A final review

Maxwell’s equations Originated from – Coulomb’s law Biot-Savert’s law Faraday’s law Field description 3/4 – E/M Gaussian eq. static modifications under dynamics displacement current 2 – (modified) Ampere’s eq. dynamic emf 1 – Faraday’s eq. Potential description more convenient in solving static E-field more convenient in solving radiation/antenna problem

valid in free-space: sourceless, homogeneous Plane wave Maxwell’s equations in arbitrary medium: FD wave equations – obtained from Faraday’s eq. + Ampere’s eq. valid in free-space: sourceless, homogeneous solution: E [V/m] H [A/m] k

Transmission line 1 The waveguide concept: identify a unique propagation direction (usually name it as the z-direction) form a resonant structure in cross-section (i.e., the x-y plane) by generating a pair of contra-propagating waves hence a standing-evanescent wave pattern is formed in the cross-sectional region, which eliminates any traveling wave in the cross-section; consequently, the wave can only propagate along the given direction without energy escaping away from a core region of the resonant structure in the cross-sectional plane A specific waveguide (transmission line) is established by: create a (2D cross-sectional) structure that supports a pair of static E and M field distributions simultaneously (which requires at least two non-connected pieces of conductors) E and M waves, therefore, can only have their k=kz along the direction normal to the cross-section (there cannot be any wave propagation in the cross-section as kx=ky=0) stretching above 2D structure along the 3rd direction (z) obtains a simplest waveguide: the transmission line (TL)

Transmission line 2 Dominant features of TL: support the TEM wave – localized plane wave, i.e., everything else the same except for the uniform E0 and H0 are replaced by (x, y) dependent E0 and H0 distributions, which satisfy the Laplace equation in cross-sectional field – the static field solution by introducing a pair of new variables: voltage (V), as the difference of the potential between the two conductors (i.e., a path integral of the E field between the two conductors), and current (I), as the conduction current on either conductor (i.e., a closed loop integral of the H field around either conductor), we no longer need to consider the (x, y) dependence, as (V, I) will have (z, t) dependence only due to the uniqueness of above path integrations in the x-y plane the governing equations to E and H fields (i.e., Maxwell’s equations) are therefore reduced to the governing equations to V and I the general solution to V and I can be obtained analytically, for they only have (z, t) dependence

Transmission line 3 Solution techniques and applications: both V and I on TL are summations of forward and backward travelling waves TL has its own featured parameters: the propagation constant β=2π/λ and the characteristic impedance Z0, they both can be expressed in terms of the distributive circuit parameters (R, L, G, C) that in turn can be found once the physical structure of TL is given for a given load impedance ZL, the (V, I) backward travelling waves are determined by their forward travelling waves through the reflection coefficient Γ the contra-propagating travelling waves on TL form a partial standing wave pattern that can be measured conveniently if the load impedance ZL is unknown, by measuring the partial standing wave pattern, we can find Γ and consequently ZL if the impedance is not matched (ZL≠Z0), the equivalent impedance on TL can be found analytically in terms of ZL, Z0, and βL – it is different at different locations on TL!

Transmission line 4 Solution techniques and applications (continued): the equivalent impedance takes special values at special points on TL, and becomes an pure reactance when TL is OC or SC since the equivalent impedance varies with the location, it is always possible to identify a location at which the conductance matches to 1/ Z0, but with a residue reactance an extra OC or SC TL stub (with a pure reactance only!) can therefore be connected (in shunt) to the main TL at above location to cancel the residue reactance, which enforces an impedance matching on TL! once the source is given, we can trace V and I distributions on TL by following the wave propagation sequence in time domain: starting the process by taking the fact that the wave initially only has a forward propagating component; the consequent backward travelling wave component will be determined by the reflection of the previous forward travelling wave component at the load end, whereas the consequent forward travelling wave component will be determined by the reflection of the previous backward travelling wave component at the source end, ….. (the treatment of the reflection on the source end is similar to that on the load end)

Plane wave reflection and transmission Solve plane wave reflection and transmission problem at a smooth boundary that separates two half free-spaces: Snell’s law (obtained from the boundary condition enforced by Maxwell’s equations) – determines the directions of the reflected wave and transmitted wave for an incoming wave with a given incident angle Snell’s law is always valid, regardless of s- or p- wave the reflected and transmitted portion of the wave (amplitude) is given by the reflection and transmission coefficient, respectively; they are, however, different for s- and p- waves when going beyond the critical angle, total reflection happens to both s- and p- waves, but total reflection doesn’t happen for external reflection – TIR only! for non-magnetic medium with identical µ, the p-wave experiences a total transmission at the Brewster angle, regardless of internal or external reflection! there is no total transmission for the s-wave

Wave in different media Ideal dielectrics – real permittivity, real propagation constant Poor dielectrics/conductors – complex or imaginary permittivity, complex propagation constant results in either decayed wave propagation (poor dielectrics) or evanescent wave with the skin effect (poor conductors) Ideal conductors – negative permittivity (< plasma frequency) or smaller than vacuum permittivity (> plasma frequency), imaginary or smaller-than-vacuum propagation constant, wave gets completely reflected in the former case, and superluminal or TIR in air (vacuum) can happen poor dielectrics: ε”<<ε’ poor conductors: ε”>>ε’

An overview Coulomb’s law, Biot-Savert’s law, Faraday’s law Maxwell’s equations Static E/M fields E quasi-static field – circuit theory M quasi-static field – power system Wave equations: full dynamic E/M fields - EM waves Plane wave in free-space Plane wave in different media Plane wave in half free-spaces (reflection and transmission) Guided wave – waveguides (metallic, dielectric, etc.) Wave radiation – antennas Wave diffraction – the transient process from source to plane wave Wave – medium interactions (physics, materials) A special waveguide: transmission line

Open problem: wave or particle? Photographs taken in dimmer light look grainier – can’t be explained by wave as the wave must be continuous at least! Very, very dim Very dim Dim Bright Very bright Very, very bright  is the "photon flux," i.e., number of photons/sec in a beam of power P If we can accept the duality of the light wave, we should see the particle behavior of the microwave and RF, …, as well!

Why it matters? Once we can handle the wave particle behavior, we can utilize the momentum of the wave directly, not only the energy of the wave. A low power, continuous wave laser that is focused through a high N.A. objective can trap particles of diameter 10m. Can move the trapped particle around, hence the laser acts as a “tweezer” by picking up and moving an individual particle. Potential applications: 1. Provide trust to push object such as plane, rocket, bullet, etc. 2. Provide trust to push micro-robot inside human body (e.g., in veins), or drive surgical robot 3. Shield human or object from being attacked by other objects such as bullet 4. Mind shift will become true?