Non-reflecting non-magnetic homogeneous polarization splitter and polarization deflector design based on transformation electromagnetics

Accepted Manuscript Title: Non-reflecting non-magnetic homogeneous polarization splitter and polarization deflector design based on transformation electromagnetics Authors: Hossein Eskandari, Mohammad Saeed Majedi, Amir Reza Attari PII: DOI: Reference:

S0030-4026(17)30101-8 http://dx.doi.org/doi:10.1016/j.ijleo.2017.01.080 IJLEO 58783

To appear in: Received date: Accepted date:

3-11-2016 26-1-2017

Please cite this article as: Hossein Eskandari, Mohammad Saeed Majedi, Amir Reza Attari, Non-reflecting non-magnetic homogeneous polarization splitter and polarization deflector design based on transformation electromagnetics, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2017.01.080 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Non-reflecting non-magnetic homogeneous polarization splitter and polarization deflector design based on transformation electromagnetics

Hossein Eskandaria,b, Mohammad Saeed Majedib,, Amir Reza Attarib

a

Communications and Computer Research Center, Ferdowsi University of Mashhad, Mashhad, Iran

b

Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran



Corresponding author. Tel.:+98-513-880-5171 E-mail Address: [email protected].

Abstract In this paper, a non-reflecting non-magnetic homogeneous polarization splitter and polarization deflector design method is proposed and investigated. Transformation electromagnetics is used as a powerful tool to achieve a desired functionality. By applying a proper linear transformation and using the fact that for different polarizations, distinct constitutive parameters appear in Maxwell equations, a non-magnetic homogeneous media is obtained that is reflectionless. Anisotropy factor of the designed media is evaluated and a brief discussion of the available tradeoff is given. To test the method, two typical polarization splitter with positive and negative lateral shift are designed. Also, by modifying the proposed polarization splitter and implementing a metallic coating, a polarization deflector is designed. Functionality of design method is validated using commercial finite element software COMSOL Multiphysics. It is seen that the proposed polarization splitters can successfully create desired separation between polarizations according to the design procedure. Also, the polarization deflector can deflect one polarization with respect to the other one by arbitrary angle.

Keywords: homogeneous, non-magnetic, polarization deflector, polarization splitter, transformation electromagnetics.

1. Introduction Since 2006 when Pendry [1] and Leonhardt [2] independently put forward the idea of transformation electromagnetics, it began to widely receive attention, because it gives the designer unprecedented ability to control the electromagnetic wave behavior. These two basic papers invented the cloak of invisibility that was able to redirect the flow of electromagnetic energy around an object. A simplified version of a cylindrical cloak was

fabricated and tested in microwave regime [3]. After the publication of these two groundbreaking papers, transformation electromagnetics rapidly found its way into lots of interesting device designing topics like field rotators [4], field concentrators [5], beam benders [6, 7], beam expanders [7], focusing lenses [6, 8, 9], polarization rotators [10], polarization splitters [10-13], wave shape transformers [14, 15] and polarization deflectors [16-18]. The material parameters that come from utilizing transformation electromagnetics are complex in general. They can be inhomogeneous and anisotropic and in general possess extreme values of  and  (very large values, negative values and near zero values). These types of media are mostly not available in nature, so using metamaterial is a must. Great deal of effort is made to experimentally verify metamaterial structures [19, 20]. These artificially-made metamaterial structures frequently contain a geometrical resonance, like split ring resonators (SRRs). These resonant structures often suffer from significant losses and high frequency dispersion which limits the operation bandwidth of the device. This can be problematic when optical regime is chosen since resonant structures have a non-negligible amount of metal loss that deteriorates the performance of the device. The problem gets worse when medium is magnetic; since SSRs have saturation effect when their structure scales at high frequency and even replacing them with other high frequency magnetic elements like coupled nanorods or nanostripes will not fully solve the problem because these are naturally lossy [21]. So when it comes to the fabrication, being non-magnetic is an advantage. Also, Homogeneous constitutive parameters are pretty much easier to fabricate since no spatial variation of media characteristics is needed. Polarization splitters are common optical devices which can separate incident unpolarized beam into two orthogonal polarizations, namely TE and TM. Usually these optical devices utilize birefringence to separate two polarizations by laterally shifting one polarization and keep the other unaffected or passing one polarization and reflecting the other [22]. On the other hand polarization deflectors separate orthogonal polarizations by creating an arbitrary angle between them as they exit the device. Here we propose a general design method of polarization splitter based on transformation electromagnetic strategy and inspired by Kwon and Werner design [10]. Generalized polarization splitter creates arbitrary lateral shift between TM and TE polarizations by tuning width of the device and the amount of anisotropy. By modifying the design of polarization splitter and using proper metallic coating, a polarization deflector is designed. It will be shown that if a suitable transformation is utilized and by affecting the TM polarization only, the resulting media will be non-magnetic and homogeneous. In addition we will show that the resulting media is reflectionless in our design method. This work is organized as follows. In section 2 theoretical basis are explained. Sub-section 2.1 covers a brief summary of transformation electromagnetics. Constitutive parameters for TE and TM polarizations in Maxwell equations are specified in sub-section 2.2. This section ends with reflectionless condition stated in 2.3. Sub-section 3.1 represents a general method for designing a reflectionless non-magnetic homogenous polarization splitter and discusses the existing tradeoff. Design method of polarization deflector of arbitrary angle is proposed in Sub-section 3.2. In section 4 anisotropy factor of proposed polarization splitter and hence polarization deflector is formulated and analyzed and the effect of design parameters is investigated. Section 5 contains simulation results obtained by COMSOL Multiphysics and device functionality validation. 2. Theoretical Basis 2.1. Transformation Electromagnetics Basically transformation electromagnetics employs the form-invariance of Maxwell equations under coordinate transformation. One can deform the space to achieve a desired wave propagation characteristics. This deformation is then realized by using the calculated transformation media parameters. Consider E and H field described by Cartesian coordinate system. Far from magnetic and electric current sources, they must satisfy Maxwell equations in order to exist. ∇ × 𝐄 = −𝑗𝜔𝐇 (1) ∇ × 𝐇 = 𝑗𝜔ε𝐄 where  and  are the electric permittivity and magnetic permeability tensors respectively.

Fig. 1. Shows the transformation electromagnetics design procedure. In Fig. 1(a) plane wave propagating in +𝑥̂ direction is shown by 3 ray paths. We apply a coordinate transformation from virtual (𝑥, 𝑦, 𝑧) space to physical (𝑥 ′ , 𝑦 ′ , 𝑧′) space described by the following equations. 𝑥 ′ = 𝑥 ′ (𝑥, 𝑦, 𝑧) (2) 𝑦 ′ = 𝑦 ′ (𝑥, 𝑦, 𝑧) 𝑧 ′ = 𝑧 ′ (𝑥, 𝑦, 𝑧) Fig. 1(b) illustrates an example of such transformed physical space. It can be observed that our physical space is perturbed in 𝑦′ direction around 𝑦 ′ = 0 axis. All constant y coordinate lines in virtual space are mapped to curved coordinate lines in (𝑥 ′ , 𝑦 ′ , 𝑧′) space. Ray paths follow these mapped coordinate lines. One can write Maxwell equations in transformed physical space using form-invariance of Maxwell equations under coordinate transformation. ∇′ × 𝐄 ′ = −𝑗𝜔μ′ 𝐇 ′ ∇′ × 𝐇′ = 𝑗𝜔ε′𝐄′ Jacobian matrix J defines the coordinate transformation between two spaces.

(3)

𝜕𝑥 ′ ⁄𝜕𝑥 𝜕𝑥 ′ ⁄𝜕𝑦 𝜕𝑥 ′ ⁄𝜕𝑧 J = (𝜕𝑦 ′ ⁄𝜕𝑥 𝜕𝑦 ′ ⁄𝜕𝑦 𝜕𝑦 ′ ⁄𝜕𝑧 ) (4) 𝜕𝑧 ′ ⁄𝜕𝑥 𝜕𝑧 ′ ⁄𝜕𝑦 𝜕𝑧 ′ ⁄𝜕𝑧 ε' and μ' in physical space are calculated using Eq. (5). These two tensors constitute the transformation media. JεJ 𝑇 det(J) (5) JμJ 𝑇 μ′ = det(J) Note that most of the time, virtual space material is taken vacuum meaning that ε = ε0 , 𝜇 = 𝜇0 . Fields in physical space are derived from fields in virtual space by Eq. (6) [23]. ε′ =

𝐄 ′ = (J 𝑇 )−1 𝐄 𝐇 ′ = (J 𝑇 )−1 𝐇

(6)

2.2. Constitutive parameters for TE and TM polarizations To derive Maxwell equations for TE and TM polarizations, we suppose the configuration is 2D i.e. media and fields do not vary with z coordinate. TE and TM polarizations are defined as polarizations that are consist of (𝐸𝑧 , 𝐻𝑥 , 𝐻𝑦 ) and (𝐻𝑧 , 𝐸𝑥 , 𝐸𝑦 ) respectively. Writing Maxwell equations in media for TM polarization, results in following equations. 𝜕𝐸𝑦 𝜕𝐸𝑥 − = −𝑗𝜔μ𝑧𝑧 H𝒛 𝜕𝑥 𝜕𝑦 𝜕𝐻𝑧 (7) = 𝑗𝜔(ε𝑥𝑥 𝐸𝑥 + ε𝑥𝑦 𝐸𝑦 ) 𝜕𝑦 𝜕𝐻𝑧 = −𝑗𝜔(ε𝑦𝑥 𝐸𝑥 + ε𝑦𝑦 𝐸𝑦 ) 𝜕𝑥 It can be clearly seen that the only constitutive parameters μ𝑧𝑧 , ε𝑥𝑥 , ε𝑥𝑦 , ε𝑦𝑥 , ε𝑦𝑦 are in Maxwell equations for TM wave propagation. Following the same procedure, constitutive parameters that affect TE wave propagation are ε𝑧𝑧 , μ𝑥𝑥 , μ𝑥𝑦 , μ𝑦𝑥 , μ𝑦𝑦 .

2.3. Reflectionless media It has been proved that if a transformation is continuous at a boundary, the resulting transformation media will be reflectionless and hence transparent to the incident wave at that boundary [24]. Reference [25] states that if device input and output boundaries in physical space can be expressed as a combination of a rotation and a displacement of the virtual space input and output boundary respectively, then the resulting media is reflectionless at these boundaries. This is more general than the statement of [24]. 3. Design method and tradeoffs 3.1. Design method of non-magnetic homogeneous polarization splitter Consider the virtual space as Fig. 2(a). The incident wave enters the medium at left boundary and leaves it at right. We want to create a desired lateral shift for TM polarization. Fig. 2(b) shows the designed physical space. Virtual space of this design is simply a rectangle with sides 𝑙 and 𝑑. While our physical space is a parallelogram with left and right sides 𝑙 and width 𝑑. Vertices points for rectangle in virtual space are indicated by (𝑎, 𝑏, 𝑐, 𝑑), while physical space parallelogram vertices are noted by (𝑎′ , 𝑏 ′ , 𝑐 ′ , 𝑑′). The angle between 𝑎′𝑐 ′ and 𝑥′ axis is 𝜃. This angle can also be negative. TM polarization can experience upward or downward lateral shift. The proposed transformation keeps 𝑥 −constant coordinate lines intact while mapping y coordinate line to constantslope lines. Desired transformation is formulated as below. 𝑥′ = 𝑥 { 𝑦 ′ = tan(𝜃) 𝑥 + 𝑦 + 𝑙 ⁄2 𝑧′ = 𝑧 Using Eq. (4) results in the following Jacobian matrix.

(8)

1 0 0 𝐽 = (𝑡𝑎𝑛(𝜃) 1 0) (9) 0 0 1 According to Eq. (5), constitutive parameters of transformation media can be calculated. Note that ε = ε0 , 𝜇 = 𝜇0 . 1 𝑡𝑎𝑛(𝜃) 0 1 𝑡𝑎𝑛(𝜃) 0 (10) ε′ = ε0 (𝑡𝑎𝑛(𝜃) 𝑠𝑒𝑐 2 (𝜃) 0) , μ′ = μ0 (𝑡𝑎𝑛(𝜃) 𝑠𝑒𝑐 2 (𝜃) 0) 0 0 1 0 0 1 It is clearly seen that using linear transformation results in a homogeneous media, since Jacobian matrix elements are independent of physical space coordinates. Also, since the determinant of Jacobian matrix of the proposed transformation is unity and the transformation keeps the area of virtual space and physical space the same, μ𝑧𝑧 will be unity. This is a needed characteristic to achieve a non-magnetic media. According to section 2.2, in order to apply the transformation to TM polarization, we use the following parameters extracted from Eq. (10). 1 tan(𝜃) 0 1 0 ε′ = ε0 (tan(𝜃) 𝑠𝑒𝑐 2 (𝜃) 0) , μ′ = μ0 (0 1 0 0 0 0 1

0 0) 1

(11)

Clearly to achieve a negative lateral shift for TM polarization, negative values of 𝜃 are needed. It is proven in Appendix. A that negative values of 𝜃 tacitly mean a 180 degree rotation of device around 𝑥′ axis. Applying the transformation presented by Eq. (8), we expect electromagnetic rays entering left side of the parallelogram to exit the right side normally. This procedure can create desired lateral shift at the output. Keep in mind that maximum allowed effective width of input Gaussian beam is considered to be less than 𝑙 to ensure separation at the output. Note that the input left boundary is exactly the same in both spaces, while the output right boundary is displaced by a vertical upward shift. Considering the reflectionless criteria in section 2.3, we expect this design to be reflectionless at the boundaries.

3.1.1. Design tradeoff of proposed polarization splitter Design procedure mentioned in sub-section 3.1 is just a basic procedure. It can be seen that this device can separate two polarization successfully if 𝑑 ≥ 𝑙 × 𝑐𝑜𝑡(𝜃). Choosing 𝑑 = 𝑙 × 𝑐𝑜𝑡(𝜃) will result in a minimum vertical separation of 𝑙 at the output. If larger amount of polarization separation is needed, one can follow two instructions as follows.  If device width 𝑑 is known, we can increase 𝜃 to achieve a larger separation distance. The created separation equals 𝑑 × 𝑡𝑎𝑛(𝜃). Note that according to section 4 calculations, increasing 𝜃 increases the anisotropy factor which may be undesirable.  If device width 𝑑 is arbitrary, since separation distance is 𝑑 × 𝑡𝑎𝑛(𝜃), designer is free to choose 𝜃 and 𝑑 to achieve the desired separation distance between polarizations. Larger amounts of 𝑑 creates more lateral shift while increases the width of polarization splitter. Hence there is a tradeoff. To achieve the desired lateral shift, one can increases 𝜃 and hence the anisotropy while keeping the width of structure low or vice versa. 3.2. Design method of non-magnetic homogeneous polarization deflector Design method of polarization splitter can be modified to achieve a polarization deflection. The main idea is to coat the 𝑎′𝑐′ side in Fig. 2.(b) with a proper metal to reflect the TE polarization from this metallic surface. This will not change the TM polarization propagation since TM polarization will not make any contact to the metallic coating surface. Since most lasers in telecommunications application operate in frequencies far below ultra violet region, using a proper metallic coating like silver, grants us reflection coefficients of more than 95% for a wide range of incident angles. Note that the deflection angle is 2𝜃. As a result, the TM polarization will face a lateral shift of 𝑑 × 𝑡𝑎𝑛(𝜃) while TE polarization exits the device at deflection angle of 2𝜃. In polarization deflector design, the amount of TM polarization lateral shift is not a target. The only parameters that should be determined are the device width and 𝜃 which is related to the required deflection angle. Choosing larger deflection angles reduces the device width but increases the anisotropy. Note that polarization deflector in [18] is homogeneous but suffers from reflection from second output boundary. Proposed deflector in [16] is non-homogeneous and has reflections at second output boundary while [17] revised the deflector in [16] and suppressed aforementioned reflections. 4. Anisotropy Evaluation Since the proposed polarization splitter and polarization deflector magnetic permeability is μ0 , One can diagonalize the relative electric permittivity tensor and calculate principal refractive indices of material (see Appendix. B). Principal refractive indices are as follows. n1 n𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 = ( 0 0

0 n2 0

0 1 2 2 0 ), {n1.2 = √2 (1 + 𝑠𝑒𝑐 (𝜃) ± 𝑡𝑎𝑛(𝜃)√3 + 𝑠𝑒𝑐 (𝜃)) n3 n3 = 1 To measure the media anisotropy, max(n1 ⁄n2 , n2 ⁄n1 ) is taken as an anisotropy factor [26].

(12)

1 (13) max(n1 ⁄n2 , n2 ⁄n1 ) = (1 + 𝑠𝑒𝑐 2 (𝜃) + |𝑡𝑎𝑛(𝜃)|√3 + 𝑠𝑒𝑐 2 (𝜃)) 2 As seen from above equation, increasing 𝜃 from near zero degree to near 90 degrees, monotonically increases the anisotropy of media. Fig. 3 illustrates this fact. Anisotropy factor represented by Eq. (13) is an even function of 𝜃. In the next section, we validate the functionality of the designed media using COMSOL Multiphysics software. This software uses finite element method and has the ability to solve Maxwell equations for arbitrary media.

5. Simulation Results Design Geometry is shown in Fig. 4. Geometry is illustrated for positive 𝜃 here. Note that the design geometry of deflector is almost the same and the only difference is that the lower side of parallelogram is coated with metallic material e.g. silver. In case of simulation of polarization splitter it is seen that using scattering boundary condition is enough for absorption of wave but in case of polarization deflector, using PML was necessary for efficient absorption of wave at the boundary of simulation domain. Simulation settings for polarization splitter are as follows. Frequency is set to 6 GHz. Simulation domain is a 30𝜆 × 30𝜆 square. Parameter 𝑙 is set to 12𝜆. The mesh element size is 𝜆⁄10 to ensure convergence and accurate results. The incident electric field is a right hand circularly polarized Gaussian wave with the spot size of 1.6𝜆 and illuminates the left boundary of polarization splitter normally. For simulation of polarization splitter, three second order scattering boundary conditions are set on the remaining walls to avoid unnecessary reflections from the outer boundary of simulation domain. At the first we intend to design a polarization splitter. Parameters 𝜃 = 𝑝𝑖/6 and 𝑑 = 𝑙 × 𝑐𝑜𝑡(𝜃) = 1.04 m are set.These parameters lead to minimum upward lateral shift equal to 𝑙 = 0.6 m. According to our design method and based on Eq. (5), the constitutive parameters are as follows. 1 ε′ = ε0 (0.577 0

0.577 1.33 0

0 0) , μ′ = μ0 1

The anisotropy factor of the designed media is 1.76. In the second design of polarization splitter, we aim for compactness and negative lateral shift. Parameters used are 𝜃 = −𝑝𝑖/4 and 𝑑 = 𝑙 × 𝑐𝑜𝑡(𝜃) = −0.6 m. The minimum downward lateral shift of 𝑙 = 0.6 m is achieved. Constitutive parameters of the second design are as follows. 1 ε′ = ε0 (−1 0

−1 2 0

0 0) , μ′ = μ0 1

Anisotropy factor of the second designed media is 2.61. Fig. 5 and Fig. 6 illustrate the propagation of E𝑧 (TE) and E𝑦 (TM) components for the first and the second design respectively. The TE component propagates as if it is propagating in vacuum. While TM component is laterally shifted upward for the first design and downward for the second one. Note that normalized real part of the fields are plotted for all cases. Fig. 7 shows the normalized magnitude distribution of poynting vector of the both designs. It can be seen that the incident wave power is equally split in two directions. Next, two polarization deflectors are designed. Here the frequency is set to 9 GHz while maintaining the resolution to avoid expansion of Gaussian beam. A 45𝜆 × 57𝜆 rectangle is chosen to avoid incidence of wave to the corners of simulation domain and provide enough space for PML layers. PML is used to absorb oblique wave that is incident on simulation domain boundary at the output. The exterior boundaries are still covered by scattering boundary condition. In the first design, deflection angle 2𝜃 is set to 𝑝𝑖/3 while the second design aims to create a deflection angle of 𝑝𝑖/2. Considering the 0.668 μm skin depth of Silver at 9 GHz, a 5𝜇𝑚 layer of silver is placed on the bottom side of parallelogram of polarization deflector. Fig. 8 and Fig. 9 illustrate the propagation of E𝑧 (TE) and E𝑦 (TM) for the first and the second deflector respectively. Fig. 10 shows the distribution of energy in the device. It is seen that the desired deflection angle is achieved.

6. Conclusion We have proposed a general design method for a reflectionless, non-magnetic homogeneous polarization splitter based on transformation electromagnetics. We have also designed a polarization deflector based on this generalized design method. This polarization splitter utilizes proper linear transformation to create arbitrary positive or negative lateral shift in TM polarization while keeping the other (TE) intact. Following the general design procedure and investigating the anisotropy factor of the designed media, proves that arbitrary separation distances can be achieved by tuning device width and the amount of anisotropy. Larger separation distances acquire higher anisotropy if device width is known. However if the device width is arbitrary, designer can create an appropriate balance between device anisotropy and width. A polarization deflector was then designed based on the presented polarization splitter and use of metal coating. These two devices can split circular or oblique-linear polarization into two orthogonal linear ones and create arbitrary lateral shift or angle between them.

Acknowledgments The authors like to thank Seyyed Saleh Seyyedmousavi for his useful opinions.

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Appendix A. Rotation of Tensor Here, we will show that rotating the tensor 180 degrees around 𝑥′ axis gives out the same result as substituting – 𝜃 in Eq. (11). Meaning that, to achieve a negative lateral shift for TM polarization, one must rotate the device around 𝑥′ axis 180 degrees. We rotate the relative electric permittivity as an example. Mathematically rotating a tensor around 𝑥′ axis is formulated as follows. 1 0 0 ε𝑟 = R 𝑥′ 𝜀R𝑇𝑥 ′ , R 𝑥′ = (0 𝑐𝑜𝑠(𝜑) −𝑠𝑖𝑛(𝜑)) (B.1) 0 𝑠𝑖𝑛(𝜑) 𝑐𝑜𝑠(𝜑) Where ε𝑟 is the rotated relative electric permittivity tensor and R 𝑥′ is the rotation matrix for rotating 𝜑 degrees around 𝑥′ axis. Applying Eq. (B.1) and substituting 𝜑 = 180° gives the following result. 1 ε𝑟 = ( 0 0

0 −1 0

1 0 0 ) (𝑡𝑎𝑛(𝜃) −1 0

𝑡𝑎𝑛(𝜃) 𝑠𝑒𝑐 2 (𝜃) 0

0 1 0 1 −𝑡𝑎𝑛(𝜃) 0 0 (B.2) ) ( ) = ( 0 −1 0 0 −𝑡𝑎𝑛(𝜃) 𝑠𝑒𝑐 2 (𝜃) 0) 1 0 0 −1 0 0 1 Which is the transformation media for negative values of 𝜃.

Appendix B. Principal refractive indices calculation

Since the media is non-magnetic, principal refractive indices can be calculated by taking square root of principal relative electric permittivities. The main task is to calculate Eigen values (principal values) of relative electric permittivity tensor. Tensor eigen values are results of solving well known𝑑𝑒𝑡(𝜆𝐼 − ε) = 0 equation. 𝜆−1 |𝜆𝐼 − ε| = 0 ⇒ |−𝑡𝑎𝑛(𝜃) 0

−𝑡𝑎𝑛(𝜃) 𝜆 − 𝑠𝑒𝑐 2 (𝜃) 0

0 2 2 0 | = 0⇒ (𝜆 − (1 + 𝑠𝑒𝑐 (𝜃))𝜆 + 1)(𝜆 − 1) 𝜆−1

(A.1)

Eigen values of relative permittivity tensor are as follows. 1 (A.2) λ1,2 = (1 + 𝑠𝑒𝑐 2 (𝜃) ± 𝑡𝑎𝑛(𝜃)√3 + 𝑠𝑒𝑐 2 (𝜃)) , 𝜆3 = 1 2 These Eigen values are in fact principal relative electric permittivities. Taking square root of these eigen values results in principal refractive indices. 1 n1,2 = √ (1 + 𝑠𝑒𝑐 2 (𝜃) ± 𝑡𝑎𝑛(𝜃)√3 + 𝑠𝑒𝑐 2 (𝜃)) , n3 = 1 2 Anisotropy factor max(n1 ⁄n2 , n2 ⁄n1 ) is expressed by the following equation. 1 max(n1 ⁄n2 , n2 ⁄n1 ) = (1 + 𝑠𝑒𝑐 2 (𝜃) + |𝑡𝑎𝑛(𝜃)|√3 + 𝑠𝑒𝑐 2 (𝜃)) 2

(A.3)

(A.4)

Fig.1. (a) Virtual Space, (b) Physical Space [23]. Fig. 2.(a) Virtual Space, (b) Physical Space Fig.3. Logarithm of the anisotropy factor versus separation angle 𝜃 Fig.4. Design geometry of simulated polarization splitter and polarization deflector for positive theta Fig. 5. Normalized real part of (a) Ez component, (b) Ey component for 𝜃 = 𝑝𝑖/6 Fig. 6. Normalized real part of (a) Ez component, (b) Ey component for 𝜃 = −𝑝𝑖/4 Fig. 7. Normalized magnitude of poynting vector (a) 𝜃 = 𝑝𝑖/6, (b) 𝜃 = −𝑝𝑖/4 Fig. 8. Normalized real part of (a) Ez component, (b) Ey component for deflection angle of pi/3 Fig. 9. Normalized real part of (a) Ez component, (b) Ey component for deflection angle of pi/2 Fig. 10. Normalized magnitude of poynting vector for deflection angles of (a) 𝑝𝑖/3, (b) 𝑝𝑖/2

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