Design of a 3D photonic band gap cavity in a diamond-like inverse woodpile photonic crystal

We theoretically investigate the design of cavities in a three-dimensional (3D) inverse woodpile photonic crystal. This class of cubic diamond-like crystals has a very broad photonic band gap and consists of two perpendicular arrays of pores with a rectangular structure. The point defect that acts as a cavity is centred on the intersection of two intersecting perpendicular pores with a radius that differs from the ones in the bulk of the crystal. We have performed supercell bandstructure calculations with up to $5 \times 5 \times 5$ unit cells. We find that up to five isolated and dispersionless bands appear within the 3D photonic band gap. For each isolated band, the electric-field energy is localized in a volume centred on the point defect, hence the point defect acts as a 3D photonic band gap cavity. The mode volume of the cavities resonances is as small as 0.8 $\lambda^{3}$ (resonance wavelength cubed), indicating a strong confinement of the light. By varying the radius of the defect pores we found that only donor-like resonances appear for smaller defect radius, whereas no acceptor-like resonances appear for greater defect radius. From a 3D plot of the distribution of the electric-field energy density we conclude that peaks of energy found in sharp edges situated at the point defect, similar to how electrons collect at such features. This is different from what is observed for cavities in non-inverted woodpile structures. Since inverse woodpile crystals can be fabricated from silicon by CMOS-compatible means, we project that single cavities and even cavity arrays can be realized, for wavelength ranges compatible with telecommunication windows in the near infrared.


I. INTRODUCTION
Many efforts are currently proceeding in the blossoming field of nanophotonics to trap light in a tiny volume in space 1,2 . Several classes of devices are pursued including micropillar and ring cavities 3,4 , point defects in two-dimensional photonic crystals 5,6 , and plasmonic structures such as metallic antennas 7,8 . Nanophotonic resonators have many interesting potential applications, such as the trapping or slowing-down of photons 1 , sensing 9 , a controlled enhancement of spontaneous emission 10 , as well as advanced cavity quantum electrodynamic control [11][12][13] . Linear arrays of wavelength-scale optical cavities are pursued for their function as waveguides with tailored properties 14,15 .
Of particular interest are cavities embedded in three-dimensional (3D) photonic crystals with a complete photonic band gap 16,17 . In the frequency range of the band gap light is forbidden to exist throughout the crystal and for all polarizations, which notably leads to the inhibition of spontaneous emission 18 . By introducing a point defect into the crystal structure, the lattice symmetry is locally broken, and a resonance appears in the band gap [19][20][21][22] , in an analogy to localized electronic defect states in a semiconductor 23,24 . A 3D photonic band gap cavity is considered to be an ultimate tool to control light down to the single photon level for several reasons. First, since the confinement is truly three-dimensional, there is no direction or dimension wherein the light will naturally leak as is the case in, e.g., a pillar or a 2D photonic crystal. Second, since in photonic band gap crystals the imaginary part of the dielectric constant of the constituent materials is minimal, the absorption of light is minimal, allowing very long storage times of light. Third, since a 3D photonic band gap effectively shields an embedded quantum system, such as an excited quantum dot, from vacuum fluctuations, an array of 3D cavities has great potential to control collective quantum systems including qubits 25 .
It is a major challenge in nanotechnology to realize optical cavities in 3D crystals 26,27 , since a controlled deviation from the periodic crystal structure must be realized deep inside the nanostructure. One demonstrated solution to this challenge are cavities in woodpile structures made with a layer-by-layer method [28][29][30][31] . In this method crystals are made by sequential stacking of layers where the central layer is modified to contain a point defect.
Unfortunately layer-by-layer stacking suffers from random fluctuations in the alignment. As a result the width of the photonic band gap is limited, hence the density of the optical states in the gap becomes filled with undesired states, thereby limiting the cavity quality factor. In a second method an optical cavity was proposed by an intriguing combination of a planar unit cell modulation (planar defect) and a waveguide (line defect) 32,33 . Relevant and interesting methods to fabricate cavities in opal -and inverse opal-based photonic crystals has been reported in references [34][35][36][37] .
In this paper, we propose and investigate a straightforward approach to realize an optical cavity in an inverse woodpile photonic band gap crystal 38 . These photonic band gap crystals have a symmetry similar to how carbon atoms are arranged in a diamond crystal 38 . Diamond-like photonic crystals stand out for their broad band gaps 39 , as a result of which an embedded cavity is optimally shielded. In addition, a broad photonic band gap offers robustness to unavoidable disorder and to inadvertent fabrication deviations 40,41 .
Among the diamond-like crystals, the inverse woodpile stand out because they are relatively straightforward to fabricate by etching two perpendicular arrays of pores in a high-refractive index material such as silicon [42][43][44][45] , as illustrated in Figure 1. Recent work on silicon inverse woodpile crystals has demonstrated the experimental signature of a broad 3D photonic band gap in reflectivity 46 , and a strong inhibition of spontaneous emission of embedded quantum emitters 18 Figure 1 illustrates the structure of an inverse woodpile photonic crystal. The orthorhombic lattice constants are a and c, and the radius of an unperturbed pore is R. If the ratio of the lattice constants equals a/c = √ 2, the crystal is cubic with a diamond-like symmetry 38 . When the pore radius is tuned to R/a = 0.24, the 3D photonic band gap has a very broad bandwidth as shown in Fig. 2(A), with a relative bandwidth ∆ω/ω c = 25.3%, with ∆ω the frequency width of the band gap, and ω c its center frequency 41,42 .

II. STRUCTURE OF CRYSTAL AND POINT DEFECT
We define three-dimensional cavities in these inverse woodpile crystals by introducing a point defect in the bulk of the crystal consisting of two intersecting perpendicular defect pores with a radius R that differs from the bulk pore radius R. A visualization of the defect is shown in Fig. 1. This intersection is the position where we expect the electricfield energy to be localized. We will discuss the confinement of light in such a cavity and explore for which defect radius R optimal confinement is achieved, quantified by a minimal  mode volume V mode for the cavity resonances. The benefits of the cavity proposed here are twofold: first, the required nanostructures can be realized with existing CMOS-compatible silicon nanofabrication techniques 44,45 , and second, no post-production steps are required to obtain a single cavity or even an array of cavities.

III. CALCULATION METHOD
We have used the well-known MIT photonic bands package to calculate the photonic bandstructures (frequency ω versus wave vector k) and the spatial electric-field energy density ε|E| 2 ( r) distributions using the plane-wave expansion 47 . To define an inverse woodpile crystal, an orthorhombic unit cell is used, as shown in Fig. 1. Throughout this paper, the dielectric constant is taken to be ε Si = 12.1, typical for silicon. More details on plane-wave calculations on inverse woodpiles are given in Ref. 41 , and in appendix A we discuss the resolution of the present calculations.
To introduce a point defect as a cavity and increase its surrounding unperturbed volume, we define the crystal by means of a supercell 48 . Since a plane-wave expansion assumes the structure under study to be infinitely extended, the supercell is replicated infinitely in all three dimensions. Since the supercell under consideration has no surrounding vacuum, a limitation of the method is that a cavity quality factor cannot be calculated precisely.
To verify that the supercell method yields correct results, we have compared the results for a 3 × 3 × 3 supercell on a perfect crystal without point defect to the results obtained with a conventional single unit cell 41,42 . We found that these calculations agree well, see appendix A, thus validating the supercell method.    It is remarkable that the appearance of donor resonances in Figure 6(A) occurs at a considerable threshold in defect radius. Such a threshold is attribute to the fact that a certain minimum dielectric volume is required to sustain a standing wave in 3D. As a result, the resonances appear at frequencies deep into the gap, as "deep donors". This behavior of donors in inverse woodpile crystals differs markedly from the occurrence of "shallow" donor resonances in direct woodpile crystals 29 , and in fcc crystals with non-spherical atoms 19 . We surmise that the difference is a result of the field distribution of the cavity resonances. In section IV above, we have seen that in the inverse woodpile cavity the field maximum appears on the sharp corners of the dielectric. Such sharp corners only appear when the defect radius differs considerably from the unperturbed radius, corresponding to a large detuning from the upper band edge, hence a "deep donor". In contrast, the cavity field distributions for the structures in Refs. 19,29 are nearly completely localized in the additional high-index material.
Therefore, the cavity resonances appear for a smaller volume of additional material, and thus at smaller detuning from the upper band edge, corresponding to "shallow donors".
In Figure 6(B) the normalized mode volume V mode /λ 3 i is presented, determined as described in appendix B. For an optical cavity a small mode volume is desirable for strong confinement of light 1,2 . The smallest mode volumes are observed for resonances 3, 4, and 5, with volumes of about V mode = λ 3 i at R /a = 0.12. At R /a = 0.11, resonance 5 even has a mode volume as small as V mode = 0.8λ 3 i . We note that resonance 3 appears to be of particular interest since it is isolated from the other resonances by the largest frequency gap.
If we combine this frequency isolation with the good confinement in real space as gauged by the small mode volume, we conclude that a defect radius of R /a = 0.12 is optimal, corresponding to an optimal defect pore radius R = 0.5R. In the next section we investigate the properties of the resonances at this optimal condition in more detail by intensive computations on a large 5 × 5 × 5 supercell.
To investigate whether inverse woodpile crystals also sustain "acceptor resonances", we have performed calculations for defect pores with radii larger than the unperturbed pores (R > R). form avoided crossings, it is likely that the field profiles of each band are orthogonal. We consider the hypothesis unlikely that each band would form a separate localized resonance, since each band varies considerably in frequency. We conclude that at large defect pore radii (R > R) no acceptor resonances appear, in contrast to the well-confined donor resonances for small defect pore radii.

VI. RESONANCE CHOICE AND PRACTICAL REALIZATION
Using a 5×5×5 supercell, extensive calculations were performed on the optimal cavity condition identified in the previous section. The bandstructure is shown in Figure 8 for  and 5 are significantly greater. When combining these figures of merit for mode volume and frequency isolation, we conclude that resonance 3 has the best potential to confine light and to be selectively addressed.
We propose to pursue the fabrication of inverse woodpiles with embedded optical cavities by modifying our existing CMOS compatible manufacturing techniques. In our realized silicon photonic crystals with typical structural properties (lattice spacings a = 680 nm, c = 492 nm, and R a = 0.24) the frequency of the target resonance corresponds to a wavelength near 1270 nm. This wavelength is in the telecommunication O-band, which makes these photonic band gap cavities relevant for applications. By slightly tuning the lattice parameters a and c, and the radii R andR , cavities can be made that operate in the C-band near 1550nm. Furthermore, it is noted that the present cavity design is also relevant for inverse woodpile crystals made from alternative high-index materials, such as GaAs, GaP, or TiO 2 . These materials would even allow the realization of photonic band gap cavities at visible wavelengths.

VII. CONCLUSIONS
We have performed supercell calculations on inverse woodpiles which contain two intersecting pores that have a different radius compared to the other pores in the crystal.
Our calculations show that isolated flat bands appear in the photonic band gap for defect radii smaller than the bulk radius, corresponding to donor levels in the band gap. We have shown that the electric-field energy is concentrated about the center of the point defect, characteristic of a resonant optical cavity. Despite the presence of the two defect pores, there are no preferential pathways for leaking of light of the cavity along each separate defect pore.
We have investigated five cavity resonances and found that the 3rd resonance at reduced frequency (ωa/2πc) = 0.534 is most promising for confinement. We report a smallest mode volume of around 0.8 cubic wavelengths, typical of a strong spatial confinement of light. By varying the radius of the defect pores, we have determined that a defect radius of R ' a = 0.12 gives the most optimal light confinement. Finally we have discussed a practical method to realize the 3D photonic band gap cavities proposed here. resolution is taken for each constituent cell. Therefore we had to reduce the grid resolution to keep calculation times tractable. Even then, the resulting computation time for the 5×5×5 supercell calculation was multiple months.
In Figure 10 a bandstructure is shown for a perfect inverse woodpile photonic crystal calculated with a resolution limited to 12 × 17 × 12. The band gap is indicated by the red bar. The band edges and the width of the band gap compare well to those found in our earlier calculations 41 with a higher grid resolution of 68 × 96 × 68, see Table I. Consequently we conclude that the results with a lower grid resolution are sufficiently accurate for the present study.
In Figure 11 the calculated bandstructure is shown of a perfect inverse woodpile photonic crystal that was defined by a 3×3×3 supercell. The geometrical properties of the crystal are equal to those used in the conventional 1×1×1 calculations. The band gap edges and band gap width found in this calculation are in excellent agreement with the conventional 1×1×1 calculation, see Table I. Because a supercell is applied, band folding occurs which causes the bands that define the band gap edges to appear flatter.

Appendix B: Calculation of the mode volume
From the calculated electric-field energy density distributions we have determined mode volumes V mode using: with W ijk the electric-field energy density in each grid-element with sizes ∆x, ∆y, and ∆z along each axis, respectively, and W max the maximum of the electric-field energy density   A band gap with a relative width ∆ω ω = 23.3% is found between reduced frequencies ω = 0.512 and 0.646. For ease of interpretation, all allowed frequencies up to the band gap edges are depicted in grey.