We propose a nano-polarization-converter made of a resonant L-shaped slot antenna in a gold film and study its optical properties using the finite-difference time-domain method. Phase retardation between the fast and slow axes of the nano-polarization-converter originates from the simultaneous excitation of both single-surface first-order magnetic plasmon resonance mode and second-order magnetic plasmon resonance mode at the working wavelength. By adjusting the size of the slot antenna, which is still much smaller than the wavelength, the working wavelength can be tuned within a large wavelength range.
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Plasmonics has achieved remarkable development over the last decade because of the exploration of the optical properties of metallic materials [1–3]. Localized surface plasmon (SP) resonance can be excited in metallic materials with subwavelength structures [4–6]. Different shapes of metallic nanostructures have been studied, including nanorods , square nanoparticles , bow-tie nanoparticles  and holes , cross-shaped holes [11,12], coaxial elliptical holes , L-shaped nanoparticles , and holes [15,16]. Most of these studies have focused on the enhancement of near-field intensity or far-field transmission.
Polarization state is a fundamental property of a light field. Metallic materials can reportedly control the polarization of light at the nanoscale. Biagioni et al. [17,18] used cross optical antennas to generate near-field polarization modulation. A circularly and elliptically polarized near field can be achieved by subwavelength apertures  and L-shaped hole arrays [15,16]. Drezet et al.  used an elliptical grating within a bull’s eye structure to obtain a plasmonic wave plate. Subwavelength nanoslits  and asymmetric crossed holes  were used to produce a quarter-wave plate. Coupled slit-hole resonator structures in a gold film were used by Zhu et al.  to fabricate a nanofilm half-wave plate. However, most of these studies have used combined asymmetric structures to tune resonances and generate phase retardation. Such tuning technique limits the working wavelength to a specific value and confers difficulty in experimentally preparing a sample.
In this paper, a nano-polarization-converter made of a resonant L-shaped slot antenna is proposed, and its optical properties are investigated using the finite-difference time-domain (FDTD) method. Interconversion between linear polarization and circular polarization in the far field is realized. Further study shows that the polarization rotation of the converter originates from the phase retardation between the single-surface first-order magnetic plasmon resonance (SFMPR) and the second-order magnetic plasmon resonance (MPR) excited in the L-shaped slot antenna. The influence of the thickness of the metal film on the performance of the antenna is also discussed.
2. Resonant modes of L-shaped slot antenna
A schematic of an L-shaped slot antenna is shown in Fig. 1(a) . An L-shaped hole made of two orthogonal rectangle holes is located in a gold film with a thickness of h = 200 nm on a silica substrate. The depth of the holes is the same as the thickness of the Au film. The arm lengths of the L-shaped slot antenna are equal and labeled L, whereas the arm width is labeled w. The x and y axes lie along the directions of the antenna arms. The origin is located at the tip of the hole on the top surface of the Au film. The m-k coordinate system has an angle of 45° with respect to the x-y coordinate system. A plane wave with an electric field amplitude of 1 V/m is incident from the bottom of the silica substrate along the + z axis. The angle between the polarization direction of the incident light and the x axis is labeled θ. The near-field resonant modes and the far-field radiation properties in the + z space are calculated using the FDTD method (FDTD solutions V7.5, Lumerical Inc., Canada). The permittivity of Au is set according to the data in Ref . The calculation region is a cubic box with dimensions of 1 μm × 1 μm × 1 μm. A cell size of 2 nm is used in the calculations and the surrounding medium is assumed to be a vacuum. The perfectly matched layer (PML) adsorbing boundary conditions are applied at the boundaries of the calculation region and the number of the PML is fixed at 12. The incident light excites the magnetic plasmon resonant mode on the slot antenna, and the polarization and phase of the scattering light depend on the resonant properties of the antenna.
Firstly, an L-shaped slot antenna with L = 360 nm and w = 140 nm is excited with the incident beam polarized along the m (θ = 45°) and k (θ = 135°) axes, respectively, considering the geometrical symmetry . The far-field transmitted intensity are calculated by recording the far-field electric field intensity |E|2 at the point F (x = 0, y = 0, and z = 1 m), as shown in Fig. 1(b). All resonant wavelengths are indicated for clarity. For a k-polarized incident beam (θ = 135°), only one resonant peak at 853 nm exists within the calculated wavelength range of 600−1600 nm. The corresponding magnetic field distributions on the top surface (z = 0 plane) and bottom surface (z = −200 nm plane) are shown in Figs. 2(a) and 2(b) respectively, which show a second-order MPR mode. Both on the two surfaces, the magnetic field distributions show two nodes along the hole. The polarization of the far-field radiation is the same as the incident beam, i.e. linearly polarized along the k-axis direction (θ = 135°). For the k-polarized incident light, the electric field in the far field at point F is labeled Ek and its phase is labeled φk.
For the m-polarized incident beam (θ = 45°), two resonant peaks at 1482 and 902 nm exist [Fig. 1(b)]. For the resonant peak at 1482 nm, the magnetic field distribution on the top surface (z = 0 plane) and bottom surface (z = −200 nm plane) are shown in Figs. 2(c) and 2(d), which indicate a first-order MPR mode. The magnetic field distributions show a node along the hole on both surfaces, i.e., the SPs on the two surfaces are in-phase. For the other resonant peak at 902 nm, the near-field magnetic field distributions on the top (z = 0 plane) and bottom (z = −200 nm plane) surfaces of the Au film are calculated and the results are shown in Figs. 2(e) and 2(f), respectively. On the top surface, the magnetic field exhibits dipolar oscillation along the hole [Fig. 2(e)]. However, on the bottom surface the magnetic field is concentrated at the corner of the Au film and does not exhibit dipolar oscillation [Fig. 2(f)]. The SPs on the two surfaces are out of phase, and only SPs on the top surface show first-order MPR.
Therefore, for the incident polarization along the m axis (θ = 45°) two kinds of first-order MPR can be excited: one is in-phase resonance that has been discussed in , and the other is out-of-phase resonance that has not yet been investigated. In this paper the latter is called the SFMPR mode, which means that SPPs on a single surface of the slot antenna is in resonance. Although the near-field distributions differ, the polarizations of the far-field radiation for the two resonant peaks are the same as the incident polarization. For the m-polarized incident light, the electric field in the far field at point F is labeled Em and its phase is labeled φm.
To study the influence of the substrate on the resonance wavelengths of the modes, we calculate another two cases in which no substrate exists (i.e., free-standing film) and the substrate is MgF2 (refractive index n = 1.38). The far-field transmitted intensity |E|2 is recorded with the change in wavelength, and the results are shown in Fig. 3 . When the refractive index of the substrate changes from 1.45 to 1.38, the resonant wavelengths and intensities vary slightly.
The resonant wavelength of the SFMPR is very close to the resonant wavelength of the second-order MPR [Fig. 1(b)] and the resonances are fairly broad. Both of them can be excited within the wavelength range of 860−910 nm. The SPs on the two surfaces of the Au film for the second-order MPR are in phase, which differs from that in the case of the SFMPR mode. The different resonant characteristics result in phase retardation Δφmk = φm − φk in the far-field, which is used to implement the polarization rotation of the incident beam in this work. The far-field electric field amplitude ratio |Em|/|Ek| and the phase retardation Δφmk are calculated with respect to the incident wavelength for the L-shaped slot antenna with L = 360 nm, w = 140 nm, and h = 200 nm, and the results are shown in Fig. 4 . With varied incident wavelength from 860 nm to 910 nm, the phase retardation Δφmk is 90° ± 2° and the far-field amplitude ratio |Em|/|Ek| is 1.00 ± 0.05. If we choose the m axis as the fast axis and the k axis as the slow axis, the L-shaped slot antenna can act as a polarization converter with phase retardation Δφmk = 90° between the fast and slow axes. Consequently, the L-shaped slot antenna acts as a nano-polarization-converter with the m axis as the fast axis and the k axis as the slow axis.
3. Polarization transformation using the nano-polarization-converter
To demonstrate that the proposed nano-polarization-converter can realize polarization transformation, the far-field electric field amplitude ratio |Ex|/|Ey| and the phase retardation Δφxy = φx−φy at point F are calculated with respect to the incident polarization angle θ and the results are shown in Fig. 5(a) . The incident wavelength is kept at 880 nm, at which the L-shaped slot antenna (L = 360 nm, w = 140 nm, and h = 200 nm) is in resonance for different incident polarizations [Fig. 1(b)]. The red solid line in Fig. 5(a) represents the relationship between the phase retardation and incident polarization angle Δφxy = 2(θ − 45°) for an ideal quarter-wave plate. The calculation results indicate that the electric field amplitude ratio |Ex|/|Ey| stays near 1.0 within the entire range, whereas the relationship between the phase retardation Δφxy and θ is fairly coincident with that in an ideal quarter-wave plate. When the incident polarization angle is set to 0°, the phase retardation between Ex and Ey in the far field is −90°, whereas the amplitude ratio |Ex|/|Ey| is close to 1.0. Thus the far-field radiation is right-circularly polarized. When the incident polarization direction is along the m or k axes (θ = 45° or 135°), the far-field radiation is linearly polarized, which is the same as the incident field. Except at θ = 0°, 45°, 90°, and 135°, the far-field radiation exhibits an elliptical polarization state.
In addition to the transformation from linear polarization to circular polarization, the reverse process from circular polarization to linear polarization can also be realized using the proposed nano-polarization-converter. We set the incident wave left circularly polarized and calculate the transmitted field Ex and Ey on a sphere 1 m in radius and whose center is located at the origin within a solid angle γ ranging from 0° to 50° with respect to the z axis. The results are shown in Figs. 5(b) and 5(c). The amplitude of Ex is much larger than that of Ey, and the ratio |Ex|/|Ey| reaches 22 at point F in the far field. Therefore the far-field radiation can be seen as linearly x polarized. Similar results exist when the incident plane wave is right circularly polarized and the amplitude ratio |Ey|/|Ex| is also 22 at point F in the far field. Consequently, transformation from circular polarization to linear polarization can be implemented using the proposed nano-polarization-converter.
4. Influence of the Au film thickness on the performance of the nano-polarization-converter
The resonant properties of the L-shaped slot antenna depend on the Au film thickness. The influence of h on the transmission spectra is investigated by calculating the far-field transmitted intensity |E|2 with the change in h. The results are shown in Fig. 6 for different incident polarizations. The length L and width w of the L-shaped slot antenna are kept at 360 and 140 nm, respectively. For the incident polarization along the m axis (θ = 45°), no significant peak corresponds to the SFMPR mode within the calculated wavelength range unless h ≥ 200 nm [Fig. 6(a)]. The increase in h results in a red shift of the resonant wavelength and in increased intensity at the resonant wavelength. The reason is that a higher h leads to weaker coupling between the SPs on the top and bottom surfaces of the L-shaped slot antenna and stronger SFMPR mode excitation. Meanwhile the effective refractive index of the L-shaped slot antenna decreases and the resonant wavelength redshifts with increased h. In addition, the influence of the thickness h on the second-order MPR is also studied and the far-field transmitted intensity is shown in Fig. 6(b). For the incident polarization along the k axis (θ = 135°), the coupling between the two surfaces of the slot antenna is weakened and the intensity of the resonance is reduced with increased h . Meanwhile, with increased h, the resonant wavelength blue shifts because the cross-sectional area S of the hole (S = w × h) increases and the slot antenna becomes less oblate .
The far-field electric field amplitude ratio |Em|/|Ek| and the phase retardation Δφmk of the L-shaped slot antenna (L = 360 nm and w = 140 nm) are calculated with the change in h and the results are shown in Fig. 7 . To make the polarization converter work in resonance, we set the incident wavelengths at the resonant wavelengths of the antenna under y-polarized excitation (θ = 90°) for each value of the thickness h. When h is small, the resonant mode excited by y-polarized excitation is only the second-order MPR mode, which results in a small Δφmk. The SFMPR mode strengthens with increased h. The phase retardation Δφmk grows and can even reach almost 180°. When h = 200 nm, the phase retardation Δφmk is 90° and the amplitude ratio |Em|/|Ek| is near 1.0 at the resonant wavelength of 880 nm under y-polarized excitation. Thus the left circularly polarized radiation is obtained in the far-field. Notably when h = 400 nm, the far-field electric field has a phase retardation of 180° ± 10° within the wavelength range of 850−1100 nm between the m- and k-axis directions. At a wavelength of 860 nm the amplitude ratio |Em|/|Ek| is near 1.0 and Δφmk = 178°, which indicates the L-shaped slot antenna can realize the function of a half-wave plate.
Apart from the thickness h of the Au film, the effective refractive index of the L-shaped slot antenna also depends on the antenna arm length L and width w. A variation in L or w within the range of ± 10 nm leads to a slight change in the working wavelength λw of the polarization converter. To obtain a large variation in the working wavelength, L and w should be simultaneously adjusted. A wide range of the working wavelength λw is obtained by carefully varying L and w while maintaining the thickness h = 200 nm, as shown in Table 1 . We can see that the L-shaped slot antenna can act as a nano-polarization-converter within the wide wavelength range of 780−980 nm, which makes it very useful in the practical applications.
Compared with other combined asymmetric structures proposed for polarization conversion [21–23], the L-shaped slot antenna proposed in this paper is an isolated structure that is easy to be fabricated experimentally. In addition, the working wavelength of the nano-polarization-converter can be extensively tuned from 780 nm to 980 nm by changing the arm length and width of the antenna. The phase retardation produced by two resonant modes can be increased by increasing the metal film thickness and even the function of a half-wave plate can be realized. When working as a wave plate, an L-shaped slot antenna array should be used to increase the transmitted intensity.
In summary, a nano-polarization-converter made of a resonant L-shaped slot antenna is proposed, and its use is demonstrated. The resonant mode of the polarization-converter is a hybrid of the SFMPR and second-order MPR modes of the L-shaped slot antenna. Changing the size of the slot antenna can enable a large range of working wavelength to be obtained. The thickness of the Au film evidently influences the optical properties of the antenna. At 400 nm thickness, even the function of a half-wave plate can be realized at 860 nm wavelength. Given the subwavelength size of the proposed nano-polarization-converter, it has promising applications in nano-photonics and plasmonic devices.
This work was supported by the National Natural Science Foundation of China under Grants 61036005 and 11074015 and the National Basic Research Program of China under Grant 2009CB623703, MOST.
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