A computer model of a three-dimensional diamond lattice reports a strongly localized light mode when one resonant scatterer is replaced by a defect, inside the crystal's photonic band gap, the frequency range in which the modeled host has no ordinary resonant states. The lowest reported in-gap decay rate, expressed relative to the model's reference rate as Γ/Γ0, was 5.49 x 10^-8 at δ = -1.02, and the corresponding mode was strongly localized near the defect.
The study is a version-one arXiv preprint dated 26 August 2026. Its authors compared analytical Green-function results for an infinite crystal with numerical finite-crystal simulations using the coupled-dipole method, allowing them to examine how the predicted modes changed when the modeled lattice had a boundary.
The defect sits inside a forbidden range
For the modeled host with lattice constant k0 a = 3.4, the calculated density of states, a measure of how many resonant states are available at each frequency, vanished within numerical resolution from δ = -2.50 to δ = -0.62. The analysis identifies this interval as a complete photonic band gap. No uncertainty interval was reported for the numerical boundaries.
To connect the defect to that gap, the study examined the crystal Green's function, a mathematical description of the lattice's response to a source. Off-site Green-function elements fell to zero within the range of two unit cells. In the infinite-crystal calculation, the condition for a defect mode allowed solutions only inside the gap and was independent of where the defect was placed.
Most resonances in the defect-free calculation were extended across the lattice, with an inverse participation ratio, or IPR, of roughly 1/N, about 10^-4. IPR is the concentration measure used in the model: higher values indicate that a mode is spread over fewer scatterers. Against that baseline, the reported minimum-decay mode was strongly localized near the defect.
Six modes in the finite model
In a finite crystal with a single defect, the calculation yielded six localized quasinormal modes, or resonances that can leak energy. Three were inside the photonic band gap and three were outside it. The in-gap eigenfrequencies agreed very well with the infinite-crystal calculation.
When the defect was strongly detuned from the host resonance, the outside-spectrum modes appeared near the defect frequency. Their normalized decay rate was about Γ/Γ0 = 1 and their IPR about 1, matching the modeled behavior of single-scatterer excitations decoupled from the photonic crystal.
The in-gap mode approached a different limit. For a strongly detuned defect, its detuning moved toward δ ≃ -2.13 and its normalized decay rate toward Γ/Γ0 ≃ 4 x 10^-5; the mode converged to the single-vacancy state. At k0 a = 3.4, the calculated vacancy state appeared at δ = -2.1, while for k0 a ≥ 4.8 the authors expected no in-gap vacancy state.
Size changes the calculated leakage
The clearest finite-size pattern came from a spherical radius series with k0 R values of 10, 15, 20, 25 and 30, corresponding to 849, 2,869, 6,851, 13,331 and 22,929 scatterers. The longest-lived mode stayed near δ ≃ -1.05 as the radius increased. Its normalized decay rate fell from 2.25 x 10^-5 at k0 R = 10 to 3.48 x 10^-13 at k0 R = 25. At k0 R = 30, four points fell below numerical precision, at Γ/Γ0 < 10^-16.
The fitted relationship was exponential, with a localization length of order the lattice spacing a. That fit comes with an important caveat: the four largest-size values were censored by numerical precision, and the study reported no fit uncertainties. The quoted rates therefore describe the model's calculated trend more securely than they pin down the exact behavior of the largest crystals.
Geometry also enters the picture
The shape comparison used a spherical sample with k0 R = 15 and 2,869 scatterers, and a cubic sample with a caption-reported k0 L = 24 and 3,059 scatterers. Along the high-frequency density-of-states plateau, defect modes in the cubic shape tended to have lower decay rates and higher IPRs than their spherical counterparts.
The study remains an idealized modeling result. Its evidence comes from the infinite-crystal calculation and finite coupled-dipole simulations of single defects in selected spherical and cubic samples, not from experimental measurements. The reported modes, decay rates and shape effects therefore remain predictions of this model rather than demonstrated device performance.
Paper data and sources
Original title: Defect states in three-dimensional diamond photonic band gap crystals
Authors: Julia Rocha, Bart A. van Tiggelen, Ad Lagendijk et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text