Interaction between fields and matters Maxwell’s equations in medium Wave equation and its solution in free-space and under guidance 1

Maxwell’s equations in medium Maxwell’s equation in vacuum Constitutive equations Maxwell’s equation in medium 17 equations, yet 16 unknowns, one equation is redundant, e.g., the 4th equation is embedded in the 1st equation Independent sources, i.e., the free charge and conduction current, have to be taken as unknowns as well for self-consistent solutions

Wave equation From Maxwell’s equations to the wave equation sourceless, harmonic homogeneous medium coupling term independent term

Solution in free space: plane wave Simplified wave equation sourceless, homogenous simplified wave equation from we have Obviously, to manipulate the polarization through coupling among different field components, the introduction of medium inhomogeneity and/or structure is necessary Free space solution of the wave equation – plane harmonic wave where or

Basic wave natures Solution to wave equation – traveling wave 1D or propagation along 3D propagation along z t The element of traveling wave and its complex expression – plane (harmonic) wave

Basic wave natures Standing wave – formed by the superposition of two contra-directionally propagating (traveling) waves Evanescent wave – spatially decayed wave Wave coupling – phase matching condition Phase velocity and group velocity

Plane (harmonic) wave Dispersion in free space Plane harmonic wave characteristics for the same reason also from we have where

Plane (harmonic) wave Therefore, the plane harmonic wave in free space has its electric field, magnetic field, and propagation direction all orthogonal – it is a condition forced by the electric and magnetic divergence free requirement E [V/m] H [A/m] k Free-space impedance, in vacuum, it is 120π~377Ω

Plane (harmonic) wave Significance of the plane harmonic wave – eigen solution of the wave equation in free space, i.e., for whatever excitation, after waiting for infinitely long, the solution at infinitely far distance from the source in free space can only be the plane harmonic wave Therefore, regardless of the source distribution, the electromagnetic wave in free space will become the plane harmonic wave after infinitely long time at infinity, the evolution process is the spatial diffraction that eventually smears out any initial nonuniformity

Guided wave For a given individual source in any form, an excited EM wave gradually turns itself into a plane wave in a free-space, and becomes difficult to be collected in a distance away by a receiver with a limited surface How can we force the EM wave to propagate along a specific direction without any spreading in the 3D world? Waveguide Multiple sources (source array)

Waveguide: basic ideas The wave has to localized in certain directions How to localize the wave? – Convert the traveling wave into the standing wave Introduce the transverse resonance z x s-wave reflection at the boundary: standing wave is formed underneath the boundary

Waveguide: basic ideas x The resonance condition for standing wave in transverse direction (x): z A necessary condition is: How to make it possible? Conductor reflection – metallic waveguide TIR – dielectric waveguide Photonic crystal – Bragg waveguide Plasma reflection – plasmonic polariton waveguide

Metallic waveguides By letting in previous derivations, we will be able to obtain the EM wave solution in metallic waveguide. 1D slab metallic waveguide – support TEM, TE, and TM waves 2D hollow metallic waveguide – support TE and TM waves, not TEM wave 1D slab or 2D dielectric waveguide – support TE and TM waves, not TEM wave

Looking for the simplest guided wave solution If the wave is guided along z, we should have: Substitute it back into the wave equation to obtain: If we are looking for a special solution having: likewise We must have:

TEM wave We also have: Therefore, neither electric field nor magnetic field can have its z-component. That’s why we call such solution transverse electromagnetic (TEM) wave – the simplest guided wave formally the same as the static field solution in a sourceless 2D cross-sectional region

Necessary condition for guiding TEM wave Minimum requirement to support a non-zero electric and magnetic field distributions in a 2D cross-section: 2 (or more) separate pieces of metals, with +/- charge distribution and in/out current flow, respectively 2. Once such a 2D structure is stretched along the 3rd dimension, one obtains a working waveguide that supports the TEM wave – such waveguides for TEM wave are called transmission lines

Dominant feature of TEM waveguides In the open area (whole space with metal excluded), both fields are curl free – both field (force) are conservative Their path integrations rely only on the starting/ending points - having nothing to do with path selection We can therefore introduce the concept of voltage (as a path integral of the electric field from one metal piece to the other) to replace E0(x, y) Likewise, we can introduce the concept of current (as a path integral of the magnetic field around one metal piece) to replace H0(x, y) As such, the dependence on (x, y) is gone. By using a pair of new variables, voltage (V) and current (I), we don’t have to deal with the cross-section (structure or variables) anymore – all information about the cross-section is lumped into V and I

Transmission lines I E H coaxial cable E H parallel lines or twisted pair V With V and I introduced, there is no need to deal with E and H fields anymore. At any cut along z (i.e., in each cross-section), there is only one value for V and one value for I, respectively. Thereby the problem is greatly simplified.

TEM wave: characteristics TEM wave – “localized” plane wave with k, E, H mutually orthogonal: k is along the direction in which the waveguide (transmission line) is extended; E and H fields are restricted in the 2D cross-section, with their longitudinal dependence identical to the plane wave, and transverse dependence identical to the static E and H fields with the same boundary condition. Propagation of the TEM wave relies on the free charge and conduction current on the metal (conductor) – dielectric surface. Namely, the TEM wave is a resonance between the EM fields and the free charge distribution. For the TEM wave, we can readily introduce the voltage and current concept to turn a field problem into a circuit problem. The TEM wave can be supported by the dual conductor transmission line: Parallel lines or twisted pair Coaxial cable Printed metal stripe lines (on PCB or other substrates) The TEM wave has no cut-off frequency, it can send DC power through.

Various electromagnetic waveguides WG Structure Guiding Pattern Applications Remarks DC Paired metal wires TEM Power transmission Transmission in free-space impossible VLF Paired metal wires or free-space TEM or plane wave Power transmission, submarine communication Huge antenna size required for broadcasting LF-MF Paired metal strips on PCB or free-space Circuits, (AM) broadcasting MF for radio broadcasting HF-VHF Coaxial cables, micro-strips on PCB, or free-space Circuits, (FM) broadcasting, wireless communication, remote control, blue tooth, etc. TV broadcasting MW Coaxial cables, hollow metallic waveguides, or free-space TEM, TE/TM, HE/EH, or plane wave Circuits, radars, space (satellite) communication Long-haul telecommunication through antenna relay LW Optical fibres, dielectric waveguides, or free-space TE/TM, HE/EH, or plane wave Optical communication, sensor systems, space (satellite) communication Li-Fi (Lightening LED for Wi-Fi)