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Contents
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| Week 1: |
Time-domain formulation of the waveguide problem. Identical reorganization of that formulation by the transverse-longitudinal decompositions. Extraction of a pair of the self-adjoint operators, which act on the transverse coordinates only, from the system of the Maxwell’s equations. |
| Week 2: |
Definition of the space of solutions as appropriate Hilbert functional space. Operator eigenvalue equation in the space of solutions. Derivation of the Dirichlet and Neumann boundary-value problems for the transverse Laplacian. |
| Week 3: |
Vector elements of a basis in the domain of the waveguide cross section. The basis for TM – time-domain waveguide modes. The basis for TE – time-domain waveguide modes. |
| Week 4: |
Normalization conditions. Orthogonality of the basis elements. Weyl theorem for the space of solutions. Proof of completeness of the sets of basis elements in the space of solutions. |
| Week 5: |
Projecting of the waveguide electromagnetic fields, sought for, onto the elements of the basis in the space of solutions. Modal decompositions of the fields where the basis elements are already known as the vectors functions of the transverse coordinates but the modal amplitudes are unknown yet scalar functions of the longitudinal coordinate z and time t. |
| Week 6: |
Projecting of the Maxwell’s equations onto the basis elements. The general case when may be filled with a medium which is specified by permittivity and permeability each of which is dependent on the axial coordinate z and time t. Derivation of a generalized Klein-Gordon equation for the modal amplitudes. |
| Week 7: |
Modal decompositions of the electromagnetic fields in a hollow lossless waveguide. Evolutionary equations for the modal amplitudes dependent on z and time t. Canonical Klein-Gordon equation. Comparison of that with the wave equation. |
| Week 8: |
Analysis of the time-domain waveguide modes within the concept of the classical time-harmonic field concept. Cut-off frequencies and cut-off wave numbers. |
| Week 9: |
Analysis of the time-domain waveguide modes within the evolutionary approach to electromagnetics. Comparison of these approaches. Causality principle. Canonical solution. |
| Week 10: |
Midterm examination |
| Week 11: |
Conservation of energy for the time-domain modes. Continuity equation for the conserved energetic characteristics of the modal fields. Energetic wave processes accompanying the phenomenon of modal field propagation. |
| Week 12: |
The properties of symmetry of the Klein-Gordon equation. Possible application of those symmetry properties for analytical studies of time-domain waveguide modes. |
| Week 13: |
Propagation of the Heaviside step signal along the waveguides. Causality principle. |
| Week 14: |
Propagation of digital signals along the waveguides. Causality principle. |
| Week 15*: |
Possible ways of further extensions of the evolutionary approach to electromagnetics. Dynamical constitutive relations for the Lorentz and Debye media. Convolution integrals. |
| Week 16*: |
Final examination. |
| Textbooks and materials: |
1. O.A. Tretyakov, “Evolutionary approach to electromagnetics in time domain,” Lecture notes, Part 2, Waveguide problem.
2. O.A. Tretyakov,.’’Essentials of nonstationary and nonlinear electromagnetic field theory,’’.Chapter 3 in book Analytical and numerical methods in electromagnetic wave theory, M. Hashimoto, M. Idemen, O. A. Tretyakov, Eds., Tokyo, Science House Co. Ltd., 1993.
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| Recommended readings: |
1. S. Aksoy and O. A. Tretyakov, .’’Evolution equations for analytical study of digital signals in waveguides,’’.JEMWA, vol. 17, no. 12, 1665-1682, 2003.
2. O.A. Tretyakov and O. Akgun, .’’Derivation of Klein-Gordon Equation from Maxwell’s Equations and Study of Relativistic Time-Domain Waveguide Modes,’’ PIER, vol. 105, pp. 171-191, 2010.
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* Between 15th and 16th weeks is there a free week for students to prepare for final exam.
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