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Abstract
We design a quantum dot (QD) embedded in a vertical-cavity photonic nanowire (NW), deterministically integrated on a silicon-on-insulator (SOI) waveguide (WG), as a novel quantum light source in a quantum photonic integrated circuit (QPIC). Using a broadband QD emitter, we perform finite-difference time domain simulations to systematically tune key geometrical parameters and to explore the coupling mechanisms of the emission to the NW and WG modes. We find distinct Fabry-Perot resonances in the Purcell enhanced emission that govern the outcoupled power into the fundamental TE mode of the SOI-WG. With an optimized geometry that places the QD emitter in a finite NW in close proximity to the WG, we obtain peak outcoupling efficiencies for polarized emission as high as eighty percent.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum photonic integrated circuits (QPIC) have emerged as a powerful platform to incorporate quantum optical elements on a single chip for driving quantum technological applications in secure data communication and distributed quantum computing [1–4]. Hereby, single photons or flying qubits are considered as key messengers and units supporting many versatile concepts, ranging from long-distance transmission of quantum information to quantum repeater networks [5,6], as well as to linear optical quantum computing [7]. Thanks to the weak decoherence of photons in transparent media, fast transmission speeds and the accessibility to highly developed nano-photonic heterogeneous integration technologies [8–10], QPICs with added scalability [11] and commercialization [12] are envisaged in the near future.
However, for such a practical QPIC, there is still the need for arrays of integrated, on-demand, deterministic and highly efficient photon sources that can generate single or entangled photons pairs. Unlike externally coupled photons generated probabilistically via spontaneous parametric down conversion sources [13], heterogeneously integrated two-level quantum emitters, such as III-V quantum dots (QDs), are the most promising solid-state based single photon sources whose brightness is intrinsically decoupled from purity. Current approaches using heterogeneously integrated III-V QDs aim primarily at QD-in-cavity-quantum electrodynamics (cavity-QED) systems, where self-assembled InAs/GaAs QDs are integrated in GaAs nanophotonic geometries that act as low-loss waveguides (e.g., self-supporting photonic crystal membranes or micropillars) [10,14–16]. In this scheme, small footprint quantum emitters with excellent single-photon emission properties and even spin-qubit operation have been demonstrated [17], however, at the expense of excessive post-growth processing and spectral fine-tuning efforts for such proof-of-principle III/V platforms. On the other hand, Si-based waveguide (WG) QPIC systems are technologically much more appealing, because of ultralow optical losses down to 0.1 dB/cm at telecommunication-compatible emission wavelengths, and the adaptability of complementary-metal-oxide-semiconductor (CMOS) technology in mainstream Si-photonics [18]. QD-in-cavity-QED systems on Si-based QPICs have remained, however, rather scarce to date [8,19,20].
Despite these advances, traditional QD-in-cavity systems [21,22] – which rely on the Purcell effect [23] that accelerates the spontaneous emission (SE) of the QD into a resonant cavity mode by nature of the high Q factor and low mode volume of the cavity – are hampered by specific intrinsic limitations such as the method to extract emission from the QDs which is limited by the resonance matching of the narrow-band cavity resonance to quasi-monochromatic emission from the QDs, making it a challenge to also achieve high extraction efficiencies from the cavity [24]. In contrast, photonic nanowires (NWs) act as excellent WGs due to the high confinement factor of their transverse modes and low-loss propagation below the material bandgap [25]. QDs embedded in NWs tackle the above limitation by enabling high broadband coupling to guided modes in the NW while also suppressing the emission into the continuum of radiation modes [26–29]. In this regard, the role of NWs as a WG has already been well understood by calculating the SE rate into the fundamental HE11 modes [30].
In addition, photonic NWs offer another huge advantage in terms of low-cost high-density heterogenous integration without the need for probabilistic post-processing. The site-selective synthesis of III-V NWs directly on proximal photonic silicon-on-insulator (SOI) WGs presents a monolithic and deterministic pathway for ultra-accurate position/geometry control for advanced emitter architectures. Optoelectronic device structures such as integrated NW lasers have been already demonstrated with remarkable position-control, and the coherent light coupling properties were studied both experimentally [31,32] and numerically [33] in the last few years. Experimentally, there have been also several demonstrations of III-V QDs embedded in simple NWs that emit (out-couple) single photons with excellent photon purity and indistinguishability into free space [21,34–37]. However, to date, there have been no reports (experimentally nor theoretically) of a direct, on-chip QD-in-NW system coupling single or entangled photons into Si-based WGs of QPICs. There were first successful attempts to demonstrate coupling of single photons by using WGs encapsulated around randomly placed NWs (picked and dropped by nanomanipulators) [38], however, these schemes are not scalable and incompatible with large scale integration.
Against this background we propose, in this article, a monolithic NW-WG architecture that efficiently couples light from a single quantum emitter embedded in a NW cavity into a proximal SOI WG. By combining the waveguiding nature of the NW and the propagation of the fundamental modes in the SOI ridge WG, we design a vertical-cavity NW and calculate numerically optimized values for the out-coupling efficiencies. We further present systematic analysis of critical geometrical parameters that affect the SE enhancement (Purcell effect) of the emitter and the coupling efficiency to the ridge WG. The simulation structure used in this work is inspired, in part, by previous [34,35] efforts to realize a quantum emitter emitting in the telecom O-band (1260-1360 nm) in a NW cavity.
2. Methods and simulation structure
We performed broadband finite-difference time-domain (FDTD) simulations, using the commercial Lumerical software, in order to calculate the in-coupling into guided modes and out-coupling efficiency of a monolithic QPIC-integrated NW quantum emitter. The frequency dependent ${n_{eff}}$ was determined using the Finite Difference Eigenmode (FDE) solver software tool MODE (Lumerical). The simulation structure consists of a single NW integrated on a Si ridge WG placed on a 4 µm thick layer of SiO2. The dimensions of the Si ridge WG are given by height H and width W, and we assume it to be infinitely long. The NW is chosen to be at the center of the WG located at W/2 from the ridge edge. Hereby, the NW is modelled as an elongated hexagon typical of [111]-oriented III-V NWs [39,40] and is placed such that the longer axis is perpendicular to the WG surface (i.e., vertical-cavity NW) as shown in Fig. 1(a). The dimensional parameters of the NW are given by the total length L and diameter 2R. To design the structure according to commonly investigated NWs for hosting single quantum emitters in the near-infrared region, we assume that the material of the NW is, for example, binary GaAs [35,37,41]. However, it is important to note that the only material dependent parameter affecting our simulations is the dielectric constant of the material used. The quantum emitter representing a single QD (e.g., InAs QD) is modelled as an electric dipole placed inside the NW along the central axis at a distance Z from the NW-Si WG interface.
Fig. 1. (a) Schematic representation of the simulation structure. Inset: In-plane dipole orientation, and the room temperature refractive index values (n) of the materials for λ = 1.3 μm [42]. (b) Illustration of the possible channels for the coupling of the QD dipole emission to guided modes in the WG: (i) the direct channel in which the dipole emission evanescently couples to the WG modes, and (ii) the indirect channel in which the dipole emission first couples to guided NW modes, and the NW modes evanescently couple to the guided WG modes. The normalized electric field intensities of these guided modes are shown along with their evanescent field overlap at the NW-WG interface. Also shown are the total SE enhancement factor ${\varGamma }$ of the dipole, the Purcell factor ${{F}_{NW}}$ for SE into the NW mode and the SE enhancement factor into radiation modes ${\varGamma }$. (c) Intensity distribution of the electric field in the longitudinal cross-section of the NW-WG system, obtained by optical 3D FDTD simulations, showing the indirect broadband coupling of the dipole’s SE to the WG TE01 mode via the NW HE11b mode. The inset shows the intensity distribution of the electric field of the HE11b and the TE01 modes, respectively.
To keep the optimization procedure widely applicable, we utilize a dipole having a broadband emission λ (0.8 µm to 1.8 µm) with a central wavelength of 1.3 µm. This frequency range covers several experimentally realized quantum emitter systems in photonic cavities [21,27,36,37,41]. All the dielectric materials have constant and real refractive indices set to their values at room temperature (300 K) at 1.3 µm [42]. For generality, the emission wavelength is normalized with a geometrical parameter, namely the radius of the NW as 2πR/λ (=ωR/c), where ω is the frequency of emission, and c is the speed of light in vacuum. All other geometrical parameters to be optimized are also normalized to the NW radius (e.g., normalized WG height H/R), to maintain the scale invariance of the physical trends.
The dipole emission couples to any mode which has an electric field intensity that spatially overlaps with the electric field of the dipole. A dipole inside the NW cavity emits both into the guided modes of the NW and the continuum of radiation modes. Therefore, there are two channels for the coupling of the emission to the guided modes in the WG, as shown in Fig. 1(b) – (i) A direct coupling by an overlap of the electric dipole vector with the electric field of the WG modes, and (ii) an indirect coupling by an overlap of the electric dipole vector with the electric field of the NW modes which then propagate to the NW-WG interface and evanescently couple with the WG modes. Since, the electric field of the WG modes decay exponentially outside the WG, for a dipole source far from the WG, we neglect the direct coupling channel (i). Figure 1(c) shows the corresponding electric-field intensity distributions for the out-coupling of the broadband dipole emission into the WG modes by the indirect channel (ii). The aim of the optimization is now to achieve the highest possible enhanced power transmission into the guided modes of the proximal ridge WG. To maximize the out-coupling into the WG modes, we focus on the indirect channel (ii), and divide the optimization into three conceptually distinct parts – (Part 1) the coupling efficiency of the dipole’s SE into NW modes, (Part 2) the evanescent coupling of the NW modes to the WG modes, and (Part 3) the Purcell enhancement of the SE of the dipole in the entire dielectric environment. This break down of the optimization problem offers clear insights into the coupling mechanisms while making the problem tractable.
3. Part 1: broadband SE enhanced coupling of QD into guided modes of NW
Semiconductor QDs embedded in NW heterostructures show near-unity radiative quantum yield at cryogenic temperatures (∼10 K) [26,43,44]. Therefore, we assume that the QD two-level system has no channels of non-radiative decay and the total SE includes only emission into guided modes of the NW and into the continuum of radiation modes. To quantitatively study the emission from the electric dipole into the guided NW modes, we considered that the NW has infinite length. The total SE enhancement factor of the dipole is given by
where ${F_M}$ is the SE enhancement factor for the dipole emission, into the transverse mode of interest M in the NW (e.g. the fundamental HE11b mode), called the Purcell factor, and $\gamma $ is the SE enhancement factor into all other modes. All three factors are normalized over the SE enhancement factor in an isotropic medium of the same dielectric material ($n = 3.41$). The coupling efficiency ${\beta _{QD - M}}$ is the fraction of photons emitted by the dipole into the guided mode M given by [28],Since the dipole is placed with an in-plane orientation ϕ at the geometric center of the cross-section of the NW, the emission couples only to the HE1 m and EH1 m classes of transverse guided modes. Note, these occur in pairs (e.g., HE11a and HE11b) and have transverse field intensity maxima at the center of the NW, as shown in the inset of Fig. 1(c). In this work, we restrict the NW geometry to support only the fundamental guided modes, which are the HE11a and HE11b modes. The propagating mode with the slightly lower ${n_{eff}}$ is conventionally referred to as the HE11a mode. But for an easier and consistent description, we rather name the modes based on the component of the electric field that has a maximum in the field profile at the NW center. Thus, HE11a is the mode with the maximum for the x-component and HE11b is the mode with the maximum for the y-component.
The in-plane orientation of the electric dipole determines the coupling efficiency into each of the HE11 modes. The projections of the dipole vector along the x- and y-axes excite the HE11 modes a and b, respectively, due to the overlap with the electric fields in the same directions. If the dipole unit vector is at an in-plane angle $\phi $, the projections along x-and y-axes are $\cos \phi $ and $\sin \phi $, respectively. Therefore, the coupling into the HE11a (HE11b) mode varies as the square of the normalized x- (y-) component ${\cos ^2}\phi \; $(${\sin ^2}\phi $). Figure 2(a) shows this dependence of the QD-NW coupling efficiency into the two HE11 modes as a function of $\phi $ for an infinite NW of fixed radius R = 180 nm.
Fig. 2. (a) Dependence of the QD-NW coupling efficiency into the two HE11 modes as a function of $\phi $ for an infinite NW of fixed R = 180 nm (b) SE enhancement factors of in-plane dipole at $\phi $ 90°: including total SE enhancement factor $\Gamma $, Purcell Factor ${F_{{HE}{{11}_b}}}$, the SE enhancement factor $\gamma $ into all modes other than HE11b, and the coupling efficiency ${\beta _{\textrm{QD} - \textrm{HE}{{11}_b}}}$ of the dipole into HE11b mode. The effective radii (top axis) are given for the central wavelength λ = 1.3µm
Next, we investigate the effect of NW radius on the coupling into one of the guided HE11 modes. Figure 2(b) shows the coupling efficiency of the QD dipole orientated at $\phi = 90^\circ $ into the HE11b mode of the NW, ${\beta _{QD - HE{{11}_b}}},$ over normalized wavelengths $2\pi R/\lambda $ (a dipole oriented at $\phi = 0^\circ \; $couples to the HE11a mode with a very similar dependence on normalized wavelengths). For small values of $2\pi R/\lambda $, ${\beta _{QD - HE{{11}_b}}}\; $is small and approaches zero because the HE11 modes are still guided but highly deconfined and the field intensity at the position of the dipole is low [39,45]. With increasing normalized wavelengths, ${\beta _{QD - HE{{11}_b}}}\; $rises very quickly to a broad maximum with values above 90% from $2\pi R/\lambda = 0.72$ to $2\pi R/\lambda = 1.01$ and a maximum coupling efficiency of 95.1% at $2\pi R/\lambda = 0.87$ (Reff $= \,\; 180 $nm). In this region, the HE11 modes are optimally confined due to an increased group index resulting in a large Purcell Factor ${F_{HE11}}_b\; ( = {F_{HE{{11}_a}}})\; $[26]. Simultaneously, the emission into radiation modes $\gamma $ is strongly inhibited [26] reaffirming the significant advantage of using the NW cavity. The coupling efficiency drops again for $2\pi R/\lambda > 1.1$ where $\gamma $ starts to dominate the SE. These values are in excellent agreement with similar calculations in literature [26,46], albeit with small deviations arising from the cylindrical NW geometries and slightly different refractive indices used in previous reports. The predicted maximum coupling efficiency (>95%) is also on par with the numerically calculated coupling efficiency of emission from a QD to slow light in photonic crystal WG architectures [14], while a coupling efficiency greater than 90% has also been experimentally measured for InAs QDs in GaAs NWs [35].
Since the dipole of an exciton in an axial QD would ideally be in-plane with any orientation $\phi \; $for an unpolarized excitation, the coupling efficiency for comparison with experiments would be the expectation value averaged over all the orientations. Assuming all orientations in-plane are equally likely for a symmetric QD, the average peak coupling efficiency into the HE11a or HE11b mode would be $ \langle{\cos ^2}\phi \rangle = \langle {\sin ^2}\phi \rangle = 0.5$ times the maximum in the $\phi = 0^\circ $/ $\phi = 90^\circ \; $cases, equal to 47.5% here. For the extraction of emission from a QD through the NW top facet, it is favourable for the randomly oriented in-plane dipoles to couple to both the HE11a and HE11b modes as shown by Claudon et.al [35]. But the presence of the WG at the bottom facet along one principal axis makes the equal average coupling to both modes undesirable, and we will show in the next section that only the HE11b mode couples to the WG’s fundamental TE mode. For all further optimization steps, we fixed the NW radius to the optimal value R = 180 nm where ${\beta _{QD - HE{{11}_b}}}$ is maximum.
4. Part 2: Evanescent coupling of NW HE modes with Si-WG TE mode
The Si-WG is considered as a lossless dielectric, thereby having a propagating mode described by a real wavevector ${k_{x,WG}} = {n_{eff,WG}} \times {k_0}$, where ${n_{eff,WG}}$ is the effective refractive index of the WG mode, and ${k_0}\; \, = \; \,2\pi /\lambda $ is the wave vector in vacuum. The transverse components ${k_y},\; {k_{}}_z$ of the guided modes are purely imaginary outside the WG leading to an exponential decay of the optical field with distance. This evanescent decay rate depends on ${n_{eff,WG}}$, and the finite electric field outside the WG allows tunneling of the mode into a dielectric nearby, such as the NW. Thus, the wavevector of the propagating mode becomes real again. The NW mode has a similar decay of the electric field inside the WG, and the evanescent field overlap of the NW and WG modes is determined, on one hand, by ${n_{eff,WG}}$. On the other hand, another necessary condition for the coupling of the NW and WG modes is their phase matching in the direction of propagation (x-axis, here). As shown by Bissinger et.al [33], this occurs when the x-components of the wavevectors of these modes are approximately equal, i.e., ${k_{x,WG}} \approx {k_{x,\; NW}}$, and here there is an implicit dependence on ${n_{eff,WG}}$ (for the definition of ${k_{x,\; NW}}$, see S3 in Supplement 1). The phase matching and the evanescent field overlap together lead to a complex frequency dependence of the total transmission from the NW mode to the WG mode.
It is also important to study the reflectivity of the NW modes at the bottom facet since, with a larger reflection back into the NW, the light that can be coupled into the WG modes is reduced. Unlike the flat top facet of the NW, which acts as a dielectric mirror for the NW modes, the WG and the substrate lead to a very different frequency dependent reflectivity at the bottom end-facet. In general, this is described as a result of phase changes picked up by the propagating and reflected modes at the NW/WG and WG/substrate interfaces and from propagation inside the WG, all of which are influenced by the WG geometry and refractive index changes [33]. Since the refractive indices of the NW ($n = 3.41$) and WG ($n = 3.5$) are very similar, the reflection occurs mainly at the WG/substrate (WG/sub) interface which has a large dielectric mismatch and a refractive index contrast of 2.33. The geometry of the WG affects ${n_{eff,\; WG}}$ through both the width W and height H, and the phase of the reflected NW mode through the path length 2H in the z-direction. Therefore, we optimize the normalized dimensions H/R and W/R separately, by calculating both the frequency dependent reflectivity for the NW HE11 modes, and the modal transmission efficiency from the HE11 modes to the WG TE modes (for details, see sec. S2 in Supplement 1).
The HE11a mode is observed to have a high transmission to the TE11 mode and negligible transmission to the TE01 mode, and it is vice-versa in the case of the HE11b mode. This is due to the orientation of the field profiles and how the phase matching condition is satisfied differently for the two HE11 modes. Since we restrict the design of the Si-WG to support only the propagation of the fundamental TE01 mode, we study only the behavior of the HE11b mode at the interface. It is important to note here that this poses a fundamental limit to the out-coupling of emission from unpolarized dipoles into the WG, since only 50% of the emission averaged over all in-plane orientations couples to the HE11b propagating mode of the NW, as calculated in Part 1.
We varied the normalized WG height between H/R = 0.3–0.75 in steps of 0.05 (ΔH = 9 nm, for R = 180 nm), keeping the normalized WG width as W/R = 3 (W = 540 nm) and investigated the frequency dependent evolution of the reflectivity ${R_{HE{{11}_b} - WG}}\; $and transmission efficiency ${T_{HE{{11}_b} - TE01}}$. For any H/R, the plot shown in Fig. 3(a) illustrates a broad maximum in reflectivity at a small value and a subsequent minimum for larger values of normalized wavelengths ($2\pi R/\lambda $) respectively. With increasing H/R, the reflectivity decreases overall (from ∼26% to less than 16%), and the features shift to smaller values of the normalized wavelength. The decrease in the overall reflectivity is due to the reduced evanescent coupling to the WG TE01 mode which supports the interaction of the HE11b mode with the substrate. On the other hand, the shift in the peak values towards smaller normalized wavelength occurs because the phase matching condition is satisfied only at larger wavelengths due to the larger$\; {n_{eff,WG}}$ and path length 2H for the total phase accumulated in the reflection process. The transmission efficiency to the TE01 mode is shown in Fig. 3(b) and is 0 for ratio of H/R = {0.3, 0.35, 0.4}, because the frequency cut-off for the TE01 mode is outside the considered range (Fig. S2 in Supplement 1). For a fixed value of H/R = 0.45, at low normalized wavelengths, ${T_{HE{{11}_b} - TE01}}$ continuously increases until it reaches a peak value of 28.4% at $2\pi R/\lambda = 0.99$. This is due to a smaller ${n_{eff,WG}}$ that leads to a larger tunneling of the WG mode into the NW and a larger spatial overlap with the HE11b mode [33]. Further increasing the normalized wavelength results in ${T_{HE{{11}_b} - TE01}}$ again tending to 0 since the phase matching condition between the HE11b and TE01 modes is no longer satisfied.
Fig. 3. Variation of reflectivity (%) (a) and transmission efficiency (%) to the TE01 mode (b) with WG height H/R, for fixed WG width W/R = 3. Peaks in ${\textrm{R}_{\textrm{HE}{{11}_b} - \textrm{WG}}}\ {\rm and}\ {\textrm{T}_{\textrm{HE}{{11}_b} - \textrm{TE}01}}$ shift to lower normalized wavelength units for larger H/R. This is due to the phase matching condition for maximal coupling. Variation of reflectivity (%) (c) and transmission efficiency (%) (d) with WG width W/R, for optimized WG height H/R = 0.45. The dimensions W/R and H/R which fall below the cutoff for the TE01 mode are shown in the plots with a transmission efficiency of 0%. All plots reveal that the highest transmission efficiencies are obtained for W/R = 3 and H/R = 0.45, both just above the cutoff size for propagation of the TE01 mode. The effective emission wavelength is given for a radius of R = 180 nm (top axis).
For H/R ratio larger than 0.45, ${T_{HE{{11}_b} - TE01}}$ decreases significantly due to the increase in ${n_{eff,WG}}$, while the peak shifts to smaller normalized wavelengths. For maximizing the transmission efficiency for a large frequency range (irrespective of an increase in reflectivity) the smallest H/R ratio with a propagating TE mode, H/R = 0.45, is ideal. As is evident for the larger H/R ratios, the peak in transmission efficiency appears at a normalized wavelength different from the position of the reflectivity minimum. This indicates that the transmission into the WG TE mode is not maximized at the reflectivity minimum due to larger transmissions into the substrate.
To tune the transmission maximum towards the reflectivity minimum, we kept H/R fixed to its optimal value of 0.45 and varied the normalized WG Width W between W/R = 2-5 in steps of 0.5 (ΔW = 90 nm for R = 180 nm), which changes ${n_{eff,WG}}\; $without affecting the path length for the reflected mode inside the WG. The reflectivities for different W/R ratios, as shown in Fig. 3(c), are very similar at large normalized wavelengths and show clear and distinct maxima for smaller values. The global maxima at larger wavelengths are due to the high lateral confinement of the WG mode leading to lower losses and therefore, a significant increase in reflectivity with increasing W/R. Considering the transmission efficiencies with W/R, as shown in Fig. 3(d), we see that the WG does not support the propagating TE01 mode for W/R = {2, 2.5}. Hereby, the transmission efficiency decreases over the entire frequency range for increasing WG width, due to larger ${n_{eff,WG}}$. Due to the phase matching condition, we also observe the familiar shift in the peak positions to smaller normalized wavelengths for larger W/R.
Crucially, the reflectivity minima and the transmission maxima move further apart with increasing W/R. Therefore, the increase in overall transmission efficiency with smaller dimensions W/R and H/R is accompanied by an increase in reflectivity, indicating that the reflected light does not get funneled into the TE01 mode anyway. While the geometry is optimized for the largest transmission efficiency, the simultaneous maximization of the reflectivity is an added benefit that leads to a stronger field for the HE11b mode inside the NW and, consequently, an enhanced interaction with the QD dipole. The optimal normalized width is W/R = 3, and in this case, supports a propagating TE01 mode with a peak transmission efficiency also of 28.4% at the same position $2\pi R/\lambda = 0.99$. In all subsequent calculations, we fix the WG dimensions to the optimized values, which are the cut-off values for the TE01 mode, in this case H/R = 0.45, and W/R = 3.
We briefly discuss the sensitivity of the out-coupling efficiency of the QD to the TE01 mode with respect to slight changes in parameters R, W and H, to provide useful insights for future experimental studies (for more details, see sec. S5 in Supplement 1). Hereby, we found that the broadband maximum in the coupling efficiency ${\beta _{QD - HE{{11}_b}}}$, as seen in Fig. 2(b), is quite insensitive to small changes in NW dimension (radius R). The WG width W is also very robust against small variations, and needs to be just above the TE01 cutoff. The WG height H, on the other hand, is the most sensitive parameter, and from Fig. S5 (Supplement 1) it is clear that the hierarchy of robustness of the parameters follows W >> R > H. However, it should be noted that since the optimal values of W and H are the cutoff dimensions for the TE01 mode, smaller values will result in a complete drop in out-coupling efficiency to zero since the TE01 mode is no longer the propagating fundamental mode in the WG.
5. Part 3: SE enhancement of QD emitter in an integrated device
So far, we have discussed broadband coupling and SE enhancement in infinite or semi-infinite wires to optimize parameters such as R, W and H. However, in a practical integrated device the NW has a finite length L and the position Z of the QD in the NW is definitive. In order to study the impact of these parameters on the coupling efficiency $\beta _{QD - TE01}^{finite}$ of the dipole emission to the WG TE01 mode, we restrict first the investigation to the case where direct evanescent coupling of the QD dipole to the WG modes (Fig. 1(b)) is negligible. By placing the electric dipole at the center of a NW of length L=3.6 µm, at a distance Z = L/2 = λmax = 1.8 μm from both end facets, we focus on the indirect coupling channel in an integrated device. This also allows us to make the assumption as in other works [28,46,47] that the emission into radiation modes is not affected by the end facets, and is the same as in the case of the infinite NW. Only the guided NW modes form longitudinal cavity modes upon successive reflections from the end facets, which act as dielectric Bragg mirrors. Thus, the NW cavity is modeled as a Fabry-Perot resonator [46,47] with frequency dependent reflectivities for the top and bottom facets for the propagating modes. The orientation of the dipole is fixed here to $\phi = 90^\circ $ so that the HE11b mode is selectively excited in the NW. The linewidth of the Fabry-Perot mode formed by the HE11b mode depends on the length of the NW cavity and the reflectivity of the HE11b at both end facets.
The normalized Purcell factor into the HE11b mode ${F_{HE{{11}_b}}}$ is calculated as,
where ${\Gamma _{finite}}$ is the normalized total SE enhancement factor calculated here for a finite NW, and ${\gamma _{infinite}}$ for an infinite NW is from the calculation in Part 1 before.First, we consider that the finite NW described now is suspended in air. The Purcell factor for the emission into the HE11b mode of the NW, shown in Fig. 4(a) (in black), describes Fabry-Perot resonances over normalized wavelength $2\pi R/\lambda $, as evidenced by the increase in distance between consecutive peaks with decreasing $\lambda $. These resonances are a result of the phase of the electric field standing waves at the dipole location, with enhancement (${F_{HE{{11}_b}}} > 1$) and inhibition (${F_{HE{{11}_b}}} < 1$) corresponding to longitudinal antinodes and nodes, respectively [47]. The Q-factor of the NW for the HE11b mode is estimated to be ∼107 from $Q = \omega /\; \mathrm{\Delta }\omega \; = ({\omega R/c} )/\mathrm{\Delta }({\omega R/c} ),\; $where $\omega R/c\; $is the FWHM of the spectral emission peak at $\omega R/c\; \, = 0.94$ obtained from a Lorentzian fit. The low Q is a result of the poor reflectivity for the HE11b mode (Fig. S4 in Supplement 1) and the small length of the NW considered here. Emission in longer NWs would result in a reduction of both the spacing between the modes, as well as their linewidths [47]. Figure 4(a) (in blue) shows the Purcell factor for the dipole in a fully integrated device (NW integrated on the Si-WG). Each peak in ${F_{HE{{11}_b}}}$ has split into two smaller peaks, one on each side of the original peak, due to two new standing wave modes in the integrated structure. A weak standing wave mode is set up at a smaller wavelength (larger normalized wavelengths) at the NW/WG interface, and a stronger standing wave mode is set up at a larger wavelength (smaller normalized wavelengths) at the WG/sub interface where the reflectivity is much higher. The reflections at these two interfaces are illustrated in the schematic of Fig. 4(d).
Fig. 4. (a) Purcell factor, ${F_{HE{{11}_b}}}\; $for the emission funneled into the HE11b mode, in a NW suspended in air and for a NW integrated on a WG. Fabry-Perot resonances are observed with a Lorentzian fit for one of the peaks revealing a Q-factor of ∼107. (b) Optimized bidirectional coupling efficiency ${\beta _{QD - TE01\; }}$from QD to WG TE01 mode for a semi-infinite NW considered in Parts 1 and 2 and a finite NW integrated on WG. Resonances are seen over the envelope curve, with peak positions corresponding to positions of anti-resonance in (a). (c) Total out-coupling into WG TE01 mode, $\mathrm{\varepsilon }$, showing peaks in transmission closely following peaks in the Purcell factor for an integrated structure in (a). The effective wavelength of emission is given for a radius of R = 180 nm (top axis). (d) Schematic of the Fabry-Perot resonances in ${F_{HE{{11}_b}}}$ for the integrated structure resulting in the splitting of the peaks in (a), and an example of the longitudinal mode at the position of a maximum in ${\beta _{QD - TE01}},$ showing an antinode at the NW-WG interface and a sub-maximum field intensity at the dipole source.
The observed decrease in maximum Purcell factors, compared to the NW in air, is due to the reduction in Q for the integrated device in which the bottom facet of the NW is now in contact with a dielectric material with $n = 3.5$ (Si), larger than both air and the NW material itself. Consequently, the original peak positions have developed into positions of anti-resonance due to a different phase change accumulated upon internal reflection at the NW bottom facet, at the same wavelength. This difference in phase change, going from air to an integrated device, is also the reason why the resonance for the same cavity length L, shifts to a shorter wavelength. It is important to note that the total internal reflections of the propagating modes at the NW end facet do not pick up a phase change of π as in the case of a plane wave incident on an optically denser medium from a rarer one [33].
Figure 4(b) shows the coupling efficiency $\beta _{QD - TE01}^{finite}$ of the dipole emission into the TE01 mode of the Si-WG over $2\pi R/\lambda $ calculated as,
The finite NW coupling efficiency, $\; \beta _{QD - TE01}^{finite}$, shows frequency dependent oscillations over the general trend in $\beta _{QD - TE01}^{semi}$, but the peak values are much higher because of successive reflections from the top facet that also propagate down in the NW and get coupled into the WG mode. The values at the anti-resonances are not exactly zero, because the reflectivity at the bottom facet is less than 1. The peaks in $\beta _{QD - TE01}^{finite}$ mainly occur at the same positions as the anti-resonances in the Purcell factor ${F_{HE{{11}_b}}}$. This is because larger transmissions into the WG are only possible when modal overlaps at the interface are maximum, which corresponds to an electric field intensity anti-node of the cavity mode right at the NW/WG interface. This situation is also depicted in the bottom schematic of Fig. 4(d). We note here that the peaks in $\beta _{QD - TE01}^{finite}$ observed at $2\pi R/\lambda \; > 1.15$ show some features dissimilar to all other peaks at smaller normalized wavelengths. At these larger normalized wavelengths, the peaks in coupling efficiency also tend to appear at the same positions as peaks in ${F_{HE{{11}_b}}}$. Since the emission is funneled dominantly into radiation modes $\gamma $ at these normalized wavelengths (see Fig. 2(b)), normalizing over ${\Gamma _{finite}}$ in Eq. (4) leads to these features, seen in comparison with ${F_{HE{{11}_b}}}$.
The total out-coupling $\varepsilon $ into the TE01 mode of the WG, which is the enhanced bidirectional power transmission at the site of integration, is plotted in Fig. 4(c). In general, the maximum out-coupling into the WG follows the coupling efficiency envelope of the semi-infinite wire but with oscillations. We observe that the exact positions of all maxima in $\varepsilon \; $are the same as the peaks in the Purcell factor ${F_{HE{{11}_b}}}$ of the QD, despite sub-maximum coupling efficiencies $\beta _{QD - TE01}^{finite}$ into the WG mode. This proves that the out-coupling to TE01 mode occurs primarily through the HE11b mode by the indirect coupling mechanism, as expected. The relatively small differences in the Purcell factors between different resonances and anti-resonances translate into large differences in the enhanced power transmission, reaching a maximum of ∼45% at $2\pi R/\lambda = 0.92$. This points at the necessity of further tailoring the geometrical parameters of the fully integrated device such that the emission is ideally at a peak position in the stronger SE enhancement of the NW + WG “extended” cavity. While R, W and H are considered to be optimized so far, the NW length, which directly affects the position and linewidth of each resonance, can still be further tailored and adjusted to ultimately bring the emission of the QD into a peak in the SE enhancement.
Another parameter that can be effectively tuned, is the optimal position, Z, of the electric dipole inside the NW. Therefore, we performed additional calculations for a dipole (QD) placed at different positions of Z = 25–100 nm, very close to the WG, keeping the same NW cavity length L = 3.6 μm. As demonstrated in the Supplement 1 section S6, we find that under such conditions, where Z is comparatively small with respect to the shortest propagating wavelength, the SE from the dipole couples both directly (evanescently) to the TE01 mode and indirectly through the coupling to the NW HE11b mode. The polarization of the dipole and the design of the WG ensure that the direct evanescent coupling of the dipole occurs selectively only to the TE01 mode. By calculating the total out-coupling into the TE01 mode ($\varepsilon $), we thereby recognize distinct increases in $\varepsilon $ for decreasing Z. This evidences the enhanced evanescent coupling of the dipole to the TE01 mode, since the overlap with the exponentially decaying field of the TE01 mode increases significantly. For example, for a dipole positioned close to the WG ($Z = 25 $nm) in a finite NW cavity as considered in the previous section, $\varepsilon $ can reach apparent values >70% for wide wavelength ranges and even as high as 85% at the peak positions of the SE enhancement at short wavelengths. We note here that this is not a fundamental limit to the out-coupling, and the trend shows that it is possible to obtain higher out-coupling with smaller Z. We further wish to point out that in this case, the wavelength dependencies of$\; \varepsilon $ are a result of complex combinatory effects, such as the phase matching condition for the direct coupling to the TE01 mode, the phase matching of the HE11b mode to the TE01 mode, and the Purcell enhancement of the dipole. Since the exact interplay of these intricate effects are not yet fully understood, this clearly motivates for follow-up modeling efforts to develop additional knowledge and ways to further control the power transmission efficiencies.
6. Conclusion
In conclusion, we have established a numerical procedure for optimizing a new vertical-cavity NW-QD architecture directly integrated on a Si photonic WG, to maximize the out-coupling of the SE of the QD into the guided modes in the NW and eventually into the fundamental TE01 mode of the WG. We have optimized key geometrical parameters (R, W and H of NW and WG) by investigating the interplay of frequency-dependent reflectivities and coupling efficiencies at the NW-WG interface. It is found that the normalized NW radius is a robust parameter whose optimal value ensures a coupling of ∼96% of emission from a QD dipole placed at the NW center and polarized at $\phi = 90^\circ $, to the propagating HE11b mode. While a randomly oriented dipole couples to both HE11 modes similarly, the polarization of the dipole is crucial for a WG supporting unimodal propagation of the fundamental mode, as amongst the two HE11 modes in the NW, only the HE11b evanescently couples to the TE01 mode. The optimal WG dimensions are values close to the cutoff for the TE01 mode. While both the WG width and height influence the evanescent field overlap and the phase matching condition for the NW and WG modes, the transmission efficiency to the TE01 mode is very sensitive to the WG height which also directly affects the reflectivity of the NW mode via the path length of propagation inside the WG. In contrast, the WG width is a very robust parameter which affects the transmission efficiency and reflectivity only through ${n_{eff,WG}}$. The frequency dependence and positions of maximum transmission are also determined by the phase matching condition. In an optimized NW-QD architecture integrated on the Si-WG, the finite NW length results in weak and strong Fabry-Perot resonances of the HE11b mode in the longitudinal cross-section of the “extended” NW-WG cavity, at the NW/WG and WG/sub interfaces, respectively, due to the refractive index contrasts. The Purcell factor, ${F_{HE{{11}_b}}}$ in the fully integrated structure poses the dominating influence on the position and magnitude of the peak power transmission, whereby the QD emission can be brought into a resonance in the enhancement by tuning the NW length. Keeping the QD closer to the WG inside the NW significantly increases the total out-coupling into the TE01 mode for a polarized dipole emission, by allowing the QD to also evanescently couple to the TE01 mode directly. These high out-coupling efficiencies point to the benefits of such prior optimization of the NW-QD architecture, and clearly motivate experimental efforts to realizing monolithically integrated NW-QD systems onto quantum photonic circuits. Since our approach is also kept general with respect to normalized NW dimensions, the results can be easily adapted to any III-V NW-QD-on-WG system and emission wavelength band of interest.
Funding
Deutsche Forschungsgemeinschaft (EXC-2111-390814868); European Research Council (771747).
Acknowledgements
The authors thank Hyowon Jeong for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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