KARIM ACHOURI, PhD, is a Postdoctoral Fellow in the Laboratory of Nanophotonics and Metrology at École Polytechnique de Montréal. He obtained his PhD from the same institution in electrical engineering in 2017. His research focus is on metamaterials, metasurfaces, photonics and optical systems.CHRISTOPHE CALOZ, PhD, is Full Professor at École Polytechnique de Montréal, the holder of a Canada Research Chair Tier-I and Head of the Electromagnetics Research Group. He has authored or co-authored over 700 technical conference, letter, and journal papers, as well as 13 books and book chapters.

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• Preface ix1 Introduction 11.1 Metamaterials 11.2 Emergence of Metasurfaces 42 Electromagnetic Properties of Materials 92.1 Bianisotropic Constitutive Relations 102.2 Temporal Dispersion 142.2.1 Causality and Kramers-Kronig Relations 152.2.2 Lorentz Oscillator Model 172.3 Spatial Dispersion 232.4 Lorentz Reciprocity Theorem 272.5 Poynting Theorem 322.6 Energy Conservation in Lossless-Gainless Systems 382.7 Classi_cation of Bianisotropic Media 413 Metasurface Modeling 433.1 E_ective Homogeneity 443.1.1 The Homogeneity Paradox 443.1.2 Theory of Periodic Structures 443.1.3 Scattering from Gratings 463.1.4 Homogenization 473.2 E_ective Zero Thickness 503.3 Sheet Boundary Conditions 533.3.1 Impedance Modeling 543.3.2 Polarizability Modeling 573.3.3 Susceptibility Modeling 603.3.4 Comparisons between the Models 664 Susceptibility Synthesis 694.1 Linear Time-Invariant Metasurfaces 694.1.1 Basic Assumptions 694.1.2 Birefringent Metasurfaces 764.1.3 Multiple-Transformation Metasurfaces 784.1.4 Relations between Susceptibilities and Scattering Parameters 814.1.5 Surface-Wave Eigenvalue Problem 924.1.5.1 Formulation of the Problem 924.1.5.2 Dispersion in a Symmetric Environments 964.1.6 Metasurfaces with Normal Polarizations 1004.1.7 Illustrative Examples 1044.1.7.1 Polarization Rotation 1044.1.7.2 Multiple Nonreciprocal Transformations 1094.1.7.3 Angle-Dependent Transformations 1124.2 Time-Varying Metasurfaces 1174.2.1 Formulation of the Problem 1174.2.2 Harmonic-Generation Time-Varying Metasurface 1204.3 Nonlinear Metasurfaces 1214.3.1 Second-Order Nonlinearity 1224.3.1.1 Frequency-Domain Approach 1234.3.1.2 Time-Domain Approach 1285 Scattered Field Computation 1335.1 Fourier-Based Propagation Method 1345.2 Finite-Di_erence Frequency-Domain Method 1415.3 Finite-Di_erence Time-Domain Method 1475.3.1 Time-Varying Dispersionless Metasurfaces 1505.3.2 Time-Varying Dispersive Metasurfaces 1565.4 Spectral-Domain Integral Equation Method 1646 Practical Implementation 1736.1 General Implementation Procedure 1746.2 Basic Strategies for Full-Phase Coverage 1786.2.1 Linear Polarization 1796.2.1.1 Metallic Scattering Particles 1796.2.1.2 Dielectric Scattering Particles 1886.2.2 Circular Polarization 1946.3 Full-Phase Coverage with Perfect Matching 1986.4 Effects of Symmetry Breaking 2076.4.1 Angular Scattering 2086.4.2 Polarization Conversion 2157 Applications 2237.1 Angle-Independent Transformation 2247.2 Perfect Matching 2297.3 Generalized Refraction 2347.3.1 Limitations of Conventional Synthesis Methods 2347.3.2 Perfect Refraction using Bianisotropy 2398 Conclusions 2459 Appendix 2499.1 Approximation of Average Fields at an Interface 2499.2 Fields Radiated by a Sheet of Dipole Moments 2529.3 Relations between Susceptibilities and Polarizabilities 255Bibliography 260

KARIM ACHOURI, PhD, is a Postdoctoral Fellow in the Laboratory of Nanophotonics and Metrology at École Polytechnique de Montréal. He obtained his PhD from the same institution in electrical engineering in 2017. His research focus is on metamaterials, metasurfaces, photonics and optical systems.CHRISTOPHE CALOZ, PhD, is Full Professor at École Polytechnique de Montréal, the holder of a Canada Research Chair Tier-I and Head of the Electromagnetics Research Group. He has authored or co-authored over 700 technical conference, letter, and journal papers, as well as 13 books and book chapters.

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• Preface ix1 Introduction 11.1 Metamaterials 11.2 Emergence of Metasurfaces 42 Electromagnetic Properties of Materials 92.1 Bianisotropic Constitutive Relations 102.2 Temporal Dispersion 142.2.1 Causality and Kramers-Kronig Relations 152.2.2 Lorentz Oscillator Model 172.3 Spatial Dispersion 232.4 Lorentz Reciprocity Theorem 272.5 Poynting Theorem 322.6 Energy Conservation in Lossless-Gainless Systems 382.7 Classi_cation of Bianisotropic Media 413 Metasurface Modeling 433.1 E_ective Homogeneity 443.1.1 The Homogeneity Paradox 443.1.2 Theory of Periodic Structures 443.1.3 Scattering from Gratings 463.1.4 Homogenization 473.2 E_ective Zero Thickness 503.3 Sheet Boundary Conditions 533.3.1 Impedance Modeling 543.3.2 Polarizability Modeling 573.3.3 Susceptibility Modeling 603.3.4 Comparisons between the Models 664 Susceptibility Synthesis 694.1 Linear Time-Invariant Metasurfaces 694.1.1 Basic Assumptions 694.1.2 Birefringent Metasurfaces 764.1.3 Multiple-Transformation Metasurfaces 784.1.4 Relations between Susceptibilities and Scattering Parameters 814.1.5 Surface-Wave Eigenvalue Problem 924.1.5.1 Formulation of the Problem 924.1.5.2 Dispersion in a Symmetric Environments 964.1.6 Metasurfaces with Normal Polarizations 1004.1.7 Illustrative Examples 1044.1.7.1 Polarization Rotation 1044.1.7.2 Multiple Nonreciprocal Transformations 1094.1.7.3 Angle-Dependent Transformations 1124.2 Time-Varying Metasurfaces 1174.2.1 Formulation of the Problem 1174.2.2 Harmonic-Generation Time-Varying Metasurface 1204.3 Nonlinear Metasurfaces 1214.3.1 Second-Order Nonlinearity 1224.3.1.1 Frequency-Domain Approach 1234.3.1.2 Time-Domain Approach 1285 Scattered Field Computation 1335.1 Fourier-Based Propagation Method 1345.2 Finite-Di_erence Frequency-Domain Method 1415.3 Finite-Di_erence Time-Domain Method 1475.3.1 Time-Varying Dispersionless Metasurfaces 1505.3.2 Time-Varying Dispersive Metasurfaces 1565.4 Spectral-Domain Integral Equation Method 1646 Practical Implementation 1736.1 General Implementation Procedure 1746.2 Basic Strategies for Full-Phase Coverage 1786.2.1 Linear Polarization 1796.2.1.1 Metallic Scattering Particles 1796.2.1.2 Dielectric Scattering Particles 1886.2.2 Circular Polarization 1946.3 Full-Phase Coverage with Perfect Matching 1986.4 Effects of Symmetry Breaking 2076.4.1 Angular Scattering 2086.4.2 Polarization Conversion 2157 Applications 2237.1 Angle-Independent Transformation 2247.2 Perfect Matching 2297.3 Generalized Refraction 2347.3.1 Limitations of Conventional Synthesis Methods 2347.3.2 Perfect Refraction using Bianisotropy 2398 Conclusions 2459 Appendix 2499.1 Approximation of Average Fields at an Interface 2499.2 Fields Radiated by a Sheet of Dipole Moments 2529.3 Relations between Susceptibilities and Polarizabilities 255Bibliography 260

KARIM ACHOURI, PhD, is a Postdoctoral Fellow in the Laboratory of Nanophotonics and Metrology at École Polytechnique de Montréal. He obtained his PhD from the same institution in electrical engineering in 2017. His research focus is on metamaterials, metasurfaces, photonics and optical systems.CHRISTOPHE CALOZ, PhD, is Full Professor at École Polytechnique de Montréal, the holder of a Canada Research Chair Tier-I and Head of the Electromagnetics Research Group. He has authored or co-authored over 700 technical conference, letter, and journal papers, as well as 13 books and book chapters.

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• Preface ix1 Introduction 11.1 Metamaterials 11.2 Emergence of Metasurfaces 42 Electromagnetic Properties of Materials 92.1 Bianisotropic Constitutive Relations 102.2 Temporal Dispersion 142.2.1 Causality and Kramers-Kronig Relations 152.2.2 Lorentz Oscillator Model 172.3 Spatial Dispersion 232.4 Lorentz Reciprocity Theorem 272.5 Poynting Theorem 322.6 Energy Conservation in Lossless-Gainless Systems 382.7 Classi_cation of Bianisotropic Media 413 Metasurface Modeling 433.1 E_ective Homogeneity 443.1.1 The Homogeneity Paradox 443.1.2 Theory of Periodic Structures 443.1.3 Scattering from Gratings 463.1.4 Homogenization 473.2 E_ective Zero Thickness 503.3 Sheet Boundary Conditions 533.3.1 Impedance Modeling 543.3.2 Polarizability Modeling 573.3.3 Susceptibility Modeling 603.3.4 Comparisons between the Models 664 Susceptibility Synthesis 694.1 Linear Time-Invariant Metasurfaces 694.1.1 Basic Assumptions 694.1.2 Birefringent Metasurfaces 764.1.3 Multiple-Transformation Metasurfaces 784.1.4 Relations between Susceptibilities and Scattering Parameters 814.1.5 Surface-Wave Eigenvalue Problem 924.1.5.1 Formulation of the Problem 924.1.5.2 Dispersion in a Symmetric Environments 964.1.6 Metasurfaces with Normal Polarizations 1004.1.7 Illustrative Examples 1044.1.7.1 Polarization Rotation 1044.1.7.2 Multiple Nonreciprocal Transformations 1094.1.7.3 Angle-Dependent Transformations 1124.2 Time-Varying Metasurfaces 1174.2.1 Formulation of the Problem 1174.2.2 Harmonic-Generation Time-Varying Metasurface 1204.3 Nonlinear Metasurfaces 1214.3.1 Second-Order Nonlinearity 1224.3.1.1 Frequency-Domain Approach 1234.3.1.2 Time-Domain Approach 1285 Scattered Field Computation 1335.1 Fourier-Based Propagation Method 1345.2 Finite-Di_erence Frequency-Domain Method 1415.3 Finite-Di_erence Time-Domain Method 1475.3.1 Time-Varying Dispersionless Metasurfaces 1505.3.2 Time-Varying Dispersive Metasurfaces 1565.4 Spectral-Domain Integral Equation Method 1646 Practical Implementation 1736.1 General Implementation Procedure 1746.2 Basic Strategies for Full-Phase Coverage 1786.2.1 Linear Polarization 1796.2.1.1 Metallic Scattering Particles 1796.2.1.2 Dielectric Scattering Particles 1886.2.2 Circular Polarization 1946.3 Full-Phase Coverage with Perfect Matching 1986.4 Effects of Symmetry Breaking 2076.4.1 Angular Scattering 2086.4.2 Polarization Conversion 2157 Applications 2237.1 Angle-Independent Transformation 2247.2 Perfect Matching 2297.3 Generalized Refraction 2347.3.1 Limitations of Conventional Synthesis Methods 2347.3.2 Perfect Refraction using Bianisotropy 2398 Conclusions 2459 Appendix 2499.1 Approximation of Average Fields at an Interface 2499.2 Fields Radiated by a Sheet of Dipole Moments 2529.3 Relations between Susceptibilities and Polarizabilities 255Bibliography 260