Beam Propagation Method for Design of Optical Wave guide Devices

<p>Preface xii</p>
<p>List of Acronyms xiv</p>
<p>List of Symbols xvi</p>
<p>1 Electromagnetic Theory of Light 1</p>
<p>Introduction 1</p>
<p>1.1 Electromagnetic Waves 21.1.1 Maxwell s Equations 2</p>
<p>1.1.2 Wave Equations in Inhomogeneous Media 5</p>
<p>1.1.3 Wave Equations in Homogeneous Media: Refractive Index 6</p>
<p>1.2 Monochromatic Waves 7</p>
<p>1.2.1 Homogeneous Media: Helmholtz s Equation 9</p>
<p>1.2.2 Light Propagation in Absorbing Media 9</p>
<p>1.2.3 Light Propagation in Anisotropic Media 11</p>
<p>1.2.4 Light Propagation in Second–Order Non–Linear Media 13</p>
<p>1.3 Wave Equation Formulation in Terms of the Transverse Field Components 16</p>
<p>1.3.1 Electric Field Formulation 16</p>
<p>1.3.2 Magnetic Field Formulation 18</p>
<p>1.3.3 Wave Equation in Anisotropic Media 19</p>
<p>1.3.4 Second Order Non–Linear Media 20</p>
<p>References 21</p>
<p>2 The Beam–Propagation Method 22</p>
<p>Introduction 22</p>
<p>2.1 Paraxial Propagation: The Slowly Varying Envelope Approximation (SVEA).Full Vectorial BPM Equations 23</p>
<p>2.2 Semi–Vectorial and Scalar Beam Propagation Equations 29</p>
<p>2.2.1 Scalar Beam Propagation Equation 30</p>
<p>2.3 BPM Based on the Finite Difference Approach 31</p>
<p>2.4 FD–Two–Dimensional Scalar BPM 32</p>
<p>2.5 Von Neumann Analysis of FD–BPM 37</p>
<p>2.5.1 Stability 38</p>
<p>2.5.2 Numerical Dissipation 39</p>
<p>2.5.3 Numerical Dispersion 40</p>
<p>2.6 Boundary Conditions 44</p>
<p>2.6.1 Energy Conservation in the Difference Equations 45</p>
<p>2.6.2 Absorbing Boundary Conditions (ABCs) 47</p>
<p>2.6.3 Transparent Boundary Conditions (TBC) 49</p>
<p>2.6.4 Perfectly Matched Layers (PMLs) 51</p>
<p>2.7 Obtaining the Eigenmodes Using BPM 56</p>
<p>2.7.1 The Correlation Function Method 58</p>
<p>2.7.2 The Imaginary Distance Beam Propagation Method 64</p>
<p>References 68</p>
<p>3 Vectorial and Three–Dimensional Beam Propagation Techniques 71</p>
<p>Introduction 71</p>
<p>3.1 Two–Dimensional Vectorial Beam Propagation Method 72</p>
<p>3.1.1 Formulation Based on the Electric Field 72</p>
<p>3.1.2 Formulation Based on the Magnetic Field 81</p>
<p>3.2 Three–Dimensional BPM Based on the Electric Field 84</p>
<p>3.2.1 Semi–Vectorial Formulation 88</p>
<p>3.2.2 Scalar Approach 96</p>
<p>3.2.3 Full Vectorial BPM 102</p>
<p>3.3 Three–Dimensional BPM Based on the Magnetic Field 113</p>
<p>3.3.1 Semi–Vectorial Formulation 116</p>
<p>3.3.2 Full Vectorial BPM 120</p>
<p>References 129</p>
<p>4 Special Topics on BPM 130</p>
<p>Introduction 130</p>
<p>4.1 Wide–Angle Beam Propagation Method 130</p>
<p>4.1.1 Formalism of Wide–Angle–BPM Based on Pad&eacute; Approximants 131</p>
<p>4.1.2 Multi–step Method Applied to Wide–Angle BPM 133</p>
<p>4.1.3 Numerical Implementation of Wide–Angle BPM 135</p>
<p>4.2 Treatment of Discontinuities in BPM 140</p>
<p>4.2.1 Reflection and Transmission at an Interface 140</p>
<p>4.2.2 Implementation Using First–Order Approximation to the Square Root 144</p>
<p>4.3 Bidirectional BPM 148</p>
<p>4.3.1 Formulation of Iterative Bi–BPM 148</p>
<p>4.3.2 Finite–Difference Approach of the Bi–BPM 151</p>
<p>4.3.3 Example of Bidirectional BPM: Index Modulation Waveguide Grating 154</p>
<p>4.4 Active Waveguides 157</p>
<p>4.4.1 Rate Equations in a Three–Level System 158</p>
<p>4.4.2 Optical Attenuation/Amplification 160</p>
<p>4.4.3 Channel Waveguide Optical Amplifier 161</p>
<p>4.5 Second–Order Non–Linear Beam Propagation Techniques 165</p>
<p>4.5.1 Paraxial Approximation of Second–Order Non–Linear Wave Equations 166</p>
<p>4.5.2 Second–Harmonic Generation in Waveguide Structures 169</p>
<p>4.6 BPM in Anisotropic Waveguides 173</p>
<p>4.6.1 TE TM Mode Conversion 175</p>
<p>4.7 Time Domain BPM 177</p>
<p>4.7.1 Time–Domain Beam Propagation Method (TD–BPM) 178</p>
<p>4.7.2 Narrow–Band 1D–TD–BPM 179</p>
<p>4.7.3 Wide–Band 1D–TD–BPM 180</p>
<p>4.7.4 Narrow–Band 2D–TD–BPM 187</p>
<p>4.8 Finite–Difference Time–Domain Method (FD–TD) 193</p>
<p>4.8.1 Finite–Difference Expressions for Maxwell s Equations in Three Dimensions 194</p>
<p>4.8.2 Truncation of the Computational Domain 198</p>
<p>4.8.3 Two–Dimensional FDTD: TM Case 199</p>
<p>4.8.4 Setting the Field Source 208</p>
<p>4.8.5 Total–Field/Scattered–Field Formulation 209</p>
<p>4.8.6 Two–Dimensional FDTD: TE Case 212</p>
<p>References 219</p>
<p>5 BPM Analysis of Integrated Photonic Devices 222</p>
<p>Introduction 222</p>
<p>5.1 Curved Waveguides 222</p>
<p>5.2 Tapers: Y–Junctions 228</p>
<p>5.2.1 Taper as Mode–Size Converter 228</p>
<p>5.2.2 Y–Junction as 1 &times; 2 Power Splitter 230</p>
<p>5.3 Directional Couplers 231</p>
<p>5.3.1 Polarization Beam–Splitter 232</p>
<p>5.3.2 Wavelength Filter 235</p>
<p>5.4 Multimode Interference Devices 237</p>
<p>5.4.1 Multimode Interference Couplers 237</p>
<p>5.4.2 Multimode Interference and Self–Imaging 239</p>
<p>5.4.3 1&times;N Power Splitter Based on MMI Devices 243</p>
<p>5.4.4 Demultiplexer Based on MMI 244</p>
<p>5.5 Waveguide Gratings 248</p>
<p>5.5.1 Modal Conversion Using Corrugated Waveguide Grating 249</p>
<p>5.5.2 Injecting Light Using Relief Gratings 250</p>
<p>5.5.3 Waveguide Reflector Using Modulation Index Grating 252</p>
<p>5.6 Arrayed Waveguide Grating Demultiplexer 257</p>
<p>5.6.1 Description of the AWG Demultiplexer 257</p>
<p>5.6.2 Simulation of the AWG 263</p>
<p>5.7 Mach–Zehnder Interferometer as Intensity Modulator 270</p>
<p>5.8 TE–TM Converters 276</p>
<p>5.8.1 Electro–Optical TE–TM Converter 277</p>
<p>5.8.2 Rib Loaded Waveguide as Polarization Converter 280</p>
<p>5.9 Waveguide Laser 282</p>
<p>5.9.1 Simulation of Waveguide Lasers by Active–BPM 283</p>
<p>5.9.2 Performance of a Nd3+–Doped LiNbO3 Waveguide Laser 286</p>
<p>5.10 SHG Using QPM in Waveguides 293</p>
<p>References 297</p>
<p>Appendix A: Finite Difference Approximations of Derivatives 300</p>
<p>A.1 FD–Approximations of First–Order Derivatives 300</p>
<p>A.2 FD–Approximation of Second–Order Derivatives 301</p>
<p>Appendix B: Tridiagonal System: The Thomas Method Algorithm 304</p>
<p>Reference 306</p>
<p>Appendix C: Correlation and Relative Power between Optical Fields 307</p>
<p>C.1 Correlation between Two Optical Fields 307</p>
<p>C.2 Power Contribution of a Waveguide Mode 307</p>
<p>References 309</p>
<p>Appendix D: Poynting Vector Associated to an Electromagnetic Wave Using the SVE Fields 310</p>
<p>D.1 Poynting Vector in 2D–Structures 310</p>
<p>D.1.1 TE Propagation in Two–Dimensional Structures 310</p>
<p>D.1.2 TM Propagation in Two–Dimensional Structures 312</p>
<p>D.2 Poynting Vector in 3D–Structures 314</p>
<p>D.2.1 Expression as a Function of the Transverse Electric Field 315</p>
<p>D.2.2 Expression as Function of the Transverse Magnetic Field 319</p>
<p>Reference 322</p>
<p>Appendix E: Finite Difference FV–BPM Based on the Electric Field Using the Scheme Parameter Control 323</p>
<p>E.1 First Component of the First Step 325</p>
<p>E.2 Second Component of the First Step 326</p>
<p>E.3 Second Component of the Second Step 327</p>
<p>E.4 First Component of the Second Step 328</p>
<p>Appendix F: Linear Electro–Optic Effect 330</p>
<p>Reference 332</p>
<p>Appendix G: Electro–Optic Effect in GaAs Crystal 333</p>
<p>References 339</p>
<p>Appendix H: Electro–Optic Effect in LiNbO3 Crystal 340</p>
<p>References 345</p>
<p>Appendix I: Pad&eacute; Polynomials for Wide–Band TD–BPM 346</p>
<p>Appendix J: Obtaining the Dispersion Relation for a Monomode Waveguide Using FDTD 349</p>
<p>Reference 350</p>
<p>Appendix K: Electric Field Distribution in Coplanar Electrodes 351</p>
<p>K.1 Symmetric Coplanar Strip Configuration 351</p>
<p>K.2 Symmetric Complementary Coplanar Strip Configuration 356</p>
<p>References 359</p>
<p>Appendix L: Three–Dimensional Anisotropic BPM Based on the Electric Field Formulation 360</p>
<p>L.1 Numerical Implementation 365</p>
<p>L.1.1 First Component of the First Step 365</p>
<p>L.1.2 Second Component of the First Step 366</p>
<p>L.1.3 Second Component of the Second Step 367</p>
<p>L.1.4 First Component of the Second Step 368</p>
<p>References 369</p>
<p>Appendix M: Rate Equations in a Four–Level Atomic System 370</p>
<p>References 372</p>
<p>Appendix N: Overlap Integrals Method 373</p>
<p>References 376</p>
<p>Index 377</p>