Mode Sources

A mode source combines electric and magnetic eigenmode profiles, a propagation direction, and a temporal signal to launch a guided wave.

The transverse eigenproblem

For a structure invariant along $z$ , write a monochromatic mode as

\mathbf E(x,y,z,t)=\operatorname{Re}\!\left\{ \mathbf e(x,y)e^{i\beta z-i\omega t}\right\},

\mathbf H(x,y,z,t)=\operatorname{Re}\!\left\{ \mathbf h(x,y)e^{i\beta z-i\omega t}\right\}.

The profiles $\mathbf e,\mathbf h$ satisfy frequency-domain Maxwell equations on the cross-section. With $\nabla_\beta=(\partial_x,\partial_y,i\beta)$ , one formulation is

\nabla_\beta\times\mu^{-1}(\nabla_\beta\times\mathbf e) =\omega^2\varepsilon\mathbf e.

At a specified $\omega$ , the solver finds allowed propagation constants $\beta$ and their profiles subject to transverse boundary conditions. The effective index is

n_{\mathrm{eff}}=\frac{\beta}{k_0},\qquad k_0=\frac{\omega}{c_0}.

A vector mode generally has longitudinal components too. For ordinary lossless index guiding, $n_{\mathrm{eff}}$ typically lies between the cladding and core indices; this check does not cover leaky or periodic structures.

Selecting the intended mode

Select by field shape, polarization, symmetry, confinement, and effective index. Labels such as “mode 0” can change physical identity as geometry or frequency changes.

BeamZ solves modes on a rasterized cross-section before FDTD stepping. Sources and modal measurements share plane extraction to use consistent material crops and field supports.

Place the source in a straight, uniform section and include the modal tails in its transverse window.

Power normalization

For a plane with forward unit normal $\hat{\mathbf n}$ , the time-averaged signed power of a phasor mode is

P=\frac12\operatorname{Re}\int_A (\mathbf e\times\mathbf h^*)\cdot\hat{\mathbf n}\,dA.

Scale both profiles by the same amplitude. If a selected forward mode has $P_0>0$ , a target power $P_{\mathrm{target}}$ requires

\mathbf e\longrightarrow\sqrt{\frac{P_{\mathrm{target}}}{P_0}}\,\mathbf e, \qquad \mathbf h\longrightarrow\sqrt{\frac{P_{\mathrm{target}}}{P_0}}\,\mathbf h.

In 2D, the line integral gives power per invariant length (W/m), unless a thickness convention is specified. The formula cannot normalize a zero-power evanescent mode.

Evaluate electric and magnetic fields at compatible locations with consistent quadrature before computing Poynting flux.

Directional injection

An electric-field-only excitation generally launches both forward and backward waves. A directional source needs the compatible magnetic profile and their relative phase. In the surface-equivalence picture, a normal pointing from the zero-field side toward the desired-field side gives equivalent sheet currents

\mathbf J_{\mathrm{sheet}}=\hat{\mathbf n}\times\mathbf H_{\mathrm{mode}}, \qquad \mathbf M_{\mathrm{sheet}}=-\hat{\mathbf n}\times\mathbf E_{\mathrm{mode}}.

The magnetic sheet current is a mathematical excitation. A discrete total-field/scattered-field construction implements these currents as curl-update corrections at the source plane.

With the chosen phasor convention, sampling at $(n+1/2)\Delta t$ adds the phase factor $e^{-i\omega\Delta t/2}$ relative to $n\Delta t$ . Spatial offsets also carry propagation phase. Ignoring either Yee offset can produce a backward wave.

BeamZ maps the solved fields onto its Yee supports and checks discrete consistency for launch.

Broadband excitation

A mode profile is computed for a particular frequency and discretized cross-section; it retains numerical error. Using it across a pulse bandwidth assumes weak profile variation, which can fail near cutoff or with strong modal dispersion.

In 3D, BeamZ can reconstruct a broadband source from several mode profiles. Set ModeSpec(num_freqs=...) to more than one: the compiler solves profiles at sampled frequencies, partitions the temporal signal into frequency contributions, and combines their injections. This reduces reliance on one carrier-frequency profile, but still requires convergence checks. num_freqs is independent of a monitor’s measurement frequencies. See the source compiler.

BeamZ capabilities

Current mode sources reject conductive material grids and explicitly supplied nonuniform rectilinear grids. GridSpec.auto() selects a compatible uniform grid when modal devices are present. The 2D and public convenience mode solvers use a cell-tensor approximation for diagonal smoothing and reject full off-diagonal coupling; the 3D launch path retains Yee-aware refinement. These limits differ from those of the rasterizer and FDTD solver. See the capability notes.

Validation

Test in a straight reference waveguide: inspect the profile, let startup transients pass, and measure forward and backward modal power away from the source. Use the same excitation for the device run; a reference ratio helps remove the temporal source spectrum from transmission.

For broadband measurements, compute modal projections at the measurement frequencies. Total flux includes radiation and other modes; modal transmission isolates the selected mode.

Return to the Theory overview or try the simulation examples.