Download Computational-Photonics-An-Introduction-With-MATLAB (2).pdf PDF

TitleComputational-Photonics-An-Introduction-With-MATLAB (2).pdf
TagsWavelength Division Multiplexing Photonics Multiplexing Optics Dispersion (Optics)
File Size5.2 MB
Total Pages468
Table of Contents
                            Contents
Preface
1 Introduction
	1.1 What is photonics?
	1.2 What is computational photonics?
		1.2.1 Methods of computational photonics. Computational electromagnetics
		1.2.2 Computational nano-photonics
		1.2.3 Overview of commercial software for photonics
	1.3 Optical fibre communication
		1.3.1 Short story of optical fibre communication
		1.3.2 Short history of communication
		1.3.3 Development of optical fibre
		1.3.4 Comparison with electrical transmission
		1.3.5 Governing standards
		1.3.6 Wavelength division multiplexing (WDM)
		1.3.7 Solitons
	1.4 Biological and medical photonics
	1.5 Photonic sensors
	1.6 Silicon photonics
	1.7 Photonic quantum information science
	References
2 Basic facts about optics
	2.1 Geometrical optics
		2.1.1 Ray theory and applications
		2.1.2 Critical angle
		2.1.3 Lenses
		2.1.4 GRIN systems
	2.2 Wave optics
		2.2.1 Phase velocity
		2.2.2 Group velocity
		2.2.3 Stokes relations
		2.2.4 Interference in dielectric film
		2.2.5 Multiple interference in a parallel plate
		2.2.6 Fabry-Perot (FP) interferometer
	2.3 Problems
	Appendix 2A: MATLAB listings
	References
3 Basic facts from electromagnetism
	3.1 Maxwell's equations
	3.2 Boundary conditions
		3.2.1 Electric boundary conditions
		3.2.2 Magnetic boundary conditions
	3.3 Wave equation
	3.4 Time-harmonic fields
	3.5 Polarized waves
		3.5.1 Linearly polarized waves
		3.5.2 Circularly and elliptically polarized waves
	3.6 Fresnel coefficients and phases
		3.6.1 TE polarization
		3.6.2 TM polarization
	3.7 Polarization by reflection from dielectric surfaces
		3.7.1 Expression for Brewster's angle
	3.8 Antireflection coating
		3.8.1 Transfer matrix approach
	3.9 Bragg mirrors
	3.10 Goos-Hanchen shift
	3.11 Poynting theorem
	3.12 Problems
	3.13 Project
	Appendix 3A: MATLAB listings
	References
4 Slab waveguides
	4.1 Ray optics of the slab waveguide
		4.1.1 Numerical aperture
		4.1.2 Guided modes
		4.1.3 Transverse resonance condition
		4.1.4 Transverse condition: normalized form
	4.2 Fundamentals of EM theory of dielectric waveguides
		4.2.1 General discussion
		4.2.2 Explicit form of general equations
	4.3 Wave equation for a planar wide waveguide
	4.4 Three-layer symmetrical guiding structure (TE modes)
		4.4.1 The algorithm
	4.5 Modes of the arbitrary three-layer asymmetric planar waveguide in 1D
		4.5.1 TE modes
		4.5.2 Field profiles for TE modes
	4.6 Multilayer slab waveguides: 1D approach
		4.6.1 TE mode
		4.6.2 Propagation constant
		4.6.3 Electric field
		4.6.4 TM modes
	4.7 Examples: 1D approach
		4.7.1 Four-layer lossless waveguide
		4.7.2 Six-layer lossy waveguide
		4.7.3 Structure by Visser
	4.8 Two-dimensional (2D) structures
		4.8.1 Effective index method
	4.9 Problems
	4.10 Projects
	Appendix 4A: MATLAB listings
	References
5 Linear optical fibre and signal degradation
	5.1 Geometrical-optics description
		5.1.1 Numerical aperture (NA)
		5.1.2 Multipath dispersion
		5.1.3 Information-carrying capacity of the fibre
		5.1.4 Loss mechanisms in silica fibre
		5.1.5 Intrinsic loss
		5.1.6 Extrinsic loss
	5.2 Fibre modes in cylindrical coordinates
		5.2.1 Maxwell's equations in cylindrical coordinates
		5.2.2 Wave equations in cylindrical coordinates
		5.2.3 Solution of the wave equation in cylindrical coordinates
		5.2.4 Boundary conditions and modal equation
		5.2.5 Mode classification
		5.2.6 Modes with m = 0
		5.2.7 Weakly guiding approximation (wga)
		5.2.8 The unified expression
		5.2.9 Universal relation for fundamental mode HE11
		5.2.10 Single-mode fibres
		5.2.11 Cutoff conditions
	5.3 Dispersion
		5.3.1 Group delay-general discussion
		5.3.2 Material dispersion: Sellmeier equation
		5.3.3 Waveguide dispersion
	5.4 Pulse dispersion during propagation
	5.5 Problems
	5.6 Projects
	Appendix 5A: Some properties of Bessel functions
	Appendix 5B: Characteristic determinant
	Appendix 5C: MATLAB listings
	References
6 Propagation of linear pulses
	6.1 Basic pulses
		6.1.1 Rectangular pulses
		6.1.2 Gaussian pulses
		6.1.3 Super-Gaussian pulse
		6.1.4 Chirped Gaussian pulse
	6.2 Modulation of a semiconductor laser
		6.2.1 Modulation formats
		6.2.2 Creation of waveforms
	6.3 Simple derivation of the pulse propagation equation in the presence of dispersion
	6.4 Mathematical theory of linear pulses
		6.4.1 One-dimensional approach
	6.5 Propagation of pulses
		6.5.1 Analytical description of the propagation of a chirp Gaussian pulse
		6.5.2 Numerical method using Fourier transform
		6.5.3 Fourier transform split-step method
	6.6 Problems
	Appendix 6A: MATLAB listings
	References
7 Optical sources
	7.1 Overview of lasers
		7.1.1 Transitions in a TLS
		7.1.2 Laser oscillations and resonant modes
	7.2 Semiconductor lasers
		7.2.1 Electron transitions in semiconductors
		7.2.2 Homogeneous p-n junction
		7.2.3 Heterostructures
		7.2.4 Optical gain
		7.2.5 Determination of optical gain
	7.3 Rate equations
		7.3.1 Carriers
		7.3.2 Photons
		7.3.3 Rate equation parameters
		7.3.4 Derivation of rate equation for electric field
	7.4 Analysis based on rate equations
		7.4.1 Steady-state analysis
		7.4.2 Small-signal analysis with the linear gain model
		7.4.3 Small-signal analysis with gain saturation
		7.4.4 Large-signal analysis for QW lasers
		7.4.5 Frequency chirping
		7.4.6 Equivalent circuit models
		7.4.7 Equivalent circuit model for a bulk laser
	7.5 Problems
	7.6 Project
	Appendix 7A: MATLAB listings
	References
8 Optical amplifiers and EDFA
	8.1 General properties
		8.1.1 Gain spectrum and bandwidth
		8.1.2 Gain saturation
		8.1.3 Amplifier noise
	8.2 Erbium-doped fibre amplifiers (EDFA)
		8.2.1 Steady-state analysis
		8.2.2 Effective two-level approach
	8.3 Gain characteristics of erbium-doped fibre amplifiers
		8.3.1 Typical EDFA characteristics
	8.4 Problems
	8.5 Projects
	Appendix 8A: MATLAB listings
	References
9 Semiconductor optical amplifiers (SOA)
	9.1 General discussion
		9.1.1 Gain formula for SOA with facet reflectivities
		9.1.2 The effect of facet reflectivities
	9.2 SOA rate equations for pulse propagation
	9.3 Design of SOA
	9.4 Some applications of SOA
		9.4.1 Wavelength conversion
		9.4.2 All-optical logic based on interferometric principles
	9.5 Problem
	9.6 Project
	Appendix 9A: MATLAB listings
	References
10 Optical receivers
	10.1 Main characteristics
		10.1.1 Receiver sensitivity
		10.1.2 Dynamic range
		10.1.3 Bit-rate transparency
		10.1.4 Bit-pattern independency
	10.2 Photodetectors
		10.2.1 Principles of photo detection
		10.2.2 Performance parameters of photodetectors
		10.2.3 Photodetector noise
		10.2.4 Detector design
	10.3 Receiver analysis
		10.3.1 BER of an ideal optical receiver
		10.3.2 Error probability in the receiver
		10.3.3 BER and Gaussian noise
	10.4 Modelling of a photoelectric receiver
	10.5 Problems
	10.6 Projects
	Appendix 10A: MATLAB listings
	References
11 Finite difference time domain (FDTD) formulation
	11.1 General formulation
		11.1.1 Three-dimensional formulation
		11.1.2 Two-dimensional formulation
		11.1.3 One-dimensional model
		11.1.4 Gaussian pulse and modulated Gaussian pulse
	11.2 One-dimensional Yee implementation without dispersion
		11.2.1 Lossless case
		11.2.2 Determination of cell size
		11.2.3 Dispersion and stability
		11.2.4 Stability criterion
		11.2.5 One-dimensional model with losses
	11.3 Boundary conditions in 1D
		11.3.1 Mur's first-order absorbing boundary conditions (ABC)
		11.3.2 Second-order boundary conditions in 1D
	11.4 Two-dimensional Yee implementation without dispersion
	11.5 Absorbing boundary conditions (ABC) in 2D
	11.6 Dispersion
		11.6.1 Material dispersion
	11.7 Problems
	11.8 Projects
	Appendix 11A: MATLAB listings
	References
12 Beam propagation method (BPM)
	12.1 Paraxial formulation
		12.1.1 Introduction
		12.1.2 Operators ^D and ^W
		12.1.3 The implementation using the Fourier transform split-step method
	12.2 General theory
		12.2.1 Introduction
		12.2.2 Slowly varying envelope approximation (SVEA)
		12.2.3 Semi-vector BPM
		12.2.4 Scalar formulation
		12.2.5 Finite-difference (FD) approximations
	12.3 The 1+1 dimensional FD-BPM formulation
		12.3.1 Simple approach
		12.3.2 Propagator approach
		12.3.3 Transparent boundary conditions
	12.4 Concluding remarks
	12.5 Problems
	12.6 Project
	Appendix 12A: Details of derivation of the FD-BPM equation
	Appendix 12B: MATLAB listings
	References
13 Some wavelength division multiplexing (WDM) devices
	13.1 Basics of WDM systems
	13.2 Basic WDM technologies
		13.2.1 Fibre Bragg grating
		13.2.2 Array waveguide grating
		13.2.3 Couplers and splitters
		13.2.4 Mathematical theory of a passive coupler
		13.2.5 Optical isolators
	13.3 Applications of BPM to photonic devices
	13.4 Projects
	Appendix 13A: MATLAB listings
	References
14 Optical link
	14.1 Optical communication system
	14.2 Design of optical link
		14.2.1 Power budget analysis
		14.2.2 Rise time budget
	14.3 Measures of link performance
		14.3.1 Eye diagram
	14.4 Optical fibre as a linear system
	14.5 Model of optical link based on filter functions
		14.5.1 Test analysis for a rectangular pulse
		14.5.2 Transmitter
		14.5.3 Fibre
		14.5.4 Receiver
		14.5.5 Implementation of link model
	14.6 Problems
	14.7 Projects
	Appendix 14A: MATLAB listings
	References
15 Optical solitons
	15.1 Nonlinear optical susceptibility
	15.2 Main nonlinear effects
		15.2.1 Kerr effect
		15.2.2 Stimulated Raman scattering
	15.3 Derivation of the nonlinear Schrodinger equation
	15.4 Split-step Fourier method
		15.4.1 Split-step Fourier transform method
		15.4.2 Symmetrized split-step Fourier transform method (SSSFM)
	15.5 Numerical results
		15.5.1 Single solitons
		15.5.2 Chirped solitons
		15.5.3 Two interacting solitons
	15.6 A few comments about soliton-based communications
	15.7 Problems
	Appendix 15A: MATLAB listings
	References
16 Solar cells
	16.1 Introduction
	16.2 Principles of photovoltaics
	16.3 Equivalent circuit of solar cells
		16.3.1 Basic model
		16.3.2 Other models
	16.4 Multijunctions
		16.4.1 Quantum dots in multijunctions
		16.4.2 Intermediate band solar cells (IBSC)
		16.4.3 Role of simulations
	Appendix 16A: MATLAB listings
	References
17 Metamaterials
	17.1 Introduction
		17.1.1 Short history of MM
	17.2 Veselago approach
		17.2.1 Wave equation
		17.2.2 Left-handed materials
		17.2.3 The refraction of a ray
	17.3 How to create metamaterial?
		17.3.1 Metamaterials with negative effective permittivity in the microwave regime
		17.3.2 Magnetic properties: split-ring resonators
	17.4 Some applications of metamaterials
		17.4.1 Perfect lenses
		17.4.2 Stopped light in metamaterials
		17.4.3 Cloaking (invisibility)
		17.4.4 Optical black holes
	17.5 Metamaterials with an active element
	17.6 Annotated bibliography
	Appendix 17A: MATLAB listings
	References
Appendix A Basic MATLAB
	A.1 Working session with m-files
	A.2 Basic rules
	A.3 Some rules about good programming in MATLAB
		A.3.1 Preallocate memory
		A.3.2 Vectorize loops
	A.4 Basic graphics
		A.4.1 Basic 2D plot
		A.4.2 Two-dimensional plots
		A.4.3 Some 3D plots
	A.5 Basic input-output
		A.5.1 Writing to a text file
		A.5.2 Reading from a text file
	A.6 Numerical differentiation
	A.7 Review questions
	References
Appendix B Summary of basic numerical methods
	B.1 One-variable Newton's method
	B.2 Muller's method
		B.2.1 Tests of Muller's method
	B.3 Numerical differentiation
		B.3.1 Numerical differentiation using Taylor's series expansion
		B.3.2 Numerical differentiation using interpolating polynomials
		B.3.3 Crank-Nicolson method
		B.3.4 Simple methods of numerical differentiation
	B.4 Runge-Kutta (RK) methods
		B.4.1 Second-order Runge-Kutta
		B.4.2 Fourth-order Runge-Kutta
	B.5 Solving differential equations
		B.5.1 Single differential equation
		B.5.2 System of differential equations
	B.6 Numerical integration
		B.6.1 Euler's rule
		B.6.2 Trapezoidal rule
		B.6.3 Simpson's rule
	B.7 Symbolic integration in MATLAB
	B.8 Fourier series
		B.8.1 Change of interval
		B.8.2 Example
	B.9 Fourier transform
	B.10 FFT in MATLAB
	B.11 Problems
	References
Index
                        
Document Text Contents
Page 1

more information - www.cambridge.org/9781107005525

http://www.cambridge.org/9781107005525

Page 234

218 Optical amplifiers and EDFA

Listing 8A.2.1 Function gain variable length.m. MATLAB function used to compute
gain versus fibre length for a fixed value of pump power.

% File name: gain_variable_length.m

% Computation of gain vs fibre length for fixed value of pump power

% User selects pump power which is controlled by P_p

% Calculations are repeated for P_p = 3,5,7,9 d-3 Watts

% User should also appropriately rename output file ’gain_P_3.dat’

% based on Iannone-book, p.86

% P_s == y(1) signal power

% P_p == y(2) pump power

%

clear all

gain = 0.0;

gain_temp = 0.0;

P_s = 100d-7; % signal power [Watts]

P_p = 9d-3; % CHANGE % pump power [Watts]

%

for d_L = 0.01:0.01:10

span = [0 d_L];

y0 = [P_s P_p]; % initial values of [P_s P_p]

[z,y] = ode45(’edfa_eqs’,span,y0); % z - distance

len = length(z);

gain_temp = y(len)/y(1);

gain = [gain, gain_temp];

end

gain_log = log(gain);

d_L_plot = 0.01:0.01:10;

d_L_plot = [0, d_L_plot];

% plot(d_L_plot, gain_log) % uncomment if you want to see plot when run

% axis([0 4 0 5])

% pause

% close all

%

d_u = length(d_L_plot);

fid = fopen(’gain_P_9.dat’, ’wt’); % CHANGE % Open the file.

for ii = 1:d_u

fprintf (fid, ’ %11.4f %11.4f\n’, d_L_plot(ii),gain_log(ii));

end

status = fclose(fid); % Close the file

Page 235

219 Appendix 8A

Listing 8A.2.2 Function edfa param.m. MATLAB function contains parameters for a
model of EDFA.

% File name: edfa_param.m

%-----------------------------------------------------------------

% Purpose:

% Contains parameters for model of EDFA based on Table 3.2

% Iannone-book

N_tot = 5.4d24; % Erbium concentration (m^-3)

Gamma_s = 0.4; % Signal overlapping integral (dimensionless)

Gamma_p = 0.4; % Pump overlapping integral (dimensionless)

s_se = 5.3d-25; % Signal emission cross-section (m^2)

s_sa = 3.5d-25; % Signal absorption cross-section (m^2)

s_p = 3.2d-25; % Pump absorption cross-section

P_ss = 1.3d-3; % Signal local saturation power (W)

P_sp = 1.6d-3; % Pump local saturation power (W)

Listing 8A.2.3 Function edfa eqs.m. MATLAB function contains equations for EDFA.

function y_z = edfa_eqs(z,y)

% Purpose:

% To establish equations for EDFA following Iannone-book, p.86

% P_s == y(1) signal power

% P_p == y(2) pump power

%

edfa_param % input of needed parameters

num = y(1)/P_ss + y(2)/P_sp;

denom = y(2)/P_sp + 2*y(1)/P_ss + 1.0;

N_me = N_tot*num/denom;

N_gr = N_tot - N_me;

%

y_z(1) = 2*pi*Gamma_s*(s_se*N_me*y(1) - s_sa*N_gr*y(1)); % derivative

y_z(2) = -2*pi*Gamma_p*s_p*N_gr*y(2);

y_z = y_z’; % must return column vector

end

Listing 8A.2.4 Function variable length plot.m. MATLAB function used to plot graph
of gain using data generated by gain variable length.m.

% File_name: variable_length_plot.m

% Plots graph of gain using data generated by ’gain_variable_length.m’

clear all

% Open files for 4 values of length of device: 5 m,10 m,15 m

fid = fopen(’gain_P_3.dat’);

a_3 = fscanf(fid,’%f %f’,[2 inf]); % It has two rows

Page 467

451 Index

propagation constant, 41, 42
in slab waveguides, 66, 95

pulse broadening, 12, 151
in optical fibres, 106

pulse half-width, 142
pulse type

chirped Gaussian, 141–2, 159
Gaussian, 139–40, 159, 163, 164, 281–6
rectangular, 138–9, 157, 340–2, 344
Super-Gaussian, 140–1, 159, 162

pulse wave, see waveform
pumping, 167, 168

quantum efficiency, 247
quantum well, 173, 232
quasi-Fermi level, 175, 176

rate equations in EDFA, 211
rate equations in laser diodes

for an electric field, 184–7
for carriers, 182–3
for photons, 183
parameters, 184

ray optics, 17
in metamaterials, 389
in slab waveguides, 64–9

Rayleigh scattering, 110
receiver, 331, 343
recombination of carriers, 183
reflection

at a plane interface, 17, 18
coefficient, 17, 45, 46, 48
external, 48, 49
internal, 48
of TE polarized waves, 44–7, 61
of TM polarized waves, 48, 61
total internal, see critical angle

refractive index, 17
in GRIN structures, 20
negative, 384
numerical values for popular materials, 18
relative difference, 65

relaxation-oscillation frequency, 189
Resonant cavity, see optical cavity
rise time, 144
rise time budget, 333–6
Runge-Kutta method

fourth order, 433
second order, 432

S-band, see transmission bands
Sellmeier equation, 31, 125–6, 135
signal-to-noise ratio (SNR), 208, 248,

336
Simpson’s rule (integration method), 437–8
slowly varying envelope approximation (SVEA), 150,

296–7

small-signal analysis (in laser diodes)
with linear gain, 188–9
with non-linear gain, 189–92

Snell’s law, 18
soliton

interactions, 363
period, 362

spectral
intensity of solar energy, 368, 369
responsivity, 247

split-ring resonator (SRR), 391–4
split-step Fourier method, 357–60
steady-state analysis

in a laser diode, 187
in an EDFA, 211–12

step index, 107
Stokes relations, 24–6
susceptibility, 186

TE modes
in optical fibres, 116–18
in slab waveguides, 71–2

three-level system, 210
threshold

carrier density, 178, 182
current, 143

time constant in photodetectors, 246, 247
time division multiplexing (TDM), 11
time-harmonic field, 39–41
TM modes

in optical fibres, 116–18
in slab waveguides, 71–2

train of pulses, see waveform
transatlantic telecommunications cable (TAT), 6
transfer function, 339
transfer matrix approach

for antireflection (AR) coatings, 51–3
for Bragg mirrors, 54–7
for slab waveguides, 79–85

transitions
in a two-level system, 169–70
in semiconductors, 174–5

transmission bands, 10
transmittance of Fabry-Perot interferometer, 29, 33
transmitter, 331, 342
transparency density, 178
transparent boundary conditions, 304–6
transverse resonance condition, 66–7

normalized form, 67–9
trapezoidal rule (integration method), 437
two-level system (TLS), 167, 169

U-band, see transmission bands

velocity
group, 23–4, 124, 126
phase, 21–3, 32

Verlet differentiation method, 432

Page 468

452 Index

wave equation, 38–9
for TE modes, 72
for TM modes, 72
in cylindrical coordinates, 112
in metamaterials, 388

waveform, 145, 160
waveguide

2D, 88–92
asymmetric slab (planar), 75–9, 95
cylindrical (optical fibre), 110–23
lossy, 86

symmetric slab (planar), 72–5
wavevector, see propagation constant
weakly guiding approximation (wga),

118–19
wire medium, 395

Y-junction, 14, 325, 327
Yee algorithm

lossless in 1D, 266–8, 282
lossless in 2D, 275–7, 285
lossy in 1D, 271–2

Similer Documents