We have theoretically investigated the characteristics of three-dimensional (3D) photonic crystal (PC) waveguides formed by the introduction of dielectric line defects. We show that the guided modes in 3D PC waveguides strongly depend on the volume, position and number of dielectric defects introduced. We have succeeded in designing a waveguide structure with a large single-mode bandwidth of 178 nm (range = 1,466 to 1,644 nm) for wavelengths used in optical communications. Our study indicates that there is great flexibility in the design of 3D PC waveguides and that a variety of desirable properties can be obtained by altering the configuration of the line defects appropriately.
©2005 Optical Society of America
Photonic crystals (PCs) [1–3] have a photonic band gap (PBG), which forbids the passage of light within a particular frequency range. As PCs confine light inside an ultra-small volume, and control the propagation and spontaneous emission of the light, it is hoped that they will ultimately form the basis of optical devices, such as ultra-compact optical circuits and zero-threshold lasers. There have been many recent studies about light propagation in two-dimensional (2D) PC slabs [4–7], which confine light in a plane by means of a PBG, and in the perpendicular direction by total internal reflection. Three-dimensional (3D) PCs have the advantage of a complete PBG that can confine light perfectly for all polarizations and directions. Techniques for fabricating such devices have significantly advanced and 3D PCs have been realized for wavelengths relevant to optical communications [3,8–11]. Moreover, the incorporation of optical cavities in a 3D PC has been investigated theoretically [12,13] and experimentally [8,11,14], and the control of spontaneous emission from such structures has been demonstrated . In addition, acceptor-type 3D PC waveguides, which are created by removing dielectric material to form line defects, have been theoretically investigated [15,16] and experimentally examined in microwave region [17,18]. Both single-mode waveguides and structures with sharp bends and large bandwidths have been shown [18–20], and properties of the coupling between nano-cavities and waveguides have been analyzed [21,22]. However, the design of donor-type 3D PC waveguides, which are created by inserting dielectric material in the bulk structure, has not yet been studied. Donor-type waveguides have the potential to provide the basis of active dispersion-control devices or waveguide lasers when non-linear materials or light emitters are introduced into line defects. The study of donor-type 3D PC waveguides is thus important both to realize a variety of active devices and to connect the functional components within an optical device. Moreover, these 3D PC waveguides do not have guided modes that cause propagation losses due to factors such as transverse electric and magnetic coupling , or leaky-mode regions — problems that are unavoidable in 2D PC slab waveguides . It is hoped that the ideal features of 3D PC waveguides will ultimately allow the complete control of light to be achieved. Therefore, it is important to clarify the characteristics of donor-type waveguides.
In this paper, we theoretically investigate the dispersion relations and electromagnetic-field distributions of various donor-type waveguides in 3D PCs, which are referred to as layer-by-layer [25,26] and woodpile , by utilizing the plane-wave expansion method [28,29].
2. Method and calculation model
The theoretical analysis was performed by the plane-wave expansion method, solving Maxwell’s equations in the frequency domain. The supercell method and dense-matrix techniques were used, where the dielectric function was computed via the inverse-matrix method . The dielectric material of the rods that constitute the 3D PC was assumed to be GaAs, with a refractive index of 3.375 that corresponds to optical communication wavelengths. Each rod had 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 5a in the z direction. The number of plane waves used was 6,475.
We first examined simple dielectric line defects, which change the width of one of the rods in the PC as shown in Fig. 1(a). The rod width W A was changed from 0.3a to 1.7a. Figure 1(b) shows the dispersion relations of guided modes for W A=0.7a and 0.8a. As W A increases, the guided modes descend to lower frequencies due to the increase in their effective refractive index. In the case of W A=0.8a, the first guided mode (mode A) has the largest single-mode bandwidth, from 0.344c/a to 0359c/a, as shown in Fig. 1(b). We define Δf and f mid as the single-mode bandwidth and its middle frequency, respectively. In this case the ratio Δf/f mid is 4.16%. For optical communication wavelengths, the corresponding bandwidth is 65 nm (range = 1,518 to 1,583 nm). This bandwidth is smaller than that of previously reported air waveguides  because higher-order guided modes might easily appear in the PBG in dielectric waveguides, which have a larger effective refractive index. We next examined the magnetic-field profile of mode A, the cross-section of which is shown in Fig. 2 for W A=0.8a. The field is strongly localized at the dielectric material of the line defects, and the volume of the guided modes is smaller than that in air waveguides  because of the large effective refractive index. From this result, it is clear that dielectric waveguides will be better suited than air waveguides for active operations when non-linear materials and light emitters are incorporated.
Waveguide structures in which an additional rod was inserted halfway between two original rods were examined, as shown in Fig. 3(a). The defect-rod width W B was varied from 0.1a to 0.7a. The dispersion relation of the guided modes for W B=0.3a, which has the largest single-mode bandwidth, is shown in Fig. 3(b). The Δf/f mid ratio of mode B is 6.15%, corresponding to a bandwidth of 95 nm (range = 1,504 to 1,599 nm). The magnetic-field profile was examined and is shown in Fig. 4. There is a node at the center of the additional rod, and two peaks located at the air regions between the defect and original rods. It should be noted that the peak field intensity in donor-type waveguides is not always located at the dielectric region. Here, using single-lobed beams, such as a Gaussian beam, losses due to coupling between the 3D PC waveguides and the external optics are significant. The waveguide shown in Fig. 3(a) has a larger single-mode region than the waveguide in which the width of an original rod is changed, and is thus inferior with respect to active operations and coupling with external optics.
We next proposed a new type of waveguide structure, where a dielectric rod crossing the original rods is introduced as shown in Figs. 5(a) and 5(b). The additional rod, of width 0.3a, is placed halfway between two rods in the upper and lower neighboring layers. The corresponding dispersion relation of this “single-cross-rod” waveguide is shown in Fig. 6(a). Two guided modes, denoted as modes C1 and C2, intersect near the middle of the PBG. Two separated single-mode regions are thus created. The single-mode bandwidths, Δf/f mid, of mode C1 are 4.50% (70 nm; range = 1,516 to 1,586 nm) and 6.45% (100 nm; range = 1,502 to 1,602 nm) for the high- and low-frequency regions, respectively. Here we have estimated the wavelengths of the two regions individually. The magnetic-field distribution of mode C1 is localized across the three rods (the cross rod, and the upper and lower neighboring rods) as shown in Fig. 6(b). It is clear that single-cross-rod waveguides have a large single-mode bandwidth and a field pattern localized in the dielectric material.
We proceeded to improve the design of the single-cross-rod waveguides to obtain simpler field distributions, which would allow more efficient connections with the external optics to be made. This was achieved by translating the upper and lower neighboring rods, such that the cross rod is sandwiched between them, forming a vertical, rather than diagonal, arrangement of the three rods [Fig. 7(a)]. The dispersion relation of the “arranged-single-cross-rod” waveguide is shown in Fig. 7(b). As can be seen from a comparison of Figs. 6(a) and 7(b), mode C1 remained essentially unchanged in the new arrangement, while mode C2 was shifted to higher frequencies. A wider single-mode bandwidth of Δf/f mid=6.90% (107 nm; range = 1,498 to 1,605 nm) was obtained. Furthermore, the magnetic-field profile of mode C1 became strongly localized and single-lobed [Fig. 7(c)]. This result demonstrates that highly efficient active operations and low-loss coupling with the external optics can be achieved with an “arranged-single-cross-rod” waveguide. Moreover, mode C1 has a frequency range with a slow group velocity (0.355 to 0.359 c/a). The realization of signal-delay devices is expected to be possible using such a frequency range.
The above study of “arranged-single-cross-rod” waveguides shows that the configuration of the cross rod and two neighboring rods determines the main properties of the guided modes. A similar structure can be created by sandwiching one original rod between a pair of inserted cross rods; the configuration of this “double-cross-rods” waveguide is shown in Figs. 8(a) to 8(c). The cross rods both have a width of 0.3a. The dispersion relation and magnetic-field distribution of the waveguide are shown in Figs. 9(a) and 9(b), respectively.
The single-mode bandwidth is the widest of any donor-type waveguide investigated, with a value of Δf/f mid=11.5% (178 nm; range = 1,466 to 1,644 nm). This bandwidth is almost as large as that of acceptor-type waveguides , although donor-type waveguides tend to be multi-mode. In addition, the magnetic-field profile is single-lobed and is essentially unchanged from that of the arranged-single-cross-rod waveguide.
We have theoretically investigated the properties of various donor-type waveguides in 3D PCs. The calculated dispersion relations and electromagnetic-field profiles indicate that the characteristics of the guided modes depend on the volume, position and number of line defects that are introduced. We have shown that donor-type waveguides with large single-mode bandwidths and small mode field can be achieved using a carefully designed arrangement of line defects. The “double-cross-rods” waveguide, which is created by sandwiching a rod of the main structure between two line-defect rods, has a wide single-mode bandwidth of Δf/f mid=11.5% (178 nm; range = 1,466 to 1,644 nm), which is the largest value for any donor-type waveguide investigated in 3D PCs. This single-mode bandwidth is almost equal to that found in acceptor-type waveguides. The magnetic field of the guided modes is localized across the original rod and the two sandwiching line-defect rods, and will thus interact strongly with light emitters and non-linear materials. In addition, the field profile is single-lobed, and a high coupling efficiency with the external optics can be realized. Our results indicate that there is great flexibility in the design of 3D PC waveguides and that a variety of desirable properties can be obtained by altering the configuration of the line-defects appropriately.
This work was partly supported by the Core Research for Evolutional Science and Technology Program from the Japan Science and Technology Agency, and by an Information Technology program and a Grant-in-Aid for Scientific Research of Priority Areas from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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