1. Introduction

Metamaterials possess extraordinary electromagnetic properties the majority of which are hardly obtainable in nature [1]. In particular, they exhibit sufficient difference in reflection (or transmission) coefficient of light with opposite circular polarization (CP). This option is of a huge interest in modern optics because it allows one to create compact elements for the controlling of light polarization and, particularly, for the generation of widely applicable circularly polarized light beams and pulses. Typical circular polarizers based on cholesteric liquid crystals (or birefrigent materials) work only in certain narrow frequency ranges [2], the central frequencies of which are strictly determined by the pitch of the helical structure of chosen liquid crystal. The operational range of such type of devices can be extended by stacking multiple polarizers [3] or by inducing a gradient of pitch of the helical structure of the crystal, which is a rather complicated task [4]. A negative factor accompanying these kind of solutions is the increase of the device thickness which makes them hard to be integrated in modern nanophotonic systems [5]. Being relatively easy to construct, planar metamaterials are appealing as a new kind of compact elements of polarization control [6]. However, planar structures exhibit much weaker difference between left circular polarization and right circular polarization transmittance compared with 3D geometries because an infinitesimally thin device is inherently achiral and excitation at oblique incidence or some form of nonreciprocal response is required [7]. In the case of normal incidence a special and rather complicated conjugation of two closely located planar layers of metamaterial is required to get a sufficient difference in transmission (reflection) coefficients of CP waves [8].

A decade ago a metamaterial composed of polymer helices organized in two-dimensional grating was constructed and its averaged transmission coefficients of left-hand and right-hand CP infrared electromagnetic waves at the incidence angles less than $7^{\circ }$ were shown to be twenty times different for the helices with eight turns [9]. This result is comparable with record characteristics of top line Faraday insulators. Using metal helices instead of polymer ones [10] provides the opportunity for the realization of selective refraction of CP radiation in a frequency range about one octave. The authors of [10] have experimentally demonstrated the applicability of such metamaterial as a wide-range circular polarizer of infrared electromagnetic radiation. The authors also mentioned the potential possibility of shifting of this range to the visible part of the spectrum by the optimization of base element parameters of the metamaterial. The properties of this structure for both cases of monochromatic [11,12] and pulsed [13] radiation were actively studied numerically.

The study of optical nonlinearities in plasmonic nanostructures and nanocomposites has attracted much attention in recent years [14–17]. Dramatic improvement of our ability to probe the optical field at deep-subwavelength and advances in nanofabrication techniques led to shift in the main research directions in nonlinear optics, from the study of coherent, phase-matched optical interactions that take place over many wavelengths to nonlinear optical processes in which the optical field interacts with matter at subwavelength scale [15]. The purpose of the present paper is to study the possible enhancement of selective refraction of CP monochromatic waves caused by nonlinear self-action provided by cubic nonlinearity of the basic element of the metamaterial (a metal helix). Its geometrical parameters were specifically chosen to make the operational range of the polarizer close to visible region of the spectrum, where the nonlinearity is the strongest. The choice of the metal (gold) is based on experimental data given in [18]. In our numerical simulations we used finite element method on hybrid unstructured mesh consisting of curvilinear elements which significantly increases the accuracy of computations, especially when dealing with nonlinear effects [19].

2. Formulation of the problem: numerical simulations

We have investigated the optical properties of a metamaterial, the basic element of which is a gold wire, curved to a helix (Fig. 1(a)). The pitch of the helix is $p=200\textrm {nm}$, the radius is $R = 50\textrm {nm}$, and the radius of the wire is $r=25\textrm {nm}$. The period of the metamaterial is $a=190\textrm {nm}$ in both $x$ and $y$ directions. The height of a helix along the $z$ directions is equal to $pN$, where $N$ is a number of turns. The quartz glass substrate(refraction index $n=1.45$) is colored gray. The complex dielectric permittivity of gold $\varepsilon$ is well described in optical range by Drude-Lorentz model:

 

Fig. 1. Sketch of the basic element of the metamaterial (a), two variants of its capturing by first-order HEX08 (b) and second-order HEX20 (c) elements and the types of elements used for hybrid unstructured mesh generation (d). The incident circularly polarized wave with wavevector $\mathbf {k}$ is schematically shown in (a).

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Let a plane CP wave $\mathbf {E}^{\mathrm {inc}}=E_0 [(\mathbf {e}_x \pm \textrm {i}\mathbf {e}_y)/\sqrt 2] \exp (-i\omega t+ikz)$ with the amplitude $E_0$ and the frequency $\omega$ is incident normally on the metamaterial along $z$-axis (see Fig. 1(a)). Here ${\mathbf {e}}_{x,y}$ are the unit vectors directed along $x$ and $y$ axes, $k=\omega /c$, $c$ is the speed of light, the signs “$+$” and “$-$” correspond to right-hand and left-hand CP wave. We used our own code written in C++ for the solution of Eq. (2) by finite element method. It allows to enforce the tangential continuity of the vector field and therefore perfectly match the continuity properties of the vector $\mathbf {E}$ at the interface between two media with different optical properties and to obtain a high order of approximation for the spatial derivatives in Eq. (2). The latter is achieved with an appropriate functional basis [21,22] and an unstructured mesh, which allows one to represent with high accuracy the shape of geometric objects with a curvilinear boundary by a small number of mesh elements (see Fig. 1(b) and 1(c)). The elements that we are using (see Fig. 1(d)) are diffeomorphic to one of the three reference elements named TET (tetrahedron), PYR (pyramid) and HEX (hexahedron). The edges of the elements of the first order (TET04, PYR05, HEX08) are straight lines, and those of the elements of the second order (TET10, PYR13, HEX20) are parabolas. The figures in their names give a number of points in space (coordinate triplets) fully constraining the corresponding mesh element. Hybrid mesh of a small number of such elements allows one to capture the helices with high accuracy (Fig. 1(b) and 1(c)). It also provides the balance between the simplicity of its own construction and the quality of obtained solution. To generate a hybrid unstructured mesh consisting of curved elements we used the program CSIMSOFT TRELIS [23] and to solve the system of algebraic equations obtained by discretization of Eq.(2) we used the Intel MKL PARDISO solver [24].

When radiation falls onto a metal object with non-smooth surface (i.e. with discontinuities in the external normal field) the electromagnetic field is localized near the edges of the object reaching local extrema in their points. The analogous phenomenon is observed in numerical simulations of radiation interacting with a metal helix with a smooth surface. This is caused by the defects of its mesh representation (see e.g. Figure 1(b)). As a result, the distribution of the absolute value of vector $\mathbf {E}$ on the surface of the helix has local extrema of numerical origin. It is especially noticeable in the presence of nonlinear response of the metal helix when the local extrema of the field sufficiently change the absolute value and the direction of nonlinear polarization vector.

The usage of the mesh with the same number of second-order elements (see Fig. 1(c)) sufficiently improves the quality of computations compared to the mesh composed of first-order elements. The local peaks of amplitude caused by the defects of the mesh disappear while the computational resources remain essentially the same. To save the resources we used the periodic boundary conditions on the $xz$ and $yz$ faces of the simulation domain and the sizes of this domain in $x$ and $y$ direction were determined by the period of the structure. On the $xy$ faces the scattering boundary conditions for the electromagnetic field were used. The length of the simulation domain along the $z$-axis was chosen in such way that the incident and transmitted field could be treated as a plane wave at the boundaries of the domain.

3. The discussion of the results

Performed numerical simulations allows one to conclude that the metamaterial composed of gold helices can selectively reflect CP waves with wavelength $650\textrm {nm}\;<\lambda \; <1000\textrm {nm}$ ($\lambda = 2\pi c/\omega$). A metamaterial composed of right-handed helices transmits left-hand CP light in this range with low losses and almost completely reflects right-hand CP radiation (Fig. 2). The bounds of the frequency range in which the selective refraction is realized are determined by the geometrical parameters of the helix and the values of dielectric permittivity of its material. For example, aluminum helices with the same geometry have the upper bound of the selective refraction range shifted to the blue part of the spectrum as compared with the gold helices. The increase of the turn number $N$ of the helices makes the selective refraction more contrast (Fig. 2).

 

Fig. 2. Dependency of transmission coefficients of right-hand (red) and left-hand (blue) CP normally incident waves on their wavelength in the case of linear optical response of the metamaterial. The basic element is a right-hand helix of $N=1$ (a), $2$ (b) and $3$ (c) turns.

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Selective refraction of CP light is caused by the interaction between the oscillations of the conduction current in metal and of the electromagnetic field near the helices. The pattern of the conduction current lines at the surface of the helix in a given moment of time are shown in Fig. 3 by black curves. The direction of the tangent vector to the lines coincides to the direction of the tangent component of the electric field vector which is continuous at the “metal-vacuum” boundary. The current lines at the surface of the metal have complicated topology with the source and sink points distributed differently in the cases of right-hand and left-hand incident light.

 

Fig. 3. Distribution of normalized absolute value of electric field strength vector on the surface of a metal helix with linear optical response and turn number $N=1$ (a and c) and $N=2$ (b and d) in the cases of left-hand (a and b) and right-hand (c and d) CP normally incident monochromatic light with wavelength $\lambda =700\textrm {nm}$. Black curves show the conduction current lines on the surface of the metal.

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To the best of our knowledge, there is no generally accepted data concerning the values of $\hat {\chi }^{(3)}$ components of gold [18]. Various published results differ from each other up to three orders of magnitude. Such a variability of the components is partially caused by essentially different ways of their measurement. For example, the part of nonlinear optical response measured in third-harmonic generation is much weaker than the more inertial part of the response, which is obtained from the measurements of the self-action effects. The latter is highly dependent on the wavelength of incident radiation and reaches its maximum near the wavelength $\lambda _0 = 550\textrm {nm}$. Our simulations have shown that transmission coefficients $T$ of CP electromagnetic waves are almost independent on the ratio $\chi _1/\chi _2$ if the sum $\chi _s = \chi _1+2\chi _2$ remains constant. Near the wavelength $\lambda _0$ in the case of defocusing nonlinearity ($\textrm {Re}\;\chi _s<0$) the frequency range in which the selective reflection of CP waves is realized expands with the increase of intensity of the incident radiation. Typical dependencies $T(\lambda )$ for the metamaterial composed of right-hand helices are given in Fig. 4(a). The increase of $|\textrm {Im}\;\chi _s|$ (i.e. the increase of dissipation) leads to decrease of $T$-coefficient for given wavelength $\lambda$ (Fig. 4(b)). The dependency of transmittance coefficients of CP waves on their intensity for $\textrm {Re}\;\chi _s<0$ and $\textrm {Im}\;\chi _s<0$ is nonlinear (Fig. 5). As the product $E_0^{2}|\textrm {Re}\;\chi _s|$ increases, the increase of $T$ goes faster (by almost two orders of magnitude) when $\lambda$ tends to blue bound of the selective refraction range (Fig. 5(a)). The impact of dissipation, however, is more noticeable as the $\lambda$ tends to the red bound of the range (Fig. 5(b)).

 

Fig. 4. Dependency of transmission coefficients of right-hand (red) and left-hand (blue) CP waves on their wavelength in the cases of linear and nonlinear (defocusing) optical response of the metal helices with turn number $N=3$. Solid curves correspond to linear response, dashed curves — $E_0^{2}|\textrm {Re}\;\chi _s| = 0.25$ (a), $0$ (b) and $E_0^{2}|\textrm {Im}\;\chi _s| = 0$ (a), $0.25$ (b), dotted curves — $E_0^{2}|\textrm {Re}\;\chi _s| = 0.5$ (a), $0$ (b) and $E_0^{2}|\textrm {Im}\;\chi _s| = 0$ (a), $0.5$ (b).

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Fig. 5. Dependency of transmission coefficients of left-hand CP wave on the product $E_0^{2}|\textrm {Re}\;\chi _s|$ (a) and $E_0^{2}|\textrm {Im}\;\chi _s|$ (b) for three different wavelengths.

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4. Conclusions

Using hybrid unstructured mesh with curvilinear elements and finite element method, we investigated the nonlinear interaction between CP light and three-dimensional metamaterial composed of gold helices. It was shown that the usage of curvilinear elements of the second order sufficiently increases the quality of simulations, especially close to the boundary of helix, while keeping the computational resources relatively low. A metamaterial, composed of right-hand helices with certain parameters, transmits left-hand CP light with low losses and almost completely reflects the light with opposite CP. The contrast of this selective reflection depends on the number of helix turns. In the case of defocusing nonlinearity of gold the frequency range in which the selective reflection is realized is widened to the visible part of spectrum. The presence of nonlinear absorption decreases the transmission coefficients of CP waves.