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Usage Guide

This guide provides detailed information on using the AnalyticEMModes.jl package for computing electromagnetic field solutions in various geometries.

General API Structure

For each geometry, the package provides a consistent set of functions following a standardized naming convention. The general pattern replaces *** with the specific waveguide type abbreviation:

Field Computation Functions

# TE mode fields (Transverse Electric: Ez = 0)
te_***_fields(coords..., geometry_params..., mode_indices..., frequency, μᵣ, εᵣ)

# TM mode fields (Transverse Magnetic: Hz = 0)
tm_***_fields(coords..., geometry_params..., mode_indices..., frequency, μᵣ, εᵣ)

Cutoff Wavenumber Functions

# Cutoff wavenumber for TE/TM modes
kc = kc_***(geometry_params..., mode_indices..., mode_type)

Mode Sorting Functions

# Get first N modes ordered by cutoff frequency
modes = first_n_modes_***(N, geometry_params...)

Coordinate Systems and Parameters

Rectangular Waveguides (rwg)

Coordinates: Cartesian (x, y)

  • x: Position along width (0 ≤ x ≤ W)
  • y: Position along height (0 ≤ y ≤ H)

Geometry Parameters:

  • W: Waveguide width (m)
  • H: Waveguide height (m)

Mode Indices:

  • m: Number of half-wave variations along width
  • n: Number of half-wave variations along height

Functions:

Ex, Ey, Ez, Hx, Hy, Hz = te_rwg_fields(x, y, W, H, m, n, f, μᵣ, εᵣ)
Ex, Ey, Ez, Hx, Hy, Hz = tm_rwg_fields(x, y, W, H, m, n, f, μᵣ, εᵣ)
kc = kc_rwg(W, H, m, n)
modes = first_n_modes_rwg(N, W, H)

Circular Waveguides (cwg)

Coordinates: Cylindrical (r, θ)

  • r: Radial distance from center (0 ≤ r ≤ R)
  • θ: Azimuthal angle (radians)

Geometry Parameters:

  • R: Waveguide radius (m)

Mode Indices:

  • m: Azimuthal mode number (angular variations)
  • n: Radial mode number (radial zeros)

Functions:

Er, Eθ, Ez, Hr, Hθ, Hz = te_cwg_fields(r, θ, R, m, n, f, μᵣ, εᵣ)
Er, Eθ, Ez, Hr, Hθ, Hz = tm_cwg_fields(r, θ, R, m, n, f, μᵣ, εᵣ)
kc = kc_cwg(R, m, n, mode_type)  # mode_type = :TE or :TM
modes = first_n_modes_cwg(N, R)

Coaxial Waveguides (coax)

Coordinates: Cylindrical (r, θ)

  • r: Radial distance from center (R2 ≤ r ≤ R1)
  • θ: Azimuthal angle (radians)

Geometry Parameters:

  • R1: Outer conductor radius (m)
  • R2: Inner conductor radius (m)

Mode Indices:

  • m: Azimuthal mode number
  • n: Radial mode number

Functions:

Er, Eθ, Ez, Hr, Hθ, Hz = te_coax_fields(r, θ, R1, R2, m, n, f, μᵣ, εᵣ)
Er, Eθ, Ez, Hr, Hθ, Hz = tm_coax_fields(r, θ, R1, R2, m, n, f, μᵣ, εᵣ)
Er, Eθ, Ez, Hr, Hθ, Hz = tem_coax_fields(r, θ, R1, R2, f, μᵣ, εᵣ)
kc = kc_coax(R1, R2, m, n, mode_type) # mode_type = :TE or :TM
modes = first_n_modes_coax(N, R1, R2)

Special: TEM mode has no cutoff (kc = 0).

Elliptical Waveguides (ewg)

Coordinates: Elliptic cylindrical (ξ, η)

  • ξ: Radial-like elliptic coordinate
  • η: Angular-like elliptic coordinate

Geometry Parameters:

  • a: Semi-major axis (m)
  • b: Semi-minor axis (m)

Mode Indices:

  • m: Angular Mathieu function order
  • n: Radial Mathieu function order
  • even: Boolean (true = even modes, false = odd modes)

Functions:

Eξ, Eη, Ez, Hξ, Hη, Hz = te_ewg_fields(ξ, η, a, b, m, n, even, f, μᵣ, εᵣ)
Eξ, Eη, Ez, Hξ, Hη, Hz = tm_ewg_fields(ξ, η, a, b, m, n, even, f, μᵣ, εᵣ)
kc = kc_ewg(a, b, m, n, even, mode_type)
modes = first_n_modes_ewg(N, a, b)

Coordinate Conversion:

ξ, η = cart2elliptic(x, y, a, b)

Radial Waveguides (radial)

Coordinates: Cylindrical (ρ, φ, z)

  • ρ: Radial distance (propagation direction)
  • φ: Azimuthal angle
  • z: Vertical position (0 ≤ z ≤ H)

Geometry Parameters:

  • H: Height between parallel plates (m)
  • m: Azimuthal mode number
  • n: Vertical mode number

Functions:

Eρ, Eφ, Ez, Hρ, Hφ, Hz = te_radial_fields(r, ϕ, z, H, m, n, Amn, Bmn, f,  μᵣ, εᵣ )
Eρ, Eφ, Ez, Hρ, Hφ, Hz = tm_radial_fields(r, ϕ, z, H, m, n, Amn, Bmn, f,  μᵣ, εᵣ )

Wedge Waveguides (wedge)

Coordinates: Cylindrical sector (ρ, φ, z)

  • Angular extent: 0 ≤ φ ≤ φ₀

Geometry Parameters:

  • H: Height between parallel plates (m)
  • φ0: Wedge angle (radians)
  • p: Angular mode number (quantized by φ₀)
  • n: Vertical mode number

Functions:

Eρ, Eφ, Ez, Hρ, Hφ, Hz = te_wedge_fields(r, ϕ, z, h, ϕ0, p, n, Amn, Bmn, f, μᵣ, εᵣ)
Eρ, Eφ, Ez, Hρ, Hφ, Hz = tm_wedge_fields(r, ϕ, z, h, ϕ0, p, n, Amn, Bmn, f, μᵣ, εᵣ)

Return Values

Field functions return a 6-tuple of complex-valued phasor components:

(E_coord1, E_coord2, E_coord3, H_coord1, H_coord2, H_coord3)

The coordinate system depends on the waveguide geometry:

  • Cartesian: (Ex, Ey, Ez, Hx, Hy, Hz)
  • Cylindrical: (Er, Eθ, Ez, Hr, Hθ, Hz)
  • Elliptic: (Eξ, Eη, Ez, Hξ, Hη, Hz)

Helper Functions

Cutoff Frequency and Propagation

using AnalyticEMModes

# Convert cutoff wavenumber to cutoff frequency
fc = cutoff_frequency(kc, μᵣ, εᵣ)

# Compute propagation constant
β = phase_constant(kc, f, μᵣ, εᵣ)

Coordinate Transformations

# Cylindrical unit vectors and metric
h_r, h_θ, e_r_x, e_r_y, e_θ_x, e_θ_y = metric_and_unit_cylindrical(r, θ)

# Elliptic unit vectors and metric
h, e_ξ_x, e_ξ_y, e_η_x, e_η_y = metric_and_unit_elliptic(ρ, ξ, η)

# Convert field components from cylindrical to Cartesian
Fx = Fr * e_r_x + Fθ * e_θ_x
Fy = Fr * e_r_y + Fθ * e_θ_y

Mode Sorting and Selection

Finding modes ordered by cutoff frequency:

# Rectangular waveguide
W, H = 0.05, 0.025
modes = first_n_modes_rwg(10, W, H)
for (mode_type, m, n, kc) in modes
    fc = cutoff_frequency(kc, 1.0, 1.0)
    println("$mode_type($m,$n): fc = $(fc/1e9) GHz")
end

Evaluate Fields

# Single point
r, θ = 0.005, π/4
Er, Eθ, Ez, Hr, Hθ, Hz = te_cwg_fields(r, θ, R, m, n, f, μᵣ, εᵣ)

# Multiple points (vectorized)
r_vec = range(0, R, length=100)
θ_vec = fill(π/4, 100)
fields = te_cwg_fields(r_vec, θ_vec, R, m, n, f, μᵣ, εᵣ)

Examples

For detailed examples with visualizations, see: