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- Herausgeber: Wiley-VCH
- Kategorie: Fachliteratur
- Sprache: Englisch
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.
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Seitenzahl: 610
Veröffentlichungsjahr: 2012
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Contents
Cover
Half Title page
Title page
Copyright page
How to Study This Textbook
Preface
List of Symbols
Acknowledgment
Chapter 1: Preliminaries
1.1 Complex Numbers
1.2 Fourier Transformation
1.3 Maxwell’s Equations
1.4 Boundary Conditions
Questions
Problems
Further Reading
Chapter 2: Light Propagation
2.1 Wave Equation
2.2 Solutions of the Wave Equation
2.3 Vectorial Description of Plane Waves
2.4 The Time-Independent Wave Equation
2.5 Paraxial Wave Equation
2.6 Gaussian Beams
2.7 The Angular Spectrum
2.8 Light Propagation in Terms of the Angular Spectrum
2.9 Evanescent Fields
2.10 Free-Space and Waveguide Propagation
Questions
Problems
Further Reading
Chapter 3: Light as Carrier of Information and Energy
3.1 Poynting Vector and Flow of Energy in a Wave Field
3.2 Flow of Information in a Wave Field
3.A Appendix: Minimal Value of the Space-Bandwidth Product
Questions
Problems
Further Reading
Chapter 4: Light Propagation in Free Space
4.1 Transmission of a Wave Field through an Object
4.2 Propagation Between Objects
4.3 Diffraction at a Single Slit
4.4 Near-Field Diffraction
4.5 Examples for Near-Field Diffraction
4.6 Far-Field Diffraction and Optical Fourier Transformation
4.7 Examples of Far-Field Diffraction
4.8 Optical Imaging
4.9 Lens Performance
Questions
Problems
Further Reading
Chapter 5: Refractive and Reflective Microoptics
5.1 Refractive Optics
5.2 Refractive Microlenses
5.3 Microprisms
5.4 Reflective Microoptics
Questions
Problems
Further Reading
Chapter 6: Diffractive Microoptics
6.1 Phase Quantization
6.2 Linear Diffraction Gratings
6.3 Diffractive Elements with Radial Symmetry
6.4 Subwavelength Gratings and Rigorous Diffraction Theory
Questions
Problems
Further Reading
Chapter 7: Micro- and Nanofabrication
7.1 Structuring and Pattern Transfer
7.2 The Lithographic Process
7.3 Exposure
7.4 Pattern Transfer
7.5 MEMS Fabrication
7.6 Nonlithographic Fabrication
7.7 Examples for the Fabrication of Multilevel and Blazed Structures
Questions
Problems
Further Reading
Chapter 8: Tunable Microoptics
8.1 Spatial Light Modulators
8.2 Tunable Microlenses Using Microfluidics
Questions
Problems
Further Reading
Chapter 9: Compound and Integrated Free-Space Optics
9.1 Microoptical Imaging
9.2 Microoptical Beam Homogenization, Beam Guiding and Steering
9.3 Integrated Free-Space Optics
9.4 MEMS-Based Integrated Free-Space Optics
Questions
Problems
Further Reading
Chapter 10: Light Propagation in Waveguides
10.1 Overview About Waveguide Mechanisms
10.2 Dielectric Waveguides
10.3 Slab Waveguides
10.4 Determining Eigenmodes in Slab Waveguides from Maxwell’s Equations
10.5 Step-Index Fibers
Questions
Problems
Further Reading
Chapter 11: Integrated Waveguide Optics
11.1 Analysis of Waveguide Circuits
11.2 Waveguide Couplers
11.3 Rectangular Waveguides
11.4 Arrayed Waveguide Gratings
Questions
Problems
Further Reading
Chapter 12: Plasmonics
12.1 Drude Model of Electrons in Metal
12.2 Surface Waves at a Metal–Dielectric Interface
12.3 Finite Height of the Metal
12.4 Three-Dimensional Plasmonic Waveguides
12.5 Enhanced Transmission Through Tiny Holes
12.6 Final Remarks
Questions
Problems
Further Reading
Chapter 13: Photonic Crystals
13.1 Introduction
13.2 Floquet–Bloch Modes
13.3 Two- and Three-Dimensional Periodic Structures
13.4 Waveguides and Bends with Photonic Crystals
13.5 Photonic Crystal Fibers
Questions
Problems
Further Reading
Chapter 14: Left-Handed Materials
14.1 Introduction
14.2 Mathematical Description of Plane Waves in Arbitrary Materials
14.3 Wave Propagation in Homogeneous Media
14.4 Practical Realization of Left-Handed Materials (Metamaterials)
14.5 Left-Handed Materials in Time Domain
Questions
Problems
Further Reading
Index
Jürgen Jahns and Stefan HelfertIntroduction to Micro- and Nanooptics
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The Authors
Prof. Dr. Jörgen JahnsFern Universität in HagenChair of Micro- and NanophotonicsUniversitätsstr. 2758084 HagenGermany
Dr. Stefan HelfertFern Universität in HagenChair of Micro- and NanophotonicsUniversitätsstr. 2758084 HagenGermany
CoverCopyright for image of insect compound eye lies with Thomas Shahan. Used with kind permission.
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.:applied for
British Library Cataloguing-in-Publication Data:A catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.
© 2012 WILEY-VCH Verlag GmbH & Co. KGaA,Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN 978-3-527-40891-7
How to Study This Textbook
The aim of this book is to make the reader familiar with the physics and mathematics of micro- and nanooptics. The book is mainly intended to serve as a textbook for senior classes at universities, typically in Master and PhD programs. However, we also hope that scientists and engineers in industry will be able to benefit from the book. In general, the skills taught in a Bachelor program in the natural sciences and engineering should be sufficient to get started. The reader should have a certain level of familiarity with basic physics, in particular, optics. Furthermore, we assume fundamental knowledge of electromagnetic theory and mathematics such as vector analysis, differential equations and Fourier theory.
Our main purpose is to provide the reader with a solid theoretical basis of micro- and nanooptical structures. The contents are organized as follows:
The fundamentals of optics like the wave equation and various aspects of light propagation are presented in the first chapters (Chapter 1–3). The basics of free-space propagation are described in Chapter 4. Specific topics of free-space microoptics are then described in Chapter 5 (Refractive and reflective microoptics), Chapter 6 (Diffractive microoptics), Chapter 8 (Tunable microoptics) and Chapter 9 (Compound and integrated free-space optics).
Fundamental aspects of waveguide propagation are presented in Chapter 10, specific examples of integrated waveguide optics in Chapter 11. In Chapters 12 and 13, we present novel areas of nanooptics are presented that have gained significant interest in recent years: Chapter 12 introduces the reader to the field of plasmonics, Chapter 13 is about photonic crystals. The list of modern topics is rounded up by Chapter 14 which deals with left-handed materials.
For a deeper understanding of the topics and for self-study, the reader will find additional material at the end of each chapter: a list of questions, that relate directly to the text, tells the reader which topics are relevant and offers the chance to test one’s comprehension. Furthermore, a few exercises are provided to be solved together in class or individually. For further reading, we suggest a few references that refer to the topic of a chapter. These also have the purpose to introduce students to the world of scientific literature. We would like to remark that the purpose of the book is not to present latest results of research. Hence, we have abstained completely from presenting photographs of research results, for example. Lecturers might add suitable material of their own for teaching a class.
This text has to be seen in conjunction with the earlier book on “Microoptics” by S. Sinzinger and J. Jahns (Wiley-VCH Verlag GmbH, 1999 and 2003, 2nd edn). In comparison to that book, here certain topics appear either in a reduced form (like fabrication, for example) or were completely omitted (like characterization and applications of microoptics). Readers interested in those topics might use both books in a complementary fashion, this one for the description of the fundamentals and the earlier book for its presentation of research trends and applications.
Preface
Microlithography has changed the world. The impact of lithographic fabrication cannot be overemphasized: it has paved the way for mass fabrication at an unprecedented level of quality and reliability. The revolutionary development of microelectronics beginning in the 1950s is the foundation of the information society that is characterized by seemingly unlimited access to information as well as capabilities to exchange and store information, symbolized and characterized most of all by the Internet. Microlithographic fabrication has, for the first time in the development of technical evolution, changed a classical pattern of experience. Usually, quality and quantity are mutually exclusive or, in other words, high performance could only be achieved at a high price. As we all know, this is not true for lithographically fabricated devices. The performance and quality have increased by many orders of magnitude (for example, the processing power of electronic computers) while the price has remained constant or has even dropped.
For a long time, people have hoped for the same development in other areas, in particular, in optics and mechanics. Both areas are strongly related, almost like twins. Improvements in fine mechanics have improved classical optics and vice versa. It was in the 1960s when scientists also started to make use of then novel digital design and fabrication techniques for optics. Based on digital design and computer-controlled plotting facilities, computer-generated holography marks the beginning of (or, at least, the forerunner to) the field now known as microoptics. This development received a tremendous push in the 1980s and 1990s. That push was, to some extent, motivated by the rapid development of computing and communications which led to a general interest in novel optical techniques and hardware. As a consequence, the field of microoptics emerged as a new branch of optics and in the course of time has gradually led to numerous useful applications. The development has not been as revolutionary as the development of microelectronics for a number of reasons that will not be discussed here. Nonetheless, microoptics has become an important area of technology that is steadily growing.
Since the 1990s, microlithographic structuring techniques have entered the submicron range. The entry into the nanoworld has the potential to lead to a development that might be as revolutionary as the initial beginnings in the microdomain. For one, it has become possible to control and interact with individual atoms rather than statistical ensembles of atoms. This allows one to observe and make use of quantum effects that are much different in their physics as compared to “macrophysics.” Furthermore, the possibility to generate synthetic nanostructures allows one to engineer material properties, for example, the refractive index of a material. The physical properties of nanostructured materials often surpass those of bulk materials. In optics, so-called quarter-wave stacks are a good example. Used as mirrors, they can be designed and fabricated to reach reflectivities very close to 100%.
The fundamental idea of micro- and nanotechnology is to define the function of a device via its structure. In this book, we deal with micro- and nanooptics. Both are related due to the common technology platform used. And yet, they are fascinatingly different areas due to the physics involved: since for many applications, the wavelengths are just around one micrometer, microoptics typically uses structures that are several or many wavelengths in size, while nanodevices are usually in the subwavelength range. Hence, microoptics is still closer to classical phenomena, while nanooptics enters a new world that we do not yet fully oversee.
Hagen, December 2011
Jürgen Jahns, Stefan Helfert
List of Symbols
In the following we give a list of the principal symbols used in this book. Some variables are used for different physical quantities. However, their meaning becomes apparent from the context.
Scalar values, vectors and matrices occur and the following notations are used:
scalar values are written italic:
E
x
or
k
0
function are written in roman: sin, cos
physical vectors are written bold and italic:
E
mathematical vectors and matrices are written in brackets: [
F
]
Scalar Quantities
Vectorial Quantities
B
magnetic flux density, magnetic induction
D
electric displacement
E
electric field
E
L
Lorentz field
F
arbitrary field
G
periodic electric field
G
reciprocal lattice vector
H
magnetic field intensity (strength)
j
electric current density
k
wave vector
M
magnetic polarization density
n
vector normal to a surface
P
electric polarization (density)
R
lattice vector
r
position vector
S
Poynting vector
S
re
real Poynting vector
ξ
ray aberration
Matrix Quantities
[
a
]
mathematical vector containg the amplitudes of various modes
[
X
]
eigenvector matrix
Greek Symbols:
Other Quantities
coefficients describing wavefront aberration
energy (of a photon)
eikonal (optical path length)
first and second derivative of eikonal
scaling factor
Specific Mathematical Functions
cylinder functions of order
m
In particular:
J
m
Bessel function of the first kind
Y
m
Bessel function of the second kind (Neumann function)
I
m
modified Bessel function of the first kind
K
m
modified Bessel function of the second kind
H
(1,2)
m
Hankel function of the first, second kind
Acronyms
FWHM
full-width at half-maximum
LC
liquid crystal
MEMS
micro-electro-mechanical system
NA
numerical aperture
psf
point spread function
SBP
space-bandwidth product
SLM
spatial light modulator
Acknowledgment
We would like to express our gratitude to Prof. em. Adolf Lohmann and Prof. em. Reinhold Pregla for many stimulating discussions throughout recent years.
We are indebted to Prof. Gladys Minguez-Vega, Universitat Jaume I, Castellón, and Prof. Stefan Sinzinger, Technische Universität Ilmenau for their suggestions and discussions regarding this book. Furthermore, we are grateful to Dr. Peter Widerin for all of his useful input.
Special thanks go to the members of the Micro- and Nanophotonics group at the University of Hagen for providing some of the results used in this book and for hints and suggestions.
Last but not least, our thanks go to Vera Palmer and Anja Tschörtner from Wiley-VCH in Berlin for their continuous interest, gentle reminders and professional help during the process of writing this book.
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)
A complex number z has a real part, denoted as (z), and an imaginary part, (z),
(1.2)
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)
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 write
(1.4)
1.2 Fourier Transformation
The Fourier transformation is probably the most important mathematical signal transformation. It is widely used for signal analysis, processing, and coding. The most prominent modern application is the encoding of streaming audio and video signals using the MPEG format as well as static images according to the JPEG standard. These formats are based on the discrete cosine transformation, a variation of the Fourier transformation suitable for discretized real-valued signals.
The Fourier transformation represents a function f(x) (which we assume to be continuous here) as a linear superposition of sine- and cosine-functions. Using the complex notation of (1.3), in the one-dimensional case, we write
(1.5)
is called the Fourier transform (also the Fourier spectrum) of f(x), that is,
(1.6)
Here, is a normalization factor which warrants that
(1.7)
As mentioned earlier, it is a matter of definition whether the exponent in (1.5) and (1.6), respectively, is written with a positive or with a negative sign. To be in agreement with the notation as in (1.1) for a single plane wave, the positive sign is used in (1.5). In contrast, when we express the Fourier transformation of a temporal signal g(t) with respect to the time coordinate t, we use the negative sign in the Fourier expansion
(1.8)
In this case, the inverse operation is
(1.9)
Sometimes, it is convenient to express the Fourier expansion in terms of the oscillation frequency νt and the spatial frequency νx, respectively, rather than the angular frequencies ω and k. In this case, the Fourier expansion of a spatial function f(x) is
(1.10)
and
(1.11)
and accordingly for g(t). When compared with (1.5) and (1.6), we note that here the normalization factor is one. This leads to the significant difference in the “DC value” of the signal given by
(1.12)
(1.13)
For further reference, Table 1.1 shows several functions relevant to this text and their Fourier transforms. First, some definitions for
Table 1.1 Fourier transformation of rect- and tri-function, exponential, Gaussian and Delta-function.
the
rect-function
rect(
x
):
(1.14)
the
sinc-function
sinc(
x
):
(1.15)
the
triangle function
tri(
x
):
(1.16)
Remark
Alternative definition of the sinc-function: In the literature, the sinc-function is often defined as
(1.17)
that is, without the factor π. Here, however, we will use the definition according to (1.15) since it has the convenient property that the zeros occur at integer values of x. The reader should be aware of the two different definitions.
With the following definitions for the (unnormalized) Gaussian function
(1.18)
and the (unnormalized) Lorentzian function,
(1.19)
keep in mind the following list of Fourier transform pairs shown graphically in Table 1.1 and listed in Table 1.2.
Table 1.2 Fourier transform pairs.
rect
↔
sinc
tri
↔
sinc
2
exponential
↔
Lorentzian
Gaussian
↔
Gaussian
Delta-function
↔
const
In optics, one usually considers functions that depend on more than one coordinate. The extension of the Fourier transformation to multidimensional functions is straightforward due to its linearity. For example, the angular spectrum of a 2D signal f(x,y) is given as
(1.20)
(1.21)
Sometimes, one encounters situations that exhibit radial symmetry. A specific example would be diffraction at a circular aperture. In that case, it may be convenient to carry out the calculations in circular coordinates r and ϕ which are given by
(1.22)
(1.23)
with and tan . With this, we can express the exponent in (1.20) as
(1.24)
Hence, for the object u(r,ϕ), the 2D Fourier transform in radial coordinates is given as
(1.25)
(1.26)
Here, we have used the identity
(1.27)
Here, is the zeroth Bessel function (of the first kind). The integral transformation in (1.26) is also known as the Hankel transformation of the function u(r).
1.2.1 Basic Fourier Rules
In the following, several useful rules for the Fourier transformation are listed which the reader may verify as an exercise. For simplicity, just the one-dimensional case will be considered. Usually, the extension to 2D is straightforward. In order to avoid the normalization factor, we express the Fourier transformation in terms of the spatial frequency variable ν (dropping the index ‘x’).
Linearity For a function which can be expressed as a linear combination of other functions, the Fourier transform is also given as the linear superposition of the individual transforms, that is,
(1.28)
Scaling If we scale a function in x-direction by a factor with a > 0, then its Fourier transform scales with 1/a, that is,
(1.29)
(1.30)
Mirror symmetry (even functions) For a symmetric function, the Fourier transform reduces to a cosine transform, that is,
(1.36)
Hermitian functions A more general statement is: if a function is Hermitian, then its Fourier transform is real-valued, for example,
(1.37)
Here, f* is the complex conjugate of f.
Shift theorem A shift of the function f(x) towards positive x-values by a distance s leads to a phase factor with a negative sign, for example,
(1.38)
The shift theorem can be very useful in conjunction with the convolution theorem, as we will see in the later example.
(1.39)
where the Fourier coefficients an are given as
(1.40)
The following properties of the Fourier transformation relate to the situations where a function f is given as the sum or the product of two other functions.
Convolution theorem If a function can be expressed as the product of two functions, then its Fourier transform is given as the convolution of the two respective Fourier transforms, that is,
(1.41)
Here, the star * denotes the convolution operation. This theorem can also be applied to the inverse case: if a function can be expressed as the convolution of two functions, then its Fourier spectrum is given as the product of the two respective Fourier transform spectra
(1.42)
(1.43)
This rather simple statement represents an important physical theorem, the Wiener–Khinchin theorem. However, we will not discuss its general significance here.
Parseval’s theorem (also known as Plancherel’s theorem) The energy of a signal in the x-domain is equal to the energy of its Fourier transform in the ν-domain, that is,
(1.44)
If instead of the frequency ν the angular frequency k is used, a normalization factor 1/2π comes in and
(1.45)
Fourier transform of the derivative function
(1.46)
This can be derived directly from (1.10). For a derivative in the Fourier domain, one gets
(1.47)
1.3 Maxwell’s Equations
An optical wave is an electromagnetic phenomenon and therefore its propagation and interaction with matter are described, in general, by Maxwell’s equations. However, there are different ways of writing Maxwell’s equations and it depends on the situation regarding which form is appropriate. In the simplest form, they may be written as
(1.48)
(1.49)
(1.50)
(1.51)
Here, E is the electric field (or electric field strength) and B is the magnetic flux density (or sometimes, typically in textbooks on physics, simply magnetic field). ε is the electric permittivity and μm is the magnetic permeability. σ denotes the electric conductivity, ρ denotes the charge density, and j is the electric current density. The symbol ∇ is the nabla operator and · denotes the vector product. In the following, we will use the notation for the first partial derivative with time, ∂E/∂t, and for the second partial derivative. Equations (1.49) and (1.50) are referred to as the inhomogeneous Maxwell’s equations since they contain the electric charge density and the current density, while the other two are called the homogeneous Maxwell’s equations.
E and B are functions of three spatial coordinates (for example, the Cartesian coordinates x, y and z) and the time coordinate t. In isotropic and homogeneous media, the “material constants” ε, μ, ρ and σ are constant. For simplicity, we assume at the beginning that they do not depend on the fields. In this case, Maxwell’s equations are linear. Linearity means that if E1 and E2 are solutions as well as B1 and B2, then all linear combinations a1E1 + a2E2 and b1B1 + b2B2 represent solutions, too. The assumption of linearity is not always justified, in particular, when the fields become very large. The generally nonlinear dependency of ε and μ shows up in describing electrooptic and magnetooptic effects.
The set of Equations (1.48)–(1.51) is complemented by the equation that relates the electric current density with the E-field, that is,
(1.52)
In this context, we have to consider the other material parameters, ε and μ. Classical optics mostly covers the case where ε > 0 and μ ≈ 1 (see Figure 1.4). In the (ε, μ)-diagram, many metals are located on the line μ ≈ 1, but for them ε < 0. The quadrant on the lower left with ε < 0 and μ < 0 represents the area of negative-index materials with very unusual electromagnetic behavior. Such materials are not known in nature (at least, so far), but they can be synthesized by subwavelength-structuring.
Figure 1.4 Diagram with the material parameters ε and μ as the coordinates.
The material parameters depend on the molecular structure of a material and/or the geometric structure of a micro- or nanodevice. For most materials and devices, the bulk quantities E and B are sufficient. However, certain aspects suggest that it may sometimes be more convenient to use new field quantities that take the material properties into account. For this purpose, one introduces the electric displacement densityD
(1.53)
P is the electric polarization (density). In a dielectric medium, an electric field causes no current flow, but the induction of dipoles. P is the dipole moment per unit volume. For a linear, homogeneous and isotropic medium, P and E are related by
(1.54)
Here, χ is the electric susceptibility and εr is the relative permittivity. By combining (1.53) and (1.54), one obtains
(1.55)
Simplified, one may say that D is the E-field in a medium with the materials properties taken into account. Expressed by D rather than E, Maxwell’s equation reads as
(1.56)
The analogous expressions for the magnetic field (strength)H are
(1.57)
(1.58)
(1.59)
M is the magnetic polarization and μr is the relative magnetic permeability.
1.4 Boundary Conditions
As mentioned, structuring of a medium, in particular, at the subwavelength scale, allows one to “engineer” the optical parameters. We shall learn about this topic in the later sections of this book. We prepare these issues by looking first at the boundary conditions for the components of the electric and magnetic field. For this purpose, we consider the interface between two media which differ in the values of the electric permittivity ε (Figure 1.5).
Figure 1.5 Path of integration along the boundary of two media.
For the derivation of the boundary conditions, one may apply the first of Maxwell’s equations. In its integral form, it reads
(1.60)
The integral on the left-hand side is a line integral along the closed path indicated in the figure by the dotted line. The integral on the right-hand side sums up across the hatched area which is enclosed by the path. dr is the path differential, and da is the surface differential normal to the surface. is the time derivative of B. By applying (1.60) to the situation of Figure 1.5, for the left-hand side, one obtains
(1.61)
The right-hand side of (1.60) can be evaluated as
(1.62)
Here, is the average value of the derivative of B. Now, we assume that we decrease Δz → 0. For a finite value of , the integral in (1.62) will go to zero so that we obtain
(1.63)
and hence
(1.64)
This means that the tangential component of the electric field is continuous at a boundary. This is not true, however, for the D-field. The tangential component Dt is discontinuous due to the surface charge at the interface. With the same arguments as above, one can show that for a linear, isotropic medium,
(1.65)
(1.66)
Here, the integration takes place over the surface indicated as a cross-section by the dotted line in Figure 1.5. q is the electrical charge contained in the integration volume. For decreasing dimensions of the integration volume (and surface, respectively), q is approximated by the surface charge σ at the interface. One can then argue that at the interface of two dielectric media, the extension of surface charges into the media is so small that for Δz → 0, the amount of surface charge between two dielectric media σ → 0 from which
(1.67)
That is, the normal component of the D-field, Dn, is continuous at an interface if surface charges can be neglected. The latter assumption is justified in the case of dielectric media. This result will be used later when we discuss the optical properties of microstructured media.
1.4.1 Method of Stationary Phase
The method of stationary phase allows one the approximate calculation of an integral given as
(1.68)
(1.69)
(1.70)
We introduce the coordinate transformation to write
(1.71)
We split up the integral on the right-hand side into its real and imaginary part, namely,
(1.72)
These integrals are known as the Fresnel integrals. Their calculation yields the values
(1.73)
Figure 1.6 Integration of the Fresnel integrals.
(1.74)
Thus, under the assumptions made earlier, we finally obtain
(1.75)
Questions
Problems
Further Reading
1 Jackson, J.D. (1998) Classical Electrodynamics, 3rd edn, John Wiley & Sons (Asia) Pte Ltd.
2 James, J.F. (2011) A Student’s Guide to Fourier Transforms, 3rd edn, Cambridge University Press.
Chapter 2
Light Propagation
Light is an electromagnetic wave caused by the mutual dependency and interaction of the electric field and the magnetic field. The mathematical description of light propagation starts with the derivation of the wave equation from Maxwell’s equations. Here, we will only consider the linear wave equation and disregard nonlinear phenomena. The basic aspects of a vectorial and a scalar description of an electromagnetic wavefield are presented. For stationary situations, the time-dependency of the general wave equation can be eliminated so that the time-independent Helmholtz equation results. As a special case of the Helmholtz equation, we consider the paraxial wave equation. A special solution of the paraxial wave equation is the Gaussian beam that is of importance to laser optics, in particular. A noteworthy property of a Gaussian beam is that its angular spectrum is also Gaussian. The angular spectrum of a wavefield is given by the Fourier transform with respect to the transverse spatial coordinates. The angular spectrum is a useful concept that allows a relatively simple calculation of the light propagation between two planes. Finally, we introduce the reader to the phenomenon of evanescent waves (better: fields) which play an important role for diffraction theory, super-resolution imaging, and novel directions in nanooptics and plasmonics.
2.1 Wave Equation
(2.1)
(2.2)
and
(2.3)
(2.4)
We notice that the equations are homogeneous. They can readily be solved by by applying the rotation operator to (2.1) and by differentiation of (2.3) with respect to time. We begin with (2.1)
(2.5)
For the evaluation of the expression ∇ × ∇ × E, the following identity is useful, namely,
(2.6)
Δ is the Laplace operator which, in Cartesian coordinates, is given as
(2.7)
(2.8)
This equation contains the derivative of the magnetic field. With (2.3), one can execute the differentiation with respect to time, that is,
(2.9)
We insert this into (2.8) to obtain
(2.10)
Similarly, it is possible to obtain, for the magnetic field B,
(2.11)
Equations (2.10) and (2.11) are the wave equations for E and B, respectively. In (2.10) and (2.10), the symbol c denotes the speed of light
(2.12)
(2.13)
Both are are exact values. Therefore, the speed of light in vacuum is exactly
(2.14)
Equation (2.12) implies the definition of one of the most important quantities of optics, the refractive index, usually denoted by the letter n, that is,
(2.15)
Dielectric materials are characterized by the fact that the relative magnetic permeability μr ≈ 1 so that for most cases n ≈ . The dielectric constant is related to the electric susceptibility of a dielectric medium. In general, the susceptibility is described by a tensor, that is, it has different values for different components of an applied electric field. In the case of an isotropic medium (amorphous materials like conventional glass, for example), however, there is no directional dependency. In lossy media, in particular metals, the refractive index is a complex quantity with the imaginary part describing absorption.
Remark
With regard to (2.15), it is necessary to add a word of caution. The underlying physical model leading to this result is based on the assumption of static behavior of the molecules. At very high frequencies, in particular, at optical frequencies that model is not adequate any more and may lead to incorrect results. A well-known counterexample to (2.15) is the case of water where εr ≈ 80 and μr ≈ 1 but n ≈ 1.33. Better models are given by the oscillator model for dielectric materials and the Drude model for metals (see Chapter 12.1) which take into account the dynamics of the interaction of the electromagnetic field with the propagation medium.
2.2 Solutions of the Wave Equation
Mathematically, the wave equations for E and E, respectively, are linear homogeneous partial differential equations of the second order. We would like to emphasize that (2.10) (and equivalently (2.11)) is a special case of the general form of the wave equation. In the general case, a source term occurs on the right side of the equation. This source term varies with the spatial coordinate. This leads to interesting phenomena to be considered later, such as photonic crystals, for example, where the refractive index varies periodically. At this point, however, we consider media with constant index of refraction and turn to (2.10).
The wave equation has many different solutions. Of special interest are those solutions that form orthogonal sets of functions. Three sets are of interest: plane waves, spherical waves and cylindrical waves. An arbitrary wave field may be superimposed from a spectrum of plane waves, for example. Mathematically, this corresponds to a Fourier transformation, as we will see a little later. Alternatively, one may describe the wave field in terms of a superposition of spherical waves or cylindrical waves. We start with a description of plane waves.
2.2.1 Plane Waves
(2.16)
The vector r describes any point lying in that plane. k is the so-called wave vector) (or simply k-vector), which is oriented orthogonally to the plane. The coefficients of the normalized vector k−1k describe the direction cosines according to
(2.17)
This means
(2.18)
Now, let us include the time-dependency. A monochromatic plane wave traveling in arbitrary direction is expressed as
(2.19)
(2.20)
so that
(2.21)
This equation expresses the fact that c is the phase velocity of an electromagnetic wave, generally referred to as the “speed of light.”
Remark
A plane wave traveling in +z-direction is mathematically also represented by the expression e−i(kz−ωt). For the physical interpretation, it makes no difference if we use a plus or a minus sign in the exponent. It is merely a matter of convention. However, it does make a difference in the formalism. An example is the case of the paraxial wave equation that we will consider later. It is helpful to be aware that both versions are found in the literature.
A harmonic wave as described by (2.19) is periodic in space and time. The spatial period is called the wavelength denoted by λ. It is defined by the condition
(2.26)
(2.27)
k is called the wave number. k is the magnitude of the wave vector k, hence
(2.28)
Equation (2.28) defines a sphere in k-space, the so-called Ewald sphere. We will learn more about it later. One implication of (2.28) is that the three components of the wave vector are not independent of each other. If, for example, kx and ky are given, then the kz-component of a plane wave is given by .
2.3 Vectorial Description of Plane Waves
Planes waves play an important role in the mathematical representation of a wave field by means of a Fourier decomposition. We will use this repeatedly throughout this book. For some problems, a vectorial description is essential. Thus, we will summarize the most important aspects in this section.
First, we consider a plane wave propagating in z-direction. This means that E is independent of x and y, and with (2.2) it follows that
(2.29)
Now, we assume the wave to be linearly polarized in y-direction, that is,
(2.30)
Following our earlier example, Ey is given as
(2.31)
With (2.1), it follows that
(2.32)
and that By and Bz are both constant and hence not of immediate interest. By simple integration, we can derive
(2.33)
or simply
(2.34)
We now derive some more fundamental and useful vectorial properties of plane waves. For this, we return to (2.19) and (2.16). The results of the exercise above, in particular (2.22) and (2.23), can be written in summary as
(2.35)
Furthermore, by explicitly writing
(2.36)
and similarly for the other spatial coordinates, we can summarize
(2.37)
Similarly, for B, we can write
(2.38)
and
(2.39)
We insert these results into (2.1) and (2.3) to obtain
(2.40)
and
(2.41)
By integration and setting the integration constants to zero, we get
(2.42)
and for the B-field
(2.43)
Forming the scalar product with k yields
(2.44)
which mathematically expresses that both E and B are orthogonal to the k-vector as shown in Figure 2.2. E, B and k obey the right-hand rule of vector mathematics: E × B||k.
Figure 2.2 Plane wave propagating in z-direction. E and B are orthogonal to k, together they obey the right-hand rule.
2.3.1 Spherical Waves
(2.45)
(2.46)
When we consider just a narrow range around the z-axis as shown in Figure 2.3 (i.e., for x2 + y2z2), then one can use a Taylor series expansion
Figure 2.3 Spherical wave emerging from a point source at the origin. With increasing distance from the source, the wavefront flattens.
(2.47)
By neglecting all but the first two terms, one obtains the paraxial approximation of a spherical wave propagating in +z-direction, that is,
(2.48)
On the right side of (2.48), the first exponential term, which depends on the transverse coordinates x and y, describes the curvature of the wavefront. Note that a plus sign in front of the exponential term denotes a diverging wave, and a minus sign represents a converging wave. Obviously, the sign changes as the wave turns from a converging to a diverging wave by passing through the focus as shown in Figure 2.3. On the optical axis, the phase change is π.
2.3.2 Waves and Rays of Light
Figure 2.5 Spherical wave and local k-vectors.
(2.52)
2.4 The Time-Independent Wave Equation
Oftentimes, one is interested in situations where the time-dependency does not play a role. In what follows, we use a scalar representation of the wave field by using a single component U(r,t) of the electric field. U(r,t) is usually a complex quantity. The wave equation for U(r,t) reads as
(2.53)
We decompose U(r,t) in its spectral components by means of a Fourier transformation
(2.54)
By inserting this integral into the wave equation (2.53), one obtains
(2.55)
(2.56)
This is the time-independent wave equation, also known as the Helmholtz equation. It is the starting point for analyzing stationary problems and represents the cornerstone of Fourier optics. The usefulness of the Helmholtz equation lies in the virtue that one can analyze every monochromatic frequency component u(r,ω) individually for its spatial properties. When all components u(r,ω) are known, then U(r,t) follows from (2.54). For simplicity, we shall use u(r) instead of u(r,ω) in what follows – at least, as no explicit knowledge of the temporal frequency is necessary. u(r
