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Abstract
We design an ultrathin water-based metasurface capable of coherent perfect absorption (CPA) at radio frequencies. It is demonstrated that such a metasurface can almost completely absorb two symmetrically incident waves within four frequency bands, each having its own modulation depth of metasurface absorptivity. Specifically, the absorptivity at 557.2 MHz can be changed between 0.59% and 99.99% via the adjustment of the phase difference between the waves. The high angular tolerance of our metasurface is shown to enable strong CPA at oblique incidence, with the CPA frequency almost independent of the incident angle for TE waves and varying from 557.2 up to 584.2 MHz for TM waves. One can also reduce this frequency from 712.0 to 493.3 MHz while retaining strong coherent absorption by varying the water layer thickness. It is also show that the coherent absorption performance can be flexibly controlled by adjusting the temperature of water. The proposed metasurface is low-cost, biocompatible, and useful for electromagnetic modulation and switching.
© 2017 Optical Society of America
1. Introduction
Metasurfaces, also known as two-dimensional metamaterials, are artificially structured layers of ordinary materials (metals, semiconductors, or dielectrics) whose thicknesses are much smaller than the operation wavelength [1–4]. They offer researchers an almost total control over their electric and magnetic responses, which can be modified by altering the subwavelength-size patterning of the metasurface. When an electromagnetic wave propagates through a metasurface, it may change its amplitude, phase, polarization, coherence, or all the four, leading to many exotic electromagnetic phenomena such as giant optical chirality [5], electromagnetically induced transparency [6], wavefront modification [7], and perfect absorption [8–10]. The latter makes metasurfaces useful in such applications as detection of electromagnetic radiation [11], sensing [12], and electromagnetic shielding [13].
One of the drawbacks of a typical metasurface absorber is that its absorptivity is determined by the internal configuration of subwavelength components and thus cannot be tuned dynamically. Recently, an original concept of time-reversed lasing — coherent perfect absorption (CPA) — was proposed for achieving total and controllable absorption in a two-port system formed by a pair of counterpropagating electromagnetic waves. [14,15] In contrast to ordinary metamaterial absorbers, coherent absorbers can change their absorptivity dynamically thanks to the interplay of absorption and interference [16–20]. This dynamic tunability makes coherent absorbers particularly attractive as transducers, modulators, and electromagnetic switches.
It is not hard to realize CPA in fully metallic metasurfaces of deeply subwavelength thicknesses. A number of recent esearch findings also show that CPA can be sustained in ultrathin layers of silicon [17], graphene [21], and molybdenum disulfide [22]. Most recently, we demonstrated the feasibility of CPA with metasurfaces made of high-permittivity ceramics [23]. Another prospective dielectric material for CPA is water. It is one of the most abundant resources on Earth, takes the shape of its container, and has numerous merits such as biocompatibility and inexpensiveness. Moreover, thanks to its high permittivity at radio frequencies [24], water can serve as a uniform material base for all-dielectric metamaterials [25–27].
In this Letter, we demonstrate through numerical simulations that CPA can be achieved in a metasurface made of water even if its thickness is several tens of times smaller than the operation wavelength. We first show that the energy of two radio waves, falling symmetrically onto different sides of the metasurface, can be almost totally absorbed at four resonant frequencies of the metasurface. We then demonstrate the possibility of deep coherent modulation and angular tolerance of our metasurface by analysing its absorptivity for each resonance and for oblique incidence. The effect of the metasurface thickness is further discussed, showing that the frequency of CPA resonance can be tuned within a wide frequency range by adjusting the metasurface thickness.
2. Permittivity of water
The permittivity of water at radio frequencies is described by the Debye formula [24]
where εs(T), ε∞(T), and τ(T) are the temperature-dependant high-frequency permittivity, static permittivity, and rotational relaxation time, τ(T) = c2exp[T2/(T + T1)], where a1 = 87.9, b1 = 0.404 K−1, c1 = 9.59 × 10−4 K−2, d1 = 1.33 × 10−6 K−3, a2 = 80.7, b2 = 4.42 × 10−3 K−1, c2 = 1.37 × 10−13 s, T1 = 133 °C, T2 = 651 °C, and T is the water temperature in °C.Figure 1 shows the variation of water’s permittivity when its temperature changes. It is seen that, for a specific temperature, the real part of water’s permittivity shows a very weak decrease at the frequency band of interest. The imaginary part shows a nearly linear increase but is significantly smaller than the real imaginary, which implies a relatively low dielectric loss. When the temperature increases from 10°C to 60°C, the real component of water’s permittivity decreases from 84 to 67, and meanwhile, the imaginary component significantly decreases by nearly 3/4. Owing to the thermal tunability of permittivity, water becomes a promising platform for constructing tunable dielectric metamaterials. Worth noting is that water can also be used as a thermally controllable substrate that enables tunability to transitional metal based metamaterials [28].
Fig. 1 Permittivities of water for different temperatures. The solid and dashed curves represent the real and imaginary components, respectively. The imaginary components are magnified by 10 time for better illustration.
3. Theoretical analysis
Figure 2(a) shows two plane electromagnetic waves of frequency f incident symmetrically from both sides on the water-based metasurface considered in this paper. The electric fields O± of the forward (+) and backward (−) outgoing waves are related to the incident fields I± = |I±|eiϕ± through the scattering matrix S via
Fig. 2 Coherent interaction of two electromagnetic waves with a water-based metasurface. Intensities of the output waves O± can be controlled by adjusting the phase difference between the incident waves I±.
Owing to the spatial symmetry and reciprocity of our metasurface, its scattering matrix is reduced to a pair of complex-valued reflection and transmission coefficients S11 = S22 = r and S12 = S21 = t.
According to the definition, the coherent absorptivity of a metasurface is given by
It is clear that the perfect absorption (Ac = 1) can be achieved if the amplitudes of forward and backward waves are equal, |I+| = |I−|, and the reflection and transmission coefficients are either equal to each other or differ only by sign, r = ±t [23]. By assuming that the first condition is met, we obtain
where ϕ = ϕ + − ϕ − is the phase difference between the incident waves.4. Perfect coherent absorption and phase modulation
Figure 2(b) shows the geometry of our metasurface, which represents a fishnet-shaped container bounding a water layer of thickness tw = 20 mm. The container is a square lattice of square holes a × a = 105 × 105 mm2, with lattice constant p = 160 mm and walls of thickness tc = 1 mm. Such a fishnet-shaped container can be easily fabricated using 3D printing technology, for example. The walls of the container are assumed to be made of polyvinyl chloride (PVC), which is widely used in 3D printing and has relative permittivity of 2.5. The electromagnetic performance of the water-based metasurface is studied via the full-wave numerical simulations, which are performed using commercial software package CST Microwave Studio. We use periodic boundary conditions in the x and y directions and assume absorbing boundaries in the z direction to simulate scattering of electromagnetic waves at an infinitely large periodic metasurface. Owing to the symmetry of the proposed metasurface, it allows us to numerically calculate the reflection and transmission coefficients using a single incident beam and then use Eq. (4) to further determine the coherent absorption performance.
We begin by finding frequencies of the highest coherent absorption at room temperature (T = 25 °C), which we refer as the CPA frequencies. This requires analyzing how our metasurface responds to a normally incident plane wave polarized along either x or y axis. Figure 3(a) shows modules (amplitudes) of the reflection and transmission coefficients, |r| and |t|, plotted as functions of the incident wave frequency. The fact that the two spectra intersect at 557.2, 820.0, and 877.6 MHz indicates the possibility of achieving CPA at these frequencies. It should also be noted that the two amplitudes have very close values for frequencies between 920 and 980 MHz. Figure 3(b) shows arguments (phases) of the reflection and transmission coefficients, arg r and arg t. According to Table 1, the difference of the two phases, ψ = arg r − arg t, is close to π at frequencies 557.2 and 820.0 MHz and to 0 at frequencies 877.6 and 931.6 MHz. This indicates that strong coherent absorption can occur at these four frequencies. Since the CPA condition r = ±t is the best met at 557.2 MHz, our metasurface is expected to most efficiently absorb radiation at this frequency.
Fig. 3 (a) Modulus and (b) argument of reflection and transmission coefficients of a plane wave incident normally on the water-based metasurface at room temperature.
Table 1. Modulation depths of coherent absorptivity for four CPA frequencies of water-based metasurface shown in Fig. 2.
Figure 4 confirms our conclusions. Its panel (a) shows the density plot of coherent absorptivity in coordinates (f, ϕ). The discovered CPA frequencies are all clearly seen from this plot. Peak absorptivities at the lower two frequencies of 557.2 and 820.0 MHz are achieved for phase delays ϕ = 2πn, n = 0, ±1, ±2, …, whereas peak absorptivities at frequencies 877.6 and 931.6 MHz require delays ϕ = (2n + 1)π. These four absorptivities as functions of ϕ are shown in panel (b) of the figure. In each case, the level of absorptivity modulation around its mean value can be characterized by the modulation index (or modulation depth)
where Ac,max and Ac,min are the maximal and minimal absorptivities. The critical absorptivities and modulation indices of the four curves in Fig. 4(b) are given in Table 1. The highest modulation index of 98.83% at f = 557.2 MHz shows that our metasurface almost totally transmits incident waves with ϕ = 2πn and almost totally absorbs them for ϕ = ±π/2+2πn. It should be emphasized that the modulation depth of 98.83% is quite large, given that the thickness of our metasurface is less than 1/24 of the operation wavelength.Fig. 4 Coherent absorptivity of water-based metasurface as a function of (a) frequency f of incident waves and phase delay ϕ between them and (b) phase delay at four CPA frequencies. The absorptivity is nearly uniform at f = 557.2 and 820 MHz in the vicinity of ϕ = 0 and at f = 877.6 and 931.6 MHz near ϕ = 180°.
If the two CPA conditions formulated earlier are satisfied, then setting r = ±t in Eq. (4) yields
This expression shows that total absorption occurs for ϕ= π(4n + 1 ± 1)/2, n = 0, ±1, ±2, …, regardless of the value of |r|. It also shows that Ac 1≥ − 4 |r|2 and one can modulate the metasurface absorptivity with any depth between 0 and 100% provided that |r| = |t| = 1/2, in which case Ac = (1 ∓ cosϕ)/2. According to Fig. 3(a), the amplitudes of the reflection and transmission coefficients are the closest to this value at 557.2 MHz. For other three resonances |t| and |r| are smaller than 1/2 and the 100% modulation depth cannot be achieved. These conclusions are also evidenced by Table 1.
We next study the absorptivity of our metasurface for oblique incidence of radiation that provides the highest modulation depth and which is therefore most attractive for applications. The CPA frequency of such radiation depends on the incident angle and corresponds to Mode I when the incidence is normal. Figure 5(a) shows how peak coherent absorptivity at this frequency changes with the incident angle of TE- and TM-polarized waves. One can see that the absorbtion of the TM wave is almost total (Ac > 97%) for incident angles of up to 60° and decreases dramatically for larger angles. In contrast to this, the TE wave is strongly absorbed only within a narrow range of about ±20° near the normal direction. The gradual absorptivity decrease outside of this range due to the diminishing excitation efficiency of the metasurface by the incident magnetic field is typical for perfect metamaterial absorbers [29]. Figure 5(b) shows that the CPA frequency is almost independent of the incident angle of the TE wave and blueshifts from 557.2 to 584.2 MHz when this angle of the TM wave is increased from zero to 80°. These features suggest usefulness of the proposed water-based metasurface for broadband modulation and applications requiring angular selectivity.
Fig. 5 Angular spectra of (a) peak coherent absorptivities at (b) the lowest CPA frequencies of TE- and TM-polarized waves.
In contrast to ordinary metamaterials, whose metallic parts do not let electromagnetic fields penetrate deep inside them, in our case electromagnetic fields can fully penetrate into the water. As a consequence, the amount of electromagnetic energy contributing to the CPA resonance grows with the thickness of the water layer and redshifts the resonance. This allows one to shift the CPA frequency over a wide domain by simply changing the metasurface thickness. The absorptivity spectra in Fig. 6(a) illustrate this opportunity by the examples of five metasurfaces illuminated with a pair of phase-matched (ϕ= 0) coherent waves. In Fig. 6, one can see that a nearly perfect absorptivity (Ac > 99.9%) can be achieved inside all the metasurfaces. As the thickness of the metasurface is increased from 10 to 30 mm, the CPA peak redshifts from 712 to 493 MHz. It is also seen that the absorption peak becomes narrower with the increase of the thickness, where the relative bandwidth decreases monotonously from 7.71% to 4.93%. This is because water is less lossy at lower frequencies, resulting in a resonance of higher quality factor and sharper absorption spectrum.
Fig. 6 (a) Coherent absorptivity spectrum, (b) peak absorptivity, and the CPA frequency for different thicknesses of water-based metasurface. Other geometrical parameters are kept unchanged.
Compared to the previously analyzed metasurfaces made of high-permittivity ceramics [23], the use of water provides extremely high tunability to the metasurface’s electromagnetic properties. As shown in Fig. 1, the permittivity of water strongly depends on its temperature. This enables thermal tuning of the absorption performance of the proposed water-based coherent absorber. In Fig. 7(a), we show the absorptivity spectra at different temperatures, where a high coherent absorption peak is achieved for each case. When the temperature of water is increased from 10 to 90 °C, the water’s permittivity decreases significantly, which is equivalent to making the metasurface thinner. Therefore, we see that the CPA peak shows a clear blueshift from 539 to 641 MHz. It is also seen that the peak coherent absorptivity is maximum at room temperature (Ac = 99.99% for T = 25 °C) and decreases significantly when the temperature shifts far away from this value (Ac = 68.79% for T = 90 °C). This is due to the fact that the proposed coherent metamaterial absorber is designed for working at the room temperature and the change of water’s permittivity via controlling temperature breaks the optimized condition for perfect coherent absorption.
Fig. 7 Thermal tunability of the coherent metamaterial absorber. (a) coherent absorptivity spectrum, (b) peak absorptivity and the CPA frequency.
5. Conclusion
We have designed and studied the performance of water-based metasurface capable of multiband CPA of radio-frequency radiation. Rigorous numerical simulations showed that our metasurface can almost totally absorb radiation at four frequency bands, and that its absorptivity can be modulated with depths of up to 98.83% by adjusting the phase difference of the incident waves. The feasibility of CPA was demonstrated for oblique incidence of TE waves of frequency 557.2 MHz and for oblique incidence of TM waves of frequencies between 557.2 and 584.2 MHz. It was also shown that the CPA frequency can be tuned over a broad frequency range by changing the metasurface thickness. We further show the feasibility of thermal tunability of the proposed coherent metasuface absorber. The proposed water-based metasurface can therefore serve as a low-cost biocompatible modulator or switcher of radio waves.
Funding
Natural Science Foundation of Shanghai (17ZR1414300), Shanghai Pujiang Program (17PJ1404100), National Natural Science Foundation of China (61675170, 61571298, and 61571289), Ministry of Education and Science of the Russian Federation (14.B25.31.0002), and Australian Research Council (DP140100883).
Acknowledgments
I.D.R. would like to thank the Ministry of Education and Science of the Russian Federation for its Grant of the President of the Russian Federation for young scientists.
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