Asymptotic Methods in Electromagnetics

1. Ray Opties.- 1.1 Introduction.- 1.1.1 The principle of localization.- 1.1.2 Determination of the rays and generalization of Fermat's principle.- 1.1.3 The laws of geometrical optics.- 1.1.4 Determination of the diffraction coefficient.- 1.1.5 Summary: The GTD as a ray method.- 1.2 Ray tracing using generalized Fermat's principle.- 1.2.1 Condition for a path L to be a ray.- 1.2.2 Laws for diffracted rays (Fig.6).- 1.2.3 Conclusions.- 1.3 Calculation of the field along a ray.- 1.3.1 Phase propagation along a ray.- 1.3.2 Conservation of the power flux in a tube of rays.- 1.3.3 Polarization conservation.- 1.3.4 Final formulas for the ray computation of fields.- 1.3.5 Summary.- 1.4 Calculation of the geometrical factors.- 1.4.1 Evolution of the geometrical parameters along a ray.- 1.4.2 Transformation of the geometrical factors in the course of interactions of the ray with the surface.- 1.4.3 Conclusions.- 1.5 Calculation of the diffraction coefficients.- 1.5.1 Reflection coefficients.- 1.5.2 Diffraction by an edge or a line discontinuity.- 1.5.3 Diffraction by a tip.- 1.5.4 Attachment coefficients, detachment coefficients, and coefficients of creeping rays.- 1.5.5 Calculation of coefficients Dsr,s.- 1.5.6 Calculation of the launching coefficients of creeping rays due source located on the surface.- 1.5.7 Coefficients of diffraction for progressive waves and edge waves.- 1.5.8 Conclusions.- 1.6 Some limitations of the ray method.- 1.6.1 The jumps at the light-shadow boundary.- 1.6.2 Infinite results on the caustics.- 1.6.3 Calculation of the field in the shadow zone of the caustic.- 1.6.4 Conclusions.- 1.7 Examples.- 1.7.1 Orders of magnitude of the contributions to the RCS.- 1.7.2 Case examples of diffraction calculation.- 1.8 Conclusions.- References.- 2. Search for Solutions in the Form of Asymptotic Expansions.- 2.1 Perturbation methods as applied to diffraction problems.- 2.1.1 Basic concepts of asymptotic expansion.- 2.1.2 Luneberg-Kline Series and the geometrical optics.- 2.1.3 Generalized Luneberg-Kline series and diffracted rays.- 2.1.4 Boundary layer method.- 2.1.5 Uniform asymptotic solution.- 2.2 Ray fields.- 2.2.1 Geometrical optics.- 2.2.2 Diffracted field by an edge.- 2.2.3 Field diffracted by a line discontinuity.- 2.2.4 Diffracted field by a tip or a corner.- 2.2.5 Field in the shadow zone of a smooth object.- 2.2.6 Conclusions.- References.- 3. The Boundary Layer Method.- 3.1 Boundary layers of creeping rays on a cylindrical surface (Fig. 3.1).- 3.1.1 Conditions satisfied by u.- 3.1.2 Choice of the form of the solution.- 3.1.3 Wave equation expressed with the coordinates (s, n).- 3.1.4 Calculation of u0.- 3.1.5 Calculation of A(s).- 3.1.6 Expression for the compatibility condition.- 3.1.7 Final result for the first term u0 of the expansion of u.- 3.1.8 Fields expressed in terms of the ray coordinates (Fig. 3.2).- 3.2 Boundary layers of creeping rays on a general surface.- 3.2.1 Introduction.- 3.2.2 Equations and boundary conditions.- 3.2.3 Form of the asymptotic expansion.- 3.2.4 Derivation of the solution of Maxwell's equations, expressed in the coordinate system (s, ?, n).- 3.2.5 Interpretation of the equation associated with the first three orders (k, k2/3, k1/3).- 3.2.6 Boundary conditions and the determination of ?.- 3.2.7 Complete determination of E0 and H0.- 3.2.8 Special case of the surface impedance given by Z = 1.- 3.2.9 Conclusions.- 3.3 Boundary layer of the whispering gallery modes.- 3.4 Neighborhood of a regular point of a caustic (Fig. 3.11).- 3.5 Neighborhood of the light-shadow boundary.- 3.6 Boundary layer in the neighborhood of an edge of curved wedge (Fig. 3.13).- 3.7 Neighborhood of the contact point of a creeping ray on a smooth surface.- 3.8 Calculation of the field in the neighborhood of a point of the shadow boundary.- 3.8.1 Calculation of the fields in the vicinity of the surface.- 3.8.2 Surface field.- 3.9 Whispering gallery modes incidents upon an inflection point (Fig.3.16).- 3.10 Matching principle.- 3.11 Matching the solution expressed in the form of a creeping ray at the contact point.- 3.12 Matching of the solution in the boundary layer in the vicinity of surface to the solution in the form of creeping rays and determination of the solution in the shadow zone.- 3.13 Matching of the boundary layer in the neighborhood of the edge of a wedge.- 3.14 The case of caustics.- 3.15 Matching in the neighborhood of the contact point (Fig. 3.17).- References.- 4. Spectral Theory of Diffraction.- 4.1 Introduction.- 4.2 The Plane Wave Spectrum (PWS).- 4.2.1 Homogeneous and inhomogeneous plane waves.- 4.2.2 Superposition of plane waves.- 4.2.3 Plane wave spectrum and Fourier transformation.- 4.2.4 Choice of the contour of integration.- 4.3 Examples of plane wave spectral representation.- 4.3.1 Surface waves.- 4.3.2 Line current.- 4.3.3 Arbitrary current source.- 4.3.4 Field diffracted by a perfectly conducting half-plane.- 4.3.5 Fock field (Fig. 4.7).- 4.3.6 Other examples.- 4.4 Diffraction of complex fields diffraction - Case examples.- 4.4.1 Diffraction of surface waves.- 4.4.2 Surface wave excitation by a line source over a perfectly conducting plane.- 4.4.3 Diffraction by two half-planes (Fig. 4.10).- 4.4.4 Grazing-incidence diffraction by a wedge with curved faces.- References.- 5. Uniform Solutions.- 5.1 Definition and properties of a uniform asymptotic expansion.- 5.2 Generalities on the research methods of a uniform solution.- 5.3 Uniform solutions through the shadow boundaries of the direct field and he field reflected by a wedge.- 5.3.1 Uniform asymptotic solutions of a perfectly conducting wedge with planar faces.- 5.3.2 Uniform asymptotic solutions for a perfectly conducting wedge with curved faces.- 5.3.3 Solution based on Spectral Theory of Diffraction (STD).- 5.3.4 Comparison between UTD, UAT and STD solutions.- 5.4 UAT solution for a line discontinuity in the curvature.- 5.4.1 Statement of the problem and details of resolution.- 5.4.2 Expression of the uniform solution.- 5.4.3 Numerical application.- 5.5 Uniform solution through the shadow boundary and the boundary layer of a regular convex surface.- 5.5.1 Uniform asymptotic solution through the shadow boundary of an imperfectly conducting surface - two-dimensional case.- 5.5.2 Uniform asymptotic expansion through the shadow boundary of an imperfectly conducting surface - three-dimensional case.- 5.5.3 Totally uniform asymptotic solution.- 5.6 Partially- and totally-uniform solutions for a wedge with curved faces, including creeping rays.- 5.6.1 Classification of asymptotic solutions for a wedge with curved faces.- 5.6.2 Solution which is valid close to the grazing incidence: Michaeli's approach (2-D case).- 5.6.3 Uniform asymptotic solution - Michaeli's approach (2-D case).- 5.6.4 Uniform asymptotic solution: Liang, Chuang and Pathak approach (2-D case).- 5.7 Uniform solutions for the saustics.- References.- Sections 5.1 and 5.2.- Section 5.3.- Section 5.4.- Section 5.5.- Section 5.6.- Section 5.7.- 6. Integral methods.- 6.1 The Maslov method.- 6.1.1 Preliminary concepts.- 6.1.2 Representation by means of a simple integral.- 6.1.3 Representation by means of a double integral.- 6.1.4 Method of spectral reconstruction.- 6.1.5 An alternate approach to handling the caustic problem in the context of the Maslov method.- 6.1.6 Extension to Maxwell's equations.- 6.1.7 Limitation of Maslov's method.- 6.2 Integration on a wavefront.- 6.2.1 Geometry of the surface of the centers of wavefront.- 6.2.2 Field computation on the caustic C.- 6.2.3 Conclusions.- References.- 7. Surface Field and Physical Theory of Diffraction.- 7.1 Uniform field.- 7.1.1 Lit zone.- 7.1.2 Transition zone - lit region.- 7.1.3 Transition zone - shadow region.- 7.1.4 Deep shadow zone.- 7.2 Fringe field.- 7.3 The physical theory of diffraction.- 7.3.1 Fringe wave.- 7.3.2 Equivalent current method.- 7.3.3 Equivalent fringe currents.- 7.3.4 Calculation of the diffracted field by using the PTD.- 7.3.5 PTD and GTD.- 7.4 Generalizations of the PTD.- 7.4.1 Extension to objects described by an impedance condition.- 7.4.2 Removal of the spurious contribution due to the fictitious jump discontinuity currents at the shadow boundary.- 7.4.3 The use of a more "realistic" uniform current.- 7.4.4 Treatment of nonconvex objects.- 7.5 PTD Application examples.- 7.5.1 The strip.- 7.5.2 Diffraction by a sharp-tipped cone with an impedance surface.- 7.6 Conclusions.- References.- 8. Calculation of the Surface Impedance, Generalization of the Notion of Surface Impedance.- 8.1 Mathematical foundations and determination of the surface impedance.- 8.1.1 Surface impedance for lossy materials with high index.- 8.1.2 Surface impedance at high frequencies.- 8.1.3 Treatment of the diffraction by edges and discontinuities.- 8.1.4 Summary.- 8.2 Direct treatment of the material.- 8.2.1 Reflected rays.- 8.2.2 Transition zone and shadow zone of a smooth obstacle.- 8.2.3 Diffraction by a wedge coated with material coating.- 8.2.4 Summary of the procedure for the direct treatment of material.- 8.3 Generalized surface impedance.- 8.4 Conclusions.- References.- Appendix 1. Canonical Problems.- A1.1 Reflection of a plane wave by a plane.- A1.2 Diffraction by a circular cylinder whose surface impedance is constant.- A1.2.1 General solution of the cylinder problem.- Al.3 Diffraction by a wedge.- References.- Appendix 2. Differential Geometry.- A2.1 Calculation of the ray lengths.- A2.2.1 Two-dimensional case.- A2.3 Geodesic coordinate system and applications.- A2.3.1 The geodesic coordinates.- A2.3.2 The surface in geodesic coordinates.- A2.4 Coordinate system of the lines of curvature.- Reference.- Appendix 3. Complex Rays.- A3.1 Complex solutions of the eikonal equation, complex rays.- A3.2 Solution of the transport equations.- A3.3 Field calculation in real space.- A3.4 Calculation of the reflected field with the method of complex rays.- A3.5 Other applications of complex rays.- References.- Appendix 4. Asymptotic Expansion of Integrals.- A4.1 Evaluation of the contributions of isolated critical points.- A4.1.1 The method of the stationary phase.- A4.1.2 The method of steepest descent.- A4.1.3 Integration by parts.- A4.1.4 Limitations of the previous methods.- A4.2 Coalescence of critical points, uniform expansions points.- Reference.- Appendix 5. Fock Functions.- A5.1 Utilization of the Fock functions.- A5.2 Definition of the Fock functions.- A5.3 Asymptotic behaviors of Airy functions.- A5.7 Conclusions.- Reference.- Appendix 6. Reciprocity Principle 523.- Reference.