Appl Phys A (2012) 107:285–291 DOI 10.1007/s00339-012-6861-2
An approach for mechanically tunable, dynamic terahertz bandstop filters Quan Li · Xueqian Zhang · Wei Cao · Akhlesh Lakhtakia · John F. O’Hara · Jiaguang Han · Weili Zhang
Received: 18 November 2011 / Accepted: 22 February 2012 / Published online: 21 March 2012 © Springer-Verlag 2012
Abstract Theoretical and experimental work was carried out on a terahertz metamaterial bandstop filter comprising an array of identical subwavelength resonators, each formed by fusing a pair of printable metallic U-shapes that have their openings pointing in opposite directions. Linear frequency tunability of the stopband electromagnetic response can be achieved by altering the overlap distance between the two fused shapes. Tuning does not significantly affect the strength or quality factor of the resonance. An approach to create mechanically tunable, dynamic terahertz filters is thereby suggested, with several functional advantages. Meanwhile, an effective equivalent circuit model based on self-inductance, mutual inductance, and capacitance has been proposed.
1 Introduction Metamaterials allow the manipulation of electromagnetic waves for numerous applications including sensing [1, 2], invisibility cloaking [3, 4], polarization filters [5, 6], modulators [7, 8], and electromagnetically switchable compoQ. Li · X. Zhang · J. Han () Center for Terahertz Waves and College of Precision Instrument and Optoelectronics Engineering, Tianjin University, Tianjin 300072, People’s Republic of China e-mail:
[email protected] W. Cao · J.F. O’Hara · W. Zhang School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, OK 74078, USA A. Lakhtakia Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USA
nents [9–11]. Although a huge spectral regime ranging from microwaves to optics has been of interest to metamaterials researchers, the terahertz regime is especially attractive because the shortage of terahertz devices is restricting exploitation of terahertz radiation for sensing, spectroscopy, imaging, security, etc. Ideally, the basic components for these devices should be simple, easy to fabricate, and dynamically tunable. Much interest has followed dynamically tunable terahertz metamaterials, in particular, since these may be the basis of future active terahertz devices. Metamaterial concepts have now spawned active terahertz filters tunable in frequency, transmission amplitude/phase, and polarization-state by means of optical, electronic, and thermal stimulation [12–15]. Terahertz filters that utilize mechanical stimulation for dynamic tuning, have received much less attention, mainly because of the more complicated fabrication process and the relatively slow response time of mechanical systems. However, mechanically tuned filters offer several functional advantages that make their continued investigations worthwhile, such as enabling strong tunable capacity, avoiding dynamic decay in resonance strength, and maintaining Q factors during tuning. Recently, experimental and theoretical studies of mechanically tunable bandstop filters based on metamaterials have attracted increasing interest [16–18]. Ozbey and Aktas numerically demonstrated magnetic-film-based cantilevers for continuous tuning over a 0.3 THz large frequency range with a resulting tuned resonance dip as well [16]. By changing the relative distance between two split-ring resonators, Fu et al. proposed a tuning scheme with a frequency range of ∼0.2 THz [17]. At optical frequencies, Ou et al. showed that the reconfigurable photonic metamaterials made from two layers of different thermal expansion coefficients exhibit reversible changes in transmission amplitude [18].
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Fig. 1 Schematics for the devised terahertz bandstop filter. (a) Two metallic U’s that merge to form the merged-U structure shown in (b). (c) Square array of subwavelength merged-U structures
In this article, we propose a terahertz bandstop filter that may be highly amenable to dynamic mechanical tunability and offers significant advantages in electromagnetic filtering behavior. It is comprised of an array of subwavelength resonators, each formed by fusing a metallic U-shape (or henceforth a “U”) with another metallic U, having their openings oriented in opposite directions. By choosing the overlap distance between the two U’s, the stopband can be continuously tuned over a broad frequency range in the terahertz regime, as we confirm using an equivalent-circuit model, full-wave simulations, and experimental measurements. We discuss the benefits and challenges of this approach.
2 Experimental and theoretical results 2.1 Experimental The devised resonator is shown in Fig. 1. A metallic U is overlaid on another metallic U, their openings pointing oppositely, as shown in Fig. 1a; parenthetically, we note that the U-shaped resonators are commonplace in metamaterials research [19, 20]. The partially overlapping U’s form an H with two cross-bars, as shown in Fig. 1b, upon being fused together. Each leg of a U is an l × w rectangle with l w, and the two legs are joined by an d × w rectangle with d w. The overlap between the two U’s is quantified by the length h, the overlap, where h = 0 occurs when the fused U’s begin to separate. We fixed l = 45 µm, d = 30 µm, and w = 5 µm, but made h variable. The fused-U resonators were made of 200-nm-thick aluminum and fabricated in a square lattice of period P = 100 µm on a 640-µm-thick ptype silicon substrate by conventional photolithography and lift-off, as shown in Fig. 1c. In order to characterize the properties of the filter experimentally, an 8-f terahertz time-domain spectroscopy (THzTDS) transmission system was employed [21, 22]. The
transmission Ein (ω) and Eout (ω) of the reference and the sample, respectively, were both measured under normal incidence with electric field parallel to the cross bars (Fig. 1b), the reference being a 640-µm-thick bare silicon substrate. The transmission magnitude t (ω) = |Eout (ω)/Ein (ω)| was then calculated, where ω is the angular frequency. Figure 2a presents measured values t (ω), ω/2π ∈ [0.4, 1.3] THz, of the bandstop filter for different values of overlap distance h ∈ [7, 19] µm. A sharp dip in the transmission amplitude, indicating performance as a bandstop filter, blueshifts gradually from 0.59 to 0.87 THz as h is increased from 7 to 19 µm. The transmission dip remains ∼0.15 during this increase in h, with Q-factors of 7.70 for h = 7 µm and 6.81 for h = 19 µm, respectively. The measured data in Fig. 2a are supported by full-wave numerical simulations using CST Microwave Studio. The unit cell shown in Fig. 1 was used in the simulations with periodic boundary conditions. The silicon substrate was modeled as a lossless dielectric material with relative permittivity εSi = 11.78, while aluminum was simulated with a conductivity of σAl = 3.72 × 107 S m−1 . The simulated spectra of t (ω) in Fig. 2b reveal good agreement with the experimental data in Fig. 2a. 2.2 Simulation Figure 3 illustrates the simulated distributions of the electric field and the surface current density in the unit cell for h ∈ {11, 15, 19} µm at the corresponding resonant frequency. The electric field is concentrated around the ends of the four legs in each case. The surface current density (shown in right-hand column): (i) is symmetric about the axis parallel to and equidistant from the two cross-bars, (ii) is antisymmetric about the central axis normal to the cross-bars, and (iii) is concentrated along the two cross-bars. As such, if we formulate an equivalent-circuit model for the merged-U structure, each bar must have a self-inductance
An approach for mechanically tunable, dynamic terahertz bandstop filters
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Fig. 2 (a) Measured values of the transmission amplitude for four different values of h. (b) Simulated spectra of the transmission amplitude of the bandstop filter
and some bar pairs must have a mutual inductance. We can ignore any inductances for the bars that are between and orthogonal to the two cross-bars because of their low current density, caused by the opposite current directions. 2.3 Theoretical modeling An equivalent self-inductance, mutual inductance, and capacitance (LMC) circuit model was established for the unit cell, as shown in Fig. 4a. Here L1 is the self-inductance of each leg and L2 is the self-inductance of each cross-bar; M1 and C1 are the mutual inductance and capacitance between the oppositely facing legs, respectively; M2 is the mutual inductance between the two cross-bars; and C2 is the capacitance between the central bars that are orthogonal to the cross-bars. To apply the LMC circuit model, it is necessary to calculate all equivalent circuit parameters. Using the line capacitance theory [23, 24], we obtained C1 = a1 ε0 εeff (0 − 2h)
K(k0 ) , K(k0 )
(1)
and C2 = a2 ε0 εeff (h − w)
K(k0 ) . K(k0 )
(2)
Here, a1 and a2 are unknown coefficients; K(k0 ) is the complete elliptic integral of the first kind with k0 = d/
(d + 2w) and k0 = 1 − k02 ; ε0 is the permittivity of free space; and εeff = f εSi + (1 − f ) is an effective relative permittivity that accounts for the contributions of both air and the substrate to the capacitance, where f ∈ (0, 1) is a factor representing the effect of the substrate. Members of the set {a1 , a2 , f } are to be fixed by fitting against three experimental data sets.
We calculate the self-inductance of each current-carrying surface as [25]: 0 μ0 0 −1 0 −1 w 2 sinh +2 sinh Λ(0 ) = 4π w w 0 2 2 2 3/2 0 (w + 0 ) 2 0 , (3) − + + 3 w w 0 w 2 where 0 is the length of the bar and μ0 is the permeability of free space. Specifically, in our structures, the selfinductances can be calculated as L1 = Λ( − 2h) and L2 = Λ(d + 2w). In addition, the mutual inductance between two parallel bars of length 0 , separated by a distance h0 , and with current densities pointed in the same direction can be expressed as [25] μ0 −1 0 20 sinh Ξ (0 , h0 ) = 4π h0 1/2
+ 2 h0 − h20 + 20 . (4) For our unit cell, M1 = −Ξ (l −2h, d +w) and M2 = Ξ (d + 2w, 2h − w); the negative sign in M1 is due to the opposite direction of the surface current density. The total impedance of the half circuit is thus given as Zm = j ω(L2 + M2 )
1 1 + + j ω(2L1 + 2M1 ) . j ωC1 j ωC2
(5)
The resonance frequency can be extracted through the relation: Im(Zm ) = 0. In Fig. 4b, we present detailed grayscale transmission-amplitude maps with respect to the frequency for h ∈ [5, 22.5] µm. The solid line represents the analytically predicted resonance frequency by the LMC model with {a1 = 0.057, a2 = 0.107, f = 0.54}, which is in good agreement with the simulated and experimental results. The obtained circuit parameters are displayed in Table 1:
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Fig. 3 Simulated distributions of the electric field and the surface current density in a unit cell, for three different values of h at the corresponding resonant frequency. The color represents the amplitude of the electric field, and the arrows indicate the direction of the surface current density
whereas C1 , L1 and M2 decrease, C2 and M1 increase, as h increases; however, L2 is independent of h. In addition, the formulated equivalent-circuit model can also well describe the dependences on other parameters. The gray-scale transmission-amplitude maps with respect to the frequency and l are shown in Fig. 4c. The solid line represents the calculated l-dependent resonance fre-
quency based on the equivalent-circuit model with {a1 = 0.044, a2 = 0.258, f = 0.54}, and is in good agreement with experimental results for all samples of different l. This further enables us to confirm the validity of LMC circuit model. Table 2 provides the circuit parameters for different values of l. Because the substrate is few orders of magnitude thicker than that of the metamaterial film, the value f
An approach for mechanically tunable, dynamic terahertz bandstop filters
Fig. 4 (a) Equivalent-circuit model of the merged-U structure. The blue lines between the parallel bars represent mutual inductances. We need only consider half of the structure, surrounded by the red dashed line, to formulate the equivalent circuit. (b) Gray-scale map of transmission amplitude with respect to the frequency and h, when Table 1 Obtained circuit parameters in the LMC model for various values of h, when l = 45 µm, d = 30 µm, and w = 5 µm
Table 2 Obtained circuit parameters in the LMC model for various values of l, when d = 30 µm, w = 5 µm, and h = 10 µm
h (µm)
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l = 45 µm, d = 30 µm, and w = 5 µm. The solid line shows the calculated resonance frequency based on the LMC model and the open circles are measured values. (c) Same as (b) except that l varies but h = 10 µm
L1 (pH)
L2 (pH)
M1 (pH)
M2 (pH)
C1 (fF)
C2 (fF)
7
19.036
26.509
−2.5980
11.178
1.0088
0.1222
11
12.883
26.509
−1.4630
7.4370
0.7485
0.3665
15
7.1948
26.509
−0.6335
5.5579
0.4881
0.6109
19
2.4457
26.509
−0.1395
4.4186
0.2278
0.8552
21
0.6801
26.509
−0.0257
4.0028
0.0976
0.9774
L2 (pH)
M1 (pH)
M2 (pH)
C1 (fF)
C2 (fF)
l (µm)
L1 (pH)
27
2.4457
26.509
−0.1395
8.1153
0.1758
0.7365
35
7.1948
26.509
−0.6335
8.1153
0.3768
0.7365
43
12.833
26.509
−1.4630
8.1153
0.5778
0.7365
51
19.036
26.509
−2.5980
8.1153
0.7787
0.7365
remains as a constant in the two sets of parameters given
3 Discussions
above. When h and l are changed, the length of the oppositely facing legs is equivalent, but the length of the central bar orthogonal to the cross-bars varies extensively, thus leading to a small change in a1 and a remarkable change in a2 .
The measurements and circuit model show that a very linear tuning of the terahertz bandstop filtering can be achieved with this approach. In Fig. 4b, the resonance frequency of the fused-U resonator varies almost perfectly linearly from
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0.5 THz at h = 5 µm to 0.8 THz at h = 16 µm, with no appreciable change in the resonance depth. Altering the parameter l, Fig. 4c, produces even better tuning results. These tuning characteristics are the result of changing the layout of the metallic resonator parts. In previous research, the metallic parts of the resonators are typically fixed, and dynamic frequency tunability is achieved by altering the complex permittivity of a dielectric or semiconducting inclusion [26, 27]. This alteration often introduces significant damping or produces relatively little tuning. By simply rearranging metallic portions of the resonator, these damping effects are eliminated, while strong tuning is enabled. The crossbars of the proposed structure, which dominate coupling to the incident field, also do not change size, thereby further avoiding dynamic decay of the resonance strength. Finally, the ratio of inductive to capacitive reactance in the proposed structure remains quite constant, fixing the Q-factor during tuning. The challenge of mechanical tunability is fabrication. To achieve dynamic tunability, the proposed structure would consist of two arrays, separately fabricated and then pressed face-to-face to achieve good electrical contact. Reliable MEMS techniques of translating one layer with respect to the other would also be necessary. These difficulties do not seem to be insurmountable. Optically flat, transparent, and thick substrates may be employed to ensure good contact between the resonator arrays. The design requires only a one-dimensional translation, which simplifies the mechanical mounting structure. The micron-scale translations are also well within the precision of modern optomechanics. We note that mechanically altering the parameter l instead of h introduces additional fabrication complexity, though the achievable tunability is improved. Indeed these modifications are currently under investigation. It is worth noting that the merged-U structure provides a good choice for terahertz bandstop filters, while the complementary design based on Babinet’s principle [28] would serve equally well as a terahertz bandpass filter.
4 Conclusions A tunable terahertz bandstop filter comprising a square array of subwavelength merged-U structures was devised, fabricated, and characterized. Comparison against a simulation and circuit modeling was good and revealed a possible dynamic tuning mechanism whereby highly linear and wide frequency tuning may be achieved without any sacrifice in filtering strength or Q-factor. The mechanism to tune the stopband relies on manipulating the overlap distance between the two metallic U’s in each unit cell. Highperformance tunability of about 0.4 THz was demonstrated. Dynamic tunability could be achieved via a one-dimensional
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translation of one array of the U’s over a second array of U’s. We reiterate that this filter exemplifies a highly functional, mechanical approach to dynamically tunable, terahertz filters—and related devices—that will be useful in future terahertz technology. While mechanical tuning is more difficult to implement, it offers functional advantages, making us very optimistic. Acknowledgements This work was partially supported by the National Science Foundation of China (Grant Nos. 61138001, 61028011, and 61007034), the U.S. National Science Foundation, and the MOE 111 Program of China (Grant No. B07014).
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