Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials

Rui Xia; Xufeng Jing; Xincui Gui; Ying Tian; Zhi Hong

doi:10.1364/OME.7.000977

1. Introduction

Polarization conversion can be applied in many areas, such as antennas, astronavigation, and communication [1, 2]. Thus, manipulating the polarization state of electromagnetic (EM) wave by polarization converters is always highly desired for researchers to efficiently control EM wave. Over the past decade, metamaterials [3], the artificial composite structures or composite materials have attracted great attention due to exotic physical properties compared with naturally occurring materials. Metamaterials have been proposed to apply in various fields, such as negative refractive index [4], perfect lens [5], and anomalous refraction/reflection [6, 7]. The traditional methods using optical activity crystals with birefringence effect always suffer from the disadvantages of narrow response frequency band and difficulties in optical system integration. Fortunately, metamaterials have opened a new way to manipulate polarization state with the advantage of ultrathin and broadband performances [8–11].

Generally, there are two kinds of metamaterial structures used for polarization conversion, anisotropic metamaterials and chiral metamaterials. Similar to the birefringent crystals, the different phase changes in two orthogonal direction of incident EM wave make the anisotropic metamaterials possible to realize polarization rotation, and the phase difference can be controlled by metamaterials design [11–14]. Other than anisotropic metamaterials, chiral metamaterials have different equivalent refractive index for right- and left-handed circularly polarized EM wave in two orthogonal directions [15–21], which leads to an extensive application in polarization conversion such as polarization rotation [15–17], linear-to-circular polarization conversion [20], and left to right-handed or right to left-handed circular polarization conversion [21]. However, it is difficult for these two kinds of metamaterial structures to keep high conversion efficiency in a wide bandwidth. Fortunately, the operating frequency band can be broadened by stacking multi-layer metamaterial structures, which has unique resonance in each layer at neighboring frequency bands. The coupling of these resonances realizes a broadband response [22, 23]. However, the broadband response is achieved at the price of bulkier devices, as multi-layer structures prejudice the device integration and complicate the fabrication process. In addition, several reports attribute polarization conversion characteristic of metamaterial structures to the plasmon resonances which are always be excited on the metal film with array of holes [10–12, 24]. Thus, these devices usually have lower transmittance, which means low efficiency in transmission mode.

Actually, some reflected metamaterial devices are researched mainly on the basis of the microstrip reflect-array antennas [25–32]. Generally, the microstrip reflect-array antennas are composed of array of elements with predetermined phase shift, which acts as a phase array to manipulate wavefront of the incident wave. Reflect-array antenna was first proposed to realize waveguides [25], but it was relatively bulky and expensive to manufacture. The concept of the microstrip reflect-array was first conceived by Malagisi [26] and patented by Munson et al. [28]. Besides, it is known that the microstrip patch antennas on a dielectric substrate over a ground plane can be highly suited for reflect-array designs [31]. As a continuation of the microstrip reflect-array antennas, the reflected metamaterial devices are mostly designed with artificial structures on similar dielectric substrate over a metal ground plane to realize exotic physical properties.

In our study, to further enhance the polarization conversion efficiency and broaden the bandwidth response, we propose a reflection mode polarization converter through anisotropic metamaterials, which is composed of two pairs of metal patches in one unit cell and works as a half wave plate in the terahertz region. This device operates in a broadband frequency range of 0.67–1.66 THz with polarization conversion rate (PCR) more than 88%, and reaches nearly 100% at three polarization conversion peaks in the case of cross polarization conversion. And the deviation of polarization rotation angle is below 10° with arbitrary incident polarization direction in the broadband frequency range. Further simulations and calculations are carefully analyzed to clarify the physical mechanism of the polarization conversion.

2. Metamaterial design

The broadband half wave plate we proposed is composed of a metal ground plane and array of two pairs of patches separated by a dielectric spacer layer, as shown in Fig. 1. By coupling the top patchs and the metal ground plane, the polarization conversion efficiency can be greatly enhanced in reflection mode. The schematic diagram is illustrated in Fig. 1(a), this device can convert the polarization direction of linearly polarized wave to an arbitrary angle by rotating the azimuth of incident wave in reflection mode. The front view of one unit sell is shown in Fig. 1(b). This center symmetric structure is composed of two pairs of patches with the same size, and each pair is two connected patches. We define u axis and v axis to analyze the anisotropy of this structure, while the x axis and y axis are used to point out the polarization direction of EM wave for cross polarization conversion. The incident electric field E_i is linearly polarized in x direction (p-polarized) with an angle β = 45° with respect to the anisotropic axis u. In a more general sense, the half wave plate can rotate the polarization direction with an angle of 2β with respect to its incident polarization direction. This structure was designed and simulated based on the three-dimensional full wave simulations of the standard finite difference time domain method. In the simulation, both the patches and metal ground plane were modeled as a 400nm gold with an electric conductivity σ = 5.810⁷ S/m and the dielectric spacer layer was modeled as polyimide with a dielectric constant ε = 3.5(1 + 0.05i). The other geometrical parameters are as follows: P = 106μm, L = 44μm, d = 70μm, w = 10μm, t = 33μm, α = 5°, ββ = 45°, incidence angle θ = 0°. All of these geometric sizes are obtained by extensive simulation and optimization from a range of size, and the changing of the sizes will lead to narrowband or lower efficiency.

Fig. 1 (a) Schematic diagram of the broadband half wave plate, where yellow stands for gold and cyan represents polyimide. And the incidence angle is θ. (b) The front view of one unit sell of the broadband half wave plate, x and y axes are used to analyze the cross polarization conversion, while u and v axes are used to mark the metamaterial anisotropic axes.

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3. Polarization conversion of metamaterial

In this work, we take the cross polarization conversion as an example to analyze the performance and physical mechanism of this half wave plate. In order to better reveal the polarization conversion, we define

R_{k j} = I_{k}^{r} / I_{j}^{i} = {(E_{k}^{r} / E_{j}^{i})}^{2} = r_{k j}^{2}

as the reflectance of incident wave to represent the ratio of converted energy and incident energy, which indicates the efficiency of the polarization converter. Here I indicates the energy of EM wave, E indicates the amplitude of EM wave, and

r_{k j}

is the reflection coefficient of incident EM wave, can be defined as

r_{k j} = E_{k}^{r} / E_{j}^{i}

. The superscripts i and r indicate incident and reflected EM wave, and the subscripts j and k indicate the polarization state of EM wave. Moreover, polarization conversion ratio (PCR) is usually used to describe conversion efficiency of the incident wave, which can be defined as:

P C R^{x} = R {}_{y x} / (R_{y x} + R_{x x})

P C R^{y} = R {}_{x y} / (R_{x y} + R_{y y}),

where the superscripts x and y indicate the polarization state of incident wave.

We simulated both x-polarized and y-polarized normal incident wave, as shown in Fig. 2. The reflectance of incident wave is shown in Fig. 2(a), both x-polarized and y-polarized incident wave can be converted to its cross polarization with a broadband and high-efficiency performance. The cross-polarized reflection carries more than 70% energy of incident power in the range of 0.67 $-$ 1.31 THz, and the co-polarized component is mostly below 8% from 0.92 to 1.6 THz. Between 0.62 and 1.6 THz, cross-polarized reflectance is higher than 50%, and the co-polarized one is mostly below 12% in the range of 0.64 $-$ 1.7 THz. This represents a broadband and high performance linear polarization converter. The broadband response results from the superposition of polarization conversion peaks around 0.71, 1.1, and 1.49 THz. Figure 2(b) shows the PCR of the incident wave, which indicates the half wave plate operate at a high conversion efficiency with both x-polarized and y-polarized incident wave. Between 0.64 and 1.67 THz, PCR is higher than 85% and reach nearly 100% at 0.69, 1.1, and 1.57 THz, where most incident wave is converted to its cross polarization with the rest loss for the metal loss and dielectric loss. The three polarization conversion peaks is achieved by the resonance of the regular quasi-TEM modes of a microstrip patch [32]. As Fig. 2 shown, the broadband half wave plate can achieve the similar performance with both x-polarized and y-polarized normal incident wave, which is caused by the symmetry of its structure.

Fig. 2 Reflectance (a) and PCR (b) of x-polarized and y-polarized normal incident wave.

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The efficiency of this device is mainly limited by the dielectric loss of dielectric spacer layer, as the dielectric spacer layer was modeled as polyimide with a dielectric constant ε = 3.5(1 + 0.05i) in the simulation, where the value of dielectric dissipation factor is taken as 0.05. Figure 3 shows the performance of the half wave plate with different dielectric dissipation factors for x-polarized incident wave. Reflectance of cross-polarization increases significantly with the decrease of dielectric dissipation factor, while the reflectance of co-polarization increases slightly as Fig. 3(a) shown. This reveals that more incident power was converted to cross-polarization rather than dielectric loss, and the efficiency of the polarization converter can be improved by reducing the dielectric loss. As a result of the simultaneous increase of both cross-polarized reflectance and co-polarized reflectance, the PCR increase slightly with the decrease of dielectric loss, and almost reach saturation when the dielectric dissipation factor decrease to 0.02 as shown in Fig. 3(b). This indicates the conversion efficiency of the half wave plate can also be improved by decreasing the dielectric loss of dielectric spacer layer.

Fig. 3 Reflectance (a) and PCR (b) on the dependence of dielectric dissipation factor for x-polarized normal incidence.

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The incident wave is converted to both co-polarized and cross-polarized component as shown in Fig. 2(a) and Fig. 3(a), which indicates the reflected wave is elliptically polarized while linearly cross-polarized wave is the desired one [24]. Generally, we use polarization rotation angle ψ and ellipticity ϕ to evaluate the quality of reflected wave [22]. When the normally incident wave is x-polarized, the polarization rotation angle ψ and ellipticity ϕ can be defined as:

Ψ = \frac{1}{2} \tan^{- 1} (\frac{2 E_{x} E_{y}}{E_{x}^{2} - E_{y}^{2}} \cos δ),

ϕ = \frac{1}{2} \sin^{- 1} (\frac{2 E_{x} E_{y}}{E_{x}^{2} + E_{y}^{2}} \sin δ),

where E_x and E_y indicate the amplitude of x-polarized (co-polarization) and y-polarized (cross-polarization) component of reflected electric field, and δ = φ_y – φ_x is the phase difference of x-polarized and y-polarized reflected wave. Figure 4(a) shows the unwrapped phase of both co-polarized and cross-polarized reflected wave for x-polarized normal incident wave. Polarization rotation angle ψ represents the angle between the main axis of the elliptically polarized reflected wave and the incident polarization direction (i.e., the x axis in cross polarization conversion) in our analysis. We define tan ϕ = E_min / E_max, where E_min is the minor axis of the elliptically polarized reflected wave and E_max is the major axis. In some degree, the ellipticity ϕ denotes the polarization state of the reflected wave. The reflected wave is linearly polarized when ϕ = 0°, circularly polarized when ϕ = 90°, and elliptically polarized when 0° < ϕ < 90°. Figure 4(b) shows the polarization rotation angle ψ and ellipticity ϕ of the x-polarized incident wave. In the range from 0.67 to 0.89 THz and 1.39 to 1.66 THz, the value of ψ is smaller than −80°, and it is larger than 80° from 0.89 to 1.39 THz and from 1.66 to 1.67 THz, and reaches 90° at 0.89, 1.39, and 1.66 THz. In the range from 0.66 to 0.69 THz and from 1.1 to 1.57 THz, the value of ϕ is smaller than 20°, and it is larger than −20° from 0.69 to 1.1 THz and from 1.57 to 1.66 THz, and reaches 0° (i.e., the reflected wave is linearly polarized) at 0.69, 1.1, and 1.57 THz. These performances mean the half wave plate can convert the x-polarized incident wave to almost linearly y-polarized reflected wave in a broadband range.

Fig. 4 (a) Unwrapped phase of both co-polarized and cross-polarized reflected wave, (b) polarization rotation angle ψ and ellipticity ϕ for x-polarized normal incident wave.

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To give an understanding of how the geometric parameters correspond to the operation of the half wave plate, we simulated the reflectance of x-polarized normal incident wave with different P, L, t, and α, respectively, as shown in Fig. 5. The operating frequency band is broadened with the increase of P at the price of lower efficiency in Fig. 5(a). Figure 5(b) and 5(c) show that the operating frequency band has an obvious redshift with the increase of L and α. The polarization conversion peaks increase from 2 to 3 with the increase of L or t in Fig. 5(b) and (d). The resonance modes contributing to the polarization conversion of the half wave plate are the regular quasi-TEM modes of microstrip patchs [30–32], which have a redshift with increasing the thickness. Thus, the cross-polarization component shows the polarization conversion peak at 1.38, 1.19, and 0.99 THz when t is 25, 30, and 35μm, respectively, in Fig. 5(d). It is worth mentioning that the interference of co-polarization component is not synchronized with the cross-polarization component. When the x-polarized normal incident wave impinges on the upper layer of the half wave plate, the patches array induces a phase variation between the emitted and the incident radiation [6]. This means there is an abrupt phase change between incident wave and co-polarized emitted radiation on the upper layer of the half wave plate, while the cross-polarized emitted radiation is induced by the surface current with continuous phase. This phase difference leads to the desynchrony interference of co- and cross-polarization component.

Fig. 5 Reflectance of x-polarized normal incident wave for different P(a), L(b), α (c), and t (d).

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As Fig. 1(b) shown, the structure we design is composed of the array of connected patches. Next, we will explore other two cases, disconnected patches and fully connected patches as shown in Fig. 6. In disconnected case as Fig. 6(a) shown, the co-polarization component of reflected wave is much larger than cross-polarization component, which indicates this structure can hardly convert the polarization state of incident wave. In the other case, eminent performance of polarization conversion is also observed with the fully connected patches as shown in Fig. 6(b), but the polarization conversion ratio decreases slightly in the high-frequency part compared with that as shown in Fig. 2(a). The surface current of disconnected patches is interrupted, which results in its failure in polarization conversion, while the surface current is nearly unchanged on the fully connected patches.

Fig. 6 Reflectance of disconnected patches (a) and fully connected patches (b). Insets: the front view of one unit sell of the explored case.

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Considering general cases, we explored the normal incident linearly polarized wave with different β. The polarization rotation angle ψ and ellipticity ϕ on β dependence for linearly polarized normal incidence are shown in Fig. 7(a) and (b), respectively. The half wave plate can rotate the polarization direction of a beam of linearly polarized normal incident wave for an angle of 2β, and the polarization state keeps a good linearity. In the range from 0.68 to 1.65 THz, the deviation of polarization rotation angle ψ is below 5° with β = 15°, 6° with β = 30°, 9° with β = 45°, 6° with β = 60°, and 5° with β = 75°. The deviation of ellipticity ϕ is below 11° with β = 15°, is about 20° with β = 30° andβ = 45°, is about 19° with β = 60°, and is 9° with β = 75°. These properties represent the half wave plate we proposed operate in a broadband frequency range with different β.

Fig. 7 Unwrapped polarization rotation angle ψ (a) and ellipticity ϕ (b) of linearly polarized normal incident wave with different β.

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4. Analysis of half wave plate

In order to further reveal underlying mechanism of broadband linear polarization conversion of the designed half wave plate, we decomposed the x-polarized incident light into two perpendicular components (E_u and E_v) to investigate the corresponding reflection response. The simulated reflectance with u- and v-polarized normal incident wave is shown in Fig. 8. Reflected electric field of u- and v-polarized incident wave are co-polarized while cross-polarized component is almost zero. So, only co-polarized component of reflected wave needs to be taken into consideration when the incident wave is u- or v-polarized electric field. Reflected fields including real part and imaginary part of u- and v-polarized incident wave are shown in Fig. 8(c) and (d).

Fig. 8 Reflectance of co-polarization for u-polarized (a) and v-polarized (b) normal incident wave. Reflected fields of u-polarized (c) and v-polarized (d) normal incident wave. Insets: the polarization direction of incident wave with respect to the structure.

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According to the electromagnetic theory of optical vector field, the linearly polarized normal incident wave can be described as $E^{i} = E e x p [- i (ω t - kz)]$ . The angle between polarization direction and u axis is β, and this incident wave can be decomposed into u and v components as:

E_{u}^{i} = \cos β E e x p [- i (ω t - kz)],

E_{v}^{i} = - \sin β E e x p [- i (ω t - kz)] .

Without considering other scattering, the u- and v-polarized reflected field can be described by the following approach:

E_{u}^{o} = \cos β r_{u u} E e x p [- i (ω t + kz + ϕ_{u})],

E_{v}^{o} = - \sin β r_{v v} E e x p [- i (ω t + kz + ϕ_{v})],

where φ_u and φ_v are the phase variation caused by the structure design of u- and v-polarized normal incident wave, respectively. Equations (8) and (9) can be expressed in a simple form as:

E_{u}^{o} = \cos β E (R e_{u} + i {Im}_{u}),

E_{v}^{o} = - \sin β E (R e_{v} + i {Im}_{v}) .

Where

R e_{u} + i {Im}_{u}

and

R e_{v} + i {Im}_{v}

are shown in Fig. 8(c) and (d), as

R e_{u}

and

I m_{u}

are the real part and imaginary part in Fig. 8(c), respectively.

R e_{v}

and

I m_{v}

are the real part and imaginary part in Fig. 8(d), respectively. By means of the principle of superposition, the co-polarized and polarization converted reflected field can be described as:

E_{co}^{o} = \cos β E_{u}^{o} - \sin β E_{v}^{o} = E [\overset{}{\cos^{2} β} {(Re}_{u} + i {Im}_{u}) + \overset{}{\sin^{2} β} {(Re}_{v} + i {Im}_{v})],

E_{con}^{o} = \cos β E_{u}^{o} + \sin β E_{v}^{o} = E [\overset{}{\cos^{2} β} {(Re}_{u} + i {Im}_{u}) - \overset{}{\sin^{2} β} {(Re}_{v} + i {Im}_{v})] .

Specifically, in the case of cross polarization conversion, β = 45°, the x- and y-polarized reflected field can be calculated as:

E_{x}^{o} = \frac{1}{2} E [({Re}_{u} + {Re}_{v}) + i ({Im}_{u} + {Im}_{v})],

E_{y}^{o} = \frac{1}{2} E [({Re}_{u} - {Re}_{v}) + i ({Im}_{u} - {Im}_{v})] .

The calculated reflectance of x-polarized normal incident wave can be obtained from Eqs. (1), (14), and (15), as shown in Fig. 9(a). The differences between simulated and calculated results stem from the neglect of scattering in other polarization state in the simulation results which were used in the calculations. Figure 9(b) shows the calculated reflectance of polarization converted reflected field for linearly polarized normal incident wave with different β.

Fig. 9 (a) Simulated and calculated reflectance of x-polarized normal incident wave. (b) Calculated reflectance of polarization converted reflected field for linearly polarized normal incident wave with different β.

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Equations (8) and (9) can be expressed in another form:

E_{u}^{o} = \cos β r_{u u} E e x p [- i (ω t + kz + ϕ_{u})],

E_{v}^{o} = - \sin β r_{v v} E e x p [- i (ω t + kz + ϕ_{u})] e x p (- i Δ ϕ) .

where

Δ ϕ = ϕ_{v} - ϕ_{u}

. Ideally, when

Δ r = r_{u u} - r_{v v} = 0

and

Δ ϕ = \pm 180^{\circ}

, the co-polarized and polarization converted reflected field can be obtained from Eqs. (12), (13), (16), and (17):

E_{co}^{o} = \cos 2 β r E e x p [- i (ω t + kz + ϕ_{u})] = cos 2 β E_{con}^{o},

E_{con}^{o} = r E e x p [- i (ω t + kz + ϕ_{u})] .

Actually, the co-polarized reflected field is a component of polarization converted reflected field in the polarization direction of incident wave, as the incident wave is converted completely. Specifically, in the case of cross polarization conversion,

E_{x}^{o} = 0,

E_{y}^{o} = r E e x p [- i (ω t + kz + ϕ_{u})] .

This indicates that, regardless of dielectric loss, the x-polarized normal incident wave can be converted completely into y-polarized reflected wave when other scattering is negligible.

Figure 11(a) shows the phases and reflections of both u-polarized and v-polarized normal incident wave. When the incident electric field is linearly polarized in the x-direction, it can be decomposed into two electric field components along the u- and v-axis. The two polarized components can excite orthogonal electric dipoles. The reflectance of u-polarized and v-polarized electric field with orthogonal electric dipoles are similar in broadband range where the designed half wave plate works. Simultaneously, the relative phase difference in Fig. 10(a) nearly reaches 180^。, leading to cross polarization rotation. Δφ and Δr are shown in Fig. 10(b). In the range from 0.65 to 0.68 THz and from 1.1 to 1.56 THz, the value of Δφ is larger than 145°, and is smaller than −130°from 0.68 to 1.1 THz and from 1.56 to 1.67 THz, and can reaches 180° at 0.68, 1.1, and 1.56 THz. In the range of 0.7 to 0.87 THz and 1.39 to 1.66 THz, the value of Δr is larger than −0.2, and is smaller than 0.1 from 0.87 to 1.39 THz and from 1.66 to 1.67 THz, and it can reaches 0 at 0.87, 1.39, and 1.66 THz. Thus, the designed half wave plate can meet the requirements of Eqs. (18) and (19) over a wide frequency range to realize a broadband 2β polarization rotation in reflection mode for linearly polarized normal incident wave.

Fig. 10 (a) Unwrapped phases and reflections of both u-polarized and v-polarized normal incident wave. (b) Wrapped Δφ and Δr of u-polarized and v-polarized normal incident wave.

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In addition, we simulated the performance of the half wave plate with different incidence angle θ in cross polarization conversion case for s-polarized and p-polarized incidence. As Fig. 11 shown, the half wave plate keeps a good performance when the incidence angle θ is below 30° for s-polarized incidence, and yields a wider incidence angle range for p-polarized incidence. At oblique incidence, several guided modes with high Q-factor are excited, leading to decrease of the bandwidth of polarization conversion. Finally, the broadband and high-efficiency performance is sustained over a wide incidence angle.

Fig. 11 Reflectance and PCR on incidence angle θ dependence for s-polarized (a), (b) and p-polarized incidence (c), (d).

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5. Conclusions

In conclusion, we propose the design specifications of a broadband half wave plate in the terahertz region in reflection mode based on anisotropic metamaterials. In the cross polarization conversion case, the linearly polarized normal incident wave can be converted to its cross-polarized reflected wave with efficiency higher than 50%, and PCR higher than 85% in a broadband frequency range, which is caused by the electromagnetic resonance and interference in the dielectric spacer layer. Due to the anisotropy of the designed structure, the half wave plate can rotate the polarization direction of a beam of linearly polarized normal incident wave for an angle of 2β with an even better performance than cross polarization conversion. In addition, the half wave plate keeps a high performance with some geometrical parameters shifting, which represent a much less strict requirement for machining.

Funding

Natural Science Foundation of Zhejiang Province (LY17F050009); National Natural Science Foundation of China (NSFC) (61405182, 61377108, and 51401197); Public Technical International Cooperation project of Science Technology Department of Zhejiang Province (2015c340009); Zhejiang Students Research and Innovation Team Funded Projects (2016R409009).

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Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials

Abstract

1. Introduction

2. Metamaterial design

3. Polarization conversion of metamaterial

4. Analysis of half wave plate

5. Conclusions

Funding

References and links

Cited By

Figures (11)

Equations (21)

Optical Materials Express