brown and Twiss effect’
+= 221 2
112 γτNNNc
γ = correlation coefficient
Disadvantage: No visibility phase information
lower SNR
Works best for high n_s: hot stars in near IR
Advantage: timescale = 1/∆ν (not 1/ν)
=> insensitive to poor optics, ‘seeing’
C. Carilli, Synthesis Summer School, 24 June 2002
16
Signal-to-Noise I: allow for A, t, ∆ν
( ) ( )
( )( ) func(A)t? ?SNRTTT:source(point)Strong
t? ?ASNRTTT:source Weak
t? ?TTT
Tt? ?
nn
SNR
:equationradiometryDickeytherecover
TTTTandAS kT :using
nrms:noise) (wave1n
AasincreasesalwaysSNR
??t)h?S
(A ??t)()n
n(SNR
nrms:stats) counting(1n
t? ? )n oftsmeasuremen (#andRx@c/Hzphotons/seh?SAn
nnntmeasuremenperVariance
1/2bgRxA
1/2bgRxA
1/2
bgRxA
A1/2
tots,
srcs,
bgRxAtot?
ss
1/2
1/2?1/2
srcs,
srcs,
1/2
srcs,s
s?
s
2
ss2s
≠∝⇒+>>
∝⇒+<<
++==
++==
∝>
∴
==
∝<
⋅=⋅=
+=∆=
C. Carilli, Synthesis Summer School, 24 June 2002
17
Signal to Noise II
For strong point source
C. Carilli, Synthesis Summer School, 24 June 2002
18
Photon occupation number I: bright radio source
source]bydominatedisTK40TT :[notedominatednoisewave
secHz20001)-(en
0005.0Tk?hK140T:VLA(25m)
0.057 zJy,1400S:ACygnus
sysbgRx
111k Th?
s
A
1.4GHz
∴≈+∴
==⇒
=⇒=
==
−−−
4
C. Carilli, Synthesis Summer School, 24 June 2002
19
Photon occupation number II: optical source
dominatednoise'counting'secHz0.0003n
Hz5e14at8?/kTh3000KT:HSTBetelgeuse
11s
B
∴=⇒
=⇒=−−
Dupree et al. 1998
C. Carilli, Synthesis Summer School, 24 June 2002
20
Photon occupation number III: faint source
?. cmatalways1nensuresK2.7Tdominates?noisewaveassumestillwedoWhy
.1n3000Tk/?hm K0.02T
4.7zmJy,0.2S:07251202
sb g
s
A
1 .4GHz
>≥
<<∴=⇒=
==−
C. Carilli, Synthesis Summer School, 24 June 2002
21
RJWien
“Even the feeble microwave background ensures that the occupation number at most radio frequencies is already high. In other words, even though the particular contribution to the signal that we seek is very very weak, it is already in a classical sea of noise and if there are benefits to be derived from retaining the associated aspects, we would be foolish to pass them up.” Radhakrishnan1998
sn
100 GHz
1n s =
C. Carilli, Synthesis Summer School, 24 June 2002
22
Quantum noise I: Commutation relations
x? P
h? P? xDh
? P
D?
?
ch?
P
?P? PD? x
x
x
z
zx
=⋅
=
=
=
==
h? t? E? ?1
? t:theorySampling
?hEPower
? ?:signaldBandlimite
=⋅
≈
∆≈∆=
C. Carilli, Synthesis Summer School, 24 June 2002
23
Quantum noise II: Coherent Amplifiers
11s
s
secHzrad1? n??2p?
? t
nh?? Eh? E ? t
:Photons
−−=⇒
=
∆==
ϕ
ϕ
k?h?2p
T
:noiseRx limitedquantumrecover h? E ? t
?2p?
? t
kT? E:etemperaturRx
sys
sys
ϕ
ϕ
=
=
=
=
e.photon/mod1of noiseminimum
hasamplifierconservingPhase
secHz1? n
rad1?:AmplifierCoherent11
s−−=⇒
<ϕ
bgQsys
bgQsys
TK75GHz)(250T
TK0.4GHz)(1.4T
>>=
<<=
Phase coherent amplifier automatically puts signal into RJ = ‘classical’ regime
C. Carilli, Synthesis Summer School, 24 June 2002
24
Quantum noise III: 2 slit paradox
Which slit does the photon enter? With a phase conserving amplifier it seems one could both detect the photon and ‘build-up’ the interference pattern (which we know can’t be correct). But quantum noise dictates that the amplifier introduces 1 photon/mode noise, such that:
I_tot = 1 +/- 1
and we still cannot tell which slit the photon came through.
5
C. Carilli, Synthesis Summer School, 24 June 2002
25
Quantum noise IV: Einstein Coefficients
[ ]
1).sspontaneou ; ?h
Tk d(stimulate
receiver limitedquantumaforexpectedas?h
TksSpontaneou
Stimulated
T?2k
Tc
?2kBI
?2hIc
nA? ?p4?h
InB?c?
?h
sSpontaneouStimulated
? ?4p?h
nAInBnB? ?c?h
xI
:TransferRadiative
BB:AbsorptionStimulated
Bc
h?8pAEmissionsSpontaneou,BemissionStimulated
B
B
B2B2
2
??
3
2
iij
iij
iijjjiiij
jiij
ij3
3
ijij
==
→
=
==→
==
+−=∂∂
=
===
C. Carilli, Synthesis Summer School, 24 June 2002
26
Quantum noise IVb: maser acts as quantum limited amplifier as dictated by the Einstein coefficients (Zmuindzinas 2000)
C. Carilli, Synthesis Summer School, 24 June 2002
27
Quantum noise V: Radio vs. Optical Interferometry
noiseRxhigh:gesdisadvanta
selectronicIFeasy
:advantagesselectronic
conservingphaseusewellasmightirrelevantisQN1n:Radio
delays optics, precise complex,
:gesdisadvanta bandwidthslarge0, is noiseRx :advantages
(CCDs)detectorsdirector
andmirrorsusetobettersdisasterouisQN1n:Optical
secHz1nnoiseQuantum
s
s
11q
SNRaffecttdoesn'polfullantennasadding
SNRlowersignal splitting means elements adding
+
∴⇒>>
⇒≈
∴⇒<<
== −−
C. Carilli, Synthesis Summer School, 24 June 2002
28
without lowering SNR!
Quantum noise Vb
C. Carilli, Synthesis Summer School, 24 June 2002
29
Quantum limit VI: Heterodyne vs. direct detection interferometry
( )
( )
( )
( ) winsHeterodyne2000NSNRSNR
:GHz1.4atVLAwithACygnus
winsDDN/3000SNRSNR
:Hz5e14at HSTwithBetelgeuse
elements#NwithNnSNRSNR
?t?Nn
SNR1/?t?nSNR
2/1
DD
Het
1/2
DD
Het
2/1s
DD
Het
2/1s
DD2/1
sHet
⇒=
⇒=
=≈
==
C. Carilli, Synthesis Summer School, 24 June 2002
30
Quantum limit VII: On the border
( ) uptoss4/NSNRSNR
1/4n1.7?/kThK10T
:2)CO(3inGHz345atOrion
2/1
DD
Het
sA
−⇒=
=⇒=⇒≈
−