Strongly Confining Light with Air-Mode Cavities in Inverse Rod-Connected Diamond Photonic Crystals

Abstract

Three-dimensional dielectric optical crystals with a high index show a complete photonic bandgap (PBG), blocking light propagation in all directions. We show that this bandgap can be used to trap light in low-index defect cavities, leading to strongly enhanced local fields. We compute the band structure and optimize the bandgap of an inverse 3D rod-connected diamond (RCD) structure, using the plane-wave expansion (PWE) method. Selecting a structure with wide bandgap parameters, we then add air defects at the center of one of the high-index rods of the crystal and study the resulting cavity modes by exciting them with a broadband dipole source, using the finite-difference time-domain (FDTD) method. Various defect shapes were studied and showed extremely small normalized mode volumes (V_eff) with long cavity storage times (quality factor Q). For an air-filled spherical cavity of radius 0.1 unit-cell, a record small-cavity mode volume of V_eff~2.2 × 10⁻³ cubic wavelengths was obtained with Q~3.5 × 10⁶.

1. Introduction

Three-dimensional (3D) photonic crystal microcavities, which are known to have high-quality factors (high-Q) and ultra-small mode volumes (V_eff), provide a novel way of trapping light (photons) [1,2,3]. Such 3D structures would also allow the observation of spontaneous emission modification (via the Purcell effect) [4], as well as the investigation of the strong coupling [5] of a small number of quantum emitters (such as quantum dots or diamond NV-color centers) in the cavity mode. The enhancement and suppression of spontaneous emission by cavities are useful for single-photon sources, for quantum information processing, while strong coupling allows the creation of all-optical switches, quantum logic gates, and quantum memories using spin-photon entanglement [6]. Furthermore, a spin-photon entangler is a fundamental quantum gate, allowing the preparation of multiple qubits (photons and spins) in complex entangled states (cluster states) that allows scalable quantum computing [7].

Most previous research on photonic crystal cavities is based on two-dimensional (2D) photonic crystals for sensors [8,9], while significant work on cavity field enhancement has been performed with a view to coupling with single emitters [10]. In this and previous works, we have focused our research on 3D photonic crystals, where we have shown partial photonic bandgaps (PBGs) in polymer photonic crystals that were fabricated using direct laser writing, exploiting two-photon polymerization (2PP)-based 3D lithography [11]. More recently, by backfilling polymer crystals with higher-index chalcogenide materials, a complete PBG has been demonstrated in technologically relevant wavelength regions (1.4–1.6 µm) [12]. This direct-writing templating technique is ideal for fabricating arbitrary 3D structures and could allow the selective writing of defects containing fluorescent material at the antinodes of cavities, i.e., the infiltration of the structure with liquid-containing quantum emitters, such as PbS colloidal quantum dots [13], single-walled carbon nanotubes (CNTs) [14], or coating with 2D materials [15,16]. Our simulation work has, thus, focused on simulating ultra-small high-index cavities by creating point defects in these crystal structures [17,18,19,20,21]. However, to get quantum emitters into such nanoscale cavities or to use such systems as nanoscale sensors will require the cavity to be a low-index air/vacuum or liquid.

Thus, in this paper, we investigate low-index cavities for high-efficiency coupling to quantum emitters [13,14,15], suspended in low-index liquids or vacuum. Specifically, we look at air-mode nanocavities [17,22,23,24] in RCD 3D photonic crystals [25,26], which exhibit the largest full PBG known to date [27,28,29]. We first optimize direct RCD and inverse RCD [19] structures for maximum PBG, using plane-wave expansion (PWE) [30] for an index contrast of 3.6:1, as reported before for an index contrast of 3.3:1 by the authors of [19], then introduce air cavities to locally enhance the electric field. The simulations were conducted using the finite-difference time-domain (FDTD) method [31], with varied low-index defect sizes and shapes. Previous work with high-index cavities identified the optimal cavity positions [19] within the unit cell. The cavity resonant wavelength λ_res, quality factor Q, and mode volume V_eff in the defect cavities are then calculated.

2. Numerical Modeling and Calculation Method

For this paper, we studied high-refractive-index contrast (3.6:1 (GaAs or Si-air)) 3D photonic crystals (PhC) with an inverse RCD lattice structure (air-rods in high-index background material). The corresponding cubic unit-cell (of size a_u) is shown in Figure 1, along with the defect position. We optimized the relative air-rod radius (r/a_u) of the inverse RCD structures to maximize their full PBG for an index contrast of n_bg:n_c = 3.6:1, by evaluating the relative gap width between bands 2 and 3 as a function of the normalized rod radius, r/a_u. The results are shown in Figure 2. An optimal radius of r_IRCD = 0.27a_u was found, with a corresponding maximum PBG of Δω/ω₀~28.71%. This radius is similar to, although slightly larger than, the one found previously (r_IRCD/a_u = 0.26) for an index contrast of 3.3:1 [19]. Figure 3 shows the band structure corresponding to the optimal radius. The maximum PBG goes from a_u/λ~0.51 to 0.69 with a mid-gap frequency of a_u/λ~0.6. The optimal radius of the air-rod, r, is about 2.7 times larger than that in the direct RCD scenario, making manufacturing more feasible [19].

Here, the finite inverse RCD structures used for the FDTD simulations were created by truncating an infinite crystal, using a cube with a size of 10a_u × 10a_u × 10a_u, centered on the defect, as previously performed by [19] for high-index defects. The grey axes shown in Figure 1 correspond to the basis used in the FDTD simulations, which was chosen so that the simulation Z-axis (Z_s) is along the [1,1,1] axis of the conventional cubic unit-cell, aligned with the inverse rod going through the defects [19]. All the simulations used the same inverse RCD crystal, with cylinders of radius r = 0.27a_u and with refractive indices of n_c = 1.0 for the cylinders and n_bg = 3.6 for the background (see Figure 1). The dimensions of the defects and the various shapes are detailed in Figure 4. Five different defect types were considered: (a) a sphere of radius, r_d, (b) a cylinder (“cylinder A”) of fixed height L = $\sqrt{3}{a}_u/4$, with a circular base of diameter D = 2r_d, (c) a cylinder (“cylinder B”) of height L_cylB = 2r_d equal to its diameter D = 2r_d, (d) a block of fixed height L = $\sqrt{3}{a}_u/4,$ with a square base of side-length W = 2r_d, and (e) a cube of side-length W_cube = L_cube = 2_rd. A wide range of defect forms has been chosen to investigate the sensitivity of the resonance modes to the cavity shape, which will be affected by fabrication limitations. For the defects “sphere”, “cylinder B” and “cube”, the size along the axis Z_s keeps increasing, while for “cylinder B” and “block”, it is restricted, leading to a maximum r_d value past which there are no more effects (until the defect starts cutting into other inverse RCD rods).

All defects have a refractive index of air n_def = 1.0. For each defect type, the FDTD simulations were run using a broadband dipole source, which was placed in the optimized location, S₂.₀ [19], within the crystal along the Γ-L or [1,1,1] axis. While the refractive index value used in [19] was n_bg = 3.3, instead of the n_bg = 3.6 used here, we assume that the field distributions would be similar for equivalent geometries and, therefore, that S₂.₀ would still be an optimal defect location for the background index used here. The dipole was excited along the [1,1,1] axis (Z_s direction).

3. Results

After calculating the amplitude of the electric field over time for an inverse RCD with these defects, the Q-factors (Q = λ_res/Δλ) can then be estimated by analyzing the resulting field decay in the frequency domain, via the fast Fourier transform (FFT) algorithm. Any perturbation disrupting the translational symmetry can act as a defect. Hence, precise tuning of the cavity resonances can be achieved by varying the amount of perturbation, through the modification of defect shapes and sizes in the 3D PhCs. We varied the size of r_d from 0.1a_u to 0.15a_u in 0.0125a_u steps and 0.15a_u to 0.5a_u in 0.025a_u steps. Figure 5 shows the normalized frequency of resonance peaks (a_u/λ) as a function of the normalized size (r_d/a_u), within a full PBG between a_u/λ~0.51 and 0.69 (mid-gap frequency a_u/λ~0.6). The normalized resonance frequency (a_u/λ) of the defect cavities increases with the defect size, except in the case of the cylinder and rectangular block defect cavities.

The corresponding Q-factors obtained for the different defect types are shown in Figure 6. They decrease with increasing defect size. Overall, the Q-factors of the various defect types behave similarly.

The appropriate selection of the shape and size of defects is critical for optimizing the confinement of the electric field within the cavities. Having determined the resonant frequency, a cavity mode on resonance can be visualized using a single frequency snapshot. The resulting normalized square of the electric field distribution (|E|²/|E|²_max) for each defect type is illustrated in Figure 7, in the case of r_d = 0.1a_u.

The isosurfaces of the dielectric material and the normalized square of the electric field distributions (using an isovalue of |E|²/|E|²_max = 0.03) around the defect regions are shown in Figure 7a,c,e, while the corresponding 1D cross-section plots along the X_s, Y_s, Z_s axes, going through the defect centers, are shown in Figure 7b,d,f. As the truncated shape of the cavity varies, the |E|²/|E|²_max no longer resembles the original electric field mode (i.e., Gaussian modulated sinusoidal mode) with a discontinuity, but instead becomes more confined to either edge of the high-index material.

Observation of modified spontaneous emission and non-linear optical effects in microcavities depend on their high-quality factors (Q) and small mode volumes (V_eff), that is, a high ratio of Q/V_eff. However, in most optical systems, there is a tradeoff between achieving a higher Q and reducing the size of the cavity mode volume. Here, we compare the mode volumes for different sizes and shapes of defect cavities within the finite inverse RCD structures. An effective mode volume (V_eff) of the cavity modes can be calculated from FDTD simulation results, via the definition of the effective mode volume, V_eff, and the dimensionless normalized mode volume, f_opt [32,33,34]:

$$V_{eff}=\frac{{{\displaystyle \iiint }}^{ }\epsilon \left(r\right){\left|E\left(r\right)\right|}^2{d}^{3}r}{\epsilon \left(r_{\max }\right){\left[{\left|E\left(r\right)\right|}^2\right]}_{\max }}$$

(1)

$$f_{opt}=\frac{V_{eff}}{{\left[\lambda /n\left(r_{\max }\right)\right]}^3}$$

(2)

where r_max is the position of the maximum electric-field amplitude. Hence, the mode volume can be minimized by increasing the mode maximum electric field and localizing the mode maximum in the low index region.

The results obtained using Equations (1) and (2) are shown in Figure 8. The mode volumes of sphere defects are smaller than in any other shape. In general, the mode volumes decrease with decreasing defect sizes, as expected, except for some small reverses at r_d = 0.1375a_u and 0.275a_u. For an air sphere (r_d = 0.1a_u), a mode volume down to V_eff~0.0022(λ/n)³ and a Q-factor up to ~3.5 × 10⁶ can be achieved (normalized resonance frequency a_u/λ~0.541). Moreover, because the bandgap is omnidirectional, it is possible to further increase the Q-factor by simply increasing the number of periods in each direction, while maintaining a small mode volume, V_eff.

4. Discussion and Conclusions

For this research, inverse RCD PhCs formed in high-index-contrast materials (3.6:1.0 (GaAs:air)) were considered. A maximum PBG of ~28.71% at (r/a_u)_opt = 0.27 was found, using the PWE method. Various defect cavities of varying sizes and shapes, placed in an optimized location within the inverse RCD structures, were studied. The Q-factors and mode volumes (V_eff) were calculated, using the FDTD method. In this paper, we report that an air sphere defect (r_d = 0.1a_u) gives the best result, with a mode volume V_eff~0.0022 (λ_res/n_air)³ and Q~3.5 × 10⁶, which corresponds to a Q/V_eff ratio ~1.59 × 10⁹ (λ_res/n_air)⁻³ with a resonance at a_u/λ~0.541. To our knowledge, this is a record-low mode volume for defect cavities in 3D photonic crystal structures. This is, at first sight, a surprising result, given the physical volume of these defects. However, the small mode volume reflects the highly peaked fields close to the high-index cavity edges. Better control of the peak field could be obtained by engineering spikes of high-index material close to the cavity center, similar to recent 2D “bow-tie” cavity designs [10].

Such high-Q cavities with ultra-small mode volume could demonstrate a universal mechanism for broad bandwidth, lossless, wavelength-scale optical circuits in a fully 3D photonic crystal microchip. Additionally, these microchips could allow the development of ultrasensitive sensing chips (guiding and confining light in air or low refractive-index materials) [35] and find applications in solar energy trapping and harvesting [36].