1. Introduction

Efficiently interfacing optical fields with numerous quantum memories is a fundamental requirement for quantum networks. Solid-state systems, especially color centers in diamond, have emerged as promising candidates for such interfaces. They are well suited for building nodes in quantum repeater networks due to their ability to transmit quantum information via spin-entangled photons [1–3]. However, a significant limitation of these color centers, including tin vacancy (SnV), silicon vacancy (SiV), and nitrogen vacancy (NV) centers, is their suboptimal emission efficiency into preferred spatial and spectral modes, particularly into the zero-phonon line (ZPL) [4,5]. Enhancing the photonic emission of color centers in the ZPL and effectively coupling these photons into network channels (such as fibers or free space) is crucial for improving entanglement generation rates, a current bottleneck in quantum network performance [5].

For spin-photon interfaces in such systems, the figure of merit, denoted as $\eta$, represents the probability that an excited state of the emitter leads to a detected photon in the ZPL. This efficiency, $\eta$, can be conceptualized as a product of two efficiencies: $\eta _{ZPL}$, the proportion of emissions into the ZPL, and $\eta _{col}$, the collection efficiency of these emissions into the desired mode [4]. Optimizing both these components is vital for maximizing $\eta$, thereby enhancing the interface efficiency essential for scalable and efficient quantum networks.

We define the figure of merit $\eta$ as the probability that an excited state of the emitter results in a photon detected in the zero-phonon-line (ZPL),

The factor $\eta _{ZPL}$ can be determined using the Purcell enhancement $F$ of the device at the targeted wavelength $\lambda _0$, and depends on the emission rate $\Gamma _\textrm { {total},0}$ and $\Gamma _\textrm { {ZPL},0}$, which represent the total unmodified emission rate and ZPL emission rate of the chosen color center, respectively [4]:

The parameter $\eta _{col}$ is calculated as the radial power in the far field collected into numerical apertures (NA) less than 0.7, which is a conservative value for an objective lens that might be used to address a color center. Prior research has demonstrated that both $\eta _{ZPL}$ and $\eta _{col}$ can be simultaneously enhanced by positioning a color center in specially designed optical cavities [4,6]. For silicon vacancy (SiV) centers, simulations predict free-space collection quantum efficiencies up to 41% [4]. However, fabricating such devices necessitates intricate diamond etching, a process challenged by diamond’s material properties. Empirical evaluations of individually designed devices have shown average quantum efficiencies up to 5.5% for nitrogen vacancy centers [6]. Recent studies suggest that SnV centers may be more advantageous for quantum networking applications [7–11], with $\Gamma _\textrm { {total},0}/\Gamma _\textrm { {ZPL},0}-1 \approx 0.25$, and are thus employed for computations in this work.

Here, we introduce a design for the vertical emission of spin-polarization entangled photons from diamond color centers, achieving quantum efficiencies up to 46% for a SnV center. This approach mitigates the precision requirements for diamond etching, leading to a design that is more scalable in terms of manufacturing yield, and also has the scalability of packing many microdisks into a small area. This design also exhibits high tolerance to many manufacturing variations.

Additionally, we present a numerically efficient optimization strategy for this multilayered diamond emission device. Our approach yields results comparable to those from a comprehensive full-wave finite difference time domain (FDTD) parameter sweep, which can be computationally intensive, particularly for resonant structures where electric fields remain trapped in the cavity for extended durations [12]. We propose a novel method for designing emitter devices that uses a simplified dipole representation of scattering from a near-field perturbation [4,13], coupled with a single control simulation performed using FDTD. This method facilitates rapid optimization of key parameters in our layered microdisk resonator, and may be applicable to photonic device design more generally.

2. Design

Microdisk resonators have high quality factors, and thus provide the Purcell enhancement necessary for $\eta _{ZPL}$ to approach unity. However, radiation loss is dominated by radial emission in the plane of the disk. In order to achieve out-of-plane emission in the vertical direction, and thus high $\eta _{col}$, our design builds on previous work showing that carefully placed perturbations in the near field mode of the microdisk can scatter photons vertically instead of radially. The difficulty of accurately etching the necessary patterns into diamond (n=2.4) prompts the exploration of alternative device designs and materials [14]. Rather than etching the diamond to enable vertical free space emission, we propose a design in which a grating layer of silicon oxynitride (Si$_{3}$O$_x$N$_{y}$, n=1.8) is etched with a triangular lattice of holes and then placed on top of an array of fabricated diamond microdisks. To avoid breaking the diamond microdisks, unetched walls could line the edge of the array to provide structural support for the grating layer. Disks would be spaced by the distance $d_p$, as shown in 1. Based on the near field in FDTD simulations, we expect that $d_p=15 \mathrm{\mu} m$ is a conservative value to avoid crosstalk between devices. A bulk silicon dioxide glass substrate (n=1.4) supports the grating layer, also shown in Fig. 1. The lattice holes scatter light from the microdisk mode so as to constructively interfere within the target numerical aperture.

 

Fig. 1. Design concept. (a) Perspective view of the disk, with a 60°cutout to show the cross-section and bulk oxide removed. The purple mark represents the point dipole source used to excite the disk mode. (b) Coordinate system for the far field emission analysis (Eqs (9)–(10)). A dipole location is highlighted in purple, and a point in the far field in black. (c) $x-z$ cross-section of a microdisk array coupled to numerical apertures. Disks would most likely be fabricated with an undercut [14], part of which is included in the FDTD simulations. (d) Microdisk with design parameters labeled.

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We choose the index of the silicon oxynitride such that the effective index of the grating layer is close to that of the glass substrate when holes are etched. This promotes transmission into the $+z$ direction. The reduced index of silicon oxynitride (n=1.8) versus that of silicon nitride (n=2.0) also provides greater index contrast between the grating layer and the diamond (n=2.4). This increases the Purcell enhancement of the device by slightly lowering the scattering rate in the grating layer.

One important aspect of our design is the capacity to overlay the grating layer onto the microdisk while accommodating reasonable alignment tolerances between the grating holes and the microdisk. The alignment between the disk and lattice is defined by a vector $(u, v)$ in the $x-y$ plane, signifying the relative position of a lattice hole’s center to the disk’s center. Due to the triangular symmetry of the lattice, all misalignments between the microdisk and the lattice can be described by the relative offset of the nearest lattice hole to the center of the microdisk. This constrains $u$ and $v$ to $0 \leq u \leq a/4$ and $0 \leq v \leq u\tan (\pi /6)$, where $a$ is the lattice constant of the lattice, as illustrated in Fig. 2.

 

Fig. 2. Triangular lattice unit cell. Point A and the white holes represent actual holes in the triangular lattice. The blue symmetry lines identify a unit cell area for $u$ and $v$, highlighted in pink. Points A and B are key alignment points inside ($u, v$) that exhibit high radial symmetry.

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Additionally, fabrication variances will inevitably lead to disparities in fabricated devices. Consequently, we aim for a design that exhibits robustness to these variations, focusing particularly on parameters such as $a$, $(u, v)$, hole radius $r_h$, and disk thickness $t$. In the design proposed, the utilization of a large microdisk array against a singular triangular lattice facilitates vertical free-space emission, even without precise alignment of the layers.

To narrow the parameter space for $a$, we compare the overlap between a given lattice and the mode of the microdisk. In a triangular lattice, the perimeter of the $n$th regular hexagon centered on any lattice point will intersect with $6n$ other lattice points, as shown in Fig. 3. If the center of the microdisk is aligned with a lattice point and radius of the disk $r_d$ and the lattice constant $a$ are chosen properly, the disk mode will strongly overlap with one of these hexagons.

 

Fig. 3. (a) Expanding hexagonal traces from a center lattice point in a triangular lattice. For $n = 3$, 18 lattice holes form the perimeter of the hexagon. Two-thirds (12) of these holes are equidistant from the center, and the circumference of the microdisk is circled in red. (b) Overlap between the lattice holes and the optical mode of the bare microdisk, with the circumference circled in red. The mode is calculated in FDTD without the lattice etched in the grating layer. Holes are highlighted in proportion to the electric field magnitude incident at their center.

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Based on previous work demonstrating that perturbations in a microdisk structure can be modeled as an array of dipole scatterers [4,13], constructive interference into lower numerical apertures will occur when the number of equally spaced azimuthal scatterers is close to the azimuthal mode number $m$ of the disk [4,15]. We choose a disk mode of $m=18$ to couple to hexagonal trace 3 in a triangular lattice, shown in Fig. 3. This choice has the distinct advantage of a having a high ratio of equidistant holes from the center (2/3) in the target hexagonal trace, creating a consistent overlap with the concentrated electric field at the edge of the disk when $u,v << a$. For the $m=18$ mode, $r_d$ is also large enough to maintain a high Purcell enhancement.

3. Simulation and modeling

In FDTD simulations, we have assumed accurate placement of the color center inside the diamond, which is necessary to maintain the high yield for scalability. In practice, a microdisk will have many color centers implanted inside of it, and this enables an experimentalist to address a color center found in the ideal location [16,17].

To optimize the device and demonstrate the robustness of the optimized design to fabrication variations, we swept parameters such as lattice constant, hole radius, and disk thickness and monitored the quantum efficiency. When fabricated, the packing distance and angular alignment between the array of microdisks and the lattice symmetry can be designed to provide structured (u,v) offsets for each microdisk in the array. Statistical sampling of 205 potential random values of (u,v) indicates that the design consistently yields high performance with respect to alignment between the disk and lattice, as shown in Fig. 4. A significant percentage of samples ( 5%) maintain far field efficiencies above 40%, and the average spectral efficiency is 99.5%. Experimentally, many color centers can be simultaenously implanted in a diamond sample and only those with desirable spectral properties may be addressed, which may increase the yield to near unity for this design.

 

Fig. 4. Manufacturing robustness according to the following parameters: (a) Disk thickness $t$ from FDTD simulation. (b) Hole radius $r_h$ from FDTD. (c) Lattice constant $a$, with results from the dipole model plotted alongside FDTD. Peak performance is denoted by a dot. (d) Cumulative fraction of samples above $\eta _{col}$ for random $(u, v)$ alignments sampled from FDTD.

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Figure 4 illustrates the robustness to manufacturing variations, showing the impact of different parameters. Panel (a) presents results from FDTD simulations for disk thickness $t$, while panel (b) shows the effects of variations in hole radius $r_h$. Panel (c) compares the optimizations of the lattice constant $a$ from both the dipole model and FDTD simulations. Panel (d) shows the cumulative fraction of samples exceeding a certain $\eta _{col}$ threshold for 205 random $(u, v)$ alignments, as sampled from FDTD simulations. It is worth noting that, given random alignment between the microdisk and lattice, simulations predict a yield of around 50% for devices with quantum efficiency of over 25% (the spectral efficiency varies little from 99.5%), with almost all devices having a yield above 14%. This allows experimentalists to fabricate many microdisks and obtain many efficient devices without concern for alignment tolerances.

The design also demonstrates robustness towards the change in resonant wavelength due to manufacturing variation. For all manufacturing variations considered in this work, the resonant wavelength changes most rapidly with disk thickness $t$, by around 0.0546% for a 1.56% change in thickness. Variations of this magnitude could be overcome by incorporating resonance tuning features such as post processing gas deposition or strain tuning of the device [18].

To decrease simulation time while optimizing $a$, we apply a dipole model as an efficient alternative to computationally expensive finite-difference time-domain (FDTD) parameter sweeps. This approach is grounded in existing research indicating that refractive index variations in the vicinity of a microdisk’s near field can effectively be represented as dipole scatterers [4,13].

The emission characteristics of the structure are derived through far-field projections from electromagnetic currents captured on a two-dimensional surface $s$. These projections are facilitated by calculating the magnetic ($\mathbf {A}$) and electric ($\mathbf {F}$) vector potentials, derived from the surface electric ($\mathbf {J}_s$) and magnetic ($\mathbf {M}_s$) currents [19]:

The electric far field $\mathbf {E}_{ff}$, magnetic far field field $\mathbf {H}_{ff}$ and far field poynting vector $\mathbf {S}_{ff}$ may then be calculated as [19]

We now examine the dipole approximation to this equation. The fields collected by our FDTD simulations will appear to have been sourced by dipoles centered on refractive-index perturbations. This is because holes in the grating layer cause large scattering in comparison to the natural losses of a microdisk with a high quality factor. Approximating each hole as a Hertzian dipole source of length $l$ leads to a magnetic vector potential of an array of Hertzian dipoles [20]:

In FDTD and with the simplified dipole model, we sum up the poynting vector over a sphere centered on the origin with a large enough radius such that $|\mathbf {r}|>> |\mathbf {r}_c|$ and $|\mathbf {r}|>>|\mathbf {r}_n'|$. For the dipole model, we sum the power collected in the glass by a numerical aperture of 0.7 and divide it by the total power passing through the sphere in order to obtain $\eta _{col}$. In FDTD, the surface currents used in the far-field projection must be evaluated in a homogeneous medium. To apply this appropriately to our layered microdisk structure, we project the surface currents collected by monitors placed only in the glass substrate. This will capture light escaping from the edge of the microdisk to around $\theta =70\circ$ in the $+z$ direction. We then divide the resulting collection power by the total power that exits the simulation in all directions to obtain $\eta _{col}$. The FDTD near and far fields are calculated using commerical software [21], and Eqs. (9)–(10) of the dipole model are implemented in Python.

We note that the simplified dipole model matches more accurately with emission profile of FDTD simulations when we assume that $I_{nz}=0$. This may be attributed to the fact that, owing to boundary conditions, light traveling in the plane of the lattice and polarized in the out-of-plane $\hat {z}$ direction will not experience a discontinuity anywhere along the boundary of a lattice hole, leading to reduced scattering relative to the in-plane components of the near field.

After running the first FDTD simulation, a single emission pattern can be calculated using the simplified dipole model within 0.32 seconds on a single AMD EPYC 7763 processor. In contrast, evaluating a single emission pattern in an FDTD sweep requires 5.1 hours on a cluster of 128 cores of the same processors, or about $7\cdot 10^{6}$ the amount of computing time. Each simulation required high resolution and size in order for the near field interaction between the disk and grating layer to converge, found by running convergence tests. The simplified model may therefore be used to search a much larger parameter space to identify optimal emission patterns. Additional FDTD simulations may then be run to verify the pattern and optimize if necessary. The simplified dipole model predicts an optical lattice constant $a=0.524/\lambda _0$ for symmetry point A, which is consistent with FDTD simulations, as shown in Fig. 4.

Table 1 shows the parameters used to obtain the optimal far-field emission. With a Purcell enhancement of 52.6 and $\eta _{col}=0.47$, we find an upper $\eta$ of 46% for a SnV center. We also list in Table 2 the fraction of light naturally emitted into the ZPL for various color centers and the associated spectral efficiency $\eta _{ZPL}$.

Table 1. Parameters of the optimized layered microdisk obtained from FDTD

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Table 2. Spectral Efficiency for various color centers in diamond.

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The best correspondence between the dipole scattering model and FDTD simulation occurrs when the center of the disk is aligned over symmetry points A and B on the lattice, as shown in Fig. 5. The correspondence between the calculated far-field emission patterns of the two models is reduced when the alignment is not near one of these symmetry points. This may be because each simulation in the dipole model uses the near field mode of a bare disk with no grating to calculate $I_{nx,ny,nz}$, which is then used in Eq. (9) to calculate the far field. When the lattice and the microdisk are not aligned at a high-symmetry point, the non-uniformites in the near-field caused by the lattice holes make this approximation less valid. Nonetheless, we find by numerical experimentation that optimization near the high symmetry points using the simplified dipole model is sufficient to optimize the far-field verified by FDTD even away from these high-symmetry points.

 

Fig. 5. Comparison of emission of the optimized layered microdisk using FDTD and the simplified dipole model. Near fields are displayed inside the grating layer, and the far field in the upper hemisphere is depicted for each model. a) High-symmetry points A and B in a triangular lattice. b) FDTD simulation omitting the etch in the grating layer. The dashed red line depicts the boundary of the microdisk. The near field inside of the grating layer is used to find $I_{nx,ny,nz}$ in the dipole model. Without scatterers, the farfield is directed radially away from the microdisk. c) FDTD simulation with the disk center aligned with point A on the lattice. FDTD near field and resulting far field. d) Geometry of the dipole model with the center of the disk aligned on point A. Dipole near field and resulting far field. e) FDTD simulation with the disk center aligned over point B on the lattice. FDTD near field and resulting far field. f) Geometry of the dipole model with the center of the disk aligned on point B. Dipole near field and resulting far field. g) Comparison of collection efficiency ($\eta _{col}$) as the target numerical aperture increases for alignment A according to FDTD simulation and the dipole model. The numerical aperture of 0.7 is marked by a blue dashed line. h) Comparison of collection for alignment B.

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Another physical effect missing from the dipole model is the consequence of the refractive index differences between the layers. The difference between the diamond microdisk and the silicon dioxide lattice is much less than the difference between the disk and the air below it, leading to preferential emission into the bulk silicon dioxide. To include this effect in our model, we phenomenologically multiply all calculated far-field emission patterns for a given geometry by a constant parameter $\alpha$. Because the monitors in FDTD simulation are placed to capture up to a $70^{\circ}$ cone of light exiting the simulation region from the microdisk, we perform a least-squares fit between the dipole model and the FDTD model up to $70^{\circ}$ to find $\alpha$, which has a root mean squared error of $0.05$ and $0.03$ for symmetry points A and B respectively.

It is notable that the dipole model demonstrates reasonable accuracy even without accounting for variations in parameters such as $t$, $r_h$, $d$, and $r_d$. As noted above, the model’s primary utility is in rapidly searching for optimal values of the lattice constant $a$. Initialization of the dipole model requires just one FDTD simulation to calculate the microdisk’s mode, influenced by $d$, $r_d$, and $t$. This helps isolate the lattice constant as the key variable for optimization. It accurately identifies the optimal lattice constant with a precision of approximately 1.5%, demonstrating the model’s potential in streamlining the design process for complex photonic structures like the layered diamond microdisk resonator.

4. Discussion

The layered microdisk resonator presented here achieves a high coupling efficiency ($\eta _{col}$) of 0.46, attributed to the oxynitride grating layer that enhances vertical emission. This layer directs most far-field emission into the upper hemisphere. However, the collection efficiency ($\eta _{col}$) is capped due to constructive interference into higher numerical apertures, a consequence of the symmetric lattice structure. Optimizing hole locations in the nitride layer could enhance lower numerical aperture interference, but would require precise fabrication alignment.

Our focus on light collection into a specific numerical aperture, like a cryogenic objective, might differ from scalable designs involving microdisk arrays coupling into optical fibers. In such designs, a spatial light modulator could adjust free-space collection to fiber mode coupling, with grating hole arrangements tailored for Gaussian beam profiles.

Our control FDTD simulation-based dipole model accurately predicts the optimal lattice constant to within 5 nm. This model’s potential extends to designing devices needing free-space coupling from near-field modes, though it currently does not fully account for asymmetric perturbations, preferential upper hemisphere emission, and lattice reflections. Refinements, such as modeling scattering at hole edges, could enhance accuracy, making this approach a valuable tool for solving various resonator scattering problems and reducing dependence on extensive FDTD simulations.

5. Conclusion

We presented a scalable alignment-free design for free-space coupling from diamond microdisks, offering a solution that alleviates the challenges of precision nanofabrication in diamond. This design demonstrates robustness across several parameters: the thickness of the diamond microdisk, the alignment of the grating and microdisk layers, the lattice constant of the nitride layer, and the dimensions of etches in the nitride. We also developed a rapid optimization method using a single FDTD simulation and a dipole model instead of comprehensive full-wave parameter sweeps. This method significantly accelerates the design process for complex structures, especially for resonant devices with high quality factors. The resulting optimized layered microdisk design allows for creating arrays of quantum emitters with high alignment yields and a maximum quantum efficiency $\eta$ of 0.46 for tin vacancy (SnV) centers in diamond. This advance in quantum emitter design holds promise for enhancing the efficiency and scalability of quantum information systems.