Photonic Waveguides

Theory and Applications

Azzedine Boudrioua

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848210271
• eBook ISBN: 9781118623787
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Waves & Wave Mechanics

This book presents the principles of non-linear integrated optics. The first objective is to provide the reader with a thorough understanding of integrated optics so that they may be able to develop the theoretical and experimental tools to study and control the linear and non-linear optical properties of waveguides.
The potential use of these structures can then be determined in order to realize integrated optical components for light modulation and generation. The theoretical models are accompanied by experimental tools and their setting in order to characterize the studied phenomenon. The passage from theory to practice makes the comprehension of the physical phenomena simple and didactic.
The book also gives a presentation of the industrial applications of the integrated optical components. The studied topics range from the theory of waveguides and the linear and non-linear optical characterization techniques to photonic crystals. This last field constitutes a major challenge of photonic technologies of the 21st century.

Chapter 1

Optical Waveguide Theory

Optical waveguides are structures with three layers controlling light confinement and propagation in a well defined direction inside the central layer (Figure 1.1).

Figure 1.1. Planar optical waveguide

Light confinement is carried out by successive total reflections on the two interface guides – substrate and guide – superstrate.

Light propagation is governed by an interference phenomenon which occurs inside the guide between two waves; one of them undergoes two successive total reflections. For a better understanding of the guided wave propagation, we will recall the main principles of these two phenomena, total reflection and interference, inside a transparent plate with parallel faces.

1.1. Principles of optics

1.1.1. Total reflection phenomenon

Let us consider an interface separating two mediums 1 and 2, which are dielectric, lossless, homogenous and isotropic with refractive indices n₁ and n₂, respectively. An electromagnetic wave propagates from 1 to 2 with an angle of incidence θi related to the normal of the interface (Figure 1.2).

Figure 1.2. Reflection on an interface (medium 1/medium 2)

The electric field of the incident wave is given by:

[1.1]

[1.2]

is the wave vector in the vacuum (λ: wavelength in the vacuum) and E_i0 is the incident wave amplitude. The electric fields of the reflected and transmitted waves can be written:

[1.3]

[1.4]

In addition, refraction law is given by:

[1.5]

[1.6]

In the case of n₁ > n₂, there is an incident angle θ_l, as:

[1.7]

For θ_i > θ_l, the incident wave is totally reflected into medium 1 (total reflection) and the angle θ_t of the transmitted wave is complex [Bor 1999, War 1988]:

[1.8]

From [1.4], the transmitted wave can be written as:

[1.9]

This wave propagates in the Oz direction with an amplitude exponentially decreasing in the Ox direction. This is called an evanescent wave. Also, according to Fresnel’s formulae, the considered wave undergoes a phase shift compared to the incident wave, given by [Bor 1999]:

[1.10]

[1.11]

Relations [1.9] and [1.10] will be used throughout this chapter in order to study the propagation of guided waves. Note that evanescent waves have been experimentally investigated and they are currently utilized in the field of integrated optics. A similar phenomenon appears at the interface between a dielectric and a metallic layer generating, under specific conditions, a surface plasmon [Rae 1997].

1.1.2. Parallel-face plate

Let us consider a transparent plate with parallel faces (Figure 1.3), with refractive index n and a thickness d, placed in air (index = 1). We will focus on the calculation of the difference of the optical path (δ) between the first two rays transmitted throughout the plate (the same approach can be applied for the first two reflected rays).

[1.12]

We can easily show that δ is given by:

[1.13]

where θ is the propagation angle within the plate.

Figure 1.3. Interference between two rays transmitted by a parallel face plate with a thickness d and a refractive index n

In these conditions the parallel-face plate introduces a phase shift between the two rays R₁ and R₂ given by the following relation:

[1.14]

The latter is at the origin of the interference between the two rays R₁ and R₂. The transmitted rays are parallel, thus the interference phenomenon is located at infinity. However, we can observe the interference fringes in the Fresnel’s field on a screen placed at the focal distance of a convergent lens [Bor 1999, War 1988, ...

Table of contents

• Cover
• Title Page
• Copyright
• Foreword
• Acknowledgments
• Introduction
• Chapter 1: Optical Waveguide Theory
• Chapter 2: Optical Waveguide Fabrication Techniques
• Chapter 3: Optical Waveguide Characterization Techniques
• Chapter 4: Non-linear Effects in Integrated Optics
• Chapter 5: The Electro-optic Effect in Waveguides
• Chapter 6: Photonic Crystal Waveguides
• Conclusion
• Index