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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 5 — Feb. 28, 2011
  • pp: 3962–3972
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Terahertz electromagnetic crystal waveguide fabricated by polymer jetting rapid prototyping

Ziran Wu, Wei-Ren Ng, Michael E. Gehm, and Hao Xin  »View Author Affiliations


Optics Express, Vol. 19, Issue 5, pp. 3962-3972 (2011)
http://dx.doi.org/10.1364/OE.19.003962


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Abstract

An all-dielectric THz waveguide has been designed, fabricated and characterized. The design is based on a hollow-core electromagnetic crystal waveguide, and the fabrication is implemented via polymer-jetting rapid prototyping. Measurements of the waveguide power loss factor show good agreement with simulation. As an initial example, a waveguide with propagation loss of 0.03 dB/mm at 105 GHz is demonstrated.

© 2011 OSA

1. Introduction

Research involving the Terahertz (THz) spectrum (100 GHz–10 THz) has experienced dramatic growth recently in both technical achievement and commercial implementation. The growth is application driven, with interest from various fields such as chemical and astronomical spectroscopy and sensing, medical and biological imaging and analysis, defense and security screening, communication networks and radars, etc [1

1. P. H. Siegel, “Terahertz Technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002). [CrossRef]

]. Many of these applications, however, utilize free-space propagation, resulting in weak beam confinement and vulnerability to environmental fluctuations. Focal spot size is diffraction limited and relatively large for the wavelengths of interest [2

2. S. Atakaramians, S. V Afshar, B. M. Fischer, D. Abbott, and T. M. Monro, “Porous fibers: a novel approach to low loss THz waveguides,” Opt. Express 16(12), 8845–8854 (2008). [CrossRef] [PubMed]

], which inconveniently requires large sample areas in material characterization applications and makes system miniaturization challenging. THz waveguides, on the other hand, provide a promising approach to overcome these drawbacks, and may lead to compact and low-cost integrated THz systems.

Various types of THz waveguides based on both electronic and photonic technologies have been proposed and studied, including metallic rectangular/circular [3

3. C. D. Nordquist, M. C. Wanke, A. M. Rowen, C. L. Arrington, M. Lee, and A. D. Grine, “Design, fabrication, and characterization of metal micromachined rectangular waveguides at 3 THz,” in IEEE AP-S Int. Symp. (San Diego, CA, 2008), pp. 1–4.

, 4

4. T. Ito, Y. Matsuura, M. Miyagi, H. Minamide, and H. Ito, “Flexible Terahertz fiber optics with low bend-induced losses,” J. Opt. Soc. Am. B 24(5), 1230–1235 (2007). [CrossRef]

], parallel-plate [5

5. R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001). [CrossRef]

], metal wire [6

6. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004). [CrossRef] [PubMed]

, 7

7. T.-I. Jeon, J. Zhang, and K. W. Goossen, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 1–3 (2005). [CrossRef]

], coaxial transmission line [8

8. T.-I. Jeon and D. Grischkowsky, “Direct optoelectronic generation and detection of sub-ps-electrical pulses on sub-mm-coaxial transmission lines,” Appl. Phys. Lett. 85(25), 6092–6094 (2004). [CrossRef]

], sub-wavelength fiber [2

2. S. Atakaramians, S. V Afshar, B. M. Fischer, D. Abbott, and T. M. Monro, “Porous fibers: a novel approach to low loss THz waveguides,” Opt. Express 16(12), 8845–8854 (2008). [CrossRef] [PubMed]

, 9

9. L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun, “Low-loss subwavelength plastic fiber for Terahertz waveguiding,” Opt. Lett. 31(3), 308–310 (2006). [CrossRef] [PubMed]

11

11. S. Atakaramians, S. V. Afshar, H. Ebendorff-Heidepriem, M. Nagel, B. M. Fischer, D. Abbott, and T. M. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17(16), 14053–15062 (2009). [CrossRef] [PubMed]

], photonic crystal fiber (PCF) [12

12. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as Terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B2B), L317-L319 (2004). [CrossRef]

-15

15. M. Skorobogaity and A. Dupuis, “Ferroelectric all-polymer hollow Bragg fibers for terahertz guidance,” Appl. Phys. Lett. 90(11), 1135141–1135143 (2007).

], etc. However, realization of a THz waveguide with acceptable performance remains a challenge in many regards such as loss, cost, coupling, and especially fabrication and component integration. For most of the reported THz waveguides, fabrication usually involves either photolithography/micromachining work flows [3

3. C. D. Nordquist, M. C. Wanke, A. M. Rowen, C. L. Arrington, M. Lee, and A. D. Grine, “Design, fabrication, and characterization of metal micromachined rectangular waveguides at 3 THz,” in IEEE AP-S Int. Symp. (San Diego, CA, 2008), pp. 1–4.

5

5. R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001). [CrossRef]

, 14

14. J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 064105 1–3 (2008).

, 16

16. W. R. McGrath, C. W. Walker, M. Yap, and Y.-C. Tai, “Silicon micromachined waveguides for millimeter-wave and submillimeter-wave frequencies,” IEEE Microwave Guided Wave Lett. 3(3), 61–63 (1993).

, 17

17. J. W. Digby, C. E. McIntosh, G. M. Parkhurst, and S. R. Davies, “Fabrication and characterization of micromachined rectagular components for use at millimeter and tetrahertz frequencies,” IEEE Trans. Microw. Theory Tech. 48(8), 1293–1302 (2000). [CrossRef]

] or high-temperature and hazardous hot-draw or CVD processes [10

10. A. Dupuis, J.-F. Allard, D. Morris, K. Stoeffler, C. Dubois, and M. Skorobogatiy, “Fabrication and THz loss measurements of porous subwavelength fibers using a directional coupler method,” Opt. Express 17(10), 8012–8028 (2009). [CrossRef] [PubMed]

, 12

12. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as Terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B2B), L317-L319 (2004). [CrossRef]

, 13

13. K. Nielsen, H. K. Rasmussen, A. J. L. Adam, P. C. M. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17(10), 8592–8601 (2009). [CrossRef] [PubMed]

, 15

15. M. Skorobogaity and A. Dupuis, “Ferroelectric all-polymer hollow Bragg fibers for terahertz guidance,” Appl. Phys. Lett. 90(11), 1135141–1135143 (2007).

]. Several of the demonstrated waveguides also required hand assembly [12

12. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as Terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B2B), L317-L319 (2004). [CrossRef]

, 14

14. J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 064105 1–3 (2008).

], which compromises fabrication accuracy and prohibits large scale production. Moreover, integration with other components or variation in waveguide parameters within a single piece of guide remains to be challenging.

In this paper, an all-dielectric waveguide operating near 112 GHz is designed and demonstrated. The waveguide is based on a hollow-core electromagnetic crystal (EMXT) structure [18

18. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: Molding the flow of Light, 2nd ed. (Princeton University Press, Princeton, NJ, 2008).

]. Both eigen-mode and driven-mode simulations are performed to verify its power transmission spectrum and propagation loss. Fabrication of the waveguide is via a novel polymer-jetting rapid prototyping technique [19

19. Z. Wu, J. Kinast, M. E. Gehm, and H. Xin, “Rapid and inexpensive fabrication of terahertz electromagnetic bandgap structures,” Opt. Express 16(21), 16442–16451 (2008). [CrossRef] [PubMed]

]. Due to the layer-by-layer printing nature of this technique, the fabrication is very convenient, accurate, mass-production capable, and highly flexible. Several waveguides of identical cross-section and various lengths were fabricated to enable THz time-domain spectroscopy characterization of the intrinsic power loss factor. Measurement results agree well with simulation, exhibiting a low power loss of 0.03 dB/mm at 105 GHz. This waveguide is quite suitable for narrow-band applications, including space and earth science spectroscopy at atmospheric window frequencies such as 94 GHz and 345 GHz [1

1. P. H. Siegel, “Terahertz Technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002). [CrossRef]

, 20

20. C. E. Honingh, M. M. Dierichs, H. H. Schaeffer, T. M. Klapwijk, and Th. de Graauw, “A 345 GHz waveguide mixer using an array of four Nb-Al-Al2O3-Nb SIS junctions,” Supercond. Sci. Technol. 4(11), 683–685 (1991). [CrossRef]

], as well as screening and imaging applications of chemical or biological agents at their signature frequencies [1

1. P. H. Siegel, “Terahertz Technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002). [CrossRef]

], such as commonly used explosives at 807 GHz [21

21. K. Yamamoto, M. Yamaguchi, F. Miyamaru, M. Tani, M. Hangyo, T. Ikeda, A. Matsushita, K. Koide, M. Tatsuno, and Y. Minami, “Noninvasive inspection of C-4 explosive in mails by terahertz time-domain spectroscopy,” Jpn. J. Appl. Phys. 43(3B3B), L414–L417 (2004). [CrossRef]

]. Moreover, under the scope of the same prototyping fabrication technique, the waveguide aperture can be easily expanded to form an electromagnetic crystal horn antenna, which would greatly enhance the coupling to THz free-space radiation [22

22. Z. Wu, W.-R. Ng, M. Gehm, and H. Xin, “Hollow-core electromagnetic band gap (EBG) waveguide fabricated by rapid prototyping for low-loss Terahertz guiding,” in IEEE MTT-S Int. Microwave Symp. (Anaheim, CA, 2010).

, 23

23. Z. Wu, W.-R. Ng, M. Gehm, and H. Xin, “Terahertz electromagnetic crystal (EMXT) based waveguide and horn antenna,” in 35th Int. Conf. on Infrared, Millimeter and Terahertz Waves, (Rome, Italy, 2010).

]. With proper transition structure from the waveguide to planar circuits, solid-state sources and detectors may be integrated with the waveguide and horn antenna to realize compact THz transceiving systems. Therefore, this waveguide serves as a critical building block in potential integrated THz systems that can be readily implemented by the proposed polymer-jetting rapid prototyping technique.

2. Polymer jetting rapid prototyping

THz electromagnetic crystal and associated components fabrication is still a major challenge remaining to be overcome, as the feature dimensions of THz EMXT components fall in a transition region between the conventional micromachining techniques used for microwave applications and the micro/nano-fabrication methods in use at optical frequencies [24

24. B. Martinez, I. Ederra, R. Gonzalo, B. Alderman, L. Azcona, P. G. Huggard, B. D. Hon, A. Hussain, S. R. Andrews, and L. Marchand, “Manufacturing tolerance analysis, fabrication, and characterization of 3-D submillimeter-wave electromagnetic-band gap crystals,” IEEE Trans. Microw. Theory Tech. 55(4), 672–681 (2007). [CrossRef]

, 25

25. V. M. Lubecke, K. Mizuno, and G. M. Rebeiz, “Micromaching for Terahertz applications,” IEEE Trans. Microw. Theory Tech. 46(11), 1821–1831 (1998). [CrossRef]

]. Several semiconductor fabrication approaches, including dicing saw machining (with minimum feature size reaching 25 μm) [26

26. R. Gonzalo, B. Martinez, C. M. Mann, H. Pellemans, P. H. Bolivar, and P. de Maagt, “A low-cost fabrication technique for symmetrical and asymmetrical layer-by-layer photonic crystals at submillimeter-wave frequencies,” IEEE Trans. Microw. Theory Tech. 50(10), 2384–2392 (2002). [CrossRef]

], wet etching (able to define ~50 μm features) [27

27. E. Öbay, E. Michel, G. Tuttle, R. Biswas, K. M. Ho, J. Bostak, and D. M. Bloom, “Terahertz spectroscopy of three-dimensional photonic bandgap crystals,” Opt. Lett. 10(15), 1155–1157 (1994).

], deep reactive iron etching (DRIE) (gaps of several μm width and ~10 μm height demonstrated) [28

28. F. Laermer and A. Urban, “Challenges, developments and applications of silicon deep reactive ion etching,” Microelectron. Eng. 67–68(1), 349–355 (2003). [CrossRef]

], deep X-ray lithography (tens of micron fabrication resolution) [29

29. G. Kiriakidis and N. Katsarakis, “Fabrication of 2-D and 3-D photonic band-gap crystals in the GHz and THz regions,” Mater. Phys. Mech. 1, 20–26 (2000).

] and laser micromachining (below 10 μm feature size, depending on the laser spot size) [24

24. B. Martinez, I. Ederra, R. Gonzalo, B. Alderman, L. Azcona, P. G. Huggard, B. D. Hon, A. Hussain, S. R. Andrews, and L. Marchand, “Manufacturing tolerance analysis, fabrication, and characterization of 3-D submillimeter-wave electromagnetic-band gap crystals,” IEEE Trans. Microw. Theory Tech. 55(4), 672–681 (2007). [CrossRef]

, 29

29. G. Kiriakidis and N. Katsarakis, “Fabrication of 2-D and 3-D photonic band-gap crystals in the GHz and THz regions,” Mater. Phys. Mech. 1, 20–26 (2000).

], have been reported. For THz micro-structured or porous fiber fabrication, hot-drawn method is generally employed, with the fiber preform prepared by various methods including stacking [14

14. J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 064105 1–3 (2008).

], drilling [13

13. K. Nielsen, H. K. Rasmussen, A. J. L. Adam, P. C. M. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17(10), 8592–8601 (2009). [CrossRef] [PubMed]

] and die extrusion [11

11. S. Atakaramians, S. V. Afshar, H. Ebendorff-Heidepriem, M. Nagel, B. M. Fischer, D. Abbott, and T. M. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17(16), 14053–15062 (2009). [CrossRef] [PubMed]

]. The achieved cross-section feature size for the hot-drawn method is usually on the order of 100 μm.

Several 3-D THz electromagnetic crystal components have been successfully fabricated by this method [19

19. Z. Wu, J. Kinast, M. E. Gehm, and H. Xin, “Rapid and inexpensive fabrication of terahertz electromagnetic bandgap structures,” Opt. Express 16(21), 16442–16451 (2008). [CrossRef] [PubMed]

]. Excellent agreement between simulated and measured THz responses of these components validated the accuracy and fabrication flexibility of this method while simultaneously demonstrating its high-speed and low-cost nature. Advanced extensions to this technique, including systematic integration of various THz components (e.g. antennas, filters and power dividers) and metallization of finished components are currently being explored as possible routes to integrated THz systems.

3. Electromagnetic crystal waveguide design

The waveguide structure of interest is based on a central air core channel in a surrounding triangular-lattice of air cylinders in a dielectric background. Because of the Bragg diffraction in the lattice, this structure exhibits electromagnetic band gaps in certain frequency bands. Within the band gaps, wave propagation is prohibited and therefore the structure is able to confine the wave propagation. If an air channel is present, this structure will support wave propagation along the channel in the band gap frequency, whereas the radiation loss along the propagation path will be greatly suppressed because of the electromagnetic band gap cladding. More importantly, the wave suffers much less loss than in conventional optical fiber because the majority of the power propagates in the central air-filled channel [18

18. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: Molding the flow of Light, 2nd ed. (Princeton University Press, Princeton, NJ, 2008).

]. Therefore, this type of waveguides is promising as a route to achieving extremely low transmission loss.

3.1 Eigen-mode simulations

Figure 1
Fig. 1 Band diagram of the complete electromagnetic crystal lattice and the hollow-core waveguide modes. For the electromagnetic crystal lattice, dark (blue online) regions denote the continuous modes region whereas in white regions no mode exists (band gaps). Inset (left) shows the schematic of the waveguide cross section when a center air core channel is introduced. Waveguide modes are marked by the dots (red) in the “Band Gap 1” region, among which the energy intensity profile of the fundamental HE 11 mode is shown in the inset (right).
is the band diagram of the complete electromagnetic crystal lattice with hollow-core waveguide mode marked, calculated using the MIT Photonic Bandgap package (MPB) which utilizes a plane-wave expansion method [30

30. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

]. Dark (blue online) regions on the band diagram consist of continuous modes of the electromagnetic crystal lattice. White regions denote where no modes exist, therefore opening several band gaps. As shown in the leftmost inset, a triangular-lattice arrangement of small air cylinders (grey) are embedded in a dielectric background (black) with dielectric constant of 2.75. An air channel is present at the center. The cladding cylinders have a radius of 0.47a with a being the lattice constant, and the air core has a radius of 1.4a. This channel results in a number of waveguide modes within the band gaps of the complete electromagnetic crystal lattice, as denoted by the dots (red online) in Fig. 1. Note that only modes beyond the light line (green online) are capable of propagation. As the band gap of the cladding rings suppresses propagation transverse to the axis, these waveguide modes create efficient energy confinement and propagation along the central core.

The rightmost inset of the figure shows the energy distribution of the fundamental HE 11 mode within the first electromagnetic band gap, obtained from an effective-index-based eigen-mode simulation using the Lumerical MODE Solutions package [31

31. Lumerical MODE Solutions package, v2.3.

]. The MPB simulation reveals that approximately 91% of the energy is concentrated within the central channel for this propagating mode, with an intensity profile following approximately Gaussian distribution. A simple mode-matching argument then indicates that efficient excitation of this propagating mode should be achievable with an incident, linearly-polarized Gaussian beam [32

32. P. F. Goldsmith, Quasioptical systems: Gaussian beam quasioptical propagation and applications (IEEE Press,Piscataway, NJ, 1997).

].

The waveguide design is scalable in frequency as the normalized axis of the band diagram in Fig. 1 illustrates. In the practical design considered here, a lattice constant of 3 mm is used, resulting in a center core radius of 4.2 mm. The cladding air cylinder radius is slightly reduced to 1.3 mm. The dielectric wall thickness between air cylinders is then 400 µm at its thinnest, providing reasonable mechanical robustness for the initial trial. The dielectric properties of the polymer used in the rapid prototyping process was previously characterized by THz time-domain spectroscopy in our earlier work. From that study, we observed that it has a dielectric constant of 2.75 and loss tangent of 0.02 near 150 GHz [19

19. Z. Wu, J. Kinast, M. E. Gehm, and H. Xin, “Rapid and inexpensive fabrication of terahertz electromagnetic bandgap structures,” Opt. Express 16(21), 16442–16451 (2008). [CrossRef] [PubMed]

]. Using these properties, the simulated operating frequency of the fundamental HE 11 mode is approximately 112 GHz. With these practical design parameters, the intrinsic radiation loss of the HE 11 mode at 112 GHz (ignoring the material loss) was calculated to be 0.012 dB/mm using Lumerical MODE Solutions.

3.2 Driven mode simulations

In order to estimate the overall propagation loss of the waveguide design including the material loss and effects from other higher-order modes, we then simulated the waveguide under two different excitations, using a finite-difference- time-domain based package (GEMS [33

33. W.-H. Yu, Y.-J. Liu, T. Su, H. Neng-Tien, and M. Raj, “A robust parallel conformal finite difference time domain processing package using MPI library,” IEEE Ant. Propag. Mag. 47(3), 39–59 (2005). [CrossRef]

]). The first excitation employed two circular perfect-electric-conductor waveguide feeds inserted into the dielectric waveguide apertures at both ends. Two wave ports excited the TE 11 mode in the feeds, which then propagated through an 84-mm long electromagnetic crystal waveguide.

Figure 2
Fig. 2 Simulated S-parameters of an 84 mm-long electromagnetic crystal waveguide.
depicts the resulting S-parameters. Four passbands are observed around 112, 130, 161 and 182 GHz, corresponding to the band gaps of the original electromagnetic crystal lattice. The field profile in the waveguide at 112 GHz resembles the desired HE 11 mode as shown in Fig. 1. The 3-dB bandwidth of the first passband centered at 112 GHz is about 15.8 GHz, giving a full bandwidth to mid-gap frequency ratio of 14.1%. The reflection coefficient S 11 is generally below −30 dB, indicating a very good coupling from the feeding TE 11 mode to the electromagnetic crystal waveguide modes. Once the S-parameters were obtained, the waveguide power loss factor was calculated by the following equation:
α=ln(|S21|21|S11|2)/(l),
(1)
where l is the waveguide length.

In a practical experiment, circular metallic waveguide feeds would be inconvenient due to cost and issues in calibrating their losses. Instead, quasi-optical measurement with an incident THz Gaussian beam is preferred. The second simulation is carried out accordingly. In this case, a Gaussian beam of 3-mm beam waist was aimed at the electromagnetic crystal waveguide aperture, and the power flux transmitted out of the other end of the waveguide was calculated. The power flux calculation was performed by integrating the Poynting vector component along the longitudinal direction, S z, over a surface covering the cross-sectional area of the waveguide. Since the coupling coefficients at the incidence and output ends are identical and independent of the waveguide length, the transmitted power flux has an exponentially decaying relation with waveguide length. A semi-log plot of the transmitted power flux (in dB) versus waveguide length should then be a straight line with a negative slope, which is the intrinsic power loss factor of the waveguide.

This conclusion only holds, however, when multiple reflections within the waveguide are excluded either via time-gating or have negligible magnitudes. Time-domain gating is in principle not quite suitable for electromagnetic crystal waveguide, because it is a highly dispersive system with narrow band features. Its time-domain responses therefore decay very slowly indicating that the multiple reflections mix together, making isolation difficult. However, the assumption that multiple reflections are of negligible magnitude is likely true in our case, as the energy is incident from free space into the air core, and couples to the waveguide propagation modes with an effective refractive index very close to 1 according to Lumerical MODE simulation.

Four waveguides of lengths of 100, 110, 120 and 130 mm were simulated under the same Gaussian beam excitation, and the transmitted power flux in the passbands does indeed show the expected exponential decay as the waveguide length increases. The extracted waveguide loss using these two simulation methods are compared in Fig. 3
Fig. 3 Power loss factors of the electromagnetic crystal waveguide extracted from the wave port (TE 11 circular perfect-electric-conductor waveguide feeds) and the Gaussian beam incidence simulations.
. The two loss spectra show very good consistency in terms of frequency and depth of the low-loss bands. The lowest propagation loss obtained is 0.022 dB/mm at 112 GHz, which corresponds to a quality (Q) factor of 463 obtained from the formula Q = β/(2α’), where β is the wave number and α’ = α/2 is the field attenuation constant, both in linear units. Therefore, the material loss and potential losses from higher-order modes bring in an extra of 0.01 dB/mm loss at this frequency, when compared to the eigen-mode simulation result of the fundamental HE 11 mode in the last section. Although 5 periods of the cladding lattice was used in the design and practically fabricated later on, simulation proves that 3 periods are adequate to achieve almost identical low-loss propagation.

4. Experiment

Fabrication of the waveguide was carried out by the aforementioned polymer jetting technique. The structure was printed vertically along the wave channel direction. In this fabrication, the glossy mode with reduced support material usage was applied to minimize surface roughness, as only limited support structure was needed. Figure 4
Fig. 4 Cross-sectional and full views of a fabricated THz waveguide, with lattice constant of 3 mm, center core radius of 4.2 mm, and cladding air cylinder radius of 1.3 mm.
shows photos of the cross-sectional and full view of a fabricated waveguide.

Figure 5
Fig. 5 Setup of the characterization experiment: from left to right: THz transmitter, parabolic mirror 1, iris, polymer lens 1, waveguide under test, polymer lens 2, parabolic mirror 2, and THz receiver. THz time-domain spectrometer used here: T-Ray 2000 turnkey THz system manufactured by Picometrix, Inc.
shows a photo of the quasi-optic setup for waveguide characterization. The iris in the photo was only for the purpose of marking the beam position and opens sufficiently for the entire beam to pass through unobstructed. As discussed previously, waveguides of several lengths needed to be measured in order to characterize the propagation loss factor, similar to the “cut-back” method used in optical fiber loss measurement [35

35. J. C. Daly, Fiber Optics (CRC Press, Boca Raton, Florida, 2000).

]. Therefore, the spacing between all elements and the waveguide ends were kept the same throughout the measurements to provide an identical optical path with the exception of the waveguide section.

Five waveguides of lengths 50, 75, 100, 125, and 150 mm and identical cross section were placed in the beam path sequentially, and their transmitted waveforms were measured. The two lenses were also positioned 35 mm (17.5 mm x 2) away without a waveguide in between to obtain a reference scan. The spectrometer was set to operate in the long scan mode, covering the entire available time delay range of 1200 pico-second (ps) with a scan speed of 5 ps per second. Each recorded scan was an average of five consecutive scans. The resulted waveforms are shown in Fig. 6(a)
Fig. 6 (a) Transmitted waveforms of five waveguides with different lengths, and the reference scan transmitted through free space. (b) Normalized power transmission of the waveguides.
. The reference scan leads in time, followed by the five waveguide scans with increased delays from the longer waveguides. Dispersion with respect to the reference waveform is observed in all five waveguide scans, which is a clear sign of guided mode resonances. Figure 6(b) plots the power transmittances of all the waveguides, which were obtained by Fourier transforming the waveforms to the frequency domain, then normalizing the spectra with respect to the reference spectrum. Four passbands centered around 105, 123, 153 and 174 GHz can be clearly seen, corresponding to the four passbands in Fig. 2 to within a downshift of a few GHz.

As shown in Fig. 7(b), the measured power loss factors agree quite well with the beam-incidence simulation results. The measured 7-GHz downshift is possibly due to fabrication tolerance, as the air cylinder diameter may not be precisely controlled due to the reduced support material usage in the glossy fabrication mode. Further, the model material has slight dielectric constant dispersion (from 2.8 to 2.73) within the 100-150 GHz range [19

19. Z. Wu, J. Kinast, M. E. Gehm, and H. Xin, “Rapid and inexpensive fabrication of terahertz electromagnetic bandgap structures,” Opt. Express 16(21), 16442–16451 (2008). [CrossRef] [PubMed]

], which was not included in the simulations and would also contribute to the discrepancy. Note that the dip at 143 GHz is not reliable because as it is at a deep band-stop frequency (as shown in Fig. 6(b)) for which the signal-to-noise-ratio is not good. Nevertheless, the measurement and the simulation agree well, demonstrating four passbands with low propagation losses at nearby frequencies. The lowest observed power loss factor was 0.03 dB/mm at 105.2 GHz, corresponding to a Q factor of 319 at this frequency. As a reference, a standard W-band rectangular waveguide (WR-10, 75-110 GHz) made of copper has a theoretical power attenuation factor of 0.0023-0.0035 dB/mm. However, the practically measured W-band rectangular waveguide loss reaches 0.01-0.015 dB/mm in [16

16. W. R. McGrath, C. W. Walker, M. Yap, and Y.-C. Tai, “Silicon micromachined waveguides for millimeter-wave and submillimeter-wave frequencies,” IEEE Microwave Guided Wave Lett. 3(3), 61–63 (1993).

] and even 0.05-0.07 dB/mm in [17

17. J. W. Digby, C. E. McIntosh, G. M. Parkhurst, and S. R. Davies, “Fabrication and characterization of micromachined rectagular components for use at millimeter and tetrahertz frequencies,” IEEE Trans. Microw. Theory Tech. 48(8), 1293–1302 (2000). [CrossRef]

], which are 5 ~ 20 times of the theoretical value. Other reported THz micro-structured waveguides working at similar frequency regime include the sub-wavelength porous fiber by Dupuis et al. in [10

10. A. Dupuis, J.-F. Allard, D. Morris, K. Stoeffler, C. Dubois, and M. Skorobogatiy, “Fabrication and THz loss measurements of porous subwavelength fibers using a directional coupler method,” Opt. Express 17(10), 8012–8028 (2009). [CrossRef] [PubMed]

], hollow-core photonic crystal fiber by Nielson et al. in [13

13. K. Nielsen, H. K. Rasmussen, A. J. L. Adam, P. C. M. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17(10), 8592–8601 (2009). [CrossRef] [PubMed]

], solid-core PCF by Goto et al. in [12

12. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as Terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B2B), L317-L319 (2004). [CrossRef]

], etc. The porous fiber demonstrated a very low power loss factor of 0.004 dB/mm at 300 GHz, whereas the solid-core PCF reported a ~0.05 dB/mm loss factor mainly due to material loss. THz micro-porous guidance was claimed to be observed at 200 GHz in [13

13. K. Nielsen, H. K. Rasmussen, A. J. L. Adam, P. C. M. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17(10), 8592–8601 (2009). [CrossRef] [PubMed]

], however the loss value was not reported. While certainly not presenting the lowest loss coefficient for waveguides at 100 – 300 GHz range, the waveguide reported in this work finds its own advantage in the ease and flexibility of the fabrication process. Moreover, with optimized polymer materials, lower loss should be achieved.

5. Conclusion

A THz waveguide based on a hollow-core electromagnetic crystal structure was designed and then fabricated by polymer jetting rapid prototyping. Quasi-optical THz lenses were also fabricated by the same technique in help with mode-matching the waveguide to free-space Gaussian beams. Results of the power loss factor characterization showed good agreement with simulation. A low transmission loss waveguide operating near 105 GHz was demonstrated with minimal fabrication complexity and cost. Scaling the waveguide to a few hundreds of GHz is feasible under the scope of this fabrication method, and is currently pursued in our lab. Because of the systematic fabrication manor of the rapid prototyping technique, this waveguide can be readily integrated with other THz components such as electromagnetic crystal based horn antennas, THz planar circuits (so that semiconductor based THz source and detector can be readily integrated), and other quasi-optical parts fabricated by the same approach. We anticipate that this will ultimately lead to integrated fully-functional THz systems.

Acknowledgments

This work was supported in part by the National Science Foundation under Awards 0823864 and 0925220, and the state of Arizona under TRIF.

References and links

1.

P. H. Siegel, “Terahertz Technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002). [CrossRef]

2.

S. Atakaramians, S. V Afshar, B. M. Fischer, D. Abbott, and T. M. Monro, “Porous fibers: a novel approach to low loss THz waveguides,” Opt. Express 16(12), 8845–8854 (2008). [CrossRef] [PubMed]

3.

C. D. Nordquist, M. C. Wanke, A. M. Rowen, C. L. Arrington, M. Lee, and A. D. Grine, “Design, fabrication, and characterization of metal micromachined rectangular waveguides at 3 THz,” in IEEE AP-S Int. Symp. (San Diego, CA, 2008), pp. 1–4.

4.

T. Ito, Y. Matsuura, M. Miyagi, H. Minamide, and H. Ito, “Flexible Terahertz fiber optics with low bend-induced losses,” J. Opt. Soc. Am. B 24(5), 1230–1235 (2007). [CrossRef]

5.

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001). [CrossRef]

6.

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004). [CrossRef] [PubMed]

7.

T.-I. Jeon, J. Zhang, and K. W. Goossen, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 1–3 (2005). [CrossRef]

8.

T.-I. Jeon and D. Grischkowsky, “Direct optoelectronic generation and detection of sub-ps-electrical pulses on sub-mm-coaxial transmission lines,” Appl. Phys. Lett. 85(25), 6092–6094 (2004). [CrossRef]

9.

L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun, “Low-loss subwavelength plastic fiber for Terahertz waveguiding,” Opt. Lett. 31(3), 308–310 (2006). [CrossRef] [PubMed]

10.

A. Dupuis, J.-F. Allard, D. Morris, K. Stoeffler, C. Dubois, and M. Skorobogatiy, “Fabrication and THz loss measurements of porous subwavelength fibers using a directional coupler method,” Opt. Express 17(10), 8012–8028 (2009). [CrossRef] [PubMed]

11.

S. Atakaramians, S. V. Afshar, H. Ebendorff-Heidepriem, M. Nagel, B. M. Fischer, D. Abbott, and T. M. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17(16), 14053–15062 (2009). [CrossRef] [PubMed]

12.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as Terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B2B), L317-L319 (2004). [CrossRef]

13.

K. Nielsen, H. K. Rasmussen, A. J. L. Adam, P. C. M. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17(10), 8592–8601 (2009). [CrossRef] [PubMed]

14.

J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 064105 1–3 (2008).

15.

M. Skorobogaity and A. Dupuis, “Ferroelectric all-polymer hollow Bragg fibers for terahertz guidance,” Appl. Phys. Lett. 90(11), 1135141–1135143 (2007).

16.

W. R. McGrath, C. W. Walker, M. Yap, and Y.-C. Tai, “Silicon micromachined waveguides for millimeter-wave and submillimeter-wave frequencies,” IEEE Microwave Guided Wave Lett. 3(3), 61–63 (1993).

17.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, and S. R. Davies, “Fabrication and characterization of micromachined rectagular components for use at millimeter and tetrahertz frequencies,” IEEE Trans. Microw. Theory Tech. 48(8), 1293–1302 (2000). [CrossRef]

18.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: Molding the flow of Light, 2nd ed. (Princeton University Press, Princeton, NJ, 2008).

19.

Z. Wu, J. Kinast, M. E. Gehm, and H. Xin, “Rapid and inexpensive fabrication of terahertz electromagnetic bandgap structures,” Opt. Express 16(21), 16442–16451 (2008). [CrossRef] [PubMed]

20.

C. E. Honingh, M. M. Dierichs, H. H. Schaeffer, T. M. Klapwijk, and Th. de Graauw, “A 345 GHz waveguide mixer using an array of four Nb-Al-Al2O3-Nb SIS junctions,” Supercond. Sci. Technol. 4(11), 683–685 (1991). [CrossRef]

21.

K. Yamamoto, M. Yamaguchi, F. Miyamaru, M. Tani, M. Hangyo, T. Ikeda, A. Matsushita, K. Koide, M. Tatsuno, and Y. Minami, “Noninvasive inspection of C-4 explosive in mails by terahertz time-domain spectroscopy,” Jpn. J. Appl. Phys. 43(3B3B), L414–L417 (2004). [CrossRef]

22.

Z. Wu, W.-R. Ng, M. Gehm, and H. Xin, “Hollow-core electromagnetic band gap (EBG) waveguide fabricated by rapid prototyping for low-loss Terahertz guiding,” in IEEE MTT-S Int. Microwave Symp. (Anaheim, CA, 2010).

23.

Z. Wu, W.-R. Ng, M. Gehm, and H. Xin, “Terahertz electromagnetic crystal (EMXT) based waveguide and horn antenna,” in 35th Int. Conf. on Infrared, Millimeter and Terahertz Waves, (Rome, Italy, 2010).

24.

B. Martinez, I. Ederra, R. Gonzalo, B. Alderman, L. Azcona, P. G. Huggard, B. D. Hon, A. Hussain, S. R. Andrews, and L. Marchand, “Manufacturing tolerance analysis, fabrication, and characterization of 3-D submillimeter-wave electromagnetic-band gap crystals,” IEEE Trans. Microw. Theory Tech. 55(4), 672–681 (2007). [CrossRef]

25.

V. M. Lubecke, K. Mizuno, and G. M. Rebeiz, “Micromaching for Terahertz applications,” IEEE Trans. Microw. Theory Tech. 46(11), 1821–1831 (1998). [CrossRef]

26.

R. Gonzalo, B. Martinez, C. M. Mann, H. Pellemans, P. H. Bolivar, and P. de Maagt, “A low-cost fabrication technique for symmetrical and asymmetrical layer-by-layer photonic crystals at submillimeter-wave frequencies,” IEEE Trans. Microw. Theory Tech. 50(10), 2384–2392 (2002). [CrossRef]

27.

E. Öbay, E. Michel, G. Tuttle, R. Biswas, K. M. Ho, J. Bostak, and D. M. Bloom, “Terahertz spectroscopy of three-dimensional photonic bandgap crystals,” Opt. Lett. 10(15), 1155–1157 (1994).

28.

F. Laermer and A. Urban, “Challenges, developments and applications of silicon deep reactive ion etching,” Microelectron. Eng. 67–68(1), 349–355 (2003). [CrossRef]

29.

G. Kiriakidis and N. Katsarakis, “Fabrication of 2-D and 3-D photonic band-gap crystals in the GHz and THz regions,” Mater. Phys. Mech. 1, 20–26 (2000).

30.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

31.

Lumerical MODE Solutions package, v2.3.

32.

P. F. Goldsmith, Quasioptical systems: Gaussian beam quasioptical propagation and applications (IEEE Press,Piscataway, NJ, 1997).

33.

W.-H. Yu, Y.-J. Liu, T. Su, H. Neng-Tien, and M. Raj, “A robust parallel conformal finite difference time domain processing package using MPI library,” IEEE Ant. Propag. Mag. 47(3), 39–59 (2005). [CrossRef]

34.

G. Gruner, ed., Millimeter and Submillimeter Wave Spectroscopy of Solids (Springer, Berlin, German, 1998).

35.

J. C. Daly, Fiber Optics (CRC Press, Boca Raton, Florida, 2000).

OCIS Codes
(220.4000) Optical design and fabrication : Microstructure fabrication
(260.3090) Physical optics : Infrared, far
(050.5298) Diffraction and gratings : Photonic crystals
(050.6875) Diffraction and gratings : Three-dimensional fabrication
(130.5460) Integrated optics : Polymer waveguides

ToC Category:
Photonic Crystals

History
Original Manuscript: November 2, 2010
Revised Manuscript: February 2, 2011
Manuscript Accepted: February 2, 2011
Published: February 15, 2011

Citation
Ziran Wu, Wei-Ren Ng, Michael E. Gehm, and Hao Xin, "Terahertz electromagnetic crystal waveguide fabricated by polymer jetting rapid prototyping," Opt. Express 19, 3962-3972 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-3962


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References

  1. P. H. Siegel, “Terahertz Technology,” IEEE Trans. Microw. Theory Tech. 50(3), 910–928 (2002). [CrossRef]
  2. S. Atakaramians, S. V Afshar, B. M. Fischer, D. Abbott, and T. M. Monro, “Porous fibers: a novel approach to low loss THz waveguides,” Opt. Express 16(12), 8845–8854 (2008). [CrossRef] [PubMed]
  3. C. D. Nordquist, M. C. Wanke, A. M. Rowen, C. L. Arrington, M. Lee, and A. D. Grine, “Design, fabrication, and characterization of metal micromachined rectangular waveguides at 3 THz,” in IEEE AP-S Int. Symp. (San Diego, CA, 2008), pp. 1–4.
  4. T. Ito, Y. Matsuura, M. Miyagi, H. Minamide, and H. Ito, “Flexible Terahertz fiber optics with low bend-induced losses,” J. Opt. Soc. Am. B 24(5), 1230–1235 (2007). [CrossRef]
  5. R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001). [CrossRef]
  6. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004). [CrossRef] [PubMed]
  7. T.-I. Jeon, J. Zhang, and K. W. Goossen, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 1–3 (2005). [CrossRef]
  8. T.-I. Jeon and D. Grischkowsky, “Direct optoelectronic generation and detection of sub-ps-electrical pulses on sub-mm-coaxial transmission lines,” Appl. Phys. Lett. 85(25), 6092–6094 (2004). [CrossRef]
  9. L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun, “Low-loss subwavelength plastic fiber for Terahertz waveguiding,” Opt. Lett. 31(3), 308–310 (2006). [CrossRef] [PubMed]
  10. A. Dupuis, J.-F. Allard, D. Morris, K. Stoeffler, C. Dubois, and M. Skorobogatiy, “Fabrication and THz loss measurements of porous subwavelength fibers using a directional coupler method,” Opt. Express 17(10), 8012–8028 (2009). [CrossRef] [PubMed]
  11. S. Atakaramians, S. V. Afshar, H. Ebendorff-Heidepriem, M. Nagel, B. M. Fischer, D. Abbott, and T. M. Monro, “THz porous fibers: design, fabrication and experimental characterization,” Opt. Express 17(16), 14053–15062 (2009). [CrossRef] [PubMed]
  12. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as Terahertz waveguide,” Jpn. J. Appl. Phys. 43(2B2B), L317-L319 (2004). [CrossRef]
  13. K. Nielsen, H. K. Rasmussen, A. J. L. Adam, P. C. M. Planken, O. Bang, and P. U. Jepsen, “Bendable, low-loss Topas fibers for the terahertz frequency range,” Opt. Express 17(10), 8592–8601 (2009). [CrossRef] [PubMed]
  14. J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92(6), 064105 1–3 (2008).
  15. M. Skorobogaity and A. Dupuis, “Ferroelectric all-polymer hollow Bragg fibers for terahertz guidance,” Appl. Phys. Lett. 90(11), 1135141–1135143 (2007).
  16. W. R. McGrath, C. W. Walker, M. Yap, and Y.-C. Tai, “Silicon micromachined waveguides for millimeter-wave and submillimeter-wave frequencies,” IEEE Microwave Guided Wave Lett. 3(3), 61–63 (1993).
  17. J. W. Digby, C. E. McIntosh, G. M. Parkhurst, and S. R. Davies, “Fabrication and characterization of micromachined rectagular components for use at millimeter and tetrahertz frequencies,” IEEE Trans. Microw. Theory Tech. 48(8), 1293–1302 (2000). [CrossRef]
  18. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: Molding the flow of Light, 2nd ed. (Princeton University Press, Princeton, NJ, 2008).
  19. Z. Wu, J. Kinast, M. E. Gehm, and H. Xin, “Rapid and inexpensive fabrication of terahertz electromagnetic bandgap structures,” Opt. Express 16(21), 16442–16451 (2008). [CrossRef] [PubMed]
  20. C. E. Honingh, M. M. Dierichs, H. H. Schaeffer, T. M. Klapwijk, and Th. de Graauw, “A 345 GHz waveguide mixer using an array of four Nb-Al-Al2O3-Nb SIS junctions,” Supercond. Sci. Technol. 4(11), 683–685 (1991). [CrossRef]
  21. K. Yamamoto, M. Yamaguchi, F. Miyamaru, M. Tani, M. Hangyo, T. Ikeda, A. Matsushita, K. Koide, M. Tatsuno, and Y. Minami, “Noninvasive inspection of C-4 explosive in mails by terahertz time-domain spectroscopy,” Jpn. J. Appl. Phys. 43(3B3B), L414–L417 (2004). [CrossRef]
  22. Z. Wu, W.-R. Ng, M. Gehm, and H. Xin, “Hollow-core electromagnetic band gap (EBG) waveguide fabricated by rapid prototyping for low-loss Terahertz guiding,” in IEEE MTT-S Int. Microwave Symp. (Anaheim, CA, 2010).
  23. Z. Wu, W.-R. Ng, M. Gehm, and H. Xin, “Terahertz electromagnetic crystal (EMXT) based waveguide and horn antenna,” in 35th Int. Conf. on Infrared, Millimeter and Terahertz Waves, (Rome, Italy, 2010).
  24. B. Martinez, I. Ederra, R. Gonzalo, B. Alderman, L. Azcona, P. G. Huggard, B. D. Hon, A. Hussain, S. R. Andrews, and L. Marchand, “Manufacturing tolerance analysis, fabrication, and characterization of 3-D submillimeter-wave electromagnetic-band gap crystals,” IEEE Trans. Microw. Theory Tech. 55(4), 672–681 (2007). [CrossRef]
  25. V. M. Lubecke, K. Mizuno, and G. M. Rebeiz, “Micromaching for Terahertz applications,” IEEE Trans. Microw. Theory Tech. 46(11), 1821–1831 (1998). [CrossRef]
  26. R. Gonzalo, B. Martinez, C. M. Mann, H. Pellemans, P. H. Bolivar, and P. de Maagt, “A low-cost fabrication technique for symmetrical and asymmetrical layer-by-layer photonic crystals at submillimeter-wave frequencies,” IEEE Trans. Microw. Theory Tech. 50(10), 2384–2392 (2002). [CrossRef]
  27. E. Öbay, E. Michel, G. Tuttle, R. Biswas, K. M. Ho, J. Bostak, and D. M. Bloom, “Terahertz spectroscopy of three-dimensional photonic bandgap crystals,” Opt. Lett. 10(15), 1155–1157 (1994).
  28. F. Laermer and A. Urban, “Challenges, developments and applications of silicon deep reactive ion etching,” Microelectron. Eng. 67–68(1), 349–355 (2003). [CrossRef]
  29. G. Kiriakidis and N. Katsarakis, “Fabrication of 2-D and 3-D photonic band-gap crystals in the GHz and THz regions,” Mater. Phys. Mech. 1, 20–26 (2000).
  30. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]
  31. Lumerical MODE Solutions package, v2.3.
  32. P. F. Goldsmith, Quasioptical systems: Gaussian beam quasioptical propagation and applications (IEEE Press,Piscataway, NJ, 1997).
  33. W.-H. Yu, Y.-J. Liu, T. Su, H. Neng-Tien, and M. Raj, “A robust parallel conformal finite difference time domain processing package using MPI library,” IEEE Ant. Propag. Mag. 47(3), 39–59 (2005). [CrossRef]
  34. G. Gruner, ed., Millimeter and Submillimeter Wave Spectroscopy of Solids (Springer, Berlin, German, 1998).
  35. J. C. Daly, Fiber Optics (CRC Press, Boca Raton, Florida, 2000).

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