A tunable terahertz filter and its switching properties in terahertz region based on a
defect mode of a metallic photonic crystal
Yong Sung Kim, Shawn-Yu Lin, Hsin-Ying Wu, and Ru-Pin Pan
Citation: Journal of Applied Physics 109, 123111 (2011); doi: 10.1063/1.3603009
View online: http://dx.doi.org/10.1063/1.3603009
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/12?ver=pdfcov Published by the AIP Publishing
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A tunable terahertz filter and its switching properties in terahertz region
based on a defect mode of a metallic photonic crystal
Yong Sung Kim,1Shawn-Yu Lin,1,a)Hsin-Ying Wu,2and Ru-Pin Pan2 1
The Future Chips Constellation and Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
2
Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan 30010, People’s Republic of China
(Received 24 January 2011; accepted 17 May 2011; published online 28 June 2011)
We theoretically investigate and discuss an electrically tunable terahertz filter design and its optical switching properties based on the defect mode of a woodpile metallic photonic crystal (MPC). The model filter design is based on a dual use of an MPC as a resonator and as electrodes with a liquid crystal used as a defect layer. The static and the dynamic responses of a realistic liquid crystal are obtained using the Oseen–Frank elastic continuum theory, and the corresponding transmission spectra are calculated using an analytic modal expansion method combined with a transfer-matrix method. The tuning range off¼ 1.4301.577 THz and the order of milliseconds switching property are observed in our design.VC 2011 American Institute of Physics. [doi:10.1063/1.3603009]
I. INTRODUCTION
Recently, terahertz (THz) technology has attracted a grow-ing attention because its potential applications such as medical and security imaging, communication, etc. THz radiation uses low energy sources (less invasive), and it can pass through many non-metallic materials such as clothing, paper, plastic, and ceramics. These make THz imaging attractive in medical and security imaging areas.1 Also, wireless short range THz communication systems are expected to be developed in near future due to the increased demand of communication band-width.2For high-resolution THz imaging applications, widely tunable high-efficient narrowband THz filters are necessary to improve image quality.1To develop a communication system operating at THz frequencies, it is necessary to develop THz optical components such as emitters, detectors, frequency fil-ters, and switches working in the THz range.2Therefore for further advances in the above applications, widely tunable THz filters and switches are commonly required.
There have been several studies to develop tunable THz filters and switches. Nemec et al. have demonstrated the thermally tunable THz filters using defect modes in a one-dimensional photonic crystal.3 Rivas et al. have demon-strated the thermal switching of THz transmission through a subwavelength two-dimensional (2 D) hole array.4Drysdale et al. have reported a continuously tunable THz filter using three-dimensional metallic photonic crystals (MPC) with mechanical shifting of two MPC plates.5 Pan et al. have demonstrated a tuning property of THz transmission using the magnetic birefringence of liquid crystal.6,7 Zhang et al. have also reported the theoretical design of THz switch and filter using the magnetic birefringence of 2 D liquid-crystal-filled photonic crystal.8 In addition, Taylor and Chen have demonstrated ultrafast switching in THz regime using metamaterials.9
In this paper, we propose a THz filter using the defect modes of a three-dimensional metallic woodpile photonic crystal. For the defect layer, realistic physical parameters of a commercially available liquid crystal, MLC-2048 (Merck), is considered.
II. DEFECT MODES OF A METALLIC PHOTONIC CRYSTAL WITH LIQUID CRYSTALS
Photonic crystals are three-dimensional periodic struc-tures that can produce a photonic bandgap where the propa-gation of electro-magnetic (EM) waves can be forbidden for a certain range of frequencies. Especially, a three-dimensional metallic photonic crystal (MPC) has a wider photonic bandgap than its dielectric counterparts due to a greatly enhanced contrast in the dielectric constant of the composite materials.10By introducing an appropriate defect layer into a photonic crystal, a mode can exist in the pho-tonic bandgap. This mode is called a defect mode and widely studied for potential applications such as lasers, res-onators, etc.11–16The frequency of the defect mode can be tuned by changing the thickness or the refractive index of the defect layer.
A liquid crystal (LC) has two principal refractive indi-ces, ordinary refractive index noand extraordinary refractive
index ne. The ordinary index is measured for the EM wave
where the electric vector vibrates perpendicular to the optical axis. The extraordinary index is measured for the EM wave where the electric vector vibrates parallel to the optical axis. An appropriately treated surface can make LC molecules align in a specific direction near the surface, and the align-ment of molecules at the surface propagates over macro-scopic distances. An applied bias voltage can change the average alignment direction of the LC molecules (director n) as shown in the inset of Fig. 4. This can eventually change the effective refractive index of the whole LC layer. Thus the frequency of the defect mode can be tuned by applying bias voltage.
a)Author to whom correspondence should be addressed. Electronic mail: sylin@rpi.
III. DESIGN FOR THZFILTERS AND EFFECTIVE INDEX
OF A LIQUID CRYSTAL
In Fig.1(b)a schematic model of a woodpile MPC with a defect layer and the calculated optical spectra are presented. In the calculation, an analytic modal expansion method com-bined with a transfer-matrix method17and a perfect conductor approximation is used.18The MPC is consists of four layers of metallic gratings that are stacked with 90 degree rotation and a half of rod-to-rod spacing shift. The rod-width,w¼ 30 lm, the rod-to-rod spacing,a¼ 65 lm, and the thickness, h ¼ 5 lm, are designed to yield a photonic bandgap in the range of 0.5 THz <f < 3.0 THz (see Fig. 1(a)) along the z-direction that is perpendicular to each metallic layer. The space between the metallic rods in each layer is assumed to be filled with a low-loss dielectric material such as high density poly-ethylene (n 1.54, k 0.003)19 to confine the liquid crystal
defect layer inside the MPC. A liquid crystal defect layer is introduced between the second and the third layers. In the cal-culation, the refractive index of the defect layer isn¼ 1.675 andk¼ 0 and the thickness is 50 lm. The thickness is
deter-mined by considering a resonance condition representing a standing wave in the defect layer. The cavity length, the thick-ness of the defect layer, is approximatelyl c/(2nf), where c is the speed of light, n is the refractive index of the defect layer, andf is the frequency of an incident EM wave.
The MPC in the design also serve as electrodes to apply bias voltage to the LC layer as depicted in Fig.2(a). To con-trol the refractive index of the LC layer appropriately, it is necessary for the MPC layers to generate a laterally uniform electric potential distribution at a certain depth in the LC defect layer. To see the uniformity of the potential distribu-tion, the electric potential is calculated using a commercial software, COMSOL MULTIPHYSICS, based on the finite
ele-ment method. For this calculation, 5 5-unit-cell-size do-main and electrostatics mode are used. The calculated lateral distribution of electric potential is nearly uniform at a fixed depth throughout the LC layer as shown in Fig.2(b).
With these calculated electric potential distribution and the parameters of an LC, the averaged effective refractive index of the liquid crystal layer, neff, can be calculated. Based
on the Oseen–Frank elastic continuum theory, the total free energy densityg is given by
g¼ ðK11cos2uþ K33sin2uÞ
du dz 2 þ D 2 e1sin2uþ e2cos2u ; (1) whereK11andK33are the Frank elastic constants of LC
mole-cules, e1 and e2 are the dielectric constants that are parallel
FIG. 1. (Color online) (a) Calculated spectra of the defectless metallic pho-tonic crystal. (b) A defect mode due to the defect layer between the second and the third layer of the metallic photonic crystal.
FIG. 2. (Color online) (a) A schematic drawing of the tunable THz filter. The liquid crystal is embedded between the second and the third layers of the me-tallic photonic crystal. (b) The distribution of electric potential as a function of spatial coordinate (x, y) at different depth z¼ 050 lm across LC layer from an electrode surface. The electric potential is almost uniform throughout the surface laterally that has the same depth (z) from an electrode.
and perpendicular to the long axis of LC molecules, respec-tively,D is the electrical displacement in the LC layer, and / is the director angle depicted in the inset of Fig.4. The initial orientation of the LC (i.e.,Vbias¼ 0) is set parallel to the MPC
surface, assuming strong surface alignment. When applying bias voltage, the LC molecules are reoriented from their initial orientations toz axis that is perpendicular to the MPC surface as depicted in the insets of Fig.3. The derivative of u can be obtained using Eq.(1)and the Euler–Lagrange equation. The corresponding effective refractive index at a certain depth and a certain electrical displacement is given by
neffðz; DÞ ¼ neno= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 esin 2uðz; DÞ þ n2 ocos2uðz; DÞ q : (2) Then, the averaged effective refractive index, neff, of the
whole LC layer can be calculated using
neffðDÞ ¼ 2 d ðd=2 0 neffðz; DÞdz: (3)
The electrical displacement D can be replaced by voltage Vbiasusing the relationD¼ CV/A, where A is the area of an
electrode and C is the voltage-dependent capacitance of the LC layer that is given by
CðVÞ ¼ 2 ðd=2 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 1e22 e2 1cos2uþ e22sin 2u s dz: (4) Based on the theories discussed in the preceding text, we cal-culate the effective refractive index of realistic LC, MLC-2048 (Merck). In the calculations, the experimentally measured opti-cal parameters of the LC are used. The optiopti-cal parameters are measured using the terahertz time-domain spectroscopy and temporal waveforms of the THz pulses are shown in Fig.3(a). The corresponding parameters are ne¼ 1.761, no¼ 1.570,
e1¼ 11.2, e2¼ 8, K11¼ 17.7 pN, and K33¼ 21.4 pN. The
effective refractive index as a function of the applied bias is in the range of neff¼ 1.5891.761 as shown in Fig.4.
To see the transmission characteristic of the filter design, a set of the transmission spectra (Fig.3) is calculated using the maximum effective refractive index (neff¼ 1.761,
k¼ 0.01) and the minimum effective refractive index (neff¼ 1.589, k ¼ 0.01). For these calculation, we employ the
imaginary part of the refractive index,k¼ 0.01, which is also measured experimentally, to take into account loss by the LC. The resulting transmissions show some loss due to the interactions with the liquid crystal during multiple reflections between the top and the bottom MPCs. From the calculated transmission spectra, the maximum tunable range of the pro-posed filter would be Df¼ 147 GHz and the full width at half maximum (C) of the transmission peaks is 16 GHz at f¼ 1.430 THz and 21 GHz at f ¼ 1.577 THz. Comparing with previously proposed designs,3–5our design is expected to have wider tunable range. In Fig.4, the effective refractive index (open circles) and the corresponding peak positions of transmission (solid circles) calculated as a function of applied bias are plotted. The positions of transmission peaks can be finely tuned in the range of f¼ 1.4301.577 THz with the operating bias voltage range of 2 15 V. The tuning sensitivity is 11.3 GHz/V and the Q-factors are 89 at f¼ 1.430 THz and 75 at f ¼ 1.577 THz. The fractional tun-ing range, defined as Df/f0with Df¼ full width of the tuning
range andf0¼ central tuning frequency, is 9.8%.
FIG. 3. (Color online) (a) Experimental waveforms of the THz pulse trans-mitted through the empty and LC cells at different alignment (ordinary vs extraordinary). (b)The calculated transmission characteristics of the filter. The two alignments of the liquid crystal are shown in the insets.
FIG. 4. The peak position (solid circle) of the filter and the refractive index (open circle) of the LC as a function of the applied bias. The director angle of the LC is shown schematically in the inset.
IV. SWITCHING PROPERTIES OF THE THZFILTERS
MLC-2048, the LC used in the filter design, is a dual-frequency LC material; this means that the refractive index of the LC can be controlled not only by changing bias volt-age (voltvolt-age-modulation) but also by changing the frequency of a fixed bias voltage (frequency-modulation).20 The fre-quency-modulation can provide shorter fall time than the voltage-modulation, and it also provides reasonably short rise time.21A short response time of a LC is critical when a LC embedded optical switch is designed. Combining these with our filter design, a THz switch can be realized. To find time-dependent refractive index change, the dynamic behav-ior u(z,t) of the LC with a certain voltage Vais solved using
the torque balance equation given by @g @u d dz @g @ _u¼ c @u @t ; (5)
where c¼ 0.3 Pa is the rotational viscosity of the LC. Then
neff(t) is solved using the same procedures used to find
neff(D). The frequency is alternating from 1 to 100 kHz, and
the operating voltage isVa¼ 50 V. The calculated time
de-pendent refractive index (dashed line) and the corresponding normalized transmission (Tr/Trat peak) at a fixed frequency
f¼ 1.430 THz (solid line) are plotted in Fig.5. The response time is defined as the time interval for the change of trans-mission from 10% to 90% or vise versa. In Fig.5(a), the fre-quency is modulated from 1 to 100 kHz, and the resulting response time (decay time) is12.5 ms. In Fig.5(b), the fre-quency is modulated from 100 to 1 kHz, and the resulting response time (rise time) is25 ms. One interesting feature in the results is that the response of transmission is faster than that of the refractive index. This is because the
trans-mission spectra are very narrow as shown in Fig.3, and thus the resulting transmission rapidly changes with a small varia-tion of the refractive index. The response times are similar to the previously reported experimental results.21 Although there is a limitation of switching time imposed by the response time of the liquid crystal, the intrinsic ability of a defect mode to tune optical response not only by changing the refractive index but also by changing volume would give more freedom to select faster responding electro-optic materials.
V. CONCLUSIONS
In conclusion, an LC embedded MPC THz tunable filter is designed, and the filter properties and the switching prop-erties of it are investigated. The tunable THz filter is based on the voltage-modulation of LC molecules, and the tunable range is calculated to be f¼ 1.4301.577 THz with bias voltage of 215 V. The switching property of the filter is based on the frequency-modulation of LC molecules and the order of milliseconds switching is observed.
ACKNOWLEDGMENTS
S.Y.L. acknowledges financial support from DOE-BES under Grant No. DE-FG02-06ER46347. H.Y.W. acknowl-edges the financial support by the National Science Council of Republic of China under the Graduate Student Study Abroad Program.
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FIG. 5. Time dependent refractive index (dashed line) and normalized trans-mission (solid line) by frequency modulation. (a) Decay response with fre-quency changed from 1 to 100 kHz and (b) rise response with frefre-quency changed from 100 to 1 kHz.
