A series of microcavities in 2D hexagonal lattice photonic crystal slabs are studied in this paper. The microcavities are small sections of a photonic crystal waveguide. Finite difference time domain simulations show that these cavities preserve high Q modes with similar geometrical parameters and field profile. Effective modal volume is reduced gradually in this series of microcavity modes while maintaining high quality factor. Vertical Q value larger than 106 is obtained for one of these cavity modes with effective modal volume around 5.40 cubic half wavelengths [(λ/2nslab)3]. Another cavity mode provides even smaller modal volume around 2.30 cubic half wavelengths, with vertical Q value exceeding105.
©2004 Optical Society of America
The discovery of photonic crystals has opened up many new methods to manipulate light [1–2]. However, three dimensional photonic crystals usually involves complex connectivity and strict alignment, which make them rather challenging for fabrication [3–4]. Alternatively, two dimensional photonic crystal slabs (2D PCS’s) have been proposed [5–6], which promise easier fabrication using present techniques.
Over the past few years, much work has been devoted to the study of microcavities in these 2D PCS’s. These microcavities exhibit attractive properties such as high quality factor (Q) and small modal volume (V), which make them potentially useful not only in miniaturized photonic devices but also in some quantum optical devices. The former applications include channel drop filter for the wavelength division multiplexer system, high efficiency light emission diodes and low threshold lasers [7–11]. Among the latter applications are controlled single photon source and entangled emitter-cavity systems [12–13].
The applications mentioned above all require microcavities to confine light strongly and densely, that is, microcavities should have both high quality factor Q and small modal volume V. The ratio Q/V is a measure of the strength of various cavity interactions and should be as large as possible.
There have been many theoretical and experimental studies to improve the quality factor of microcavities in 2D PCS’s. It is known that total Q value (Qtot) can be separated into two parts, namely the in-plane Q value (Q ⊲) and the vertical Q value (Q⊥) by a simple relation . In a 2D PCS microcavity, light is confined within the defect region by two combined mechanisms, namely distributed Bragg reflection (in-plane) and total internal reflection (vertically). The in-plane confinement is determined by the number of periods of the host lattice surrounding the cavity, with the assumption that the resonant frequency of the defect mode lies within the in-plane guided mode bandgap. Thus, the more periods surrounding the defect, the stronger the in-plane light confinement and the larger the in-plane Q value (Q⊲). The vertical confinement is due to the standard waveguiding by total internal reflection. In momentum (k) space analysis, vertical radiation losses occur when the cavity modes have in-plane momentum components (k⊲) that lie within a region called ‘light cone’ [14–15]. The larger the fraction of in-plane momentum components that lie within this light cone, the larger the vertical radiation loss and consequently the smaller the vertical Q value.
As we have seen, one of the key issues in the design of high Q microcavity in 2D PCS’s is to reduce the k⊲ field components that lie within the light cone and thus enhance the vertical confinement. One group has achieved this by carefully modifying the air hole radii into a graded pattern . This leads to some cavity modes with Q⊥ larger than 105. Another group has come up with a design of simpler structure . The cavity is constructed by removing three air holes in a row and shifting the left, right adjacent holes outwards by 0.15 a, where a is the lattice constant. Experiments have shown that the cavity supports a resonant mode with a high quality factor Q=4.5×104 and a small modal volume V=7.0×10-14 cm3. Recently a group has reported a hexapole cavity mode with Q larger than 106 and small modal volume on the order of cubic wavelengths in material . However, this high Q hexapole mode is very sensitive to its surrounding geometrical parameters, which might bring up difficulties for fabrication.
In this paper, we start with a cavity similar to that in Ref. , but consider it as a short section of a line defect (waveguide), ended with perfect 2D photonic crystal lattices (as shown in Fig. 1). We shorten this waveguide section gradually to reduce the modal volume and study how its Q value varies. Contrary to the acceptor mode in Ref. , whose resonant frequency lies slightly above the lower bandedge , we concentrate on the donor mode inside this encapsulated waveguide section cavity that has a higher resonant frequency just below the upper bandedge.
The paper is organized as follows: We first introduce the methods we use, including the setup for the finite difference time domain (FDTD) simulations, how to calculate Q value and effective modal volume. In section 3, we begin with a three missing hole cavity (named as an M3 cavity). We first modify its surrounding geometrical parameters to achieve highest Q value for this mode and then calculate its modal volume. Next, we run the same procedures on two missing hole cavities (M2), one missing hole cavities (M1) and finally cavities with no missing hole at all (M0). In section 4, we give a brief comparison of the cavity modes.
2. Calculation methods
We use the finite difference time domain method for our simulations . The structure size is 23a×23a for the entire cavity modes calculated in this paper, which is capable of providing sufficiently large Q⊲ so that Qtot is primarily limited by Q⊥. The grid size is 0.05a in x, z direction and 0.0433a in y direction. Perfect matched layers are used as absorbing boundaries .
The Qtot value of the cavity modes is calculated using a combination of FDTD techniques and Padé approximation with Baker’s algorithm [22–23], which is capable of achieving ultra accurate Q value using a much smaller number of time steps compared to other methods (typically <0.5% numerical errors and ~15,000 time steps for Q>105). For comparison reasons, some high Q values of certain cavity modes are also calculated using conventional method: , where f0 is the resonant frequency of the cavity mode, U is the time averaged energy stored in the cavity, and P is the power loss. Q⊥ and Q⊲ are calculated separately using this method.
The modal volume is calculated using the following equation ,
where ε (x, y, z) is the spatial distribution of dielectric constant and
3. Waveguide-section microcavities
3.1 Three missing hole waveguide-section cavity (M3)
The structure of our 2D PCS cavity is shown in Fig. 1, with three missing holes in the center. The refractive index of the slab is chosen to be 3.4, which corresponds to silicon at 1.55 µm. The thickness of the slab is 0.7a, where a is the lattice constant. The radius of the air holes is set to be R=0.29a. These two parameters are also the same for M2, M1 and M0 cavities.
As mentioned earlier, we have to reduce the k⊲ components of the cavity mode that lie inside the leaky region in order to achieve high Q value. The first step we take is similar to that described in Ref. . Basically, we apply symmetries to enforce special boundary conditions on the momentum space representation of the mode, so that the in-plane electric field polarizations at k⊥=0 (DC) are eliminated. Since all the cavities in this paper support multiple modes, we confine our discussion only to the donor modes whose H𝑧 field distribution is even in both x and y direction. These modes, as we shall see later, can achieve extremely high Q and ultra small modal volume when the waveguide section is well shortened.
To further reduce the in-plane momentum components inside the leaky region, we modify the geometries of the lattice structure surrounding the defect. This, in essence, changes the Bragg reflection conditions at the cavity edges. By doing so, we allow the light to penetrate deeper inside the surrounding lattice and be reflected gently and perfectly. The spatial variation of the envelope function is tailored into a Gaussian profile, which significantly reduces the momentum components in the leaky region while keeping the modal volume small . As shown in Fig. 1(b), we modify the two geometrical parameters d and R1 to improve the Q value.
For simplicity’s concern, we will use the same name (M3, M2 and etc.) for the cavity and its donor mode discussed in this paper, although all these cavities support multiple modes.
After careful scanning of d and R1, we find the highest Q value 8,000 can be achieved at d=0.24a and R1=0.23a. Figure 2(a) shows the H𝑧 field distribution of the M3 mode. The field is spread out into the surrounding lattice of the cavity and since the resonant frequency 0.3077 a/λ is near to the upper bandedge (~0.32 a/λ), the in-plane quality factor Q⊲ is limited to 35,000 even for a large size (23a×23a).
For the k space analysis, we take the 2D (kx, ky) Fourier transform of Ex, Ey and Hx, Hy field distribution in the air layer one FDTD grid cell above the slab surface, denoted by FT 2(Ex), FT 2(Ey), FT 2(Hx) and FT 2(Hy), respectively. The total radiated power P of the cavity mode can be estimated by the following equations .
where , and
The distribution of I is shown in Fig. 2(b). We can see that there is still a considerable amount of k⊲ component lying inside the light cone, which limits Q⊥ to about 10,000. Using Eqs. (1) and (2), the modal volume V is calculated to be 7.90(λ/2nslab) 3 for the M3 mode.
The M3 mode is not very exciting compared to the acceptor mode in Ref. , which has Qtot=9.91×104 at resonant frequency 0.2618 a/λ when d=0.2a and slab thickness t=0.6a, according to our simulations. For d=0.2a and t=0.7a, Qtot equals 2.16×104 with resonant frequency 0.2549 a/λ. However, the donor mode becomes interesting when the waveguide-section is further shortened. Next, we study cavities with only two air holes removed but its surrounding geometrical structures retained.
3.2 Two missing hole waveguide-section cavity (M2)
For the two missing hole cavity, shown in Fig. 3(a), we tune d=0.23a and R1=0.2a. Qtot is calculated to be 5.4×104, with Q⊥=6.0×104 and Q⊲=4.5×105. Both Q⊥ and Q⊲ are much improved compared to the previous case. The former improvement is due to the reduced k space components inside the leaky region, as shown in Fig. 3(c) and the latter improvement is due to the reduced resonant frequency at 0.2955 a/λ, which is now lying closer to the middle of the bandgap.
For the M2 mode, the effective modal volume is calculated to be 9.52 (λ/2nslab)3.
3.3 One missing hole waveguide-section cavity (M1)
To further reduce the modal volume, we move on to M1 cavity with only one missing hole, shown in Fig. 4(a). As with the M3 and M2 cavities, we vary d and R1 to search for the setup of the highest Q donor mode for the M1 cavity. We find when d is tuned to 0.21a and R1 to 0.22a, Qtot=5.85×105 can be achieved at resonant frequency 0.2902 a/λ. Since M1 cavity is more compact, we vary more geometrical parameters hoping to improve the Q value. We find that by modifying the radius of four air holes from R=0.29a to R2=0.25a [also shown in Fig. 4(a)], we can largely enhance the Q value for the M1 mode to Qtot≈Q⊥=1.06×106 and Q⊲>2×107. The Resonant frequency is further reduced to 0.2881 a/λ and modal volume reduced to 5.40 (λ/2nslab)3.
It is interesting to note that M3, M2 and M1 modes all reach their highest Q value when d and R1 are tuned to around 0.22a. We can conclude that tuning d to around 0.22a allows light to penetrate deeper inside the surrounding lattice in the x direction and be reflected gently. Similarly, tuning R1 (and also R2 for M1 cavity) allows the reflections in the y direction to be smoother. The combined effect is that the spatial variation of the envelope function is tailored into a Gaussian profile so that the k⊲ components inside the light cone are reduced and thus Q⊥ improved. On the other hand, increasing d and decreasing R1 helps pull down the resonant frequency closer to the middle of the bandgap , which causes Q⊲ to increase.
3.4 Zero missing hole waveguide-section cavity (M0)
There is still room to further reduce the modal volume. We move on to M0 cavity, which involves no missing holes at all. The geometric structure is shown in Fig. 5(a). The structure is very simple, with only two adjacent holes (labeled ‘A’) shifted outward in x direction by a distance of d and the radius of the two air holes nearest to the cavity center in the y direction (labeled ‘B’) is modified to R1. After careful scanning, we find that d=0.14a and R1=0.27a gives the highest Q. We also modified other geometric parameters, such as the radius of the ‘A’ holes and the ‘C’ holes, but this only reduces the Q value of the M0 mode, so we have skipped them in this paper. The final Q value is 1.35×105 with resonant frequency 0.2888 a/λ.
The Hz field distribution is shown in Fig. 5(b). The Hz field profile is now very simple and the number of the nodal points is much reduced. Figure 5(c) shows the k space electric intensity profile. There are very few k⊲ components inside the leaky region, which accounts for a large Q⊥ (≈1.9×105) of this mode.
The effective modal volume for the M0 mode is extremely small, only 2.30 cubic half wavelengths. For a lattice spacing of 450 nm, which corresponds to resonant wavelength of 1564 nm for the M0 mode, the modal volume is only 2.76×10-14 cm3.
For another configuration of the M0 cavity, we set d=0.14a and R1=R=0.29a, that is, we only move ‘A’ holes outward by 0.14a, without changing any other surrounding lattice. The modal volume is 2.25 cubic half wavelengths, with Q⊥=1.12×105. Since only the position of two air holes needs to be changed for this modified M0 cavity, without affecting any air hole radius, it is relatively easier to fabricate than other cavities discussed above.
We also calculated the Q value of the other modes supported by the M2, M1 and M0 cavity. The donor modes discussed in this paper have the highest Q value for each cavity respectively.
Figure 6 shows the electric field intensity profile for each of the four modes discussed above. From Fig. 6(a) to (d) we can easily see that as the number of missing holes decreases, the electric field intensity maxima in the x direction are pulled closer to each other. The number of nodal points becomes smaller and the field gets more compressed in both x and y directions, leading to reduced modal volumes.
Figure 7(a) compares the Qtot value and the effective modal volume of the four modes. We can clearly see that the M1 mode offers the highest Q value and the M0 mode has the smallest modal volume while keeping a relatively large Q. Figure 7(b) compares the Qz and the radiation factor RF of the four modes. , where P is calculated using equation (3) and (4). W is the total energy stored in the cavity . We can clearly see the inverse relation between Q⊥ and RF in Fig. 7(b).
To conclude, we have numerically demonstrated a series of waveguide-section microcavity modes in 2D hexagonal lattice photonic crystal slabs. High Q factor larger than 106 is obtained for the M1 mode by carefully tuning some structural parameters. Modal volume as small as 2.30 cubic half wavelengths is achieved for the M0 mode. The profile of the M0 mode is simple, yet with Q factor exceeding 105. We believe that M0 and M1 cavities are well suited for large-scale photonic integration and single-photon sources for quantum optics operations.
This work was supported by the Swedish Foundation for Strategic Research (SSF) on Photonics and the Swedish Research Council (VR) under project 2003-5501.
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