1. Introduction

In systems that display optical bistability, the spectra are strongly dependent on the input intensity, and might even display a hysteresis loop. As such, the property of optical bistability could be explored to develop a variety of nonlinear functional devices, such as all optical switches, transistors, diodes, memories, and logical gates. Recently, bistable wave transmission, or optical switch effect, has been extensively investigated in nonlinear photonic (or electromagnetic) bandgap (PBG, or EBG) structures [1–4], for example, by utilizing the defect mode [1], band edge [2], or resonance of optical microcavity [3,4]. However, PBGs are usually constructed by periodically arranged artificial discontinuities that satisfy the Bragg condition [5–7]. As a result of the multiple scattering at interfaces in conventional photonic crystal (PC), the bandgap is greatly influenced by the periodic number and the contrast of dielectric constants of PCs. These two factors will affect either the scattering times or efficiency at interfaces significantly. Simply increasing the number of PBG cells, as well as the wave impedance at interfaces, will lead to enlarged device size and lower transfer efficiency of passband. All these negative features certainly restrain the application range of nonlinear PBGs devices, especially at the low-frequency region where the corresponding wavelength is large [8,9]. In order to circumvent these problems, some exotic concepts must be introduced.

In recent few years, meta-atoms that are utilized to mimic some quantum optical phenomena, like electromagnetically induced transparency (EIT), have attracted massive interests from scientists. EIT is a special quantum optical phenomenon originated from the destructive interference between “bright” and “dark” states [10]. As a classical-light analogy to quantum EIT, the EIT-like effects occurring in metamaterials is a type of linear interference phenomenon of light [11–23]. Owing to the interesting physics and potential applications, such as slow light and optical storage [12,13], sensing [14–16], and low-loss metamateials [17], EIT in classical electromagnetic system has been investigated intensively, covering a wide range of spectral region from microwave [12], terahertz [18,19], to infrared and visible radiation [20], and even X-rays [21]. The EIT phenomenon in metamaterials provide a way for strong localization of electromagnetic fields at the subwavelength scale, which is benefit to the realization of extremely compact optical components with high integration efficiency [22–24]. Moreover, the nonlinear EIT phenomenon in metamaterials provides more opportunities for the nonlinear optical device when the nonlinear medium is embedded in the metamaterials.

In this paper, EIT-like meta-atoms are introduced into a microstrip PC cavity and its influence on the nonlinear properties is investigated intensively. As is known, when the resonance frequency of EIT-like meta-atoms coincides with that of the PC cavity, highly electromagnetic (EM) localization and related slow group velocity can be achieved, which provides better interaction between EM waves and nonlinear medium [25]. Here, varactor-loaded SRR of the EIT-like meta-atoms is considered as the nonlinear medium inclusion. Nonlinear responses, including the shift of resonant frequency as well as hysteresis effects, are demonstrated experimentally in distinct ranges of input powers. Ultra-low switching energy is realized (−6.1 dBm), and the transmission contrast exceeds 4.0 dB. Such low-threshold and high-contrast transmission action comes from the composite PC-EIT mechanism, which possesses higher Q-factor and stronger localized EM field than either PC cavity or EIT-like meta-atoms. As these advantages above are not at a cost of extra device volume, the design principle of composite PC-EIT structure is promising to be applied in integrated nanocircuits or metactronics.

2. Results and discussions

The photograph of the microstrip PC cavity is shown in Fig. 1(a). The dielectric constant (${\epsilon }_r$) and thickness (h) of the FR4 substrate are 4.4 and 1.6 mm respectively. The width w of the conductor strip at the top plane is 3.0 mm in order to match the characteristic impedance Z₀ = 50.0Ω. The microstrip PC cavity is made by periodically etching rectangles copper strips with linewidth 0.5 mm and dimension 10.0 mm × 15.0 mm. Rectangles copper strips are linked by capacitive fingers. The widths, gaps and lengths of the fingers are 0.5 mm, 0.5 mm, and 140.5 mm, respectively. A defect is introduced by changing the length of the finger at the middle of the waveguide from 140.5 to 249.0 mm, leading to defect mode with frequency f₀ = 0.87 GHz located at the center of PC bandgap. The photograph of the EIT-like meta-atoms fabricated on the top plane of the substrate is shown in Fig. 1(b). To mimic a three level atomic EIT system, our EIT-like meta-atoms are composed of an open-ended comb line (l₁ = 25.0 mm, l₂ = 25.2 mm) side-coupled with the microstrip and a split ring resonator (SRR) far away from the microstrip. Thus, the open-ended comb line with linewidth 0.3 mm can be excited directly by the input wave, playing a role of bright meta-atom, like the radiative state in atomic EIT molecules strongly coupled with the incident field. The SRR acts as a dark meta-atom that cannot be excited directly by the input wave through the microstrip, mimicking an atomic dark state coupled weakly with the incident field [11]. The SRR with a 1.0 mm slit etched at one side is made of 2.0 mm width copper strip, and its total dimension is 11.2 mm × 11.2 mm. The distance from the SRR to the microstrip is 13.3 mm. The closer distance from the SRR to the open-ended comb line is 0.2 mm. In order for the EIT-like meta-atoms to function as a highly dispersive material, it is important to guarantee that its resonance frequency coincides with that of the PC cavity. A silicon hyperabrupt varactor (Infineon BBY52) is loaded in the slit of SRR, which acts as the nonlinear medium. The photograph of the composite PC-EIT structure is shown in Fig. 1(c). The EIT-like meta-atoms are embedded within the PC cavity, to realize Q-factor and EM field enhancement, and get the ultimate goal of nonlinear properties improvement.

 

Fig. 1 Photograph of (a) microstrip PC cavity, (b) EIT-like meta-atoms, and (c) composite PC-EIT structure. The inset shows the detailed schematic of EIT meta-atoms.

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We first investigate the linear transmission spectrums with numerical simulations (CST Microwave Studio). In this case a fixed value capacitor of C = 2.65 pF is loaded instead of the varactor. For the purpose of comparison, the transmissions of individual microstrip PC cavity, individual EIT-like meta-atoms, and the composite PC-EIT structure are all shown in Fig. 2(a) as the black dotted line, the red dash line, and the blue solid line, respectively. The transmission spectrum of the PC bandgap structure is also depicted as the green curve in the inset. In the inset of Fig. 2(a), the transmission of a photonic crystal without cavity layer is given, which indicated clearly that from about 0.70 to 1.05 GHz, there exists a stop band almost overlapping with the stop band of PC-cavity structure. It is clear that, for the two individual structures, microstrip PC cavity and EIT-like meta-atoms, both Q-factors, 21.6 and 9.3, are quite limited. For the composite PC-EIT structure, however, the Q-factor is enlarged to 126.5. To validate the above simulations, microwave experiments are also carried out by a network analyzer, Agilent 8722ES. Figure 2(b) shows the measured transmissions, which confirm that the way of embedding EIT-like meta-atoms into the PC cavity can indeed enhance the Q-factor effectively. Comparing the transmissions in Fig. 2(b) with those shown in Fig. 2(a), despite a frequency shift less than 5%, the experimental results coincide with the simulated results. The frequency shift is due possibly to the approximated dispersions for the FR4 substrate used in the simulations, as well as the accumulated errors during fabrication. Note that, the cavity figure of merit improvements are not at costs of extra device volume and transmission losses, which is highly desired but difficult to achieve for conventional nonlinear devices based on PC cavities.

 

Fig. 2 Linear transmission spectra for microstrip PC cavity, EIT-like meta-atoms and PC-EIT structure: (a) simulation, (b) experiment. The inset shows the transmission spectrum of the PC bandgap structure.

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To shed light on the physical mechanism of this Q-factor enhancement, electric energy density distributions of microstrip PC cavity, EIT-like meta-atoms and composite PC-EIT structure at an operating frequency of 0.87 GHz are calculated. Figure 3(a) shows the electric energy density of the microstrip PC cavity. One can observe clearly that there is some EM localization along the propagating direction. However, the confinement is not that strong for the reason that the number of periods of the microstrip PC cavity is quite limited. EM fields can also reside outside the defect owing to weak confinement. The electric energy density distribution of the EIT-like meta-atoms is shown in Fig. 3(b). Note also that there is EM field confinement between the bright and dark meta-atoms. Figure 3(c) shows the electric energy density distribution for the composite PC-EIT structure. The electric energy density of the composite PC-EIT structure can be as high as 0.407 J/m³ at the spacing of dark meta-atoms, about one or two orders higher than that of the microstrip PC cavity (0.0143 J/m³) and the EIT-like structure (0.0463 J/m³). That is, the PC cavity confines the EM waves along the propagating direction, as well as the EIT-like meta-atoms provide extra EM localization along direction perpendicular to the propagating direction. As a result of the cooperation of these two EM localization mechanisms, the combined PC-EIT structure cannot only improve the cavity Q-factor, but also enhance the localized EM field. Hence, the combined PC-EIT structure must benefit the PC cavity nonlinear responses.

 

Fig. 3 Top view of electric energy density distributions at 0.87 GHz in (a) microstrip PC cavity, (b) EIT-like meta-atoms, and (c) composite PC-EIT structure.

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In the following, we experimentally investigated the bistable wave transmission effect by loading the varactor acting as nonlinear medium in the sample. Figure 4 presents the transmission spectra measured at different levels of input power from −25 dBm to 8 dBm. As expected, the transmission around the PC-EIT mode is strongly depended on the input intensity of power. With the increase of input power, the local electric field on the varactor is enhanced greatly, and hence PC-EIT mode shifts to lower frequency. Quantitatively, the self-tuned frequency could change by 3% when the input power increases from 6 dBm to 8 dBm by 1 dBm. In the meanwhile, the transmission peak becomes more and more asymmetric, which is unambiguous evidence of bistable response.

 

Fig. 4 Measured transmission for the combined PC-EIT structure with respect to the input power.

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For the purpose of finding the minimum bistable threshold, the measured transmission spectra with respect to forward and backward frequency sweep at different input level are plotted in Fig. 5. Hysteresis loops are observed at the range around the working frequency, owing to the dynamic feedback in a nonlinear resonant system. The hysteresis effect disappears until the power value decrease to −6.1 dBm. It should be mentioned that −6.1 dBm can be achieved for normal microwave equipment without power amplifier, such as vector network analyzer and signal generator. For the input power of −6.1 dBm, the contrast in transmission between the two bistable states has a ratio of about 4.0 dB, which is sufficient high to meet the general application.

 

Fig. 5 Hysteresis effects in frequency sweeping at different input power levels. The lowest bistable threshold is only −6.1 dBm.

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To further demonstrate the bistability response of the sample system, we use a monochromatic input signal at f₀ = 0.88 GHz with the power strength in the range of −6.0 dBm to −4.0 dBm. The bistable hysteresis loop with respect to bidirectional sweep of input power is shown in Fig. 6. Clearly, there exist two threshold levels at −5.4 dBm and −5.0 dBm for forward and backward sweeps, and the contrast in transmission between the two bistable states has a ratio of about 4 dB. When the increasing (decreasing) input power meets its threshold level, the transmission is abruptly up (down). The two threshold levels define a regime of input power to trigger the multi-valued transmission at the operating frequency of 0.88 GHz, which can be used for a subwavelength EM switch or transistor.

 

Fig. 6 Measured transmission spectra at 0.88 GHz as a function of input power from −6 dBm to −4 dBm. The black triangular and red inverse-triangular are respected to forward (increasing) and reverse (decreasing) power sweep, respectively.

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It should be mentioned that, if the EIT resonance frequency is shifted away from the cavity frequency, the situation will be more complex. However, it can still be understood under a Fano-type resonance scheme. The stop band of PC can be regarded as a continuous spectrum, as well as the EIT-like meta-atoms provide multiple discrete resonances, leading to complex Fano-type spectrum.

3. Summary

To sum up, a microstrip PC cavity embedded with nonlinear EIT-like meta-atoms in an on-chip configuration is proposed. The investigation on the linear transmission spectrums demonstrate that the PC cavity Q-factor can be efficiently enhanced by the embedded EIT-like meta-atoms. The study on the electric energy density distributions reveals that this Q-factor enhancement is mainly due to the cooperative contributions from both the microstrip PC cavity and the side-coupled meta-atoms. Importantly, high Q-factor and strong confinement of EM field are essential to the cavity nonlinear performance. Detailed microwave experiments confirm that the threshold of low to −6.1 dBm and the transmission contrast of up to 4.0 dB between the two bistable states can be achieved, with a varactor as the nonlinear medium inclusion. It is worth mentioned that all these improvements are not at a cost of device volume. As the technologies for fabricating multilayer thin films and solid-based meta-atoms are well developed, our structure could be extended to the optical region.