As shown in Fig. 1(a), the basic building block is a GaN nano-fin on a silica substrate. The nano-fin had its basic dimensions fixed (length: 300 nm, width: 100 nm, and period: 330 nm), while its thickness (d1) was varied for the analysis of corresponding optical properties. All simulations were performed using the finite-difference time domain method, which is used to simulate the PCE and transmittance. In this work, incident light was assumed to be normally incident at a wavelength of 633 nm. The polarization state was x-linear polarization (XLP) propagating along the z-direction from bottom to up, as shown in Fig. 1(a). The inclined angle between the x-axis and the long axis of the nano-fin was set at 45°. Under this assumption, the PCE and transmittance were identical to a circular polarized light normally incident on a nano-fin at an arbitrary rotation angle. The XLP incident light was denoted by Ex−in. After passing through the nano-fin, the polarization state varied because of the anisotropy of the nano-fin. Therefore, the output component of the E-field (denoted by Eout) consisted of Ex−out and Ey−out, which represented Ex and Ey components at the output plane, respectively. The PCE can be calculated as follows:
$$\frac{{|{E_{y - out}}{|^2}}}{{|{E_{out}}{|^2}}} \times 100\% =\frac{{|{E_{y - out}}{|^2}}}{{|{E_{x - out}}{|^2}+|{E_{y - out}}{|^2}}} \times 100\%$$
1
For PB-phase metasurfaces, the PCE is occasionally used to evaluate the efficiency. However, emphasis is mostly on the overall efficiency, which is the ratio of the signal light intensity to the incident light intensity. Therefore, transmittance and overall PCE must be determined.
The transmittance is defined as follows:
$$\frac{{{\text{|}}{E_{out}}{|^2}}}{{|{E_{x - in}}{|^2}}} \times 100\% =\frac{{|{E_{x - out}}{|^2}+|{E_{y - out}}{|^2}}}{{|{E_{x - in}}{|^2}}} \times 100\%$$
2
Finally, the overall PCE can be defined as PCE multiplied by transmittance, as given below:
$$\frac{{|{E_{y - out}}{|^2}}}{{|{E_{x - in}}{|^2}}} \times 100\%$$
3
The overall PCE is more appropriate for describing the overall efficiency of a PB-phase unit cell, which considers both transmittance and PCE. Figure 1(b) depicts the transmittance, PCE, and overall PCE as a function of the thickness (d1) of the GaN nano-fin. The square, circle, and triangle represent the transmittance, PCE, and overall PCE, respectively. The peak of the PCE is at d1 = 1050 nm, corresponding to the half-waveplate condition for polarization conversion. However, due to the yellow-band absorption of GaN, caused by Ga vacancies or their complexes33,34, GaN is llossy at λ = 633 nm and the corresponding refractive index is n + ik = 2.29 + 0.061i. Therefore, transmittance decreased with an increase in thickness. After considering the contribution of transmittance, the highest overall PCE was believed to appear at d1 = 950 nm.
Here, we considered three different cases of the MgF2-based antireflection structure for reducing the reflection loss of GaN nano-fins. The first was the GaN nano-fin deposited on a flat MgF2 film over a silica substrate, which is the most intuitive antireflection structure (denoted by the green square in Figure (2). The second (denoted by the red circle) was a GaN nano-fin stacked on a MaF2 nano-fin, which was fabricated through standard lithography and reactive-ion etching. This is a hetero-nano-fin. For convenience, the second nano-fin is called the GaN/MgF2 nano-fin. Moreover, the MgF2 nano-structure had the same geometric parameters as the nano-fin in the x and y directions. Finally, the third nano-fin (denoted by the blue triangle) was a MgF2 nano-fin stacked on the GaN nano-fin and was called a MgF2/GaN nano-fin. Same as the previous case, the geometric parameters of MgF2 and GaN nano-fins were identical in the x and y directions. The simulation results of the overall PCE, PCE, and transmittance as a function of MgF2 thickness are displayed in Fig. 2(a), (b), and (c), respectively. For d1 = 1050 nm, the flat MgF2 film (green square) barely contributed to the PCE, overall PCE, and transmittance even with an increase in the thickness. Therefore, this case was used as a reference. For the GaN/MgF2 nano-fin (red circle), a noticeable oscillation of overall PCE was observed with an increase in MgF2 thickness. When MgF2 thickness increased, the PCE curve oscillated and became lower, which was because the anisotropic OPD of nano-fin is away from the optimal optimized thickness of the half-waveplate (as mentioned in Fig. 1(b)). At the same time, the GaN/MgF2 nano-fin positively contributed to transmittance. Although a 0.5% increase in transmittance was observed for MgF2 = 60 nm, the transmittance was suppressed for most thicknesses. Notably, the overall PCE was significantly improved for the MgF2/GaN nano-fin. The overall PCE increased from 52.7–54% when MgF2 thickness increased from 0 to 140 nm. Although the overall PCE oscillated with varying MgF2 thickness, the overall PCE constantly improved, compared with a bare GaN nano-fin. Because the GaN nano-fin with a thickness of 1050 nm already had the highest PCE, only a tiny increment was observed in the PCE after the addition of the MgF2 nano-structure. Therefore, improvement in the overall PCE is believed to be mainly contributed by transmittance. Figure 2(c) shows a good agreement that transmittance improves from 54.4–55.7%. Figure 2(d), (e), and (f) depict the overall PCE, PCE, and transmittance for d1 = 950 nm, which is the optimal thickness of the bare GaN nano-fin for the overall PCE. Compared with d1 = 1050 nm, both GaN/MgF2 and MgF2/GaN nano-fins showed an improved overall PCE for d1 = 950 nm, as shown in Fig. 2(d). Both PCE and transmittance exhibited significant improvement. The MgF2 nano-structure simultaneously played a role in polarization conversion and antireflection. The increasing thickness allowed the anisotropic OPD to match the optimal half-waveplate condition. Therefore, the PCE for both GaN/MgF2 and MgF2/GaN nano-fins improved with an increase in MgF2 thickness. For transmittance, behaviors of GaN/MgF2 and MgF2/GaN differed with increasing MgF2 thickness. Although GaN/MgF2 enhanced transmittance to 1.3% at a MgF2 thickness of 260 nm, the transmittance was generally reduced for other thicknesses of MgF2. By contrast, the transmittance of MgF2/GaN was constantly improved regardless of the variation in MgF2 thickness. The flat MgF2 layer made no contribution to overall PCE for both d1 = 1050 or 950 nm. This means that the hetero-structure is necessary for antireflection.
For a homogeneous medium, the Goos–Hanchen phase is π when the incident angle is less than the total reflection angle35. In our case, an optical anisotropic nano-fin was considered. The reflected phase is no more exactly equal to π owing to anisotropy, depolarization, and scattering. Here, a simple optical isolator was used for comparison. Right-circularly polarized (RCP) light normally impinges on an optical homogeneous and anisotropy film, that is, a half-waveplate. After traveling to the top of the film–air interface, part of the light reflects and back travels to the input plane. An antireflection structure allows incident and reflection beams to deconstructively interfere. The Jones matrix of the input and reflected light can be represented as follows:
$${E_{out}}={\text{R}}\left( { - \theta } \right)\left( {\begin{array}{*{20}{c}} {{e^{ - i\frac{\Gamma }{2}}}}&0 \\ 0&{{e^{i\frac{\Gamma }{2}}}} \end{array}} \right){\text{R}}\left( \theta \right)\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right){\text{R}}\left( { - \theta } \right)\left( {\begin{array}{*{20}{c}} {{e^{ - i\frac{\Gamma }{2}}}}&0 \\ 0&{{e^{i\frac{\Gamma }{2}}}} \end{array}} \right){\text{R}}\left( \theta \right)\frac{1}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}} 1 \\ { - i} \end{array}} \right)$$
4
where θ is the phase retardation and R is the rotation matrix:
$${\text{R}}\left( { - \theta } \right)=\left( {\begin{array}{*{20}{c}} {\left. {{\text{cos}}(\theta } \right)}&{\left. {{\text{sin}}(\theta } \right)} \\ {\left. { - {\text{sin}}(\theta } \right)}&{\left. {{\text{cos}}(\theta } \right)} \end{array}} \right)$$
5
As the RCP travels at the quarter-waveplate position, the polarization state converts to XLP. At the top of the film–air interface, polarization states of the incident and reflected light are left-circularly polarized (LCR) and RCP, respectively. When light back travels to the quarter-waveplate position, the polarization state converts to y-linear polarization. Finally, at the input plane, the back-reflected light is LCP, which is orthogonal to the input light. Thus, the vector product of the input and reflected light is zero. Therefore, the input and reflected light cannot interfere, and thus, a conventional thin-film antireflection coating is not suitable for reducing the Fresnel reflection loss of a homogeneous and anisotropy film.
In our nano-fin, the Goos–Hanchen reflection phase from the structure is not only equal to π but also has phase anisotropy. Therefore, polarization states of forward and backward propagation light are not completely orthogonal in our discussed PB-phase nano-fin system. We thus can realize antireflection in the PB-phase system.
Figure 3 presents the phase distribution at the upper interface of the nano-fin coated with four antireflection structures: bare nano-fin, nano-fin with a flat MgF2 film, GaN/MgF2 nano-fin, and MgF2/GaN nano-fin. The corresponding phase distributions are presented in Fig. 3(a), (b), (c), and (d). Here, the thickness of GaN and MgF2 was fixed at 950 and 140 nm, respectively. To analyze phase distribution intuitively, we calculated the standard deviation of the phase and marked it as σ. Figure 3(a) depicts that the σ of the bare nano-fin is 0.7201, which can be considered as a reference. We first observed that the σ of the nano-fin with the flat MgF2 film was 0.7212, as shown in Fig. 3(b). Compared with the reference, the flat MgF2 film contributed inconspicuously to the antireflection effect. The σ of the GaN/MgF2 nano-fin was 0.7565. Compared with the flat MgF2 film, phase distribution on the interface changed drastically, corresponding to the decreases in efficiency. As shown in Fig. 2(d), the GaN/MgF2 nano-fin exhibited lower overall PCE at a MgF2 thickness of 140 nm. Finally, the σ of the MgF2/GaN nano-fin was 0.3277, as shown in Fig. 3(d). Compared with the flat MgF2 film, the variation in phase distribution on the interface was mitigated and transmittance was increased.
As mentioned above, GaN suffers from absorption loss in the visible range. The optical characteristics of material loss make improving the overall PCE difficult. Therefore, materials without absorption loss in the visible range must be identified. For example, Choudhury et al. comprehensively surveyed a dielectric material for a dielectric metasurface for visible and IR spectral ranges36. They suggested silicon nitride (Si3N4) and titanium oxide (TiO2) as good metasurfaces in visible range applications.
In addition to absorption loss, the refractive index is crucial for fabrication. Both the propagation phase and anisotropy are positively related to the refractive index. Therefore, a high aspect ratio is required to accumulate sufficient phase modulation for materials with a relatively low refractive index. We thus focused on dielectric materials that can be applied in the visible range and have a high refractive index. Figure 4 depicts the optimized thickness (d2) of nano-fins based on various materials. As shown in the inset of Fig. 4, geometric parameters were all fixed (length, 300 nm; width, 100 nm; and period, 330 nm). Recently, Prof. Tsai’s group demonstrated a GaN structure with an aspect ratio as high as 10–20 for high efficiency18,37,38. However, fabricating a nano-fin with such a high aspect ratio is extremely challenging. Thus, we used the geometric parameters of the GaN nano-fin as a benchmark for the state-of-the-art fabrication. Nano-fins with an optimized thickness higher than that of GaN nano-fin were excluded as candidates for a high-efficiency dielectric metasurface. According to this criterion, Si3N4, tantalum pentoxide (Ta2O5), and sputtering TiO2 are not suitable materials. Compared with GaN, both anatase and rutile TiO2 have the advantages of higher efficiency and lower aspect ratio. However, the crystalline phase control of TiO2 during deposition is severe. Thus, crystalline TiO2 is also excluded. Amorphous silicon (a-Si) is a material with a high refractive index and can be easily processed using standard semiconductor-compatible manufacturing techniques. However, it suffers from huge absorption loss in the visible range, which makes it inappropriate. Finally, Nb2O5 offers a fair refractive index and low absorption (n + ik = 2.32 + 0i), making it suitable for visible applications. Therefore, we believe that Nb2O5 is suitable for high-efficient dielectric metasurfaces.
We simulated the optical response of a nano-fin composed of Nb2O5 patterned on the same substrate at an incident wavelength of 633 nm, as shown in Fig. 5(a). The polarization state was XLP propagating along the z-direction. Figure 5(b) depicts the transmittance, PCE, and overall PCE as a function of the thickness (d3) of the Nb2O5 nano-fin. Corresponding to the half-waveplate condition for polarization conversion, the peak of the PCE appeared at d3 = 1000 nm. Because Nb2O5 does not suffer absorption loss in the visible range, which can be easily observed, transmittance did not decrease with an increase in d3. Therefore, same as the PCE, the highest overall PCE appeared at d3 = 1000 nm. As expected, the overall PCE of Nb2O5 was considerably higher than that of the GaN nano-fin. By contrast, an obvious dip appeared at d3 = 1100 nm, which is caused by guided-mode resonance39.
We also considered three different cases of the MgF2-based antireflection layer for reducing the reflection loss of the Nb2O5 nano-fin: a flat MgF2 film under the nano-fin, Nb2O5/MgF2 nano-fin, and MgF2/Nb2O5 nano-fin, which are denoted by a green square, red circle, and blue triangle, respectively, in Fig. 6. Here, the thickness (d3) of Nb2O5 for all cases was 1000 nm. The simulation results of the overall PCE, PCE, and transmittance as a function of MgF2 thickness are presented in Fig. 6(a), (b), and (c), respectively. First, a flat MgF2 film in the Nb2O5 system barely contributed to the PCE, overall PCE, and transmittance. Therefore, this case was marked as a reference. Both Nb2O5/MgF2 and MgF2/Nb2O5 nano-fins exhibited obvious enhancement of the overall PCE (Fig. 6(a)). The highest overall PCE of both Nb2O5/MgF2 and MgF2/Nb2O5 nano-fins was 94.6% when MgF2 thickness was 160 and 140 nm, respectively. The thickness of the Nb2O5 nano-fin (d3 = 1000 nm) was already an optimal optimized half-waveplate. For both Nb2O5/MgF2 and MgF2/Nb2O5 nano-fins, an additional MgF2 layer forced the nano-fin away from the optimal optimized thickness of the half-waveplate and decreased the PCE (Fig. 6(b)). Thus, transmittance enhancement contributed to all the improvement in the overall PCE. Figure 6(c) shows a transmittance of 94.7% for both Nb2O5/MgF2 and MgF2/Nb2O5 nano-fins. Although both these nano-fins exhibited a good enhancement of the overall PCE, the MgF2/Nb2O5 nano-fin attained thickness efficiency at a lower MgF2 thickness. Thus, the antireflection layer located on the top of a nano-fin is considered more suitable.
As mentioned above, MgF2 contributed negatively to the PCE and reduced the overall PCE when the nano-fin thickness was under an ideal half-waveplate condition. Therefore, we further optimized the hetero-nano-fin to ensure that the Nb2O5 nano-fin was slightly thinner than the ideal half-waveplate condition. Figure 7 presents the overall PCE as a function of MgF2 and Nb2O5 thicknesses. The color represents the overall PCE. Warm and cold colors represent the enhanced and suppressed overall PCE, respectively, compared with the bare Nb2O5 nano-fin. In this calculation, MgF2 was located on the top of the Nb2O5 nano-fin. As the thickness of Nb2O5 was 950 nm, the overall PCE improved from 91–96% when MgF2 thickness increased from 0 to 140 nm. At this time, MgF2 made up for a shortage in the ideal half-waveplate condition. MgF2 plays a role in both polarization conversion and antireflection. Moreover, the overall PCE of the MgF2/Nb2O5 nano-fin was 1.7 times higher than that of the MgF2/GaN nano-fin. Therefore, we believe that the MgF2/Nb2O5 hetero-nano-fin is a highly efficient candidate for dielectric metasurfaces.