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Abstract
In this work, we design a structure of metamaterials that consists of double sliver-ring resonators, in which highly-dispersive unidirectional reflectionlessness and absorption are achieved based on high-order plasmon resonance. Reflections of +z and -z directions at 461.34 THz (456.68 THz) are $\sim$0 (0.82) and $\sim$0.85 (0) when the distance $d=222.9$ nm (259.8 nm), respectively. High absorption of $\sim$0.97 and the quality factor of $\sim$435 can be obtained in the loss metal structure at room temperature. What’s more, unidirectional reflectionlessness is investigated at low temperature.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metamaterials are artificial composites in periodic or aperiodic structures, whose cell structure are much smaller than the incoming wavelength. They have extraordinary physical properties that cannot be achieved in natural materials, such as left-handed materials [1,2], metamagnetic materials [3], invisibility [4], electromagnetically induced absorption [5–8], electromagnetically induced transparency [9–11], perfect absorption [12], unidirectional transmission [13–19], unidirectional reflectionlessness [20–34], and so on.
In recent years, unidirectional reflectionlessness has attracted more and more attentions. For example, some researchers [21–26] demonstrate unidirectional reflectionless phenomenon at exceptional points (EPs) in plasmonic waveguide side-coupling to cavity system. However, Zhao et al. [27] demonstrate double-band unidirectional reflectionless phenomenon by using two nanohole resonators in plasmonic waveguide end-coupling to cavity system. Their quality factor (Q-factor) is about 130, which is 2 times higher than that in the previous schemes [21–26]. Besides, unidirectional reflectionless phenomenon has also been demonstrated in metamaterials. For example, Kang et al. [28] explore unidirectional zero reflection producing topologically stable phase dislocation for the occurrence of EP in a hybridized ultrathin metamaterial configuration based on near-field coupling. Later, Kang et al. [29] realize not only unidirectional zero reflection, but also a single EP point that can be converted into a ring of EPs based on the previous structure [28]. Bai et al. achieve the single-band unidirectional reflectionlessness in symmetric stacked metamaterial based on far-field coupling [30] and asymmetric stacked metamaterial based on near-field coupling [31], respectively. Gu et al. [32] achieve a controllable unidirectional reflectionless phenomenon in a metasurface system composed of two nanoring resonators based on far-field coupling. What’s more, Han et al. [33] confirm the dual-band unidirectional reflectionlessness by using two gold resonators of circular hole with Q-factor of $\sim$20. A large number of studies have shown that the loss in metal material is unavoidable. Especially, the self-loss in metamaterial comprising of metal is large, which can result in a low Q-factor. In order to reduce the loss of metamaterial, Yin et al. [34] propose a non-Hermitian metamaterial structure consisting of double layered silicon resonators to achieve single-band unidirectional reflectionless phenomenon. Their Q-factor is about 83, much higher than the previous schemes [30–32]. However, the size of their unit cell is relatively larger. All the above studies are performed at eigenfrequency of resonator. Schemes of highly-dispersive unidirectional reflectionlessness based on high-order resonance mode (have low loss in metal structure) are rarely reported.
In this work, we investigate the highly-dispersive unidirectional reflectionless phenomenon based on high-order plasmon resonance in metamaterial that consists of double silver-ring resonators embedded in photopolymer. In our scheme, high absorption of $\sim$0.97 can be obtained and the Q-factor for absorption spectrum at EP is $\sim$435, $10$ times higher than those in [30–32] in room temperature. Besides, we investigate the unidirectional reflectionlessness and absorption for different low temperatures.
2. Results and discussion
Figure 1 shows the schematic of the non-Hermitian metamaterial structure. The unit cell of structure consists of a upper silver ring (USR) and a lower silver ring (LSR) resonators embedded in photopolymer that placed on a glass substrate. The heights of USR and LSR resonators are both $h$ = 30 nm. The identical inner radius of double resonators is $r$ = 70 nm, while the outer radiuses of double resonators are $R_{1}$ = 185 nm and $R_{2}$ = 164 nm, respectively. The periods of the unit cell are $L$ = 500 nm in $x$ and $y$ directions. The thickness of glass substrate is $H$ = 150 nm. The dielectric constants of glass and photopolymer are 2.25 and 2.4025, respectively, and the dielectric constant of silver is obtained by Drude model with collision frequency $\omega _{c}~=~3.07~\times ~10^{13}$ Hz and plasmon frequency $\omega _{pl}~=~1.366~\times ~10^{16}$ rad/s [35]. Numerical simulation is carried out by employing a finite-integration package (CST Microwave Studio).
In order to demonstrate the highly-dispersive unidirectional reflectionless phenomenon, we analyze the properties of structure by scattering matrix S. Corresponding to the structure in Fig. 1, the scattering properties of metamaterial structure for the incident wave in a certain frequency $\omega$ can be expressed by transfer matrix ${\textrm T}_{all}$ [36]
Figures 2(a) and 2(b) delineate the numerical simulation (solid line) and analytical calculation (dotted line) of reflection spectra in +z and -z directions with distance $d$ = 222.9 nm and 259.8 nm, respectively. We should mention that the accuracy of the analytical model is limited, as can be seen in Figs. 2(a) and 2(b). Even so, the reflection spectra based on numerical simulation and analytical calculation have a good consistency to some extent. From Fig. 2(a), reflections of +z and -z directions at 461.34 THz are $\sim$0 and $\sim$0.85 when $d$ = 222.9 nm according to numerical simulation, respectively. While, in Fig. 2(b), reflections of +z and -z directions at 456.68 THz are $\sim$0.82 and $\sim$0 when $d$ = 259.8 nm according to numerical simulation, respectively. This is to say, unidirectional reflectionless phenomenon appears at 461.34 THz and 456.68 THZ, respectively. In addition, the absorption and transmission spectra for +z and -z directions are shown in Figs. 2(c) and 2(d) when distance $d$ = 222.9 nm and 259.8 nm, respectively. By using the formula $A$ = 1-$T$-$R$ ($T$ and $R$ are transmission and reflection), absorptions at 461.34 THz and 456.68 THz of $\sim$0.95 and $\sim$0.97 with Q-factors of $\sim$282 and $\sim$435 can be obtained, respectively.
Fig. 1. Schematic of the non-Hermitian metamaterial structure. Illustrations on the right are the cross sections of the upper and lower silver rings in $x-y$ plane, respectively. Distance $d$ is alterable and the incident wave is along +z or -z directions.
Fig. 2. (a) and (b) are analytical and simulated reflection spectra for +z and -z directions when $d$ = 222.9 nm and 259.8 nm, respectively. (c) and (d) are absorption and transmission spectra for +z and -z directions when $d$ = 222.9 nm and 259.8 nm, respectively. The relevant parameters are $\Gamma _{1}$ = 1.141 THz (1.14 THz), $\Gamma _{2}$ = 0.5351 THz (0.4013 THz), $\gamma _{1}$ = 10.219 THz (12.22 THz), $\gamma _{2}$ = 0.713 THz (0.894 THz) and $\phi$ = 0.8603$~\pi$ (1.125$~\pi$), respectively.
To further illustrate the unidirectional reflectionlessness at two EPs (461.34 THz and 456.68 THz), z-component electric-field distributions of USR and LSR are described in Fig. 3. According to Fig. 3, we can see that high-order plasmon resonance appears in double resonators. From Figs. 3(a)–3(d), directions of the induced currents between USR and LSR at 461.34 THz are same and opposite in +z and -z directions when $d$ = 222.9 nm, respectively. Therefore, phase difference between double resonators is close to $2\pi$ ($\pi$) in +z (-z) direction, which means a low (high) reflection appears at 461.34 THz based on Fabry-P$\acute {\textrm e}$rot (FP) resonance. This is consistent with the reflection spectra of red and blue solid lines in Fig. 2(a), respectively. From Figs. 3(e)–3(h), directions of the induced currents between USR and LSR at 456.68 THz are opposite and same in +z and -z directions when $d$ = 259.8 nm, respectively. Phase differences between double resonators are close to $\pi$ in +z direction and $2\pi$ in -z direction, which corresponding to a high and a low refections at 456.68 THz, respectively. It is consistent with the reflection spectra of red and blue solid lines in Fig. 2(b), respectively. That is, unidirectional reflectionless phenomenon can be achieved at EPs.
Fig. 3. z-component distributions of electric field for USR and LSR in +z and -z directions when $d$ is 222.9 nm (a)–(d) and 259.8 nm (e)–(h) at 461.34 THz and 456.68 THz, respectively.
We plot the reflection spectra for +z and -z directions by changing the distance $d$, as shown in Fig. 4. From Figs. 4(a) and 4(b), near-zero reflection peaks have red-shifts for +z and -z directions when increasing distance $d$. Moreover, Fig. 4(a) shows the low reflection area in the range of frequency $f$ from 454 THz to 459 THz and distance $d$ from 238 nm to 280 nm in +z direction. Figure 4(b) shows the low reflection occurs in frequency $f$ range of 460 THz $\sim$ 462 THz and the corresponding variation range of distance $d$ are 221 nm $\sim$ 231 nm in -z direction. Comparing Figs. 4(a) with 4(b), low (high) reflection area in +z direction corresponds to high (low) reflection area in -z direction. It turns out that the unidirectional reflectionless phenomenon can be implemented in a wide range of distance $d$ in our system.
Fig. 4. Reflections for +z (a) and -z (b) directions as the functions of distance $d$ and frequency.
Then scattering matrix S is used to analyze the unidirectional reflectionless phenomenon of further, which can be given by Eq. (4), as
And the corresponding eigenvalues of S-matrix can be written as where $r_{+z}$ and $r_{-z}$ are the reflection coefficients in +z and -z directions, respectively, and $t$ represents the transmission coefficient. When $\sqrt {r_{+z}r_{-z}}$ = $0$, two eigenvalues coalesce and EP appears, that is to say, when $r_{+z}$ or $r_{-z}$ is zero, unidirectional reflectionlessness occurs at EP.Figures 5(a)–5(d) delineate the real and imaginary parts of eigenvalues $E_{\pm }$ of S-matrix varied with frequency when different distance $d$ = 222.9 nm, 259.8 nm and phase shift $\phi$ = 0.8603 $\pi$, 1.125 $\pi$. The real and imaginary parts of two eigenvalues merge (Figs. 5(a) and 5(c))and cross (Figs. 5(b) and 5(d)) at 461.34 THz and 456.68 THz, respectively. That is, unidirectional reflectionlessness appears at EPs (461.34 THz and 456.68 THz). In addition, the imaginary parts of eigenvalues $E_{\pm }$ at two EPs are not zero, so S-matrix is non-Hermitian.
Fig. 5. (a)–(d) Real and imaginary parts of eigenvalues $E_{\pm }$ when $d$ = 222.9 nm, 259.8 nm and $\phi$ = 0.8603 $\pi$, 1.125 $\pi$.
Next, we investigate the unidirectional reflectionlessness for different temperature $T$. Figures 6(a)-6(g) show the highly-dispersive reflection and absorption spectra for +z and -z directions at different low temperature $T$. When temperature $T$ = 200 K, 150 K, 100 K, 80 K and 60 K [37], unidirectional reflectionless phenomenon occurs at frequencies 470.68 THz, 470.52 THz, 470.38 THz, 470.26 THz and 470.1 THz, respectively, according to distance $d$ = 166 nm, 169 nm, 174 nm, 175 nm and 177 nm from Figs. 6(a)–6(e). Here, the collision frequency $\omega _{c}$ of silver decreases with the decreasing of temperature $T$. With decreasing the temperature $T$, bandwidths of reflection and absorption spectra gradually become narrow, while absorption remains almost the same value of $\sim$0.8 although the loss of metal decreases with decreasing the temperature. Figure 6(h) shows the Q-factors of unidirectional reflectionlessness versus different temperature $T$. Obviously, Q-factor increases with decreasing the temperature $T$. Especially, when $T$ = 60 K, Q-factor reaches $\sim$4274. In addition, near-zero reflection only appears in -z direction when temperature $T$ is equal to 200 K, 150 K, 100 K, 80 K and 60 K, respectively. And that, when temperature is lower than 60 K (eg, example 40 K and 20 K), unidirectional reflectionless phenomenon can no longer be found in our structure, and the reflections for +z and -z directions are nearly the same. That is to say, the phase differences between two resonators are nearly the same in +z and -z directions. In addition, we notice that more resonance dips appear when temperature is lower than 60 K.
Fig. 6. (a)–(g) Reflection and absorption spectra of +z and -z directions for $T$ = 200 K, 150 K, 100 K, 80 K, 60 K, 40 K and 20 K, respectively. (h) Q-factor of unidirectional reflectionlessness at $T$ = 300 K, 200 K, 150 K, 100 K, 80 K and 60 K.
3. Conclusion
Based on high-order plasmon resonance, we have demonstrated the highly-dispersive unidirectional reflectionless phenomenon in a non-Hermitian metamaterial system composed of double sliver-ring resonators. Reflections are $\sim$0 (0.82) and $\sim$0.85 (0) at 461.34 THz (456.68 THz) for +z and -z directions when the distance $d=222.9$ nm (259.8 nm), respectively. Moreover, high absorption of $\sim$0.97 can be achieved and the corresponding Q-factor is $\sim$435 at room temperature. With decreasing the temperature from 200 K to 60 K, near-zero reflection only occurs in -z direction. When temperature is 60 K, Q-factor of the unidirectional reflectionlessness reaches $\sim$4274. What’s more, when temperature of the system is lower than 60 K, the unidirectional reflectionlessness does not occur. We believe that our structure will have potential applications to filters, sensors, optical switching and optical diode-like devices.
Funding
National Natural Science Foundation of China (11364044, 11864043); Science and Technology Development Foundation of Jilin Province (STDF) (20180101342JC).
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