Using receiver eigenmodes, we perform a timedependent analysis of optical receivers whose optical inputs are corrupted by the amplified spontaneous emission. We use Gaussian receivers for the analysis with Gaussian input pulses. We find the number of contributing eigenmodes increases as the measurement time moves from the pulse center towards the pulse edges at the output of the optical receiver’s electrical filter. This behavior is dependent on the bandwidth ratio between the optical and the electrical filters as well as the input pulse’s time width.
I. INTRODUCTION
Optical receivers are basic building blocks in various kinds of optical communication systems. To increase the transmission distances, optical receivers suffer from various electrical and optical noises. In early optical communication systems, shot and thermal noises limit the receiver performances dominantly
[1
,
2]
. After the appearance of optical amplifiers, the amplifiedspontaneous emission (ASE) becomes the most dominant noise source that mixes with the optical channel at the photodetector
[3

5]
. For the analysis of optical receivers in the presence of the ASE, there are exact methods using receiver eigenmodes in the time domain
[6

9]
and in the optical spectral domain
[10

13]
. The time domain analyses have been developed initially for radiofrequency receivers.
Conventional analyses find the receiver eigenmode contributions at a specific time where the decision is made. Actually, the receiver eigenmode contributions change as a function of time within a bit period. These behaviors are affected by the optical and the electrical filters within the optical receiver. Accordingly, the distribution of the electrical voltage after the electrical filter will undergo similar changes within a pulse. Until now, however, no timedependent analysis has been performed yet which is important for understanding the physics of optical receivers.
In this paper, we provide a timedependent analysis of optical receivers whose optical inputs are corrupted by the ASE. We use the receiver eigenmodes in the optical spectral domain. Our analysis explains the physics about the receiver output signal and noise distributions as a function of time using receiver eigenmodes. We choose Gaussian optical receivers
[14]
that are good approximations to practical optical receivers yielding quantitative results with closedform eigenfunctions.
II. TIMEDEPENDENT OPTICAL RECEIVER ANALYSYS
We assume the optical receiver has an optical filter in front of a photodetector to select an optical channel and also to filter out the ASEs from optical amplifiers. After the photodetector, there is a lowpass electrical filter to filter out high frequency electrical beat noises.
For simplicity, we neglect the polarization component perpendicular to that of the received signal. It will be included as an additive term after the signal contributions are fully evaluated. Just before the optical filter, the complexelectric field amplitude is denoted as
ε_{in}
(
ω
) in the optical frequency domain. After the optical filter, the complexelectricfield amplitude is
ε_{out}
(
ω
)=
ε_{in}
(
ω
)
H_{o}
(
ω
), where
H_{o}
(
ω
) is the optical filter’s transmittance for the complexelectricfield amplitude. After the optical detection, the photodetector output current is (
k
/2)
E_{out}
(
t
)
^{2}
, where
k
is a proportional constant and
E_{out}
(
t
) is the inverse Fourier transform of
ε_{out}
(
ω
). After the electrical filter having a transfer function of
H_{e}
(
ω
), we have the voltage waveform at the output of the electrical filter as
where the kernel
K
(
ω
,
ω'
) is given by
We expand
ε_{in}
(
ω
)exp(
jωt
) in (1) as
where
ϕ_{m}
(
ω
) is the eigenfunction of the
m
th receiver eigenmode and
V_{m}
(
t
) is its expansion coefficient. The eigenfunctions satisfy the second kind of homogeneous Fredholm integral equation
λ_{m}
is the eigenvalue of the mth receiver eigenmode. The eigenvalues are real since the kernel is Hermitian. The eigenfunctions satisfy the orthonormal relation
where the asterisk symbol represents the complex conjugation.
Using (3) for (1),
y
(
t
) can be simplified greatly as a summation of receiver eigenmode contributions
The amplitude of the
m
th eigenmode
V_{m}
(
t
) can be decomposed into signal and noise components denoted as
S_{m}
(
t
) and
N_{m}
(
t
), respectively, as
Without loss of generality, we may set
t
= 0 in (3). Then we have
The inverse Fourier transform of (7) gives
where
ψ_{m}
(
t
) is
times the inverse Fourier transform of
ϕ_{m}
(
ω
) and also satisfies the orthonormal relation in time domain
Inserting (7) into (3), we find
From this relation,
V_{m}
(
t
) is found as
where
C_{km}
(
t
) is a correlation function
Equation (12) shows that the expansion coefficient set {
V_{m}
(
t
)} at
t
≠0 can be found from {
V_{m}
(0)} with
m
= 0, 1, 2...
Multiplying both sides of (3) with their complex conjugates and integrating over
ω
, we obtain
[14]
Equating the signal parts from both sides of (14), we have
where
ε_{in,s}
(
ω
), is the signal part of
ε_{in}
(
ω
). It means that the sum of 
S_{m}
(
t
)
^{2}
is a constant proportional to the total signal energy assumed to be finite here. Note that (15) holds independent of time. Thus, if the amplitude of a dominant eigenmode decreases, the amplitudes of other eigenmodes increase making the sum of 
S_{m}
(
t
)
^{2}
over m unchanged.
If the other polarization component of the received channel is included, (5) may be rewritten as
where we have used a prime notation for the contributions from the newly added polarization component. The polarization vector for the second term of (16) has only noise components and
V'_{m}
(
t
)=
N'_{m}
(
t
). All the real and imaginary parts of {
N_{m}
(
t
)} and {
N'_{m}
(
t
)} at a fixed time are mutually independent Gaussian random variables with zero mean and an identical variance of
[10]
.
In dense wavelengthdivision multiplexing (DWDM) systems
[15
,
16]
, the optical filter’s bandwidth is comparable to the modulation bandwidth. In this case, the magnitude of the eigenvalue
λ_{m}
increases rapidly as m increases from zero
[10
,
14]
. This can be understood by the decrease in the freedom of mixing between optical frequency components in DWDM systems. As a result, the first term of (16) will be dominant as long as 
S
_{0}
(
t
)>>
S_{m}
(
t
) for
m
≠0 such that
If we use
ψ
_{0}
(
t
) for a single mark signal transmission, it means
V_{m}
(0)=0 for
m
≠0.
ψ
_{0}
(
t
) has a single maximum and the corresponding lowestorder eigenfunction in the spectral domain
ϕ
_{0}
(
ω
) occupies the smallest bandwidth of all eigenfunctions. We can prove that, in this case, the signaltonoise ratio is maximized at
t
= 0
[14]
and the decision time should be chosen here to have the voltage
In DWDM systems, (18) may be approximated as
It is straightforward that the probability distribution function (pdf) of
y
(0) from (19) has a noncentral chisquare distribution
[17
,
18]
. As 
t
 increases from zero, higherorder receiver eigenmode contributions to
y
(
t
) increase from the relation
S_{m}
(
t
)=
S
_{0}
(0)
C
_{0m}
(
t
). At the same time, the magnitude of
S
_{0}
(
t
) decreases since the total sum of 
S_{m}
(
t
)
^{2}
is conserved as
Therefore, according to the centrallimit theorem
[19]
, the pdf of
y
(
t
) becomes closer to Gaussian as the number of nonnegligible or contributing eigenmodes increases. Note that the pdf of
y
(0) becomes more symmetric also when its mean value increases. If we use pulses somewhat different from
ψ
_{0}
(
t
) to send a mark signal, normally a small number of eigenmodes would contribute to
y
(0) from the lowest order and the pdf of
y
(
t
) would resemble the Gaussian more closely. These behaviors can be explained more quantitatively using Gaussian optical receivers.
III. ANALYSIS USING GAUSSIAN OPTICAL RECEIVERS
We use Gaussian optical receivers
[14]
for our timedependent analysis since they are good approximations to various optical receivers and have closed form eigenfunctions. We will show that, when the received optical pulse is a Gaussian pulse, the electrical pulse after the electrical filter is also a Gaussian pulse. Then our results will be compared with our foregoing timedependent analysis.
The optical filter of a Gaussian optical receiver has a Gaussian impulse response as
where
t_{o}
is the time delay of the optical filter.
h_{0}
(
t
) is the inverse Fourier transform of
H_{o}
(
ω
). Also, the electrical filter of a Gaussian optical receiver has a Gaussian impulse response
where
t_{e}
is the time delay of the electrical filter. The 3dB bandwidth of 
H_{o}
(
ω
)
^{2}
is
Similarly, the 3dB bandwidth of 
H_{e}
(
ω
)
^{2}
measured from the origin is
The Gaussian receiver’s eigenfunction
ϕ_{m}
(
ω
) is given as a Hermite function times exp(
jωt_{d}
) as follows:
where,
t_{d}
(=
t_{o}
+
t_{e}
) is the total time delay of the receiver.
H_{m}
(
ω
/
a
) is the Hermite polynomial. The 3dB bandwidth of 
ϕ
_{0}
(
ω
)
^{2}
is
The parameters,
a
,
α
, and
β
, are related as
α
^{2}
=
a
^{2}
(1+
q
)/(1−
q
) and
β
^{2}
=
a
^{2}
(1−
q
^{2}
)/2
q
, where
q
is a positive quantity less than 1 given by
r
= 2
α
/
β
is the 3dB bandwidth ratio of the optical and the electrical filters. Note that the Fourier transform of the Hermite function is also proportional to a Hermite function. Thus we have
The corresponding eigenvalues are given as
where
H_{c}

H_{o}
(0)
^{2}
H_{e}
(0).
We assume the received signal is a Gaussian pulse without the ASE such that
where
b
is a constant. Then its Fourier transform is
and the 3dB bandwidth of 
ε_{in}
(
ω
)
^{2}
is
When
b
=
a
,
E_{in}
(
t
) is proportional to
ψ
_{0}
(
t
). From (7), we find
In particular, we have for
m
=0
The
E_{out}
(
t
) after the optical filter is found as
The photodetector output current is (
k
/2)
E_{out}
(
t
)
^{2}
. The output voltage of the electrical filter can be evaluated exactly as follows:
The output waveform of the Gaussian receiver for the Gaussian input (25) is also a Gaussian with its maximum at
t
= 0.
From (5), the output of the Gaussian receiver has the following expression:
Comparing (30) with (31), we can find the receiver eigenmode contributions at arbitrary times. For example, setting
t
=0, where the output voltage of the Gaussian receiver is maximized, we may expand
y
(0) obtained from (30) as a Taylor series about the
q
=0 point except the
factor
Only the even eigenmodes contribute and the lowestorder eigenmode (
m
= 0) contribution becomes dominant when
q
is much less than 1. In particular, when b=a, only the lowestorder mode term is present at
y
(0) independent of the
q
value.
We can find the lowestorder mode contribution to
y
(
t
), denoted as
y_{L}
(
t
), after setting
q
=0 except the
factor,
The ratio between
y_{L}
(
t
) and
y
(
t
) is
This ratio has a maximum at
t
= 0 where the signal is also maximized. In other words, the contribution of the lowestorder eigenmode is maximized at the pulse center
t
= 0. It decreases as 
t
 departs from zero and higherorder eigenmode contributions increase. The lowestorder eigenmode contribution also decreases as
q
increases. When
b
=
a
, we have
y_{L}
(
t
)/
y
(
t
) = exp(
qa
^{2}
t
^{2}
/2). There are no higherorder eigenmode contributions at
t
= 0 irrespective of
q
or
r
.
The correlation functions for the Gaussian optical receiver can be derived as
where
is the associated Laguerre polynomials
[20]
. Especially, when
k
= 0, we have
When
b
=
a
, (30) reduces to
Comparing (38) with (31), we find
This result matches with (37).
In
Fig. 1
(a), we plot
y
(
t
) and
y_{L}
(
t
) for
r
= 2 (
q
= 0.268) for several
b
values.
y
(
t
) and
y_{L}
(
t
) are normalized by
y_{dc}
which is the dc response of
y
(
t
) with
b
=0. Between the two traces for a given b value, the lower one corresponds to
y_{L}
(
t
). At
b
=0, the signal is unmodulated and about 4% of
y
(
t
) is the contributions from higherorder eigenmodes. As
b
increases from zero to
a
, the higherorder eigenmode contributions decrease to zero at
t
=0, where the signal power is maximized. When
b
=
a
, only the lowestorder eigenmode contributes or
y
(0)=
y_{L}
(0) As
b
increases further from
a
, the contributions of higherorder eigenmodes increase at
t
=0 but more slowly. In
Fig. 1
(b), we use
r
=4 (
q
=0.5). At
The traces of y(t) and y_{L}(t) normalized by y_{dc} which is the dc response of y(t) with b=0. y_{L}(t) is obtained from y(t) neglecting all the eigenmode contributions except m=0. For each value of b, the upper trace is y(t)/y_{dc} and the lower trace is y_{L}(t)/y_{dc}. (a) r=2 (q=0.268). (b) r=4 (q=0.5).
b
=0, about 13% of
y
(
t
) belongs to higher order eigenmodes. Overall, the higherorder eigenmode contributions are larger than
Fig. 1
(a). When
b
=
a
, only the lowestorder eigenmode contributes at
t
=0 as
Fig. 1
(a).
In
Fig. 2
(a), we show the pdfs of
ȳ
(
t
)=
y
(
t
) /
y_{sp}
for
b
=
a
and for
r
=2 (
q
=0.268) at
t
=0 and
T
/4.
T
is the bit period chosen to be 100 ps assuming the bit rate as 10 Gb/s.
y_{sp}
is the average value of
y
(
t
) per polarization in the absence of the signal given as
We have set
A
_{0}
(0)=81.3 and
A
_{m≠0}
that gives 10
^{9}
biterror rate at
t
=0
[14]
. The pdfs are found as an inverse Fourier transform of the characteristic function of
y
(
t
)
[10]
. We have used 30 eigenmodes in the full eigenmode analysis (FEA) and used only the lowestorder eigenmode for the singleeigenmode analysis (SEA). We have chosen
as 7 GHz. At
t
=0, the FEA and the SEA give almost the same pdfs, which means the effects from {
N
_{m≠0}
(0)} are negligible. At
t
=
T
/4, the FEA gives more symmetric pdf with respect to its peak than that of the SEA. Note that the pdf at
t
=0 is more symmetric than the pdf obtained by the FEA at
t
=
T
/4. We show in
Fig. 2
(b) the pdfs of
ȳ
(
t
) for
r
=4 (
q
=0.5) with all other conditions the same as
Fig. 2
(a). In this case, as
Fig. 1
shows, higher order eigenmodes become more important at
t
=
T
/4 than
Fig. 2
(a). At
t
=0, the FEA and the SEA still give almost the same pdfs. At
t
=
T
/4, the FEA gives more symmetric pdf than the SEA. The pdf at
t
=0 has a similar symmetry with respect to its peak as the pdf obtained by the FEA at
t
=
T
/4.
Probability density functions of ȳ(t)=y(t)/y_{sp} at t=0 and at t=T/4, where T is the bit period which is equal to 100 ps. FEA: fulleigenmode analysis using 30 receiver eigenmodes. SEA: singleeigenmode analysis using the lowestorder receiver eigenmode. (a) r=2 (q=0.268). (b) r=4 (q=0.5).
IV. CONCLUSION
We have analyzed the timedependent behaviors of optical receivers corrupted by the ASE using receiver eigenmodes. The correlation functions between receiver eigenmodes in time domain determine the amplitudes of eigenmodes at different times. Gaussian optical receivers are used for our analysis that have Gaussian optical and electrical filters with closedform receiver eigenfunctions. The edge parts of the Gaussian electrical pulse have larger contributions of higherorder eigenmodes than the pulse center where the lowestorder eigenmode is dominant. This effect is dependent on the bandwidth ratio between the optical and the electrical filters as well as the input pulse’s time width. It is enhanced when the timewidth of the received Gaussian optical pulse decreases or when the bandwidth ratio increases. According to the central limit theorem, as the number of contributing eigenmodes increase, the voltage distribution of the electrical pulse at that instant becomes more symmetric and close to Gaussian. Our analysis explains the timedependent signal and noise properties within the electrical pulses at the output of the receiver electrical filter. For Gaussian optical receivers, we have shown that the best decision time is at the peak of the electrical pulse. For other cases of nonGaussian filters and of nonGaussian input pulses, our analysis can be used also to get the best decision timing.
Acknowledgements
This research was supported partly by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2013R1A1A2012918) and also partly supported by the Research Grant of Kwangwoon University in 2012.
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