Radiation in Materials

The interaction of electromagnetic radiation and matter is examined, specifically electric and magnetic fields in materials with real and imaginary responses: under certain conditions the fields move through the material as a wave and under others they diffuse. The movement of a pulse of radiation in dispersive materials is described in which there are two wave velocities: group versus phase. The reflection of light from a sharp interface is analyzed and the Fresnel reflection/transmission equations derived. The response of materials to applied electric and magnetic fields in the time domain are correlated to their frequency response of the material’s polarization. The generalized Kramers–Kronig equations are derived and their applicability as a fundamental relationship between the real and imaginary parts of any material’s polarizability is discussed in detail. Finally, practical measurement techniques for extracting the real and imaginary components of a material’s index of refraction are introduced.

General Relations between Fields and Sources

The general relationship of changes in source current, charge and/or position and the fields that they produce are examined in the context of the development of equations that are known as “Jefimenko’s Equations.” These expressions give the fields at a point removed from the source in terms of the charge and current distributions evaluated at the “retarded time.” In this development, the finite speed of light is shown to connect the time rate of change in source conditions to the spatial variations of the potential at the field point. Using a graphical argument, the transverse nature of radiation fields is demonstrated based on electric field lines as envisioned by Faraday.

Radiation Fields in Constrained Environments

The vacuum field energies and their connection to the force between two plates are presented while deriving the Casimir force. Thermal radiation and the Planck spectrum are introduced and the Einstein A and B coefficients and their properties under thermodynamic equilibrium examined. Microwave cavities and wave guides are considered and their general properties derived with special consideration given to conducting waveguides and transmission lines. Optical waveguides are analyzed using the ray optic picture. Topics include planar waveguides, variable index circular waveguides, single mode fibers, and dispersion. Finally, photonic crystals are discussed and photonic band structures are shown to be analogous to electronic bands.

Relativistic Electrodynamics

The concept of action is introduced using Lagrangian and Hamiltonian mechanics, and is used to describe the relativistic mechanics of a free particle: free particle canonical 4-momentum and angular momentum 4-tensor. The problem of a charged particle in an external field is considered in detail, resulting in the relativistic version of the Lorentz force law. The electromagnetic field is described using the action principle: The Lagrange density function and the recovery of Maxwell’s equations and charge conservation. The simplest Lagrangian density that can be constructed from a four-vector field is known as the “proca Lagrangian,” but it is shown to predict a massive photon. Finally, the canonical stress-energy tensor is derived along with conservation laws.

Fields in Terms of the Multipole Moments of the Source

Many if not most evaluations of electric and magnetic fields arising from source configurations are performed using multipole moments of the source distribution. As an example, Jefimenko’s equations are evaluated in the radiation zone by expansion of the charge and current distributions in their lowest order moments. The important moments, electric and magnetic dipole, and the electric quadrupole are examined in detail. The electric and magnetic fields for all regions of space relative to the source location (near, intermediate, and radiation) are evaluated using the multipole expansion of the vector potential in terms of these moments. Finally, the power radiated by the multipole moments of the source is presented.

The Potentials

The concepts of scalar and vector potentials are introduced and the electric and magnetic fields are shown to be derived from specific forms of these potentials. The choice of these forms is restricted by gauge considerations, and the Lorenz gauge is introduced as the one most applicable for radiation. Using this, the wave equations prescribing the potentials in terms of the source conditions are presented. The modifications of vector and scalar potentials to account for speed of light and causality lead to the concept of “retarded time.” The potentials can be expressed in terms of moments of the source along with concepts of “near,” “intermediate,” and “far” zones to facilitate derivation of approximate expressions for the potentials evaluated at appropriate distances from the source. Finally, expressions for the vector potential in terms of the electric and magnetic dipole, and electric quadrupole moments of the source in the approximation zones are presented.

Diffraction and the Propagation of Light

Geometric optics is considered and the eikonal equation is introduced. Krirchoff’s diffraction theory is presented with his integral theorem. Rayleigh–Sommerfeld diffraction is discussed and Fresnel’s approximation for the Kirchoff integrals and Babinet’s principle are given. Fraunhoffer diffraction is considered in detail, specifically diffraction by a rectangular and circular aperture. Special emphasis is given to the angular spectrum representation and its applications, including Gaussian beams, Fourier optics, and tight focusing of fields. Finally, the fields and modes of a tightly focused Gaussian beam are considered and the diffraction limits on microscopy are given.

Scattering of Electromagnetic Radiation in Materials

The formulation of generalize electromagnetic scattering is given. Previously derived multipole expansions using the language of scattering are presented and applied to resonant and plasmon resonances. Formal scattering theory is introduced, and the integral scattering equation is derived and used to find the Born expansion and to prove the optical theorem. Partial wave analysis for the scaler scattering problem is discussed with connections between quantum (wave theory) and classical views. Vector spherical harmonics and the extension of partial wave analysis to the scattering of vector fields of electromagnetic waves are presented. Finally, Mie scattering is considered in detail with applications including glory scattering and whisper gallery mode resonances.

Field Reactions to Moving Charges

The concept of self-force on a moving charge is introduced along with the idea of electromagnetic mass. Lorentz’s calculation of the self-force is presented that yields an analytical form of the self-force on an accelerating sphere of charge. The equations of motion for a charge including self-force are derived, known as the Abraham–Lorentz formula. This formula is applied to sinusoidal motion and constant acceleration and its general solution is shown to result in runaway and pre-acceleration. The Landau–Lifshitz approximation alleviates these issues and the self-force is shown to act only over an experimentally extremely short characteristic time. Additionally, issues include the problem that the dynamically calculated electromagnetic mass is 4/3 the statically calculated one; the introduction of Poincare stresses reconciles this difference. Finally, the infinite electromagnetic mass problem of a point-like particle is shown to be manageable by considering advanced as well as retarded waves in the potentials.

Radiation from Charges Moving at Relativistic Velocities

The evaluation of the electric scalar and magnetic vector potentials for a source moving relative to the observer is considered. With retarded time taken into account, the potentials have a form known as the Lienard–Wiechert potentials. Using these and accounting for retarded time in space and time derivatives, electric and magnetic fields due to a charge undergoing acceleration while moving at an arbitrary velocity are derived. The power and its spectral content from radiation of an accelerated charge for accelerations parallel and perpendicular to its velocity are given, with application to synchrotron radiation. The special case of a charge moving with constant velocity is examined in detail. The spectral density of the fields of a charge moving at constant velocity is derived with an approximate calculation of the equivalent number of virtual photons. Finally, Bremsstrahlung radiation of an electron colliding with an ion is calculated using virtual photons and the Weizacker and Williams method.