# A Wave Splitter with Simple Structure Based on Biaxial Anisotropic Medium

Authors: Chushun Zhang; Hui Huang; Luchen Huang; Yaru Dong; Xiaoyu Liu
Journal: International Journal of Antennas and Propagation
 (2017)

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## Abstract

A TE/TM wave splitter based on biaxial anisotropic slab is designed. According to the total reflection and total transmission of TE/TM waves incident from isotropic medium into biaxial anisotropic medium, we propose a wave splitter with simple structure and little limitation on both constitutive parameters and incident angle. All the results are verified by simulation.


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## Body

## 1. Introduction

The studies of metamaterial have been the research focus in the field of electromagnetic and materials since Veselago proposed left-handed medium in 1968 [1] and Smith et al. made the first metamaterial sample which has negative index with permittivity and permeability in 2001 [2]. It is well known that the propagations and characteristics of metamaterial are significantly different from those of isotropic materials [3–15]. Metamaterial has many unusual properties and may lead to important applications, such as absorber [16], wave splitter [17], and subwavelength rectangular cavity [18].Theoretically, metamaterial can be considered biaxial anisotropic medium; that is, its permittivity and permeability are both diagonal tensors:(1)ε̿=εx000εy000εz,(2)μ̿=μx000μy000μz.Generally, εx≠εy≠εz and μx≠μy≠μz. The studies of biaxial anisotropic medium have received much interest and attention. Reference [19] has shown that anomalous negative refraction can occur at an interface associated with a metamaterial, which does not necessarily require that all tensor elements of ε̿ and μ̿ have negative values. The propagation of plane-wave in biaxial anisotropic medium with single-sheeted hyperboloid dispersion relation is investigated in [20]. Characteristics of surface wave in biaxial anisotropic left-handed materials are discussed in [21]. Negative refraction and cross polarization effects of biaxial anisotropic medium are summarized in [22]. The double-reflection and light interference pattern in a biaxial anisotropic thin film is numerically investigated in [23]. Under such circumstances, this paper focuses on the application of biaxial anisotropic medium.Because of different dispersion relations of TE waves and TM waves in biaxial anisotropic medium, they propagate in different directions and phase velocities. Thus, the propagation of electromagnetic waves can be controlled by designing constitutive parameters. In this paper, we present the theoretical analysis of a wave splitter based on biaxial anisotropic medium.Nowadays, available technology can separate electromagnetic waves by their different power, mode, or frequency according to different engineering applications [24–28]. The wave splitter based on biaxial anisotropic medium we design in this paper can separate electromagnetic waves by polarization. More specifically, this slab with specific constitutive parameters can make TE waves totally reflected and TM waves totally transmitted, while, with the other specific constitutive parameters, we can get TM waves totally reflected and TE waves totally transmitted. Compared with traditional wave splitters, such a wave splitter with simple structure has little limitation on both constitutive parameters and incident angle. Meanwhile, it can be designed with any chirality of biaxial anisotropic medium and any width. This provides a theoretical basis to the engineering practice.The paper is organized as follows: in Section2, we present a brief review on the total reflection and total transmission in biaxial anisotropic medium. In Section 3, we discuss two ways to separate TE waves and TM waves. Finally in Section 4, we summarize the advantages of this TE/TM wave splitter.


## 2. Theoretical Analysis

### 2.1. Total Reflection of TE Waves

Considering the configuration in Figure1, a TE wave is incident from a semi-infinite isotropic medium into a biaxial anisotropic medium at an oblique angle θi. Region 0 is the isotropic medium, with permittivity ε0 and permeability μ0. Region t is the biaxial anisotropic medium, with permittivity ε̿ and permeability μ̿ which are shown in (1) and (2), respectively.Figure 1
Reflection and transmission of a TE wave at an interface between a semi-infinite isotropic medium (region 0) and a biaxial anisotropic medium (regiont).For Figure1, with wave vectors k-iTE=x^k0xTE+y^kyTE (incident) and k-rTE=-x^k0xTE+y^kyTE (reflection) in the isotropic medium and k-tTE=x^ktxTE+y^ktTE (transmission) in the biaxial anisotropic medium, the dispersion relations can be expressed as (3)k0xTE2+kyTE2=ω2ε0μ0,(4)ktxTE2μy+kyTE2μx=ω2εz.Based on the dispersion relations of two mediums and the boundary conditions, we can get the reflection coefficient for TE waves(5)RE=μyk0xTE-μ0ktxTEμyk0xTE+μ0ktxTE.When ktxTE becomes imaginary, that is, (6)μyμxkyTE2>ω2εzμy,total reflection occurs at the boundary. Together with (3), we can get the total reflection conditions in Table 1.Table 1
Conditions of total reflection of TE waves.
The range of constitutive parameters
The condition of total reflection
(A)μxμy>0∩εzμy>0
(a) Ifεzμx>ε0μ0, there is no total reflection at the boundary
(b) Ifεzμx<ε0μ0, total reflection occurs when incident angle θi> critical angle θc, where θc=arcsin⁡εzμx/ε0μ0
(B)μxμy>0∩εzμy<0
Total reflection occurs for all TE waves
(C)μxμy<0∩εzμy<0
(a) Ifεzμx>ε0μ0, total reflection occurs for all TE waves no matter what incident angle θi is
(b) Ifεzμx<ε0μ0, total reflection occurs when incident angle θi< critical angle θcGenerally, we choose two biaxial anisotropic mediums: medium 1 is left-handed medium with the constitutive parameters:ε̿1=0.60000.5000-0.5; μ̿1=-0.5000-0.90000.7; medium 2 is right-handed medium with the constitutive parameters: ε̿2=0.60000.50001; μ̿2=0.250000.30000.2. Both of them satisfy the condition b of case (A) shown in Table 1 and the critical angles are both θc=30°. In Figure 2, we show z component of electric field in the xy plane generated by the TE incident wave with two different incident angles. The incident angles in Figures 2(a) and 2(c) are both 25°. We can see that there is nontotal reflection. Differently, the transmission wave and the incident wave are on the same side of the normal in medium 1 (left-handed) while they are on the different side in the medium 2 (right-handed). Shown as Figures 2(b) and 2(d), when the incident angles are 45° (θi>θc), total reflection occurs at the boundary. All the results are in accordance with the theoretical derivation summarized in Table 1.Figure 2
E
z of a TE wave incident from isotropic medium into biaxial anisotropic medium; (a) medium 1 (left-handed) with incident angle θi=25°; (b) medium 1 (left-handed) with θi=45°, total reflection; (c) medium 2 (right-handed) with θi=25°; (d) medium 2 (right-handed) with θi=45°, total reflection.
(a)
(b)
(c)
(d)


### 2.2. Total Transmission of TM Waves

Similarly, consider a TM incident case. According to the dispersion relations of TM waves in two mediums,(7)k0xTM2+kyTM2=ω2ε0μ0,(8)kxTM2εy+kyTM2εx=ω2μz,and the boundary conditions, the reflection coefficient for TM waves can be written as(9)RM=εyk0xTM-ε0ktxTMεyk0xTM+ε0ktxTM.When total transmission occurs at the boundary, let RM=0; we can get the total transmission condition in terms of Brewster Angle θB(10)sin2⁡θB=εxμ0εy-μzε0μ0εxεy-ε02.Since 0<sin2⁡θB<1, the existence conditions of Brewster Angle θB can be written as follows:Ifεx>0,(11)μ0εy-μzε0μ0εxεy-ε02>0μzεx-ε0μ0μ0εxεy-ε02>0.Ifεx<0,(12)μ0εy-μzε0μ0εxεy-ε02<0μzεx-ε0μ0μ0εxεy-ε02<0.We can see that total transmission at the boundary between an isotropic medium and a biaxial anisotropic medium for TM waves is up to the value of εx, εy, and μz. Together with (7), we can get the total transmission conditions in Table 2.Table 2
Conditions of total transmission of TM waves.
The range of constitutive parameters
The condition of total transmission
(D)εyμz=ε0μ0∩εxεy≠ε02
Total transmission occurs only when TM waves are incident vertically
(E)εyμz=ε0μ0∩εxεy=ε02
Total transmission occurs for all TM waves no matter what incident angleθi is
(F)εyμz≠ε0μ0∩εxεy≠ε02
Total transmission occurs only when the incident angle is Brewster AngleθBGenerally, we choose the constitutive parameters of biaxial anisotropic medium to match condition (E) in Table2: ε̿=20000.50000.6; μ̿=0.90000.80000.5. In Figure 3, we show z component of magnetic field with a TM incidence with two different incident angles. The incident angles are 30° and 60°, respectively. We can see that, for both cases, total transmission occurs at the boundary. All the results match with the theoretical derivation listed in Table 2.Figure 3
H
z of a TM wave incident from isotropic medium into biaxial anisotropic medium, total transmission occurs. (a) The incident angle θi=30°. (b) θi=60°.
(a)
(b)Besides, in order to demonstrate the condition that total transmission occurs only when the incident angle is Brewster AngleθB, we choose condition (F) in Table 2: ε̿=20000.50000.6; μ̿=0.90000.80000.5. Its Brewster Angle θB=50.63°. In Figure 4, we show the z component of magnetic field with a TM incidence with different incident angles. The incident angles are 30°, 50.63°, and 70°, respectively. We can see that total transmission only occurs when the incident angle is Brewster Angle θB=50.63°. All the results match with the theoretical derivation listed in Table 2.Figure 4
H
z of a TM wave incident from isotropic medium into biaxial anisotropic medium. (a) The incident angle θi=30°, no total transmission. (b) θi=70°, no total transmission. (c) The incident angle is Brewster Angle θB=50.63°, total transmission occurs.
(a)
θ
i
=
30
°
(b)
θ
i
=
70
°
(c)
θ
i
=
θ
B
=
50.63
°Other situations can also be simulated and verified.


## 2.1. Total Reflection of TE Waves

Considering the configuration in Figure1, a TE wave is incident from a semi-infinite isotropic medium into a biaxial anisotropic medium at an oblique angle θi. Region 0 is the isotropic medium, with permittivity ε0 and permeability μ0. Region t is the biaxial anisotropic medium, with permittivity ε̿ and permeability μ̿ which are shown in (1) and (2), respectively.Figure 1
Reflection and transmission of a TE wave at an interface between a semi-infinite isotropic medium (region 0) and a biaxial anisotropic medium (regiont).For Figure1, with wave vectors k-iTE=x^k0xTE+y^kyTE (incident) and k-rTE=-x^k0xTE+y^kyTE (reflection) in the isotropic medium and k-tTE=x^ktxTE+y^ktTE (transmission) in the biaxial anisotropic medium, the dispersion relations can be expressed as (3)k0xTE2+kyTE2=ω2ε0μ0,(4)ktxTE2μy+kyTE2μx=ω2εz.Based on the dispersion relations of two mediums and the boundary conditions, we can get the reflection coefficient for TE waves(5)RE=μyk0xTE-μ0ktxTEμyk0xTE+μ0ktxTE.When ktxTE becomes imaginary, that is, (6)μyμxkyTE2>ω2εzμy,total reflection occurs at the boundary. Together with (3), we can get the total reflection conditions in Table 1.Table 1
Conditions of total reflection of TE waves.
The range of constitutive parameters
The condition of total reflection
(A)μxμy>0∩εzμy>0
(a) Ifεzμx>ε0μ0, there is no total reflection at the boundary
(b) Ifεzμx<ε0μ0, total reflection occurs when incident angle θi> critical angle θc, where θc=arcsin⁡εzμx/ε0μ0
(B)μxμy>0∩εzμy<0
Total reflection occurs for all TE waves
(C)μxμy<0∩εzμy<0
(a) Ifεzμx>ε0μ0, total reflection occurs for all TE waves no matter what incident angle θi is
(b) Ifεzμx<ε0μ0, total reflection occurs when incident angle θi< critical angle θcGenerally, we choose two biaxial anisotropic mediums: medium 1 is left-handed medium with the constitutive parameters:ε̿1=0.60000.5000-0.5; μ̿1=-0.5000-0.90000.7; medium 2 is right-handed medium with the constitutive parameters: ε̿2=0.60000.50001; μ̿2=0.250000.30000.2. Both of them satisfy the condition b of case (A) shown in Table 1 and the critical angles are both θc=30°. In Figure 2, we show z component of electric field in the xy plane generated by the TE incident wave with two different incident angles. The incident angles in Figures 2(a) and 2(c) are both 25°. We can see that there is nontotal reflection. Differently, the transmission wave and the incident wave are on the same side of the normal in medium 1 (left-handed) while they are on the different side in the medium 2 (right-handed). Shown as Figures 2(b) and 2(d), when the incident angles are 45° (θi>θc), total reflection occurs at the boundary. All the results are in accordance with the theoretical derivation summarized in Table 1.Figure 2
E
z of a TE wave incident from isotropic medium into biaxial anisotropic medium; (a) medium 1 (left-handed) with incident angle θi=25°; (b) medium 1 (left-handed) with θi=45°, total reflection; (c) medium 2 (right-handed) with θi=25°; (d) medium 2 (right-handed) with θi=45°, total reflection.
(a)
(b)
(c)
(d)


## 2.2. Total Transmission of TM Waves

Similarly, consider a TM incident case. According to the dispersion relations of TM waves in two mediums,(7)k0xTM2+kyTM2=ω2ε0μ0,(8)kxTM2εy+kyTM2εx=ω2μz,and the boundary conditions, the reflection coefficient for TM waves can be written as(9)RM=εyk0xTM-ε0ktxTMεyk0xTM+ε0ktxTM.When total transmission occurs at the boundary, let RM=0; we can get the total transmission condition in terms of Brewster Angle θB(10)sin2⁡θB=εxμ0εy-μzε0μ0εxεy-ε02.Since 0<sin2⁡θB<1, the existence conditions of Brewster Angle θB can be written as follows:Ifεx>0,(11)μ0εy-μzε0μ0εxεy-ε02>0μzεx-ε0μ0μ0εxεy-ε02>0.Ifεx<0,(12)μ0εy-μzε0μ0εxεy-ε02<0μzεx-ε0μ0μ0εxεy-ε02<0.We can see that total transmission at the boundary between an isotropic medium and a biaxial anisotropic medium for TM waves is up to the value of εx, εy, and μz. Together with (7), we can get the total transmission conditions in Table 2.Table 2
Conditions of total transmission of TM waves.
The range of constitutive parameters
The condition of total transmission
(D)εyμz=ε0μ0∩εxεy≠ε02
Total transmission occurs only when TM waves are incident vertically
(E)εyμz=ε0μ0∩εxεy=ε02
Total transmission occurs for all TM waves no matter what incident angleθi is
(F)εyμz≠ε0μ0∩εxεy≠ε02
Total transmission occurs only when the incident angle is Brewster AngleθBGenerally, we choose the constitutive parameters of biaxial anisotropic medium to match condition (E) in Table2: ε̿=20000.50000.6; μ̿=0.90000.80000.5. In Figure 3, we show z component of magnetic field with a TM incidence with two different incident angles. The incident angles are 30° and 60°, respectively. We can see that, for both cases, total transmission occurs at the boundary. All the results match with the theoretical derivation listed in Table 2.Figure 3
H
z of a TM wave incident from isotropic medium into biaxial anisotropic medium, total transmission occurs. (a) The incident angle θi=30°. (b) θi=60°.
(a)
(b)Besides, in order to demonstrate the condition that total transmission occurs only when the incident angle is Brewster AngleθB, we choose condition (F) in Table 2: ε̿=20000.50000.6; μ̿=0.90000.80000.5. Its Brewster Angle θB=50.63°. In Figure 4, we show the z component of magnetic field with a TM incidence with different incident angles. The incident angles are 30°, 50.63°, and 70°, respectively. We can see that total transmission only occurs when the incident angle is Brewster Angle θB=50.63°. All the results match with the theoretical derivation listed in Table 2.Figure 4
H
z of a TM wave incident from isotropic medium into biaxial anisotropic medium. (a) The incident angle θi=30°, no total transmission. (b) θi=70°, no total transmission. (c) The incident angle is Brewster Angle θB=50.63°, total transmission occurs.
(a)
θ
i
=
30
°
(b)
θ
i
=
70
°
(c)
θ
i
=
θ
B
=
50.63
°Other situations can also be simulated and verified.


## 3. Result and Discussion

Since biaxial anisotropic medium is a reciprocal medium, if total transmission occurs at the boundary when a TM wave is incident from isotropic medium into biaxial anisotropic medium, total transmission will also occur when this TM wave is incident from biaxial anisotropic medium into the same isotropic medium. Therefore, TM waves can full-pass the biaxial anisotropic slab, independent of the thickness of the slab as long as constitutive parameters match the condition in Table2.Besides, the dispersion relations of TE waves and TM waves are dual (according to (4) and (8)), so we can get the other dispersion relation by exchanging the elements of ε̿ and μ̿ in either relation (μx, μy, and εz exchange with εx, εy, and μz). Also because of the dual property, although we only discussed the total reflection with TE waves incidence above, we can get the TM case if we replace μx, μy, and εz with εx, εy, and μz. In the similar way, for total transmission, we can we can get the condition for TE waves by replacing εx, εy, and μz with μx, μy, and εz.Hence, if biaxial anisotropic medium’s permittivityε̿ and permeability μ̿ are set appropriately for some special value, it will simultaneously satisfy both the condition of TE (TM) waves’ total reflection and TM (TE) waves’ total transmission. In this way, the separation of TE waves and TM waves can be realized easily. Specifically, there are two ways to design such a wave splitter.


### 3.1. Total Reflection for TE Waves and Total Transmission for TM Waves

We can choose constitutive parameters to satisfy one of the conditions in Tables1 and 2 to separate TE waves and TM waves, which are combined both in incidence waves. In this way, TE waves will totally reflect while TM waves will totally transmit.For instance, we choose biaxial anisotropic medium’s constitutive parameters to satisfy condition (b) of case (A) in Table1 and case (E) in Table 2: ε̿=20000.50001; μ̿=0.250000.80000.5; in this way, its critical angle is θc=30° and the total reflection condition of TE waves is satisfied. In Figure 5, we show the electric field and the magnetic field when a Gaussian beam is incident from air into the biaxial slab with the incident angle of 45°. We can see from Figure 5(a) that when θi>θc, total reflection occurs for TE waves. Figure 5(b) shows that TM waves can totally transmit.Figure 5
The total reflection for TE waves and total transmission for TM waves of a Gaussian beam incident from air into a biaxial anisotropic slab with the incident angleθi=45°. (a) Ez of an incident TE wave, total reflection. (b) Hz of an incident TM wave, total transmission.
(a)
(b)


### 3.2. Total Reflection for TM Waves and Total Transmission for TE Waves

For this case, we choose biaxial anisotropic medium’s constitutive parameters:ε̿=-0.5000-0.70000.5; μ̿=20000.50000.8. These constitutive parameters satisfy both the condition of TM waves’ total reflection and TE waves’ total transmission. Figure 6(a) shows that there is total reflection for TM waves. Figure 6(b) shows that there is always total transmission for TE waves.Figure 6
The total reflection for TM waves and total transmission for TE waves of a Gaussian beam incident from air into a biaxial anisotropic slab with the incident angleθi=45°. (a) Hz of an incident TM wave, total reflection. (b) Ez of an incident TE wave, total transmission.
(a)
(b)


## 3.1. Total Reflection for TE Waves and Total Transmission for TM Waves

We can choose constitutive parameters to satisfy one of the conditions in Tables1 and 2 to separate TE waves and TM waves, which are combined both in incidence waves. In this way, TE waves will totally reflect while TM waves will totally transmit.For instance, we choose biaxial anisotropic medium’s constitutive parameters to satisfy condition (b) of case (A) in Table1 and case (E) in Table 2: ε̿=20000.50001; μ̿=0.250000.80000.5; in this way, its critical angle is θc=30° and the total reflection condition of TE waves is satisfied. In Figure 5, we show the electric field and the magnetic field when a Gaussian beam is incident from air into the biaxial slab with the incident angle of 45°. We can see from Figure 5(a) that when θi>θc, total reflection occurs for TE waves. Figure 5(b) shows that TM waves can totally transmit.Figure 5
The total reflection for TE waves and total transmission for TM waves of a Gaussian beam incident from air into a biaxial anisotropic slab with the incident angleθi=45°. (a) Ez of an incident TE wave, total reflection. (b) Hz of an incident TM wave, total transmission.
(a)
(b)


## 3.2. Total Reflection for TM Waves and Total Transmission for TE Waves

For this case, we choose biaxial anisotropic medium’s constitutive parameters:ε̿=-0.5000-0.70000.5; μ̿=20000.50000.8. These constitutive parameters satisfy both the condition of TM waves’ total reflection and TE waves’ total transmission. Figure 6(a) shows that there is total reflection for TM waves. Figure 6(b) shows that there is always total transmission for TE waves.Figure 6
The total reflection for TM waves and total transmission for TE waves of a Gaussian beam incident from air into a biaxial anisotropic slab with the incident angleθi=45°. (a) Hz of an incident TM wave, total reflection. (b) Ez of an incident TE wave, total transmission.
(a)
(b)


## 4. Conclusion

This paper firstly discusses the total reflection and total transmission when electromagnetic waves are incident from a semi-infinite isotropic medium into a biaxial anisotropic medium. Furthermore, we design a TE/TM wave splitter based on a biaxial anisotropic slab. The structure of the wave splitter is simple and has little limitation on constitutive parameters, incident angle, and chirality. Meanwhile, such wave splitters can separate TE waves and TM waves as long as constitutive parameters are chosen in a wide range. The results of simulation verify the theoretical derivations.



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