API Reference

wavenumber(f, μᵣ, εᵣ)

Computes the wavenumber ω√(μ⋅ε).

Arguments

• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

phase_constant(kc, f, μᵣ, εᵣ)

Computes β given the cutoff wavenumber kc.

Arguments

• kc: Cutoff wavenumber
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

attenuation_factor(kc, f, μᵣ, εᵣ)

Computes α, the attenuation constant of a given mode for frequencies below the cutoff.

Arguments

• kc: Cutoff wavenumber
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

propagation_constant(kc, f, μᵣ, εᵣ)

Computes γ = α + jβ given the cutoff wavenumber kc.

Arguments

• kc: Cutoff wavenumber
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

cutoff_frequency(kc, μᵣ, εᵣ)

Computes the cutoff frequency of a mode given the cutoff wavenumber kc.

Arguments

• kc: Cutoff wavenumber
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

mode_wavelength(β)

Returns the guided wavelength (mode wavelength) of the propagating mode given the phase constant β.

Arguments

• β: Phase constant

mode_impedance(T, fcutoff, f, μᵣ, εᵣ)

Computes the mode impedance of a T ∈ (:TE, :TM) mode given the cutoff frequency.

Arguments

• T: Mode type (:TE or :TM)
• fcutoff: Cutoff frequency
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

kc_rwg(a, b, m, n)

Calculates the cutoff wave number of the m,n mode for a rectangular waveguide with dimensions a, b where a >= b.

Arguments

• a: Width of the waveguide (larger dimension)
• b: Height of the waveguide (smaller dimension)
• m: Mode index in x-direction
• n: Mode index in y-direction

te_rwg_fields(x, y, a, b, m, n, c_e, c_h)
te_rwg_fields(x, y, a, b, m, n, f, μᵣ, εᵣ)

Calculate the electric and magnetic field components of a TE_{m,n} mode.

Arguments

• x, y: Coordinates where x ∈ [0, a] and y ∈ [0, b]
• a: Width of the waveguide
• b: Height of the waveguide
• m, n: Mode indices
• c_e, c_h: Field coupling coefficients (from te_coefficients), or
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

Returns

(Ex, Ey, Ez, Hx, Hy, Hz)

tm_rwg_fields(x, y, a, b, m, n, c_e, c_h)
tm_rwg_fields(x, y, a, b, m, n, f, μᵣ, εᵣ)

Compute the electric and magnetic field components of a TM_{m,n} mode.

Arguments

• x, y: Coordinates where x ∈ [0, a] and y ∈ [0, b]
• a: Width of the waveguide
• b: Height of the waveguide
• m, n: Mode indices
• c_e, c_h: Field coupling coefficients (from tm_coefficients), or
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

Returns

(Ex, Ey, Ez, Hx, Hy, Hz)

Example

c_e, c_h = tm_coefficients(kc, β, f, μᵣ, εᵣ)
tm_rwg_fields(x, y, a, b, m, n, c_e, c_h)

te_normalization_rwg(a, b, m, n, kc, β, f, μᵣ, εᵣ)

Normalization factor for TE modes to achieve unit power.

The normalization is derived from integrating the Poynting vector over the waveguide cross-section, giving the total transmitted power. In TE modes, Hₒ is normalized. δm, δn account for mode indices being 0 or non-zero.

Arguments

• a: Width of the waveguide
• b: Height of the waveguide
• m, n: Mode indices
• kc: Cutoff wavenumber
• β: Phase constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

tm_normalization_rwg(a, b, m, n, kc, β, f, μᵣ, εᵣ)

Normalization factor for TM modes to achieve unit power.

The normalization is derived from integrating the Poynting vector over the waveguide cross-section, giving the total transmitted power. In TM modes, Eₒ is normalized. In this case there is no δm, δn because m and n >= 1.

Arguments

• a: Width of the waveguide
• b: Height of the waveguide
• m, n: Mode indices
• kc: Cutoff wavenumber
• β: Phase constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

kc_cwg(r, m, n, T; tol = 1e-9, max_iters = 10)

Calculates the cutoff wave number of the m,n mode for a circular waveguide with radius r. Uses tabulated data or asymptotic expansions of Bessel functions to calculate the zero.

Arguments

• r: Radius of the circular waveguide
• m: Azimuthal mode index
• n: Radial mode index
• T: Mode type (:TE or :TM)
• tol: Tolerance for zero-finding (default: 1e-9)
• max_iters: Maximum iterations for Newton-Raphson (default: 10)

Example

kc_cwg(1.0, 2, 1, :TE)
kc_cwg(1.0, 2, 1, :TM)

te_cwg_fields(r, θ, m, kc, c_e, c_h)
te_cwg_fields(r, θ, a, m, n, f, μᵣ, εᵣ)

Computes the electric and magnetic field components of a TE_{m,n} mode in a circular waveguide.

Arguments

• r: Radial coordinate
• θ: Angular coordinate
• m: Azimuthal mode index
• kc: Cutoff wavenumber, or
• a: Radius of the waveguide
• n: Radial mode index
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity
• c_e, c_h: Field coupling coefficients (from te_coefficients)

Returns

(Er, Eθ, Ez, Hr, Hθ, Hz) in cylindrical coordinates centered at (0,0).

tm_cwg_fields(r, θ, m, kc, c_e, c_h)
tm_cwg_fields(r, θ, a, m, n, f, μᵣ, εᵣ)

Computes the electric and magnetic field components of a TM_{m,n} mode in a circular waveguide.

Arguments

• r: Radial coordinate
• θ: Angular coordinate
• m: Azimuthal mode index
• kc: Cutoff wavenumber, or
• a: Radius of the waveguide
• n: Radial mode index
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity
• c_e, c_h: Field coupling coefficients (from tm_coefficients)

Returns

(Er, Eθ, Ez, Hr, Hθ, Hz) in cylindrical coordinates centered at (0,0).

te_normalization_cwg(a, m, kc, β, f, μᵣ, εᵣ)

Normalization factor for TE modes to achieve unit power.

The expression can be derived by integrating the Poynting vector over the cross-section of the guide.

Arguments

• a: Radius of the waveguide
• m: Azimuthal mode index
• kc: Cutoff wavenumber
• β: Phase constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

tm_normalization_cwg(a, m, kc, β, f, μᵣ, εᵣ)

Normalization factor for TM modes to achieve unit power.

The expression can be derived by integrating the Poynting vector over the cross-section of the guide.

Arguments

• a: Radius of the waveguide
• m: Azimuthal mode index
• kc: Cutoff wavenumber
• β: Phase constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

metric_and_unit_cylindrical(r, ϕ)

Computes the metric coefficients (scale factors) and the Cartesian components of the unit vectors (êᵣ, êᵩ) for the 2D cylindrical (polar) coordinate system (r, ϕ).

This function is used for transforming differential operators (like the gradient or Laplacian) and vectors between the cylindrical and Cartesian systems.

Arguments

• r: The radial coordinate (distance from the z-axis).
• ϕ: The azimuthal coordinate (angle with respect to the positive x-axis, in radians).

Returns

A tuple containing the six components:

• h_r: Metric coefficient in the radial direction . Always 1.0.
• h_ϕ: Metric coefficient in the azimuthal direction . Equal to r.
• e_r_x: X-component of the radial unit vector
• e_r_y: Y-component of the radial unit vector
• e_ϕ_x: X-component of the azimuthal unit vector
• e_ϕ_y: Y-component of the azimuthal unit vector

kc_coax(a, b, m, n, T)

Compute the cutoff wave number for a coaxial waveguide.

Arguments

• a: Outer radius
• b: Inner radius
• m: Azimuthal mode index
• n: Radial mode index
• T: Mode type (:TE, :TM, or :TEM)

Note

For TEM mode, returns 0.0 (no cutoff).

te_coax_fields(r, θ, Am, Bm, m, kc, c_e, c_h)
te_coax_fields(r, θ, a, b, m, n, f, μᵣ, εᵣ)

Arguments

• r: r polar coordinate.
• θ: angle in polar coordinates.
• Am: First value returned by coax_boundary_coeff_te. Is always 1.0
• Bm: Second value retuened by coax_boundary_coeff_te.
• m: The azimuthal mode number.
• n: radial mode number.
• kc: cutoff wavenumber.
• c_e: imωμ/kc^2
• c_h: -im*β/kc^2
• a: outer radius.
• b: inner radius
• f: frequency.
• μᵣ: relative permeability
• εᵣ: relative permittivity

Returns

• (Er, Eθ, Ez, Hr, Hθ, Hz)

Calculate the electric and magnetic field components in polar coordinates of a TE_{m,n} mode given the dimensions of the circular section with radius r. It is assumed that the center of the circular section is at (0,0).

tm_coax_fields(r, θ, Am, Bm, m, kc, c_e, c_h)

Arguments

• r: r polar coordinate.
• θ: angle in polar coordinates.
• Am: First value returned by coax_boundary_coeff_te. Is always 1.0
• Bm: Second value retuened by coax_boundary_coeff_te.
• m: The azimuthal mode number.
• n: radial mode number.
• kc: cutoff wavenumber.
• c_e: imωμ/kc^2
• c_h: -im*β/kc^2
• a: outer radius.
• b: inner radius
• f: frequency.
• μᵣ: relative permeability
• εᵣ: relative permittivity

Returns

• (Er, Eθ, Ez, Hr, Hθ, Hz)

Calculate the electric and magnetic field components in polar coordinates of a TM_{m,n} mode given the dimensions of the circular section with radius r. It is assumed that the center of the circular section is at (0,0).

tm_coax_fields(r, θ, a, b, m, n, f, μᵣ, εᵣ)

Arguments

• r: r polar coordinate.
• θ: angle in polar coordinates.
• a: outer radius.
• b: inner radius
• m: azimuthal mode number.
• n: radial mode number.
• f: frequency.
• μᵣ: relative permeability
• εᵣ: relative permittivity

Returns

• (Er, Eθ, Ez, Hr, Hθ, Hz)

Calculate the electric and magnetic field components in polar coordinates of a TM_{m,n} mode given the dimensions of the circular section with radius r. It is assumed that the center of the circular section is at (0,0).

tem_coax_fields(r, a, b, μᵣ, εᵣ)

Calculate the electric and magnetic field components of the TEM mode in a coaxial waveguide.

Arguments

• r: Radial coordinate
• a: Outer radius
• b: Inner radius
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

Returns

• (Er, Eθ, Ez, Hr, Hθ, Hz)

te_normalization_coax(a, b, m, kc, β, f, μᵣ, εᵣ)

Normalization factor for TE modes to achieve unit power.

The expression can be derived by integrating the Poynting vector over the cross-section of the guide.

Arguments

• a: Outer radius
• b: Inner radius
• m: Azimuthal mode index
• kc: Cutoff wavenumber
• β: Phase constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

tm_normalization_coax(a, b, m, kc, β, f, μᵣ, εᵣ)

Normalization factor for TM modes to achieve unit power.

The expression can be derived by integrating the Poynting vector over the cross-section of the guide.

Arguments

• a: Outer radius
• b: Inner radius
• m: Azimuthal mode index
• kc: Cutoff wavenumber
• β: Phase constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

characteristic_coax_equation_te(m, kc, a, b)

The zeros of this function correspond to the cutoff wave number of the TE modes.

Arguments

• m: Azimuthal mode index
• kc: Cutoff wavenumber
• a: Outer radius
• b: Inner radius

characteristic_coax_equation_tm(m, kc, a, b)

The zeros of this function correspond to the cutoff wave number of the TM modes.

Arguments

• m: Azimuthal mode index
• kc: Cutoff wavenumber
• a: Outer radius
• b: Inner radius

kc_radial(βz, f, μᵣ, εᵣ)

Computes the radial propagation constant kᵨ (analogous to kc in other geometries) for a radial waveguide.

Arguments

• βz: Axial phase constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

Reference

Advanced engineering electromagnetics by Constatine A. Balanis. Chapter 9.5.1

phase_constant_radial(h, n)

Phase constant (βᶻ) of the axial component of the wavenumber in a radial waveguide.

This constant arises from the PEC boundary conditions imposed at z = 0 and z = h, which constrain the field variation along the axial direction as:

Eρ(0 <= ρ <= ∞, 0 <= ϕ <= 2π, z = 0) = Eρ(0 <= ρ <= ∞, 0 <= ϕ <= 2π, z = h) = 0 Eϕ(0 <= ρ <= ∞, 0 <= ϕ <= 2π, z = 0) = Eϕ(0 <= ρ <= ∞, 0 <= ϕ <= 2π, z = h) = 0

Arguments

• h: Height of the radial waveguide
• n: Axial mode index

Reference

Advanced engineering electromagnetics by Constatine A. Balanis. Chapter 9.5.1

cutoff_frequency_radial(βz, μᵣ, εᵣ)

Like cutoff_frequency(kc, μᵣ, εᵣ) but using βz instead of kc for the particular case of the radial waveguide.

Arguments

• βz: Axial phase constant
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

te_radial_fields(r, ϕ, z, m, Amn, Bmn, kc, β, c_e, c_h)
te_radial_fields(r, ϕ, z, h, m, n, Amn, Bmn, f, μᵣ, εᵣ)

Compute the electric and magnetic field components of a TE_{m,n} mode in a radial waveguide.

Arguments

• r: Radial coordinate
• ϕ: Angular coordinate
• z: Axial coordinate where z ∈ [0, h]
• m: Azimuthal mode index
• Amn, Bmn: Wave amplitudes (outgoing/incoming)
• kc: Radial propagation constant, or
• h: Height of the waveguide
• n: Axial mode index
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity
• β: Axial phase constant
• c_e, c_h: Field coupling coefficients

Returns

(Er, Eϕ, Ez, Hr, Hϕ, Hz)

Notes

Outgoing: Amn = 1, Bmn = 0; Incoming: Amn = 0, Bmn = 1

tm_radial_fields(r, ϕ, z, m, Amn, Bmn, kc, β, c_e, c_h)
tm_radial_fields(r, ϕ, z, h, m, n, Amn, Bmn, f,  μᵣ, εᵣ )

Compute the electric and magnetic field components of a TM_{m,n} mode given the dimensions radial waveguide. It is assumed that the radial section is centered at (0, 0, z) with height z ∈ [0, h].

te_zw(m, r, kc, Amn, Bmn, f, μᵣ)

Impedance of the wave in the ρ direction for a TE mode.

Arguments

• m: Azimuthal mode index
• r: Radial coordinate
• kc: Radial propagation constant
• Amn, Bmn: Wave amplitudes
• f: Frequency in Hz
• μᵣ: Relative permeability

tm_zw(m, r, kc, Amn, Bmn, f, μᵣ)

Impedance of the wave in the ρ direction for a TM mode.

te_normalization_radial(r, h, m, Amn, Bmn, kc, f, μᵣ, εᵣ)

Normalization factor for TE modes to achieve unit power.

The expression can be derived by integrating the Poynting vector over the cross-section of the guide.

Arguments

• r: Radial coordinate
• h: Height of the waveguide
• m: Azimuthal mode index
• Amn, Bmn: Wave amplitudes
• kc: Radial propagation constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

tm_normalization_radial(r, h, m, Amn, Bmn, kc, f, μᵣ, εᵣ)

Normalization factor for TM modes to achieve unit power.

The expression can be derived by integrating the Poynting vector over the cross-section of the guide.

Arguments

• r: Radial coordinate
• h: Height of the waveguide
• m: Azimuthal mode index
• Amn, Bmn: Wave amplitudes
• kc: Radial propagation constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

kc_wedge(βz, f, μᵣ, εᵣ)

Computes the radial propagation constant for a wedge waveguide (same as kc_radial).

Arguments

• βz: Axial phase constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

te_wedge_fields(r, ϕ, z, ϕ0, p, Amn, Bmn, kc, β, c_e, c_h)
te_wedge_fields(r, ϕ, z, h, ϕ0, p, n, Amn, Bmn, f, μᵣ, εᵣ)

Compute the electric and magnetic field components of a TE_{p,n} mode in a wedge waveguide.

Arguments

• r: Radial coordinate
• ϕ: Angular coordinate where ϕ ∈ [0, ϕ0]
• z: Axial coordinate where z ∈ [0, h]
• ϕ0: Wedge angle
• p: Angular mode index
• Amn, Bmn: Wave amplitudes
• kc: Radial propagation constant, or
• h: Height of the waveguide
• n: Axial mode index
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity
• β: Axial phase constant
• c_e, c_h: Field coupling coefficients

Returns

(Er, Eϕ, Ez, Hr, Hϕ, Hz)

tm_wedge_fields(r, ϕ, z, ϕ0, p, Amn, Bmn, kc, β, c_e, c_h)
tm_wedge_fields(r, ϕ, z, h, ϕ0, p, n, Amn, Bmn, f, μᵣ, εᵣ)

Compute the electric and magnetic field components of a TM_{p,n} mode in a wedge waveguide.

Arguments

• r: Radial coordinate
• ϕ: Angular coordinate where ϕ ∈ [0, ϕ0]
• z: Axial coordinate where z ∈ [0, h]
• ϕ0: Wedge angle
• p: Angular mode index
• Amn, Bmn: Wave amplitudes
• kc: Radial propagation constant, or
• h: Height of the waveguide
• n: Axial mode index
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity
• β: Axial phase constant
• c_e, c_h: Field coupling coefficients

Returns

(Er, Eϕ, Ez, Hr, Hϕ, Hz)

te_normalization_wedge(r, h, ϕ0, p, Amn, Bmn, kc, f, μᵣ, εᵣ)

Normalization factor for TE modes to achieve unit power.

The expression can be derived by integrating the Poynting vector over the cross-section of the guide.

Arguments

• r: Radial coordinate
• h: Height of the waveguide
• ϕ0: Wedge angle
• p: Angular mode index
• Amn, Bmn: Wave amplitudes
• kc: Radial propagation constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

tm_normalization_wedge(r, h, ϕ0, p, Amn, Bmn, kc, f, μᵣ, εᵣ)

Normalization factor for TM modes to achieve unit power.

The expression can be derived by integrating the Poynting vector over the cross-section of the guide.

Arguments

• r: Radial coordinate
• h: Height of the waveguide
• ϕ0: Wedge angle
• p: Angular mode index
• Amn, Bmn: Wave amplitudes
• kc: Radial propagation constant
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity

kc_ewg(a, b, m, n, even, T)

Computes the cutoff wavenumber k_c for the (m, n)-th mode in an elliptical waveguide (EWG).

This function determines the characteristic values of the Mathieu functions corresponding to the given mode symmetry (even) and type (T = :TE or :TM). The cutoff wavenumber is derived from the eigenvalues qₘₙ obtained by locating the zeros of the appropriate Mathieu function or its derivative.

Arguments

• a, b: Semi-major and semi-minor axes of the ellipse.
• m, n: Mode indices.
• even::Bool: Whether the mode is even (true) or odd (false).
• T::Symbol: Mode type. Either :TE or :TM.

Returns

• kc::Float64: Cutoff wavenumber for the specified mode.

Notes

• For TM modes, the zeros of the Mathieu functions Ceₘ or Seₘ are used.
• For TE modes, the zeros of their derivatives Ceₘ′ or Seₘ′ are used.

te_ewg_fields(ξ, η, ρ, m, even, coeff, q, c_e, c_h)
te_ewg_fields(ξ, η, a, b, m, n, even, f, μᵣ, εᵣ)

Computes the electric and magnetic field components of a TE_{c/s,m,n} mode in an elliptical waveguide.

Arguments

• ξ: Radial elliptical coordinate
• η: Angular elliptical coordinate
• ρ: Focal distance √(a²-b²), or
• a: Semi-major axis
• b: Semi-minor axis
• m: Mode order
• n: Radial mode index (used with a, b)
• even: true for even modes (c), false for odd modes (s)
• coeff: Mathieu function coefficients
• q: Parameter q = (kc*ρ)²/4
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity
• c_e, c_h: Field coupling coefficients

Returns

(Eξ, Eη, Ez, Hξ, Hη, Hz) in elliptical coordinates centered at (0,0).

tm_ewg_fields(ξ, η, ρ, m, even, coeff, q, c_e, c_h)
tm_ewg_fields(ξ, η, a, b, m, n, even, f, μᵣ, εᵣ)

Computes the electric and magnetic field components of a TM_{c/s,m,n} mode in an elliptical waveguide.

Arguments

• ξ: Radial elliptical coordinate
• η: Angular elliptical coordinate
• ρ: Focal distance √(a²-b²), or
• a: Semi-major axis
• b: Semi-minor axis
• m: Mode order
• n: Radial mode index (used with a, b)
• even: true for even modes (c), false for odd modes (s)
• coeff: Mathieu function coefficients
• q: Parameter q = (kc*ρ)²/4
• f: Frequency in Hz
• μᵣ: Relative permeability
• εᵣ: Relative permittivity
• c_e, c_h: Field coupling coefficients

Returns

(Eξ, Eη, Ez, Hξ, Hη, Hz) in elliptical coordinates centered at (0,0).

cart2elliptic(x, y, a, b)

Transform from cartesian coordinates to elliptic coordinates.

Arguments

• x, y: Cartesian coordinates
• a: Semi-major axis of the ellipse
• b: Semi-minor axis of the ellipse

Returns

(ξ, η) elliptic coordinates

metric_and_unit_elliptic(ρ, ξ, η)

Computes the metric factor and unit vectors for the 2D elliptic coordinate system.

ce_m(m, q, z)

Even angular Mathieu function of order m with parameter q evaluated at z and its derivative.

Arguments

• m: Order of the Mathieu function
• q: Parameter related to the geometry (q = (kc*ρ)²/4)
• z: Coordinate (typically η)

se_m(m, q, z)

Odd angular Mathieu function of order m with parameter q evaluated at z and its derivative.

Arguments

• m: Order of the Mathieu function
• q: Parameter related to the geometry (q = (kc*ρ)²/4)
• z: Coordinate (typically η)

Ce_m(m, q, z)

Even Radial Mathieu function and its derivative. It's equivalent to the Modified Mathieu Function of first kind of order m with parameter q evaluated at z.

Arguments

• m: Order of the Mathieu function
• q: Parameter related to the geometry (q = (kc*ρ)²/4)
• z: Coordinate (typically ξ)

Se_m(m, q, z)

Odd Radial Mathieu function and its derivative. It's equivalent to the Modified Mathieu Function of first kind of order m with parameter q evaluated at z.

Arguments

• m: Order of the Mathieu function
• q: Parameter related to the geometry (q = (kc*ρ)²/4)
• z: Coordinate (typically ξ)

first_n_modes_rwg(N, a, b)

Returns the first N modes for a rectangular waveguide with width a and height b.

Returns

A vector of tuples (kind, m, n, kc) where:

• kind: Mode type (:TE or :TM)
• m, n: Mode indices (number of half-wavelength variations in x and y directions)
• kc: Cutoff wavenumber

Mode restrictions

• TE modes: m + n > 0 (at least one index must be non-zero)
• TM modes: m ≥ 1 and n ≥ 1 (both indices must be positive)

The dominant mode is TE₁₀ (or TE₀₁ if b > a).

Performance

This implementation uses a heap-based algorithm and is much faster than brute-force sorting over all possible (m,n) combinations, especially for large N.

first_n_modes_cwg(N, r)

Returns the first N modes for a circular waveguide of radius r.

Returns

A vector of (kind, m, n, kc) where:

• kind: Mode type (:TE, :TM, :TEc, :TEs, :TMc, :TMs)
• m, n: Mode indices
• kc: Cutoff wavenumber

Mode notation

For m > 0, there are two degenerate polarizations with the same cutoff frequency:

• :TEc, :TMc: Cosine modes (cos(mθ) variation)
• :TEs, :TMs: Sine modes (sin(mθ) variation)

For m = 0, only :TE and :TM are returned (no polarization degeneracy).

first_n_modes_coax(N, r1, r2)

Returns the first N modes for a coaxial waveguide with inner radius r2 and outer radius r1.

Returns

A vector of(kind, m, n, kc) where:

• kind: Mode type (:TEM, :TE, :TM, :TEc, :TEs, :TMc, :TMs)
• m, n: Mode indices
• kc: Cutoff wavenumber

Mode notation

The TEM mode (:TEM) has zero cutoff frequency and is the dominant mode.

For m > 0, there are two degenerate polarizations with the same cutoff frequency:

• :TEc, :TMc: Cosine modes (cos(mθ) variation)
• :TEs, :TMs: Sine modes (sin(mθ) variation)

For m = 0, only :TE and :TM are returned (no polarization degeneracy).

first_n_modes_radial(N, h)

Returns the first N modes for a radial waveguide with height h between parallel plates.

Returns

A vector of (kind, m, n, kc) where:

• kind: Mode type (:TEM, :TE, :TM)
• m: Azimuthal mode index
• n: Vertical mode index
• kc: Cutoff wavenumber (radial propagation constant)

Note

The TEM mode (m=0, n=0) has zero cutoff frequency and corresponds to radial propagation between parallel plates.

first_n_modes_ewg(N, a, b)

Returns the first N modes for an elliptic waveguide with semi-major axis a and semi-minor axis b.

Returns

A vector of (kind, m, n, kc) where:

• kind: Mode type (:TEe, :TEo, :TMe, :TMo)
• m, n: Mode indices
• kc: Cutoff wavenumber

Mode notation

Unlike circular or coaxial waveguides, elliptic waveguides have even and odd modes that are not degenerate (they have different cutoff frequencies):

• :TEe, :TMe: Even modes (using even Mathieu functions, cosine-elliptic)
• :TEo, :TMo: Odd modes (using odd Mathieu functions, sine-elliptic)

For m = 0, only even modes exist (:TEe, :TMe). For m > 0, both even and odd modes exist with different cutoff frequencies.