1. Introduction

A three-dimensional (3D) photonic crystal can have a complete photonic band gap (PBG) [1, 2], a frequency range in which light is forbidden to propagate in all directions. This unique feature enables three-dimensional confinement of light in a small space. While the mode volume (V) is kept to be small, the quality factor (Q) can be infinitely large for a passive, infinitely-large photonic crystal nano-resonator in a complete photonic band gap. Therefore, actual Q factors would be limited by other loss mechanisms, including the material absorption and the fabrication inaccuracy. In the photonic crystal community, the Q factor exceeding one million is often recognized as an ultra-high-Q while 100 millions or greater is typically considered to be an ultra-high-Q in the microcavity community. Ultra-high-Q nano-resonator modes [3, 4] would be useful for many applications, including cavity quantum electrodynamics (QED) [5], nonlinear optics [6–8] and thresholdless lasers [9]. In a strong-coupling cavity-QED system [10, 11], large ratios os g/κ∝Q/√V and g/γ∝1/√V are desired for improving the quantum coherence, where g is the emitter-cavity field coupling rate, and κ and γ are decay rates of cavity and emitter. Vacuum Rabi splitting was observed between a single quantum dot (SQD) and a compact resonator [5, 12, 13]. The observed g was as high as 20.6 GHz for strong coupling experiments [5]. We can expect to reduce the threshold of Raman lasing [7] and optical parametric oscillation [8] as well as the optical switching energy [6], by a factor of V/Q ². A two-dimensional (2D) photonic crystal slab cavity [14] confines light by two-dimensional Bragg reflection in plane and by total internal reflection in perpendicular direction to the slab surfaces. Due to an escaping light cone determined by a slab-air interface, the Q factor is limited and saturates even for a structure with an infinite number of layers. Many designs [3, 4, 15–18] have been reported to suppress the power loss in vertical direction, and the Q factor of 7×10⁷ has been reported for V~12 (λ₀/2n)³ in their modeling [3], where λ₀ and n are the wavelength of light in vacuum and the refractive index in a maximum field position. However, the Q factor increases at a price of a larger mode volume [18]. In order to increase the figure of merit, we often compromise the mode volume. Zhang et al [17] obtained the Q factor of 10⁵ for 2.3 (λ₀/2n)³ mode volume. It is still a challenge to build ultra-high-Q resonators as a mode volume approaches (λ₀/2n)³. A complete photonic band gap may resolve this challenge. Three-dimensional photonic crystal nano-cavity modes may be constructed in a complete photonic band gap, so ultra-high Q factors can be obtained even at a small mode volume. The Q factor might not become a factor limiting the figure of merit for each application. In addition, the investigation of light localization in a complete photonic band gap would be insightful for understanding the fundamental limits in light localization.

With the progress in 3D photonic crystal fabrication technology, 3D photonic crystals have been realized at optical wavelength [19, 20], and the spontaneous emission control has also been demonstrated [21] in a 3D photonic crystal. The past 3D cavity designs [22–24] are made by introducing a three-dimensionally-localized disorder. In this work, we construct ultra-high-Q 3D photonic crystal nano-resonators consisting of straight rods only. We confirm that ultra-high Q factors can be obtained at a small mode volume, and they will not saturate in the range analyzed. In 2D photonic crystal slab geometries, the double-heterostructure designs achieved the Q factors exceeding one million, and this concept is applied to construct 3D photonic crystal nano-cavities in this work. We use a layered woodpile structure [25]. A line defect waveguide [26] provides large disorder of lattice, and waveguide modes are formed in the complete band gap. The unit cell size is modulated along the line defect in order to excite 3D localized modes in a waveguide mode gap, which is located in the complete band gap. As a result, a compact cavity mode can be formed. Here, we show how to construct ultra-high-Q 3D photonic crystal nano-resonators in a complete photonic band gap by using the modulation of unit cell size.

2. Method

Sub-wavelength-scale optical waveguides and resonators are modeled by 3D plane-wave expansion (PWE) [27] and 3D finite-difference time-domain (FDTD) methods [28]. The dispersion relation is calculated by the PWE method. A waveguide-supercell analyzed has 5×1×5 unit cells, and 8.1×10⁵ plane waves are used for dispersion calculation. Three-dimensional FDTD is then used to calculate resonant frequencies and Q factors, and to visualize field profiles of the cavity modes. Mode volumes are calculated by the definition given n by V=∫∫∫ε(r̄)|E(r̄)|² dV/max[ε(r̄)|E|²]. The perfect matching layer (PML) boundary condition [29] is applied to walls of a computational space for our resonator modeling. Three-dimensional FDTD method with Bloch boundary condition is also utilized to calculate the waveguide propagation loss. The waveguide-supercell size is N×1×N unit cells, where N is an integer number. In all calculations, the resolution is 30 points per lattice constant, corresponding to approximately 75 points per wavelength.

The dielectric rods that constitute the woodpile cavity have a refractive index of 3.4, corresponding to that for GaAs in optical communication wavelength. In one unit cell, both w/a and h/a are set to 0.3, where w is the width of each rod in xy plane, h is the height in z direction, and a is the center-to-center distance of the rods in the same layer, also known as a unit cell length in x and y directions. The unit cell volume is a×a×a_z, where a_z=4h is the unit cell length in z direction. The frequency in this work is represented by a normalized frequency a/λ₀. For a woodpile unit cell with the parameters above, the complete photonic band gap ranges from 0.3526 to 0.4252. N_x, N_y and N_z stand for the total number of unit cells in each direction.

3. Results and discussion

A three-dimensional photonic crystal waveguide [26] used to construct compact resonators is formed by increasing the width of one rod in a woodpile structure, as shown in Fig. 1(a). The width of the defect rod is wx, and y direction is defined as the propagation direction. Fig. 1(b) illustrates the dispersion relation of each guiding mode inside the complete photonic band gap for wx/a=0.6. Note that there is a mode gap region in which no modes exist between the forth (mode-b) and the fifth lowest mode (mode-c) within the bandgap. In this frequency range, light propagation is not allowed in any direction. There is another large mode gap below the mode-a. The width and position of the mode gaps change when the width of the line defect is varied, and our modeling results are summarized in Fig. 1(c). The mode gaps also shift when the unit cell is distorted or the rod width is varied along the light propagation direction. In order to construct resonators, we use two types of waveguides with different unit cell length or rod width in waveguide direction, but both have the same unit cells in xz directions. The unit cell sizes are a×a×a_z in waveguide I, and a×a ^′×a_z in waveguide II. A double heterostructure is constructed by sandwiching waveguide II with two waveguide I geometries, and the central waveguide II has only one unit cell in y direction as shown in Fig. 2. The lattice matching condition is satisfied because the unit cells in these two types of waveguides are same in xz plane. If a guiding mode of the waveguide II is inside a mode gap of the waveguide I, the mode defined by the waveguide II cannot propagate into waveguide I region, and thus a resonator is formed in a complete photonic band gap. The waveguide II is seen as a core region and the waveguide I is considered as a cladding region.

 

Fig. 1. Woodpile 3D photonic crystal waveguide, dispersion relations, mode gap ranges and the waveguide Q factor. (a) A waveguide formed by a wider dielectric rod along y direction. One unit cell in y direction is shown, and Block boundary condition is applied in calculation. (b) The dispersion relations for the w_x/a=0.6 waveguide. Note that there are two mode gaps: the mode gap A∈(0.3576, 0.3834) and the mode gap B∈(0.4028, 0.4089) in a dimensionless frequency unit (a/λ₀). (c) The mode gap frequency ranges as a function of the line-defect waveguide width. (d) The waveguide Q factor as a function of the number of unit cells used to confine each mode. Squares and circles show the Q factor for the mode-c with WG0.7 and that for the mode-a with WG0.6, respectively.

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In order that a mode gap of the cladding region confines a waveguide mode in the core, the unit cell length along y direction is modulated. Using mode-a, -b, or -c, we can consider three cases to bring an optical waveguide mode into a mode gap: (1) pull the mode-a down into the gap A, (2) push mode-b up into the gap B, and (3) pull mode-c down into the gap B. In this work, we show the formation of compact resonators based on the first and third cases. We evaluate the propagation loss by calculating the Q factor of the waveguide mode-a with defect width w_x/a=0.6, and that of the mode-c in a waveguide with w_x/a=0.7, assuming an infinite number of unit cells along the waveguide direction; see Fig. 1(d). The analyzed supercell size is (Na×a×Na_z) for each point, where N is a positive, integer number of unit cells along x and z directions. The Q factor exponentially increases as N increases. This waveguide mode would also be useful as a low-loss optical waveguide. In the analyzed N range, the Q factor of the mode-a is about one order of magnitude higher than that of the mode-c, which means the power losses toward x and z directions are relatively small for the mode-a. The slope of the Q-factor increase is steeper for the w_x/a=0.7 waveguide than that for the w_x/a=0.6 because the mode is close to a midgap point.

 

Fig. 2. 3D photonic crystal nano-resonator geometry formed by sandwiching a waveguide II with two waveguide I structures. The waveguides I and II use the same unit cell length in xz plane.

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First, we use the w_x/a=0.7 waveguide (WG0.7) and pull the mode-c into the mode gap B by increasing the unit cell length and the rod width of the waveguide II along the waveguide direction. We induce a small amount of unit-cell-length modulation in order to preserve the waveguide mode and also observe the transition from loose confinement to tight confinement of light. At first, the unit cell length a ^’ is increased to 1.033a, and the rod width in the core region remains the same. One mode is found at a/λ ₀=0.3928. It has a similar field distribution as in the waveguide (Fig. 3(a)), but due to the mode gap effect, the field decays in the cladding waveguide region; see Fig. 3(b). The resonant frequency is very close to an upper edge of the mode gap B. As a result, the mode can still be coupled to the guided mode in the cladding waveguide, and the field distribution is elongated along y direction. The Q factor does not increase rapidly as the number of unit cells increases.

To improve the confinement of light for this mode, we increase the unit cell size modulation in the core region so that the mode can be pulled into the mode gap further. We choose the center unit cell size a ^′=1.067a, and width w_y ^′=0.333a. This way, a mode with lower resonant frequency a/λ ₀=0.3898 is formed. The mode frequency is located at the middle of the mode gap (a/λ₀=0.3886). The field profile becomes localized in the center region (Fig. 3(c)), and it appears to be unchanged as the increase in the number of unit cells in the cladding regions. The Q value is calculated for each photonic crystal structure size as shown in Fig. 3(d). In order to evaluate the confinement of light in each direction, we calculate the power loss in each direction. For this mode, the confinement of light per one unit cell is weak along the waveguide direction. When the numbers of cladding layers in x and z directions are fixed at N_x=N_z=7, the Q factor first grows exponentially as N_y increases, and then becomes saturated when N_y≥23. The confinement of light in y direction becomes strong and, instead, power losses in x and z directions become dominant. Therefore, in order to increase the Q factor further, we need to increase N_x and N_z. This was confirmed in our modeling. The Q factor saturates around 8486 even with 35 unit cells in y direction if N_x and N_z are not increased, but increases from 6,974 to 31,941 by adding four more unit cells in x direction and two more in z direction. The modeling results also show that the saturated Q factor, for constant N_x and N_z values, is close to the waveguide Q factor for the same N_x and N_z. Waveguide Q factors are calculated by neglecting power losses along light propagation direction. The perturbation in the photonic crystal is so small that the losses toward directions normal to the waveguide geometry are considered to remain unchanged. Thus, the cavity Q factor approaches the waveguide Q factor. This way, we can estimate the maximum Q factor that can be reached for certain N_x and N_z values.

 

Fig. 3. Properties of the cavity modes constructed on a w_x/a=0.7 waveguide, and the comparison of loose and tight confinement. (a) The profile of E_x component in the central xy plane of the waveguide. (b) The profile of E_x component in the central xy plane of the cavity with a ^’=1.033a. (c) The profile of E_x component in the same plane of the cavity with a ^’=1.067a, and w_y ^’=0.333a. The total number of unit cells of these structures is 7×11×7. (d) The development of the Q factor with the number of unit cells in the cavity with a ^’=1.067a, and w_y ^’=0.333a.

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The mode profiles of electrical field magnitude in the waveguide and the designed cavity are shown in Fig. 4 (a and b). Inside the core region of the cavity, the field profile appears to be similar with that of the waveguide. However, the field decays quickly when penetrating into the cladding unit cells because the mode frequency is indeed within a waveguide mode gap in the cladding region. We check 3D vector plots of the electric field, and confirm that this is a quadrupole mode. The vector plots in the central xy and in an xz plane are shown in Fig. 5(a) and Fig. 5(b), respectively. The calculated mode volume is 8.32 (λ₀/2n)³, which is similar or slightly smaller than typical one for 2D photonic crystal slab double-heterostructure nano-cavities [4].

 

Fig. 4. Profiles of the magnitude of electric field in the central xy plane of waveguide or cavity. One unit cell length PML is applied to all walls except ones in the waveguide propagation direction. (a) Mode-c in WG0.7. Supercell size is 7×1×7. (b) The cavity mode formed by inducing waveguide mode-c into the mode gap of WG0.7. a ^’=1.067a, and w_y ^’=0.333a. N_x=N_z=7, N_y=15. (c) Mode-a of WG0.6 in one unit cell in y direction. Supercell size is 7×1×7. (d) The cavity mode formed by inducing waveguide mode-a into mode gap A in WG0.6. a ^’=1.2a, and w_y ^’=0.4a. N_x=N_z=7, N_y=11.

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Fig. 5. Vector plots of electric field on (a) xy plane and (b) xz plane for the designed quadrupole mode (based on WG0.7, a ^’=1.067a, and w_y ^’=0.333a), and ones on (c) xy plane and (d) xz plane for the designed dipole mode (based on WG0.6, a ^’=1.2a, and w_y ^’=0.4a).

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Next, we use the mode-a, which can be pulled down into the mode gap A by increasing the unit cell length and rod width of the waveguide region II in y direction. There are two advantages of using this mode. First, the mode gap A is wider than the mode gap B, so we are able to pull down the mode deeper to the middle of the gap, and the coupling of the cavity mode into the cladding waveguide would be small. Thus, the power loss in y direction is expected to be reduced for the same photonic crystal unit cells number in y direction with that for the quadrupole mode. Secondly, the waveguide mode-a has much higher Q factors than others, indicating small power losses in x and z directions. As a result, we would not need a large photonic crystal to obtain an ultra-high Q factor.

 

Fig. 6. The properties of the cavity mode based on the waveguide of w_x/a=0.6. In the core region, a ^’=1.2a, and w_y ^’=0.4a. (a) The E_x distribution in the central xy plane of the cavity. N_x=N_z=7, N_y=11. (b) The Q factor for different unit cell numbers. The Q factor increases exponentially with N, and no saturation is observed.

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In this design, the waveguide with w_x/a=0.6 is used, and the parameters in the core region are: a ^’/a=1.2, and w_y ^’/a=0.4. The mode frequency is 0.3800. The confinement of light per one unit cell is estimated to be very similar for all directions, so we equally increase the unit cell numbers in all directions (N_x=N_y=N_z=N). The Q factor is calculated for each N; see Fig. 6 (b). It increases exponentially with N, and no sign of saturation is observed. In the modeled range, the Q factor can be approximated by the following equation: Q=10^{1.5214+0.4201N}. We notice that the computational error grows for a structure with a large number of unit cells. The error becomes significant when N is increased to more than 17. The precise evaluation of ultra-high Q factor requires large computational resource. However, we are still able to obtain the Q factor of 2.9×10⁸ for N=17, and the Q factor is confirmed to significantly increase.

The comparison of a mode profile of the waveguide with that of the cavity is shown in Fig. 4 (c and d). As the unit-cell-length modulation becomes large, the field penetration into the cladding unit cells becomes small, and the mode becomes more confined within the core region. The vector plots of this mode in the central xy and xz planes are shown in Fig. 5 (c and d), and it is confirmed to be a dipole mode. As a result, this mode has a small mode volume, and the mode volume is estimated to be 2.88 (λ₀/2n)³.

 

Fig. 7. The resonator Q factor and mode volume for a different amount of unit cell size modulation. The waveguide has a line defect w_x/a=0.5, and the parameters (a ^’, w_y ^’) in the core region are shown on the top axis. The unit cell size of these structures is 9×9×9.

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We then study the relationship of Q factor and mode volume with the extent of the modulation. The waveguide with w_x/a=0.5 is used because it has the largest mode gap A, which ranges from 0.3576 to 0.4035, in the defect width range analyzed. As the unit cell modulation increases gradually, the resonant frequency decreases as expected (Fig. 7). The Q factor is high for the modes located deep inside the mode gap, and decreases quickly as the mode frequency become close to one of mode gap edges. Modes formed by small modulation, as explained above, can still be coupled to the cladding waveguide, so the loss along the waveguide direction is dominant. The mode is elongated, and the mode volume is large as we can see for the mode with resonant frequency of 0.4028 in Fig. 7. As the mode frequency decreases, the coupling to the cladding waveguide also decreases, and the mode becomes well confined in a small mode volume. For large modulation, the mode is near the lower edge of the mode gap, the Q factor deceases for three possible reasons. First, the coupling to the lower propagation bands in the cladding waveguide might increase. Secondly, the mode is also close to the lower edge of the complete band gap, so the loss in the vertical direction of the waveguide might increase. Thirdly, the unit cells in the core region are significantly distorted, so the losses into the x and z directions in the core region might also grow. Except for the highest frequency mode, all the other modes have very similar mode volume, which is about 3 (λ₀/2n)³. The volume of higher Q factor modes is slightly larger than that of the modes with lower Q factor.

Two ultra-high-Q nano-resonator modes are designed. The one with 2.88 (λ₀/2n)³ mode volume is pulled down into a middle of a wide mode gap so that it has a steeper slope of the Q factor per one unit cell increment than the other mode. Our study shows that, for the double heterostructure cavity design, the properties of waveguide modes and the mode gap value appear to be determinant factors. It appears to be essential to study low-loss 3D photonic crystal waveguides with a large mode gap inside a complete photonic band gap for tight 3D confinement of light. The designed resonator geometries are unique in that only straight rods are needed. We are working on the fabrication and testing to evaluate the Q factors.

4. Conclusion

In this work, we show that a complete photonic band gap is a possible solution of resolving the challenge of building ultra-high-Q nano-resonators with the mode volume close to the fundamental mode. Three-dimensional photonic crystal nanocavities based on unit cell modulation along an optical waveguide are modeled. The calculated Q factors and electromagnetic field profiles indicate that the properties of the cavity modes strongly depend upon original waveguide modes and mode-gap differences between the core and cladding regions. Compact, ultra-high-Q modes are confirmed in our calculation. The modeling results confirm that the Q factor increases exponentially with the number of unit cells in the range analyzed. In the cavity mode induced from a low-loss waveguide mode into the lower wide mode gap in the waveguide with defect width w_x/a=0.6, ultra-high Q factors are obtained with a small photonic crystal volume, and the mode volume is as small as 2.88 (λ₀/2n)³. Photonic crystal volumes required for the Q of one million are approximately 11³a²a_z=1331a2az and 11×35×9a²a_z=3465a²a_z for the designed dipole and quadrupole modes, respectively, where a2az is one unit cell volume of the cladding region. As expected, 3D photonic crystal cavities formed in a complete band gap are extremely low-loss even for a small mode-volume mode. These designs would be useful for improving the performance in many applications such as the improvement of the quantum coherence in cavity-QED experiments, the reduction of optical switching energy, and the reduction of threshold in lasers such as Raman lasers. For example, the interaction rate for a strong-coupling cavity-QED system would still be increased, and the photon decay rate could be minimized.