Cover
Contents
About the editors
Introduction
References
1 New insights in integral representation theory for the solution of complex canonical diffraction problems
1.1 Representations of spectral function in frequency and time domain, for the scattering by a polygonal region
1.1.1 Basic elements in Sommerfeld–Maliuzhinets representation and properties
1.1.1.1 Basic integral representation
1.1.1.2 Basic properties of the total field and its spectral function
1.1.2 Spectral functions f± attached to the radiation of a single face and simple relation to f
1.1.2.1 Radiation of a single face of a wedge-shaped region
1.1.2.2 Spectral function attached to the Sommerfeld–Maliuzhinets representation of H(2)0 (kR)
1.1.2.3 Simple exact expression of single-face spectral function
1.1.2.4 Exact expression of f from diffraction coefficient F and consequences
1.1.3 Spectral function f from far field radiation of one face with arbitrary shape
1.1.3.1 Simple exact expression of the spectral function f derived from fields on a single planar face and properties
1.1.3.2 Deformation and simple exact expression of the spectral function f from fields on a piecewise smooth single face
1.1.4 Exact causal time domain representation of a field above a dispersive wedge-shaped region
1.1.4.1 Causality of Fc(α + ϕ, τ )
1.1.4.2 Spectral causal expression of the field in time domain
1.2 Several orders asymptotic representation for scattering by a curved impedance wedge
1.2.1 Asymptotic representation in a region without creeping waves
1.2.1.1 Maliuzhinets type representation and asymptotic boundary conditions on curved faces
1.2.1.2 The first term of f influenced by the curvatures
1.2.2 Several orders asymptotic expressions in a region with creeping waves terms
1.2.2.1 Plane wave illumination and observation at infinity
1.2.2.2 Plane wave illumination and observation at finite distance
1.2.2.3 Point source illumination and observation at finite distance
1.2.3 Some validations concerning the expression of f = ∑ n≥0fn/kn for arbitrary wedge angle
1.2.3.1 Perfectly conducting case: curved half plane to discontinuity of curvature
1.2.3.2 Discontinuity of curvature in an impedance surface
1.3 A novel expression of the field for arbitrary bounded sources above a passive or active impedance plane
1.3.1 Formulation of the problem
1.3.2 An expression of potentials (E inc,Hinc) for bounded
sources J and M
1.3.3 Expression of the potentials (E s,H s) for an impedance plane
1.4 Spectral representation of the field for 3D conical scatterers
1.4.1 Formulation
1.4.2 Expression of the potentials with Kontorovich–Lebedev integrals
1.4.2.1 Integral and nonintegral terms
1.4.2.2 The equality we,h(ω, ω0) = ge,h(ω, ω0,1/2), and the compatibility conditions on ge,h(ω, ω0,1/2)
1.4.3 Potentials and properties for an incident plane wave
1.4.3.1 An efficient expression
1.4.3.2 Complex properties of gie,h
References
2 Scattering of electromagnetic surface waves on imperfectly conducting canonical bodies
2.1 Introduction and survey of some known results
2.1.1 Electromagnetic surface waves on impedance surfaces
2.1.1.1 Electromagnetic surface waves supported by planar impedance surfaces
2.1.1.2 Electromagnetic surface waves on a curved surface with varying surface impedance in an inhomogeneous medium
2.1.2 Electromagnetic surface waves on a right circular conical surface
2.2 Excitation of an electromagnetic surface wave by a dipole located near a plane impedance surface
2.3 Scattering of a skew incident surface wave by the edge on an impedance wedge
2.3.1 Integral equations for the spectra
2.3.2 Far-field expansion
2.3.3 Reflection and refraction of an incident surface wave at the edge of an impedance wedge
2.3.4 Beyond the critical angle of edge diffraction
2.4 Conclusion
Appendix A
A.1 Brewster angles
Acknowledgement
References
3 Dielectric-wedge Fourier series
3.1 Introduction
3.2 The diffraction problem
3.2.1 The Hilbert-space problem
3.2.2 The singular-field problem
3.3 Integral equations
3.4 The solution of the integral equation
3.5 The Bessel–Hankel Fourier series
3.6 Incident plane waves
3.7 Numerical results
3.8 Summary
Appendix A
Acknowledgment
References
4 Green’s theorem, Green’s functions and Huygens’ principle in discrete electromagnetics
4.1 Introduction
4.2 Green’s theorem for adjacency matrices
4.2.1 Adjacency matrix
4.2.2 Weighted adjacency matrix
4.2.3 Matrix of adjacency 1
4.2.4 The vertex Laplacian matrix
4.2.5 Incidence matrices
4.3 Green’s theorem on topological vector space
4.4 Difference forms and discrete exterior calculus
4.4.1 Simplicial decomposition
4.4.2 Dual forms
4.4.3 Hypercube decomposition
4.4.4 Contextual algebraic notation of forms
4.4.5 Manifolds, graphs and lattices
4.4.6 Essentials of cell decomposition
4.5 Higher order Green’s theorem and Green’s functions
4.5.1 Green’s theorem for r-forms
4.5.2 Kirchhoff’s theorem for r-forms
4.5.3 Green’s function for r-forms
4.6 Dynamical systems on topological vector spaces: Maxwell’s equations
4.6.1 Discrete Maxwell equations
4.6.2 Electromagnetic fields as differential forms
4.7 Time-domain Green’s functions for dynamical systems
4.8 Discrete time
4.9 Discrete Green’s theorem and Green’s functions in computational field theory
4.9.1 Exterior–interior connection
4.9.2 Diakoptics
4.10 Conclusion
References
5 The concept of generalized functions and universal properties of the Green’s functions associated with the wave equation in boun
5.1 A short historical background
5.1.1 A remark
5.2 Some basic properties of the δ distribution
5.2.1 Distributions involving δ(R) and δ(R1 − R2)
5.3 Green’s functions associated with the wave equation in bounded partially homogeneous domains
5.3.1 The outgoing Green’s function
5.3.2 The ingoing Green’s function
5.3.3 Some universal properties of the outgoing and ingoing Green’s functions
5.4 Proofs of the theorems
5.4.1 Proof of Theorem 1
5.4.2 Proof of Theorem 2
5.4.3 Proof of Theorem 3
5.4.4 Proof of Theorem 4
5.4.5 Proof of Theorem 5
5.4.6 Proof of Theorem 6
5.5 Application:An inverse initial-value problem connected with the photoacoustic tomography in bounded non-homogeneous domains
5.5.1 Extension of the inverse initial value problem to the range (−∞) < t < ∞
5.5.2 Solution of the extended problem
5.5.3 Proof of (5.64)
References
6 Elliptic cylinder with a strongly elongated cross-section: high-frequency techniques and function theoretic methods
6.1 Introduction
6.2 Asymptotic currents on an elliptic cylinder with a truncated strongly elongated cross-section
6.2.1 Analysis of the interactions
6.2.2 Asymptotic field in the boundary layer due to a magnetic line current
6.2.2.1 The incident field
6.2.2.2 Representation of the incident field in terms of Whittaker functions
6.2.3 Radiated field and total field
6.2.4 Spectral decomposition of the field in the boundary layer
6.2.5 Diffraction by the edge of a truncated elliptic cylinder
6.2.6 Asymptotic field in the boundary layer due to an incident plane wave
6.2.6.1 The incident field
6.2.6.2 Representation of the incident field in terms of Whittaker functions
6.2.6.3 Total field in the boundary layer and spectral decomposition
6.2.6.4 Diffraction of a plane wave with a small incident angle by the edge of a truncated elliptic cylinder
6.2.7 Asymptotic currents
6.3 Asymptotic currents on a cylinder with an ogival cross-section composed of two symmetric arcs of a strongly elongated ellips
6.3.1 Presentation of the geometry and analysis of the problem
6.3.2 Asymptotic currents outside grazing incidence
6.3.2.1 Illuminated edge
6.3.2.2 Edge excited currents on shadowed edge
6.3.3 Grazing incidence
6.3.3.1 Fringe current contribution to the diffracted field
6.3.3.2 Radiation of the PO current
6.3.3.3 Asymptotic current for grazing incidence
6.4 Conclusion
References
7 High-frequency hybrid ray–mode techniques
7.1 Introduction
7.2 Ray–mode conversion technique
7.3 Modal excitation at the aperture
7.3.1 Formulation
7.3.2 Numerical results
7.4 Diffraction by a slit on a thick conducting screen
7.4.1 Background
7.4.2 Formulation
7.4.2.1 Edge-diffracted rays
7.4.2.2 Modal excitation
7.4.2.3 Modal reradiation and reflection coupling
7.4.2.4 Total diffracted field
7.4.3 Diffraction by a thin slit
7.4.4 Diffraction by a thick and loaded slit
7.4.5 Diffraction by a trough
7.5 Conclusions
Acknowledgments
References
8 Scattering and diffraction of scalar and electromagnetic waves using spherical-multipole analysis and uniform complex-source bea
8.1 Introduction
8.2 Solution of Maxwell’s equations in sphero-conal coordinates
8.2.1 Sphero-conal coordinates
8.2.2 Solution of the Helmholtz equation in sphero-conal coordinates
8.2.3 Vector spherical-multipole expansion of the electromagnetic field in the presence of a PEC semi-infinite elliptic cone
8.3 Complex-source beams
8.3.1 Converging and diverging CSB
8.3.2 Uniform CSB
8.4 Green’s function of the semi-infinite elliptic cone for an incident uniform complex-source beam
8.4.1 Scalar Green’s function
8.4.2 Dyadic Green’s function
8.5 Numerical evaluation
8.5.1 Convergence analysis
8.5.2 Numerical results for an acoustically soft or hard semi-infinite elliptic cone
8.5.3 Numerical results for a perfectly conducting semi-infinite elliptic cone
8.6 Conclusions
References
9 Changes in the far-field pattern induced by rounding the corners of a scatterer: dependence upon curvature
9.1 Problem formulation
9.2 Numerical results and discussion
9.3 Analytic bounds for the far-field difference
9.3.1 Integral equations for the difference in surface quantities
9.3.2 Approximate integral equation for the difference
9.3.3 The far-field difference
9.4 Conclusion
References
10 Radiation from a line source at the vertex of a right-angled dielectric wedge
10.1 Introduction
10.2 Formulation of the boundary value problem
10.3 Singular integral equation for the double Laplace transform of the electric field
10.4 Approximate solution of the singular integral equation
10.5 Calculation of E(1)(0,0)
10.6 Radiated far field
10.7 Conclusions
References
11 Wiener–Hopf analysis of the diffraction by a thin material strip
11.1 Introduction
11.2 The case of E polarization
11.2.1 Formulation of the problem
11.2.2 Factorization of the Kernel functions
11.2.3 Formal solution of theWiener–Hopf equation
11.2.4 Asymptotic solution of a certain integral equation in the complex plane
11.2.5 High-frequency asymptotic solution
11.2.6 Scattered far field
11.3 The case of H polarization
11.4 Numerical results and discussion
11.5 Conclusions
Acknowledgment
References
12 TheWiener–Hopf Fredholm factorization technique to solve scattering problems in coupled planar and angular regions
12.1 Introduction
12.2 TheWH equations of the problem
12.3 Reduction of theWH equations to FIEs
12.3.1 The Fredholm equation of the region (c)
12.3.2 The Fredholm equations of the region (b)
12.3.3 The Fredholm equation of the angular region (a)
12.4 Solution of the FIE
12.5 Analytical continuation of the numerical solution
12.6 A novel test case
12.7 Conclusion
Appendix A
References
13 On the analytical regularization method in scattering and diffraction
13.1 Introduction
13.2 Instability in the numerical solution of infinite algebraic systems
13.3 The ARM: when is it necessary?
13.4 Potentials and their pseudodifferential representations
13.5 Solution of the key diffraction problems
13.5.1 Dirichlet BVP
13.5.2 Neumann BVP
13.6 Diffraction by a semi-transparent obstacle
13.6.1 The BVP description
13.6.2 Integral representation for us(+) and
13.6.3 Reduction of the BVP to a system of integral equations
13.6.4 Reduction of the system of integral equations to an infinite system of linear algebraic equations
13.7 Diffraction of waves with complex frequencies and spectral theory of open cavities
13.7.1 Description of the BVP
13.7.2 Dirichlet BVP for complex-valued wave numbers
13.7.3 Qualitative features of the Dirichlet BVP
13.7.4 Numerical calculation of complex-valued eigen-wavenumbers and eigenmodes
13.8 ARM: considerations for implementation
13.9 ARM: various applications and conclusion
References
14 Resonance scattering of E-polarized plane waves by two-dimensional arbitrary open cavities: spectrum of complex eigenvalues
14.1 Introduction
14.1.1 Preliminary remarks
14.1.2 Development of a systematic approach
14.2 Mathematical background
14.2.1 Schematic description of the MAR
14.2.2 Scheme for finding the complex eigenvalues
14.3 Computation of the complex eigenvalues for various open cavities
14.3.1 Circular cylinder with longitudinal slit
14.3.2 Elliptic cavity with moveable longitudinal slit
14.3.3 Open rectangular cavity with finite flanges
14.4 Resonance response of slotted cavities
14.4.1 Surface current calculations
14.4.2 Far-field calculations
14.5 Conclusion
References
15 Numerical solutions of integral equations for electromagnetics
15.1 The EFIE and MFIE for perfectly conducting bodies
15.2 Some alternative formulations to remediate fictitious internal resonances
15.3 Integral equations for homogeneous dielectric bodies
15.4 Formulations that remediate fictitious internal resonances for dielectric targets
15.5 Single-source integral equations for dielectric bodies
15.6 Low-frequency breakdown of integral equations
15.7 Numerical solution of integral equations
15.8 Vector basis functions
15.9 Interpolatory and hierarchical vector basis functions
15.10 Singular vector basis functions
15.11 Summary
References
16 Electromagnetic modelling at arbitrarily low frequency via the quasi-Helmholtz projectors
16.1 Introduction
16.2 Notation and background
16.2.1 Frequency domain
16.2.1.1 Definition of the EFIE
16.2.1.2 Definition of the RWG elements
16.2.1.3 Definition of the LS matrices
16.2.1.4 Discretization strategy for the EFIE
16.2.2 Time domain
16.2.2.1 Definition of the TD-EFIE equation
16.2.2.2 Spatial discretization of the TD-EFIE
16.2.2.3 Temporal discretization of the TD-EFIE
16.3 The low-frequency breakdown in the FD
16.3.1 Illustration of the problem
16.3.2 Analysis of the low-frequency breakdown
16.3.2.1 Numerical instability
16.3.2.2 Low-frequency ill conditioning
16.3.3 Traditional LS decomposition
16.4 The large time step breakdown in the TD
16.5 DC instabilities
16.6 The qH projectors
16.7 An effective solution to the low-frequency breakdown for the EFIE
16.7.1 Leveraging the qH projectors
16.7.2 Implementation details
16.8 Solution to the large time step breakdown and the DC instability for the TD-EFIE
16.8.1 Preconditioning
16.8.2 Time discretization
16.8.3 Numerical results
16.9 Conclusions
References
17 Resistive and thin dielectric disk antennas with axially symmetric excitation analyzed using the method of analytical regulariza
17.1 Introduction
17.2 Formulation and GBC
17.3 Singular IEs and solution by MAR
17.3.1 Hyper-singular IE for a VED-excited resistive disk in free space
17.3.2 Eigenfunctions of the IE operator static limit for VED-excited PEC and resistive disks
17.3.3 Matrix equation and DIE for a VED-excited disk
17.3.4 Log-singular IE for a VMD-excited resistive disk in free space
17.4 Resistive disk MSA excited byVED
17.4.1 Dual IEs for a resistive disk MSA
17.4.2 Matrix equation for a resistive disk MSA
17.5 Thin disk DA excited byVED
17.5.1 Coupled set of DIEs for a thin disk DA
17.5.2 Matrix equation for a thin disk DA
17.6 Numerical results
17.6.1 Radiation characteristics of resistive MSA
17.6.2 Radiation characteristics of thin disk DA
17.7 Conclusions
References
18 Scattering and guiding problems of electromagnetic waves in inhomogeneous media by improved Fourier series expansion method
18.1 Introduction
18.1.1 Formulation
18.1.1.1 Scattering problem
18.1.1.2 Guiding problem
18.1.2 Numerical results
18.1.2.1 Characteristic of incident angle
18.1.2.2 Characteristic of normalized frequency
18.1.2.3 Propagation characteristics
18.1.3 Conclusions
18.2 Slanted layer and rhombic media with strips
18.2.1 Slanted layer
18.2.2 Rhombic media with strips
18.2.3 Conclusions
18.3 Elliptically layered, columnar, and rectangular media
18.3.1 Elliptically layered and columnar media
18.3.2 Rectangular media
18.3.3 Conclusions
18.4 Energy distribution of defect layers
18.4.1 Conclusions
18.5 Mixed positive and negative media
18.5.1 Conclusions
References
19 Methods and fast algorithms for the solution of volume singular integral equations
19.1 Introduction
19.2 Formulation of the problems
19.3 Spectrum of integral operator
19.3.1 Spectrum for low-frequency case
19.4 Stationary iteration methods
19.4.1 Generalized simple iteration method
19.4.2 Generalized Chebyshev iteration method
19.5 Nonstationary iteration methods
19.6 Discretization of integral equations
19.7 Fast algorithms
19.8 Numerical results
19.9 Conclusion
References
20 Herglotz functions and applications in electromagnetics
20.1 Introduction
20.2 Basics about Herglotz functions
20.3 Passive systems
20.4 Sum rules and physical bounds
20.5 Convex optimization and physical bounds
20.6 Conclusions
Acknowledgments
References
21 Scattering and guidance by layered cylindrically periodic arrays of circular cylinders
21.1 Introduction
21.2 Formulation of the problem
21.2.1 Field expressions
21.2.2 Calculation of the scattering amplitudes
21.2.3 Reflection and transmission matrices
21.2.4 Hertzian dipole source radiation in the layered cylindrical structure
21.2.5 Plane wave scattering by the layered cylindrical structure
21.2.6 Guidance in the layered cylindrical structure
21.3 Numerical results and discussions
21.3.1 Directivity of radiation of a dipole source coupled to the cylindrical EBG structure
21.3.2 Light scattering by the metal-coated dielectric nanocylinders with angular periodicity
21.3.3 Modal analysis of specific microstructured optical fibers
21.4 Conclusion
Acknowledgment
References
22 Analytical and numerical solution techniques for forward and inverse scattering problems in waveguides
22.1 Introduction
22.2 Inverse problems
22.2.1 General statement for inverse problems
22.3 Inverse problems. Class I (Isotropic case)
22.3.1 Statement of the inverse problem for isotropic one-sectional diaphragm (Class I)
22.3.2 Explicit solution to the inverse problem
22.4 Inverse problems. Class AnI (Anisotropic case)
22.4.1 Statements of inverse problems for anisotropic one-sectional diaphragm (Class AnI)
22.4.2 Explicit formulas for the transmission coefficient
22.4.3 The existence and uniqueness of the solution to the inverse problem
22.5 Inverse problem for multi-sectional diaphragm. Class M
22.6 Numerical results
22.6.1 Example 1. Inverse problem for one-sectional anisotropic diaphragm
22.6.2 Example 2. Extraction of the complex permittivity of each section of three-sectional isotropic diaphragm
22.6.3 Example 3. Extraction of permittivity and permeability of one-sectional anisotropic diaphragm
22.6.4 Example 4. Extraction of permittivity tensor of two-sectional anisotropic diaphragm
22.6.5 Example 5. Inverse problem PCe1
22.7 Conclusion
Acknowledgments
References
23 Beam-based local diffraction tomography
23.1 Introduction and overview
23.2 The UWB-PS-BS method
23.2.1 BS methods: an overview
23.2.2 The UWB-PS-BS method: a frequency domain formulation
23.2.2.1 WFT frame
23.2.2.2 BS representation of the radiation field
23.2.2.3 UWB considerations
23.2.2.4 The beam frames
23.2.3 The phase-space pulsed BS method: a TD formulation
23.2.3.1 The plan-wave spectrum in the TD
23.2.3.2 TheWRT frame
23.2.3.3 Phase-space pulsed BS representation for the field
23.2.3.4 The pulsed beam frames
23.3 UWB tomographic inverse scattering
23.3.1 Tomographic inverse scattering: frequency domain formulation
23.3.1.1 Problem statement
23.3.1.2 The DT identity
23.3.1.3 Object reconstruction via angular diversity (monochromatic tomography)
23.3.1.4 Object reconstruction via frequency diversity (UWB-DT)
23.3.2 Time-domain diffraction tomography
23.4 Beam-based TD-DT
23.4.1 The beam-domain data
23.4.2 The beam-domain data-object relation within the Born approximation
23.4.3 Backpropagation and local reconstruction of O(r)
23.4.4 Numerical examples
23.5 Conclusions
Acknowledgments
References
24 Modal expansions in dispersive material systems with application to quantum optics and topological photonics
24.1 Introduction
24.2 Electrodynamics of dispersive media
24.3 Hermitian formulation in the time domain
24.4 Poynting theorem and stored energy
24.5 Canonical momentum
24.6 Modal expansions
24.7 Green’s function
24.8 Positive and negative frequency components of the Green function
24.9 Application to topological photonics
24.10 Application to quantum optics
24.11 Summary
Acknowledgments
References
25 Multiple scattering by a collection of randomly located obstacles distributed in a dielectric slab
25.1 Introduction
25.2 The geometry
25.3 Integral representation
25.4 Exploiting the integral representations
25.5 Expansions of surface fields
25.6 The transmitted and reflected fields
25.7 Statistical problem—ensemble average
25.8 Approximations
25.9 Conclusions
Appendix A Spherical vector waves
Appendix B The translation matrices
Appendix C Planar vector waves
Appendix D The Green dyadic
Appendix E Probability density functions
References
26 Electromagnetics of complex environments applied to geophysical and biological media
26.1 Introduction
26.2 Stochastic wave theories
26.3 Time-reversal imaging
26.4 Imaging through random multiple scattering clutter
26.5 Geophysical remote sensing and imaging, and super resolution
26.6 Wigner distribution function and specific intensity
26.7 Biomedical electromagnetism and optics
26.8 Heat diffusion in tissues
26.9 Ultrasound in tissues and blood
26.10 Low coherence interferometry and optical coherence tomography (OCT)
26.11 Waves in metamaterials and electromagnetic and acoustic Brewster’s angle
26.12 Coherence in multiple scattering
26.13 Porous media
26.14 Seismic coda
26.15 Conclusion
Acknowledgments
References
27 Innovative tools for SI units in solving various problems of electrodynamics
27.1 Introduction
27.2 Novel format of Maxwell’s equations in SI units: Energetic and mechanical field characteristics
27.2.1 Novel format of Maxwell’s equations in SI units
27.2.2 Energetic characteristics of the electromagnetic field
27.2.3 Mechanical equivalents of the energetic field characteristics
27.3 Exact solutions for polarization of Lorentz media associated with a signal of finite duration
27.3.1 Rearrangement of the motion equation (27.10) to its
equivalent matrix format and solving a vector Cauchy
problem
27.3.1.1 Formulas for calculation of e−τQ
27.3.1.2 Matrix exponential e−τQ for three different cases
27.3.2 Exact explicit solutions for the amplitudes of the polarization vector
27.4 Upgrading the evolutionary approach to electrodynamics (EAE)
27.4.1 Comparison of two alternative approaches to the electromagnetic field theory
27.4.2 Separation of a self-adjoint operator from the vectorial Maxwell’s equations
27.4.3 Normalization of the eigenvectors of operator
27.4.4 Configurational orthonormal modal basis in the space of solutions
27.4.5 Projecting the field vectors and Maxwell’s equations onto the modal basis
27.4.5.1 Projecting the field vectors onto the modal basis
27.4.5.2 Projecting Maxwell’s equations onto the same basis elements
27.4.5.3 Exact explicit solutions for the modal amplitudes
27.5 Present state of art and recent advances
27.6 Ongoing and future research
References
Index
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