Upload
pranab-kumar-bandyopadhyay
View
3.894
Download
10
Tags:
Embed Size (px)
DESCRIPTION
This text tries to give a brief ideal about the SOA, its realization based on matlab simulation with the reservoir model and cross-gain modulation
Citation preview
Performance of
Semiconductor Optical Amplifier
A report submitted for the partial fulfilment of the 4th
year syllabus of the four
year B.tech. course under West Bengal University of Technology
by
Pranab Kumar Bandyopadhyay (univertsy roll no : 071690103020)
Md. Taushif (univertsy roll no : 071690103039)
Samadrita Bhattacharyya (univertsy roll no : 071690103040)
Sanghamitra Bhattacharjee (univertsy roll no : 071690103046)
Prakash Kumar (univertsy roll no : 071690102033)
Acknowledgement
i
It is a pleasure to thank the many people who made this project work possible
for us. It is difficult to overstate our gratitude to our guide, Prof. Suranjana
Banerjee, Lecturer, Dept. of Electronics & Communication, Academy Of
Technology. With her enthusiasm, her inspiration and her great efforts to
explain things simply and clearly, she has helped to make this project work
convenient for us. Throughout my project work period, she provided
encouragement, sound advice, good teaching, good company and lots of good
ideas. We would have been lost without her.
We would like to thank our director Prof. Santu Sarkar, Head of The Dept.
Electronics & Communication Engg., Academy Of Technology, for giving us an
opportunity to carry out the project work here. We are indebted to our teachers
for providing a stimulating and challenging environment in which to learn and
grow.
Last, but by no means least, we thank our friends for their support and
encouragement throughout.
Date:-
Signature of students
Certificate by the Supervisor
ii
This is to certify that this technical report “Performance of Semiconductor Optical Amplifier” is a record
of work done by Pranab Kumar Bandyopadhyay, Md. Taushif, Samadrita Bhattacharyya, Sanghamitra
Bhattacharjee & Prakash Kumar, during the time from August 2010 to April 2011as a partial fulfillment
of the requirement of the final year project at Academy of Technology, affiliated under West Bengal
University of Technology.
These candidates have completed the total parameters and requirement of the entire project.
This project has not been submitted in any other examination and does not from a part of any other
course undergone by the candidates.
______________________________
(Prof. Suranjana Banerjee)
Lecturer,
Dept. of Electronics & Communication Engineering,
Academy of Technology,
West Bengal
Preface
iii
In this report, we are going to discuss, simulate and realize an popularly know optical
amplifier, the SOA. SOAs have been in use for the purpose of cheap, reliable and
environment suitable optical amplifiers in the field of long distance optical communication.
In the practical field, where the distance between the two successive optical amplifiers are
more than 100 km , SOAs have been very useful to provide a low maintenance, low cost and
less fragile system for signal boosting.
Our report on the project continuous to discuss on the performance of SOA on the aspect
of gain, cross-gain modulation & BER as well as power penalty for the system comprising of
a WDM ring network.
All the necessary theories to derive or to simulate the SOA features are tried to be
described on the following chapter.
With a grateful heart we are expressing our feelings of gratude to our respected teacher
Prof. Mrs. Suranjana Banerjee for her kind help and guide to us in the simulation throught
the span of the project, without which this work was almost impossible.
index
Chapter no. Topic Page no.
1 Introduction 1
2 History 4
3 Why SOA? 5
4 Basic Principle 10
5 Fundamental device characteristics & Materials used in SOA 15
6 Modelling of SOA 21
7 Cross-gain modulation 46
8 Work done 51
9 Power penalty & BER in SOA receiver 88
10 Summary 94
11 Bibliography 95
Introduction Chapter1
1
Communications can be broadly defined as the
transfer of information from one point to
another. In optical fiber communications, this
transfer is achieved by using light as the
information carrier. There has been an
exponential growth in the deployment and
capacity of optical fiber communication
technologies and networks over the past
twenty-five years. This growth has been made
possible by the development of new
optoelectronic technologies that can be
utilized to exploit the enormous potential
bandwidth of optical fiber. Today, systems are
operational which operate at aggregate bit
rates in excess of 100 Gb/s. Such high
capacity systems exploit the optical fiber
bandwidth by employing wavelength division
multiplexing.
Optical technology is the dominant carrier of
global information. It is also central to the
realization of future networks that will have
the capabilities demanded by society. These
capabilities include virtually unlimited
bandwidth to carry communication services of
almost any kind, and full transparency that
allows terminal upgrades in capacity and
flexible routing of channels. Many of the
advances in optical networks have been made
possible by the advent of the optical amplifier.
In general, optical amplifiers can be divided
into two classes: optical fiber amplifiers and
semiconductor amplifiers. The former has
tended to dominate conventional system
applications such as in-line amplification used
to compensate for fiber losses. However, due
to advances in optical semiconductor
fabrication techniques and device design,
especially over the last five years, the
semiconductor optical amplifier (SOA) is
showing great promise for use in evolving
optical communication networks. It can be
utilized as a general gain unit but also has
many functional applications including an
optical switch, modulator and wavelength
converter. These functions, where there is no
conversion of optical signals into the electrical
domain, are required in transparent optical
networks.
In this chapter we begin with the reasons why
optical amplification is required in optical
communication networks. This is followed by a
brief history of semiconductor optical amplifiers
(SOAs), a summary of the applications of SOAs
and a comparison between SOAs and optical
fiber amplifiers (OFAs).
WHY WE NEED OPTICAL
AMPLIFICATION? :-
Optical fiber suffers from two principal limiting
factors: Attenuation and dispersion. Attenuation
leads to signal power loss, which limits
transmission distance. Dispersion causes optical
pulse broadening and hence inter symbol
interference leading to an increase in the system
bit error rate (BER). Dispersion essentially
limits the fiber bandwidth. The attenuation
spectrum of conventional single-mode silica
fiber, shown in Fig. 1.1, has a minimum in the
1.55 µm wavelength region. The attenuation is
somewhat higher in the 1.3 µm region. The
dispersion spectrum of conventional single-
mode silica fiber, shown in Fig. 1.2, has a
minimum in the 1.3 µm region. Because the
attenuation and material dispersion minima are
located in the 1.55 µm and 1.3 µm ‘windows’,
these are the main wavelength regions used in
commercial optical fiber communication
systems. Because signal attenuation and
dispersion increases as the fiber length increases,
at some point in an optical fiber communication
link the optical signal will need to be
regenerated. 3R (reshaping-retiming-
retransmission).Regeneration involves detection
(photon-electron conversion), electrical
amplification, retiming, pulse shaping and
retransmission (electron-photon conversion).
2
Fig 1.1: Typical attenuation spectrum of low-
loss single-mode silica optical fiber.
3
This method has some disadvantages- ►Firstly, it involves breaking the optical link and so is not optically transparent.
►Secondly, the regeneration process is dependent on the signal modulation format and bit rate and so is not electrically transparent. This in turn creates difficulties if the link needs to be upgraded. Ideally link upgrades should only involve changes in or replacement of terminal equipment (transmitter or receiver).
►Thirdly, as regenerators are complex systems and often situated in remote or difficult to access location, as is the case in undersea transmission links, network reliability is impaired. In systems where fiber loss is the limiting factor, an in-line optical amplifier can be used instead of a regenerator. As the in-line amplifier has only to carry out one function (amplification of the input signal) compared to full regeneration, it is intrinsically more reliable and less expensive device. Ideally an in-line optical amplifier should be compatible with single-mode fiber, impart large gain and be optically transparent (i.e. independent of the input optical signal properties). In addition optical amplifiers can also be useful as power boosters, for example to compensate for splitting losses in optical distribution networks, and as optical preamplifiers to
improve receiver sensitivity. Besides these basic system applications optical amplifiers are also useful as generic optical gain blocks for use in larger optical systems. The improvements in optical communication networks realized through the use of optical amplifiers provides new opportunities to exploit the fiber bandwidth. There are two types of optical amplifier: The SOA and the OFA. In recent times the latter has dominated; however SOAs have attracted
renewed interest for use as basic amplifiers and also as functional elements in optical communication networks and optical signal processing devices.
HISTORY Chapter2
4
The first studies on SOAs were carried out around the time of the invention of the semiconductor laser in
the 1960’s. These early devices were based on GaAs homo-junctions operating at low temperatures. The arrival of double hetero-structure devices spurred further investigation into the use of SOAs in optical
communication systems. In the 1970’s Zeidler and Personick carried out early work on SOAs. In the
1980’s there were further important advances on SOA device design and modeling. Early studies concentrated on AlGaAs SOAs operating in the 830 nm range. In the late 1980’s studies on InP/InGaAsP
SOAs designed to operate in the 1.3 µm and 1.55 µm regions began to appear.
Developments in anti-reflection coating technology enabled the fabrication of true travelling-wave SOAs.
Prior to 1989, SOA structures were based on anti-reflection coated semiconductor laser diodes. These devices had an asymmetrical waveguide structure leading to strongly polarization sensitive gain.
In 1989 SOAs began to be designed as devices in their own right, with the use of more symmetrical waveguide structures giving much reduced polarization sensitivities. Since then SOA design and
development has progressed in tandem with advances in semiconductor materials, device fabrication,
antireflection coating technology, packaging and photonic integrated circuits, to the point where reliable
cost competitive devices are now available for use in commercial optical communication systems. Developments in SOA technology are ongoing with particular interest in functional applications such as
photonic switching and wavelength conversion. The use of SOAs in photonic integrated circuits (PICs) is
also attracting much research interest.
WHY SOA? Chapter3
5
As optical technology has become an integral
part of telecommunications, the need for reliable
optical signal transmission has become more and
more pronounced. In order to transmit over long
distances, it is necessary to account for
attenuation losses. Initially, this was done
through an expensive conversion from optical to
electrical and back. This was soon remedied
with the creation of optical amplifiers.
The optical amplifiers we have today are
1.EDFA.
2. SOA.
3. LOA.
One of the first widely adopted optical
amplifiers was the Erbium Doped Fiber
Amplifier (EDFA). This revolutionized the
optical communications industry. The next big
step in optical amplifiers came with
semiconductor optical amplifiers (SOA).
Although these didn’t perform as well as the
EDFAs in some conditions, they had many
advantages including smaller size and the ability
to easily integrate with semiconductor lasers.
The latest step in semiconductor amplifiers came
with the introduction of a SOA that operated as a
linear amplifier (LOA). Thus far this has
eliminated many of the downfalls of SOAs such
as cross talk and high signal to noise ratio.
1. EDFA: Erbium doped fiber amplifiers are
commonly used optical amplifier. An EDFA consists of a pump laser coupled to an input
signal and passed through an optical fiber
slightly doped with erbium ions. The pump laser is used to excite erbium ions which emit photons
in phase with the input signal which acts to
amplify it. EDFA’s amplify in the 1520-1600
nm range which corresponds to the energy difference between the excited and ground states
of the erbium ions.
2. SOA: The semiconductor optical amplifier
is an amplifier with a laser diode structure that is
used to amplify optical signals passing through its optical region. Amplification occurs through
stimulated emission in the active region as input
6
signal energy propagates through the wave
guide. This can be seen below
3. LOA: The linear
optical amplifier (LOA) is actually a SOA with an integrated
vertical cavity surface emitting laser (VCSEL).
The amplifier and the VCSEL share the same active region, which causes the VCSEL to act as
a feedback device, preventing carrier depletion
even when the input power varies. This can be seen in Figure
Why SOA is better?
1. In the practical applications in the rigorous
field of the industry, it is
easier to use SOA, because it
uses direct electrical drive current as its energy pump
that is more robust in
structure than the laser as used as the energy pump in
EDFA.
2.The switching
characteristics of EDFA is not
very good. SOAs & LOAs
show better switching
properties under continuous
on& on signal. SOA are seen to be tolerant upto
a switching speed varying from 0.5 to 5 GHz.
7
3. The
Bit-error
rate characteristics of the SOAs are much better
than the EDFA. In the EDFA, the BER
progressively gets worse from
channel to
channel, which is
unlikely in SOAs. SOAs can operate at the
lowest Bi- error rate of 10-15.
8
4. One of the main
drawbacks of SOA
devices is the need for
9
polarization matching. The
polarization of the incident
laser must match the
polarization of the
semiconductor.
From the above
discussion we can be sure to
choose SOA instead of the of
the other device, i.e. EDFA or
LOA.
Basic Principle Chapter 4
10
An SOA is an optoelectronic device that
under suitable operating conditions can
amplify an input light signal. A schematic
diagram of a basic SOA is shown in Fig. 2.1.
The active region in
the device imparts
gain to an input
signal. An external
electric current
provides the energy
source that enables
gain to take place.
An embedded waveguide
is used to confine the
propagating signal wave to the active region.
However, the optical confinement is weak so
some of the signal will leak into the
surrounding lossy cladding regions. The output
signal is accompanied by noise. This additive
noise is produced by the amplification process
itself and so cannot be entirely avoided. The
amplifier facets are reflective causing ripples
in the gain spectrum.
SOAs can
be classified
into two main
types shown
in Fig. 4.02:
The Fabry-
Perot SOA
(FP-SOA)
where
reflections
from the end
facets are
significant(i.e.
the signal
undergoes
many passes
through the
amplifier) and
the travelling-
wave SOA
(TW-SOA)
where
reflections are negligible (i.e. the signal
undergoes a single-pass of the amplifier).
Anti-reflection coatings can be used to create
SOAs with facet reflectivities <10-5
.The TW-
SOA is not as sensitive as the
FP-SOA to fluctuations in
bias current, temperature and
signal polarisation.
Principles of Optical
Amplification:-
In an SOA electrons (more commonly
referred to as carriers) are injected from an
external current source into the active region.
These energised region material, leaving holes
in the valence band (VB). Three radiative
mechanisms are possible in the semiconductor.
These are shown in Fig 2.3 for a material with
an energy band structure consisting of two
discrete energy levels.
11
In stimulated absorption an incident light
photon of sufficient energy can stimulate a
carrier from the
VB to the CB.
This is a loss
process as the
incident photon
is
extinguished.
If a photon
of light of
suitable energy
is incident on
the
semiconductor,
it can cause
stimulated
recombination
of a CB carrier
with a VB hole.
The recombining carrier loses its energy in the
form of a photon of light. This new stimulated
photon will be identical in all respects to the
inducing photon (identical phase, frequency
and direction, i.e. a coherent interaction). Both
the original photon and stimulated photon can
give rise to more stimulated transitions. If the
injected current is sufficiently high then a
population inversion is created when the
carrier population in the CB exceeds that in the
VB. In this case the likelihood of stimulated
emission is greater than stimulated absorption
and so semiconductor will exhibit optical gain.
In the spontaneous emission process, there
is a non-zero probability per unit time that a
CB carrier will spontaneously recombine with
a VB hole and thereby emit a photon with
random phase and direction. Spontaneously
emitted photons have a wide range of
frequencies. Spontaneously emitted photons
are essentially noise and also take part in
reducing the carrier population available for
optical gain. Spontaneous emission is a direct
consequence of the amplification process and
cannot be avoided; hence a noiseless SOA
cannot be created. Stimulated processes are
proportional to the intensity of the inducing
radiation whereas the spontaneous emission
process is
independent of
it.
Spontaneous and induced transitions:-
The gain properties of optical
semiconductors are directly related to the
processes of spontaneous and stimulated
emission. To quantify this relationship we
consider a system of energy levels associated
with a particular physical system. Let N1 and
N2 be the average number of atoms per unit
volume of the system characterised by the
average number of atoms by energies E1 and
E2 respectively, with E2 > E1 .If a particular
atom has energy E2 then there is a finite
probability per unit time that it will undergo a
transition from E2 to E1 and in the process emit
a photon. The spontaneous carrier transition
rate per unit time from level 2 to level 1 is
given by
where A21 is the spontaneous emission
parameter of the level 2 to level 1 transition.
Along with spontaneous emission it is also
possible to have induced transitions. The
4.1
12
induced carrier transition rate from level 2 to
level 1 (stimulated emission) is given by
where B21 is the stimulated emission
parameter of the level 2 to level 1 transition
and ρ(v) the incident radiation energy density
at frequency v. The induced photons have
energy hv = E2 – E1 The induced transition
rate from level 1 to level 2 (stimulated
absorption) is given by
where B12 is the stimulated emission
parameter of the level 2 to level 1 transition. It
can be proved, from quantum-mechanical
considerations [1,2], that
B12 = B21
where ηr is the material refractive index
and the speed of light in a vacuum. Inserting
(4.5) into (4.2) gives
In the case where the inducing radiation is
monochromatic at frequency v, then the
induced transition rate from level 2 to level 1
is
where ρv is the energy density (T/m3) of the
electromagnetic field inducing the transition
and l(v) is the transition lineshape function,
normalised such that
l(v)dv is the probability that a particular
spontaneous emission event from is level 2 to
level 1 will result in a photon with a frequency
between v and v+dv. The inducing field
intensity (w/m3) is
So (4.7) becomes
Absorption and amplification :-
By using the expression for the stimulated
transition rates developed in previously, it is
now possible to derive an equation for the
optical gain coefficient for a two level system.
We consider the case of a monochromatic
plane wave propagating in the z-direction
through a gain medium with cross-section area
A and elemental length dz. The net power dPv
generated by a volume Adz of the material is
simply the difference in the induced transition
rates between the levels multiplied by the
transition energy hv and the elemental volume
i.e.
This radiation is added coherently to the
propagating wave. This process of
amplification can then be described by the
differential equation
gm(v) is the material gain coefficient given
by
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.9
4.11
4.12
4.13
13
(4.13) implies that to achieve positive gain
a population inversion (N2 > N1) must exist
between level 2 and level 1. It also shows, by
the presence of A21, that the process of optical
gain is always accompanied by spontaneous
emission, i.e. noise.
Spontaneous emission noise :-
As shown above, spontaneous emission is a
direct consequence of the amplification
process. In this section an expression is
derived for the noise power generated by an
optical
amplifier. We
consider the
arrangement of
Fig. 4.4, which
shows an input
monochromatic
signal of
frequency v
travelling
through a gain
medium having
the energy level
structure of Fig
4.03. A
polariser and
optical filter of
bandwidth B0
centred about v
are placed
before the
detector. The
input beam
is focussed
such that its waist occupies the gain medium.
If the beam is assumed to have a circular
cross-section with waist diameter D then the
beam divergence angle is
where λ0 is the free space wavelength. The net
change in the signal power due to coherent
amplification by an elemental length dz of the
gain medium is
A volume element, with cross-section area A
and length dz at position z, of the gain medium
spontaneously emits a noise power
This noise is emitted isotropically over a 4π
solid angle. Each spontaneously emitted
photon can exist with equal probability in one
of two mutually orthogonal polarisation states.
The polariser passes the signal, while reducing
the noise by half. Hence the total noise power
emitted by the volume element into a solid
angle dΩ and bandwidth B0 is
The smallest solid angle that can be used
without losing signal power is
4.14
4.15
4.16
4.17
14
This solid angle can be obtained by the use of
a suitably narrow output aperture. In this case
(4.17) can be rewritten as
The total beam power P (signal and noise) can
then be described by
where the spontaneous emission factor nsp is
given by
The solution of (2.20), assuming that gm is
independent of z, is
where Pm is the input signal power. If the
amplifying medium has length L then the total
output power is
where G = egmL
is the single-pass signal gain.
The amplifier additive noise power is
(4.24) shows that increasing the level of
population inversion can reduce SOA noise.
The noise can also be reduced by the use of a
narrowband optical filter.
4.18
4.19
4.20
4.21
4.22
4.23
4.23
Fundamental Device Characteristics & Materials Used in SOA
Chapter 5
15
The most common application of SOAs is
as a basic optical gain block. For such an
application, a list of the desired properties is
given in Table 2.1. The goal of most SOA
research and development is to realise these
properties in practical devices.
Table 5.01: Desirable Properties of a practical SOA
Small-signal gain and gain bandwidth
In general there are two basic gain
definitions for SOAs. The first is the intrinsic
gain G of the SOA, which is simply the ratio
of the input signal power at the input facet to
the signal power at the output facet. The
second definition is the fibre-to-fibre gain,
which includes the input and output coupling
losses. These gains are usually expressed in
dB. The gain spectrum of a particular SOA
depends on its structure, materials and
operational parameters. For most applications
high gain and wide gain bandwidth are
desired. The small-signal (small here meaning
that the signal has negligible influence on the
SOA gain coefficient) internal gain of a Fabry-
Perot SOA at optical frequency v is given by
Where R1 and R2 are the input and output
facet reflectivities and Δv is the cavity
longitudinal mode spacing given by
v0 is the closest cavity resonance to v. Cavity
resonance frequencies occur at integer
multiples of Δv. The sin2 factor in (5.1) is
equal to zero at resonance frequencies and
equal to unity at the anti-resonance frequencies
(located midway between
successive resonance
frequencies). The effective
SOA gain coefficient is
where Γ is the optical mode
confinement factor (the
fraction of the propagating
signal field mode confined to the active
region) and α the absorption coefficient.
Gs=egl is the single-pass amplifier gain.
An uncoated SOA has facet reflectivities
approximately equal to 0.32. The amplifier
gain ripple Gr is defined as the ratio between
the resonant and non-resonant gains. From
(5.1) we get
From (5.4) the relationship between the
geometric mean facet reflectivity
and Gr is
Curves of Rgeo versus Gs are shown in Fig.
5.02 with Gs as parameter. For example, to
obtain a gain ripple less than 1 dB at an
amplifier single-pass gain of 25 dB requires
that Rgeo < 3.6 x 10-4. Facet reflectivities of this
order can be achieved by the application of
anti-reflection (AR) coatings to the amplifier
facets. The effective facet reflectivities can be
5.1
5.2
5.3
5.4
5.5
16
reduced further by the use of specialised SOA
structures.
A typical TW-SOA small-signal gain
spectrum is shown in Fig. 5.01. The gain
bandwidth Bopt of the amplifier is defined as
the wavelength range over which the signal
gain is not less than half its peak value. Wide
gain bandwidth
SOAs are
especially useful
in systems where
multichannel
amplification is
required such as
in WDM
networks. A wide
gain bandwidth
can be achieved in
an SOA with an
active region
fabricated from
quantum-well or
multiple quantum-
well (MQW)
material. Typical
maximum internal
gains achievable
in practical
devices are in the
range of 30 to 35 dB.
Typical small-signal
gain bandwidths are in
the range of 30 to 60 nm.
Polarisation
sensitivity
In general the gain of
an SOA depends on the
polarisation state of the
input signal. This
dependency is due to a
number of factors
including the waveguide
structure, the polarisation
dependent nature of anti-
reflection coatings and the gain material.
Cascaded SOAs accentuate this polarisation
dependence. The amplifier waveguide is
characterised by two mutually orthogonal
polarisation modes termed the Transverse
Electric (TE) and Transverse Magnetic (TM)
modes. The input signal polarisation state
usually lies
Fig 5.02: Geometric mean facet reflectivity
17
somewhere between these two extremes. The
polarisation sensitivity of an SOA is defined as
the magnitude of the difference between the
TE mode gain GTE and TM mode gain GTM i.e.
Signal gain saturation
The gain of an SOA is
influenced both by the
input signal power and
internal noise generated
by the amplification
process. As the signal
power increases the
carriers in the active
region become depleted
leading to a decrease in
the amplifier gain. This
gain saturation can cause
significant signal
distortion. It can also limit
the gain achievable when
SOAs are used as
multichannel amplifiers. A
typical SOA gain versus output signal power
characteristic is shown in Fig. 5.03. A useful
parameter for quantifying gain saturation is the
saturation output power Po,sat which is defined
as the amplifier output signal power at which
the amplifier gain is half the small-signal gain.
Values in the range of 5 to 20 dBm for are
typical of practical devices.
Noise figure
A useful parameter for quantifying optical
amplifier noise is the noise figure. F, defined
as the ratio of the input and output signal to
noise ratios, i.e.
The signal to noise ratios in (5.7) are those
obtained when the input and output powers of
the amplifier are detected by an ideal
photodetector.
In the limiting case where the amplifier
gain is much larger than unity and the
amplifier output is passed through a
narrowband optical filter, the noise figure is
given by
The lowest value possible for nsp is unity,
which occurs when there is complete inversion
of the atomic medium, i.e. N1=0, giving F = 2
(i.e. 3 dB). Typical intrinsic (i.e. not including
coupling losses) noise figures of practical
SOAs are in the range of 7 to 12 dB. The noise
figure is degraded by the amplifier input
coupling loss. Coupling losses are usually of
the order of 3 dB, so the noise figure of typical
packaged SOAs is between 10 and 15 dB.
Dynamic effects
SOAs are normally used to amplify
modulated light signals. If the signal power is
high then gain saturation will occur. This
would not be a serious problem if the amplifier
gain dynamics were a slow process. However
in SOAs the gain dynamics are determined by
the carrier recombination lifetime (average
time for a carrier to recombine with a hole in
the valence band). This lifetime is typically of
a few hundred picoseconds. This means that
the amplifier gain will react relatively quickly
5.6
5.7
5.8
18
to changes in the input signal power. This
dynamic gain can cause signal distortion,
which becomes more severe as the modulated
signal bandwidth increases. These effects are
further exacerbated in multichannel systems
where the dynamic gain leads to interchannel
crosstalk. This is in contrast to doped fibre
amplifiers, which have recombination
lifetimes of the order of milliseconds leading
to negligible signal distortion.
Nonlinearities
SOAs also exhibit
nonlinear behaviour. In
general these nonlinearities
can cause problems such as
frequency chirping and
generation of second or third
order intermodulation
products. However,
nonlinearities can also be of
use. in using SOAs as
functional devices such as
wavelength converters.
BULK MATERIAL PROPERTIES
An SOA with an active region whose
dimensions are significantly greater than the
deBroglie wavelength λB=h/p.( where p is the
carrier momentum) of carriers is termed a bulk
device. In the case where the active region has
one or more of its dimensions (usually the
thickness) of the order of λB the SOA is
termed a quantum-well (QW) device. It is also
possible to have multiple quantum-well
(MQW) devices consisting of a number of
stacked thin active layers separated by thin
barrier (non-active) layers.
Bulk material band structure and gain
coefficient
The active region of a bulk SOA is
fabricated from a direct band-gap material. In
such a material the VB maximum and CB
minimum energy levels have the same
momentum vector. Direct bandgap
semiconductors are used because the
probability of radiative transitions from the CB
to the VB is much greater than is the case for
indirect bandgap material. This leads to greater
device efficiency, i.e. conversion of injected
electrons into photons. A simplified energy
band structure of this material type is shown in
Fig. 5.04, where there is a single CB and three
VBs. The three VBs are the heavy-hole band,
light-hole band and a split-off band. The heavy
and light-hole
bands are
degenerate;
that is their
maxima have
the same
energy and
momentum.
Fig 5.04: Carrier and optical confinement in DH SOA
Fig 5.05: Energy band structure of direct band
gap semiconductor
19
In this model the energy of a CB electron
or VB hole, measured from the bottom or top
of the band respectively is given by
Ea = ħ2∗𝑘𝑝 ^2
2∗𝑚𝑐
and
𝐸𝑏 =ħ2∗𝑘𝑝 ^2
2∗𝑚𝑣
where kp is the magnitude of the
momentum vector, mc the CB electron
effective mass and mv VB hole effective mass.
Under bias conditions the occupation
probability f(c)of an electron with energy E in
the CB is dictated by Fermi-Dirac statistics
given by
Where Efc is the quasi-Fermi level of the
CB relative to the bottom of the band, k is the
Boltzmann constant and T the temperature.
Similarly the occupation probability of an
electron in the VB with energy E, increasing
into the band, is given by
where Efv is the quasi-Fermi level of the
VB relative to the top of the band. The quasi-
Fermi levels can also be estimated using the
Nilsson approximation
𝐸𝑓𝑐 = 𝑙𝑛𝛿 + 𝛿 64 + 0.05524𝛿 64 +
𝛿 −1
/4}𝑘𝑇
Efv = -{ ln ε+ ε [64 +0.05524ε (64+ 휀)]^-
1/4}KT
Where δ = 𝑁
𝑛𝑐 and ε =
𝑝
𝑛𝑣
Where nc and nv are constants given by
and
where mhh and mlh and are the VB heavy
and light-hole effective masses.
For a two-level system we have from an
expression for the optical gain coefficient at
frequency υ
This expression applies to any particular
transition. Without lack of generality we can
apply it to transitions, having the same
momentum vector, between a CB energy level
Ea and VB energy level Eb where
Thus we obtain the relations:
Ea= (hυ-Eg(n))*(𝑚ℎℎ
𝑚𝑒 +𝑚ℎℎ ))
Eb = -(h(υ)-Eg(n))*(𝑚𝑒
𝑚𝑒 +𝑚ℎℎ)
Where mhh is the effective mass of heavy
hole and me is the effective mass of electrons.
It is assumed that heavy-holes dominate over
light-holes due to their much greater effective
mass.
5.9
5.10
5.11
5.12
5.18
5.17
5.19
5.13
5.14
5.16
5.15
5.20
5.21
20
Thus the optical gain coefficient of the
amplifier is given by
The above equations are used to compute
the fitting parameters in farther calculations.
5.22
Modeling of SOA CHAPTER6
21
6.5
6.6
6.7
6.8
6.1. MODELING
Models of SOA steady-state and dynamic behavior are important tools that allow
the SOA designer to develop optimized devices
with the desirable characteristics. They also allow the applications engineer to
predict how an SOA or cascade of SOAs
behaves in a particular application. Some models are amenable to analytical solution
while others require numerical solution. The
main purpose of an SOA model is to relate the
internal variables of the amplifier to measurable external variables such as the output signal
power, saturation output power and amplified
spontaneous emission (ASE) spectrum.
In this chapter two important model of SOA are
discussed.
Steady state numerical model proposed
by M.J. Connelly or Connelly model
Dynamic model of SOA or Reservoir
model
6.1.1. STEADY STATE NUMERICAL
MODEL
This model uses a comprehensive wideband
model of a bulk InP–InGaAsP SOA. The model can be applied to determine the steady-state
properties of an SOA over a wide range of
operating regimes. A numerical algorithm is
described which enables efficient implementation of the model.
A. The InGaAsP direct band gap bulk-material active region has a material
gain coefficient gm(υ) given by
The band gap energy Eg can be expressed as
Where Eg0 the band gap energy with no injected
carriers, is given by the quadratic approximation
Where a, b and c are the quadratic coefficients and e is the electronic charge. ΔEg (n) is the
band gap shrinkage due to the injected carrier
density given by
where Kg is the band gap shrinkage coefficient.
The Fermi-Dirac distributions in the CB and VB are given by
Efc is the quasi-Fermi level of the CB relative to
the bottom of the band. It is the quasi-Fermi
level of the VB relative to the top of the band. They can be estimated using the Nilsson
approximation.
6.1
6.2
6.3
6.4
22
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
9
6.21
𝐸𝑓𝑐 = 𝑙𝑛𝛿 + 𝛿 64 + 0.05524𝛿 64 +
𝛿 −1
/4}𝑘𝑇
Efv = -{ ln ε+ ε [64 +0.05524ε (64+ 휀)]^-
1/4}KT
Where δ = 𝑁
𝑛𝑐 and ε =
𝑝
𝑛𝑣
Where nc and nv are constants given by
And
Where mhh and mlh and are the VB heavy and
light-hole effective masses.
For a two-level system we have from an expression for the optical gain coefficient at
frequency υ
This expression applies to any particular
transition. Without lack of generality we can
apply it to transitions, having the same
momentum vector, between a CB energy level
Ea and VB energy level Eb where
Thus we obtain the relations:
Ea= (hυ-Eg(n))*(𝑚ℎℎ
𝑚𝑒 +𝑚ℎℎ ))
Eb = -(h(υ)-Eg(n))*(𝑚𝑒
𝑚𝑒 +𝑚ℎℎ)
Where mhh is the effective mass of heavy hole
and me is the effective mass of electrons. It is assumed that heavy-holes dominate over light-
holes due to their much greater effective mass.
Thus the optical gain coefficient of the amplifier is given by
The above equations are used to compute the
fitting parameters in farther calculations.
gm (υ) is composed of two components one is the gain coefficient
And another is the absorption coefficient
So
Plot for gm and gm ́is given in the fig.6.1.
23
6.22
6.23
6.24
6.25
6.26
6.27
6.28
Figure.6.1. Typical InGaAsP bulk semiconductor gain spectra.
The SOA parameters used in Connelly model is
given in the table
The material loss coefficient α is modeled as a linear function of carrier density
K0 and K1 are the carrier-independent and
carrier-dependent absorption loss coefficients,
respectively.
B. TRAVELLING WAVE EQUATION
FOR SIGNAL FIELD
In the model, signals are injected with optical frequencies υk ( k=1 to Ns) and power Pink
before coupling loss. The signals travel through
the amplifier, aided by the embedded
waveguide, and exit at the opposite facet. The SOA model is based on a set of coupled
differential equations that describe the
interaction between the internal variables of the amplifier, i.e., the carrier density and photon
rates. The solution of these equations enables
external parameters such as signal fiber-to-fiber gain and mean noise output to be predicted. In
the following analysis, it is assumed that
transverse variations in the photon rates and
carrier density are negligible. This assumption is
valid for SOAs with narrow active regions. In
the model, the left (input) and right (output) facets have power reflectivity R1 and R2,
respectively. Within the amplifier, the spatially
varying component of the field due to each input
signal can be decomposed into two complex traveling-waves Es+ and Es-, and, propagating
in the positive and negative directions,
respectively lies along the amplifier axis with its origin at the input facet. The modulus squared of
the amplitude of a traveling-wave is equal to the
photon rate (s) of the wave in that direction, so
The light wave representing the signal must be
treated coherently since its transmission through the amplifier depends on its frequency and phase
when reflecting facets are present Esk+ and Esk-
obey the complex traveling-wave equations
And
Boundary conditions
Where the k-th input signal field to the left of
the input facet is
The k-th output signal field to the right of the output facet is
24
6.29
6.30
6.31
6.32
6.33
6.34
6.35
The k-th output signal power after coupling loss
is
ηin and ηout are the input and output coupling
efficiencies, respectively.
The amplitude reflectivity coefficients are
The kth signal propagation coefficient is
neq is the equivalent index of the amplifier
waveguide
n2 is the refractive index of the InP material surrounding the active region. neq is modeled as
a linear function of carrier density
neq0 is the equivalent refractive index with no
pumping. The Differential in given
C. TRAVELING-WAVE EQUATIONS
FOR THE SPONTANEOUS
EMISSION
The amplification of the signal also depends on the amount of spontaneously emitted noise
generated by the amplifier. This is because the
noise power takes part in draining the available
carrier population and helps saturate the gain.
However, it is not necessary to treat the spontaneous emission as a coherent signal, since
it distributes itself continuously over a relatively
wide band of wavelengths with random phases
between adjacent wavelength components. When reflecting facets are present, the
spontaneously emitted noise will show the
presence of longitudinal cavity modes. For this reason, it may be assumed that noise photons
only exist at discrete frequencies corresponding
to integer multiples of cavity resonances. These frequencies are given by
Where the cutoff frequency at zero injected carrier density is given by
Δυc is a frequency offset used to match υ0 to a
resonance. Km and Nm are positive integers. The
values of Km and Nm chosen depend on the gain bandwidth of the SOA and accuracy
required from the numerical solution of the
model equations. The longitudinal mode frequency spacing is
This technique can be applied to both resonant
and near-traveling-wave SOAs and greatly
reduces computation time. It can be shown that averaging the coherent signal over two adjacent
cavity resonances is identical to treating the
signal coherently in terms of traveling-wave
power (or photon rate) equations. It is sufficient to describe the spontaneous emission in terms of
power, while signals must be treated in terms of
waves with definite amplitude and phase. Nj+ and Nj- and are defined as the spontaneous
emission photon rates (s) for a particular
polarization [transverse electric (TE) or
transverse magnetic (TM)] in a frequency spacing centered on frequency, traveling in the
positive and negative directions, respectively.
And obey the traveling-wave equations
25
6.36
6.37
6.38
6.39
6.40
6.41
6.42
6.43
6.44
6.45
6.46
6.47
6.48
Subject to the boundary conditions
The function Rsp(vj,n) represents the
spontaneously emitted noise coupled into Nj+ or
Nj- . An expression for Rsp can be derived by a
comparison between the noises outputs from an
ideal amplifier obtained using with the quantum
mechanically derived expression. An ideal amplifier has no gain saturation (which implies a
constant carrier density throughout the
amplifier), material gain coefficient, and zero loss coefficient, facet reflectivities, and coupling
losses. In this case, is obtained from the solution
to
The output noise power at the single frequency
band
The equivalent quantum mechanical expression
The traveling-wave power equations describing
and assume that all the spontaneous photons in spacing are at resonance frequencies. In a real
device the injected spontaneous photons,
originating from, are uniformly spread over. The noise is filtered by the amplifier cavity. To
account for this, and are multiplied by a
normalization factor which is derived as follows.
If the single-pass gain is at , then the signal gain
for frequencies within spacing Δυm around υj
Where the single-pass phase shift is
At resonance, the signal gain is
Let the amplifier have a noise input spectral
density (photons/s/Hz) distributed uniformly over centered. The total output noise (photons/s)
in is then
If the input noise power were concentrated at
(resonance), then the output noise photon rate would be
Where
where
Kj is equal to unity for zero facet reflectivities.
26
6.49
6.50
6.51 6.52
D. CARRIER DENSITY RATE
EQUATION
The carrier density at obeys the rate equation
Where I is the bias current and R (n(z)) is the
recombination rate given by
Rrad(n) and
Rnrad(n) carrier recombination rates, respectively,
both of which can be expressed as polynomial
functions
Arad and Brad are the linear and bimolecular
radioactive recombination coefficients.
E. STEADY STATE NUMERICAL
SOLUTION OF CONNELLY
MODEL
As the SOA model equations cannot be solved
analytically, a numerical solution is required. In
the numerical model the amplifier is split into a number of sections labeled from i=1 to Nz as
shown in Fig.6.2. The signal fields and
spontaneous emission photon rates are estimated at the section interfaces. In evaluating Q (i) in
the i-th section the signal and noise photon rates
used are given by the mean value of those quantities at the section boundaries. In the
steady-state Q (i) is zero. To predict the steady-
state a characteristic, an algorithm is used which
adjusts the carrier density so the value of throughout the amplifier approaches zero. A
flowchart of the algorithm is shown in Fig. 6.3.
Figure.6.2. the ith section of the SOA model. Signal fields and spontaneous emission are
estimated at the section boundaries. The carrier
density is estimated at the center of the section
The first step in the algorithm is to initialize the
signal fields and spontaneous emission photon
rates to zero. The initial carrier density is obtained from the solution of carrier density rate
equation with all fields set to zero, using the
Newton–Raphson technique. The coefficients of the traveling-wave equations are computed. In
the gain coefficient calculations, the radiative
carrier recombination lifetime is approximated
by
Next, the signal fields and noise photon densities
are estimated. The noise normalization factors are then computed. Q (i) is then calculated. This
process enables convergence toward the correct
value of carrier density by using smaller carrier density increments. The iteration continues until
the percentage change in the signal fields, noise
photon rates and carrier density throughout the
SOA between successive iterations is less than the desired tolerance. When the iteration stops,
the output spontaneous emission power spectral
density is computed using the method of Section VII and parameters such as signal gain, noise
figure and output spontaneous noise power are
calculated. The algorithm shows good
convergence and stability over a wide range of operating conditions. A flowchart of the
algorithm is shown in Fig. 6.3.
27
Figure.6.3. SOA steady-state model algorithm
28
6.53
F. ESTIMATION OF THE OUTPUT
SPONTANEOUS EMISSION
POWER SPECTRAL DENSITY
The average output noise photon rate spectral density (photons/ s/Hz) after the coupling loss
over both polarizations and Bandwidth KmΔυm
centered on υj is
Figure.6.4. SOA output spectrum. Resolution
bandwidth is 0.1 nm. The input signal has a
wavelength of 1537.7 nm and power of -25.6
dBm. Bias current is 130 mA. The predicted and experimental fiber-to-fiber signal gains are both
25.0 dB. The experimental gain ripple of 0.5 dB
is identical to that predicted. The difference between the predicted and experimental ASE
level is approximately 2.5 dB.
G. OUTPUT OF THE CONNELLY
MODEL
Figure 6.6. predicted and experimental SOA
fiber-to-fiber gain versus bias current
characteristics. The input signal has a
wavelength of 1537.7 nm and power of -25.6 dBm.
Figure 6.7. predicted SOA noise figure spectrum. Input parameters are as for Fig.
5. A noise figure of 11.4_0.5 dB at 1537.7 nm is
predicted compared to an experimental value of 8.8_0.3 dB.
29
Figure 6.8. SOA predicted fiber-to-fiber gain
and output ASE power versus input signal power. Signal wavelength is 1537.7 nm
and bias current is 130 mA.
30
Figure 6.10. predicted SOA output ASE spectra with the input signal power as parameter,
showing non-linear gain compression. Signal
wavelength is 1537.7 nm and the bias current is
130 mA. Resolution bandwidth is 0.1 nm.
A wideband SOA steady-state model and numerical solution has been described. The
model predictions show good agreement with
experiment. The model can be used to investigate the effects of different material and
geometrical parameters on SOA characteristics
and predict wideband performance under a wide range of operating conditions.
31
SOA PARAMETERS USED IN STEADY
STATE CONNELLLY MODEL
32
6.2. RESERVIOR MODEL
Another important SOA model is the Reservoir
model proposed by Walid Mathlouthi, Pascal
Lemieux, Massimiliano Salsi, Armando Vannucci, Alberto Bononi, and Leslie A.
Rusch.
This model is the dynamic version of the steady state Connelly model. We are interested in
analyzing the response of SOAs to optical
signals that are modulated at bit rates not exceeding 10 Gb/s, such as those planned for
next-generation metropolitan area networks.
Therefore, ultrafast intra band phenomena such
as carrier heating (CH) and spectral hole burning (SHB) can be neglected, and only carrier
induced gain dynamics need to be included, as
was done in several SOA models developed in the past. Such models can be divided into two
broad categories: 1) space-resolved numerically
intensive models, which take into account facet reflectivity as well as forward and backward
propagating signals and amplified spontaneous
emission (ASE) and offer a good fit to
experimental data simplified analytical models with a coarser fit to experimental data but
developed to facilitate conceptual understanding
and performance analysis. For the purpose of carrying out extensive Monte Carlo simulations
for statistical signal analysis and bit-error rate
(BER) estimation, the accurate space-resolved
models are ruled out because of their prohibitively long simulation times. However, a
simplified model with a satisfactory fit to
experimental results would be highly desirable. Most simplified models can be derived from the
work of Agrawal and Olsson. Under suitable
assumptions, Agrawal and Olsson managed to reduce the coupled propagation and rate
equations into a single ordinary differential
equation (ODE) for the integrated gain. The
simplicity of the solution is due to the fact that waveguide scattering losses and ASE were
neglected. ASE has an important effect on the
spatial distribution of carrier density and
saturation, and it may significantly affect the
SOA steady-state and dynamic responses. Scattering losses also have an impact on the
dynamic response of the SOA.
Moreover, Agrawal and Olsson’s model was
originally cast for single-wavelength-channel amplification, although it can be extended to
multi wavelength operation by assuming that the
channels are spaced far enough apart to neglect FWM beating in the co propagating case. Saleh
arrived independently at the same model as
Agrawal and Olsson’s coincides with and then introduced further simplifying approximations to
get to a very simple block diagram of the single-
channel SOA, which was exploited for a
mathematically elegant stochastic performance analysis of single-channel saturated SOAs. The
loss of accuracy due to Saleh’s extra
approximations with respect to Agrawal’s model was quantified in Saleh’s model was later
extended to cope with injection current
modulation, scattering losses, and ASE. In addition, Agrawal’s model was extended to
include ASE in both and ASE was added
phenomenologically at the output of the SOA
and did not influence the gain dynamics, thereby limiting the application to very small saturation
levels.
In this paper, we first develop a dynamic version of the steady-state wideband SOA Connelly
model which is shown to fit quite well with our
dynamic SOA experiments with OOK channels.
The Connelly model was selected because it derives the SOA material gain coefficient from
quantum mechanical principles without the
assumption of linear dependence on carrier density that was made in.
Our dynamic Connelly model serves then as a
benchmark to test the accuracy and computational-speed improvement of a novel
state-variable SOA dynamic model, which
represents the most important contribution of
this paper. The novel model is an extension of Agrawal’s model, with the inclusion of
approximations for scattering loss and ASE to
better fit the experimental results and the dynamic Connelly model predictions. In such a
model, the SOA dynamic behavior is reduced to
the solution of a single ODE for the single state variable of the system, which is proportional to
the integrated carrier density, which, for WDM
33
operation is a more appropriate variable than the
integrated gain used in. Once the state-variable dynamic behavior is found, the behavior of all
the output WDM channels is also obtained. The
state variable is called ―reservoir‖ since it plays
the same role as the reservoir of excited erbium ions in an erbium-doped fiber amplifier (EDFA).
Quite interestingly, then, the SOA for WDM
operation admits almost the same block diagram description as that of an EDFA suggested by
Such a novel SOA block diagram is shown in
Fig. 6.11 (without ASE for ease of drawing) and will be derived in the next sections. Note that
this model treats the intensity of the electrical
field, but the field phase can be indirectly
obtained since it is a deterministic function of the reservoir. In the SOA, the role of the optical
pump for EDFAs is played by the injected
current I. The most striking difference between the two kinds of amplifiers is the fluorescence
time τ, which is of the order of milliseconds in
EDFAs and of a fraction of nanosecond in SOAs. Such a huge difference accounts for most
of the disparity in the dynamic behavior between
the two kinds of amplifiers and explains why
SOAs have not been used for WDM applications for a long time]. However, recent cheap gain-
clamped SOAs] are likely to promote the use of
SOAs for WDM metro applications. As already mentioned, the reservoir model requires the (co-
propagating) WDM channels to have minimum
channel spacing in excess of a few tens of
gigahertz, in order to neglect the carrier-induced FWM fields generated in the SOA. This should
not be a problem for channels allocated on the
International Telecommunications Union grid with 50 GHz spacing or more. However, an
intrinsic limit of the reservoir model is its
neglecting SHB and CH, which generate FWM and XPM interactions among WDM channels
even when the minimum channel spacing is
large enough to rule out any carrier-induced
interaction. The predictions of the reservoir model will be accurate whenever the carrier
induced XGM mechanism dominates over FWM
and XPM. It is worth mentioning that state-variable amplifier block diagrams are very
important simulation tools that enable the
reliable power propagation of WDM signals in optical networks with complex topologies;
therefore, the present reservoir SOA model
provides a new entry aside from the already
known models for EDFAs and for Raman amplifiers .A challenge in our reservoir model,
as in all simplified SOA models, is to correctly
choose the values of the wavelength-dependent
coefficients that give the best fit to the experimental results. We propose and describe
here a methodology to extract the needed
wavelength-dependent coefficients from the parameters of the dynamic Connelly model.
This paper is organized as follows. In Section II,
the dynamic Connelly model is introduced, and a procedure to derive its parameters from
experiments is described. In Section III, the
SOA reservoir model is derived first without
ASE and then with ASE that is resolved over a large number of wavelength bins. Simulations
show good accordance between the reservoir
model predictions and experiments, and good improvement in calculation time with respect to
the Connelly model. However, inclusion of
many ASE wavelength channels makes even the reservoir model too slow for the BER
estimations we have in mind. Hence, in order to
further simplify the model, we introduce the
reservoir model with a single equivalent ASE channel. The ASE can be seen as an independent
input-signal channel (with proper input power
and wavelength) that depletes the reservoir of a noiseless SOA. Results show that this last model
is the most efficient one since it can be made to
accurately predict experimental results with an
execution time that is 20 times faster than that of the dynamic Connelly model for single-channel
operation, with the savings increasing with the
number of WDM signal channels. In Section III-C, we examine a model that was obtained by
dividing the SOA into several sections, each
characterized by its own reservoir. Here again, the ASE can be modeled as a single channel that
propagates through the different reservoir stages.
Results show better precision, although the
increase in precision is not worth, in most cases, the loss in execution time. Most of the numerical
results are reported in Section IV. Finally,
Section V summarizes the main findings of this paper.
34
6.54
6.55
6.56
Figure6.11. Block diagram of the reservoir
model. ASE contribution not shown for ease of drawing.
6.2.2 DYNAMIC CONNELLY MODEL
A. Theory
In this paper, we adopt the wideband model for a bulk SOA proposed in Connelly model, which is
based on the numerical solution of the coupled
equations for carrier-density rate and photon
flux propagation for both the forward and backward signals and the spectral components of
ASE. At a specified time t and position z in the
SOA, the propagation equation of photon flux Q±k [photons/s] of the kth forward (+) or
backward (−) signal is
where Γ is the fundamental mode confinement factor, gk is the material gain coefficient at the
optical frequency νk of the kth signal, α is the
material-loss coefficient, and both are functions of carrier density N(z, t). The power of the
propagating signal is related to its photon flux as
P±k = hνkQ± k (in watts), where h is Planck’s
constant. The ASE photon flux on each ASE wavelength channel obeys a similar propagation
equation given by
where Rsp,j(N) is the spontaneous emission rate coupled into the ASE channel at frequency νj.
The expression of Rsp,j(N) will be used in
Section III-B to develop a reservoir model
equation that takes ASE into account. The carrier density at coordinate z evolves as
where I is the bias current; q is the electron
charge; d, L, andW are the active-region
thickness, length, and width, respectively, and R(N) is the recombination rate. The reservoir
model of Section III uses a linear approximation
for R (N) in (9); nsig is the number of WDM signals; nASE is the number of spectral
components of the ASE; and Kj is an ASE
multiplying factor, which equals 1 for zero facet
reflectivity [12]. The factor 2 in accounts for two ASE polarizations. Note that equation contains
an important approximation: it is the sum of the
signals and ASE powers (fluxes), instead of—more correctly—the power of the sum of the
signals and ASE fields, which depletes carrier
density N. Therefore, (3) neglects the carrier-
density pulsations due to beating among WDM channels that generate FWM and XPM in SOAs
[9]. Although such an approximation is
inappropriate for extremely dense or high-power WDM channels, it is accurate for typical
wavelength spacing of 0.4 nm or more. The
material gain gk(N) ≡ g(νk,N) is calculated as in Connelly model. Fig.6.12 plots the material gain
N versus wavelength λk = c/νk (with c being the
speed of light) using the SOA parameters.
Figure.6.12. Gain coefficient g(λ,N) versus
wavelength and carrier density
35
B. Parameterization
In order to fit the experimental results that we obtained with a commercial Optospeed SOA
model 1550MRI X1500, we used the SOA
parameters provided in the Table in Connelly
model, except for a subset of different values reported in Table I in this paper; the most critical
of such parameters were determined as follows.
1) The active-region length L was determined by
measuring the frequency spacing between two
maxima of the gain spectrum ripples: L = λ20 /2nrΔλ, where λ0 is the central wavelength
(1550 nm), nr is the average semiconductor
refractive index, and Δλ is the ripple wavelength
spacing.
2) The band gap energy Eg0 was set so that the
experimental cutoff wavelength of the gain spectrum (which was about 1605 nm) matched
the simulated one.
3) The parameters of the carrier-dependent
material-loss coefficient, i.e.
α (N(z)) = K0 +ΓK1N
where chosen so that the maximum simulated gain matched the measured one.
4) The active-region thickness and width were set so as to match the experimental and
simulated curves of gain as a function of the
injection current.
5) The band gap shrinkage coefficient Kg was
set so that the peak gain wavelength equals the
measured value of 1560 nm at an injection current of 500 mA.
36
Figure.6.13. Fiber to
fiber unsaturated gain
versus wavelength. Measured (dashed)
and simulation (solid)
results using Connelly model.
C. Simulations with Connelly Model
We present simulation results obtained with the Connelly model and compare them against
experimental measurements.
The experiment consisted in amplifying a
tunable continuous wave (CW) laser whose wavelength was varied around the Optospeed
SOA peak gain wavelength. Laser polarization
was controlled so as to obtain maximum gain.
1) Unsaturated Gain Spectrum: Fig. 3 shows the
simulated and measured unsaturated gain spectra at a signal input power of −30 dBm and an
injection current of 500 mA. A good match
between the simulations and experiments was
obtained when using the values of Table I. In the
ensuing Fig. 4
fiber to fiber gain versus input
optical power. Measured (dashed) and Connelly
model (solid). Experiments and simulations, the input signal will be fixed at the gain peak
wavelength of 1560 nm.
2) Gain Saturation: Fig. 6.13. shows the fiber-to-fiber gain as a function of the input power.
The wavelength of the input laser was 1560 nm,
and the injection current was 500 mA.
3) Dynamic Response: The experimental setup is
depicted in Fig. 5. The input laser at 1560 nm was externally modulated at 1 Gb/s. The laser
power was varied from −25 to −10 dBm in steps
of 5 dB. The measured photo receiver
responsively was 400 mV/mW. The injection
37
6.57
6.58
6.59
6.60
6.61
current was 500 mA. Since we are interested in
testing the action of the SOA on the propagating signal power in this paper, no optical filter was
inserted before detection.
The measured experimental input pulses to the
SOA were replicated in the simulator. The length of the input-signal time series was 1350
points over a 2-ns time window. In Fig. 6, we
plot the experimental and the simulated output pulses at an input power of −18 dBm. At this
power level, the SOA is not heavily saturated by
the signal; thus, the ASE-induced saturation significantly contributes to the dynamic
response.
Fig. 6.15 demonstrates that the dynamic
Connelly model is also able to accurately predict the amplified output pulse shape.
Similar results were also obtained for many
different input powers and signal wavelengths.
4) Computation Time: The major drawback of
the Connelly model is its long execution time. Our Matlab code, which was run on a 3-GHz
Intel processor, took about 12 s to calculate an
output bit resolved over 1350 points. Similar
calculations for a time series of 50 000 points (37 bits) took about 432 s. This presents a major
limitation when typical Monte Carlo BER
estimations are sought, which require transmission of millions of bits. A drastic
simplification of the gain dynamics calculation
is required in order to significantly decrease
execution time. Reduced computation time and the facility of analysis motivate our introduction
of the reservoir model.
Figure.6.15. Response to square wave input (see
inset representing optical input power in dBm). Measured (dashed) and dynamic Connelly
model (solid).
6.3. RESERVOIR MODEL We now derive the reservoir model for a
traveling-wave
SOA (zero facet reflectivity) fed by WDM signals. For k =1, . . . , nsig, the propagation and
carrier density update
where A and V = AL are the active waveguide
area and volume, respectively, and we introduced the propagation direction variable uk,
which equals +1 for forward signals and −1 for
backward signals. · QASE j stands for an equivalent ASE flux that accounts for the impact
of both forward and backward ASE on the
carrier-density update equation. The formal
solution of the propagation equation is obtained by multiplying both sides by uk, dividing them
by Qk, integrating both sides in dz from z = 0 to
z = L for each k, and obtain an equivalent equation of the form Qout k = Qin k Gk, where
the gain
is independent of the signal propagation direction. For convenience, we will let
denote the net gain coefficient per unit length in
the SOA. Now, define the SOA reservoir as
which physically represents the total number of
carriers in the SOA that are available for
38
6.62
6.64
6.63
6.65
6.66
6.67
6.68
6.69
6.70
conversion into signal photons by the stimulated
emission process. If one approximates both the recombination rate and the material gain as
linear functions of N then
where τ is the fluorescence time and σk[m2] and
N0k[m−3] are wavelength-dependent fitting
coefficients, then one obtains
Where
are two dimensionless parameters. In addition, one can multiply both sides of the second
equation in (5) by A and integrate in
dz to obtain
For the time being, the contribution of ASE will
be neglected.
It will be tackled in Section III-B. Now, integrating in dz both sides of the first equation
in (5) gives
the ―reservoir dynamic equation‖ given by
Note that the reservoir dynamic equation is quite
similar to the EDFA reservoir equation.
A. Extraction of Reservoir Parameters from
Connelly Model We next explain how to extract the fitting
parameters of the gain linearization from the
Connelly gain g (λ,N), whose plot versus
wavelength and carrier density was already given in Fig.6.12 for our Optospeed SOA. A
plot of gnet k (λ,N) would have a similar form;
in particular, a rigid shift downward would result if K1 = 0, i.e., if α did not depend on N.
Fig. 7 gives a slice of the surface in Fig. 2 at a
wavelength of 1560 nm, which was plotted over
a wide range of carrier density N. As shown, a linear approximation of the gain coefficient is
well justified especially as the physically
achievable range of carrier densities is much smaller than the range shown. Our task is now to
provide good estimates of the wavelength-
dependent coefficients σk and N0k. First, we identify the achievable range of N over which
we will restrict our linear fit. To this aim, using
the steady-state Connelly model, we calculated
the maximum and minimum values of the ―average carrier density,‖ i.e.,
which were obtained for the extreme cases of a
single input signal at very low (−40 dBm) and very high (0 dBm) input power at 1560 nm.
These extremes cover the small-signal regime
and saturation at an injection current of 500 mA without ASE was used to find N (z) at steady
state (dN/dt = 0) for a small signal and saturation
at λk. The carrier density was integrated across z
to give the extreme values Nmax,k and Nmin,k, which are depicted in Fig. 7. The process was
repeated at each wavelength from 1450 to 1600
nm in intervals of 5 nm. The parameters of the gain coefficient linear fit were then extracted
from the extreme values as follows:
where gmax,k_= g(λk,Nmax,k) and gmin,k is similarly defined.
In Fig. 8, we provide the wavelength
dependence of the extracted fitting parameters σk and N0,k for our Optospeed SOA. Once the
liberalized gain parameters are calculated, we
can investigate the steady state and the dynamic
behavior predicted by the reservoir model and, as explained in the Appendix, look for the value
of τ that best fits the steady-state and dynamic
experimental curves. However, before doing so, the fundamental role of spontaneous emission in
39
6.71
6.72
6.73
6.74
the rate equation must be properly accounted
for.
Figure.6.16. Connelly gain coefficient g (dashed) and net gain coefficient gnet in
(7) (solid) versus carrier density N for λ = 1560
nm. SOA parameters as in
Table I. Dotted is the linear approximation used in the reservoir model.
Figure.6.17. Coefficients σk (squares solid) and
N0, k (triangle solid) of the linearization of the gain coefficient g versus wavelength for our
Optospeed SOA. Also shown are the coefficients
γk and N1,k of the linearization of the emission gain coefficient g_
B. Including ASE
We now take into account the ASE-induced saturation term in (5) that was neglected in the
previous section. The ASE flux at z is obtained
by solving the propagation (2) with zero initial condition
where Gj(z) = exp[_z 0 Γgnetj (N(z ))dz] is the gain from 0 to z. If, for this calculation, we
assume that the carrier density is constant along
z at the average carrier density N = r/V, then the preceding equation simplifies to
Such an expression can now be used to evaluate
the ASE
Integrals
where G(r) = exp{Γgnet j (N)L} is the gain and is a function of the reservoir only.
If we linearize g
j(N) ∼=γj(N − N1j) and use the linearization
where r1j_= N1jV . As a dimensional check, γj and A are measured in [m2], while aj is
dimensionless so as to correctly obtain a
dimensionless nsp,j . Fig. 6.17. also shows the values of the wavelength-dependent coefficients
γj and N1j in the linearization of g , which
were obtained using exactly the same procedure that yields the linearization coefficients of g
detailed in Section III-A. Finally the reservoir
dynamic equation including ASE becomes
C. Multistage Reservoir Model The multistage reservoir model consists of
subdividing the SOA into several cascaded
40
6.74
sections or ―stages,‖ each characterized by its
own reservoir (Fig. 10). Let ns be the number of stages. Then, the reservoir equation for each
stage i is
where ri is the reservoir of the ith stage with
length Li = L/ns and Gk(ri) is its gain given in (10) and (11) (where Li is used instead of L), and
nsp,j is the spontaneous emission factor in (21).
For the signal channels, the flux Qin k,i+1 input
to the (i + 1)th stage is the output flux of the ith stage, which is in turn equal to the ith reservoir
gain Gk(ri) multiplied by its input flux Qin k,i.
For the ASE channels, the first-stage input flux is zero. The output ASE of one stage becomes
an input ASE signal to the next stage, which is
accounted for in (23) by the second summation
term. The third summation term is, as usual, the ASE generated inside stage i. considering
forward ASE only has the advantage of
simplicity, but the approximation brought into a multistage scenario is evident: Each stage is
saturated by forward ASE from the upstream
stages. Modeling the SOA with multiple stages is similar to the algorithm used in the space
resolved models, which provide the carrier-
density evolution N(t, zi) at discrete positions zi
along the SOA. Hence, the multistage reservoir model is expected to give similar results to the
Connelly model.
Fig. 6.18. . Variation of the total output ASE
power for a square input pulse train simulation results with dynamic Connelly model (solid) and
with reservoir model including ASE (dashed).
Figure.6.19. Multi stage reservoir model.
D. Reservoir Model with Single-Channel ASE
Consider the single-stage reservoir model. In
order to further speed up calculations, we now introduce a single fictitious CWinput ASE
channel. Once its wavelength is fixed, the power
of such a CW channel should be chosen so that the time behavior of reservoir r (t) in a noiseless
SOA is as close as possible to r(t) in the actual
SOA that is saturated by signals and ASE. We
call such an input channel the ―ASE depleting channel’
6.4. RESULTS
The purpose of this paper is to demonstrate that calculations using the SOA reservoir model are
much faster than the space resolved Connelly
model and hence, are suitable for Monte Carlo
simulations. We also demonstrate that using the correct wavelength-dependent parameters, the
reservoir model is sufficiently accurate. In this
section, we first compare the computation speed of both models. Then, we assess the accuracy of
the reservoir models that were developed in the
previous sections by comparing gain spectrum,
gain saturation, and dynamic response with the predictions of our experiments.
A. Calculation Speed We present the calculation times required for
different models, namely, the dynamic Connelly
model presented in Section II, the reservoir model with multiple ASE channels in Section
III-B, and the reservoir model with a single ASE
channel in Section III-D. For the reservoir
model, we determined the computation time for
41
a single-stage SOA, as well as three multistage
SOAs (two, five, and ten stages). The calculation times in Table II refer to the
response to a single input pulse with a duration
of 2 ns that was resolved over 1350 temporal
points. As a reference, the execution time for the Connelly model was 11.95 s. The calculation
times in Table III refer to the response to a string
of multiple pulses with the same time step as before, for a total of 50 000 temporal points. As
a reference, the execution time for the Connelly
model was 432.54 s. In the Connelly model, we always used a space resolution of 43.33 μm,
with the ASE resolved over 30 channels in bins
of 2.5 nm each, which were symmetrically
arranged around the gain peak. As shown, the reservoir model with single-stage ASE is always
the fastest model. The simulation is 20 times
faster than the Connelly model when a single ASE channel is used. However, when several
reservoir stages are used, the calculation speed
of the single-ASE model becomes of the same order as that of the multiple-ASE case. In this
case, the use of multiple ASE channels is better
for accuracy.
The improvement in computation time in all reservoir models with respect to the Connelly
model is predicted to significantly increase when
increasing the number of propagated WDM signal channels.
Fig. 6.20. Fiber to fiber gain spectrum versus
wavelength. Measured (dashed) and simulation (solid) results using the single (squares) and
five-stage (circles) reservoir with 20 ASE
channels
Fig. 6.21. Fiber to fiber gain versus input
optical power. Measured (dashed) and
simulation (solid) results using the single (squares) and five-stage (circles) reservoir with
20 ASE channels.
Fig. 6.22. Response to square wave input.
Measured (dashed) and simulation (solid)
results using the single (squares) and five-stage (circles) reservoir with 20 ASE channels.
B. Single-Stage Reservoir with ASE
1) Gain Spectrum: Fig. 6.20 shows both
simulated (solid lines with markers) and
experimental fiber-to-fiber (dashed dotted line) gain versus wavelength. The input laser power
was −25 dBm. We can see a reasonable match
between simulations and experiments. A slight gap between simulation and experiment is
42
observed at shorter wavelengths. The peak gain
wavelength was the same in both simulations and experiment. We also see that the five-stage
reservoir model is slightly more accurate than
the single-stage reservoir one.
2) Gain Saturation: Fig. 6.21 shows both
simulated and experimental fiber-to-fiber gains
versus input power at a signal wavelength of 1560 nm. We see that the simulations reasonably
predict the small-signal gain. A slight
discrepancy is observed when saturation sets in. This is attributed to the fact that the ratio gk/gnet
k is larger than one in deep saturation since the
denominator tends to zero. In such cases, it is
preferable to include the term gk(r)/gnet k (r) in the reservoir equation rather than set it to 1 and
play with the fitting parameter τ, as we did in
this paper. Here again, we see that the five-stage reservoir is closer to the experimental data.
3) Dynamic Response: Fig. 6.22 shows the simulated (solid with squared markers) and
experimental (dashed) response in mill watts to a
square-wave input (see inset in Fig. 6).
Simulations include the ASE total detected power, which plays a fundamental role in the
reservoir equation, since it partially saturates the
amplifier, hence reducing the amplifier gain. We see that the CW levels (zero and one levels) are
well predicted in Fig. 6.22. This suggests that
the approximation that we used to calculate the
ASE power is valid, although the simulated and experimental output pulses are slightly different.
We see that the pulse’s overshoot and
undershoot are better predicted by the five-stage reservoir model.
Fig. 6.23. fiber to fiber gain versus optical input
power. Measured (dashed) and simulated (solid)
results using a three-stage reservoir with single
channel ASE.
Fig. 6.24. Response to a square wave input. Measured (dashed) and simulated (solid) results
using a three-stage reservoir with ASE depleting
channel.
C. Multistage Reservoir With ASE
Figs. 6.20–6.22 show the gain spectrum, gain
saturation at 1560 nm, and the output pulse power for the five-stage (solid with circle
markers) reservoir with ASE, respectively. The
parameters used in the simulations are the same as those considered in the single-stage reservoir
model. Increasing the stage number beyond five
does not increase the accuracy, which might be
attributed to the neglect of ASE that propagates backward across the stages. In these figures, we
see that the dynamic and steady-state fits are
43
more accurate than those in the single stage.
Particularly, the simulated gain spectrum shape (Fig. 6.20) is closer to the experimental one
compared with the single-stage reservoir.
Moreover, the overshoot and undershoot of the
output pulse are much closer to the experimental one. However, this precision comes at the price
of simulation speed: The larger the number of
stages, the longer the execution time. An advantage of the multistage model is that it
allows trading execution time for precision,
eventually reaching a comparable precision (and a comparable computational burden) as the
space-resolved Connelly model.
D. Multistage Reservoir With ASE Depleting
Channel
In order to fit the experimental results, we
arbitrarily fixed the ASE-depleting-channel wavelength at 1520 nm and then found the value
of its input flux, which gave the minimum mean
square error fit with the prediction of the Connelly model. As shown in Figs. 14 and 15,
the simulation results are not far from the
experimental ones, but they are less accurate
than those in the multichannel ASE case. To investigate the dynamic response, we cascaded
three reservoir stages and propagated both the
signal and the ASE depleting channel. We note from the figures that simulations fit
measurements in a way comparable to the
multistage reservoir with ASE, which proves the
effectiveness of the ASE-depleting-channel approach. Moreover, the advantage of this
approach is the computation speed. In fact, ASE-
depleting-channel simulations are twice as fast as those for the multichannel ASE case (see
Tables II and III). The use of more than three
stages does not improve accuracy.
E. WDM Amplification
In order to verify the efficiency of our model for
a wider range of simulation scenarios, we investigated the case of WDM-signal
amplification. We recall that both the Connelly
and the reservoir models are not able to reproduce carrier induced nonlinear effects such
as FWM and XPM and can only model the
effects of carrier-induced self-gain modulation and XGM.
Fig. 16 shows the measured power at the output
of our Optospeed SOA as well as simulation results using the Connelly model and (a) a one-
stage reservoir model and (b) a three-stage
model, in which the fluorescence τ was set at the
value of 360 ps to best match the measurements. The SOA was fed with four synchronously
OOK-modulated WDM signals with a
wavelength spacing of 3 nm (λ1 = 1550 nm, λ2 = 1553 nm, λ3 = 1556 nm, and λ4 = 1559 nm).
The SOA output is optically filtered so that the
ASE is eliminated, and the desired channel is selected. The optical filter is 1.2 nm wide, so its
effect on the pulse shape is negligible at an
experimental bit rate of 1 Gb/s. The average
input power of each channel is −20 dBm (experimentally, lower input power showed
noisy pulses). Under such conditions, we
observe a good match between the measurements and the Connelly model
predictions. A reasonable match is also obtained
between the experiment and the reservoir models. However, we verified that at lower input
powers, the simulations give a less exact fit.
The lack of accuracy during the transients is due
to the linear approximations of the gain and recombination rate.
Note the different slopes of measured and
simulated pulses after the overshoot in Fig. 15. We believe the reason for this to lie in the linear
approximation of R(N) is when the signal
reaches a maximum (and the carrier density
reaches a minimum), the actual time constant of the SOA is larger than that employed in (9).
Moreover, ultrafast phenomena (neglected in
this paper) will have an increasing impact for overshoots and undershoots on the order of a
few picoseconds.
44
Fig. 6.25. Response of four WDM channels (with
a spacing of 3 nm) to a square wave input (see inset showing input optical powers in dBm). (a)
Measured (dashed) and dynamic Connelly
model (solid). (b) Measured (dashed), one stage
reservoir with single channel ASE (solid with squares) and three-stage reservoir with single
channel ASE(solid with circles).
6.5. CONCLUSION
A novel state-variable SOA model that is
amenable to block diagram implementation for WDM applications and with fast execution times
was presented and discussed. We called the
novel model the reservoir model, in analogy with similar block oriented models for EDFAs
and Raman amplifiers. While ASE self-
saturation can be simply included in the EDFA
reservoir model [28], an added complexity in SOAs with respect to EDFAs is that scattering
losses cannot be neglected. These increases the
difficulty in developing a reservoir model for SOAs, and we proposed innovative solutions to
tackle the problem.
A critical step in the SOA reservoir model is the
appropriate selection of the values of its
wavelength-dependent parameters that provide a good fit with the experiments. We proposed and
described at length a procedure to extract such
parameters from the parameters of a detailed and accurate space-resolved SOA model due to
Connelly, which we extended to cope with the
time-resolved gain transient analysis. It is
important to note that our reservoir model is not entirely dependent on the space resolved
simulator. The key wavelength-dependent
parameter for the reservoir model is the material gain as a function of both wavelength and
inversion. A detailed knowledge of this
dependence allows accurate linearization around
45
the working point and hence, more accuracy for
the reservoir model. A procedure to extract the model parameters directly from the
measurements would be of great practical value.
A number of other issues remain to be explored
and deserve further research. The presence of nonzero facet reflectivity was not considered
and would be important for modeling reflective
SOAs with the reservoir. In addition, a different approximation for the recombination rate,
accounting for a reservoir-dependent time
constant, could increase the reliability of the model. In this paper, we assumed a linear
dependence of this parameter on the inversion.
A better approximation (R(r) = a1(r) + a2r2 +
a3r3 + . . .) could be obtained if we assume a constant inversion
N = r/V over the SOA length (as we did for ASE
calculations in Section III-B). However, the accuracy obtained with such approximations will
be at the cost of slower execution time.
The raison d’être of the reservoir model is to find a tradeoff between accuracy and calculation
speed. To achieve this goal, we considered
several variations of the model, with increasing
complexity, which allow the accurate inclusion of both scattering losses and gain saturation
induced by ASE. To speed up the emulation of
transmission of long bit sequences in the reservoir model, we introduced a single
equivalent input ASE channel with appropriate
power and gain parameters, which feeds a
noiseless reservoir model to give equivalent dynamics.
We showed that at a comparable accuracy, the
reservoir model with the single ASE channel can be 20 times faster than the
Connelly model in single-channel operation and
much more significant time savings are expected for WDM operation. The accuracy of the model
is limited to modulation rates per channel not
exceeding 10 Gb/s since ultrafast phenomena
such as CH and SHB are neglected. However, such rates are of interest for next-generation
metropolitan optical networks. In addition,
beating-induced carrier gratings that generate FWM and XPM in SOAs are not captured by the
reservoir model, which then is reliable whenever
XGM dominates over such effects. The true value of the SOA reservoir model is that
together with block diagram descriptions of
EDFA and Raman amplifiers, it provides a
unique tool with reasonably short computation times.
Cross-gain modulation Chapter 7
46
The material gain spectrum of an SOA is
homogenously broadened. This means that
carrier density changes in the amplifier will affect all of the input signals. The carrier
density temporal response is dependent on the
carrier lifetime. As
discussed in the
preceding chapter,
carrier density changes
can give rise to pattern
effects and interchannel
crosstalk in
multiwavelength
amplification. The most
basic cross-gain
modulation (XGM)
scenario is shown in Fig.
7.01 where a weak CW
probe light and a strong pump light, with a
small-signal harmonic modulation at angular
frequency ω are injected into an SOA. XGM
in the amplifier will impose the pump
modulation on the probe. This means that
the amplifier is acting as a wavelength
converter, i.e. transposing information at one
wavelength to another signal at a different
wavelength.
The most useful figure of merit of the
converter is the conversion efficiency η
which is defined as the ratio between the
modulation index of the output probe to the modulation index of the input pump.
Semiconductor optical amplifiers (SOA) display nonlinear optical response on short
time scales which arises from the changes
induced by the injected optical field in both the
total carrier density and its distribution over the energy bands. These ultrafast optical
nonlinearities may allow for efficient all-
optical signal processing. Actually, all-optical wavelength conversion of the signal, data-
format translation and add-drop functionalities
have been demonstrated by using SOAs via cross-gain modulation (XGM), cross-phase
modulation, or four-wave mixing. SOAs are
usually of the travelling-wave type, which
maximizes the optical bandwidth by strongly suppressing the ripples due to facet
reflectivities.
In order to suppress the sensitivity to
polarization inherent to planar structures,
especially when quantum-well active regions are used, SOAs require specific designs that
make their coupling efficiency to optical fibers
quite low.
All-optical wavelength converters are
expected to become key components in future
broadband networks. Wavelength conversion
techniques include cross-gain modulation (XGM) or cross-phase modulation (XPM) in
semiconductor optical amplifiers (SOA), four-
wave mixing (FWM) in passive waveguides, SOAs, or semiconductor lasers, gain-
suppression mechanism in the semiconductor
lasers such as DBR lasers, and T-Gate lasers,
laser-based wavelength conversion, and difference frequency generation (DFG).
Optical XGM in SOAs has been
intensively studied in the past. However, there are relatively few papers on XGM in
semiconductor lasers, especially small-signal
modulation. An intensity-modulated input signal at a
pump wavelength λ2 is used to modulate the
carrier density and consequently also the gain
of a test laser due to gain saturation. In the test laser, a continuous wave (CW) beam at desired
test wavelength λ1 (called the test signal) is
modulated by the gain variation. In this way, information is transferred from the pump
wavelength to the test wavelength. The XGM
response, which is obtained by pumping in the
gain region of the quantum wells (QWs), is of great practical significance for wavelength
conversion. The modulation response in this
case will suffer virtually no adverse transport
Fig 7.01 (Simple Wavelength converter using XGM in SOA)
47
7.1
effects; hence, the response is practically
intrinsic in nature, and shows a clear picture of the physical interactions taking place in the
semiconductor laser. Our theoretical model
also focuses on small-signal analysis, which is
used to study the modulation bandwidth or wavelength conversion speed. If one is
interested in bit-error rate, however, a large-
signal approach is required. Several groups have measured the optical-absorption
modulation response of a semiconductor laser
for optical pumping within the QW region, where the pump photons create electron-hole
pairs as they are absorbed. The newly created
carriers relax into the lower states of the QW,
modulating the QW carrier density and the laser output. When the optical pump
wavelength is within the gain region of the test
laser, the pump signal will be amplified through stimulated recombination of carriers
rather than the creation of carriers through
absorption. The amplification of the pump signal will have two major effects. First, the
carrier lifetime will decrease because of
stimulated recombination. Second, the test-
laser intensity will decrease at a given bias when the pump signal is injected. The test-
laser photon density and carrier lifetime
significantly impact the modulation response of the laser. Moreover, there are effects which
arise from cross-gain saturation due to the
presence of more than one intense laser field
which can also influence the modulation response.
Consider a pump laser (denoted by the
subscript 2) with a photon density S2 competing for the gain with a test laser
(denoted by the subscript 1) with a photon
density S1. The rate equations for the carrier density N(1/cm
3 ) and the photon density
S1(1/cm ) of the lasing mode (test signal) are
where, I test-laser current;
V volume of the active region;
q unit charge of the carrier;
τn carrier lifetime;
νg group velocity;
τp photon lifetime;
Γ optical confinement factor;
G1,2 gain at the test and pump laser wavelength,
respectively.
In order to take into account the effects of nonlinear gain suppression with cross-gain-
saturation, we include ε11 and ε22,which are
the self-nonlinear gain saturation coefficients, and ε12 and ε21, which are the cross-nonlinear
gain saturation coefficients. The cross-
saturation properties of the gain due to pump-test-laser interactions describe how the pump
and test signals interact with each other in the
active region. The gain suppression at a
wavelength λ1 will be due to the presence of both the test and pump photon densities,
although not necessarily to the same degree.
The spontaneous emission term has been neglected because the test laser is above
threshold.
A. Steady-State Solution In the steady state, the time-varying terms
are set to zero in the rate equations 7.1 and 7.2.
The equation for the photon density is used to define the steady-state gain–loss relation
For simplicity in notation, capital letters S1
and S2 stand for steady-state values. The equation for the carrier density can also be
used to solve for the light–current (L-I)
characteristics of the test laser, after setting the time-varying terms to zero
where, is the original
threshold current without an external pump.
With cross saturation, the L-I relationship may not behave as a simple, linear function. For a
given test-laser current I, the photon density of
the test-laser S1 will be less than what it would
be if S2 were not present, since the pump competes for carriers, causing both a shift in
threshold for the test laser and a change in the
slope of its L-I curve.
7.2
7.3
7.4
48
7.8
7.10
7.11
7.12
7.14
7.15
7.16
7.17
7.9
7.18
B. Small-Signal Solution
In this section, the changes in the lasing mode photon densities and carrier density due
to the pump signal variation are assumed to be
much smaller than the steady-state value of the
photon and carrier densities. To solve for the small-signal modulation response, the
expressions for carrier and photon densities are
and by linearizing the gain function
where g’1,2 is the differential gain at wavelength λ1 or λ2. For the small-signal
analysis, the quantity N-N0 will equal the
small-signal change in carrier density, denoted
by η. Taylor’s series expansion is used to simplify
the small-signal form of the rate equations.
Note that the source of modulation is the pump photon density. Terms containing products of
steady-state and small-signal components are
linearized, and only first-order terms are
retained. The small-signal rate equations can be expressed as follows:
After eliminating the carrier density n and
solving for s1/s2, the response is obtained
Where, the numerator N(ω) is
in which the effective carrier lifetime τn’ due
to stimulated recombination by the pump S2 is defined as
and the cross-gain saturation term X is
Now the damping factor can be defined,
after simplification, as
and the resonant frequency squared may be
written as
or, replacing 1/τn’ by 1/τn using 7.13
where,
The expression for the damping factor remains almost the same, except for the
reduced carrier lifetime. The relaxation
frequency ( ωr = 2πfr), however, depends on
pump laser photon density S2. The overall response is simply the “intrinsic” form of the
response in the denominator, but with different
values defining the relaxation frequency ωr and the damping factor γ. Equations 7.13, 7.15,
and 7.16 indicate new analytical results on the
effective inverse carrier lifetime (1/τn’), γ and ωr respectively. The numerator N(ω) remains
almost constant within the frequency range of
7.5
7.6
7.7
7.13
49
7.12
interest. As a final step, the overall response is
normalized, and the magnitude is written as
The equations are summarized in Table
7.01. The expressions for the conventional intrinsic small-signal modulation response are
also listed in table 7.01for comparison. I
should be noted that the two sets of modulation responses are identical when the
photon density S2 approaches zero. Therefore,
the expressions for the small-signal optical
gain modulator response are actually the intrinsic modulation response of the
semiconductor laser and are useful in studying
the physics of XGM.
Table 7.01: COMPARISON OF INTENSITY MODULATION RESPONSES: INTRINSIC AND XGM
50
Table 7.02: Structure of a Common Test Laser fro lab use
Work Done Chapter 8
51
Analysis of the performance of the
Semiconductor Optical Amplifier includes
several phases of realization of the behaviour
of the same under different input signal
condition.
In our case as the final result should
have been based on the simulation under input
as 4 signals WDM multiplexed with one
reference wave and the whole signal path
including 3 SOAs connected as a ring network
with standard difference between tow SOA of
about 200 KM, we divided our simulation
development into 3 phases to ease out the
difficulties arising due to programming
complexities.
The three phase are:-
1. Simulation with 1 signal, 1 continuous
wave (as reference1), and signal
passing through only 1 SOA.
2. Simulation with 4 signals (WDM
multiplexed), 1 continuous wave and
signal passing through 1 SOA.
3. Simulation with 4 signals (WDM
multiplexed), 1 continuous wave as
reference and signal path consisting of
a ring network containing 3 SOAs.
Now, while realizing the SOA based
WDM ring network with identical SOAs the
SOA, though show identical performance for
each of the stages, each SOA show different
behaviour for different signals input onto it,
possibly caused due to the cross-gain
modulation. This cross-gain modulation can be
observed as soon as several signals are fed into
the SOA. Now, on the way to describe each of
the phases to the simulation we need to give
the theory used by us to generate the code for
the simulation, and then to make the procedure
understandable, we are going to describe it
through folw charts of each of the simulation
stages.
1. SOAs describe the effect cross-gain modulation the variation
of gain for one signal due amplification of the other, which is
easily determined by feeding an extra continuous wave onto
the SOA as reference signal.
Phase 1
In this phase we are only sending one
signal of single pulse through a single
SOA along with a continuous wave
reference signal. So, before realizing the
SOA for the simulation we need to define
some parameters used for defining the
SOA using the reservoir model of SOA as
described earlier. These are as follows:-
Table 8.01
symbol Parameter name value
Length of the SOA 1300 µm
Number of sections 3
Velocity of light 3 x 108 m/s
Planck‟s constant 6.626068 x
10-34
Active region width 0.7 µm
Input facet reflectivity 0.9 x 10-6
Output facet reflectivity 0.5 x 10-6
Carrier independent
absorption loss
coefficient
6000 m-1
Active region thickness 0.7 µm
Carrier dependant
absorption loss
coefficient
6000 x 10-24 m2
Linear radiative
recombination coefficient
3.5 x 108 s-1
Bimolecular radiative
recombination coefficient 4 x 10-16 m3s-1
Linear non-radiative
recombination coefficient 7.5 x 108 s-1
Bimolecular non-
radiative recombination
coefficient
7.5 x 10-16 m3s-1
Band gap energy 0.773 eV
52
The program flowchart is as follows :-
Fit Parameters:-
Now, the primary parameters of the SOAs
are defined, not our primary objective is to
describe the SOA by the equations given in the
reservoir model. On this step, the first way is
to get the fit parameters out the given data for
the SOA simulation:-
Wavelength of the launched power: - 1550nm
Length of the SOA: - 10-4
m Pump Current: - 0.25 A
Fundamental mode confinement factor: -0.36
The given can be implemented using the following matlab function:-
function [sigma N0k gamma N1k]=fit_parameter(Wavel,L,I,Con_F)
Carr_Den = 0.5:.1:3.5; inguess = 1; Pin_low=-40; Pin_high=0;
for i=1:length(Carr_Den) gain_res(i) = matgain_res (Wavel,Carr_Den(i));
end;
fit_data=polyfit(Carr_Den,gain_res,1);
Nmax =fzero(@(N)calculate_avg(N,Pin_low,Wavel,fit_data,I,L,Con_F),inguess)
Nmin =fzero(@(N)calculate_avg(N,Pin_high,Wavel,fit_data,I,L,Con_F),inguess)
sigma = ( matgain_res (Wavel,Nmax*1E-24) - matgain_res (Wavel,Nmin*1E-24) )
/(Nmax-Nmin);
53
N0k = Nmax - (matgain_res(Wavel,Nmax*1E-24)/sigma); gamma = ( mat_gbar_res (Wavel,Nmax*1E-24,L,Con_F) - mat_gbar_res
(Wavel,Nmin*1E-24,L,Con_F) ) / ( Nmax-Nmin ) ;
N1k = Nmax - mat_gbar_res(Wavel,Nmax*1E-24,L,Con_F)/gamma;
As the function signifies, the function,
known by the name “ fit_parameter”, has 4 arguments, namely:-
1. Wavel :- the wavelength of the input
signal; 2. L :- the length of the SOA;
3. I :- input pump current;
4. Con_F :- fundamental mode confinement factor;
The function outputs the fit parameter for the given input signal. Now, we can define the
parameters by the previously derived
equations by Connelly as:-
σk = ( matgain_res (Wavel,Nmax*1E-24) - matgain_res (Wavel,Nmin*1E-24) )
/(Nmax-Nmin);
where,
Nmax= fzero(@(N)calculate_avg(N,Pin_low,Wavel,fit_data,I,L,Con_F),inguess);
&
Nmin= fzero(@(N)calculate_avg(N,Pin_high,Wavel,fit_data,I,L,Con_F),inguess);
Here, the function “fzero” gives the initial
value of the function defined within it, in this case be it “calculate_avg.”
“calculate_avg” is a function that is used to
calculate the average value of the dN(z, t)/dt , described in the Connelly model of SOA. The
function “calculate_avg” is defined as under:-
function out = calculate_avg(N,PindB,Wavel,fit_data,I,L,Con_F);
Pin = dbtoc(PindB); q = 1.602177E-19; % Electronic charge (Coulomb) d = 0.7E-6; % Active region thickness (m) W = 0.7E-6; % Central active region width (m)
Arad = 3.5E8; % Linear radiative recombination coefficient
(S^-1) Brad = 4E-16; % Bimolecular radiative recombination
coefficient (m^3*s^-1) Anrad = 7.5E8; % Linear non-radiative recombination
coefficient due to traps (S^-1) Bnrad = 7.5E-16; % Bimolecular non-radiative recombination
coefficient (m^3*s^-1) Caug = 0.2E-42; % Auger recombination co-efficient (m^6*s^-1)
h = 6.6260755E-34; % Planck's constant (J*s) vel_light = 2.99792458E8; % Velocity of light (m/s) Freq_use = vel_light/(Wavel*1E-9); Qin = Pin/(h*Freq_use);
% ========= Calculation of Wavelength Dependent Gain ========= % % ============================================================ % K = 1.3806505E-23; % Boltzmann constant (Joule/kelvin)
54
T = 300; % Absolute temperature (kelvin) n1 = 3.22; % InGaAsP active region refractive index neq0 = 3.22; % Equivalent refractive index at 0 carrier density delneq_n = -1.34E-26; % Differential of equivalent refractive index WRT
carrier density (m^-3) me = 4.10E-32; % Effective Mass of Electron in CB (Kg) mhh = 4.19E-31; % Effective Mass of a heavy hole in VB (Kg) mlh = 5.06E-32; % Effective Mass of a light hole in VB (Kg) Kg = 0.1E-10; % Bandgap shrinkage coefficient (eVm) R1 = 0.9E-6; % Input facet reflectivity R2 = 0.5E-6; % Output facet reflectivity Eg0 = 1.237E-19; % Bandgap Energy deln1_n = -1.8E-26; % Differential of active region
refractive index WRT carrier density (m^-3)
h1 = h/(2*pi); mdh = (mhh^(3/2) + mlh^(3/2))^(2/3); Nc = 2 * (((me*K*T) / (2*pi*h1^2)) ^ (3/2)); Nv = 2 * (((mdh*K*T) / (2*pi*h1^2)) ^ (3/2)); Delta = N/Nc; Efsilon = N/Nv;
Efc = (log(Delta)+Delta*(64+0.05524*Delta*(64+sqrt(Delta)))^(-1/4))*K*T; Efv = -(log(Efsilon)+Efsilon*(64+0.05524*Efsilon*(64+sqrt(Efsilon)))^(-
1/4))*K*T;
Del_EgN = q*Kg*(N^(1/3)); EgN = Eg0 - Del_EgN;
Ea = (h*Freq_use - EgN) * (mhh/(mhh + me)); Eb = - (h*Freq_use - EgN) * (me/(mhh + me)); fcF = (exp((Ea - Efc)/(K*T))+1)^-1; fvF = (exp((Eb - Efv)/(K*T))+1)^-1;
Tow = 1/(Arad + N*Brad); % From Equation 52 [Connelly] gain = ((vel_light^2)/(4*sqrt(2)*(pi^(3/2))*(n1^2)*Tow*(Freq_use^2)))... *(((2*me*mhh)/(h1*(me+mhh)))^(3/2))*sqrt(Freq_use-EgN/h)*(fcF-fvF); % ============================================================ %
f1 = I/(q*L*d*W); % First Part of the Steady-state Equation f2 = N * (Arad+Anrad+Brad*N+Bnrad*N+Caug*N^2); % Second Part of the
Steady-state Equation
f4 = (Con_F/(d*W))*Qin*polyval(fit_data,N)*1E-24; %Third Part of the
Steady-state Equation f3 = (Con_F/(d*W))*Qin*matgain_res(1560,2);
matgain_res(1560,2);
out = f1-f2-f4;
and the function “matgain_res” gives the material gain value of the current active
medium, under the given particular condition
given as the argument. This function is defined as :-
function out = matgain_res(Wavel,Carr_Den);
55
N = Carr_Den*1E24; wavel_use = Wavel*1E-9;
% Important parameters needed to calculate the material gain % ========================================================== h = 6.6260755E-34; % Planck's constant (J*s) me = 4.10E-32; % Effective Mass of Electron in CB (Kg) mhh = 4.19E-31; % Effective Mass of a heavy hole in VB (Kg) mlh = 5.06E-32; % Effective Mass of a light hole in VB (Kg) K = 1.3806505E-23; % Boltzmann constant (Joule/kelvin) T = 300; % Absolute temperature (kelvin) e = 1.602177E-19; % Electronic charge (Coulomb) Kg = 0.1E-10; % Bandgap shrinkage coefficient (eVm) vel_light = 2.99792458E8; % Velocity of light (m/s) Arad = 3.5E8; % Linear radiative recombination coefficient (S^-1) Brad = 4E-16; % Bimolecular radiative recombination coefficient (m^3*s^-1) n1 = 3.22; % InGaAsP active region refractive index Eg0 = 1.237E-19; % Bandgap energy
% ==========================================================
h1 = h/(2*pi); mdh = (mhh^(3/2) + mlh^(3/2))^(2/3); Nc = 2 * (((me*K*T) / (2*pi*h1^2)) ^ (3/2)); Nv = 2 * (((mdh*K*T) / (2*pi*h1^2)) ^ (3/2)); Delta = N/Nc; Efsilon = N/Nv;
Efc = (log(Delta)+Delta*(64+0.05524*Delta*(64+sqrt(Delta)))^(-1/4))*K*T; Efv = -(log(Efsilon)+Efsilon*(64+0.05524*Efsilon*(64+sqrt(Efsilon)))^(-
1/4))*K*T;
Del_EgN = e*Kg*(N^(1/3)); EgN = Eg0 - Del_EgN;
Freq_use = vel_light/wavel_use; Ea = (h*Freq_use - EgN) * (mhh/(mhh + me)); Eb = - (h*Freq_use - EgN) * (me/(mhh + me)); fcF = (exp((Ea - Efc)/(K*T))+1)^-1; fvF = (exp((Eb - Efv)/(K*T))+1)^-1;
Tow = 1/(Arad + N*Brad); % From Equation 52 [Connelly] out = ((vel_light^2)/(4*sqrt(2)*(pi^(3/2))*(n1^2)*Tow*(Freq_use^2)))... *(((2*me*mhh)/(h1*(me+mhh)))^(3/2))*sqrt(Freq_use-EgN/h)*(fcF-fvF);
In this way, we can also define the other three fit parameters as :-
N0k = Nmax - (matgain_res(Wavel,Nmax*1E-24)/sigma);
gamma = (mat_gbar_res(Wavel,Nmax*1E-24,L,Con_F)-mat_gbar_res(Wavel,Nmin*1E-
24,L,Con_F))/(Nmax-Nmin); N1k = Nmax - mat_gbar_res(Wavel,Nmax*1E-24,L,Con_F)/gamma;
56
Input Signal :-
Here, we need to parameterize the input
signal to be fed into the SOA. Now the several
specification we are abiding by are :-
1. Modulation frequency(Fm)=0.5E9 2. Samples per bit = 80
3. Duty ratio = 1
The number of bits for the simulation is
chosen to be 1 with the consent of the project
guide
We define the input signal with the following
coding:-
[ti,sig]=signalin(no_of_bits,Fm,samples_per_bit,Duty_ratio,effective_power) sig_voltage1 = sig1;
the function “signalin” is a
user defined function defined
to generate a two dimensional
array of time and signal
amplitude that take input
argument as number of bits,
modulation frequency of the
input wave, samples to be
drawn per bit, duty ratio of
the step pulse to be
generated and the effective power. The function
“signalin” can realize as:-
function
[ti,sig]=fn1(a1,a2,a3,
a4,a5);
% Initialization of
the parameters for
different Modulation
Formats Fig: - 8.01 % ================================================================= no_of_bits = a1; Fm = a2; %Modulation frequency in bps samples_per_bit = a3; Duty_ratio = a4; effective_power = a5; %Power launched in dBm bit_period = 2/Fm; %Bit period of the modulated signal N = no_of_bits*samples_per_bit; %Total no of samples sample_interval = bit_period/samples_per_bit; %Sampling period (1/Fs)
P_av = (10^(effective_power/10))*(10^(-3));%Average launched power in watts P_peak = P_av/Duty_ratio;
57
%%%%%% GENERATION OF SIGNAL %%%%%%%%%%%%% %=======================================% % bit_pattern = pnseq7(no_of_bits); T0 = 0.5E-9; m = 20; %Defines the super-gaussianity of the envelope
point_array=[-(bit_period/2):sample_interval:(bit_period/2)-
sample_interval]; gauss_env = exp(-0.5*((point_array./T0).^(2*m))); %&&&&&&&&&&&&&&&&&&&&&&&& figure; plot(point_array,gauss_env);
T = 0; sample_point = 1; A = zeros(1,N); DUTY = round(samples_per_bit*Duty_ratio); Amp=P_peak; %Amplitude of the envelope of the lunched electric field
for (n=1:no_of_bits) du = 1; for(t = T:sample_interval:T + bit_period - sample_interval if (du <= A(1,sample_point) = Amp*gauss_env(1,du); sample_point = sample_point + 1; else
A(1,sample_point) = 0; sample_point = sample_point + 1; end; du = du + 1; end; T = T + bit_period; end;
ti=0:sample_interval:(sample_interval*N)-sample_interval; sig=A;
the generated wave look like one in the Fig:-
8.01. The figure, based on a Gaussian envelop
defined as :- e (-0.5*((point_array./T0).^(2*m)));
where, the point_array is defined over the bit
period with a resolution of the sample interval.
Introduce Continuous Wave:-
Now, here we are to introduce a continuous
wave signal so that we can witness the
phenomenon of the cross-gain modulation, as:- PindB_cw = -6; pav= (10^(PindB_cw/10))*1E-3; L4=length(p1); for (k=1:L4-1) p_cw(k)= pav; end
Solve Rate Equation :- As the continuous wave is defined already, we
noe can proceed towards the solution of the
rate equation defined in the Reservoir model, as:-
The solution of this rate equation gives us the
rate of generation of the carrier in the SOA. To
solve the equation with the matlab function “ode45” we need to first determine the value
of the function at t=0; i.e., the initial condition.
To determine that we use a matlab function
58
named “fzero”. This is used in the simulation
as :- inguess=1;
y1=fzero(@(r)sol_ase(r,Lz,p1(1),pin2(1),pav,lamdak,sigmaK,nok,gammaK,n1k,gamma,I,
lifetime), inguess);
Here, the function sol_ase gives the value of the of the rate equation. This is defined by the codes below :-
function reservoir=sol_ase(r,Lz, Pin,Pin2, p_cw,lamdak,sigmaK,
nok,gammaK,n1k,gamma,I,lifetime) deltanu = 30*(1E-9)/130 ; % assumed SOA BW 30 nm %---------------- q= 1.602E-19; % electronic charge in [C] c= 2.99E8; % velocity of light [m/s] h= 6.626068E-34 ; % Planck Constant in [m^2-kg/s] A= ( 0.7^2)*1E-12; % area in [m^2] ko= 6000; %absorption constant indep of n in [m^-1] k1= 6000*1E-24 ; %k1=6000*1E-24 ; in m^2 ak= (gamma*(sigmaK-k1))/A ;% gain constant of the reservoir model,unitless. rok= ((gamma*sigmaK*nok + ko)*Lz)/ak ; % in [m^-2] r1k= n1k *(A*Lz); Qin=(lamdak*Pin*1E-9)/ (h*c); Qin2=(lamdak*Pin2*1E-9)/ (h*c); Qin_cw=(lamdak*p_cw*1E-9)/ (h*c); nsp = ( gamma*gammaK*(r-r1k))/(A*ak*(r-rok)); Gr=exp(gamma*r*(sigmaK -k1)/A - (gamma*sigmaK*nok+ko)*Lz );
reservoir= I/q - r/lifetime- (Qin +Qin2+ Qin_cw)*( exp(ak*(r-rok))-1 )-
4*deltanu * nsp*(Gr -1-log(Gr));
Now, as the initial value of the is derived, we can proceed towards the solution
of it. We shall solve it for every instance of
time over the timespan of the wave for each stage of the SOA separately (we devided the
SOA in three stages or sections of same
length). The approach passes the arguments, i.e. the power readings of the signal, the same
of the continuous wave, the length of each
section, the fit parameters, the pump current, the lifetime of the carriers and the initial value
to
the function “solve_rateq_ase” and the
function returns the in a mere time versus amplitude array. The amplitude signifies the
rate of generation of carrier in the bulk
material of SOA, which can be simply passed on to a separate user-defined function named
“single_pass_gain_ase” to calculate the gain.
This function calculates the gain of the SOA for a single pass or stage. The codes to realize
the solution of the rate equatin can be
described as follows :-
j=1; for (i= 1:1:T1-1) a=0; b=0; [a,b] =
solve_rateeq_ase(p(i),p_cw(i),Lz,lamdak,sigmaK,nok,gammaK,n1k,gamma,I,
lifetime,t1(i),t1(i+1),y1); % return 'a' as time and 'b' as r values L2=length(b); % length of the each time division y1=b(L2); % y1=b(L2);final value of the solution of differential
eq.for that time division r1(j)=b(1); % r1(j)=b(1);collecting one value of 'r' in each pass
of the solution of diff. Eqn
59
x2(j)=a(1); % x2(j)=a(1);collecting time instants. arrayr(k,j)=r1(j); j=j+1; end
The rate of generation of carrier is outputted by the array „r1‟. Now the function
“single_pass_gain_ase” shall calculate the gain
including the noise power of ASE. The
function can be realized as follows :-
% calculation of gain function value1 = single_pass_gain_ase(r,Lz,sigmaK, nok,gamma); A= ( 0.7^2)*1E-12; ko= 6000; % 6000 m^-1 k1= 6000*1E-24 ; %k1=6000*1E-24 ; in m^2 ak= (gamma*(sigmaK-k1))/A ; rok= ((gamma*sigmaK*nok + ko)*Lz)/ak ; % in [m^-2] value1 = exp(ak*(r - rok)); % Value1 is Gk, which is a function of time 't'
at wavelength lamda 'k'
Gain Calculation:- Once the gain is out it is easy to calculate
the power output
for each of the stage of SOA by just
multiplying the gain
to the input power
as the input power to each stage is
signified by the
power out by the previous stage.
As the described in
the Fig 8.02, the graph gives the
generated carriers
in the reservoir, for
three consecutive stages. Similarly,
the next figure Fig
8.03 gives the amount of photons
generated in the
reservoir, which
give the idea of the gain for that signal.
In the figure the
photons generated due to the signal is described by the blue graph while the red one
gives the amount of photons generated due to
the inserted continuous wave. Now, we can define an three dimensional array to store the
values of the gain versus time for three stage
simultaneously, multiplying which to the input
signal of each stage we get the output power for each stage.
As shown in the Fig 8.04, 8.05 & 8.06, the figures give the amount of power outputted
from the 1st, 2
nd, 3
rd stage respectively.
Fig 8.02
60
Fig 8.03
Fig 8.04
61
Fig 8.05 Hence, we are successful to realize a simple
SOA with 1 signal & 1 continuous reference
wave as input, and complete the Phase 1 of our
simulation.
Fig 8.06
62
Phase 2
Our second phase of simulation continues
with the simulation with 4 input signals,
instead of only one for the previous one, the
same previously added continuous wave signal
that is used for reference to the effect of cross-
gain modulation and a single SOA to pass by.
The simulation starts with those previously
described coding but the only difference is the
input. The input is WDM multiplexed.
In fiber-optic communications, wavelength
division multiplexing (WDM) is a technology
which multiplexes a number of optical carrier
signals onto a single optical fiber by using
different wavelengths (colours) of laser light.
This technique enables bidirectional
communications over one strand of fiber, as
well as multiplication of capacity.
The term wavelength-division multiplexing
is commonly applied to an optical carrier
(which is typically described by its
wavelength), whereas frequency-division
multiplexing typically applies to a radio carrier
(which is more often described by frequency).
Since wavelength and frequency are tied
together through a simple directly inverse
relationship, the two terms actually describe
the same concept. A WDM system uses a
multiplexer at the transmitter to join the
signals together, and a demultiplexer at the
receiver to split them apart. With the right type
of fiber it is possible to have a device that does
both simultaneously, and can function as an
optical add-drop multiplexer. The optical
filtering devices used have traditionally been
etalons, stable solid-state single-frequency
Fabry–Pérot interferometers in the form of
thin-film-coated optical glass.
Now, here we are multiplexing 4 signals in
the SOA, with there carrier optical signal
wavelength as 1548 nm, 1552 nm, 1556 nm,
1560 nm.
So, the coding includes several stages of
the previous codings edited.
1. Firstly we are to derive the fit
parameters for different wavelength by
passing them on to the function
“fit_parameter” and saving the fit
parameters for different fit parameters
for different wavelength under different
variable length, like the following :-
[a11, a12, a13, a14]=
fit_parameter(lamdak1,L,I,gamma)
; sigmaK1= a11 ; nok1= a12 ; gammaK1= a13 ; n1k1= a14;
[a21, a22, a23, a24]=
fit_parameter(lamdak2,L,I,gamma)
; sigmaK2= a21 ; nok2= a22 ; gammaK2= a23 ; n1k2= a24;
[a31, a32, a33, a34]=
fit_parameter(lamdak3,L,I,gamma)
; sigmaK3= a31 ; nok3= a32 ; gammaK3= a33 ; n1k3= a34;
[a41, a42, a43, a44]=
fit_parameter(lamdak4,L,I,gamma)
; sigmaK4= a41 ; nok4= a42 ; gammaK4= a43 ; n1k4= a44;
2. as the input is ready with for signals,
the next problem comes with the
solution of the rate equation.
We can remember the rate equation that we
have already used for the coding during the
first phase of the work, there we used the
equation:-
63
The equation already contained the term
which signified input of n
number of signals numbering from 1 to
nsig. The difference was that we used only
1 signal for that phase. This time we are
using 4 signals (and the continuous wave
signal as well).
So the function to derive changes to:-
function reservoir=sol_ase(r,Lz,
Pin1,Pin2,Pin3,Pin4,p_cw,lamdak1,lamdak2,lamdak3,lamdak4,sigmaK1,sigmaK2,si
gmaK3,sigmaK4, nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma,I,lifetime)
deltanu = 30*(1E-9)/130 ; % assumed SOA BW 30 nm q= 1.602E-19; % electronic charge in [C] c= 2.99E8; % velocity of light [m/s] h= 6.626068E-34 ; % Planck Constant in [m^2-kg/s] A= ( 0.7^2)*1E-12; % area in [m^2] ko= 6000; %absorption constant indep of n in [m^-1] k1= 6000*1E-24 ; %k1=6000*1E-24 ; in m^2
ak1= (gamma*(sigmaK1-k1))/A ; ak2= (gamma*(sigmaK2-k1))/A ; ak3= (gamma*(sigmaK3-k1))/A ; ak4= (gamma*(sigmaK4-k1))/A ;% gain constant of the reservoir model,
unitless.
rok1= ((gamma*sigmaK1*nok1 + ko)*Lz)/ak1 ; % in [m^-2] rok2= ((gamma*sigmaK2*nok2 + ko)*Lz)/ak2 ; % in [m^-2] rok3= ((gamma*sigmaK3*nok3 + ko)*Lz)/ak3 ; % in [m^-2] rok4= ((gamma*sigmaK4*nok4 + ko)*Lz)/ak4 ; % in [m^-2]
r1k1= n1k1 *(A*Lz); r1k2= n1k2 *(A*Lz); r1k3= n1k3 *(A*Lz); r1k4= n1k4 *(A*Lz);
Qin1=(lamdak1*Pin1*1E-9)/ (h*c); Qin2=(lamdak2*Pin2*1E-9)/ (h*c); Qin3=(lamdak3*Pin3*1E-9)/ (h*c); Qin4=(lamdak4*Pin4*1E-9)/ (h*c);
Qin_cw=((lamdak1*p_cw*1E-9)+(lamdak2*p_cw*1E-9)+(lamdak3*p_cw*1E-
9)+(lamdak4*p_cw*1E-9))/ (h*c);
nsp1 = ( gamma*gammaK1*(r-r1k1))/(A*ak1*(r-rok1)); nsp2 = ( gamma*gammaK2*(r-r1k2))/(A*ak2*(r-rok2)); nsp3 = ( gamma*gammaK3*(r-r1k3))/(A*ak3*(r-rok3)); nsp4 = ( gamma*gammaK4*(r-r1k4))/(A*ak4*(r-rok4));
Gr1=exp(gamma*r*(sigmaK1 -k1)/A - (gamma*sigmaK1*nok1+ko)*Lz ); Gr2=exp(gamma*r*(sigmaK2 -k1)/A - (gamma*sigmaK2*nok2+ko)*Lz ); Gr3=exp(gamma*r*(sigmaK3 -k1)/A - (gamma*sigmaK3*nok3+ko)*Lz ); Gr4=exp(gamma*r*(sigmaK4 -k1)/A - (gamma*sigmaK4*nok4+ko)*Lz );
reservoir= I/q - r/lifetime- (Qin1+Qin_cw)*( exp(ak1*(r-rok1))-1 )-
(Qin2+Qin_cw)*( exp(ak2*(r-rok2))-1 )-(Qin3+Qin_cw)*( exp(ak3*(r-rok3))-1
)-(Qin4+Qin_cw)*( exp(ak4*(r-rok4))-1 )- 4*deltanu * nsp1*(Gr1 -1-
64
log(Gr1))- 4*deltanu * nsp2*(Gr2 -1-log(Gr2))- 4*deltanu * nsp3*(Gr3 -1-
log(Gr3))- 4*deltanu * nsp4*(Gr4 -1-log(Gr4));
Now, once we have determined the initial
value of the we can proceed toward
deriving its solution to get the value of „r‟ of the rate of generation of the carriers in the
reservoir. This is again done by the function
“solve_rateq_ase”, where the different parameters and 4 different signals are passed.
The function can be called as following:-
for (i= 1:1:T1-1) a=0; b=0; [a,b] =
solve_rateeq_ase(pin1(i),pin2(i),pin3(i),pin4(i),p_cw(i),Lz,lamdak1,lamdak2
,lamdak3,lamdak4,sigmaK1,sigmaK2,sigmaK3,sigmaK4,nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma,I,lifetime,t
1(i),t1(i+1),y1); % return 'a' as time and 'b' as r values L2=length(b); % length of the each time division y1=b(L2); % y1=b(L2);final value of the solution of differential
eq.for that time division r1(j)=b(1); % r1(j)=b(1);collecting one value of 'r' in each pass
of the solution of diff. Eqn x2(j)=a(1); % x2(j)=a(1);collecting time instants. arrayr(k,j)=r1(j); j=j+1; end
and realized as following:-
function [t,r] =
solve_rateeq(pw1,pw2,pw3,pw4,pcw,Lz,lamdak1,lamdak2,lamdak3,lamdak4,sigmaK1
,sigmaK2,sigmaK3,sigmaK4,nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma,I,lifetime,d
t1,dt2,yi);
deltanu = 30*(1E-9)/130 ; % assumed 30 nm SOA bw % bw=1.5 times bitrate. A= ( 0.7^2)*1E-12; % area in [m^2] ko= 6000; %absorption constant indep of n in [m^-1] k1= 6000*1E-24 ; %k1=6000*1E-24 ; in m^2 q= 1.602E-19; % electronic charge in [C] c= 2.99E8; % velocity of light [m/s] h= 6.626068E-34 ; % Planck Constant in [m^2-kg/s] ak1= (gamma*(sigmaK1-k1))/A ; % gain constant of the reservoir model,
unitless. ak2= (gamma*(sigmaK2-k1))/A ; % gain constant of the reservoir model,
unitless. ak3= (gamma*(sigmaK3-k1))/A ; % gain constant of the reservoir model,
unitless. ak4= (gamma*(sigmaK4-k1))/A ; % gain constant of the reservoir model,
unitless. %----------------------------- %finding rok rok1= ((gamma*sigmaK1*nok1 + ko)*Lz)/ak1 ; % in [m^-2] rok2= ((gamma*sigmaK2*nok2 + ko)*Lz)/ak2 ; % in [m^-2] rok3= ((gamma*sigmaK3*nok3 + ko)*Lz)/ak3 ; % in [m^-2] rok4= ((gamma*sigmaK4*nok4 + ko)*Lz)/ak4 ; % in [m^-2] %finding r1k r1k1= n1k1 *(A*Lz);
65
r1k2= n1k2 *(A*Lz); r1k3= n1k3 *(A*Lz); r1k4= n1k4 *(A*Lz);
%----------------------------- % finding Qin from input power pw Qin1 = (lamdak1*(pw1+pcw)*1E-9)/ (h*c); Qin2 = (lamdak2*(pw2+pcw)*1E-9)/ (h*c); Qin3 = (lamdak3*(pw3+pcw)*1E-9)/ (h*c); Qin4 = (lamdak4*(pw4+pcw)*1E-9)/ (h*c); %----------------------------- tspan = [dt1 dt2]; y0=yi; % initial value of the differential equation, updated in each time
interval [t,r]=ode45(@rateeq ,tspan,y0 ); function drdt = rateeq(t,r) nsp1 = ( gamma*gammaK1*(r-r1k1))/(A*ak1*(r-rok1)); Gr1=exp(gamma*r*(sigmaK1 -k1)/A -
(gamma*sigmaK1*nok1+ko)*Lz ); nsp2 = ( gamma*gammaK2*(r-r1k2))/(A*ak2*(r-rok2)); Gr2=exp(gamma*r*(sigmaK2 -k1)/A -
(gamma*sigmaK2*nok2+ko)*Lz ); nsp3 = ( gamma*gammaK3*(r-r1k3))/(A*ak3*(r-rok3)); Gr3=exp(gamma*r*(sigmaK3 -k1)/A -
(gamma*sigmaK3*nok3+ko)*Lz ); nsp4 = ( gamma*gammaK4*(r-r1k4))/(A*ak4*(r-rok4)); Gr4=exp(gamma*r*(sigmaK4 -k1)/A -
(gamma*sigmaK4*nok4+ko)*Lz ); drdt = [-r(1)/lifetime + I/q - Qin1* ( exp( ak1*(r(1)-rok1))-1) -
Qin2* ( exp( ak2*(r(1)-rok2))-1) - Qin3* ( exp( ak3*(r(1)-rok3))-1) - Qin4*
( exp( ak4*(r(1)-rok4))-1)- 4*deltanu * nsp1*(Gr1 -1-log(Gr1))- 4*deltanu *
nsp2*(Gr2 -1-log(Gr2))- 4*deltanu * nsp3*(Gr3 -1-log(Gr3))- 4*deltanu *
nsp4*(Gr4 -1-log(Gr4)) ];
end figure(2) plot(t,r); % plot for each time division hold on; % holding the plot for each time division xlabel('time in seconds'); ylabel('Number of carriers in reservoir'); end
now, we calculate the gain likewise we did in
the previous phase and determine the output power for each stage of the SOA. The
generated carrier vs. Time graph will look like
Fig 8.07, whereas the output power vs. Time
graph for 1st, 2
nd and 3
rd stage will be like Fig
8.08, 8.09 and 8.10 respectively.
Effect of Cross-gain Modulation
so far:-
From the first phase of simulation, we have observed the effect of cross-gain modulation
on the inputted signal. On reviewing the Fig
8.03, we can observe that the amplitude if the
constant amplitude continuous wave signal, that is inputted, has changed during there is a
high amplitude in the inputted optical message
signal. The amplitudes can be analysed by the
following table 8.02.
Time(10-8
sec)
Photons/sec
(x1.018
)(signal)
Photons/sec
(x1.020
)(cont.
wave)
0 0 3.2725
0.005 0 0.0134
0.01 0 0.0207
0.015 0 0.0209
0.02 0 0.0210
66
0.025 0 0.0210
0.03 0 0.0210
0.035 0 0.0210
0.04 0 0.0210
0.045 0 0.0210
0.05 0 0.0210
0.055 0 0.0210
0.06 0 0.0210
0.065 0 0.0210
0.07 0 0.0210
0.075 0 0.0210
0.08 0 0.0210
0.085 0 0.0210
0.09 0 0.0210
0.095 0 0.0210
0.1 0 0.0210
0.105 0 0.0210
0.11 0 0.0210
0.115 0 0.0210
0.120 0 0.0210
0.125 0 0.0210
0.13 0 0.0210
0.135 0 0.0210
0.14 0 0.0210
0.145 0 0.0210
0.15 5.7278 0.0210
0.155 1.3498 0.0030
0.16 1.5272 0.0034
0.165 1.8586 0.0041
0.17 1.8163 0.0040
0.175 1.8072 0.0040
0.18 1.8056 0.0040
0.185 1.8053 0.0040
0.19 1.8053 0.0040
0.195 1.8053 0.0040
0.2 1.8053 0.0040
0.205 1.8053 0.0040
0.21 1.8053 0.0040
0.215 1.8053 0.0040
0.22 1.8053 0.0040
0.225 1.8053 0.0040
0.23 1.8053 0.0040
0.235 1.8053 0.0040
0.24 1.8051 0.0040
0.245 1.7921 0.0040
0.25 1.1057 0.0040
0.255 0 0.0076
0.26 0 0.0514
0.265 0 0.0221
0.27 0 0.0206
0.275 0 0.0208
0.28 0 0.0209
0.285 0 0.0209
0.29 0 0.0210
0.295 0 0.0210
0.3 0 0.0210
0.305 0 0.0210
0.31 0 0.0210
0.315 0 0.0210
0.32 0 0.0210
0.325 0 0.0210
0.33 0 0.0210
0.335 0 0.0210
0.34 0 0.0210
0.345 0 0.0210
0.35 0 0.0210
0.355 0 0.0210
0.36 0 0.0210
0.365 0 0.0210
0.37 0 0.0210
0.375 0 0.0210
0.38 0 0.0210
0.385 0 0.0210
0.39 0 0.0210
Tabale 8.02: Reading of the photons per second graph
Cross-phase modulation can be relevant under different circumstances:
It leads to an interaction of optical
pulses in a medium, which allows e.g.
the measurement of the optical intensity of one pulse by monitoring a
phase change of the other one (without
absorbing any photons of the first beam).
The effect can also be used for
synchronizing two mode-locked lasers using the same gain medium, in which
the pulses overlap and experience
cross-phase modulation.
In optical fiber communications, cross-phase modulation in fibers can
lead to problems with channel
crosstalk. Cross-phase modulation is also
sometimes mentioned as a mechanism
for channel translation (wavelength
conversion), but in this context the term typically refers to a kind of cross-
phase modulation which is not based
on the Kerr effect, but rather on changes of the refractive index via the
carrier density in a semiconductor
optical amplifier.
67
So, the effect of the cross-gain modulation
is clearly visible.
Phase 3
On the third phase of our simulation we
will continue to edit of matlab coding by inserting 4 waves in the input along with the
previously present continuous wave and pass
the signal through a ring network consisting of 3 SOAs.
Let us first describe a bit about a ring
network, in brief.
The following Fig 8.07 best describes a
ring network. A ring network is a
network topology in which each node
connects to exactly two other nodes,
forming a single continuous pathway for
signals through each node - a ring. Data
travels from node to node, with each node
along the way handling every packet.
Because a ring topology provides only one
pathway between any two nodes, ring
networks may be disrupted by the failure
of a single link. A node failure or cable
break might isolate every node attached to
the ring.
A ring network has some advantages as:-
Very orderly network where every device has access to the token and the
opportunity to transmit
Performs better than a bus topology
under heavy network load Does not require network server to
manage the connectivity between the
computers
But it also has some disadvantages as :-
One malfunctioning workstation or
bad port in the MAU can create
problems for the entire network
Moves, adds and changes of devices can affect the network
Network
adapter cards and MAU's are much
more expensive
than Ethernet cards and hubs
Much
slower than an
Ethernet network under normal load
Now, for an SOA
based ring network
simulation, we
assume that the
signal is flowing from 1 SOA to
another but, an
optical path realized in this
way does not
seem to have the same loosy
characteristics
like a practical
one. So, we add some attenuators in
between the SOAs to practically
realize the network.
68
In case we assume the optical signal to have a
bandwidth of 1550 nm, or more generically said to be
operating in the
third window,
most of the fibres present
industrially
show an attenuation of
0.2 dB/km. So,
if we assume a gap of 200 km
between two
consecutive
fibres, there will be total 40
dB loss of the
signal to reach
another SOA, or the signal amplitude will be 1/10000 time
that of the signal launched. So the actual ring
network we are using to simulate can be
represented as Fig 8.08.
Now, we can move forward
to briefly describe our final
coding to achieve this goal. Our algorithm of the total simulation can
be described as :-
69
As the coding is a replication of the
previously used coding, we do not go into detail of the description and start writing the
actual coding used:-
% Cross gain modulation %---------------------------------------- % if confinement factor is increased to say 0.3 there is a sharp transient % in rise and fall time. Also power output considerably increases with the % increase of gamma. %-------------------------------- clear all; clc; % dividing SOA length into number of sections L= 10E-4; % 10E-4length of SOA in [m] sections = 3; % number of sections i.e. number of loops to be
run Lz= L/sections ; % length of each subdivisions %--------------------------- c= 3E8; % velocity of light [m/s] h= 6.626068E-34 ; lamdak1=1548; % wavelength in [nm] lamdak2=1552; % wavelength in [nm] lamdak3=1556; % wavelength in [nm] lamdak4=1560; % wavelength in [nm] gamma= 0.36; %Confinement factor; Note gamma has been made
to 0.8 to fit with the curve in Fig 11. A= ( 0.7^2)*1E-12; % area in [m^2] ko= 6000; %absorption constant indep of n in [m^-1] k1= 6000*1E-24 ; I= 0.25; % 0.25 A lifetime= 0.310E-9; % in [s], lifetime 310 pS , %&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& % parameter extraction [a11, a12, a13, a14]= fit_parameter(lamdak1,L,I,gamma); sigmaK1= a11 ; nok1= a12 ; gammaK1= a13 ; n1k1= a14; [a21, a22, a23, a24]= fit_parameter(lamdak2,L,I,gamma); sigmaK2= a21 ; nok2= a22 ; gammaK2= a23 ; n1k2= a24; [a31, a32, a33, a34]= fit_parameter(lamdak3,L,I,gamma); sigmaK3= a31 ; nok3= a32 ; gammaK3= a33 ; n1k3= a34; [a41, a42, a43, a44]= fit_parameter(lamdak4,L,I,gamma); sigmaK4= a41 ; nok4= a42 ; gammaK4= a43 ; n1k4= a44;
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& % Control Parameters for constructing supergaussian pulse no_of_bits = 1;
70
Fm=0.5E9; samples_per_bit = 80; % 1350 number of time domain points
Duty_ratio1 = 1; Duty_ratio2 = 1; Duty_ratio3 = 1; Duty_ratio4 = 1;
effective_power1 = -3; %peak power launched in -20 dBm (4,5,15,-
12) effective_power2 = -3; effective_power3 = -3; %peak power launched in -20 dBm (4,5,15,-
12) effective_power4 = -3; %peak power launched in -20 dBm (4,5,15,-
12)
[ti,sig1]=signalin(no_of_bits,Fm,samples_per_bit,Duty_ratio1,effective_powe
r1); sig_voltage1 = sig1;
[ti,sig2]=signalin(no_of_bits,Fm,samples_per_bit,Duty_ratio2,effective_powe
r2); sig_voltage2 = sig2;
[ti,sig3]=signalin(no_of_bits,Fm,samples_per_bit,Duty_ratio3,effective_powe
r3); sig_voltage3 = sig3;
[ti,sig4]=signalin(no_of_bits,Fm,samples_per_bit,Duty_ratio4,effective_powe
r4); sig_voltage4 = sig4;% converting peak power in terms of voltage for 1/0 bit
pattern
figure(1); % plot of single bit
plot(ti(1:samples_per_bit),sig1(1:samples_per_bit), 'b'); hold on;
plot(ti(1:samples_per_bit),sig2(1:samples_per_bit), 'g'); hold on;
plot(ti(1:samples_per_bit),sig3(1:samples_per_bit), 'k'); hold on;
plot(ti(1:samples_per_bit),sig4(1:samples_per_bit), 'r'); hold on; grid on; xlabel('Time in Seconds'); ylabel('Input Pulse Amplitude in volts');
B1=10*log10(sig1*(10^3)); B2=10*log10(sig2*(10^3)); B3=10*log10(sig3*(10^3)); B4=10*log10(sig4*(10^3));
figure(7);
plot(ti(1:samples_per_bit),B1(1:samples_per_bit), 'b'); % comment this for
bit stream hold on;
71
plot(ti(1:samples_per_bit),B2(1:samples_per_bit), 'g'); % comment this for
bit stream hold on;
plot(ti(1:samples_per_bit),B3(1:samples_per_bit), 'k'); % comment this for
bit stream hold on;
plot(ti(1:samples_per_bit),B4(1:samples_per_bit), 'r'); % comment this for
bit stream hold on; grid on; xlabel('Time in Seconds'); ylabel('Input Pulse Power in dBm');
t1=ti(1:samples_per_bit); % time division in array T1 = length(t1);
p1 = sig1(1:samples_per_bit); % pulse power in array p2 = sig2(1:samples_per_bit); % pulse power in array p3 = sig3(1:samples_per_bit); % pulse power in array p4 = sig4(1:samples_per_bit); % pulse power in array
%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& % Average power of the CW input optical power in dBm PindB_cw = -6; pav= (10^(PindB_cw/10))*1E-3; %&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
% finding out initial condition inguess=1; y1 =fzero(@(r) sol_ase(r,Lz,
p1(1),p2(1),p3(1),p4(1),pav,lamdak1,lamdak2,lamdak3,lamdak4,sigmaK1,sigmaK2
,sigmaK3,sigmaK4, nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma,I,lifetime),
inguess); % p1(1)is in non dB and first value of the array p1 will be
supplied to the steadystatesol, %calculation of rate equation
L4=length(p1);
pin1=p1(1:L4-1); % eleminate first component of the array 'p1' to make its
length equal to L5 pin2=p2(1:L4-1); pin3=p3(1:L4-1); pin4=p4(1:L4-1); % making pav an array of same length as p
for (k=1:L4-1) p_cw(k)= pav; end
for(k=1:1:sections) j=1;
for (i= 1:1:T1-1) a=0;
72
b=0; [a,b] =
solve_rateeq_ase(pin1(i),pin2(i),pin3(i),pin4(i),p_cw(i),Lz,lamdak1,lamdak2
,lamdak3,lamdak4,sigmaK1,sigmaK2,sigmaK3,sigmaK4,nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma,I,lifetime,t
1(i),t1(i+1),y1); % return 'a' as time and 'b' as r values
L2=length(b); % length of the each time division y1=b(L2); % y1=b(L2);final value of the solution of differential
eq.for that time division r1(j)=b(1); % r1(j)=b(1);collecting one value of 'r' in each pass
of the solution of diff. Eqn x2(j)=a(1); % x2(j)=a(1);collecting time instants. arrayr(k,j)=r1(j); j=j+1; end
Gk1= single_pass_gain_ase(r1,Lz,sigmaK1, nok1,gamma); % Gk is function of
time Gk2= single_pass_gain_ase(r1,Lz,sigmaK2, nok2,gamma); % Gk is function of
time Gk3= single_pass_gain_ase(r1,Lz,sigmaK3, nok3,gamma); % Gk is function of
time Gk4= single_pass_gain_ase(r1,Lz,sigmaK4, nok4,gamma); % Gk is function of
time Qkout1 =Gk1.*((lamdak1*pin1*1E-9)/ (h*c)); % output signal photons per
sec Qkout2 =Gk2.*((lamdak2*pin2*1E-9)/ (h*c)); % output signal photons per
sec Qkout3 =Gk3.*((lamdak3*pin3*1E-9)/ (h*c)); % output signal photons per
sec Qkout4 =Gk4.*((lamdak4*pin4*1E-9)/ (h*c)); % output signal photons per
sec
Qkout_cw = (Gk1.*((lamdak1*p_cw*1E-9)/ (h*c)))+(Gk2.*((lamdak2*p_cw*1E-
9)/ (h*c)))+(Gk3.*((lamdak3*p_cw*1E-9)/ (h*c)))+(Gk4.*((lamdak4*p_cw*1E-9)/
(h*c)));
Pkout1 =Gk1.*pin1; % output power for each multistage Pkout2 =Gk2.*pin2; % output power for each multistage Pkout3 =Gk3.*pin3; % output power for each multistage Pkout4 =Gk4.*pin4; % output power for each multistage
arrayPkout1(k,:)=Pkout1; arrayPkout2(k,:)=Pkout2; arrayPkout3(k,:)=Pkout3; arrayPkout4(k,:)=Pkout4;
Pkout_cw =(Gk1.*p_cw)+(Gk2.*p_cw)+(Gk3.*p_cw)+(Gk4.*p_cw); arrayPkout_cw(k,:)=Pkout_cw;
73
Pase =
asepower(r1,Lz,lamdak1,lamdak2,lamdak3,lamdak4,sigmaK1,sigmaK2,sigmaK3,sigm
aK4,nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma); %
output ase power in watt
pin1=Pkout1; pin2=Pkout2; pin3=Pkout3; pin4=Pkout4;
end
L8=length(x2); L9=length(Qkout1); L10=length(Qkout2); L11=length(Qkout3); L12=length(Qkout4); L17=length(Qkout_cw);
L13=length(Pkout1); L14=length(Pkout2); L15=length(Pkout3); L16=length(Pkout4); L18=length(Pkout_cw);
Pkout_voltage1= (Pkout1); % converting optput power in voltage Pkout_voltage2= (Pkout2); % converting optput power in voltage Pkout_voltage3= (Pkout3); % converting optput power in voltage Pkout_voltage4= (Pkout4); % converting optput power in voltage Pkout_voltage_cw= (Pkout_cw);
figure(4) subplot(5,1,1); plot(x2(20:L8),arrayPkout1(1,20:L13), 'b'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,2) plot(x2(20:L8),arrayPkout2(1,20:L14), 'g'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,3) plot(x2(20:L8),arrayPkout3(1,20:L15), 'k'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,4)
74
plot(x2(20:L8),arrayPkout4(1,20:L16), 'r'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw(1,20:L18), 'm'); axis auto; grid on;
figure(5) subplot(5,1,1); plot(x2(20:L8),arrayPkout1(2,20:L13), 'b'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
subplot(5,1,2) plot(x2(20:L8),arrayPkout2(2,20:L14), 'g'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output voltage from second stage in volts');
subplot(5,1,3); plot(x2(20:L8),arrayPkout3(2,20:L15), 'k'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
subplot(5,1,4); plot(x2(20:L8),arrayPkout4(2,20:L16), 'r'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw(2,20:L18), 'm'); grid on; axis auto;
xlabel('time in seconds'); ylabel('Output voltage from second stage in volts');
figure(6) subplot(5,1,1); plot(x2(20:L8),arrayPkout1(3,20:L13), 'b'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
75
subplot(5,1,2) plot(x2(20:L8),arrayPkout2(3,20:L14), 'g'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,3); plot(x2(20:L8),arrayPkout3(3,20:L15), 'k'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,4); plot(x2(20:L8),arrayPkout4(3,20:L16), 'r'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw(3,20:L18), 'm'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
%%second SOA
pin2_1=arrayPkout1(3,1:L13); pin2_2=arrayPkout2(3,1:L14); pin2_3=arrayPkout3(3,1:L15); pin2_4=arrayPkout4(3,1:L16);
for(i= 1:1:79)
pin2_1(i)=pin2_1(i)/10000;
pin2_2(i)=pin2_2(i)/10000;
pin2_3(i)=pin2_3(i)/10000;
pin2_4(i)=pin2_4(i)/10000;
end;
p_cw_2=p_cw(1:79); for(k=1:1:sections) j=1;
76
for (i= 1:1:T1-1) a=0; b=0;
[a,b] =
solve_rateeq_ase1(pin2_1(i),pin2_2(i),pin2_3(i),pin2_4(i),p_cw(i),Lz,lamdak
1,lamdak2,lamdak3,lamdak4,sigmaK1,sigmaK2,sigmaK3,sigmaK4,nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma,I,lifetime,t
1(i),t1(i+1),y1); % return 'a' as time and 'b' as r values
L2=length(b); % length of the each time division y1=b(L2); % y1=b(L2);final value of the solution of differential
eq.for that time division r1(j)=b(1); % r1(j)=b(1);collecting one value of 'r' in each pass of
the solution of diff. Eqn x2(j)=a(1); % x2(j)=a(1);collecting time instants. arrayr(k,j)=r1(j); j=j+1;
end
Gk1= single_pass_gain_ase(r1,Lz,sigmaK1, nok1,gamma); % Gk is function of
time Gk2= single_pass_gain_ase(r1,Lz,sigmaK2, nok2,gamma); % Gk is function of
time Gk3= single_pass_gain_ase(r1,Lz,sigmaK3, nok3,gamma); % Gk is function of
time Gk4= single_pass_gain_ase(r1,Lz,sigmaK4, nok4,gamma); % Gk is function of
time
Pkout2_1 =Gk1(1:L13).*pin2_1; % output power for each
multistage Pkout2_2 =Gk2(1:L14).*pin2_2; % output power for each
multistage Pkout2_3 =Gk3(1:L15).*pin2_3; % output power for each
multistage Pkout2_4 =Gk4(1:L16).*pin2_4; % output power for each
multistage
arrayPkout2_1(k,:)=Pkout2_1; arrayPkout2_2(k,:)=Pkout2_2; arrayPkout2_3(k,:)=Pkout2_3; arrayPkout2_4(k,:)=Pkout2_4;
Pkout_cw_2
=(Gk1(1:L13).*p_cw_2)+(Gk2(1:L14).*p_cw_2)+(Gk3(1:L15).*p_cw_2)+(Gk4(1:L16)
.*p_cw_2); arrayPkout_cw2(k,:)=Pkout_cw_2;
Pase =
asepower(r1,Lz,lamdak1,lamdak2,lamdak3,lamdak4,sigmaK1,sigmaK2,sigmaK3,sigm
aK4,nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma); %
output ase power in watt
pin2_1=Pkout2_1;
77
pin2_2=Pkout2_2; pin2_3=Pkout2_3; pin2_4=Pkout2_4;
end
L8=length(x2); L9=length(Qkout1); L10=length(Qkout2); L11=length(Qkout3); L12=length(Qkout4); L17=length(Qkout_cw);
L13=length(Pkout2_1); L14=length(Pkout2_2); L15=length(Pkout2_3); L16=length(Pkout2_4); L18=length(Pkout_cw_2);
figure(8) subplot(5,1,1); plot(x2(20:L8),arrayPkout2_1(1,20:L13), 'b'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,2) plot(x2(20:L8),arrayPkout2_2(1,20:L14), 'g'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,3) plot(x2(20:L8),arrayPkout2_3(1,20:L15), 'k'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,4) plot(x2(20:L8),arrayPkout2_4(1,20:L16), 'r'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw2(1,20:L18), 'm'); axis auto; grid on;
78
figure(9) subplot(5,1,1); plot(x2(20:L8),arrayPkout2_1(2,20:L13), 'b'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
subplot(5,1,2) plot(x2(20:L8),arrayPkout2_2(2,20:L14), 'g'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output voltage from second stage in volts');
subplot(5,1,3); plot(x2(20:L8),arrayPkout2_3(2,20:L15), 'k'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
subplot(5,1,4); plot(x2(20:L8),arrayPkout2_4(2,20:L16), 'r'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw2(2,20:L18), 'm'); grid on; axis auto;
xlabel('time in seconds'); ylabel('Output voltage from second stage in volts');
figure(10) subplot(5,1,1); plot(x2(20:L8),arrayPkout2_1(3,20:L13), 'b'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,2) plot(x2(20:L8),arrayPkout2_2(3,20:L14), 'g'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,3); plot(x2(20:L8),arrayPkout2_3(3,20:L15), 'k'); grid on; axis auto; xlabel('time in seconds');
79
ylabel('Output from Third Stage');
subplot(5,1,4); plot(x2(20:L8),arrayPkout2_4(3,20:L16), 'r'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw2(3,20:L18), 'm'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
%%third SOA
pin3_1=arrayPkout2_1(3,1:L13); pin3_2=arrayPkout2_2(3,1:L14); pin3_3=arrayPkout2_3(3,1:L15); pin3_4=arrayPkout2_4(3,1:L16);
for(i= 1:1:79)
pin3_1(i)=pin3_1(i)/10000;
pin3_2(i)=pin3_2(i)/10000;
pin3_3(i)=pin3_3(i)/10000;
pin3_4(i)=pin3_4(i)/10000;
end;
p_cw_3=p_cw(1:79); for(k=1:1:sections) j=1;
for (i= 1:1:T1-1) a=0; b=0;
[a,b] =
solve_rateeq_ase2(pin3_1(i),pin3_2(i),pin3_3(i),pin3_4(i),p_cw_3(i),Lz,lamd
ak1,lamdak2,lamdak3,lamdak4,sigmaK1,sigmaK2,sigmaK3,sigmaK4,nok1, nok2,
nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma,I,lifetime,t
1(i),t1(i+1),y1); % return 'a' as time and 'b' as r values L2=length(b); % length of the each time division
80
y1=b(L2); % y1=b(L2);final value of the solution of differential
eq.for that time division r1(j)=b(1); % r1(j)=b(1);collecting one value of 'r' in each pass of
the solution of diff. Eqn x2(j)=a(1); % x2(j)=a(1);collecting time instants. arrayr(k,j)=r1(j); j=j+1;
end
Gk1= single_pass_gain_ase(r1,Lz,sigmaK1, nok1,gamma); % Gk is function of
time Gk2= single_pass_gain_ase(r1,Lz,sigmaK2, nok2,gamma); % Gk is function of
time Gk3= single_pass_gain_ase(r1,Lz,sigmaK3, nok3,gamma); % Gk is function of
time Gk4= single_pass_gain_ase(r1,Lz,sigmaK4, nok4,gamma); % Gk is function of
time
Pkout3_1 =Gk1(1:L13).*pin3_1; % output power for each
multistage Pkout3_2 =Gk2(1:L14).*pin3_2; % output power for each
multistage Pkout3_3 =Gk3(1:L15).*pin3_3; % output power for each
multistage Pkout3_4 =Gk4(1:L16).*pin3_4; % output power for each
multistage
arrayPkout3_1(k,:)=Pkout3_1; arrayPkout3_2(k,:)=Pkout3_2; arrayPkout3_3(k,:)=Pkout3_3; arrayPkout3_4(k,:)=Pkout3_4;
Pkout_cw_3
=(Gk1(1:L13).*p_cw_3)+(Gk2(1:L14).*p_cw_3)+(Gk3(1:L15).*p_cw_3)+(Gk4(1:L16)
.*p_cw_3); arrayPkout_cw3(k,:)=Pkout_cw_3;
Pase =
asepower(r1,Lz,lamdak1,lamdak2,lamdak3,lamdak4,sigmaK1,sigmaK2,sigmaK3,sigm
aK4,nok1, nok2, nok3,
nok4,gammaK1,gammaK2,gammaK3,gammaK4,n1k1,n1k2,n1k3,n1k4,gamma); %
output ase power in watt
pin3_1=Pkout3_1; pin3_2=Pkout3_2; pin3_3=Pkout3_3; pin3_4=Pkout3_4;
end
L8=length(x2); L9=length(Qkout1); L10=length(Qkout2); L11=length(Qkout3); L12=length(Qkout4);
81
L17=length(Qkout_cw);
L13=length(Pkout3_1); L14=length(Pkout3_2); L15=length(Pkout3_3); L16=length(Pkout3_4); L18=length(Pkout_cw_3);
figure(11) subplot(5,1,1); plot(x2(20:L8),arrayPkout3_1(1,20:L13), 'b'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,2) plot(x2(20:L8),arrayPkout3_2(1,20:L14), 'g'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,3) plot(x2(20:L8),arrayPkout3_3(1,20:L15), 'k'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,4) plot(x2(20:L8),arrayPkout3_4(1,20:L16), 'r'); axis auto; grid on;
xlabel('time in seconds'); ylabel('Output voltage from first stage in volts');
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw3(1,20:L18), 'm'); axis auto; grid on;
figure(12) subplot(5,1,1); plot(x2(20:L8),arrayPkout3_1(2,20:L13), 'b'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
82
subplot(5,1,2) plot(x2(20:L8),arrayPkout3_2(2,20:L14), 'g'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output voltage from second stage in volts');
subplot(5,1,3); plot(x2(20:L8),arrayPkout3_3(2,20:L15), 'k'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
subplot(5,1,4); plot(x2(20:L8),arrayPkout3_4(2,20:L16), 'r'); axis auto; grid on; xlabel('time in seconds'); ylabel('Output voltage second stage volts');
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw3(2,20:L18), 'm'); grid on; axis auto;
xlabel('time in seconds'); ylabel('Output voltage from second stage in volts');
figure(13) subplot(5,1,1); plot(x2(20:L8),arrayPkout3_1(3,20:L13), 'b'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,2) plot(x2(20:L8),arrayPkout3_2(3,20:L14), 'g'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,3); plot(x2(20:L8),arrayPkout3_3(3,20:L15), 'k'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
subplot(5,1,4); plot(x2(20:L8),arrayPkout3_4(3,20:L16), 'r'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
83
subplot(5,1,5) plot(x2(20:L8),arrayPkout_cw3(3,20:L18), 'm'); grid on; axis auto; xlabel('time in seconds'); ylabel('Output from Third Stage');
All the functions are the same as used in the phase 2. The runtime of this matlab code is
a bit high (nearly 5 minutes), because we are
using arrays for saving the amplitude and the corresponding time instances of any signal, but
the arrays are dynamically defined.
Preallocating may become a solution to this problem but preallocating such an array may
increase the the program complexity and also
the difficulty to dbug. The coding gives output
to the amount of generated carriers in each SOA for the corresponding time and the output
of each stage of each SOA.
The network was first fed by a four signals
of wavelength 1548 nm, 1552 nm, 1556 nm, 1560nm and each of them of duty ratio 1. So,
they look like Fig 8.09:-
Thereafter, the Fig 8.10 gives the generated
carriers in the first SOA.
84
The figure Fig 8.11- Fig 8.13 gives the
amplitude (in Volts) vs. time graph for the output of each stage of the 1
st SOA :-
85
The output of third stage
is attenuated 1/10000 times and passed onto the next SOA which gives the response graph
Fig 8.14 – Fig 8.17.
Fig 8.15: output voltage 1st
stage 2nd SOA
86
and,
lastly, the
signal is
attenuated again to be fed to the
last SOA, which
gives the
response of the graphs Fig 8.18
– Fig 8.21.
Fig 8.18:
number of carrier in 3
rd
SOA
87
Power Penalty and BER in SOA Receiver CHAPTER 9
88
9.1. Optical communication systems can be
classified broadly as non-coherent and coherent transmission. In non-coherent transmission
only the intensity of an optical carrier signal is
modulated. At the receiver the signal is directly detected, a process that is only sensitive to the
signal intensity. Such systems are termed
Intensity Modulation-Direct Detection (IM-
DD).
A schematic diagram of a basic IM-DD receiver
is shown in Fig. 9.1. In this scheme intensity
modulated optical carrier signal is detected by a
photo detector (p-i-n diode or avalanche
photodiode (APD)). The resulting photocurrent
is amplified and passed to a decision circuit that
determines whether each received bit is a mark
or space.
Figure..9.1. IM-DD Receiver in optical
communication
To make decisions on the received waveform
the received waveform is sampled every bit
period, usually at the centre of the bit and compared the sampled value to a threshold
level. If the sampled value is less than the
threshold level the received bit is interpreted as a space and vice versa.
The usual figure of merit for an optical receiver
is the bit-error-rate (BER). Apart from BER the other figure of merits are
Power penalty
Quality factor
Another important figure of merit for optical
amplifier is noise figure.
9.1.1. Bit Error Rate
The IM-DD receiver can be analyzed as
follows. We assume on-off keyed (OOK)
modulation where spaces and marks are represented by input powers of zero and 2Ps
respectively.
Ps is the average received power assuming that the transmission probabilities of a mark or a
space are equal.
The photocurrent id
And the responsively of the receiver is
where η is the detector quantum efficiency.
Apart from the signal detection current there are
noise due to dark current, shot noise and receiver circuit current.
Due to these noises and interference of the
adjacent pulses, the receiver cannot always detect the digital signal correctly.
BER is the measuring of rate of errors.
It is defined as
BER= 𝑵𝒆
𝑵𝒕
Where Ne is the number of erroneous bits and Nt
is the number of bits received at a certain
interval t.
For a conventional OOK receiver, if it is
assumed that the noise currents have Gaussian
probability density functions, the BER is given by
BER=
To an approximation it can be written as
89
9.1.2. Q-Factor
Q-factor is widely used to specify the
receiver performance. It is related to the OSNR (optical signal to noise ratio).
If we assume that the receiver threshold be
optimized for the minimum BER then it is called Q-factor. It is defined as
Where S1=Is^2 and S2=0 signal power for mark and space respectively.
In ideal case where dark current and cicuit noise
is neglected then
Where Be is the electrical BW of the photo
detector.
9.1.3. Power penalty Optical extinction ratio
Power penalty Optical extinction ratio (re) is
defined as re=I1/I0=P1/P0 where P0 and P1 are
the power of bit ‘0’ and ‘1’ respectively. Ideally P0=0 making re infinity. If the extinction
ratio is not optimum the transmitted power must
be increased to maintain the same BER at the
receiver. This increase in power is called Power
penalty. It is the excess optical power required
to account for the degradation due to ISI,
reflections, mode partition etc.
9.2. Noise figure
The addition of spontaneous emission (i.e. noise) is an inevitable consequence of the
amplification of light. The use of an optical filter
at the amplifier output can greatly reduce this noise; however it is impossible to eliminate it
entirely. When the signal and accompanying
noise are detected by a photo detector the
square-law detection process gives rise to beat-noise currents in addition to the usual shot-noise.
A useful figure of merit for an optical amplifier
is the electrically equivalent noise figure F, defined as the ratio between the amplifier input
and output electrical SNRs
The SNRs are calculated by assuming that the
amplifier input signal and output signal plus
ASE are passed through a narrowband optical
filter prior to detection by an ideal photo
detector (i.e. unity quantum efficiency). In this
case the only photocurrent noise terms that need
to be taken into account are the signal shot noise and the signal-spontaneous beat noise.
And
where G is the amplifier gain.
So the noise figure is
9.3. To evaluate the SOA receiver parameter
We now simulate the Q-factor, BER and other
performance parameters. We also calculate the
beat current due to the shot noise in the receiver.
The parameters of the SOA receiver are given as follows:
90
Parameter symbol Parameter name Value
g0 Intrinsic gain of SOA 30dB
λ Input wavelength 1550 nm
B0 BW of the optical filter 126 GHz
nsp Spontaneous emission factor 4
M Number of SOA in ring network 3
R Responsivity 0.8
Psat Saturated output power 10 dBm
9.3.1 PROGRAM OF Q-FACTOR AND BER
OF A 4-CHANNEL SOA
9.3.1.a. SUBPROGRAM
function gain= sol_gain(g,g0,Pin,Psat,B0,nsp,h,fs, M) gain = g0*exp(-(g-1)*(g*Pin+(M-1)*2*nsp*(g-1)*h*fs*B0)/(g*Psat))-g;
The parameters are passed from the main
program. This program has a subprogram which
calculates the gain of the SOA and the optical
network before reception of the signal.
9.3.1.b. The BER of the received signal is
calculated from the Q-factor. Two signals are
considered here. Pin1 is the desired signal and
Pin2 is the signal with noise. Here only the shot
noise is incorporated. The number of amplifier
used in the ring network is 3 and 4 WDM
channels are taken. Responsivity of the receiver
assumed to be 0.8.
Gain of the network is calculated by calling the
subprogram. The power spectral density of ASE
noise is calculated with and without shot noise.
The photo detector current is given by
Is =Responsivity* Incident photon power.
91
The ASE beat-noise variance (σ) with and
without transient is calculated. Q-factor is given
by Q=Is /σ.
BER is calculated using the formula BER=
Power penalty is the difference in the input
signal (in dB) to establish a particular BER with
and without noise.
% finding out initial condition nsp=4; %spontaneous emission factor h=6.62E-34; %unit is in J.s c=3*1E8; lamda1= 1550*1E-9; number_of_ch = 4;
fs = c/lamda1; B0=126E9; %unit is in Hz g0_dB= 30; %unit is in dB g0= (10^(g0_dB/10)); Psat_dbm= 10; % in dBm Psat = (10^(Psat_dbm/10))*1E-3; %unit is in watt Pin_dbm= [-30 -25 -20 -15 -12 -9 -7 -5 -3 0 3 5 7]; L1 =length(Pin_dbm); Pin1= (10.^(Pin_dbm/10)).*1E-3.*number_of_ch; %unit is in watt; signal input power Pin1 Pin_Tr= [ 0.1*1e-4 0.7*1e-4 1.39*1e-4 4.41*1e-4 8.81*1e-4 0.0018 0.0028 0.0044 0.0070 0.0140 0.0279 0.04 0.070 ];% Transient power in watt Pin2 = Pin1 + Pin_Tr;
Pin2 Pin2_dbm = 10*log10(Pin2); M=3; % number of amplifiers in the ring inguess=1; for (i=1:1:L1) gain1(i) =fzero(@(g) sol_gain(g,g0,Pin1(i),Psat,B0,nsp,h,fs, M), inguess);
gain1 gain2(i) =fzero(@(g) sol_gain(g,g0,Pin2(i),Psat,B0,nsp,h,fs, M), inguess); gain2 end Pin_temp = Pin1 + Pin_Tr./gain2; PintempdB = 10*log10(Pin_temp); Nase1=(gain1-1).*(nsp*h*fs);
Nase2=(gain2-1).*(nsp*h*fs); e=1.6E-19; % electron charge n=1; R=0.8; Is1=R*Pin1; Is1 Is2= R*Pin_temp; Is2
92
Pase1=2*B0*Nase1; % ASE power for the input signal only Pase2=2*B0*Nase2; %ASE power for the input signal with receiver shot noise Iase1=(e*n*lamda1*Pase1)./(h*c);
Iase1 Iase2=(e*n*lamda1*Pase2)./(h*c); Iase2 Be=1.25*1E9; % unit is in Hz ase_beat1=(4*(R^2)*Be).*Nase1.*Pin1; %Beat current sigma1=15*sqrt(ase_beat1);
sigma1 ase_beat2=(4*(R^2)*Be).*Nase2.*Pin2; %Beat current with transient sigma2=15*sqrt(ase_beat2); sigma2 Q1=Is1./sigma1; Q1 Q2=Is2./sigma2; Q2 y1=(exp(-(Q1.^2)/2)./(Q1.*sqrt(2*pi))); % BER for the signal only y2=(exp(-(Q2.^2)/2)./(Q2.*sqrt(2*pi) ));% BER for signal and noise y1 y2 y1_dB=10*log10(exp((-Q1.^2)/2)./(Q1.*sqrt(2*pi))); % BER in dB for the signal only
y2_dB=10*log10(exp((-Q2.^2)/2)./(Q2.*sqrt(2*pi) ));% BER in dB for signal and transient figure(1) %plot(Pin_dbm,y1(1,:),'k'); plot(Pin_dbm,y1_dB,'k', Pin_dbm,y2_dB,'r'); xlabel('Input power in dBm'); ylabel('BER'); %plot(Pin_dbm,y2,'r'); grid on; axis([-35, -5, -10, 1]);
a. The resultant graph shown in the figure 9.1.
93
Figure.9.1. Input power vs. BER graph
From the graph we can show the power penalty i.e. the difference in input power (dBm) to maintain a
constant BER with and without noise. At -10 dB BER i.e. 0.1 BER
Figure.9.2. Magnified output at -10.5 dB BER
Power penalty = (-13.7-(-14.2)) = 0.5 dB
94
b. Now for 10^-9 BER i.e.-90 dB BER the graph is
Figure.9.3. Input power vs. BER graph for 10E-9 BER
From the graph the power penalty for 10E-9 BER is shown
Figure.9.4. Magnified output for 10e-9 BER
Power penalty = (-6.9-(-7.4)) = 0.5 dB
Power penalty = 0.5 dB. This power serves no additional purpose but is an extra requirement to
compensate the noise interference due to non-ideal extinction ratio. Less the power penalty, more
efficient is the system.
Summary Chapter 10
94
At the end of the report describing our project, we come to conclude about the performance of the
Semiconductor Optical Amplifier.
In the first and second chapter, we have introduced the SOA wit its brief history. In the third
chapter, we have justified why we have selected the SOA. In the fourth chapter, we have given the
basic principle of the SOA. The fifth chapter describes the fundamental device characteristics and the
material used in the SOA. The modelling of SOA is described in the sixth chapter, where wideband
SOA steady-state model and numerical solution has been described, while the seventh chapter gives
the description about the cross-gain modulation . The whole of the eight and ninth chapter gives
description about our work done on the simulation and the power penalty calculation including the
BER calculation.
Bibliography Chapter 11
95
1. Studies on Placement of Semiconductor Optical Amplifiers in Wavelength Division
Multiplexed Star and Tree Topology Networks by Yatindra Nath Singh submitted in
fulfilment of the requirement of degree of Doctor of Philosophy (Ph.D.) to Electrical
Engineering Department Indian Institute of Technology, Delhi Hauz Khas, New Delhi 110016
India September 1996
2. Theory and Experiment of High-Speed Cross-Gain Modulation in Semiconductor Lasers by
X. Jin, T. Keating, and S. L. Chuang
3. Investigation of Pulse Pedestal and Dynamic Chirp Formation on Pico second Pulses After Propagation Through an SOA by A. M. Clarke, M. J. Connelly, P. Anandarajah, L. P. Barry,
and D. Reid 4. SOA-Based WDM Metro Ring Networks With Link Control Technologies by T. Rogowski, S.
Faralli, G. Bolognini, F. Di Pasquale, Member, IEEE, R. Di Muro, and B. Nayar, Member, IEEE
5. Optical Amplifiers by Bala Ramasamy and Robert Stacey 6. Optical Fibre Communication by Gard Kaiser, international edition, 1991
7. Nonlinear Fibre Optics, Third Edition, by Govind P. Agrawal, The Institute of Optics,
University of Rochester
8. Optical Fibre communication, by J. M. Senior, 1985
9. Wideband Semiconductor Optical Amplifier Steady-State Numerical Model, byMichael J.
Connelly, Member, IEEE 10. Semiconductor Optical Amplifiers– High Power Operation, by Boris Stefanov, Leo
Spiekman David Piehler Alphion Corporation,IEEE 802.3av Task Force Meeting, Orlando,
13-15 March 2007. 11. Fast and Efficient Dynamic WDM Semiconductor Optical Amplifier Model, Walid
Mathlouthi, Pascal Lemieux, Massimiliano Salsi, Armando Vannucci, Member, IEEE,
Alberto Bononi, and Leslie A. Rusch, Senior Member, IEEE.