Manipulation of polarization and spatial properties of light beams with chiral metafilms

Two-dimensional lattices of chiral nanoholes in a plasmonic film with lattice constants being slightly larger than light wavelength are proposed for effective control of polarization and spatial properties of light beams. Effective polarization conversion and strong circular dichroism in non-zero diffraction orders in these chiral metafilms are demonstrated by electromagnetic simulations. These interesting effects are found to result from interplay between radiation pattern of single chiral nanohole and diffraction pattern of the planar lattice, and can be manipulated by varying wavelength and polarization of incoming light as well as period of metastructure and refractive indexes of substrate and overlayer. Therefore, this work offers a novel paradigm for developing planar chiral metafilm-based optical devices with controllable polarization state, spatial orientation and intensity of outgoing light.


Introduction
Control over spatial and polarization characteristics of light is important for many applications in optical devices. Nowadays miniaturization of such devices becomes crucial for such applications, and making use of plasmonic nanostructures seems to be promising in this direction. In particular, polarization conversion of electromagnetic waves by periodic nanostructures as well as by metasurfaces has been investigated theoretically and experimentally [1][2][3][4][5]. For example, Gordon et. al. observed strong polarization dependence in light transmission through a square lattice of nanoholes in metal on the ellipticity and orientation of the holes [1]. Fedotov et. al. demonstrated normal incidence transmission asymmetry of circularly polarized waves through a lossy anisotropic planar chiral structure [2]. Gansel et. al. reported significant extinction of circularly polarized plane waves by a square lattice of three-dimensional gold helices with the same handedness as of incident wave, while light waves of the other helicity pass the structure with high transmission [3]. In [4], asymmetrical transmission of linearly polarized light through a planar metamaterial composed of three-dimensional chiral meta-atoms with rotational symmetry was demonstrated. Finally, Zhao et. al. recently showed that planarized ultrathin broadband circular polarizers fabricated by planar technologies could show functionalities previously provided only by three-dimensional geometries [5].
Optical metamaterials considered in [1][2][3][4][5] have lattice constants being smaller than light wavelength. Consequently, only zero diffraction order is excited by incident light. In other words, these systems do not show diffraction effects. To control light more effectively, therefore, the concept of metasurfaces such as gradient metasurfases was recently put forward (see, e.g., [6][7][8][9][10][11][12][13][14]). The structure of gradient metasurfaces has two effective periods. One is the distance between nearest neighboring elements (meta-atoms) which is smaller than operating wavelength. The other effective period is the distance between similar groups of meta-atoms and is bigger than wavelength. Within this paradigm, metasurfaces which exhibit anomalous reflection and refraction which are nevertheless in agreement with generalized laws derived from Fermat's principle, was presented in [6]. Planar holey-metal lens made of concentric circular arrays of nanoscaled holes, which are used as a phase-shifting element, was demonstrated in [8]. Analog computing using reflective plasmonic metasurfaces was also reported recently in [9][10][11]. Metasurfaces that transform linearly polarized incident waves to circularly polarized outgoing waves in a wide wavelength range was demonstrated in [14].
We note that similar results can be obtained if subwavelength periodic features are replaced by individual meta-atoms (nanoparticles or nanoholes) of a special shape. As a result, interplay between radiation pattern of a single meta-atom and radiation pattern of the array can result in desirable behaviors of light beams. As an example of meta-atoms of sophisticated form, we can consider chiral meta-atoms with gammadion shape. This geometry is popular at present because it allows to effectively convert the polarization state of incident light (see, e.g., [15][16][17][18][19][20]).
In the present work, therefore, in order to control both polarization and spatial distribution of light beams, we consider, as an example, a square lattice of chiral nanoholes in a plasmonic film with the period being slightly larger than the wavelength (see Fig. 1) and analyze the polarization states of light scattered into different diffraction orders. To this end, we develop a formulation for evaluating polarization state-decomposed and diffraction channel-resolved transmissions and also perform numerical simulations for the metafilm with different sizes of chiral nanoholes. The rest of this article has the following structure. Section 2 contains basic formulae derived to analyze state of polarization and transmission coefficient of light wave scattered into various diffraction orders. In section 3, the results of numerical simulations for a planar lattice of gammadion-shaped nanoholes in a gold film with various nanohole sizes as well as calculated polarization conversion efficiency, circular dichroism and power spatial distribution are presented. Finally, conclusions drawn from this work are given in section 4. Fig. 1. A square lattice of chiral nanoholes in a gold layer. The period of the lattice is made slightly bigger that wavelength W λ > .

Analysis of polarization state of diffracted waves
Let us consider a square lattice of chiral nanoholes, which is excited by a normally incident plane wave ( Fig. 2) with: Due to the periodicity of the system, the electric field of transmitted light can be expressed as sum of infinite number of plane waves: components of the wavevector that correspond to , n m diffraction order, k is the absolute value of the wavevector in lower half space, W is the lattice constant of the square lattice, and , , x y z are the Cartesian coordinates. nm E is the contribution to total electric field from the wave of , n m diffraction order. The Fourier coefficients nm C can be calculated from electric field in, e.g., the plane of z = 0 by: Here the prime represents the vector and its components in the local coordinate system. Clearly, in the local coordinate system, wavevector nm ′ k has only the z component. Electric field of the plane wave corresponds to the , n m diffraction order is: In the local coordinate system, the electric field nm ′ E can be written as: Equation (6) explicitly shows that the electric field in the local coordinate system is purely transverse. The corresponding Poynting vector is: where ( ) This definition correspond to clockwise(counterclockwise) rotation of electric field vector from the point of view of source for RCP(LCP) wave. Therefore, the electrical field components of RCP and LCP waves for the , n m diffraction order in the local coordinate system can be written as:  (7), the Poynting vector can be written as: where R,nm S and L,nm

Numerical results and discussion
Let us now apply the formalism described in the preceding section to the structure shown in Fig. 3. The gold film of 220 nm thickness is deposited on a quartz substrate and is then covered with immersion oil in the other side. The hole is also filled with immersion oil. The optical dielectric constants of gold from Weber [21] are used. Refractive indexes of quartz and oil are set to 1.443 and 1.51, respectively. Frequency dispersion of the dielectrics is neglected. The gammadion geometry is defined by a single parameter s, the size of the 17 identical squares which make up the gammadion, as shown in Fig. 3. Here we set 86 nm s = . The edges of the squares sitting at the ends and corners of the gammadion are smoothed with curvature radius of s/5. The smoothing of edges is rather important because this makes the problem more definite from mathematical point of view and also increases the accuracy of calculations. For more detailed analysis of the problem of sharp edges, see [22].
The square lattice constant is taken to be 1 W m μ = . The wavelength considered is in the range of 760 820 nm λ = − . The maximum order of diffraction peaks that appear in this system is 1, The system is illuminated normally by a plane wave with either type of circular polarizations. Nanoholes with both left and right twists are considered. Twist of gammadions is defined from the point of view of incident wave. Right-handed (left-handed) gammadion has clockwise (counterclockwise) twist. In Fig.3, only the left twisted gammadion is shown.

Convers
The   2 n m + = is more effective than in orders of 2 2 1 n m + = (see, e.g., Fig. 4). Furthermore, our simulations also reveal that bigger nanoholes give a more effective polarization conversion. Also, if the helicity of the incident wave matches the handedness of the nanoholes, the polarization conversion in orders 2 2 2 n m + = is more effective than the mismatched case. Indeed, Fig. 4 shows that when a LCP wave irradiates left twisted nanoholes, there is a wavelength range in which the diffracted waves of the 2 2 2 n m + = order, has more RCP content than that of LCP in the wavelength from ~783 to ~802 nm. And at λ = 786 nm, the conversion of the LCP to RCP light is nearly perfect.  To better understand this interesting effect, the asymmetry of LCP and RCP components versus wavelength in diffraction order 2 2 2 n m + = is displayed in Fig. 6. Here the polarization asymmetry is defined as where the transmission coefficients   , and this means nearly full conversion of incident LCP to RCP in n = m =1 lobe. Moreover, the polarization can also become purely linear at wavelengths of 765 nm and 803 nm (Fig. 6). The results for all orders with 2 2 2 n m + = are the same. The effect of polarization conversion in non-zero diffraction lobes is due to interplay between radiation patterns of single chiral nanohole and a square lattice of point sources. First of all, radiation pattern of single chiral nanohole has directions in which polarization of scattered light differs from polarization of incident light. This effect is of purely geometrical nature and, strictly speaking, can be observed even for light scattering by metallic sphere [24]. However, handedness of nanoholes leads to substantial increase of efficiency of polarization conversion. In particular, our simulations for the square lattice of cylindrical holes with the same cross-section area as that of 86 nm s = gammadions show that the effect is much weaker. For example, maximum value of nm P (Eq. 12) is equal to 0.13 at 799 nm wavelength while, for gammadion-shaped holes, it is 0.9 at 786 nm. Radiation pattern of the whole system can be found as multiplication of radiation pattern of single chiral nanohole with radiation pattern of a square lattice of point sources [25]. Therefore, when lobes of radiation pattern of the square lattice of point sources coincide with angle of maximum polarization conversion of single chiral nanohole, an effective polarization conversion into this lobe occurs. Interestingly, superposition of maximum of polarization conversion with lobes of radiation pattern can be tuned by changing wavelength λ , lattice constant W and refractive index of surrounding media.  (Fig. 3). Results for three diffraction orders, namely, n=0, m=0 (blue line, standard CD), n=0, m=1 (red line) and n=1, m=1 (yellow line), are shown.

Circular dichroism in transmission
One may expect that the system under consideration would exhibit circular dichroism (CD). One can characterize this CD in transmission for all diffraction orders in natural way, i.e., are given by Eq. (10). In Fig. 7, the wavelength dependence of the calculated L CD nm for all considered diffraction orders for the square lattice of left twisted gammadions (Fig. 3) is displayed. Figure 7 shows clearly that our system exhibits CD and hence is a truly chiral one despite of a small difference in refractive index between the substrate (SiO 2 ) and superstrate (immersion oil) (see Fig. 3). This is interesting because in the case when substrate and superstrate have the same refractive indices, the system would possess mirror symmetry and hence exhibit no chiral property. Moreover, given the small thickness of the gold film (220 nm), the chirality and CD showed by our system is large. Another useful definition of circular dichroism is related to the fact, that a linearly polarized incident light beam would become elliptically polarized after passing through the square lattice of chiral nanoholes. Therefore, circular polarization of transmitted light can also be characterized with ellipticity θ [26] R, L, where R,nm E′ and L,nm E′ are, respectively, the magnitudes of the electric field vectors of the RCP and LCP light when a linear polarized wave irradiates the system. The usefulness of this classical definition is the fact that θ = 0 corresponds to the absence of CD and chirality, while θ = 45° corresponds to the full absorption of one component and maximal chirality. In Fig. 8, the wavelength dependence of the calculated θ for all considered diffraction orders for the square lattice of left twisted gammadions (Fig. 3) for light polarized along y-axis is displayed. Since ellipticity is defined for linearly polarized incident light, ellipticity in the diffraction orders with fixed 2 2 n m + is no longer identical, as was the case for circularly polarized incident light. Now only diffraction orders , n m and , n m − − have equal ellipticity. angles confirm again that our system has rather strong chirality. Moreover, further optimizations of the system could result in even stronger ellipticity.

Spatial distribution of transmitted power and polarization state
To understand an overall power distribution over different diffraction channels, we display polarization state for each diffraction maximum for two wavelengths 765 nm λ = and 786 nm λ = in Figs. 9(a) and (b) for parameter s = 86 nm (Fig. 4). It is clear from Fig. 9(a) that for wavelength 765 nm λ = , power diffracted to a side channel of the first-order is roughly equal to 1.2 %, while power going into the principal lobe is 2 times larger. Also, the total power transmitted to all side lobes is more than 3 times greater than power going into the main lobe. Another interesting feature is that polarization states of all orders with 2 2 2 n m + = are purely linear despite of LCP incident wave [ Fig. 9(a)]. In contrast, Fig. 9(b) shows that for 786 nm λ = , nearly no energy is transmitted to 2 2 1 n m + = diffraction orders, while roughly equal amount of energy goes to the zero diffraction order and nonzero diffraction orders of 2 2 2 n m + = . Remarkably, polarization of the waves in diffraction orders of 2 2 2 n m + = is almost purely RCP, while that of the zero diffraction order is the same as that of incident wave, as it should [ Fig. 9(b)]. The energy efficiency of the polarization conversions reported in Sec. 3.1. is not high (i.e., about 1% of the incoming energy). Nevertheless, we expect that optimizations by making use of optical Tamm states can enhance this efficiency substantially [27][28].  (Fig. 3). Arrows indicate rotation directions of the electric field vector (helicity). Clockwise rotation corresponds to RCP and counterclockwise rotation corresponds to LCP. In fact, this figure also represents spatial distribution of polarization and transmittance.
It would also be important to know relations between transmittances to different diffraction orders. To this end, one can define transmittance ratios I  I  I  I  I  I   I  I  I   I  I  I  I  I  I   L  L  L  L  L  L  ,11 ,11 ,10 ,10 ,11 ,11 L L L ,2/0 ,1/0 ,2/1 L L L L L L , 0 0 , 0 0 , 0 0 , 0 0 , 1 0 , 1 0 , , , P R P L P R P L P R P L P P P P R P L P R P L P R P L   T  T  T  T  T  T  q  q  q  T  T  T  T  T where upper index L denotes twist of gammadions and ( R or L) denotes polarization state of incident wave. The wavelength dependence of these ratios for the array of left twisted nanoholes for LCP and RCP incident waves are shown in Figs. 10 and 11, respectively.    (Fig. 4). Interestingly, situation is different when the twists of incident wave and gammadion are opposite (see Fig. 11). Although average transmittance into zero diffraction order is still approximately two times larger than transmittance to order We have also carried out the calculations for light entering the system from the immersion oil side. Nevertheless, in this case, the system exhibits similar properties of polarization conversion, although the polarization conversion is weaker. In particular, the maximum value of the asymmetry coefficient [Eq. (12)] is 0.7 and is slightly smaller than 0.9 for the case of light incident from SiO 2 side. Also, the corresponding total transmittance to 2 2 2 n m + = orders is only 0.3 %, being significantly smaller than 1.4 % for light incident from SiO 2 side.

Conclusions
A formulation for analysis of polarization conversion in different diffraction orders in a square lattice of chiral nanoholes has been developed. It allows us to calculate polarization dependent transmittance to each diffraction channel. Generalization of this formulation to other types of planar lattices (e.g., hexagonal lattice) is straightforward. Application of this formulation to a planar array of gammadion-shaped nanoholes in a gold film reveals that CD ellipticity in non-zero diffraction orders can be strong (up to 36.5°) and also that polarization conversion in non-zero diffraction orders can be effective. Indeed, in certain wavelength ranges, LCP incident light waves can be almost completely converted into RCP waves scattered into diffraction orders 2 2 2 n m + = and 2 2 0 n m + = by the planar array of left twisted gammadions. This interesting effect has been attributed to the interplay of radiation pattern of single chiral nanohole cell and diffraction pattern of a square lattice. Furthermore, it is found that significant power redistribution among different diffraction orders can be engineered by adjusting wavelength and polarization of incident light as well as period of metastructure and refractive indexes of surrounding media. These interesting findings suggest a novel route for developing optical devices made of planar chiral metafilms with controllable polarization state, spatial direction and intensity of outgoing light.