904
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 9, NO. 7, JULY 1997
Experimental Study of the Statistical
Properties of Nonlinearly Amplified Signals
in Semiconductor Optical Amplifiers
M. Shtaif and G. Eisenstein, Senior Member, IEEE
Abstract— In this letter, we examine the effect of nonlinear
amplification on the noise properties of an amplified signal.
We show that the output noise statistics can be approximated
as Gaussian over a wide range of practical parameters. In
addition, we show that nonlinear amplification of noise carrying
signals may result in an enhancement of the signal-to-noise ratio
(SNR). Basic principles dictate that this can not improve the bit
error rate performance of a communication link, as long as an
optimal detection scheme is used. Hence, we conclude that for
communication systems that exploit nonlinear amplification in
semiconductor optical amplifiers, evaluation of performance on
the basis of the SNR can be misleading and should be made with
great care.
recovery time of the amplifier and is of the order of several
GHz. The narrow-band intensity noise compression as well as
the frequency dependence of the PDS were recently confirmed
experimentally and reported recently [6].
In this letter, we report experimental examinations of nonlinear amplification for noise carrying input signals. Two issues
are addressed: First the effect of nonlinear amplification on the
signal-to-noise ratio (SNR) of an injected signal is discussed
in Section II. Second, the effect of nonlinear amplification on
the statistics of the amplified signal is examined in Section III.
Index Terms— Amplifier noise, noise distribution, nonlinear
optical amplifier.
II. THE EFFECT OF NONLINEAR AMPLIFICATION ON THE SNR
I. INTRODUCTION
T
HE NOISE properties of linear semiconductor optical
amplifiers have been studied for many years [1]–[3]. It
is well known that in the linear regime, when the number of
detected photons is large, the output noise can be described
semiclassical in terms of an additive white Gaussian process
noise. The power density spectrum (PDS) of that additive
noise is given by the expression
with being
the optical frequency,
the inversion factor [1] and
the amplifier gain. This simple description is not valid for
nonlinear amplification where the noise properties are modified
due to two effects [4]. First, the gain which is signal dependent
in the nonlinear regime varies along the amplifier axis as do the
carrier density and the inversion factor [4]–[5], all affecting the
noise. Second, and more important, the saturating signal and
the amplifier noise interact nonlinearly as they propagate along
the amplifier axis. This interaction correlates different spectral
components of the noise so that it can no longer be white.
Both these effects have been analyzed in [4] where we have
shown analytically that the nonlinear interaction between the
noise and the signal results in a significant suppression of the
intensity noise over some narrow frequency band. This effect
manifests itself as a dip around zero frequency in the PDS
function of the electronically detected signal. The bandwidth
of this noise suppression mechanism depends on the gain
Manuscript received December 2, 1996; revised March 28, 1997.
The authors are with the Department of Electrical Engineering, Barbara and
Norman Seiden Advanced Optoelectronics Center, Technion, Haifa 32000,
Israel.
Publisher Item Identifier S 1041-1135(97)05020-9.
In this section, we examine the effect of nonlinear amplification on the SNR of the amplified signal. Unlike for additive
Gaussian channels [7], in the case of optical communication
systems the detected signals that corresponds to a transmitted
“zero” or a “one” differ not only in their mean value but also
in the probability distributions. For this reason the definition of
SNR in optical systems is not unique. In this paper we adopt
a commonly used definition according to which the SNR is
the ratio between the square of the mean value of the detected
signal and its variance in the case in which the detected signal
consists only of “ones” [5]. This is an easily measurable
parameter on the basis of which performance evaluation of
systems is frequently made [8].
We treat the general case of an injected signal that may carry
noise. Unlike the situation of a noiseless input signal (treated in
[6]), where the output noise properties are determined only by
the amplified spontaneous emission (ASE), here the incident
noise and its interaction with the signal along the amplifier
play an important role in dictating the noise properties at the
output.
While the addition of the ASE increases the noise power
and reduces the SNR, the nonlinear interaction between the
signal and the incident noise tends to suppress the intensity
fluctuations over some narrow frequency band. This effect
results from the fact that the amplifier gain is a monotonically
decreasing function of the incident intensity, so that when the
intensity increases (decreases) due to noise, the gain is reduced
(increased) thereby suppressing the intensity fluctuations [4],
[6]. The efficiency of this effect is limited to a bandwidth
in which the intensity variations are sufficiently slow for the
amplifier gain to adjust as explained in [4] and demonstrated
in [6]. Consequently, in those cases where the added ASE
1041–1135/97$10.00  1997 IEEE
SHTAIF et al.: EXPERIMENTAL STUDY OF THE STATISTICAL PROPERTIES
Fig. 1. Experimental setup.
power at the output is negligible relative to the noise due to the
incident noise, the overall effect is of a decrease in the narrowband intensity noise power and an increase in the SNR of
the amplified signal relative to the SNR before amplification.
Since it is well known that no amplification process can
reduce the error probabilities in optimal detection schemes
[7], the enhancement of the SNR can not be translated into
a corresponding improvement in the best obtainable bit-errorrate (BER) performance of an optical communication system.
This means that in systems exploiting nonlinear amplification,
the evaluation of the performance on the basis of the SNR
may cause significant mistakes and should be done with great
care.
To demonstrate the enhancement of SNR we use the experimental setup described in Fig. 1. The noise carrying input
signal was obtained from a DFB laser whose output was
attenuated and then amplified by a 30-dB EDFA fiber amplifier
followed by a 1-nm-wide optical filter. The 1550-nm nonlinear
semiconductor optical amplifier had a small signal gain of 26
dB, a 3-dB output saturation power of 8 mW and mirror
reflectivities lower than below 0.01%. The output of the
semiconductor optical amplifier was filtered using a second
1-nm-wide filter and coupled to an ac-coupled receiver, which
fed an RF spectrum analyzer. The fiber amplifier operated in
the linear regime so that its noise contribution can be properly
described in terms of an additive white Gaussian noise.
The effect of nonlinear amplification on the SNR of the
amplified signal is demonstrated in Fig. 2. It is based on an
examination, at the amplifier output, of both the detected PDS
and the optical spectrum, each for two input levels (with the
same input SNR) yielding two levels of gain compression:
13.5 and 4 dB. Fig. 2(a) describes the ratio between the PDS
of the detected output signal and the PDS of the input signal,
which is white. The absolute optical powers of the respective
signals were adjusted to be identical before detection and
the electronic noise floor was subtracted from each measured
spectrum. The spectra were measured in the frequency range
of 0.5–2 GHz. Fig. 2(a) reveals, a frequency dependent normalized PDS (defined as the output PDS divided by the input
PDS, which is white) ratio with a low-frequency dip for both
levels of saturation, consistent with the results described in
[6]. The broken lines describe a fit to a theoretical prediction
based on the model presented in [4]. The main result shown
in the figure is that the values of the normalized PDS differ in
the two cases, however. For deep saturation (
13.5
dB), it is well below unity, indicating a significant noise
compression and consequently an enhancement of the SNR.
905
(a)
(b)
Fig. 2. (a) The measured power density spectrum at the output of the
amplifier normalized the measured spectrum of the input signal, for two levels
of gain saturation and (b) The respective optical spectra at the semiconductor
amplifier output, measured before the 1 nm output filter.
The optical spectrum, measured before the 1 nm output optical
filter, is shown in Fig. 2(b). We note that the added amplified
spontaneous emission (ASE) is negligible and that the total
output noise is completely dominated by the noise originating
from the input signal. The most important process taking
place is then the nonlinear interaction between the input
noise and the saturating signal as they propagate along the
amplifier. This interaction that compresses the low-frequency
noise at the output of the amplifier as predicted in [4]. In
contrast, the moderate saturation case (
4 dB) yields a
normalized PDS level which is larger than unity indicating
an increase in output noise due to the added ASE. This
is also seen in the optical spectrum, Fig. 2(b), where the
added ASE noise dominates. The nonlinear interaction, while
obviously taking place between the signal, the input noise and
the ASE is less significant, resulting in a degraded output
SNR.
III. THE EFFECT OF NONLINEAR AMPLIFICATION
ON THE OUTPUT NOISE STATISTICS
The evaluation of the nonlinearly amplified noise statistics
is very important for the estimation of system performance
when nonlinear amplification is exploited. When an amplifier
is operated in the nonlinear regime, the statistics of the output
electric field does not, in principle, stay Gaussian. The degree
to which the statistics deviate from Gaussian depends on the
intensity of the nonlinear mechanism and is determined by
three main factors. The first is the SNR. When the SNR is large
so that the amplitude of the noise is much lower than that of the
signal, the effect of the noise on the gain of the amplifier can be
considered as a small perturbation of steady state conditions.
In such cases, the effect of the amplifier on the noise is almost
linear and the deviation from Gaussian statistics is small.
The second factor is the electronic bandwidth of the receiver.
Due to the narrow frequency band of the nonlinear effect we
consider, increasing the electronic receiver bandwidth results
in a decrease of the relative part of the intensity noise that is
affected by the nonlinear mechanism and the deviation from
Gaussian statistics is reduced. The third factor is the level of
gain saturation. Unlike the first two factors, its effect on the
intensity of the nonlinear mechanism is not monotone. At very
low levels of gain saturation, the nonlinearity is negligible. At
moderate and large saturation levels the effect is strong but it
906
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 9, NO. 7, JULY 1997
(a)
(b)
Fig. 3. The measured BER curves as a function of the decision threshold.
The transmitted sequence was assumed to consist of an infinitely long
sequence of “zeros.”
reduces in very deep saturation because the response of the
amplifier gain to fluctuation in the optical intensity becomes
small.
Our goal is to estimate the range of applications in which
the deviation from Gaussian statistics is negligible. Clearly, it
is impossible to measure the complete statistical properties of
the amplified signal. However, for all practical applications in
optical communication, the interesting parameter is the first
order distribution of samples of the detected signal which
are taken after matched electronic filtering whose bandwidth
equals the bandwidth of the incident data stream [7]. In cases
where the deviation of the optical field from Gaussian statistics
is negligible, the distribution of the detected signal samples
is noncentral chi square, since they are proportional to the
optical intensity [9] (for reasonably high values of the SNR,
the difference between the noncentral chi square and Gaussian
distributions is negligible [10]). To measure this distribution,
we coupled the signal whose RF spectrum is described in
Fig. 2(a) (which is the output intensity noise in the range
of 0.5–2 GHz) into the receiver unit of a bit-error-rate test
set (BERT). The receiver unit contains both the electronic
filtering and the sampler. Since the signal fed into the BERT
system does not contain the dc component, it is operated as if
the detected signal corresponds to an infinitely long sequence
of “zeros,” whose bit rate has been arbitrarily defined as 1
Gb/s. The measured BER in this case was the relative number
of events in which the measured voltage crossed a given
decision threshold level,
. The BER versus
function so
, where
is the cumulative
obtained, equals to
probability function defined as the probability that the voltage
that corresponds to a sample of the detected signal is lower
than
. The derivative of
with respect to the threshold voltage renders, by definition, the
probability density function of the measured signal samples
[11]. At this point we should clarify that the BERT served
only as a sampler and an accurate, variable thresholding tool.
The BER-like curves versus threshold voltage generated in the
experiments have no direct connection to data transmission or
detection.
Fig. 3(a) and (b) describe these measured BER curves
for the two levels of gain compression. The solid curves
correspond to the signal measured at the output of the semi-
conductor amplifier while the dashed curves correspond to the
input signals. The dashed dotted curves describe the reference
distribution determined by the electronic circuits with no
incident light. For each saturation level, the input and output
intensities were adjusted to identical levels before they were
coupled into the photodetector. In the deep saturation case,
13.5 dB the output probability distribution function is
narrower than the corresponding input distribution everywhere,
including the tails. In the moderate saturation case,
4
dB, the output distribution is measurably wider than the input
distribution due to the dominance of the added ASE. However,
the most important result is that, apart from a certain scaling
factor there is no noticeable functional difference between the
distribution curves that correspond to the signal before and
after the nonlinear amplification process. In fact, all curves
can be fit by the noncentral chi square distribution function
and the scaling difference between the curves is consistent with
the difference in the respective SNR’s described in Fig. 2. This
implies that for all practical purposes in optical communication
(in the relevant range of error rates 10 ), the distribution
of the amplified optical field can be efficiently approximated
as Gaussian.
Obviously, the above conclusion is based on the specific
setup used in the presented experiment. However, a more
general statement can be made regarding configurations in
which the efficiency of the nonlinear mechanism is larger
than in the experiment presented above. This means that the
output noise statistics can be approximated as Gaussian in
all applications in which the bit rates are larger than 1 Gb/s,
where the incident noise results from linear preamplification
with gain of less than 30 dB and where the level of gain
saturation is lower than 13.5 dB.
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