Near-unity efficiency in ridge waveguide-based, on-chip single-photon sources

First, we investigate the dependence of ε_xy on the waveguide geometry for an infinitely long and pristine waveguide (with no holes), whose cross-section and top view are depicted in figures 2(a) and (b). We consider a rigde waveguide of height h and width w in the ranges $h\in[100\, \mathrm{nm},300\, \mathrm{nm}]$, $w\in[200\, \mathrm{nm},580\,\mathrm{nm}]$ respectively, with a step-size $\Delta h = \Delta w = 10$ nm. Transparent boundary conditions are implemented using perfectly matched layers [33] to perform the infinitely spatial scattering simulations. Then, we compute the mode overlap and the emitted power by $x/y$-polarized dipole for each set of parameters to calculate the Purcell factors and efficiencies according to equations (1)–(3), and the results are shown in figures 2(c)–(g). The white areas correspond to the specific sets where no guided mode can be found.

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Comparing figures 2(c) and (d), the Purcell factor of y-polarized dipole F_y is more sensitive to the dimensional variation than that of x-polarized dipole F_x . This is because the oscillation axis of the y-polarized emitter (y-axis) is perpendicular to the waveguide cross-section (xz-plane) [37]. Next, we consider the sum of the power extracted from the top and bottom outputs, $P_ {\mathrm {M},\mathrm{top}}$ and $P_{\mathrm {M},\mathrm{bot}}$, as shown by the arrows in figure 2(b). In principle, $P_{\mathrm {M}}$ = $P_{\mathrm {M},\mathrm{top}}$+$P_{\mathrm {M},\mathrm{bot}}$. Figure 2(e) shows $P_{\mathrm {M}}$ normalized by its maximum value. $P_{\mathrm {M}}$ increases by reducing the waveguide dimensions to an optimum mode confinement.For a purely x-polarized emitter, the coupling efficiency ε_x can reach as high as 95%. In addition, the area with a small cross-sectional size can maintain high efficiency of over 90%, as shown in the dark-red area in figure 2(f), where we can also see ε_x diagonally drops, as indicated by the arrow. The reason is that a wider structure supports additional guided modes, thus decreasing the relative emission into the fundamental TE mode. The white cross in figure 2(g) represents the geometric parameters h = 140 nm and w = 220 nm yielding a maximum coupling efficiency ε_xy of 70% for an xy-polarized emitter. The structures that we investigate in the following will inherit this set of dimensions.

Comparing figures 2(d), (f) and (g), higher values of ε_xy coincide with high ε_x and low F_y . Moreover, the bright/dark areas in figure 2(g) almost perfectly correspond to the dark/bright areas in figure 2(d). This is due to $P_{\mathrm {M},y} = 0$ and the strong dependence of efficiencies on the dipole orientation (x). Therefore, the distribution of ε_xy relies on the interplay of ε_x and F_y , and a relative suppression of F_y will significantly contribute to a higher ε_xy . However, a bare waveguide always features $P_{\mathrm {M}, \mathrm{top}} = P_{\mathrm {M}, \mathrm{bot}}$, which means we will always lose 50% of the light if we only collect from the top. Therefore, in the following we will try to make $P_{\mathrm {M}, \mathrm{top}} \gg P_{\mathrm {M}, \mathrm{bot}}$ by engineering a highly reflective bottom mirror.

As depicted in figure 3(a), the DBR mirror is composed of cylindrical air holes with a number of $N_{\mathrm{bottom}}$. The hole radius r and the distance between successive hole centers, periodicity p_h , should be carefully selected to ensure a good reflection from the bottom mirror at the designed SE wavelength 940 nm. We find the correct periodicity by performing a photonic band structure calculation for 3D unit cell of the PC in a 1D-periodic arrangement for different values of p_h and r, as shown in the inset of figure 3(b). The yellow shade indicates the bandgap [38] of the periodic structure with parameters r = 20 nm, $p_h = 237$ nm, in the center of which a red dashed line denotes its center wavelength $\lambda_\mathrm c$ precisely at 940 nm. For the highest possible reflectivity in DBR, the band gap should be engineered exactly around the SE frequency. After scanning the periodicities for three radii that we would investigate in the following and extracting all the corresponding center wavelengths, we finally get the impact of p_h on $\lambda_\mathrm c$, as shown in figure 3(b). It reveals that a longer period is necessary for a larger hole or a longer wavelength to achieve the greatest reflection. The intersections of the dashed lines in figure 3(b) represent the optimal p_h of three periodic structures at the designed wavelength 940 nm. They are $p_h = 237\,\mathrm{nm} (r = 20\, \mathrm{nm}) p_h = 260\, \mathrm{nm} (r = 40\, \mathrm{nm}), p_h = 290\, \mathrm{nm} (r = 60\, \mathrm{nm})$, respectively. By applying these parameters, we can expect a fairly high bottom reflection to appear in this structure.

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Next, a stronger Purcell enhancement can be achieved by the constructive interference between the forward-propagating photons and the ones reflected at the bottom mirror [37], which requires the dipole to be precisely at an antinode relative to the mirror. We first place the emitter at a random distance L to the bottom mirror to obtain a field profile, where we can observe the position of the antinodes. As shown in figure 3(c), there are three antinodes between the dipole and the mirror, indicated by three bright standing wave patterns. The second step is to choose one of the antinodes and relocate the emitter there. Here, we avoid the first antinode in case there is any influence induced by the close distance between QD and DBR. In this structure, we choose the third antinode to locate the dipole, and the field profile is shown in figure 3(d), where the output signal is much stronger than that in figure 3(c). As mentioned above, this results from a perfect constructive interference when the emitter is precisely at the antinode. The quantitative analysis for the dipole misalignment will be discussed in section 3.4. For different radii, the distances of QD to the bottom mirror used in this work are given as follow: $L = 289\, \mathrm{nm} (r = 20\, \mathrm{nm}), L = 298\, \mathrm{nm} (r = 40\, \mathrm{nm}),$ $L = 312\, \mathrm{nm} (r = 60\, \mathrm{nm})$. Although we observed an enhanced SE rate and $P_{\mathrm {M}, \mathrm{top}} \gg P_{\mathrm {M}, \mathrm{bot}}$ by only implementing a bottom mirror, we are still pursuing a better Purcell factor and higher source efficiency, as shown in the next section.

The third step in the design procedure is to construct a complete cavity, in which two sets of DBR sandwich the QD in the center, and thus cavity length is twice as long as L. By applying the parameters designed in the previous step, the cavity will further enhance the QD SE through a more substantial Purcell effect. Figure 4(a) is the sketch of a QD-nanobeam cavity waveguide coupling system. Here, we design a top mirror with a higher transmission than the bottom one, which is achieved by using less holes. For a purely x-polarized dipole, part of the total emitted power $P_\mathrm T$ funnels into the waveguide fundamental guided mode M, denoted by $P_\mathrm M$. The rest lost into radiation, named $P_\mathrm L$, mainly appears at the interfaces of the cavity and the DBRs.

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Figures 4(b)–(e) show the impact of the number of periods in top mirror $N_\mathrm{top}$ on the SE properties and coupling efficiencies under four sets of configurations. Generally, when $N_\mathrm{top}$ increases, the top reflectivity increases as well. For each data point, $N_{\mathrm{bottom}}$ is accordingly increased to ensure the bottom reflectivity stays higher. Here, we state that the parameters used in the design of r = 30 nm are $p_h = 248$ nm, L = 291 nm. Figure 4(b) demonstrates the normalized $P_\mathrm T$ emitted by an x-polarized dipole as a function of $N_\mathrm{top}$. In the range of $N_\mathrm {top}\in[0,5]$, the large size design (r = 60 nm) has already reached the highest Purcell factor. In contrast, the curves for 20, 30 and 40 nm are still going up but will not increase infinitely. Due to the scattering taking place at the cavity-DBR interface, there is a trade-off between the enhancing quality factor and the scattering loss when increasing $N_\mathrm{top}$, and therefore those curves will reach a maximum at a larger $N_\mathrm{top}$. As $N_\mathrm{top}$ increases, the reflectivity from the top becomes comparable to that from the bottom, and thus more light into the free space leads to a rise in $P_\mathrm L$, as shown in figure 4(c). Here, the structure with the smallest size (r = 20 nm) yields the lowest $P_\mathrm L$. Figures 4(d) and (e) confirm indeed that a smaller radius is beneficial for suppressing $P_\mathrm L$, as both ε_x and ε_xy achieve much higher values in this regime.

We find a maximum efficiency ε_xy = 97.7% for r = 20 nm and $N_\mathrm{top} = 12$ (and a corresponding $N_{\mathrm{bottom}} = 85$), accompanied by a Purcell factor of $F_x = 38.6$, and a Q factor of 668. As the supplementary information to figure 4(e), we show the maximum efficiency ε_xy for four different radii and the corresponding period configurations in table 1. The optimum efficiency increases with a smaller hole radius thanks to the reduction in scattering losses. Still, it pays a cost for more periods due to the reduced reflection from a single hole. A single computation in the parameter scan is performed with an accuracy setting yielding an estimated numerical relative error of $0.5\%$ for Q and of $0.1\%$ for ε_xy . The corresponding discrete problem has a dimension of N = 14 836 908 and is solved on with a RAM consumption of 500GB and with a computation time of roughly 4 hours, using 6 nodes on a standard workstation. The numerical error has been estimated using a numerical convergence study with gradually increasing numerical accuracy settings.

Table 1. Maximum coupling efficiency ε_xy under different configurations.

Radius (nm)	$N_{\mathrm{bottom}}$	$N_\mathrm{top}$	Maximum εxy
20	85	12	$97.7\%$
30	35	5	$94.6\%$
40	15	3	$91.3\%$
60	10	1	$80.6\%$

To make a more fabrication-friendly design, the last step in the design procedure is to find a structure with fewer periods but which can keep a near-unity coupling efficiency simultaneously. On one hand, the r = 20 nm platform effectively suppresses the scattering loss into the free space but takes a large number of periods because a single period contributes very little to the overall reflectivity. On the other hand, the 40 nm platform achieves high bottom reflection with few holes, but features stronger scattering losses due to a significant mode mismatch between the Bloch mode in the DBR mirror section and the cavity mode. One way to take advantage of both configurations is to implement a taper section [39–41] in the bottom mirror, where the hole radius gradually increases from 20 nm in the vicinity of the QD to 40 nm. We thus consider a bottom mirror consisting of $N_{\mathrm{regular}}$ holes of radius 40 nm and a taper section including $N_{\mathrm{taper}}$ holes, as shown in figure 5(a), and the new variable distance between two holes p^ʹ is given by $p^{^{\prime}} = (p_{h1}+p_{h2})/2$, where $p_{h1}$ and $p_{h2}$ are the individual periodicities of two adjacent periods. The principal power loss in this platform still appears at the interfaces of the cavity and two mirrors, as depicted by the thick arrows.

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However, the increased size in the taper section also introduces additional scattering loss, denoted by the small arrows in figure 5(a). We investigate the impact of $N_{\mathrm{taper}}$ on the normalized $P_\mathrm L$, as shown in figure 5(b). We can find increasing $N_{\mathrm{taper}}$ will effectively suppress $P_\mathrm L$. In our investigations, the cavity effect is indispensable, and therefore we employ a large $N_{\mathrm{taper}}$ in the design. Then, we propose an optimum new design yielding a coupling efficiency $\epsilon_{xy} = 97.6\%$, a remarkable Purcell factor of 38.3, and a Q factor of 616 with the parameters $N_{\mathrm{regular}} = 9$, $N_{\mathrm{taper}} = 21$, $N_\mathrm{top} = 12$. Compared with the previous result in this section, which is obtained via 20 nm-hole regular DBR nanobeam cavity, we achieve identical performance but with only 1/3 of the total periods in the bottom mirror, contributing to a smaller mode volume and, in turn, a broader bandwidth.

In principle, the figure of merit ε_xy can be further improved by using a longer taper section and smaller holes, but these come at the price of much longer structure and heavier calculations. Under the premise that the minimum design radius is 20 nm, we do not implement a similar taper section in the top mirror and instead keep the regular DRB. This avoids the additional power loss introduced by the increased radius in the taper.

Considering the imperfections that will appear in the actual situations, the tolerance study for the optimum QD-taper cavity platform is carried out by randomly changing the radii of the well-designed holes. Figures 5(c) and (d) demonstrate the Purcell factor and efficiencies as a function of the maximum radius deviation $dr_ \mathrm {max}$ between the designed radii and the possible imperfections in fabrication, which means that each hole radius is changed by an individual random amount in the interval (0, $dr_\mathrm {max}$). Even under a 5 nm deviation, our platform is still robust on ε_x and ε_xy , keeping the performances of over 90% with a standard deviation of 1%. However, the Purcell factor is more sensitive to imperfect situations. This is because a misalignment of the new antinode with respect to QD position leads to poor constructive interference.

In section 3.2, the field strength comparison between figures 3(c) and (d) implies the QD-cavity coupling might highly depend on the accuracy of the QD position. To offer a more quantitative analysis, we investigate the impact of QD misalignment on the SPS performance. In this subsection, we still consider the on-axis dipole but change its position along the axis. We define ρ as the deviation of QD position from the ideal center in the cavity, and this value varies in the range [−(L−r_b ),(L−r_t )] when we assume ρ = 0 is the cavity center. Here, L is the half cavity length, and r_b (r_t ) is the radius of the hole closest to the cavity in the bottom (top) mirror. The peaks and valleys in figure 6(a) correspond to the QD at the cavity's antinodes and nodes. The wave shape is the consequence of walking in the cycle between constructive and destructive interference. The green shade area reveals that a relative QD position deviation of 50 nm is tolerable with a cost of 50% Purcell enhancement. In figure 6(b), the efficiencies ε_x and ε_xy present an even broader spatial linewidth, denoted by the blue shade area. Within a 90 nm QD-cavity center deviation, our platform can still perform a more than 90% coupling efficiency.

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