Harnessing the Diamond-Air Interface as an Efficient Photon Antenna for Solid-State Emitters

TL;DR

This work investigates photon extraction from NV centers just below the diamond surface by treating the diamond–air interface as a dielectric antenna. Using back focal plane imaging and an analytical dipole-emission model, the authors show that back-side oil-immersion collection can achieve high collection efficiencies and photon rates, approaching those of nanofabricated optics, with measured saturated rates around $5.11\times10^5\,\mathrm{s}^{-1}$ and a maximum CE near $0.54$; they also identify spherical aberrations as a limiting factor that reduces the effective NA. The study provides a method to quantify the quantum efficiency of shallow defect centers (approx. $2\%$ in this work) and discusses possible reasons for discrepancies with higher QE values reported in the literature. Broadly, the results highlight a practical route to efficient, non-nanofabricated photon collection for solid-state emitters and offer a framework to compare QE across emitters and geometries, with implications for sensing and quantum networks.

Abstract

Extracting photons from defect centers is challenging due to the high refractive index of typical substrates. For nitrogen-vacancy centers in diamond, reaching saturation count rates above $2.5\times10^5\,\mathrm{counts}/\mathrm{s}$ so far requires nanofabricated optics like diamond waveguides or solid immersion lenses. Here we present an experimental and theoretical study of defect center emission at unmodified planar dielectric surfaces by quantitative back focal plane imaging and analytical modeling. Our results indicate that photon count rates approaching those of nanofabricated optics can also be achieved by oil-immersion optics. This is due to a dielectric antenna effect which directs the majority of the emission into the substrate within a narrow angle window suitable for back-side collection. By quantifying the collection efficiency of back-side detection, our work also enables a novel measurement method for the quantum efficiency of shallow defect centers. Its result challenges established values. Possible reasons for the discrepancy are discussed.

Figures (4)

• Figure 1: Experimental setup. a) Schematic of the optical setup. Obj.: Objective, DM: dichroic mirror, LP filter: longpass filter, BFP: back focal plane b) Confocal photoluminescence image of the diamond sample with color-coded crameri_misuse_2020 photon rate. The NV center used for the quantitative analysis in Fig. \ref{['fig:quantitative']} is encircled in white. c) Photoluminescence intensity in the image plane of a single NV center. Scale bar width corresponds to 20 or $9.1$ optical units. d) Image of the BFP for the emission of a single NV center. Scale bar width corresponds to 400.
• Figure 2: Measurement and simulation of angular emission intensity. a, b) BFP images for two different NV centers. Scale bar width corresponds to 400. c) Schematic of emission intensity simulation. d) Simulation of the angular emission intensity. The black, dashed (gray, dash-dotted) lines indicate the angle of total internal reflection at the diamond-air (diamond-oil) interface. e) BFP image converted to angular emission intensity. The red, dotted line marks the cut-off angle due to the NA of the microscope objective. f) Simulated and measured angular emission intensity separately integrated over the azimuthal coordinate for the upper and lower part of the images in d, e.
• Figure 3: Quantitative photon rate analysis in image plane and BFP. a) Emission intensity in NV center image plane with logarithmic color scale. The green, dotted outline indicates region of interest (ROI) with $1.4\,\mathrm{mm}$ diameter. Scale bar width corresponds to 300. b) Zoom into the image in a), but with linear color scale. The circular ROI with yellow, dashed outline shows the edge of the Airy disk with 27 diameter. Scale bar width corresponds to 20. c) Unsaturated photon rate within circular ROIs with different diameter in the NV center image plane with the uncertainty illustrated by the shaded region. The yellow, dashed (green, dotted) line marks the photon rate for a 27 ($1.4\,\mathrm{mm}$) ROI (cf. a and b). The x axis is scaled quadratically. d) Azimuthal average of the photon rate when defocusing the detection plane from the pinhole plane. e)-h) Photon rate in the BFP for different diameters of the variable aperture. Scale bar width corresponds to 400. i) Photon rate as a function of the diameter of the variable aperture. Yellow, dashed line: photon rate for the inner ROI in the BFP, as indicated in e)-h). Green, dotted line: total photon rate in the BFP, as indicated in e)-h). Red, solid line: total photon rate within the Airy disk, measured with confocal setup (see SI), value corresponds to intensity encircled in the ROI in b). Note: The stated photon rates are for an unsaturated NV center and correspond to count rates for a detector with unity photon detection efficiency. See the main text for the saturation count rate achievable with typical single photon detectors.
• Figure 4: Simulation of the photon collection efficiency. a) Angular emission density. The blue line on the upper part shows the emission intensity towards the top, through the diamond-air interface. The yellow (black, dashed) line show the emission towards the back side including (excluding) the Fresnel transmission coefficients at the diamond-oil interface. b) Collection efficiency for different detection geometries. For oil immersion with $n=1.51$ ($n=1.78$) a numerical aperture of $1.45$ ($1.7$) was used.