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4.3.3. Data recovery
The data recovery section consists of a decision circuit and a clock recovery circuit. The purpose of
clock recovery is to isolate a frequency component f = B from the data to be used to synchronize
the decision process. For the RZ modulation format, a component at B is always present in
the received spectrum, while for the NRZ format, there is no frequency component at B. The
component at B/2 is used and frequency doubled.
The decision circuit compares the output from the linear channel to a threshold level in order
make a binary decision. Sampling times are determined by the clock recovery circuit and are
targeted to occur at the center of each bit slot. The output of the data recovery unit is a clean
digital signal. However, because of noise in the system, there is always some probability that
decisions are incorrent. Digital receivers are usually designed to keep the error probability below
1 error in 109 bits, or a bit-error rate of 10−9 .
4.4. Receiver Noise
There are two dominant sources of noise in optical detection - shot noise and thermal noise.
4.4.1. Shot noise
Shot noise is the result of quantized energy and manifests in both optical signals and electrical
signals. An electric current is a stream of electrons generated at random times, and can be written
I(t) = Ip + is (t)
where Ip = RPin is the average current, and is (t) is the noise current. For shot noise, is is a
stationary random process with Poisson statistics. We will approximate is with Gaussian statistics.
The autocorrelation of the noise current is
his (t)is (t + τ )i=
autocorreclation
Z∞
Ss (f )e2πif τ df,
−∞
where Ss = qIp is the power spectral density. Shot noise is white noise, which means that the
noise power is constant across frequency. The noise variance is obtained by using τ = 0, and is
written
Z∞
σ 2s = hi2s (t)i =
Ss (f )df = 2qIp ∆f.
−∞
Here, ∆f is the effective noise bandwidth of the receiver, and depends on the receiver design. In
general, the effective noise bandwidth is given by
Z ∞
∆f = 2qIp
|HT (f )|2 df.
0
The photodetector dark current also has shot noise, so that the total RMS noise current due to
shot noise is given by
σ 2s = 2q (Ip + Id ) ∆f.
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4.4.2. Thermal noise
Thermal (or Johnson or Nyquist) noise is the result of random thermal motion of electrons in a
resistor, and generates a noise current even in absence of an applied voltage. Thermal noise is
modeled as a stationary gaussian random process with spectral density independent of frequency,
up to about 1 THz. In the presence of thermal noise only, the total current
I(t) = Ip + iT (t)
and the power spectral density
ST (f ) =
2kB T
.
RL
The noise variance is then
σ 2s =< i2T (t) >=
Z∞
ST (f )df =
−∞
4kβ T
∆f,
RL
which does not depend on average current like shot noise.
In general, the thermal noise in an optical receiver is more complicated than than just the
noise generated in a load resistor due to the presence of noise in other components. The primary
components are the pre-amplifier and amplifier. We can account for added amplifier noise by
including a factor Fn , which is called the amplifier noise figure. With this modification, the
thermal noise current variance becomes
σ 2T =
4kB T
Fn ∆f.
RL
The total noise current variance can be obtained by adding the variances due to shot and
thermal noise, or
4kB T
Fn ∆f.
σ 2 = σ 2s + σ 2T = 2q (Ip + Id ) ∆f +
RL
The total noise current is then σ, and the signal to noise power ratio Ip2 /σ 2 .
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4.4.3. pin receiver noise
The signal to noise ratio (SNR) is defined
Ip2
Ip2
average signal power
= 2 = 2
average noise power
σ
σ s + σ 2T
2
R2 Pin
=
.
2q (RPin + Id ) ∆f + 4 kRBLT Fn ∆f
SNR =
This expression is valid for a receiver using a pin photodetector. For an uncooled pin detector
at room temperature, thermal noise usually dominates (unless the incident optical power is very
large). Under thermal noise only,
2
RL R2 Pin
,
SNR =
4kB T Fn ∆f
and it is desirable to have large RL . However, large RL decreases receiver bandwidth.
The noise equivalent power, or NEP, is the minimum optical power per unit bandwidth that
achieves SNR=1. In the case of thermal noise,
2
RL R2 Pin
P2
4kB T Fn
⇒ in =
4kB T Fn ∆f
∆f
RL R 2
r
Pin
2 kB T Fn
NEP = √
.
=
R
RL
∆f
1=
√
The detectivity is defined as NEP−1 . For a typical pin receiver, the NEP∼ 1 − 10 pW/ Hz. As
an example, for 1 MHz receiver bandwidth, the incident optical power must be in the 1-10 nW
range.
For cooled pin detectors or in situatations where the incident optical power is large, then shot
noise dominates. This is called the shot-noise limit.
SNR =
2
R2 Pin
RPin
=
,
2qRPin ∆f
2q∆f
where detector dark current was ignored. We can also express SNR in terms of the number of
incident photons comprising a “1” bit Np . The energy in a bit of duration 1/B is Ep = Pin /B.
The number of photons is then
Ep
Pin
Np =
=
.
hν
Bhν
Using a receiver bandwidth ∆f = B/2, the signal to noise ratio becomes
SNR =
RPin
RNp Bhν
RNp hν
=
=
= Np .
qB
qB
q
Therefore, for an SNR of 20 dB, we need Np = 100 incident photons. At a wavelegnth of 1.55 µm,
and bit-rate B = 10 Gb/sec, Np = 100 corresponds to Pin ∼ 130 nW. Note that in the thermal
noise limit, several thousand photons are needed, and incident powers in the µW range are required.
Since shot noise is a fundamental manifestation of quantum mechanics, receiver operation in the
shot noise limit is the best possible circumstance.
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4.4.4. APD receiver noise
Using an APD detector, the photocurrent
Ip = M RPin ,
and we would expect an increase in SNR by the factor M 2 for thermal noise, but no change for
shot noise. However, due to the avalanch process, shot noise is enhanced. This effect is described
by
σ 2s = 2qM 2 FA (RPin + Id ) ∆f,
where FA is the APD excess noise factor defined by
FA = kA M + (1 − kA ) (2 − 1/M ) .
When FA = 1, Ip and σ s are increased by same factor and the SNR is unchanged in the shot noise
limit.
In general, the SNR for APD receivers is worse than that for pin receivers when shot noise
dominates, but can be considerably better when thermal noise dominates (because Ip ∝ M ). The
total SNR for an APD receiver is
SNR =
Ip2
σ 2s + σ 2T
(M RPin )2
=
.
2qM 2 FA (RPin + Id ) ∆f + 4 (kB T /RL ) Fn ∆f
In the shot-noise limit,
RPin
(M RPin )2
=
SNR =
2
2qM FA (RPin ) ∆f
2qFA ∆f
and the SNR is reduced by the factor FA . In the thermal-noise limit
2
(M RPin )2
RL R2 Pin
SNR =
=
M 2,
4 (kB T /RL ) Fn ∆f
4kB T Fn ∆f
which is a factor M 2 better than for a pin.
In the general expression for APD SNR, it is clear that the SNR depends strongly on the value
of FA , which itself depends on kA and the current gain M. As a result, the is an optimal value of
M for which SNR is maximized. This value is obtained from the expression
3
kA Mopt
+ (1 − kA ) Mopt =
4kB T Fn
.
qRL (RPin + Id )
For Si APD’s for which kA ≪ 1, Mopt ∼ 100, while for InGaAs APD receivers for which kA ∼ 0.7,
Mopt ∼ 10.