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NUMERICAL SIMULATIONS ON STIMULATED RAMAN SCATTERING FORFIBER RAMAN AMPLIFIERS AND LASERS USING SPECTRAL METHODS
A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OFMIDDLE EAST TECHNICAL UNIVERSITY
BY
HALIL BERBEROGLU
IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR
THE DEGREE OF DOCTOR OF PHILOSOPHYIN
PHYSICS
JULY 2007
Approval of the thesis
NUMERICAL SIMULATIONS ON STIMULATED RAMAN
SCATTERING FOR FIBER RAMAN AMPLIFIERS AND LASERS
USING SPECTRAL METHODS
submitted by HALIL BERBEROGLU in partial fulfillment of the require-ments for the degree of Doctor of Philosophy in Physics Department,Middle East Technical University by,
Prof. Dr. Canan OzgenDean, Gradute School of Natural and Applied Sciences
Prof. Dr. Sinan BilikmenHead of Department, Physics
Assoc. Prof. Dr. Serhat CakırSupervisor, Physics Dept., METU
Assoc. Prof. Dr. Hakan TarmanCo-supervisor, Engineering Sciences Dept., METU
Examining Committee Members:
Prof. Dr. Sinan BilikmenPhysics Dept., METU
Assoc. Prof. Dr. Serhat CakırPhysics Dept., METU
Assoc. Prof. Dr Hakan TarmanEngineering Sciences Dept., METU
Asst. Prof. Dr. Behzat SahinElectrical and Electronics Engineering Dept., METU
Prof. Dr. Arif DemirPhysics Dept., Kocaeli University
Date:
I hereby declare that all information in this document has been obtainedand presented in accordance with academic rules and ethical conduct. Ialso declare that, as required by these rules and conduct, I have fully citedand referenced all material and results that are not original to this work.
Name, Last Name: HALIL BERBEROGLU
Signature :
iii
ABSTRACT
NUMERICAL SIMULATIONS ON STIMULATED RAMAN SCATTERING FORFIBER RAMAN AMPLIFIERS AND LASERS USING SPECTRAL METHODS
Berberoglu, Halil
Ph.D., Department of Physics
Supervisor : Assoc. Prof. Dr. Serhat Cakır
Co-Supervisor : Assoc. Prof. Dr. Hakan Tarman
July 2007, 97 pages
Optical amplifiers and lasers continue to play its crucial role and they have become
an indispensable part of the every fiber optic communication systems being installed
from optical network to ultra-long haul systems. It seems that they will keep on to be
a promising future technology for high speed, long-distance fiber optic transmission
systems.
The numerical simulations of the model equations have been already commercialized
by the photonic system designers to meet the future challenges. One of the challenging
problems for designing Raman amplifiers or lasers is to develop a numerical method
that meets all the requirements such as accuracy, robustness and speed.
In the last few years, there have been much effort towards solving the coupled differ-
ential equations of Raman model with high accuracy and stability. The techniques
applied in literature for solving propagation equations are mainly based on the finite
differences, shooting or in some cases relaxation methods. We have described a new
method to solve the nonlinear equations such as Newton-Krylov iteration and per-
iv
formed numerical simulations using spectral methods. A novel algorithm implement-
ing spectral method (pseuodspectral) for solving the two-point boundary value prob-
lem of propagation equations is proposed, for the first time to the authors’ knowledge
in this thesis. Numerical results demonstrate that in a few iterations great accuracy
is obtained using fewer grid points.
Keywords: Optical Fibers, Raman Amplifiers, Raman Lasers, Numerical Methods,
Spectral Methods.
v
OZ
FIBER RAMAN YUKSELTECLERI VE LASERLARINDAKI UYARILMISRAMAN SACILIMI ETKISININ SPEKTRAL METOT KULLANILARAK
YAPILAN NUMERIK SIMULASYONLARI
Berberoglu, Halil
Doktora, Fizik Bolumu
Tez Yoneticisi : Assoc. Prof. Dr. Serhat Cakır
Ortak Tez Yoneticisi : Assoc. Prof. Dr. Hakan Tarman
Temmuz 2007, 97 sayfa
Optik yukseltecleri ve lazerleri kritik rolunu surdurmekte ve ayrıca optik aglardan ultra uzun-
luktaki sistemlere kadar yer alan butun fiber optik komunikasyon sistemlerinin vazgecilmez
bir parcası olmaktadır.
Gelecekteki ihtiyacların karsılanabilmesi icin model denklemlerinin simulasyonları fotonik sis-
tem tasarımcıları tarafından coktan ticarilestirilmistir. Raman yukseltec ve lazer tasarımındaki
en onemli problemlerden biri kesinlik, hız ve guvenilirlik gibi unsurları yerine getiren numerik
metodların gelistirilmesidir.
Son yıllarda, Raman model diferansiyel denklemlerinin yuksek kesinlik ve kararlılıkta cozulebilmesi
icin onemli cabalar sarf edilmektedir. Literaturde uygulanan teknikler genellikle sınırlı fark,
tahmini atıs ve hafifletme yontemleridir. Bu noktada, yeni bir metot olarak Newton-Krylov
uygulandı ve spektral metot kullanılarak similasyon yapıldı. Bu tezde, bilindigi kadarıyla ilk
defa, sınır deger problemi cozumunde spektral eleman metodu uygulandı. Numerik sonuclar
birkac iterasyonda cok daha az noktada cok yuksek dogruluk gostermektedir.
vi
Anahtar Kelimeler: Optik Fiberler, Raman Yukseltecleri, Raman Lazerleri, Numerik Metotlar,
Spektral Metotlar.
vii
To my parents ...
viii
ACKNOWLEDGMENTS
Each committee member has been wonderful in their support and patience throughout
my research. First of all, I am very thankful to Prof. Dr. Sinan Bilikmen for his endless
support and encouragement during my Ph.D program in METU. I would like to thank
my supervisor Assoc. Prof. Dr. Serhat Cakır for his moral support, encouragement
and guidance throughout the whole study. I would also like to express my gratitude to
Assoc. Prof. Dr. Hakan Tarman for his contribution that has given the shape of my
thesis and for deriving the formulation of spectral methods as well. My sincere thanks
go to Prof. Dr. Arif Demir for his stimulus questions and curiosity to the subject. I
also would like to thank Asst. Prof. Dr. Behzat Sahin for the valuable discussions
and the friendship throughout the research.
Particular thanks go to two of my colleagues from USA, Dr. Aydın Yeniay in Photon-x
LLC. and Dr. Tamer Coskun in ASML Inc. for their time, suggestions and enlight-
ening me in finding solutions to the problem.
Finally, heartfelt thanks go to all of my family for their love and encouragement.
ix
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
OZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
DEDICATON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
CHAPTERS
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Properties of Silica Fiber . . . . . . . . . . . . . . . . . . . . . 2
1.2 Optical Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Theory of Raman Scattering . . . . . . . . . . . . . . . . . . . 10
1.3.1 Spontaneous and Stimulated Raman Scattering . . . 11
2 FIBER RAMAN AMPLIFIERS AND LASERS . . . . . . . . . . . . 18
2.1 Fiber Raman Amplifier . . . . . . . . . . . . . . . . . . . . . . 18
2.1.1 Raman Threshold . . . . . . . . . . . . . . . . . . . . 25
2.2 Fiber Raman Laser . . . . . . . . . . . . . . . . . . . . . . . . 26
3 NUMERICAL FORMULATION . . . . . . . . . . . . . . . . . . . . . 29
3.1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Alternative Forms . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Exponential Change of Variables . . . . . . . . . . . 32
3.2.2 Integration Factor (IF) Method . . . . . . . . . . . . 33
3.2.3 Hybrid Method . . . . . . . . . . . . . . . . . . . . . 33
x
3.3 Pseudospectral Discretization . . . . . . . . . . . . . . . . . . 34
3.4 Weak and Strong Forms . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Strong Form (SF) . . . . . . . . . . . . . . . . . . . . 43
3.4.2 Weak Form (WF) - Spectral Element Formulation . . 43
3.4.3 Domain Decomposition . . . . . . . . . . . . . . . . . 45
3.5 Quasi-Linearization - Newton Method . . . . . . . . . . . . . . 48
3.5.1 Initial Estimate . . . . . . . . . . . . . . . . . . . . . 49
3.6 Solving Nonlinear Equations . . . . . . . . . . . . . . . . . . . 50
3.6.1 Newton-Krylov Iteration . . . . . . . . . . . . . . . . 51
3.6.2 Armijo Rule . . . . . . . . . . . . . . . . . . . . . . . 54
3.6.3 Inexact Newton Condition . . . . . . . . . . . . . . . 55
3.6.4 Preconditioning . . . . . . . . . . . . . . . . . . . . . 56
4 NUMERICAL IMPLEMENTATION . . . . . . . . . . . . . . . . . . . 59
4.1 Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Simulation Results for FRA . . . . . . . . . . . . . . . . . . . 61
4.3 Simulation Results for FRL . . . . . . . . . . . . . . . . . . . . 65
4.3.1 Basic Design of Fiber Raman Laser . . . . . . . . . . 65
4.3.2 Design of The n-th Order FRL . . . . . . . . . . . . 70
5 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 Major Contributions . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Suggestions for Future Research . . . . . . . . . . . . . . . . . 80
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
APPENDICES
A EFFICIENCY OF THE NONLINEARITIES IN BULK AND OPTI-CAL FIBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B NOTES ON KRONECKER PRODUCTS AND MATRIX CALCULUS 89
B.1 The Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . 92
B.1.1 Strong Formulation (SF) . . . . . . . . . . . . . . . . 93
B.1.2 Weak Formulation (WF) . . . . . . . . . . . . . . . . 94
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
xi
LIST OF TABLES
TABLES
Table 1.1 Progress on attenuation of optical fiber [9]. . . . . . . . . . . . . . . 4
Table 1.2 Bands used for fiber optic systems. . . . . . . . . . . . . . . . . . . . 6
Table 4.1 Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 73
Table 4.2 Raman gain coefficients used in the simulation (kmW)−1. . . . . . . 73
xii
LIST OF FIGURES
FIGURES
Figure 1.1 Observed loss spectrum of GeO2 doped single-mode fiber [12]. . . . 5
Figure 1.2 Typical spectrum of light scattering . . . . . . . . . . . . . . . . . . 11
Figure 1.3 The radiation pattern of dipole oscillator. . . . . . . . . . . . . . . 14
Figure 1.4 Quantum mechanical description of Raman scattering. . . . . . . . 16
Figure 2.1 Schematic of various fiber Raman phenomena. a) Stimulated Raman
scattering, b) Raman amplification, c) Raman laser. . . . . . . . . . . . . 19
Figure 2.2 Raman gain spectrum for fused SiO2 at a pump wavelength λP = 1µm. 21
Figure 2.3 Evolution of ASE noise along the fiber from the 1 W input pump
power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Figure 2.4 ASE noise spectra at the end of the fiber (50 km) for 1 W input power. 22
Figure 2.5 ASE noise spectra at the end of the fiber (50 km) for 2 and 3 Watt
input power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 2.6 Spontaneous emission factor versus frequency shift for T = 300 and
30 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 2.7 Temperature effect for the pump power of 1 and 2 W. . . . . . . . 25
Figure 2.8 Transmitted pump and integrated Stokes power versus input power
for 50 km fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 2.9 Pump and integrated ASE power along fiber . . . . . . . . . . . . . 27
Figure 3.1 The seven Lagrange polynomials. . . . . . . . . . . . . . . . . . . . 37
Figure 3.2 Chebyshev Gauss-Lobatto nodes for N=14. . . . . . . . . . . . . . . 39
Figure 3.3 Domain decomposition for E=3, M=2. . . . . . . . . . . . . . . . . 46
xiii
Figure 4.1 The error of pseudospectral method and Runge-Kutta for different
grid numbers N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Figure 4.2 The errors for domain decomposed PS. . . . . . . . . . . . . . . . . 62
Figure 4.3 Illustration of energy transfer among channels. . . . . . . . . . . . . 63
Figure 4.4 Simulation of signal evolution under two backward pumps of 0.5 W. 64
Figure 4.5 Simulation of signal with Rayleigh effect evolution under three back-
ward pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Figure 4.6 Schematic diagram of first-order FRL and definition of the problem. 66
Figure 4.7 Power distribution of forward pump and Stokes radiation in FRL
for P0 = 1 W and Rr = 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Figure 4.8 Output lasing power versus input power for the reflectivity Rr = 50%. 68
Figure 4.9 Output lasing power versus reflectivity of output coupler for P0 = 1
W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Figure 4.10 Power distribution of backward pump and Stokes radiation in FRL
for P0 = 1 W and Rr = 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Figure 4.11 Output lasing power versus reflectivity of output coupler for back-
ward pump of P0 = 1 W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Figure 4.12 Configuraiton of nth-order FRL and definition of the problem. . . . 72
Figure 4.13 Unilateral configuration. Powers of pump and Stokes components
propagating in the cavity of fifth-order FRL as a function of z: (a) pump
wave and (b), (c), (d), (e), (f) are the 1st, 2nd, 3rd, 4th, 5th Stokes waves,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Figure 4.14 Unilateral configuration. Power evolutions of pump and Stokes. . . 75
Figure 4.15 Bilateral configuration. Power evolutions of pump and Stokes. . . . 76
Figure 4.16 The output power versus total input pump power. . . . . . . . . . . 77
Figure 4.17 The output power versus total input pump power. . . . . . . . . . . 77
Figure 4.18 The output power versus fiber length. . . . . . . . . . . . . . . . . . 78
Figure A.1 Geometry of experiment for the bulk medium . . . . . . . . . . . . 86
Figure A.2 Experiment of geometry for fiber . . . . . . . . . . . . . . . . . . . 87
xiv
CHAPTER 1
INTRODUCTION
Today, fiber optics is an interdisciplinary branch of science encompassing fields as
diverse as medicine, sensing, telecommunications and lasers. In communication ap-
plications, optical fibers are considered to be the optimum solution as a waveguide
carrying high capacity signals in the range of terabits per second (Tbps) for hundreds
of kilometers.
Optical communication systems use near-infrared (NIR) wavelengths (700-1700 nm)
in optical fibers as carrier of information. By modulating the carrier, one can encode
the information to be transmitted. According to theory of communication, it is well-
known that the amount of information that can be modulated onto a carrier wave is
roughly proportional to the carrier wave’s frequency. Therefore, the higher the carrier
frequency, the more information a signal can hold. However, if one wants to compute
the classic capacity of a communication channel which can be defined as the maximum
rate of information transferred through the channel without an error, the well-known
Shannon formula might be used to obtain a rough estimate
C = Wlog2
(1 +
S
N
)(1.1)
Here W is the spectral bandwidth and S/N is the signal to noise ratio in power. With
the elimination of OH peaks in loss spectrum of silica fiber, W might be as wide as
400 nm (1200-1600 nm) at most. This wide range around minimum loss of 1550 nm
corresponds to 50 THz. if we assume 20 dB gain for signals, that makes S/N 100.
Finally, the capacity of communication would be as much as 330 Tbps. Indeed this
number is not true, in optical communication there are many limitations and factors
1
affecting capacity of information [1].
Although the complexity of system design in optical communication stands by the
large number of components with different parameters and operational characteristic,
the main constituents of optical communication can be summed up as follows: prop-
agation medium as a waveguide (fiber), transmitter, amplifiers and receivers. To be
consistent with content of thesis, the properties of the fibers and amplifiers will be
discussed in this chapter. In addition the theory of Raman scattering will also be
presented here in summary.
1.1 Properties of Silica Fiber
Optical fiber is the central point of an optical transmission. The main material for
optical fiber is fused silica glass (SiO2) which is dielectric material. Due to the material
property, it prevents any interferences which may cause cross-talk between the fibers.
Also, the signal can not be easily tapped from a fiber since the signals inside the fibers
does not radiate. Therefore, the signal is relatively secure and clear.
Moreover, fused silica glass is also a nonlinear material because its refractive index de-
pends on the light intensity (see Appendix A). Nonlinear properties become important
when the intensity is increased. Nonlinearity limit the power injected into the fiber
and cause noise and cross-talk especially in wavelength division multiplexing systems.
Although the nonlinearity is usually harmful in optical fiber communication, it may
be utilized as favorable application, e.g., fiber Raman amplifiers.
The band gap of fused silica glass is around 10 eV which matches the ultraviolet
photon of wavelength 125 nm. Therefore, this forbids the absorption of photon in the
visible region, as it is stated in the quotation below.
Unlike metals, glasses are transparent because the electrons of the atomicshells are bound to the individual atom. With metals there are free elec-trons which can oscillate with the frequency of an incident electromagneticwave. Therefore the light is strongly reflected. Although the electrons ofthe glass are bound they can be excited to higher energy levels by UVradiation [2].
Glass structure has been for years the subject of many theoretical, experimental and
2
simulation works. It is both a fundamental subject related to nature of the glass
transition but also an active field in applied sciences for example for telecommunication
fibers transmission improvement [3]. Fiber glass is assembled as an amorphous network
of Si-O-Si with a covalent bonds. Each silicon atom bonds tetrahedrally to four oxygen
atoms. Each oxygen atom bonds to two silicones, thus linking adjacent tetrahedra to
one another [4].
Dopants such as GeO2 and P2O5 serve to confine the light by raising the index of
refraction in the core region. Therefore, the light is enforced to travel in the core of an
optical fiber by total internal reflection at the boundary of the lower-index cladding [5].
Silica glass has excellent optical transparency, or very small optical loss, over a wide
range of wavelength between the NIR and the ultraviolet (UV) [6]. To appreciate this
quotation a brief history on development of fiber must be given.
Although early measurements indicated that the attenuation of NIR light in glass
was about 1000 dB per km (or 1 dB/m) which was fine for medical imaging but
too much for communications, researchers challenged to improve the transparency of
glass fibers. In 1966, that challenge was partially overcome by K. Charles Kao and
George Hockham of Standard Telecommunications Laboratories, Harlow England who
suggested that the high attenuation was due to the impurities, and not an intrinsic
property of the glass. In a paper [7], they forecasted that the glasses with a loss about
20 dB/km at 600 nm would be obtained as the impurities were eliminated, although
the lowest loss coefficient for glass was approximately 200 dB/km at that time.
After almost four years of trial and error, experiment and learning, the breakthrough
ultimately came early in August 1970, when the measurements indicated that the
three Corning Glass Works scientists Dr. D. Deck, Dr. B. Maurer and Dr. P. Schultz
had demonstrated a loss of less than 20 dB/km by the chemical vapor deposition
(CVP) method [8]. Later, they announced that they succeeded to make a single-mode
fiber with loss of 16 dB/km at He-Ne laser line of 633 nm by doping titanium into
fiber core. With this low loss, fiber could potentially transmit 65,000 times as much
information as cooper wire. This low loss is the driving force behind the fiber optic
revolutions. The chronological progress of optical fibers is shown in Table 1.1.
3
Table 1.1: Progress on attenuation of optical fiber [9].
Years Loss(dB/km) \Wavelength(µm) References
1970 20\0.633 Corning [7]
1974 2-3\1.06 ATT,Bell Lab [10]
1976 0.47\1.2 NTT,Fujikura [11]
1979 0.20\1.55 NTT [12]
1986 0.154\1.55 Sumitomo [13]
2002 0.1484\1.57 Sumitomo [14]
The principal source of attenuation in optical fibers can be broadly classified into two
categories: absorption and scattering; and they depend on the wavelength of light
and the material. Silica glass has electronic resonance in UV region and vibrational
resonances in the far infrared (FIR) region beyond 2000 nm but absorbs little light in
the wavelength region 500-2000 nm [15]. Absorption is intrinsic to the pure material
properties that needs not to be any density variations or inhomogeneities. In the UV
region, incoming light is subjected to the absorption when photon excite electrons to a
higher energy level. Because the photon energy gradually decreases as the increase of
photon wavelength, absorption of this type which is associated with UV region will be
gradually weakened. In the IR region, another absorption process begins to appear.
The vibrational modes of silica and dopants are responsible for the IR absorption [16].
The absorption occurring IR region is the result of energy transfer from the light to
the vibrational modes of silica. The energy transfer increases as the wavelength of
light increases and so does absorbtion. Another source of absorption is the amount
of water present in silica glass fiber having a strong peak near 1380 nm and a smaller
peak near 1230 nm as seen in Figure 1.1 . The water absorbs light strongly when H
and O atoms vibrate. However, by using a special production method like modified
chemical vapor deposition process, the OH content is almost eliminated. Current
values of OH-ion are less than a few parts per billion.
The second dominant factor leading to signal attenuation are the scattering losses.
The scattering of the signal in optical fiber involves the physical processes known as
Rayleigh scattering and Raman and Brillouin scattering among which Rayleigh scat-
4
Figure 1.1: Observed loss spectrum of GeO2 doped single-mode fiber [12].
tering accounts for more than 85 % of the scattering in silica glass [17]. Therefore,
the scattering loss is mainly dominated by Rayleigh scattering arising from refractive-
index fluctuation due to the microscopic nonuniformity of density and dopant concen-
tration and is increased by doping of Ge-ions. The intensity of Rayleigh scattering
and hence corresponding optical power loss is inversely proportional to the fourth
power of the wavelength of light [18]. Figure 1.1 shows the measured losses as a func-
tion of wavelength. The transmission window for fused silica lies between absorption
due to the electronic transitions and Rayleigh scattering at short wavelength side of
spectrum, and vibration transitions on the long wavelength side of the spectrum.
The International Telecommunications Union (ITU) defines the spectral bands in order
to clarify terminology that is used for fiber optic systems as in Table 1.2. The range
is not strict, it may vary some.
The wavelength windows used by single-mode fiber are the second (1250-1350 nm) and
the third wavelength windows (1450-1620 nm). Dry fiber manufactured by removing
the OH-ion (peak at 1383 nm) from fiber opens up the 1350-1450 nm wavelength range
in use, and the attenuation for Raman pumps around that peak drops to 0.31 dB/km
5
Table 1.2: Bands used for fiber optic systems.
Name Meaning Wavelength Range (nm)
O-band Original 1260-1360
E-band Extended 1360-1460
S-band Short 1460-1530
C-Band Conventional 1530-1565
L-Band Long 1565-1625
U-Band Ultralong 1625-1575
from 1 dB/km. That has been produced by Lucent Technology, known as AllWave1
which has widened the bandwidth from 1260 t0 1675 nm.
Silica fiber is mainly categorized as either multi-mode fiber which is used for short-
distance connections, or single-mode fiber which is used for long-distance connections.
While multi-mode fiber has relatively large diameter core (50-80 µm), single-mod fiber
has a small diameter core (7-10 µm). The most important recommendations defined
by ITU which describes the geometrical and transmissive properties are follows:
• ITU G.652: Characteristics of a single-mode optical fiber also known as standard
single-mode fibers (SSMF). It is the first single-mode optical fibers manufactured
as step-index low loss silica fiber. It covers single-mode non-dispersion shifted
fiber (NDSF). Transmission takes place in the 1310 nm range where there is
minimal signal dispersion. The intrinsic loss at 1310 nm and 1550 nm is about
0.4 dB/km and 0.2 dB/km, respectively.
• ITU G.653: Characteristics of a dispersion-shifted single-mode optical fiber ca-
ble. It was designed to overcome the dispersion limit which was shifted from
1310 nm to 1550 nm.
• ITU G.655: Characteristics of a non-zero dispersion shifted single-mode optical
fiber cable. The name comes from the fact that their dispersion is shifted to a
value that is low (not zero) in the 1550 nm. It was designed to prevent especially
1 AllWave is a trade mark of Lucent Technology.
6
four-wave mixing nonlinear effect which distorts the signal in wavelength division
multiplexing systems.
Corning and Lucent Technologies are the major providers of long-haul cables. Lucent’s
TrueWave and AllWave cables are made of single-mode non-zero dispersion fibers that
support all the wavelength windows. TrueWave is specifically designed for optically
amplified, high-powered long-distance DWDM networks operating in both the C-band
and the L-band.
Corning’s LEAF is a single-mode NZ-DSF fiber designed for DWDM systems. It
combines low attenuation and low dispersion with an effective area that is 32 percent
larger than non- NZ-DSF fiber.
It has been almost 25 years since the commercial usage of fiber optic communications
launched and the technology has been maturing constantly since then. Just very
recently, NTT (Nippon Telegraph and Telephone) has accomplished optical transmis-
sion of 14 Tbps (spectral efficiency of 2 b/s/Hz) over a single 160 km long optical
fiber. The result was reported in a post deadline paper at the European Conference
on Optical Communication 2006, Cannes, France between 24-28 Septembers [19]. The
company aims to design 10 Tbps large capacity core optical network. To put this in
perspective, fiber optics very easily accommodates 1 Tbps (i.e., 1000 billion bits per
second). But what does that really mean? At 1 Tbps it is possible to transmit all 32
volumes of the Encyclopedia Britannica in 1/1000 second (ms) anywhere in the world.
That is an incredible speed and no other medium is capable of this rate of transmission
at such distances. Without optical amplifiers, that would not be possible.
1.2 Optical Amplifiers
Today, a single-mode fiber can carry more than 1 Tbps over more than 10,000 km. The
signals need amplification as they travel through fiber cable. At this stage, optical
amplifiers come into play which follow directly after the transmitter and boost the
optical power. Before optical amplifiers, attenuation along the fiber was eliminated
by the regenerators which amplify the signal only in electrical domain, therefore, they
introduces some complexity and lack of flexibility. In addition regenerators are only
7
capable of caring only one channel and also are bit-rate dependent as well. With the
advent of optical amplifiers, those difficulties were overcome.
Some examples of widely used optical amplifiers can be written as follows: semi-
conductor optical amplifiers (SOAs), Erbium-doped fiber amplifiers (EDFAs), fiber
Raman amplifiers (FRAs), and optical parametric amplifiers (OPAs). Among them,
EDFA was the first commercially available amplifiers which can provide 35-40 nm
gain bandwidth. OPAs amplify the signal based on a nonlinear process called the
four-wave-mixing (FWM).
Although Raman amplification in optical fiber was first observed and measured in
1973 [20], it had to wait to be deployed until high-power diode lasers became available
in late 1990s. The research on Raman amplification was shadowed by the successful
implementation of EDFA during the early 1990s. However, by realizing that distrib-
uted amplifiers provide better system performance in terms of noise in WDM systems,
it was time to reconsider Raman amplifiers and inclination toward it was unavoidable
in the mid to late 1990s. Today, almost all long-haul and ultra long-haul transmission
system uses Raman amplification. One of the main advantages of the FRAs over other
types of optical amplifiers is that gain medium becomes the transmission line itself
which is a conventional low-loss silica fibers. Next, gain is achievable at any wave-
length by proper choice of pump wavelength. By using this property and adjusting
their pump powers, flat Raman gain can be obtained over a wide band [21]. This
saves one less component such as flattening filter and also saves money as well.
Optical amplifiers are not perfect devices for optical communication. They produce
noise as they amplify the signals. One of the major sources of noise in Raman am-
plifier is the amplified spontaneous emission (ASE) generated by spontaneous Raman
scattering mechanism. When the signal undergoes to optical amplification, the spon-
taneous emission noise introduced by the optical amplifiers rolls up along the fiber
length and is amplified by the same mechanism. Present discussion is continued and
extended in Chapter 2.
In fact, spontaneous emission noise is fundamental source of noise in nature and it
can not be avoided, so it is also called quantum noise. The definition of noise given by
A. Yariv: “what exists in a given communication channel when no signal is present”.
8
It is known that quantum effects dominate over thermal effects when hν > kT [22].
At frequencies below the kT, thermal noise, shot noise, etc. dominate. In optical
frequencies, quantum noise is dominant. Therefore it must be taken seriously.
It is known that distributed Raman amplification gives better noise performance than
lumped amplification. It occurs during optical signal amplification in the amplifier.
ASE noise in FRAs is intrinsically low because it acts as a fully inverted system. To
have some insight, it is good to extent the discussion little further. ASE spectral
density is constant and exists at all frequencies (white noise). By assuming that the
noise exists only over the amplifier bandwidth and can be further reduced by placing
an optical filter at the amplifier output, total ASE power after the amplifier would
be [23]
PASE(ν) = 2SASE∆ν (1.2)
where SASE and ∆ν are ASE spectral density and filter bandwidth or optical spectrum
analyzer (OSA) resolution bandwidth. The factor 2 appears in this equation to account
for the two polarization modes of the fiber. ASE spectral density is given as [24]
SASE(ν) = nsp(G− 1)hν (1.3)
where G is the optical gain of the amplifier, nsp is the population inversion factor
(or spontaneous emission factor), and h is Planck’s constant. The amplifier noise
performance in a frequency interval ∆ν is defined by the noise figure (Feq)
Feq(ν) =1
Gon/off
(1 +
PASE
hν∆ν
)(1.4)
The noise figure in distributed FRA is introduced by the equivalent noise figure which
represents the noise figure that a discrete amplifier placed at the receiver end of the
transmission line would need, in the absence of Raman amplification, to maintain the
same gain and noise as that obtained using distributed FRA [15]. Because the net gain
in distributed FRA is typically much smaller than the on/off gain, it is appropriate
to use the on/off gain which is the ratio of the signal output power with the pump on
to that with the pump off [25]. Note that the Raman gain in optical communication
systems is often described by the on/off gain and it is given by the expression
Gon/off = 10 log10Sout(Pump : on)Sout(Pump : off)
(1.5)
9
It is assumed in Eq. (1.4) that signal-ASE noise is the dominant noise source over
ASE-ASE noise at the output of amplifier [26]. If one can make the gain much larger
than 1 (Gon/off À 1), then
FASE(ν) = 2nsp (1.6)
Here nsp can be written for a two level model as
nsp =σe(λ)N2
σe(λ)N2 − σa(λ)N1≈ N2
N2 −N1(1.7)
This is related to wavelength dependent absorption and emission cross-section and
population of the lower N1 and upper level N2 for a doped fiber amplifier. As N2/N1
increases, nsp decreases. When the upper level population is much greater than lower
level population which is the case for FRA, nsp reaches its minimum (nsp ≈ 1). This
fundamental result is stated in as [27] minimum amplifier noise output is obtained
when complete population inversion is achieved in the amplifying medium. In thermal
equilibrium, the relation between N2 and N1 is given by the Boltzmann’s formula
(mean photon occupation number)
N2
N1= exp(−hΩ/kBT ) (1.8)
from which we write
nsp(Ω) =N2
N2 −N1=
11− exp(−hΩ/kBT )
(1.9)
where Ω = |µ − ν| is the Raman shift of optical frequencies of µ and ν [23]. At 0 K,
for all frequencies, nsp becomes 1. At room temperature, if Ω = 13 THz, nsp ≈ 1. As
stated at the beginning that FRAs behaves as a fully inverted system. Then, if the
attenuation of fiber is ignored, noise figure of Raman amplifiers would go to theoretical
quantum limit of 2 (or 3 dB). In reality, attenuation of fiber causes the lesser gain,
and so degradation of noise figure is unavoidable.
1.3 Theory of Raman Scattering
Coupled intensity equation based on the theory of stimulated Raman scattering (SRS)
that governs the propagation of light fields in the fiber has been discussed in various
text and articles [28, 29]. Here, we only give some summary and make discussion to
appreciate the physics behind.
10
Figure 1.2: Typical spectrum of light scattering
1.3.1 Spontaneous and Stimulated Raman Scattering
The spectrum of light scattering of solids, liquids and gases by monochromatic light
of frequency ω0 has the form of the Figure 1.2, which consists of an intense light (like
laser) at ω0 due to the elastic scattering known as Rayleigh scattering as well as all
types of scattered lights due to the inelastic scattering such as Raman, Brillouin, and
Rayleigh-wing [30]. The present discussion is confined to Raman scattering process
which results from the interaction of light with the vibrational modes of the medium.
To begin with, the classical picture of spontaneous Raman scattering will be depicted
first. By spontaneous, it is meant that the intensity of light field is less than some
certain threshold level so that the optical properties of the medium are unmodified.
The spontaneous Raman scattering (SpRS) which was first observed and discovered
by C. V. Raman in 1928 can be related to inelastic light scattering by oscillating
electrons due to the presence of light field (laser) and the fluctuation of fields because
of molecular vibrations. Therefore, the classical theory of SpRS is based on the electric
dipole moment induced on a molecule by the incident laser field [31]. When light goes
into the medium, electric or magnetic field can interact with the constituent atoms
of that medium and the interaction takes place between light and electron clouds of
atom. Since electron clouds is much more sensitive than the atomic nucleus to applied
field, it is easily shifted along applied field. Therefore, the center of positive nuclear
charge is not in the same position as the center of negative charge of atom’s electron
cloud. That is how dipole moment is formed and it is the key to understand the origin
of Raman scattering. Although emphasized in [31], the direction of dipole moment
generally is not that of E-field, it is assumed that E-field propagating along the z-axis
11
(denoted as subscript L) induces dipole moment parallel to E, which is actually true
for isotropic systems:
P (z, t) = αEL(z, t), (1.10)
where α is optical polarizability of the molecule being a function of charge distribution
and EL(z, t) is the electric field of the incident laser. Since α (not constant) depends
on the charge configuration and so internuclear distance, it is constantly modulated
by the nuclear vibrations. For sufficiently small displacement Q of nuclei from the
equilibrium positions, α can be described by a Taylor series expansion
α(z, t) ≈ α0 +(
dα
dQ
)
0
Q(z, t) (1.11)
where α0 is the polarizability in the equilibrium position (Q = 0). Then the induced
dipole moment for single molecule takes the form of the following:
p(z, t) =[α0 +
(dα
dQ
)
0
Q(z, t)]
EL(z, t) (1.12)
Then, assume the forms of incident light fields EL(z, t) oscillating with ωL and Q(z, t)
for a molecule vibrating with ωV
E(z, t) = ELei(kLz−ωLt) + adj.,
Q(z, t) = Q0ei(qz−ωV t) + adj. (1.13)
plugging them into (1.12) and separating according to frequencies as following.
pL(z, t) = α0
[ELei(kLz−ωLt) + adj.
]
= 2α0cos(kLz − ωLt),
(1.14)
shows that dipole moment radiates at the frequency of laser field which corresponds
to Rayleigh scattering effect. Elastic scattering takes place in this process.
pS(z, t) = ELQ0
(dα
dQ
)
0
[ELei(kL−q)zei(ωL−ωV )t + adj.
]
= 2ELQ0
(dα
dQ
)
0
cos [(kL − q)z − (ωL − ωV )t] (1.15)
and
pAS(z, t) = ELQ0
(dα
dQ
)
0
[ELei(kL+q)zei(ωL+ωV )t + adj.
]
= 2ELQ0
(dα
dQ
)
0
cos [(kL + q)z − (ωL + ωV )t] (1.16)
12
and the total dipole moment is
p(z, t) = pL + pS + pAS (1.17)
Eq. (1.15) and (1.16) shows that the induced oscillating dipole moment also contains
the sum and difference frequency terms between laser and vibrational frequencies,
ωL ± ωV . They are called as Stokes and anti-Stokes Raman scattering. From the
energy conservation, in the Stokes process the laser field loses energy to the vibrating
molecule (lattice is heated), while in the anti-Stokes process energy is gained by the
laser field from the molecule (lattice is cooled). The Stokes and anti-Stokes frequencies
emanate from the modulation of the electronic polarizability α by the vibration of
atoms.
Now we have time dependent polarization electric dipole moment as seen above. Elec-
tric dipole moment per unit volume of the medium gives the polarization which is the
only term in the Maxwell equations relating directly to the medium. According to the
inhomogeneous wave equation, therefore, dipole moment induced by the electric field
can serve as the source of new electric field. The intensity of radiation emitted by the
dipole moment (1.17) per unit solid angle dΩ = sinθdθdφ is given by
I(t) =(
14πε0
)(1
4πc3
)sin2θ
(d2p
dt2
)2
= C
(d2p
dt2
)2
(1.18)
where
C =(
14πε0
)(1
4πc3
)sin2θ (1.19)
The radiation is greatest in direction normal to the dipole axis (θ = π/2, and falls to
zero in parallel to dipole axis as shown in Figure 1.3. Now put (1.17) into the (1.18)
and ignoring the spatial parts, we end up the intensity of scattered light per unit solid
angle is given [31]
I(t) = CE2L
[c0cos
2(ωLt) + c1cos2(ωL − ωV )t + c2cos
2(ωL + ωV )t]+ crossterms
(1.20)
13
Figure 1.3: The radiation pattern of dipole oscillator.
where
c0 = α20ω
4L,
c2 =14
(dα
dQ
)2
0
Q20(ωL − ωV )4
c3 =14
(dα
dQ
)2
0
Q20(ωL + ωV )4 (1.21)
then, one can get the ratio of intensities of the anti-Stokes to Stokes line as
IAS
IS=
(ωL + ωV )4
(ωL − ωV )4(1.22)
which misleads that anti-Stokes intensity is greater than Stokes intensity, contradict-
ing to experiments. Classical theory presents the frequencies correctly, but it fails
to give correct intensities. In fact, classical or semi-classical approach can not ex-
plain the difference between the spontaneous and stimulated processes. On the other
hand, quantum mechanical treatment of the process show that according to Placzek’s
theory [31],IAS
IS=
(ωL + ωV )4
(ωL − ωV )4e−~ωV /kBT (1.23)
where ~ is Planck’s constant, kB is Boltzmann factor, and T is the temperature in
Kelvin. Boltzmann distribution of vibrational energy shows up which causes the Stokes
line is more intense than the corresponding anti-Stokes line. This is the opposite
of the classical result in (1.22). As the temperature goes to 0, the ani-Stokes line
disappears. As the temperature increases, anti-Stokes intensity reaches the Stokes
intensity. Therefore, to be precise, quantum treatment of the Raman process is to be
14
used. In fact, one of the most striking feature of quantum theory is that the concept
of the virtual energy level is used, and it represents the intermediate quantum state
occupied by the combine system of photon field and the medium [32]. Virtual level
is not a real eigenstate, so the electron can not be stable in this state for a definite
duration, but it lives there only very short time as long as Heisenberg’s uncertainty
relation allows [33].
To explain the process quantum mechanically, when a photon at angular frequency ωL
is inelastically scattered by a quantized molecular vibration, it excites one quantum
of vibrational energy ωV and generates a downshifted scattered photon of frequency
ωS = ωL − ωV . This process known as Stokes is associated with a creation of phonon
of frequency ωV . In the inverse process, the molecule may already have vibrational
energy, then the incident photon can absorb that energy and producing an upshifted
photon of frequency ωAS = ωL + ωV . In this case, the scattering molecule ends up in
the ground state. This process known as anti-Stokes is associated with a annihilation
of phonon of frequency ωV . These two processes may be spontaneous or stimulated.
For the stimulated process, Stokes or anti-Stokes photons must be incident on the
scattering medium together with the original photon. The scattering diagrams for
Stokes and anti-Stokes processes are given in the Figure 1.4.
For the quantum mechanical description of interaction between optical fields and mole-
cular phonons in single-mode optical fibers, we refer to the paper [34] which corrects
the Raman propagation equation by including group velocity of pump and signals
and also the anti-Stokes spontaneous emission terms which were usually omitted. In
addition we also refer to the paper [35] which explains the how the fiber temperature
effects amplifier performance by applying quantum mechanical description and the
derivation of Raman gain coefficients using a classical electromagnetic model.
SRS is a nonlinear process which can be also utilized in Raman amplifiers and lasers.
When the incident laser intensity is lower than a certain threshold, the presence of
other light fields can be ignored. On the contrary, when the intensity is higher than
certain threshold, a very strong highly directional coherent scattering takes place, the
existence of Stokes and anti-Stokes fields can not be ignored. Therefore, the total
15
Figure 1.4: Quantum mechanical description of Raman scattering.
electric field in (1.10) must be modified to following.
E(z, t) = ELei(kLz−ωLt) + ESei(kSz−ωSt) + EASei(kASz−ωASt) (1.24)
which give rise to the nonlinear interaction between fields so that the spontaneous
Raman scattering can evolve into a stimulated form.
The classical theory that describes SRS uses the model of a simple diatomic molecule.
It approximates the binding effect of the electronic charge distribution by a spring
model between its nuclei. Therefore, the vibrational mode can be described as a simple
harmonic oscillator with damping. The key assumption of the classical theory is that
the optical polarizability of the molecule is not constant but depends on distance
between nuclei as given in Eq. (1.11). Although the Raman scattering process is
nonresonant, interaction between the applied electric field and the molecular vibration
occurs through the fact that polarizability will be modulated in time, and hence the
index of refraction will be modulated in time. At this point, we end up the discussion
and refer to the various reference such as [28, 30, 36] for the comprehensive study of
the Raman scattering.
SRS can be both detrimental and beneficial in optical fiber communication. For
instance, FRAs make use of SRS to enhance the optical signal level by transferring
energy from a higher frequency pump to lower frequency signal. On the other hand,
16
SRS has some side effects because the Raman gain spectrum is very broad (about 40
THz), the energy transfer causes a tilt in the output spectrum (refer to the Figure
4.3) and also optical noise (ASE) was produced and amplified by the same mechanism
which can absorb energy from the pump and signals.
17
CHAPTER 2
FIBER RAMAN AMPLIFIERS AND LASERS
Fiber Raman amplifiers (FRA) and lasers (FRL) have been continuing to attract sci-
entists involving in optical communications all over the world since the discovery of
low loss optical fibers. The research was accelerated after observing stimulated Raman
effect in silica glass optical fiber [37]. Early works focused on the utilization of stim-
ulated Raman scattering for frequency conversion. The FRL played a critical role in
the initial experiments demonstrating Raman amplification in optical communications
[23].
Figure 2.1 represents schematically the process of SRS, FRA, and FRL [38]. Some
of the fiber requirements for the efficient Raman process are low loss, small effective
area and glass that has a proper Raman spectrum. The efficiency of the Raman
process is increased by the amount of germania dopant in silica core. Note that
Raman gain coefficient of silica glass is low (gR ≈ 1×10−13 m/W) [20], but it is about
eight times higher for germanosilica glass than for silica glass fiber. Therefore, the
germanosilicafiber is used extensively in the FRL and FRA [39].
2.1 Fiber Raman Amplifier
Optical amplifiers can be distributed or discrete (lumped). Distributed Raman ampli-
fiers (DRA) possess channel interaction over the 10’s of kilometers of the transmission
fiber. If the interaction takes place over the limited fiber length, which is not the part
of the original transmission line, it becomes the lumped Raman amplifiers (LRA).
Distributed type uses the standard single mode fiber as a gain medium. For the dis-
crete case, few kms of dispersion compensating fibers are used. EDFA are also called
18
Figure 2.1: Schematic of various fiber Raman phenomena. a) Stimulated Raman
scattering, b) Raman amplification, c) Raman laser.
discrete amplifiers. DRA alone or together with LRA offer better noise figure and
reduce the nonlinear effect.
FRAs are based on SRS in fiber. Long-haul telecommunication systems have utilized
stimulated Raman scattering for the amplification of the signals. Some of the ad-
vantages of FRAs (may be distributed or lumped) might be summarized as follows.
First, gain can be obtained in any spectral region (gain is nonresonant) and over a
wide spectral band [40]. Second, Raman gain exists in every fiber, thus it is very easy
to upgrade the existing fiber link. Lastly, adjustment of gain flatness can be simply
made adjusting the pump powers and wavelength. This problem can be numerically
solved several methods such as genetic algorithm.
Some of the discussions given on this chapter was adapted by the C. M. McIntoshs’
PhD dissertation [41].
Theoretical and experimental studies on the ASE of FRA have started in 1980’s [42].
Since then, the number of channels was increased enormously, so the new issues have
emerged such as temperature effect, nonlinear mutual interaction among channels.
19
Raman amplifiers provide gain for the input optical signal via the process of stimu-
lated scattering process. In addition to that, Raman amplifiers generate spontaneous
emission noise through the spontaneous Raman scattering process which is amplified
along the amplifier leading to ASE noise effect in fiber. The noise properties of Raman
amplifiers is very important for the system performance and margins. The amplifier
noise performance is formulated by the noise figure as given in Chapter 1.
Fi =1Gi
(1 +
PASE,i(L)
hνi∆ν
)(2.1)
where Gi is the on/off gain for the i-th channel, hνi is the channel photon energy, and
PASE,i is the output noise power for the i-th channel in the bandwidth ∆ν.
Propagation of generated ASE noise can be given by the following system of coupled
equation [43]
dPASE,i
dz= −αiPASE,i +
n+m∑
j=1
gjiPj(PASE,i + hνi∆νFji), i = 1, 2, ..., n + m (2.2)
where PASE,i is the ASE noise power in the bandwidth ∆ν. The temperature depen-
dent term contributing to ASE noise power is given by the Fji:
Fji =
(nV,ji + 1), νj > νi, for Stokes
−nV,ji, νj < νi, for anti− Stokes.
(2.3)
which gives the effective inversion population factors of the Raman transition between
j-th and i-th wave [43]. Here, nV is the thermal occupation number of the phonons
that is known to follow the Bose-Einstein distribution is given in Eq. (2.4)
nV,ji =1
eh|νi−νj |
kBT − 1= nsp − 1 (2.4)
where kB is The Boltzmann constant, T is the temperature of the system degree in
K, and h is the Planck constant.
It is well known that the Raman gain coefficient is quite low in silica glass, however,
SRS is one of the dominant nonlinearities observed in optical fiber. As explained in
Appendix A, nonlinearities are the results of the confinement of light in a very small
core area and the long interaction lengths of the fiber. Raman amplifiers which make
use of SRS can be both detrimental and beneficial in optical communication systems.
Simply, it is beneficial because it is relatively easier especially for distributed Raman
20
amplifiers (DRA) to keep the signal level preventing to exceed both amplifier noise
and fiber nonlinearities which are the the principal limitations in optically amplified
transmission systems [40]. It is detrimental because as the pump gets close to signal
wavelength which is the case for multi-pump (FRA), the noise figure of the amplifier
increases because of thermal noise.
To begin with, we first apply (2.2) to simulate the interaction of pump wave with the
ASE noise generated along fiber. For the sake of simplicity, we consider that only a
pump wavelength of 1500 nm is launched at the the input end of the fiber. Then,
the systems of differential Eq. (2.2) is integrated using Matlab ”ode45” solver which
is based on explicit fourth/fifth order Runge-Kutta. The Raman gain is halved to
approximate polarization randomization effects. Other parameters are taken to be
α = 0.2 dB/km, Aeff = 50 km2, ∆ν = 150 GHz and the temperature is always set to
300 K if not stated otherwise. In the Figure 2.3, it is seen how the ASE noise from 1
Watt is evolved. There is no input at the Stokes frequency, and all the Stokes power
is from ASE. Figure 2.4 shows the noise spectra at 50 km for the same power.
0 5 10 15 20 25 30 35 40 450
0.2
0.4
0.6
0.8
1x 10
−16
Frequency Shift (THz)
Ram
an G
ain
Coe
f. (m
/W)
λP = 1 µm
Figure 2.2: Raman gain spectrum for fused SiO2 at a pump wavelength λP = 1µm.
There are several vibrational modes occurring in silica glass which brings about Raman
21
Figure 2.3: Evolution of ASE noise along the fiber from the 1 W input pump power.
Figure 2.4: ASE noise spectra at the end of the fiber (50 km) for 1 W input power.
22
Figure 2.5: ASE noise spectra at the end of the fiber (50 km) for 2 and 3 Watt input
power.
scattering as shown in Figure 2.2. As seen, the gain bandwidth is over 40 THz due to
the amorphous nature of fused silica and GeO2. The strongest Raman modes are at
the 13.2 THz due to transverse optical mode the and 14.7 THz due to the longitudinal
optical modes. The other relatively weak modes are located at the 24 and 32.25 THz.
Although the mode at 13.2 THz line is the main contributor, the relative strength
of 14.7 THz line increases with increasing pump power [16]. It is also worth to note
that Raman gain coefficient gR strongly depends on the dopants and the fiber core
composition.
It can be seen in Figure 2.3 and 2.4 how the 14.7 THz line gives its energy to the modes
at 24 THz as the light propagates in the fiber . This second order SRS becomes more
visible as the pump power is increased up to 2 and 3 Watt as shown in Figure 2.5. As
the input pump wave propagates down the fiber, it excites ASE noise due to the SRS,
and the first amplification process starts at 13.2 THz where the gain is maximum.
Then it reaches a level where it begins to pump around 24-25 THz frequency shift
which gives rise to second order Stokes. In fact, this feature might be useful for higher
order fiber Raman amplifiers and lasers [44].
23
Figure 2.6: Spontaneous emission factor versus frequency shift for T = 300 and 30 K.
It should be pointed out that at smaller frequency shift between channels makes
the amplifier not fully inverted, and temperature dependent effects come into play
(see Figure 2.6). For instance, the occupation number of phonons at much lower
temperatures is reduced and spontaneous emission factor approaches 1 [45]. However,
at room temperature, ASE noise in the distributed Raman amplifier where the pump
and signal lie close to each other degrades the noise figure and system performance.
The temperature effect can be seen in the Figure 2.7. It is also important to notice
that anti-Stokes spontaneous emission is sometimes omitted [46] when designing fiber
Raman amplifiers.
Input pump power is injected to the fiber to see the temperature effect clearly. At
small frequency shifts (< 5) THz, the noise amplification at 300 K is higher than the
that of 30 K. That becomes very dramatic as the input pump power is increased (see
Figure 2.7).
24
Figure 2.7: Temperature effect for the pump power of 1 and 2 W.
2.1.1 Raman Threshold
Spontaneous scattered Stokes light is amplified by the input pump power to generate a
stimulated Raman output. The total Stokes power is integrated over all frequencies at
the end of the fiber. Then, in order to find SRS threshold or critical power where the
Stokes power equals to the pump power, integrated Stokes power is plotted against
input pump power shown in Figure 2.8. Limited amount of energy transfers to
the Stokes light below the input power of 1 Watt. As the input power increases, the
SRS is assembled very quickly, and it reaches the threshold power near input power of
1234 mW. Stimulate Raman threshold power was first derived by Smith [47]. To bring
out the essential feature undepleted pump approximation was used and the thermal
occupation number was also neglected in [47]. The condition for a critical power for
the unpreserved polarization becomes
gRP cr0 Leff
Aeff= 32 (2.5)
Using the same parameter that is used in simulation, critical pump power predicted
by (2.5) is 1307 mW, and this value is good enough to approximate SRS threshold.
Figure 2.8 is plotted for the fixed input pump power of 2 Watt to see the length
dependence of input power and integrated Stokes power. If the input pump power is
25
Figure 2.8: Transmitted pump and integrated Stokes power versus input power for 50
km fiber.
below the computed critical pump power, lines would never cross each other.
2.2 Fiber Raman Laser
Almost two decades have passed since the first demonstration of a fiber Raman lasers
(FRL) based on fiber Bragg gratings (FBG). Technology has been constantly maturing
since then. Particularly, multi-wavelength (FRL) have been very attractive subject in
recent years due to their potential applications in WDM, sensing, testing systems and
even industry [48]. As a lasers, they have found one of the major applications in the
fiber optic communication systems where they are employed as a pump sources for
the EDFA and FRA [49]. Their high power and customizable operating wavelength
makes them even more appealing.
Nonlinear effects in optical fiber offer the possibility of generating new frequencies. Al-
though the nonlinearity may not be always desired in photonic application, it becomes
useful for multi-wavelength fiber Raman lasers (FRL) which has been very attractive
subject in recent years due to their potential applications, e.g., pump sources for Ra-
man and rare-earth fiber amplifiers. Fiber Raman lasers are based on a well-known
26
Figure 2.9: Pump and integrated ASE power along fiber
nonlinear optical process called stimulated Raman Scattering resulting in frequency
down-shifted Stokes light. This affect can be used in various ways. For example, in
the case of FRA, SRS is used to amplify signals. Similar idea but different approach
is used to generate light at desired wavelength for FRL. Therefore, Raman scattering
process in the glass fiber can not only be used to amplify signals, but it can also be
used to form a Raman laser.
Raman laser is made up three parts: a laser pump source, a gain medium and a
feedback mechanism. The choice of fiber as a gain medium is based on several factors
for the Raman laser: higher Raman gain coefficient, lesser effective area, lesser fiber
attenuation, reduced splice losses, and the ability to form Bragg gratings (periodic
dielectric structures). One can readily form a FRL by only mounting two FBG onto
a piece of fiber. In Figure 2.1.c, cavity can be formed by two FBGs denoted as M1,2
which provides a means of feedback in the cavity. Fiber Bragg gratings are formed
by changing the refractive index in the fiber core under the ultraviolet radiation as to
reflect light back to the cavity at the Bragg wavelength. FBGs act as high reflectivity
mirrors and serve as the feedback mechanism. For instance, by using the intrinsic
property of glass fiber as an amplifying medium, quasilossless transmission over 75-
27
km is demonstrated by transforming optical fiber into an ultra long cavity laser [50].
In their experiment, they have used symmetric design of the cavity for the secondary
pump and carefully selected pump wavelengths and power levels to provide virtually
lossless transmission optical media.
There are basically three quantities which describes the performance of the lasers [23]:
the slope efficiency (ηs), pump threshold power (Pth) and the overall (total) efficiency
(ηt) which can be obtained from the linear fit of graph of the input pump power (Pin)
versus output power (Pout) as
Pout = ηs(Pin − Pth)
ηt =Pout
Pin(2.6)
At high powers there are some limitation factors that appear in fiber lasers. First,
optical damage might occur to the fiber facets. A practical solution may be to use the
bilateral pumping configuration for the input pump powers. Second, nonlinear effects
such as the optical Kerr effects and stimulated Brillouin and Raman scattering may
disturb the lasing operation at high power densities.
28
CHAPTER 3
NUMERICAL FORMULATION
Numerical methods are the indispensable tools in many branch of sciences. Many
science and engineering applications require solving a number of governing equations
simultaneously over the computational domain. In literature, the numerical solution
methods of Boundary Value Problems (BVPs) may be categorized into two main
groups: Shooting and Finite Difference methods [51]. Shooting methods exploit the
close theoretical relationship between BVPs and Initial Value problems (IVPs) to
construct a numerical method for a given BVP by relating it to corresponding IVPs.
Finite difference methods, on the other hand, use local representations by low-order
polynomials to discretize the solution over the entire interval of interest. Shooting
methods are advantageous due to conceptual simplicity and the ability to make use
of the excellent, widely available, adaptive initial value ODE solvers. But there are
fundamental disadvantages as well, mainly in that the algorithm inherits its stability
properties from the stability of the IVPs that it solves, not just the stability of the
given BVP. The (simple) shooting method may run into stability problems. These
drawbacks are alleviated by more complex methods like multiple shooting. However,
the attractive simplicity of simple shooting is lost along the way. The use of low order
polynomial approximations to approximate the solution in Finite difference methods
necessitates the use of fine grid discretization of the solution domain for a satisfactory
level of approximation. This in turn determines the size of the resulting nonlinear
system after discretization, thus, putting high demand on memory and computing
resources.
29
In contrast, spectral methods make use of global representation, usually by high-order
polynomials and achieve a high degree of accuracy that local methods cannot match.
In fact, while popular finite differences can only achieve algebraic convergence, the
use of much coarser mesh are needed in spectral methods to achieve the same accu-
racy. The adequacy of a coarser mesh associated with the higher accuracy implies
a smaller number of data values to store and operate upon. Spectral collocation (or
pseudospectral) methods are most suited for nonlinear problems as they offer the sim-
plest treatment of nonlinear terms. A good choice of the collocation points (nodes
- grid) may improve the approximation error of the numerical solution of a BVP,
such as the collocation points associated to Gaussian quadrature methods. They are
also associated with the zeros and extrema of the ultraspherical polynomials arising
as eigensolutions to the singular Sturm-Liouville problem in finite domain. The two
typical examples are the Chebyshev and Legendre polynomials and the associated
collocation points are called Chebyshev and Legendre points, respectively. The ap-
proximations based on these collocation points are the most efficient in resolving data.
As a measure of resolving power, equispaced points achieve convergence if there are
at least 6 points per wavelength, while for Chebyshev points that is only π points
per wavelength on average. This family of collocation methods is also referred to as
orthogonal collocation in the engineering literature [52].
The fundamental principle of PS method as presented in [53] that in a given set of grid
points, interpolate the data globally, then evaluate the derivative of the interpolant
on the grid. The choice for the trial functions would be either trigonometric poly-
nomials (Fourier series) in equispaced points for the periodic problems or orthogonal
polynomials of Jacobi type in unevenly spaced points for the nonperiodic problems.
3.1 Model Equations
In order to present the numerical formulation we consider the boundary value problem
of interest in the representative index form
dyk
dz=
N∑
n=1
aknyn + yk
N∑
n=1
bknyn, 0 < z < L (3.1)
30
for k = 1, 2, ..., N or in the representative matrix form
dY
dz= A ∗ Y + diag(Y ) ∗B ∗ Y, 0 < z < L (3.2)
where A = [akn], B = [bkn], Y (z) = [y1(z) y2(z) · · · yN (z)]T and diag(Y) is the
N×N diagonal matrix with vector Y as its diagonal elements.
The data of the problem, namely, the two N×N matrices A and B, take different forms
depending on the model equations under consideration. For example, in the Raman
Amplifier case the matrix B is full while A is a diagonal matrix. In the Raman Laser
case, the matrix B is in block tri-diagonal form while A is in block diagonal form as
follows:
A =
D1
D2
. . .
Dk
and B =
O X
X O. . .
. . . . . . X
X O
(3.3)
where each O and X denote 2×2 zero and nonzero block matrices, and
Dj =
−αj r
r αj
(3.4)
if Rayleigh backscattering is not ignored (r6=0). Here we are assuming that the ele-
ments of vector Y stand for
Y (z) =[P+
1 P−1 · · · P+
j−1 P−j−1 P+
j P−j P+
j+1 P−j+1 · · · P+
K P−K
]T. (3.5)
The model equations are subjected to N two-point boundary conditions
C (Y (0), Y (L)) = 0. (3.6)
The functional form of vector function C is bilinear in the vectors Y(0) and Y(L). In
the particular case of Raman Laser, it is
C (Y (0), Y (L)) =
C01
. . .
C0K
︸ ︷︷ ︸C0
Y (0) +
CL1
. . .
CLK
︸ ︷︷ ︸CL
Y (L)−
Pin
...
0
︸ ︷︷ ︸N×1
(3.7)
31
with 2×2 block diagonals
C01 =
0 0
0 0
, C0
j =
1 −R−
j
0 0
, CL
1 =
0 0
−R+1 1
, CL
j =
0 0
−R+j 1
, (3.8)
for j = 2, 3, ..., K. Note that R+K = R+
OC . In the Raman Amplifier case, the mixed
specified boundary conditions for the pump yp and for the signal ys components of
the vector Y are
ys(0) = αs and yp(L) = βp (3.9)
for k = 1, 2, ..., K with s = k1, ..., kt and p = kt+1, ..., kt+q where t + q = K.
3.2 Alternative Forms
Before discretizing the model equations for numerical approximation, various other
alternative forms of the equations may be considered. These forms mainly aim to
reduce the equations to more convenient forms by rearranging or transforming.
3.2.1 Exponential Change of Variables
As it follows from the physics behind the model, the positivity of the dependent vari-
able Y as well as its amplification or attenuation behavior resembling an exponential
pattern with the propagation distance z motivates the exponential change of variable
in the form uk = ln(yk), yk = exp(uk). Substitution into the model equations yields
the form
duk
dz=
N∑
n=1
akn exp(un − uk) +N∑
n=1
bkn exp(un), 0 < z < L (3.10)
for k = 1, 2, ..., N . In particular, for A a diagonal matrix, A = Λ = diag(λi), the
equations become
duk
dz= λk +
N∑
n=1
bkn exp(un) (3.11)
32
3.2.2 Integration Factor (IF) Method
The idea is to make a change of variable arising from solving the linear part of the
equation exactly [54]. Considering the model equations, we define
V = exp(−Az) ∗ Y. (3.12)
The term exp(−Az) is known as the integrating factor and is associated with the
linear part of the equations. Introducing the new variable V into the equations gives
dV
dz= exp(−Az) ∗ diag(exp(AZ) ∗ V ) ∗B ∗ (exp(Az) ∗ V ). (3.13)
Here, we assume A is nonsingular. For A a nonsingular diagonal matrix, A = Λ =
diag(λi), the equations become in matrix form
dV
dz= diag(V ) ∗B ∗ (exp(Λz) ∗ V ), (3.14)
or in the index formdvk
dz= vk
N∑
n=1
bkn exp(λnz)vn (3.15)
for k = 1, 2, ..., K. In the case that A is not a diagonal matrix but instead it is
diagonalizible under a similarity transformation, A = SΛS−1, the matrix exponential
appearing in the equations may be evaluated by
exp(−Az) = S exp(−Λz)S−1 = S diag(exp(−λiz))S−1, (3.16)
where Λ = diag(λi) is the diagonal matrix whose elements are the eigenvalues of A
and the matrix S has the corresponding eigenvectors of A as its column vectors.
3.2.3 Hybrid Method
The form of the last equation
1vk
dvk
dz=
N∑
n=1
bkn exp(λnz)vn (3.17)
suggests that the exponential change of variable in the form wk=ln(vk), vk=exp(wk)
may be applied in a hybrid manner following the application of the integration factor
method to the original equations to get
dwk
dz=
N∑
n=1
bkn exp(λnz)exp(wn) (3.18)
33
This form may be obtained directly from the original form of the equations for the
case A is a nonsingular diagonal matrix, A = Λ = diag(λi), by the change of variables
wk+λkz=ln(yk), yk=exp(wk+λkz).
3.3 Pseudospectral Discretization
The formulation of any numerical schemes for solving differential equations involves
two essential steps [55]:
• Choice of a finite dimensional space, QN , approximating the continuous space,
Q and providing the space in which approximate solutions are sought.
• Definition of a projection operator, PN : Q → QN .
Here, N is the dimension of the dense subspace, QN ⊂ Q. While the solution u is
assumed to belong to the Hilbert space Q, the numerical solution uN = PNu belongs
to QN . Note that the projection of a function in QN is the identity operation, i.e.
PNuN = uN .
The first step in spectral methods is to assume that the unknown function (solution),
u(x)∈ Q, can be expressed as a series expansion of global and smooth polynomial trial
functions, φn(x), defined on D such that
u(x) =∞∑
n=0
unφn(x), (3.19)
with the truncated approximation
PNu(x) = uN (x) =N∑
n=0
unφn(x). (3.20)
The choice of the trial functions defines the subspace QN in which we seek the ap-
proximate (numerical) solution uN (x). In this work, we assume that φn ∈ Q belongs
to a polynomial family, that is complete in Q and orthogonal under the associated
inner product. With this assumption, the finite dimensional subspace, QN , is of di-
mension N+1 and is spanned by a subset of polynomial family dense in QN , that is,
QN=spanφnNn=0.
34
The way to find the expansion coefficients, un, actually determines the projection
operator PN . Assuming that the trial functions φn(x) form a complete and orthogonal
system with respect to the weight, w(x), we recover
un = (u, φn)w ≡ 1γn
∫
Du(x)φn(x)w(x) dx (3.21)
where the orthogonality property is
(φn, φm)w = γnδnm =
γn if n = m
0 if n 6= m.
(3.22)
Since, we are working in a grid free continuous framework, un are named the contin-
uous expansion coefficients.
Thus, the computation of the continuous expansion coefficients involves the integration
of the function u(x). In the general case, this may be very hard and certainly very
impractical for real and nonlinear problems. To circumvent this problem, we introduce
a number of discrete and distinct grid (collocation) points, xj , and discrete weights,
wj , such that the discrete weighted inner product
[u, φn]w ≡N∑
j=0
u(xj)φn(xj)wj (3.23)
is identical to the continuous inner product
[u, v]w = (u, v)w, u, v ∈ QN (3.24)
for all functions in QN . The validity of this assumption is provided by the theory
of Gauss integration. Under this assumption we immediately recover the discrete
expansion coefficients un in the expansion
INu(x) = uN (x) =N∑
n=0
unφn(x) (3.25)
on the form
un =1γn
[u, φn]w =1γn
N∑
j=0
u(xj)φn(xj)wj (3.26)
where [φn, φm]w = γnδnm and γn = [φn, φn]w for 0 ≤ n,m ≤ N . It can be shown that
INu(xj) = uN (xj) = u(xj), j = 0, 1, ..., N and hence the projection operator IN with
respect to the discrete inner product is an interpolation operator with INu being the
35
interpolant of u. The resulting approximation scheme is called spectral collocation or
pseudospectral method. Note that excepting u ∈ QN , in general,N∑
n=0
unφn(x) = INu(x) 6= PNu(x) =N∑
n=0
unφn(x), (3.27)
where the difference between the two approximations is known as the aliasing error.
The collocation method may also be formulated in a more convenient form. Since
the approximating polynomial uN (x) of order N corresponds exactly to the function
at the collocation points, i.e. INu(xj) = u(xj) and since an Nth order polynomial,
specified at N+1 distinct points, is unique, we may express the approximation using
an interpolating polynomial
INu(x) =N∑
j=0
u(xj)Lj(x) =N∑
n=0
u(xj)
(wj
N∑
n=0
1γn
φn(xj)φn(x)
)(3.28)
where we require that Lj(xk) = δjk. A polynomial satisfying this requirement is the
interpolating Lagrange polynomial of order N on the form
Lj(x) =N∏
k=0,k 6=j
x− xk
xj − xk=
Π(x)(x− xj)Π
′(xj), Π(x) =
N∏
k=0
(x− xk) (3.29)
also known as the Cardinal function. Figure 3.1 verifies graphically that these poly-
nomials have the property of Lj(xk) = δjk at the Chebyshev nodes as shown. This
approach allows for computing derivatives such that we may differentiate the interpo-
lating polynomial to obtain
d(INu)dx
∣∣∣∣x=xk
=N∑
j=0
u(xj)dLj(x)
dx
∣∣∣∣x=xk
(3.30)
with the derivative of the Lagrange polynomial at the collocation points being given
as
Dkj ≡ dLj(x)dx
∣∣∣∣x=xk
=
Π′(xk)
(xk−xj)Π′ (xj)
j 6= k
12
Π′′(xj)
Π′(xj)
j = k
(3.31)
where D is called the differentiation matrix.
In the search of suitable polynomial basis families, it is crucial to identify those which
result in rapidly convergent spectral expansions, independent of the boundary condi-
tions. In fact, the truncation error resulting from the truncated approximation
‖u− Pnu‖w =
( ∞∑
n=N+1
γnu2n
)1/2
, (3.32)
36
1 2 3 4 5 6 7
−1−0.5
00.5
1−0.5
0
0.5
1
1.5
INDEX
LAGRANGE INTERPOLANTS
X
L
Figure 3.1: The seven Lagrange polynomials.
where ‖.‖2w = (., .)w, indicates that the approximation error depends on the decay
of the expansion coefficients which depends on the actual orthogonal basis family
selected. Such a family is given by the polynomial eigensolutions (Jacobi Polynomials)
to the singular Sturm-Liouville problem. It can be shown that they achieve exponential
convergence for the approximation of smooth functions on the interval [-1,1]. This is
also known as spectral accuracy and implies that the expansion coefficients un decay
faster than any algebraic power of 1/n as n → ∞. Two typical examples within
the ultraspherical polynomials subclass of Jacobi polynomials are the Legendre and
Chebyhev polynomials.
Legendre Polynomials, Pn(x), are the specific Jacobi polynomial which minimize
the unweighted w(x) = 1 least square error
minun
∥∥∥∥∥u(x)−N∑
n=0
unPn(x)
∥∥∥∥∥w
for x ∈ [−1, 1] . (3.33)
They form a complete orthogonal family with respect to weight w(x) = 1 and achieve
exponential convergence in approximating smooth functions in the interval [-1,1].
They can be constructed by the three-term recurrence relation
xPn(x) =n
2n + 1Pn−1(x) +
n + 12n + 1
Pn+1(x) (3.34)
37
where the few are
P0(x) = 1, P1(x) = x, P2(x) =12(3x2 − 1), P3(x) =
12(5x3 − 3x) (3.35)
Chebyshev Polynomials, Tn(x), are the specific Jacobi polynomial which minimize
the uniform approximation error
maxx∈[−1,1]
∣∣xN+1 − TN (x)∣∣ . (3.36)
They form a complete orthogonal family with respect to weight w(x) =(√
1− x2)−1
and achieve exponential convergence in approximating smooth functions in the interval
[-1,1]. They can be constructed by the three-term recurrence relation
xTn(x) =12Tn−1(x) +
12Tn+1(x) (3.37)
where the few are
T0(x) = 1, T1(x) = x, T2(x) = 2x2 − 1, T3(x) = 4x3 − 3x (3.38)
Another useful definition of Chebyshev polynomials is given by the expression
Tn(x) = cos(n arccos(x)). (3.39)
The development of the spectral collocation method is now completed with the intro-
duction of special collocation points through the basic theory of Gauss quadrature for
Jacobi polynomials. One particular case, Gauss Lobatto quadrature is the most suited
for enforcing boundary conditions when solving boundary value problems due to the
inclusion of the endpoints x = ±1 in the quadrature nodes as well as their clustering
close to the boundaries. It is based on the nodes obtained from the derivatives of the
Jacobi polynomials. The main result is stated by the following theorem:
Theorem 3.3.1 Assume that the Gauss Lobatto collocation points (nodes), xi, and
the N+1 weights wi are given, then
∫ 1
−1p(x)w(x) dx =
N∑
j=0
p(xj)wj (3.40)
38
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Figure 3.2: Chebyshev Gauss-Lobatto nodes for N=14.
is exact for all p(x) ∈ Q2N−1.
Legendre Gauss-Lobatto Quadrature points xi are found as the roots of the
polynomial
q(x) = (1− x2)d
dxPN (x). (3.41)
No explicit formula is known for these points. The weights wi appear as
wi =
2N(N+1) i = 0, N
2N(N+1) [PN (xi)]−2 i = 1, ..., N − 1.
(3.42)
Chebyshev Gauss-Lobatto Quadrature points xi appear as the roots of the poly-
nomial
q(x) = (1− x2)d
dxTN (x), (3.43)
yielding
xi = − cos( π
Ni)
, i = 0, ..., N (3.44)
where a typical distribution of these points are shown in Figure 3.2. The corre-
sponding weights wi are given as
wi =
π2N i = 0, N
πN i = 1, ..., N − 1.
(3.45)
The development of the quadrature rules, now, opens the way to devise accurate
methods for the computation of the expansion coefficients based on summations rather
than integrations. The resulting discrete expansion is the key development in the
implementation of the pseudospectral method.
The Discrete Legendre Expansion associated with the Gauss-Lobatto points is
given as
INu(x) =N∑
n=0
unPn(x), un =1γn
N∑
j=0
u(xj)Pn(xj)wj (3.46)
39
where
γn =
22N+1 n = 0, ..., N − 1
2N n = N.
(3.47)
Since INu(x) is given as the interpolant at the Gauss Lobatto quadrature points, we
may also express the approximation as
INu(x) =N∑
j=0
u(xj)Lj(x) (3.48)
where the interpolating Lagrange polynomial is obtained on the form
Lj(x) =−1
N(N + 1)(1− x2)P
′N (x)
(x− xj)PN (xj). (3.49)
We recover the entries of the differentiation matrix
Dij =
−N(N+1)4 , i = j = 0
PN (xi)PN (xj)
1(xi−xj)
, i 6= j
0 , i = j = 1, ..., N − 1
N(N+1)4 , i = j = N.
(3.50)
such that differentiation is performed as
INd
dxINu(xi) =
N∑
j=0
Diju(xj). (3.51)
The Discrete Chebyshev Expansion associated with the Gauss-Lobatto points is
given as
INu(x) =N∑
n=0
unTn(x), un =2
Ncn
N∑
j=0
1cj
u(xj)Tn(xj) (3.52)
where
cn =
2 , n = 0, N
1 , n = 1, ..., N − 1.
(3.53)
The explicit forms of the Chebyshev polynomials and Chebyshev Gauss Lobatto
quadrature points given, respectively, by Eq. (3.39) and (3.44), allow the expression
INu(xj) =N∑
n=0
un cos( π
Nnj
), un =
2Ncn
N∑
j=0
1cj
u(xj) cos( π
Nnj
)(3.54)
40
Hence, the discrete Gauss-Lobatto Chebyshev expansion is a Cosine series and the
expansion coefficients as well as the interpolation can be computed using Fast Fourier
Transform. This is the reason for the widespread use of Chebyshev polynomials.
Since INu(x) is given as the interpolant at the Gauss Lobatto quadrature points,
the approximation can be expressed as Eq. (3.48) where the interpolating Lagrange
polynomial is obtained on the form
Lj(x) =(−1)N+j+1
cj
(1− x2)T′N (x)
N2(x− xj). (3.55)
Associated with the interpolating Lagrange polynomial is the differentiation matrix
Dij =
2N2+16 , i = j = 0
cicj
(−1)i+j
(xi−xj), i 6= j
− xi
2(1−x2i )
, i = j = 1, ..., N − 1
−2N2+16 , i = j = N.
(3.56)
such that differentiation can be performed by the Eq. (3.51).
Now, we demonstrate the pseudospectral discretization of a typical BVP
du(z)dz
= G(z, u), a < z < b, subject to g(u(a), u(b)) = 0. (3.57)
Since the natural interval [-1,1] for x in which the Gauss-Lobatto collocation points
are located may be adapted to the problem at hand by a linear transformation z =12(b− a)(x− 1) + b to the interval [a,b] for z, we proceed in the variable x.
We seek solutions, uN (x) ∈ QN , of the form
uN (x) =N∑
n=0
unφn(x) =N∑
j=0
un(xj)Lj(x), (3.58)
where the space, QN , in which we seek solutions is given as
QN =
spanLj(x)Nj=0 | g(uN (x0 = −1), uN (xN = 1)) = 0
. (3.59)
The discrete expansion coefficients, un, may be evaluated using the Gauss-Lobatto
quadrature rule
un =1γn
N∑
j=0
u(xj)φn(xj)ωj , (3.60)
41
or we may express the polynomial uN (x) using the interpolating Lagrange polynomial
Lj(x) based on the Gauss-Lobatto quadrature points, i.e. we have the identity
INu(xj) = u(xj) . (3.61)
Introducing the remainder,
RN (x) =dun
dx−G(x, uN ) , (3.62)
we proceed by requiring that this vanishes exactly at the interior grid points as
INRN (xj) = 0 for j = 1, ..., N − 1 (3.63)
leading to the N-1 equations
DijuN (xj) = G(xj , uN (xj)) for j = 1, ..., N − 1 (3.64)
with the additional requirements that
g(uN (x0), uN (xN )) = 0, (3.65)
resulting in N+1 equations for the N+1 unknowns uN (xj), j = 0, ..., N . Here Dij
stands for the differentiation matrix with the first and the last rows are deleted.
The system resulting from the pseudospectral discretization has much less sparsity
than do systems induced by low-order methods, such as finite difference methods. In
the one-dimensional case, the matrix is full. In the multi-dimensional case it is banded,
but the bandwidth is very large. The resulting system becomes very ill-conditioned
with increasing size of the grid. These properties do not prevent orthogonal collocation
methods from giving extremely accurate solutions to regular problems [52]. In fact,
while popular finite differences can only achieve algebraic convergence, fewer terms
are needed in orthogonal collocation methods to achieve the same accuracy.
3.4 Weak and Strong Forms
Before the spectral discretization of the model differential equations, they may be cast
into two broad forms, namely, strong and weak forms. Strong form is the pointwise
approach to the differential equations, while weak form is an approach in distributions
42
sense in which the equations are put into integral form by means of a set of test
functions. In the following, the original form of the equations introduced in Section
3.1 will be used. For the other alternative forms, the corresponding change of variable
can easily be performed.
3.4.1 Strong Form (SF)
The pseudospectral discretization under this form starts with the truncated represen-
tation of the dependent variables
yMk (z(x)) =
M∑
j=0
ykjLj(x), k = 1, ..., N (3.66)
in terms of Lagrange interpolants where ykj = yk(z(xj)) are the collocated values of
the dependent variables at the Gauss-Lobatto-Chebyshev points xj = − cos(πj/M)
and z(x) = L/2(x + 1) is the map from [−1, 1] onto [0, L]. The residual associated to
the differential equation
R(z) ≡ d
dz(yM
k )−N∑
n=1
aknyMn −
N∑
n=1
bknyMk yM
n (3.67)
is set to zero at the collocation points zi = z(xi), i = 0, 1, ..., M , to get the collocation
equations2L
M∑
j=0
Dijykj −N∑
n=1
aknyni −N∑
n=1
bknykiyni = 0 (3.68)
where Dij = L′j(xi) is the differentiation matrix. These equations corresponding to
the indices i = 0 and i = M are modified to accommodate the boundary conditions
Ck(yi0, yjm) = 0, i, j, k,= 1, ..., N (3.69)
at z = 0 or z = L, respectively corresponding to these indices.
3.4.2 Weak Form (WF) - Spectral Element Formulation
If we multiply the differential equation by a test function Ψ(z), that vanishes on the
corresponding Drichlet boundaries, and integrate over the domain [0,L], we obtain the
weak form∫ L
0
dyk
dzψ(z) dz =
N∑
n=1
akn
∫ L
0ynψ(z) dz +
N∑
n=1
bkn
∫ L
0ykynψ(z) dz. (3.70)
43
Legendre collocation is selected for its computational efficiency over Chebyshev collo-
cation for a weak formulation. Thus, the function yk(z) is similarly approximated by
Lagrange polynomial interpolation based on the set of Gauss Lobatto-Legendre points
xj ,
yMk (z(x)) =
M∑
j=0
ykjLj(x), (3.71)
where ykj = yk(z(xj)). The Lagrange interpolant discretizes the interval [0,L] using
the mapped Lobatto-Legendre-points, 0 = z0, z1, ..., zM = L, where zj = z(xj), with
the associated Gauss-quadrature weights ω0, ω1, ..., ωM .
The test functions are selected as the Lagrange interpolants, ψ(z) = Lj(x(z)) and
introduced into the weak form to get
∫ L
0
d
dz(yk)Lj dz =
N∑
m=1
akm
∫ L
0ymLj dz +
N∑
m=1
bkm
∫ L
0ykymLj dz (3.72)
for k = 1, 2, ..., N and j = 1, ..., M or j = 0, ..., M − 1, depending on the location of
the specified boundary conditions for each variable yk.
Now, we integrate by parts
ykLj(z)|L0 −∫ L
0ykL
′j dz =
N∑
m=1
akm
∫ L
0ymLj dz +
N∑
m=1
bkm
∫ L
0ykymLj dz (3.73)
and introduce expansion to get
ykMδjM − yk0δj0 −∫ L
0
(M∑
i=0
ykiLi
)L′j dz =
N∑
m=1
akm
∫ L
0
(M∑
s=0
ymsLs
)Lj dz +
N∑
m=1
bkm
∫ L
0
(M∑
i=0
ykiLi
)(M∑
s=0
ymsLs
)Lj dz(3.74)
In order to evaluate the integrals, Gauss-Lobatto quadrature is used to get
∫ L
0LiL
′j dz =
∫ 1
−1Li(x)L
′j(x) dx =
M∑
p=0
ωpLi(xp)L′j(xp) = ωiDji,
∫ L
0LsLj dz =
L
2
∫ 1
−1Ls(x)Lj(x) dx =
L
2
M∑
p=0
ωpLs(xp)Lj(xp) =L
2ωjδsj ,
∫ L
0LiLsLj dz =
L
2
∫ 1
−1Li(x)Ls(x)Lj(x) dx =
L
2
M∑
p=0
ωpLi(xp)Ls(xp)Lj(xp) =
L
2ωjδijδsj , (3.75)
44
where we use the cardinality property Lj(xp) = δjp. Rewriting in terms of these
integrals gives the equations
ykMδjM − yk0δj0 −M∑
i=0
ykiDjiωi =L
2
N∑
m=1
akmymjωj +L
2
N∑
m=1
bkmykjymjωj . (3.76)
3.4.3 Domain Decomposition
Weak formulation of the equations allows the introduction of the boundary conditions
into the system in a natural way and further facilitates the use of domain decomposi-
tion strategy. Domain decomposition is a useful strategy in adapting the discretization
process to changing behavior of the solution, such as rapid or slow variation, over dif-
ferent parts of the domain interval. For higher dimensional problems, it has the added
adaptivity to irregular geometries. The solution to the problem in consideration ex-
hibits rapid changes near the boundaries while being relatively smooth in the interior
of the domain interval [0, L]. A three subdomain approach, namely, Ω1 = [0, Γ1],
Ω2 = [Γ1, Γ2], Ω3 = [Γ2, L], to Ω = [0, L] will result in a better resolution control
by applying higher resolutions only in subdomains 1 and 3 where needed. Since, the
above weak formulation still holds within the subdomains, except that the range of
the index j should be adjusted to obey the Dirichlet boundaries within the individ-
ual subdomains, in the following, we will only concentrate on the formulation of the
continuity across subdomains.
The global domain Ω is decomposed into E nonoverlapping subdomains (or elements)
Ωe, e = 1, ..., E. Thus, the representation takes the local form
yM (x)∣∣Ωe
=M∑
i=0
yei Li(x) (3.77)
where we concentrate on a typical dependent variable y (among yk’s) as a function
of the natural local variable x ∈ [−1, 1] and M denotes a generic grid size within a
typical element e. As an advantage of using Lagrangian basis, the continuity across
elements is enforced simply by equating coincident nodal values, i.e.
xei = xe
i ⇒ yei = ye
i (3.78)
If M is the number of distinct nodes in Ω, then the above equation represents (M +
1)E − M constraints on the choice of the local nodal values yei . It is convenient to
45
Figure 3.3: Domain decomposition for E=3, M=2.
cast the continuity constraint in matrix form [52]. Let Y denote the vector of nodal
values associated with a global numbering of the distinct nodes in all of Ω. Let Y e
denote the vector of local values associated with Ωe:
Y e = (ye0, y
e1, ..., y
eM )T , e = 1, ..., E, (3.79)
and let YL be the collection of these local vectors
YL = (Y 1, Y 2, ..., Y e, ..., Y E)T = (y10, y
11, ..., y
1M , y2
0, ..., yEM )T (3.80)
The continuity of y implies the existence of a Boolean connectivity matrix Q that maps
Y to YL. The operation YL = QY is referred to as a scatter from the global Y to local
YL vector. Since Q is not invertible, the converse is closely related gather operation
Y = QT YL. The output of this operation is denoted with a different notation Y ,
because the action of QT is to sum entries from corresponding nodes while the action
of Q is to copy entries of Y to YL.
Consider as an example E = 3, M = 2 case as follows (see Figure (3.3)) : The
scatter operation YL = QY then becomes
YL =
y10
y11
y12
y20
y21
y22
y30
y31
y32
=
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
y1
y2
y3
y4
y5
y6
y7
= QY (3.81)
46
while the gather operation Y = QT YL results in
Y =
y10
y11
y12 + y2
0
y21
y22 + y3
0
y31
y32
=
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 1 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
y10
y11
y12
y20
y21
y22
y30
y31
y32
= QT YL (3.82)
To develop the spectral element operators, we need to extend the notion of local and
global vector representations to system matrices. For a single domain Ωe, we consider
the integral ∫
Ωe
dy
dzΨ dx = (Ψe)T DeY e (3.83)
where D is the local matrix evaluated using quadrature. The unassembled matrix is
the block diagonal collection of local matrices
DL =
D1
D2
. . .
DE
. (3.84)
Clearly, the local integral can be extended to the whole domain via∫
Ω
dy
dzΨ dx =
E∑
e=1
(Ψe)T DeY e = ΨTLDLYL . (3.85)
Introducing the scatter operations ΨL = QΨ and YL = QY , we arrive at∫
Ω
dy
dzΨ dx = ΨT QT DLQY (3.86)
where the M×M matrix D = QT DLQ is referred to as the assembled Neumann opera-
tor, since Dirichlet boundary conditions is yet to be enforced. Dirichlet boundary con-
ditions can be enforced by using a local mask array ML, which restricts nodal values to
Dirichlet boundary to zero, yielding an invertible system matrix D = QT MLDLMLQ.
Mask array is essentially the identity matrix whose diagonal entries corresponding to
nodes on Dirichlet boundary to zero.
47
3.5 Quasi-Linearization - Newton Method
Pseudospectral discretization eventually leads to a nonlinear system of equations. The
most popular method for solving nonlinear equations is Newton’s method. It is based
on the repeated quasilinearization of the system around increasingly improved estimate
of the solution. In order to demonstrate the process, let us consider the typical BVP:
du(z)dz
= G(z, u), a < z < b, subject to g(u(a), u(b)) = 0. (3.87)
Let u0(z) be an initial solution profile (initial guess). In an attempt to arrive at
the exact solution, let u(z) = u0(z) + h(z) be the unknown exact solution with the
correction h(z) to be determined. Introducing into the BVP and expanding in Taylor
series for small h(z) yields
du0(z)dz
+dh(z)
dz= G(z, u0 + h) = G(z, u0) +
∂G(z, u0)∂u
h + O(‖h‖2)
g(α, β) +∂g
dαh(a) +
∂g
dβh(b) + O(|h(a)|2) + O(|h(b)|2) = 0 (3.88)
where α = u(a) and β = u(b). Truncating the higher order terms in this expansion
gives a linear equation to be solved for an estimate of the correction function h(z) ≈h0(z) [
d
dz− ∂G(z, u0)
∂u
]h0 = G(z, u0)− du0(z)
dz(3.89)
subject to∂g
dαh(a) +
∂g
dβh(b) = −g(u(a), u(b)) (3.90)
so that a new estimate can be constructed u1(z) = u0(z)+h0(z). Repeated application
of this procedure leads to the Newton’s iteration
un+1(z) = un(z) + hn(z) (3.91)
with[
d
dz− ∂G(z, un)
∂u
]hn = G(z, un)− dun(z)
dz
∂g
dαhn(a) +
∂g
dβhn(b) = −g(un(a), un(b)) (3.92)
for n = 0, 1, .... For the problem under consideration, it becomes
yn+1k (z) = yn
k (z) + hnk(z) (3.93)
48
with
dhnk
dz−
N∑
i=1
akihni −hn
k
N∑
i=1
bkiyni − yn
k
N∑
i=1
bkihni =
N∑
i=1
akiyni + yn
k
N∑
i=1
bkiyni −
dynk
dz(3.94)
and
C0Hn(0) + CLHn(L) = −C(Y n(0), Y n(L)) (3.95)
where
Hn(z) = [hn1 (z) hn
2 (z) · · · hnN (z)]T . (3.96)
Similar Newton iterative formulations can be obtained for the other alternative forms
by performing the corresponding change of variable. Subsequent pseudospectral dis-
cretization along the lines of section 4 reduces the problem to a linear matrix form.
3.5.1 Initial Estimate
The local convergence theory for the Newton method requires that the initial solution
profile y0k(z) be near the solution. In order to facilitate the construction of such
an initial profile, we propose an estimation (training) process. For this purpose, we
suggest to modify the problem under consideration by introducing a small coefficient
ε > 0 to the nonlinear terms as follows
dyk
dz=
N∑
n=1
aknyn + εyk
N∑
n=1
bknyn, 0 < z < L (3.97)
whose solution is to be denoted by yk(z; ε). Starting from a small value ε0 such that
y0k = yk(z; ε = 0) (the linear solution) is an acceptable initial solution profile for
the Newton iteration improving y0k towards yk(z; ε0), we next propose to set ε = ε1
(ε0 < ε1 < 1) and use the improved solution at ε = ε0 as the initial profile for the
Newton iteration towards yk(z; ε1). Repeating this process as ε → 1− is expected to
result in an acceptable initial profile y0k = yk(z; ε = 1) for the actual BVP (ε = 1).
The quadratic speed of Newton iterations, when the initial guess is near the solution,
is expected to speed up the overall iteration process.
49
3.6 Solving Nonlinear Equations
The main ingredient of nonlinear solvers is to use iterative methods to approximate
the solution to F (x) = 0 numerically in the form of a sequence of approximations
xn. Newton’s method under some standard assumptions can achieve rapid con-
vergence quantified as q-quadratic, namely, the number of significant figures in the
approximates doubles with each iteration. A sequence of the computation of a Newton
iteration requires [56, 57],
• evaluation of F (xn) and a test for termination,
• approximate solution of the equation F′(xn)s = −F (xn) for the Newton step s,
• construction of xn+1 = xn +λs, where the step length λ is selected to guarantee
decrease in ‖F‖.
The computation of the Newton step in item 2 consumes most of the work. Direct
method of computing the step requires storing and factoring the Jacobian matrix that
may be difficult or impractical for very large problems. In multi-dimensional problems,
in addition to large size of the resulting nonlinear system, some degree of sparcity is
also introduced in the process of discretization. In such cases matrix-free iterative
methods are the choice. These are termed Newton iterative methods and realize step
2 by applying a linear iterative method to the equation for the Newton step. This
linear iteration is referred to as inner iteration while the nonlinear Newton iteration
is called the outer iteration. A basic algorithm is given below:
Newton(x,F,τa, τr)Evaluate F (x); τ ← τr|F (x)|+ τa.while F (x) > τ doFind d such that ‖F ′
(x)d + F (x)‖ ≤ ηF (x)If no such d can be found, terminate with failureλ = 1while ‖F (x + λd)‖ > (1− αλ)‖F (x)‖ doλ ← σλ, where σ ∈ [1/10, 1/2] is computed byminimizing the polynomial model of ‖F (x + λd)‖2
end whilex ← x + λdend while
50
In this algorithm, the outer iteration is terminated when
‖F (x)‖ ≤ τr‖F (x0)‖+ τa (3.98)
where x0 is the initial guess and τr and τa are relative and absolute error tolerances,
respectively. The norm of F (x) is used as a reliable indicator of the rate of decay in
the error as the outer iteration progresses. The computation of the Newton direction d
is done iteratively using Newton-GMRES inner iterations. This is a Krylov subspace
based linear solver and belongs to the class of the Newton-Krylov methods. As a
termination criterion for the inner iterations, the inexact Newton condition
‖F ′(xn)d + F (xn)‖ ≤ η‖F (xn)‖ (3.99)
is used. The forcing term η can be varied as the Newton iteration progresses. Choosing
a small value of η will make the iteration more like Newton’s method, therefore leading
to convergence in fewer outer iterations, on the other hand, increasing the number of
inner iterations to satisfy a stringent termination criteria. Following the computation
of the Newton direction d, we compute a step length λ and a step s = λd so that the
sufficient decrease condition
‖F (xn + λd)‖ < (1− αλ)‖F (xn)‖ (3.100)
holds. The introduction of the step length parameter λ is an attempt to achieve global
convergence in Newton’s method by relaxing the requirement in the local convergence
theory that the initial iterate be near the solution. A line search method known as
Armijo rule is implemented to compute a proper step length λ.
3.6.1 Newton-Krylov Iteration
Iterative methods for solving a given linear system Ax = b are concerned with im-
proving an approximation for x in a systematic way. In doing this, the given system is
replaced by some nearby system that can be more easily solved [58]. That is, instead
of Ax = b we solve the simpler system Px0 = b and take x0 as an approximation for
x. We want the correction h that satisfies
A(x0 + h) = b. (3.101)
51
This leads to a new linear system
Ah = b−Ax0. (3.102)
Again, we replace this system by a nearby system and most often P or a cycle of
different approximations P are taken
Ph0 = b−Ax0. (3.103)
This leads to the new approximation x1 = x0 +h0. The correction procedure can now
be repeated for x1, which constitutes an iterative method
xi+1 = xi + hi
= xi + P−1(b−Axi). (3.104)
Here, P−1 is only for notation. z = P−1b actually means z is solved from Pz = b. The
iterative formulation can be interpreted as the basic iteration for the preconditioned
linear system
P−1Ax = P−1b (3.105)
with preconditioner P−1. If we consider P = I or the system Ax = b as preconditoned
system, we arrive at the well-known Richardson iteration
xi+1 = xi + hi = xi + (b−Axi) = b + (I −A)xi = xi + ri (3.106)
with the residual ri = b−Axi. Further, by introducing iteration parameters we get
xi+1 = xi + αiri (3.107)
nonstationary iteration form. This leads to the error reduction formula
ri+1 = (I − αiA)ri. (3.108)
By repeating the Richardson iteration, we observe that
xi+1 = x0 + α0r0 + α1r1 + α2r2 + · · ·+ αiri
= x0 +i∑
j=0
(I − αjA)jr0
︸ ︷︷ ︸spanr0,Ar0,...,Air0≡Ki+1(A;r0)
. (3.109)
The m-dimensional space spanned by a given vector v, and increasing powers of A
applied to v, up to the (m-1)-th power, is called the m dimensional Krylov subspace,
52
generated with A and v, denoted by Km(A; v). This is the subspace in which the
successive approximate solutions are located. Attempts to generate better approxi-
mations from the Krylov subspace are often referred to as Krylov subspace methods
or Krylov projection methods. The Krylov subspace methods are distinguished into
classes for identifying suitable approximation x ∈ Kk(A; r0).
The identification procedure for x ∈ Kk(A; r0) require a suitable basis for the Krylov
subspace of increasing dimension. The raw basis r0, Ar0, ..., Ak−1r0 for x ∈ Kk(A; r0)
is not very attractive from a numerical point of view, since the vectors Ajr0 point
more and more in the direction of dominant eigenvector for increasing j (recall Power
method) and hence the basis vectors become dependent in finite precision arithmetic.
Thus, all the approaches start with the derivation of an orthogonal basis that spans
the Krylov subspace.
The minimum norm residual approach, that leads to the generalized method of resid-
uals (GMRES), is based on the minimization of the residual in the Euclidean norm
‖b−Axk‖2 is minimal over Kk(A; r0). GMRES iterates are constructed as
xk = x0 + α0v0 + α1v1 + α2v2 + · · ·+ αk−1vk−1 (3.110)
where the coefficients αj are chosen to minimize the residual norm ‖b−Axk‖2. Here,
the vjk−1j=0 is the orthonormal basis for Kk(A; r0) constructed through a modified
Gram-Schmidt orthogonalization process referred to as Arnoldi method. GMRES is
designed to solve nonsymmetric linear systems. The major drawback to GMRES
is that the amount of work and storage required per iteration rises linearly with
the iteration count. The usual way to overcome this limitation is by restarting the
iteration after a chosen number (m) of iterations. The accumulated data before restart
are cleared and the intermediate results are used as the initial data for the next m
iterations. This procedure is repeated until convergence is achieved. Unfortunately,
there are no definite rules governing the choice of m, except that, if m is “too small”,
GMRES(m) may be slow or fail to converge, if m is “too large”, it involves excessive
work and storage. If no restarts are used, GMRES will, in theory, converge in no more
than N steps for an N × N linear system. Of course this is not practical when N is
large.
When storage is restricted or the problem is very large, GMRES may not be prac-
53
tical. GMRES(m), for a small m, may not converge rapidly enough to be useful,
Bi-CGSTAB (Bi-Conjugate Gradient Stabilized method) and TFQMR (Transpose-
free Quasi-Minimal Residual method) should be considered in all cases where there is
not enough storage for GMRES(m) to perform well.
3.6.2 Armijo Rule
Methods like Armijo rule are called line search methods because one searches for a
decrease in ‖F‖ along the line segment [xn, xn + d] where d is the Newton direction
d = −F′(xn)−1F (xn). This is an attempt to globalize the Newton method despite
the requirement in the local convergence theory that the initial iterate be near the
solution. In order to motivate the strategy, consider the typical example of applying
Newton’s method to find the root x∗ = 0 of the function F (x) = arctan(x) with initial
iterate x0 = 10 which is too far from the root for the local convergence theory to hold.
This is reflected in the step
s =F (x0)F ′(x0)
≈ 1.5−0.01
≈ −150 , (3.111)
that is far too large in magnitude while being in the correct direction towards the root
from . In fact, the initial and the four subsequent iterates are
10, − 138, 2.9× 104, − 1.5× 109, 9.9× 1017 (3.112)
and they point in the correct direction but overshoots by larger and larger amounts.
A simple reduction of the step by half until ‖F‖ has been reduced will put the iterates
in the right track.
In order to introduce the procedure, we make a distinction between the Newton direc-
tion d and the Newton step s = λd. A full step refers to λ = 1 and so s = d. In order
to keep the step going too far, we let the step be s = 2−md and xn+1 = xn + 2−md,
and find the smallest integer m ≥ 0 such that
‖F (xn + 2−md)‖ < (1− α2−m)‖F (xn)‖ (3.113)
where the parameter α ∈ (0, 1) is a small number intended to ease the satisfaction
of the above inequality. In order to make the reduction procedure more adaptive,
54
following two unsuccessful reductions by halving, we base the next reduction on a
three-point parabolic model of
φ(λ) = ‖F (xn + λd)‖2 (3.114)
fitted to the data φ(0), φ(λm) and φ(λm−1) where λm and λm−1 are the most recently
rejected values of λ. In this approach, the next λ is the minimum of this parabola
p(λ) = φ(0)+λ
λm − λm−1
[(λ− λm−1)(φ(λm)− φ(0))
λm+
(λm − λ)(φ(λm−1)− φ(0))λm−1
].
(3.115)
If p′′
> 0, then we set λ to the minimum of p, λt = −p′(0)/p
′′(0) and apply the
safeguarding step
λm+1 =
λm/10 if λt < λm/10,
λm/2 if λt > λm/2,
λt otherwise.
(3.116)
If p′′(0) ≤ 0, set λm+1 = λm/2. So the algorithm generates a sequence of candidate
step-length factors in the interval
1/10 ≤ λm+1/λm ≤ 1/2. (3.117)
3.6.3 Inexact Newton Condition
The main goal in varying the forcing term η in the inexact Newton condition
‖F ′(xn)d + F (xn)‖ ≤ η‖F (xn)‖ (3.118)
as the Newton iteration progresses, is to achieve just enough precision in the satisfac-
tion of the linear equation F′(xn)s = −F (xn) for the Newton step s in order to make
good progress when far from a solution, but also to obtain quadratic convergence when
near a solution [57]. That is to protect against oversolving. A basic choice may be
based on the residual norms, i.e.
ηResn = γ‖F (xn)‖2/‖F (xn−1)‖2 (3.119)
where γ ∈ (0, 1] is a parameter. If ηResn stays bounded away from 1 for the entire
iteration, the choice ηn = ηResn is sufficient. Otherwise, to make sure that ηn stays
55
well away from 1, we can simply limit its maximum size by
ηn = min(ηmax,max(ηsafen , 0.5τt/‖F (xn)‖)) (3.120)
where τt = τr‖F (x0)‖ + τa is the termination tolerance for the nonlinear (outer)
iteration and
ηsafen =
ηmax for n = 0,
min(ηmax, ηResn ) for n > 0, γη2
n−1 ≤ 0.1,
min(ηmax,max(γη2n−1, η
Resn )) for n > 0, γη2
n−1 > 0.1.
(3.121)
Here, ηmax is an upper limit on the forcing term ηn and typical choices for the para-
meters are γ = ηmax = 0.9. This strategy of safeguarding to avoid volatile decrease
in ηn is based on the idea that if ηn−1 is sufficiently large, then ηn is not allowed to
decrease too much, thus limiting the decrease to a factor of ηn−1, namely, γη2n−1.
3.6.4 Preconditioning
The idea of preconditioning is introduced earlier as the idea of being able to solve
the simpler system Px = b instead of Ax = b. While P (or P−1) is referred to as
the preconditioner, P−1Ax = P−1b is the preconditioned system. The ideal precon-
ditioner should be close to the inverse of A or, in the context of the Newton method,
close to the inverse of the Jacobian. A conditioning of the Jacobian is an important
factor in achieving high speed of convergence in iterative (inner iterations) solution
of the Newton linear equation. We will mainly focus on the part of the Jacobian
associated with the differential operator. The discretized differential operator under
pseudospectral discretization is known to get very ill-conditioned with increasing size
of the grid. Thus, we expect the differential operator part of the Jacobian to be an
important source of ill-conditioning that needs to be preconditioned.
Low order discretization methods, such as finite difference methods, are a good source
of preconditioners for discretized differential operators. They fulfill the first require-
ment to produce simpler systems due to being sparse (banded). The second re-
quirement, that R = P−1A is to be well conditioned, i.e. its condition number
κ(R) = ‖R‖‖R−1‖ is O(1), is a research area. The condition number κ(R) gives
56
an idea of the distribution of the eigenvalues of R in the complex plane. When κ(R) is
large, we expect scattered eigenvalues with considerable variations in their magnitude.
When κ(R) is close to 1, the moduli of eigenvalues are gathered together in a small
interval. Defining π(R) as a measure of scatter of eigenvalues
π(R) =max1≤k≤n|λk|min1≤k≤n|λk| , (3.122)
where the λk’s are the eigenvalues of R, we have 1 ≤ π(R) ≤ κ(R). Generally, π(R)
and κ(R) display more or less the same behavior. Thus, an effective preconditioner
reduces the spread of eigenvalues of the preconditioned matrix.
Using finite differences over the Gauss-Lobatto grid xjNj=0, one can construct an
(N + 1) × (N + 1) bidiagonal preconditioner P for first-order differential operator
du/dx subject to u(x0 = −1) = σ on the natural interval [-1,1] as follows
[P ]ij =
1 if i = j = 0,
−1/h(N)i if 1 ≤ i = j + 1 ≤ N,
1/h(N)i if 1 ≤ i = j ≤ N,
0 elsewhere
(3.123)
where h(N)j = x
(N)j −x
(N)j−1, 1 ≤ j ≤ N . The spread of eigenvalues of the preconditioned
matrix P−1D is reduced. Here, D denotes the pseudospectral differentiation matrix
based on Gauss-Lobatto points.
A more effective preconditioner is proposed in [59] in the form of ZP where
[Z]ij =
1 if i = j = 0,
L(N)j (y(N)
i ) if 1 ≤ i, j ≤ N,
0 elsewhere
(3.124)
The coefficients L(N)j (y(N)
i ), 1 ≤ i, j ≤ N are the entries of N × N matrix relative
to the mapping T . The linear operator T:RN→ RN maps the vector p(yj)Nj=1 into
the vector p(xj)Nj=1 for any polynomial p of degree ≤ N − 1. Here yjN
j=1 are the
Gauss quadrature points as the roots of the Jacobi polynomials q(x) = φN (x) and T
is described by the Lagrange interpolant representation of the polynomial p based on
57
Gauss points
p(x(N)i ) =
N∑
j=1
p(y(N)j )L(N)
j (x(N)i ), 1 ≤ i ≤ N. (3.125)
The effectiveness of the preconditioner ZP is explained by noting that the finite dif-
ference operator P is a good approximation of the operator Z−1D, which maps the
value p(xj)Nj=0 into the values p
′(y(N)
i ) ≈ (p(x(N)i ) − p(x(N)
i−1))/h(N)i , 1 ≤ i ≤ N .
Thus, (ZP )−1D = P−1(Z−1D) is very close to the identity operator. In fact, in the
Chebyshev case, the eigenvalues λi of (ZP )−1D fall into 1 ≤ λi < π/2, 0 ≤ i ≤ N .
58
CHAPTER 4
NUMERICAL IMPLEMENTATION
Make brief introduction.
4.1 Numerical Experiment
As a prototype problem, let’s use the following coupled nonlinear equations to test
the spectral method:
±dPp
dz= −αpPp − c1PpPs
dPs
dz= −αsPs + c2PsPp (4.1)
where Pp,s denote the powers of pump and signal signal and αp,s accounts for loss for
pump and signal wave, and c1,2 are coefficient of nonlinear terms. Parameters used
for test equations are αp = αs = 0.2 dB/km and c1 = 0.670 and c2 = 0.626. If the
sign of pump equation is (+) the forward pumped case, Eq. (4.1) becomes an initial
value problem (IVP). The exact solution of Eq. (4.1) is known as an IVP and initial
values for the pump and signal are chosen to be 1 and 10−3 watts, respectively. For
the implementation of spectral method, it does not not make any difference whether
the problem is an initial or a boundary value problem, so let’s compare pseudospectral
method with other methods for the particular case of (4.1) as an initial value systems.
To solve the Eq. (4.1) numerically, some well-known numerical methods was applied
such as linear multistep up to even sixth-order, and Runge-Kutta up to even fifth-
order, predictor-corrector. The best results was obtained by the fifth-order Runge-
Kutta method which was modified from classical form to increase accuracy as follows.
59
Given the first order ODE
y′ = f(x, y), y(x0) = y0 (4.2)
the value of y(x) at the point x0 + h may be approximated by a weighted average
of values of f(x, y) taken at different points in the interval x0 ≤ x ≤ x0 + h. The
fifth-order Runge-Kutta method for the problem
y(x0 + h) = y(x0) exp[
790
hk1 +3290
hk3 +1290
hk4 +3290
hk5 +790
hk6
],
where
k1 = f(x0, y0),
k2 = f(x0 + h/4, y0) exp (hk1/4),
k3 = f(x0 + h/4, y0) exp (hk1/8 + hk2/8),
k4 = f(x0 + h/2, y0) exp (−hk2/2 + hk3),
k5 = f(x0 + 3h/4, y0) exp (3hk1/16 + 9hk4/16),
k6 = f(x0 + h, y0) (4.3)
× exp[−3
7hk1 +
27hk2 +
127
hk3 +−127
hk4 +87hk5
]
where the numbers in fraction comes from Butcher array for fifth-order Runge-Kutta [60,
61]. The error is defined as eN = ||(y− y∗)||∞ where y∗ is approximation of exact y.
Figure 4.1 shows the remarkable achievement of PS method over fifth-order Runge-
Kutta. For the Runge-Kutta method, as N increases, the error typically decreases
like O(N−m) where m=4.7 which depends on the smoothness of the solution. The
errors for PS method decrease very rapidly until such high precision is achieved and
the convergence at a rate O(cN ) is achieved where c=0.7 for our case.
Moreover, Eq. (4.1) was solved as BVP where the (±) becomes as (−) sign which
corresponds to backward pumped case. Similar error graphic was obtained by the PS
method as in Figure 4.1. Then,the equations was solved by “bvp4c” which is Matlab
solver for BVP. bvp4c is a finite difference code which uses collocation formula. It has
fourth-order accuracy uniformly in [a, b]. Mesh selection and error control are based on
the residual of the continuous solution1. bvp4c uses a finite difference approximation
for the Jacobian which requires additional evaluations of the right-hand side of (4.2).
Although bvp4c is very effective solver, the underlying method is not appropriate1 See Matlab Help for more information.
60
101
102
103
10−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
N
Err
or
Pseudo Spectral Mod. Runge−Kutta
Convergence Rate ≈ N−4.7
Figure 4.1: The error of pseudospectral method and Runge-Kutta for different grid
numbers N.
for high accuracy nor for problems with extremely sharp changes in their solutions.
Therefore, it takes more than 4000 grid points to reach the accuracy of PS which takes
less than 100 grids.
It is worth to note that domain decomposition makes PS much powerful method. It
provides better accuracy as seen from the Figure 4.2 since N corresponds to N/2 for
first and N/2 for the second domain. Here, domain was decomposed into 2 region,
but it can be as many as we need.
4.2 Simulation Results for FRA
In designing FRA, wave propagations along the fiber of the time average value of
optical power in steady state are characterized by a variety of major physical effects
such as nonlinear mutual interaction between waves due to the SRS effect and the
attenuation of waves. Major considerations are the interaction of the pump to pump
(referred to as pump interaction), signal to signal (referred to as Raman crosstalk)
and pump to signal (referred to as pump depletion). Therfore, if we ignore the the
61
101
102
103
10−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
N
Err
or
PS (domain decomposed)PS (w/o domain decomp.)Mod. Runge−Kutta
Figure 4.2: The errors for domain decomposed PS.
negligible effect of Rayleigh scattering amplified spontaneous emission (ASE), then
the coupled nonlinear Raman process in the fiber can be expressed as
±dPi
dz= −αiPi +
n+m∑
j=1
gjiPiPj , i = 1, 2, ..., n + m (4.4)
where n, m is the number of pump and signal waves, respectively. Pi, αi describes
the average power and the attenuation for the channel of νi frequency, respectively.
The plus and minus signs on the left-hand sides of the equation denotes forward and
backward propagating waves. Note that the frequencies are specified in decreasing
order, so νi > νj accounts for i < j and the frequency ratio νiνj
describes the vibrational
losses. Finally, the gain coefficients gji are described as following:
gji =1
AeffΓgj(νj − νi), νj > νi
gji = − 1AeffΓ
νi
νjgi(νi − νj), νj < νi (4.5)
which describes the power transfer by SRS between j-th and i-th waves. gji is the
Raman gain coefficient between the frequency νj and νi. In the simulation, normalized
Raman gain (gN ) is used. The magnitude of peak Raman gain (gpeak) is scaled to
reference pump frequency ν0. In terms of frequency, the scaling can be done by using
62
the following equation.
gR(ν, νS) =(
ν
ν0
)gR(ν0, νS) (4.6)
Here, gR(ν, νS) is the Raman gain coefficient at any desired pump frequency ν and
gR(ν0, νS) is the Raman gain coefficient measured at a reference pump frequency ν0.
Its unit is m/W.
In FRA, higher frequency channels deplete their power and transfer it to the lower
frequency channels through stimulated Raman scattering process as illustrated in
Figure 4.3. The energy transfer produces a tilt in the optical spectra which must be
offset in the transmission. The tilt becomes worse as the input powers increase. The
Figure 4.3: Illustration of energy transfer among channels.
strong nonlinear interactions between channels inside the FRA makes the computation
harder and, the major contributions come from the interaction of pumps and signals.
Particularly, the interaction that takes place between pump waves make it difficult
to predict the required pump power sets for the target gain profile. In literature,
simulated annealing and genetic algorithms are widely used for flat gain among which
genetic algorithms are the most common and applied successfully [62, 63].
In the first simulations, following assumptions are made that fiber losses are α = 0.22
and 0.35 dB/km for signals and pumps, respectively. Length of the fiber is 80 km.
Two pump powers of 0.5 W, 1426 nm and 1453 nm are injected from output end of
the fiber to compensate the loss of 34 signals in C band (1538-1565). The simulation
results is shown in Figure 4.4. The elapsed time for the simulation is about 4 sec.
for the strong formulation and 10 sec. for the weak formulation. We used Newton-
Raphson method to solve the nonlinear systems of equations (F (x) = 0). The number
of iteration to finish simulation is 5 and very a high accuracy (norm of error approaches
almost to machine zero) is obtained with only 24 number of grids. The same model is
simulated by using Matlab solver ‘bvp4c’. To meet the maximum residual of around
63
0 10 20 30 40 50 60 70 8010
−2
10−1
100
101
102
103
Lengthn (km)
Pow
er (
mW
)
Gnet
Gon/off
withoutRaman
Signals
Pumps
Figure 4.4: Simulation of signal evolution under two backward pumps of 0.5 W.
10−6, about 200 mesh points are used. Therefore, our method is better than ‘bvp4c’
in terms of accuracy, grid numbers and even elapsed time. We note that ‘on-off gain’
of 22 dB was computed by using Eq. (1.5) which is one of the important quantity
characterizing Raman amplifier. The net gain (Gnet) of signals was also computed as
4.7 dB.
For the next simulation, we also consider the Rayleigh backscattering effect by adding
the term to the Eq. 4.4(
dP+i
dz
)
Ray
= rP−i (4.7)
Backward signals denoted as (-) superscript are produced due to the single Rayleigh
backscattering. As an example, in this simulation we have used three backward pumps
wirt power 0.30, 0.25, 0.20 Watt. The number of signals are 32. Here, we have used
spectral method for the pump and signals. The computation time is about 8 sec. For
the Rayleigh scattering components, we have used Matlab bvp4c, but the accuracy is
not satisfactory.
Many simulations are performed by the spectral method without much trouble.
64
0 10 20 30 40 50−40
−30
−20
−10
0
10
20
30
Distance (km)
Pow
er (
dBm
)
Pumps
Signals
Backward Signals
Figure 4.5: Simulation of signal with Rayleigh effect evolution under three backward
pumps
4.3 Simulation Results for FRL
From the point of numerical aspects, it is quite challenging paradigm due to coupled
boundary conditions. However, this problem is well-suited to the numerical method
introduced previous chapter.
4.3.1 Basic Design of Fiber Raman Laser
Figure 4.6 shows the setup of a conventional single cavity FRL (first-order). In pre-
vious chapters, equations describing fiber Raman amplifiers were presented. These
equations have to be modified to account waves in the cavity for FRL. The equations
in the steady state which describes the first-order FRL can be written as
±dP±p
dz= −αpP
±p ± rP∓
p − νp
νsgR(P+
s + P−s )P±
p
±dP±s
dz= −αsP
±s ± rP∓
s + gR(P+p + P−
p )P±s (4.8)
65
Figure 4.6: Schematic diagram of first-order FRL and definition of the problem.
with the boundary condition given
P+p (0) = P0
P−p (L) = RpP
+p (L)
P+s (0) = RlP
−s (0)
P−s (L) = RrP
+s (L) (4.9)
where subscripts (s and p) denotes for Stokes and pump wave. α, r and gR are the
attenuation, Rayleigh backscattering and Raman gain coefficients, respectively. L is
length of the FRL cavity. Rl and Rr denotes the reflectivity of the FBG at z = 0
and z = L and assumed to be lossless. Here we neglect the ASE noise because it may
affect the threshold slightly.
The reflectivity Rl of the FBG is chosen to be 99% as usual. For the sake of sim-
plicity, we consider only single-pass pumping, so (Rp = 0). In order to find optimum
configuration which releases maximum output power, the parameters, reflectivity of
output coupler Rr and fiber length L, can be changed.
66
Now, let’s simulate a Raman laser with the pump at 1460 nm and the Stokes wave
at 1550 nm. The fiber used in simulation is dispersion shifted fiber (DSF). The
attenuation coefficients for pump and Stokes wave are assumed to be 0.30 and 0.22
dB/km. Raman gain coefficient used in the simulation is 0.69× 10−14 m/W and the
Rayleigh coefficient is 5×10−5 1/km. Figure 4.7 shows the simulation result of power
waves in the cavity with input forward pump power of P0 = 1 Watt and the FBG
reflectivity of 50% at the output port (Rr). Then, the emitted radiation after
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
Cavity Length (km)
Pow
er (
Wat
t)
Pump (FW)Stokes Sig. (FW)Stokes Sig. (BW)
Stekes (FW)
Figure 4.7: Power distribution of forward pump and Stokes radiation in FRL for
P0 = 1 W and Rr = 50%.
passing through the FBG can be computed approximately as little less than 0.5 W.
Output pump power (1550 nm) is plotted versus the input pump power (1460 nm) in
Figure 4.8. Threshold pump power for lasing from Figure 4.8 appears to be a little
bit above 0.20 W.
We can also explore the relationship between reflectivity of output coupler and output
lasing power. This is illustrated in Figure 4.9. Input pump power is fixed at 1 W
which is much larger than threshold power. It can be seen that 1550 nm laser power
reaches the maximum output power when the reflectivity is around 20%. Moreover,
any desired output power can be obtained by only varying the reflectivity.
67
Figure 4.8: Output lasing power versus input power for the reflectivity Rr = 50%.
Eq. (4.8), with the boundary condition defined in Eq. (4.9) describes that FRL is
basically a boundary value problem and requires numerical modeling due to complex
nonlinear and multiple boundary power transfer between forward and backward trav-
eling wave and between Stokes for the n-th order FRL as well. In literature, as in
the case of FRA, the nonlinear systems of equations of this type have been mostly
solved by shooting or finite difference method, [64], [65], [66], for example. Another
choice might be the relaxation method which can be very fast, but it has a lack of
some flexibility by its nature. Shooting methods is a trial and error approach which
requires special effort for reasonable guesses. Therefore, those numerical methods fol-
low rather cumbersome and confusing ways to accelerate the computation speed and
ensure the stability. That might be one of the reason why they seek approximate
analytical solutions of the equations to keep up the convergence. Again as in the pre-
vious chapters, spectral method have been applied to solve FRL equations with the
imposed boundary conditions. All the simulations have been performed with great
accuracy and stability as well. Once spectral method is implemented, it offers several
advantages from the point of numerical considerations. For example, it can be easily
converted to the backward pumping case, and it provides the same accuracy as the
68
Figure 4.9: Output lasing power versus reflectivity of output coupler for P0 = 1 W.
forward pumping problem.
Simulation for the backward pumping case with the same parameters as forward pump-
ing is shown in the Figure 4.10. Here, the emitted radiation after passing through
the FBG can be computed approximately as little more than 0.5 W. Approximately,
output power of backward pumping is 14 mW is higher than that of forward pumping.
A plot, similar with Figure 4.9 can be obtained for the reversed pumping situation.
One of the measures of the FRL performance used is the the overall (total) efficiency
(ηT ) which are defined as
ηT =Pout
Pin(4.10)
Figure 4.9 and 4.11 can be considered as efficiency plots since Pin = 1 W. Efficiencies
for both configuration gives almost same result, but backward configuration is just a
little bit more profitable.
69
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
Cavity Length (km)
Pow
er (
Wat
t)
Pump (BW)Stokes Sig. (FW)Stokes Sig. (BW)
Stokes (FW)
Figure 4.10: Power distribution of backward pump and Stokes radiation in FRL for
P0 = 1 W and Rr = 50%.
4.3.2 Design of The n-th Order FRL
Numerical analysis of n-th order continuous-wave FRL is mathematically described by
a system of 2(n + 1) ordinary differential equations with the corresponding boundary
conditions at cavity mirrors where n is the the number of Stokes wave:
±dP±0
dz= −α0P
±0 ± rP∓
0 − ν0
ν1g1(P+
1 + P−1 )P±
0
±dP±j
dz= −αjP
±j ± rP∓
j − νj−1
νjgj+1(P+
j+1 + P−j+1)P
±j + gj(P+
j−1 + P−j−1)P
±j
±dP±n
dz= −αnP±
n ± rP∓n + gn(P+
n−1 + P−n−1)P
±n . (4.11)
Although the Rayleigh backscattering coefficient is small, it is preferred to be included.
The reason might be that the high efficiency Raman fiber that is typically used in fiber
lasers has very high numerical aperture and so much more backscattered light can be
captured in the fiber due to Rayleigh scattering effect.
The boundary conditions are given by the the launched pump power and by the Bragg
70
Figure 4.11: Output lasing power versus reflectivity of output coupler for backward
pump of P0 = 1 W.
gratings at each end of the fiber as:
P+0 (0) = P0, P−
0 (L) = R+0 P+
0 (L)
P+j (0) = R−
j P−j (0), P−
j (L) = R+j P+
j (L)
(4.12)
for j = 1, ..., n.
As an example, let’s design a fifth-order Ge-doped RFL. This example is adapted from
the latest reference [65]. In this paper, they have to use approximate analytic results
as the initial values for shooting method in order to reduce computation time from
several hours to a few minutes. Same computation takes just a few seconds in spectral
methods with a satisfactory accuracy as expected.
Fifth-order RFL is simulated for three different pumping configuration without having
any difficulty. The laser cavity for n-th order FRL and the definition of problem is
illustrated in Figure 4.12.
The values of simulation parameters for the 5-th order FRL and related Raman gain
71
Figure 4.12: Configuraiton of nth-order FRL and definition of the problem.
coefficients are given by Table 4.1 and 4.2, respectively.
We begin simulation with unilateral pump configuration. In this configuration, al-
though there is a possibility of damaging the coating layer of fiber as the heat produced
by the injected power increases to a relatively high level, we ignore it for the sake of
numerical demonstration. The length of the fiber is 180 meters and the pump power
for unilateral (forward pumping) configuration is 10 W. The reflectivity of all fiber
Brag gratings is 99%, but output coupler is taken to be 16%. Finally, the Rayleigh
backscattering coefficient is 10−4 1/km.
Figure 4.13 shows the field power distributions of pump and Stokes waves in the
cavity. The elapsed time for this computation is about 30 seconds. The output
power for the forward pumping case would be computed as 6.15 Watt if we solved for
the overheating problem.
72
Table 4.1: Simulation parameters.
Stokes Wavelength (nm) Loss (1/km)
0(pump) 1117 2.12
1 1174 1.75
2 1238 1.54
3 1309 1.17
4 1389 2.63
5 1480 0.76
Table 4.2: Raman gain coefficients used in the simulation (kmW)−1.
g1 g2 g3 g4 g5
4.72 4.31 3.95 3.53 3.03
Next, we can easily switch to another unilateral (backward pumping) case with our
numerical model. The same parameters are used for this simulation. The result of
power evolutions for backward pumping is shown in Figure 4.14 The computation
time is about 30 seconds. The output power for backward pumping is same that of
forward pumping.
Finally, we plot the power distributions of bilateral pump configuration as shown in
Figure 4.15. The computation time takes less than 20 seconds, and the output
power for bilateral configuration is 6.13 Watt. Overheating problem can be avoided
by this configuration as we split high pump powers into equal two parts and launch
them from both ends of fiber. Note that the symmetric pump power evolution makes
the Stokes components possesses a good symmetry.
The slope efficiencies ηs are obtained by the Figure 4.16, input power versus output
power. We used Matlab linear fitting tool to compare the slope efficiencies of lateral
and bilateral pump configuration. As seen from the figure, there is almost linear
relation ship between the input and output pump power. Threshold powers can also
be obtained by this figure. It is 1.3 W for unidirectional pump and 2.1 W for bilateral
pump configuration.
73
0 50 100 150 2000
5
10
Cavity Length (m)
Pow
er (
W)
(a)
0 50 100 150 2000
5
10
Cavity Length (m)
Pow
er (
W)
(b)
0 50 100 150 2000.5
1
1.5
Cavity Length (m)
Pow
er (
W)
(c)
0 50 100 150 2003.1
3.2
3.3
Cavity Length (m)
Pow
er (
W)
(d)
0 50 100 150 2000.5
1
1.5
Cavity Length (m)
Pow
er (
W)
(e)
0 50 100 150 2000
5
10
Cavity Length (m)Po
wer
(W
)
(f)
FW BW
Figure 4.13: Unilateral configuration. Powers of pump and Stokes components prop-
agating in the cavity of fifth-order FRL as a function of z: (a) pump wave and (b),
(c), (d), (e), (f) are the 1st, 2nd, 3rd, 4th, 5th Stokes waves, respectively.
The output power as a function of output coupler reflectivity for unilateral (forward)
pump whose power is fixed at 10 W and bilateral pump power whose power is fixed at
5 W at each ends is shown in Figure 4.17. From the Figure 4.17, the output power
reaches a maximum when the reflectivity is around 10%.
Additionally, the optimum fiber length can be investigated. Figure 4.18 shows output
power versus fiber length. Unilateral pump gives the better output power at length
70 meters, but in practice it is not applicable due to overheating problem. Bilateral
pump configuration can be more practical and its optimum length appears to be
around 130 meters. Using longer fiber length causes loss to dominate over gain.
In all simulations, effect of splice and insert losses are not considered which would
make the computation even simpler. Note that we did not use any analytical solution
in the computation to lead the convergence and increase the accuracy. It is just the
spectral method that accomplishes the task given.
74
0 50 100 150 2000
5
10
Cavity Length (m)
Pow
er (
W)
(a)
0 50 100 150 2000
5
10
Cavity Length (m)
Pow
er (
W)
(b)
0 50 100 150 2000.5
1
1.5
Cavity Length (m)
Pow
er (
W)
(c)
0 50 100 150 2003.1
3.2
3.3
Cavity Length (m)
Pow
er (
W)
(d)
0 50 100 150 2000.5
1
1.5
Cavity Length (m)
Pow
er (
W)
(e)
0 50 100 150 2000
5
10
Cavity Length (m)Po
wer
(W
)
(f)
FW BW
Figure 4.14: Unilateral configuration. Power evolutions of pump and Stokes.
In Eq. (4.11), the coupling between adjacent Stokes waves comes simply from the fact
that a Stokes wave order j is pumped by the Stokes wave of order j − 1 and act as
a pump for the Stokes wave of order j + 1. Here, only the maximum of the Raman
gain curve is used since FBG are placed in the model. This means that each Stokes
wave is assumed to be monochromatic and separated by around 13 THz shift from
one another. All the calculations made in this chapter assume that the anti-Stokes
wave is negligible and there is no phase matching between the fields so that no FWM
terms are present. Therefore, within those approximations, each Stokes wave can only
be pumped by a single wave and be a pump for a single Stokes wave.
In the above analysis, we neglected the linewidth of the laser and Four-wave Mixing
(FWM) interactions. Such a semiempirical model has been proposed in which rep-
resenting the FWM interaction between the laser modes and spectral widths of the
reflectors was taken into account( [67]). For low powers near the threshold lasing
occurs only at the peak reflectivity. As the power is increased modes with smaller re-
flectivities also reach threshold and start lasing. This results an increase in line-width.
75
0 50 100 150 2000
5
10
Cavity Length (m)
Pow
er (
W)
(a)
0 50 100 150 2002
3
4
Cavity Length (m)
Pow
er (
W)
(b)
0 50 100 150 2000.86
0.88
0.9
Cavity Length (m)
Pow
er (
W)
(c)
0 50 100 150 2003.1
3.2
3.3
Cavity Length (m)
Pow
er (
W)
(d)
0 50 100 150 2000.5
1
1.5
Cavity Length (m)
Pow
er (
W)
(e)
0 50 100 150 2000
5
10
Cavity Length (m)Po
wer
(W
)
(f)
FW BW
Figure 4.15: Bilateral configuration. Power evolutions of pump and Stokes.
The detailed models may include more Stokes order within the a few nm of ∆λ = 0.01
to 0.1 depending on the desired accuracy. Therefore, Eq. 4.11 must be modified to
account those effects.
To reach the desired accuracy and to overcome the heavy numerical modelings like the
FRL and FRA, spectral methods would be a right decision among any other numerical
methods.
76
Figure 4.16: The output power versus total input pump power.
Figure 4.17: The output power versus total input pump power.
77
Figure 4.18: The output power versus fiber length.
78
CHAPTER 5
CONCLUSIONS
5.1 Major Contributions
The main motivation in this work is to construct a numerical procedure that handles
Raman equations modeling amplifiers and lasers with high accuracy and flexibility. In
order to achieve this, we propose a spectral method (pseudospectral) together with a
Newton-Krylov type nonlinear solver to eliminate some of the deficiencies present in
the available methods.
First of all, very high accuracy can be obtained by fewer grid points which translates
into considerable savings in computing time and memory. In need of high accuracy, the
contest between the proposed pseudospectral procedure and other low order methods,
such as finite differences, finite elements, is not even a battle but a rout, spectral
methods wins without any doubt [68]. Spectral methods are more difficult as far as
their implementations are concerned, in fact it is more costly per degree of freedom
compared to finite differences. I believe those drawbacks are much more compensated
by their advantages. Therefore, spectral methods should be preferred in fields where
high accuracy is required, especially when benchmarking is of interest. Second, the
flexibility is of much interest in some models. Spectral methods provide high flexibility
in responding to modifications in models.
In this dissertation, pseudospectral method together with the Newton method as a
nonlinear solver is used as a computational tool to investigate FRA and FRL models
numerically. Most of the numerical simulations are performed without much trouble,
79
but there have been some challenges to overcome. It is very ironic that although spec-
tral methods are very efficient in memory management, we have encountered storage
problems when the size of the problem is increased. Most of the techniques presented
in the numerical formulation are mainly introduced to alleviate these problems we
faced, however, due to the time constraints they are not fully utilized in the numerical
implementation. Therefore, we restricted our attention to those cases with pumps
and signals only. Due to the high overhead in the numerical implementation of the
Newton-Krylov nonlinear solver, we have used Matlab’s bvp4c solver to complete our
numerical implementation for the time being. Those techniques will be fully imple-
mented in the subsequent publications. Therefore, we used spectral method to solve
equations with only pump and signals which size of the problem is not big, then we
plug this solution into Matlab bvp4c solver to complete the full model of Raman
amplifier accounting for ASE noise and Rayleigh Backscattering effects.
5.2 Suggestions for Future Research
First of all, we surely need to improve the spectral method to compete with other
conventional methods such as finite difference and finite element methods. Relatively
speaking, even with this form, it is often competitive when the size is small because
it gets very high accuracy with extremely less grid numbers.
Secondly, gain flattening can be studied by the help of spectral method, and I believe
it provides a comprehensive solution to the problem.
Next, the model used for fiber Raman lasers must also be improved so that more
realistic problems can be studied. We have seen that spectral method fits very well to
this problem. Pseudospectral methods is also successfully used with a small computing
time for a sufficient accuracy to solve nonlinear Schrodinger equation (NLSE) in optical
communication. Optical pulse propagation inside a fiber is governed by the NLSE
which takes into account the dispersive properties of the fiber and nonlinear Kerr
effects. Numerically, I would like to assemble Raman equations and NLSE into one
algorithm and solve it with spectral methods. That provides much better analysis of
optical communication systems.
80
There is no doubt that silica glass is one of the key element of optical communication.
Although silica glass offers great advantages and has been installed as fibers for all over
the world, the research on the necessity of novel materials suitable for the replacement
of SiO2 has proceeded with constant craving. For instance, the peak Raman response
of tellurite-based glasses have been proven to be over 50 times that of fused silica [69].
The numerical simulation for these new glasses have been already performed [70].
Spectral method would be the right choice for anyone who desires accuracy, reliability
and more from the algorithm. Another particular area that we intend to perform
simulations is photonic crystal fibers.
81
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84
APPENDIX A
EFFICIENCY OF THE NONLINEARITIES IN BULK
AND OPTICAL FIBERS
Fused silica glass is called nonlinear because its index of refraction depends on light
intensity which can be expressed as (also known as optical Kerr effect).
n(ω, |E|2) = n0(ω) + n2|E|2, (A.1)
where n0(ω) is the linear part that can be well approximated by the Sellmeier equation,
and also related with the first order susceptibility as
n0(ω) = (1 + χ)1/2, (A.2)
and the n2 is the nonlinear-index coefficient (sometimes called as the second order
index of refraction) and is related to third order susceptibility by
n2 =3
8n0Re(χ(3)), (A.3)
where ‘Re’ stands for the real part and |E|2 is the optical intensity inside the fiber.
This effect is due to the deformation of the electron cloud of molecules by the strong
electric field of the light wave.
For silica fibers, n0 = 1.444 and n2 ≈ 2.6 × 10−20 m2/W near 1500 nm. Measured
values are found to vary in the range 2.2−3.9×10−20 m2/W. Different dopant materials
such as GeO2 and Al2O3 affect the measured value of n2. n2 is small compared to
most other nonlinear media by at least two orders of magnitude. In spite of the
intrinsically small values of the nonlinear coefficients in fused silica, nonlinear effects
in optical fibers can be observed at relatively low power levels [15]. To give an example,
consider an optical pump power of 100 mW propagating through an optical fiber of
85
Figure A.1: Geometry of experiment for the bulk medium
effective area with 80 µm2 produces index change ∆n = 3 × 10−11. So what makes
fiber a suitable medium for which the observation of various nonlinearities is notable?
Two characteristics of the fiber can strongly enhance optical nonlinearities: the small
core size and the length of the fiber [71].
In order to get the efficiency of a nonlinear process, we need to consider the product
of (ILeff ) where I is the intensity of light wave and Leff is the effective length of
interaction region. Now let’s compare this efficiency for the bulk and fiber medium:
For the bulk material (no waveguiding), the geometry of the experiment is shown in
Figure A.1.
Note that the beam amplitude has a Gaussian distribution with a spot size of ω0 at
z = 0. The spot size w(z) grows with distance according to formula which has a
minimum value at z = 0.
ω(z) = ω0
√1 +
z2
z20
, (A.4)
z0 is called focal beam parameter or Rayleigh range and given as
z0 =πω2
0n
λ(A.5)
where n is refractive index of the bulk medium and λ is the wavelength of the incident
laser beam. Intensity distribution of a Gaussian beam is given by
I(r, z) = I0
(ω0
ω(z)
)2
exp(−2r2
ω2(z)
)(A.6)
86
Figure A.2: Experiment of geometry for fiber
Considering only longitudinal distribution (r = 0), we get
I(z) = I0
(ω0
ω(z)
)2
(A.7)
Efficiency of nonlinear process can be computed by the integral
(ILeff )bulk =∫ L/2
−L/2I(z) dz <
∫ ∞
−∞I(z) dz
≤∫ ∞
−∞I0
(ω0
ω(z)
)2
dz (A.8)
where I0 = P/πω20 and the integral becomes
(ILeff )bulk =P
πω20
∫ ∞
−∞
z20
z20 + z2
dz =Pπn
λ(A.9)
which can be approximated in the range [−L/2, L/2] to
(ILeff )bulk ≈ P
λ(A.10)
For the fiber, the geometry of experiment is depicted in Figure A.2, In the single-
mode fibers, transverse distribution of the laser beam remains unchanged along the
fiber; therefore the same spot size
omega0 can be maintained in the fiber core radius of a. In this case, the nonlinear
efficiency for the dielectric waveguide becomes
(ILeff )fiber =∫ L
0I(z)exp(−αz)dz =
P
πω20α
[1− exp(−αL)] (A.11)
where we used I(z) = I0exp(−αz), note that the change in optical signal intensity
which propagates through the optical fiber is given as
dI
dz= −αI (A.12)
87
If we assume that αL À 1, then
(ILeff )fiber =P
πω20α
(A.13)
Obviously, efficiency of optical nonlinearities can be greatly enhanced by the small
core radius and low loss which make silica fiber a suitable medium for the observation
of number of nonlinearities at relatively low power level. The enhancement can be as
large as 109. As a result, no matter how small n2 is, silica-fibers have a big impact on
a wide variety of nonlinear effects.
Practically, the actual parameter, γ that determines the magnitude of the correspond-
ing nonlinear effect can be calculated as follows. First recall that propagation constant
can be given as
β =2πn
λ(A.14)
i.e., the variations in the refractive index due to the Kerr effect will modify the prop-
agation constant. By using the Kerr effect,
n(P ) = n0 + n2P
Aeff(A.15)
multiplying both sides by 2π/λ, then we get
2π
λn(P ) =
2π
λn0 +
2π
λ
P
Aeffn2
β(P ) = β0 + γP (A.16)
where β0 is a linear part of linear refractive index n0 and γ is the nonlinear parameter
given as
γ =2πn2
λAeff(A.17)
It is clear that in (A.17) γ inversely depends on effective cross-sectional area and signal
wavelength. Typical values of γ for conventional single-mode optical fibers operating
at wavelengths around 1550 nm (Aeff = 80µm2, n2 = 2.6× 10−20) is 1.32 (Wkm)−1.
Note that the experimental parameter that is measured when investigating the non-
linearities in optical fibers is n2/Aeff . This is known as the nonlinear coefficient.
Experimentally, nonlinear coefficient and the effective area are measured first, and
then using the parameter above, n2 is obtained.
88
APPENDIX B
NOTES ON KRONECKER PRODUCTS AND
MATRIX CALCULUS
Some of the useful matrix calculations are presented by using the reference [72]. First,
we begin with an example of vectorizing a matrix, because it has a notable use of vec
operator. Therefore, consider the matrix A of order m×n, the vec operator is defined
as
vec A =
A.1
A.2
...
A.n
(B.1)
where A.i denotes i-th column of matrix A whose dimension is mn. For example, if
A =
a d
b e
c f
(B.2)
then
vec A =
a
b
c
d
e
f
(B.3)
From the definition we see that vec A is a vector of order 6.
Kronecker product, also known as a tensor product is an another concept used in
matrix calculus. The Kronecker product of two matrices, denoted by A⊗B is defined
89
as
A⊗B =
A11B A12B · · · A1nB
A21B A22B · · · A2nB...
......
Am1B am2B · · · AmnB
(B.4)
where matrix A is of order m×n and matrix B is of order r× s. A⊗B is an mr×ns.
For example, let
A =
A11 A12
A21 A22
, B =
B11
B21
(B.5)
then
A⊗B =
A11B A12B
A21B A22B
=
A11B11 A12B11
A11B21 A12B21
A21B11 A22B11
A21B21 A22B21
(B.6)
Some properties of Kronecker product:
1. If α is scalar, then
A⊗ (αB) = αA⊗B
2. The product is distributive, that is
(a) (A + B)⊗ C = A⊗ C + B ⊗ C
(b) A⊗ (B + C) = A⊗B + A⊗ C
3. The product is associative
A⊗ (B ⊗ C) = (A⊗B)⊗ C
4. The Mixed Product Rule:
(A⊗B)(C ⊗D) = AC ⊗BD
5. Let Im denote the unit matrix of order (m×m) (square matrices) :
Imn = Im ⊗ In
Note that the same holds for zero matrices.
6. An important relationship between the Kronecker product and vec operator is
vec AB = (I ⊗A)vec B or
= (B′ ⊗A)vec I
90
where A and B are both order of (n × n) and the prime means ‘the transpose
of’.
Turning now to calculus, the derivative of a matrix is often encountered in matrix
operation. Given the matrix
A(t) = [aij(t)] (B.7)
For example, consider the simplest derivative with respect to a scalar variable t.
d
dtA(t) =
[d
dtaij(t)
](B.8)
Some of the important properties are as follows:
1. Given conformable matrices A(t) and B(t)
d
dt(AB) =
dA
dtB + A
dB
dt(B.9)
2. let C = A⊗B, then
dC
dt=
dA
dt⊗B + A⊗ dB
dt(B.10)
Turning now to calculus, a number of applications involve the derivative of matrix.
To define the derivatives for vectors x of order n with a components of x1, x2, · · · , xn
and y of order m with a components of y1, y2, · · · , xm, first consider the derivative of
vector y with respect to x,
∂y
∂x=
∂y1
∂x1
∂y2
∂x1· · · ∂ym
∂x1
∂y1
∂x2
∂y2
∂x2· · · ∂ym
∂x2
......
...∂y1
∂xn
∂y2
∂xn· · · ∂y2
∂xn
(B.11)
which gives the matrix of order (n×m). The vec is omitted for the sake of simplicity.
Second, the derivatives of a scalar y with respect to a vector x can be written as
follows
∂y
∂x=
∂y∂x1
∂y∂x2
...∂y
∂xn
(B.12)
91
Lastly, the derivative of a vector y with respect to a scalar is
∂y
∂x=
[∂y1
∂x∂y2
∂x · · · ∂ym
∂x
](B.13)
Note that the derivatives of matrices with respect to matrices can be taken by vec-
torizing the matrices.
Here, we include the numerical procedure that we have used in the simulation of FRAs
and FRLs. For the time being, we have only implemented the matrix formulations of
Strong and Weak Forms. The only problem with this is when the size is increased (i.e,
number of equations), we run out of memory. This problem is primarily overcome in
Chapter 3.
B.1 The Proposed Algorithm
In order to describe the numerical algorithm, we consider the boundary value problem
of interest in the representative form
dyk
dz= akyk + yk
N∑
n=1
bknyn, 0 < z < L, k = 1, 2, ..., N (B.14)
with mixed specified boundary conditions
ys(0) = αs and yp(L) = βp (B.15)
where s = k1, ..., kt and p = kt+1, ..., kt+q with t + q = N . Another convenient form of
the differential equation is
d
dz[exp(−akz)yk] = exp(−akz)
N∑
n=1
bknykyn (B.16)
obtained using the integrating factor exp(−akz). In a more compact form Eq. (B.16)
becomes
d
dz(Yk) =
N∑
n=1
bknexp(anz)YkYn (B.17)
where Yk = exp(−akz)yk.
92
B.1.1 Strong Formulation (SF)
We consider the Chebyshev collocation or equivalently Lagrange polynomial interpola-
tion approximation of the function Yk(z) based on the set of Gauss Lobatto-Chebyshev
points xj = cos(πj/M),
Y Mk (z(x)) =
M∑
j=0
Ykjhj(x), (B.18)
where Ykj = Yk(z(xj)) and z(x) = L/2(x+1) is the map from [−1, 1] onto [0, L]. The
Lagrange interpolant hj(x) is the polynomial of degree defined by
hj(x) =(−1)j−1(1− x2)T
′M
cjM2(x− xj)(B.19)
where
cj =
2, if i=0,M
1, otherwise.(B.20)
The residual associated to the differential equation
R(z) =d
dzY M
k −N∑
n=1
bknexp(anz)Y Mk Y M
n (B.21)
is set to zero at the collocation points zi = z(xi), i = 0, 1, ..., M , to get the collocation
equations2L
M∑
j=0
DijYkj −N∑
n=1
bknexp(anzi)YkiYni = 0 (B.22)
which are modified for i = 0 and i = M to accommodate the respective boundary
conditions
YsM − αs = 0 and Yp0 − βpexp(−apL) = 0 (B.23)
Here Dij = L′j(xi) is the differentiation matrix defined by Eq. (3.56). The collocation
equations written as a nonlinear system of N(M + 1) equations
F (Y ) ≡
2L
[IN ⊗D]− diag[(B ⊗ IM+1)diag(E)Y ]
Y = 0 (B.24)
can be solved by Newton-Raphson iteration sheme
Y (l+1) = Y (l) + H and J(Y (l))H = −F (Y (l)) (B.25)
starting from
Y(0)sj = αs and Y
(0)pj = βpexp(−apL) (B.26)
93
for all j = 0, ...,M , where J is the Jacobian matrix defined by
J(Y ) =2L
(IN ⊗D)− diag[(B ⊗ IM+1)diag(E)Y ]
− diag(Y )[(B ⊗ IM+1)diag(E)]. (B.27)
Here, ⊗ denotes the Kronecker product, diag(A) the diagonal matrix form of a vector
A, IN the identity matrix of size N , Bij = bij ,
Y = [Y1, Y2, ..., YN ]T , and E = [E1, E2, ..., EN ]T , (B.28)
where Eij = exp(aizj). During actual implementation, the boundary conditions are
incorporated into F and J is modified accordingly.
B.1.2 Weak Formulation (WF)
If we multiply the differential equation by a test function V (z), that vanishes on the
corresponding Drichlet boundaries, and integrate over the domain [0,L], we obtain the
weak form ∫ L
0
d
dz(Yk)V (z) dz =
N∑
n=1
bkn
∫ L
0exp(anz)YkYnV (z) dz. (B.29)
Legendre collocation is selected for its computational efficiency over Chebyshev collo-
cation for a weak formulation. Thus, the function Yk(z) is similarly approximated by
Lagrange polynomial interpolation based on the set of Gauss Lobatto-Legendre points
xj ,
Y Mk (z(x)) =
M∑
j=0
Ykjhj(x), (B.30)
where Ykj = Yk(z(xj)). The Lagrange interpolant discretizes the interval [0, L] us-
ing Lobatto-Legendre-points, 0 = z0, z1, ..., zM = L with the associated weights
ω0, ω1, ..., ωM so that
yk(z) ≈M∑
i=0
ykiLi(z), (B.31)
where Li(zj) = δij are the Lagrangian interpolants and yki = yk(zi), and∫ zM
z0
f(z) dz ≈ ω0f(z0) + ω1f(z1) + ... + ωMf(zM ). (B.32)
Integrate after multiplication with the Lagrange interpolants to get
∫ zM
z0
d
dz
(e−akzyk
)Lj(z) dz =
N∑
m=1
bkm
∫ zM
z0
e−akzykymLj(z) dz (B.33)
94
for j = 0, 1, ..., M and k = 1, 2, ..., N .
Integrate by parts
e−akzykLj(z)∣∣zM
z0−
∫ zM
z0
e−akzykL′j(z) dz =
N∑
m=1
bkm
∫ zM
z0
e−akzykymLj(z) dz (B.34)
and introduce expansion to get
e−akzM ykMδjM − e−akz0yk0δj0 −∫ zM
z0
e−akz
(M∑
i=0
ykiLi(z)
)L′j(z) dz =
N∑
m=1
bkm
∫ zM
z0
e−akz
(M∑
i=0
ykiLi(z)
)(M∑
s=0
ymsLs(z)
)Lj(z) dz (B.35)
or equivalently
e−akzM ykMδjM − e−akz0yk0δj0 −M∑
i=0
yki
∫ zM
z0
e−akzLi(z)L′j(z) dz =
N∑
m=1
bkm
M∑
s=0
M∑
i=0
ykiyms
∫ zM
z0
e−akzLi(z)Ls(z)Lj(z) dz. (B.36)
Introduce the Gaussian quadrature to approximate the integrals
e−akzM ykMδjM − e−akz0yk0δj0 −M∑
i=0
yki
M∑
n=0
ωne−akznLi(zn)L′j(zn) =
N∑
m=1
bkm
M∑
s=0
M∑
i=0
ykiyms
M∑
n=0
ωne−akznLi(zn)Ls(zn)Lj(zn) (B.37)
or equivalently
e−akzM ykMδjM − e−akz0yk0δj0 −M∑
i=0
yki
M∑
n=0
ωne−akznδinL′j(zn) =
N∑
m=1
bkm
M∑
s=0
M∑
i=0
ykiyms
M∑
n=0
ωne−akznδinδsnδjn. (B.38)
Finally, it then takes the form of a system of N × (M + 1) nonlinear equations
e−akzM ykMδjM − e−akz0yk0δj0 −M∑
i=0
ykiωie−akziL
′j(zi) =
N∑
m=1
bkmykjymjωje−akzj (B.39)
or equivalently
ykMδjM − e−ak(z0−zM )yk0δj0 −M∑
i=0
ykiωie−ak(zi−zM )L
′j(zi) =
ωje−ak(zj−zM )ykj
N∑
m=1
bkmymj (B.40)
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for any given j = 0, 1, ...,M and k = 0, 1, ..., N .
Then, for each k = 0, 1, ..., N ,
e−akz0 0 0
0 0 0
0 0 −e−akzM
yk0
yk1
ykM
+
L′0(z0) L
′0(z1) L
′0(z2)
L′1(z0) L
′1(z1) L
′1(zM )
L′M (z0) L
′M (z1) L
′M (zM )
+
N∑
m=1
bkm
ym0
ym1
ymM
:
ω0e−akz0 0 0
0 ω1e−akz1 0
0 0 ωMe−akzM
yk0
yk1
ykM
=
0
0
0
(B.41)
This can be written in the form of an N(M + 1) nonlinear system:
F (Y ) ≡
(IN ⊗ diag(W0)) +[(IN ⊗DT ) + diag((B ⊗ IM+1)Y )
]
(IN ⊗ diag(W ))
diag(E) Y = 0 (B.42)
where
E =[e−a1z0 , e−a1z1 , . . . , e−a1zM , . . . , e−aNz0 , e−aNz1 , . . . , e−aNzM
],
W = [ω0, ω1, . . . , ωM ] , W0 = [1, 0, . . . , 0, − 1] of size M + 1,
B =
b11 . . . b1n
.... . .
...
bN1 . . . bNN
, D =
L′0(z0) . . . L
′M (z0)
.... . .
...
L′0(zM ) . . . L
′M (zM )
,
Y = [y10, y11, . . . , y1M , . . . , yN0, yN1, . . . , yNM ]T . (B.43)
Similarly, Jacobian of this system can be written as an N(M +1)×N(M +1) matrix:
J ≡ [Jij ] ≡[∂Fi
∂Yj
]
≡
(IN ⊗ diag(W0)) +[(IN ⊗DT ) + diag((B ⊗ IM+1)Y )
](IN ⊗ diag(W ))
diag(E) + (IN ⊗ diag(W )) diag(E) diag(Y ) (B ⊗ IM+1) (B.44)
96
VITA
Author was born in Denizli on August 3, 1970. He received a B.Sc. degree in Engi-
neering Physics from the Ankara University in 1992. He graduated from the Physics
department of Lehigh University in 1996 with M.Sc. degree on Plasma Physics under
the supervisor of Prof. Dr. Y. Kim. He has been a Ph.D. student at the Physics de-
partment of METU since 1997. He can be contacted via e-mail: [email protected]
97