1. Introduction

Electromagnetic metamaterials, which are artificially structured materials with periods much smaller than operating wavelengths, are attracting much attention [1]. The extraordinary electromagnetic phenomena can be exhibited by metamaterials owing to their effective permittivity and permeability not found in naturally obtained materials and have brought about a renewal of classical electromagnetics. Although one of the recent concerns for those researches of metamaterials is the realization of three-dimensional structures, planar metamaterials are still an important research target, particularly in the terahertz region. Novel planar terahertz devices based on metamaterial concepts such as electrical amplitude and phase modulators for terahertz waves have been demonstrated [2–5]. In split-ring resonator (SRR)-type metamaterials, desired electromagnetic responses are obtained by arranging “meta-atoms” separated spatially with each other [6, 7]. These responses are modified when there are interactions between “meta-atoms” to form “metamolecules” or band structures [8, 9]. Furthermore, when the metal meta-atoms are randomly connected with each other, the effective optical constants are expected to be explained by the theory similar to the percolation theory for the random composite materials composed of conductors and insulators [10]. The electromagnetic responses of metamaterials can be designed by incorporating randomness of connection, which increases the flexibility of the design of optical devices such as filters.

The critical nature of the percolation threshold appears in composite materials composed of conductors and insulators. One of the simplest systems of such composites in two dimension is a metallic checkerboard pattern (MCP), shown in Fig. 1 [11–13]. The MCPs show complementary electromagnetic responses depending on whether the adjacent metal squares contact with each other or not, as briefly summarized in the following section. In the present study, it is revealed that a drastic but continuous change in the transmission spectra of the MCPs occurs with the change of the proportion of the connection, which implies that the transmission spectra can be designed by the proportion of connected meta-atoms even for other systems. The experimental results also suggest that the characteristic change of transmission spectra of the MCP with the proportion of the connection is analogous to the absorption spectra of liquid solutions containing molecular species with two types of absorption spectra. Finally, as an interesting application, revealing of an optically hidden pattern made of near-checkerboard patterns by terahertz beams is demonstrated.

 

Fig. 1 Schematics of metallic checkerboard patterns.

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2. Babinet’s principle and equivalent circuit model

Babinet’s principle implies that a perfect conductive screen and its complementary screen have complementary transmission spectra [14]. When the screen has π/2 rotational symmetry, the complex amplitude transmission coefficients of the original screen t₁ and complementary screen t₂ satisfy the following equation:

We consider two structures quite near the ideal MCP: a structure in which adjacent metal squares are in contact with each other (inductive MCP, or I-MCP) and one in which they are not in contact with each other (capacitive MCP, or C-MCP). When the wavelength of the incident electromagnetic wave λ is much larger than the lattice constant p, the impedance Z_I and Z_C of the I-MCP and C-MCP are described by parallel and series combinations of inductances and capacitances, respectively [16]:

Switching between the responses of the I-MCPs and C-MCPs is abrupt depending on d around the crossover point $d_0=p/\sqrt{2}$, where d is the side length of the metallic squares. For example, as $\omega \to 0$, Z_I becomes 0, and Z_C diverges to $-i\infty$. If we increase d while keeping the lattice constant the same, the reactance changes abruptly from capacitive to inductive at the crossover point d₀. The above discussion shows that the electromagnetic response critically depends on the small structural changes near the crossover point.

3. Finite-difference time-domain simulations

The transmission spectra of free-standing C-MCPs and I-MCPs are calculated using a finite-difference time-domain (FDTD) method [17]. Figure 2(a) shows the simulated transmission spectra of the near self-complementary MCPs, in which d = d₀ + Δd (I-MCP) and d = d₀ – Δd (C-MCP). Here, p = 1 mm, Δd = 0.01 mm, and the metal thickness is 0.01 mm. The metal is assumed to be a perfect conductor. Although Δd is far smaller than both the incident wavelength and d₀, the transmission spectra of the C-MCPs and I-MCPs are opposite below the first-order diffraction frequency f_d = p/c = 0.3 THz, where c is the speed of light. Near the crossover point d₀, a small structural variation drastically changes the electromagnetic responses. Above f_d, the transmittance approaches the fractional area of the opening following the geometrical optics, and consequently, the transmittances are near 0.5.

 

Fig. 2 Simulated transmission spectra of I-MCPs (solid) and C-MCPs (dashed) for (a) a perfect conductor and (b) a lossy metal. The inset is a unit cell from the simulated model. (c) Refractive index dispersion of the lossy metal used for simulation. The solid and dashed curves in (c) indicate the real and imaginary parts of the refractive index, respectively.

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In practice, the Drude model describes the relative permittivity of metals as $\epsilon ={\epsilon }_{\infty }-{\omega }_{\text{p}}^2{\omega }^{-1}{(\omega -i/\tau )}^{-1}$. Here, ${\epsilon }_{\infty }$ is the high-frequency permittivity, ω_p is the plasma frequency, and τ is the relaxation time. Absorption by the metal adds the resistance term R on the right-hand side of Eq. (3). Figure 2(b) shows the simulated transmittances of the MCPs consisting of a lossy metal with ${\epsilon }_{\infty }=1$, f_p = ω_p /2π = 1.51 THz, and τ = 2.72 ps. This metal has a refractive index (Fig. 2(c)) similar to that of real metals in the visible region. For example, the complex refractive index $n=\sqrt{\epsilon }=0.49-i4.84$ at 0.3 THz is that of aluminum at 400 nm [18]. The switching from inductive to capacitive response near the crossover point is clearly seen even for the lossy metal. The transmission spectra of the MCPs with a lossy metal imply that when the MCPs are scaled down to the visible region, the C-MCP exhibits a color quite different from the I-MCP. This color manipulation is different from that by subwavelength metal hole arrays, in which the period is changed to change the color [19].

The electromagnetic responses of the MCPs have been investigated in the visible and infrared regions for the nonlinear optical application of the strong local electric field rather than for color (transmission and reflection spectra) manipulation [20–22]. C-MCPs have attracted attention for enhancing the intensity of the local electric fields. The enhancement of the electric component of an electromagnetic field is expected in the gap of the SRRs at the LC resonant frequency [7]. Compared to SRRs, the electric field enhancement in C-MCPs occurs in a broad frequency region. In [20–22], C-MCPs were fabricated by electron-beam lithography, and nonlinear optical effects were successfully enhanced in the optical region. Although their main interest was the electric field enhancement in C-MCPs, their samples are sure to exhibit the exotic transmission and reflection spectra mentioned above.

4. Near self-complementary metallic checkerboard patterns printed on paper

The near self-complementary MCPs were fabricated with a color printer using metallic color ink [23]. Figure 3 shows the typical micrographs of the near self-complementary MCPs. The MCPs were printed on substrate papers with the thickness of 0.07 mm and a refractive index of ~1.5 – i0.05. The thickness of the metallic ink, h_m, is approximately 2.5 μm. The permittivity of the metallic color ink was measured by a terahertz time-domain spectroscopy (THz-TDS) system [24]. The dipole and bowtie-type photoconductive antennas were used for the emitter and detector, respectively. The permittivity spectra are fitted well by the Drude model with ${\epsilon }_{\infty }$ = 1, f_p = 57.2 THz, and τ = 0.1 ps. The period p of the metal square arrays is 1 mm, so that the MCP is self-complementary when the side length d of the squares has the value $d_0=p/\sqrt{2}$ ≈0.707 mm.

 

Fig. 3 Typical photographs of the near self-complementary MCPs printed on the paper.

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The terahertz responses of the MCPs were measured by the THz-TDS. Figures 4(a) and (b) show the real and imaginary parts, respectively, of the complex amplitude transmission coefficient t(ω), which are normalized by that of the substrate paper. As d varies from 0.66 to 0.76 mm, the transmission spectra change systematically below the first-order diffraction frequency of 0.3 THz. The dashed lines in Figs. 4(c) and 4(d) are the real and imaginary parts of t simulated with d = 0.70 ($t_{d=0.70}$) and 0.71 mm ($t_{d=0.71}$), respectively. In our experiments, the switching from capacitive to inductive responses is not as abrupt as the switching in the microwave region reported by Edmunds et al. [12] and switching in our simulations. This is probably due to the limited spatial resolution of the printer (2400 dpi), which brings about some randomness in the structure. However, the small structural variation still induces very large spectral variations, and between d = 0.72 and 0.73 mm, these variations are much larger than those for other d values; this implies the existence of a percolation threshold.

 

Fig. 4 (a) Real and (b) imaginary parts of the amplitude transmission coefficients of the near self-complementary MCPs. (c) Real and (d) imaginary parts of $t=A{t}_{d=0.71}+(1-A)t_{d=0.70}$, where $t_{d=0.70}$ and $t_{d=0.71}$ are the amplitude transmission coefficients simulated by the FDTD method.

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It should be noted that as shown in Figs. 4(a) and 4(b), there are frequencies with invariant values of the real and imaginary parts of t as a function of d. The real parts of t have invariant values near 0.13 and 0.24 THz [ω₁ and ω₃ in Fig. 4(a)], and the imaginary parts of t have them near 0.20 and 0.28 THz [ω₂ and ω₄ in Fig. 4(b)]. The invariant points imply the existence of two components whose densities are linearly interrelated with different spectra. Typically, the invariant points are observed in the absorption spectra of two kinds of absorbers whose densities are linearly interrelated in a liquid solution, which are called the isosbestic points [25]. The isosbestic points are observed at the frequencies where the absorption spectra of the two absorbers cross each other. In the case of the near self-complementary MCP, both regions, which exhibit the amplitude transmission coefficients of C-MCPs, t_C, and I-MCP, t_I, are mixed together owing to the randomness of the structures caused by the limited spatial resolution of the printer. Similar to the solution showing the isosbestic points of two absorbers, the transmission coefficient of the near self-complementary MCPs may be described by $t=A{t}_{\text{I}}+(1-A)t_{\text{C}}$, $(0\le A\le 1)$. Here, we take the area ratio A of the I-MCP as the density of the absorber like the liquid solution. The area ratio A of the I-MCP increases with d in the vicinity of d₀, causing the invariant values of the real and imaginary parts of t as well as the isosbestic point. Figures 4(c) and 4(d) show the real and imaginary parts of $t=A{t}_{d=0.71}+(1-A)t_{d=0.70}$, which reproduce the experimental values of t for the near self-complementary MCPs semi-quantitatively. In the experimental results, the invariant points are somewhat broadened probably owing to the fluctuation of p and d in fabrication. We believe that the strict invariant points may be obtained if the ratio of the connecting to nonconnecting corners of the squares is varied with keeping the period p rigorously. Further, very flat spectra will be obtained when the ratio is 0.5 like the spectra for A = 0.5 in Figs. 4(c) and 4(d).-This property of near self-complementary MCPs implies that the transmission spectra can be designed by the ratio of the connected to unconnected meta-atoms. A similar phenomenon has been observed for the thin metallic films. At the metal insulator transition, the frequency-independent transmission spectra are observed and such properties can be used to anti-reflection coatings [26–28]. Recently, Nakata et al. investigated theoretically the resistivity-loaded checkerboard patterns and showed that the specific value of the resistance loaded between the perfect conductor patterns makes the transmission spectra flat [29].

Another characteristic feature of the electromagnetic responses of the MCPs is seen in Fig. 4. The frequencies at the extreme points (around ω₂ and ω₄ for the real part, and around ω₁ and ω₃ for the imaginary part) and inflection points (around ω₁ and ω₃ for the real part, and around ω₂ and ω₄ for the imaginary part) in the transmission coefficients are correlated with each other. Further, at the frequencies of the invariant points at ω₁ and ω₃ in the real part, the corresponding imaginary part varies largely with d. Conversely, at the frequencies of the invariant points at ω₂ and ω₄ in the imaginary part, the corresponding real part varies largely with d. These features indicate that the complex amplitude transmission coefficient of the MCPs, which operate as a linear and passive system to the incident terahertz pulse, obeys the Kramers-Kronig relation;

5. Effective permittivities

In our previous article, a metal-to-insulator transition in a metamaterial composed of metallic wire arrays was discussed [30]. It was shown that the Drude-like (metallic) frequency dispersion of the effective permittivity of the metamaterials changes into a Lorentz-like (insulating or dielectric) dispersion when small cuts are introduced in the metallic wires. Then, it was pointed out that the electromagnetic responses of conducting polymers, which change critically depending on the sample quality, could be modeled by such metamaterials.

The MCPs can be regarded as metamaterials extended from wire arrays so that they are isotropic for the incident polarizations. Figure 5 shows the effective relative permittivities of the MCPs when they are assumed to be homogeneous media with a thickness of h_m deduced from the experimental complex transmittance in Figs. 4(a) and 4(b). Here it should be noted that the effective permittivities are meaningful below f_d. The Drude-like (free-electron-like) frequency dispersion (d = 0.76 mm) changes into Lorentz -like (localized-electron-like) dispersions (typically, d = 0.66 mm) with decreasing d. The effective plasma frequency is about 0.2 THz for d = 0.76 mm, which is quite small compared with the plasma frequency of 57.2 THz for the metallic ink. As well as the case of the metamaterials of the wire arrays, the transition from a Drude-like to a Lorentz-like frequency dispersion in MCPs can be induced by a very small structural variation. Once the metallic squares are separated, the gaps works as the capacitance in series and the impedance diverges with $\omega \to 0$.

 

Fig. 5 (a) Real and (b) imaginary parts of the effective permittivities of the MCPs.

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6. Terahertz imaging of an optically hidden image

Finally, we propose a potential application for detecting small structural variations in near self-complementary MCPs. For example, our eyes can barely distinguish between the MCPs with d = 0.72 and 0.70 mm in Fig. 3, but the terahertz spectra can enable us to do so clearly. The above property of the transmission spectra of the MCPs can be applied to view an almost visually imperceptible image by terahertz imaging. We fabricate a sample made of metal squares by printing, as shown in Fig. 6(a). The characters “THz” are drawn with an MCP of d = 0.70 mm in the background of an MCP with d = 0.72 mm. The sample is scanned by terahertz pulses with a beam diameter of 4 mm in steps of 2 mm. Spectroscopic images are obtained by Fourier transforming the time-domain waveform measured at each point. Figure 6(b) shows the spectroscopic image at 0.08 THz. Although we can barely see the characters in the photograph, the characters “THz” appear clearly in the terahertz image. This example demonstrates that when near self-complementary MCPs are fabricated on a flexible substrate, small structural variations of the substrate may be detected by their electromagnetic responses.

 

Fig. 6 (a) Photograph of the MCP and (b) its transmission image at 0.08 THz, in which the characters “THz” are imbedded as regions with a square size slightly different from that of the background.

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7. Conclusion

In conclusion, we have investigated the electromagnetic responses of near self-complementary MCPs using an FDTD simulation and an experiment using the samples fabricated by the ink-jet printing method in order to clarify the critical nature of these structures. The terahertz responses of the near self-complementary MCPs exhibited drastic spectral changes with small structural variations depending on whether the adjacent metal squares are in contact with each other. The experiments for the samples fabricated by the color printer show that randomness produces invariant points in the transmission spectra of near self-complementary MCPs for the variation of the nominal side length d of the metal square. This property is analogous to two interrelated absorbers in a liquid solution and implies that the transmission spectra can be designed by the connection ratio of the meta-atoms. Additionally, we have proposed a potential application of terahertz imaging for detecting small structural variations, which takes advantage of the large spectral change of MCPs near their self-complementary or percolation threshold. We demonstrated that a nearly visually imperceptible pattern could be viewed by terahertz imaging.