In a strip-loaded waveguide, the mode is confined within a plane by a high-index thin film and is loosely confined in the lateral direction by a lower index strip, as shown in Figure 1Ai. We choose an HSQ strip, titanium dioxide core, and silicon dioxide substrate, and optimise the waveguide dimensions for maximum lateral confinement of light with a vacuum wavelength of approximately 600 nm. The optimal dimensions are as follows: strip height of 180 nm, strip width of 300 nm, and core thickness of 70 nm. A Bragg cavity is constructed based on this waveguide geometry, with the Bragg reflectors being twice as long as the waveguide is wide, as shown in the inset of Figure 1Aii. The transmission spectrum of this cavity is shown in Figure 1Aii for different values of the total number of Bragg reflectors N. For N = 100, the quality factor is found to be 230 and the outcoupling efficiency, defined as the cavity-waveguide coupling ratio, is 90%. The cavity mode is shown in Figure 1C, and the mode volume is found to be 6.96 λ / n 3 . We also show the stopband in Figure 1Aiii and, for comparison, in Figure 1B the equivalent results for a plasmonic cavity with the same HSQ strip [20].

The quantum emitter should be embedded in the cavity at the position where the cavity mode electric field is maximal. Emitted photons need to have both high indistinguishability, defined as the ability of two photons to interfere on a beam-splitter in a Hong-Ou-Mandel setup, and high quantum efficiency, defined as the fraction of photons emitted into the cavity. In the narrowband-enhancement weak-coupling regime (see Figure 2), the indistinguishability and efficiency of emitted photons are given by = F P + 1 F P + 1 + 2 γ Γ , and η = B 2 F P B 2 F P + 1 , respectively, where γ is the dephasing rate and B , known as the Frank-Condon factor, is the square-root probability of photon emission into ZPL [10]. For example, for a quantum dot at 4K the optimal values of Q and V eff — corresponding to κ ∼ ξ / 10 and g ∼ ξ / 20 —are given by Q = ω 2 κ , and V eff = ω μ 2 2 ℏ ϵ 0 n 2 g 2 , where ω is the angular frequency and μ the dipole moment. Taking μ = 27.2 D [21], 2 π c ω = 600 nm and ξ = 1.45 meV, we find the optimal values Q = 7,100 and V eff = 7   λ / n 3 . We should note that the wavelength used in these calculations is that of our proposed cavity, which differs from that of typical QDs by a factor of 1.5.

To determine the indistinguishability, we follow ref. [10]: we assume that the dominant contribution to the dephasing γ is due to phonons. This assumption holds in electrically contacted structures, where charge and spin noise are minimized [22]. We also assume a bulk phonon spectrum, achievable through clamping [23]. The dephasing then depends on g and κ through Equation 2: γ = 2 π g B κ 2 J 2 g B coth ℏ g B k B T (2) where J ν = α ν 3 ⁡ exp − ν ξ 2 , α = 0.03 p s 2 is the exciton–phonon coupling strength, T is the temperature, and we take B = 0.95 [10]. In Figure 2, we show the combined value I η as a function of V eff and Q . The regions outside the narrowband-enhancement weak-coupling regime have been excluded. At the point where F P is at its maximum, the indistinguishability is I = 0.97 and the efficiency is η = 0.98 .

The concept of the directional coupler, introduced in the mid-20th century, has been foundational for many advancements in optical and microwave technologies [24]. While the fundamental idea remains relevant, the design of the entire structure—including S-bends and a chain of materials tailored for specific applications—continues to hold significant importance in contemporary engineering. More recently, the directional coupler has become a crucial component in photonic quantum integrated circuits, demonstrating the capability to facilitate quantum computing logic operations when employed with single photons [25–27].

In this section, we present two platforms for directional couplers: one based on dielectric-loaded surface plasmon polariton (DLSPP) waveguides and the other on strip-loaded waveguide configurations. We then compare the two structures in terms of their applicability for single-photon routing. Notably, the overall dimensions of both platforms are consistent with the Bragg grating cavity structure discussed in the previous section.

The directional coupler is designed for both parallel straight waveguides and S-bends based on sine curves, enabling continuous bend curvature and adiabatic modification [28–30] of the DLSPP waveguide and the strip-loaded waveguide mode throughout the bend, as shown in Figures 3Ai, Bi, respectively. One crucial aspect to consider is the design of the S-bend. As illustrated in Figure 3A, the losses associated with the S-bend can be categorised into transition losses and radial losses.

Radial losses occur due to the limited guiding of the propagation mode caused by speed limitations beyond the outer end of the bent waveguide. Transition losses occur when there are discontinuities in the curvature of the waveguide, where sudden changes in modal propagation characteristics take place. The guided modes in curved waveguides are broader compared to straight waveguides and change outward along the curve. The normalised transition loss due to mismatch between the straight waveguide mode and the curvature can be calculated as follows in Equation 3 [29]. Γ T = 1 − ∬ E i n x , y E o u t * x , y d y d x 2 ∬ E i n x , y 2 ∬ E o u t x , y 2 d x d y (3) where E i n and E o u t represent the electrical input and output mode fields, respectively. The normalised radial loss, related to the radial attenuation coefficient α R , l per unit length l in a bend radius R , can be expressed as [29]: Γ R = exp − ∫ L o α R l ′ , l ′ d l ′ .

In this study, we utilised the commercial software Lumerical FDTD to calculate the minimum bending radius R_min. The design of the S-bends was based on sine curves [31, 32], allowing for continuous bend curvature and thereby adiabatic modification of the mode throughout the bend. For the DLSPP structure, due to its high optical contrast and strong confinement of the DLSPP mode, a curvature radius as small as 2 μm still transmitted a significant portion of the DLSPP mode (>50%). This finding is consistent with experimental measurements using scanning near-field optical microscopy (SNOM) [31].

For the strip-loaded structure, where the effective mode index with HSQ cladding differs from that with air by a small amount of 0.02 (from 1.64 to 1.62, as shown in Figure 3Bii), the minimum bending radius must be sufficiently large to allow the light to bend effectively. We calculated this value from simulations to be 100 µm [33], which is close to the value obtained from the analytical expression in reference [34]. From this expression, R min ≈ 0.1325 λ 0 / n 1 − n 2 / n 1 3 / 2 , we estimated the minimum bending radius to be approximately 70 μm, where n 1 and n 2 are the effective indices with and without HSQ cladding, respectively.”

This characteristic is crucial for designing optical devices with tight bends and for sensitive optical applications, such as single-photon routing. However, in applications like single-photon sources, the overall propagation length is limited by structural and environmental factors when the emitter is coupled to the waveguide. Although the photon propagation length is generally longer than that of SPPs, issues may arise concerning the structure’s overall length. Additionally, in these structures, distinguishability may be significantly compromised at very high dimensions. Nevertheless, the strip-loaded structure remains a strong option for applications where minimising loss is the primary concern.

The interaction length required to achieve the desired power coupling to the second waveguide is calculated using L c = λ / 2 n e f f + − n e f f − sin − 1 √ P o u t 2 / P i n ¯ . For a π phase shift, the relation simplifies L c = λ / 2 n e f f + − n e f f − [35, 36]. Figure 3Aiii and Figure 3Biv show the numerical calculations of the coupling length for both DLSPP and strip-loaded structures, using symmetric (Figure 3Aii top and Figure 3Biii top) and asymmetric modes (Figure 3Aii bottom and Figure 3Biii bottom) at a wavelength of 700 nm. While strip-loaded structures generally exhibit lower loss compared to DLSPP structures, their large radius of curvature can still limit their practical application in certain scenarios.

Another approach to enhancing the Purcell effect is to use DLSPP waveguide-ring resonators (WRRs). A WRR, consisting of a straight waveguide laterally coupled to a ring resonator, typically provides more pronounced wavelength selection and a high Q -factor, which is crucial in integrated optics. It is important to note that the extinction ratio, which indicates the contrast in transmission through a WRR between resonant and non-resonant wavelengths, is affected by both the coupling efficiency and the internal loss within the resonator [37]. In plasmonic WRRs, coupling efficiency is usually low due to the tight confinement in plasmonic waveguides, while internal loss is high due to propagation loss. As a result, achieving critical coupling for a high extinction ratio can be challenging.

Figure 4A shows a 3D schematic of the DLSSP-WRR structure, designed with the same parameters discussed in the previous section. Figure 4B presents the transmission spectrum for a ring with a radius of 2.5 µm and three different separations (g) of 200 nm, 150 nm, and 100 nm. A 100 nm gap provides a higher extinction ratio compared to the other gaps, though this enhanced coupling efficiency also leads to mode broadening. Figure 4C depicts the full profile of the structure at a resonant wavelength of 616.4 nm and a non-resonant wavelength of 623 nm. The bandwidth of the WRR is determined by the radius of the ring resonator and the wavelength-dependent effective index of the bent waveguide. Figure 4D displays the transmission spectra for a 100 nm separation and ring radii ranging from 2 μm to 3.5 µm in 0.5 µm increments. Increasing the ring radius improves the maximum resolution of the DLSSP, resulting in a higher quality factor. Figure 4E shows the mode area of the 250 nm width HSQ as a function of HSQ thickness (h). The mode area definition related to the Purcell effect, based on [10], indicates that as HSQ thickness increases from 160 nm to 240 nm, the effective mode area remains unchanged. However, for h < 160 nm, mode leakage into the air cladding increases the mode area. Additionally, we have computed the mode area, normalized Purcell factor, and quality factor of the structure for various ring radii. As the ring radius increases, the bandwidth decreases, leading to an increase in the quality factor. For a ring with a radius of 2.5 µm, the Purcell factor and quality factor are ∼10 and ∼133, respectively, which are relatively high compared to a strip-loaded waveguide.