Reflection and transmission of Gaussian beam by a uniaxial anisotropic slab

1. Introduction

The electromagnetic properties of anisotropic media have aroused increasing interest over the years, owing to their wide applications in optical signal processing, optimum design of optical fibers, radar cross section controlling, and microwave device fabrication, etc. A number of studies have been devoted to the analysis of the reflection and transmission of electromagnetic waves at a plane interface separating an isotropic and anisotropic medium. The reflection and transmission of an electromagnetic plane wave incident on a biaxially anisotropic–isotropic interface are treated by Graham et al. [1], and on a uniaxial chiral slab by Dong et al. [2]. For the case of an incident shaped beam, Stamnes et al. present the formulations and numerical results of focused paraxial field intensities inside a uniaxial or biaxial crystal [3–7]. In one previous paper [8], within the generalized Lorenz-Mie theory (GLMT) framework we have obtained the expansion of an incident Gaussian beam (focused TEM₀₀ mode laser beam) in terms of cylindrical vector wave functions (CVWFs). In this paper, the use of such an expansion enables us to construct an analytical solution to the reflection and transmission of a Gaussian beam by a uniaxial anisotropic slab.

The body of this paper proceeds as follows. Section 2 provides the theoretical procedure for the determination of the reflected, internal as well as transmitted fields for a Gaussian beam incident on a uniaxial anisotropic slab. In Section 3, numerical results of the normalized field intensity distributions are displayed. Section 4 is the conclusion.

2. Formulation

As shown in Fig. 1, an incident Gaussian beam propagates in free space and along the positive $z^{\prime }$ axis in the plane $xOz$ of the Cartesian coordinate system $Oxyz$, with the middle of its beam waist located at origin $O^{\prime }$ on the axis $O^{\prime }{z}^{\prime }$. The planes $z=0$ and $z=d$ are the interfaces between free space and an infinite uniaxial anisotropic slab of thickness $d$. Origin $O$ has a coordinate $z_0$ on the axis $O^{\prime }{z}^{\prime }$, and the angle of incidence or the angle made by the axis $O^{\prime }{z}^{\prime }$ with the axis $Oz$ is $\beta$. In this paper, the time-dependent part of the electromagnetic fields is assumed to be $\exp (-i\omega t)$.

 

Fig. 1 Geometry of an incident Gaussian beam from free space on a uniaxial anisotropic slab.

Download Full Size | PPT Slide | PDF

In [8], an expansion has been obtained of the electromagnetic fields of an incident Gaussian beam (focused TEM00 mode laser beam) in terms of the CVWFs with respect to the system $Oxyz$, which, for the sake of subsequent applications of boundary conditions, can be written as

In Eq. (2) $\lambda =k_0\sin \zeta$, $h=k_0\cos \zeta$, $k_0$ is the free-space wavenumber, and $I_{n,TE}^m$, $I_{n,TM}^m$ are the Gaussian beam shape coefficients. For a TE polarized Gaussian beam, the coefficients $I_{n,TE}^m$, $I_{n,TM}^m$ are

For a TM polarized Gaussian beam, the corresponding expansions can be obtained only by replacing $I_{n,TE}^m$ by $i{I}_{n,TM}^m$, and $I_{n,TM}^m$ by $i{I}_{n,TE}^m$.

Equation (1) can be interpreted as an incident Gaussian beam being expanded into a continuous spectrum of cylindrical vector waves, with each cylindrical vector wave having a propagation vector $k_0=\lambda \widehat{r}+h\widehat{z}$. The angle between $k_0$ and the positive $z$ axis is $\zeta$, so that only $E_1^i$ represents the cylindrical vector waves that are incident on the interface $z=0$.

By following Eq. (2), the reflected beam is expanded as

Let’s consider that a uniaxial anisotropic medium is characterized by a permittivity tensor $\overline{\epsilon }=\widehat{x}\widehat{x}{\epsilon }_t+\widehat{y}\widehat{y}{\epsilon }_t+\widehat{z}\widehat{z}{\epsilon }_z$ in the system $Oxyz$ and a scalar permeability ${\mu }_0$.

In [12], based on the eigen plane wave spectrum representation of the fields and the Fourier expansion for the unknown angular spectrum amplitude, the eigenfunction representations of the fields in uniaxial anisotropic media are developed in terms of the CVWFs, as follows:

Due to the phase continuity over the interfaces $z=0$ and $z=d$ when using the boundary conditions, we have ${\lambda }_q=k_q\sin {\theta }_k=\lambda =k_0\sin \zeta$ ($q=1,2$ and $\zeta$ is from $0$ to $\pi /2$). Then, Eq. (7) is transformed into

Let ${\theta }_k=\pi -{\psi }_k$ in Eq. (7). Then, by following the same theoretical procedure as in obtaining Eq. (14), i.e., having ${\lambda }_q=k_q\sin {\psi }_k=\lambda =k_0\sin \zeta$ ($q=1,2$ and $\zeta$ is from $0$ to $\pi /2$), Eq. (7) becomes

It should be pointed out that $E_1^w$ represents the electromagnetic waves within the uniaxial anisotropic slab (internal beam) that propagate towards the interface $z=d$, and that $E_2^w$ towards the interface $z=0$.

An appropriate expansion of the transmitted beam in terms of the CVWFs can be given by

For the sake of brevity, only the expansions of the electric fields are written, and the corresponding expansions of the magnetic fields can be obtained with the following relations

The unknown expansion coefficients $a_m(\zeta )$, $b_m(\zeta )$ in Eq. (6), $E_{mq}(\zeta )$, $F_{mq}(\zeta )$ in Eqs. (14) and (20) as well as $c_m(\zeta )$, $d_m(\zeta )$ in Eq. (21) can be determined by using the following boundary conditions

By virtue of the fields expansions, the boundary conditions in Eq. (23) can be written as

The expansion coefficients $a_m(\zeta )$, $c_m(\zeta )$, $E_{m1}(\zeta )$ and $F_{m1}(\zeta )$ can be determined from Eqs. (25), (27), (29) and (31), and $b_m(\zeta )$, $d_m(\zeta )$, $E_{m2}(\zeta )$ and $F_{m2}(\zeta )$ from Eqs. (26), (28), (30) and (32). By substituting them into Eqs. (6), (14), (20) and (21), the reflected, internal and transmitted beams are obtained.

3. Numerical results

In this paper, we will focus our attention on the normalized field intensity distributions, which are defined, respectively, by

By substituting the $r$, $\varphi$ and $z$ components of the CVWFs into Eqs. (1), (6), (14), (20) and (21), the $r$, $\varphi$ and $z$ components of the electric fields of the incident Gaussian beam, reflected beam, internal beam as well as transmitted beam can be obtained [8, 12], and then the explicit expressions of Eqs. (33)–(35).

Figures 2 and 3, sharing the same colorbar, show the normalized field intensity distributions in the $xOz$ plane for a uniaxial anisotropic slab, illuminated by a TM and TE polarized Gaussian beam, respectively. The thickness of the slab $d$ and the beam waist radius $w_0$ are assumed to be ten and two times the wavelength of the incident Gaussian beam, and $a_1=\sqrt{3}{k}_0$, $a_2=\sqrt{2}{k}_0$, $\beta =\pi /3$, $z_0=0$.

 

Fig. 2 ${\left|(E^i+E^r)/E_0\right|}^2$, ${\left|(E_1^w+E_2^w)/E_0\right|}^2$ and ${\left|E^t/E_0\right|}^2$ for a uniaxial anisotropic slab illuminated by a TM polarized Gaussian beam.

Download Full Size | PPT Slide | PDF

 

Fig. 3 Same model as in Fig. 2 but illuminated by a TE polarized Gaussian beam.

Download Full Size | PPT Slide | PDF

From Figs. 2 and 3 we can see that the middle of the transmitted beam for a TM polarized Gaussian beam is at the left of that for a TE polarized one, showing that the former is refracted more strongly, which does not appear for the case of a dielectric isotropic slab. Another noticeable difference lies in the fact that the normalized field intensity of the reflected beam for a TM polarized Gaussian beam becomes negligibly small, as a result of the Brewster angle phenomenon (Brewster's angle of a dielectric medium with a relative refractive index $\sqrt{3}$ is $\pi /3$).

4. Conclusion

An approach to compute the reflection and transmission of an incident Gaussian beam by a uniaxial anisotropic slab is given. Numerical results of the normalized field intensity distributions demonstrate that, compared with a TE polarized Gaussian beam, a TM polarized one is reflected less and refracted more strongly. As a result, this study provides an exact analytical model for interpretation of Gaussian beam propagation phenomena through a uniaxial anisotropic slab, and can also be extended to other cases such as chiral, biaxial and gyrotropic anisotropic slabs.