Optimized Designs for Telecom-Wavelength Quantum Light Sources Based on Hybrid Circular Bragg Gratings

We present a design study of quantum light sources based on hybrid circular Bragg Gratings (CBGs) for emission wavelengths in the telecom O-band. The evaluated CBG designs show photon extraction efficiencies>95% and Purcell factors close to 30. Using simulations based on the finite element method, and considering the influence of possible fabrication imperfections, we identify optimized high-performance CBG designs which are robust against structural aberrations. In particular, full 3D simulations reveal that the designs show robustness regarding deviations of the emitter position in the device well within reported positioning accuracies of deterministic fabrication technologies. Furthermore, we investigate the coupling of the evaluated hybrid CBG designs to single-mode optical fibers, which is particularly interesting for the development of practical quantum light sources. We obtain coupling efficiencies of up to 77% for off-the-shelf fibers, and again proof robustness against fabrication imperfections. Our results show prospects for the fabrication of close-to-ideal fiber-coupled quantum light sources for long distance quantum communication.

wavelengths. In addition, a detailed understanding how the CBG geometry affects the optical performance of fabricated devices is missing.
In this work, we perform an in-depth analysis on the design parameters of hybrid CBG-based quantum light sources emitting in the telecom O-band. Using simulations based on the finite element method (FEM), we evaluate designs for GaAs-based CBGs with an embedded quantum emitter integrated on a broadband reflective mirror. The evaluated device designs show Purcell factors as high as 30, while maintaining extraction efficiencies of > 95% of the embedded dipole's power. Furthermore, by investigating the impact of common fabrication imperfections on the optical properties of the CBGs, we prove the robustness of the evaluated designs against structural aberrations including QD positioning errors within reported deterministic fabrication uncertainties in the 5-50 nm range. Finally, we address the coupling efficiency of selected CBG device designs directly attached to single-mode optical fibers, representing an important setting for the development of plug-and-play photonic devices. Our results show prospects for the realization of close-to-ideal quantum light sources at telecom wavelengths.
To investigate the influence of various geometrical parameters on the performance of hybrid CBGs in the telecom O-band, we employ FEM simulations in the frequency domain using the software package JCMsuite [26]. The simulations utilize geometries discretized with a non-uniform-mesh triangulation and, unless stated otherwise, exploit the rotational symmetry of the CBG-structure by reducing the computational problem to a two-dimensional cross-section, thereby reducing the computational efforts [27]. The simulation domain is surrounded by perfectly matched layer (PML) boundaries. For the simulations, the room temperature refractive indices were used for all materials at a wavelength of λ = 1320 nm, already accounting for an expected blue shift of the resonant modes (of about 10 nm) due to refractive index changes at cryogenic temperatures (<10 K) required for non-classical light generation with highest quantum-optical quality [28]. The quantum emitter (QD) inside the structure is modeled as a TE dipole source at the center of the central disc of the CBG. The collection efficiencies of the evaluated designs are quantified by the dipole power collection efficiency (DCE), corresponding to the integrated power in the far-field emitted into NA=0.8 divided by the total power of the embedded dipole. For a better comparison with other reports [23], we also calculate the photon collection efficiency (PCE) defined as the dipole power emitted into NA = 0.8 divided by the power emitted into the upper half space. Figure 1(a) shows a schematic cross-section of the hybrid CBG device, where the GaAs (nGaAs = 3.3885) center disc and the grating are located on a silicon dioxide (SiO2, nSiO2 = 1.4500) layer, combined with a gold (Au, nAu = 0.3970, kAu = 8.9523) layer for upward reflection of photons emitted into the lower halfspace. The design parameters of the device are the thicknesses t(Au), t(SiO2) and t(GaAs) of the Au, the GaAs and the SiO2 layer, respectively, the radius R of the inner disk, the period length P and gap-width W of the surrounding Bragg-grating and the number of grating rings. Performing a detailed parameter study to optimize the extraction efficiency and the Purcell enhancement (considering the targeted operation wavelength), we found the following set of design parameters: t(Au) = 100 nm, t(SiO2) = 300 nm, t(GaAs) = 240 nm, R = 550 nm, P = 500 nm, W = 160 nm and 11 rings forming the CBG. A top view of the corresponding hybrid CBG structure and the near-field profile of the supported fundamental optical mode are depicted in Figs. 1(b) and 1(c), respectively. The far-field intensity distribution of the same optical mode is shown in Fig. 1(d) for a NA of 0.8. Fig. 1(e) presents the Purcell factor FP and the collection efficiencies of the optimized CBG device. The DCE into NA = 0.8 reaches values exceeding 95% in this case. Noteworthy, although objectives with NA of 0.8 are commercially available for the O-band spectral range, even for NA = 0.4 this CBG device shows a DCE of 88% due to its high directionality (see Fig. 5 in the Supplementary Material). The PCE even reaches 98% for this NA, showing the effectiveness of the bottom gold mirror. The corresponding Purcell factor is close to 30 at the operation wavelength with a bandwidth (full width at half maximum) of Δλ = 3.3 nm.
Interestingly, our design study revealed that it is possible to achieve similar high performance in terms of the DCE for other sets of parameters, which might be easier to achieve with currently available in-situ lithography techniques. Figure 1(f) exemplarily presents simulations for a CBG device with a gap-width W = 300 nm, which is almost a factor of two enlarged compared to the device in Fig. 1(e). Optimization of the remaining parameters to R = 520 nm and P = 630 nm for maximum Purcell enhancement and extraction efficiency results in excellent performance in terms of DCE-values above 95% with operation at about λ = 1323 nm. Although the maximum Purcell factor of ~15 is somewhat lower in this case, the Purcell enhancement extends over a broader spectral range (Δλ = 11.9 nm). This is important for applications, such as the generation of polarization entangled photon-pairs via the biexcition-exciton (XX-X) radiative cascade in QDs. Here, the relevant excitonic states, i.e. X and XX, have a spectral separation of a few nanometers in case of InGaAs/GaAs QDs emitting in the telecom O-band [29]. In addition, the DCE shows an almost wavelength independent behavior, making the device more robust for applications such as the generation polarization-entangled photon pairs. Since the larger gap-width promises to be less demanding for lithography processes, we expect this large-W design most suitable for device fabrication. The device's design parameters can be systematically changed in the following way: An increase in gap-width W leads to a blue shift of the operation wavelength of maximum FP. An increase in ring-size (by increasing P, while keeping W constant) causes a red-shift in operation wavelength of maximum FP. This is due to the effective refractive index experienced by the mode, and thus by changing P and W with respect to each other, the operation wavelength can be systematically tuned to a desired value. Similarly, the effects of P and W on the out-coupling properties are systematic, where an increase in P (or decrease in W) leads to a blue-shift of the operation wavelength compared to the spectral regions with high DCE-values.  Next, we consider fabrication imperfections potentially affecting the optical performance of hybrid CBG devices. A typical issue in the device fabrication is an insufficient etching of the grating gaps, resulting in tilted side walls of the grating. This tilt effectively leads to a reduced gap-width W and an enlarged diameter of the central disc. Such problems arise for instance from insufficient development (under-development) during lithography, which negatively affects subsequent etch steps. Figure 2(a) illustrates such imperfections with a target gap-width of W = 300 nm and a variable tilt of the sidewalls described by ΔW. In this notation, a discrepancy of ΔW = 60 nm corresponds to an imperfection of dW = ΔW/W of 20% or a deviation of about 8° from a vertical side wall. As seen from Figs. 2(b) and 2(c), the operation wavelength is significantly affected by the fabrication imperfections, while the DCE remains at a high level (about 95% for NA = 0.8) almost independent of the tilt. The gray shaded region in Fig. 2(b) corresponds to a spectral window of 23 nm over which a moderate Purcell enhancement with FP = 2 can be achieved, despite the shift in operation wavelength, if the imperfections can be limited to dW = 6.67%. Moreover, a slight decrease in the maximal FP and an increase in the bandwidth of the Purcell enhancement is observed with increasing tilt. The high sensitivity of the operation wavelength to the tilt is attributed to the fact that the latter effectively leads to an enlarged size of the central disc (increased R). The size of the central disc in turn governs the confinement of the optical mode, due to relatively high near-field intensities close to the disc edge, as seen in Fig. 2(d). A change in disc size causes a noticeable change in the effective refractive index experienced by the optical mode, resulting in a shift of the operation wavelength. To verify this explanation, we performed simulations with fixed disc radius and tilt (R = 500 nm, dW = 20%), but variable effective gap-width W*.  operation wavelengths close to 1320 nm. Consequently, if the size of the central disc can be controlled well enough (within ΔR = ±20 nm), the constraints for the remaining design parameters are relaxed, and Purcell enhancement is achieved at the desired wavelength. Even if the disc size is controlled less precise, the outcoupling efficiency remains largely unaffected by these imperfections. That means that a high optical performance can be achieved, despite of structural aberrations in terms of sidewall tilts and varying gapwidths W. This finding is very promising for the robust fabrication of near-optimal telecom wavelength quantum light sources based on hybrid CBG devices.
Another possible fabrication imperfection is the lateral displacement of the quantum emitter with respect to the center of the device. To investigate the influence of this imperfection, full 3D simulations of CBG devices with off-center dipole emitters were performed (cf. Fig. 3(a)). Taking advantage of the fact, that the broadband design from Fig. 1(f) requires only 3 Bragg rings as compared to 11 rings for the narrowband design of Fig. 1(e), allowed us to significantly reduce the size of the computational domain to 6.04 µm × 6.04 µm × 0.84 µm, resulting in convenient simulation times of about 20 minutes (per emitter displacement) despite the significantly increased computational costs of the 3D setting. The broadband design with only 3 rings still shows FP-values close to 15 and DCE-values close to 92% (cf. Fig. 5(a) in Supplementary Material), Fig. 3(b) and 3(c) show the Purcell factor (normalized to its maximum FP,max~15) and DCEvalues for lateral emitter positions on a 200 nm × 200 nm grid around the center of the device. A x-and ydisplacement of the dipole emitter of 34 nm, which corresponds to the reported positioning accuracy of insitu electron-beam lithography [30], results in a reduction of FP by only 5.4% in x-direction and 15.6% in y-direction, respectively, where the asymmetry results from the linearly polarized dipole chosen to mimic a quantum emitter. The out-coupling efficiency is even more robust, with DCE-values for NA of 0.8 reduced by only 0.02% and 0.05% for a 34 nm displacement in x-and y-direction, respectively. Considering the simulation results above and the fact, that device positioning accuracies down equal or better than 50 nm [30][31][32] have been already demonstrated experimentally, the broadband design shows robust behavior also for a finite lateral displacement of the quantum emitter inside the device. We also want to emphasize, that the observed asymmetry in the x-and y-dependency originates from the polarized emission of the embedded dipole source. Quantum emitters such as QDs typically emit a mixture of both polarizations and the mode near-field in this case would be circularly symmetric and extents its field further into the y-direction. Noteworthy, the observed behavior of FP follows the distribution of the electric field-intensity of the confined optical mode. Since the spatial extend of the optical mode is larger for longer wavelengths, we therefore expect in general a higher robustness for a lateral emitter displacements of CBG devices operating at telecom-wavelengths compared to devices optimized for shorter wavelengths.
For the practical use of quantum light sources in photonic quantum technologies, a fiber-coupling of the emission is highly beneficial. Therefore we investigate the direct coupling of telecom-wavelength hybrid CBG devices to optical single-mode fibers using FEM simulations in a rotational symmetric setting as introduced earlier. We considered two different types of commercially available fibers: first 980HP with a core diameter of dcore = 3.6 µm and refractive indices for core and cladding of ncore = 1.460 and ncladding = 1.447 (at λ = 1320 nm), and second SMF28, assuming dcore = 4.07 µm, ncore = 1.452 and ncladding = 1.447. Calculations were carried out by assuming vacuum or UV adhesive (NOA81, n = 1.56 [33]) inside the gap between fiber and CBG perfectly aligned in axial direction. Using adhesives, a permanent and stable coupling can be achieved between a photonic device and a fiber [34,35]. The schematic in Fig.  4(a) shows the setting including the optical fiber and the hybrid CBG device, which was used to simulate the mode coupling efficiency (MCE) defined as the fraction of the dipole power which is coupled to the two-fold degenerate fundamental mode of the fiber [27]. Figure. 4(b) presents the MCE as a function of the distance d between fiber and CBG for the narrowband and broadband design presented in Figs. 1(e) and 1(f), respectively. For the coupling to 980HP fiber in vacuum, a maximum MCE of 76.0% and 50.3% is observed for the narrowband and broadband design, respectively. With UV adhesive, a maximum MCE of 54.6% is found for the narrowband design. In this case the radius of the central disc was reduced from R = 550 nm to R = 540 nm to account for the red shift induced by the present adhesive due to its higher refractive index compared to air, and to achieve λ close to 1320 nm. All three simulated curves show a periodic oscillation of MCE as a function of d, due to interference of the emitted light field inside the gap region. Moreover, both designs show a plateau-like behavior of the mean MCE at small distances d < 4 µm, followed by an almost linear decrease. This behavior is similarly observed for the simulation with adhesive, the MCE, however, is less sensitive to d in this case. The plateau reflects the high directionality of the emission of the hybrid CBG device, and is promising for the experimental realization of direct fibercoupling. To optimize the fiber-coupling efficiency for both designs, Fig. 4(c) presents the maximum MCEs (at d = 1350 nm) under variation of W around its initial values. For the narrowband design in vacuum, a slightly higher MCE of 76.9% can be achieved (at W = 170 nm) for the 980HP fiber, while a significant improvement is possible for the broadband design which shows up to 69.6% for this fiber-type in gap-width W around its initial value for different narrowband (NB-) and broadband (BB-) designs, considering the coupling to 980HP and SMF28 fiber. For the narrowband design, the MCE is also calculated with an adhesive (NOA81) in-between fiber and CBG. For the NB-design with adhesive, the central disc radius was reduced to R = 540 nm to account for a redshift caused by the adhesive with higher refractive index compared to air. (d) 2D fiber-mode profiles and CBG-mode profile at the distance of maximum MCE for narrowband and broadband designs.
vacuum (at W = 345 nm). With UV adhesive, the narrowband design shows a maximum MCE of 61.4% at W = 180 nm (d = 650 nm).
In addition, Fig. 4(c) depicts results obtained from a design with imperfect sidewalls (ΔW = 60 nm) coupled to the 980HP fiber in vacuum. Even in this case, coupling efficiencies close to 80% are reached, confirming that also the fiber-coupling of hybrid CBG devices is robust against fabrication imperfections. Additionally, calculations for the coupling of CBGs to SMF28 fibers in vacuum are presented for the narrowband and broadband design under variation of W (assuming ideal sidewalls). In this case, the MCE reaches values of 56.4% (at W = 175 nm) and 54.8% (at W = 350 nm). In general, all curves for the MCE show a less sensitive behavior for shorter gap-widths W (for about 60 nm), but an abrupt degradation towards longer W. We attribute the observed dependencies of the MCE for different W to the degree of overlap between the near-field of the CBG at the respective distance and the Gaussian fiber modes. As suggested by the field intensity distribution for W = 170 nm of the narrowband design in Fig. 4(d), highest MCE is achieved for an almost symmetrical Gaussian shape of the near-field distribution in X-and Ydirection. For W = 345 nm of the broadband design in Fig. 4(d), the field distribution shows larger asymmetries and deviations from a Gaussian shape, resulting in a reduced mode-overlap between CBG and fiber. This mode-mismatch becomes even more pronounced for W = 190 nm and W = 360 nm, since volume and lateral extend of the CBG mode become large compared to the fiber mode (see Fig. 8 in the Supplementary Material). Importantly, the coupling to the 980HP fiber is significantly more efficient (by a factor of 1.2 to 2.2 for specific W for narrowband and broadband design) compared to the SMF28 fiber, which results from a better mode-matching for the smaller core-size fiber. This becomes even more important in the presence of UV adhesive, as the already highly directional CBG mode-field is further squeezed in lateral dimensions.
In summary, the predicted coupling efficiencies to single-mode fibers surpass values reported for other systems, including microlenses [27], photonic trumpets [36], and photonic crystal cavities [37], while being only marginally smaller compared to coupling efficiencies reported for tapered photonic crystal waveguides [38] and micropillar structures [35]. Further improvements in the coupling performance are expected, e.g. by optimizing the combined design parameters of hybrid CBG devices using optimization algorithms [39] and specialty fibers with tailored geometry and index profiles. Last but not least, the influence of surface roughness (random disorder) and lateral asymmetries on the performance of CBG devices with and without fiber-coupling is interesting for future studies.
In conclusion, we presented a detailed design study for the fabrication of quantum light sources based on hybrid CBG devices operating in the telecom O-band. Using FEM simulations, we evaluated optimized device designs for highest performance in terms of photon-extraction efficiency, enabling Purcell-Factors as high as 30 and dipole power extraction efficiencies exceeding 95%. Furthermore we showed, that the excellent performance of the proposed hybrid CBG designs can be maintained and optimized for technically less demanding feature sizes. Additionally, our designs proved to be robust against common fabrication issues in terms of imperfect sidewall etching and lateral displacement of the emitter's position inside the device. Finally, we demonstrated the prospects for a highly efficient and robust direct fiber-coupling of hybrid CBG devices, revealing mode-coupling efficiencies close to 80% for off-the-shelf single-mode optical fibers. Further improvements are anticipated using customized/tailored specialty fibers in combination with optimized CBG design parameters. Our results are encouraging towards the fabrication of near-optimal and practical quantum light sources at telecom wavelength for various applications in photonic quantum technologies.  Fig.  1(f)). The Purcell enhancement follows again a bell-shaped curve with varied SiO2 thickness. (b) and (d) show clearly that this change in FP arises from the influence of t(SiO2) on Q, whereas VM is constant over the varied thickness range, except for very thin dielectric layers. A maximum FP = 54 can be obtained at t(SiO2) = 190 nm. Regarding the origin of this resonance, we found that the 190 nm of SiO2 form a λ/2 distance between the position of the dipole and the gold mirror, corresponding to (1/2 t(GaAs) ⋅ nGaAs)/λ + (t(SiO2) ⋅ nSiO2)/λ, t(GaAs) = 240 nm ; nGaAs = 3.3885 ; nSiO2 = 1.45 ; λ = 1314.5 nm. This means, that the emitted light travels a λ-distance back and forth to the mirror and is able to constructively interfere with the emitter again. On the other hand, a SiO2 thickness of at least 250 nm is required to achieve DCEs larger 90%.(e) and (f) Influence of the central disc radius R and CBG membrane thickness t(GaAs) on the optical properties. Increasing R and t(GaAs) both red-shift the operation wavelength, which can be used to fine-tune λ. The larger shift of λ with R is attributed to the high mode intensity at the edges of the disc.  Fig. 3: 980HP fiber-mode profiles and CBG field-intensity distributions at a distance d of maximum fiber-mode coupling efficiency for the narrowband and broadband designs with W around 160 nm and 300 nm. The intensity cross-cuts and 2D intensity distributions indicate, that highest coupling efficiency is achieved at optimum overlap of CBG field-and fiber-mode profile.