A Highly Efficient Plasmonic Polarization Conversion Metasurface Supporting a Large Angle of Incidence

Abstract

The angle of incidence of the compact polarization conversion device is crucial for practical use in integrated miniaturized optical systems. However, this index is often ignored in the design of quarter-wave plate based on metasurface. Herein, it is shown that a thick metallic cross-shaped hole array supports extraordinary optical transmission peaks controlled by a Fabry–Pérot (FP) resonator mode. The positions of these peaks have been proven to be independent over a large range of incidence angles. We numerically design a miniatured quarter-wave plate (QWP) with an 80 nm bandwidth (840~920 nm) and approximately 80% average efficiency capable of effectively functioning as a linear-to-circular (LTC) polarization converter at an incidence inclination angle of less than 30°. This angle-insensitive compact polarization conversion device may be significant in a new generation of integrated metasurface-based photonics devices.

1. Introduction

Polarization of light is an important branch of light that has an important place in many fields that cannot be ignored, such as photonic communications [1], satellite remote sensing [2], and bio-optical imaging [3]. The polarizer and the wave plate that provides a phase delay are two fundamental components in polarized optics. In conventional optics, researchers use the feature that birefringent crystals can support optical modes with different propagation speeds to modulate the phase difference, but unfortunately, this difference generally requires a huge size of the element in order to obtain a notable phase difference. Therefore, the search for an ultrathin wave plate fabrication technique is urgently needed for miniaturized and integrated optical systems [4].

A metasurface [5,6,7,8] is a periodic array of metallic or dielectric particles that has a wide degree of freedom to manipulate the phase and polarization information of light, and it can provide a compelling idea in the design of ultrathin wave plates [6,9,10,11,12,13,14,15,16,17,18,19]. In addition, when the metasurface focuses on achieving active metamaterial functionality, it attains greater degrees of freedom in terms of tunability as well as shaping and modulation of electromagnetic waves [20,21]. Numerous quarter-wave plates based on metasurfaces have been reported. Reflective metasurfaces [22,23,24,25,26], typically comprising a top metal particle array, a dielectric layer, and a metallic back reflection layer, can function as a QWP by optimizing the parameters on a large scale. The transmissive plasmonic metasurface has achieved efficient and broadband LTC polarization conversion by leveraging the anisotropy of its unit cell [27,28,29,30,31,32,33,34,35]. In 2012, Yu et al. [27] proposed an optically thin QWP based on metasurfaces consisting of an array of plasmonic antennas, which can produce high-quality circularly polarized light. In 2015, Li et al. [30] designed a linear-to-circular device consisting of arrays of gold nanorods, which owes its excellent performance to the tailoring of optical interference at subwavelength scales. In 2019, Wang et al. [34] presented an ultrathin quarter-wave plate based on a 2 × 2 rectangular array of air holes, where the transmittance can be increased to 0.44 by the diffraction enhancement effect of small holes. In 2021, Song et al. [35] designed a plasmonic QWP with a large bandwidth of 600 nm based on a Fabry–Perot (FP) resonator, which shows great potential for applications in miniaturized optical polarization detection systems. However, all the above devices only discuss the case of the normal incidence of light and do not analyze the optical mode of the device under the oblique incidence.

In this paper, we numerically simulate a metal quarter-wave plate made up of a gold film etched with a cross-shaped air hole array with an 80 nm bandwidth (840~920 nm) and about 80% average efficiency. The metasurface supports the FP resonance mode [36,37,38,39,40], which can surpass the theoretical limit of 50% transmittance for thin plasmonic quarter-wave plates and is insensitive to the incidence angle. The metasurface can maintain a good function of converting the linearly polarized light into the circularly polarized light under an incidence inclination angle of less than 30°. This angle-insensitive quarter-wave plate may be crucial for practical use in integrated miniaturized optical systems.

2. Materials and Methods

Figure 1a shows the free-standing structure with a gold cross-shaped hole array, and the upper and lower media of the gold grating are air. The electric field of the incident light is projected in two orthogonal directions, denoted as ‘o’ and ‘★’. The incidence inclination angle is θ. The azimuthal angle is set as 45° to ensure equal projection values of the electric field of the incident light in the ‘o’ and ‘★’ directions. The optical constants of gold are sourced from the reference [41]. Meanwhile, the geometry of the unit cell of the metasurface is shown in Figure 2b. Line A is the red dashed line in the xy plane, which makes an angle of 45 degrees with the x axis. The plane of incidence is formed by the line A and the z axis, and the angle of incidence θ is the angle between the incident light and the z axis, and the electric field is in the plane of incidence. When the angle of incidence becomes 0 degree, the direction of incidence is the direction opposite to the z axis, ‘o’ becomes the positive direction of the x axis, and ‘★’ becomes the positive direction of the y axis. The full-wave numerical simulation model, referred to as the optical model A, utilizes the commercial software COMSOL Multiphysics 5.6 to calculate the intensity and phase information for the ultrathin QWPs. Meanwhile, a periodic boundary condition, which belongs to the Floquet mode, is applied in the x and y directions of the model to truncate the electromagnetic far field. A Perfectly Matched Layer (PML), which absorbs the reflected electromagnetic waves, is applied in the z direction of the model to reduce the simulation area. In addition, in the COMSOL model A, the physical field interface is The Electromagnetic Waves, Frequency Domain Interface, and the numerical method is The MUMPS Solver in the Linear System Solver Handbook of COMSOL. The distance from the incident port to the metasurface is one wavelength. The maximum grid size for gold is 20 nm, and the maximum grid size for the space medium is the free-space wavelength divided by the refractive index divided by 8. Typical wavelength options in PML are vacuum wavelength divided by the cosine of the angle of incidence.

3. Results

3.1. Optical Mode Analysis

As shown in Figure 2a, a two-dimensional numerical model, referred to as the optical simulation model B, is utilized to analyze the characteristics of propagation modes, which are supported by an isolated plasmonic waveguide (IPW) consisting of cross-shaped air holes and a gold coating extending infinitely in xy plane. In this model, a length of five wavelengths that is far beyond the skin depth of gold is used to replace the optical infinity. The Perfect Electric Conductor (PEC) boundary condition is applied at the outer boundary to truncate the simulation region. The mode analysis command in COMSOL is then utilized to solve the eigenvalues of the waveguide equation. Considering the metasurface as a two-dimensional array of weakly coupled IPWs is beneficial for analyzing its phase difference spectrum. Figure 2b displays the real part of the effective refractive index of the propagation mode of the IPW under x- and y-polarized normal incident light, respectively. Figure 2c illustrates the imaginary part of the effective refractive index, which characterizes the attenuation characteristics of the waveguide modes. Moreover, the lower cutoff wavelengths of the attenuation TE and TM modes are 890 nm and 1000 nm, respectively. The phase difference of the plasmonic metasurface (Δϕ) depends on the equivalent refractive index of the optical material in two orthogonal directions, and it satisfies the following:

$$\Delta \varphi =(n_x-n_y)k_{0}d$$

(1)

where $n_x$ and $n_y$ are the real parts of the effective refractive index for two orthogonal waveguide modes, respectively, d is the thickness of the metasurface, and $k_0$ is the vacuum wave vector. Figure 2d depicts the phase difference spectrum of the devices. The red line represents the phase difference data extracted from the full-wave numerical simulation model A under normal incidence. The black line refers to the phase difference calculated using Equation (1), where $n_x$ and $n_y$ are obtained from Figure 2b. A good agreement between the two curves occurs in the 750 to 930 nm wavelength range, indicating that the phase manipulation of the metasurface can be explained by the propagation mode calculated by model B. However, the deviation in the two curves in the 930 to 1200 nm wavelength band is attributed to the severe ohmic loss of the plasmon waveguide.

Figure 3a shows the transmission peaks of the plasmonic metasurface in the near infrared band under the TM and TE normal incidence, respectively. The peak intensities of these transmission peaks exceed 75%, which is significantly higher than the theoretical efficiency limit of 50% for the thin plasmon quarter-wave plate [42,43]. To gain insight on the primary channels involved in the transmission mechanism, it is necessary to have access to electromagnetic near-field distribution diagrams of the device. Figure 3b−e show the electric field intensity of the metasurface at x–y and x–z cross-sections at the transmission peak wavelength. As shown in Figure 3b,d, the electric field is mainly localized in the vertical rectangular cavity for the TE incidence mode. A very strong local enhancement of the electric field occurs at the four points produced by the overlap of two rectangular cavities. Figure 3c,e show the standing waves in the narrow rectangular cavity in the z direction at 816 nm and 890 nm wavelengths. Meanwhile, we speculate that these two significant transmission peaks can be explained by the typical FP resonance cavity mode with m = 0, 1, respectively. These strongly localized modes in the plasmonic metasurface configuration consisting of the cross-shaped air holes and gold coating and satisfy the following condition:

$$n_{eff}{k}_0{h}_1+{\Phi }_R=m\pi$$

(2)

where m is an integer,$n_{eff}$ is the effective refractive index, and ${\Phi }_R$ is an additional phase produced at each opening. For the TE incidence mode, $n_{eff}$ corresponds to the black line in Figure 2b. It is very easy to obtain the value of ${ \Phi }_R$ of different orders (m = 0, 1) through Equation (2), and ${\mathsf{\Phi }}_{\mathrm{R}}$ is −0.6224 rad and −1.0121 rad for $\mathsf{\lambda }$ = 890 nm and 816 nm, respectively. Figure 3f–i show the distribution of electric field intensity in the TM mode incidence. The physical origin of the strong resonance mode in the metasurface in the optical regime in the TM mode incidence is the same that gives rise to the strong confinement phenomenon of the electrical near-field and high transmittance in the case of TE mode incidence.

3.2. Phase Difference

The origin of the transmission peak can also be verified by a simple inspection of the height-scanning experiment. In the previous part of the text, we assume that the resonance peak of the metasurface is controlled by the FP resonance satisfying Equation (2). When the thickness of the device increases gradually, there will be a higher-order FP resonance mode. Figure 4a,b show the relationship between metasurface thickness and intensity (transmission and absorption) at normal incidence wavelengths of 816 nm and 890 nm for TE polarization. The transmittance results calculated by model A are depicted by the solid line. The vertical blue line is derived from Equation (2), where ${\Phi }_R$ corresponding to each transmission peak is derived from the previous paragraph. As depicted in Figure 4a, the blue line coincides with the transmission peaks, which further substantiates the rationality of the existence of the FP resonance mode. However, the higher-order FP modes at 890 nm resonance wavelength are absent in Figure 4b. This anomaly can be attributed to the cutoff wavelength of the attenuation mode caused by the ohmic loss. In a nutshell, the combined effects of long propagation length and loss characteristics of waveguide mode lead to a sharp increase in device absorptivity, resulting in the disappearance of higher-order transmission peaks. Figure 4c,d show the impacts of device thickness on transmittance at wavelengths of 888 nm and 1020 nm for the TM mode incidence. The conclusions drawn from Figure 4c,d are similar to those observed for the TE mode incidence. Even if the ohmic loss may cause the resonant to disappear, the FP mode still manipulates the optical properties of the metasurface with a finite thickness.

The transmission of the device versus both wavelength and the sine of the incidence angle are shown in Figure 5, for different orientations of the incidence angle and electric field. The corresponding computed band diagram of the FP resonance modes in the plasmonic metasurface are depicted with discrete point symbols. The most distinguishing feature of the periodic array of the considered cross-shaped holes is that transmission peaks based on the strongly localized cavity depend weakly on the size of the incidence angle. The pentagram dots represent the zero-order FP cavity mode, whereas the high-order mode without transmission loss is rendered with the other dot symbol. The green dotted line existing only in Figure 5a,c represents surface plasmon polaritons (SPPs), which conform to the polarization selection rule of the SPP mode [44,45,46]. The dispersion relation of the SPPs between air and a gold layer is $\mathsf{\beta }=k_0\sqrt{{\displaystyle \frac{{\epsilon }_1{\epsilon }_2}{{\epsilon }_1+{\epsilon }_2}}}$, where ${\epsilon }_1$ and ${\epsilon }_2$ represent the electric permittivity of dielectric and metal material, respectively, and $k_0$ is the free space wave vector. The coupling effect between the FP resonance modes and the SPPs at a large tilt angle damages the angle insensitivity of the device, especially for higher-order modes. This is corroborated by the calculations, and the bending of the photonic band of the m = 1 order mode occurs when the incidence angle is greater than 30 degrees.

4. Discussion

In the preceding sections, the optical properties of the cross-shaped hole configuration have been thoroughly studied under TE and TM incidence, and it emphasize the angle insensitivity of the waveguide mode that supports the construction of a compact plasmonic wave plate. In this section, we describe a transmissive plasmonic device utilizing the nanostructure dimensions shown in Section 2 with an average efficiency of 80%, and it exhibits the characteristics of an optical QWP with a weak incidence angle dependence for the near infrared band. Figure 6a illustrates the dependence of the intensity of transmission peaks 1 to 4 on the incidence angle. There is almost no change in the transmission peak 4, and the central wavelength and intensity of No. 1 to No. 3 transmission peaks do not change obviously when the incidence angle is small (θ < 30 degree). The phase difference (Δφ) between the components of the transmitted light ray colliding with metasurfaces should comply with the requirement of 90°± 10° for an excellent quarter-wave plate. Under the normal incidence condition and within the ±10° degrees tolerance rule, the device exhibits an effective bandwidth of 80 nm (840 to 920 nm), as shown in Figure 6b. However, when the incidence angle in the model is larger than 30 degrees, the effective bandwidth is sharply reduced to 15 nm.

An alternative path to characterize the figure of merit of the plasmonic QWP is by utilizing the ellipticity angle ξ and ellipticity χ for the 45° linear polarized incident light with different incidence angles:

$$\xi =0.5arcsin{\displaystyle \frac{2\left|E_x\right|\left|E_y\right|sin(\Delta \phi )}{{\left|E_x\right|}^2+{\left|E_y\right|}^2}}$$

(3)

$$\mathsf{\chi }={\displaystyle \frac{\left|E_x\right|}{\left|E_y\right|}}$$

(4)

From Figure 6c, it can be seen that ξ approaches 45° within the wavelength range from 840 nm to 920 nm for incidence angles ranging from 0 to 30 degrees, indicating that the transmitted light is the circularly polarized light. Figure 6d shows that the sharp decrease in the effective bandwidth of the plasmonic quarter-wave plate occurs when the incidence angle exceeds 30 degrees. These results further demonstrate that the metasurface can convert the linearly polarized light into the circularly polarized light when the incidence angle is in the range of 0 to 30 degrees.

For an acceptable QWP, it should also convert the circularly polarized light into a linearly polarized wave. The angle of linear polarization (AoLP) and the degree of linear polarization (DoLP) of device defined by the Stokes parameters can fully describe the polarization characteristics of linearly polarized light:

$$DoLP={\displaystyle \frac{\sqrt{{s_1}^2+{s_2}^2}}{s_0}}$$

(5)

$$AoLP=0.5\times {\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left({\displaystyle \frac{s_2}{s_1}}\right)$$

(6)

where $s_0={\left|E_x^T\right|}^2+{\left|E_y^T\right|}^2$, $s_1={\left|E_x^T\right|}^2-{\left|E_y^T\right|}^2$, and $s_2=E_x^T{\left(E_y^T\right)}^{\mathrm{*}}+E_y^T{\left(E_x^T\right)}^{\mathrm{*}}$. Here, ‘*’ represents the complex conjugate operator dealing with complex numbers, and superscript ‘T’ is the electric field in transmission domain, and the subscripts ‘x’ and ‘y’ correspond to the projection component of the electric field in two orthogonal directions. As shown in Figure 6a,b, the DoLP of the transmitted light ray in the entire operation bandwidth (840~920 nm) for the circularly polarized incident light under an incidence angle of less than 30° is bigger than 0.92, and its polarization characteristic belongs to linear polarization. It is seen that the AoLP with respect to the wavelength within the bandwidth under an incidence angle of less than 30° is insensitive. The AoLP is an almost invariant constant at $\pm 45°$, whose variation is below $5°$. A quantitative comparison is necessary by extracting the average errors for $S_1$, $S_2,$ and $S_3$. The errors for the degree of linear polarization (DoLP) and the degree of circular polarization (DoCP) are defined as ${\left|\right|\sqrt{S_1^2+S_2^2}/S_0-\sqrt{D_1^2+D_2^2}/D_0\left|\right|}^2$ and ${\left|\right|S_3/S_0-D_3/D_0\left|\right|}^2$, respectively. The ($S_1,S_2,S_3$) is the full Stokes parameters of the transmitted light ray obtained by utilizing the COMSOL numerical simulations, and ($D_1{,D}_2,D_3$) is the theoretical Stokes parameter defined as (0, 1, 0) and (0, −1, 0) for the different circular polarized light, respectively. We found that the average errors of DoLP and DoCP are lower than 0.008 and 0.002, respectively, at operation bandwidths of 840~920 nm under an incidence angle of less than 30° for the circular polarized incident light, as shown in Figure 7c,d. Meanwhile, we have created a table to strengthen the connection with the previous research. As shown in

Table 1. Comparison of the characteristics of QWPs.

Structure Design	Wavelength (nm)	Efficiency	Maximum Incidence Angle (deg)
Metal grating [32]	1400	0.9	~10
Ag bars [47]	1250	~8.5	6
Rectangular hole [34]	1500	~0.4	NA
Broken annular [31]	1550	~0.4	NA
U-shaped [38]	1600	NA	<10
Proposed	880 nm	~0.8	30

, the QWPs we designed support the largest range of incidence angles while maintaining high efficiency.

5. Conclusions

In conclusion, a numerically achievable gold film with a cross-shaped air hole array supporting the FP resonance mode in the near-infrared band is considered. The FP resonance mode has two main characteristics: (1) breaking the theoretical limit of 50% transmittance of thin plasmon quarter-wave plate and (2) insensitivity to the incidence angle. Based on the above two characteristics, we numerically designed a high-quality plasmonic device with an 80 nm operation bandwidth (840~920 nm) and about 80% average efficiency, which can maintain the effective conversion of linearly polarized light into circularly polarized light under an incidence inclination angle of less than 30°. This angle-insensitive compact device may be helpful for practical application in integrated miniaturized optical systems.