Wave Optics: Lecture about the principles of wave optics

Wave Optics: Lecture about the principles of wave optics

Introduction

The quest for the nature of light is centuries old and today there can be at least three answers to
the question what light is depending on the experiment which is used to investigate the nature of
light: (i) light consists of rays which propagate e.g. rectilinear in homogeneous media, (ii) light
is an electromagnetic wave, (iii) light consists of small portions of energy, the so called photons.
The first property will be treated in the lecture about Geometrical Optics and geometrical
optics can be interpreted as a special case of wave optics for very small wavelengths. On the
other hand the interpretation as photons is unexplainable with wave optics and first of all also
contradicting to wave optics. Only the theory of quantum mechanics and quantum field theory
can explain light as photons and simultaneously as an electromagnetic wave. The field of optics
which treats this subject is generally called Quantum Optics and is also one of the lecture
courses in optics.
In this lecture about Wave Optics the electromagnetic property of light is treated and the
basic equations which describe all electromagnetic phenomena which are relevant for us are
Maxwell’s equations. Starting with the Maxwell equations the wave equation and the Helmholtz
equation will be derived. Here, we will try to make a trade–off between theoretical exactness
and a practical approach. For an exact analysis see e.g. [1]. After this, some basic properties
of light waves like polarization, interference, and diffraction will be described. Especially, the
propagation of coherent scalar waves is quite important in optics. Therefore, the chapter about
diffraction will treat several propagation methods like the method of the angular spectrum
of plane waves, which can be easily implemented in a computer, or the well–known diffraction
integrals of Fresnel–Kirchhoff, Fresnel and Fraunhofer. In modern physics and engineering lasers
are very important and therefore the propagation of a coherent laser beam is of special interest.
A good approximation for a laser beam is a Hermite–Gaussian mode and the propagation of a
fundamental Gaussian beam can be performed very easily if some approximations of paraxial
optics are valid. The formula for this are treated in one of the last chapters of this lecture script.
It is tried to find a tradeoff between theoretical and applied optics. Therefore, practically
important subjects of wave optics like interferometry, optical image processing and filtering
(Fourier optics), and holography will also be treated in this lecture.

Contents

1 Maxwell’s equations and the wave equation 1
1.1 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Energy conservation in electrodynamics . . . . . . . . . . . . . . . . . . . 3
1.1.3 Energy conservation in the special case of isotropic dielectric materials . . 3
1.1.4 The wave equation in homogeneous dielectrics . . . . . . . . . . . . . . . . 5
1.1.5 Plane waves in homogeneous dielectrics . . . . . . . . . . . . . . . . . . . 6
1.1.6 The orthogonality condition for plane waves in homogeneous dielectrics . 7
1.1.7 The Poynting vector of a plane wave . . . . . . . . . . . . . . . . . . . . . 8
1.1.8 A time–harmonic plane wave . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 The complex representation of time–harmonic waves . . . . . . . . . . . . . . . . 11
1.2.1 Time–averaged Poynting vector for general time–harmonic waves with
complex representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Material equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Discussion of the general material equations . . . . . . . . . . . . . . . . . 16
1.3.1.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1.2 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Specialization to the equations of linear and non–magnetic materials . . . 17
1.3.3 Material equations for linear and isotropic materials . . . . . . . . . . . . 18
1.4 The wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.1 Wave equations for pure dielectrics . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Wave equations for homogeneous materials . . . . . . . . . . . . . . . . . 21
1.5 The Helmholtz equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.1 Helmholtz equations for pure dielectrics . . . . . . . . . . . . . . . . . . . 22
1.5.2 Helmholtz equations for homogeneous materials . . . . . . . . . . . . . . . 22
1.5.3 A simple solution of the Helmholtz equation in a homogeneous material . 24
1.5.4 Inhomogeneous plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Polarization 26
2.1 Different states of polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.1 Linear polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.2 Circular polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.3 Elliptic polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 The Poincar´e sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 The helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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