The Purcell factor and emitters near nanostructures

The factor by which an emitter's spontaneous decay rate changes because of its surroundings: F = γ / γ₀, its rate near a structure over its rate in a uniform medium. Edward Purcell pointed out in 1946 that a resonant cavity can strongly enhance emission; nanoparticles and antennas do the same through their near fields.

For an emitter in a resonant cavity, on resonance and aligned with the field maximum: F_P = (3/4π²) (λ/n)³ (Q/V), with the cavity's quality factor Q and mode volume V. Near a nanoparticle the mode picture fails (the "cavity" is very lossy and open), and the rate is computed directly from the field the emitter's own dipole creates at its position: γ/γ₀ = 1 + (6πε₀ε_m/|p|² k³) Im(p* · E_s), where E_s is the field scattered back by the particle.

From Fermi's golden rule, the decay rate is proportional to the local density of optical states at the emitter's position and orientation. Classically, the same ratio equals the power radiated by a dipole near the structure over the power of the dipole alone, which is how it is calculated here.

Place a point dipole at the emitter's position and compare the power it emits with and without the structure: the total power (flux through a small closed surface around the dipole) gives the total decay rate, the power reaching a large surface gives the radiative rate. Mesh finely near the dipole; for spheres, spheroids and dimers the exact calculator gives the reference values.

Quantum efficiency is the share of decays that produce light: radiative rate over total rate. Within about 10 nm of a metal surface, much of the energy is absorbed in the metal instead, so the total rate rises but the emitted light can drop (quenching). The best distance balances the two, typically 5–20 nm for gold.

Yes. High-index particles have Mie resonances but hardly absorb: a dipole 10 nm from a 220 nm TiO₂ sphere decays about 6 times faster at 704 nm with a quantum efficiency close to 100 %.

Both concentrate the local density of states: the tip of a plasmonic nanorod at its longitudinal resonance and, even more, the gap between two particles or between a particle and a mirror, where enhancements of hundreds to thousands are possible for an emitter oriented across the gap.