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Beam Propagation Method

Paraxial wave equation solved with the Crank–Nicolson scheme. Launch a Gaussian beam into a step-index waveguide and watch the intensity evolve along the propagation axis.

v1.0.0·Updated 2026-06-10
Model & Assumptions Beta
Model
Paraxial scalar beam propagation method (BPM). The paraxial Helmholtz equation is solved with a Crank–Nicolson finite-difference scheme — unconditionally stable and second-order accurate.
Assumptions
Paraxial approximation (small angles relative to the propagation axis), scalar field (no polarization), 2D cross-section (x–z plane), step-index waveguide profiles.
Validity
Accurate for weakly guiding structures and beam propagation angles below ~15°. Not valid for high-contrast waveguides, backward reflections, or resonant cavities.
Limitations
No vectorial effects, no backward-propagating waves, no nonlinear media. Absorbing boundary conditions (quadratic imaginary-n ramp) may introduce small artifacts near the window edges.
References
K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006), Chapter 7.

Parameters

Propagation
Beam
Grid
Waveguide
Boundary conditions

Crank–Nicolson BPM. Solves the paraxial Helmholtz equation ∂zu = (i/2km) (∂2xu + (k²−km²) u) with a second-order implicit scheme that is unconditionally stable. Absorbing BCs add a quadratic imaginary-n ramp at the window edges to damp outgoing waves.

Intensity |v(x,z)|²