Wave Optics
The quest to understand the nature of light is centuries old and today there can be at least three answers to the single question of what light is depending on the experiment used to investigate lightʼs nature: (i) light consists of rays that propagate, e
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Wave Optics
3. Wave Optics
can be performed very easily if some approximations of paraxial optics are valid. The formulae for this are treated in the last section of this chapter.
3.1
3.2
3.3
3.4
3.5
Maxwell’s Equations and the Wave Equation......................... 3.1.1 The Maxwell Equations ................. 3.1.2 The Complex Representation of Time-Harmonic Waves .............. 3.1.3 Material Equations ....................... 3.1.4 The Wave Equations ..................... 3.1.5 The Helmholtz Equations............... Polarization ......................................... 3.2.1 Different States of Polarization ...... 3.2.2 The Poincaré Sphere ..................... 3.2.3 Complex Representation of a Polarized Wave ...................... 3.2.4 Simple Polarizing Optical Elements and the Jones Calculus..................
88 88 94 95 98 99 102 105 105 106 106
Interference ......................................... 3.3.1 Interference of Two Plane Waves .... 3.3.2 Interference Effects for Plane Waves with Different Polarization States ... 3.3.3 Interference of Arbitrary Scalar Waves................................ 3.3.4 Some Basic Ideas of Interferometry
108 108
Diffraction ........................................... 3.4.1 The Angular Spectrum of Plane Waves ............................ 3.4.2 The Equivalence of the Rayleigh–Sommerfeld Diffraction Formula and the Angular Spectrum of Plane Waves ............................ 3.4.3 The Fresnel and the Fraunhofer Diffraction Integral ....................... 3.4.4 Numerical Implementation of the Different Diffraction Methods......... 3.4.5 The Influence of Polarization Effects to the Intensity Distribution Near the Focus .............................
123
111 115 119
123
125 126 135
138
Gaussian Beams ................................... 143 3.5.1 Derivation of the Basic Equations ... 143 3.5.2 The Fresnel Diffraction Integral and the Paraxial Helmholtz Equation .... 145
Part A 3
The quest to understand the nature of light is centuries old and today there can be at least three answers to the single question of what light is depending on the experiment used to investigate light’s nature: (i) light consists of rays that propagate, e.g., rectilinear in homogeneous media, (ii) light is an electromagnetic wave, (iii) light consists of small portions of energy, or so-called photons. The first property will be treated in the chapter about geometrical optics, which 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, above all, contradictory to wave optics. Only the theory of quantum mechanics and quantum field theory can simultaneously explain light as photons and electromagnetic waves. The field of optics treating this subject is generally called quantum optics. In this chapter about wave optics the electromagnetic property of light is treated and the basic equations describing all relevant electromagnetic phenomena are Maxwell’s equations. Starting with
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