1. Introduction

A metamaterial (MM) is a periodically inhomogeneous structure that can be considered as an effectively homogeneous medium, and which exhibits characteristics that are otherwise unobservable in nature. Research on MMs was highlighted in 2001, following the realization that such MMs could have a negative refractive index [1]. Early studies focused on these extraordinary characteristics and their possible application, including the fabrication of negative refractive index materials, cloaking, light-trapping, etc [2]. Soon, their application to various other fields spread, including proposals for a MM absorber (MMA) [3]. Various experimental setups were used to verify the unusual characteristics of MMs. For MMs operating in the GHz range, measurements utilizing a vector network analyzer (VNA) and a pair of horn antennas became popular, since this experimental setup enables the extraction of a MM’s effective parameters [4,5]. Although a VNA is capable of a time-domain measurement, which can simplify the measurement process, the frequency-domain measurement is still widely used. Indeed, the measured spectra were obtained using this experimental setup in order to measure the S-parameter [5–10]. In this arrangement, both antennas are held at fixed positions, while one antenna works as a radiation source and the other as a receiver. In order to determine the reflectivity of a MM, the parameter S₂₁ is first measured with a perfect mirror, and then measured for the MM sample. The reflectivity is then given by |S₂₁(sample)/S₂₁(mirror)|². When measuring the transmission, the two antennas are positioned along a straight line, and S₂₁ is measured twice, with and without the sample present. The transmittivity of the sample is then given by |S₂₁(with sample)/S₂₁(without sample)|².

Errors are unavoidable in real measurements, and measurements of MMs are no exception. There are two kinds of errors: random and systematic. Random errors are generally unavoidable; however, they can be reduced by improving measurement techniques, employing more accurate devices, and increasing the data-acquisition time and repetition cycles. On the other hand, systematic errors can be avoided simply by removing the source of the error, once identified. However, even though the sources may be identified, often the systematic errors remain unavoidable because of current measurement technique limitations. In these cases, the source-identified systematic errors are simply ignored.

The absorption and reflectance spectra of MMAs operating in the GHz range frequently exhibit small periodic fluctuations (PFs), in addition to their main features. These PFs are usually ignored because of their negligible amplitude [3, 5–8]. However, depending on the experimental setup, the PF can assume different features. In [3], Rhee et al., the periodicity of the PF appearing in the reflectance spectrum is about 0.7 GHz. More significantly, the amplitude of the PF is considerably smaller than the amplitudes observed other works. Unlike these other works, the authors placed a beam splitter between the sample and the source, in order to avoid a large voltage-standing-wave ratio. Additionally, their horn antennas appear to have been placed rather close to the sample. In [8], Cheng et al., the periodicities of the experimental and finite-difference time-domain (FDTD) simulation spectra were approximately 0.16 GHz and 0.5 GHz, respectively. In [7], Yi et al., the PF periodicity of the measured experimental spectrum was only 0.087 GHz. Finally, in [9], Lam et al., a very complex form of PF resulting from the interference of two (or more) radiation sources was shown.

Simulations with periodic boundary conditions (PBC) performed in the frequency-domain do not exhibit such PF. Meanwhile, the time-domain-based FDTD simulations may, or may not, produce spectra with PFs irrespective of the boundary conditions. In this work we report our results of an investigation into the source of the PF. Our results reveal that the cause of the PF was interference between the electromagnetic (EM) wave propagating directly from the radiation source to the receiver, and the EM wave reflected from the sample. Since the PF is an unavoidable systematic error due to the limitations of current measurement techniques and the experimental setup, we can safely ignore it.

A detailed description of the sample preparation and the measurement system has been presented elsewhere [6]. The commercial software suite, CST Microwave Studio®, was used for our simulations. In the FDTD simulation, a Gaussian pulse was used for the excitation signal. The absorption spectra of the studied MMA features two peaks. Instead of using PBCs, our FDTD simulation was conducted with perfectly matched layer boundaries for all boundaries. Further details about the simulation process are provided in [7], Yi et al. We note that in this preceding research, the PF was intentionally eliminated by time gating in order to clearly distinguish the minor peak from the 3^rd-harmonic peak.

2. Results and discussion

Periodic fluctuations were observed in both experiment and simulation. Figure 1 shows two FDTD-simulation absorption spectra with incident angles of 5° and 19°. From this Fig., we see that not only does the amplitude of the fluctuations increase, but their periodicity also increases, as the angle of incidence (AoI) is increased. The simulated spectra for angles of incidence between 5° and 19° (not shown) exhibit the same tendency. This tendency is exactly applicable to the experimental results (see Fig. 6 of [7], Yi et al.). From this tendency, we can clearly identify that the PF is a systematic error, and has its origin in the interference of two EM waves arriving at the receiver; one directly from the source, and the other reflected from the sample. It implies that there are two paths from the radiation source to the receiver, and that their path lengths are, naturally, different. As the receiver, the radiation source, and the sample (mirror) are placed at fixed positions, the phase difference between the two paths depends only on the frequency (or equivalently, wavelength) of the EM wave emitted by the radiation source, and the resulting phase change upon reflection from the sample (mirror).

 

Fig. 1 Simulated absorption spectra at two angles of incidence.

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The experimental spectra were not obtained in the time domain, but in the frequency domain. In order to obtain the transmission or reflection spectrum in the frequency domain from the time-domain signal of the receiver, one needs to transform the signal with a Fourier transform (FT). Throughout the simulation, an appropriate time-gating is necessary in order to obtain a simulated spectrum that matches the experimental measurement well. Figure 2 shows a typical time-domain signal of the receiver obtained by our FDTD simulation. The two pulses correspond to the two paths. In [7], Yi et al., the simulated spectra were obtained by excluding the minor pulse. This kind of signal process is called time-gating. There, since PF was not the major concern of the study, the minor pulse of Fig. 2 was excluded when taking the FT, because it was evident that the minor pulse was due to the propagation of the EM wave from the radiation source directly to the receiver. As a consequence, we were able to obtain the simulated spectra without PF. The smoothed spectra can be justified further because the main concern of [7], Yi et al. was the role of Wood’s anomaly responsible for the appearance of the minor peak located just above the major peak, which was due to the third-harmonic magnetic resonance. Even though the removal of the PF by excluding the minor pulse of the time-domain signal can be partially justified, here we have to reconsider the exclusion of the minor pulse of the time-domain signal more carefully because the smoothed spectra cannot reproduce the experimental spectra perfectly.

 

Fig. 2 Time-domain signal of Port 2 (Receiver) throughout the duration of the simulation. When t = 0, Port 1 (Source) emits radiation. After 2 ns, a minor packet arrives at Port 2. At 14 ns, the major packet arrives at Port 2, after being reflected from the sample.

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By allowing the inclusion of the minor pulse of the time-domain signal when taking the FT, we can argue that the PF is due to the systematic error, and the interference between two EM waves travelling along different paths. Furthermore, one may argue that the minor pulse of the time-domain signal can be excluded during the FT since it arrives at the receiver before the major pulse, and hence, there is no opportunity for it to interfere with the major pulse. However, as mentioned above, the experiment was performed in the frequency domain with the radiation source emitting EM waves continuously, and not in the form of Gaussian pulse. This implies that the EM wave directly propagating from the radiation source kept impinging on the receiver during the arrival of the EM wave reflected from the sample, resulting in their interference. Therefore, it is necessary to include the minor pulse of the time-domain signal when taking the FT.

By comparing the simulated spectra shown in Fig. 1 with those of [7], Yi et al., it is apparent that the PF is due to the inclusion of the minor pulse of the time-domain signal of the receiver. However, the appearance of the PF on inclusion of the minor pulse of the time-domain signal during the FT cannot completely justify the interference between the two EM waves as the origin of the systematic error producing the PF. We need some further quantitative justification. In other words, the dependence of PF upon AoI variation should be explained quantitatively using interference theory.

Figure 3 shows the PF of simulated and experimental spectra. The agreement between the simulation and the experiment is reasonably good. Figures 3(a) and 3(b) show the period and amplitude of the PF as a function of the AoI, respectively. Both period and amplitude increase as the AoI increases. Discrete Fourier transform (DFT) was used to extract the period and the amplitude of the PF. We note that the major and minor peaks were ignored in the DFT results because their impact was fully discussed in [7], Yi et al. It is also evident that there was unavoidable interference in our experimental setup. The observed period depends on the path difference of the two waves, and is defined as the frequency difference, Δf, between two consecutive maxima or minima. The period should satisfy the relation $\mathrm{\Delta }f=\frac{c}{(\text{Path}\text{diff}.)}$, where c is the speed of light in vacuum. As the AoI increases, the path difference between the two waves in our experiments is shortened continuously. The path differences observed in our experiment are listed in Table I. The path difference is then given by 4(1 − sin θ) meters. This result implies that the period is relatively insensitive to the variation of the AoI, when the AoI ≤ 11°. If the AoI > 11°, as the distance between the two antennas is fixed, the sample must be moved forward to increase the AoI, resulting in a rapid decrease of the path difference. The calculated periods of PF are listed in Table I, and are also plotted in Fig. 3(a).

 

Fig. 3 Angle of incidence dependency of the (a) period, and (b) amplitude of the experimental and simulated spectra. In (a), gray line shows the period of PFs calculated from the experimental setup, shown in Table I. In (c), the blue line shows the absorption spectrum calculated from the squared magnetic field. In the experimental spectrum, the major peak of the 3rd harmonic magnetic resonance, and the accompanying minor peak, were eliminated using Lorentzian peak-fitting, and were shifted downward by 0.2 for ease of comparison

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Table 1. Geometrical parameters of experimental setup of [7], Yi et al. L_aa is distance between two antennas, and L_as is distance between one antenna and the sample.

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The variation in amplitude can also be explained by the change of the distance and directional angle between the radiating antenna and the receiving antenna. When AoI ≤ 11°, the distance between the radiation source and the receiver increases as AoI increases. Moreover, the facing angle of the two antennas also increases, resulting in an increased amount of EM waves propagating directly from the source to the receiver, which in turn, enhances the interference effect and increases the amplitude of the PF. Meanwhile, up to an AoI of 11°, the distance between two antennas increases, leading to reduced amplitude. As a result, these two effects cancel out each other when the AoI ≤ 11°, and the amplitude is relatively insensitive to the variation of AoI. On the other hand, when AoI ≥ 13°, the amount of directly impinging radiation increases rapidly. Since the distance between the two antenna is fixed and the facing angle is increased, the amount of interference also increases, and, hence, the amplitude of the PF is enhanced. This trend is clearly observed in Fig. 3(b). However, as evident in Fig. 3, some discrepancies between simulation and experiment exist. The period of the experiment with an AoI = 13° shown in Fig. 3(a) is significantly larger than that of simulation. Furthermore, the PF shown in Fig. 3(c) clearly indicates that there are multiple periods. After some additional study, we concluded that the multiple periods do not imply the existence of three or more paths, and rather that the frequency dependence of the radiation characteristics of the horn antenna caused the complex fluctuations shown in this study. This result will be presented in a separate publication.

We performed an additional simulation in order to further support our claim that the source of the PF is the interference between the two EM radiations. In this simulation, we inserted a perfect mirror between the source and the receiver in order to effectively block the radiation propagating directly from the emitter. Our simulation results are shown in Fig. 4. It is clear that the PF is significantly suppressed by the insertion of a perfect mirror between the emitter and receiver, since the mirror effectively reduces the radiation impinging on the receiver, propagated directly from the emitter.

 

Fig. 4 Absorption spectra with (shielding) and without (5°) a perfect mirror between the emitter and the receiver. Here, time gating was not used in the transformation of FDTD signal to frequency-domain spectrum.

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Although we have successfully identified that the source of PF is the interference, we need one more proof. PF was also observed in other works [3,6–10], however, all the simulated spectra, except for that in [8], Cheng et al., did not display PF because PBC was employed, in which there was no direct propagation from emitter to receiver because they were located at the same place. According to our argument, another origin of PF should exist in simulation result of [8], Cheng et al. We paid attention to a couple of aspects revealed in the spectra displayed in [8], Cheng et al. The first aspect is the period of the PF. The period of the PF in experiment listed there is three or four times shorter than that of the simulated one. Furthermore, in the experiment, the amplitude of the PF depends on the polarization, while that of the simulated result does not. Some polarization dependence is common in the spectra measured by rectangular horn antenna. Another interesting point is the difference among their simulation results. Figure 4 of [8], Cheng et al. shows the AoI dependence in simulated results, but none of the spectra exhibits any PF. We reproduced the experimental results of [8], Cheng et al. by simulation. Figure 5 shows our simulation results. The spectrum obtained using a common FDTD setup with PBC exhibits no PFs unless we use a rough convergence condition. In the MMA, the energy of the incident wave is converted into a plasmonic resonance first, and diminishes rather slowly, being absorbed as another form of energy. This implies that we should run the FDTD simulation for a longer duration than the common EM-wave scattering system. Figure 6(a) shows the received signal in the time domain. As shown in Fig. 6(b), if one stops the simulation too early PFs appear. The simulated spectra reproducing the experimental results of [8], Cheng et al. by a non-PBC FDTD simulation are shown in Fig. 6(c). We could successfully reproduce both the PFs of the simulation, and time-gated results, which do not contain PFs, as was done in [7], Yi et al. As we cannot access simulation and experimental details, our argument about [8], Cheng et al. is purely a speculation. Also, we could see the relation between period of PF and the convergence condition.

 

Fig. 5 FDTD simulation of MMA used in [8], Cheng et al. in the case of normal incidence. The periodicity of the PF was about 0.084 GHz, which is about twice as large as that of the experiment. With time-gating, we could obtain the spectrum without PFs, as was done in [7], Yi et al.

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Fig. 6 Reproduction of the simulated spectrum of [8], Cheng et al. (a) Total energy of simulated time-domain spectrum in dB scale. The maximum is reached when the excitation is over, after which it decreases exponentially. (b) Receiver signal. In our reproduction, the total energy was chosen as a convergence condition. (c) Spectra corresponding to the different convergence conditions. As we choose stricter convergence conditions, PFs can be completely suppressed. Inset shows the spectrum in the whole range.

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3. Conclusion

We successfully reproduced the periodic fluctuation appearing in the experimental absorption spectra of an MMA. The periodic fluctuation is due to interference between the radiation directly propagating from the source to the receiver, and reflected radiation from the sample. From this result, we can conclude that the periodic fluctuations were systematic errors, and thus, that they can be ignored in spectral analysis. The cause of variations in the period and amplitude of the periodic fluctuation with varying angles of incidence were also successfully elucidated.