Detection of Electromagnetic Detection of Electromagnetic Radiation III:Radiation III:Photon NoisePhoton Noise

Phil Mauskopf, University of Phil Mauskopf, University of RomeRome

19 January, 200419 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 byspecifying all of the ports and their impedances andthe complex coefficients that give the coupling betweeneach port and every other port.

For an optical system with N ports, there are NxN coefficientsnecessary 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 calledS-parameters.

S11 S12 S13 S14 ...

S =

Scattering matrix: Lossless networks - unitarity condition,conservation of energy

For a network with no loss, the S-matrix is unitary:

SS = 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 =

This is because R = (Z1-Z2)/(Z1+Z2)

Going from lower to higher impedance Z1 Z2 givesthe opposite sign as going from higher to lower impedance.

T R-R T

Scattering matrix: Power divider

How about 3-port networks? Can we make an opticalelement that divides the power of an electromagneticwave in half into two output ports?

Guess:

What is the S-matrix for this circuit? What is the opticalanalogue?

Z1

2 Z1

2 Z1

Scattering matrix: 4-port networks - 90 degree hybrid

A

B

(A+iB)/2

(A-iB)/2

S =

0 0 1 10 0 i -i1 1 0 0i -i 0 0

Scattering matrix: 4-port networks - 90 degree hybridOptical analogue: Half power beam splitter

A

B

(iA+B)/ 2

(A+iB)/ 2

S = = =

0 0 1 i0 0 i 11 i 0 0i 1 0 0

0 2

2 0

0 0 1 10 0 i -i1 1 0 0i -i 0 0

With a 90 phase shift on port 2

Scattering matrix: 4-port networks - 180 degree hybrid

A

B

(A+B)/ 2

(A-B)/ 2

S = =

0 0 1 10 0 1 -11 1 0 01 -1 0 0

0 3

3 0

In general lossless scattering matrices SU(n)

Resistive elements in transmission line - loss:

C

R

G

L

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+iL)/(G+iC)

Z has real part and imaginary part. Imaginary part givesloss

Resistive elements in transmission line - loss:

You can replace loss terms in the scattering matrix(which makes it non-unitary) with additional portsthat account for the lost signal.

C

R

G

L

C G

L

Z0Z0

Z=R

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. thinmetal 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 R=Z0

Optics: Direct coupling to detectors (simplest)

Without an antenna connected to a microstrip line, theminimum size of an effective detector absorber is limitedby diffraction

Single mode - size ~ 2

The number of modes in an optical system is limited bythe total throughput:

n(modes) = A/2

The throughput is limited by the coupling between opticalelements

Two types of mm/submm focal plane architectures:

SCUBA2PACSSHARC2

BOLOCAMSCUBAPLANCK

Filter stack

Bolometer array

IR Filter

Antennas (e.g. horns)

X-misson line

Detectors

Bare array Antenna coupled

Microstrip Filters

Mm and submm planar antennas:

You can have single mode andmulti-mode antennas -e.g. scalar feed vs. winston

Quasi-optical (require lens):

Twin-slot - small number of modesLog periodic - multimode

Coupling to waveguide (require horn):

Radial probeBow tie

Optics: Modes and occupation number

A mode is defined by its throughput: A = 2

The occupation number of a mode is the number ofphotons in that mode per unit bandwidth

For a single mode source emitting a power, P in abandwidth , with an emissivity,

The occupation number is:

N = (P/2h)(1/ )

For a blackbody source at temperature, T, this is just theBose-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 ~ 1so it is in between low and high occupation number!

1990s: SuZIE, SCUBA, NTD/composite

1998: 300 mK NTD SiN

PLANCK: 100 mK NTD SiN

Wave noiseShot 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 scatteringmatrix, 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 methodapplies):

bi() = j Sij() aj() or b = Sa

S = probability amplitude for photon entering port j to exitat port i

Noise: Derivation

Start with simplest network - single port = two terminals

Z

R

Port, P has characteristic impedence = Z = R. Thereforethere are no reflections. We can think of this as a transmissionline terminated at infinity with another resistor, R

P

Noise: Derivation

Start with simplest network - single port = two terminals

Z

R

PZ

RRp

=

Where Rp is the port impedence

In fact, if you use simulation packages such as ADS, theyrequire that you terminate all ports with a characteristicimpedance.

If Rp is infinite = open circuit then we have voltage noiseIf Rp = 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 electrons2. Transmission line model and thermodynamics

Both methods give classical solutions that are modified byquantum effects

We’ll consider only the transmission line model - from Nyquist

Z

R2R1

Based on the principle that in thermal equilibrium there isno average power flow

Noise: Derivation

R2R1

If the voltage noise from R1 is given by V1 then the powergenerated 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:Vi

2 Ri and Vi2 T. Define power spectrum, SV() = Vi

2

V12 R2/(R1 + R2)

2

V22 R1/(R1 + R2)

2

Noise: Derivation l

Z

RR

Suppose R1 = R2 = ZZ is a lossless transmission line = L/CWave velocity in the transmission line v = 1/LC

The thermal power delivered to the transmission line fromeither 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 thetransmission line is equal to the power emitted x travel time = l/v:

dE = dP t 2 = (l/2Rv) SV() d/2

Noise: Derivation

Z

RR

If we suddenly cut the lines at the end of the transmissionline, a certain amount of energy is trapped in standingwaves:

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 permode is given by the energy per photon times the photonoccupation 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:

Z SV1(t)

I2(t)I1(t)

V2(t)

Impedance matrix, Z: Scattering matrix, S:

V1 Z11 Z12 I1 b1 S11 S12 a1

V2 Z21 Z22 I2 b2 S21 S22 a2

=

a1

b1

a2

b2

=

Where ai represents the amplitude of incoming wavesand bi represents the amplitude of outgoing waves

Noise: Derivation: 4-terminals = 2 ports

Generalize to multiple ports: Obtain noise correlation matrix

Zij

ei

Sij

i

V1(t)

I1(t)

a1

b1

V2(t)

I2(t)

Vn(t)

In(t)

a2

b2

an

bn

Sij*() = (1-SS)ij kTSeiej

*() = 2(Z+Z)ij kT

Noise: Equations

Include Bose-Einstein statistics and obtain the so-called‘Classical’ formulae for noise correlations:

Sij*() = (1-SS)ij kT (1-SS)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

orVi = 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 thereis still energy.

Add this energy to the ‘Semiclassical’ noise correlation matrixand we obtain:

Seiej*() = 2 (Z+Z)ij coth(/2kT) = 2 R (2nth +1)

Sij*() = (1-SS)ij coth(/2kT) = (2nth +1)

Noise: Derivation - Quantum mechanics

This is where the Scattering Matrix formulation is moreconvenient than the impedance method:

Replace wave amplitudes, a, b with creation andannihilation operators, a, a, b, b and impose commutationrelations:

[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 representloss in the scattering matrix by a set of additional portsthat 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 - SS)ijnth

So we have lost the zero point energy term again...

Noise: Quantum Mechanics IV: Detection operators

An ideal photon counter can be represented quantummechanically by the photon number operator for outgoingphotons on port i:

di = b i b

i

which is related to the photon number operator forincoming 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

an am = nth(m,) nm which is the occupation number of

incoming photons at port m

Noise: Quantum Mechanics IV: Detection operators

Thereforedi = m S*

imSim nth(m,) + ci ci = d Bii()

Where: ci ci = (I - SS)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 - SS)ijnth(T,)

Assuming that nth(m,) refers to occupation number of incomingwaves, am , and nth(T,) refers to occupation number of internallossy components all at temperature, T

Noise: Example 1 - single mode detector

No loss in system, no noise from detectors, only signal/noiseis from port 0 = input single mode port:Sim = 0 for i, m 0S0i = Si0 0

di = d S*i0Si0 nth(0,) + c

i ci = d Bii()ii

2 = di dji - di di = d Bii() ( Bii()+ ii )

For lossless system - ci ci = 0 and

ii2 = d Bii() ( Bii()+ ii ) = d Si0

2 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 pointfluctuation - 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 inputS12 = G gain (amplitude not power)S22 = 0 no reflection at amplifier outputS21 = 0 isolated output

Signal and noise at output port 2:d2 = d S*

12S12 nth(1,) + c2 c2 = d B22()

222 = d2 d2 - d2 d2 = d B22() ( B22()+ 1 )

c2 c2 = (1 - (SS)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

Noise: Gain

0 0 0 G 0 0 G 0 0 0 0 G2

c2 c2 = -(1 - (SS)22) = (G2 - 1)

d2 = d S*12S12 nth(1,) + c

2 c2 = d B22()22

2 = 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

SS = =

Noise: Gain

22 ~ (nth (1,)+ 1)

expressed in power referred to amplifier input, multiply by theenergy 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: Pin = n hand in = h n(1+n) /( )

(S/N)0 = Pin /in = n/(1+n)

With gain: Pin = n hand 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 incomingradiation?

Two ways:1. Divide signal into N frequency bands using filters and detect the photons with N detectors2. Divide signal power by N and detect autocorrelation of the input signal with N lags in N detectors