OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 2 — Jan. 22, 2007
  • pp: 695–703
« Show journal navigation

Characteristics of photonic band gaps in woodpile three-dimensional terahertz photonic crystals

Huan Liu, Jianquan Yao, Degang Xu, and Peng Wang  »View Author Affiliations


Optics Express, Vol. 15, Issue 2, pp. 695-703 (2007)
http://dx.doi.org/10.1364/OE.15.000695


View Full Text Article

Acrobat PDF (987 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

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

1. Introduction

Terahertz (THz) waves that exist between infrared and microwave are considered as one of the last unexplored frontiers of the spectrum [1

1. P. H. Siegel, “Terahertz technology,” IEEE Trans. Microwave Theory and Tech. 50,910–928 (2002). [CrossRef]

]. 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 [2

2. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417,156–159 (2002). [CrossRef] [PubMed]

]. 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.

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. [20

20. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89,413–416 (1994). [CrossRef]

] and was produced by the “layer-by-layer” method [21

21. S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and Jim Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394,251–253 (1998). [CrossRef]

] or holographic lithography [18

18. S. Shoji, H. B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. 83,608–610 (2003). [CrossRef]

]. 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 [20

20. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89,413–416 (1994). [CrossRef]

]. 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 [20

20. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89,413–416 (1994). [CrossRef]

].

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 [22

22. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65,3152–3155 (1990). [CrossRef] [PubMed]

] for computing the complete band gaps.

Fig. 1. Schematics of woodpile 3-D lattice structures. (a) Diagram of a woodpile lattice with fct symmetry, which is made of layers of rectangular dielectric rods with a stacking sequence that is repeated every four layers with a repeat distance of c, corresponding to a single unit cell. (b) Diagram of a novel woodpile lattice structure. In a unit cell, the width of the rods of the first and third layers (w1) is not equal to the width of the rods of the second and fourth layers (w2). In Fig. 1(b), w2>w1.

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:

lx=6×periodx+w1
ly=7×periodz+w2
(1)

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 [23

23. C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77,2949–2952 (1996). [CrossRef] [PubMed]

]. 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%.

Fig. 2. Band gap diagrams for the woodpile PCs composed of Si rods piled up in an air background. (a) the variation of the complete PBGs with w. (b) the variation of the complete PBGs with w2. (c) the variation of the complete PBGs with w1.

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.

Fig. 3. Band gap diagrams for the woodpile PCs composed of air rods piled up in a Si substrate. (a) the variation of the complete PBGs with w. (b) the variation of the complete PBGs with w2. (c) the variation of the complete PBGs with w1.

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.

Fig. 4. Band gap diagram as a function of the rod height for the woodpile fct PC composed of Si rods piled up in an air background.
Fig. 5. Band gap diagram as a function of the rod height for the woodpile fct PC composed of air rods piled up in a Si substrate.

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.

Fig. 6. Band gap diagrams as a function of the refractive index difference (nrod-nair) for the woodpile PCs composed of dielectric rods stacked up in an air background. (a) fct symmetry, w=27.8 μm and h=30.5 μm. (b) a novel woodpile lattice, w1=27.8 μm, w2=32.6 μm and h=30.5 μm. (c) a novel woodpile lattice, w1=25 μm, w2=27.8 μm and h=30.5 μm.
Fig. 7. Band gap diagrams as a function of the refractive index of the background material (nbackground) for the woodpile PCs composed of air rods stacked up in a dielectric substrate. (a) fct symmetry, w=73.3 μm and h=38.1 μm. (b) a novel woodpile lattice, w1=73.3 μm, w2=69 μm and h=38.1 μm. (c) a novel woodpile lattice, w1=75.5 μm, w2=73 μm and h=38.1 μm.

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.

4. Conclusion

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.

Acknowledgments

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.

References and links

1.

P. H. Siegel, “Terahertz technology,” IEEE Trans. Microwave Theory and Tech. 50,910–928 (2002). [CrossRef]

2.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417,156–159 (2002). [CrossRef] [PubMed]

3.

N. Jukam and M. S. Sherwin, “Two-dimensional terahertz photonic crystals fabricated by deep reactive ion etching in Si,” Appl. Phys. Lett. 83,21–23 (2003). [CrossRef]

4.

S. W. Wang, W. Lu, X. S. Chen, Z. F. Li, X. C. Shen, and W. J. Wen, “Two-dimensional photonic crystal at THz frequencies constructed by metal-coated cylinders,” J. Appl. Phys. 93,9401–9403 (2003). [CrossRef]

5.

A. Di Falco, C. Conti, and G. Assanto, “Terahertz pulse generation via optical rectification in photonic crystal microcavities,” Opt. Lett. 30,1174–1176 (2005). [CrossRef] [PubMed]

6.

H. Kurt and D. S. Citrin, “Photonic crystals for biochemical sensing in the terahertz region,” Appl. Phys. Lett. 87,041108 (2005). [CrossRef]

7.

C. Lin, C. Chen, G. Schneider, P. Yao, S. Shi, A. Sharkawy, and D. Prather, “Wavelength scale terahertz two-dimensional photonic crystal waveguides,” Opt. Express 12,5723–5728 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-23-5723. [CrossRef] [PubMed]

8.

T. D. Drysdale, R. J. Blaikie, and D. R. S. Cumming, “Calculated and measured transmittance of a tunable metallic photonic crystal filter for terahertz frequencies,” Appl. Phys. Lett. 83,5362–5364 (2003). [CrossRef]

9.

A. L. Reynolds, H. M. H. Chong, I. G. Thayne, J. M. Arnold, and P. De Maagt, “Analysis of membrane support structures for integrated antenna usage on two-dimensional photonic-bandgap structures,” IEEE Trans. Microwave Theory and Tech. 49,1254–1261 (2001). [CrossRef]

10.

A. R. Weily, K. P. Esselle, T. S. Bird, and B. C. Sanders, “Experimental woodpile EBG waveguides, bends and power dividers at microwave frequencies,” Electron. Lett. 42,32–33 (2006). [CrossRef]

11.

K. Takagi, K. Seno, and A. Kawasaki, “Fabrication of a three-dimensional terahertz photonic crystal using monosized spherical particles,” Appl. Phys. Lett. 85,3681–3683 (2004). [CrossRef]

12.

H. Liu, J. Q. Yao, E. B. Li, W. Q. Wen, Q. Zhang, and P. Wang, “Theoretical analysis of optimum parameters for complete forbidden bands of three-dimensional photonic crystals with typical lattice structures,” Acta Phys. Sin. 55,230–238 (2006).

13.

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys. 78,1415–1418 (1995). [CrossRef]

14.

P. Kopperschmidt, “Tetragonal photonic woodpile structures,” Appl. Phys. B 76,729–734 (2003). [CrossRef]

15.

E. Özbay, E. Michel, G. Tuttle, R. Biswas, K. M. Ho, J. Bostak, and D. M. Bloom, “Terahertz spectroscopy of three-dimensional photonic band-gap crystals,” Opt. Lett. 19,1155–1157 (1994). [PubMed]

16.

A. Chelnokov, S. Rowson, J.-M. Lourtioz, L. Duvillaret, and J.-L. Coutaz, “Terahertz characterisation of mechanically machined 3D photonic crystal,” Electron. Lett. 33,1981–1983 (1997). [CrossRef]

17.

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83,4480–4482 (2003). [CrossRef]

18.

S. Shoji, H. B. Sun, and S. Kawata, “Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference,” Appl. Phys. Lett. 83,608–610 (2003). [CrossRef]

19.

Y. Lin, D. Rivera, and K. P. Chen, “Woodpile-type photonic crystals with orthorhombic or tetragonal symmetry formed through phase mask techniques,” Opt. Express 14,887–892 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-887. [CrossRef] [PubMed]

20.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89,413–416 (1994). [CrossRef]

21.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and Jim Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394,251–253 (1998). [CrossRef]

22.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65,3152–3155 (1990). [CrossRef] [PubMed]

23.

C. M. Anderson and K. P. Giapis, “Larger two-dimensional photonic band gaps,” Phys. Rev. Lett. 77,2949–2952 (1996). [CrossRef] [PubMed]

OCIS Codes
(220.0220) Optical design and fabrication : Optical design and fabrication
(300.6270) Spectroscopy : Spectroscopy, far infrared

ToC Category:
Photonic Crystals

History
Original Manuscript: October 12, 2006
Revised Manuscript: November 20, 2006
Manuscript Accepted: December 7, 2006
Published: January 22, 2007

Citation
Huan Liu, Jianquan Yao, Degang Xu, and Peng Wang, "Characteristics of photonic band gaps in woodpile three-dimensional terahertz photonic crystals," Opt. Express 15, 695-703 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-695


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. P. H. Siegel, "Terahertz technology," IEEE Trans. Microwave Theory and Tech. 50, 910-928 (2002). [CrossRef]
  2. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, "Terahertz semiconductor-heterostructure laser," Nature 417, 156-159 (2002). [CrossRef] [PubMed]
  3. N. Jukam, and M. S. Sherwin, "Two-dimensional terahertz photonic crystals fabricated by deep reactive ion etching in Si," Appl. Phys. Lett. 83, 21-23 (2003). [CrossRef]
  4. S. W. Wang, W. Lu, X. S. Chen, Z. F. Li, X. C. Shen, and W. J. Wen, "Two-dimensional photonic crystal at THz frequencies constructed by metal-coated cylinders," J. Appl. Phys. 93, 9401-9403 (2003). [CrossRef]
  5. A. Di Falco, C. Conti, and G. Assanto, "Terahertz pulse generation via optical rectification in photonic crystal microcavities," Opt. Lett. 30, 1174-1176 (2005). [CrossRef] [PubMed]
  6. H. Kurt, and D. S. Citrin, "Photonic crystals for biochemical sensing in the terahertz region," Appl. Phys. Lett. 87, 041108 (2005). [CrossRef]
  7. C. Lin, C. Chen, G. Schneider, P. Yao, S. Shi, A. Sharkawy, and D. Prather, "Wavelength scale terahertz two-dimensional photonic crystal waveguides," Opt. Express 12, 5723-5728 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-23-5723. [CrossRef] [PubMed]
  8. T. D. Drysdale, R. J. Blaikie, and D. R. S. Cumming, "Calculated and measured transmittance of a tunable metallic photonic crystal filter for terahertz frequencies," Appl. Phys. Lett. 83, 5362-5364 (2003). [CrossRef]
  9. A. L. Reynolds, H. M. H. Chong, I. G. Thayne, J. M. Arnold, P. De Maagt, "Analysis of membrane support structures for integrated antenna usage on two-dimensional photonic-bandgap structures," IEEE Trans. Microwave Theory and Tech. 49, 1254-1261 (2001). [CrossRef]
  10. A. R. Weily, K. P. Esselle, T. S. Bird, and B. C. Sanders, "Experimental woodpile EBG waveguides, bends and power dividers at microwave frequencies," Electron. Lett. 42, 32-33 (2006). [CrossRef]
  11. K. Takagi, K. Seno, and A. Kawasaki, "Fabrication of a three-dimensional terahertz photonic crystal using monosized spherical particles," Appl. Phys. Lett. 85, 3681-3683 (2004). [CrossRef]
  12. H. Liu, J. Q. Yao, E. B. Li, W. Q. Wen, Q. Zhang, and P. Wang, "Theoretical analysis of optimum parameters for complete forbidden bands of three-dimensional photonic crystals with typical lattice structures, " Acta Phys. Sin. 55, 230-238 (2006).
  13. S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Theoretical investigation of fabrication-related disorder on the properties of photonic crystals," J. Appl. Phys. 78, 1415-1418 (1995). [CrossRef]
  14. P. Kopperschmidt, "Tetragonal photonic woodpile structures," Appl. Phys. B 76, 729-734 (2003). [CrossRef]
  15. E. Özbay, E. Michel, G. Tuttle, R. Biswas, K. M. Ho, J. Bostak, and D. M. Bloom, "Terahertz spectroscopy of three-dimensional photonic band-gap crystals," Opt. Lett. 19, 1155-1157 (1994). [PubMed]
  16. A. Chelnokov, S. Rowson, J.-M. Lourtioz, L. Duvillaret, and J.-L. Coutaz, "Terahertz characterisation of mechanically machined 3D photonic crystal," Electron. Lett. 33, 1981-1983 (1997). [CrossRef]
  17. A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, "Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses," Appl. Phys. Lett. 83, 4480-4482 (2003). [CrossRef]
  18. S. Shoji, H. B. Sun, and S. Kawata, "Photofabrication of wood-pile three-dimensional photonic crystals using four-beam laser interference," Appl. Phys. Lett. 83, 608-610 (2003). [CrossRef]
  19. Y. Lin, D. Rivera, and K. P. Chen, "Woodpile-type photonic crystals with orthorhombic or tetragonal symmetry formed through phase mask techniques," Opt. Express 14, 887-892 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-887. [CrossRef] [PubMed]
  20. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994). [CrossRef]
  21. S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, "A three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998). [CrossRef]
  22. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
  23. C. M. Anderson, and K. P. Giapis, "Larger two-dimensional photonic band gaps," Phys. Rev. Lett. 77, 2949-2952 (1996). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited