Each metal nanorod is modelled as a point dipole with a resonant, Lorentzian polarizability. Only the central feed rod is driven directly (by a local emitter placed in its gap); the reflector and director rods are passive — they radiate only because the near field of their neighbours induces a dipole moment in them. This is the discrete-dipole / coupled-dipole approximation used to model plasmonic nanoparticle arrays:
p_i = α_i · ( E_inc,i + Σ_j≠i G(r_ij) p_j )
α_i = 1 / (δ_i + iΓ) (driven-oscillator resonance)
G(r) = e^(ikr)·( k²/r + ik/r² − 1/r³ ) (dipole near/far-field coupling)
δ_i is how far rod i's length sits from resonance: δ=0 excites in phase with the field that drives it, δ>0 (longer rod, inductive) lags in phase, δ<0 (shorter rod, capacitive) leads. That is exactly the trick behind a Yagi-Uda antenna — a lagging reflector behind the feed and leading directors in front of it make every rod's re-radiated field arrive in phase along the forward axis and out of phase backward, so the lobes add up front and cancel behind.
The far-field pattern in the plane perpendicular to the rods is the array factor of the solved dipole moments:
AF(φ) = Σ_i p_i · e^(i k x_i cos φ), I(φ) = |AF(φ)|²
- Rod spacing d/λ — sets the phase delay light picks up travelling between rods; too small and the near-field coupling dominates, too large and grating lobes appear.
- Directors — how many leading (shorter) parasitic rods sit in front of the feed; more directors narrow and strengthen the forward lobe, up to a practical limit.
- δR / δD — how far the reflector/directors are detuned from resonance; the classic Yagi-Uda optimum is a modest positive reflector detuning and a smaller negative director detuning, which "Snap to optimal" sets directly.
Real-world relevance: single-molecule and quantum-dot optical antennas built exactly this way (Curto et al. 2010; Novotny & van Hulst) redirect a nanoemitter's normally isotropic light into a narrow forward beam — the optical analogue of a TV rooftop Yagi-Uda antenna, useful for single-photon sources and on-chip photonic links.