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

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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.

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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

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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

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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

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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.