Field Decomposition

Why can we decompose our fields?

The systems are invariant in the structure of propagation, longitudinal media, and both constant.

How does field decomposition work?

Split up vectorial quantities, like fields or maxwells equations, like fields or maxwells equations, in a component longitudinal to the direction of the waveguide (generally z-dir) and components perpendicular to it, typically shown by index t for transversal.

What does field decomposition mean for the modes?

TE mode: transverse electric Ez = 0, only Hz left in longitudinal direction
TM mode: transverse magnetic Hz = 0, only EZ left in the longitudinal direction
TEM mode: transverse electromagnetic Ez = Hz = 0, no longitudinal fields

Why do we want to use field decomposition

If we find the longitudinal fields we can calculate the transversal ones from a 1D wave

How do we handle change in media? What do we assume?

Impose boundary conditions at the interface between the media. Still helpful because most applications except gradient fiber have abrupt change in the media, that we have an interface.
Assume no losses, and PEC. Waveguide in z-direction and media constant in z time harmonic fields.

What is a waveguide? Boundary conditions?

Arbitrary metallic form in xy plane, but close form without inner conductor. Dirichlet BC for TM modes, Neumann BC for TE modes, TEM modes are Dirichlet but dt0 = 0 where phi0 obeys the laplace equation.

What is the meaning of eigenfunctions?

Describe the field distribution of allowed modes inside a waveguide.

What is the meaning of eigenvalues?

Correspond to the wave propagation constant and cutt off frequency of the mode.

Rectangular waveguide

Dominant mode is TE10, cartesian coordinates (x,y,z). Use dirirchlet bc.

Circular waveguide

Use cylindrical coordinates. Idea is to separate rho and phi dependence for the eigenfunction and impose BC (rho) or periodicity (phi) on them to find the solution.

Mode degeneracy

If different modes have the same cutoff frequency they degenerate. This happens in symmetric waveguides (for example) like square or circular waveguides. This means these modes have the same phase velocity but different orientation/ polarization. In reality this can become problematic because the smallest discontinuity can cause them to mix. Degeneracy also simplifies calculations and allows arbitrary orientation in symmetric waveguides. In circular waveguides for example for each azimuthal mode number, m, the eigenfunctions can have either a cos(mphi) or sin(mphi) dependence, leading to a +-pi/2 field pattern rotation.

What is resonance?

Resonance occurs when electromagnetic waves constructively interfere with themselves, forming a standing wave pattern in a bounded structure (like a cavity)

Q-factor

Quality factor measures energy retention in a resonator.
Q = omega/deltaomega
higher Q, lower energy loss

Modes in Optical Fibers

Optical fibers support EH and HE modes which are hybrid due to weakly guiding properties.
Core and cladding interactions define HE,EH,TE, and TM modes
V-parameter is the number of supported modes: v = a*sqrt(k1²-k2²), where k1 and k2 are wave numbers in core and cladding

How do surface losses affect the Q-factor of a cavity resonator?

The q-factor measures energy storage vs loss in a resonator. Surface impedance (due to imperfect conductors) causes power dissipation. The curved wall has higher losses than flat PEC walls.

Why do lower conductivity materials lead to a lower Q-factor?

Lower conductivity results in higher resistive losses at the conductor surface, increasing dissipation and reducing energy storage, lowering the Q-factor.

Quasi-TEM mode

Nearly TEM but it has small longitudinal fields (approximation). The surrounding dielectric affects mode velocity and impedance.

How does the dielectric constant impact quasi-TEM velocity?

A higher dielectric constant decreases phase velocity by increasing the effective permittivity, slowing down wave propagation.

Hybrid modes

mix TE and TM properties, appearing in structures like optical fibers

Multi conductor systems

Multi conductor waveguides introduce additional coupled modes due to cross talk and modal interference

Impedance Boundary Condition

The impedance boundary condition relates the tangential electric and magnetic fields at a surface Et = Zs*Ht

Boundary Impedance

Boundary impedance describes how a material interacts with an incident wave, affecting reflection and transmission.

Modes excited by a current

A current can excite TE, TM, or TEM depending on the source configuration