Tunable Duplex Metalens Based on Phase-Change Materials in Communication Range

Metalenses recently have attracted attention because of their more compact size in comparison with conventional lenses; they can also achieve better optical performance with higher resolution. Duplexer is an interesting function of a metalens that can distinguish different sources and divide them into two parts for specific purposes. In this article, we design tunable duplex metalenses with phase-change material Ge2Sb2Te5 for the first time. Two types of special unit cells are designed to modulate the incident lights, and four metalenses are designed based on the two types of unit cells. Specific phase profiles are calculated for different sections of metalens in which the corresponding unit cells are settled; accordingly, the metalenses can focus the incident lights at any positions according to our design. Moreover, the metalenses become selectable via tuning the state of phase-change material, which means that the output light field can be actively controlled. The proposal of our tunable duplex metalenses will offer new opportunities for active three-dimensional imaging or optical coding.


Introduction
Recently, researchers have begun to pay increasing attention to metasurfaces due to their extraordinary performance in electromagnetism fields . As 2D, artificial, photonic metamaterials, metasurfaces can manipulate the electromagnetic wave freely. By shaping the subwavelength resonators of metasurfaces, the amplitude, phase, polarization, and propagation direction of light can be easily controlled [1][2][3][4]. With the completion of new theories and technologies in this field, they have been applied to various functions, such as to ultrathin metalens [5,6], cloaking [7,8], nonlinear device [9,10], holograms [11,12], surface plasmon launcher [13], computing [14,15], biosensing [16,17], switching [18,19] and many novel photonic devices [20,21]. Metalens has become an exciting research topic that holds great promise to the applications of metasurfaces. Metalens can not only show better optical performance but also be smaller and lighter, which is more suitable for compact devices in comparison with conventional lens. Recently, excellent research about metalens have been reported. For example, Khorasaninejad et al. verified the exceptional ability of metalens experimentally [5]. Also, other functions such as chromatic aberration correction [6,[22][23][24] and active tuning [25][26][27] have been proposed or proved. However, there are still various functions to be achieved, owing to the subwavelength manipulation to the electromagnetic field.
Duplexer can divide a light source into two parts for distinct uses. Using this as inspiration, we can design duplex metalens that can separate light sources and focus or image them at designed positions, which is impossible for conventional lenses. This is especially important in beam splitter or

Single-Wavelength Tunable Duplex Metalenses
The structure of the unit cell is depicted in Figure 1a. A nanocuboid waveguide GST was placed on the transparent SiO 2 substrate. The orientation of GST formed an angle θ with x axis as shown in Figure 1b; Figure 1c is the side view of the unit cell. A 30-nm thin transparent film of indium tin oxide (ITO) was sandwiched between the GST and the SiO 2 substrate. This film was treated as a conductive layer to tune the state of GST by electrically heating the structured GST.
In this work, P-B phase shift is adopted to achieve phase control. A left/right circularly polarization (LCP/RCP) incident light passes through the unit cell, propagating along +z direction; by applying the Jones matrix theory, the output field can be simplified as [28]: E E E (1) Here, the incident light ⁄ = 1 √2 ⁄ ( ± * ) is the normalized representation of Jones matrix for LCP and RCP separately, and = ± , where to and te represent the complex transmission coefficients along the optical fast axis and slow axis which is determined by the parameters of the unit cell (p, l, w, h, ε(ω)), and the symbol "±" distinguishes the co-polarization and cross-polarization light, respectively. According to the formula, the incident CP light is divided into two cross polarization outputs. The co-polarization light only receives the amplitude modulation but the cross-polarization light has the 2 phase modulation simultaneously. 2π phase tuning can be achieved for the cross-polarization light if the angle rotates from 0 to π. When the state of the GST is changed, the transmission is changed simultaneously; thus the output of the unit cell can be actively controlled.
The simulation of all the unit cells was performed using finite-element method in frequency domain with the unit cell boundary in xy direction and the open boundary in z direction; the minimum size of the mesh was 20 nm. All unit cells were settled in free space. The information of the system used for computing this structures is as follows: The CPU of the system was Intel(R) Xeon(R) CPU E5-2420 v2; the RAM was 24 GB; the duration of the calculation for one unit cell was approximately several minutes; the HDD space was 1.72 TB. In the simulation, the incident LCP plane wave propagated along +z direction through the substrate. The GST data were obtained from the Here, the incident light E L/R = 1/ √ 2 e x ± i * e y is the normalized representation of Jones matrix for LCP and RCP separately, and α = t o ± t e , where t o and t e represent the complex transmission coefficients along the optical fast axis and slow axis which is determined by the parameters of the unit cell (p, l, w, h, ε(ω)), and the symbol "±" distinguishes the co-polarization and cross-polarization light, respectively. According to the formula, the incident CP light is divided into two cross polarization outputs. The co-polarization light only receives the amplitude modulation but the cross-polarization light has the 2θ phase modulation simultaneously. 2π phase tuning can be achieved for the cross-polarization light if the angle θ rotates from 0 to π. When the state of the GST is changed, the transmission is changed simultaneously; thus the output of the unit cell can be actively controlled.
The simulation of all the unit cells was performed using finite-element method in frequency domain with the unit cell boundary in xy direction and the open boundary in z direction; the minimum size of the mesh was 20 nm. All unit cells were settled in free space. The information of the system used for computing this structures is as follows: The CPU of the system was Intel(R) Xeon(R) CPU E5-2420 v2; the RAM was 24 GB; the duration of the calculation for one unit cell was approximately several minutes; the HDD space was 1.72 TB. In the simulation, the incident LCP plane wave propagated along +z direction through the substrate. The GST data were obtained from the experimental data in reference [47], in which the dielectric function ε(ω) was investigated by infrared spectroscopy and spectroscopic ellipsometry. The conversion efficiency from LCP to RCP and the phase modulation of the unit cell were calculated from the simulations on the unit cell. After optimization at the wavelength of 1.31 µm, the parameters of the unit cell U 1 were determined to be p = 600 nm, h = 700 nm, l = 230 nm and w = 180 nm. According to the theoretical analysis from Equation (1), a part of incident light will be changed into RCP light with phase modulation of ϕ = 2θ. When θ = 0, the conversion efficiency from LCP to RCP for two different GST states is shown in Figure 2a. When the GST remained at amorphous state, the conversion efficiency was as high as 87% at the wavelength of 1.31 µm; in contrast, the conversion efficiency was only 16% at the wavelength of 1.55 µm (used for another purpose). When the GST was in crystalline state, the ε(ω) changed and the conversion efficiency was tuned lower than 5%, which can be treated as turning off in the whole range from 1.31 µm to 1.55 µm. Next, the incident light was fixed at 1.31 µm, at which point the angle of structure was rotated from 0 to π; the corresponding phase modulation is shown in Figure 2b. For amorphous GST, the simulation showed no difference with theoretical analysis; however, the phase modulation linearly covered the entire 2π range. Figure 2c shows that the conversion efficiency remained largely unchanged when the angle was rotated from 0 to π, which indicates that the angle had little influence in the conversion efficiency for two different states of GST. All simulations demonstrated that the unit cell is a good candidate for tunable metalens. experimental data in reference [47], in which the dielectric function ε(ω) was investigated by infrared spectroscopy and spectroscopic ellipsometry. The conversion efficiency from LCP to RCP and the phase modulation of the unit cell were calculated from the simulations on the unit cell. After optimization at the wavelength of 1.31 μm, the parameters of the unit cell U1 were determined to be p = 600 nm, h =700 nm, l = 230 nm and w = 180 nm. According to the theoretical analysis from Equation (1), a part of incident light will be changed into RCP light with phase modulation of = 2 . When = 0, the conversion efficiency from LCP to RCP for two different GST states is shown in Figure 2a. When the GST remained at amorphous state, the conversion efficiency was as high as 87% at the wavelength of 1.31 μm; in contrast, the conversion efficiency was only 16% at the wavelength of 1.55 μm (used for another purpose). When the GST was in crystalline state, the ε(ω) changed and the conversion efficiency was tuned lower than 5%, which can be treated as turning off in the whole range from 1.31 μm to 1.55 μm. Next, the incident light was fixed at 1.31 μm, at which point the angle of structure was rotated from 0 to π; the corresponding phase modulation is shown in Figure 2b. For amorphous GST, the simulation showed no difference with theoretical analysis; however, the phase modulation linearly covered the entire 2π range. Figure 2c shows that the conversion efficiency remained largely unchanged when the angle was rotated from 0 to π, which indicates that the angle had little influence in the conversion efficiency for two different states of GST. All simulations demonstrated that the unit cell is a good candidate for tunable metalens. To focus like a conventional spherical lens, the phase profile of the metalens should satisfy: Where λ is the designed wavelength, f is the focal length. (x, y, 0) represents the position of the unit cell, and (xd, yd, f) represents the coordinate of an arbitrary focal point. According to the simulation result of = 2 , the rotating angle of the unit cell in (x, y, 0) should be: The designed metalens is divided into two equal sections marked as A (x > 0) and B (x < 0). Unit cells U1 were used to construct the metalens M1. Specially, unit cells in section A were arranged as xd = 5 μm, yd = 0, f = 20 μm; unit cells in section B were arranged as xd = -5 μm, yd = 0, f = 20 μm. The radius of the metalens was fixed at 20 μm. The overall structure layout is depicted in Figure 3a. Figure 3b shows the imperfect phase distribution of metalens M1 along the x axis sampling at the center of each unit cell. To focus like a conventional spherical lens, the phase profile of the metalens should satisfy: Where λ is the designed wavelength, f is the focal length. (x, y, 0) represents the position of the unit cell, and (x d , y d , f ) represents the coordinate of an arbitrary focal point. According to the simulation result of ϕ = 2θ, the rotating angle of the unit cell in (x, y, 0) should be: The designed metalens is divided into two equal sections marked as A (x > 0) and B (x < 0). Unit cells U 1 were used to construct the metalens M 1 . Specially, unit cells in section A were arranged as The radius of the metalens was fixed at 20 µm. The overall structure layout is depicted in Figure 3a. Figure 3b shows the imperfect phase distribution of metalens M 1 along the x axis sampling at the center of each unit cell.
The results of all the metalenses were simulated by using finite integrity in time domain and open boundary condition in all directions; the size of the uniform mesh was set at 20 nm along all axes to minimize numerical errors. The entire devices were settled in free space. The duration of the calculation for one metalens was about 24 h (different metalenses spend different time). The light source was set at 1.31 µm LCP plane wave in the simulation. The spatial dimension of incident light is 40 × 40 µm 2 and the LCP plane wave can be expressed as: The incident light was settled at the position lower 1 µm from the metalens. The thickness of the substrate, ITO layer, and GST was 0.2 µm, 0.03 µm, and 0.7 µm, respectively. The output light field was obtained from the position upper 1 µm from the metalens. The spatial dimension of the simulation domain for the full metalens is 40 × 40 × 2.93 µm 3 . The light field was then exported after finite integrity in time domain and calculated by Matlab with Angular Spectrum Diffraction method in the spatial dimension of 40 × 40 × 40 µm 3 .   When the GST in both sections remained in an amorphous state, the incident light was separated into two parts: one focused at the position (5.29, 0.32, 19.9) µm, the other focused at the position (−4.64, 0.32, 20.3) µm. The result is illustrated in Figure 4a. Full width at half maximum (FWHM) of the spots in focal plane were 0.95 µm and 0.97 µm, respectively, which indicates that the metalens achieved subwavelength resolution. The deviance from the expected result (±5, 0, 20) µm came from the discrete phase distribution, as depicted in Figure 3b. By applying appropriate electrical current pulse through the ITO layer, the state of the GST can be changed. When the GST in section A remained in an amorphous state and the GST in section B was changed into a crystalline state, the incident light was focused at the position (−4.64, 0.32, 20.3) µm with FWHM of 0.95 µm, as shown in Figure 4b. When the GST in section A was changed into a crystalline state while the GST in section B was in an amorphous state, the incident light was focused at the position (5.29, 0.32, 19.9) µm with FWHM of 0.97 µm, as shown in Figure 4c. The result of the GST in both sections changing to a crystalline state is shown in Figure 4d. There was no light focusing because the conversion efficiency from LCP to RCP is very low when GST remains in a crystalline state. The results prove that the focusing effect of our tunable duplex metalens can be actively controlled in horizontal direction and will bring some convenience to integrated optical systems.    Figure 5a. FWHM of the spot at z = 15.2 µm was 0.83 µm and FWHM of the spot at z = 24.7 µm was 1.03 µm. This is reasonable because longer focal length means smaller numerical aperture (NA), resulting in a larger focal spot. When the GST in section A stayed in an amorphous state and the GST in section B changed to a crystalline state, the incident light was focused at the position (0.27, 0.27, 15.2) µm with FWHM of 0.83 µm, as shown in Figure 5b. When the GST in section A was changed into a crystalline state but the GST in section B remained in an amorphous state, the incident light was focused at the position (0.35, 0.35, 24.7) µm with FWHM of 1.03 µm, as shown in Figure 5c. When the GST in both sections was changed into crystalline state, the metalens was turned off completely, as seen in Figure 5d. These results demonstrate that our tunable duplex metalens can actively control the focusing in vertical direction, and it may have potential in three-dimension imaging.  Figure 5a. FWHM of the spot at z = 15.2 μm was 0.83 μm and FWHM of the spot at z = 24.7 μm was 1.03 μm. This is reasonable because longer focal length means smaller numerical aperture (NA), resulting in a larger focal spot. When the GST in section A stayed in an amorphous state and the GST in section B changed to a crystalline state, the incident light was focused at the position (0.27, 0.27, 15.2) μm with FWHM of 0.83 μm, as shown in Figure 5b. When the GST in section A was changed into a crystalline state but the GST in section B remained in an amorphous state, the incident light was focused at the position (0.35, 0.35, 24.7) μm with FWHM of 1.03 μm, as shown in Figure 5c. When the GST in both sections was changed into crystalline state, the metalens was turned off completely, as seen in Figure  5d. These results demonstrate that our tunable duplex metalens can actively control the focusing in vertical direction, and it may have potential in three-dimension imaging.

Dual-Wavelength Tunable Duplex Metalenses
When incident light becomes a mixed light of 1.55 μm and 1.31 μm, one single structure apparently can not accomplish the task to distinguish the two different frequencies of lights. Accordingly, another unit cell, U2, needs to be designed to modulate the phase at wavelength of 1.55

Dual-Wavelength Tunable Duplex Metalenses
When incident light becomes a mixed light of 1.55 µm and 1.31 µm, one single structure apparently can not accomplish the task to distinguish the two different frequencies of lights. Accordingly, another unit cell, U 2 , needs to be designed to modulate the phase at wavelength of 1.55 µm. The parameters of U 2 are determined to be p = 600 nm, h = 700 nm, l = 505 nm, and w = 100 nm after optimization. The same period and height as unit cell U 1 was selected to make sure that the different unit cells can be fabricated in the same steps. In this simulation, the LCP plane wave passed through U 2 along +z direction. When the angle θ = 0, the conversion efficiency from LCP to RCP for two different states of GST is shown in Figure 6a. When GST stayed at amorphous state, the conversion efficiency was as high as 97% at the wavelength of 1.55 µm; however, the conversion efficiency was only 23% at the wavelength of 1.31 µm. The two unit cells U 1 and U 2 were designed specially; U 1 let the 1.31 µm light pass but turned off the 1.55 µm light whereas U 2 let the 1.55 µm light pass but turned off the 1.31 µm light, which will be very useful to separate multi-wavelength incident lights. When the GST stayed in crystalline state, the conversion efficiency was lower than 23% in the range from 1.31 µm to 1.55 µm, which can be treated as turning off in comparison with the effect of the amorphous GST. Next, the incident light was fixed at the wavelength of 1.55 µm. The angle of the structure was rotated from 0 to π, affecting the corresponding phase modulation, as depicted in Figure 6b. The results fit well with theoretical analysis because the phase modulation covers the whole 2π range linearly when the GST stays in amorphous state. Figure 6c demonstrates the relationship between conversion efficiency and the rotating angle, and the results prove that the rotating angle has no influence on the conversion efficiency for unit cell U 2 .
Metalens specially; U1 let the 1.31 μm light pass but turned off the 1.55 μm light whereas U2 let the 1.55 μm light pass but turned off the 1.31 μm light, which will be very useful to separate multi-wavelength incident lights. When the GST stayed in crystalline state, the conversion efficiency was lower than 23% in the range from 1.31 μm to 1.55 μm, which can be treated as turning off in comparison with the effect of the amorphous GST. Next, the incident light was fixed at the wavelength of 1.55 μm. The angle of the structure was rotated from 0 to π, affecting the corresponding phase modulation, as depicted in Figure 6b. The results fit well with theoretical analysis because the phase modulation covers the whole 2π range linearly when the GST stays in amorphous state. Figure 6c demonstrates the relationship between conversion efficiency and the rotating angle, and the results prove that the rotating angle has no influence on the conversion efficiency for unit cell U2.  Figure 3d is the overview of the metalens, in which the green structures represent U1 and the red structures represent U2. The phase distribution of the metalens M3 along the x axis is depicted in Figure 3e.
In this simulation, the conditions remain unchanged except the incident light source was changed into mixed light of two wavelengths at 1.55 μm and 1.31 μm. When the GST in both sections stayed in an amorphous state, the unit cells in section A and B could allow the light of 1.55 μm and 1.31 μm to pass through, respectively. The incident 1.55 μm light was focused at the position (4.93, 0.38, 20.2) μm with FWHM of 1.12 μm and the incident 1.31 μm light was focused at the position (−4.96, 0.38, 20.1) μm with FWHM of 1.00 μm, which have been marked as red spot and green spot, respectively, as seen in Figure 7a. The size of the spot was proportional to the wavelength while the focal lengths were equal. When the GST in section A stayed in an amorphous state but the GST in section B changed into a crystalline state, only the 1.31 μm incident light was focused at the position (−4.96, 0.38, 20.1) μm with FWHM of 1.00 μm, as shown in Figure 7b. When the GST in section A changed to a crystalline state but the GST in section B was in an amorphous state, the 1.55 μm incident light was focused at the position (4.93, 0.38, 20.2) μm with FWHM of 1.12 μm. Simultaneously, the 1.31 μm incident light dissipated, as shown in Figure 7c. When the GST in both sections changed to a crystalline state, the metalens completely turned off without any focusing, as shown in Figure 7d. The results show that metalens M3 can separate two different lights and focus them at different  Table 1.  Considering metalens M3 separate the incident in horizontal, here we designed another metalens M4 to separate the incident light in vertical. M4 was also constructed by putting unit cells U1 and U2 together. Unit cells U1 were arranged in section A as xd = 0, yd = 0, f =15 μm, and unit cells U2 were arranged in section B as xd = 0, yd = 0, f =25 μm. The corresponding phase distribution of metalens M4 along the x axis is depicted in Figure 3 Figure 8d shows that the metalens M4 was turned off. These results prove that the tunable duplex metalens M4 can divide the two different lights in a vertical direction and actively control them if necessary. The proposal of the metalens M4 may bring some reference value to three-dimensional colored imaging. Four metalenses (M1, M2, M3 and M4) have been designed to accomplish different functions in this article. The comparison of them can be seen in Table 1.

Conclusions
In this paper, by utilizing phase change material GST, we designed two types of unit cells to modulate the phase of incident light: when the GST stays in amorphous state, U 1 responds to 1.31 µm light but shuts off at the wavelength of 1.55 µm. In contrast, U 2 responds to 1.55 µm light but shuts off at the wavelength of 1.31 µm. When the GST stays in crystalline state, the two unit cells are shut off at either wavelength. On the basis of two other types of unit cells, four metalenses were designed with different functions. Metalens M 1 and M 2 can divide the single-wavelength light into two parts and focus them at different positions along horizontal and vertical directions, respectively. In comparison, the effects of metalenses can be actively controlled by tuning the state of GST on demand. Metalens M 3 and M 4 can separate the dual-wavelength light and focus them at designed positions along horizontal and vertical directions, respectively. By tuning the state of GST, the two metalenses can also achieve different focusing effects as necessary. In summary, our tunable duplex metalenses would make metalens more applicable in photonic devices.