Introduction to Micro- and Nanooptics

Jürgen Jahns, Stefan Helfert

• Publisher: Wiley-VCH
• Year: 2012
• Print ISBN: 9783527408917
• eBook ISBN: 9783527670895
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

This first textbook on both micro- and nanooptics introduces readers to the technological development, physical background and key areas.
The opening chapters on the physics of light are complemented by chapters on refractive and diffractive optical elements. The internationally renowned authors present different methods of lithographic and nonlithographic fabrication of microoptics and introduce the characterization and testing of microoptics.
The second part of the book is dedicated to optical microsystems and MEMS, optical waveguide structures and optical nanostructures, including photonic crystals and metamaterials.
Each chapter includes exercises illustrating a sample approach to new and complex topics, making the textbook suitable for lectures on optics as part of a physics or electrical engineering course.

Chapter 1

Preliminaries

We begin with a brief sampler of some mathematical topics that are useful for reading the later chapters. The description does not aim at being rigorous nor comprehensive. Rather, the purpose is to allow the reader to quickly update his and her knowledge and also it serves the purpose of establishing the notation used in this book.

1.1 Complex Numbers

For the mathematical description of oscillations and waves, the use of complex exponential functions is very practical. For example, a plane wave traveling in x-direction can be represented mathematically by

(1.1)

This is the notation we will use in this book. Here, i denotes the imaginary unit defined by i² = −1. In engineering, quite often, the letter j is often used instead of i to avoid confusion with the symbol for the electric current. It is also common to write

. This has no physical consequence, of course. However, it does make a difference in the mathematical formalism, when the first derivative (or, more general, uneven-order derivatives) occur, as it is the case, for example, in the paraxial wave equation.

A complex number z has a real part, denoted as

(z), and an imaginary part,

(z),

(1.2)

Here, a =

(z) and b =

(z) are real-valued numbers. Using them like Cartesian coordinates, z is represented graphically by its position in the complex plane (Figure 1.1).

Figure 1.1 Graphical representation of a complex number in the complex plane.

For the description of a wave that is a harmonic oscillation in space and time, the use of complex exponential functions using polar coordinates is convenient as in (1.1). The exponential form of a complex number is introduced by Euler’s equation

(1.3)

Here, |z| is the modulus of z with |z|² = a² + b². ϕ is called the argument or the phase of z (Figure 1.2). It is ϕ = arg(z) = arctan(b/a). In turn, one obtains the Cartesian coordinates from the polar coordinates by a = |z| cos ϕ and b = |z| sin ϕ. By varying ϕ, z moves on a circle in the complex plane with a periodicity of 2π. Hence, there is an ambiguity in the polar representation: for a specific point in the complex plane described by the pair of coordinates (a,b), all polar coordinates of the form (r,ϕ + m2π) with m = 0, ±1, ±2, … also represent the same point. This 2π-phase ambiguity is an important aspect of all wave phenomena.

Figure 1.2 Graphical representation of a complex number z using polar coordinates. z* is the conjugate of z.

Finally, we introduce the conjugate of a complex number. Two numbers z₁ and z₂ are conjugate to each other if their real parts are the same and their imaginary parts differ by a minus sign. The complex conjugate number is denoted either by a bar,

, or by a star, z*. Here, we use the latter notation. Thus, we can wri...

Table of contents

• Cover
• Half Title page
• Title page
• Copyright page
• How to Study This Textbook
• Preface
• List of Symbols
• Acknowledgment
• Chapter 1: Preliminaries
• Chapter 2: Light Propagation
• Chapter 3: Light as Carrier of Information and Energy
• Chapter 4: Light Propagation in Free Space
• Chapter 5: Refractive and Reflective Microoptics
• Chapter 6: Diffractive Microoptics
• Chapter 7: Micro- and Nanofabrication
• Chapter 8: Tunable Microoptics
• Chapter 9: Compound and Integrated Free-Space Optics
• Chapter 10: Light Propagation in Waveguides
• Chapter 11: Integrated Waveguide Optics
• Chapter 12: Plasmonics
• Chapter 13: Photonic Crystals
• Chapter 14: Left-Handed Materials
• Index