We studied the electromagnetic interaction between two asymmetric terahertz nano resonators, rectangular holes which have a few hundred micron lengths but nanoscale widths. We report that the dominant resonant transmission of the structures can be modulated by the horizontal distance between two rectangles due to the different oscillation strength of the asymmetric coupling at two different resonance frequencies. Our results are significant for an optimum design of rectangular holes in terahertz frequency regime for applications such as sensitive nanoparticle detection and terahertz filters.
©2012 Optical Society of America
The transmission characteristics of electromagnetic (EM) waves through various sub-wavelength holes have been studied theoretically and experimentally since the extraordinary optical transmission (EOT) phenomenon was reported over a decade ago by Ebbesen et al. [1–5]. It has been shown that the interaction between individual elements in an array structure strongly affects the resonance properties, resulting in spectral peak position shifts and the linewidth changes relative to that of a single structure [6–10]. Since then, strongly coupled systems with various geometries have attracted considerable attention in the past few years in a wide spectral range from terahertz (THz) to visible frequencies [11–22]. Furthermore, by breaking the geometrical symmetry of the coupled structure, new electromagnetic properties, such as achieving resonances with high-quality factors and directional scattering and chirality, were observed due to asymmetric mutual interaction [23–30].
Recently, we have already studied, theoretically and experimentally, the longitudinal  and transverse  coupling between identical THz nano resonators, which are hundreds of microns long rectangular holes with nanosized widths. According to these works, the resonance properties strongly depend on the distances which control the coupling strength. In particular, for the transverse coupling, a closely packed periodic array of rectangular holes was found to exhibit ultra-broadband transmission due to the strong interaction between the rectangles. To understand the physical origin of these collective behaviors of extended systems, such as chains and arrays, a better understanding of the constituent coupling between two single holes is necessary.
In this work, we investigate transmission characteristics of coupled two THz nano resonators transversely aligned. For symmetric structures, the transmission resonance reveals oscillatory behavior depending on the horizontal distance between the rectangles. Furthermore, we consider two asymmetric structures to distinguish the coupling effects which affect each resonance differently. Compared with transmissions of symmetric structures, asymmetric systems show modulations of the transmissions at two resonant frequencies, caused by locally modified coupling strengths depending on the distance. We interpret our experimental results by using a fully analytical model based on coupled-mode formalism.
2. Samples and methods
We have fabricated pairs of rectangular holes in a 200-nm-thick gold film deposited onto a silicon (Si) substrate with a thickness of 500 μm, as illustrated in Fig. 1(a) . The horizontal distance between two rectangles, d, was varied from 200 μm down to 2 μm. Symmetric samples are rectangular hole pairs which have an equal length of 100 μm and a fixed width of 300 nm. Asymmetric samples are formed by rectangles with two different lengths of 150 μm and 100 μm and the same width of 350 nm. In order to obtain high signal-to-noise ratio, our samples consist of six (nine) vertically aligned two asymmetric (symmetric) rectangles with a separation (center to center) of 160 μm (110 μm), in which the vertical coupling is negligible. Figure 1(b) shows scanning electron microscope (SEM) images of symmetric (left) and asymmetric (right) rectangles with a distance of 100 μm, patterned by electron beam lithography using a negative photoresist and the single layer lift-off process.
We perform THz time-domain spectroscopy (THz-TDS) to measure transmitted amplitudes through various samples in the frequency range from 0.3 THz to 1 THz [33, 34]. As shown in Fig. 1(c), a single-cycle THz pulse of about 2 ps duration is generated from a 2 kV cm−1 biased semi-insulating gallium arsenide, GaAs emitter illuminated by a femtosecond Ti:sapphire laser pulse of 150 fs duration, centered at a wavelength of 780 nm with a repetition rate of 76 MHz and the average power of 600 mW. The generated THz pulse is collected and focused using off-axis parabolic mirrors and the diameter of THz spot is about 3 mm. Samples are located at the THz focus center, and the transmitted THz waves are focused onto an electro-optic detection crystal (500-μm-thick zinc telluride, ZnTe (110)) for detecting the horizontal electric field. The THz-TDS system is capable of detecting the transmitted signal through subwavelength holes due to a high signal-to-noise ratio (SNR) (up to 10,000:1). We only consider the case of normal incidence, and the electric field of the incident wave is polarized along the short edge of the rectangle. The transmission properties of the samples are experimentally characterized by the normalized transmitted amplitude defined as35, 36].
3. Results and discussion
3.1 Single terahertz nano resonators
We begin with the investigation of transmission properties of single THz nano resonators. Figure 2(a) shows experimental results of the transmission through two kinds of single resonators with the fixed width of 350 nm and different lengths of 150 μm and 100 μm. SEM images of these samples are shown in the inset of same figure. The maximum of transmission through two samples show about 4.3% and 2.6% even though the ratios of total area covered by the holes are 0.03% and 0.02%, respectively. This is because the field inside rectangular hole with nano-sized width can be enhanced up to two orders of magnitude higher than the incident field . The resonance frequency is approximately the cutoff waveguide frequency, ~c/(2neffl), where neff is the effective refractive index of the substrate [38, 39]. Normalized-to-area amplitudes obtained by the numerical calculations based on the modal expansion are shown in Fig. 2(b) and in good agreement with the experimental results. The normalized-to-area amplitude is the average enhancement of electric field inside the aperture [32, 38]. The discrepancy between the experimental and theoretical spectra may be caused by several reasons such as sample imperfections (gold roughness and structure profile) and the perfect electrical conductor (PEC) approximation in the calculations.
3.2 Coupled-mode formalism
To understand transmission properties of two rectangular holes, we have applied a theoretical coupled-mode formalism based on the modal expansion of the EM fields in the different regions of the structure. A detailed account of theoretical formalism can be found in references [5, 40]. The modeling of the system ends up with solving a 4 by 4 matrix for and which are the modal amplitudes of the electric field at the input and output sides of the αth (α = 1, 2) aperture, respectively:32]. The interference of the radiated field in the far-field region makes the transmission to be effectively enhanced () or reduced (), depending on the distance between two holes.
3.3 Two symmetric terahertz nano resonators
We investigate the symmetric coupling in pairs of THz nano resonators. The symmetric system consists of two identical rectangular holes with width w = 300 nm and length l = 100 μm. We fabricated a set of seven samples with different distance d which were varied from 200 μm down to 5 μm, as shown in the left column of Fig. 3(a) . Notice that the nanometer scale width of the resonators enables reaching a few micrometer distance which is a deep subwavelength distance (d = 5 μm ~λres / 100). It allows us to realize the strong coupling effects. The experimentally acquired transmitted amplitude spectra for these different samples are presented in the middle column. The magnitude of the transmission peak slightly decreases and returns to increase as the distance is reduced from 200 μm to 80 μm. For the sample with the smallest distance (d = 5 μm), however, the resonant transmitted amplitude dramatically reduces due to the strong interaction between two nano resonators. The calculated total normalized-to-area amplitudes are shown in the right column for a comparison. The discrepancy between the experimental and theoretical spectra can be caused by the calculation based on PEC approximation, as mentioned before. Nevertheless the spectral features and tendencies observed in the experimental results are well reproduced by the theoretical calculations.
To investigate the evolution of the symmetric coupling behavior, we have normalized the maximum transmitted amplitude of each sample by that of a single nano resonator, (see Fig. 3(b)). A distance-dependent coupling effect is clearly observable when using the normalization: one would expect that the normalized maximum transmission is 2 for two non-coupled resonators. First concentrating on the overall tendency of the maximum transmission, the resonance transmission is not a monotonic function of d. The origin of the non-monotonic behavior can be traced to the oscillatory and slowly decaying trend of EM coupling, . Specifically, the total resonant transmission through the pair of rectangles for d = 100 μm is enhanced 3 times more than that of a single hole. It means that the maximum transmission through each resonator of the pair increases by a factor 1.5 due to the EM coupling with respect to that of a single hole. When d < 100 μm, the transmission is decreasing as d is reduced and the normalized transmission is smaller than 2 times of single hole case for d < 50 μm. This short-range interaction can be understood by the strong overlapping of two large transmission cross-sections characterizing rectangular holes with extremely large aspect ratio.
3.4 Two asymmetric terahertz nano resonators
Next, we investigate transmission properties of asymmetric THz nano resonator pairs. The sample structures consist of two rectangular holes with different lengths, l1 = 150 μm and l2 = 100 μm but a same width w = 350 nm. The respective resonant frequencies of these rectangles are located at 0.4 THz and 0.6 THz, as demonstrated in Fig. 2. The left column of Fig. 4(a) shows the SEM images of different samples with the decreasing d, from 200 μm down to 2 μm. The measured spectra are shown in the middle column of the figure. When the distance between two holes is sufficiently large for weak coupling (d = 200 μm), the transmission spectrum shows two well-separated resonances centered at about 0.4 THz (I) and 0.6 THz (II). As the distance is decreased from 200 μm, 170 μm and 130 μm, the only resonance II slightly decreases and returns to increase. For d = 100 μm and 80 μm, it is seen that the resonance II is more enhanced while the resonance I is slightly reduced, so that the resonance II is dominant. By further decreasing d down to 2 μm, the resonance II is drastically suppressed, and the resonance I is more enhanced, resulting in a dominant resonance at 0.4 THz. The strong coupling in the asymmetric system leads to the suppression of higher frequency resonance. As a result, the dominant resonance in the asymmetric system can be modulated and chosen by varying the distance between two nano resonators. The modulation nature of the observed results is confirmed by theoretical calculations, which is in good agreement with the experimental results (see the right column of Fig. 4(a)). In Fig. 4(b), we show the transmission peak at two different resonances normalized by the respective maximum transmission through a single hole, as a function of d. As expected, the maximum transmissions at two resonances have oscillatory behavior as changing the distance. However, more enhanced resonance oscillation amplitude at the resonance II is clearly observed, while the resonance oscillation amplitude at the resonance I is not so much changed. This is because the high frequency resonance can be excited at the long rectangular hole, while it is difficult to excite the low frequency resonance at the short rectangular hole due to the asymmetric coupling effect. Furthermore, the in-phase (out-of-phase) excitation of two rectangular holes can enhance (reduce) the maximum transmission of the high frequency resonance with changing of the distance. Remarkably, at the resonance II the transmitted amplitude reaches up to about 2 times when compared to the case of a single hole. It leads to the change of the dominant resonance from the resonance I to the resonance II.
To interpret the modification of the oscillation strength at two resonance frequencies, the EM coupling between two holes normalized by are plotted as a function of the distance in Fig. 5 . The oscillatory and slowly decaying trend and the period of the coupling terms at two resonances are similar with the change of maximum transmission plotted in Fig. 4(b). As we mentioned before, the negative (positive) of is accompanied by the enhanced (suppressed) transmission. In addition, a strong oscillation is observed at resonance II, while the oscillation at resonance I has a much weaker amplitude. Particularly, when the distance is decreasing down to the short-range interaction regime, increases up to 2 times of self-energy of single hole, , which leads to the suppression of the resonance I. As a results, the asymmetric oscillation strength of the EM coupling leads to the modulation of the dominant resonance peak.
In conclusion, we have experimentally and theoretically demonstrated the coupling effects in symmetric and asymmetric terahertz nano resonator pairs. For the symmetric case, we obtained an optimum distance for the maximum resonant transmission through two coupled rectangular holes. In addition, the resonant transmission shows an oscillatory behavior with the distance between two holes, due to the oscillatory electromagnetic coupling. By breaking the symmetry of the structure, the oscillation strengths at two different resonances can be modified, showing that the oscillation amplitude at the higher (lower) frequency resonance is enhanced (suppressed). These results lead to the modulation and interchange of the dominant resonance peak. Our findings provide valuable insight into the design and optimization of rectangular holes with desirable optical properties, such as maximum enhanced electric fields.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (SRC, No: 2008-0062255) (GRL, No: K20815000003) (Others, No: 2010-0029648, 2011-0019170, 2011-0020209), the Global Frontier R&D Program on Center for Multiscale Energy System, and Hi Seoul Science / Humanities Fellowship from Seoul Scholarship Foundation. We would like to thank Prof. Paul C. M. Planken, Prof. Aurèle J. L. Adam, and Nishant Kumar (TU Delft) for discussions about near-field.
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