Polarization-insensitive tunable terahertz polarization rotator

Shaoxian Li; Minggui Wei; Xi Feng; Qingwei Wang; Quan Xu; Yuehong Xu; Liyuan Liu; Chunmei Ouyang; Wentao Zhang; Cong Hu; Xueqian Zhang; Jiaguang Han; Weili Zhang

doi:10.1364/OE.27.016966

1. Introduction

As a particular kind of optical element, polarization rotators, which can rotate the incident polarization by a certain angle, have been sufficiently studied and are now widely applied in modern optics [1–5]. Functionalities of many composite devices, such as modulators, isolators, and polarization controllers, are realized based on polarization rotation. Conventionally, polarization rotation is based on birefringence [6,7], total internal reflection [8–10] and Faraday effect [11,12]. Among them, birefringence rotators, such as half wave plates, are commonly used. They rely on accumulated phase difference between two orthogonal linear polarizations to change the polarization angle. Therefore, precise control of the material thickness is required. Meanwhile, the rotation angle of such rotators is strongly polarization dependent. Prism rotators, which rely on multiple total internal reflections to rotate the polarization, could realize polarization-insensitive polarization rotation over a broadband range. However, the design of the optical path and the fabrication of the reflection interfaces are quite complex. Faraday rotators based on magneto-optic effect could also realize polarization-insensitive rotation of the incident polarization. Meanwhile, the rotation angle can be magnetically tunable. However, they require external magnetic field, certain magneto-optical materials, and more important, careful match between them, making the devices difficult to fabricate and also very expensive. This is also true for other types of polarization rotators based on electro-optic effect [13] and acousto-optic effect [14]. Beside, all of the three kinds of rotators are quite bulky and thus hard to be integrated. Apart from the above, another effect to realize polarization rotation is the optical activity in chiral materials [15]. The origin of this polarization rotation is due to the fact that the crystalline structures are lack of mirror symmetry, which is known as chirality. However, this effect is quite weak in natural materials and cannot achieve large rotation.

Metasurface, a planar surface composed of subwavelength structures, has received extensive attention over the past few years, owing to its fantastic ability in freely manipulating the phase, amplitude, polarization and wavefront of electromagnetic waves [16,17]. Based on metasurface, a series of functional devices have been reported, including polarization rotators and converters [18–23]. To realize polarization rotation, multi-layer plasmonic structures are mostly applied. Forming a longitudinal cavity, multi-layered structures can gradually rotate the incident polarization to desired angle based on the Fabry-Pérot effect, thus greatly increasing the rotating efficiency. However, to achieve the desired performance, fabrication of such structures often face great challenges in precise alignment and accurate control of the thickness of the dielectric spacer. More importantly, the rotation is usually designed for certain polarized wave, other polarized waves will get degraded or even wrong responses. Another route to realize polarization rotation using multi-layer structures is to compose chiral structures with larger rotation symmetry, which can generate giant chiral response [24,25]. In this case, the rotator is polarization-insensitive. However, for both the above two methods, the corresponding polarization rotation angle is fixed after the structures are fabricated, thus lacking tunabilities.

In this study, we proposed a rotation-angle-variable polarization rotator and demonstrated it in the terahertz regime. The proposed rotator is composed of double-layer all-dielectric metasurfaces. Here, the declared polarization rotation is designed at 1.0 THz and realized by cascading two dielectric metasurfaces back to back with a controllable relative angle. By changing this angle, the polarization rotation angle can be arbitrarily tuned from 0° to 360°. Meanwhile, the rotation angle is insensitive to the input state of polarization. This feature makes the device able to meet different application requirements. Such design may find broad applications in polarization control for both terahertz and optical waves.

2. Design strategy

The basic structure is schematically illustrated in Fig. 1(a), which is an anisotropic rectangular-shaped silicon pillar etched in a silicon substrate.

Fig. 1 (a) Structural unit cell. (b) Relation of the two metasurfaces’ sp coordinates and the xy coordinate, as well as diagram of the polarization rotation. (c) Schematics of the polarization-insensitive tunable terahertz polarization rotator. The geometric parameters of the designed silicon pillar are: Λ = 150 μm, a = 48 μm, b = 82 μm, and h = 200 μm, respectively.

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According to the symmetry, the main axes of the pillar are along its width and length directions, which can be denoted as s and p directions, respectively. These two directions form a local Cartesian coordinate, which has a relative in-plane angle α with regards to the xy coordinate, as illustrated in Fig. 1(a). When the incident polarization is along either of the two main axes, there will be no cross-polarized output. So, the transmission matrix $T^{s p}$ of such a pillar in the sp coordinate can be expressed as:

T^{s p} = [\begin{matrix} t_{s} e^{i φ_{s}} & 0 \\ 0 & t_{p} e^{i φ_{p}} \end{matrix}],

where t_s (t_p) and φ_s (φ_p) are the corresponding transmission amplitude and phase under the s-polarized (p-polarized) incidence, respectively. The transmission matrix in the xy coordinate can thus be seen as taking an α rotation operation to Eq. (1):

T_{α}^{x y} = [\begin{matrix} t_{s} e^{i φ_{s}} \cos^{2} (α) + t_{p} e^{i φ_{p}} \sin^{2} (α) & (t_{s} e^{i φ_{s}} - t_{p} e^{i φ_{p}}) \sin (α) \cos (α) \\ (t_{s} e^{i φ_{s}} - t_{p} e^{i φ_{p}}) \sin (α) \cos (α) & t_{s} e^{i φ_{s}} \sin^{2} (α) + t_{p} e^{i φ_{p}} \cos^{2} (α) \end{matrix}] .

To realize polarization-insensitive polarization rotation, we consider a doublet configuration, which contains two layers of silicon pillar structures of same dimensions but different orientation angles. As shown in Fig. 1(b), we make the silicon pillar in the other layer has a relative in-plane angle α + θ with regards to the xy coordinate. In this case, the transmission matrix can be calculated as $T_{α + θ}^{x y}$ . Therefore, the overall transmission matrix T_M of the doublet structure can be expressed as:

T_{M} = T_{α + θ}^{x y} T_{α}^{x y} = [\begin{matrix} T_{x x} & T_{x y} \\ T_{y x} & T_{yy} \end{matrix}] .

The elements of T_M are shown as following:

\begin{array}{l} T_{x x} = \frac{1}{2} [t_{s}^{2} e^{i 2 φ_{s}} - t_{p}^{2} e^{i 2 φ_{p}}] \cos (2 α + θ) \cos (θ) + (^{t_{s} e^{i φ_{s}} - t_{p} e^{i φ_{p}}) 2} \frac{\cos (2 θ)}{4} + \frac{1}{4} (^{t_{s} e^{i φ_{s}} + t_{p} e^{i φ_{p}}) 2}, \\ T_{x y} = \frac{1}{4} (t_{s} e^{i φ_{s}} - t_{p} e^{i φ_{p}}) [(t_{s} e^{i φ_{s}} - t_{p} e^{i φ_{p}}) \sin (- 2 θ) + 2 (t_{s} e^{i φ_{s}} + t_{p} e^{i φ_{p}}) \sin (2 α + θ) \cos (θ)], \\ T_{y x} = \frac{1}{4} (t_{s} e^{i φ_{s}} - t_{p} e^{i φ_{p}}) [(t_{s} e^{i φ_{s}} - t_{p} e^{i φ_{p}}) \sin (2 θ) + 2 (t_{s} e^{i φ_{s}} + t_{p} e^{i φ_{p}}) \sin (2 α + θ) \cos (θ)], \\ T_{y y} = \frac{- 1}{2} (t_{s}^{2} e^{i 2 φ_{s}} - t_{p}^{2} e^{i 2 φ_{p}}) \cos (2 α + θ) \cos (θ) + \frac{1}{4} (^{t_{s} e^{i φ_{s}} - t_{p} e^{i φ_{p}}) 2} \cos (2 θ) + \frac{1}{4} (^{t_{s} e^{i φ_{s}} + t_{p} e^{i φ_{p}}) 2}, \end{array}

where the former and latter subscripts represent the output and incident polarizations, respectively. It can be seen that, if t_s = t_p and φ_s – φ_p = ± 180°, Eq. (3) is simplified as:

T_{M} = t_{s}^{2} e^{i 2 φ_{s}} [\begin{matrix} \cos (2 θ) & - \sin (2 θ) \\ \sin (2 θ) & \cos (2 θ) \end{matrix}] .

Clearly, T_M becomes a simple rotation matrix that is only related to twice of the relative angle θ between this two silicon pillars. This matrix indicates that any incident polarized (except circularly polarized) wave, irrespective to its orientation angle, can be rotated by Δ = 2θ. Figure 1(c) illustrates a schematic of such a double layer polarization rotator. Two identical silicon pillar metasurfaces are placed back to back with each other. By mechanically changing the relative angle θ between the two metasurfaces, the rotation angle can thus be actively controlled. In fact, the conditions of t_s = t_p and φ_s – φ_p = ± 180° just correspond to the response of a half-wave plate. In a simple picture, we suppose the incident wave E_in is linearly polarized, whose orientation angle is β with respect to the x axis, as illustrated in Fig. 1(b). When it interacts with the rotator, the polarization orientation will first rotates across the p axis of the top layer structure p₁ to (2α – β + 180°) (E_temp), then rotates across the p axis of the bottom layer structure p₂ to (2θ + β) (E_out).

To realize such a silicon pillar, we run numerical simulations using CST Microwave Studio. Here, the working frequency is set to be 1.0 THz. In the simulation, the dielectric constant of silicon is set to be 11.9. Figures 2(a) and 2(b) show the distributions of the transmission amplitudes t_s, and t_p by sweeping the width a and the length b of the pillar, respectively. The period Λ and the height h are fixed to be 150 μm and 200 μm, respectively. The transmission is normalized to that of air. To find the best pillar satisfying the above mentioned designing requirements, we calculated the distributions of the transmission ratio t_s /t_p, and the phase difference φ_s – φ_p, as illustrated in Figs. 2(c) and 2(d), respectively. By calculating the minimum of $| t_{s} / t_{p} \exp [i (φ_{s} - φ_{p})] - (- 1) |$ , the corresponding optimized pillar can be extracted, as indicated by the star markers inside the Fig. 2. Here, $t_{s} / t_{p} \exp [i (φ_{s} - φ_{p})] = - 1$ is the perfect choice for the polarization rotator. The geometric parameters of the selected pillar using this method are: Λ = 150 μm, a = 48 μm, b = 82 μm, and h = 200 μm, respectively. It is found that the corresponding transmission amplitudes are nearly the same while the phase difference is π at 1.0 THz.

Fig. 2 The simulated parameter sweep data of the silicon pillar at 1.0 THz. Transmission amplitude distributions (a) t_s and (b) t_p as a function of a and b, respectively. (c) Transmission ratio distribution t_s /t_p and (d) phase difference distribution φ_s – φ_p as a function of a and b, respectively. The star markers indicate the chosen geometric parameters in the design.

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Here, the origin of the phase difference φ_s – φ_p is from the anisotropy of silicon pillars. Figures 3(a) and 3(b) show the simulated distributions of the propagation phases of the selected structure at 1.0 THz under the s- and p-polarized incidences from the silicon substrate across the pillar to the air, respectively. It is seen that the phase changes are induced in the pillar region, and the corresponding phase change values are different. To give in-depth understanding of the effect, we plot the corresponding simulated distributions of the maximum amplitudes of the electric fields in Figs. 3(c) and 3(d), respectively, where different propagation modes are observed. Under the s-polarized incidence, the electric fields are mainly localized at the side interfaces between the pillar and the air. However, under the p-polarized incidence, the electric fields are mainly confined inside the pillar. According to the effective medium theory, the two different propagation modes have different effective dielectric constants, owing to the duty cycles of the electric fields in (outside) the pillar are different. A π phase difference can thus be realized by appropriately choosing the width and the length of the pillar.

Fig. 3 Simulated distributions of the propagation phases of the selected pillar under the (a) s-polarized and (b) p-polarized incidences, respectively. Simulated distributions of the maximum amplitudes of E-fields of the selected pillar under the (c) s-polarized and (d) p-polarized incidences, respectively. All the distributions are plotted at 1.0 THz at the cross-sections formed by the incident polarization and the propagation direction.

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3. Results and discussion

Two metasurfaces composed of same silicon pillars were fabricated on 1 mm-thickness high-resistivity float-zone silicon wafer using conventional photolithography and deep reactive ion etching. Figures 4(a) and 4(b) illustrate the top-view and side-view microscope images of part of the fabricated metasurface sample. Figures 4(c) and 4(d) illustrate the simulated and measured transmission spectra of the selected pillar, respectively, where the main features are agreeing well with each other. The amplitude differences between the simulation and fabrication may be mainly attributed to the uneven structural surfaces and residual contaminants in the fabrication process, which may cause unexpected loss effect in the pillar regions. Besides, it is seen that the measured phase difference shifts towards the higher frequency compared with the simulated result. This can be attributed to the fabrication deviations, where smaller size of the pillar and smaller dielectric constant of the applied silicon wafer (11.5) can cause such effect.

Fig. 4 (a) Top-view ( × 50 magnification) and (b) side-view ( × 20 magnification) microscope images of part of the fabricated metasurface sample. (c) Simulated and (d) measured transmission amplitude and phase difference spectra of the designed metasurface under the s- and p-polarized incidences, respectively.

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Then, the two metasurfaces were combined together to form the polarization rotator, as shown in Fig. 1(c). The composite real device can be seen in Fig. 5(a). To demonstrate that the proposed device can realize polarization-insensitive tunable terahertz polarization rotation, we applied linear polarization incidence using a home-build photoconductive-antenna-based terahertz time-domain spectroscopy (THz-TDS) system, which is pumped by a mode-locked Ti:sapphire laser (Mantis, Coherent, Inc.) with 80 MHz repetition frequency, 800 nm central wavelength, and 20 fs pulse width. Figure 5(b) schematically illustrates the experimental setup, where the THz-TDS system comprises four 90° off-axis parabolic mirrors aligned in an 8-F confocal geometry, which enables good collection, collimation, and focusing of the terahertz beam. The effective focal length of the four parabolic mirrors are 4 inch, 2 inch, 2 inch and 4 inch in sequence from the transmitter to the detector, respectively. Four broadband terahertz wire grid polarizers were used to realize polarization-resolved transmission measurement. The polarizers P1 and P4 were used to ensure the transmitted terahertz waves to be horizontally polarized. The polarizers P2 and P3 were placed either 45° or −45° with respect to P1 and P4 to make sure equal polarization projections. The polarizer P2 was used to control the incident polarization towards the rotator, while P3 was used to analyze the output polarization. All the polarizers have a clearance of 50 mm, which is big enough for the terahertz beam to pass through without being blocked. Here, we define the coordinate formed by ± 45° directions to be the xy coordinate, as indicated by the red arrows in Fig. 5(b). With this setup, each component (t_xx, t_yy, t_xy, t_y_x) of the transmission matrix T_M of the sample can thus be measured, which could be applied to fully characterize the polarization responses of the rotator.

Fig. 5 (a) Photograph of the doublet metasurface polarization rotator mounted in a rotator, in which one metasurface is fixed while the other can be rotated. (b) Schematic of the experimental setup for the polarization rotator characterization.

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Figure 6(a) illustrates the measured polarization rotation angle at different θ from 0° to 180° with a step of 15° under the x- and y-polarized incidences at 1.0 THz, respectively. The polarization rotations under the x- and y-polarized incidences can be calculated by

Δ_{x} = ψ = \frac{1}{2} \arctan \frac{2 | t_{x x} t_{y x} | \cos δ}{{| t_{x x} |}^{2} - {| t_{y x} |}^{2}},

Δ_{y} = ψ - 90 ° = \frac{1}{2} \arctan \frac{2 | t_{x y} t_{y y} | \cos δ}{{| t_{x y} |}^{2} - {| t_{y y} |}^{2}} - 90 °,

respectively, where ψ represents the polarization orientation angle, δ represents the phase difference between the output y and x polarization components. It can be seen that both the incident polarizations were rotated by 2θ, which agrees well with the theoretical calculation. Figure 6(b) illustrates the corresponding ellipticity angle χ as a function of θ. It is seen that ellipticity angles were all very close to the theoretical value 0° with an only ± 5° maximum deviation, which means that the transmitted waves were also well linearly polarized. These features demonstrate that the device could realize excellent tunable polarization rotation. As any polarization can be seen as a combination of the x and y polarizations. The same rotation feature to the x- and y- polarized incidences demonstrates that the rotator could work well for other polarization incidences, i.e. elliptical polarizations.

Fig. 6 Experimentally measured and theoretically calculated (a) polarization rotation angle and (b) ellipticity angle χ of the polarization rotator at 1.0 THz as a function of θ under the x- and y-polarized incidences, respectively. Inset in (a): Definition of the polarization orientation angle and the ellipticity angle of an arbitrary polarization state.

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Efficiency is an important parameter in describing the performance of a rotator. Here, we define the efficiency as the ratio between the output and incident wave intensities. Due to multiple interfaces of the design (Fig. 1(c)) and large refractive index of the silicon substrate, the device has a large reflection loss. The measured results show a relative small efficiency of ~9%. To increase the efficiency, one could reduce the number of the interfaces. For example, making the backsides of the two metasurface contact with each other or fabricating anti-reflection layers onto the backsides of the metasurfaces [26,27], which will reduce the reflection loss by approximately 50%. The corresponding simulated efficiency can reach to ~45%. To further increase the efficiency, one could design other all-silicon structures with different shapes which exhibits higher transmission. One could also use substrates with low refractive indexes as the silicon pillar supporters to reduce the reflection loss, such as silica [28].

Beside polarization-insensitive rotation, the device could also serve as a phase modulator for the circularly polarized incidences. Here, we define the Jones matrices of the left-handed circular polarization (LCP) and right-handed circular polarization (RCP) as $[1, - i] / \sqrt{2}$ and $[1, i] / \sqrt{2}$ , respectively. According to Eq. (5), the corresponding transmission matrix in the circular polarization basis can be expressed by:

T_{M}^{l r} = \frac{t_{s}^{2} e^{i 2 φ_{s}}}{\sqrt{2}} [\begin{matrix} e^{i 2 θ} & 0 \\ 0 & e^{- i 2 θ} \end{matrix}] .

Equation (8) indicates that the device could keep the circular polarization state of the incident circularly polarized wave. More importantly, the transmission amplitude is irrelevant to the relative angle θ, while the phase will linearly change with ± 2θ in the whole 360° range. Figures 7(a) and 7(b) illustrate the measured transmission amplitudes and phase differences under the LCP and RCP incidences at 1.0 THz as a function of θ, respectively. They were calculated by transforming the results in the xy basis in Figs. 6(a) and 6(b) into those in the circular polarization basis. It can be seen that the transmitted amplitudes of the LCP and RCP components are nearly the same (around 0.3) at different θ, while the corresponding phase differences (normalized to the corresponding values at θ = 0°) follow a + 2θ and a −2θ dependences, respectively. These results are consistent well with the theoretical prediction, showing a good performance of our device as a circular-polarization-dependent phase modulator.

Fig. 7 Measured (a) transmission amplitudes and (b) phase differences of the polarization rotator as a function of θ at 1.0 THz under the RCP and LCP incidences, respectively. Here, the phase differences were normalized to the corresponding values at θ = 0. The solid lines in (b) represent the corresponding theoretical results.

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4. Summary

We proposed, designed, fabricated and demonstrated a polarization-insensitive terahertz polarization rotator with variable rotation angles based on an all-dielectric metasurface doublet. By designing each metasurface to function as a half-wave plate, the composite device could rotate any incident polarization by twice of the relative angle formed by the two metasurfaces. Furthermore, the rotator could also function as a polarization-maintained phase modulator for the circularly polarized incidence. Though the present rotator is only designed at 1.0 THz, the working bandwidth can be increased by using broadband metasurface half-wave plate design. The proposed device is promising in realizing mechanical based fast phase modulation, and may find applications related to interference, i.e. coherent control. Although the work is demonstrated at terahertz frequencies, the proposed methodology is also applicable to other frequency ranges.

Funding

National Natural Science Foundation of China (61605143, 61735012, 61875150, 61775159, 61705163, 61427814, 61420106006); Guangxi Key Laboratory of Automatic Detecting Technology and Instruments (YQ17203, YQ18205).

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Polarization-insensitive tunable terahertz polarization rotator

Abstract

1. Introduction

2. Design strategy

3. Results and discussion

4. Summary

Funding

References

Cited By

Figures (7)

Equations (8)

Optics Express