## Abstract

We report on newly-designed H1-type photonic crystal (PhC) nanocavities that simultaneously exhibit high *Q* factors, small mode volumes, and high external coupling efficiencies (*η*_{⊥}) of light radiated above the PhC membrane. Dipole modes of the H1 PhC nanocavities, which are doubly-degenerate and orthogonally-polarized in theory, are investigated both by numerical calculations and experiments. Through modifying the sizes and positions of the air-holes near to the defect cavity, a *Q* factor of 62,000 is achieved, accompanied with an improved *η*_{⊥} of 0.38 (assuming an objective lens with a numerical aperture of 0.65). A further increase of *η*_{⊥} to more than 0.60 is observed at the expense of slight degradation of *Q* factor (down to 50,000). We further experimentally confirm the increase of both *Q* and *η*_{⊥}, using micro-photoluminescence measurements, and demonstrate high *Q* factors up to 25,000: the highest value ever reported for dipole modes in H1 PhC nanocavities.

© 2012 Optical Society of America

## 1. Introduction

Two dimensional (2D) Photonic crystal (PhC) nanocavities have been widely investigated as an excellent platform for developing novel optoelectronic devices, including micro-lasers [1] and all optical switches [2], and, have also been studied as testbeds for cavity quantum electrodynamics (c-QED) in the solid state [3, 4]. Among various types of PhC cavities, H1-type nanocavities, which are formed by a single missing air-hole in a triangular 2D PhC lattice, show distinguished features: a wide variety of confined cavity modes [5, 6], high *Q* factors [7, 8] and small mode volumes *V*. Especially, doubly-degenerate, orthogonally-polarized dipole modes in H1 PhC nanocavities [9–13] have been receiving much attention due to their various potential applications, including the cavity-assisted generation of entangled photons [9,10], spin-photon media conversion [14] and all optical switches [15].

The realistic implementation of these exciting devices will require a high external coupling efficiency toward the out-of-plane direction, *η*_{⊥}, as well as high *Q* and small *V*. In general, however, high *Q* 2D PhC cavities provide small *η*_{⊥}. This has resulted in most previous studies trying to increase *η*_{⊥} at the expense of significantly decreasing the cavity *Q* [16–19]. Less attention has been given to simultaneously trying to achieve high *Q*, small *V* and large *η*_{⊥}, especially for doubly-degenerate orthogonally-polarized cavity modes.

In this work, we report on design and fabrication of H1 PhC nanocavities that simultaneously exhibit high *Q*, small *V* and large *η*_{⊥} for the dipole modes. These superior properties have been achieved by shifting and shrinking the first to third closest neighbor holes around the defect. This modification maintains the six-fold rotational symmetry, so that the degeneracy of the two dipole modes is not broken. Through numerical simulations, we calculated this modification improves *Q* factors up to 62,000: the highest value ever reported for the dipole modes of the H1 PhC nanocavities. Remarkably, the same cavity with the high *Q* design exhibits a large *η*_{⊥} of 0.38 (assuming an objective lens with a moderate numerical aperture of 0.65). Under other design parameters, we demonstrate that *η*_{⊥} can be improved to over 0.6, while keeping high *Q* factors over 50,000. Analysis of the cavity field distribution in momentum space reveals that the observed increase of both cavity *Q* and *η*_{⊥} is accompanied by a reduction of the total leaky components [20–22]. Indeed, the leaky components become redistributed around the center of momentum space.

The nanocavities were fabricated into a GaAs wafer including InAs quantum dots (QD), and the increase of both *Q* and *η*_{⊥}, relative to unoptimized H1 cavities fabricated in the same way, was experimentally confirmed. At optimum design parameters, the maximum experimental cavity *Q* of 25,000, which is the highest value ever reported for the dipole modes of H1 PhC nanocavities, is observed.

## 2. Cavity design and numerical simulations

#### 2.1. Model and calculation method

We investigated the fundamental cavity modes of H1 PhC nanocavities, namely dipole modes, with numerical simulations using three dimensional finite differential time domain (FDTD) method. The nanocavities are modeled into air-suspended planar PhC slabs of refractive index *n* = 3.46. In the simulation, proper (anti-)symmetric boundary conditions at the *x* = 0, *y* = 0, and *z* = 0 planes are applied, so that only one polarization-mode is excited in each calculation [16, 23]. Here, the origins of the *x* and *y* axes are chosen to be the center of the defect, and that of *z* axis is set at the middle of the PhC slab. We used commercially-available FDTD software (RSOFT fullwave) for the calculations of *Q*, *V*, and *η*_{⊥}. The grid size in the calculation was set to be *a*/16, where *a* (= 260 nm) is the lattice constant of the PhC.

Figure 1(a) shows the main design parameters for optimizing the nanocavity. Shifts and changes in size of the nearest neighbor six air-holes around the defect are applied, which are known to increase the *Q* factor of H1 cavities [6, 7, 11, 23]. For achieving a higher *Q* together with large *η*_{⊥}, we introduce additional modifications for the second and the third nearest neighbor holes along the Γ-K direction. These modifications do not break the six-fold rotational symmetry, and therefore the degeneracy of the two dipole modes is retained. This design strategy may therefore find use in applications that require the degeneracy of the two modes, such as cavity-assisted entangled photon generators [9, 11].

As a starting point of the design optimization, we chose an H1 PhC nanocavity possessing optimized first nearest neighbor holes with following parameters: hole radii *r* = 0.3*a*, slab thickness *d* = 0.5*a*, nearest hole shift Δ_{1} = 0.12*a*, and nearest hole radius *r*′_{1} = 0.24*a*. With these design parameters, *x*- and *y*-dipole modes are found at the normalized frequency (*a*/*λ*) of 0.273. The *Q* factors and mode volumes for both modes are calculated to be 19,000 and 0.47 ((*λ*/*n*)^{3}), respectively. The *Q* factors can be further increased to 29,000, by shrinking the radii of the second nearest neighbor holes (*r*′_{2}) to 0.27*a*. The calculated field distributions of these cavity modes are shown in Fig. 1(b) and (c). In the following sections, we investigate the dependence of *Q* and *η*_{⊥} on additional shifts of the third nearest neighbor holes (Δ_{3}). We note that the mode volume *V* remains almost constant during the optimization process, and since the two dipole modes exhibit negligible differences in *Q*, *V*, and *η*_{⊥} in the numerical simulations, only the results obtained for the *x*-dipole modes are shown below.

#### 2.2. Calculation results

Figure 2(a) shows a plot of calculated *Q* factors (*Q _{calc}*) of the

*x*-dipole cavity modes as a function of Δ

_{3}. The sign of Δ

_{3}is set to positive in the direction moving away from the defect. Surprisingly, the

*Q*factor significantly increases as Δ

_{3}is tuned towards the negative direction, as opposed to the case of L3-type PhC nanocavities [22]. The

*Q*factor reaches a maximum value of 62,000 at Δ

_{3}= −0.26

*a*. This

*Q*value is the highest ever reported for dipole modes of H1 PhC nanocavities [24].

_{calc}Next, we calculated the coupling efficiency (*η*_{⊥}) into an objective lens (numerical aperture, *N.A.* = 0.65) placed above the PhC membrane, by simulating far-field emission patterns of the cavity modes with the aid of a near-to-far-field projection method (RSOFT fullwave). The near field pattern used for the projection are evaluated at a plane of *z* = 10*a*/13. We define *η*_{⊥} as a ratio of collected light by the objective lens to the total light emitted upwards from the cavity, which also emits light downwards with the same intensity. A plot of the calculated *η*_{⊥} for various Δ_{3} is overlaid in Fig. 2(a). For positive Δ_{3} the widely-observed relation between *Q* and *η*_{⊥} is apparent [16–19]: *η*_{⊥} increases as *Q* decreases, and vice versa. In contrast, for negative Δ_{3}, simultaneous increase of both *Q* and *η*_{⊥} are observed up to Δ_{3} = −0.26*a*, where *η*_{⊥} reaches a value of 0.38, and the *Q* factor rises to 62,000. Further reduction of Δ_{3} from −0.26*a* to −0.3*a*, results in an increasing *η*_{⊥}, which exceeds 0.7 at Δ_{3} = −0.3*a*. Although the *Q* slightly drops in this region, we found a good compromise can be reached at Δ_{3} = −0.28*a*, where both the *Q* is larger than 50,000 and *η*_{⊥} exceeds 0.6. This high *Q* design with high *η*_{⊥} will be very useful in various applications requiring vertical beaming of the doubly-degenerate cavity modes. The observed increase of *η*_{⊥} can be visualized in far-field emission patterns, as shown in Fig. 2(b) and (c). For no shift (Δ_{3} = 0), the dominant field is emitted at large angles to the plane normal (high N.A. region). For the case of Δ_{3} = −0.26*a*, a strong beaming of the leak field in the low N.A. region is observed. These drastic changes of the emission pattern result in the observed high *η*_{⊥} when using an objective lens with a moderate N.A. of 0.65. In Fig. 2(d), we plot calculated *η*_{⊥} as a function of N.A. for five different Δ_{3}s. We observed a significant change in the behavior of *η*_{⊥} by negatively increasing Δ_{3}. With zero and insufficient shifts (Δ_{3} = 0, −0.15*a*), a sharp increase of *η*_{⊥} starts around N.A. = 0.6, in the both cases. Meanwhile, at the same N.A., high *η*_{⊥}s and its gradual increases are observed for three calculation results using Δ_{3} of −0.26*a*, −0.28*a*, and −0.3*a*. The latter observation is a result of the concentration of radiation field around the center of the objective lens.

#### 2.3. Momentum space analysis

In order to further elucidate the observed simultaneous increase of both *Q* and *η*_{⊥}, we investigated the near field distributions of the cavity modes in momentum space. The momentum space field distributions are calculated by two dimensional Fourier transformation (FT) of the dominant electric field (*E _{y}*) for the

*x*-dipole mode, at the upper slab-air interface. This provides significant information about light confinement [20, 21]. A fraction of light in the cavity mode leaks out from the slab, when the amplitude of it’s in-plane momentum component, $\left|{k}_{\left|\right|}\right|=\sqrt{\left({k}_{x}^{2}+{k}_{y}^{2}\right)}$, is smaller than

*ω*/

*c*[20, 21], where

*k*and

_{x}*k*are wave vectors of the light propagating in

_{y}*x*and

*y*directions, respectively,

*ω*is the angular frequency of the mode, and

*c*is the speed of light in a vacuum. The line defined by

*k*

_{||}=

*ω*/

*c*is called a light line, below which is the leaky region.

The calculated momentum space distributions of
${\left|{E}_{y}^{FT}\right|}^{2}$ for *x*-dipole modes are shown in Fig. 3(a) and (b), for Δ_{3} = 0 and Δ_{3} = −0.26*a*, respectively. We note that, for the *y*-dipole mode, the momentum space distribution of the dominant field component,
${\left|{E}_{x}^{FT}\right|}^{2}$, shows similar distribution upon a 90 degree rotation in the plane of the cavity. The ratio of the leaky components to the whole Fourier components, defined by

_{3}from 0 to −0.26

*a*. This reduction corresponds to the increase of

*Q*.

_{calc}More interestingly, the distributions of the leaky components show drastic changes by the introduction of the shift of Δ_{3} = −0.26*a*. Without the shift, the components mainly distribute near the rim of the leaky region. On the other hand, they gather to the center of 2D momentum space (*k*_{||} ∼ 0) as Δ_{3} becomes increasingly negative towards −0.26*a*. Since the leaky components conserve their momentum when they escape from the cavity, their distribution in the leaky region is related to the direction of radiation, namely, their far field emission patterns. Considering an equation, (*ω*/*c*)^{2} = |*k*_{||}|^{2} + |*k*_{⊥}|^{2}, the radiation angle of a leaky component, *θ _{k}*, can be related to its in-plane momentum by

*θ*= sin

_{k}^{−1}(

*k*

_{||}/(

*ω*/

*c*)), where

*k*is the wave vector in the

*z*direction. Therefore, a concentration of leaky components near |

*k*

_{||}| = 0 is expected to cause an increase of vertical emission with small

*θ*, and hence, an improvement of the coupling efficiency,

_{k}*η*

_{⊥}.

For a quantitative discussion, an effective coupling efficiency ${\eta}_{\perp}^{FT}$ is calculated from the distribution of ${E}_{y}^{FT}$. Here, ${\eta}_{\perp}^{FT}$ is defined by

*k*

_{||}| < 0.65

*ω*/

*c*) is divided by sum of whole components inside the light line. The factor cos

*θ*projects ${E}_{y}^{FT}$ onto the components which contributes the net leakage of light in the vertical direction. Figure 3(c) summarizes the calculated ${\eta}_{\perp}^{FT}$ at various Δ

_{k}_{3}. The plot is overlaid with the curve of

*η*

_{⊥}, previously calculated from the far field emission patterns. These two curves show a good agreement, demonstrating the validity of our method that used Eq. (2) for estimating the external efficiency.

This agreement can also be visualised by plotting distributions of
${\left|{E}_{y}^{FT}\right|}^{2}\text{cos}{\theta}_{k}$ with the units of *θ* sin*ϕ* and *θ* cos*ϕ*, as shown in Fig. 3 (d) and (e). These two figures are separately plotted for Δ_{3} = 0 and −0.26*a* and respectively show good agreement with the far-field distribution in Fig. 2(b) and (c). This demonstrates that the distribution of leakage power in the vertical direction can be well reproduced by calculating
${\left|{E}_{y}^{FT}\right|}^{2}\text{cos}{\theta}_{k}$. We consider that remaining imperfections in the distribution plots in Fig. 3(d) and (e) arise from absence of the effects of phase and of contributions from other field components. From the observed agreements, we conclude that the increase of *η*_{⊥} is due to the redistribution of
${E}_{y}^{FT}$ near the center of momentum space.

As a summary of this section, from the momentum space distribution we confirmed that the simultaneous increase of *Q* and *η*_{⊥} arise from the reduction of the leaky components below the light line and a redistribution of them in small |*k*_{||}| region, where radiation with small *θ _{k}* occur. This result suggests that

*η*

_{⊥}can be engineered by controlling the distribution of leaky components below the light line, while retaining a constant

*Q*by keeping the amount of leaky components.

## 3. Sample fabrication and optical measurements

#### 3.1. Sample fabrication and experimental setup

We fabricated the newly-designed H1 PhC nanocavities, and experimentally investigated the behavior of their *Q* factors and *η*_{⊥} as a function of Δ_{3}. All design parameters other than Δ_{3} were kept the same as those used in the numerical simulations. The nanocavities were fabricated into a 130-nm-thick GaAs wafer containing a layer of InAs QDs in the middle. Areal density of the QDs is ∼ 1 × 10^{9} cm^{−2}. A typical photoluminescence (PL) spectrum taken at 4K is shown in Fig. 4(a). This PhC layer was grown on a 1000-nm-thick Al_{0.7}Ga_{0.3}As sacrificial layer. The PhC lattice constant, *a*, was 260 nm, and the resonant wavelengths of the cavity modes (∼ 950 nm) are within an inhomogeneously-broadened emission range of the QDs. The PhCs were patterned by a combination of electron-beam lithography and inductive coupled plasma reactive ion etching (ICP-RIE) using a Cl_{2}/Ar mixture. Finally, the sacrificial layer underneath the GaAs slab was removed with HF:H_{2}O (1:9) acid solution to form air-bridge structures.

The fabricated samples were set in a temperature-controlled liquid-helium-flow cryostat, cooled to 4K, and investigated with a micro-PL measurement system. Optical pumping was performed with a continuous-wave Ti:sapphire laser operating at *λ* = 800 nm, which was focused onto the sample surface by a 50x objective lens with a numerical aperture of 0.65 (the same value used in the calculations above). Emission from the sample was collected by the same objective lens and sent to a spectrometer equipped with a liquid-nitrogen cooled CCD sensor. The spectral resolution of our measurement setup is ∼ 20*μ*eV at *λ* = 950 nm. All the measurements were performed at a fixed excitation power of 20 nW (measured before the objective lens). We note that measured PhCs in this study are distributed within a small region (< 1 mm^{2}), in which the PhC fabrication and the QD density are well controlled and uniform.

#### 3.2. Experimental results

Figure. 4(b) shows a typical PL spectrum of the fabricated cavity with Δ_{3} = −0.26*a*. The two peaks arise from *x*- and *y*-dipole cavity modes, the degeneracy of which was lifted by imperfections in the fabrication process. Several methods for compensating the observed split of the modes [12,13,25] may be applicable to these nanocavities. The red line indicates a fitting result of the measured cavity emission by two Lorentzian peak functions. From the fit, a *Q* factor of 25,000 for both peaks was obtained, which is the highest value ever reported for the dipole modes in any H1 PhC nanocavities. Here, the experimental cavity *Q* factor (*Q _{exp}*) is lower than the theoretical value of

*Q*= 62, 000, most probably owing to imperfections in the fabrication process. This situation is well described by the equation, ${Q}_{\mathit{exp}}^{-1}={Q}_{\mathit{calc}}^{-1}+{Q}_{\mathit{fab}}^{-1}$ [26], where

_{calc}*Q*expresses the fabrication limited

_{fab}*Q*factor. The

*Q*for our sample was estimated from high-design-

_{fab}*Q*(

*Q*∼ 200,000) L3 nanocavities fabricated together with the H1 nanocavities, and was evaluated to be 35,000. From this

_{calc}*Q*, the expected

_{fab}*Q*for the H1 nanocavities with Δ

_{exp}_{3}= −0.26

*a*was 22,500, which agrees reasonably with the measured

*Q*factor of 25,000.

Figure 4(c) shows value of *Q _{exp}* for the fabricated cavities with various values of Δ

_{3}.

*Q*at each value of Δ

_{exp}_{3}is derived from an average of 6 peaks measured from 4 samples (after excluding the maximum and minimum values). We did not distinguish the

*x*- and

*y*-dipole modes in this measurement because cavity

*Q*for the two dipole modes are identical in the numerical calculations. A clear increase of

*Q*near Δ

_{exp}_{3}= −0.26

*a*was observed, together with a flat region of the

*Q*(0 < Δ

_{exp}_{3}< 0.2

*a*) and a reduction of the

*Q*(Δ

_{exp}_{3}> 0.2

*a*). These behaviors agree well with the FDTD calculations. The red dashed line in Fig. 4(c) expresses the modified theoretical

*Q*factors (

*Q*′

*) including the effects of fabrication errors through an equation: ${Q}_{\mathit{calc}}^{\prime -1}={Q}_{\mathit{calc}}^{-1}+{Q}_{\mathit{fab}}^{-1}$ (using the fixed*

_{calc}*Q*of 35,000). A good reproduction of the experimental result was observed with this simple equation.

_{fab}Next, we investigated the behavior of *η*_{⊥} as a function of Δ_{3}. The cavity emission intensity, *I _{cav}*, is used as a measure of

*η*

_{⊥}(the two are linearly related).

*I*at each Δ

_{cav}_{3}was derived through an average of several measurements, as we did for

*Q*, so that fluctuations of the intracavity power was considered to be averaged out. Thus, measurement of

_{exp}*I*should well reflect the relative changes of

_{cav}*η*

_{⊥}. The obtained result is summarized in Fig. 4(d). The red dashed line indicates the theoretical

*η*

_{⊥}calculated in the previous section. Again, the experiments well reproduce the prediction of the calculation: the emission intensities show a sharp increase near Δ

_{3}= −0.25

*a*. The PL intensities of the cavity modes with a shift Δ

_{3}= −0.26

*a*become roughly twice as strong as those with Δ

_{3}∼ 0, while the

*Q*at Δ

_{exp}_{3}= −0.26

*a*was also still higher than that at Δ

_{3}= 0.

These two experimental results demonstrate that both *Q* and *η*_{⊥} in these H1 PhC nanocavities can be simultaneously improved for Δ_{3} = −0.25*a* ∼ −0.3*a*, as predicted in the numerical simulations. At Δ_{3} = −0.26*a*, the average *I _{cav}* (corresponding to

*η*

_{⊥}) becomes two times larger than that without the shift (Δ

_{3}= 0), while the

*Q*is increased to over 20,000.

_{exp}## 4. Conclusion

We have designed H1 PhC nanocavities that simultaneously exhibit both high *Q* and large *η*_{⊥}, through optimizing up to the third nearest neighbor air-holes around the cavity. A high theoretical *Q* factor of 62,000 is demonstrated whilst acquiring a high *η*_{⊥} of 0.38 (for N.A. = 0.65) for doubly-degenerate orthogonally-polarized cavity modes (dipole modes). Further theoretical increase of *η*_{⊥} to values over 0.6 (whilst keeping *Q* > 50,000) is also obtained by additional optimization of the design parameters. These behaviors are well explained by changes of the field distribution in momentum space. We have also experimentally verified the predictions by performing PL measurements, and observed the highest experimental *Q* factor of 25,000 for the dipole mode in the H1 PhC nanocavities ever reported.

Authors would like to thank M. Holmes for fruitful discussion. This work was supported by Project for Developing Innovation Systems of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, and by Japan Society for the Promotion of Science (JSPS) through its “Funding Program for world-leading Innovation R&D on Science and Technology (FIRST Program)”.

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