1. Introduction

Quantum photonic integrated circuits (qPICs) are functionalized based on the quantum nature of photons [1–3]. They can find applications in new fascinating fields related to quantum information processing and computing [4,5], quantum communication [6], or quantum metrology [6]. As in classical photonic integrated circuits (PICs), chip-scale integration is a crucial factor offering miniaturization and thus a small system footprint, scalability, mechanical stability, low energy as well as material consumption, and low manufacturing costs [1,2,7,8]. To achieve full application capabilities, the qPIC requires integrating various passive and active optical components within a single wafer-scale platform allowing for non-classical light generation, photon routing, control over the whole qPIC elements, and detection at a single photon level, or light outcoupling to the external photonic environment [1,3,7,8]. Many existing material approaches are currently considered for developing a genuinely functional qPIC, including major platforms like silicon-on-insulator (SOI), silica-on-insulator (SiO₂OI), silicon nitride (Si₃N₄), silicon oxynitride, aluminum nitride (AlN), silicon carbide (SiC), lithium niobate (LN), diamond (C), polymers, tantalum pentoxide (Ta₂O₅), gallium arsenide (GaAs), or indium phosphide (InP) [3,7,8]. Nevertheless, within all the mentioned, monolithic fabrication of a functional qPIC-based device is currently very challenging or impossible, routing towards hybridization of many different materials [1].

Silicon-on-insulator is a very attractive major material platform that can be used for the qPICs integration [9–11]. It has been successfully utilized in classical PICs, offering advantages such as compatibility with mature complementary metal-oxide-semiconductor (CMOS) technologies, thus allowing for electronic control over the qPICs elements [12], providing high index contrast necessary for the light routing [13] and assuring low-cost, maturity, and high processing-yield [9–11]. Within the Si platform, one can fabricate various monolithic components, including splitters [14], crossings, phase shifters, filters, electro-optical switches, waveguides [15], multiplexers [16], micro-ring resonators [17] and photon detectors [18]. Notably, the silicon platform offers ultralow losses for light routing within a chip [15,19] supported by a high refractive index [13]. For the scalability of quantum photonic devices, low photon losses within a chip are of primary importance, being the scalability limiting factor [1,2]. However, the main hurdle is the lack of a compatible non-classical photon source. It seems that, as in the case of other material platforms, hybridization of a non-classical photon generator with the Si platform might be a challenging but necessary solution [20–22].

One of the attractive solutions would be a heterogeneous integration of a III-V semiconductor quantum dot (QD) with a Si-based platform [2,20,22]. The challenge of this approach is that photons must be effectively distributed within both materials of different refractive indices [13,23]. Therefore, an efficient light field coupling must be provided between the QD, its environment, and the Si photonic platform.

It has already been demonstrated that a single InAs/GaAs QD placed in the host GaAs waveguide (WG) can be effectively integrated with a Si₃N₄ WG through a direct wafer bonding scheme [24]. The hybridization scheme has a significant potential for scalability, which has already been proven for the application to classical PICs [25]. The efficient light coupling of 90% has been achieved between GaAs and Si₃N₄ WGs through the WGs tapering [24]. This integrated GaAs/Si₃N₄ system also demonstrated on-demand indistinguishable photons distributed over the Si₃N₄ WG [26]. Despite the successful presentation of the proof-of-concept, the hybrid GaAs/Si₃N₄ material system suffers from limitations related to the useful spectral range of the transmitted photons. For example, having a single photon source compatible with the telecom industry standards exploiting 1.3 µm or 1.55 µm optical transmission windows of silica fibers eliminate the problem of frequency converters that can be an essential source of losses for a hybrid III-V/Si qPIC used in distant quantum communication or distributed quantum computation applications [5]. While InAs/GaAs QDs can be forced to emit at near-infrared by utilizing strain tuning [27–31], InAs/InP QDs [22,32–35]can naturally emit photons in the mentioned spectral ranges. A particular interest is paid to the 1.55 µm transmission channel. It promises low photon losses and single-mode propagation due to a high refractive index contrast [36] and thus strong field confinement providing effective transmission in Si WGs and silica fibers at large distances.

In this work, we propose hybridizing an InAs/InP QD embedded in an InP WG with the Si WG fabricated on the SOI platform for future Si-based qPICs. We provide a process flow chart which is fully feasible and compatible with today's technology by utilizing a direct wafer bonding approach. The main problem tackled in this work is how to efficiently transfer photons originating from an InAs/InP QD emitting at 1.55 µm from the host InP WG to the Si WG. Our studies are realized numerically employing the finite-difference time-domain (FDTD) method implemented in Lumerical – a commercially available FDTD solver [37]. Lumerical has proved its capabilities in optimizing of various nanophotonic and optoelectronic devices [38–41], and is also recognized as a potent tool for studying light propagation in nanophotonic structures.

In the first part of the work, we investigate the transmission of the tapered hybridized InP/Si WG section, as the vision is to provide the highest coupling of InAs/InP QD-confined dipole emission to Si WG via maximized light transmission through the tapered geometry. It is realized by scanning an extensive parameter space of the taper geometry and then simulating the multimode propagation along the number of taper structures. Next, we evaluate a dipole, directional emission coupling to guided modes confined in a specified hybrid InP/Si WG using a centered dipole position, and as a function of cross-sectional displacements. Finally, we study the possibility of emission outcoupling from the Si WG to free space that simulates an interlink between the qPIC and the off-chip transmission channel, e.g., an optical fiber [42]. Each component's transmission/coupling efficiency is examined with respect to the transmitted photon wavelength in the range of 1.5-1.6 µm, which is important for the device tolerance related to the imperfection of the component's fabrication process. All of the components communicating on-chip could compose an envisioned nanophotonic circuit for quantum information processing, as schematically demonstrated in Fig. 1.

 

Fig. 1. Concept of a nanophotonic circuit for quantum information processing utilizing InAs/InP quantum dot emitters embedded in hybrid InP/Si waveguides on a common SOI platform. A typical nanophotonic chip may consist of photon sources, couplers, splitters, ring resonators, and light outcoupling systems off the PIC towards in-plane or out-of-plane directions of the chip.

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2. Method and model structure

The geometry of InP and Si WGs and a hybrid WG are modeled within the Lumerical 3D CAD environment. Although the model is rather conceptual, our main concern is to find a realistic and a consistent model with the well-established fabrication and processing steps. There are a few possible approaches to realize these structures, including (i) in-situ lithography [43,44], (ii) transfer printing [45,46], or the (iii) pick-and-place method [47]. For each of these techniques, at least one challenge can be identified: (i) the complex equipment needed, (ii) the low fabrication precision, and (iii) the low process yield, respectively. On the other hand, direct wafer bonding followed by electron beam lithography (EBL) was successfully applied to fabricate GaAs/Si₃N₄ waveguides [24]. The benefits of the bonding approach are a relatively low complexity that leads to much better scalability and high fabrication accuracy (<50 nm [48], as compared with ∼1 µm for transfer printing [49]). Notably, the results of the presented geometry optimization are independent of the fabrication method, however, to prove the usability of our approach, we give the details of the proposed fabrication based on direct wafer bonding. It requires the integration of an epitaxially grown InP wafer containing an array of self-assembled InAs quantum dots with an SOI wafer and subsequent WGs processing. The processing would start with patterning Si waveguides and alignment marks in the top layer of the SOI wafer. The next step would be plasma-activated direct bonding of InP to the patterned SOI chip [50], followed by removal of the InP substrate. Then, uncovered alignment marks previously defined in the Si layer would be used as a reference for deterministic patterning of InP WGs with a QD emitter on top of Si WGs. The final pattern of hybrid WG would then be transferred to the InP layer by dry etching (using the interferometry-based metrology to control the etch rate or deposition of a thin oxide layer as an etch stop). The overview of the fabrication steps is demonstrated schematically in Fig. 2.

 

Fig. 2. The proposed fabrication workflow to realize a quantum photonic integrated circuit with heterogeneously integrated InAs/InP quantum dot system on a silicon-on-insulator (SOI) platform: a) preparation of InP wafer with quantum dots and SOI wafer with a define Si waveguide, b) molecular wafer bonding process of flipped InP wafer, c) wet chemical etching process to remove InP substrate, and d) dry etching to define InP waveguide upon Si waveguide.

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We find that the main requirement for the hybrid InP/Si WG geometry, enabling confinement of the fundamental mode in the InP part, is that the Si part must be of a smaller width than the InP part. However, we expect the structure to be mechanically unstable, so the subsequent results would be considered for the opposite constraints of our conceptual design, i.e. a narrower and higher InP part is placed on top of a broader and thin Si part. The proposed fabrication method allows for achieving the hybrid InP/Si WG, as depicted in Fig. 3. It consists of three elements that can be used to develop a simple qPIC demonstrator operating at a single photon level using an embedded InAs/InP quantum emitter. The first element is a hybrid waveguide structure containing an InP WG (with a single InAs QD embedded) on top of a Si WG of different widths. The second element is a tapered structure in which both the InP and Si WGs have tapered geometries. In the case of the InP WG, the taper shrinks along the propagation direction of the mode, whereas the inverted taper geometry is used for the Si WG. Here, we consider only the case where the taper length of the InP WG is equal to or smaller than the taper length of the Si WG. Both tapers have a linear dependence on the width along the waveguide direction. The third component is the Si WG section, which is an extension of the Si WG taper end without the InP WG on top. This component is terminated by the outcoupler ring structure, which can be used to scatter the transferred light to the off-chip detection system, normal to the structure plane. Optionally, the side detection system can be used for the cleaved facet of the sample. In both cases, photon outcoupling can be realized using a microscope objective or a lensed optical fiber to enable high collection efficiencies.

 

Fig. 3. Elements of the integrated quantum photonic circuit with the integrated quantum dot in a hybrid InP-Si waveguide system: a) quantum dot coupled in waveguide, b) linear taper structure, and c) grating outcoupler. All of these elements are feasible by using the proposed fabrication method in Fig. 2.

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The essential material parameters for the simulations are refractive indices: n_InP = 3.167, n_Si = 3.475 and n_SiO2 = 1.44, which are taken for the 1550 nm photon wavelength. The size of the simulation domain in FDTD is defined as a 3 × 40 × 1 µm³ box in the x-y-z Cartesian space and a mesh cell size is defined as 40 × 40 × 40 nm³ (see Supplement 1 for more details about FDTD settings). The height of the InP (Si) WG is fixed at 0.58 (0.22) µm, which gives the total height of 0.8 µm for the hybrid WG structure. In the case of Si WG, the height is determined by one of the commercially available SOI wafers. In the case of InP WG, the exact value is set after analysis of the preliminary calculations. The preliminary studies were performed for a QD embedded in the center of a 1 µm-wide InP part of the hybrid WG on top of a 1.5 µm-wide Si part, using a fully sharp (w2_InP = 0 nm), 20 µm-long taper structure. The outcoupler had a grating period R = [λ₀· (1−1/2)/n_InP + λ₀/2]. The geometries of the taper and outcoupler were not optimized here. Varying the height of InP WG in the range of 0.4 µm to 0.8 µm, we found that for the value of 0.58 µm, one can achieve the best vertical outcoupling, reaching roughly 6%. An initial test was performed for the emission wavelength of 1.55 µm. We should note here that by using the optimized system, as it is described in Results section 3 below, and performing similar calculations in dependence on the InP height, we found that for the InP height of 0.58 µm, again, one can get the maximum efficiency for the vertical outcoupling (see Supplement 1 and the related Figures S1-S3 for more details).

Although calculations are performed within a broad spectral range of 1.5 µm to 1.6 µm of photon wavelengths, the geometry optimization offering the best InP to Si WG coupling is tuned to the 1.55 µm wavelength.

3. Results

3.1 Light transmission from the InP to Si waveguide

In this section, we study the transmission efficiency of a propagating light field from the hybrid waveguide section, through the taper region, to the Si WG section. To simulate the light field generated by a single InAs quantum emitter embedded in the InP WG, we use a Gaussian beam source (waist radius of 0.50-0.55 µm is used depending on w1_InP) providing directional propagation of the field along the waveguide. The directional propagation of the Gaussian beam along the waveguide simplifies and generalizes the problem of finding efficient taper structure. First, the source excites a multimode propagation similarly to a random set of electric dipoles, which is related to a position uncertainty inside the InP WG expected for random in-plane distribution of self-assembled QDs in the InAs layer, and to the vertical shift of the QD layer determined during the growth process. Second, the directionality of the Gaussian beam source allows one to evaluate the taper transmission efficiency T for half of symmetric bidirectional dipole coupling. Third, due to the not preferred order of refractive index contrast, i.e. n_InP < n_Si, the Gaussian beam is located in InP circumventing excitation of the fundamental mode confined in Si, favouring higher-order modes relevant for the QDs embedded in InP. In this approach a single-shot simulation for a given geometry of the taper structure provides reliable results instead of time and resources consuming single-dipole modelling, and then by a parameter scan of the taper geometry we can search for optimized system with a maximum efficiency of the light transfer from hybrid WG to Si WG in a more efficient way.

The light confined in the Si WG after the coupler section (tapered area) is collected at the end of the last section of the structure (the end of Si WG). Therefore, the respective light transmission can be calculated as $T = {P_{Si\; WG}}/{P_{source}}$, where ${P_{Si\; WG}}$ is the optical power collected at the end of the Si WG and ${P_{source}}$ is the optical power injected into the hybrid InP/Si WG. According to the proposed hybrid WG geometry (see Fig. 3), we evaluated the optical transmission efficiency by examining the space of the following hybrid WG parameters: ${L_{InP}}$, ${L_{Si}}$, $w{1_{InP}},$ $w{2_{InP}}$, $w{1_{Si}}$ $w{2_{Si}}$. These parameters are assigned according to Fig. 3(b) and span the ranges given in Table 1:

Table 1. Range of geometrical parameters of the taper structure used in FDTD simulations.

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We also impose the following conditions: ${L_{InP}} \le {L_{Si}}$, $w{1_{InP}} \ge w{2_{InP}}$, $w{1_{Si}} \le w{2_{Si}}$ and $w{1_{InP}} \le w{1_{Si}}$ to fulfil the hybrid WG geometry requirements in Fig. 3(b). This constraint reduces the total number of iterations in our FDTD simulations from 2880 to 1614.

In Fig. 4(a), we demonstrate the distribution histogram of the evaluated transmission coefficient T. The statistics of the results show T above 60% for the vast majority (74.8% cases) of a relatively simple taper structure and for a coarse scan of the geometries. It indicates that the tolerance in processing such tapered WGs would be relatively high, requiring the fabrication precision on the order of a few tens of nanometers for the tapered InP and a few hundreds of nanometers for inversely tapered Si WGs, respectively. Thus, while electron-beam or deep-UV lithography should be used for fabricating InP WGs, the requirements for the fabrication of Si WGs are relaxed, and a simpler UV lithography technology could be employed.

 

Fig. 4. (a) Histogram of the transmission coefficients T evaluated for the tapers calculated for 1614 geometries (see text for details), (b) Evaluation of taper transmission for fine tuning of taper tip widths w2InP and w2Si around geometry #4 (see Table 2) with the highest transmission found in (a).

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Table 2. Range of geometrical parameters of the taper structure used in FDTD simulations.

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Next, a performance as high as 85% is found. The respective geometric parameters and the transmission evaluated for each case from the coarse scan are presented in Table 2.

At this stage, it is worth noting that all the best cases are obtained for the following:

Next, we performed numerical studies for each of the 5 cases presented in Table 2 using a finer parameter space. We varied both w2_InP and w2_Si by ±0.05 ${\mathrm{\mu} \mathrm{m}}$, ± 0.10 ${\mathrm{\mu} \mathrm{m}}$, ± 0.15 ${\mathrm{\mu} \mathrm{m}}$. The distributions of the results for all 5 cases are shown in Supplement 1 (Fig. S6) and show no local extrema with a slight variation in T within ${\pm} 3\; \%$. The highest transmission reaching 87.1% is found for the taper, like in case #4, for $- $0.1 µm detuning of the w2_Si parameter, as shown in Fig. 4(b). On the contrary, a maximum 5% reduction in T is expected when changing one of the parameters by +0.15 µm. In this context, a typical 40-50 nm EBL precision [48], which is better than <1 µm one expected for micro-transfer printing [49], favouring the suggested processing based on wafer bonding to realize the taper structure.

Moreover, we observe that the maximum values in T are obtained for the longest tapers of a parameter space given in Table 1, therefore we performed additional simulations in dependence on tapers length from 10 to 38 ${\mathrm{\mu} \mathrm{m}}$ for the most efficient taper shown in Fig. 4(b). In principle, due to the minimized field divergence along the taper, higher transmission is expected for a longer linear taper [51]. Our results presented in Fig. 5(a) are consistent with that expectation, and we observe an increase of transmission with the taper length reaching a maximum value of 89.1% for the 36 ${\mathrm{\mu} \mathrm{m}}$ long taper, however, the trend is not continuous and a further increase of the taper results in a reduced transmission. The discontinuities have a form of local minima likely related to the reflection from the taper tip. According to our above observation (i) from a coarse scan, a finite tip width is preferred to obtain the highest T, and therefore the system is susceptible to unwanted reflections. From the technological point of view, fabrication of longer tapers would also lead to fluctuations of their geometry deteriorating the field transfer to a Si-WG. Therefore, a taper length limited to about 25 ${\mathrm{\mu} \mathrm{m}}$ could be more practical to obtain a highly efficient light transfer from the InP/Si to Si WG.

 

Fig. 5. Evaluation of taper transmission in dependence on (a) a taper length and (b) a relative shift between the principal axes of the InP waveguide with respect to the Si waveguide.

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Last, we evaluated the taper transmission in dependence on the relative shift between the principal axes of the InP WG and the Si WG. The corresponding results that account for up to 200 nm mismatch are demonstrated in Fig. 5(b). One can see that taper transmission is reduced only by 5.3% within the considered mismatch range. Consequently, considering the accuracy of EBL (40-50 nm) used for the proposed fabrication method, the taper transmission should still be at a very high level, above 85.8%.

Furthermore, we took a closer look at the evaluated propagation of the optical field along the hybrid WG structure. Figure 6 presents the field distribution of the hybrid WG structure, which exhibits the highest transmission to the Si WG. In Fig. 6(a) we plot the Gaussian input beam in the x-z plane in the InP region. The propagation of the field along the taper region, the y-z plane (integrated over ‘x’ dimmension), is shown in Fig. 6(b). Finally, the field distribution at the Si WG output is depicted in Fig. 6(c). Interestingly, Fig. 6(b) shows the field oscillations between the InP and Si WG regions along the entire area of the taper. Such a field propagation character over the hybrid WG structure indicates that the taper length must be carefully adjusted to avoid reflections from the InP finite tip and to obtain a highly efficient transfer to the Si region. In addition, it suggests a multimode character of the field distribution in the tapered waveguide section.

3.2 Mode analysis and dipole coupling

The analysis of propagating optical field modes is essential to determine the coupling between an emitting dipole and a WG. The in-plane-oriented dipole (excitation in a QD characterized by the in-plane-oriented dipole moment) is placed in the hybrid InP/Si WG section, in the center of the InP part, and the resulting in-plane linear polarization coincides with the TE-like propagating modes. The analysis of the InP/Si WG mode starts with a fixed Si WG width of 1.5 µm, scanning the InP width from 0.5 µm to 1.5 µm. The assumed 1.5 µm width of the Si WG already supports the propagation of the fundamental TE-like mode. Figure 7(a) displays the calculated real part of effective refractive indices (n_eff) plotted as a function of InP width. It shows that the mode structure evolves, and for a broader InP WG, the n_eff increases. Interestingly, although one would expect that a higher n_eff would be translated into a longer effective propagation because of stronger mode confinement, it turns out that this is not the case for the fundamental mode (TE-like). Increasing the InP width increases the mode transfer from Si to InP. The latter has a lower refractive index, so the excess loss, related to a propagation leakage along the WG, is getting higher. However, we note that for the higher order modes, as n_eff increases, the excess loss decreases with increasing the InP width from 1.0 µm to 1.5 µm, as the field is distributed mode towards InP, so the leakage is effectively reduced for wider WGs. Detailed characteristics of the excess loss due to leakage for fundamental and higher-order modes are illustrated in Supplement 1 (Fig. S5).

 

Fig. 6. Visualization of light transmission through a tapered hybrid WG structure (a) Normalized input Gaussian |E|² field centered in the InP WG, (b) normalized integrated |E|² field distribution over the ‘x’ axis in the tapered region, and c) output |E|² field distribution in the Si WG.

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Fig. 7. (a) Real part of the effective refractive index (n_eff) for the hybrid InP/Si waveguide structure with fixed w1_Si = 1.5 µm demonstrated as a function of w1_InP. The selected region is the case analyzed in (b) where the field distribution of the propagating modes is displayed (modes #1-#8 are top to bottom in (a)). The results shown in (c) contain the dipole (QD) coupling to the specific mode (see the article text), the TE-fraction of the field and the effective refractive index together with the excess loss.

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Now, in the case of 1.5 µm-wide InP WG, for which the excess loss is minimized, we analyze the electric field distribution for the modes. Figure 7(b) shows the normalized integrated intensity for the first eight propagating modes (#1 - #8). The mode character – TE or transverse magnetic (TM), a QD (dipole) to the hybrid WG mode coupling, and excess losses are recovered, and the output results are summarized in Fig. 7(c). The mode character is evaluated by calculating the TE field fraction in the total distributed field by:

The dipole-WG mode coupling is examined by simulations performed for the 35 µm-long hybrid WG structure (convergence is observed around 30 µm) with emission from the in-plane polarized dipole source embedded in the center of the InP WG (x = 0 µm; z = 0.51 µm). The light field propagates over 35 µm, and is finally collected by the monitor at y = 35 µm. Then, the overlap $\eta $ between the scattered field {E,H} and the guided mode {E_f,H_f} is calculated by using [52]:

The total overlap of dipole emission is calculated as a sum for of all propagating modes. As the light emission from a point source embedded in the center of the InP WG is bidirectional, the effective dipole coupling evaluated for our system is $\eta $/2. On the other hand, applying a grating system on one side of the WG [53] or setting a dipole in a chiral position [54] could significantly enhance the directionality of emission and improve the on-chip coupling of QD emission.

Based on the evaluated TE-fraction of the modes, we conclude that in this case, we have 4 x TE-like, 2 x TM-like modes, and 2 x hybridized modes. First, we need to note that a fundamental mode (TE-like) is partially localized in both InP and Si WG sections (see Fig. 7(b)). In the set of higher order modes, one can find the modes localized more in the InP WG, like the already discussed mode #3 (also TE-like). As described in the analysis above, the dipole coupling between the InAs QD emitter inside the InP WG and the fundamental mode is less effective (∼6%) than the coupling with mode #3 (∼20%). In Fig. 7(c), we also show the losses for all the confined modes in the system presented in Fig. 7(b), which are calculated as

We also evaluate a possible change of the dipole coupling to relevant TE modes of the hybrid WG employing the geometry used in Fig. 7(b), (c). Here, we consider the vertical displacement $\Delta $z and the horizontal displacement $\Delta $x. The results are shown in Fig. 8. In the first case, we find that the overall coupling increases when the dipole is shifted upward along the ‘z’ axis of the InP WG, reaching 42.3% at $\Delta $z = 0.1 µm. A more detailed analysis of the coupling to individual modes (see Supplement 1, Fig. S7) shows that from $\Delta $z = $ - $0.2 µm to $\Delta $z = 0.1 µm the coupling to higher order modes, i.e. #3 and #6 increases, while the coupling to fundamental mode, #1, is less effective. A further increase of $\Delta $z > 0.1 µm results in decrease of the coupling due to less effective coupling to mode #3. On the other hand, the displacement along the ‘x’ axis of the WG up to $\Delta $x = 0.5 µm reveals an oscillatory-like behaviour in coupling efficiency. The minimum observed at $\Delta $x = 0.25 µm corresponds to a significantly reduced coupling to a HE mode (#6), and a reduced coupling to both fundamental modes, #1 and #3, although, it is compensated by an activated coupling of about 10% to mode #4. A further change along the axis ‘x’ to $\Delta $x = 0.4 µm results in a still high overall coupling of 37.5% due to a restored coupling to HE mode (#6) and a still significant contribution of the coupling to mode #4.

3.3 Vertical outcoupler

Here, we examine the possibility of outcoupling the light field from the Si WG following its transfer from the hybrid tapered WG section. We note that by using a multimode hybrid WG system, the Si WG of 0.22 µm heigth itself operates likewise in a multimode manner when w2_Si is larger than 1.2 µm. In the context of long-range on-chip information processing, it would be favourable to apply another taper that shrinks w2_Si toward below 1.2 µm, however, a multimode character could also be beneficial in the context of quantum interference realized with the use of spatial mode beam splitter [55]. To evaluate the field transfer from the Si WG to the external detection system we use a directional Gaussian beam source localized in Si propagating towards the intersection with a grating system. The Si WG is terminated by the outcoupler based on a circular Bragg grating (CBG) geometry (see Fig. 3(c)) fabricated in InP on planar Si. Another design, based on standard components available for 220 nm Si could be used for high outcoupling efficiency, like non-uniform grating couplers burried in SiO2, which provides 70% of outcoupling efficiency directly to optical fiber at 1550 nm wavelength [56]. Alternatively, the outcoupler rings could be realized in InP on top of the planar Si region during a dry etching step, which is consistent with the same process flow applied for the taper fabrication (see Fig. 2). Therefore, the InP-made CBG height H in the following analysis is also set to 0.58 µm. The outcoupler design is based on half-CBG to modulate the refractive index with a radial period R, defined as a sum of the InP material radial width W and the width of the etched part, as schematically presented in Fig. 3(c). In this approach, we look for the optimal geometry focusing on the highest collection efficiency along the normal direction to the sample surface at λ₀ = 1.55 µm wavelength, within the numerical aperture of NA = 0.65, characteristic for typical microscope objectives used in experimental setups. Therefore, the outcoupling efficiency (OE) is defined as OE = P_NA0.65/P_SiWG, where P_NA0.65 is the power collected within a given numerical aperture and P_SiWG is the power transferred along the Si WG.

 

Fig. 8. Evaluated directional dipole emission coupling for a hybrid InP-Si waveguide system for a dipole displacement along (a) the vertical axis ‘z’ and (b) the horizontal axis ‘x’. The position assigned to Δz = 0 µm and Δx = 0 µm corresponds to the center of InP WG.

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First, considering terminated Si WG and the beginning of the InP/Si outcoupler section, we evaluated high modes matching between fundamental modes on the level of 60%, which, at first, leads to high transmission between sections, secondly, to efficient photon field transfer from the Si WG to the InP/Si grating. The reason is mainly related to a mismatch in refractive indices between InP and Si (n_InP < n_Si), which leads to a strong localization of the mode in the Si of the outcoupler section. We note that contribution of higher order modes coupling in the intersection bewteen the Si WG and CBG is negligible (see also Supplement 1, Sec. VII, Fig. S8); however, by using different SOI wafers the transfer to InP via higher order modes could possibly be enhanced.

Next, optimal collection efficiency along the normal direction to the sample plane is expected if the Bragg condition is fulfilled, namely, when the grating period R is related to the propagating photon wavelength λ₀: n·2π/R = 2π·n_eff/λ₀, where n = 0, 1, …, thus R = λ₀/n_eff. Therefore, we need to reformulate the Bragg condition using n_eff, which leads to R = λ₀·(1−1/m)/n_eff+ λ₀/m, where ‘m’ gets a value of 2 or 4. The first term is assigned to the InP radial width (W) and the second term to the etched region. In this framework, we obtained OE of roughly 1% at 1.55 µm for both m = 2 and m = 4, suggesting that in the case of 3D scattered field simulations, this approximation is far from optimal. Subsequently, we redefined the grating period to R = [λ₀·(1−1/m)/n_eff + λ₀/m]·S, where the additional parameter S is used to scale the size of the grating system. In the case of m = 2, we performed simulations using S in the range of 0.8–1.8. We found that the maximum value of outcoupling efficiency is roughly 11%, for S = 1.75. Next, we followed the simulation procedure for m = 4 and S spanned the range of 0.6–2.2. The obtained results of OE as a function of S in a wavelength range of 1.5–1.6 µm are demonstrated in Fig. 9(a). The maximum OE at 1.55 µm amounts to 26%. It is obtained for S = 1.925, corresponding to the ring period R ≅ 1.453 µm. Interestingly, OE > 20% is obtained for R roughly ranging from 1.396 µm to 1.566 µm, demonstrating that the tolerance in fabrication accuracy of such a CBG outcoupler is approximately 85 nm. In addition, Fig. 9(b) shows the calculated optical far-field distribution in the cross-section of the outcoupler and the evaluated far-field distribution of the scattered field along the top direction. Simulations are performed for the highest outcoupling at 1.55 µm. In the cross-sectional view (bottom), one can see an effective field transfer from the Si planar structure to InP rings and the gradually decreasing field intensity along the direction determined by the waveguide structure. The projected far-field distribution is well-focused, leading to high collection efficiency with the NA = 0.65 microscope objective as well as by optical setups equipped with lower NA objectives (e.g., 0.4), for which OE amounts to 17%, that is, a still reasonably high value.

 

Fig. 9. a) Vertical outcoupling efficiency for a circular Bragg grating structure plotted as a function of the scaling parameter of the grating size (S) and wavelength. b) Far-field distribution of the scattered optical field collected above the outcoupler and the integrated E-field distribution in the outcoupler vertical cross-section.

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3.4 Broadband operation

The idea of using a self-assembled single quantum dot as an emitting source in photonic integrated circuits usually suffers from some uncertainty in the QD operation wavelength due to randomness in the nanostructure size and composition. In this respect, it is required that the on-chip photonic system would have sufficient tolerance for the variability in the QD emission wavelength. The concept presented in this work is based on the elements where no cavity effects are needed. Therefore, efficient broadband operation is expected. We examined all the components introduced here regarding wavelength dependence between 1.5 µm to 1.6 µm. These components are: i) the hybrid tapered waveguide structure used for the transfer of emission from the InP to Si WG, ii) dipole coupling to TE modes confined in the hybrid WG section, and iii) the outcoupler based on a CBG for outcoupling of the field from the Si WG. Figure 10 shows the dispersion of transmission efficiencies and dipole coupling for the best taper structure (see Fig. 4(b)). Interestingly, on-chip optical field transfer maintains a very uniform overall efficiency of about 33${\pm} $3% in the entire considered spectral range, while it could be increased twice by using a directional QD-WG system by using a grating structure [53] or chiral coupling [54]. Considering the light outcoupling from the chip with the field collection optics of a high numerical aperture (NA = 0.65), one can attain a total transmission efficiency of 9 ± 1%, which equivalently is the on-chip transfer efficiency of bidirectional dipole emission multiplied by the outcoupling efficiency of the CBG-based outcoupler.

 

Fig. 10. Wavelength dependence of: I - transmission coefficient of the taper structure, II - dipole (QD) coupling efficiency to the TE modes and III - efficiency of the vertical outcoupler together with a combined total efficiency of about 8-10% offered in the range of 1.5-1.6 µm.

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In addition to out of plane outcoupling, one can consider the in-plane outcoupling method. In this case, a Si WG fiber output port along the Si WG can be applied. With this solution, the Si WG should be terminated with an inverted taper structure, i.e. spot size converter, to enhance the in-plane directionality of outcoupling. For this type of in-plane outcoupler, we used FDTD to evaluate the incoming light scattering from the Si WG on a cleaved facet of the PIC. To simulate the collection efficiency, we used the far-field monitor placed at various distances ranging from 50 µm to 200 µm. The far-field pattern analysis using NA = 0.65 demonstrates collection efficiency ranging from 10% to 2%, resulting in circa 5% of the total efficiency of the side detection. This value can be increased by utilizing the lensed optical fiber as a far-field collector aligned to the cleaved facet of the qPIC (here, the spot size converter should be used to taper down the Si WG tip, optionally burried in SiO2). It increases the in-plane collection efficiency to 26% [24], resulting in roughly 8% of the total in-plane photon field transfer efficiency in the hybrid WG system with QDs.

4. Conclusions

In this work, we focus on the numerical investigation of the heterogeneously integrated hybrid waveguide system developed on the silicon-on-insulator platform to demonstrate its capability when used as a scalable quantum on-chip photonic integrated circuit. The structure can be fabricated by standard photo- or electron-beam lithography techniques, utilizing a bonding step to join two wafers of different materials. One of the materials can be based on the III-V compound, e.g. InP, which allows for the fabrication of high-quality quantum dot emitters, using, e.g. InAs, and operating at the 1550 nm photon wavelength. Therefore, we studied the dipole coupling in the hybrid InP/Si waveguide, the mode structure depending on the cross-sectional geometry, the linear taper structure for efficient transfer of guided light from the InP/Si hybrid system to the Si waveguide, and the possible outcoupling to interconnect with the external detection system. First, we found that the dipole coupling to the InP/Si waveguide modes amounts to 38%. In addition, applying a reflector on one side of the waveguide [53] or employing a chiral coupling effect [54] can ensure unidirectional emission and the dipole coupling can reach up to 76%. This number can still be significantly increased using a quantum dot-in-cavity design, where the cavity is incorporated into the InP/Si waveguide, nonetheless, for the price of narrowing the spectral bandwidth of the device. Next, we found that the photon field transfer between the InP/Si waveguide and the Si waveguide reaches 87%. Therefore, the overall on-chip coupling, considering the light field transfer from an emitting dipole to the Si waveguide, can reach 33%. Again, this number is highly controlled by the dipole-waveguide directional coupling and can be increased at the cost of spectral narrowing of the device's transmission bandwidth. However, we show that our conceptual device can operate in the broadband spectral range of 1.5-1.6 µm. It eliminates the problem of precise control over the quantum dot emission wavelength, which is critical for the cavity fabrication process. Finally, we investigated outcoupling efficiency realized by the circular Bragg grating-based system for the out-of-plane collection and by the spot size conversion for the in-plane collection. The results show the efficiency of the vertical outcoupling at the level of 26% and up to 10% of the in-plane outcoupling, both within NA of 0.65, and both showing broadband operability.