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

1. 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)

2. Acknowledgement
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
i

3. Certificate by the Supervisor
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
ii

4. Preface
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.
iii

5. 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

6. Introduction Chapter1
Communications can be broadly defined as the domain, are required in transparent optical
transfer of information from one point to networks.
another. In optical fiber communications, this
transfer is achieved by using light as the
information carrier. There has been an In this chapter we begin with the reasons why
exponential growth in the deployment and optical amplification is required in optical
capacity of optical fiber communication communication networks. This is followed by a
technologies and networks over the past brief history of semiconductor optical amplifiers
twenty-five years. This growth has been made (SOAs), a summary of the applications of SOAs
possible by the development of new and a comparison between SOAs and optical
optoelectronic technologies that can be fiber amplifiers (OFAs).
utilized to exploit the enormous potential
bandwidth of optical fiber. Today, systems are WHY WE NEED OPTICAL
operational which operate at aggregate bit AMPLIFICATION? :-
rates in excess of 100 Gb/s. Such high
capacity systems exploit the optical fiber Optical fiber suffers from two principal limiting
bandwidth by employing wavelength division factors: Attenuation and dispersion. Attenuation
multiplexing. leads to signal power loss, which limits
transmission distance. Dispersion causes optical
Optical technology is the dominant carrier of
global information. It is also central to the pulse broadening and hence inter symbol
realization of future networks that will have interference leading to an increase in the system
the capabilities demanded by society. These bit error rate (BER). Dispersion essentially
capabilities include virtually unlimited limits the fiber bandwidth. The attenuation
bandwidth to carry communication services of spectrum of conventional single-mode silica
almost any kind, and full transparency that fiber, shown in Fig. 1.1, has a minimum in the
allows terminal upgrades in capacity and 1.55 µm wavelength region. The attenuation is
flexible routing of channels. Many of the somewhat higher in the 1.3 µm region. The
advances in optical networks have been made
dispersion spectrum of conventional single-
possible by the advent of the optical amplifier.
mode silica fiber, shown in Fig. 1.2, has a
In general, optical amplifiers can be divided minimum in the 1.3 µm region. Because the
into two classes: optical fiber amplifiers and attenuation and material dispersion minima are
semiconductor amplifiers. The former has located in the 1.55 µm and 1.3 µm ‘windows’,
tended to dominate conventional system these are the main wavelength regions used in
applications such as in-line amplification used commercial optical fiber communication
to compensate for fiber losses. However, due systems. Because signal attenuation and
to advances in optical semiconductor dispersion increases as the fiber length increases,
fabrication techniques and device design, at some point in an optical fiber communication
especially over the last five years, the link the optical signal will need to be
semiconductor optical amplifier (SOA) is
regenerated. 3R (reshaping-retiming-
showing great promise for use in evolving
optical communication networks. It can be retransmission).Regeneration involves detection
utilized as a general gain unit but also has (photon-electron conversion), electrical
many functional applications including an amplification, retiming, pulse shaping and
optical switch, modulator and wavelength retransmission (electron-photon conversion).
converter. These functions, where there is no
conversion of optical signals into the electrical
1

7. Fig 1.1: Typical attenuation spectrum of low-
loss single-mode silica optical fiber.
2

8. This method has some disadvantages- improve receiver sensitivity. Besides these basic
►Firstly, it involves breaking the optical link system applications optical amplifiers are also
and so is not optically transparent. useful as generic optical gain blocks for use in
larger optical systems. The improvements in
►Secondly, the regeneration process is optical communication networks realized
dependent on the signal modulation format and through the use of optical amplifiers provides
bit rate and so is not electrically transparent. new opportunities to exploit the fiber
This in turn creates difficulties if the link needs bandwidth.
to be upgraded. Ideally link upgrades should There are two types of optical amplifier: The
only involve changes in or replacement of SOA and the OFA. In recent times the latter has
terminal equipment (transmitter or receiver). dominated; however SOAs have attracted
►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 renewed interest for use as basic amplifiers and
amplifier should be compatible with single- also as functional elements in optical
mode fiber, impart large gain and be optically communication networks and optical signal
transparent (i.e. independent of the input processing devices.
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
3

9. HISTORY Chapter2
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.
4

10. WHY SOA? Chapter3
As optical technology has become an integral advantages including smaller size and the ability
part of telecommunications, the need for reliable to easily integrate with semiconductor lasers.
optical signal transmission has become more and The latest step in semiconductor amplifiers came
more pronounced. In order to transmit over long with the introduction of a SOA that operated as a
distances, it is necessary to account for linear amplifier (LOA). Thus far this has
attenuation losses. Initially, this was done eliminated many of the downfalls of SOAs such
through an expensive conversion from optical to as cross talk and high signal to noise ratio.
electrical and back. This was soon remedied
with the creation of optical amplifiers. 1. EDFA: Erbium doped fiber amplifiers are
commonly used optical amplifier. An EDFA
The optical amplifiers we have today are consists of a pump laser coupled to an input
signal and passed through an optical fiber
1.EDFA. slightly doped with erbium ions. The pump laser
is used to excite erbium ions which emit photons
2. SOA. in phase with the input signal which acts to
amplify it. EDFA’s amplify in the 1520-1600
3. LOA. nm range which corresponds to the energy
difference between the excited and ground states
One of the first widely adopted optical of the erbium ions.
amplifiers was the Erbium Doped Fiber
Amplifier (EDFA). This revolutionized the
optical communications industry. The next big 2. SOA: The semiconductor optical amplifier
step in optical amplifiers came with is an amplifier with a laser diode structure that is
semiconductor optical amplifiers (SOA). used to amplify optical signals passing through
Although these didn’t perform as well as the its optical region. Amplification occurs through
EDFAs in some conditions, they had many stimulated emission in the active region as input
5

11. signal energy propagates through the wave a feedback device, preventing carrier depletion
guide. This can be seen below 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.
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
6

12. 3. The channel to
Bit-error channel, which is
rate characteristics of the SOAs are much better unlikely in SOAs. SOAs can operate at the
than the EDFA. In the EDFA, the BER lowest Bi- error rate of 10-15.
progressively gets worse from
7

13. 4. One of the main
drawbacks of SOA
devices is the need for
8

14. 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.
9

15. Basic Principle Chapter 4
An SOA is an optoelectronic device that reflections are negligible (i.e. the signal
under suitable operating conditions can undergoes a single-pass of the amplifier).
amplify an input light signal. A schematic Anti-reflection coatings can be used to create
diagram of a basic SOA is shown in Fig. 2.1. SOAs with facet reflectivities <10-5.The TW-
The active region in SOA is not as sensitive as the
the device imparts FP-SOA to fluctuations in
gain to an input bias current, temperature and
signal. An external signal polarisation.
electric current
provides the energy
source that enables
gain to take place. Principles of Optical
An embedded waveguide Amplification:-
is used to confine the
propagating signal wave to the active region. In an SOA electrons (more commonly
However, the optical confinement is weak so referred to as carriers) are injected from an
some of the signal will leak into the external current source into the active region.
surrounding lossy cladding regions. The output These energised region material, leaving holes
signal is accompanied by noise. This additive in the valence band (VB). Three radiative
noise is produced by the amplification process mechanisms are possible in the semiconductor.
itself and so cannot be entirely avoided. The These are shown in Fig 2.3 for a material with
amplifier facets are reflective causing ripples an energy band structure consisting of two
in the gain spectrum. discrete energy levels.
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
10

16. In stimulated absorption an incident light proportional to the intensity of the inducing
photon of sufficient energy can stimulate a radiation whereas the spontaneous emission
carrier from the process is
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
independent of
with a VB hole.
it.
The recombining carrier loses its energy in the
form of a photon of light. This new stimulated Spontaneous and induced transitions:-
photon will be identical in all respects to the
inducing photon (identical phase, frequency The gain properties of optical
and direction, i.e. a coherent interaction). Both semiconductors are directly related to the
the original photon and stimulated photon can processes of spontaneous and stimulated
give rise to more stimulated transitions. If the emission. To quantify this relationship we
injected current is sufficiently high then a consider a system of energy levels associated
population inversion is created when the with a particular physical system. Let N1 and
carrier population in the CB exceeds that in the N2 be the average number of atoms per unit
VB. In this case the likelihood of stimulated volume of the system characterised by the
emission is greater than stimulated absorption average number of atoms by energies E1 and
and so semiconductor will exhibit optical gain. E2 respectively, with E2 > E1 .If a particular
atom has energy E2 then there is a finite
In the spontaneous emission process, there probability per unit time that it will undergo a
is a non-zero probability per unit time that a transition from E2 to E1 and in the process emit
CB carrier will spontaneously recombine with a photon. The spontaneous carrier transition
a VB hole and thereby emit a photon with rate per unit time from level 2 to level 1 is
random phase and direction. Spontaneously given by
emitted photons have a wide range of
frequencies. Spontaneously emitted photons 4.1
are essentially noise and also take part in
reducing the carrier population available for
where A21 is the spontaneous emission
optical gain. Spontaneous emission is a direct
parameter of the level 2 to level 1 transition.
consequence of the amplification process and
Along with spontaneous emission it is also
cannot be avoided; hence a noiseless SOA
possible to have induced transitions. The
cannot be created. Stimulated processes are
11

17. induced carrier transition rate from level 2 to l(v)dv is the probability that a particular
level 1 (stimulated emission) is given by spontaneous emission event from is level 2 to
level 1 will result in a photon with a frequency
4.2
between v and v+dv. The inducing field
where B21 is the stimulated emission intensity (w/m3) is
parameter of the level 2 to level 1 transition
and ρ(v) the incident radiation energy density 4.9
at frequency v. The induced photons have
energy hv = E2 – E1 The induced transition
rate from level 1 to level 2 (stimulated So (4.7) becomes
absorption) is given by
4.9
4.3
where B12 is the stimulated emission
parameter of the level 2 to level 1 transition. It
can be proved, from quantum-mechanical Absorption and amplification :-
considerations [1,2], that
By using the expression for the stimulated
B12 = B21 4.4 transition rates developed in previously, it is
now possible to derive an equation for the
optical gain coefficient for a two level system.
4.5 We consider the case of a monochromatic
plane wave propagating in the z-direction
where ηr is the material refractive index through a gain medium with cross-section area
and the speed of light in a vacuum. Inserting A and elemental length dz. The net power dPv
(4.5) into (4.2) gives generated by a volume Adz of the material is
simply the difference in the induced transition
rates between the levels multiplied by the
4.6 transition energy hv and the elemental volume
i.e.
In the case where the inducing radiation is
4.11
monochromatic at frequency v, then the
induced transition rate from level 2 to level 1
is This radiation is added coherently to the
propagating wave. This process of
amplification can then be described by the
4.7
differential equation
where ρv is the energy density (T/m3) of the
4.12
electromagnetic field inducing the transition
and l(v) is the transition lineshape function,
normalised such that gm(v) is the material gain coefficient given
by
4.8
4.13
12

18. (4.13) implies that to achieve positive gain
4.15
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 A volume element, with cross-section area A
gain is always accompanied by spontaneous and length dz at position z, of the gain medium
emission, i.e. noise. spontaneously emits a noise power
4.16
Spontaneous emission noise :-
This noise is emitted isotropically over a 4π
As shown above, spontaneous emission is a solid angle. Each spontaneously emitted
direct consequence of the amplification photon can exist with equal probability in one
process. In this section an expression is of two mutually orthogonal polarisation states.
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. The polariser passes the signal, while reducing
If the beam is assumed to have a circular the noise by half. Hence the total noise power
cross-section with waist diameter D then the emitted by the volume element into a solid
beam divergence angle is angle dΩ and bandwidth B0 is
4.17
4.14
The smallest solid angle that can be used
where λ0 is the free space wavelength. The net without losing signal power is
change in the signal power due to coherent
amplification by an elemental length dz of the
gain medium is
13

19. The noise can also be reduced by the use of a
narrowband optical filter.
4.18
This solid angle can be obtained by the use of
a suitably narrow output aperture. In this case
(4.17) can be rewritten as
4.19
The total beam power P (signal and noise) can
then be described by
4.20
where the spontaneous emission factor nsp is
given by
4.21
The solution of (2.20), assuming that gm is
independent of z, is
4.22
where Pm is the input signal power. If the
amplifying medium has length L then the total
output power is
4.23
where G = egmL is the single-pass signal gain.
The amplifier additive noise power is
4.23
(4.24) shows that increasing the level of
population inversion can reduce SOA noise.
14

20. Fundamental Device Characteristics & Chapter 5
Materials Used in SOA
The most common application of SOAs is
as a basic optical gain block. For such an
application, a list of the desired properties is v0 is the closest cavity resonance to v. Cavity
given in Table 2.1. The goal of most SOA resonance frequencies occur at integer
multiples of Δv. The sin2 factor in (5.1) is
research and development is to realise these
equal to zero at resonance frequencies and
properties in practical devices. equal to unity at the anti-resonance frequencies
(located midway between
successive resonance
frequencies). The effective
SOA gain coefficient is
5.3
where Γ is the optical mode
confinement factor (the
fraction of the propagating
Table 5.01: Desirable Properties of a practical SOA signal field mode confined to the active
region) and α the absorption coefficient.
Small-signal gain and gain bandwidth
Gs=egl is the single-pass amplifier gain.
In general there are two basic gain
An uncoated SOA has facet reflectivities
definitions for SOAs. The first is the intrinsic
approximately equal to 0.32. The amplifier
gain G of the SOA, which is simply the ratio
gain ripple Gr is defined as the ratio between
of the input signal power at the input facet to
the resonant and non-resonant gains. From
the signal power at the output facet. The
(5.1) we get
second definition is the fibre-to-fibre gain,
which includes the input and output coupling
losses. These gains are usually expressed in 5.4
dB. The gain spectrum of a particular SOA
depends on its structure, materials and
operational parameters. For most applications From (5.4) the relationship between the
high gain and wide gain bandwidth are geometric mean facet reflectivity
desired. The small-signal (small here meaning
and Gr is
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 5.5
Curves of Rgeo versus Gs are shown in Fig.
5.02 with Gs as parameter. For example, to
5.1 obtain a gain ripple less than 1 dB at an
amplifier single-pass gain of 25 dB requires
Where R1 and R2 are the input and output
that Rgeo < 3.6 x 10-4. Facet reflectivities of this
facet reflectivities and Δv is the cavity
order can be achieved by the application of
longitudinal mode spacing given by
anti-reflection (AR) coatings to the amplifier
facets. The effective facet reflectivities can be
5.2
15

21. reduced further by the use of specialised SOA Cascaded SOAs accentuate this polarisation
structures. dependence. The amplifier waveguide is
characterised by two mutually orthogonal
A typical TW-SOA small-signal gain polarisation modes termed the Transverse
spectrum is shown in Fig. 5.01. The gain Electric (TE) and Transverse Magnetic (TM)
bandwidth Bopt of the amplifier is defined as modes. The input signal polarisation state
the wavelength range over which the signal usually lies
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. Fig 5.02: Geometric mean facet reflectivity
16

22. somewhere between these two extremes. The In the limiting case where the amplifier
polarisation sensitivity of an SOA is defined as gain is much larger than unity and the
the magnitude of the difference between the amplifier output is passed through a
TE mode gain GTE and TM mode gain GTM i.e. narrowband optical filter, the noise figure is
given by
5.6
5.8
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 The lowest value possible for nsp is unity,
typical SOA gain versus output signal power which occurs when there is complete inversion
characteristic is shown in Fig. 5.03. A useful of the atomic medium, i.e. N1=0, giving F = 2
parameter for quantifying gain saturation is the (i.e. 3 dB). Typical intrinsic (i.e. not including
saturation output power Po,sat which is defined coupling losses) noise figures of practical
as the amplifier output signal power at which SOAs are in the range of 7 to 12 dB. The noise
the amplifier gain is half the small-signal gain. figure is degraded by the amplifier input
Values in the range of 5 to 20 dBm for are coupling loss. Coupling losses are usually of
typical of practical devices. the order of 3 dB, so the noise figure of typical
packaged SOAs is between 10 and 15 dB.
Noise figure
Dynamic effects
A useful parameter for quantifying optical
amplifier noise is the noise figure. F, defined SOAs are normally used to amplify
as the ratio of the input and output signal to modulated light signals. If the signal power is
noise ratios, i.e. high then gain saturation will occur. This
would not be a serious problem if the amplifier
gain dynamics were a slow process. However
5.7
in SOAs the gain dynamics are determined by
the carrier recombination lifetime (average
The signal to noise ratios in (5.7) are those time for a carrier to recombine with a hole in
obtained when the input and output powers of the valence band). This lifetime is typically of
the amplifier are detected by an ideal a few hundred picoseconds. This means that
photodetector. the amplifier gain will react relatively quickly
17

23. to changes in the input signal power. This momentum vector. Direct bandgap
dynamic gain can cause signal distortion, semiconductors are used because the
which becomes more severe as the modulated probability of radiative transitions from the CB
signal bandwidth increases. These effects are to the VB is much greater than is the case for
further exacerbated in multichannel systems indirect bandgap material. This leads to greater
where the dynamic gain leads to interchannel device efficiency, i.e. conversion of injected
crosstalk. This is in contrast to doped fibre electrons into photons. A simplified energy
amplifiers, which have recombination band structure of this material type is shown in
lifetimes of the order of milliseconds leading Fig. 5.04, where there is a single CB and three
to negligible signal distortion. VBs. The three VBs are the heavy-hole band,
light-hole band and a split-off band. The heavy
and light-hole
bands are
Nonlinearities
degenerate;
SOAs also exhibit that is their
nonlinear behaviour. In maxima have
general these nonlinearities the same
can cause problems such as energy and
frequency chirping and momentum.
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. Fig 5.04: Carrier and optical confinement in DH SOA
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
Fig 5.05: Energy band structure of direct band
The active region of a bulk SOA is gap semiconductor
fabricated from a direct band-gap material. In
such a material the VB maximum and CB
minimum energy levels have the same
18

24. In this model the energy of a CB electron Where nc and nv are constants given by
or VB hole, measured from the bottom or top
of the band respectively is given by
5.15
ħ2 ∗������������ ^2
Ea = 2∗������������ 5.9
and
5.16
ħ2 ∗������������ ^2
������������ = 2∗������������
5.10
and
where kp is the magnitude of the
momentum vector, mc the CB electron 5.17
effective mass and mv VB hole effective mass.
where mhh and mlh and are the VB heavy
Under bias conditions the occupation
probability f(c)of an electron with energy E in and light-hole effective masses.
the CB is dictated by Fermi-Dirac statistics For a two-level system we have from an
given by expression for the optical gain coefficient at
frequency υ
5.11
5.18
Where Efc is the quasi-Fermi level of the This expression applies to any particular
CB relative to the bottom of the band, k is the transition. Without lack of generality we can
Boltzmann constant and T the temperature. apply it to transitions, having the same
Similarly the occupation probability of an momentum vector, between a CB energy level
electron in the VB with energy E, increasing Ea and VB energy level Eb where
into the band, is given by
5.19
5.12 Thus we obtain the relations:
������ ℎℎ
where Efv is the quasi-Fermi level of the Ea= (hυ-Eg(n))*( ������������ +������ ℎℎ )) 5.20
VB relative to the top of the band. The quasi-
Fermi levels can also be estimated using the
Nilsson approximation ������������
Eb = -(h(υ)-Eg(n))*(������������ +������ ℎℎ )
������������������ = ������������������ + ������ 64 + 0.05524������ 64 +
−1
5.21
������ /4}������������ 5.13
Where mhh is the effective mass of heavy
Efv = -{ ln ε+ ε [64 +0.05524ε (64+ ������)]^- hole and me is the effective mass of electrons.
1/4}KT 5.14 It is assumed that heavy-holes dominate over
light-holes due to their much greater effective
������ ������
Where δ = ������������ and ε = ������������ mass.
19

25. Thus the optical gain coefficient of the
amplifier is given by
5.22
The above equations are used to compute
the fitting parameters in farther calculations.
20

26. Modeling of SOA CHAPTER6
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 The band gap energy Eg can be expressed as
external variables such as the output signal
power, saturation output power and amplified
spontaneous emission (ASE) spectrum. 6.2
In this chapter two important model of SOA are Where Eg0 the band gap energy with no injected
discussed. carriers, is given by the quadratic approximation
 Steady state numerical model proposed
by M.J. Connelly or Connelly model
 Dynamic model of SOA or Reservoir 6.3
model Where a, b and c are the quadratic coefficients
and e is the electronic charge. ΔEg (n) is the
6.1.1. STEADY STATE NUMERICAL band gap shrinkage due to the injected carrier
MODEL density given by
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 6.4
operating regimes. A numerical algorithm is
described which enables efficient where Kg is the band gap shrinkage coefficient.
implementation of the model.
The Fermi-Dirac distributions in the CB and VB
A. The InGaAsP direct band gap bulk- are given by
material active region has a material
gain coefficient gm(υ) given by 6.5
6.6
6.7
6.8
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.
6.1 They can be estimated using the Nilsson
approximation.
21

27. ������������������ = ������������������ + ������ 64 + 0.05524������ 64 +
−1
������ /4}������������
6.9 6.15
Thus we obtain the relations:
Efv = -{ ln ε+ ε [64 +0.05524ε (64+ ������)]^- ������ ℎℎ
Ea= (hυ-Eg(n))*( ������������ +������ ℎℎ )) 6.16
1/4}KT
6.10 ������������
Eb = -(h(υ)-Eg(n))*(������������ +������ ℎℎ ) 6.17
������ ������
Where δ = ������������ and ε = ������������ 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-
Where nc and nv are constants given by holes due to their much greater effective mass.
Thus the optical gain coefficient of the amplifier
is given by
6.11
6.12
6.18
The above equations are used to compute the
fitting parameters in farther calculations.
And gm (υ) is composed of two components one is
the gain coefficient
And another is the absorption coefficient
6.13
So
Where mhh and mlh and are the VB heavy and
light-hole effective masses. 6.19
For a two-level system we have from an
expression for the optical gain coefficient at
frequency υ
6.14 6.20
9
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 6.21
Ea and VB energy level Eb where Plot for gm and gm´ is given in the fig.6.1.
22

28. 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
Figure.6.1. Typical InGaAsP bulk treated coherently since its transmission through
semiconductor gain spectra. the amplifier depends on its frequency and phase
when reflecting facets are present Esk+ and Esk-
The SOA parameters used in Connelly model is obey the complex traveling-wave equations
given in the table
6.23
The material loss coefficient α is modeled as a
linear function of carrier density
And
6.22
K0 and K1 are the carrier-independent and 6.24
carrier-dependent absorption loss coefficients,
respectively.
Boundary conditions
B. TRAVELLING WAVE EQUATION
FOR SIGNAL FIELD 6.25
6.26
In the model, signals are injected with optical
frequencies υk ( k=1 to Ns) and power Pink Where the k-th input signal field to the left of
before coupling loss. The signals travel through
the input facet is
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 6.27
interaction between the internal variables of the
amplifier, i.e., the carrier density and photon
rates. The solution of these equations enables The k-th output signal field to the right of the
external parameters such as signal fiber-to-fiber output facet is
gain and mean noise output to be predicted. In
6.28
the following analysis, it is assumed that
transverse variations in the photon rates and
carrier density are negligible. This assumption is
23

29. The k-th output signal power after coupling loss carrier population and helps saturate the gain.
is 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
6.29 between adjacent wavelength components.
When reflecting facets are present, the
ηin and ηout are the input and output coupling
spontaneously emitted noise will show the
efficiencies, respectively.
presence of longitudinal cavity modes. For this
The amplitude reflectivity coefficients are
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
The kth signal propagation coefficient is
Where the cutoff frequency at zero injected
carrier density is given by
6.30
6.34
neq is the equivalent index of the amplifier
Δυc is a frequency offset used to match υ0 to a
waveguide
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
6.31
n2 is the refractive index of the InP material 6.35
surrounding the active region. neq is modeled as
a linear function of carrier density 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
6.32 cavity resonances is identical to treating the
signal coherently in terms of traveling-wave
neq0 is the equivalent refractive index with no power (or photon rate) equations. It is sufficient
pumping. The Differential in given 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
6.33 emission photon rates (s) for a particular
polarization [transverse electric (TE) or
C. TRAVELING-WAVE EQUATIONS transverse magnetic (TM)] in a frequency
FOR THE SPONTANEOUS spacing centered on frequency, traveling in the
EMISSION positive and negative directions, respectively.
The amplification of the signal also depends on And obey the traveling-wave equations
the amount of spontaneously emitted noise
generated by the amplifier. This is because the
noise power takes part in draining the available
24

30. If the single-pass gain is at , then the signal gain
for frequencies within spacing Δυm around υj
6.36
6.42
6.37
Subject to the boundary conditions
Where the single-pass phase shift is
6.38
6.43
The function Rsp(vj,n) represents the At resonance, the signal gain is
spontaneously emitted noise coupled into N j+ 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 6.44
constant carrier density throughout the Let the amplifier have a noise input spectral
amplifier), material gain coefficient, and zero density (photons/s/Hz) distributed uniformly
loss coefficient, facet reflectivities, and coupling over centered. The total output noise (photons/s)
losses. In this case, is obtained from the solution in is then
to
6.39
The output noise power at the single frequency 6.45
band If the input noise power were concentrated at
(resonance), then the output noise photon rate
would be
6.40 Where 6.46
The equivalent quantum mechanical expression
where 6.47
6.41
The traveling-wave power equations describing
and assume that all the spontaneous photons in 6.48
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 Kj is equal to unity for zero facet reflectivities.
account for this, and are multiplied by a
normalization factor which is derived as follows.
25

31. D. CARRIER DENSITY RATE
EQUATION
The carrier density at obeys the rate equation
Figure.6.2. the ith section of the SOA model.
Signal fields and spontaneous emission are
6.49 estimated at the section boundaries. The carrier
Where I is the bias current and R (n(z)) is the density is estimated at the center of the section
recombination rate given by
The first step in the algorithm is to initialize the
signal fields and spontaneous emission photon
rates to zero. The initial carrier density is
Rrad(n) and obtained from the solution of carrier density rate
Rnrad(n) carrier recombination rates, respectively, equation with all fields set to zero, using the
both of which can be expressed as polynomial Newton–Raphson technique. The coefficients of
functions the traveling-wave equations are computed. In
the gain coefficient calculations, the radiative
6.50 carrier recombination lifetime is approximated
by
6.51
Arad and Brad are the linear and bimolecular 6.52
radioactive recombination coefficients. Next, the signal fields and noise photon densities
are estimated. The noise normalization factors
E. STEADY STATE NUMERICAL are then computed. Q (i) is then calculated. This
SOLUTION OF CONNELLY process enables convergence toward the correct
MODEL value of carrier density by using smaller carrier
density increments. The iteration continues until
As the SOA model equations cannot be solved the percentage change in the signal fields, noise
analytically, a numerical solution is required. In photon rates and carrier density throughout the
the numerical model the amplifier is split into a SOA between successive iterations is less than
number of sections labeled from i=1 to Nz as the desired tolerance. When the iteration stops,
shown in Fig.6.2. The signal fields and the output spontaneous emission power spectral
spontaneous emission photon rates are estimated density is computed using the method of Section
at the section interfaces. In evaluating Q (i) in VII and parameters such as signal gain, noise
the i-th section the signal and noise photon rates figure and output spontaneous noise power are
used are given by the mean value of those calculated. The algorithm shows good
quantities at the section boundaries. In the convergence and stability over a wide range of
steady-state Q (i) is zero. To predict the steady- operating conditions. A flowchart of the
state a characteristic, an algorithm is used which algorithm is shown in Fig. 6.3.
adjusts the carrier density so the value of
throughout the amplifier approaches zero. A
flowchart of the algorithm is shown in Fig. 6.3.
26

32. Figure.6.3. SOA steady-state model algorithm
27

33. F. ESTIMATION OF THE OUTPUT G. OUTPUT OF THE CONNELLY
SPONTANEOUS EMISSION MODEL
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
6.53
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.6.4. 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 Figure 6.7. predicted SOA noise figure
experimental fiber-to-fiber signal gains are both spectrum. Input parameters are as for Fig.
25.0 dB. The experimental gain ripple of 0.5 dB 5. A noise figure of 11.4_0.5 dB at 1537.7 nm is
is identical to that predicted. The difference predicted compared to an experimental value of
between the predicted and experimental ASE 8.8_0.3 dB.
level is approximately 2.5 dB.
28

34. 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.
29

35. 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.
30

36. SOA PARAMETERS USED IN STEADY
STATE CONNELLLY MODEL
31

37. 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
6.2. RESERVIOR MODEL amplification, although it can be extended to
multi wavelength operation by assuming that the
Another important SOA model is the Reservoir channels are spaced far enough apart to neglect
model proposed by Walid Mathlouthi, Pascal FWM beating in the co propagating case. Saleh
Lemieux, Massimiliano Salsi, Armando arrived independently at the same model as
Vannucci, Alberto Bononi, and Leslie A. Agrawal and Olsson’s coincides with and then
Rusch. introduced further simplifying approximations to
This model is the dynamic version of the steady get to a very simple block diagram of the single-
state Connelly model. We are interested in channel SOA, which was exploited for a
analyzing the response of SOAs to optical mathematically elegant stochastic performance
signals that are modulated at bit rates not analysis of single-channel saturated SOAs. The
exceeding 10 Gb/s, such as those planned for loss of accuracy due to Saleh’s extra
next-generation metropolitan area networks. approximations with respect to Agrawal’s model
Therefore, ultrafast intra band phenomena such was quantified in Saleh’s model was later
as carrier heating (CH) and spectral hole burning extended to cope with injection current
(SHB) can be neglected, and only carrier modulation, scattering losses, and ASE. In
induced gain dynamics need to be included, as addition, Agrawal’s model was extended to
was done in several SOA models developed in include ASE in both and ASE was added
the past. Such models can be divided into two phenomenologically at the output of the SOA
broad categories: 1) space-resolved numerically and did not influence the gain dynamics, thereby
intensive models, which take into account facet limiting the application to very small saturation
reflectivity as well as forward and backward levels.
propagating signals and amplified spontaneous In this paper, we first develop a dynamic version
emission (ASE) and offer a good fit to of the steady-state wideband SOA Connelly
experimental data simplified analytical models model which is shown to fit quite well with our
with a coarser fit to experimental data but dynamic SOA experiments with OOK channels.
developed to facilitate conceptual understanding The Connelly model was selected because it
and performance analysis. For the purpose of derives the SOA material gain coefficient from
carrying out extensive Monte Carlo simulations quantum mechanical principles without the
for statistical signal analysis and bit-error rate assumption of linear dependence on carrier
(BER) estimation, the accurate space-resolved density that was made in.
models are ruled out because of their Our dynamic Connelly model serves then as a
prohibitively long simulation times. However, a benchmark to test the accuracy and
simplified model with a satisfactory fit to computational-speed improvement of a novel
experimental results would be highly desirable. state-variable SOA dynamic model, which
Most simplified models can be derived from the represents the most important contribution of
work of Agrawal and Olsson. Under suitable this paper. The novel model is an extension of
assumptions, Agrawal and Olsson managed to Agrawal’s model, with the inclusion of
reduce the coupled propagation and rate approximations for scattering loss and ASE to
equations into a single ordinary differential better fit the experimental results and the
equation (ODE) for the integrated gain. The dynamic Connelly model predictions. In such a
simplicity of the solution is due to the fact that model, the SOA dynamic behavior is reduced to
waveguide scattering losses and ASE were the solution of a single ODE for the single state
neglected. ASE has an important effect on the variable of the system, which is proportional to
spatial distribution of carrier density and the integrated carrier density, which, for WDM
32

38. operation is a more appropriate variable than the provides a new entry aside from the already
integrated gain used in. Once the state-variable known models for EDFAs and for Raman
dynamic behavior is found, the behavior of all amplifiers .A challenge in our reservoir model,
the output WDM channels is also obtained. The as in all simplified SOA models, is to correctly
state variable is called ―reservoir‖ since it plays choose the values of the wavelength-dependent
the same role as the reservoir of excited erbium coefficients that give the best fit to the
ions in an erbium-doped fiber amplifier (EDFA). experimental results. We propose and describe
Quite interestingly, then, the SOA for WDM here a methodology to extract the needed
operation admits almost the same block diagram wavelength-dependent coefficients from the
description as that of an EDFA suggested by parameters of the dynamic Connelly model.
Such a novel SOA block diagram is shown in This paper is organized as follows. In Section II,
Fig. 6.11 (without ASE for ease of drawing) and the dynamic Connelly model is introduced, and
will be derived in the next sections. Note that a procedure to derive its parameters from
this model treats the intensity of the electrical experiments is described. In Section III, the
field, but the field phase can be indirectly SOA reservoir model is derived first without
obtained since it is a deterministic function of ASE and then with ASE that is resolved over a
the reservoir. In the SOA, the role of the optical large number of wavelength bins. Simulations
pump for EDFAs is played by the injected show good accordance between the reservoir
current I. The most striking difference between model predictions and experiments, and good
the two kinds of amplifiers is the fluorescence improvement in calculation time with respect to
time τ, which is of the order of milliseconds in the Connelly model. However, inclusion of
EDFAs and of a fraction of nanosecond in many ASE wavelength channels makes even the
SOAs. Such a huge difference accounts for most reservoir model too slow for the BER
of the disparity in the dynamic behavior between estimations we have in mind. Hence, in order to
the two kinds of amplifiers and explains why further simplify the model, we introduce the
SOAs have not been used for WDM applications reservoir model with a single equivalent ASE
for a long time]. However, recent cheap gain- channel. The ASE can be seen as an independent
clamped SOAs] are likely to promote the use of input-signal channel (with proper input power
SOAs for WDM metro applications. As already and wavelength) that depletes the reservoir of a
mentioned, the reservoir model requires the (co- noiseless SOA. Results show that this last model
propagating) WDM channels to have minimum is the most efficient one since it can be made to
channel spacing in excess of a few tens of accurately predict experimental results with an
gigahertz, in order to neglect the carrier-induced execution time that is 20 times faster than that of
FWM fields generated in the SOA. This should the dynamic Connelly model for single-channel
not be a problem for channels allocated on the operation, with the savings increasing with the
International Telecommunications Union grid number of WDM signal channels. In Section III-
with 50 GHz spacing or more. However, an C, we examine a model that was obtained by
intrinsic limit of the reservoir model is its dividing the SOA into several sections, each
neglecting SHB and CH, which generate FWM characterized by its own reservoir. Here again,
and XPM interactions among WDM channels the ASE can be modeled as a single channel that
even when the minimum channel spacing is propagates through the different reservoir stages.
large enough to rule out any carrier-induced Results show better precision, although the
interaction. The predictions of the reservoir increase in precision is not worth, in most cases,
model will be accurate whenever the carrier the loss in execution time. Most of the numerical
induced XGM mechanism dominates over FWM results are reported in Section IV. Finally,
and XPM. It is worth mentioning that state- Section V summarizes the main findings of this
variable amplifier block diagrams are very paper.
important simulation tools that enable the
reliable power propagation of WDM signals in
optical networks with complex topologies;
therefore, the present reservoir SOA model
33

39. 6.56
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
Figure6.11. Block diagram of the reservoir model of Section III uses a linear approximation
model. ASE contribution not shown for ease of for R (N) in (9); nsig is the number of WDM
drawing. signals; nASE is the number of spectral
components of the ASE; and Kj is an ASE
multiplying factor, which equals 1 for zero facet
6.2.2 DYNAMIC CONNELLY MODEL reflectivity [12]. The factor 2 in accounts for two
A. Theory ASE polarizations. Note that equation contains
In this paper, we adopt the wideband model for a an important approximation: it is the sum of the
bulk SOA proposed in Connelly model, which is signals and ASE powers (fluxes), instead of—
based on the numerical solution of the coupled more correctly—the power of the sum of the
equations for carrier-density rate and photon signals and ASE fields, which depletes carrier
flux propagation for both the forward and density N. Therefore, (3) neglects the carrier-
backward signals and the spectral components of density pulsations due to beating among WDM
ASE. At a specified time t and position z in the channels that generate FWM and XPM in SOAs
SOA, the propagation equation of photon flux [9]. Although such an approximation is
Q±k [photons/s] of the kth forward (+) or inappropriate for extremely dense or high-power
backward (−) signal is 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
6.54
N versus wavelength λk = c/νk (with c being the
where Γ is the fundamental mode confinement
speed of light) using the SOA parameters.
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
6.55
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 Figure.6.12. Gain coefficient g(λ,N) versus
carrier density at coordinate z evolves as wavelength and carrier density
34

40. B. Parameterization 3) The parameters of the carrier-dependent
In order to fit the experimental results that we material-loss coefficient, i.e.
obtained with a commercial Optospeed SOA
model 1550MRI X1500, we used the SOA α (N(z)) = K0 +ΓK1N
parameters provided in the Table in Connelly
model, except for a subset of different values where chosen so that the maximum simulated
reported in Table I in this paper; the most critical gain matched the measured one.
of such parameters were determined as follows.
4) The active-region thickness and width were
1) The active-region length L was determined by set so as to match the experimental and
measuring the frequency spacing between two simulated curves of gain as a function of the
maxima of the gain spectrum ripples: L = λ20 injection current.
/2nrΔλ, where λ0 is the central wavelength
(1550 nm), nr is the average semiconductor 5) The band gap shrinkage coefficient Kg was
refractive index, and Δλ is the ripple wavelength set so that the peak gain wavelength equals the
spacing. measured value of 1560 nm at an injection
current of 500 mA.
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.
35

41. Figure.6.13. Fiber to
fiber unsaturated gain
versus wavelength.
Measured (dashed)
and simulation (solid)
results using Connelly
model.
ensuing Fig. 4
fiber to fiber gain
versus input
C. Simulations with Connelly Model optical power. Measured (dashed) and Connelly
We present simulation results obtained with the model (solid). Experiments and simulations, the
Connelly model and compare them against input signal will be fixed at the gain peak
experimental measurements. wavelength of 1560 nm.
The experiment consisted in amplifying a
tunable continuous wave (CW) laser whose 2) Gain Saturation: Fig. 6.13. shows the fiber-
wavelength was varied around the Optospeed to-fiber gain as a function of the input power.
SOA peak gain wavelength. Laser polarization The wavelength of the input laser was 1560 nm,
was controlled so as to obtain maximum gain. and the injection current was 500 mA.
1) Unsaturated Gain Spectrum: Fig. 3 shows the 3) Dynamic Response: The experimental setup is
simulated and measured unsaturated gain spectra depicted in Fig. 5. The input laser at 1560 nm
at a signal input power of −30 dBm and an was externally modulated at 1 Gb/s. The laser
injection current of 500 mA. A good match power was varied from −25 to −10 dBm in steps
between the simulations and experiments was of 5 dB. The measured photo receiver
obtained when using the values of Table I. In the responsively was 400 mV/mW. The injection
36

42. current was 500 mA. Since we are interested in Figure.6.15. Response to square wave input (see
testing the action of the SOA on the propagating inset representing optical input power in dBm).
signal power in this paper, no optical filter was Measured (dashed) and dynamic Connelly
inserted before detection. model (solid).
The measured experimental input pulses to the
SOA were replicated in the simulator. The 6.3. RESERVOIR MODEL
length of the input-signal time series was 1350 We now derive the reservoir model for a
points over a 2-ns time window. In Fig. 6, we traveling-wave
plot the experimental and the simulated output SOA (zero facet reflectivity) fed by WDM
pulses at an input power of −18 dBm. At this signals. For k =1, . . . , nsig, the propagation and
power level, the SOA is not heavily saturated by carrier density update
the signal; thus, the ASE-induced saturation
significantly contributes to the dynamic
response.
Fig. 6.15 demonstrates that the dynamic 6.57
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.
6.58
4) Computation Time: The major drawback of where A and V = AL are the active waveguide
the Connelly model is its long execution time. area and volume, respectively, and we
Our Matlab code, which was run on a 3-GHz introduced the propagation direction variable uk,
Intel processor, took about 12 s to calculate an which equals +1 for forward signals and −1 for
output bit resolved over 1350 points. Similar backward signals. · QASE j stands for an
calculations for a time series of 50 000 points equivalent ASE flux that accounts for the impact
(37 bits) took about 432 s. This presents a major of both forward and backward ASE on the
limitation when typical Monte Carlo BER carrier-density update equation. The formal
estimations are sought, which require solution of the propagation equation is obtained
transmission of millions of bits. A drastic by multiplying both sides by uk, dividing them
simplification of the gain dynamics calculation by Qk, integrating both sides in dz from z = 0 to
is required in order to significantly decrease z = L for each k, and obtain an equivalent
execution time. Reduced computation time and equation of the form Qout k = Qin k Gk, where
the facility of analysis motivate our introduction the gain
of the reservoir model.
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is independent of the signal propagation
direction. For convenience, we will let
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denote the net gain coefficient per unit length in
the SOA. Now, define the SOA reservoir as
6.61
which physically represents the total number of
carriers in the SOA that are available for
37