Practical Electrodynamics with Advanced Applications

PRELIMS.pdf
Preface
References
Acknowledgement
Author biography
Sergey Leble
Epigraph
CH001.pdf
Chapter 1 Introduction
1.1 General remarks: units
1.2 Inertial reference frames
1.2.1 Reference body: basic variables in mechanics
1.2.2 Observation of Galileo: relativity
1.2.3 Summarizing: Poincare group
1.3 Tensor fields
1.3.1 Definition of a tensor with respect to a group
1.3.2 Tensor field derivatives
1.3.3 Exercises
References
CH002.pdf
Chapter 2 Basic notions and equations of electrodynamics
2.1 Electrodynamics in vacuum
2.1.1 Definition of electric field vector E⃗
2.1.2 Lorentz force: definition of magnetic induction field B⃗
2.2 Maxwell’s equations in integral form
2.2.1 I Maxwell’s equation from Coulomb’s law
2.2.2 II Maxwell equation: absence of magnetic charges
2.2.3 Faraday’s law: III Maxwell equation
2.2.4 Ampère–Maxwell law: IV Maxwell equation
2.2.5 System of Maxwell’s equations in differential form
2.3 Initial-boundary problem for Maxwell system in vacuum
2.4 Vector and scalar potentials
2.4.1 What possibilities give us the gauge choice?—Theoretic investigation
2.4.2 Hertz vector
2.5 Conservation principles: Poynting theorem
References
CH003.pdf
Chapter 3 Electromagnetic waves in vacuum
3.1 Wave equations
3.2 Harmonic plane wave in vacuum without charges
3.3 Wave packets
3.4 Cauchy Problem in 1 + 1 space–time
3.4.1 The initial (Cauchy) problem for plane wave with a fixed polarization
3.4.2 On projection method
3.5 Discussion and exercises
3.6 Inhomogeneous wave equation: wave generation
3.6.1 On the Green function method for wave equations
3.6.2 General formulas for electromagnetic field and wave emission
3.7 Emission of the isolated charged point particle
3.8 Emission of oscillating charged system of particles: multipole expansion
3.8.1 General remarks
3.8.2 Multipole expansion
3.8.3 The emission phenomenon in the dipole approximation
3.8.4 Next approximation: quadrupole momentum
3.8.5 Dipole magnetic moment definition and corresponding wave emission
3.8.6 Minidiscussion and exercises
References
CH004.pdf
Chapter 4 Theory of relativity
4.1 Lorentz transformation
4.2 Space–time geometry
4.3 Relativistic kinematics and four-vectors
4.4 Relativistic mechanics
4.5 Discussion
4.6 Exercises
4.7 A historical note: about a birth of new mechanics (theory of relativity)
References
CH005.pdf
Chapter 5 Electromagnetic field in a matter
5.1 Definition of vectors: polarization, electric induction, magnetization and magnetic field strength—Maxwell’s equations for electromagnetic field in a matter
5.1.1 Implicit definition of the vectors: polarization, electric induction
5.1.2 On implicit definition of magnetization vector and magnetic field strength
5.1.3 Maxwell’s equations for electromagnetic field in matter
5.1.4 Explicit definition of vectors: polarization, electric induction
5.1.5 On explicit empiric definition of vectors: magnetic field strength H⃗ and magnetic induction B⃗
5.1.6 The fields measurements in experimental practice
5.1.7 Material equations as thermodynamic equations of state
5.2 Macroscopic Maxwell’s equations, links to microscopic parameters
5.2.1 Resume, the empiric definition of the fields: macroscopic Maxwell’s equations
5.2.2 A link between polarization vector and dipole momentum
5.2.3 A link between magnetization vector and magnetic dipole momentum
5.2.4 More comments on the definitions
5.2.5 Energy conservation/balance law derivation for matter
5.3 Classification of substances with respect to electric and magnetic properties
5.3.1 Classification of continuum matter on a base of material equations
5.3.2 Dielectrics
5.3.3 Magnetics
5.3.4 Ferromagnetism
5.3.5 Combined action: multiferroics
5.3.6 On dielectrics
5.3.7 Metamaterials
5.3.8 Exercises
References
CH006.pdf
Chapter 6 Dispersion and transport
6.1 Dispersion account, operator material relations
6.1.1 Maxwell’s equations: operators of dielectric permittivity and magnetic permeability
6.1.2 Energy density of wave packets in a dispersive medium
6.1.3 On Maxwell–Lorentz equations
6.2 Discussion
6.2.1 Polarization vector via a potential from coupled point charges
6.3 Dispersion in dielectrics, conductors and plasma
6.3.1 Lorentz model
6.3.2 Drude theory of metals: Ohm’s law
6.4 Back to Ohm’s law: Hall effect
6.4.1 On the DC Ohm law
6.4.2 Magnetic field account: Hall effect
6.4.3 Basic relations for 2D Hall resistance
6.4.4 On quantum electrodynamics manifestation of Hall effect
6.5 EM waves in isotropic conducting matter case
6.5.1 The linear equation and plane wave
6.5.2 The 1D nonlinear model outline
6.5.3 On dynamic projection method in linear problem
6.5.4 Dynamical projecting in a nonlinear problem
References
CH007.pdf
Chapter 7 Plasma
7.1 Plasma types
7.1.1 General remarks: a matter as potential plasma
7.1.2 Atmosphere plasma
7.1.3 Solid state physics: electron–hole plasma
7.1.4 On stability of plasma
7.1.5 Tokamak plasma
7.2 Propagation of waves in a plasma: example of helicoidal waves
7.2.1 General remarks on plasma theoretical description
7.2.2 Simplifications and the linearized system: plasma waves
7.2.3 The initial problem formulation in matrix form: Fourier transform
7.2.4 The initial problem solution by dynamical projecting
7.3 The nonlinear case
References
CH008.pdf
Chapter 8 Metamaterials
8.1 Research on metamaterials
8.1.1 Introduction: on the chapter content
8.2 Statement of problem: dispersion operator
8.2.1 Maxwell’s equations: operators of dielectric permittivity and magnetic permeability
8.2.2 Boundary regime problem
8.3 Projecting operators
8.3.1 Projecting operators approach
8.3.2 Projecting operators construction
8.4 Separated equations and definition for left and right waves
8.5 Nonlinearity account
8.6 Wave propagation in a metamaterial within the lossless Drude dispersion and Kerr nonlinearity
8.6.1 Drude model for dispersion
8.6.2 Interaction of left and right waves with Kerr effect
8.6.3 Stationary problem solutions
8.7 Discussion and conclusion
References
CH009.pdf
Chapter 9 Problems of electromagnetism in a piecewise continuous matter
9.1 Electro- and magneto-statics
9.2 Boundary conditions
9.2.1 Absence of surface charges and currents case
9.2.2 Conditions for magnetic and electric moments
9.2.3 Magnetic moment: spins contribution
9.2.4 Surface charges and currents
9.3 Demagnetization field
9.3.1 Instructive example
9.3.2 General demagnetization
9.4 Stray fields
9.4.1 On definition
9.4.2 Landau–Lifshits–Gilbert (LLG) equation and domain wall (DW) motion model
9.5 Microwire: DW and observations
9.5.1 Wire DW model
9.6 The stray field of the planar DW
9.6.1 Observation of domain walls in ferromagnetic microwires
References
CH010.pdf
Chapter 10 Reflection and refraction of electromagnetic waves at a boundary
10.1 Reflection and transmission of a plane wave on a border
10.1.1 General textbook relations
10.1.2 Ampère–Maxwell equation for a wave with given frequency
10.1.3 Reflection and transmission of wave propagating orthogonally to a boundary
10.2 Problem of a plane wave with fixed frequency refraction
10.3 Boundary conditions impact
10.3.1 Snell law
10.3.2 To Fresnel formulas generalization
10.4 Energy density flux
10.4.1 Preparation: Pointing vector for a matter
10.4.2 Polarization choice
10.4.3 The generalized Snell law
10.5 Discussion
References
CH011.pdf
Chapter 11 New dielectric guides techniques
11.1 Planar waveguides
11.1.1 Novel experiments in dielectric guides
11.1.2 Dielectric slab as a waveguide
11.2 Cylindrical dielectric waveguides
11.2.1 On the problem
11.2.2 Linear problem
11.2.3 Transformation to frequency domain
11.2.4 Projection operators in time domain
11.3 Including nonlinearity
11.3.1 Application of projection operators
References
CH012.pdf
Chapter 12 Propagation of electromagnetic waves in exclusive dispersive media such as metamaterials
12.1 Electromagetic waves in metamaterial
12.1.1 On dispersion in 1D metamaterial
12.2 Directed modes in rectangular waveguides: polarization, dispersion, nonlinearity
12.2.1 Maxwell’s equations for matter inside a waveguide
12.3 Boundary conditions: the transversal waveguide modes evolution
12.3.1 The boundary regime problem formulation for the transversal modes
12.3.2 Projecting operators
12.4 Rectangular waveguide filled with metamaterial: nonlinearity account
References
CH013.pdf
Chapter 13 Plasma basic equations, waveguide formation
13.1 Maxwell-kinetic system
13.1.1 Joint electrodynamics—particles kinetics description
13.1.2 Kinetic equation: Vlasov plasma
13.2 Waves in homogeneous plasma
13.2.1 Cold plasma: general dispersion equation
13.2.2 Maxwell distribution background: Langmuir waves
13.2.3 More roots of the dispersion equation
13.3 Weakly inhomogeneous plasma
13.4 Plasma waveguides
13.4.1 On plasma confinement
13.4.2 The confined plasma perturbations
13.4.3 Hydrodynamic equations approach: flute instability
References
CH014.pdf
Chapter 14 Helicoidal and other plasma wave phenomena
14.1 Helicoidal waves interactions
14.1.1 Basic equations
14.1.2 Introducing the projectors P+ and P−
14.1.3 Model with nonlinear term: the three-waves equation
14.1.4 The three-wave system in 1 + 1 case
14.2 Algebraic method of three-wave systems solution: solitons
14.2.1 Solutions derived using dressing by two-fold Darboux transformation (TfDT)
14.2.2 Solutions plots and discussion
14.3 Interaction of plasma waves
14.3.1 Interaction of Langmuir and ion waves
References
CH015.pdf
Chapter 15 Diffraction in the presence of conductivity, x-rays manipulation and focusing
15.1 General remarks
15.2 Basic equations
15.3 Propagation of x-rays in vacuum
15.4 Approximation of electromagnetic field as a superposition of Gaussian beams
15.4.1 Paraxial equation for Kshevetskii–Wojda beam
15.4.2 Superposition of Gaussian beams
15.4.3 Quasi-exact solution of the Helmholtz equation
15.5 Oriented Gaussian beams method application to x-rays propagation through optical elements
15.6 Study of accuracy and efficiency of Gaussian beam methods
15.6.1 Estimation of convergence rate of solution obtained with superposition of oriented Gaussian beams to electric field described by boundary condition, which is a fast oscillating function
15.6.2 Propagation of x-rays through a lens and its aperture boundaries
15.7 Numerical calculations scheme
15.7.1 Implicit Runge–Kutta scheme
15.7.2 Algorithm: parameters of integration choice
15.8 The numerical simulations
15.8.1 General description
15.8.2 The first case, 33 aluminium lenses with 15 keV x-ray
15.8.3 The choice of space steps and errors
15.9 Results for ideal lenses and the bulk defects influence
15.9.1 Space steps choice and plots
15.9.2 Final remarks
References
CH016.pdf
Chapter 16 Magnetic field dynamics, novel aspects of a theory based on Landau–Lifshitz–Gilbert equations
16.1 An exchange interaction concept
16.2 Heisenberg network dynamics
16.2.1 Heisenberg model: anisotropy
16.2.2 General continuum LLG equations
16.3 Walker theory
16.3.1 Walker solution of 1D LLG
16.3.2 Walker solution instability
16.3.3 Nanowires as Heisenberg chain
16.4 Propagation of domain wall in cylindrical amorphous ferromagnetic microwire
16.4.1 Introductory remarks
16.4.2 The LLG equation for cylindric microwires
16.4.3 LLG transforms: statement of problem
16.4.4 Basic equation in quadratic-linear approximation and its general solution
16.5 Average magnetization fields and DW dynamics
16.5.1 The third order nonlinearity account
16.5.2 Stationary background introduction
16.5.3 A linearization on a nonzero background
16.5.4 Averaging procedure and DW mobility
16.5.5 Velocity and acceleration
16.6 Exact particular solutions of LLG equation
16.6.1 Anisotropy coefficient coordinate dependence impact
16.6.2 The illustrations of DW form in 3D
16.6.3 Velocity of DW propagation: anisotropy constants determination
16.6.4 The field strength and induced magnetization by a coil
References
CH017.pdf
Chapter 17 Condensed matter electrodynamics: equations of state by partition function
17.1 On derivation ab initio of an equation of state
17.1.1 The first law of thermodynamics forms by classical statistical physics
17.1.2 The first law of thermodynamics by quantum statistical physics
17.1.3 The scheme for two subsystems
17.2 Spin system and equations of state
17.2.1 Classical Langevin theory
17.2.2 Brillouin theory: space quantization
17.3 Heisenberg theory
17.3.1 Partition function
17.3.2 On Heisenberg equation solution
17.4 Para-, and ferro-magnetic matter
17.4.1 The magnetization curve for a paramagnetic
17.5 Problem of ferromagnetic state
17.5.1 Towards the Curie law
17.6 Multiferroics
17.6.1 Electric field action: Stark effect
17.6.2 Exchange integrals
17.6.3 Magneto-electric effect: material equation of state
17.6.4 On ferroelectricity
17.7 Fine particles case
17.7.1 Energy distribution
17.7.2 Back to statistical sum
17.7.3 Partition functions for a tiny particle
17.7.4 Numerical estimations and plot
References
CH018.pdf
Chapter 18 More general material relations
18.1 A concept
18.1.1 E–D–B–H relations
18.1.2 Hydrodynamics–electrodynamics material relations
18.1.3 Continuum medium-electrodynamics material relations
18.1.4 Energy balance
18.2 Symmetry and groups
18.2.1 Crystallographic symmetry and groups
18.2.2 Tensorial symmetry with respect to indexes transpositions
18.2.3 Pauli symmetry with respect to electrons permutations
18.3 Euclidean and Lorentz symmetry
18.3.1 Euclidean group covariance
18.3.2 Lorentz group covariance
18.3.3 General tensors relations
18.4 Active dielectrics
18.4.1 Ferroelectrics
18.4.2 Piezoelectricity
18.4.3 Paraelectricity
18.4.4 Magnetoelasticity
18.5 Flexoelectricity
18.6 Ferroelasticity
References
CH019.pdf
Chapter 19 On direct and inverse problems of electrodynamics
19.1 Direct problem of plane wave propagation in a layered medium
19.2 On inverse problem
19.2.1 Remarks
19.2.2 Some details of problem formulation
19.2.3 Equations of the inverse problem
19.3 Data collection methods: examples
19.3.1 Plasma Langmuir probe
19.3.2 Radar
19.3.3 Huygens’ and Kirchhoff’ formulas
19.3.4 Direct and inverse problems for a radar/lidar
19.4 Inverse problems as ill-posed one
19.4.1 The Tikhonov regularization
References
CH020.pdf
Chapter 20 Advanced exercises
20.1 Short list of useful vector and tensor relations
20.2 A few definitions: curves, surfaces, integrals, etc
20.2.1 Curves
20.2.2 Surfaces
20.2.3 Integrals
20.2.4 Dirac delta-function
20.3 Projecting operators
20.4 Dressing method
20.5 Dielectric waveguides
20.5.1 Dielectric slab
20.5.2 Dielectric cylinder—optical fibers
20.5.3 Rectangular waveguide
20.6 Electromagnetic waves in metamaterials
20.6.1 Electromagnetic waves in metal rectangular waveguide, system derivation
20.7 Plasma confinement
20.8 Wave propagation at plasma
20.9 Refraction in presence of conductivity
20.10 Magnetism, a novel aspect
20.11 Condensed matter electrodynamics: equations of state by partition function
20.11.1 Paramagnetics and ferromagnetics
20.11.2 Multiferroics
20.12 General material relations
20.12.1 Piezoelectricity
20.13 Inverse problems of electrodynamics
References