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Fundamentals of Modern Optics Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 1 Fundamentals of Modern Optics Winter Term 2017/2018 Prof. Thomas Pertsch Abbe School of Photonics, Friedrich-Schiller-Universität Jena Table of content 0. Introduction ............................................................................................... 4 1. Ray optics - geometrical optics (covered by lecture Introduction to Optical Modeling) .............................................................................................. 15 1.1 Introduction ......................................................................................................... 15 1.2 Postulates ........................................................................................................... 15 1.3 Simple rules for propagation of light ................................................................... 16 1.4 Simple optical components ................................................................................. 16 1.5 Ray tracing in inhomogeneous media (graded-index - GRIN optics) .................. 20 1.5.1 Ray equation ............................................................................................ 20 1.5.2 The eikonal equation ................................................................................ 22 1.6 Matrix optics ........................................................................................................ 23 1.6.1 The ray-transfer-matrix ............................................................................. 23 1.6.2 Matrices of optical elements ..................................................................... 23 1.6.3 Cascaded elements.................................................................................. 24 2. Optical fields in dispersive and isotropic media ....................................... 25 2.1 Maxwell’s equations ........................................................................................ 25 2.1.1 Adaption to optics ..................................................................................... 25 2.1.2 Temporal dependence of the fields .......................................................... 29 2.1.3 Maxwell’s equations in Fourier domain .................................................... 30 2.1.4 From Maxwell’s equations to the wave equation ...................................... 30 2.1.5 Decoupling of the vectorial wave equation ............................................... 32 2.2 Optical properties of matter ................................................................................. 33 2.2.1 Basics ....................................................................................................... 33 2.2.2 Dielectric polarization and susceptibility ................................................... 36 2.2.3 Conductive current and conductivity......................................................... 38 2.2.4 The generalized complex dielectric function ............................................. 39 2.2.5 Material models in time domain ................................................................ 43 2.3 The Poynting vector and energy balance ........................................................... 45 2.3.1 Time averaged Poynting vector ................................................................ 45 2.3.2 Time averaged energy balance ................................................................ 46 2.4 Normal modes in homogeneous isotropic media ................................................ 49 2.4.1 Transverse waves .................................................................................... 50 2.4.2 Longitudinal waves ................................................................................... 51 2.4.3 Plane wave solutions in different frequency regimes ............................... 52 2.4.4 Time averaged Poynting vector of plane waves ....................................... 58 2.5 The Kramers-Kronig relation ............................................................................... 59 2.6 Beams and pulses - analogy of diffraction and dispersion .................................. 62 2.7 Diffraction of monochromatic beams in homogeneous isotropic media .............. 64 2.7.1 Arbitrarily narrow beams (general case) ................................................... 65 2.7.2 Fresnel- (paraxial) approximation ............................................................. 72 2.7.3 The paraxial wave equation ..................................................................... 76 2.8 Propagation of Gaussian beams ......................................................................... 78 2.8.1 Propagation in paraxial approximation ..................................................... 79 2.8.2 Propagation of Gauss beams with q-parameter formalism ....................... 84 2.8.3 Gaussian optics ........................................................................................ 84 Script "Fundamentals of Modern Optics", FSU Jena, Prof. T. Pertsch, FoMO_Script_2017-10-10s.docx 2 2.8.4 Gaussian modes in a resonator ............................................................... 87 2.9 Dispersion of pulses in homogeneous isotropic media ....................................... 93 2.9.1 Pulses with finite transverse width (pulsed beams) .................................. 93 2.9.2 Infinite transverse extension - pulse propagation ................................... 100 2.9.3 Example 1: Gaussian pulse without chirp ............................................... 101 2.9.4 Example 2: Chirped Gaussian pulse ...................................................... 104 3. Diffraction theory .................................................................................. 108 3.1 Interaction with plane masks ............................................................................. 108 3.2 Propagation using different approximations ...................................................... 109 3.2.1 The general case - small aperture .......................................................... 109 3.2.2 Fresnel approximation (paraxial approximation) .................................... 109 3.2.3 Paraxial Fraunhofer approximation (far field approximation) .................. 110 3.2.4 Non-paraxial Fraunhofer approximation ................................................. 112 3.3 Fraunhofer diffraction at plane masks (paraxial) ............................................... 112 3.4 Remarks on Fresnel diffraction ......................................................................... 117 4. Fourier optics - optical filtering .............................................................. 119 4.1 Imaging of arbitrary optical field with thin lens .................................................. 119 4.1.1 Transfer function of a thin lens ............................................................... 119 4.1.2 Optical imaging using the 2f-setup ......................................................... 120 4.2 Optical filtering and image processing .............................................................. 122 4.2.1 The 4f-setup ........................................................................................... 122 4.2.2 Examples of aperture functions .............................................................. 125 4.2.3 Optical resolution ................................................................................... 126 5. The polarization of electromagnetic waves ........................................... 129 5.1 Introduction ....................................................................................................... 129 5.2 Polarization of normal modes in isotropic media ............................................... 129 5.3 Polarization states ............................................................................................ 130 6. Principles of optics in crystals ............................................................... 132 6.1 Susceptibility and dielectric tensor .................................................................... 132 6.2 The optical classification of crystals .................................................................. 134 6.3 The index ellipsoid ............................................................................................ 135 6.4 Normal modes in anisotropic media .................................................................. 136 6.4.1 Normal modes propagating in principal directions .................................. 137 6.4.2 Normal modes for arbitrary propagation direction .................................. 138 6.4.3 Normal surfaces of normal modes ......................................................... 143 6.4.4 Special case: uniaxial crystals ................................................................ 145 7. Optical fields in isotropic, dispersive and piecewise homogeneous media ........................................................................................................... 148 7.1 Basics ............................................................................................................... 148 7.1.1 Definition of the problem ........................................................................ 148 7.1.2 Decoupling of the vectorial wave equation ............................................. 149 7.1.3 Interfaces and symmetries ..................................................................... 150 7.1.4 Transition conditions .............................................................................
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