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Abstract
We experimentally and theoretically investigate the nonlinear electromagnetic properties of light-tunneling heterostructures embedded with a varactor-loaded electromagnetically-induced-transparency (EIT)-like highly dispersive meta-molecule. We illustrate that when an EIT-like meta-molecule is hired at the interface of ε-negative medium and µ-negative medium, the Q-factor and corresponding local fields of tunneling modes can be greatly boosted. Further study reveals that the electromagnetic (EM) field confinement along the propagating direction provided by the heterostructures, and the in-plane localization originated from the EIT-like meta-molecule, give rise to the three-dimensional enhancement of sub-wavelength EM localization corporately. Moreover, such a configuration can generate an extremely high transmission contrast up to 12 dB between two bistable states in its transmission with a bistability threshold low to −4.7 dBm. These advantages are not at the cost of extra device volume and drastic reduction of transmittance.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical bistability (OB) is a nonlinear optical phenomenon characterized by a hysteresis loop response to the input power intensity. The importance of OB stems from its exotic light control in optoelectronic applications such as high-speed signal processing and optical computing. Low input intensity for a sizeable nonlinear response and high transmission coefficient for a miniaturized optical component are two instrumental factors for bistable device performance [1–4]. However, governed by photon-photon interactions, optical nonlinearities are inherently weak. Considerable efforts have been devoted during last decade to develop optical bistable elements and different schemes have been proposed to achieve the above mentioned two goals within one structure [5–9]. Among them, an unparalleled way is to use the light tunneling effect occurring in a heterostructure constructed by an epsilon-negative (ENG) material and mu-negative (MNG) material [10–14]. Light tunneling heterostructures provide confinement along the propagation direction which results in enhancement of the tunneling effect and corresponding electromagnetic (EM) localization. Based on the strong EM localization at the interface of the heterostructures, potential applications of light tunneling effects have been suggested, such as polariton lasers [15], enhancement of Faraday rotation [16], and various nonlinear effects [17]. However, the EM confinement for these structures is determined only by the propagating direction of EM waves. To boost the EM localization further, the in-plane confinement also may be involved [18–20].
Recently, quantum optical phenomena, including electromagnetically induced transparency (EIT), have attracted enormous attention from scientists [21–32]. EIT is a special quantum optical phenomenon originated from the strong destructive interference between super-radiant and sub-radiant modes, resulting in “bright” and “dark” states, respectively. As a novel way of optical quantum control, EIT has some major properties, for example, slow light, steep phase dispersion, and spatial localization of light. In 2008, Zhang and associates proposed a plasmonic analogy of EIT by the coupling of bright and dark “meta-atoms” [21]. Soon after, Giessen et al. validated this EIT design by state-of-the-art experiments in optics [22]. Some other theoretical and experimental works based on “meta-atoms” or “meta-molecules” in different systems also were carried out to demonstrate the EIT-like effect, such as metallic fish-scales [23], U-shaped resonators [24–26], dipole antennas [27], trapped-mode patterns [28], array of nanoparticles [29], and so on. Compared to cold atomic gas, usually in terms of three-level quantum optical systems, this sort of solid-state-based configuration is more stable, low cost, and easily integrated. Specially, the local resonances of these EIT-like structures can also boost the nonlinear interactions with the assistance of strong in-unit localization of EM fields at the subwavelength scale [30–32]. Active control of the EIT effect is of great significance for miniaturized and versatile devices in practical applications, and the tunable EIT-like effect of metamaterials integrated with active elements, such as PIN diodes [33], has also been demonstrated. Up to now, nonlinear light tunneling effect enhanced by an EIT-like meta-molecule has not been reported.
In this paper, nonlinear light tunneling behaviors influenced by a meta-molecule with EIT-like dispersion are studied intensively. Both measured and simulated results demonstrate that when a three-level molecule-like system with EIT-like properties is introduced at the interface of ENG-MNG pair, the Q-factor and corresponding local fields of the tunneling mode can be enhanced significantly. As a result, an extremely high transmission contrast up to 12 dB between two bistable states in its transmission is achieved with bistability threshold low to −4.7 dBm. Furthermore, these advantages are not at costs of extra device volume and drastic reduction of transmittance, which are highly desired but difficult to achieve for traditional nonlinear devices based on light-tunneling heterostructures.
2. Results and discussions
According to Ref. 11, photonic crystal (PC) takes advantage of its stop band properties, can be thought to be equivalent to an effective single negative material under some well-designed geometric and EM parameters. Here, an effective light-tunneling heterostructure is realized through a microstrip PC heterostructure (ABA)N-(BAB)N, with periodic number N = 4, as shown in Fig. 1. The heterostructure is fabricated and deposited on a printed circuit board (PCB), which has a copper layer attached to the bottom. The 1.6 mm thick PCB substrate used here is FR4, with its relative permittivity 4.4 and loss tangent 0.002, respectively. The width of input and output microstrip transmission line is 2.9 mm and the characteristic impedance is Z0 = 50.0 Ω. The detailed structural parameters are as follows: a = 15.0 mm, b = 5.0 mm, c = 114.0 mm, w = 0.5 mm, where a and b are the outside length and inside width of the rectangle, respectively, c is the length of the junction line, and w is the uniform width of the line. Local details are shown in the inset in Fig. 1. In this case, the PCs (BAB)4 and (ABA)4 play the role of ENG and MNG mediums, respectively. All numerical simulations are carried out by a commercial finite-integration-technique based EM solver from CST Microwave Studio. In experiments, transmission spectra are measured by an Agilent N5222A microwave vector network analyzer.
Fig. 1 Photograph of the microstrip light-tunneling heterostructure (ABA)4-(BAB)4. The local detailed structural parameters are shown in the inset.
In Fig. 2, the transmission spectra of the left PC (ABA)4, the right PC (BAB)4, and the paired light-tunneling heterostructure (ABA)4-(BAB)4 are shown by the blue dashed lines, the red dotted lines, and the solid black lines, respectively. As is known, although the EM waves in ENG or MNG are always decaying, they can still tunnel through the ENG-MNG heterostructure when the matching conditions are satisfied [10]. The shadow area in Fig. 2 indicates that first band gaps of two PCs overlap from 700 MHz to 750 MHz. Therefore, (ABA)4 and (BAB)4 can only support evanescent waves and they are all opaque to incident light in this frequency range. However, for PC heterostructure (ABA)4-(BAB)4, a transparent tunneling mode is observed at 726 MHz in simulation and 758 MHz in the experimental measurement, as indicated by the green dotted lines. Comparing the measured results with the simulated results, despite a frequency blue shift less than 5%, they coincide with each other.
Fig. 2 The simulated and measured transmissions for individual MNG medium, individual ENG medium, and the composite ENG-MNG heterostructure are shown by the red dotted lines, the blue dashed lines, and the black solid lines, respectively.
The photograph of ENG-MNG heterostructure with embedded EIT-like meta-molecule is shown in Fig. 3. The geometrical parameters of the ENG-MNG heterostructure are the same as the above mentioned structure, as shown in Fig. 1. The EIT-like meta-molecule is embedded in the interface of the ENG-MNG heterostructure. To mimic a three-level molecule-like EIT system, our EIT-like meta-molecule is composed of an open-ended comb line connected to the transmission line and a split ring resonator (SRR) far away from the microstrip. Local details of the EIT-like meta-molecule are displayed in the inset in Fig. 3. The comb line (l1 = 30.0 mm, l2 = 30.3 mm) is made of 0.3 mm width copper strip, and has a resonant mode at 726 MHz. As a branch of the microstrip, the comb line can be excited directly by the input wave, playing a role of bright meta-atom. The square SRR with a 1.0 mm slit etched at one side can be considered as a dark meta-atom that cannot be excited directly through the microstrip. The edge length of SRR is 10.6 mm and the width is 1.5 mm. The working frequency of SRR is also located at 726 MHz. The gap between SRR and the comb line is 0.3 mm. The separation between SRR and microstrip is set to be 10.1 mm. A silicon hyperabrupt varactor (Infineon BBY52), acting as the nonlinear medium inclusion, is mounted on the FR4 substrate in the slit of SRR. Due to the geometrical parameters we chosen, the operating frequency of the EIT-like meta-molecule is in accordance with that of the light-tunneling heterostructure. Note that, in order for the EIT-like meta-molecule to function as a highly dispersive material, it is important to guarantee that its resonance frequency coincides with that of the light-tunneling heterostructure.
Fig. 3 Photograph of light-tunneling heterostructure with embedded EIT-like meta-molecule. The inset shows the detailed schematic of the EIT-like meta-molecule.
Firstly, the EIT-like meta-molecule is studied at the weak-power situation. In this case, we load an ideal capacitor of C = 2.65 pF instead of the varactor. All simulations are performed by a finite-integration-technique (FIT) based EM solver from CST Microwave Studio. The calculated transmission spectra of bright meta-atom and dark meta-atom are provided in Fig. 4 as red dash line and blue dotted line, respectively. Obviously, the bright meta-atom exhibits strong broad linewidth response, while the dark meta-atom exhibits weak narrow linewidth response. Thus, the narrow response interferes destructively with the broad response the meta-molecule exhibits an EIT-like line shape, as the black solid line shown in Fig. 4.
Fig. 4 Calculated transmission spectra for individual bright meta-atom, individual dark meta-atom, and the EIT-like meta-molecule.
Now we investigate the linear transmission spectrum of ENG-MNG heterostructure with EIT-like meta-molecule embedding. For the purpose of comparison, the simulated transmission spectra for individual ENG-MNG heterostructure (blue dashed line), individual EIT-like meta-molecule (red dotted line), and ENG-MNG heterostructure with loaded EIT-like meta-molecule (dark solid line) are all shown in Fig. 5(a). It is found that the Q-factors of the two individual structures, ENG-MNG heterostructure and EIT-like meta-molecule, 18.6 and 9.1, are quite limited. However, for the ENG-MNG heterostructure with EIT-like meta-molecule loading, the Q-factor is enlarged to 114.6, about a six-fold enhancement. To validate the above simulations, microwave experiments are also carried out. Figure 5(b) shows the measured transmissions, which confirm that the Q-factor of tunneling mode can indeed be enhanced remarkably after the EIT-like meta-molecule is loaded. It is worth mentioning that, unlike the traditional way of Q-factor enhancement [34], our method takes the advantages of miniaturized device size and low transmission loss. In addition, the operating frequency of EIT-embedded heterostructure is 726 MHz in simulation and 765 MHz in the experimental measurement as indicated by the green dotted lines in Fig. 5.
Fig. 5 The simulated and measured transmission spectra for individual ENG-MNG heterostructure (blue dashed line), individual EIT-like meta-molecule (red dotted line), and ENG-MNG heterostructure with EIT-like meta-molecule loading (dark solid line).
In order to reveal the underlying physics of this Q-factor enhancement, electrical energy density distributions of the conventional ENG-MNG heterostructure, the individual EIT-like meta-molecule and the ENG-MNG heterostructure with embedded EIT-like meta-molecule are simulated and shown in Fig. 6. For heterostructure with detailed structure of (ABA)4(BAB)4 shown in Fig. 6(a), confinement along the propagating direction can be achieved with a maximum electrical energy density of 0.011 J/m3 around their interface. It is clear that for ENG-MNG heterostructure only the confinement along the propagating direction, i.e., one dimensional confinement is effective. It can also be observed that notable EM energy is still residing outside of the interface area due to weak confinement. The electric energy density distribution of the EIT-like meta-molecule is shown in Fig. 6(b). Note that there is in-unit EM field confinement between the adjacent bright and dark meta-atoms, and the achieved maximum electrical energy density is only 0.071 J/m3. However, for the ENG-MNG heterostructure with EIT-like meta-molecule embedding, as shown in Fig. 6(c), the maximum electrical energy density is 0.556 J/m3, about one order higher than that of the conventional ENG-MNG heterostructure and the EIT-like meta-molecule. Interestingly, in the heterostructure with EIT element loading, EM energy is further confined at some special points around the ENG-MNG interface in three-dimensional space. Thus, we can conclude that the confinement along the propagating direction provided by ENG-MNG heterostructure and the in-plane localization originated from EIT-like meta-molecule give rise corporately to the enhancement of tunneling effect and the corresponding EM localization. Besides the EIT-like meta-molecule is etched at the interface of ENG-MNG heterostructure, no extra volume is added to the whole device, which is crucial for the miniaturization of modern nonlinear optical devices.
Fig. 6 Normalized electrical energy density distributions at 726 MHz of (a) individual ENG-MNG heterostructure, (b) individual EIT-like meta-molecule (c) ENG-MNG heterostructure with EIT-like meta-molecule loading (the above distributions are depicted from top view).
Let us now investigate the nonlinear properties by introducing the varactor in the sample system. The measured transmissions under different input power changing from −20 dBm to 9 dBm are presented in Fig. 7. As predicted, the enhanced Q-factor tunneling mode is very sensitive to the input intensity of power. For not powered or low input power situations, for example the input power of −20 dBm, the transmission spectrum displays a peak around 765 MHz. Then we increased the −20 dBm on the sample, it can be seen that the transmission peak frequency exhibits a prominent red-shift. Quantitatively, with the rise in input power from −20 dBm to 9 dBm, the self-adjusting frequency can change by 14%. Especially when the input power is 9 dBm, a prominent transmission contrast of about 20 dB can be achieved at 677 MHz, as indicated by the green dotted lines in Fig. 7. From this point of view, the ENG-MNG heterostructure with EIT-like meta-molecule loading can act as a switching for the microwave controlled by the input intensity.
Fig. 7 Measured transmission of the ENG-MNG heterostructure with EIT-like meta-molecule loading under different input power changing from −20 dBm to 9 dBm.
In the following, the bistable responses of the sample system are investigated in details. The measured transmission curves with respect to bidirectional frequency sweeping at disparate incident wave intensities are plotted in Fig. 8. The evolution of transmission takes different paths as a function of the input intensities is typically the hysteresis effect. A minimum bistable threshold of −4.7 dBm is required for producing the hysteresis loop for the ENG-MNG heterostructure with EIT-like meta-molecule embedding. Importantly, −4.7 dBm can be achieved without power amplifier, which is most easily done by signal generator. Moreover, at the switching intensity of 0 dBm, the transmission contrast between two bistable states exceeds 7 dB, which is sufficient for the general microwave application needs.
Fig. 8 The bistable hysteresis loops in bidirectional frequency sweeping (from 700 MHz to 780 MHz) at four typical input power intensities. The achieved threshold of bistability is only −4.7 dBm.
To further clarify this bistable behavior, we introduce six monochromatic input signals at 640 MHz, 660 MHz, 680 MHz, 700 MHz, and 740 MHz into the sample. The measured transmission intensities versus forward and backward sweep of input power in the range of −5.0 dBm to 15.0 dBm are presented in Fig. 9. Clearly, with the decrease of frequency, the threshold of bistable state will increase, but the transmission contrast may increase. As the input power increases, taking 660 MHz as an example, the transmission intensity firstly follows the lower level and then suddenly jumps to the higher level at 6.9 dBm. With sweeping input power back, the system maintains on the higher level and then makes a transition to the lower level at −4.7 dBm. Thus, a hysteresis loop and a modulation contrast of up to 12 dB are achieved on the transmission through the sample. Generally speaking, the transmission at 660 MHz can be drastically modulated under a small change in the input power intensity thanks to the enhanced subwavelength EM field confinement along the propagating direction provided by the light-tunneling heterostructure and the in-plane localization originated from the EIT-like meta-molecule.
Fig. 9 Measured transmission (in dB) at 640 MHz, 660 MHz, 680 MHz, 700MHz, 720 MHz, and 740MHz versus input power on the sample (from −5 dBm to 15 dBm). The black triangle and red inverted-triangle are respected to increasing and decreasing sweep of input power, respectively.
3. Summary
In summary, the nonlinear properties of light-tunneling heterostructures are investigated by introducing a meta-molecule with EIT-like dispersion. Both numerical simulations and microwave experiments are performed, showing that when an EIT-like meta-molecule is hired at the interface of ENG-MNG pair, the Q-factor and corresponding EM fields of tunneling mode can be enhanced greatly. As a consequence, a transmission contrast up to 12 dB between two bistable states in its transmission is achieved with the bistability threshold low to −4.7 dBm. As the advantages above are not at costs of extra device volume and drastic reduction of transmittance, this design is promising to be applied in miniaturized integrated photonic nano-circuits or meta-electronics.
Funding
National Natural Science Foundation of China (NSFC) (Nos. 51607119, 11704273, and 11674247); Natural Science Foundation of Jiangsu Province, China (No. BK20170375); Natural Science Foundation of Jiangsu Higher Education Institutions, China (Nos. 17KJB140021, 17KJA140001); Jiangsu Province Key Discipline of China’s 13th five-year plan (No. 20168765).
Acknowledgments
The authors thank Dr. J. Jiang and L. He for useful discussions and help in experiments.
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