Goos-Hänchen Lateral Displacements at the Interface between Isotropic and Gyroelectric Media

Abstract

A detailed study on Goos-Hänchen (GH) lateral displacements of the reflected and transmitted waves propagating at the interface between an isotropic medium and a gyroelectric medium in Voigt configuration is presented. After the reflection coefficient and transmission coefficient are derived, based on the stationary phase approach, GH lateral displacements are obtained analytically. The numerical results for a specific gyroelectric medium are also given. It shows that with the existence of an applied magnetic field, the GH effect occurs not only during total reflection but also during nontotal reflection, which is not true for isotropic media. Moreover, due to the nonreciprocal property of the gyroelectric medium, the sign of the incident angle also influences the displacements. Finite-element method simulations have verified the theoretical results.

1. Introduction

Gyroelectric medium is an electron plasma with an applied magnetic field. The characteristics of electromagnetic waves propagation in gyroelectric plasmas have been theoretically investigated in many literatures. The magnetoplasma modes in Voigt, perpendicular, and Faraday configurations have been studied by Kushwaha and Halevi [1–3]; Gillies and Hlawiczka have done some researches on gyrotropic waveguides in detail [4–8], and dyadic Green's functions for gyrotropic medium have been investigated by both Eroglu and Li [9–11]. There are also some studies focusing upon the effects of magnetic field on semiconducting plasma slab and negatively refracting surfaces [12, 13]. Furthermore, propagation and scattering characteristics in gyrotropic systems [9, 14] and surface modes at the interface of a special gyroelectric medium [15] have been investigated extensively.

The Goos-Hänchen (GH) effect [16, 17] has been studied for many years. GH lateral displacement refers to the spatial displacement of a reflected wave from the position expected by geometrical reflection. Traditionally, GH shift is always a phenomenon during total reflection, when an electromagnetic beam is reflected at the interface between media with different reflective indexes. The lateral displacement has the well-known expression 𝑆=−d𝜙/d𝑘_𝑦, which was proposed by Artmann using the stationary phase method [18]. The GH effect has been analyzed both theoretically [19, 20] and experimentally [21–23]. The lateral shifts were found to be positively or negatively large for both reflected and transmitted beams in different media such as dielectric surfaces or slabs [24–26], metal surfaces [27], dielectric-chiral surface [28–31], absorptive media [32–34], and metamaterial [35–38].

In this paper, we focus on the reflection and transmission characteristics of an electromagnetic beam propagating at the interface between an isotropic medium and a gyroelectric medium in Voigt configuration. After obtaining the analytical expressions for the reflection and transmission coefficients, we get the mathematical result for the GH lateral displacement by the stationary phase approach. Based on the results, we discuss some unique phenomena.

2. Reflection, Transmission, and Goos-Hänchen Lateral Displacements

Considering the configuration in Figure 1, a plane wave is incident from a semi-infinite isotropic medium into a gyroelectric medium at an oblique angle 𝜃_𝑖 with respect to the normal of the interface. Region 1 is an isotropic medium, with permeability 𝜇₁ and permittivity 𝜀₁. In region 2, an external magnetic field 𝐵₀ is applied in +𝑧 direction, parallel to the interface and perpendicular to the direction of the wave propagation (i.e., gyroelectric medium in Voigt configuration). Its permeability is 𝜇₂, while permittivity 𝜀₂ is a tensor, which takes the following form: 𝜀₂=𝜀_𝑥𝑥𝑖𝜀_𝑔0−𝑖𝜀_𝑔𝜀_𝑦𝑦000𝜀_𝑧𝑧, (1) where the matrix elements are given by𝜀_𝑥𝑥=𝜀_𝑦𝑦=𝜀_∞𝜔1−^2𝑝𝜔²−𝜔²_𝑐,𝜀_𝑧𝑧=𝜀_∞𝜔1−^2𝑝𝜔²,𝜀_𝑔=𝜀_∞−𝜔²_𝑝𝜔_𝑐𝜔𝜔²−𝜔²_𝑐. (2a)(2b)(2c)Here, 𝜔_𝑝=𝑁𝑞²_𝑒/𝑚_eff𝜀_∞ and 𝜔_𝑐=𝑞_𝑒𝐵₀/𝑚_eff  are the plasma and cyclotron frequencies, respectively, 𝜀_∞ is the background permittivity, 𝑁 is the electron density, 𝑚_eff is the effective mass, and 𝑞_𝑒 is the electron charge.

Figure 1 

Reflection and transmission of  TM waves at an interface between a semi-infinite isotropic medium (region 1) and a gyroelectric medium in the Voigt configuration (region 2). In region 2, applied magnetic field 𝐵0 is in +𝑧 direction.

It is known that for gyroelectric medium in the Voigt configuration, waves can be decoupled into TE and TM modes, and only the TM mode is affected by the gyrotropy [13, 14]. Thus, we focus on the TM case.

For Figure 1, with wave vectors 𝑘_𝑖=𝑦𝑘_𝑦+𝑥𝑘_1𝑥 (incident) and 𝑘_𝑟=𝑦𝑘_𝑦−𝑥𝑘_1𝑥 (reflection) in the isotropic medium, 𝑘_𝑡=𝑦𝑘_𝑦+𝑥𝑘_2𝑥 (transmission) in the gyroelectric medium, the dispersion relations can be expressed as [13, 14] 𝑘²_𝑦+𝑘²_1𝑥=𝜔²𝜇₁𝜀₁,𝑘²_𝑦+𝑘²_2𝑥=𝜔²𝜇₂𝜀_𝑉. (3)(4) Here, 𝜀_𝑉=(𝜀²_𝑥𝑥−𝜀²_𝑔)/𝜀_𝑥𝑥 is the equivalent permittivity of the gyroelectric medium in the Voigt configuration for TM waves.

2.1. Nontotal Reflection Case

According to the Maxwell equations and boundary conditions, when there is no total reflection at the boundary, the transmission and reflection coefficients for TM waves can be written as 𝑅=𝜎𝑘_1𝑥−𝑘_2𝑥+𝑖𝜏𝑘_𝑦𝜎𝑘_1𝑥+𝑘_2𝑥−𝑖𝜏𝑘_𝑦,𝑇=2𝜎𝑘_1𝑥𝜎𝑘_1𝑥+𝑘_2𝑥−𝑖𝜏𝑘_𝑦, (5) where the parameters 𝜎 and 𝜏 are both dimensionless, defined as 𝜀𝜎=^𝑉𝜀₁𝜀,𝜏=^𝑔𝜀_𝑥𝑥. (6)

According to (5), both reflection coefficient 𝑅 and transmission coefficient 𝑇  are not real. They always have an imaginary part due to 𝜏 as a result of the gyrotropy. Hence, the GH angle 𝜙 of either reflection or transmission is not zero. Therefore, there always exists a GH lateral displacement, although there is no total reflection at the boundary.

Noting that the denominators of both coefficients are the same, the GH lateral displacements of the reflection and transmission for TM waves can be expressed as 𝑆_𝑡1=−|𝑇|²[𝑇][𝑇]Re𝜕Im𝜕𝑘_𝑦[𝑇][𝑇]−Im𝜕Re𝜕𝑘_𝑦=−𝜏𝑘_𝑦𝜎𝑘_1𝑥+𝑘_2𝑥²+𝜏𝑘_𝑦²×𝜎𝑘_1𝑥+𝑘_2𝑥𝑘_𝑦+𝜎𝑘_𝑦𝑘_1𝑥+𝑘_𝑦𝑘_2𝑥,𝑆_𝑟1=−|𝑅|²[𝑅][𝑅]Re𝜕Im𝜕𝑘_𝑦[𝑅][𝑅]−Im𝜕Re𝜕𝑘_𝑦=𝑆_𝑡+Δ𝑆_𝑟𝑡, (7) where Δ𝑆_𝑟𝑡=−𝜏𝑘_𝑦𝜎𝑘_1𝑥−𝑘_2𝑥²+𝜏𝑘_𝑦²×𝜎𝑘_1𝑥−𝑘_2𝑥𝑘_𝑦+𝜎𝑘_𝑦𝑘_1𝑥−𝑘_𝑦𝑘_2𝑥. (8)

It is interesting to note that both 𝑆_𝑡 and 𝑆_𝑟 are proportional to 𝜏, the unique parameter of the gyroelectric medium, and 𝜏 is a direct manifestation of the applied magnetic field 𝐵₀ (see (6)). In fact, if the external magnetic field is zero, that is, 𝜔_𝑐=0, according to the permittivity tensor (see (1) and (2a), (2b), and (2c)), it is reasonable to consider the gyroelectric medium as a dispersive isotropic medium. Furthermore, without the external magnetic field, both 𝜀_𝑔 and 𝜏 are zero, and according to (7)-(8), the GH lateral displacements are also zero, which are consistent with the fact that GH displacement is always a phenomenon along with total reflection for isotropic media case.

2.2. Total Reflection Case

When 𝑘²_𝑦>𝜔²𝜇₂𝜀_𝑉, according to (4), 𝑘_2𝑥 becomes imaginary, that is, total reflection occurs at the boundary. Let 𝑘_2𝑥=𝑖𝛼, the dispersion relation of gyroelectric medium becomes 𝑘²_𝑦−𝛼²=𝜔²𝜇₂𝜀_𝑉, and (5) can be rewritten as 𝑅^′=𝜎𝑘_1𝑥−𝑖𝛼+𝑖𝜏𝑘_𝑦𝜎𝑘_1𝑥+𝑖𝛼−𝑖𝜏𝑘_𝑦,𝑇^′=2𝜎𝑘_1𝑥𝜎𝑘_1𝑥+𝑖𝛼−𝑖𝜏𝑘_𝑦. (9)

Since the transmitted wave is evanescent in the 𝑥 direction, our focus is only upon the GH lateral displacement of the reflection, which can be expressed as 𝑆^′_𝑟1=−𝑅^′2Re𝑅^′𝜕Im𝑅^′𝜕𝑘_𝑦−Im𝑅^′𝜕Re𝑅^′𝜕𝑘_𝑦=2𝜎𝑘_1𝑥𝜎𝑘_1𝑥²+𝛼−𝜏𝑘_𝑦²𝑘_𝑦𝛼−𝜏𝑘_𝑦𝑘²_1𝑥+𝑘_𝑦𝛼−𝜏. (10) We can see that if total reflection occurs, the lateral displacement 𝑆^′_𝑟 is not zero, even for the case without applied magnetic field (𝜏=0). It means that for isotropic media, there is GH effect during the total reflection, which is the fact we have known.

3. Result and Discussion

Here, we consider an indium antimony (InSb) with an external magnetic field as the gyroelectric medium. The isotropic medium is a vacuum. The material parameters used in the computation are 𝜇₁=𝜇₂=𝜇₀, 𝜀₁=𝜀₀, 𝜀_∞=15𝜀₀, 𝑁=10²² m⁻³, and 𝑚_eff=0.015𝑚₀=0.13664×10⁻³¹ kg [39–41]. Hence, 𝜔_𝑝=1.19×10¹³ rad/s and 𝜔_𝑐/𝜔_𝑝=0.98𝐵₀.

For TM waves, since 𝜀_𝑥𝑥 and 𝜀_𝑔 change with frequency, the equivalent permittivity 𝜀_𝑉 is also dependent on frequency, as shown in Figure 2. The applied magnetic field 𝐵₀ splits 𝜀_𝑥𝑥 into 2 branches with the boundary of cyclotron frequency 𝜔_𝑐, and the case of 𝜀_𝑔 is similar. Thus, the equivalent permittivity 𝜀_𝑉 also has 2 branches with another boundary of 𝜔_𝐻, and each branch has its own plasma frequency, denoted by 𝜔_𝑝− and 𝜔_𝑝+, shown as [42] 𝜔_𝐻=𝜔²_𝑐+𝜔²_𝑝,𝜔_𝑝±=12±𝜔_𝑐+𝜔²_𝑐+4𝜔²_𝑝. (11) We plot the GH lateral displacements in the (𝜔,𝜃_𝑖) plane with a color scale proportional to the magnitude, as shown in Figure 3. The horizontal axis is the frequency, normalized to the plasma frequency 𝜔_𝑝, and the vertical axis is the incident angle, from −85° to +85°. We define the angle to be positive when the incident wave comes from the lower left (as shown in Figure 1) and to be negative when the incident wave comes from the upper left. The displacements are normalized to 𝜆₀=2𝜋𝑐/𝜔, the wavelength in the vacuum.

(a)

(b)

Figure 2 

The frequency dependence of the permittivity tensor and the corresponding equivalent permittivity 𝜀𝑉 for gyroelectric medium (in the Voigt configuration). The external magnetic field is (a) 𝐵0=0 and  (b) 𝐵0=+𝑧0.4T, respectively.

(a)

(b)

(c)

(d)

Figure 3 

Goos-Hänchen lateral displacements of  TM waves transmitted and reflected from the interface of vacuum and gyroelectric medium. The frequencies are normalized to 𝜔𝑝, and the displacements are normalized to 𝜆0=2𝜋𝑐/𝜔, the wavelength in the vacuum. (a) The lateral displacement of reflected waves 𝑆𝑟 with no applied magnetic field. (b) The lateral displacement of transmitted waves 𝑆𝑡 with no applied magnetic field. (c) The lateral displacement of reflected waves 𝑆𝑟 with an applied magnetic field 𝐵0=+𝑧0.4T, and the Brewster angle is also shown (in white dashed line). (d) The lateral displacement of transmitted waves 𝑆𝑡 with the same applied magnetic field as (c).

For the plasma, when there is no applied magnetic field, since both 𝜀_𝑔 and 𝜏 are zero, the medium can be considered as a dispersive isotropic medium. Corresponding to the change of permittivity 𝜀_𝑉, there are two regions (A and B) in Figures 3(a) and 3(b). In region A, total reflection occurs at the interface. Thus, there is GH effect, and the lateral displacement of the reflected wave is not zero. But the transmitted wave is evanescent in the 𝑥 direction, making GH lateral displacement for transmitted waves meaningless, leaving a blank in Figure 3(b). In region B, there is no total reflection and no GH lateral displacement for either reflected or transmitted waves. This is consistent with the fact of isotropic media case.

For TM waves, the existence of the applied magnetic field splits each region up into two, marked with subscript 1 and 2, shown in Figures 3(c) and 3(d). In regions A₁ and A₂, total reflection occurs at the interface. In regions B₁ and B₂, although there is no total reflection at the interface, GH lateral displacements for both reflection and transmission are not zero, which is different from the isotropic media case.

It is also of interest to note that in regions B₁ and B₂ of Figure 3(c), there is a large displacement of reflected TM waves. It is due to the Brewster angle, shown in the white dashed line in the figure. For TM waves, there is an abrupt phase change near the Brewster angle [41] which causes a large lateral displacement for the reflected beam. However, there is no abrupt change for the phase of the transmission coefficient at the Brewster angle. Therefore, it is reasonable to expect a large lateral displacement of the reflected beam near the Brewster angle but no such phenomenon for the transmitted beam.

It is important to note that the displacements with applied magnetic field are asymmetric with respect to the incident angle 𝜃_𝑖 (see Figures 3(c) and 3(d)), but they are symmetric without applied magnetic field (see Figures 3(a) and 3(b)). For example, in region B₂, 𝑆_𝑟 is positive for a positive incident angle while it becomes negative for a negative incident angle. This asymmetric phenomenon is corresponding to the nonreciprocal property of the gyroelectric medium. If we only change the sign of incident angle 𝜃_𝑖 from positive to negative, that is, the wave is incident from the upper left instead of the lower left, the field in gyroelectric medium changes even for the same absolute value of incident angle. Thus, the reflection and transmission coefficients change and so do the GH lateral displacements.

We show the reflection and transmission of a Gaussian TM beam incident from the vacuum to gyroelectric medium in Figure 4, which is the numerical result based on the finite-element method. All the parameters used to calculate are the same except the sign of the incident angle. To be specific, 𝜔=𝜔_𝑝 and |𝜃_𝑖|=80^∘. The only difference is that in Figure 4(a), the TM wave incidents from the lower left and 𝜃_𝑖 are positive, while in Figure 4(b), the wave comes from the upper left, and 𝜃_𝑖 is negative. From Figure 4, we can see that the fields are different and so are the lateral displacements, which are consistent with our analysis as shown in Figure 3.

(a)

(b)

Figure 4 

The reflection and transmission of a Gaussian TM beam incident from the vacuum to gyroelectric medium. The frequencies are both 𝜔𝑝, and the absolute values of the incident angles are both 80°. All the parameters are the same except the sign of incident angle. (a) The wave is incident from the lower left (𝜃𝑖=+80∘); (b) The wave is incident from the upper left (𝜃𝑖=−80∘).

4. Conclusion

This paper investigates the Goos-Hänchen lateral displacements of a TM beam propagating at the interface between an isotropic medium and a gyroelectric medium in Voigt configuration. As different from the isotropic case, there are always GH effects, not only when total reflection occurs at the boundary but also when there is no total reflection. Furthermore, due to the nonreciprocal property of the gyroelectric medium, the sign of the incident angle also influences the displacements. The results here may have potential applications in magnetic modulations and Terahertz researches.

Acknowledgments

This work is sponsored by National Natural Science Foundation of China (51107003) and Beijing Jiaotong University (2009JBM092, 2011RC049, and 2011JBM110).