Based on plane wave expansion method, complete photonic band gaps (PBGs) of a woodpile three-dimensional (3-D) terahertz (THz) photonic crystal (PC) with face-centered-tetragonal (fct) symmetry are optimized by varying structural parameters and the highest band gap ratio can reach 26.71%. In order to further optimize the complete PBGs, we propose a novel woodpile lattice with comparatively decreased symmetry and the highest band gap ratio can be increased to 27.61%. The woodpile THz PCs with two different symmetries both have a wide range of filling ratios to gain high quality complete PBGs, making the manufacturing process convenient. Woodpile 3-D PCs will be very promising materials for THz functional components.
© 2007 Optical Society of America
Terahertz (THz) waves that exist between infrared and microwave are considered as one of the last unexplored frontiers of the spectrum . With the recent developments of THz wave generation and detection techniques, THz wave technology has been applied in various scientific fields, such as medical imaging, astronomy, and chemical detection . However, most current THz systems are based on free space propagation and processing due to the absence of low dispersion and low loss materials. How to construct an integrated optical circuit for THz generation, propagation, manipulation and measurement is becoming a brand-new focus of study.
Recently, many researchers have realized that photonic crystals (PCs) would play a significant role in the THz region because of their non-loss, low dispersion and transparency in the THz wave regime [3–6]. Some studies on novel THz PC components, such as waveguides, filters, and antennas, have been reported [7–9]. However, most of them are based on two-dimensional (2-D) PCs because of their comparatively simple structures and easy fabrication. As we know, for complete control of the design, three-dimensional (3-D) PCs are preferred because of their complete 3-D band gaps . However, only some researchers have paid attention to 3-D THz PCs in the aspects of theory and experiment [10–11] since until now there is no suitable fabrication process for 3-D THz PCs . Therefore it is very important to develop the manufacturing techniques for 3-D THz PCs and design novel THz functional components based on 3-D PCs.
The woodpile lattice is one of the most popular 3-D lattice structures. Compared with other lattice structures, the woodpile lattice can generate higher quality band gaps in a wider range of filling ratios at a fixed dielectric constant ratio [12–14]. The first 3-D THz PC reported in 1994 was based on the woodpile lattice structure . With the maturity of the semiconductor technology and holographic lithography, the woodpile lattice becomes the most feasible lattice structure in the THz region with the advantages of being large-area or defect-free [16–19]. As a result, it is of significance for us to investigate the band gap properties of woodpile 3-D THz PCs in detail.
In this letter, based on plane wave expansion method, we analyze the band gap characteristics of woodpile 3-D THz PCs with face-centered-tetragonal (fct) symmetry by varying structural parameters. In order to achieve higher band gap ratios, we design a novel woodpile lattice whose spatial symmetry is lower than fct symmetry. All the theoretical results provide useful guidance for the favorable design of 3-D THz functional devices.
2. Model analyses
Figure 1(a) shows a woodpile lattice with fct symmetry, which was first proposed by K. M. Ho et al.  and was produced by the “layer-by-layer” method  or holographic lithography . It is made of layers of dielectric rods with a stacking sequence that is repeated every four layers with a repeat distance of c, corresponding to a single unit cell . Within every layer, the rods are parallel to each other and separated from each other by a distance of a; in each successive layer the rods are rotated by 90°. In a unit cell, the third and fourth layers are shifted by 0.5a relative to the first and second layers respectively. All the dielectric rods have a width of w. The periodicities along the x, z and y directions are periodx=a=100 μm, periodz=a=100 μm and periody=c=122 μm respectively. If c/a=1.414, the lattice can be regarded as a face-centered-cubic (fcc) unit cell with a basis of two rods, otherwise, the woodpile lattice symmetry is fct .
Based on fct symmetry, we propose a novel woodpile lattice, which is illustrated in Fig. 1(b). In a unit cell, the rods at the first and third layers have a width of w1 and the rods at the second and fourth layers have a width of w2. The only difference between Fig. 1(a) and Fig. 1(b) is that w1 is not equal to w2. In Fig. 1(b), w2>w1. Through the moderate change, the spatial symmetry of the new woodpile lattice is lower than fct symmetry.
In this letter, we choose silicon (Si) with a dielectric constant (ε) of 11.7 as the material of the rods or the background because of its high transparency in the THz region. We adopt plane wave expansion method  for computing the complete band gaps.
3. Numerical results and discussion
Firstly, we calculate the variation of the complete PBGs for woodpile 3-D lattice structures with the rod width.
In our computational models, the rod length along the x or z direction is decided by the following equation:
where lx is the rod length along the x direction and ly is the rod length along the y direction. lx and ly will change with w1 and w2, respectively. However, the latter calculations indicate that the rod length is not one of the factors affecting the filling ratio. The rod width can directly decide the filling ratio and affect the complete PBGs.
The woodpile lattice with fct symmetry is composed of Si rods stacked up in a background of air and the rod height (h) is equal to 30.5 μm, that is, the rods of every layer just touch each other along the stacking direction. The relation between the complete PBGs and the rod width (w) is shown in Fig. 2(a). At w=27.8 μm, the band gap ratio is as high as 18.71% and the complete PBG exists in the 1.0653–1.2852 THz range. When w ranges between 17.2 μm and 38.2 μm, the band gap ratios are above 15%.
As we know, the high spatial symmetry of PCs can result in the degeneracy of photonic bands . Reducing the spatial symmetry of PCs is one of the most effective methods for increasing the band gap ratios and the width of the complete PBGs. In a unit cell, if w1 is not equal to w2, the spatial symmetry of the woodpile lattice will be reduced moderately and the complete PBGs can be optimized. Firstly, we fix w1=27.8 μm and vary w2. The results shown in Fig. 2(b) are in good agreement with expectation. When w2=32.6 μm, the highest band gap ratio of 19.59% is obtained and the corresponding complete band gap is in the range of 1.0305–1.2543 THz. As w2 varies between 20 μm and 43 μm, the band gap ratios surpass 15%. Secondly, we fix w2=27.8 μm and change w1. The dependence of the complete PBGs on w1 is shown in Fig. 2(c). At w1=25 μm, the highest band gap ratio is 19.63% and the corresponding complete PBG stands between 1.0749 THz and 1.3089 THz. Varying w1 in the range of 18.4–36.2 μm, the band gap ratios exceed 15%.
The above results and analyses indicate that the decrease of the spatial symmetry can benefit the optimization of the complete PBGs, i.e., the increase of the band gap ratios and the improvement of the regulating range of filling ratios at which high quality complete band gaps can be obtained.
Now we reverse the dielectric configuration. The woodpile lattice is composed of air rods stacked in a Si background. The computed process and results are similar to those of the unreversed woodpile lattice.
First, we set h=38.1 μm and change the width of all the air rods. The results are shown in Fig. 3(a). At w=73.3 μm, the highest band gap ratio of 18.89% is obtained and the corresponding complete PBG exists in the range of 1.0791–1.3035 THz. As the rod width varies between 61.5 μm and 82.5 μm, the band gap ratios are above 15%. Second, we fix w1=73.3 μm and change w2. The results are shown in Fig. 3(b). At w2=69 μm, the highest band gap ratio of 21.44% is obtained. When w2 varies in the range of 55.5–81.5 μm, the band gap ratios can exceed 15%. Last, we fix w2=73.3 μm and change w1. The results are indicated in Fig. 3(c). At w1=75.5 μm, the highest band gap ratio of 20.72% is gained and when w1 ranges between 60 μm and 82.5 μm, the band gap ratios are above 15%. From these results, we can easily obtain the same conclusion as that of the unreversed woodpile lattice.
Secondly, we calculate the variation of the complete PBGs for woodpile 3-D lattice structures with the rod length and the rod height, respectively.
First, we discuss the relation between the complete PBGs and the rod length. The results show that the variation of the rod length has no effect on the complete PBGs no matter whether the woodpile lattice structure is composed of Si rods stacked in an air background or of air rods stacked in a Si substrate. The reason is that the rod length is not among the factors that affect the filling ratio. In fact, when the rod length varies, the PBG width and the gap center frequency both remain the same.
Then, we discuss the dependence of the complete PBGs on the rod height (h). The periodicity along the stacking direction (periody) is 122 μm and when h varies, periody remains the same, that is, the relation between them dissatisfies the equation: periody=4h. If h>30.5 μm or h<30.5 μm, the rods of four layers in a unit cell will be overlapped or separated to a certain extent along the y direction and the gap between the separated rods is filled by the high or low dielectric materials.
In the first case, the woodpile fct lattice is constructed with Si rods stacked in an air background. The width of the Si rods is fixed as 27.8 μm and the other parameters are the same as before. The results are shown in Fig. 4. When h=30.5 μm, the highest band gap ratio 18.71% is obtained and the corresponding complete band gap exists between 1.0653 THz and 1.2852 THz. In this optimal condition, the Si rods just touch each other and are not overlapped or separated. As h<30.5 μm, the gap between the rods is filled with air. This hypothesis does not come into existence in practice and we use air as the background material only in order to compute the PBGs conveniently.
In the second case, the woodpile fct lattice is designed to be composed of air rods piled up in a Si substrate and prepared through the ion beam etching method. The rod width is 73.3 μm and the other structural parameters are not changed. The results are shown in Fig. 5. At h=38.1 μm, the highest band gap ratio is 20.21% and the corresponding complete band gap exists in the range of 1.1529–1.4121 THz.
The above results indicate that the structural parameters of the woodpile fct PC all have a wide range of tunability to gain high quality complete PBGs. The novel woodpile lattice can achieve higher band gap ratios and have better parameter flexibility than the original one under the same conditions.
Finally, we calculate the variation of the complete PBGs for woodpile 3-D lattice structures with the refractive index contrast. Generally speaking, the higher the refractive index contrast, the stronger the Bragg diffraction, and the easier the generation of the complete PBGs. Increasing the refractive index contrast is one of the most effective methods for improving the band gap quality.
In the first case, the woodpile fct lattice is composed of dielectric rods stacked up in an air background. We set w=27.8 μm and h=30.5 μm. The other parameters remain the same. Here what we need to explain is that in Fig. 6 nrod is the refractive index of the dielectric rods, nair is the refractive index of air, and the abscissa is the differential value nrod-nair. The results shown in Fig. 6(a) indicate that a complete PBG appears at a refractive index contrast as low as 2.26. The band gap ratio and the width of the complete PBG both increase step by step with the rise of the differential value. When nrod=4 (Ge, ε=16), the band gap ratio reaches as high as 24.02% and the PBG stands in the range of 0.9219–1.1736 THz. If nrod continues increasing, the band gap ratios can be further optimized. However, the larger the differential values are, the more difficult it is to find the dielectric materials. So during the calculation, it is enough that we choose germanium as the material of the rods or the background with the highest refractive index. When w1 is not equal to w2, the results shown in Fig. 6(b) and Fig. 6(c) demonstrate that the quality of the complete PBGs is more enhanced. At nrod=4, w1=27.8 μm, and w2=32.6 μm, the highest band gap ratio is 24.61% and as nrod=4, w1=25 μm, and w2=27.8 μm, the highest band gap ratio is 25.1%.
In the second case, we reverse the dielectric configuration and hypothesize that the woodpile lattice is made up of air rods piled up in a dielectric background. The calculated process and results shown in Fig. 7 are similar to the former calculations. At nbackground=4, the band gap ratios of 26.71%, 27.61% and 27.13% can be obtained respectively.
In this letter, using plane wave expansion method, we calculate the complete PBGs of woodpile 3-D THz PCs under different structural parameters. By varying the filling ratio, the refractive index contrast and decreasing the lattice symmetry, the highest band gap ratio can reach 27.61%. We propose a novel woodpile lattice where the spatial symmetry is moderately reduced in comparison to fct symmetry. The calculated results show that the novel woodpile lattice has a wider range of filling ratios to obtain high quality complete PBGs and can reach higher band gap ratios than the original one, which provides great flexibility for the fabrication of 3-D THz PCs. Our results and analyses demonstrate that woodpile 3-D PCs will play an important role in THz functional components and integrated systems.
This research is supported by National Science Foundation of China, under the grant numbers of 10474071 and the Ph.D. Programs Foundation of Ministry of Education of China, under the grant numbers of 20040056010.
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