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Switchable tri-function terahertz metasurface based on polarization vanadium dioxide and photosensitive silicon

Jiu-Sheng Li and Xiang-Jun Li

Open Access Open Access

Abstract

A terahertz switchable metasurface with the function of absorption and polarization conversion is proposed. It consists of metal pattern layer - dielectric layer - VO2 layer - dielectric layer - metal pattern layer, and the photosensitive silicon is embedded in the metal pattern. When VO2 is in insulated state, the metasurface behaves as a linear polarization converter. The polarization conversion rate (PCR) is more than 90% at two frequency bands of 1.64 THz ∼ 1.91 THz and 2.35 THz ∼ 2.75 THz. The polarization converter has good asymmetric transmission ability. Moreover, the polarization conversion performance can be dynamically controlled by changing the conductivity of the photosensitive silicon. When VO2 is in metallic state, the metasurface becomes a terahertz bidirectional absorber, which exhibits different absorption properties under TE and TM waves with the maximum absorptance reaching to 100%. In addition, the absorption of TE and TM terahertz waves can be controlled at the specific frequency by changing the conductivity of photosensitive silicon. We also explore the application of dynamic control of polarization waves in the near-field image display.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent years, various types of terahertz functional devices have been proposed, such as absorbers, modulators, polarization converters and so on [1–6]. By aiding graphene, photosensitive silicon, and phase change materials [7–9], the metasurface can realize the dynamic control of terahertz waves [10–12]. For example, Ye et al [13] proposed a broadband terahertz absorber based on graphene, whose absorptance varies with the change of graphene chemical potential. Yuan et al [14] designed a switchable absorber between single band and dual band by using photosensitive silicon. Dong et al [15] demonstrated a hybrid metal-VO2 metasurface to realize terahertz wave asymmetric transmission switch by controlling the state of VO2. Shabanpour et al [16] proposed a metamaterial absorber consisting of honeycomb and VO2 films. With the transition of VO2 from the insulated state to the metallic state, the absorber exhibited a strong absorptivity above 90% up to the incidence angle of 87°. However, these reported structures have only a single function regulation. In 2020, Song et al [17] verified a bifunctional metasurface with the function from absorber to reflective polarization converter via VO2. Zhang et al [18] released a bifunctional metasurface based on VO2 and Dirac semimetal, which realized the polarization conversion for transmission and reflection modes. In 2021, Li et al [19] proposed a dual-functional terahertz absorber based on VO2 and graphene, which can simultaneously realize absorption for linear and circular terahertz wave incidence. But there are few studies about terahertz metasurfaces with asymmetric transmission and absorption switching.

In this article, a switchable multifunctional metasurface is designed to realize asymmetric transmission and absorption in terahertz region. Compared with the phase change material GST, the phase change temperature of VO2 is lower (only 68°), which makes it easy to control. Here, we employ VO2 as the hybrid metasurface phase change medium. When VO2 is in insulated state, the metasurface works as the linear polarization converter, which can realize asymmetric transmission in two bands. When VO2 is in metallic state, the metasurface becomes as the bidirectional absorber, which can achieve good absorption for incident terahertz waves with different polarization states. In addition, the dynamic regulation of polarization conversion and absorption can also be achieved by external light irradiation. It is worth noting that the absorption difference between TE and TM wave at the specific frequency can be controlled by changing conductivity of photosensitive silicon, and it realize the dynamic near-field image display. The designed metasurface shows potential advantages in terahertz communication and radar monitoring.

2. Structure design and theory calculation

The proposed structure includes top metal pattern layer, upper dielectric layer, VO2 film, lower dielectric layer and bottom metal pattern layer, as shown in Fig. 1. The VO2 state switching can be realized by employing resistive heater electrode for Joule heating actuation. The metallic via hole is drilled through the bottom dielectric layer to connect the VO2 layer and the bottom metal split ring, and the bottom metal split ring is connected to the ground layer as the negative pole. In the meantime, the VO2 film and the top metal split ring are also connected by the metallic via hole drilled through the top dielectric layer. The top metal split ring acts as the positive electrode. Therefore, the metasurface function switching can be achieved by applying the DC bias current. The material of metal pattern layer is copper with the conductivity σ = 5.8 × 107 S / m, and photosensitive silicon is embedded in the split ring. The dielectric layer is SiO2 with the constant ɛ of 3.75. The structural parameters optimized by CST Microwave Studio software are as follows: P =58µm, a = 48µm, w =12µm, g = 8µm, metal pattern thickness of 0.5µm, SiO2 layer thickness of 4.5µm, and VO2 thickness of 1µm. The bottom metal pattern is obtained by the clockwise rotation of the top structure by 90°. Both x- and y-directions are set to periodic boundary conditions, while z-direction is arranged floquet port. The permittivity of VO2 can be described by the Drude model [20]

$$\varepsilon (\omega ) = {\varepsilon _\infty } - \frac{{{\omega _p}^2(\sigma )}}{{{\omega ^2} + i\gamma \omega }}$$
where the ɛ∞= 12 is the dielectric permittivity at the infinite frequency, γ = 5.75×1013 rad/s is the collision frequency. And the plasma frequency ωp can be approximately defined as
$${\omega _p}^2(\sigma ) = \frac{\sigma }{{{\sigma _{ref}}}}{\omega _p}^2({\sigma _{ref}})$$
where σref = 3.0×105 S/m and ωp(σ)= 1.4×1015 rad/s.

 figure: Fig. 1.

Fig. 1. Diagram of the proposed metasurface structure based on metal-VO2-dielectric. (a) 3Dview of the unit cell, (b) Side view of the unit cell, (c) Top view of the unit cell, (d) Bottom view of the unit cell.

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The metasurface is placed on the x-y plane, so the incident and transmitted electric fields of terahertz wave in the + z direction can be expressed as

$$\left\{ \begin{array}{l} {E_i}(r,t) = \left( {\begin{array}{{c}} {{I_x}}\\ {{I_y}} \end{array}} \right){e^{i(kz - \omega t)}}\\ {E_t}(r,t) = \left( {\begin{array}{{c}} {{T_x}}\\ {{T_y}} \end{array}} \right){e^{i(kz - \omega t)}} \end{array} \right.$$
where ω is the frequency of terahertz wave, k is the wave vector, Ix and Iy are the complex amplitude of incident wave in x and y direction, Tx and Ty are the complex amplitude of transmitted wave in x and y direction. For linearly polarized waves, the relationship between the complex amplitudes of the incident wave and the transmitted wave can be described by Jones matrix [21]
$$\left( {\begin{array}{{c}} {{T_x}}\\ {{T_y}} \end{array}} \right) = \left( {\begin{array}{{cc}} {{T_{xx}}}&{{T_{xy}}}\\ {{T_{yx}}}&{{T_{yy}}} \end{array}} \right)\left( {\begin{array}{{c}} {{I_x}}\\ {{I_y}} \end{array}} \right) = {T_{\textrm{lin}}}^F\left( {\begin{array}{{c}} {{I_x}}\\ {{I_y}} \end{array}} \right)$$
Txx represents the polarization of incident wave and transmitted wave along x direction; Tyy refers to the polarization of incident wave and transmitted wave along y direction; Txy depicts the polarization of incident wave along y direction and transmitted wave along x direction; Tyx denotes the polarization of incident wave along x direction and transmitted wave along y direction. The subscript “lin” represents linear polarization, while the superscript “F” and “B” are both forward and backward incidence, respectively. According to the reciprocity theorem, the Jones matrix of the backward direction can be described as
$${T_{\textrm{lin}}}^B = \left( {\begin{array}{{cc}} {{T_{xx}}}&{ - {T_{yx}}}\\ { - {T_{xy}}}&{{T_{yy}}} \end{array}} \right)$$
Here the total transmission of x-polarized wave along the forward and backward direction can be written as
$$\left\{ \begin{array}{l} {T^F}\textrm{ = }{|{{T^F}_{xx}} |^2} + {|{{T^F}_{yx}} |^2}\\ {T^B}\textrm{ = }{|{{T^B}_{xx}} |^2} + {|{{T^B}_{yx}} |^2} \end{array} \right.$$
So the asymmetric parameters of the polarized waves can be derived from Jones matrix
$$\left\{ \begin{array}{l} {\triangle_{\textrm{lin}}}^\textrm{X}\textrm{ = }{|{{T^F}_{xx}} |^2} + {|{{T^F}_{yx}} |^2} - {|{{T^B}_{xx}} |^2} - {|{{T^B}_{yx}} |^2} = {T^F} - {T^B}\\ {\triangle_{\textrm{lin}}}^\textrm{Y}\textrm{ = }{|{{T^F}_{yy}} |^2} + {|{{T^F}_{yx}} |^2} - {|{{T^B}_{yy}} |^2} - {|{{T^B}_{yx}} |^2} ={-} {\triangle_{\textrm{lin}}}^\textrm{X} \end{array} \right.$$

3. Results analysis

When the VO2 is in insulated state (the conductivity is 0 S / m), the designed metasurface behaves as a transmissive polarization converter. Figure 2 shows the forward (+ z) and backward (-z) transmission coefficients for terahertz wave illumination on the proposed metasurface. It can be seen from Fig. 2(a) that when the terahertz wave is forward transmission, tyx is maintained above 0.6 in two frequency bands of 1.58 THz – 1.85 THz and 2.49 THz – 2.85 THz. In addition, the tyx reaches 0.87 at 1.77 THz and 2.6 THz. On the contrary, txy is less than 0.1 in the whole working frequency band. When the terahertz wave is backward transmission, the transmission curves of txy and tyx are exactly opposite compared with the results of the forward transmission, which indicate that the structure realize polarization conversion for both x and y polarized wave and has good asymmetric transmission effect.

 figure: Fig. 2.

Fig. 2. Transmission curves for terahertz wave illumination on the proposed metasurface, (a) forward transmission (+ z), (b) backward transmission (-z).

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The calculated asymmetric transmission curve is shown in Fig. 3. When the terahertz wave propagates along forward direction, the asymmetric transmission curve maintains a high transmission coefficient in two bands with the transmission coefficient reaching 0.76 at frequency of 1.77 THz and 2.6 THz. It proves that only the polarization conversion is allowed from x-polarized wave to y-polarized wave. When the terahertz wave propagates backward direction, the curves of Δy are exactly opposite to that of Δx. It indicates that the polarization conversion is allowed from y-polarized wave to x-polarized wave within two bands. Figure 4 displays the PCR curve of the designed metasurface. As can be seen from the figure, PCR is more than 90% in the two bands of 1.64 THz ∼ 1.91 THz and 2.35 THz ∼ 2.75 THz when the incident terahertz wave is forward transmission. It indicates that the transmission wave is mainly composed of cross polarization wave, and x-polarized wave completely transforms into y-polarized wave. When the incident terahertz wave is backward transmission, the PCR curve of y-polarized wave is completely consistent with that of x-polarized wave, which conforms to the principle of asymmetric transmission.

 figure: Fig. 3.

Fig. 3. Asymmetric transmission curves of the designed metasurface under forward incidence of x-polarized wave and backward incidence of y-polarized wave.

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 figure: Fig. 4.

Fig. 4. PCR curve, (a) forward incidence for x-polarized wave, (b) backward incidence for y-polarized wave.

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In order to analyze the polarization conversion effect of the metasurface, Fig. 5 gives the current distribution of the top and bottom metal rings at 1.7 THz, 2.6 THz and 2.74 THz as the incident terahertz wave propagates forward direction. When f = 1.7 THz, the currents at the top and bottom of the two metal rings are opposite, so two excited current loops between the top and bottom metal rings cause two magnetic dipole moments (m1) and (m2), which induce magnetic fields (H1) and (H2). Because the induced magnetic field (H1) is parallel to the incident electric field (E), and the cross coupling between them produces the polarization conversion effect [22]. At the same time, the electric dipole moment (p1) at the bottom metal ring is also excited, resulting in an induced electric field (E1). Since the incident electric field (E) is parallel to the induced electric field (E1), it cannot lead to cross coupling and has no effect on polarization conversion. When f = 2.6 THz, the magnetic dipole moment (m1) is induced by the opposite current at the bottom of the double-layer metal ring. The induced magnetic field (H1) is perpendicular to the incident electric field E, which cannot produce polarization conversion effect. In the meantime, the electric dipole moment (p1) and (p2) cause the induced electric field (E1) and (E2). The induced electric fields (E2) are excited by two mutually opposite p2 (see Fig. 5b). The induced electric field (E1) is parallel to the incident electric field E and the induced electric field (E2) is perpendicular to the incident electric field (E), which cause the cross-coupling. Therefore, at the frequency of 2.6 THz, both the cross coupling of the induced electric field (E2) and the incident electric field (E) produce polarization conversion effect. Similarly, when f = 2.74 THz, the induced magnetic field (H1) is parallel to the incident electric field, and the induced electric field (E2) is perpendicular to the incident electric field, which contribute to producing the polarization conversion effect together.

 figure: Fig. 5.

Fig. 5. Current distribution of double layer metal ring nuder different operating frequencies, (a) f = 0.87 THz, (b) f = 1.15 THz, (c) f = 2.3 THz.

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The dynamic control of the polarization conversion is studied by changing the conductivity of the photosensitive silicon. Figure 6 exhibits the curves of PCR and asymmetric transmission as the conductivity of photosensitive silicon changes. When the conductivity of photosensitive silicon varies from 1 S/m to 50000 S/m, PCR is gradually inhibited. The peak value at the first band is reduced from 100% to 20%. The PCR tends to 0 in the second band when the conductivity of photosensitive silicon increases to 50000 S/m. At this time, the absolute value of asymmetric transmission decreases from 0.76 to 0 with the increase of the conductivity of photosensitive silicon. It reveals that the asymmetric transmission is realized by changing the conductivity of photosensitive silicon.

 figure: Fig. 6.

Fig. 6. Influence of photosensitive silicon conductivity on polarization conversion performance, (a) PCR, (b) asymmetric transmission coefficient.

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Next, when the VO2 film is converted to metallic state (The conductivity changes from 0 S/m to 200000 S/m), the proposed metasurface behaves as terahertz bidirectional absorber, which can achieve high efficiency absorption for both forward transmission and backward transmission terahertz wave. Figure 7 illustrates the absorption curves of TE and TM waves for forward and backward directions without pump laser irradiation (i.e. σSi = 0 S/m). When the terahertz wave is forward transmission, the absorption of TE and TM waves at 1.23 THz and 1.86 THz are 95% and 98%, respectively. When the terahertz wave is backward transmission, the absorption of TE and TM waves at 1.23 THz and 1.86 THz are 98% and 95%, respectively.

 figure: Fig. 7.

Fig. 7. Absorption curves of TE and TM waves without external laser irradiation (σSi = 0 S/m), (a) forward incidence, (b) backward incidence.

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Figure 8 shows the electric field distribution of the split ring and VO2 film with the photosensitive silicon conductivity of σSi = 0 S/m. As shown in Fig. 8(a), a pair of positive and negative induced charges accumulate in the upper and lower sides of the split ring under the TE wave forward incidence at f = 1.23 THz, which forms the galvanic resonance. The charge distribution flow direction in the VO2 film is opposite to that of the split ring. Therefore, the strong coupling between the two layers generates the magnetic dipole resonance, which consumes the electromagnetic energy and induces an absorption peak of 1.23 THz. Figure 8(b) depicts the electric field distribution of TM wave at f = 1.23 THz. In Fig. 8(c), at f = 1.86 THz, it can be noted that two pairs of positive and negative induced charges are generated on the split ring and VO2 film under the TM wave forward incidence, which forms an electric quadrupole, resulting in second harmonic magnetic resonance. Figure 8(d) shows the electric field distribution of f = 1.86 THz under TE wave backward incidence. The absorption peak is generated at frequency of 1.86 THz for the high-order electro-magnetic resonance (electric quadrupole and second-order harmonic magnetic resonance). Figure 9 displays the comparison of the electric field distribution of the proposed metasurface under normal incidence of TE and TM terahertz waves with the conductivity changing of the photosensitive silicon. Figure 9(a) gives the electric field distribution of TE wave forward incidence with σSi= 0 S/m and 500000 S/m. It can be found that the electric field distribution at 1.23 THz under σSi = 0 S/m is similar to that of 1.42 THz under σSi = 500000 S/m, indicating that the increase of the conductivity of photosensitive silicon promotes blue shift of the absorption peak. The absorption peaks at 1.23 THz and 1.42 THz are both induced by the same electromagnetic resonance. Figure 9(b) shows the electric field distribution of TM wave forward incidence at σSi= 0 S/m and 500000 S/m. Compared with the electric field distribution of 1.86 THz at σSi= 0 S/m, the electric field energy of 2.09 THz at σSi= 500000 S/m is weakened, which proves that the absorption performance is suppressed and the absorption peak decreases under this state. It is worth noting that a new absorption peak appears at 1.28 THz due to the increase in the conductivity of photosensitive silicon, resulting in the conduction state of the left and right charges in the split ring. The new electromagnetic resonance mode leads to the absorption phenomenon at 1.28 THz.

 figure: Fig. 8.

Fig. 8. Electric field distribution of split ring layer and VO2 layer when σSi = 0 S/m, (a) forward incidence TE wave when f= 1.23 THz, (b) backward incidence TM wave when f= 1.23 THz, (c) forward incidence TM wave when f= 1.86 THz, (d) backward incidence TE wave when f= 1.86 THz.

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 figure: Fig. 9.

Fig. 9. Comparison of electric field distribution between split ring and VO2 layer when σSi = 0 S/m and 500000 S/m, (a) forward incidence of TE wave, (b) forward incidence of TM wave.

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Figure 10 studies the effect of oblique incidence angle on absorption. When σSi= 0 S/m, the absorptance remains above 0.9 with the incident angle of TE wave varying from 0° to 40° at frequency of 1.23 THz, as shown in Fig. 10(a). As the oblique incidence angle is larger than 40°, the absorption of the proposed metasurface decreases obviously due to the continuous weakening of the magnetic field component along the Y axis and the magnetic resonance cannot be excited any longer. When σSi = 0 S/m, the incident angle of TM wave changes from 0° to 80°, the absorptance maintains above 0.8 at 1.86 THz and the absorption peak has a slight blue shift, as displayed in Fig. 10(b). This is because that the magnetic field component along the Y axis is almost unchanged at different angles, resulting in no influence on the absorption of TM wave. Similarly, when σSi= 500000 S/m, the incident angle of TE wave increases from 0° to 50°, and the absorptance keeps above 0.9 at frequency of 1.42, as shown in Fig. 10(c). Figure 10(d) gives the absorption change of TM wave at σSi= 500000 S/m. When the incident angle changes from 0° to 80°, the absorptance holds above 0.8 at 1.28 THz. It can also be noted that the absorption peak at high frequency band increases under large incident angle, which is caused by the high-order resonant mode. The results show that the designed metasurface still has good absorption performance under large oblique incidence angle.

 figure: Fig. 10.

Fig. 10. Absorption spectrum with various incident angles, (a) TE wave at σSi= 0 S/m, (b) TM wave at σSi = 0 S/m, (c) TE wave at σSi= 500000 S/m, (d) TM wave at σSi= 500000 S/m.

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Figure 11(a) and 11 (b) exhibit the absorption curves under TE and TM wave incidence with the conductivity of photosensitive silicon σSi= 0 S/m and σSi= 500000 S/m, respectively. When σSi= 0 S/m, one can see that TE wave is completely reflected while the TM wave is perfect absorptance at 1.86 THz. There is a huge absorption rates difference between TE/TM polarization wave incidence. When σSi= 500000 S/m, the absorptance of both TE and TM polarization waves are less than 0.2. Based on the phenomenon, the near-field imaging effect can be achieved by using the polarization absorption differentiation characteristics. Here, we design a “CJLU” pattern by using two different types of unit cells arranged in the inner and outer regions of the alphabetic box. The pattern arrangement of the top and bottom metal layers is shown in Fig. 12(a). The monitor observation distance is 20 µm from the surface of the proposed metasurface. The results in Fig. 12 are derived from full-wave simulation of the overall structure. When σSi= 0 S/m, there will be obvious reflection differences in different regions under TE and TM waves incidence. When σSi = 500000 S/m, the “CJLU” pattern will be blurred at f = 1.86 THz. From Fig. 12(b) and 12(d), it can be found that the top layer images appear complementary states under the normal incidence of TE and TM waves when σSi = 0 S/m. Moreover, under normal incidence of the same polarized wave, the top layer pattern image and the bottom layer pattern image are also opposite. From Fig. 12(c) and 12(e), in can be found that the pattern image has disappeared obviously when σSi = 500000 S/m. That is to say, the “CJLU” pattern image can be switched by changing the conductivity of photosensitive silicon.

 figure: Fig. 11.

Fig. 11. Absorptance of TE and TM waves (a) at σSi = 0 S/m, (b) at σSi = 500000 S/m.

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 figure: Fig. 12.

Fig. 12. Pattern design and near-field images, (a) Top and bottom array layouts, (b) Top image display at σSi = 0 S/m, (c) Top image display at σSi = 500000 S/m, (d) Bottom image display at σSi = 0 S/m, (e) Bottom image display at σSi = 500000 S/m.

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4. Conclusion

A multifunctional terahertz metasurface based on VO2 and photosensitive silicon is designed. The switching function between asymmetric transmission and absorption can be achieved by utilizing the phase transition characteristics of VO2. When VO2 is in the insulated state, the PCR is above 90% in two bands of 1.64 THz ∼ 1.91 THz and 2.35 THz∼ 2.75 THz. The designed metasurface has good asymmetric transmission ability. Moreover, the polarization conversion performance can be controlled by changing pump light intensity illumination on photosensitive silicon. When VO2 is in metallic state, the absorptance of forward incidence TE wave (backward incidence TM wave) is 95% at 1.23 THz and that of forward incidence TM wave (backward incidence TE wave) is 98% at 1.86 THz. With the increase of photosensitive conductivity, the absorption difference between TE wave and TM wave is gradually reduced at the specific frequency. Based on this phenomenon, the application of dynamic near-field image display is studied. The designed structure provides a new idea for future multifunctional metasurface device.

Funding

National Natural Science Foundation of China (61831012, 61871355); Talent project of Zhejiang Provincial Department of science and technology (2018R52043); Zhejiang Key R & D Project of China (2021C03153, 2022C03166); Research Funds for the Provincial Universities of Zhejiang (2020YW20, 2021YW86); Zhejiang Lab (2019LC0AB03).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Figures (12)
Fig. 1.
Fig. 1. Diagram of the proposed metasurface structure based on metal-VO2-dielectric. (a) 3Dview of the unit cell, (b) Side view of the unit cell, (c) Top view of the unit cell, (d) Bottom view of the unit cell.

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Fig. 2.
Fig. 2. Transmission curves for terahertz wave illumination on the proposed metasurface, (a) forward transmission (+ z), (b) backward transmission (-z).

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Fig. 3.
Fig. 3. Asymmetric transmission curves of the designed metasurface under forward incidence of x-polarized wave and backward incidence of y-polarized wave.

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Fig. 4.
Fig. 4. PCR curve, (a) forward incidence for x-polarized wave, (b) backward incidence for y-polarized wave.

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Fig. 5.
Fig. 5. Current distribution of double layer metal ring nuder different operating frequencies, (a) f = 0.87 THz, (b) f = 1.15 THz, (c) f = 2.3 THz.

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Fig. 6.
Fig. 6. Influence of photosensitive silicon conductivity on polarization conversion performance, (a) PCR, (b) asymmetric transmission coefficient.

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Fig. 7.
Fig. 7. Absorption curves of TE and TM waves without external laser irradiation (σSi = 0 S/m), (a) forward incidence, (b) backward incidence.

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Fig. 8.
Fig. 8. Electric field distribution of split ring layer and VO2 layer when σSi = 0 S/m, (a) forward incidence TE wave when f= 1.23 THz, (b) backward incidence TM wave when f= 1.23 THz, (c) forward incidence TM wave when f= 1.86 THz, (d) backward incidence TE wave when f= 1.86 THz.

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Fig. 9.
Fig. 9. Comparison of electric field distribution between split ring and VO2 layer when σSi = 0 S/m and 500000 S/m, (a) forward incidence of TE wave, (b) forward incidence of TM wave.

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Fig. 10.
Fig. 10. Absorption spectrum with various incident angles, (a) TE wave at σSi= 0 S/m, (b) TM wave at σSi = 0 S/m, (c) TE wave at σSi= 500000 S/m, (d) TM wave at σSi= 500000 S/m.

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Fig. 11.
Fig. 11. Absorptance of TE and TM waves (a) at σSi = 0 S/m, (b) at σSi = 500000 S/m.

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Fig. 12.
Fig. 12. Pattern design and near-field images, (a) Top and bottom array layouts, (b) Top image display at σSi = 0 S/m, (c) Top image display at σSi = 500000 S/m, (d) Bottom image display at σSi = 0 S/m, (e) Bottom image display at σSi = 500000 S/m.

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Equations (7)

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ε ( ω ) = ε ∞ − ω p 2 ( σ ) ω 2 + i γ ω
ω p 2 ( σ ) = σ σ r e f ω p 2 ( σ r e f )
{ E i ( r , t ) = ( I x I y ) e i ( k z − ω t ) E t ( r , t ) = ( T x T y ) e i ( k z − ω t )
( T x T y ) = ( T x x T x y T y x T y y ) ( I x I y ) = T lin F ( I x I y )
T lin B = ( T x x − T y x − T x y T y y )
{ T F  =  | T F x x | 2 + | T F y x | 2 T B  =  | T B x x | 2 + | T B y x | 2
{ △ lin X  =  | T F x x | 2 + | T F y x | 2 − | T B x x | 2 − | T B y x | 2 = T F − T B △ lin Y  =  | T F y y | 2 + | T F y x | 2 − | T B y y | 2 − | T B y x | 2 = − △ lin X
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