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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)
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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
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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
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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
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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
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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).
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Johnson Noise (cont.)Johnson Noise (cont.)
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- 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
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(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
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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
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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
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Unified Derivation (3)Unified Derivation (3)
.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
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Unified Derivation (4)Unified Derivation (4)
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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).
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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
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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
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GG--R Noise as a Shot NoiseR Noise as a Shot Noise
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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!
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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
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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
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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