Download Photon counting FIR detectors

Detection of Electromagnetic
Radiation III:
Photon Noise
Phil Mauskopf, University of Rome
19 January, 2004
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
Scattering matrix: S-parameters
The components of the scattering matrix are called
S-parameters.
S11 S12
S=
S13 S14 ...
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
Scattering matrix: Examples - two-port networks
Dielectric interface:
S=
T
-R
R
T
This is because R = (Z1-Z2)/(Z1+Z2)
Going from lower to higher impedance Z1  Z2 gives
the opposite sign as going from higher to lower impedance.
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:
2 Z1
Z1
2 Z1
What is the S-matrix for this circuit? What is the optical
analogue?
Scattering matrix: 4-port networks - 90 degree hybrid
A
(A+iB)/2
B
(A-iB)/2
0
0
S= 1
i
0
0
1
-i
1
i
0
0
1
-i
0
0
Scattering matrix: 4-port networks - 90 degree hybrid
Optical analogue: Half power beam splitter
(iA+B)/ 2
A
(A+iB)/ 2
B
0
0
S= 1
i
0
0
i
1
1
i
0
0
i
1
0 =
0
0
0
1
i
With
0 1 1
0 2
0 i -i
=  0
1 0 0
2
-i 0 0
a 90 phase shift on port 2
Scattering matrix: 4-port networks - 180 degree hybrid
A
(A+B)/ 2
B
(A-B)/ 2
0
0
S= 1
1
0
0
1
-1
1
1
0
0
1
-1
0
0
=
0 3
3 0
In general lossless scattering matrices  SU(n)
Resistive elements in transmission line - loss:
R
L
C
G
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 = (R+iL)/(G+iC)
Z has real part and imaginary part. Imaginary part gives
loss
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.
R
Z0
Z=R
L
C
G

Z0
L
C
G
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
+
R=Z0
Z0
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
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
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
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
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)
Optics: Modes and occupation number
N = (P/2h)(1/ ) = 1/(exp(h/kT)-1)
Low frequencies (R-J limit): h/kT << 1
N  kT/h >> 1 = High photon occupation number
Wave noise dominated = Zero point fluctuations
High frequencies (Wein limit): h/kT >> 1
N  exp(-h/kT) << 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!
1990s: SuZIE, SCUBA, NTD/composite
1998: 300 mK NTD SiN
PLANCK: 100 mK NTD SiN
Shot noise
Wave noise
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 >> 1  p = h N/ = Pd/(  )
N << 1  p = h  N / =  Pd h/ /
Noise: Derivation
Take an N-port optical system with an NxN scattering
matrix, Sij():
Port labels are i = 1…N
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
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
Noise: Derivation
Start with simplest network - single port = two terminals
P
Z
Z
=
R
Rp
R
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 R = 0 = short circuit then we have current noise
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
We’ll consider only the transmission line model - from Nyquist
Z
R1
R2
Based on the principle that in thermal equilibrium there is
no average power flow
Noise: Derivation
V12 R2/(R1 + R2)2
R1
V22 R1/(R1 + R2)2
R2
If the voltage noise from R1 is given by V1 then the power
generated by R1 and dissipated in R2 is given by:
2
V1  R2/(R1 + R2)
2
and the power generated by R2 and dissipated in R1 is given by:
2
V2  R1/(R1 + R2)
2
Thermodynamics says these must be equal at all frequencies so:
Vi2  Ri and Vi2  T. Define power spectrum, SV() = Vi2
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
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)
Equipartition theorem: average energy per mode = kT
dE = mkT = (d l/v)kT = (l/2Rv) SV() d/2  SV() = 4kTR
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
Noise: Derivation: 4-terminals = 2 ports
Impedence representation and S-matrix representation:
I1(t)
V1(t)
I2(t)
b1
Z
V2(t)
a1
b2
S
a2
Impedance matrix, Z:
Scattering matrix, S:
V1
Z11
=
V2
Z21
b1
S11
=
b2
S21
Z12
Z22
I1
I2
S12
S22
Where ai represents the amplitude of incoming waves
and
bi represents the amplitude of outgoing waves
a1
a2
Noise: Derivation: 4-terminals = 2 ports
Generalize to multiple ports: Obtain noise correlation matrix
I1(t)
b1
V1(t)
I2(t)
V2(t)
In(t)
a1
b2
Zij
ei
a2
bn
Vn(t)
Sij
i
an

Seiej*() = 2(Z+Z )ij kT
Sij*() = (1-SS)ij kT
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(Z+Z)ij kT  2(Z+Z)ij /(exp(/kT)-1)
Relations between voltage current and input/output waves:
1/4Z0 (Vi+Z0Ii) = ai
1/4Z0 (Vi - Z0Ii) = bi
or
Vi = Z0 (ai + bi)
Ii = 1/Z0 (ai - bi)
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, T=0 there
is still energy.
Add this energy to the ‘Semiclassical’ noise correlation matrix
and we obtain:
Seiej*() = 2  (Z+Z)ij coth(/2kT) = 2  R (2nth +1)
Sij*() =  (1-SS)ij coth(/2kT) =  (2nth +1)
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)
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...
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 S*inan )(m Simam)  +  c i ci  = d Bii()
 (n S*inan)(m Simam)  = n,m S*in Sim  an am 

 a n am  = nth(m,) nm which is the occupation number of
incoming photons at port m
Noise: Quantum Mechanics IV: Detection operators
Therefore
di = m S*imSim 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 S*imSjm nth(m,) +  ci cj 
= m S*imSim 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
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 S*i0Si0 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
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
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 S*12S12 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)
Noise: Gain
SS = 0
G
0
0
0
0
G
0
=0
0
0
G2
 c2 c2  = -(1 - (SS)22) = (G2 - 1)
d2 =  d S*12S12 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  >> 1 and expressed in uncertainty in number of photons
In other words, there is an uncertainty of 1 photon per unit 
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!)
Noise: Gain
What happens to the photon statistics?
No gain:
and
Pin = n h
in = h n(1+n) /( )
(S/N)0 = Pin /in = n/(1+n)
With gain:
and
Pin = n h
in = h (1+n) /( )
(S/N)G = Pin /in = [n/(1+n)]
(S/N)0/(S/N)G = (1+n)/n
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