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We designed three-dimensional (3D) photonic crystal (PC) waveguides by simultaneously introducing one acceptor-type and two donor-type line defects. The waveguides have an extremely large single-mode bandwidth, which covers more than 90% of the complete photonic band gap. The relatively large group velocity and the mode-field localization in the air core should prevent unintended nonlinear phenomena for ultra-short pulse propagation. These promising characteristics could only be achieved by using 3D PCs, which have the advantages of complete light confinement and no restrictions of the light cone.
©2006 Optical Society of America
1. Introduction
Photonic crystals (PCs) [1] which have periodic refractive-index distributions, have a photonic band gap (PBG) that prevents the passage of light within a particular frequency range. PCs might ultimately form the basis of light control on the submicron scale and allow the production of novel optical devices, such as ultra-compact optical circuits and zero-threshold lasers. Several previous reports have addressed the control of light-propagation in two-dimensional (2D) PC slabs [2,3], which confine light in a plane due to a 2D PBG and in the third dimension by total internal reflection. Three-dimensional (3D) PCs, which have the advantage of a complete PBG that can perfectly confine light of all polarizations in all directions, have been realized for optical-communication wavelengths [4,5] due to significant advances in fabrication techniques. Both the introduction of point-defect cavities [5,6] and the control of spontaneous emission [6] have been demonstrated. In addition, several groups have investigated the light propagation phenomena in line-defect waveguides, both theoretically [7–12] and experimentally [13,14]; these authors have reported on the optimal structures with sharp bends [7], the design of waveguide structures with wide single-mode bandwidths [8,11,12], the light coupling phenomena between cavities and waveguides [9,13], and the polarization-independent linear waveguides [10]. 3D PC waveguides do not have guided modes that cause propagation losses due to factors such as transverse electric and magnetic coupling, nor do they have leaky-mode regions — problems that are unavoidable in 2D PC slab waveguides. These characteristics of 3D PC waveguides might ultimately allow the complete control of light. In previously reported 3D PC waveguides [7–14], either acceptor-type or donor-type line defects were individually introduced to the structures; the resulting waveguides have larger single-mode bandwidths than those achieved in 2D PC slabs, despite using part of the complete PBG as a guided frequency range. Extremely wide single-mode bandwidths — as wide as the whole PBG — might be obtained by optimizing line-defect structures in 3D PCs. In this paper, we present the designs of new waveguides based on woodpile PCs [4,6,7,9,12–17] by simultaneously introducing acceptor-type and donor-type line defects, and theoretically analyze the resulting structures.
2. Method and calculation model
The theoretical analysis was performed by the plane-wave-expansion method [18,19], solving Maxwell’s equations in the frequency domain. The supercell method and dense-matrix techniques were used; the dielectric function was computed via the inverse-matrix approach [20]. The dielectric material of the rods constituting the 3D PC was assumed to be GaAs, with a refractive index of 3.375 corresponding to optical-communication wavelengths. Each rod have a width of 0.3a and a height of 0.3125a, where a represents the center-to-center spacing of the rods. We defined the y direction as the guided direction. The dimensions of the supercell were 6a in the x direction, 1a in the y direction and 6.25a in the z direction. The number of plane waves used was 9,583.
3. Results
We first designed a novel 3D PC waveguide structure, as shown in Figs. 1(a) and 1(b). The waveguide is composed of three defect layers and outer 3D PC cladding. The middle-defect layer has an acceptor-type line defect created by removing an original rod. The upper-defect and lower-defect layers have donor-type line defects created by filling a space between the original rods with the same dielectric material. The additional-dielectric line defects have a width of 0.3a; the three defects are jointly oriented in the out-of-plane direction, as shown in Fig. 1(c).
Fig. 1. (a) Schematic representation of a novel 3D PC waveguide composed of three defect layers and outer 3D PC cladding. The line defects created by simultaneously introducing both acceptor-type defect (white rod) and donor-type line defects (red rods). (b) Top view of the upper and lower layers with additional dielectric defects (left), and top view of the middle layer with an air defect (right). (c) Schematic vertical cross-section of the waveguide structure.
Fig. 2. (a) The dispersion relations of the 3D PC waveguide. (b) Vertical cross-section, orthogonal to the guided direction, of the magnetic-field component (Hx) at ky=0.375 (2π/a).
Figure 2(a) shows the dispersion relations of the guided modes in the waveguide. Three guided modes appeared in the complete PBG, and one guided mode has a large single-mode bandwidth. We defined Δf and f mid as the single-mode bandwidth and its middle frequency, respectively. To our knowledge, the ratio Δf/f mid=14.2% is the largest value for all PC waveguides. The bandwidth corresponds to 222 nm (range=1,447 to 1,669 nm) in optical-communication wavelengths.
Next, we investigated the magnetic-field distribution of the guided mode. The magnetic field is strongly localized at the acceptor-type defect, called the air area, as shown in Fig. 2(b).
We predicted that several guided modes derived from each line defect might appear in the complete PBG, and so this structure might be a multimode waveguide. However, in practice, the one acceptor-type and the two donor-type line defects seem to function as a “compound defect waveguide,” that is to say, a series of parallel line defects couple such that we can interpret them as a single line-defect waveguide. We successfully obtained a 3D PC waveguide with the characteristics of both a wide single-mode bandwidth, occupying 86% of the PBG, and a small mode-field, localized in a narrower air area than that of the previously reported acceptor-type waveguide [7].
Fig. 3. (a) Schematic vertical cross-section of the modified waveguide structure, showing the rods (green) adjacent to either side of the air-line defect. d and W denote the shifts and widths of the modified rods, respectively. (b) Changes in the normalized single-mode bandwidth Δf/f mid with changing values of d and W.
The fact that light is confined in an extremely small air area implies that the effective refractive index is small. In fact, the group velocity of the guided mode is larger than the air waveguide [7], as illustrated by the divergence of the dispersion relation in Fig. 2(a). In addition, although guided modes in PC waveguides generally have large group-velocity dispersion (GVD) [12], which can be applied to signal modulation devices, our advanced 3D PC waveguide has a relatively small GVD. These results suggest that light propagating in the waveguide would have minimal interaction with the surrounding dielectric materials, because electromagnetic waves would rapidly propagate in the small air core, through the wide frequency range. We therefore predict the absence of unintended third-order nonlinear phenomena for ultra-short and strong pulse propagation.
In order to obtain a larger ratio Δf/f mid, we next modified the positions and the widths of the two rods immediately adjacent to either side of the air-line defect. The two rods were shifted away from the air-line defect, and the widths were increased from 0.3a to 0.4a, as shown in Fig. 3(a). The shifts and widths of the two rods are denoted d and W, respectively. Figure 3(b) illustrates the changes in the ratio Δf/f mid with changing values of d and W. An extremely wide single-mode bandwidth was obtained for d=0.1a and W=0.35a, because the other two guided modes — which lay around the lower band edge and interfered with the broadening of Δf — were shifted to a lower frequency. The ratio Δf/f mid for this waveguide is 15.1%; the bandwidth corresponds to 235 nm (range=1,441 to 1,677 nm) in optical-communication wavelengths, and occupies more than 90% of the PBG. This waveguide has a 27% larger single-mode bandwidth than previous air waveguides [7]. The mode profile of the guided mode shows little change after the modifications to d and W.
4. Conclusion
We successfully designed 3D PC waveguides that confined light to a small air area. These advanced 3D PC waveguides were created by simultaneously introducing one acceptor-type and two donor-type line defects. The three line defects seem to operate as a compound line defect, and we obtained a large single-mode bandwidth (Δf/f mid=14.2%). The magnetic field of the guided mode is localized in the air area between the two donor-type line defects. We predict no unintended third-order nonlinear phenomena for ultra-short and strong pulse propagation, because of the relatively large group velocity and the mode field localization in the extremely small air core. In addition, we obtained a 3D PC waveguide with an extremely wide single-mode bandwidth (Δf/f mid=15.1%), which almost covers the whole frequency range of the complete PBG, by altering the width and position of the two rods adjacent to either side of the air-line defect. The waveguide characteristics we obtained could only be achieved using 3D PCs, which have the advantages of complete light confinement in all directions and polarizations, and no restrictions of the light cone, which are unavoidable in 2D PC slab waveguides.
Acknowledgments
This work was partly supported by the Core Research for Evolutional Science and Technology Program from the Japan Science and Technology Agency, by an Information Technology program, by a Grant-in-Aid for Scientific Research of Priority Areas from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and by the Japan Society for the Promotion of Science (JSPS) the 21st Century COE Program.
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