Modal Coupling of Single Photon Emitters Within Nanofiber Waveguides

Nanoscale generation of individual photons in confined geometries is an exciting research field aiming at exploiting localized electromagnetic fields for light manipulation. One of the outstanding challenges of photonic systems combining emitters with nanostructured media is the selective channelling of photons emitted by embedded sources into specific optical modes and their transport at distant locations in integrated systems. Here, we show that soft-matter nanofibers, electrospun with embedded emitters, combine subwavelength field localization and large broadband near-field coupling with low propagation losses. By momentum spectroscopy, we quantify the modal coupling efficiency identifying the regime of single-mode coupling. These nanofibers do not rely on resonant interactions, making them ideal for room-temperature operation, and offer a scalable platform for future quantum information technology.


Purcell studies
Light sources placed inside a subwavelength nanofibre suffer from suppressed emission rates due to a smaller than unity Purcell factor at the high-low index (PMMA-air) interface coming from the reduced local density of optical states (LDOS) [S1]. We measured the decay rate Γ distribution of individual quantum dots inside free-standing nanofibres as shown in Figure 1d, and compared them to a reference PMMA film. While both systems show a large Γ distribution with a full width at half maximum around ~50%, when the emitters are inside the free-standing nanofibre the average decay rate decreases from 1210 -3 ns -1 to 910 -3 ns -1 ( Figure S1a). Assuming that the quantum dots dipoles randomly oriented in the nanofibre, the data agrees well with the theoretical predictions that we obtained through the finite difference time domain (FDTD) method obtained by considering the intensity emitted by a dipolar source in different position inside the nanofibre, plotted in Figure S1b and S1c. Figure S1b and S1c show the LDOS maps normalized to the bulk material for two dipolar orientations inside the PMMA sub-wavelength nanofiber (D = 300 nm). The Purcell factor is close to unity (in the range 0.8 to 1) in the case of a longitudinal dipole ( Figure S1b) and conversely, it is reduced to around 0.5 for a transverse dipole ( Figure S1c). For a transverse dipole in a subwavelength Silicon fibre (n = 3.4, D =130nm) the Purcell factor is ~0.1 Figure S2 shows the emitted intensity from a single quantum dot embedded in a freestanding nanofibre as a function of the illumination power. A typical saturation curve is, with a saturation at an average pump intensity of ~0.5 μW, where we collect 25 kphotons/s from the nanofibre. Well above saturation, the maximum measured count-rate is 38 kphotons/s, when the dot emission is expected to approach 2.5 Mphotons/s, given our laser repetition rate of 2.5 MHz, a value which is limited by the less than unitary quantum efficiency (~0.7) and 3 by the off-state of the quantum dot (~ 30% on-state due to blinking) to a value of ~0.5

Quantum dot saturation
Mphotons/s. The line in Figure S2 is a fit to the data using the saturation curve of a two-level system: S = S∞ (I/Is)(1 + I/Is) -1 [1 -exp{-(1 + I/Is) τp/τf}] where I and Is are respectively the excitation and the saturation intensities, respectively, and τp the excitation laser pulse width (100 ps) and τf the decay time of the quantum dots [S2]. From the fit we obtain a saturation count rate S∞ = 42 kphotons/s and Is = 7 KWcm -2 .

Modal coupling calculations
The total and modal coupling of a quantum dot to the nanofibre modes is calculated by FDTD simulations using a commercial package (Lumerical). Figure S3 shows both the total coupling (β) and modal (β01) to a specific fibre mode, the first excited one LP11. The total coupling efficiency was obtained by calculating the intensity transmitted through a monitor crossing the free-standing nanofibre 4 µm away from the source; the modal coupling efficiency was obtained by mode projection of the same intensity onto the independently calculated nanofibres modes. The total coupling remains roughly constant for larger diameters, with oscillations due to the emergence of the different modes. Instead, the coupling to the fundamental mode β01 peaks at / = 0.55 and then quickly drops to zero for larger diameters, indicating that selective coupling to the fundamental mode is best achieved for subwavelength nanofibres.

Momentum spectroscopy
We confirm numerically the accuracy of the estimation of the quantum dot -nanofibre coupling by momentum spectroscopy by FDTD calculations. Angular patterns were obtained by means of far field projections of the electro-magnetic fields in a plane 300 nm outside the nanofibre inside the substrate of a nanofibre laying on glass. 4 In Figure S4 the coupling to a nanofibre lying on glass calculated by far-field projections is compared to the value obtained by mode projection of the light transmitted in a free-standing nanofibre . The agreement is within 10% for both dipole orientations, for the considered diameter range D = 300-1000 nm. The discrepancy is explained due to the variable transmissivity of different k-components to the far field. For small diameters, the overestimation is also explained in terms of light emitted in the upper direction and not collected by the objective. In the real experiment, the collection is limited by the objective NA and response, which for large k-vectors further lowers the apparent coupling.

Broad-band light coupling and transport
The nanofibre has a broadband response which encompasses the emission spectrum of the quantum dots. In Figure S5 we report the spectrum of the light emitted at the edge of an intentionally cut nanofibre once a distant quantum dot is excited. The quantum dot emission is well transported in its entirety, for a bandwidth of at least ~ 100 nm.

Nanofibre morphology
The morphology of the realized PMMA nanofibers embedding quantum dots was investigated by atomic force microscopy (AFM). Supporting Figure S6a shows a typical AFM topographic image of a single nanofibre. The nanofibre has a circular cross section (ratio between the height H and the diameter D, is H/D=0.96 in Fig. S6b). The AFM analysis performed on several fibers showed that H/D ratios are in the range 0.9-1, evidencing the uniformity of the cross-sectional shape of the produced fibers. Moreover, the variation of the nanofibre diameter over a length of few micrometers is of the order of 1% or less (Fig. S6c).
To investigate this issue more in depth, we also measured the variation of the nanofibre 5 diameter over a length of the order of 1 mm (Fig. S6d-h). A maximum variation of the diameter, ΔD, of the order of 70 nm was measured over a length L=0.8 mm, providing a ratio ΔD/L=9×10 -5 . Another important fiber morphological property for efficient light transport is related to the surface roughness. Indeed, Rayleigh scattering from surface defects is proportional to the variance of the surface roughness squared [S3]. AFM data provided a nanofibre surface roughness of about 3 nm, allowing for estimating a light transport length in the range 100-1000 µm.