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Ultra-Fast Fiber lasers Principles and Applications with MATLAB® Models Ultra-Fast Fiber lasers Principles and Applications with MATLAB® Models Le Nguyen Binh Nam Quoc Ngo MATLAB® and Simulink® are trademarks of The MathWorks, Inc and are used with permission The MathWorks does not warrant the accuracy of the text of exercises in this book This book’s use or discussion of MATLAB® and Simulink® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® and Simulink® software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number: 978-1-4398-1128-3 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging‑in‑Publication Data Binh, Le Nguyen Ultra-fast fiber lasers : principles and applications with MATLAB models / authors, Le Nguyen Binh, Nam Quoc Ngo p cm Includes bibliographical references and index ISBN 978-1-4398-1128-3 Fiber optics Laser pulses, Ultrashort I Ngo, Nam Quoc II Title TA1800.B56 2010 621.382’7 dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 2010006603 Contents Preface xiii Acknowledgments xv Authors xvii Introduction 1.1 Ultrahigh Capacity Demands and Short Pulse Lasers 1.1.1 Demands 1.1.2 Ultrashort Pulse Lasers 1.2 Principal Objectives of the Book 1.3 Organization of the Book Chapters 1.4 Historical Overview of Ultrashort Pulse Fiber Lasers 1.4.1 Overview 1.4.2 Mode-Locking Mechanism in Fiber Ring Resonators 11 1.4.2.1 Amplifying Medium and Laser System 12 1.4.2.2 Active Modulation in Laser Cavity 14 1.4.2.3 Techniques for Generation of TerahertzRepetition-Rate Pulse Trains 15 1.4.2.4 Necessity of Highly Nonlinear Optical Waveguide Section for Ultrahigh-Speed Modulation 16 References 17 Principles and Analysis of Mode-Locked Fiber Lasers 23 2.1 Principles of Mode Locking 23 2.2 Mode-Locking Techniques 25 2.2.1 Passive Mode Locking 25 2.2.2 Active Mode Locking by Amplitude Modulation 27 2.2.3 Active Medium and Pump Source 28 2.2.4 Filter Design 30 2.2.5 Modulator Design 30 2.2.6 Active Mode Locking by Phase Modulation 32 2.3 Actively Mode-Locked Fiber Lasers 36 2.3.1 Principle of Actively Mode-Locked Fiber Lasers 36 2.3.2 Multiplication of Repetition Rate 37 2.3.3 Equalizing and Stabilizing Pulses in Rational HMLFL 39 2.4 Analysis of Actively Mode-Locked Lasers .42 2.4.1 Introduction 42 2.4.2 Analysis Using Self-Consistence Condition with Gaussian Pulse Shape 43 2.4.3 Series Approach Analysis 46 v vi Contents 2.4.4)>> Mode Locking 49 2.4.4.1)>> Mode Locking without Detuning 49 2.4.4.2)>> Mode Locking by Detuning 54 2.4.5)>> Simulation 60 2.5)>> Conclusions 65 References 66 Active Mode-Locked Fiber Ring Lasers: Implementation 71 3.1)>> Building Blocks of Active Mode-Locked Fiber Ring Laser 71 3.1.1)>> Laser Cavity Design 72 3.1.2)>> Active Medium and Pump Source 73 3.1.3)>> Filter Design 74 3.1.4)>> Modulator Design 75 3.2)>> AM and FM Mode-Locked Erbium-Doped Fiber Ring Laser 76 3.2.1)>> AM Mode-Locked Fiber Lasers 76 3.2.2)>> FM or PM Mode-Locked Fiber Lasers 78 3.3)>> Regenerative Active Mode-Locked Erbium-Doped Fiber Ring Laser 81 3.3.1)>> Experimental Setup 82 3.3.2)>> Results and Discussion 84 3.3.2.1)>> Noise Analysis .84 3.3.2.2)>> Temporal and Spectral Analysis 85 3.3.2.3)>> Measurement Accuracy 87 3.3.2.4)>> EDF Cooperative Up-Conversion 88 3.3.2.5)>> Pulse Dropout 88 3.4)>> Ultrahigh Repetition-Rate Ultra-Stable Fiber Mode-Locked Lasers 91 3.4.1)>> Regenerative Mode-Locking Techniques and Conditions for Generation of Transform-Limited Pulses from a Mode-Locked Laser 92 3.4.1.1)>> Schematic Structure of MLRL 92 3.4.1.2)>> Mode-Locking Conditions 93 3.4.1.3)>> Factors Influencing the Design and Performance of Mode Locking and Generation of Optical Pulse Trains 94 3.4.2)>> Experimental Setup and Results 96 3.4.3)>> Remarks 100 3.5)>> Conclusions 102 References 102 NLSE Numerical Simulation of Active Mode-Locked Lasers: Time Domain Analysis 105 4.1)>> Introduction 105 4.2)>> The Laser Model 106 Contents vii 4.2.1)>> Modeling the Optical Fiber 106 4.2.2)>> Modeling the EDFA 107 4.2.3)>> Modeling the Optical Modulation 107 4.2.4)>> Modeling the Optical Filter 108 4.3)>> The Propagation Model 109 4.3.1)>> Generation and Propagation 109 4.3.2)>> Results and Discussions 111 4.3.2.1)>> Propagation of Optical Pulses in the Fiber 111 4.4)>> Harmonic Mode-Locked Laser 118 4.4.1)>> Mode-Locked Pulse Evolution 118 4.4.2)>> Effect of Modulation Frequency 122 4.4.3)>> Effect of Modulation Depth 123 4.4.4)>> Effect of the Optical Filter Bandwidth 123 4.4.5)>> Effect of Pump Power 127 4.4.6)>> Rational Harmonic Mode-Locked Laser 128 4.5)>> FM or PM Mode-Locked Fiber Lasers 131 4.6)>> Concluding Remarks 134 References 136 Dispersion and Nonlinearity Effects in Active Mode-Locked Fiber Lasers 139 5.1)>> Introduction 139 5.2)>> Propagation of Optical Pulses in a Fiber 140 5.2.1)>> Dispersion Effect 141 5.2.2)>> Nonlinear Effect 144 5.2.3)>> Soliton 145 5.2.4)>> Propagation Equation in Optical Fibers 146 5.3)>> Dispersion Effects in Actively Mode-Locked Fiber Lasers 147 5.3.1)>> Zero Detuning 147 5.3.2)>> Dispersion Effects in Detuned Actively Mode-Locked Fiber Lasers 150 5.3.3)>> Locking Range 153 5.4)>> Nonlinear Effects in Actively Mode-Locked Fiber Lasers 154 5.4.1)>> Zero Detuning 154 5.4.2)>> Detuning in an Actively Mode-Locked Fiber Laser with Nonlinearity Effect 157 5.4.3)>> Pulse Amplitude Equalization in a Harmonic Mode-Locked Fiber Laser 159 5.5)>> Soliton Formation in Actively Mode-Locked Fiber Lasers with Combined Effect of Dispersion and Nonlinearity 160 5.5.1)>> Zero Detuning 160 5.5.2)>> Detuning and Locking Range in a Mode-Locked Fiber Laser with Nonlinearity and Dispersion Effect 163 5.6)>> Detuning and Pulse Shortening 165 5.6.1)>> Experimental Setup 165 viii Contents 5.6.2)>> Mode-Locked Pulse Train with 10↜GHz Repetition Rate 166 5.6.3)>> Wavelength Shifting in a Detuned Actively Mode-Locked Fiber Laser with Dispersion Cavity 169 5.6.4)>> Pulse Shortening and Spectrum Broadening under Nonlinearity Effect 171 5.7)>> Conclusions 173 References 173 Actively Mode-Locked Fiber Lasers with Birefringent Cavity 177 6.1)>> Introduction 177 6.2)>> Birefringence Cavity of an Actively Mode-Locked Fiber Laser 178 6.2.1)>> Simulation Model 180 6.2.2)>> Simulation Results 182 6.3)>> Polarization Switching in an Actively Mode-Locked Fiber Laser with Birefringence Cavity 185 6.3.1)>> Experimental Setup 185 6.3.2)>> Results and Discussion 186 6.3.2.1)>> H-Mode Regime 186 6.3.2.2)>> V-Mode Regime 188 6.3.3)>> Dual Orthogonal Polarization States in an Actively Mode-Locked Birefringent Fiber Ring Laser 189 6.3.3.1)>> Experimental Setup 189 6.3.3.2)>> Results and Discussion 191 6.3.4)>> Pulse Dropout and Sub-Harmonic Locking 197 6.3.5)>> Concluding Remarks 198 6.4)>> Ultrafast Tunable Actively Mode-Locked Fiber Lasers 200 6.4.1)>> Introduction 200 6.4.2)>> Birefringence Filter 201 6.4.3)>> Ultrafast Electrically Tunable Filter Based on Electro-Optic Effect of LiNbO3 202 6.4.3.1)>> Lyot Filter and Wavelength Tuning by a Phase Shifter 202 6.4.3.2)>> Experimental Results 203 6.4.4)>> Ultrafast Electrically Tunable MLL 206 6.4.4.1)>> Experimental Setup 206 6.4.4.2)>> Experimental Results 207 6.4.5)>> Concluding Remarks 209 6.5)>> Conclusions 210 References 212 Ultrafast Fiber Ring Lasers by Temporal Imaging 215 7.1)>> Repetition Rate Multiplication Techniques 215 7.1.1)>> Fractional Temporal Talbot Effect 216 Contents ix 7.1.2)>> Other Repetition Rate Multiplication Techniques 217 7.1.3)>> Experimental Setup 218 7.1.4)>> Results and Discussion 219 7.2)>> Uniform Lasing Mode Amplitude Distribution 222 7.2.1)>> Gaussian Lasing Mode Amplitude Distribution 224 7.2.2)>> Filter Bandwidth Influence 225 7.2.3)>> Nonlinear Effects 225 7.2.4)>> Noise Effects 227 7.3)>> Conclusions 229 References 230 Terahertz Repetition Rate Fiber Ring Laser 233 8.1)>> Gaussian Modulating Signal 233 8.2)>> Rational Harmonic Detuning 240 8.2.1)>> Experimental Setup 241 8.2.2)>> Results and Discussion 243 8.3)>> Parametric Amplifier–Based Fiber Ring Laser 251 8.3.1)>> Parametric Amplification 251 8.3.2)>> Experimental Setup 252 8.3.3)>> Results and Discussion 252 8.3.3.1)>> Parametric Amplifier Action 252 8.3.3.2)>> Ultrahigh Repetition Rate Operation 253 8.3.3.3)>> Ultra-Narrow Pulse Operation 260 8.3.3.4)>> Intracavity Power 261 8.3.3.5)>> Soliton Compression 262 8.4)>> Regenerative Parametric Amplifier–Based Mode-Locked Fiber Ring Laser 263 8.4.1)>> Experimental Setup 263 8.4.2)>> Results and Discussion 263 8.5)>> Conclusions 264 References 265 Nonlinear Fiber Ring Lasers 267 9.1)>> Introduction 267 9.2)>> Optical Bistability, Bifurcation, and Chaos 268 9.3)>> Nonlinear Optical Loop Mirror 273 9.4)>> Nonlinear Amplifying Loop Mirror 276 9.5)>> NOLM–NALM Fiber Ring Laser 277 9.5.1)>> Simulation of Laser Dynamics 277 9.5.2)>> Experiment 280 9.5.2.1)>> Bidirectional Erbium-Doped Fiber Ring Laser 280 9.5.2.2)>> Continuous-Wave NOLM–NALM Fiber Ring Laser 285 9.5.2.3)>> Amplitude-Modulated NOLM–NALM Fiber Ring Laser 287 Appendix B 391 )>> end )>> if (ii>N_pass 20) ỗãồ in = round(ii/20) + 1; ỗãồ ARR2(in,:)=abs(Usol).2 ; ỗãồ Up(in,:) = Usol; )>> end % % close(h1); end % Eout = Eout/atten; % ======================================================= figure(1); plot(tstep,abs(Usol).ˆ2) colormap(‘default’) mesh (real(abs(Up).ˆ2),‘meshstyle’,‘row’,‘facecolor’,‘none’); % waterfall(ARR2); % waterfall(abs(Up).ˆ2); view(17.5,42); axis tight; % delete solpair.mat save solpair5 Up tstep Po TotDisp Gamma Ld To Ci r teta Tfwhm Tb Vm Vpi Pav = 3*max(max(ARR2))/(2*qr) NN = size(Up,1); figure(2); PP = angle(Up(1,:)); Pha_tinput = unwrap(PP); ff = -diff(Pha_tinput)/Ts; chrate = diff(ff)/Ts; subplot(221); % plot (tstep*1e12,Pha_tinput);grid; plot (tstep(1:Ns−1)*1e12,ff);grid; % plot (tstep(2:Ns−1)*1e12,chrate);grid; PP = angle(Up(5,:)); Pha_tinput = unwrap(PP); ff = -diff(Pha_tinput)/Ts; chrate = diff(ff)/Ts; subplot(222); % plot (tstep*1e12,Pha_tinput);grid; plot (tstep(1:Ns−1)*1e12,ff);grid; % plot (tstep(2:Ns−1)*1e12,chrate);grid; in = round(NN/2); PP = angle(Up(in,:)); Pha_tinput = unwrap(PP); ff = -diff(Pha_tinput)/Ts; chrate = diff(ff)/Ts; subplot(223); % plot (tstep*1e12,Pha_tinput);grid; plot (tstep(1:Ns−1)*1e12,ff);grid; % plot (tstep(2:Ns−1)*1e12,chrate);grid; 392 PP = angle(Up(NN,:)); Pha_tinput = unwrap(PP); ff = -diff(Pha_tinput)/Ts; chrate = diff(ff)/Ts; subplot(224); % plot (tstep*1e12,Pha_tinput);grid; plot (tstep(1:Ns−1)*1e12,ff);grid; % plot (tstep(2:Ns−1)*1e12,chrate);grid; figure(3); title(‘The phase evolution of soliton pairs’); subplot(221); plot (tstep*1e12,ARR2(1,:));grid; subplot(222); plot (tstep*1e12,ARR2(5,:));grid; subplot(223); in = round(NN/2); plot (tstep*1e12,ARR2(in,:));grid; subplot(224); plot (tstep*1e12,ARR2(NN,:));grid; figure(4); Kmag = 1; Nplot = 100; Uf = Up.’; Eoutfreq = fft(Uf,Ns);)>> %(:,N1) Eoutfreq1 = fft(Uf,Ns*Kmag); %(:,N1) Ioutfreq = Eoutfreq1.*conj(Eoutfreq1)/(Ns*Kmag)ˆ2; ind = (- Nplot/2 : Nplot/2)’; freq = ind/Ts/Ns/Kmag; ind = mod(↜(ind + Ns*Kmag),Ns*Kmag)+1; title(‘The spectrum evolution of pulse’); subplot(221) % plot(freq,Ioutfreq(ind,1)); plot(freq,10*log10(Ioutfreq(ind,1))+30); Xlabel(‘Freq (Hz)’); Ylabel(‘P (W)’); subplot(222) % in = round(N1/4); % plot(freq,Ioutfreq(ind,5)); plot(freq,10*log10(Ioutfreq(ind,5))+30); Xlabel(‘Freq (Hz)’); Ylabel(‘P (W)’); subplot(223) in = round(NN/2); % plot(freq,Ioutfreq(ind,in)); plot(freq,10*log10(Ioutfreq(ind,in))+30); Xlabel(‘Freq (Hz)’); Ylabel(‘P (W)’); Appendix B Appendix B 393 subplot(224) % plot(freq,Ioutfreq(ind,NN)); plot(freq,10*log10(Ioutfreq(ind,NN))+30); Xlabel(‘Freq (Hz)’); Ylabel(‘P (W)’); figure(5);grid; Pha = Vw*pi./Vpi; % Phase change in time dff = -diff(Pha)/Ts; subplot(211);plot(tstep*1e12,Pha);xlabel(‘Time (ps)’);ylabel(‘Phase (rad)’);axis tight; subplot(212);plot(tstep(1:Ns−1)*1e12,dff*1e-12);xlabel(‘Time (ps)’);ylabel(‘Chirping (THz)’); function [Eout,gain] = AmpSimpNoise(Ein,GssdB,PoutsatdB,NF) % amp_simp(Ein,GssdB,PoutsatdB,NF) % simple model of optical amplifier The model includes the gain % saturation without noise % written by Lam Quoc Huy % Amplifier parameters: %)>> small signal gain: GssdB (dB) %)>> output saturation power: PoutsatdB (dBm) % % The input is a column vector containing block N samples of the optical signal sampling at the % rate 1/Ts % The output is calculated using %)>> Eout = Ein*sqrt(G) % where: G is the saturated gain %)>> G = Gss*exp(-(G−1)Pin/Psat))>> (eq1) global Ts f = 193.1e12; hplank = 6.6261*1e-34; Gss = 10ˆ(GssdB/10); Poutsat = (10ˆ(PoutsatdB/10))*1e-3; Psat = Poutsat*(Gss−2)/Gss/log(2); % Pinsat = 2* Poutsat/Gss; N = size(Ein,1); % Pin = (sum(Ein.*conj(Ein))/N); Pin = mean(↜(Ein.*conj(Ein))↜); % numerical calculation of G from the equation G = (Gss lnG)*Psat/Pin + tol = 0.05;)>> % tolerance for G calculation step = Gss/2; G = Gss; err = 10; while (err > tol) )>> G1 = Gss*exp(-(G−1)*Pin/Psat); )>> err = G1 - G; 394 Appendix B )>> if err>0 ỗãồif step > step = -step/2; ỗãồend )>> else ỗãồif step >0 ỗãồ)>> step = -step/2; ỗãồend ỗãồerr = -err; )>> end )>> G = G + step; end G = G - step; % Eout = sqrt(G)*Ein; gain = G; Egain = sqrt(G)*Ein; dt = Ts; Bsim = 1/dt; FigNoise = 10ˆ(NF/10); nsp = (FigNoise*G−1)/(2*(G−1)); % Pase = hplank.*opfreq.*nsp*(OGain−1)*Bsim Pase = hplank.*f.*nsp*(G−1)*Bsim/1000; PasedB = 10*log10(Pase); % afout = fft(Egain) + (randn(size(Egain))+i*randn(size (Egain))↜)*sqrt(Pase)./sqrt(2); % Eout = ifft(afout); % Eout = Egain + (randn(size(Egain))+i*randn(size (Egain))↜)*sqrt(Pase)./sqrt(2)./1; Eout = Egain + wgn(N,1,PasedB,‘complex’); % Eout = Egain; function [Atout] = GaussLPfilt(t,At,N,f3dB) % global BitRate % if nargin == % f3dB = 0.7*BitRate; % end S = length(t); dt = abs(t(2)-t(1)); fin = fftshift(1/dt/1*(-S/2:S/2−1)/(S)); % k = (1:S)−1; % k(S/2+1:S) = k(S/2+1:S) - S; % % k = k’; % fin = k/dt/S; Afin = fft(At); T = exp(-log(sqrt(2))*(fin/f3dB).ˆ(2*N)); % Ham truyen bo loc fout = fin; Afout = T.*Afin; 395 Appendix B Atout = ifft(Afout); % figure(4); % plot(fin,T); function [As_out,A3d,z] = hconst(As_in,L,h,b2,b3,a,g,p) % Symmetrized Split-Step Fourier Method % with constant step % Input: %)>> L = Length of fiber %)>> h = step size %)>> As_in = Input field in the time domain % % Output: %)>> As_out = Output field in the time domain % %)>> wrtten by N D Nhan - PTIT global global global global global global beta2 beta3 w alpha gam pmdmode beta2 = b2; beta3 = b3; alpha = a; gam = g; pm = p; pmdmode = 0; Atemp = As_in; M = round(L/h);)>> n = 1; z = 0; l = 1; A3d = []; for )>> )>> )>> )>> )>> end % number of steps k = 1:M Atemp = ssfm(Atemp,h); z(1,n+1) = z(1,n)+h; n = n+1; % A3d(l,:) = At; l = l + 1; As_out = Atemp; )>> % ============Testing % c = 3e8; % nin = 1.5; % vg = c/nin; % Tr = 50/c/nin; 396 % % % % % % Nc Tw Tm Th Appendix B =size(As_in,1); = Nc*1e-13; = 1/10e9; = Tr/1000; As_out = As_out.*exp(j*2*pi*(Tm-Th)/Tm); function Eout = hpha_mod(Ein,Vm,Vbias,Vpi,fm,st,ph0) % phase modulator parameters % m: phase modulation index % bias: DC phase shift (rad) % modulation frequency: fm % ph0: initial phase %% written by Nguyen Duc Nhan global tstep; global Ts; global Vw; % N = size(Ein,1); % k = (1:N)’; % tstep = Ts*(k-N/2); mrad = Vm/Vpi*pi; Norder = 1; % Vw = Vbias+Vm*cos(2*pi*fm*(tstep-st)+ph0)+0.27*Vm*cos (2*pi*2*fm*(tstep-st)+ph0−0.6*pi/1)… %)>> +0.001*Vm*cos(2*pi*3*fm*(tstep-st)+ph0−0.6*pi/1)+0.0001*Vm *cos(2*pi*4*fm*(tstep-st)+ph0−0.6*pi/1); Vw = Vbias+Norder*Vm*cos(2*pi*Norder*fm*(tstep-st)+ph0+ 0.0*pi)+0.48*Norder*Vm*cos(2*pi*2*Norder*fm* (tstep-st)+2*ph0−1.4*pi/1)… +0.003*Norder*Vm*cos(2*pi*3*Norder*fm*(tstep-st)+ 3*ph0−0.1*pi/1)+0.0001*Norder*Vm*cos(2*pi*4*fm* (tstep-st)+4*ph0−0.1*pi/1); Â� Eout = Ein.*exp(j*Vw); function Eout = pha_mod(Ein,Vm,Vbias,Vpi,fm,ph0) % phase modulator parameters % m: phase modulation index % bias: DC phase shift (rad) % modulation frequency: fm % ph0: initial phase % % written by Nguyen Duc Nhan global tstep; global Ts; global Vw; N = size(Ein,1); % k = (1:N)’; % tstep = Ts*(k-N/2); Appendix B 397 T0 = 1/fm; modeph = 0; if modeph == )>> Vw = Vbias+Vm*cos(2*pi*fm*tstep+ph0); elseif modeph == )>> Vw = real(synth_sig(Vbias,Vm,fm,tstep,ph0*T0/(2*pi),2)); else )>> Vw = real(synth_sig(Vbias,Vm,fm,tstep,ph0*T0/(2*pi),1)); end % delta_phi = pi/4*(2− bias*2 - ext*Vm); Phi = Vw*pi/Vpi; Eout = Ein.*exp(j*Phi); % Eout = Ein.*Vm; % solplot.m % plotting the soliton – soliton pairs load solpair5 Ts = tstep(2)-tstep(1); Ns = size(Up,2); ARR2 = abs(Up).ˆ2; figure(1); colormap(‘default’) mesh (real(abs(Up).ˆ2),‘meshstyle’,‘row’,‘facecolor’,‘none’); %waterfall(abs(Up).ˆ2); view(17.5,42); axis tight; NN = size(Up,1); figure(2); PP = angle(Up(1,:)); Pha_tinput = unwrap(PP); ff = -diff(Pha_tinput)/Ts; chrate = diff(ff)/Ts; subplot(221); % plot (tstep*1e12,Pha_tinput);grid; % plot (tstep(1:Ns−1)*1e12,ff);grid; plot (tstep(2:Ns−1)*1e12,chrate);grid; PP = angle(Up(5,:)); Pha_tinput = unwrap(PP); ff = -diff(Pha_tinput)/Ts; chrate = diff(ff)/Ts; subplot(222); % plot (tstep*1e12,Pha_tinput);grid; % plot (tstep(1:Ns−1)*1e12,ff);grid; plot (tstep(2:Ns−1)*1e12,chrate);grid; in = round(NN/2); PP = angle(Up(in,:)); Pha_tinput = unwrap(PP); ff = -diff(Pha_tinput)/Ts; chrate = diff(ff)/Ts; 398 subplot(223); % plot (tstep*1e12,Pha_tinput);grid; % plot (tstep(1:Ns−1)*1e12,ff);grid; plot (tstep(2:Ns−1)*1e12,chrate);grid; PP = angle(Up(NN,:)); Pha_tinput = unwrap(PP); ff = -diff(Pha_tinput)/Ts; chrate = diff(ff)/Ts; subplot(224); % plot (tstep*1e12,Pha_tinput);grid; % plot (tstep(1:Ns−1)*1e12,ff);grid; plot (tstep(2:Ns−1)*1e12,chrate);grid; figure(3); title(‘The phase evolution of soliton pairs’); subplot(221); plot (tstep*1e12,ARR2(1,:));grid; subplot(222); plot (tstep*1e12,ARR2(5,:));grid; subplot(223); in = round(NN/2); plot (tstep*1e12,ARR2(in,:));grid; subplot(224); plot (tstep*1e12,ARR2(NN,:));grid; figure(4); Kmag = 1; Nplot = 100; Uf = Up.’; %(:,N1) Eoutfreq = fft(Uf,Ns);)>> Eoutfreq1 = fft(Uf,Ns*Kmag); %(:,N1) Ioutfreq = Eoutfreq1.*conj(Eoutfreq1)/(Ns*Kmag)ˆ2; ind = (- Nplot/2 : Nplot/2)’; freq = ind/Ts/Ns/Kmag; ind = mod(↜(ind + Ns*Kmag),Ns*Kmag)+1; title(‘The spectrum evolution of pulse’); subplot(221) % plot(freq,Ioutfreq(ind,1)); plot(freq,10*log10(Ioutfreq(ind,1))+30); Xlabel(‘Freq (Hz)’); Ylabel(‘P (W)’); subplot(222) % in = round(N1/4); % plot(freq,Ioutfreq(ind,5)); plot(freq,10*log10(Ioutfreq(ind,5))+30); Xlabel(‘Freq (Hz)’); Ylabel(‘P (W)’); subplot(223) Appendix B Appendix B 399 in = round(NN/2); % plot(freq,Ioutfreq(ind,in)); plot(freq,10*log10(Ioutfreq(ind,in))+30); Xlabel(‘Freq (Hz)’); Ylabel(‘P (W)’); subplot(224) % plot(freq,Ioutfreq(ind,NN)); plot(freq,10*log10(Ioutfreq(ind,NN))+30); Xlabel(‘Freq (Hz)’); Ylabel(‘P (W)’); function [Ato] = ssfm(Ati,h) % Symmetrized Split-Step Fourier Method % used for single channel % Input: % L = Length of fiber % h = Variable simulation step % Ati = Input field in the time domain % % Output: Ato = Output field in the time domain %)>> % written by N D Nhan - PTIT %)>> global global global global global % % % % % % beta2 beta3 w alpha gam c = 3e8; nin = 1.5; vg = c/nin; Tr = 50/c/nin; Nc =size(Ati,1); Tw = Nc*1e-13; D = -i/2*beta2.*(i*w).ˆ2+1/6*beta3.*(i*w).ˆ3-alpha/2; % linear operator N1 = i*gam.*(abs(Ati).ˆ2); % nonlinear operator N2 = N1; %Propagation in the first half dispersion region, z to z +h/2 At1 = ifft(exp(h/2.*D).*fft(Ati)); % % At1 = ifft(exp(1/Tr*h/2.*D).*fft(Ati)); % % ======================================================= % Iteration for the nonlinear phase shift (2 iterations) % ======================================================= for m = 1:4 )>> At1temp = ifft(exp(h/2.*D).*fft(Ati)); )>> At2 = exp(h/2*(N1+N2)).*At1temp; 400 Appendix B )>> At3 = ifft(exp(h/2.*D).*fft(At2)); )>> %At3 = At3.’; )>> N2 = i*gam.*(abs(At3).ˆ2); %)>> At1temp = ifft(exp(1/Tr*h/2.*D).*fft(Ati)); %)>> At2 = exp(1/Tr*h/2*(N1+N2)).*At1temp; %)>> At3 = ifft(1/Tr*exp(h/2.*D).*fft(At2)); %)>> %At3 = At3.’; %)>> N2 = i*gam.*(abs(At3).ˆ2); end At4 = exp(h/2.*(N1+N2)).*At1; % Propagation in the second Dispersion region, z +h/2 to z + h Ato = ifft(exp(h/2.*D).*fft(At4)); % Ato = Ato.*exp(j*Tr/Tw); function Vout = synth_sig(Vbias,Vac,fm,t,initphase,opt) % Synth_sig is a function to synthesize an arbitrary waveforms % to generate the signal driving a phase modulator % Vbias - DC voltage % Vac - amplitude of the ac component % fm - modulation frequency % t - vector of times Nharm = 38; period = 1/fm; y = 0; if opt == )>> step = 2; )>> Vdc = Vbias; )>> for ii = -Nharm:step:Nharm ỗãồif ii == ỗãồ)>> y = y + 0; ỗãồelse ỗãồ)>> = 2/(pi*ii)2; ỗãồ)>> y = y + ai*exp(j*2*pi*ii*(t-initphase)/period); ỗãồend end )>> Va = Vac*2*y; elseif opt == )>> step = 1; )>> Vdc = Vbias + 0;%15/20; )>> Va = Vac/2; )>> for ii = -Nharm:step:Nharm ỗãồif ii == ỗãồ)>> y = y + 0; ỗãồelse ỗãồ)>> = 2/(pi*ii)2; ỗãồ)>> y = y + ai*exp(j*2*pi*ii*(t-initphase)/period); ỗãồend end Appendix B 401 Va = Va*2*y; else )>> % Vdc)>>= Vbias; )>> % pp)>> = initphase*2*pi/period; )>> % ff)>> = cos(2*pi*fm*t+pp); )>> % Va)>> = Vac*exp(ff); )>> Vdc)>> = Vbias; )>> pp)>> = initphase*2*pi/period; )>> mm)>> = 0.02; )>> Tfwhm)>>= mm*period; )>> ti = t; % ff = Vac*cos(2*pi*fm*t+pp); )>> % ti = rem(t,period) ; % fraction of time within BitPeriod )>> n = fix(t/period)+1 ; % extract input sequence number )>> Va = Vac*exp(-log(2)*1/2*(2*ti./Tfwhm).ˆ(2*1)); end Vout = Vdc + Va; Appendix C: Abbreviations ACF)>> Autocorrelation function AM)>> Amplitude modulation AMLM-EDFLỗãActively mode-locked multiwavelength erbium-doped fiber laser AOM)>> Acousto-optic modulator APE)>> Annealed proton exchange APL)>> Additive pulse limiting APM)>> Addictive pulse mode-locking ASE)>> Amplified spontaneous emission AWG)>> Arrayed waveguide grating BER)>> Bit error rate BPG)>> Bit pattern generator BPF)>> Bandpass filter ccw)>> Counter clockwise CFBG)>> Chirped fiber Bragg grating CSA)>> Communications signal analyzer CSRZ)>> Carrier suppressed return-to-zero cw)>> Clockwise CW)>> Continuous-wave DC)>> Direct current DCF)>> Dispersion compensating fiber DFB)>> Distributed feedback DM)>> Dispersion management DSF)>> Dispersion shifted fiber DWDM)>> Dense wavelength division multiplexing E/O)>> Electrical-optical conversion EDF)>> Erbium-doped fiber EDFA)>> Erbium-doped fiber amplifier EDFL)>> Erbium-doped fiber laser Electro-optic EO)>> ESA)>> Excited state absorption FBG)>> Fiber Bragg grating Fabry–Perot filter FFP)>> FFT)>> Fast Fourier transform Inverse fast Fourier transform IFFT)>> FM)>> Frequency modulation FP)>> Fabry–Perot Fiber ring laser FRL)>> FS)>> Frequency shifter Frequency shift keying FSK)>> 403 404 FSR)>> FTTH)>> FTTx)>> FWHM)>> FWM)>> GVD)>> HDTV)>> HiBi)>> HMLFL)>> HNLF)>> IPTV)>> KLM)>> LCFG)>> LD)>> LHS)>> LPFBG)>> MBS)>> ML)>> MLFL)>> MLFRL)>> MLL)>> MZ)>> MZIM)>> NALM)>> NF)>> NLSE)>> NOLM)>> NPR)>> NTT)>> NZ-DSF)>> O/E)>> OE)>> OEO)>> OPO)>> OSA)>> OSC)>> OTDM)>> OTDR)>> PA)>> PC)>> PCF)>> PD)>> PDM)>> PM)>> PMD)>> Appendix C Free spectral range Fiber to the home Fiber to the x Full width at half maximum Four wave mixing Group velocity dispersion High definition television Hi-birefringence Harmonic mode-locked fiber laser Highly nonlinear fiber Internet protocol television Kerr lens mode-locking Linearly chirped fiber grating Laser diode Left-hand side Long period fiber Bragg grating Multi-bound solitons Mode locked Mode-locked fiber laser Mode-locked fiber ring laser Mode-locked laser Mach–Zehnder Mach–Zehnder intensity or interferometric modulator Nonlinear amplifying loop mirror Noise figure Nonlinear Schrödinger equation Nonlinear optical loop mirror Nonlinear polarization rotation Nippon Telegraph and Telephone Company Nonzero dispersion–shifted fiber Optical-electrical conversion Opto-electronic Opto-electrical oscilloscope Optical parametric oscillator Optical spectrum analyzer Oscilloscope Optical time division multiplexing Optical time domain reflectometer Parametric amplifier Polarization controller Photonic crystal fiber Photodetector Polarization division multiplexing Phase modulation Polarization mode dispersion 405 Appendix C PM-EDF)>> PMF)>> PO)>> Pol)>> PS)>> PZT)>> RF)>> RFA)>> RHML)>> RHMLFL)>> RHS)>> RMLFRL)>> RMS)>> SAW)>> SMF)>> SNR)>> SOA)>> SPM)>> SRD)>> SRS)>> SSFM)>> TBP)>> TBRRM)>> TDM)>> TPA)>> VRC)>> WDM)>> XPM)>> Polarization-maintaining erbium-doped fiber Polarization-maintaining fiber Parametric oscillator Polarizer Phase shifter Piezoelectric transducer Radio frequency Radio frequency amplifier Rational harmonic mode locking Rational harmonic mode-locked fiber laser Right-hand side Regenerative mode-locked fiber ring laser Root mean square Surface acoustic waves Single-mode fiber Signal-to-noise ratio Semiconductor optical amplifier Self-phase modulation Step recovery diode Stimulated Raman scattering Split step Fourier method Time-bandwidth product Talbot-based repetition rate multiplication Time division multiplexing Two-photon absorption Variable ratio coupler Wavelength-division multiplexing Cross-phase modulation .. .Ultra-Fast Fiber lasers Principles and Applications with MATLAB® Models Ultra-Fast Fiber lasers Principles and Applications with MATLAB® Models Le Nguyen Binh Nam Quoc Ngo MATLAB® and Simulink®... trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging‑in‑Publication Data Binh, Le Nguyen Ultra-fast fiber lasers : principles and applications. .. Electron., 8, 506–519, 2002 Principles and Analysis of Mode-Locked Fiber Lasers This chapter deals with the principles of mode locking in fiber ring lasers and then the analyses and some simulation results

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