Detection of Electromagnetic Radiation III: Photon Noise

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Detection of Electromagnetic Radiation
IIIPhoton Noise

• Phil Mauskopf, University of Rome
• 19 January, 2004

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Example - impedances of transmission
lines Parallel strips - 5 ?m wide, 0.3 ?m
separation, ? 4.5 (SiO) Z h/w 377/? ? 10
? Microstrip line - one plate inifnitely wide
therefore impedance is slightly lower Z 5
? Coaxial cable - Radius of outer conductor R,
inner r Z ln(R/r) (1/2?) 377/? ? 60 ?
ln(R/r) /? ? For R/r 2.3, ? 1, Z 50 ?
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Scattering matrix Ports Ports are just points
of access to an optical system. Each port has a
characteristic impedance Any optical system can
be described completely by specifying all of the
ports and their impedances and the complex
coefficients that give the coupling between each
port and every other port. For an optical system
with N ports, there are NxN coefficients necessary
to specify the system. This NxN set of
coefficients is called the Scattering Matrix
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Scattering matrix S-parameters The components
of the scattering matrix are called S-parameters.
S11 S12 S13 S14 ... S
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Scattering matrix Lossless networks - unitarity
condition, conservation of energy For a network
with no loss, the S-matrix is unitary S?S
I This is just the expression of conservation of
energy, For a two port network 1 R T
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Scattering matrix Examples - two-port
networks Dielectric interface S This is
because R (Z1-Z2)/(Z1Z2) Going from lower to
higher impedance Z1 ? Z2 gives the opposite sign
as going from higher to lower impedance.
T R -R T
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Scattering matrix Power divider How about
3-port networks? Can we make an optical element
that divides the power of an electromagnetic wave
in half into two output ports? Guess What
is the S-matrix for this circuit? What is the
optical analogue?
2 Z1
Z1
2 Z1
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Scattering matrix 4-port networks - 90 degree
hybrid
A
(AiB)/?2
(A-iB)/?2
B
0 0 1 1 0 0 i -i 1 1 0 0 i -i
0 0
S
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Scattering matrix 4-port networks - 90 degree
hybrid Optical analogue Half power beam splitter
(iAB)/? 2
A
(AiB)/? 2
B
0 0 1 i 0 0 i 1 1 i 0 0 i 1
0 0
0 0 1 1 0 0 i -i 1 1 0 0 i -i
0 0
0 ?2 ?2 0
S
With a 90? phase shift on port 2
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Scattering matrix 4-port networks - 180 degree
hybrid
A
(AB)/? 2
(A-B)/? 2
B
0 0 1 1 0 0 1 -1 1 1 0 0 1 -1
0 0
0 ?3 ?3 0
S
In general lossless scattering matrices ? SU(n)
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Resistive elements in transmission line - loss
R
L
G
C
R represents loss along the propagation path
can be surface conductivity of waveguide or
microstrip lines G represents loss due to finite
conductivity between boundaries 1/R in a
uniform medium like a dielectric Z
?(Ri?L)/(Gi?C) Z has real part and imaginary
part. Imaginary part gives loss
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Resistive elements in transmission line -
loss You can replace loss terms in the
scattering matrix (which makes it non-unitary)
with additional ports that account for the lost
signal.
ZR
R
L
L
?
Z0
G
G
Z0
C
C
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Optics Direct coupling to detectors
(simplest) Need to match detector to free space
- 377 ? One way to do it is with resistive
absorber - e.g. thin metal film Transmission
line model Converts radiation into heat -
detect with thermometer the famous
bolometer! How about other detection techniques?
Impedance mismatch? - Non-destructive sampling
- sample voltage - high input Z - sample
current - low input Z Both cases the signal is
reflected 100 E.g. JFET readout of NTD, SQUID
readout of TES
Z0
-
RZ0
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Optics Direct coupling to detectors
(simplest) Without an antenna connected to a
microstrip line, the minimum size of an effective
detector absorber is limited by
diffraction Single mode - size ?2 The number
of modes in an optical system is limited by the
total throughput n(modes) A?/?2 The
throughput is limited by the coupling between
optical elements
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Two types of mm/submm focal plane architectures
Bare array
Antenna coupled
IR Filter
Filter stack
Bolometer array
Antennas (e.g. horns)
X-misson line
SCUBA2 PACS SHARC2
Microstrip Filters
Detectors
BOLOCAM SCUBA PLANCK
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Mm and submm planar antennas You can have
single mode and multi-mode antennas - e.g. scalar
feed vs. winston Quasi-optical (require
lens) Twin-slot - small number of modes Log
periodic - multimode Coupling to waveguide
(require horn) Radial probe Bow tie
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Optics Modes and occupation number A mode is
defined by its throughput A? ?2 The
occupation number of a mode is the number
of photons in that mode per unit bandwidth For a
single mode source emitting a power, P in
a bandwidth ??, with an emissivity, ? The
occupation number is N (P/2h?)(1/? ??) For a
blackbody source at temperature, T, this is just
the Bose-Einstein term N 1/(exp(h?/kT)-1)
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Optics Modes and occupation number N
(P/2h?)(1/? ??) 1/(exp(h?/kT)-1) Low
frequencies (R-J limit) h?/kT ltlt 1 N ? kT/h? gtgt
1 High photon occupation number Wave noise
dominated Zero point fluctuations High
frequencies (Wein limit) h?/kT gtgt 1 N ?
exp(-h?/kT) ltlt 1 Low photon occupation
number Shot noise dominated Johnson-Nyquist
noise CMB at millimetre-wavelengths h?/kT
1 so it is in between low and high occupation
number!
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1990s SuZIE, SCUBA, NTD/composite
1998 300 mK NTD SiN
PLANCK 100 mK NTD SiN
Shot noise
Wave noise
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Noise Formulae The 1? uncertainty in the
optical power is ?p h??? ??N(1 ?N) /( ?????
) N mode occupation number ? efficiency ?
integration time ? central frequency ??
bandwidth Limits ?N gtgt 1 ? ?p h? N???/?
Pd/(???? ? ) ?N ltlt 1 ? ?p h? ? N
??/?? ? Pd h?/? /?
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Noise Derivation Take an N-port optical system
with an NxN scattering matrix, Sij(?) Port
labels are i 1N Incoming wave amplitudes are
given by ai(?) Outgoing wave amplitudes are
given by bi(?) Considering only linear systems
(for which the S-matrix method applies) bi(?)
?j Sij(?) aj(?) or b Sa S probability
amplitude for photon entering port j to exit at
port i
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Noise Derivation Start with simplest network -
single port two terminals
P
Z
R
Port, P has characteristic impedence Z R.
Therefore there are no reflections. We can think
of this as a transmission line terminated at
infinity with another resistor, R
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Noise Derivation Start with simplest network -
single port two terminals
P
Z
Z

R
R
Rp
Where Rp is the port impedence In fact, if you
use simulation packages such as ADS, they require
that you terminate all ports with a
characteristic impedance. If Rp is infinite
open circuit then we have voltage noise If Rp 0
short circuit then we have current noise
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Noise Derivation Formula for noise can be
derived in (at least) two ways 1. Brownian
motion or random walk of electrons 2.
Transmission line model and thermodynamics Both
methods give classical solutions that are
modified by quantum effects Well consider only
the transmission line model - from Nyquist
Z
R2
R1
Based on the principle that in thermal
equilibrium there is no average power flow
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Noise Derivation
?V12? R2/(R1 R2)2
R2
R1
?V22? R1/(R1 R2)2
If the voltage noise from R1 is given by V1 then
the power generated by R1 and dissipated in R2 is
given by ?V12? R2/(R1 R2)2 and the power
generated by R2 and dissipated in R1 is given by
?V22? R1/(R1 R2)2 Thermodynamics says these
must be equal at all frequencies so ?Vi2? ? Ri
and ?Vi2? ? T. Define power spectrum, SV(?)
?Vi2?
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Noise Derivation l
Z
R
R
Suppose R1 R2 Z Z is a lossless transmission
line ? L/C Wave velocity in the transmission
line v 1/?LC The thermal power delivered to
the transmission line from either R1 or R2 in a
frequency interval d?/2? is dP (1/4R) SV(?)
d?/2? For a transmission line of length, l the
energy stored in the transmission line is equal
to the power emitted x travel time l/v dE
dP ? t ? 2 (l/2Rv) SV(?) d?/2?
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Noise Derivation
Z
R
R
If we suddenly cut the lines at the end of the
transmission line, a certain amount of energy is
trapped in standing waves dE dP ? t ? 2
(l/2Rv) SV(?) d?/2? Expanding the standing
waves in modes gives m (d?/2?)/(v/2l) Equipar
tition theorem average energy per mode kT dE
mkT (d? l/?v)kT (l/2Rv) SV(?) d?/2? ? SV(?)
4kTR
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Noise Derivation Quantum Mechanics I Include
Bose-Einstein statistics Quantum mechanically,
the average thermal energy per mode is given by
the energy per photon times the photon occupation
number dE m ?? nth (d? l/?v)
??/(exp(??/kT)-1) Setting this equal to the
energy stored in the transmission line dE
(l/2Rv) SV(?) d?/2? gives, SV(?) 4
??R/(exp(??/kT)-1) 4 ?? R nth
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Noise Derivation 4-terminals 2
ports Impedence representation and S-matrix
representation
I2(t)
I1(t)
b1
b2
Z
S
V2(t)
V1(t)
a1
a2
Impedance matrix, Z Scattering matrix, S V1
Z11 Z12 I1 b1 S11 S12
a1 V2 Z21 Z22 I2 b2
S21 S22 a2


Where ai represents the amplitude of incoming
waves and bi represents the amplitude of
outgoing waves
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Noise Derivation 4-terminals 2
ports Generalize to multiple ports Obtain noise
correlation matrix
I1(t)
b1
Zij ei
Sij ?i
V1(t)
a1
I2(t)
b2
V2(t)
a2
In(t)
bn
Vn(t)
an
S?i?j(?) (1-S?S)ij kT
Seiej(?) 2(ZZ?)ij kT
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Noise Equations Include Bose-Einstein
statistics and obtain the so-called Classical
formulae for noise correlations S?i?j(?)
(1-S?S)ij kT ? (1-S?S)ij ??/(exp(??/kT)-1)
Seiej(?) 2(ZZ?)ij kT ? 2(ZZ?)ij
??/(exp(??/kT)-1) Relations between voltage
current and input/output waves ?1/4Z0 (ViZ0Ii)
ai ?1/4Z0 (Vi - Z0Ii) bi or Vi ?Z0 (ai
bi) Ii 1/?Z0 (ai - bi)
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Noise Derivation Quantum Mechanics II Include
zero point energy Zero point energy of quantum
harmonic oscillator ??/2 I.e. on the
transmission line, Z at temperature, T0 there is
still energy. Add this energy to the
Semiclassical noise correlation matrix and we
obtain Seiej(?) 2 ?? (ZZ?)ij coth(??/2kT)
2 ?? R (2nth 1) S?i?j(?) ?? (1-S?S)ij
coth(??/2kT) ?? (2nth 1)
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Noise Derivation - Quantum mechanics This is
where the Scattering Matrix formulation is
more convenient than the impedance
method Replace wave amplitudes, a, b with
creation and annihilation operators, a, a?, b, b?
and impose commutation relations a, a?
1 Normalized so that ? a? a ? number of
photons a, a? ?? Normalized so that ? a? a
? Energy Quantum scattering matrix b? ?a?
c? Since b, b? a, a? ?? then the
commutator of the noise source, c is given
by c, c ? ??(I - ?2)
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Noise Quantum Mechanics III Calculate Quantum
Correlation Matrix If we replace the noise
operators, c, c? that represent loss in the
scattering matrix by a set of additional
ports that have incoming and outgoing waves, a?,
b? c? i ?? ?i? a? ? and (I - ?2)ij ??
?i? ??j? Therefore the quantum noise
correlation matrix is just ? c? i c i ? (I -
?2)ij nth (I - S?S)ijnth So we have lost
the zero point energy term again...
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Noise Quantum Mechanics IV Detection
operators An ideal photon counter can be
represented quantum mechanically by the photon
number operator for outgoing photons on port
i di ? b? i b i ? which is related to the
photon number operator for incoming photons on
port j by ? b? i b i ? ? (?n Sinan?)(?m
Simam) ? ? c?i ci ? ?d? Bii(?) ? (?n
Sinan?)(?m Simam) ? ?n,m Sin Sim ? a?n am ?
? a?n am ? nth(m,?) ?nm which is the
occupation number of incoming photons at port m
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Noise Quantum Mechanics IV Detection
operators Therefore di ?m SimSim nth(m,?) ?
c?i ci ? ? d? Bii(?) Where ? c?i ci ? (I -
S?S)iinth The noise is given by the variance in
the number of photons ?ij2 ? di dj ? - ? di
?? di ? ? d? Bij(?) ( Bij(?) ?ij ) Bij(?)
?m SimSjm nth(m,?) ? c?i cj ? ?m
SimSim nth(m,?) (I - S?S)ijnth(T,?) Assuming
that nth(m,?) refers to occupation number of
incoming waves, am , and nth(T,?) refers to
occupation number of internal lossy components
all at temperature, T
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Noise Example 1 - single mode detector No loss
in system, no noise from detectors, only
signal/noise is from port 0 input single mode
port Sim 0 for i, m ? 0 S0i Si0 ? 0 di ?
d? Si0Si0 nth(0,?) ? c?i ci ? ? d?
Bii(?) ?ii2 ? di dji? - ? di ?? di ? ? d?
Bii(?) ( Bii(?) ?ii ) For lossless system - ?
c?i ci ? 0 and ?ii2 ? d? Bii(?) ( Bii(?)
?ii ) ? d? Si02 nth(?) (Si02 nth(?)
1) Recognizing Si02 ? as the optical
efficiency of the path from the input port 0 to
port i we have ?ii2 ? d? ?nth(?) (?nth(?) 1)
express in terms of photon number
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Noise Gain - semiclassical Minimum voltage
noise from an amplifier zero point fluctuation
- I.e. attach zero temperature to input SV(?)
2 ?? R coth(??/2kT) 2 ?? R (2nth 1) when nth
0 then SV(?) 2 ?? R Compare to formula in
limit of high nth SV(?) 4 kTN R where TN ?
Noise temperature ? Quantum noise minimum TN
??/2k
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Noise Gain Ideal amplifier, two ports, zero
signal at input port, gain G S11 0 no
reflection at amplifier input S12 G gain
(amplitude not power) S22 0 no reflection at
amplifier output S21 0 isolated output Signal
and noise at output port 2 d2 ? d? S12S12
nth(1,?) ? c?2 c2 ? ? d? B22(?) ?222 ? d2
d2? - ? d2 ?? d2 ? ? d? B22(?) ( B22(?) 1 ) ?
c?2 c2 ? (1 - (S?S)22)nth(T,?) What does T,
nth mean inside an amplifier that has gain? Gain
Negative resistance (or negative
temperature) namp(T,?) -1/ /(exp(-??/kT)-1) ?
-1 as T ? 0
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Noise Gain 0 0 0 G 0
0 G 0 0 0 0 G2 ? c?2 c2
? -(1 - (S?S)22) (G2 - 1) d2 ? d? S12S12
nth(1,?) ? c?2 c2 ? ? d? B22(?) ?222 ? d2
d2? - ? d2 ?? d2 ? ? d? B22(?) ( B22(?) 1 )
? d? (G2 nth (1,?) G2 - 1)(G2 nth (1,?)
G2) If the power gain is ? G2 then we
have ?222 ? d? (?nth (1,?) ? - 1)(?nth
(1,?) ?) ?2(nth (1,?) 1)2?? for ? gtgt 1 and
expressed in uncertainty in number of photons In
other words, there is an uncertainty of 1 photon
per unit ?
S?S
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Noise Gain ?22 ?(nth (1,?) 1)??? expressed
in power referred to amplifier input, multiply by
the energy per photon and divide by gain, ?22
h?(nth (1,?) 1)??? Looks like limit of high nth
Amplifier contribution - set nth 0 ?22 h?
??? kTn ??? or Tn h?/k (no factor of 2!)
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Noise Gain What happens to the photon
statistics? No gain Pin n h??? and ?in
h??? ?n(1n) /(???? ) (S/N)0 Pin /?in
?n???/(1n) With gain Pin n h??? and ?in
h??? (1n) /(???? ) (S/N)G Pin /?in
????n/(1n) (S/N)0/(S/N)G ?(1n)/n
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Noise Interferometry What if we want to
measure the spectrum of incoming radiation? Two
ways 1. Divide signal into N frequency bands
using filters and detect the photons with N
detectors 2. Divide signal power by N and detect
autocorrelation of the input signal with N
lags in N detectors