FDTD Near-to-Far-Field (NTFF)

Balanis — Antenna Theory: Analysis and Design, 4th ed., §6.7 (Wiley 2016).

cad-sim::antenna ships the only MIT/Apache implementation of the Huygens-equivalence Near-to-Far-Field transform in the open-source ecosystem. Every existing OSS FDTD with a built-in NTFF is GPL:

• openEMS — GPL-3.0
• MEEP (NanoComp / MIT) — GPL-2.0
• gprMax — GPL-3.0-or-later

cad-sim::antenna::ntff_far_field_at re-derives the integral from textbook Balanis so openie-cad's MIT/Apache posture stays clean.

The problem

A finite-difference time-domain simulation gives you E and H fields everywhere on the Yee grid, but radiation patterns are properties of the far field — the field at distances large compared to the radiator and the wavelength. Computing the far field by extending the FDTD grid out to many wavelengths is wasteful: most of that volume is empty propagation.

The Huygens equivalence principle reduces the problem to a closed surface integral: pick any closed surface enclosing the radiating structure, record the tangential E and H fields on that surface, and the far-field anywhere outside is determined.

The transform

Define equivalent surface currents on the Huygens surface:

J = n̂ × H        (electric surface current, A/m)
M = −n̂ × E      (magnetic surface current, V/m)

The far-field at observation direction r̂(θ, φ) is:

E_θ(θ, φ) ∝ Σᵢ [Jᵢ·θ̂ + (1/η) Mᵢ × r̂·θ̂] · Aᵢ · e^{jk r̂·rᵢ}
E_φ(θ, φ) ∝ Σᵢ [Jᵢ·φ̂ + (1/η) Mᵢ × r̂·φ̂] · Aᵢ · e^{jk r̂·rᵢ}

with the leading factor −jωμ / 4π.

The phase factor e^{jk r̂·rᵢ} is the only direction-dependent piece; everything else is precomputed per sample. The whole transform is a matrix-vector multiply.

openie-cad implementation

The Wave 69 substrate tests verify:

• A single Hertzian dipole sample reproduces the analytic sin θ pattern shape with consecutive-angle ratios within 10⁻⁶ of the expected sin θ ratio.
• Two unit elements at z = ±λ/4 in phase show the classic broadside pattern (|E_θ| at θ = 90° dominates near-axis values by >5×).
• The pattern sweep peak lands near θ = 90° (broadside) for a z-oriented dipole, gain normalised to 0 dB at peak.

The Wave 69.B substrate tests verify:

• A box surface (2..=5)³ emits 96 cells with correct ±unit outward normals; opposing-face normals sum to zero.
• A static Ex = 1 V/m field, sampled at t = 0, accumulates exactly (dt, 0) in the DFT phasor — pure cosine projection.
• At t = T/4 the same field accumulates (0, −dt) — quarter-period phase rotation correct.
• End-to-end: an FDTD grid with non-trivial Ez / Hy fields, sampled once, produces finite E_θ / E_φ when fed to the NTFF integral.

Calling it

The NTFF integral is a library function. There's no MCP tool yet — Wave 69.C will add one. For now, hosts that want antenna patterns run cad-sim FDTD locally and call the NTFF function in-process.

Reference

Balanis — Antenna Theory: Analysis and Design, 4th ed., §6.7 (Wiley 2016). The Huygens-equivalence integral form used here.