1. Introduction

Since E. Yablonovitch and S. John firstly proposed the concept of photonic bandgap (PBG) in 1987 [1,2], the photonic crystal (PhC) cavity has attracted particular attention owing to its ability to confine photons within wavelength dimensions by the PBG. Especially two-dimensional (2D) PhC slab cavities have been widely studied by utilizing well established fabrication processes in the semiconductor industry [3–18]. In such 2D PhC slab structures, photons can be confined horizontally by the 2D PBG and vertically by total internal reflection (TIR). However, the quality factor (Q) in the 2D PhC cavity is mainly limited by the leaky components due to insufficient TIR.

An important design rule has been proposed to obtain high Qs in 2D PhC slab cavities [4,5]: The electric field in a cavity should slowly vary like a Gaussian function to eliminate leaky components out of the slab. A photonic heterostructure cavity with different lattice constants along the PhC waveguide was proposed to satisfy this design rule and showed the remarkable high experimental Qs of 2.5×10⁶ in Si [6–8] and 10⁵ in GaAs [9]. Kuramochi et al. proposed a different heterostructure by locally modulating the waveguide width and also demonstrated a high Q of 1.3×10⁶ experimentally [10,11]. Recently, other types of photonic heterostructure cavities have been studied by a local index modulation of a waveguide using fluidic infiltration or photosensitive materials [12–14]. On the other hand, it was reported that PhCs with different air hole radii can confine photons by shifting PBG [15].

In this paper, we propose a new heterostructure PhC cavity by the local modulations of the radii of the air holes along a line defect waveguide, which completes the series of the heterostructure cavities realizable by lithography. For the proposed heterostructure cavity, we investigate the mode patterns in the real and the wavevector space and the mode behaviors such as Q and mode volume (Vm) by three-dimensional (3D) finite-difference time-domain (FDTD) method. We discuss the effects of the length and the air hole size offset of the tapered regions on Q and Vm.

2. Design of heterostructure cavity using air hole modulation

Our cavity is based on a single line defect waveguide (W1) in a triangular lattice of air holes drilled in a slab surrounded by air. In our cavity, the photon is mainly concentrated between two mirror regions in which the radius of air holes, r_wg, is enlarged compared to the background PhC, r_pc. In order to obtain more gently varying electric field, the tapered regions with linearly increasing air holes are introduced between the cavity and the mirror regions, as shown as red boxes in Fig. 1(a) and (b). The air hole radii in the tapered region are linearly interpolated between r_PC and r_wg. Here the air hole size offset is the differences between r_PC and r_wg. The heterostructure cavity in this paper is described by two structure parameters, the length of tapered regions and r_wg of the mirror region. For example of our nomenclature, S5 and S3 cavities have 5 and 3 layers from the cavity center to the mirror region with r_wg~0.28a, including 4 and 2 tapered layers, respectively, as shown in Fig. 1. The photons are confined in the cavities by mode-gap along the waveguide, PBG in the photonic crystal, and TIR in the vertical direction of the slab. The refractive index of a slab is 3.4, which corresponds to indexes of Si and GaAs at λ=1.55 µm. We assumed a lattice constant a of 400 nm, a slab thickness of 250 nm, and a hole radius in the PhC region, r_PC, of 0.25a, respectively. The size of surrounding PhC is sufficiently large, 48(x axis)×24(y axis) lattice periods. Under these conditions, the total Q is almost equal to the vertical Q in most of the cases.

 

Fig. 1. Proposed photonic crystal cavities. Dotted purple lines indicate the center of the cavities. (a) S5 cavity. (b) S3 cavity. For both cavities, the mirror regions are represented by yellow boxes, the tapered regions by red boxes. S5 and S3 cavities have 4 and 2 tapered layers. (c) Air hole radius along the waveguide. The air hole radius of the photonic crystal and the middle of the waveguides is r_pc=0.25a and the radius in the mirror regions is r_wg. The lattice constant is denoted by a.

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Dispersion characteristics of the PhC W1 waveguide mode are plotted in Fig. 2(a), which is calculated by two-dimensional (2D) plane wave expansion method. The mode in the white area is guided inside the PhC slab by TIR. As r_wg is enlarged, the effective waveguide width decreases so that the frequency of W1 bandedge blue-shifts. When a waveguide with r_wg~0.25a is connected to a waveguide with r_wg~0.28a, the mode near the bandedge of the waveguide with r_wg~0.25a is placed within the mode-gap of the waveguide with r_wg~0.28a, as shown in Fig. 2(a). Therefore, the waveguide with enlarged r_wg can work as a mirror for the waveguide mode with smaller r_wg near the bandedge. The magnetic field pattern of the bandedge mode, which is the origin of the heterostructure cavity mode, is illustrated in Fig. 2(d). As expected by the wavevector 0.5(2π/a) of the bandedge, the field oscillates with a period with 2a along the Γ-K.

 

Fig. 2. (a) The band structure for a line defect waveguide with r_wg~0.25a and 0.28a. The gray area and the yellow box indicate leaky mode and mode-gap regions. The simulated structures with r_wg~0.25a and 0.28a are shown in (b) and (c), respectively, indicated by the yellow colored circles. (d) Magnetic field pattern at the bandedge, wavevector~0.5(2π/a), of the waveguide with r_wg~0.25a. Field is orthogonal to the slab surface.

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Fig. 3. Field patterns and far-field patterns of the S5 cavity with r_wg~0.27a. (a) Electric field intensity profile. The black circles indicate air holes. (b) Vertical magnetic field (H_z). (c) Far-field pattern. The mapping (x, y) is (sinθcosϕ, sinθsinϕ). θ is the angle from the vertical direction to slab surface so that the center indicates vertical emission direction. Black solid line circle represents horizontal emission direction. Black dotted circle shows 30° emission angle. (d) E_x field. (e) E_y field. (f) Polarization of the collected electric fields within NA=0.6 in the simulated far-field.

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The typical mode patterns and the far-field patterns of the heterostructure cavity mode are illustrated in Fig. 3, which are obtained for the S5 cavity with the largest vertical Q~8.8×10⁷ in the case of r_wg~0.27a. This value is a very high theoretical Q value for PhC nanocavities, which is comparable with the numerical Q obtained by Kuramochi et al. [10]. Here the Vm is 1.62 (λ/n)³ and the normalized frequency is 0.255 (a/λ). Because all heterostructure cavities are based on the idea that the near-bandedge mode of the W1 waveguide with lower frequency is confined by the mode-gap of a W1 waveguide with higher frequency, the mode patterns shown in Fig. 3 have almost the same symmetries and distributions for all investigated cavities in this paper as well as other heterocavities reported by other groups [6–14]. Hence the mode behaviors for the length and the air hole size offset of tapered regions discussed in the paper can be commonly applied for all other heterostructure cavities. Of course the air hole size offset corresponds to the lattice constant, the waveguide width, and the refractive index changes in other cavities.

Figure 3(a) shows the calculated electric field intensity profile. It shows that the mode is strongly confined near the center, even for the cavity in the image where the change of the air hole radii along the waveguide is not recognizable, which prove that the mode-gap due to the modulation of the radii of the air holes works as a mirror. The vertical magnetic field profile (H_z) of Fig. 3(b) is almost same with the one of the W1 bandedge mode shown in Fig. 2(d).

In general, the electric field symmetry of the PhC cavity mode explains how the mode can be collected in the vertical direction of the slab by an objective lens, which is the most common way to investigate the photoluminescence of PhC cavities. Figures 3(d) and (e) show E_x and E_y field distributions. In the E_x field distribution, the vertical emission is prohibited because of the odd mirror symmetries for both x=0 and y=0. However, in the E_y field distribution, since both mirror symmetries are even, the cavity mode can emit y-polarized photons vertically. Actually, the collected electric field within the NA(Numerical Aperture)=0.6 in the far-field simulation shows a strong y-directional polarized emission, as shown in Fig. 3(f). The polarization of the mode can be used to identify the heterostructure cavity mode in an experiment.

The calculated far-field pattern is plotted in Fig. 3(c). Here the far-field radiation pattern, which can be observed sufficiently far (r≫λ) from a cavity, is calculated by Fourier-transforming the in-plane field components obtained by 3D FDTD [16]. The cavity mode emits most of photons with ±75° emission angle from the vertical direction of the slab and the amounts of the emitted photons within ±30° (NA=0.6) is negligible. Because the heterostructure cavity mode is designed to minimize the components within the light cone so that the mode components near at k_x=k_y=0 in wavevector space ought to be negligible. Therefore, as long as the ultrahigh Q is maintained in the cavity, the PhC heterostructure cavity is not appropriate as a photon source with vertical emission. However, since the mode in the heterostructure cavity is based on the W1 waveguide mode, the mode is expected to be able to couple to the W1 waveguide very well, which will be illustrated later in this paper.

3. Effect of the length and the air hole size offset of the tapered region

For a slowly changing cavity structure, the tapered region can be introduced for any heterostructure cavity. It was reported that the cavity with one or two tapered layers showed a higher Q [10]. However, in our knowledge, the optimal length and the taper offset in the tapered region have not yet been investigated for a high Q cavity. In this section, we investigate the effect of the length and the air hole size offset of the tapered region on Q and Vm, which can be utilize to enhance any heterostructure cavity using tapered regions.

3.1 Effect of the length of the tapered region

The Q and the Vm of S5, S3, S2 and S1 cavities are investigated over a broad range of r_wg. Here, S5, S3, S2, and S1 cavities have 4, 2, 1, and 0 tapered layers from the center to the mirror regions. The Vm is defined in terms of the electric field density in the cavity [3]. Cavities with smaller Vm have broader spreading in the distribution of the wavevector space leading to higher losses into the vertical direction. Consequently, such cavities with smaller Vm tend to have a lower Q [17,18]. If we compare the cavities with the same Vm to exclude the effect of Vm among various cavities, we can focus on the effect of the length of tapered regions.

The normalized frequency, Q, and Vm for the cavities are shown in table 1. The Qv (vertical Q) is inversely proportional to the optical losses out of photonic crystal slab, which limits total Q in most of the cases. The Vm of the cavities is 1.45~1.46 (λ/n)³ and the normalized frequency is around 0.255~0.257 (a/λ). The normalized frequency does not show any significant change for all cavities with various r_wg because the mode is originated from the bandedge of W1 waveguide with r_wg~0.25a so that the frequency of the cavity is always close to it of the waveguide. The Qv of the cavities increases dramatically with the length of the tapered region. For example, the Qv of the S5 cavity is 3.0×10⁷, which is 100 times larger than 2.8×10⁵ of the S1 cavity.

Table 1. Vertical Q factors (Qv) and mode volumes (Vm) of S5, S3, S2, S1 cavities

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In order to inspect the behavior of the larger Qv for the cavity with longer tapered regions, the mode profiles of the heterostructure cavities are investigated for the different lengths of the tapered regions. As the amount of the mode within the light cone determines the vertical loss in PhC slabs, it is easier to analyze the behaviors of Q through the mode distribution in the wavevector space. The mode distributions of the cavities S1 and S5 in the wavevector space are illustrated in Fig. 4(a) and (b), which are obtained through 2D Fourier transforms of E_x and E_y fields calculated by the 3D FDTD method. The modes of the investigated cavities are localized with some spreading at the four points, indicated by red color in Fig. 4(a) and (b). The fact that k_x of the mode is near ±0.5(2π/a) means that the mode is very close to the bandedge of W1 waveguide mode. The spreading of k_y is broader than that of k_x because the mode is confined in a shorter distance along the y-direction compared with the confinement along the x-direction, as shown in Fig. 3(a).

On the other hand, in contrast to the distribution of the S5 cavity in Fig. 4(b), fringes over the wavevector space are observed in the distribution of the S1 cavity in Fig. 4(a). These fringes result in non-negligible components within the light cone (indicated by a yellow circle) in the mode distribution of S1 cavity. The components inside the light cone correspond to the vertical loss which cannot satisfy TIR condition in PhC slabs. Therefore the S1 cavity with the shortest tapered regions has a smaller Qv than the S5 cavity with the longest tapered regions. Since the leaky components within the light cone are generated along the x-axis in real space [5], we investigated mode distribution of S1, S2, S3, and S5 cavities along k_x direction in Fig. 4(c).

 

Fig. 4. Mode patterns of (a) S1 cavity and (b) S5 cavity in the wavevector space (Log scale). |FT(Ex)|²+|FT(Ey)|² is plotted. The yellow circle indicates the light cone. (c) Mode distributions of S1, S2, S3, S5 cavities along k_x direction in the wavevector space (Log scale). The yellow box corresponds to the light cone. The r_wg of each cavity is as given in table.1. (d) Qv of the cavities as a function of the length of the tapered region.

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S2, S3, and S5 cavities have similar spreading near k_x=±0.5(2π/a) but not S1 since the mode volumes are almost same. The spreading of the S1 cavity is not clearly observable due to large fringes. Although the magnitudes of the fringes are different for each cavity, there always exist some fringes between k_x=±0.5(2π/a) points. As the length of tapered regions increases (S1→S5), the fringes inside the light cone (indicated a yellow box) become smaller which results in a higher Qv. The fringes corresponding to higher order wavevector components are originated from the abrupt change of the electric field envelop [5]. Therefore, the smoothly changing cavity structure building by the longer tapered regions can make a more gently varying electric field envelop, which contributes to a higher Qv.

In Fig. 4(d), Qv increases exponentially as the length of the tapered region, but increases only 1.5 times from the S3 to S5 cavities. According to this graph, it is expected that the Qv would be saturated as the tapered region becomes longer than that of S5 cavity. Because the change of the air hole radii in the tapered region of the S5 cavity is gentle enough to make slowly varying electric field envelop for high Q cavity. On the other hand, in the real fabrication, less modification of air holes makes it simpler to fabricate a cavity. Therefore S3 cavity which has comparable Q with S5 cavity and less modulation of radii can be one of good candidates for an ultrahigh Q cavity.

3.2 Effect of the air hole size offset of the tapered region

In order to investigate the effect of the air hole size offset in tapered regions, we have simulated Q, Vm, and normalized frequency of the S3 cavity for a range of r_wg from 0.26a to 0.36a. As the air holes in the tapered region increases linearly from r_pc to r_wg and r_pc is fixed to 0.25a, larger r_wg gives a larger air hole size offset in the tapered region. The total Q factor (Qt) decreases exponentially from 2.0×10⁷ to 1.0×10⁵ with larger r_wg after a maximum at 0.27a, as shown in Fig. 5(a). Therefore, smaller air hole size offset in the tapered regions can give a higher Qv. It will be discussed in Fig. 6 why Qv always decreases with larger r_wg. The total Q and Vm of the S5 cavity show similar behaviors with the S3 cavity, as shown in Fig. 5(d). But Q and Vm of the S5 cavity are always larger than S3 cavity due to more gently changing air hole radii in the tapered region.

 

Fig. 5. Mode characteristics of the S3 cavities with various r_wg. (a) Total Q and mode volume. (b) Q factors for each loss channel. (c) The normalized frequencies (a/λ) of the cavity mode (black line) and the bandedge mode (red line) of mirror waveguide as a function of r_wg. (d) Total Q and mode volume of the S5 cavities with various r_wg.

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On the other hand, Vm decreases linearly with larger r_wg from 2.0 to 0.9 (λ/n)³. It should be noted that Qt is always higher than 10⁶ for a broad range of r_wg from 0.27a to 0.32a while Vm is kept on the order of a cubic wavelength. It means the cavity has a large tolerance with respect to fabrication errors causing overall air hole size changes. Qv, Qx, and Qy are estimated from the energy flow out of the slab surface, along the waveguide, and through the PhC region, respectively, and are shown in Fig. 5(b). In most of the cases, Qv limits Qt over various r_wg except for the cavity with r_wg~0.26a where Qx limits Qt.

At the cavity with r_wg~0.26a, Qx drops to 50,000. Even though we put many PhC layers along the x-direction, Qx was still small enough to limit Qt because the bandedge frequency of the mirror region with r_wg~0.26a is very close to the frequency of the cavity mode resulting in a small mode-gap and weak photon confinement. In contrast to that the frequency of the bandedge mode of the mirror region increases rapidly with r_wg, the frequency of the heterostructure cavity mode increases slowly with r_wg since only a part of the mode experiences the mirror region with larger r_wg, as shown in Fig. 5(c). Thus the mode-gap corresponding to the region below the bandedge mode frequency of the mirror waveguide with smaller r_wg becomes smaller so that photon confinement is getting weaker. Therefore, a cavity consisting of two kinds of waveguides with too small difference of refractive index, lattice constant, waveguide width, or air hole radius is not suited for a high Q. Since the fabrication error can broaden the dispersions of PhCs, the small mode-gap would disappear in that case. On the other hand, Vm becomes smaller in the cavity with larger r_wg due to stronger confinement by the larger mode-gap.

 

Fig. 6. Electric field intensity patterns (linear scale) of S3 cavities (a) with r_wg~0.27a and (b) r_wg~0.36a. Wavevector distributions (Log scale) of S3 cavities (c) with r_wg~0.27a (d) r_wg~0.36a. (e) Mode distributions along k_x direction in the wavevector space (Log scale). The yellow box corresponds to the light cone.

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In Fig. 6, we compare the mode patterns of the S3 cavities with r_wg~0.27a and r_wg~0.36a in the real space and the wavevector space to analyze the reason for the higher Qv in the cavity with smaller r_wg. Qv and Vm are 2.0×10⁷ and 1.46 (λ/n)³ for the cavity with r_wg~0.27a, 1.0×10⁵ and 0.93 (λ/n)³ for the cavity with r_wg~0.36a. The Qv of the cavity r_wg~0.27a is 200 times larger than one in the case of r_wg~0.36a. The real space mode pattern for the cavity with smaller r_wg in Fig. 6(a) spreads more along the waveguide direction than one for the cavity with larger r_wg in Fig. 6(b), which gives larger mode volume. As mentioned previously, the mirror region with larger r_wg has a larger mode-gap which results in stronger confinement in the real space.

The mode patterns in the wavevector space show two differences in Fig. 6(c) and (d), which becomes even clearer in the mode distributions along k_x direction in Fig. 6(e). First, wavevector spreading along the k_x at the k_x=±0.5(2π/a) for the cavity with r_wg~0.27a is apparently smaller than it is for the cavity with r_wg~0.36a. This smaller spreading due to larger Vm induces less leaky components within light cone, which contributes partly to larger Qv in the cavity with smaller r_wg. Second, there are relatively larger fringes inside light cone in the cavity mode with r_wg~0.36a in Fig. 6(e). It is explained by that larger air hole size offset in the cavity with larger r_wg makes rapid change of the cavity structure so that higher order wavevector components appear within light cone. It also partly results in larger Qv in the cavity with smaller r_wg.

4. Light extraction using waveguides

As shown in the far-field of Fig. 3(c), since a heterostructure cavity emits photons with large emission angles, only small portion of the emitted photons in the cavity can be extracted to the vertical direction of the slab. On the other hand, the heterostructure cavity would be expected to be a good candidate as an in-plane light source since the cavity is realized by modifying a waveguide. Qt and extraction efficiency, η, are calculated in the S3 cavity with r_wg~0.28a as the length of mirror layer connected to output waveguide increases. The extraction efficiency is defined as the fraction of the energy coupled into the output waveguide of total emitted energy. The simulated structure is a S3 cavity with finite number of mirror layers connected with the output waveguide regions with r_wg~0.25a at the both sides.

 

Fig. 7. (a) Qs for each loss channel of the S3 cavity with r_wg~0.28a vs. number of mirror layers. (b) Total Q and extraction efficiency, η, vs. number of mirror layers.

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Qy and Qv are kept to be almost constant, however, Qx, which is inversely proportional to the energy portion coupling to the output waveguides, decreases with a smaller number of mirror layers, as shown in Fig. 7(a). This means that the output waveguide works as a pathway of photons which does not induce any vertical scattering or loss through PhC region. This coupling behavior is due to the fact that the mode of the output waveguide with r_wg~0.25a basically has the same mode pattern of the heterostructure cavity mode so that the most of photons couple to the output waveguide in the case of a small number of mirror layers, the extraction efficiency becomes larger, as shown in Fig. 7(b). When the S3 cavity is connected to the output waveguide with 5 mirror layers, Qt is over 10⁶ and 80% of the photons emitted from the cavity can couple to the output waveguide.

5. Conclusions

In conclusion, we demonstrate a new type of heterostructure PhC cavity by the local modulation of air hole radii along a W1 waveguide. Introducing the tapered regions in the heterostructure cavity increases Qv greatly. Owing to the smoothly changing cavity structure, Qv of the heterostructure cavity increases by a factor of 100 as the tapered regions increases from 0 to 4 lattice periods. The cavity with a two lattice periods long tapered regions has an ultrahigh Q of 2.0×10⁷. The smaller air hole size offset of the tapered regions is also able to give a higher Qv due to a gently varying envelop of the electric field. In contrast to the small vertical emission of the heterostructure cavity, the cavity mode can couple into the output waveguide with 80% coupling efficiency while maintaining Qt~10⁶ in the in-plane waveguide coupling. Thus the heterostructure cavity can be a good candidate for an in-plane photon source. The mode behaviors of the cavity with air hole modulation discussed in this paper can give clues to design any heterostructure cavity with same mode distributions based on W1 waveguide and mode-gap.