## Abstract

In this paper, we propose a simple model, a combination of effective medium theory and the Jones matrix, to analyze the optical properties of a dielectric PB-phase metasurface with an arbitrary incident polarization state. The optical properties, such as the polarization conversion efficiency spectrum, rotation-angle-dependent phase modulation, and phasor diagram, shows a fair agreement with the finite difference time domain method results. This model provides a fast and sufficient accuracy compared to the time-consuming finite element methods. Moreover, the shortness of the proposed model is also discussed.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

A metasurface, one kind of two-dimensional metamaterial, has presented exceptional abilities for controlling the properties of electromagnetic waves. It has been widely presented exceptional abilities for engineering the properties of electromagnetic waves various designs of the unit cell, for example, split ring resonator (SRR) [1,2], Mie resonance [3], Huygens metasurface [4], coupled antenna [5], and so on. The above-mentioned unit cells of metasurfaces are based on the resonance, which usually performs a limited spectral range. Pancharatnam-Berry phase (PB-phase) not based on the resonance but based on the geometric phase of light contribution was first discussed by Pancharatnam. It may be related to the Berry phase under certain conditions [6]. Pancharatnam showed that PB-phase of light depends on the trajectory of polarization on the Poincare sphere. For a cyclic evolution of polarization, the trajectory of polarization on the Poincare sphere is a closed curve. In this case, PB-phase equals half of the solid angle subtended at the origin by the area enclosed by the closed curve on the Poincare sphere. Besides the PB-phase, the EM-wave also acquires an additional optical path contributed to a phase delay, named dynamic phase. A metasurface based on the PB-phase structure is named PB-phase metasurface. The PB-phase metasurface, especially the dielectric PB-phase metasurface, attracts a lot of research interests and is demonstrated for many fascinating applications, such as metalens [7–10], polarization multiplexing imaging [11,12], etc.

Usually, the input polarization for PB-phase metasurfaces is circular polarized [7–12]. Under this condition, the dynamic phase of the metasurface is seldom discussed and can be considered a constant. However, for most conditions, the polarization state evolves during waves propagate. In general, the final polarization state changes, which is a typical case of a PB-phase metasurface with a non-perfect polarization conversion efficiency (PCE). Moreover, for the polarization-multiplexing PB-phase metasurface, the overall phase modulation is essential information. For non-resonance metasurfaces, the overall phase modulation includes dynamic phase and geometric phase [13]. In general, the optical properties of the dielectric metasurface can be precisely analyzed by using the finite-difference time-domain (FDTD) method. However, the FDTD is time-consuming. Here, we propose a fast analytic method for PB-Phase metasurface to provide an efficient and fast design tool for anisotropic PB-phase metasurfaces.

As light passes through a PB-phase metasurface, it acquires the geometric phase modulation and changes its original polarization state to the cross-polarization. Therefore, the PB-phase metasurface can also be regarded as an array of tiny waveplates. Conventional waveplate is made of birefringent materials. About three decades ago, the structure-induced artificial birefringence, named form-birefringence [14], was proposed, and a series of polarization-multiplexing devices was demonstrated, for example, polarization-dependent computer-generated hologram [15], radial polarizer [16], etc. The unit cell of a dielectric PB-phase metasurface can also be regarded as a waveplate with a specific rotation angle. The schematic of the dielectric PB-phase metasurface, consisting of a nano-fin on a glass substrate, is shown in Fig. 1. *θ _{p}* indicates the incline angle between the

*x*-axis and the long axis of the nano-fin. The

*w*,

*l*and

*d*represents the width, length, and thickness of the nano-fin, respectively. The birefringence can be thus created by the asymmetric geometric of the unit cell. According to the PB-phase theory, the output light carries a geometric phase of 2

*θ*.

_{p}It is known that as a high efficiency dielectric PB-phase metasurface, the unit cell of the metasurface must act as a half-wave plate. For x-linear-polarization (XLP) input light, the polarization conversion efficiency (PCE) is defined as the intensity of the electric field y-component, E_{y}^{2} at the output plane normalized to the power of the total electric field at the output plane, E_{out}^{2}. However, generally, the phase modulation of a dielectric PB-phase metasurface may not be a perfect half-wave plate. Therefore, we would like to analyze the output complex amplitude of an EM wave with an arbitrary input polarization state passing through a phase-retarder with an arbitrary anisotropic phase delay. Let us start considering the Jones matrix of a waveplate with phase retardation of Γ to introduce the working principle of the dielectric PB-phase metasurface which can be characterized as following:

*R*is rotation matrix;

*W*

_{0}is the Jones matrix of a waveplate with a phase retardation of Γ.

*θ*is the azimuth angle of the waveplate and is also the incline angle of a nano-fin metasurface. Expending the Eq. (1), it can be represented as follow:

_{p}Now consider a circularly polarized light with Jones vector ${\textstyle{{\sqrt 2 } \over 2}}{({1, \pm i} )^T}$, the Jones vector of the output light can be calculated as:

From the Eq. (3), it can be seen that the cross-polarization term carries a geometric phase of 2*θ _{p}* and is independent of Γ. The co-polarization term carries a constant dynamic phase and is independent of

*θ*. As the phase-retardation is π, the output electric field, E

_{p}_{out}is:

This is a standard form of a circular polarization (CP) converted to its orthogonal polarization state. For a CP-light input, it experiences a constant phase delay of -π/2 and the geometric phase of 2*θ _{p}*. At this time, the PCE is 100%. For a non-reflection and non-absorption case, the polarization conversion efficiency is only function of Γ and can be calculated as PCE = sin

^{2}Γ. For example, for Γ = λ/4, the PCE = 50%.

## 2. Example for x-polarization analysis

As stated before, we have considered the geometric phase modulation and the polarization conversion efficiency for an input light with circular polarization. For the following, we are going to consider the complex amplitude for x-linear polarized (XLP) light passing through a wave plate with an arbitrary phase retardation of Γ. The following formulation is also valid for an arbitrary incident polarization, ${\textstyle{{\sqrt 2 } \over 2}}{({A,B + Ci} )^T}$.

For convenience, we consider an input light with x-polarization which can be treated as the linear combination of RCP and LCP light.

Figure 2 shows the concept to analyze overall phase modulation of an XLP light passing through a waveplate with arbitrary phase retardation of Γ. The XLP can be represented as a superposition of two mutually orthogonal circular polarization, right- and left-hand circular polarization (RCP and LCP). The RCP component of the XLP impinges a waveplate, part of the RCP converts to LCP and then carries the geometric phase, named R-LCP, while part of the RCP remains its original polarization and carries dynamic phase, named R-RCP. Similarly, a part of the LCP component converts to RCP (L-RCP) and carries the geometric phase, while part of LCP carries dynamic phase (L-LCP). The overall phase modulation of the RCP-component at the output plane is the combination of R-RCP with dynamic phase and L-RCP with geometric phase. This term is named X-RCP which means the overall phase modulation of the RCP component for XLP input light. Similarly, the X-LCP indicates the overall phase modulation of the LCP component for XLP input light. Moreover, the final output polarization state, which usually is an elliptical polarization, can be obtained by the superposition of the X-RCP and X-LCP.

## 3. Effective index of nano-fin

According to the above analysis model, one can easily obtain the output complex amplitude for arbitrary input polarization state as long as the Γ is known. In this section, we use effective medium theory (EMT) to analyze the birefringence of a typical unit cell of metasurfaces, an optically anisotropic nano-fin, and the corresponding Γ can thus be quickly obtained. A nano-fin array can be regarded as a two-dimensional grating with two different filling factors, *f*_{x} and *f*_{y}, where filling factors are referred to as the fraction of the grating period that is filled with the grating material, and subscript *x* and *y* indicates the faction in x- and y-direction, respectively. By using a one-dimensional EMT theory twice, we are able to simplify the nano-fin to be an effective homogeneous anisotropic medium [17]. Via the EMT, we are able to obtain the magnitude of the birefringence, i.e., the refractive index difference of the effective anisotropic medium.

For the polarization-dependency, one needs to choose the material and geometrical parameters of the optically anisotropic nano-fin. This can be quickly done with the EMT. Figure 3 shows the *Δn*, *Δn* = *n _{TE}* -

*n*, as a function of

_{TM}*f*and

_{x}*f*. On the x- and y-axis is a 1D grating. Beside of the geometric parameters, the form birefringence of nano-fin depends on the refractive index of the material, which makes choosing the right (highly refractive) material crucial for the design process. This is similar to the requirement of an appropriate material for dielectric metasurface. A high Δ

_{y}*n*is preferable for less fabrication challenge. The higher Δ

*n*, the less etching depth of the nano-fin eventually need to be etched in order to achieve the desired phase difference of

*π*for the orthogonal polarization directions. The maximum

*Δn*= 0.4633 is at (

*f*,

_{x}*f*) = (0.4124, 1) and (

_{y}*f*,

_{x}*f*) = (1, 0.4124). Alternatively, one reaches a maximal

_{y}*Δn*for (

*f*

_{x},

*f*

_{y}) = (0.407, 1) for TiO

_{2}, (

*f*

_{x},

*f*

_{y}) = (0.4114, 1) for Nb

_{2}O

_{5}, and (

*f*

_{x},

*f*

_{y}) = (0.4484, 1) for fused silica. To directly compare the potential sets of parameters in terms of manufacturability, one needs to calculate the aspect ratio. Considering the critical dimension limitation of the fabrication process, the filling factor of GaN nano-fin should away from 1 or 0. For example, the critical dimension limitation of commercial KrF 248 nm light source stepper is 120 nm. The minimum and maximum filling factor are 0.3636 and 0.6364, respectively. Under this condition, the

*Δn*is 0.1825 which is significantly lower than a GaN grating with

*f*or

_{x}*f*= 0.4124. Here, we choose (

_{y}*f*

_{x},

*f*

_{y}) = (0.9090, 0.3030), corresponding to

*w*= 100 nm and

*l*= 300 nm, which can be fabricated by using E-beam lithography. The corresponding

*Δn*is 0.4115 which is slightly lower than that of a grating structure.

## 4. Results

Once the geometry parameter of the nano-fin is decided, one can quickly obtain the polarization-dependent refractive index. Then, via the Jones matrix, the PCE as a function of nano-fin thickness can be calculated. The input polarization is XLP, and the rotation angle of nano-fin, *θ _{p}* is 45°. Here, the PCE is defined as the I

_{y}= E

_{y}

^{2}component normalized to the total intensity at the output plane. The PCE as a function of nano-fin thickness predicted by EMT and FDTD are represented by red solid line and black circle symbol, respectively. The peaks of PCE correspond to the half-waveplate thickness. It is shown that both EMT and FDTD results have a similar trend. Nevertheless, the half-waveplate thicknesses predicted by EMT and FDTD are 550 nm and 950 nm, respectively. The corresponding Δ

*n*is respectively 0.4115, and 0.3332 predicted EMT and FDTD. According to Ref. [18], the first order EMT only leads to good results with deviations in reflectivity of the effective medium <1% for small periods (Λ < λ/40). In our case, we utile second-order EMT for large periods (Λ ∼ λ/2). Under our condition, the

*Δn*predicted by FDTD is 19% smaller than that predicted by second-order EMT. The EMT assumes that the E-field is uniform over the structure layer, which is not sufficiently rigorous for Λ ∼ λ/2. Therefore, the scattering and resonance induced phase modulation is also ignored. Moreover, the EMT ignores the tiny absorption coefficient at λ = 633 nm. Therefore, the PCE of the second peak at

*d*= 2850 nm calculated by EMT is identical to that at

*d*= 950 nm. By multiplying a factor of 0.81, the EMT curve is similar to the FDTD, as presented by the red dashed line in Fig. 4.

Besides the optimized geometric parameters of nano-fins and the corresponding anisotropic index, the PCE spectrum can also be calculated by using EMT. Figure 5(a) shows the PCE spectrum of GaN nano-fin simulated by EMT (red solid line) and FDTD (black circle symbol). The geometric parameters of the GaN nano-fin are *w* = 100 nm, *l* = 300 nm and d = 950 nm, respectively. As mentioned, the prediction of the Δ*n* by EMT and FDTD has a difference of 19%. After multiplying a factor of 0.81, the PCE spectrum calculated by EMT shows a fair agreement with that simulated by FDTD. It can be found there is a peak at 633 nm, which is the design target wavelength. The PEC decreases as the wavelength is away from 633 nm. It can be seen a dip at λ = 455 nm. For λ = 455 nm, the corresponding Γ is 2π. Therefore, the PCE is zero. It is worth emphasizing that the factor is a constant for the entire spectral range. Therefore, we can conclude that no other arisen effect is different from λ = 633 nm. For example, no wavelength-dependent scattering effect, guided-mode resonance (GMR), and Mie resonance effect significantly affect the effective index and phase modulation. These effects are not considered in the EMT. For comparison, GaN nano-fin with *d* = 550 nm is shown in Fig. 5(b). As expected, the peak blueshift to λ = 434 nm owing to the phase difference decreasing proportionally to the thickness. Moreover, it can be seen that the EMT didn’t match well with the FDTD result at the blue light range, even for a factor is multiplying to fix the *Δn* difference between EMT and FDTD. This is because the EMT ignores the absorption coefficient. The absorption coefficient of GaN at the blue range is so significant that it cannot be ignored. Nevertheless, compared to the time-consuming FDTD method, the EMT can still provide a quick analysis with acceptable accuracy.

Usually, people consider the PB-phase as a function of *θ _{p}*, regraded as the operation curve to assign the corresponding

*θ*at each unit cell of the meta-device. Figure 6 shows the overall phase modulation as a function of the

_{p}*θ*for XLP incident light. A series of difference

_{p}*d*is simulated. The solid line and symbols represent the results of EMT and FDTD, respectively. It can be seen that as the

*d*is away from the λ/2 waveplate thickness, for example,

*d*= 200 nm and

*d*= 1400 nm, the overall phase is not simply proportional to the

*θ*. Moreover, as the

_{p}*d*is thinner than λ/4 waveplate thickness, full 2π phase modulation cannot be achieved.

Here, we use a phasor diagram to represent the total phase modulation as an *x*-polarization light passing through a nano-fin array with arbitrary *d*. As mentioned, the overall phase modulation is the combination of dynamic phase and geometric phase. The overall phase modulation is the phase term of the superposition of light modulated by dynamic effect and light modulated by the geometric effect. Moreover, not only the phase but also the amplitude of the modulated light is crucial information for the designation of a meta-device. Therefore, the phasor diagram can represent the overall modulated complex electric field, including the phase and amplitude information. The complex electric-field of the dynamic term, geometric term as well as the corresponding superposition term are shown in Fig. 7. The complex electric field modulated by the dynamic effect is represented by green arrows. For CP, the dynamic phase is a constant. Therefore, for convenience, we set the dynamic phase as zero. The complex electric field modulated by the geometric effect is represented by black arrows. Consequently, the black arrows are arranged with radial symmetry. The resulting output electric-field can be easily obtained by adding vectors and is characterized by red arrows.

As shown in Fig. 7(a), the red arrows for *d* = 200 nm, i.e., the corresponding superposition term, located only at the 1st and the 4th quadrant. The red arrows compose a cone shape. The vertex angle is 40°, which means the maximum phase modulation for *d* = 200 nm is only 40°. Apparently, such a thin thickness of a nano-fin is not suitable for a high efficient dielectric metasurface. Figure 7(b) shows the output electric-field for *d* = 560 nm. Compared to *d* = 200 nm, the magnitude of the dynamic term is decreased, and the magnitude of the geometric term is increased. The magnitude of dynamic and geometric terms are very close. According to Eq. (4), the Γ is π/2, i.e., a quarter wave-plate. Under this condition, the corresponding red arrows still only located in the 1st and 4th quadrant, and the maximum of the phase modulation for *d* = 560 nm is 180°. For *d* = 1000 nm, this thickness is close to the optimized thickness for maximized PCE. As shown in Fig. 7(c), the magnitude of the dynamic term is small compared to the geometric term. At this time, the resulting electric field is dominated by the geometric term. The corresponding red arrows evenly distribute in the phase diagram. Therefore, we can obtain a 2*θ _{p}* phase modulation. For comparison, the FDTD simulation results for

*d*= 200 nm, 560 nm and 1000 nm are shown in Fig. 7(d), 7(e) and 7(f), respectively. It can be seen that the EMT shows a fair agreement with FDTD.

Although we have shown that it is convenient to utilize the Jones matrix and EMT to analyze the optical properties of PB-phase metasurfaces with arbitrary input polarization. However, the assumption of EMT is the electric field is uniform inside the structure. Therefore, some physical phenomena are ignored. Here, we use Nb_{2}O_{5}, a lossless dielectric material with an index of 2.32, nano-fin as an example. The PCE as a function of Nb_{2}O_{5} nano-fin thickness is shown in Fig. 8(a). For the EMT results (red solid line), the behavior is similar to Fig. 4. However, the FDTD results show a significant dip in PCE for *d* = 1100 nm. This dip is caused by the GMR. Generally, GMR is a kind of anomaly diffraction of the grating. The mechanism is that the incident light is phase-matched with the effective planar waveguide structure to be coupled to the waveguide. The out-coupled light experiences a multi-beam interference, and thus at specific wavelength and thickness, it shows a destructive interference dip. The effective refractive index of Nb_{2}O_{5} nano-fin is higher than that of the substrate silica substrate. This makes the Nb_{2}O_{5} nano-fin can be regarded as a planar waveguide. The period of the unit cell plays a role of grating providing additional momentum, and coupling the incident light into the Nb_{2}O_{5} effective planar waveguide. Figure 8(b) shows the E-field distribution for *d* = 1100 nm. It is shown two hot spots, which corresponds to a waveguide mode. As a comparison, Fig. 8(c) shows the E-field distribution for *d* = 1000 nm. It can be seen multiple standing waves along z-direction, just like a typical non-resonance dielectric PB-phase metasurface.

## 5. Conclusions

In summary, we propose an analytic method for analyzing the optical properties of a dielectric PB-phase metasurface with an arbitrary incident polarization state. The optical properties, such as PCE spectrum, rotation-angle-dependent phase modulation, and phasor diagram, predicted by the proposed model, show a fair agreement with the FDTD method. It is shown that the overall phase modulation of a PB-phase metasurface with a phase retardation Γ < π/4 cannot achieve 2π phase modulation, which is essential for a meta-device with high efficiency. This model provides a fast and sufficient accuracy compared to the time-consuming finite element methods. We believe this model will benefit the optimization of the geometric structure of an optically anisotropic metasurface.

## Funding

Ministry of Science and Technology, Taiwan (MOST 109-2124-M-008-002-MY3, MOST107-2628-E-008-004-MY3).

## Acknowledgments

The authors would like to acknowledge financial support from the Ministry of Science and Technology, Taiwan (grant no. MOST107-2628-E-008-004-MY3 and MOST 109-2124-M-008-002-MY3).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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