## Abstract

Using a quasinormal mode (QNM) theory for open cavity systems, we present detailed calculations and designs of a photonic crystal nanocavity (PCN) side-coupled to a photonic crystal waveguide (PCW) for on-chip single photon source applications. We investigate various cavity-waveguide geometries using an L3 PCN coupled to a W1 PCW, obtaining the quality factors, effective mode volumes, and single photon Purcell factors of the complete loaded cavity-waveguide system as a function of spatial separation between the two. We also show that the quality factor does not monotonically increase with increasing separation between the PCN and PCW, and we identify a particular hole/defect which acts as the key structural parameter in the cavity-waveguide coupling.

© 2016 Optical Society of America

## 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 0.04*a* (*a* is the lattice constant of the PC) and a shift of 0.19*a* 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

*. 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*

_{h}_{x0}(

*a*/2

*π*)=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].

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**

*), where*

_{d}**n**

*is the QD dipole vector and*

_{d}**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**

*). From Figs. 1(b) and 1(c), the amplitude of E*

_{d}*(*

_{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 (a

_{1}and a

_{2}) vectors. Both a

_{1}and a

_{2}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, a

_{2}= 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 a

_{1}=2, a

_{2}=1 or (2,1); Figs. 2(b) and 2(c) represent (2,0) and (2,2) devices, respectively, and we vary a

_{1}from 0 to 2 (3 diagonals in total) and a

_{2}from 0 to 6.

## 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,

**G**(

**r**,

**r′**,

*ω*) is the electric-field Green function and $\text{Im}\left[{\mathbf{G}}_{0}(\mathbf{r},\mathbf{r},\omega )\right]=\frac{{\omega}^{3}{n}_{b}}{6\pi {c}^{3}}\mathbf{I}$ 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

*ω̃*=

_{c}*ω*−

_{c}*iω*/2

_{c}*Q*is the complex eigenfrequency, and

**f̃**

*is the normalized QNM, which is obtained from a source free solution to Maxwell’s equations with outgoing boundary conditions, namely from*

_{c}*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

**h̃**

*=*

_{c}*ω̃*

_{c}/ic**∇**×

**f̃**

*(obtained for the actual dissipative cavity structure) and*

_{c}*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]

*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

*k̃*=

_{c}*ω̃*, and ${\text{I}}_{a}({x}_{0})=\frac{1}{2}{\int}_{\perp}{\int}_{{x}_{0}}^{{x}_{0}+a}\epsilon (\mathbf{r}){\tilde{\mathbf{f}}}_{c}(\mathbf{r})\cdot {\tilde{\mathbf{f}}}_{c}(\mathbf{r})-{c}^{2}{{\mu}_{0}}^{2}{\tilde{\mathbf{h}}}_{c}(\mathbf{r})\cdot {\tilde{\mathbf{h}}}_{c}(\mathbf{r})\text{d}{\mathbf{r}}_{\perp}dx$.

_{c}/cHere *I*_{cav} is evaluated over the volume,V_{cav}, bounded by y_{span}, z_{span} (see below) and between PML and *x*_{0} 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, **r**_{0}, which is obtained from

## 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*/2

_{c}*Q*. 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

*=3.5 at 950 nm) with the following structural parameters: membrane thickness of*

_{b}*t*=164 nm, lattice constant

*a*=240 nm and radius of bulk hole r

*=0.28*

_{b}*a*; r

*= 0.2*

_{h}*a*was used unless otherwise stated. Spatial mesh steps of

*a*/20=12 nm, $a\left(\sqrt{3}/2\right)/20=10.39\hspace{0.17em}\text{nm}$ 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 10

*t*. It was found that increasing the value of x

_{wg}beyond 15.5

*a*had no effect on the loaded

*Q*

_{L}factor so

*Q*

_{L}has saturated to its maximum value; we also define

*Q*as the

_{V}*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

*x*

_{0}= 10

*a*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}.

## 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 a_{2} 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 a_{2}, 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 (a_{1} = 2) do not increase monotonically with a_{2} 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 a_{1} = 1 are shown in Fig. 4(b); the *Q*_{L} decrease as the PCW is moved from the first to third row (a_{2} =1 & 2 respectively) and begin to increase for devices which are farther than the third row (a_{2} > 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.

In order to understand the peak-like feature in the *Q*_{L} for a_{2} = 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*factors for the devices with a

_{L}_{1}= 2 for three different

*r*values of 0.125

_{h}*a*, 0.2

*a*and 0.28

*a*. For

*r*= 0.28

_{h}*a*= bulk radius, the

*Q*

_{L}factors for devices with a

_{2}> 2 are largest indicating overall less coupling. Devices with

*r*= 0.2

_{h}*a*exhibited smaller

*Q*

_{L}for a

_{2}> 2 showing an increased coupling with smaller

*r*but still maintaining a similar trend in

_{h}*Q*

_{L}as a

_{2}increased. However, as

*r*was further decreased to 0.125

_{h}*a*, the peak feature shifted to a lower a

_{2}value of 2. Besides, at this smaller value of

*r*, the

_{h}*Q*

_{L}factors for all the devices with

*a*

_{2}> 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,
${Q}_{\text{L}}^{-1}$ is the total new decay rate of the loaded cavity,
${Q}_{\text{V}}^{-1}$ is the decay rate along the vertical channel and
${Q}_{\text{WG}}^{-1}$ is the coupling rate of the cavity into the WG then,
${Q}_{\text{L}}^{-1}={Q}_{\text{WG}}^{-1}+{Q}_{\text{V}}^{-1}$. The *β* factor is then evaluated as

*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 ∞, 10

^{5}, 10

^{4}. 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}=10

^{4}, with the devices located along the farthest diagonal (

*a*

_{1}=2) being influenced the most. Detailed calculations have shown that the intrinsic

*Q*from intrinsic structural disorder is likely around 10

^{5}–10

^{6}[30, 31], so the values of 10

^{4}are likely somewhat pessimistic.

## 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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