# Nanostructured and Subwavelength Waveguides: Fundamentals and Applications

# Nanostructured and Subwavelength Waveguides: Fundamentals and Applications

ISBN: 978-1-118-34324-1 May 2012 304 Pages

**E-Book**

$132.99

## Description

Optical waveguides take a prominent role in photonics because they are able to trap and to transport light efficiently between a point of excitation and a point of detection. Moreover, waveguides allow the management of many of the fundamental properties of light and allow highly controlled interaction with other optical systems. For this reason waveguides are ubiquitous in telecommunications, sensing, spectroscopy, light sources, and high power light delivery. Nanostructured and subwavelength waveguides have additional advantages; they are able to confine light at a length scale below the diffraction limit and enhance or suppress light-matter interaction, as well as manage fundamental properties of light such as speed and direction of energy and phase propagation.This book presents semi-analytical theory and practical applications of a large number of subwavelength and nanostructured optical waveguides and fibers operating in various regions of the electromagnetic spectrum including visible, near and mid-IR and THz. A large number of approximate, while highly precise analytical expressions are derived that describe various modal properties of the planar and circular isotropic, anisotropic, and metamaterial waveguides and fibers, as well as surface waves propagating on planar, and circular interfaces. A variety of naturally occurring and artificial materials are also considered such as dielectrics, metals, polar materials, anisotropic all-dielectric and metal-dielectric metamaterials.

Contents are organized around four major themes:

- Guidance properties of subwavelength waveguides and fibers made of homogeneous, generally anisotropic materials
- Guidance properties of nanostructured waveguides and fibers using both exact geometry modelling and effective medium approximation
- Development of the effective medium approximations for various 1D and 2D nanostructured materials and extension of these approximations to shorter wavelengths
- Practical applications of subwavelength and nanostructured waveguides and fibers

*Nanostructured Subwavelengths and Waveguides* is unique in that it collects in a single place an extensive range of analytical solutions which are derived in various limits for many practically important and popular waveguide and fiber geometries and materials.

**Series Preface xiii**

**Preface xv**

**1 Introduction 1**

1.1 Contents and Organisation of the Book 2

1.2 Step-Index Subwavelength Waveguides Made of Isotropic Materials 3

1.3 Field Enhancement in the Low Refractive Index Discontinuity Waveguides 5

1.4 Porous Waveguides and Fibres 6

1.5 Multifilament Core Fibres 7

1.6 Nanostructured Waveguides and Effective Medium Approximation 8

1.7 Waveguides Made of Anisotropic Materials 9

1.8 Metals and Polar Materials 10

1.9 Surface Polariton Waves on Planar and Curved Interfaces 12

1.10 Metal/Dielectric Metamaterials and Waveguides Made of Them 16

1.11 Extending Effective Medium Approximation to Shorter Wavelengths 18

**2 Hamiltonian Formulation of Maxwell Equations for the Modes of Anisotropic Waveguides 21**

2.1 Eigenstates of a Waveguide in Hamiltonian Formulation 21

2.2 Orthogonality Relation between the Modes of a Waveguide Made of Lossless Dielectrics 23

2.3 Expressions for the Modal Phase Velocity 26

2.4 Expressions for the Modal Group Velocity 27

2.5 Orthogonality Relation between the Modes of a Waveguide Made of Lossy Dielectrics 29

2.6 Excitation of the Waveguide Modes 30

**3 Wave Propagation in Planar Anisotropic Multilayers, Transfer Matrix Formulation 39**

3.1 Planewave Solution for Uniform Anisotropic Dielectrics 39

3.2 Transfer Matrix Technique for Multilayers Made from Uniform Anisotropic Dielectrics 41

3.3 Reflections at the Interface between Isotropic and Anisotropic Dielectrics 44

**4 Slab Waveguides Made from Isotropic Dielectric Materials. Example of Subwavelength Planar Waveguides 47**

4.1 Finding Modes of a Slab Waveguide Using Transfer Matrix Theory 47

4.2 Exact Solution for the Dispersion Relation of Modes of a Slab Waveguide 50

4.3 Fundamental Mode Dispersion Relation in the Long-Wavelength Limit 53

4.4 Fundamental Mode Dispersion Relation in the Short-Wavelength Limit 55

4.5 Waveguides with Low Refractive-Index Contrast 57

4.6 Single-Mode Guidance Criterion 57

4.7 Dispersion Relations of the Higher-Order Modes in the Vicinity of their Cutoff Frequencies 57

4.8 Modal Losses Due to Material Absorption 58

4.9 Coupling into a Subwavelength Slab Waveguide Using a 2D Gaussian Beam 64

4.10 Size of a Waveguide Mode 69

**5 Slab Waveguides Made from Anisotropic Dielectrics 75**

5.1 Dispersion Relations for the Fundamental Modes of a Slab Waveguide 75

5.2 Using Transfer Matrix Method with Anisotropic Dielectrics 77

5.3 Coupling to the Modes of a Slab Waveguide Made of Anisotropic Dielectrics 78

**6 Metamaterials in the Form of All-Dielectric Planar Multilayers 81**

6.1 Effective Medium Approximation for a Periodic Multilayer with Subwavelength Period 81

6.2 Extended Bloch Waves of an Infinite Periodic Multilayer 82

6.3 Effective Medium Approximation 84

6.4 Extending Metamaterial Approximation to Shorter Wavelengths 86

6.5 Ambiguities in the Interpretation of the Dispersion Relation of a Planewave Propagating in a Lossy Metamaterial 89

**7 Planar Waveguides Containing All-Dielectric Metamaterials, Example of Porous Waveguides 91**

7.1 Geometry of a Planar Porous Waveguide 91

7.2 TE-Polarised Mode of a Porous Slab Waveguide 91

7.3 TM-Polarised Mode of a Porous Slab Waveguide 99

**8 Circular Fibres Made of Isotropic Materials 103**

8.1 Circular Symmetric Solutions of Maxwell’s Equations for an Infinite Uniform Dielectric 104

8.2 Transfer Matrix Method 107

8.3 Fundamental Mode of a Step-Index Fibre 110

8.4 Higher-Order Modes and their Dispersion Relations Near Cutoff Frequencies 115

8.5 Dispersion of the Fundamental *m* = 1 Mode 122

8.6 Losses of the Fundamental *m* = 1 Mode 123

8.7 Modal Confinement and Modal Field Extent into the Cladding Region 125

**9 Circular Fibres Made of Anisotropic Materials 137**

9.1 Circular Symmetric Solutions of Maxwell’s Equations for an Infinite Anisotropic Dielectric 137

9.2 Transfer Matrix Method to Compute Eigenmodes of a Circular Fibre Made of Anisotropic Dielectrics 139

9.3 Fundamental Mode of a Step-Index Fibre 141

9.4 Linearly Polarised Modes of a Circular Fibre 146

**10 Metamaterials in the Form of a Periodic Lattice of Inclusions 155**

10.1 Effective Dielectric Tensor of Periodic Metamaterials in the Long-Wavelength Limit 156

10.2 Bloch Wave Solutions in the Periodic Arrays of Arbitrary-Shaped Inclusions, Details of the Planewave Expansion Method 164

**11 Circular Fibres Made of All-Dielectric Metamaterials 167**

11.1 Porous-Core Fibres, Application in Low-Loss Guidance of THz Waves 167

11.2 Multifilament Core Fibres, Designing Large Mode Area, Single-Mode Fibres 175

11.3 Water-Core Fibres in THz, Guiding with Extremely Lossy Materials 182

**12 Modes at the Interface between Two Materials 185**

12.1 Surface Modes Propagating at the Interface between Two Positive Refractive Index Materials 185

12.2 Geometrical Solution for the Bound Surface Modes 188

12.3 Modes at the Interface between a Lossless Dielectric and an Ideal Metal, Excitation of an Ideal Surface Plasmon 191

12.4 Modes at the Interface between a Lossless Dielectric and a Lossy Material (Metal or Dielectric) 194

**13 Modes of a Metal Slab Waveguide 209**

13.1 Modes of a Metal Slab Waveguide Surrounded by Two Identical Dielectric Claddings 210

13.2 Long-Range Plasmon Guided by Thin and Lossy Metal Slab 221

13.3 Modes of a Metal Slab Surrounded by Two Distinct Lossless Claddings. Leaky Plasmonic Modes 226

**14 Modes of a Metal Slot Waveguide 233**

14.1 Odd-Mode Dispersion Relation Near the Light Line of the Core Material *n*eff ∼ *n*o. Visible–Mid-IR Spectral Range 235

14.2 Odd-Mode Dispersion Relation near the Mode Cutoff *n*eff ∼ 0. Visible–Mid-IR Spectral Range 238

14.3 Fundamental Mode of a Metal Slot Waveguide. Visible–Mid-IR Spectral Range 240

14.4 Fundamental Mode Dispersion Relation at Low Frequencies *ω* → 0. Far-IR Spectral Range 243

**15 Planar Metal/Dielectric Metamaterials 247**

15.1 Extended Waves in the Infinite Metal/Dielectric Periodic Multilayers (Long-Wavelength Limit) 247

15.2 Extending Metamaterial Approximation to Shorter Wavelengths 250

**16 Examples of Applications of Metal/Dielectric Metamaterials 253**

16.1 Optically Transparent Conductive Layers, Case of *ε*_ *>* 0, *ε*⊥ *>* 0 253

16.2 Perfect Polarisation Splitter, Case of *ε*_ *>* 0, *ε*⊥ *<* 0 256

16.3 Surface States at the Interface between Lossless Dielectric and Metal/Dielectric Metamaterials 260

16.4 Surface Plasmons in a Two-Material System *ε*i = *ε*d 262

16.5 Practical Application of Surface Plasmons Supported by Metamaterials 1, 2, 3 271

**17 Modes of MetallicWires, Guidance in the UV–near-IR, Mid-IR and Far-IR Spectral Ranges 281**

17.1 Guidance by the Metallic Wires with Diameters Smaller than the Metal Skin Depth 281

17.2 Guidance by the Metallic Wires with Diameters Much Larger than the Metal Skin Depth 285

17.3 Wire Plasmons in the Visible–Near-IR Spectral Range 286

17.4 Wire Plasmons in the Mid-IR–Far-IR Spectral Range 291

**18 Semianalytical Methods of Solving Nonlinear Equations of Two Variables 301**

18.1 Polynomial Solution of a Nonlinear Equation in the Vicinity of a Known Particular Solution 301

18.2 Method of Consecutive Functional Iterations 302

18.3 Method of Asymptotics 304

**References 307**

**Index 311**

“Coverage of material is both rigorous and transparent and thus this volume is likely to be used extensively by researchers in these rapidly developing subject areas.” (*Optics & Photonics News*, 9 November 2012)

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