Lesson-04 Noise Sources Relevant to Detectors

8/12/2019 Lesson-04 Noise Sources Relevant to Detectors

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1. Photon noise

a. Signal photon noise

b. Background photon noise

2. Noise generated in detector

a. Johnson

b. Shot

c. Generation-Recombination

d. 1/f

e. Temperature fluctuations

3. Noise of interface

a. Pre-amp. electronics, A/D, display

Noise Sources Relevant to Detectors(3 Major Classes)



2

Noises can be expressed in VOLTAGE or

CURRENT for this context.

The variance around the mean is defined as the

noise2 of this random variable.

Consider a voltage source

Ergodic: time average equals the

ensemble average.

Time

2)( V

Noise (Noise (concon’’tt))

2 2

2

0

1( )V V V V V dt

2

V



3

Variances add, so to obtain total noise, one adds σ 2 not σ .

The square root is the root mean square (rms).

Noises add in a fashion. 2

(total noise)2 = Johnson2 + shot2

(all in the same units)

i.e.

At a given frequency: 10sin 20sin

Power adds but voltage adds in phasor addition.

V t t

2 2

2 2 22

1 2 3

= noise variance

i.e. - for 3 independent noise sources

( ) v v v

i

i

V

2where ( V) mean square voltage fluctuation is proportional

to power.

2( )V

2 2 2 2

1 2 3rmsV V V V V

2 2

6

10

6 10 11.7

J

s

T

V V

V V

V V V



4

White

If noise is not white, one must integrate over

electrical bandwidth.

Noise in some frequency range ;

power spectral density

f df f S V t v )(

2

2

2

2

1

2 noise f V f V )(2 f V

f 1 f 2

spectrumelectricalnoisefor white-f noise

f

f



5

Sources of Noise for Detectors

Not represented as a voltage or current because it manifests

itself differently for different detector types.

Photodiode - shot noisePhotoconductive - gen.-recomb. noise

Photo emissive - shot noise

Thermal - Temp. fluctuations

relation]ansformFourier tr [from 21

bandwidthelectronic

n timeobservatio

c photons/seof #average

- Noise

whiteFTfndelta

Why?freq.)(elect.of indep.-noisete Whi

statisticsPoisson

NoisePhoton

t f

f

t

t n

f

q

qrms



6

The fluctuations caused by the thermal motion of the charge

carriers in a resistive element. Local random thermal motion

of carriers set up fluctuating charge gradients, yet charge

neutrality exists on the whole.

Johnson NoiseJohnson Noise

Consider the following circuit

R V nC

This circuit has one thermodynamic degree of freedom, Vn.

In equilibrium, Vn is fluctuating and has an average energy

of kT/2. The capacitor stores energy:

kT V C

CV E

n

cap

2

1

2

1 thatso

2

1

2

2

G.H. Rieke, “Detection of Light: from the Ultraviolet to the

Submillimeter” (Cambridge University Press, 1994).



Johnson Noise (cont.)Johnson Noise (cont.)

7

- Boltzmann's Constant - temperature

- resistance

- Noise Equivalent Bandwidth

4 J B

B

V k TR f

k T

R

f

The randomly fluctuating potential energy on the capacitor

must have a fluctuating kinetic energy component. i J . This

energy is of the form

RC kT t P where,2

1

The maximum power that can be delivered to a device

connected across the terminals is

R

f kT i

f Ri

P

J

J

4 Thus

response.lexponentiafor4

1 and

2

2

2

In voltage terms



8

(1) Voltage source

VJ

noiseless

resistor

(2) Current source

R noiselessi J

What is the noise of two resistors connected in parallel

as shown below?

Model the Circuit EquivalentModel the Circuit Equivalent

23 3

J

-9

V = 4 1.38 10 300 5 10 100 Volts (rms)

= 90 10 Volts (rms)

10

@300 K

K 10

@300 K

5

@300 K

K

10 K

EXAMPLE: ( 100 , 1000 , 1100 )?low high f Hz f Hz f Hz

4 J BV k TR f

4 B J

k T f i R

kT qV e /1



Unified Derivation of Johnson andUnified Derivation of Johnson andShot NoiseShot Noise

G

g B A

i(t)

V

The model consists of a tunnel junction.

G is an ideal constant voltage generator.

i(t) is the current flowing through the

junction. Between metal contacts A and

B is tunneling gap, g .

Assume that the current is sampled over discrete intervals and

that is sufficiently short that only three possibilities can occur:

1. An electron tunneled from A B; Probability = P AB ;

2. An electron tunneled from B A; Probability = P BA ;

3. No electron tunnels; Probability= 1-( P AB+P BA) ;

/qi

/qi

0i

L. Callegaro, Am. J. Phys. 74, 438 (2006).

)(

]1)[0(1

1

BA AB

AB BA AB BAik

P P q

P q

P P P q

pii

The average current is



Unified Derivation (2)Unified Derivation (2)

The mean square (2nd moment) value of the current is

)()0(222

2

BA AB AB BA P P q

P q

P q

i

Note that

RV i

The states A and B have occupation numbers n A and n B. The

average number of transition events per unit time in the AB

direction = average number per unit time in the BA direction.

In steady state , detailed balance requires

B BA A AB n P n P

The states A and B have energies E A and E B with difference

E A-E B= qV . Boltzmann distribution implies

kT qV

AB

BAkT qV

B

A

e P

P

en

n //

or




.2

1

2 s f

f

Thus

kT qV AB

kT qV

AB BA AB

eqR

V

P

eqP P P q R

V i

/

/

1or

]1[)(

Also

kT qV

AB BA AB e P q

P P q

i /22

2 1

Remembering that from Nyquist’s sampling theorem

And replacing P AB in the equation for <i2>, we obtain

NoiseJohnson4

lim 0 and 0For

NoiseShot2 0 and 0For

1

12

2

0

2

/

/2

f R

kT iV T

f iqiV T

f e R

eV qi

V

kT qV

kT qV




12

Caveats:

Derivation not rigorous. However, a full quantum-

mechanical derivation can apparently be found in: Y.M. Blanter

and M. Büttiker, Phys. Rep. 336, 1-166 (2000).

Simplifications:

1. Sampling procedure assumes only three possibleevents

2. Noise power is identified with <i2> rather than the

variance, thus includes DC power.

3. Boltzmann approximation

Still gives correct result for f < 100 GHz at room temperature with

a reference to B. Abbott et al ., IEEE Trans. Educ. 39, 1-13(1996).



13

Shot noise:

Johnson noise :

Total noise:

Typically associated with the dc current flowing across a potential barrier - current through a diode has this noise.

NOTE: Current through a resistor does not have shot noise.

Example: What is the total noise current of a photodiode at

zero bias of 106 Ω resistance at room temperature (300 K)

with 1 μA of photocurrent in a noise bandwidth of 1000 Hz?

Shot NoiseShot Noise

2 s dci qi f

pA9.17A1079.1

)1000)(101)(106.1(2

2

11

619

s

dc s

i

f qii

pA1.410

)1000)(300)(104(1.38

4

6

23-

J

B J

i

R

f T k i

pA4.181.49.17 22



14

Statistical fluctuation in the rate of generation and

recombination of charged particles. These variations

may be caused by variation in carrier lifetimes or the

random generation

The detector may be cooled so thatThe detector may be cooled so that ggthth 0. Then,0. Then,

If the photons determine the fluctuation dominant, it

is photon noise limited -- this occurs in

photoconductors

2/12 f g f A E qGi thd q gr

G = photoconductive gain

g t h = thermal generation

GenerationGeneration--Recombination NoiseRecombination Noise

2 gr q d i qG E A f



GG--R Noise as a Shot NoiseR Noise as a Shot Noise

15

f iqG f iiqGi

or

f iiqG f qGi f qGii

A E qGi

qGg i

f g Gq f A E Gqi

f g f A E qGi

total dark photo gr

dark photodark photo gr

d q photo

thdark

thd q gr

thd q gr

)(4)(4

)(444

thatso and

but

44

or

2

2

22222

2/1

Compare to the shot noise formula!



16

Fluctuation in temperature of the sensitive element due to

the radiative exchange with the background or conductance

with the heat sink .

It can be derived that the variance in optical radiant power

is:

This noise is important in bolometers, thermistors, etc.

Note: The heat capacity ( H ) is not present in the noise

expression (independent of detector material and volume).

The spectrum of the mean-square fluctuation in T is

5.111)Eq.B&(D 222

22

H K T

e

k B = Boltzmann’s constant

K = Thermal conductance

T = Temperature

H = Heat capacity

Temperature NoiseTemperature Noise

5.130)Eq.B&(D 4 22 KT k Be

5.131)Eq.B&(D

2

4222

22

H f K

f KT k T B

Detailed derivation: Dereniak & Boreman, Sect. 5.3.4



1/f Noise1/f Noise

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Causes are non-ohmic contacts at electrodes, surface state traps -

• Always present in photoconductors and bolometers

– I d always

• Can be eliminated in P.V. - why

• Causes drift in detectors – calibration changes

• Limits how much signal averaging one can do

Microphonic

Caused by mechanical displacement due to changes in

the interelectrode wire capacitance caused by

displacement from their position relative to

ground.

f

f BI i f

2

possiblenot;0,i : f f NOTE

Log f

2log f i

B = proportionality constant

~ 2

~ 1



18

This is either current and voltage noise which is controlled

by selection of components for particular detector types.

detector noise Pre - amp noise

Low Noise Devices

Bipolar JFET MOSFET Oper. Amp2N4403 2N6484 3N160 OPA111;0p05

Quantization noise or digital noise associated with A/D

conversion.

A/D

PADetector

max

122

1V V

nrms

V max = A/D voltage range

n = # of bits

PrePre--Amplifier NoiseAmplifier Noise

Lesson-04 Noise Sources Relevant to Detectors

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