1. Introduction

There has been great interest in the properties of photonic crystals (PCs), which prohibit the propagation of light with frequencies that lie within a photonic band gap (PBG) [1,2]. In the last twenty years many studies had been reported on the fundamental properties and potential applications of two-dimensional (2D) PCs [3–8], in which light is confined in the plane direction by means of the PBG, and in the perpendicular direction by means of total internal reflection. Meanwhile, three-dimensional (3D) PCs, possessing a complete PBG (cPBG), have also been receiving growing attention for their ability to control light of all polarizations in all directions [1,2]. 3D photonic crystal waveguides (PCWs) are a key optical component for the implementation of highly-integrated 3D optical circuits due to their capability of creating sharp bends with high transmission efficiencies [9,10]. In principle, thanks to the cPBG, 3D PCWs do not suffer from losses due to inter-mode coupling between TE and TM modes [11], nor do they experience leaky-mode region problems that are unavoidable in 2D PC slab waveguides. 3D PCs also offer new possibilities for directly funneling the light output from the 3D PC laser cavity [12,13] and subsequently guiding it through 3D waveguides to other optical components on the same chip with low losses [14–16]. In addition, they are also an important element for incorporating other functionalities such as channel dropping [17] in 3D optical circuits.

In most of the previous reports on 3D PC waveguides [18–22], either simple donor- or acceptor-type line defects, which are created by respectively adding or removing dielectric material to or from a regular 3D PC, were employed as the waveguide. In these types of waveguides, the frequency regions for single-mode propagation within the cPBG are generally limited because several modes that partly overlap exist within the cPBG. For several applications, it is desirable to have a wider available frequency range for single-mode light propagation, which will allow for a more flexible design of 3D PC components and circuits. Indeed, several designs of 3D PCWs for single-mode operation in a wider frequency range had been reported using GaAs as the material [23–25]. In reference [25], a compound waveguide structure consisting of an acceptor-type waveguide sandwiched by two donor-type ones has been examined. With an optimized design, a frequency range for single-mode operation covering ~90% of the cPBG was achieved. However, it is still meaningful to further enlarge the available frequency range for single-mode light guiding in the cPBG. Interestingly, so far there are not many reports on the design of 3D PCWs made of silicon [26,27], which is CMOS-compatible and could be a platform for future optical circuits, utilizing large bandwidth (BW) for single-mode propagation.

Here, we report on the design of silicon 3D PCWs with an enlarged BW for single-mode operation within the cPBG by utilizing two single guided modes. In an optimized structure, the BW covers the entire range of the cPBG. The paper is organized as follows: First we briefly describe the simulation method and calculation conditions in section 2. Then, we report the waveguide design which exhibits an ultra large BW by controlling the separation of modes in section 3. In section 4, several issues on the operation of the designed waveguide will be discussed.

2. Simulation method and calculation conditions

The theoretical calculations were performed by the plane-wave expansion method, using RSoft Photonics Component Design Suite version 9.0 (RSoft Inc.). The supercell method was used. The dimensions of the supercell were 6a, 1a, and 7.2a in x, y and z directions, respectively, where the y direction was defined as the guided direction, and a represents the center-to-center spacing of the rods that constitute the 3D PC with woodpile lattice. The dielectric material of the rods was assumed to be Si, with a refractive index of 3.55. Each rod had a width of 0.24a and a height of 0.3a as denoted in Fig. 1(a) . The number of plane waves used was 2359296. The calculation of the PBG was carried out with the same method for a defect-free woodpile structure. The validity of our method was confirmed by reproducing the results in previous literatures.

 

Fig. 1 (a) Schematic of a 3D PCW composed with three defect layers and outer 3D PC cladding. White, blue and black rods denote the original rods, mid-rod and cross rods, respectively. (b) Dispersion relation of the waveguide modes when W_mid = 0.24a.

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3. Design of single mode waveguides and results

Figure 1(a) shows our starting structure. The waveguide in a 3D woodpile PC is composed of three defect layers and an outer 3D PC cladding. The middle-defect layer has an acceptor-type line defect created by narrowing an original rod to the width W_mid. The upper-defect and lower-defect layers have donor-type line defects created by adding a cross rod just above and below the narrowed rod in the middle-defect layer. The additional-dielectric line defects have the same width and height as the original rods. These three defect rods work as a waveguide along the y direction.

The width of the middle line defect W_mid was varied from 0.24a (the original rod width) to 0a (the mid-rod is completely removed). We note that the cases of W_mid = 0.24a and 0a have already been discussed in [24] and [25], respectively, using GaAs as a constituting material. From the dispersion relation when W_mid = 0.24a shown in Fig. 1(b), the PBG is mainly filled with two guided modes designated as modes A and B. In this case, mode B does not contribute to the single-mode operation of the waveguide, but instead diminishes the BW due to the crossing of the two modes. In order to circumvent this issue, the width of the central rod, W_mid, was reduced. As W_mid is decreased, all the guided modes move upward to higher frequencies due to a decrease in the effective refractive index of the waveguide. However, modes A and B move differently as W_mid changes. This difference can be understood by looking at the electric field distribution of these two modes (shown in Fig. 2 when W_mid = 0.08a as an example). As mode A is more confined in the mid-rod region, it is more sensitive to the rod width than mode B. Consequently, these two modes tend to separate gradually with decreasing W_mid, while always being connected at wave vector k = π/a. This behavior can be observed in Fig. 3(a) . When W_mid = 0.12a, the overlapping frequency region of these two modes is much smaller than the case W_mid = 0.24a. This enables one to use a part of mode B for expanding the frequency range of single-mode propagation.

 

Fig. 2 Vertical cross-section, orthogonal to the guided direction, of E_z of modes A and B when W_mid = 0.08a.

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Fig. 3 (a) Dispersion relation of the waveguide modes when W_mid = 0.12a. (b) Relation between single-mode operation occupancy of PBG and the width of the mid-rod. (c) Dispersion relation of the waveguide modes when W_mid = 0.08a (optimized structure). (d) Dispersion relation of the waveguide modes when W_mid = 0.02a.

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In order to quantitatively discuss the effect of changing W_mid, we introduce the total BW for single-mode propagation, which is the summation of the BWs for mode A and B (BW_A + BW_B). BW_A and BW_B are defined as shown in Fig. 3(a). The PBG occupancy ratio of the total BW for single-mode propagation (BW_A + BW_B)/PBG as a function of W_mid is shown in Fig. 3(b). As W_mid is decreased from 0.24a, the PBG occupancy ratio is increased due to the separation of the two modes, reaching the highest value of 98.7% when W_mid = 0.06a and 0.08a. The dispersion diagram for W_mid = 0.08a is shown in Fig. 3(c). This result suggests that almost full frequency region of PBG can be utilized for efficient light propagation usingtwo guided modes A and B. The optimized W_mid corresponds to ~50 nm for optical-communication wavelengths (1.55 μm). This could be realized by typical planar processes and by layer-stacking techniques [12,16,27–29].

This wide total BW for single-mode propagation is attributed to a large mode separation between A and B as W_mid was reduced from 0.24a. On the other hand, with further decrease of W_mid, BW_A gets much smaller as part of it moves out of the PBG. Although BW_B gets larger and partly compensates for the diminished BW_A, the total BW for single-mode propagation is reduced by the emergence of additional modes entering the PBG from the low frequency region as denoted with red arrow in Fig. 3(d). Note that adopting the same design conception of [24] and [25] in our Si 3D PC only gives PBG occupancy of 76.9% and 87.2%, respectively (these correspond to the cases with W_mid = 0.24a and W_mid = 0a, respectively).

As shown in the dispersion relation when W_mid = 0.08a in Fig. 3(c), an additional mode appears in the small k region denoted with a red arrow, which limits the total BW for single-mode propagation. In order to investigate the possibility of expanding the frequency region for single-mode guiding over the entire PBG, we next tuned the thickness of the neighboring layers, where the donor-type line defects are located, denoted as T_nl in Fig. 4(a) . When T_nl is decreased from the original value of 0.3a to 0.2925a, the additional guided mode at the small k region is shifted out of the PBG, resulting in 100% single-mode occupancy of the PBG as shown in Fig. 4(b). The optimized T_nl for optical-communication wavelengths (1.55 μm) is ~181 nm, while original layer thickness is ~186 nm. Advanced semiconductor growth technologies such as molecular beam epitaxy could grow such layers with enough accuracy to exploit this effect.

 

Fig. 4 (a) Schematic of a 3D PCW composed with three defect layers and outer 3D PC cladding. W_mid = 0.08a, yellow rods denote the rods in neighboring layers, where the donor-type line defects are located. (b) Dispersion relation of the waveguide modes when T_nl = 0.2925a.

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4. Discussion

We demonstrated wide frequency ranges in the PBG available for single mode propagation based on the combination of two guided modes A and B. Here we will discuss several issues regarding the use of proposed WGs.

The first issue is the effect of difference in coupling efficiencies for the two modes, especially at around the Brillouin zone edge (k = π/a), where the two modes have the same frequency. When an optical pulse is injected at the frequency corresponding to that at the Brillouin zone edge, strong distortion of the pulse might occur due to the difference. Supposing an optical pulse at 1.55 μm with a pulse duration of one to a few picoseconds which is a typical optical pulse used in various experiments, the spectral width is ~0.1-1% of the center frequency. In this small frequency range, the change of mode distributions is not significant. In addition, both modes have relatively similar field distributions at k = π/a except for the inversion with respect to the center line (see in Fig. 5 ). Therefore, the distortion due to the difference in coupling efficiencies is expected not to be significant for these pulses. Even on the same band, the coupling efficiency will change in frequency [27]. Design of efficient coupler for 3D PCWs will be needed.

 

Fig. 5 Vertical cross-section, orthogonal to the guided direction, of E_z of modes A and B at the transition point k = π/a for the optimized structure with W_mid = 0.08a and T_nl = 0.2925a.

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Secondly, the frequency range for single mode propagation contains slow light regions. Small group velocities make the efficient external coupling difficult. Injector structures efficient for slow lights in 2D PCWs have been reported [30,31]. The knowledge will be available for designing an efficient injector even for 3D PCWs. Once efficient couplers and injectors for 3D PCWs are invented, a multi-port configuration with add-drop functions [17] as illustrated in Fig. 6 will enable to utilize the wide single-mode BW more effectively.

 

Fig. 6 Schematic illustration of a prospective configuration for an efficient use of the wide single-mode operation frequency range in the 3D PCW discussed in this report.

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Increased propagation loss [32,33] and enhanced optical nonlinear effects [34] in the slow light region will be other concerns. These are important general issues for 3D PCWs. In other reports [21,24–27], slow light regions are also observed. Detail characteristics of propagation loss in 3D PCWs, especially at slow light regions, are expected to be revealed in the near future. Enhanced optical nonlinear effects are unwanted effect for signal transmission. However, these could be utilized to incorporate several functionalities in 3D PC circuits using the slow light region.

5. Conclusion

We have successfully designed waveguides possessing an ultra-large BW for single-mode operation in silicon 3D PCs. The waveguides support two main guided modes, and thus the combination of these two modes can provide an ultra-large BW for single-mode propagation. By finely tuning the width of the acceptor-type line defect and the thickness of the neighboring layers where the donor-type line defects are located, the two guided modes can be made to cover an extremely large single-mode BW that occupies the entire range of the cPBG of the 3D PC. These wide ranges of available frequencies for single mode propagation enable to flexibly design of 3D PC components and circuits. The designed 3D PCW with such a large BW for single-mode operation is CMOS-compatible and would provide a route towards future 3D photonic circuits. We also found that the modulation scheme of W_mid and T_nl are also effective for improving single mode BW of 3D PCWs using GaAs as the dielectric material.