Quasinormal mode theory and design of on-chip single photon emitters in photonic crystal coupled-cavity waveguides

T. Malhotra; R.-C. Ge; M. Kamandar Dezfouli; A. Badolato; N. Vamivakas; S. Hughes

doi:10.1364/OE.24.013574

1. Introduction

A quantum single photon source (SPS) is highly desirable for applications in quantum information processing and secure communication. An ideal SPS should emit single photons deterministically or on-demand, and each photon should be indistinguishable with a high repetition rate and a high β factor, which is the the collection efficiency of the output photon into a desired mode. Photonic crystals (PCs) offer a platform for an all-integrated solid state SPS, and prototype QDs embedded in PC nanocavities (PCNs) [1–3] have been demonstrated. In the weak-to-intermediate coupling regime, the spontaneous emission (SE) rate of the QD is enhanced into a specific PCN mode, and large Purcell factors (F_P) have been realized due to the high quality factor (Q factor) and small effective mode volume, V_eff of the cavity. The bandwidth in these devices is limited by the Q factor and the far-field emission profile of the PCN dictates the maximum achievable β-factor. Single photon source from QDs embedded in a PC waveguide (PCW) have also been proposed [4–6] and realized [7–12]. The PCWs support slow light modes near the mode edge, and the large group index results in large local density of optical states (LDOS) [4, 13]; when a QD exciton is tuned in resonance with a slow light Bloch mode, enhanced SE can be observed. Besides SE enhancement, the photon can be emitted into a propagating mode, offering possibility for further on-chip manipulation. Even though these PCWs offer large β factors, they suffer from fabrication imperfections limiting the maximum achievable group index and F_P [14].

A possible hybrid approach using a coupled PCN-PCW has been proposed [15, 16], where a SPS is realized by having a QD inside a PCN evanescently coupled to an output PCW. The QD emits in the modified LDOS of the PCN and the emitted photons are then coupled into the propagating Bloch mode of PCW. The PCN can be designed to have a high Q factor (large F_P) and the coupling between the cavity and PCW can be optimized to give a large β factor as well. One of the advantages of working with high Q cavities is the ability to implement quantum optical manipulations, such as photon Blockade and strong coupling effect in a waveguide configuration. However, operating in the regime of large Q is not a requirement for single photon emission [17], and may actually be problematic for the various figures-of-merit such as the single photon indistinguishability. For this work we will mainly focus on the Purcell factor in in the weak coupling regime, though one can easily extend the work to explore regimes of anharmonic cavity-QED and strong coupling.

The choice of optimum geometry for the PCN-PCW coupling is, however, a complex problem, and the theoretical complexity of cavity-waveguide systems requires extra care in obtaining the normalized mode profiles and effective mode volume of the hybrid device. These open cavity systems can be rigorously described in the language of quasinormal modes (QNMs) [18–22], which are the solution to the Helmholz equation with open boundary conditions. However, as shown recently, the periodic PCW presents an extra challenge because of the nonlocal boundary conditions of the output waveguide [23]. In this work, we study an L3 PCN coupled to a W1 PCW in a membrane-based PC with a triangular lattice of air holes. The L3 PCN is formed by removing 3 holes along the nearest neighbor direction and the W1 PCW is formed by removing a row of holes along the nearest neighbor direction; such architectures have been experimentally realized [24] using an L3 PCN coupled to a W1 PCW located on the cavity axis. Here, we explore a set of coupling geometries between an L3 PCN and W1 PCW. Using a QNM approach for coupled waveguide-cavity systems [23], we calculate the Q factors and F_P factors of the loaded cavity as the spatial separation between the PCN and PCW is changed and find them to be sensitive to the structural modifications of the coupling region. We present calculations of the effective mode volume for the complete structure, without recourse to approximate coupled mode theories. Our calculations also show that the loaded Q factor, which is inversely proportional to the coupling rate, does not increase monotonically with increasing separation between the PCN and PCW. In particular, we identify and show the effect of changing the radius of an h-hole [25], explained later, on the Q factors of the coupled cavity-waveguide devices.

2. Isolated cavity and isolated waveguide properties

Figure 1(a) shows the schematic of an L3 PCN. The edge-holes of the PCN are marked in bold black outline with a radius reduction of 0.04a (a is the lattice constant of the PC) and a shift of 0.19a away from the center of the cavity. This modification increases the Q factor of the fundamental cavity mode [3] by redistributing the spatial Fourier components of the mode outside the light cone (leaky mode region). The real parts of the the electric field components, E_x(r) and E_y(r), of the fundamental L3 cavity mode are shown in Figs. 1(b) and 1(c), respectively; E_y(r) is evenly symmetric about the y-axis. A schematic of the semi-infinite W1 PCW terminated at the end closest to the L3 PCN is shown in Fig. 1(d). The hole shown in bold black outline, next to which the first missing hole of the W1 PCW is located, is identified as the h-hole with radius r_h. Figure 1(e) plots the band structure of an infinite W1 PCW. The blue dashed circled line is the dispersion of the fundamental Bloch mode of the W1 PCW, which is confined in a TE like band gap of PC; this mode has even symmetry about the y-axis. The dispersion goes flat near the waveguide mode edge allowing slow light propagation in the W1 PCW. The resonant frequency for our L3 cavity is shown as a dashed horizontal line in the same figure and it intersects the waveguide band at k_x0 (a/2π)=0.375. This coupling region is encircled in the Fig. 1(e). The intensity of the x-component and the y-component of the Bloch mode profile of the W1 PCW at k_x0 are shown in Figs. 1(f) and 1(g), respectively. Note that we focus on coupling the L3 PCN to the W1 PCW away from the slow light region, to minimize the problem of disorder-induced losses [14].

Fig. 1 (a) & (d) Schematic of an L3 PCN with modified edge holes shown in bold black outline and of a W1 PCW with the h-hole marked in bold black, respectively; mode profiles for the fundamental mode of L3 PCN (b) Real E_x, (c) Real E_y along with 4 different h-holes and their location in the (a₁,a₂) basis (see text); (e) Band structure of W1 PCW calculated for the even parity mode; intensity of electric field mode profiles for W1 PCW at the spectral location of L3 PCN (k_x0 (a/2π)=0.375) (f) x-component and (g) y-component.

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A single QD can be confined inside the PCN by first locating an isolated QD on a wafer and fabricating the PCN structure around it [26]. The coupling between a QD dipole and the PCN mode depends on n_d · E_c(r_d), where n_d is the QD dipole vector and E_c(r) is the PCN mode electric field. Thus, for maximum coupling, the QD must be near a field antinode point with its dipole moment oriented along E_c(r_d). From Figs. 1(b) and 1(c), the amplitude of E_x(r) is zero and the amplitude of E_y(r) is maximum at the center of the L3 PCN. Therefore, a QD in a L3 PCN must be positioned either near the center of the cavity where E_y(r) has an antinode or near the holes where E_x(r) has an antinode. Hereafter we will study an ideal coupling between the QD (simulated as a polarization dipole) and the E_y(r) of the PCN fundamental mode by keeping the dipole moment of the QD parallel to y-axis and positioning it at the center of the PCN. Figure 1(c) also shows that the E_y(r) cavity mode spatially extends in greater magnitude along 60 degrees to cavity axis as compared to its value along the cavity axis. Thus in order to characterize the coupling between the L3 PCN and W1 PCW as a function of spatial separation between the two, the position of the h-hole was moved along this direction (diagonally away from the L3 PCN). We study the separation between the L3 PCN and the W1 PCW along three diagonals. For example, Fig. 1(c) also shows four different locations of the h-hole marked in white outline, along one diagonal, depicting four possible devices. Each of the designs is referred to by a name describing the position of the h-hole in the basis of (a₁ and a₂) vectors. Both a₁ and a₂ are measured from the edge hole of the L3 PCN and represent the number of holes along the cavity axis and the number of rows respectively to the position of the h-hole. For example, a₂= 0 refers to all the devices with the PCW located on the cavity axis. Figure 2(a) shows a schematic of the device with the h-hole located two holes horizontally right and one row up from the PCN, which is referred to by its coordinates as a₁=2, a₂=1 or (2,1); Figs. 2(b) and 2(c) represent (2,0) and (2,2) devices, respectively, and we vary a₁ from 0 to 2 (3 diagonals in total) and a₂ from 0 to 6.

Fig. 2 Schematic of the different coupled geometries studied. The basis vectors (a₁ and a₂) used to identify each of the geometries are also shown. Note the origin for these vectors is the edge-hole of the cavity. The devices are identified by the position of the h-hole. (a) (a₁=2, a₂=1); (b) (2, 0), (c) (2, 2).

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3. Modified spontaneous emission, Green function and QNM theory of coupled cavity-waveguides

When a PCW is brought in the vicinity of the PCN, the cavity fields find an additional decay channel by evanescently coupling to the PCW Bloch modes. This modifies its Q factor, which is inversely related to the rate of decay of stored energy inside the cavity, to Q_L, where Q_L is the loaded or complete Q factor of the coupled device.

The modified SE rate relative to a homogeneous medium can be obtained from the generalized Purcell factor,

F_{P} = \frac{Im [n \cdot G (r, r, ω) \cdot n]}{Im [n \cdot G_{0} (r, r, ω) \cdot n]},

where G(r, r′, ω) is the electric-field Green function and

Im [G_{0} (r, r, ω)] = \frac{ω^{3} n_{b}}{6 π c^{3}} I

is the homogeneous Green function [16], with I the unit dyadic.

The transverse part of the Green function can be expanded in the basis of the QNMs [22], and with one cavity mode contribution, takes the form

G (r, r^{'}, ω) \approx \frac{ω^{2}}{2 {\tilde{ω}}_{c} ({\tilde{ω}}_{c} - ω)} \frac{{\tilde{f}}_{c} (r) {\tilde{f}}_{c} (r^{'})}{〈 〈 {\tilde{f}}_{c} | {\tilde{f}}_{c} 〉 〉},

where ω̃_c = ω_c − iω_c/2Q is the complex eigenfrequency, and f̃_c is the normalized QNM, which is obtained from a source free solution to Maxwell’s equations with outgoing boundary conditions, namely from

\nabla \times \nabla \times \tilde{f} - {(\frac{{\tilde{ω}}_{c}}{c})}^{2} ε (r) \tilde{f} = 0,

where c is the speed of light and ε is the dielectric constant.

For high Q cavities, and assuming a lossless dielectric, it is common in cavity optics to adopt “normal mode” theory, whose modes are normalized through

{〈 {\tilde{f}}_{c} | {\tilde{f}}_{c} 〉}^{NM} = \frac{1}{2} \int_{V} (ε (r) {| {\tilde{f}}_{c} (r) |}^{2} + c^{2} {μ_{0}}^{2} {| {\tilde{h}}_{c} (r) |}^{2}) d r,

where h̃_c =ω̃_c/ic∇ × f̃_c (obtained for the actual dissipative cavity structure) and V → ∞. However, this normal mode expression is incorrect in general as it fails to account for dissipation, and the norm diverges exponentially as a function of space [20]. Instead, in the absence of dispersion, the QNMs can be normalized from [21, 23]

{〈 〈 {\tilde{f}}_{c} | {\tilde{f}}_{c} 〉 〉}^{unreg} = \frac{1}{2} \int_{V} [ε (r) {\tilde{f}}_{c} (r) \cdot {\tilde{f}}_{c} (r) - c^{2} {μ_{0}}^{2} {\tilde{h}}_{c} (r) \cdot {\tilde{h}}_{c} (r)] d r,

and once again V → ∞ and 〈〈···〉〉 now denotes the true QNM norm. This (“unregularized”) expression still requires regularization of the integral in general, which for an open cavity surrounded by a homogeneous medium can be implemented in a number of ways [27], including use of PMLs (perfectly matched layers). For PCWs, however, special care is required in obtaining a semi-infinite periodic waveguide properties since the boundary condition is nonlocal. It was recently shown [23] that regularizing this integral is elegantly achieved by separating this integral into cavity and waveguide parts, through

{〈 〈 {\tilde{f}}_{c} | {\tilde{f}}_{c} 〉 〉}^{reg} = I_{cav} + I_{wg},

where

I_{cav} = \frac{1}{2} \int_{V_{cav}} (ε (r) {\tilde{f}}_{c} (r) \cdot {\tilde{f}}_{c} (r) - c^{2} {μ_{0}}^{2} {\tilde{h}}_{d} (r) \cdot {\tilde{h}}_{d} (r)) d r

, and

I_{wg} = \frac{I_{a} (x_{0})}{1 - exp (2 i {\tilde{k}}_{c} a)}

, where k̃_c = ω̃_c/c, and

I_{a} (x_{0}) = \frac{1}{2} \int_{⊥} \int_{x_{0}}^{x_{0} + a} ε (r) {\tilde{f}}_{c} (r) \cdot {\tilde{f}}_{c} (r) - c^{2} {μ_{0}}^{2} {\tilde{h}}_{c} (r) \cdot {\tilde{h}}_{c} (r) d r_{⊥} d x

.

Here I_cav is evaluated over the volume,V_cav, bounded by y_span, z_span (see below) and between PML and x₀ along the x-axis, chosen at a position far enough away from the cavity so that the waveguide Bloch mode is well defined [23]. The PCW waveguide integration is carried out through use of a convergent series [23]. This normalized QNM, is used to evaluate the effective mode volume (V_eff) of this coupled system, evaluated at field maximum point, r₀, which is obtained from

\frac{1}{V_{eff}} = Re (\frac{1}{v_{Q}}), v_{Q} = \frac{〈 〈 {\tilde{f}}_{c} | {\tilde{f}}_{c} 〉 〉}{ε (r_{0}) {\tilde{f}}_{c} (r_{0}) \cdot {\tilde{f}}_{c} (r_{0})},

whereas the (incorrect) normal mode volume would be obtained from

V_{eff}^{NM} = \frac{〈 {\tilde{f}}_{c} | {\tilde{f}}_{c} 〉}{ε (r_{0}) {| {\tilde{f}}_{c} (r_{0}) |}^{2}}

.

4. Numerical calculations of the QNM effective mode volume and Purcell factor for a coupled cavity-waveguide geometry

To compute the QNMs and complex eigenfrequencies, we use 3D finite-difference time-domain (FDTD) techniques from Lumerical [28]. The cavity mode is excited using a dipole source which emits a Gaussian pulse in time. The fields build up while the source is emitting and start to decay after the source dies. Specifically for the structures studied here, a dipole source emitting at 952 nm with a bandwidth of 33 nm (pulse width of 40 fs) was placed at the center of the L3 PCN coupled to W1 PCW. For maximum coupling, this dipole was oriented along the y-axis and the electric fields at different spatial locations inside the cavity were recorded using time monitors. The simulation was run for t_sim= 3000–4000 fs. The rate of decay of the recorded fields gave the Q factor and resonant frequency of the mode, ω_c, was computed from the Fourier transform of the fields. The complex eigenfrequency of the structure was then computed using ω̃_c = ω_c − iω_c/2Q. The FDTD fields for the coupled system were obtained using 3D field monitors and temporal window was applied to compute the run-time Fourier transform.The fields were normalized using the above regularization technique to yield the QNM of the coupled device. Simulations were done for GaAs membranes designed to incorporate QDs emitting at 950 nm (refractive index n_b=3.5 at 950 nm) with the following structural parameters: membrane thickness of t=164 nm, lattice constant a=240 nm and radius of bulk hole r_b=0.28a; r_h= 0.2a was used unless otherwise stated. Spatial mesh steps of a/20=12 nm, $a (\sqrt{3} / 2) / 20 = 10.39 nm$ and h/10=16.4 nm was used along x, y and z direction, respectively; PML boundary conditions were employed along in-plane directions with 100 PML layers. The location of the PML is shown in Fig. 3(a) for the (2,0) device: x_c and y_c refer to the location of PML boundary from the center of the cavity and x_wg and y_wg measures the location from the h-hole, whose values are given in the caption; this set the x_span and y_span for the simulations, and z_span was set at 10t. It was found that increasing the value of x_wg beyond 15.5a had no effect on the loaded Q_L factor so Q_L has saturated to its maximum value; we also define Q_V as the Q factor for an isolated cavity. The mode profile, real(Ẽ_y(r)), of the coupled (1,0) device is computed and shown in Fig. 3(b). Also shown is the surface x₀ = 10a in white vertical line which was used for regularizing the integral. We stress that this mode is the true QNM for the entire coupled device. In order to better highlight the need for regularization with the normalized QNM, the computed mode volume was compared using both the regularized (V^reg) and unregularized (V^unreg) normalizations as a function of the number of the unit cells along the x-axis. The results are plotted in Fig. 3(d) which show V^reg in solid blue line and V^unreg in blue circles. The V^unreg oscillates and gradually increases as can be seen in Fig. 3(d). Also plotted is the normal mode volume (V^NM) which clearly diverges as more number of unit cells are added to the computation domain demonstrating the need to use the regularized norm for accurate evaluation of f̃_μ(r) and V_eff. To verify the accuracy of this approach, we also compared the computed F_P value as a function of frequency for a nominally coupled device (1,0) (Q_L factor ≈ 6400) using Eq. (2) and and a full 3D dipole computation of the Green function [16]. Figure 3(c) shows the excellent match between the two computation techniques validating the use of QNM theory for F_P value calculations. We thus use this approach below to compute and analyze our coupled cavity-waveguide designs. We emphasize that QNM computation makes the calculations much faster than a direct dipole simulation (since the Q factor can be obtained without a full dipole simulation), and allows one to adopt an intuitive modal approach to obtain important parameters such as a rigorous definition of the effective mode volume for the entire structure, and the Purcell factor as a function of space and frequency, without doing any more calculations (which would be required, e.g., if doing by direct FDTD for different dipole positions). The time benefit depends on the Q_L of the structure under investigation; for the (1,0) device shown in Fig. 3(c), regardless of the memory requirements, the full dipole calculation for obtaining the accurate F_P needs a 20-fold longer run time. This becomes even worse for higher Qs. In addition, full dipole calculations allow less use of symmetry and therefore add up to the memory requirements for a typical simulation which in turn can increase the time difference mentioned earlier, at the very least, by a factor of 2. We assume the Purcell coupling regime, but the theory can easily account for strong coupling effects if desired [15]. Figure 3(e) shows the effective mode volumes for the (2,0) device, which exhibits the same trend as (1,0) with a slower divergence of the V^NM.

Fig. 3 (a) Schematic showing the location of the PML (red) from the h-hole and edge hole for (1,0) device. x_c=15a; y_c=15a_y; x_wg=15.5a; y_wg ≥ 12.5a_y were used for this work. (b) The regularized QNM, real(Ẽ_y(r)), is also shown for the same device. The vertical white line shows the regularization boundary x₀ = 10a. (c) Purcell factor computed for (1,0) device using both QNM and full dipole calculations (see text). The inset shows the zoom in of the curves around the maximum Purcell factor values. 3D Dipole method gave a maximum value of F_P as 395.378 at a frequency of 312.12217 THz whereas the QNM method gave the maximum F_P value of 394.038 at a frequency shifted by 950MHz (312.12122 THz). (d) Mode volume of the (1,0) device computed using three definitions as a function of x-span. Blue horizontal line shows V^reg [using Eq. (6)], the blue circles show the V^unreg[using Eq. (5)] oscillating around V^reg and gradually increasing and green squares show the rapidly diverging V^NM[using Eq. (4)]. (e) As in (d), but for a (2,0) cavity-waveguide system which has a higher Q factor (note the different y scale range).

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5. Investigation and design of various off-axis cavity-waveguide coupling geometries

Physically, the complex coupling between the PCN and PCW mode depends on the polarization of the fields, the field profiles, spatial overlap between the fields and the spectral location of the modes of the two structures involved [29]. As the PCW is moved away from the PCN, the spatial overlap between the fields changes. Figures 4(a)–4(c) shows the computed Q_L factors and F_P values for all the studied devices as a function of separation between the PCN and PCW. The Q_L factors are plotted on the left vertical axis in Figs. 4(a)–4(c) as a₂ on the horizontal axis increased from 0 to 6 (the PCW is moved from the 0th to 6th row). For the devices along the closest diagonal (Fig. 4(a)), the Q_L factors increased monotonically with a₂, i.e. as the separation between the PCN and PCW was increased. Note that there is no h-hole in the (0,0) or the first device in the 0th row along this diagonal. Interestingly from Fig. 4(c), the Q_L factors for the farthest diagonal (a₁ = 2) do not increase monotonically with a₂ and show an abrupt decrease for the devices in the 1st (2,1) and 4th row (2,4) indicating an increase in coupling between the PCN and PCW with (2,4) having a higher Q_L than the (2,1) device. For further separation of 5 and 6 rows [(2,5) and (2,6)], the Q_L factors increase. Calculations for the second diagonal along a₁ = 1 are shown in Fig. 4(b); the Q_L decrease as the PCW is moved from the first to third row (a₂ =1 & 2 respectively) and begin to increase for devices which are farther than the third row (a₂ > 3). The right vertical axis on Figs. 4(a)–4(c) show the computed F_P values. Note that as the separation between the cavity and PCW is changed along all three diagonals, the F_P values and the Q factors change in a similar fashion. This is expected because changing the location of the PCW around the cavity is mainly affecting the decay rate of the cavity into the PCW with the mode volume of the cavity being unaffected. None of the devices explored here have PCW close enough to the cavity so as perturb its mode volume drastically.

Fig. 4 Results: (a–c) Simulated Q_L factors (left vertical axis) and F_P (right vertical axis) values as a function of a₂ for the devices along three different diagonals, a₁ = 0, 1 and 2, respectively. (d) Q_L factors for the devices with a₁ = 2 for three different r_h values. Also shown are schematics for three different coupled structures with a₁= 0,1,2 and a₂ = 3.

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In order to understand the peak-like feature in the Q_L for a₂ = 1, we further investigated the effect of changing the radius of h-hole, r_h, for the devices along this third diagonal. These simulations were done with lower spatial resolution to make them less computationally intensive. Figure 4(d) shows Q_L factors for the devices with a₁ = 2 for three different r_h values of 0.125a, 0.2a and 0.28a. For r_h = 0.28a = bulk radius, the Q_L factors for devices with a₂ > 2 are largest indicating overall less coupling. Devices with r_h = 0.2a exhibited smaller Q_L for a₂ > 2 showing an increased coupling with smaller r_h but still maintaining a similar trend in Q_L as a₂ increased. However, as r_h was further decreased to 0.125a, the peak feature shifted to a lower a₂ value of 2. Besides, at this smaller value of r_h, the Q_L factors for all the devices with a₂ > 2 are lower indicative of an overall increase in coupling. The h-hole is a structural parameter shared by the PCN fields and the PCW fields probably facilitating scattering of the fields and affecting the coupling. Its position and radius therefore also affect the coupling mechanism.

Next, the waveguide β factors were computed using the decay rates of the cavity into the various channels as described below. If, $Q_{L}^{- 1}$ is the total new decay rate of the loaded cavity, $Q_{V}^{- 1}$ is the decay rate along the vertical channel and $Q_{WG}^{- 1}$ is the coupling rate of the cavity into the WG then, $Q_{L}^{- 1} = Q_{WG}^{- 1} + Q_{V}^{- 1}$ . The β factor is then evaluated as

β = \frac{Q_{WG}^{- 1}}{Q_{L}^{- 1} + {Q_{dis}}^{- 1}},

where Q_dis includes processes such as disorder, background dots, material losses through a complex n, etc. Figure 5 shows the computed β factors along the three different diagonals for three Q_dis values of ∞, 10⁵, 10⁴. As can be seen, the β factors qualitatively maintain their functional dependence on the separation for different Q_dis values. Also note that β factors for the all devices get smaller for Q_dis=10⁴, with the devices located along the farthest diagonal (a₁=2) being influenced the most. Detailed calculations have shown that the intrinsic Q from intrinsic structural disorder is likely around 10⁵–10⁶ [30, 31], so the values of 10⁴ are likely somewhat pessimistic.

Fig. 5 Waveguide β factors computed from Eq. (8) are plotted as a function of separation (a₂) for the devices along three different diagonals, a₁ = 0, 1 and 2. Each subfigure was computed using a different loss rate (a) Q_dis=∞, (b) Q_dis=10⁵ and (c) Q_dis=10⁴.

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6. Summary

We have employed a QNM theory to calculate and design efficient PC coupled cavity-waveguides systems for SPS applications, and introduced important modal properties of the coupled system, including the effective mode volumes and quality factors for the complex 3D coupled cavity infinite waveguide system. We also demonstrated the critical role that the h-hole plays in coupling a PCN and PCW, and found, somewhat counter intuitively, that for intermediate PCN-PCW separations the loaded Q factor exhibits a local minimum. The precise size of the h-hole determines for which PCN-PCW separation this local minimum is achieved. In PC based SPS devices that aim to optimize Purcell factors and the collection of single photons emitted into a waveguide, the crucial role played by the h-hole must therefore be carefully taken into account. We anticipate a complete optimization of the h-hole for maximum coupling for the devices studied can possibly be accomplished using topological optimization algorithms [32].

Acknowledgments

We thank the National Science Foundation (NSF) and the Natural Sciences and Engineering Research Council of Canada. A.B. acknowledges support from NSF under CAREER Grant No. ECCS- 1454021 and Grant No. DMR- 1309734.

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Quasinormal mode theory and design of on-chip single photon emitters in photonic crystal coupled-cavity waveguides

Abstract

1. Introduction

2. Isolated cavity and isolated waveguide properties

3. Modified spontaneous emission, Green function and QNM theory of coupled cavity-waveguides

4. Numerical calculations of the QNM effective mode volume and Purcell factor for a coupled cavity-waveguide geometry

5. Investigation and design of various off-axis cavity-waveguide coupling geometries

6. Summary

Acknowledgments

References and links

Cited By

Figures (5)

Equations (8)

Optics Express