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Two types of ultra-high-Q monopole modes are designed in a woodpile three-dimensional photonic crystal. The unit cell size modulation is applied to a woodpile photonic crystal waveguide in a complete photonic band gap. A monopole mode overlapping with a dielectric rod is designed for solid-state sub-wavelength-scale light-matter interaction devices such as nanolasers, cavity-QED and optical switches, whereas another type of monopole mode overlapping with vacuum is designed for optical trapping experiments. For the mode overlapping with vacuum, the mode volume is as small as 0.4 cubic half-wavelengths.
©2009 Optical Society of America
1. Introduction
A photonic crystal [1, 2] is a major focus of localizing light in sub-wavelength-dimensions. The most popular photonic crystal system has been a two-dimensional (2D) photonic crystal slab cavity [3], thanks to the development of simple, high quality (Q) factor resonator designs [4-7] since the demonstration of photonic crystal laser in 1999 [8]. Vacuum Rabi splitting was demonstrated on a single quantum dot coupled to a single-mode 2D photonic crystal microresonator [9]. The observed interaction rate g ∝ 1/V1/2 was as high as 20.6 GHz for Q~10,000 and V~8 (λ/2n)3, where V, λ and n are the mode volume, the wavelength of light in vacuum and the refractive index in a maximum field position. However, due to the imperfect optical confinement resulting from an escaping light cone, the Q factor is limited even for a structure with an infinite number of layers. The maximum Q factor is dependent upon the mode volume [10]. Thus, it is still a challenge to build ultra-high-Q (UHQ) resonators as the mode volume approaches (λ/2n)3. For example, the design by Zhang et al [11] is one of the best ones; Q~105 and V~2.3 (λ/2n)3. Apparently, there is a technological or fundamental limit in light localization for a 2D photonic crystal slab cavity. One promising solution is to use a complete photonic band gap (PBG) provided by 3D photonic crystal [12-15]. Due to its property of omni-directional confinement of light, the Q factor can increase as an increase in the passive photonic crystal size unless other physical mechanism limits it. Here, we show two types of UHQ monopole mode designs in a woodpile 3D photonic crystal [12]. For the first one designed for light-matter interaction devices, light is localized in a center dielectric rod. The second design offers a monopole mode localized in air or vacuum, and potentially provides ultra-compact devices trapping atoms, ions and particles by optical fields.
2. Monopole mode overlapping with dielectrics
Fig. 1. Design of woodpile photonic crystal monopole mode overlapping with dielectrics. The unit cells size is 5×5×5.
At first, we show the monopole mode overlapping with dielectrics. As shown in [16], we apply the unit cell size modulation to a woodpile photonic crystal waveguide. The rod width and height are w and h. The waveguide is introduced by adding a rod of width wx along the y axis. Then, central unit cells representing the core region are widened along the waveguide (y) direction so that the size of the central unit cells becomes Ma × a’ × Maz, where a and az are unit cell length in x (y) and z directions, respectively, and M is a positive integer. The width of rods along y direction is also increased to wy. This core region is sandwiched by two cladding woodpile regions whose size is Ma × ma × Maz, where m is a positive integer. We choose M = 2m + 1 in the calculation. The structure is shown in Fig. 1; m = 2, M = 5, w = 0.3a, h = 0.3a, a’ = 1.2a, wx = 0.2a, and wy = 0.5a.
Fig. 2. Electric field energy distribution of monopole mode overlapping with dielectrics on xy, yz, and zx planes intersecting the middle point of the structure with m=2.
Three-dimensional finite-difference time-domain (FDTD) method [17] is used to analyze monopole modes in the structure. The mesh resolution is twenty. The indices of refraction for the dielectric rods and background media are 3.4 and 1.0, respectively. Even mirror symmetry condition is applied to x = 0 and y = 0 planes, and monopole modes can easily be identified. Figure 2 shows some electric field energy profiles of the monopole mode, which has the maximum field in the central dielectric rod. Figure 3 shows our analysis results of the monopole mode. As seen in the panel (a), there is an optimum wy/a range; at wy = 0.6a, the Q factor is maximized, and V is as small as 0.36 (λ/n)3 = 2.88 (λ/2n)3. The normalized resonance frequency is 0.372. The quality factor dependence on the size parameter m is presented in the panel (b). For m = 11 and wx = 0.2a, the Q factor is 900 millions. Therefore, this monopole mode forms an ultra-high-Q resonator.
Fig. 3. Modeling results for the monopole mode overlapping with dielectrics. (a) The Q factor, mode volume and normalized frequency as a function of wy/a for wx = 0.1a and m = 7. (b) The quality factor vs. the size parameter m for some different size variables with the monopole mode overlapping with dielectrics. The inset shows Ez component on the central xy plane.
This design outperforms any 2D photonic crystal resonator designs, and would enhance the interaction rate in a solid-state cavity QED system. The interaction rate for an InGaAs quantum dot would be as high as by replacing the L3 cavity used in [9] with this monopole cavity. At the same time, the photon decay rate can be suppressed. The dephasing rate of a (quasi-)atom can also be suppressed by the use of appropriate ones such as a nitrogen vacancy center in a diamond. This way, the quantum coherence would significantly be improved so that quantum information devices, including single-photon-level switches, quantum repeaters, and single photon sources, may become more practical. The classical nonlinear optics devices also benefit from the developed design. The switching energy is proportional to V/Q2, and atto-joule-level optical switching would be possible.
3. Monopole mode overlapping with vacuum
Next, we show the monopole mode overlapping with air or vacuum. A waveguide is constructed by removing one rod from a woodpile photonic crystal, and the unit cell size modulation is applied to the waveguide. The same monopole mode is found for different extent of modulation. Here, we only show the analysis results for one modulation. Both the lattice constant (a’) and the rod width in the core region (wy’) are widened along the waveguide direction to be 1.1a and 0.35a; see Fig. 4 for some views of the resonator structure. The mode has a normalized frequency of 0.361. The electric field is confirmed to have the maxima overlapping with vacuum. The electric field energy distributions are displayed in Fig. 5(a), and the Q factor dependence on the size parameter m is presented in Fig. 5(b). For m = 13, the Q factor is 65 millions. The mode volume is as small as 0.4 (λ/2)3.
Fig. 4. Design of woodpile photonic crystal monopole mode overlapping with air or vacuum. The unit cells size is 5×5×5. The parameters in the core region are: a’= 1.1a, and wy’ = 0.35a.
Fig. 5. Properties of the monopole mode overlapping with vacuum. (a) Electric field energy distributions on xy, yz, and zx planes intersecting the middle point of the structure with m=3. (b) The Q factor vs. the size parameter m. The inset shows Ez component on the central xy plane.
A majority portion of fields resides in an n = 1 medium. All rods forming the monopole photonic crystal are straight, and we may drop particles from a rod opening on a side of the photonic crystal toward the cavity center. In order to evaluate the optical trapping capability, we calculate the forces induced by the monopole mode field. With the monochromatic field and dipole approximations, the cycle-averaged force [18] is represented by
where the dipole moment, and electric field are
The modal field is assumed to be unchanged due to the existence of a particle. The time-independent electric field Ê(r⃗) can be expressed as a product of the amplitude E 0(r⃗) ∈ ℝ1 and a unit polarization vector n⃗(r⃗) ∈ ℂ3. So that the force can be written as
where the polarizability α = αR + iαI (αR, αI ∈ ℝ). The first and second terms in the right hand side are dipole and scattering forces, respectively. In the analyzed mode, the dipole force is dominant, so only dipole force is considered in the following. This mode has two positions with maximum electric field. The dipole force is directing to these two points, so a passive particle nearby can be trapped around one point; see Fig. 6 as an example of the optical force near one maximum electric field point. We confirmed this force direction three-dimensionally.
Fig. 6. Normalized electric field intensity and dipole force vector plot for monopole mode overlapping with vacuum near a maximum electric field position in the central yz plane. The grey region shows the lower-right corner of the central dielectric rod in the middle panel of Fig.5 (a). There is another trapping point on the lower-left corner of the central rod.
4. Conclusion
In summary, two types of compact monopole modes in a woodpile 3D photonic crystal are designed and analyzed. The quality factor exponentially increases as an increase in the photonic crystal size and can exceed one million, thanks to its complete photonic band gap. The mode overlapping with dielectrics is potentially useful for thresholdless nanolasers, quantum information devices based on a single quantum dot, and classical nonlinear optics devices. On the other hand, the monopole mode overlapping with air or vacuum is potentially applied to optical trapping technology and atom-optics experiments.
Acknowledgements
The authors acknowledge NSF for financial support, TeraGrid for computation support, and Will Kane for some calculations.
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