Dual-Band Unidirectional Reflectionless Propagation in Metamaterial Based on Two Circular-Hole Resonators

Dual-band unidirectional reflectionless propagation at two exceptional points is investigated in metamaterial, which is composed of only two gold resonators with circular holes, by simply manipulating the angle of incident wave and distance between two resonators. Furthermore, the dual-band unidirectional reflectionless propagation can be realized in the wide ranges of incident angle from 0∘ to 50∘ and distance from 255 nm to 355 nm between two resonators. In addition, our scheme is insensitive to polarization of incident wave due to the circular-hole structure of the resonators.

In this work, we propose a novel scheme to realize dual-band unidirectional reflectionlessness in metamaterial based on phase coupling between two circular-hole resonators. The metamaterial system consists of an upper and lower circular-hole gold resonators with different radii. The reflectivities for +z (−z) direction at 262 THz and 276.24 THz are ∼0.63 (∼0) and ∼0 (∼0.62) at two EPs, respectively, and high absorptions of ∼0.86 and ∼0.90 for +z and −z directions can be obtained with both quality factors of ∼20, respectively.

Structure
The unit cell of the metamaterial structure in Figure 1a consists of an upper and a lower gold resonators with circular holes. The thicknesses of the two resonators are h = 30 nm, and the upper and lower radii are r = 86 nm and R = 112 nm, respectively. Periods t are 600 nm in both x and y directions. Two gold resonators are embedded in photopolymer (thickness 405 nm) with the permittivity of 2.4025. The entire structure is placed on glass substrate (thickness 300 nm) with the permittivity of 2.25. Distance s between two gold resonators is variable. The permittivity of gold is complied with Drude model with plasmon frequency ω pl = 1.366 × 10 16 rad/s and collision frequency ω c = 12.24 × 10 13 s −1 [33,34]. Numerical simulation is carried out by using the commercial finite difference time domain softwave package (CST Microwave Studio). The boundary condition is the unit cell in x and y directions, while is open in z direction. Figure 1b simply describes the scattering property of the structure, where the amplitudes of +z and −z directions propagating waves are denoted by S +ij and S −ij (i, j = 1,2), respectively.  Here, the incident wave is in x-z plane and has a incident angle θ with +z axis, and the electric field is along y axis.

Results and Discussions
The proposed structure ( Figure 1) can be investigated by the temporal coupled-mode theory (CMT) [35][36][37]. The evolutions of amplitudes a 1 (upper gold resonator) and a 2 (lower gold resonator) can be expressed as where ω 1 and ω 2 are resonant frequencies of the upper and lower gold resonators, and γ 1 and γ 2 are decay rates of the upper and lower gold resonators due to intrinsic loss, respectively. Γ 1 and Γ 2 are dissipation rates of the upper and lower gold resonators due to energy escaping into outside space, respectively. κ is the coupling coefficient between two gold resonators. With the conservation of energy, the amplitudes S +ij and S −ij (i, j = 1, 2) satisfy the following relations In addition, the propagation waves should satisfy the formulas S +21 = S +12 e iϕ and S −12 = S −21 e iϕ with phase difference ϕ = ωRe(n e f f )s/c, n e f f and c the effective refractive index of incident wave and light velocity in vacuum, respectively. Hence, we can obtain the complex coefficients for +z (−z) direction transmission t +z (t −z ) and reflection r +z (r −z ), respectively, as So, transmissivity, +z and −z direction reflectivities are T = |t| 2 , R +z = |r +z | 2 and R −z = |r −z | 2 , respectively. Accordingly, absorptivities for +z and −z directions can be expressed as Figure 2 shows the reflection, absorption and transmission spectra for +z and −z directions, respectively. Figure 2a is the reflection spectra for +z and −z directions based on numerical simulation (solid line) and analytical calculation (dotted line), respectively. Obviously, the results obtained by analytical calculation are in good agreements with that by numerical simulation. According to numerical simulation, reflectivities in +z (−z) direction are ∼0.63 (∼0) and ∼0 (∼0.62) at 262 THz and 276.24 THz, respectively. Therefore, dual-band unidirectional reflectionlessness is realized in metamaterial based on two gold resonators with circular holes. Moreover, high absorptions of ∼0.86 and ∼0.90 in +z and −z directions can be obtained, respectively, at 276.24 THz and 262 THz, where the transmissions are very low, as shown in Figure 2b. Additionally, the corresponding quality factors of ∼20 can be obtained by formula f /∆ f with f and ∆ f the resonant frequency and full width half maximum, respectively. The similar approach has been employed in Reference [38], in which the planar array of resonant metallic cross-shape structure has been used and the absorption properties have been discussed. In our unit-cell, two-circular-hole resonators structure is chosen in view of the insensitivity to polarization and high efficiency on realizing dual-band absorption in two directions.
Obviously, absorptions in Reference [38] and our scheme are both insensitive to polarization of the incident wave. A comparison of the work in Reference [38] and ours shows that absorption in one direction is obtained in metallic cross-shape structure and two-direction absorption is obtained in two-circular-hole resonators structure. To clearly demonstrate the dual-band unidirectional reflectionlessness, we investigate the z-component distributions of electric field of two gold resonators with circular holes for +z (−z) direction at two EPs, as shown in Figure 3. At 262 THz, two gold resonators are excited simultaneously in +z (−z) direction and the induced currents are in the opposite (same) directions, as shown in Figure 3a-d, which means that the phase differences between two gold resonators are ∼π (Figure 3a   We also plotted the z-component distributions of magnetic field of the two gold resonators for +z (−z) direction at two EPs, as shown in Figure 4. It is obvious to see that the induced magnetic fields of the two gold resonators are opposite (same) at 262 THz in +z (−z) direction, as shown in Figure 4a-d and the induced magnetic fields are same (opposite) at 276.24 THz in +z (−z) direction, as shown in Figure 4e-h. Obviously, dual-band unidirectional reflectionless phenomenon appears at two EPs based on Fabry-Pérot resonance coupling between two gold resonators. As a result, high two-band absorption can be obtained in view of the low transmission at two EPs, as shown in Figure 2b.  Next, based on scattering matrix [21] S = t r −z r +z t , we continue to discuss the relevant physics phenomena at two EPs. According to scattering matrix S, we obtain the eigenvalues of the scattering matrix S as λ ± = t ± √ r +z r −z . Here, when √ r +z r −z = 0, two eigenvalues λ ± coalesce and EP appears. In other words, when r +z or r −z is 0, two eigenvalues coalesce and unidirectional reflectionlessness appears. Real and imaginary parts of the eigenvalues λ ± as the functions of frequency ω are depicted in Figure 5. From Figure 5a,b, the real parts of the two eigenvalues λ ± coalesce, while the imaginary parts cross at 262 THz and 276.24 THz, respectively. Clearly, the real and imaginary parts of eigenvalues are nonzero at 262 THz and 276.24 THz. In this case, t is complex and √ r +z r −z is 0. Therefore, the dual-band unidirectional reflectionlessness can be obtained at 262 THz and 276.24 THz. (ω − ω 1(2) ) (γ 1(2) + Γ 1 (2) ) .
Then, we investigate the influences of incident angle and frequency on reflections (absorptions) in +z and −z directions based on numerical simulation. Figure 6a-d shows reflections (absorptions) in +z and −z directions versus the incident angle and frequency when s = 275 nm, respectively. Clearly, low reflection (high absorption) region occur blue-shifts with increasing the incident angle in +z and −z directions, respectively. Moreover, low reflection (high absorption) region in +z direction shown in Figure 6a,c corresponds to high reflection (low absorption) region in −z direction shown in Figure 6b  Next, we further study the influence of different distance s between two gold resonators on reflections (absorptions) in +z and −z directions. Figure 7 shows the reflections and absorptions as the functions of distance s and frequency ω in +z and −z directions when the incident angle is zero. From Figure 7a,c, low reflection and high absorption peaks occur red-shifts with increasing the distance s from 255 nm to ∼315 nm and do not occur shift when the distance s over ∼315 nm. While low reflection and high absorption peaks do not occur shift with increasing the distance s from 255 nm to ∼295 nm and occur red-shifts with increasing the distance s from ∼295 nm to 355 nm, as shown in Figure 7b,d. Moreover, low reflection (high absorption) region for distance s from 255 nm to 355 nm in +z direction shown in Figure 7a,c corresponds to high reflection (low absorption) region in −z direction shown in Figure 7b,d around 280 THz. Also, low reflection (high absorption) region for distance s from 255 nm to 355 nm in −z direction shown in Figure 7b,d corresponds to high reflection (low absorption) region in +z direction shown in Figure 7a,b around 260 THz. Obviously, dual-band unidirectional reflectionlessness and dual-band absorption can be realized in a wide distance range of 255 nm∼355 nm.

Conclusions
In summary, we have demonstrated the dual-band unidirectional reflectionless propagation at two EPs in metamaterial by using only two gold resonators with circular holes. Reflectivities for +z and −z directions are ∼0.63 (∼0) and ∼0 (∼0.62) at 262 THz (276.24 THz). Moreover, the dual-band unidirectional reflectionless propagation can be realized in the wide ranges of incident angle from 0 • to 50 • and distance between two gold resonators from 255 nm to 355 nm. Through using the circular-hole structure of the resonators, the scheme is insensitive to polarization of incident wave. Our structure is quite suitable for experimental fabrication using standard nano-fabrication procedures [39]. These results will provide a good platform to realize the extraordinary properties of metamaterial systems with potential applications in the integrated nanophotonic devices, such as optical filters, sensors, and diodes. Especially, it can also be applied to refractive index measurements. In our structure, refractive index of surface medium will impact on the resonance frequency of the structure and the variation of resonance frequency will cause the displacement of the position of unidirectional reflecionlessness. Thus, the change of refractive index can be measured by analyzing the variation of spectrum. In the future, we will study the methods to raise the quality factor of absorption, such as by adopting the high-Q resonator.