Optical chiral metamaterial based on the resonant behaviour of nanodiscs

Circular dichorism and optical activity have been achieved by chiral metamaterials in the optical spectrum, but for the case of negative index of refraction, remarkable achievements have not been obtained in this region so far. We employ nanoparticles to shift the resonant frequency of a chiral metamaterial based on twisted cross wires to optical domain. Our proposed structure provides giant optical activity, strong circular dichorism and also negative refractive index in the optical wavelengths. Optical activity in our structure has a rotary power similar to a gyrotropic crystal of quartz, but in a thickness which is four orders of magnitude smaller. The foundation of our method for realizing such an optical chiral metamaterial is based on creating a different coupling between longitudinal modes of localized surface plasmons for right and left circularly polarized incident waves.


Introduction
Metamaterials are artificial materials that their unusual properties, such as negative permeability and negative refractive index, depend on their geometries not their components. These unusual properties can offer special applications such as electromagnetic absorbers, electrically small antennas (1) and the cloak of invisibility (2). In 2003 (3) and later in 2004 (4), chiral structures were used as an alternative to create metamaterials. In general, chiral structures have no symmetry planes. In addition to negative index of refraction, chiral metamaterials also show optical activity and circular dichorism. Optical activity can be used to manipulate polarization, and circular dichorism can serve as circular wave polarizer. Several different designs have been proposed and investigated for chiral metamaterials having all three features (optical activity, circular dichorism and negative index of refraction) in the microwave (5)(6)(7)(8) and THz regimes (9)(10)(11). Also, at optical frequencies, various designs are reported for chiral metamaterials, but only optical activity and circular dichorism have been achieved, not negative index of refraction (12)(13)(14)(15). In (16) and (17), chiral metamaterials with negative refractive index in the near infrared and optical spectrums are reported, respectively.
For realizing chiral metamaterials, unlike conventional methods, there is no need for the permittivity and the permeability to be negative simultaneously. Using this feature, metamaterials can be realized easier. Chiral  right circularly polarized (RCP) and left circularly polarized (LCP) waves. One of the parameters used to show this difference is the chirality coefficient. In natural materials, this parameter is very weak, but in chiral metamaterials, it is so large that makes one of the circularly polarized waves to have negative refractive index.
At high frequencies, metals do not behave as a perfect conductor. As a result, a type of propagating surface mode that is called surface plasmon polariton, is created. The dispersion diagram, i.e. frequency versus propagation constant diagram, of the surface plasmon polaritons has a negative slope that can be used to make metamaterials (18)(19)(20). Also, as another result of the finite conductivity, when a metallic nanoparticle is exposed to an electric field, it resonates and creates a strong field around itself that can be up to several hundred times of the applied field. Alu and coworkers have used this point to create a metamaterial in the visible range of the spectrum (21).
In this paper, we use nanoparticles and their surface plasmon frequencies to shift the resonance frequency of a chiral metamaterial to the optical region. In Section 2, the proposed structure is described in detail. The simulation results and the retrieved effective parameters are provided in Section 3. The last section of the paper is devoted to the conclusions.

Proposed structure
In (6), a chiral metamaterial based on twisted cross wires is reported in which, resonance occurs when the lengths of the wires are around a quarter of the wavelength and at this resonance, negative refractive index is obtained. Indeed, the lengths of the wires determine the resonant frequency at which the negative index of refraction occurs. To obtain negative refractive index materials at optical frequencies, one should scale down the structure to optical wavelengths. But, with respect to the ohmic losses and the plasmonic effects, this is not easy. For example, in (13), the dimensions of the cross wires are decreased to optical wavelengths, but only optical activity and circular dichorism are obtained, not negative index of refraction. In this paper, we propose a novel structure based on bilayer cross wires (6) but, in our case, wires are replaced by arrays of aligned nanodiscs. Therefore, in our structure, the resonant frequency is determined by the resonant frequency of the surface plasmons that is in the optical region.
The unit cell of the proposed structure with two and three layers of nanodisc crosses are shown in Figures 1  and 2, respectively. In these figures, The radii of nanodiscs are 30 nm and their heights are 20 nm. Each line has seven nanodiscs and the distance between the centres of adjacent nanodiscs is 70 nm. To increase the visibility of the structure in these figures, various nanodisc crosses are sketched by different colours but in fact, all of them are made of silver. In our simulations, Johnson and Christy's database is used for the silver (22). The used dielectric is a type of glass, FK51A, with n ∼ = 1.487 and negligible loss at the frequency band of our simulations, i.e. around f = 485 THz.
We present a design method to select the frequency band of negative refractive index in a wide range of optical spectrum. One can make major changes to this frequency band by changing the radii of the nanodiscs or minor changes by changing the thickness of the dielectric spacer or the size of the unit cell, but three points should be noticed in this regard.
As the first point, the twist angle of the nanodisc crosses between successive layers, ϕ , should be proportional to the thickness of the dielectric spacer. It might seem that a 22.5 • twist angle would produce the most chirality in the structure, at least in one with two layers of nanodisc crosses, but, the simulations show different results. If this twist angle is equal to the retardation phase of a circularly polarized wave going through the dielectric spacer, longitudinal modes of localized surface plasmons (23) are excited efficiently. Since in our structure, the direction of rotation of nanodisc crosses between successive layers is left handed, this excitation could happen easier for the case of LCP illumination. This event could have occurred for RCP illumination, if the direction of rotation is right handed. Therefore, the twist angle should be where d and n are the thickness and the refractive index of dielectric spacer, respectively, and λ is the wavelength that corresponds to the resonant frequency of nanodisc crosses. Let T + and T − be the transmission coefficients of the structure for the incident RCP and LCP waves, respectively. If Equation (1) is satisfied, the deepest notch is created in the magnitude diagram of T + or T − (in our case, T − ). The maximum depth means the maximum difference between the two different circular polarization and consequently maximum circular dichorism, maximum optical activity and also maximum chirality. Therefore, ϕ could be seen as a tunning parameter for the amount of circular dichorism. In order to reach negative index of refraction, one does not need maximum chirality, but the chirality must be large enough to overcome the first term on the right side of the following equation where μ and are the effective permeability and effective permittivity of the medium, respectively, κ is the chirality coefficient of the structure and finally, (+) and (−) refer to RCP and LCP waves, respectively. However, larger chirality leads to more bandwidth for negative index of refraction. On the other hand, for the negative refractive indices, n = −1 is the best case and only under this condition, perfect reconstruction of the image is possible (25). As the second point, our simulations on various designs show that the designed structure should have fourfold rotational symmetry, i.e. the rotation of the structure by 90 • , results in similar structure. Under this constraint, the two linear components of the incident circularly polarized wave (e.g. E x and E y , if the propagation direction of the incient wave is z-axis) see similar structures. In other words, it helps that the model of structure follows the model of the reciprocal chiral mediums, i.e. when the incident wave is RCP (LCP), the outgoing wave remains RCP (LCP) too and the reflected wave would be LCP (RCP).
As the third point, since the foundation of chirality is based on the difference between T + and T − , the magnitude of one of them which corresponds to the negative refractive index, should be small. As a result, the area of the nanoparticles on the surface of the unit cell, should not be small. Indeed, there is a trade-off between transmission coefficient and, as mentioned earlier, an easier way to reach the negative index materials. On the other hand, relatively low transmission coefficient means relatively high insertion loss.
We simulated the structure with two and three layers of nanodisc crosses. In the structure with two layers of nanodisc crosses, large circular dichorism and also optical activity were achieved but not large enough to create a noticeable negative refractive index. By increasing the number of layers to three, circular dichorism, optical activity and negative index of refraction were all obtained.

Simulation and retrieving the effective parameters
Both of our proposed structures, with two and three layers of nanodisc crosses are classified in the reciprocal chiral mediums and their electromagnetic behaviours are characterized by the following constitutive relations (26) where c 0 is the speed of light in vaccum. We used Lumerical FDTD Solutions to simulate our proposed structures. All simulations were done for normal incidence of circularly polarized waves propagating in the +z-direction. Our simulations show that the resonant wavelength of a single nanodisc cross placed above a 20-nm thick dielectric, is about 600 nm. So, from Equation (1), ϕ = 17.85 • , but as mentioned above different parameters can affect the resonant wavelength of nanoparticles. After optimization, we set ϕ = 17 • and ϕ 1 = ϕ 2 = 17.5 • for the structures with two and three layers of nanodisc crosses, respectively.
The transmission coefficients of RCP and LCP waves are shown in Figure 3 and the ellipticity angle, η, for the linearly polarized light are calculated using the following relations (27) The results are shown in Figures 4(a) and 5(a) for the structures with two and three layers of nanodisc crosses, respectively. As can be seen in Figure 3 We used the method that is introduced in (29) for retrieving the effective parameters of our proposed chiral metamaterials. The real and imaginary parts of chirality and also the real and imaginary parts of refractive indices of the RCP and LCP waves for the case of the structure with two and three layers of nanodisc crosses are shown in Figures 4(b)-(d) and 5(b)-(d), respectively. As can  be seen in Figure 4(c)-(d), the real part of refractive indices are approaching to zero and even goes below zero for RCP incident wave in a part of the simulation spectrum, but, this is not a noticeable achievement. For the structure with three layers of nanodisc crosses, the refractive index is negative for incident RCP wave from f = 468 to f = 476 THz and for incident LCP wave from f = 504 to f = 512 THz, as can be seen in Figure  5 We have studied the proposed structure with four and five layers of nanodisc crosses and found that for a fourlayer structure, the chirality coefficient is less than the three-layer case and more than the two-layer case. In the four-layer case, only one of the circular polarization has negative index of refraction. In the case of five-layer structure, the chirality coefficient is reduced further, such that none of the two circularly polarization has negative index of refraction. The explanation for these observations is that if the number of layers is increased beyond a number (herein 3), the structure asymmetry is weakened and the structure becomes more symmetric. This results in decreasing the chirality coefficient. Therefore, increasing the number of layers beyond three, makes the structure more complex and also corresponds to weaker results. For example, a five-layer structure is shown in Figure 6. To clarify the above explanation, the middle dielectrics are not shown in the figure.
In the case of closely spaced nanoparticles, these nanoparticles can be modelled as electric point dipoles (23). On the other hand, the surface charges are related to normal components of electric fields. So, for better understanding the excitation of the longitudinal modes corresponding to localized surface plasmons, we set some monitors in our simulations to investigate the normal components of electric fields. As an example, for the structure with two layers of nanodisc crosses, the real part of the normal component of electric field on the two nanodisc crosses are sketched in Figure 7, for LCP illumination at its resonant frequency, f = 508 THz.
At last, we make a comparison between the proposed structure and a conventional structure. The conventional structure is similar to our proposed structure but, it has continuous strips instead of arrays of aligned nanodiscs. The results of this comparison are described in detail in a supplementary file. In summary, the resonant frequency of the conventional structure cannot increase beyond a limit. In fact, by shortening the lengths of strips, they gradually act as nanoparticles. In this case, the structure can be considered as an array of nanoparticles with one element. The resonant behaviour of this one-element array is not strong enough to create negative refraction.

Conclusion
In this paper, we used the resonant frequency of surface plasmons in nanoparticles to shift the resonant frequency of metamaterial based on bilayer cross wires (6) to optical domain. In fact, we used crosses of nanodiscs instead of continuous wires. Our proposed chiral metamaterial provides significant optical activity, large circular dichorism and also negative refractive index in the optical spectrum. First, we realized the proposed structure by two layers of nanodisc crosses but, only strong circular dichorism and optical activity were obtained and the results about the refractive index were not remarkable. When we added another layer, negative refractive index was achieved too. In a thickness of only 100 nm, our proposed structure provides the same rotary power as a gyrotropic crystal of quartz of 1-mm thickness. The foundation of the proposed structure to obtain negative refractive index is based on the creation of different coupling between longitudinal modes of localized surface plasmons under the incidence of RCP and LCP waves. Meanwhile, we developed a formula for the twist angle of the nanodisc crosses between adjacent layers under which the maximum chirality is achieved. Also, we mentioned some points for changing the resonant frequency of a chiral metamaterial in a wide range of optical spectrum.