1. Introduction

Metamaterials [1,2] are a class of synthetic structural materials with the feature of subwavelength and have many novel properties not found in naturally materials, including the abnormal refraction [3–5] and sub-wavelength focusing [6,7], etc. Metamaterials have opened the pandora's box to study the novel and exciting electromagnetic responses [8,9] and have achieved the arbitrarily controlling abilities of incident and transmitted/reflected electromagnetic waves on demand [10–13]. At the beginning of metamaterial development, most of the designs cannot achieve the dynamic controlling for the electromagnetic responses. Thereafter, researchers proposed several methods to realize the real-time electromagnetic response controlling/adjusting features by integrating different active components and/or specific materials, such as the diode/varactor [14], graphene [15], MEMS device [16], liquid crystal [17], ferrite [18], and phase change materials [19], etc., and by altering the biasing DC or low frequency voltage, current, magnetic field, temperature. Quit recently, researchers have developed several exciting sub-research directions based on the well-established active/reconfigurable metamaterial techniques, that is the digital-coding, programmable, intelligent, self-adaptive, and even cognitive metamaterials/metasurfaces [20–26].

On another hand, among the different realization methods for the active/reconfigurable metamaterials, the nonlinear metamaterial is one of sub-area which can also be used to achieve the reconfigurable features [27,28]. Moreover, nonlinear metamaterials can also be used to achieve several new functions, such as the high-order harmonics generations and wave mixings [29–31]. In this nonlinear metamaterial areas, again, the diode/varactor, graphene, MEMS device, liquid crystal, and some other nonlinear substrate can be used [32–34], and the nonlinear response are achieved by controlling the incident/pump electromagnetic wave power levels. Most interestingly, one kind of the nonlinear metamaterial did not integrate any additional active components or nonlinear materials. It is realized by constructing the metamaterial unit on a mechanical deformable substrate, and the nonlinear response is achieved by the multi-physics interaction between the electromagnetic wave and the mechanical potential energies [35]. Specifically, it generates the field-structure interaction and results in a force under a certain intensity of electromagnetic wave excitation, and in turn causes the electromagnetic characteristics changes of the metamaterial [35,36]. This new kind of metamaterial has been called as optomechanical metamaterial [37]. Different from the conventional nonlinear metamaterials, this kind of optomechanical metamaterial has the advantages of, for examples, the simple construction, easy manipulation of electromagnetic wave characteristics and fast response tuning speed, because only the mechanical feature of the substrate is taken into considerations.

In the past several years, the forces driving the substrate of the optomechanical metamaterials into mechanical shift/deformation, vibration, or even oscillation mainly include optical gradient force [37–40], electromagnetic induced forces [35,41–46], and photothermal expansion forces [47,48] generated by electromagnetic wave coupling. Specifically, the electromagnetic induced forces driving for the optomechanical metamaterial include the Ampère force, Coulomb forces and Lorentz forces [35,41–46]. However, it should be noticed that, to drive the optomechanical metamaterial worked at lower electromagnetic wave energy, the as high as passible of the opto-mechanical coupling rate should be considered, and unfortunately, most of the reported optomechanical metamaterials did not focus on this point. Similar to the cavity optomechanics research area [49–54], to achieve the highest opto-mechanical coupling rate, the electromagnetic resonance in electromagnetic metamaterials should have the as high as possible of the quality factor, and the electromagnetic resonator and mechanical vibrator should be the same structure as well. Specifically, with higher quality factor within smaller resonator structure, the electromagnetic energy can be highly confined and concentrated, and based on the theoretical equation g_om= g₀(n_cav)^1/2, the opto-mechanical rate can be enhanced accordingly [49,50]. Here g₀ presents the opto-mechanical single-photon coupling strength and n_cav relates to the intracavity electromagnetic energy. On another hand, in recent years, several metamaterial unit configurations have been proved that can exhibit very high quality factors, including the asymmetry Fano resonator [55,56], toroidal resonator [57,58], anapole resonator [59,60], and the reflect-type electric inductive-capacitive (ELC) resonator based metamaterial absorber [61,62].

Based on this basic design guide, therefore in this paper, we propose a conceptual optomechanical metamaterial by using the basic ELC resonator deposed on a flexible printed circuit board (FPCB) and backed by a fully-sized metallic reflector with four movable support beams to form the high quality factor conditions. Specifically, we study the wave-absorbing properties of the new optomechanical metamaterial and the mechanical deformation/moving properties under multi-physics coupling by simulations. The nonlinear coupling characteristics of the optomechanical metamaterial driven by the electromagnetic induced force are theoretically analyzed and experimentally verified as well.

Compared with the previously reported optomechanical metamaterials drove by electromagnetic induced forces [35,41–46], the reflection-type ELC-based optomechanical metamaterial proposed in this paper can show better opto-mechanical coupling performances with higher resonant quality factor and more sensitive resonant frequency shift under smaller electromagnetic wave energy changes. Also, different from the previously reported reflection-type optomechanical metamaterial absorber configuration which drove by photothermal expansion forces [48], our proposed reflection-type optomechanical metamaterial is drove by, as mentioned before, the electromagnetic induced forces, so the lower wave energy is needed. Moreover, the optomechanical coupling state for each unit cell can be independently controlled in this paper because each ELC unit is supported by four movable beams, and therefore it can be used to realize more flexible reflective-type reconfigurable metasurface for abnormal reflection controlling in the near future.

2. Structure design and operating mechanism analysis

Figures 1(a) and (b) show the schematic array and unit structure diagrams, respectively, for the proposed optomechanical metamaterial with detailed structural parameter definitions. Each optomechanical metamaterial unit consists of four functional layers: (I) The top layer is the regular metallic ELC resonance structure, (II) the second layer is a FPCB flexible substrate with a thickness of h, (III) the third layer is an air layer with thickness of b, and (IV) the bottom layer is a metal backplane of optomechanical metamaterial with a thickness of t. It is well known that the ELC resonator backed by a full-sized metallic reflector can be considered as a regular metamaterial absorber [18,63]. Therefore, assuming that when the electromagnetic waves acts on each optomechanical unit structure, a strong induced current will be generated on the ELC resonator at the resonance frequency. Simultaneously, an inverse induced current will be generated on the metal backplane which has been widely confirmed at all the ELC based metamaterial absorbers [18,63]. As an example, the simulated absorptivity curve is shown in Fig. 1(c) with the induced current distributions at the ELC resonator and metal backplane. As a result, there should have mutually exclusive Ampere’s forces between the ELC and the metal backplane. When the intensity of the incident electromagnetic wave is large enough, it will cause mechanical deformation or displacement for the ELC resonator which deposed on the FPCB flexible substrate and suspended by the support beams, which in turn affects the induced current amplitude and interaction force. On another hand, due to the elasticity of the FPCB flexible substrate, the inherent elastic force will prevent the mechanical deformation or displacement and reach to a balance state once the induced Ampere’s forces and elastic force are equal.

 

Fig. 1. Schematic presentation and basic electromagnetic and mechanical responses of the optomechanical metamaterial. (a) 3D schematic diagram of optomechanical metamaterial array. (b) Optomechanical metamaterial unit cell and its top view and side view with parameter definitions. (c) Simulated absorptivity for the optomechanical metamaterial unit. The inset figures are the current density distributions on the ELC and backplane at resonance frequency. The parameters of the unit cell used here are p = 5 mm, d = 4.5 mm, m = 2.5 mm, g = 0.4 mm, w = 0.3 mm, s = 1 mm, b = 0.205 mm, and t = 0.8 mm. (d) Simulated absorptivity curves for different air gap b. (e) The electromagnetic resonance frequency shift curve as the function of air gap b. (f) The absorptivity curve of the optomechanical metamaterial unit as the function of air gap b at a fixed incident frequency 10.32 GHz. (g) The simulated mechanical deformation feature of the ELC resonator deposed on the FPCB flexible substrate, when the incident electromagnetic wave power is set as 5 W. The inset figures are the displacement diagrams of the structure at the mechanical fundamental resonance frequency.

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Specifically, to better understand the operating mechanism for the proposed optomechanical metamaterial, following we explain such energy coupling progress based on a semi-simulation method. As shown in Fig. 1(c), when the air gap b is initially 0.205 mm, under the action of a certain intensity of incident electromagnetic waves, the absorption rate of the optomechanical metamaterial is close to 1 at the center frequency of 10.29 GHz, which is almost perfect absorption, and the electromagnetic energy has been transferred into the optomechanical metamaterial unit cell. So it has to be coupled to other physics field somehow. Then, as shown in the inset of Fig. 1(c), the induced current densities at resonance frequency on the two loops of ELC resonator are symmetrical, meaning that the currents pass through the central branch and opposite in the two outer branches. So the (instantaneous) current is clockwise in one loop and counterclockwise in the other loop and produce a displacement current in the gap.

Since the displacement current must flow in the same direction through the capacitor compared with the induced currents on the loop, the opposite magnetic moments produced by the two current loops will cancel out and prevent the existence of an axial magnetic dipole moment. That means theoretically the ELC resonator cannot be driven by a uniform axial time-varying magnetic field [63]. However, an electric dipole moment still exists on the plane along the gap direction, and the resonator can be excited by a uniform time-harmonic electric field applied in that direction. This is the basic resonance mechanism for the ELC resonator and this result will be used to explain the following nonlinear optomechanical coupling mechanism in next section.

Moreover, before we discuss the mechanical deformation of the optomechanical metamaterial, the electromagnetic absorptivity changing features under different air gaps are numerically investigated. As can be seen in Fig. 1(d), as the air gap increases, the absorptivity curves show the right shift and their amplitude values gradually increase. It means that with the increase of the air gap, the electromagnetic resonance frequency of the optomechanical metamaterial increases, as shown in Fig. 1(e) for the collected resonance frequency shifting features. And if a fixed-frequency electromagnetic wave acted on the optomechanical metamaterial unit, the absorptivity will first increase and then decrease with the increase of air gap. At a certain air gap value, the absorptivity of the optomechanical metamaterial is the best, as shown in Fig. 1(f). This resonance feature will be also used to the nonlinear optomechanical coupling mechanism in next section.

On another hand, we also numerically reveal the multi-physics coupling phenomenon for the proposed optomechanical metamaterial in COMSOL. In simulation environment, the flexible substrate is fixed with the cantilevers and the displacement diagram under the mechanical characteristic frequency of the optomechanical metamaterial is simulated as shown in the first inset of Fig. 1(g). In the optomechanical metamaterial, there is a background high-frequency magnetic field under the action of electromagnetic waves and an induced magnetic field due to the induced current between the metal ELC resonator and the metal backplane. The deformation of the flexible substrate in the optomechanical metamaterial is only considered by the Ampere’s force of electromagnetic induction under the effect of the induced magnetic field. Using the four-field coupling of electromagnetic waves, magnetic fields, solid mechanics, and thermoviscous acoustics in COMSOL, the mechanical frequency is used as the modulation wave to modulate the electromagnetic resonance frequency of the optomechanical metamaterial, so that the modulated wave maintains the characteristics frequency of the high-frequency carrier. The envelope waveform formed by the amplitude change is the waveform of the modulating signal. The flexible substrate between the metal ELC resonator ring and the metal backplane is deformed as a unified whole, as shown in the second inset of Fig. 1(g). The main panel of Fig. 1(g) shows the mechanical spectrum curve of the optomechanical metamaterial under a specific electromagnetic wave power, indicating that the energy of the optomechanical metamaterial is coupled, and the energy is transferred from the electromagnetic mode to the mechanical mode, causing the flexible substrate to deform and produce displacement.

3. Theoretical analysis of the nonlinear coupling mechanism

3.1 Theoretical equation evolution

According to the above briefly operating mechanism analysis with the help of numerical simulations, firstly, here we theoretically calculate the Ampere’s force between the ELC and the metal backplane based on the established coordinate system as shown as Fig. 1(b). Specifically, the ELC resonator can be divided into two symmetrical parts. An electromagnetic wave acts on the optomechanical metamaterial will induce an electrodynamic potential, which can be written in the form of [35]

Here ${\mu _0} = 4\pi \times {10^7}\textrm{ H/m}$ is the vacuum permeability, $\omega$ and $H_{0}$ are the frequency and magnetic field strength of the incident electromagnetic wave, and $d$ is the side length of the ELC resonator. The complete impedance equation for the ELC resonator is given by

Here $\Phi {}_{ext} = 1/2 \cdot {d^2}{\mu _0}{H_0}$ is the magnetic flux generated by the applied magnetic field strength, and ω₀ = (LC)^1/2 is the resonance frequency of the optomechanical metamaterial unit. It should be known that the frequency of the incident electromagnetic wave is much higher than the mechanical eigen-frequency of the optomechanical metamaterial. So we have assumed that the currents induced on the rings are uniform when talk about the opto-mechanical interactions.

Then we can establish the coordinate system as shown in Fig. 1(b), and divide the two loops of the ELC resonator in Fig. 1(a) into left and right parts. We first calculate the magnetic induction intensity of the right part. Specifically, the magnetic induction intensity at point P generated by the line BC with current I can be calculated initially. At this time, the field point P is located on the metal back plate directly below the BC side of the ELC [44].

In this case, l₁ is half of BF length and l₂ is half of BC length. The same method can be used to calculate the magnetic induction intensity generated at point P by the currents on the other three sides of the right half part,

It is obviously that the magnetic induction intensity generated at point P in the left half part is the same as the calculation method in the right half part.

Next, we consider the magnitude of the Ampere’s force between the ELC and the metal back plate. The force acting on the entire ELC ring is given by the line integral. For example, we can find the Ampere’s force between the up side of the metal back plate which directly below AD and the left half part of the metal ELC ring. Because the AD side produces only the magnetic flux density in the x and z directions.

Then the magnetic induction intensity in the x direction generated by the current on the AD side corresponds to the magnetic induction Ampere’s force in the negative z axis [45],

Then the magnetic induction intensity in the z direction generated by the current on the AD side corresponds to the Ampere’s force in the negative x-axis direction,

Similarly, the magnetic induction intensity in the x direction generated by the EF side current corresponds to Ampere’s force in the negative z-axis direction,

According to this method, the force of the left half part of the metal ELC on the other three sides can be obtained, and the Ampere’s force of the left half part of the metal ELC and the left half part of the metal back plate can be obtained by adding up. Thus the total Ampere’s force can be obtained. Based on the established Ampere’s force calculation method, next we can analysis the mutual coupling appeared in the proposed reflection-type ELC-based optomechanical metamaterial.

3.2 Nonlinear coupling characteristic analysis

Based on the above established force conditions, we firstly analyze the dynamic coupling progress when the incident electromagnetic wave power level is increased from zero to a sufficiently large value and then decreased. Firstly, assume that there is no electromagnetic wave in initial state, and the Ampere’s force and elastic force are zero. As the intensity of the electromagnetic wave increases, the optomechanical metamaterial maintain a stable state near the initial distance b, for example, b = 0.2 mm. This state can be called as ground state, as shown in Fig. 2(a), marked with red solid dots in the left bottom corner. In the process of gradually increasing the magnetic field strength of incident electromagnetic wave, the magnitude of the Ampere’s force increases, as shown by curve 1 to curve 8 in Fig. 2(a), and there is a tangent point on the left side (marked with a red cross point in curve 7). Although the Ampere’s force equal to the elastic force, this state is not a truly stable state. We can call it as a critical state and reach the threshold point of electromagnetic wave intensity. In this state, when the intensity of the electromagnetic wave increases slightly, the optomechanical metamaterial will jump to the actual stable state. According to Fig. 2(a), the optomechanical metamaterial expands suddenly and reaches another steady state, which is marked by a red solid circle on the right side of the curve 7 (this is the right stable state). This dynamic process is the basic operating principle of the optomechanical metamaterial drove by the electromagnetic induced forces. On another hand, when we reduce the intensity of incident electromagnetic waves, the optomechanical metamaterial remains on the right side, as shown by the black points on the curves 8, 7, 6, 5, 4, and 3 in Fig. 2(a). With the magnetic field decreased further, the optomechanical metamaterial will reach a critical state (marked with the black cross) again and then jump to the left stable state (red circles).

 

Fig. 2. The theoretical response analysis of the ELC resonator based optomechanical nonlinear metamaterial under actions of incident electromagnetic waves. (a) The force balance relationship between Ampere’s force under different incident electromagnetic wave intensities (curves 1–8) and the elastic force (straight line) as a function of distance b changings. (b) The variation of distance b with increasing and decreasing H at different incident frequencies, green: 10.45 GHz, purple: 10.46 GHz, pink: 10.47 GHz, and black: 10.48 GHz. (c) The schematically representation for explaining the unreachable stable state. (d) and (e) The force balance relationship under different incident electromagnetic wave frequencies. (f) The variation of distance b with increasing and decreasing frequencies at different magnetic field strengths, red: 0.5 A/m, blue: 0.27 A/m, black: 0.1A/m.

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Based on the above theoretical analysis, we can study the steady state change characteristics of the optomechanical metamaterial when the intensity/frequency of the incident electromagnetic wave increases or decreases. Specifically, as shown in Fig. 2(b), we call the path where the electromagnetic wave intensity increases as the ascending path, and the path where the electromagnetic wave intensity decreases as the descending path. In the initial stage of increasing the intensity of the incident electromagnetic wave, the steady state of the optomechanical metamaterial will be continuously kept in a nonlinear manner as parameter b changed from the ground state (for example 0.2 mm as shown in the start point in Fig. 2(b)) until the Ampere’s force curve is tangent to the elastic force curve (the red cross point in Fig. 2(a)). We call the stable state at the tangent point of the Ampere’s force curve and the elastic force curve as a metastable state. In the metastable state, the direction in which the distance b increases corresponding to the stable state of the metamaterial discontinuously changes, and this change is called steady-state jump. After the metastable state, the stable state of optomechanical metamaterial will be continuously kept in a nonlinear manner as parameter b changed from the jump point (the red point shown on the red line of Fig. 2(a)).

On another hand, when the intensity of the electromagnetic wave gradually decreases, the induced Ampere’s force also gradually decreases. Therefore, the stable state of the optomechanical metamaterial changes in a nonlinear manner similar to the power increasing path, see the descending path feature shown in Fig. 2(b). In the above process of increasing and decreasing incident electromagnetic waves, the stable state of the optomechanical metamaterial has undergone two processes of continuous change and steady-state jump. The state loop composed of the steady state and the metastable state obtained in the above way can be called as the steady state variation loop of the optomechanical metamaterial under different electromagnetic wave intensities.

In addition, there is a third special stable state. Although the electromagnetic induced Ampere’s force is equal to the elastic force at this time, it can only be achieved through external stimulation or temporarily increase magnetic field. We call it the unreachable state. As shown in the Fig. 2(c), when a force between F1 and F2 is applied, the structure will reach state 1, which is called unreachable state point. When a force exceed F2 is applied, the structure will return to the ground state and state 2 will become unreachable. If the intensity of electromagnetic wave is temporarily increased, the structure will change to state 3 and finally stabilize to state 2.

As shown in Fig. 2(d), on anther hand, when we fix the strength of the incident magnetic field, and increase or decrease the frequency of electromagnetic waves, the peak coordinates of the Ampere’s force curve will move to the right or left, and the amplitude of the Ampere’s force will decrease or increase, as shown in Figs. 2(d) and (e). Specifically, as the frequency of the incident electromagnetic wave gradually increases, the peak value of the electromagnetic induction Ampere’s force gradually decreases, and the distance b gradually increases at this time. Since all these stable states are on the right side of the peak of the Ampere's force curve, we collectively call these stable points as the right stable state. When the electromagnetic induction force is equal to the elastic force, the frequency of the incident electromagnetic wave is continuously increased, and the optomechanical metamaterial will jump and eventually return to the ground state. When the electromagnetic wave frequency decreases, the optomechanical metamaterial will continuously change from the ground state, and then jump to the right stable state. The reason for why it does not jump from the ground state point here is that the Ampere’s force is greater than the elastic force under the strength of the incident magnetic field here. When the magnetic field strength is too small, the optomechanical metamaterial will maintain the ground state (also can be found in Fig. 2(e)). Specifically, Fig. 2(f) shows the variation feature of the distance b with increasing and decreasing frequencies at different magnetic field strengths.

4. Experimental demonstrations

4.1 Basic demonstrations of the optomechanical metamaterial units

Firstly, in order to experimentally investigate the absorption characteristics of the proposed optomechanical metamaterials, we use a closed rectangular waveguide measurement system to measure reflection parameters. The fabricated sample is placed in the center of an X-band (8–12 GHz) rectangular waveguide. A vector network analyzer (Agilent N5230A) and a coaxial to rectangular waveguide transition operated at X-band are used in this measurement system. A metal plate is used to tightly cover the waveguide port. The unit structure size of this sample is 5 mm × 5 mm, and the cantilever arm size is 2 mm × 1 mm. The gasket material used in experiments is Teflon with a radius of 1 mm, which is used to support the optomechanical metamaterial to simulate an air gap. Here the fabricated samples have three different unit structures: 4-unit, 2-unit and 1-unit. Figure 3(a) shows such 3 different units structure samples. The sample size of 4-unit is g = 0.6 mm, d = 4.6 mm, b = 0.4 mm, the sample size of 2-unit is g = 0.4 mm, d = 4.7 mm, b = 0.4 mm, and the sample size of 1-unit is g = 0.4 mm, d = 4.7 mm, b = 1.5 mm. All the samples are placed in the center of the metal plate. Figure 3(b) shows the assembly of the sample in the rectangular waveguide, and Fig. 3(c) shows the measurement environment. It should be noticed that here we analyze three structure units with different support beams and anchor methods as presented in Fig. 3(a). This is mainly to investigate the elastic force amplitude conditions for different support configurations and to see the electromagnetic resonance strength maintaining abilities for different unit numbers. So we can find a better optomechanical metamaterial configuration to investigate clearly the nonlinear dynamic process in next subsection.

 

Fig. 3. Measurement system of reflection characteristics for the fabricated optomechanical metamaterial samples. (a) 3 different units structure samples. (b) the assembly of the sample in the rectangular waveguide. (c) Test platform for resonance characteristics of optomechanical metamaterials. Experimentally demonstrated reflectivity of optomechanical metamaterials in an X-band rectangular waveguide compared with simulated results under (d) 1-unit structure, (d) 2-unit structure, and (f) 4-unit structure.

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The measured results compared with corresponding simulated results in the same rectangular waveguide conditions are shown in Figs. 3(d) to (f), and generally the simulations and measurements are matched to each other very well. Specifically, as shown in Fig. 3(d), the return loss of the 1-unit optomechanical metamaterial sample reach to maximum at 10.37 GHz, means the electromagnetic wave energy at such frequency is absorbed well. The results shown in Fig. 3(e) show that the 2-unit optomechanical metamaterial sample has the similar maximum return loss at 10.32 GHz. And simultaneously the result shown in Fig. 3(f) indicates that the maximum return loss of the 4-unit optomechanical metamaterial sample is appeared at 10.55 GHz. The resonance frequency and resonance depth features of the three samples are basically the same. There are some differences between measured and simulated results in each condition, including the resonance frequency offset and the quality factor reduction. The resonance frequency offset for the measurement result is mainly because of the fabrication tolerance, and the quality factor reduction in measurement is mainly due to the dielectric loss of the FPCB and the effects of practical measurement environment tolerances. Those obtained measured results shown in Fig. 3 prove that the fabricated optomechanical metamaterial units can work well and very high-Q (the numerical Q-factor is above 1000 and the measured Q-factor is about 460) resonance features are reached. Next, we will experimentally investigate the nonlinear responses for those samples.

4.2 Nonlinear response demonstrations

In order to verify the nonlinear characteristics of proposed reflection-type ELC-based optomechanical metamaterials driven by electromagnetic induction force, we conducted the experiments as shown in Fig. 4. Specifically, Fig. 4(a) shows the connection diagram showing the setup of the test platform, while Fig. 4(b) shows the test platform and measurement setup in the actual environment. In this experimental demonstration, we use a 1-unit structure sample to clearly see the nonlinear dynamic process based on the obtained initial measurement results shown in above subsection. As shown in Fig. 4(b), the coaxial single-port rectangular waveguide working in the X-band is connected to the Agilent N5230A vector network analyzer (VNA) through port 1 of the coaxial circulator (8.0–12.0 GHz 60W, SMA-K). Then we set the vector network analyzer within the allowable range (up to 5 dBm) and add a low-noise power amplifier (Mini-circuits ZVE-3W-183, gain: 35 dB) to change the applied power level up to 40 dBm, and capture the obtained transmission power, with connected attenuator (attenuation: 40 dB) to ensure instrument safety. The entire system was calibrated using the thru-reflect-line (TRL) calibration procedure to eliminate the system errors before the measurements. In addition, in order to obtain a more obvious resonance frequency shift when applying different incident power levels, we measured and collected all the transmission curves of different incident electromagnetic field strengths with the input power of the VNA varying from −20 dBm to 5 dBm in steps of 0.01 dBm. In order to obtain as accurate and clear resonant frequency deviation as possible, we set 16001 samples and narrowed the scan bandwidth (9.5 GHz-10.5 GHz) in the VNA.

 

Fig. 4. Experimental demonstration of nonlinear response for the reflection-type ELC-based optomechanical metamaterial. (a) Schematic diagram of the connection of the device for testing the electromagnetic resonance frequency shift of the optomechanical metamaterial. (b) Schematic view of the measurement setup for testing the electromagnetic resonance frequency shift of optomechanical metamaterials in a real environment. (c) Two-dimensional plot for the metamaterial transmission spectra under different incident power levels from 5 dBm to 30 dBm. (d) Results of the transmissions under three power levels of optomechanical metamaterials. (e) The resonance frequency shift with increasing powers ranged from 5 dBm to 30 dBm.

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Figure 4(c) shows the two-dimensional results of all the measured and collected transmission curves under different incident electromagnetic wave intensities, which can clearly show the change trend of the resonance frequency and the changes of the transmission performance. Figure 4(d) shows the transmission curves of three different VNA input power levels selected from Fig. 4(c). As the power of the incident wave increases, the resonance frequency moves toward the high frequency. Figure 4(e) compares the theoretical and experimental results of the change trend of the resonant frequency for the optomechanical metamaterial under different incident electromagnetic wave intensities. It can be concluded that when the input power of the vector network increases from −20dBm to 5dBm (the estimated power levels after the power amplifier, circulator, coaxial cable, and the waveguide-coaxial converter are from 5 dBm and 30 dBm), the transmission dip (the electromagnetic resonance frequency) shifts significantly and shows an increasing trend. The resonance frequency of the optomechanical metamaterial changes with an increase or decrease of the incident magnetic field due to changes of the distance between the metal ELC and the metal back plate (air gap) in the optomechanical metamaterial, which directly verified the nonlinear characteristics of the optomechanical metamaterial driven by electromagnetic force. It can be seen from the results of the theoretical fitting that the characteristics of the measured electromagnetic resonance offset are consistent with the analysis of the theoretical results. Moreover, compared with previously reported optomechanical metamaterial [35] operated at the similar microwave range, our proposed reflection-type ELC-based optomechanical metamaterial shows much larger resonant frequency shift slop (almost two-order larger) under the same power increasing range, which means the larger opto-mechanical rate.

Moreover, for the proposed reflective-type ELC-based optomechanical metamaterial in this paper, the distance (air gap) between the ELC resonator and metal plat is enlarged under increased incident electromagnetic wave actions, so theoretically the larger resonance frequency can be obtained if we further increase the power levels. However, due to the mechanical deformation limit of the FPCB support beams, the air gap cannot be enlarged unlimited. The upper bound is decided by the mechanical deformation limit. On another hand, when the concentrated electromagnetic energy within the optomechanical unit cell is large enough, the ELC resonator would be burned due to the limited power capacity of the FPCB.

5. Conclusion

In summary, we propose a new type of optomechanical metamaterial based on a flexible substrate planar absorbing structure with the ELC resonators. The theoretical and experimental results in this paper show the nonlinear characteristics of optomechanical metamaterials driven by electromagnetic force. It also shows the mechanical deformation of the flexible substrate in the optomechanical metamaterial by the Ampere’s force under multi-field coupling. Our results will help researchers to further study the operating mechanism of optomechanical nonlinear metamaterials, and can be widely used for the reconfigurable metasurface for dynamic wave manipulations.