1. Introduction

Efficient and fruitful manipulation of the polarization state of light is highly desirable in practical applications. One of the most conventional polarization manipulation device is the polarizer—an optical filter that only the light of a specific polarization can pass through, e.g., linear polarizer or circular polarizer. Metamaterials [1–5] are composed of periodic arrays of subwavelength artificial atoms, the electromagnetic properties of which are mainly governed by their specific structures and corresponding resonances. Metamaterials can be arbitrarily tailored for complex electromagnetic responses, including huge anisotropy, bianisotropy, chirality, and magneto-electric coupling, making meatamaterials good candidates for efficient polarization manipulation [6]. Recently, it is seen a considerably increasing interest on the polarization related exotic optical phenomena and polarization manipulation using metamaterials [7–24]. Starting from asymmetric transmission only for circular polarization [7], polarization related phenomena in metamaterials have been widely expanded for light of both circular and linear polarization, and theoretically and/or experimentally demonstrated from microwave to terahertz and even optical frequencies [8–14]. In the area of polarization manipulation using metamaterials, great interest has been paid on the realization of polarization rotators [15–17] and circular polarizers [18–24] of subwavelength thickness—a property hardly be available when using ordinary anisotropic materials. However, the goal to pursue a multiple operation band circular polarizer with single resonator metamaterial structure is still challenging. Such a polarizer with a compact size is very useful and urgently required for integrated optics systems.

In this paper, we propose a simple hybridized metamaterial to realize a dual-band unidirectional circular polarizer. Interestingly, this metamaterial polarizer shows opposite handedness polarization filtration at the two operational frequency bands, which ensures its applications for both left-handed and right-handed light. Analytical models are employed for capturing the main optical responses around each resonance, which are in good accordance with numerical simulations, evidently verifying the circular polarizer performance with nearly uniform contrast ratio. Our design is impressive to expand the exotic functionality of existing metamaterials and is sure to find extensive applications in polarization manipulation systems.

2. Requirements and design

We start by considering the criterion for realizing a unidirectional circular polarizer as depicted in Fig. 1(a). For sake of simplicity, we focus on the transmission process under normal incidence. The incident and transmitted light are related by a (2 × 2)-dimensional Jones matrix T. Considering an incident light E₀ propagating along the +z direction, the Jones matrix in the circular polarization basis can be simply described by

 

Fig. 1 (a) Schematic view of metamaterial circular polarizer under normal incidence and (b) unit cell of the proposed metamaterial design. Inset shows geometric details of the unit cell. LH(RH) indicates the left(right)-handed circular polarization incidence.

Download Full Size | PDF

According to the Jones matrix, a left-handed circular polarizer which blocks the light with right-handed circular polarization, can be realized by

Similarly, the opposite handedness circular polarizer can be satisfied by

When the propagation direction is reversed for a system satisfying reciprocity theorem, the off-diagonal elements t_xy and t_yx not only interchange their values but also get an additional π phase shift. Asymmetric factor Δ, representing the degree of asymmetric transmission, is defined as the difference between the transmittances in the two opposite propagation directions (+z and −z). For the linear polarization case (x or y), we have Δ_lin = ||t_yx|² − |t_xy|²|, while for the circular polarization case, Δ_circ = ||t₊₋|² − |t₋₊|²|. To characterize the circular polarizer performance, we also introduce the visibility factor ${\mathrm{\Lambda }}_{\pm z}=\frac{{\left|t_{++}\right|}^2\pm {\left|t_{-+}\right|}^2-{\left|t_{--}\right|}^2\mp {\left|t_{+-}\right|}^2}{{\left|t_{++}\right|}^2+{\left|t_{--}\right|}^2+{\left|t_{+-}\right|}^2+{\left|t_{-+}\right|}^2}$ to indicate the degree of polarizer for incident light from the +z and −z directions. For the special solution satisfying the requirements of Eq. (2), the asymmetric factor is Δ = |t_xx|² for both linear polarization and circular polarization. There indeed exists asymmetric transmission for both circular and linear polarization, and more importantly, the asymmetric factors are equal. The visibility Λ_+z is 1 for positive direction and Λ_−z is zero for negative direction, implying that the system can be regarded as a unidirectional circular polarizer. For the special solution satisfying the requirements of Eq. (3), Λ_+z = −1 and Λ_−z = 0 indicate the system can be regarded as the opposite handedness unidirectional circular polarizer. If the optical response of investigated metamaterial can be fulfilled by the special solutions satisfying the requirements Eq. (2) and Eq. (3) at different frequency bands, a dual-band circular polarizer with opposite handedness can be definitely realized.

As an example, we propose a metamaterial design to realize such a unidirectional circular polarizer. As sketched in Fig. 1(b), the top layer is a metallic z-shaped antenna with thickness t_t = 30 nm, width w_t = 50 nm, length a = 180 nm, and arm length b = 150 nm; the middle layer is glass with thickness t_d = 30 nm; the bottom layer is a continuous metallic strip with width w_b = 120 nm and thickness t_b = 100 nm. Worth noting that no planar chirality exists for each single metallic layer as shown in Fig. 1(b). However, strong chirality can be achieved by hybridizing the top and bottom layers through near-field coupling, which is critical for the realization of a circular polarizer. The periodicity d of the unit cell is only 250 nm, which guarantees no high-order diffraction mode exists below 1200 THz. The dielectric substrate we utilized is glass with permittivity ε_d = 2.16. The permittivity dispersion of gold follows a Drude model, ${\epsilon }_{\text{m}}={\epsilon }_{\infty }-f_{\text{p}}^2/(f^2+i{\gamma }_{\text{p}}f)$, where f_p = 2.18 × 10¹⁵ Hz, γ_p = 1.62 × 10¹² Hz, and ε_∞ = 9 [25].

3. Results and discussion

In the wide frequency range from 50 to 450 THz, we vary the geometric parameters of the unit cell and calculate the Jones matrix elements with rigorous finite-difference time-domain (FDTD) method. With the optimized geometric parameters defined in Fig. 1(b), the hybridized metamaterial shows two obvious resonance modes as shown in Fig. 2. The Jones matrix elements t_yy and t_yx preserve small amplitude, therefore the optical performance of the investigated metamaterial is mainly dominated by t_xx and t_xy. Around each resonance, t_xx is mainly responsible for the dipole oscillation along the x direction, while t_xy originates from the projection to the x direction from the dipole oscillation along the y direction for the top layer. Thus, we can develop a simple analytical model to understand the physics behind the exotic response of our proposal near each resonant frequency.

 

Fig. 2 Amplitudes of the Jones matrix elements for the hybridized metamaterial along the forward (+z) direction in the linear polarization base.

Download Full Size | PDF

The Jones matrix element t_xx around the n-th resonance (n = 1, 2) can be written by $t_{xx}=\left[(f_{n,xx}-f)+i({\gamma }_{n,xx}^{\text{s}}-{\gamma }_{n,xx})\right]/(f_{n,xx}-f-i{\gamma }_{n,xx})$, where f_n,xx is the resonant frequency, γ_n,xx is the total damping width, and ${\gamma }_{n,xx}^{\text{s}}$ is the scattering width. The off-diagonal Jones matrix element t_xy around the n-th resonance (n = 1, 2) is in the form of $t_{xy}=i{\gamma }_{n,xy}^{\text{s}}/(f_{n,xy}-f-i{\gamma }_{n,xy})$, where f_n,xy is the resonant frequency, γ_n,xy is the total damping width, and ${\gamma }_{n,xy}^{\text{s}}$ is the scattering width. The fitting parameters of the 1st resonance are f_1,xx = 173.2 THz, γ_1,xx = 30 THz, ${\gamma }_{1,xx}^{\text{s}}=17\hspace{0.17em}\text{THz}$, f_1,xy = 184.4 THz, γ_1,xx = 20 THz, and ${\gamma }_{1,xx}^{\text{s}}=12\hspace{0.17em}\text{THz}$. The analytical model finds good agreement with the numerical results, as shown in Figs. 3(a) and 3(c). Around the 1st resonance, the nearly zero contribution from t_yy and t_yx, and f_1,xx < f_1,xy provide the feasibility to realize the special solution satisfying the requirement of Eq. (2). For the 2nd resonance, the fitting parameters are taken as f_2,xx = 387.2 THz, γ_2,xx = 50 THz, ${\gamma }_{2,xx}^{\text{s}}=35.6\hspace{0.17em}\text{THz}$, f_2,xy = 364.4 THz, γ_2,xy = 30 THz, and ${\gamma }_{2,xy}^{\text{s}}=19\hspace{0.17em}\text{THz}$. The good agreements between analytical model and numerical results are also observed in Figs. 3(b) and 3(d). Around the 2nd resonance, besides the nearly zero contribution from t_yy and t_yx, f_2,xx > f_2,xy provides the possibility to satisfy the requirement of Eq. (3), which can make the opposite handedness circular polarizer in contrast with the 1st resonance.

 

Fig. 3 Analytical (dashed curve) and numerical (solid curve) amplitudes of the Jones matrix elements t_xx and t_xy around each resonance.

Download Full Size | PDF

In Fig. 4, we show the Jones transmission matrix in the circular polarization base. The elements t₋₋ and t₊₋ are reduced to almost zero close to the frequency f_c1 = 168.4 THz while t₊₊ and t₋₊ exhibit transmission amplitudes around 0.5. It is clearly seen that the critical requirements of Eq. (2) are approximatively matched using the proposed metamaterial, directly revealing the potential application of this metamaterial as a circular polarizer. Interestingly, at frequencies close to the 2nd resonance, the transmission in the same metamaterial can satisfy the requirements of circular polarizer with an opposite handedness. The transmission elements t₊₊ and t₋₊ are reduced to near zero around the frequency f_c2 = 363.6 THz, while t₋₋ and t₊₋ exhibit high transmission amplitudes around 0.5, supporting the opposite handedness circular polarizer as described in Eq. (3).

 

Fig. 4 Amplitudes of the Jones matrix elements for the hybridized metamaterial along the forward (+z) direction in the circular polarization base.

Download Full Size | PDF

For better illustration of the performance of the metamaterial circular polarizer, we calculate the visibility Λ of this metamaterial in Fig. 5(a). It is seen that our metamaterial can be used as a unidirectional circular polarizer with good performance in transmission process, which is not a good candidate for reflection process. Besides this, our metamaterial also shows asymmetric transmission for both linear and circular polarized light, as seen in Fig. 5(b). The degree of asymmetric factor Δ for circular or linear polarization can be effectively manipulated using our simple metamaterial design. When considering to experimentally demonstrate the exotic performance to visible light, we need to optimize the metamaterial structure to obtain the desired performance, and take into account the influence of the substrate.

 

Fig. 5 (a) Visibilities of the circular polarizer for the positive (+z) and negative (−z) propagation directions. (b) Degree of asymmetric factor for circular and linear polarization base.

Download Full Size | PDF

4. Conclusions

In conclusion, we have presented a clear demonstration of dualband unidirectional circular polarizer using a simple hybridized metamaterial design. The developed analytical model, in good agreement with rigorous numerical simulation, reveals the main physics of the investigated metamaterial near each resonance. We believe that our proposal can be practically achieved, which has the promising applications in optical polarization manipulation and is highly valuable for the development of nanophotonic devices.