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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

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