1 Introduction

The AT is a Lorentz reciprocal phenomenon, which is mainly attributed to the different polarization selection and conversion properties of the medium for two orthogonal polarized waves impinging from opposite directions. The reciprocal route in AT is similar to the unidirectional transmission feature of the diodes, which offers a scheme to selectively block or transmit certain electromagnetic waves. Therefore, the AT has potential applications in the design of nonreciprocal light devices such as isolator, circulator and frequency selector or filter [1,2,3,4,5]. Conventional approaches for realizing AT utilize magnetic-optical materials [6], nonlinear materials [7] or time-dependent media [8]. However, these materials are not suitable for on-chip integration due to their large size or strict working conditions (e.g. high power). In 2006, Fedotov first demonstrated the AT effect for circularly polarized (CP) wave in MMs [9], since then, various kinds of MM structures have been proposed for realizing polarization conversion and AT effect owing to their flexible design, ultra-thin, and broadband polarization conversion performances [10].

According to polarization state of controlling waves, AT in MMs can be divided into two kinds, LP AT and CP AT [11,12,13]. Recently, great efforts have been devoted to achieving AT for LP waves [2, 4, 5, 14,15,16,17,18,19,20]. For example, based on 3D low-symmetry chiral MM, Menzel et al. [2] reported the first experimental observation and theoretical analysis of AT for LP wave in infrared regime. Serebryannikov et al. [16] realized multiband AT in MM comprising two twisted complementary split-ring resonators (SRRs) and a sandwiched metallic mesh by combining chirality and tunneling effects. Han et al. [17] proposed a MM composed of symmetric rectangular SRRs and a sandwiched metal strip with stepped widths to achieve AT with two operation bands, 2.53–2.56 GHz and 3.87–4.78 GHz. In addition, various other types of MM structures, such as cross wire pairs [18], twisted slanted complementary metal strips [19] and chiral twisted multi-gap rings [20] were also proposed to realize AT for LP waves.

Actually, MM-based polarization converters and AT are particularly attractive in THz frequency regime due to the lack of naturally responding THz materials. Singh et al. [21] presented experimental and numerical evidences of AT for CP THz wave in a planar chiral MM. M. Stolarek et al. [22] proposed a double-layer metallic gratings with different periods for unidirectional transmission in the THz frequency range. Dai et al. [23] designed a chiral, Dirac semimetals based double-T MM to realize AT for LP wave in the THz regime, and the achieved AT parameter remains > 0.35 in frequency band 1.38–1.63 THz. Zhao et al. [24] reported the numerical investigation of AT for LP wave based on hybrid metal-graphene metasurface, where the AT parameter remains > 0.5 in band 0.5–1.46 THz. However, the above mentioned AT effects still suffer from the drawbacks of low efficiency or relatively narrow bandwidth, which might restrict their practical applications in THz systems. Meanwhile, although single-layer THz MMs are easy to fabricate, they often have low AT parameter or narrow bandwidth. Therefore, in recent years, multi-layer MM composed of sub-wavelength gratings and metallic array have been proposed as substitutes to realize efficient, broadband polarization conversion and AT effect. For example, Cheng et al. [25] proposed a tri-layer MM composed of a split disk array sandwiched by two orthogonal metallic sub-wavelength gratings to realize AT, and the acquired AT parameter for LP wave is beyond 0.9 in the range of 0.23–1.17 THz, equivalent to 134.3% relative bandwidth. Based on anisotropic MM composed of bi-layer metal gratings sandwiched with symmetrical dual-radar array, Li et al. [26] presented a dual-band AT, in which the achieved AT parameter is above 0.8 in bands 0.60–1.12 THz and 1.61–1.98 THz with the relative bandwidths of 60.47% and 20.61%, respectively. Pan et al. [27] numerically simulated a broadband THz polarization conversion with significant AT effect based on a double L array sandwiched by two orthogonal metallic gratings, and the acquired polarization conversion rate (PCR) is larger than 0.99 from 0.2 to 1.97 THz, and the enhanced AT parameter is beyond 0.8 from 0.37 to 1.73 THz with the relative bandwidth of 129.52%. In 2013, Grady et al. [28] proposed a three-working-layer MM comprising a cut-wire (CW) array layer and two orthogonal grating layers to realize broadband polarization conversion, where the conversion efficiency exceeds 50% in band 0.52–1.82 THz. However, in their report, no AT effect was discussed. Obviously, the polarization conversion and AT effect in MM structures composed of dual-layer metal gratings can have relatively high efficiency and bandwidth comparing with the former mentioned ones, however, the MM structures with simple sandwiched structure, larger AT bandwidth, and higher efficiency are still in need. Moreover, the mechanisms for broadband polarization conversion are still need to be deeply investigated.

In this paper, a three-working-layer MM structure is designed to achieve ultra-broadband, highly efficient polarization conversion and AT effect for LP wave in THz regime. To this end, the MM structures are composed of two orthogonal metallic sub-wavelength grating layers and a sandwiched cross-shaped array layer. The orthogonal grating layers are used as polarization filters to enhance polarization conversion efficiency, while the cross-shaped structure is utilized as a polarization converter. Owing to the coupling between two different CWs in the cross-shaped structure and the excitation of higher-order plasmon mode, the polarization conversion bandwidth is greatly expanded comparing with Ref. [28]. The physical mechanisms of the ultra-broadband and polarization conversion of the MM structure are studied via electric field and surface current distributions, respectively. Moreover, the influence of structure parameters on the AT parameter and bandwidth is investigated. To highlight the novelty and validity of this work, comparisons between our results and recently published results are made.

2 Structure, theory and results

The multi-layer MM we designed is comprised of three metallic layers and two polyimide dielectric spacer layers, shown as Fig. 1a. Figure 1b presents the detailed structures of the former, middle and bottom metallic layers in one unit cell, respectively. As shown, the former and latter layers are two sub-wavelength gratings with orientations orthogonal to each other, which are used as polarization filters to selectively block or transmit wave according to its polarization state and propagating direction. The middle metallic layer is a periodic cross-shaped array, which functions as a 90° polarization converter. The cross-shaped structure can be seen as a combination of two orthogonal CWs, the long and short CWs are marked as LCW and SCW, respectively. LCW and SCW are respectively titled 45° relative to the y- and x-axis. In the design, the grating layers are used to enhance conversion efficiency, while the cross-shaped array layer helps to broaden conversion bandwidth. Actually, the proposed MM structure can be seen as two cascaded Fabry–Perot-like resonant cavities, which further improves the conversion bandwidth and efficiency [29, 30].

Fig. 1
figure 1

A Schematic diagram of the designed multi-layer MM structure for AT effect and its working theory, b the former layer, middle layer, and latter layer in one unit cell, where yellow parts stand for metal copper and blue parts represent dielectric polyimide

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The AT process of the designed MM is shown in Fig. 1a. Firstly, an x-polarized THz wave propagating along the forward (− z-axis) direction permits to transmit through the former grating layer, then converts to y-polarized wave by the middle cross-shaped array layer, and finally transmits through the latter grating layer with low loss. However, for the same propagating direction, the incident y-polarized wave is totally reflected by the first grating layer. In case of illuminating from backward (+ z-axis) direction, the incident y-polarized wave is converted to transmitted x-polarized wave, while the incident x-polarized wave is blocked.

The finite difference time domain method based on CST Microwave Studio is employed to perform the numerical simulations. In simulations, all metallic layers are made of copper with thickness 200 nm and electric conductivity σ = 5.8 × 107 S/m. The dielectric constant of the polyimide spacers is set to be ε = 3(1 + 0.05i) and thickness is h = 26 μm. To obtain broadband and highly efficient AT effect, the optimum geometrical parameters of the MM structure are set as follows: L1 = 88 μm, L2 = 34 μm, w = 4 μm, a = 10 μm. Periodicities of one unit cell along x and y axis are Px = Py = 70 μm. As the proposed structure is a uniformly periodic structure, periodic boundary conditions are applied in the x and y directions, while the perfect match layer (PML) boundary conditions are imposed at the boundaries in z direction.

The MM is normally illuminated by two orthogonal LP THz waves with electric field vector along x- and y axis, respectively. For forward propagation, the complex electric fields of the incident and transmitted waves are related to each other by Jones transmission matrix [1]:

$$ \left( \begin{gathered} E_{{{\text{tr}}}}^{x} \hfill \\ E_{{{\text{tr}}}}^{y} \hfill \\ \end{gathered} \right) = \left( \begin{gathered} T_{xx}^{f} \quad T_{xy}^{f} \hfill \\ T_{yx}^{f} \quad T_{yy}^{f} \hfill \\ \end{gathered} \right)\left( \begin{gathered} E_{{{\text{in}}}}^{x} \hfill \\ E_{{{\text{in}}}}^{y} \hfill \\ \end{gathered} \right), $$
(1)

where the superscript f denotes forward propagation; \(E_{{{\text{in}}}}^{x} \left( {E_{{{\text{tr}}}}^{x} } \right)\) and \(E_{{{\text{in}}}}^{y} \left( {E_{{{\text{tr}}}}^{y} } \right)\) indicate complex electric fields of the incident (transmitted) x- and y-polarized waves, respectively. The polarization conversion performance of the MM is represented by the 2 × 2 transmission matrix, where \(T_{ij}^{f}\) represent i-polarized transmission coefficient from j-polarized incidence. Txx and Tyy are co-polarized transmission coefficients, while Txy and Tyx are cross-polarized transmission coefficients. For backward propagation, superscript f is replaced by b.

The AT performance of the MM is characterized by AT parameter Δ, which is defined as the difference between the transmissions in two opposite propagating directions. For describing the AT for LP wave, the AT parameters for x- and y-polarized incident waves are respectively represented by:

$$ \Delta_{{{\text{lin}}}}^{x} = \left| {T_{yx}^{f} } \right|^{2} - \left| {T_{yx}^{b} } \right|^{2} = \left| {T_{xy}^{f} } \right|^{2} - \left| {T_{xy}^{b} } \right|^{2} = - \Delta_{{{\text{lin}}}}^{y} $$
(2)

To ensure AT effect occurring, the transmission coefficients Tij for each propagating direction should satisfy the following two conditions:

$$ \left| {T_{yx}^{{}} } \right| \ne \left| {T_{xy}^{{}} } \right|,\;\;\left| {T_{xx}^{{}} } \right| = \left| {T_{yy}^{{}} } \right|. $$
(3)

Moreover, to measure the polarization conversion performance of the proposed MM structure, the PCRx and PCRy for forward x- and backward y- polarized incident waves are respectively defined as:

$$ {\text{PCR}}_{x}^{f} = \frac{{\left| {T_{yx} } \right|^{2} }}{{\left| {T_{yx} } \right|^{2} + \left| {T_{xx} } \right|^{2} }},\;\quad {\text{PCR}}_{y}^{b} = \frac{{\left| {T_{xy} } \right|^{2} }}{{\left| {T_{xy} } \right|^{2} + \left| {T_{yy} } \right|^{2} }}. $$
(4)

Figure 2 presents the simulated transmission spectra, PCRs, and AT parameters. Figure 2a and b illustrates the amplitudes of four transmission coefficients of the proposed MM for forward and backward propagations, respectively. In Fig. 2a, for forward propagation and x-polarized incidence, the cross-polarized transmission Tyx experiences a broadband resonance from 0.50 to 2.41 THz with amplitude larger than 0.7 (3 dB), while its co-polarized transmission Txx is smaller than 0.1 in this frequency range. It means that the incident x-polarized wave is converted to y-polarized wave and the polarization conversion is realized due to the difference between the amplitudes of Tyx and Txx. Contrarily, for y-polarized incidence, the cross-polarized transmission Txy and co-polarized transmission Tyy are lower than 0.1, which means that the incident y-polarized wave is forbidden to transmit through the proposed MM along forward direction. For the case of backward propagation, as shown in Fig. 2b, the transmission Txy is much larger than Tyy, Txx and Tyx, meanwhile, Tyy = Txx, it means that the incident y-polarized wave can transmit through the MM and convert to x-polarized wave, while the incident x-polarized wave is strictly forbidden. Figure 2c further shows the PCRs for forward x-polarized incidence and backward y-polarized incidence. In the range of 0.41—2.38 THz, the PCRx and PCRy are above 0.99 and equal to each other with a relative bandwidth of 141.22%, which demonstrates that the polarization conversion performances of the proposed MM for both two opposite directions are the same, and each incident waves is completely converted into its cross-polarized waves. Therefore, the proposed MM can realize x- to y-polarized conversion for forward direction and y- to x-polarized conversion for backward direction. Moreover, the transmission amplitudes for both two propagating directions satisfy the two AT conditions in Eq. (3), thus, the proposed MM can also implement AT effect for LP waves.

Fig. 2
figure 2

Simulated transmission spectra for incidences in a forward and b backward directions; and c PCR and d AT parameters for LP and CP waves under the illumination from forward and backward, respectively

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The AT parameters for x- and y-polarized waves respectively propagating along opposite directions are plotted in Fig. 2d. In the range of 0.50–2.41 THz, AT parameters for both two polarized waves are larger than 0.64, and the relative bandwidth is 131.27%. Particularly, the enhanced AT parameter larger than 0.8 is achieved at two frequency bands, 0.53–1.48 THz and 1.80–2.06 THz, with relative bandwidth of 94.53% and 13.47%, respectively. Moreover, the curves of \(\Delta_{{{\text{lin}}}}^{x}\) and \(\Delta_{{{\text{lin}}}}^{y}\) are exactly contrary to each other, demonstrating that the polarization conversion performances of the MM along two opposite directions are the same. Apparently, the proposed MM reveals a one-way transmission for LP waves, one direction for x-polarized wave and the other direction for y-polarized wave.

Similarly, the direction dependent propagating property can also be used to describe incident CP waves. With CP waves, the relation between the incident and transmitted electric fields changes to:

$$ \left( \begin{gathered} E_{{{\text{tr}}}}^{x} \hfill \\ E_{{{\text{tr}}}}^{y} \hfill \\ \end{gathered} \right) = \left( \begin{gathered} T_{ + + }^{f} \quad T_{ + - }^{f} \hfill \\ T_{ - + }^{f} \quad T_{ - - }^{f} \hfill \\ \end{gathered} \right)\left( \begin{gathered} E_{{{\text{in}}}}^{x} \hfill \\ E_{{{\text{in}}}}^{y} \hfill \\ \end{gathered} \right) = \left( \begin{gathered} T_{xx}^{f} { + }T_{yy}^{f} + i\left( {T_{xy}^{f} - T_{yx}^{f} } \right)\quad T_{xx}^{f} - T_{yy}^{f} - i\left( {T_{xy}^{f} + T_{yx}^{f} } \right) \hfill \\ T_{xx}^{f} - T_{yy}^{f} + i\left( {T_{xy}^{f} + T_{yx}^{f} } \right)\quad T_{xx}^{f} { + }T_{yy}^{f} - i\left( {T_{xy}^{f} - T_{yx}^{f} } \right) \hfill \\ \end{gathered} \right)\left( \begin{gathered} E_{{{\text{in}}}}^{x} \hfill \\ E_{{{\text{in}}}}^{y} \hfill \\ \end{gathered} \right), $$
(5)

where + and − represent the right- and left-handed CP waves propagating along forward direction.

Correspondingly, AT parameter for right- and left-handed CP waves is written as:

$$ \Delta_{{{\text{cir}}}}^{ + } = \left| {T_{ - + }^{f} } \right|^{2} - \left| {T_{ - + }^{b} } \right|^{2} = \left| {T_{ + - }^{f} } \right|^{2} - \left| {T_{ + - }^{b} } \right|^{2} = - \Delta_{{{\text{Cir}}}}^{ - } . $$
(6)

By introducing two orthogonal circular illuminations in the simulation and substituting the corresponding co- and cross-polarized transmissions into Eqs. (5) and (6), the AT parameters for CP waves are acquired. It is found that within the discussed frequency range, they are equal to zero, indicating that the proposed MM structure cannot achieve AT effect for CP waves.

3 Discussions

In the proposed MM structure, the two sub-wavelength gratings function as two polarization filters and can only transmit the incident waves with polarization perpendicular (or parallel) to the grating grids (or grating vector), thus the former grating permits x-polarized wave to pass and blocks y-polarized wave. If there is no middle layer, the transmitted x-polarized wave from the former grating cannot transmit through the latter grating. Thus, the realization of polarization conversion is mainly attributed to the cross-shaped array. To explore the physical mechanism of broadband polarization conversion performance in the proposed MM structure, we study the function of the cross-shaped array in the middle layer.

In Fig. 3a, defining u and v axis as the directions along 45° with respect to x-axis, for forward propagation, the incident electric field Ex can be divided into two equal components along u- and v axis. Meanwhile, the cross-shaped structure can be seen as a combination of two mutually orthogonal copper rectangles, LCW and SCW. The cross-polarized transmission spectra and the AT parameters \(\Delta_{{{\text{lin}}}}^{x}\) for the original MM, and two simplified MMs with the original crossed-shaped array replaced by individual LCW array and SCW array are plotted in Fig. 3b–d, respectively. Obviously, none of the simplified individual MMs can achieve ultra-broadband transmission spectra or AT effect, which indicates that the excellent polarization conversion performance and AT effect of the cross-shaped MM are the results of combination of two individual MM structures. In Fig. 3c, resonance peaks of the combined MM structure are located at four frequency points: f1 (0.72 THz), f2 (1.26 THz), f3 (1.94 THz) and f4 (2.36 THz). At the former three points, the AT parameters are larger than 0.90, while at the fourth point, it decreases to 0.70.

Fig. 3
figure 3

a View of the cross-shaped structure, where u- and v axis are used to mark the microstructure anisotropic axis; Cross-polarized transmission spectra b and AT parameters c, d for the simplified LCW MM, SCW MM, and the original MM when forward propagation with x-polarized incidence

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To understand the physical origin of broadband conversion performance of our proposed polarization converter, we investigate the electric field distributions on x–y plane at above four frequency points. As shown in Fig. 4a, at frequency f1, the electric field is mainly distributed at two ends of the LCW. In this case, the LCW is equivalent to a dipole resonator and plasmon resonance is excited on it. Therefore, the LCW array determines the polarization conversion at f1. In Fig. 4c, at f3, the electric field is mainly concentrated on two ends of the SCW. However, in Fig. 3c, at f3, the spectrum also shows AT effect when the SCW is not in, but this AT parameter is smaller than that in Fig. 3d when only SCW MM exists, thus, we conclude that the polarization conversion at f3 is mainly dominated by the SCW array. In Fig. 4b, at f2, the electric field distributes on both two CW resonators, implying that the hybridized coupling between two dipole resonance modes is responsible for the polarization conversion at his point. In Fig. 4d, at f4, the electric field still distributes on both two CWs and higher-order resonance modes are excited. Obviously, the broadband conversion performance of our proposed MM structure originates from the coupling of two CWs in the cross-shaped structure, meanwhile, the excitation of higher-order mode further expands the conversion bandwidth.

Fig. 4
figure 4

Electric field distributions for the MM structure at resonance frequencies: a 0.72 THz, b 1.26 THz, c 1.94 THz and d 2.36 THz

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Previous studies have shown that the cross-coupling between the incident electric field and induced magnetic field by the surface currents is responsible for polarization conversion and AT effect [17, 31, 32]. Therefore, to understand the physical origin of the designed MM for AT effect, we simulate the surface current distributions on three layers at four resonance frequencies, shown as Fig. 5a–d. In Fig. 5a1–a3, the currents on the former and latter gratings flow along the corresponding grating grips, and the current on the middle layer flows along the LCW. For the sake of analysis, the current on the middle layer can be treated as a combination of two orthogonal components along x- and y axis, respectively, shown as Fig. 5a4. The current along y axis oscillates oppositely to that on the former layer, so these two antisymmetric currents enables the formation of a strong magnetic resonance on the x–y plane middle between the former and middle layers, and the corresponding magnetic dipole moment is represented by m1. Similarly, magnetic dipole moment m2 is also excited on the x–y plane middle between the middle and latter layers by the anti-symmetric currents on the middle and latter layers. As the induced magnetic moment m2 is perpendicular to the incident electric field Ein, there is no cross-coupling between m2 and Ein, m2 has no contribution to polarization conversion. On the contrary, m1 is along x-axis and parallel to the incident electric field, thus, the cross-coupling between the electric and magnetic fields leads to the polarization conversion at resonance frequency 0.72 THz. The similar method can be utilized for analyzing the conversion at other resonance frequencies, shown as Fig. 5b4–d4. Each magnetic dipole moment contributes to a magnetic field, among which m6, m8, m9 and m10 are perpendicular to the incident electric field Ein and have no contribution to the polarization, while m3, m4, m5 and m7 are along the x-axis and parallel to the incident electric field Ein, thus these parallel magnetic fields contribute to the polarization conversions at the corresponding resonance frequencies. The similar method can be utilized for analyzing the polarization conversion of the incident y-polarization in the backward direction.

Fig. 5
figure 5

Surface current distributions and the induced magnetic fields at resonance frequencies: a 0.72 THz, b 1.26 THz, c 1.94 THz, and d 2.36 THz

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Next, we investigate the influence of CW lengths on AT parameter when the incident wave is x-polarization. In Fig. 6a, as SCW length l2 is fixed on 34 μm and LCW length l1 increases from 82 to 88 μm, the conversion bandwidth and AT parameter improve. Meanwhile, resonance frequency f1 redshifts, but f3 keeps almost unchanged. Similarly, in Fig. 6b, as l1 is fixed on 88 μm and l2 increases from 34 to 40 μm, f3 redshift much, but f1 keeps almost unchanged. These results further confirm that the resonance on f1 is mainly determined by LCW and the resonance on f3 is mainly determined by SCW. The redshifts of f1 and f3 can be explained by plasma resonance condition. Along a CW, the resonance angular frequency can be simplified as \(\omega_{{\text{L}}} = \frac{\pi - \delta }{{\sqrt {\varepsilon_{{{\text{eff}}}} } L_{{{\text{eqv}}}} }}\), where εeff, δ and Leqv represent the effective dielectric constant, phase change and equivalent resonance length of the CW, respectively [33, 34]. Obviously, the resonant frequency is inversely proportional to Leqv, so the length dominated resonance frequencies f1 and f3 redshift with the increase of length. From Fig. 6a and b, it is seen whether l1 or l2 changes, the f2 shifts. As resonance f2 is the coupling results of the LCW and SCW, the frequency shift of f2 is not as much as that of f1 in Fig. 6a and f3 in Fig. 6b. In addition, for the higher-order resonance mode at f4, length l1 has more effect on AT parameter than length l2.

Fig. 6
figure 6

AT parameters of the MM with respect to lengths of a LCW and b SCW

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4 Conclusion

In this paper, a three-working-layer MM structure composed of two orthogonal sub-wavelength grating layers and a simple cross-shaped array layer is designed to realize a broadband LP conversion and near-perfect AT effect in THz regime. Results show that the PCR of the polarization converter exceeds 0.99 in frequency range from 0.41 THz to 2.38 THz, with a relative bandwidth of 141.22%. The AT parameters are larger than 0.64 in frequency range of 0.50 to 2.41 THz with a relative bandwidth of 131.27%, and the near-perfect AT effect with AT parameter beyond 0.8 occurs in two frequency bands, 0.53–1.48 THz and 1.80–2.06 THz, with a relative bandwidth of 94.53% and 13.47%, respectively. By analyzing the electric field distributions, we deduce that the broadband conversion performance originates from hybridized coupling of the two CWs and excitation of higher-order modes in the cross-shaped structure. By analyzing the surface current distributions, we prove that the physical origin of polarization conversion is the cross-coupling between the induced magnetic and incident electric field. Moreover, the influence of structure parameters on the AT parameter and bandwidth are investigated. The comparisons of our results with other reported results [25, 26, 35, 36]are provided in Table 1, which shows that our proposed MM structure can realize relative wider operation bandwidth, higher PCR and AT efficiency. Due to the excellent AT performance, the proposed MM structure has potential applications in THz diodes and system.

Table 1 Performance comparison with other reported results
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