Omnidirectional constant transmission and negative Brewster angle at planar interfaces associated with a uniaxial medium

Hongyi Chen; Shixiang Xu; Jingzhen Li

doi:10.1364/OE.17.019791

1. Introduction

The phenomena of reflection and refraction of light at the interface of two transparent media are widely used for controlling the intensity, phase, polarization and direction of light in many optical devices [1]. Such phenomena in anisotropic media may be much different from those in isotropic media thus have recently attracted increasing amount of attention [2–11]. Negative refraction, which was of great interests these years [12–14], can be induced by anisotropy [2–5] under sufficiently small incidence. It is well known that the amplitude reflection and transmission coefficients [15] at a planar interface of two isotropic media depend on the incident angle. And the refraction requires refractive index mismatch, which seems to inevitably result in a finite reflection loss except at particular oblique incidence like Brewster angle. However, it is not always true for the case of a planar interface associated with a uniaxial medium. Recent experiment [6] demonstrated a very interesting phenomenon, omnidirectional total transmission, which means that the wave is totally transmitted for arbitrary incoming direction at a unique type of the interface of twinning structures using uniaxial crystals. The phenomenon can also occur at interface between isotropic and uniaxial media [7] by appropriate refractive index matching and optical arrangement of the optical axis of a uniaxial medium. Another property in anisotropic system is that the Brewster angle can be tuned from 0° to 90° under certain conditions, which is much different from in isotropic case [8–10]. Abnormal total reflection can also occur in anisotropic media [11]. However, all these properties can be derived by investigating the amplitude reflection and transmission coefficients. In this paper, we show theoretically that omnidirectional constant transmission is available at planar interfaces associated with a uniaxial medium. In addition, although negative refraction caused by anisotropy can only occur for small incident angle, it may occur without reflection loss under Brewster condition.

2. Theory

The incident wave is assumed to be a monochromatic uniform electromagnetic (EM) plane wave with an optical frequency ω and a space-time dependence specified by exp(i k·r-iωt) which has been omitted in the following as for simplification. As shown in Fig. 1 , the relation between the principal axes of the anisotropic medium (x,y,z) and the surface coordinates (x^’,y^’,z^’) can be expressed as

x = x^{'} \cos θ + z' \sin θ, y = y^{'}, z = - x^{'} \sin θ + z' \cos θ,

where θ is the inclined angle of the optical axis to the surface. Since the media are isotropic in the y–z plane, the wave vectors can be assumed to be in the x–z plane without losing any generality. In this section we will only examine the case of incidence from the uniaxial medium 2, and medium 1 is isotropic.

Fig. 1 Principal axes of the anisotropic medium (x,y,z) and the surface coordinates (x^’,y^’,z^’).

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In a uniaxial medium, when extraordinary TM waves (E_y = H_x = H_z = 0) are of interest, in the principal axes’ coordinates, the EM field can be expressed as

H = (0, H_{0}, 0), E = (k_{z} H_{0} / ω ε_{0} ε_{x}, 0, k_{x} H_{0} / ω ε_{0} ε_{z}),

and the wave vector components k_x and k_z satisfy the following dispersion relation:

k_{x}^{2} / ε_{z} + k_{z}^{2} / ε_{x} = k_{0}^{2},

where k_x and k_z are, the x and z components of the wave vectors respectively, and k ₀=ω/c with c is the speed of light in vacuum.

The wave vector components of the incident, the reflected and the transmitted rays parallel to the interface must satisfy the boundary condition:

k_{i x}^{'} = k_{r x}^{'} = k_{t x}^{'} \equiv B,

where the subscripts i, r and t denote the incident, reflected and transmitted ray respectively, and the superscript “’” denotes the surface coordinates. For isotropic medium 1 with refractive index n_t, B can be deduced from B = n_tsinθ^’_t k ₀, where θ^’_t is the refractive angle.

From Eq. (1), the dispersion relation in the surface coordinates becomes:

α B^{2} + β B k_{z}^{'} + γ k_{z}^{'}^{2} = k_{0}^{2} ε_{z} ε_{x},

where

α = ε_{x} \cos^{2} θ + ε_{z} \sin^{2} θ, β = (ε_{x} - ε_{z}) \sin 2 θ, γ = ε_{x} \sin^{2} θ + ε_{z} \cos^{2} θ .

Solutions of k^’_z in Eq. (5) are

k_{z \pm}^{'} = - β B / 2 γ \pm A

with

A = {[ε_{z} ε_{x} (γ k_{0}^{2} - B^{2})]}^{1 / 2} / γ .

Both k^’_iz and k^’_rz satisfy Eq. (5). To take simplify but without lose of generality, we set k^’_iz = k^’_z- and k^’_rz = k^’_z+ on the following.

It is easy to get the expressions of the EM wave in the surface coordinates:

H = {(0, H_{0}, 0)}^{'}, E = {(γ k_{z}^{'} + β B / 2, 0, - α B - β k_{z}^{'} / 2)}^{'} H_{0} / ω ε_{0} ε_{z} ε_{x} .

The amplitude reflection coefficient r and transmission coefficient t can be obtained using the boundary condition. From Eqs. (7) and (9), the incident and reflected EM waves can be written as

H_{i} = {(0, H_{0}, 0)}^{'}, E_{i} = {(γ A, 0, - α B - β k_{i z}^{'} / 2)}^{'} H_{0} / ω ε_{0} ε_{z} ε_{x}

and

H_{r} = r {(0, H_{0}, 0)}^{'}, E_{r} = r {(- γ A, 0, - α B - β k_{r z}^{'} / 2)}^{'} H_{0} / ω ε_{0} ε_{z} ε_{x}

separately. Assuming medium 1 is isotropic with permittivity ε ₀ ε ₁, the corresponding transmitted EM waves is expressed as

H_{t} = t {(0, H_{0}, 0)}^{'}, E_{t} = t (k_{t z}^{'}, 0, - B) H_{0} / ω ε_{0} ε_{1} .

The EM waves components parallel to the interface must be continuous to satisfy the boundary condition, so the amplitude reflection and transmission coefficients can be deduced as

r = H_{r} / H_{i} = (γ A / ε_{z} ε_{x} - k_{t z}^{'} / ε_{1}) / (γ A / ε_{z} ε_{x} + k_{t z}^{'} / ε_{1}),

t = H_{t} / H_{i} = (2 γ A / ε_{z} ε_{x}) / (γ A / ε_{z} ε_{x} + k_{t z}^{'} / ε_{1}) .

From Eq. (9), the time-averaged Poynting vector in the surface coordinates is given as

S^{'} = Re [{(α B + β k_{z}^{'} / 2, 0, γ k_{z}^{'} + β B / 2)}^{'} H_{0}^{2} / 2 ω ε_{0} ε_{z} ε_{x}] .

Using Eqs. (7) and (15), the incident and reflected angles of ray vectors can be proved to be

tg {θ^{'}}_{s i} = S_{i x}^{'} / S_{i z}^{'} = β / 2 γ + B ε_{x} ε_{z} / A γ^{2},

tg {θ^{'}}_{s r} = - S_{r x}^{'} / S_{r z}^{'} = - β / 2 γ + B ε_{x} ε_{z} / A γ^{2} .

3. Results and discussions

For the case of incidence from the uniaxial medium, comparing with the isotropic cases, it can easily find that k^’_iz/ε_i is replaced by γA/ε_zε_x in the amplitude reflection and transmission coefficients. In a similar way, the coefficients can be easy written down by same replacements in cases of wave incident from the isotropic medium and at an interface of two uniaxial media. For a fixed incident angle, the coefficients can be tuned by γ with changing θ, which gives a new way to control the amplitudes of reflected and transmitted waves as well as the Brewster angle.

It should be noted that omnidirectional total transmission has been discussed in Ref [6, 7]. without presenting the amplitude reflection and transmission coefficients. From Eq. (13) it can easy be seen that omnidirectional total transmission occurs at the surface between a uniaxial medium and an isotropic medium under condition of γA/ε_zε_x = k^’_tz/ε ₁, which means

γ = ε_{1}

and

ε_{z} ε_{x} = ε_{1}^{2} .

Equations (13) and (14) also show that under condition of γ=(ε_zε_x)^1/2, the coefficients have the same forms as isotropic media on both sides with refractive indexes (ε_zε_x)^1/2 and ε ₁. So with a proper θ giving γ=(ε_zε_x)^1/2, a uniaxial medium can be considered as an equivalent isotropic medium of ε_eq=(ε_zε_x)^1/2 in some means. If ε_eq=ε ₁, omnidirectional total transmission yields at the interface as if the same media on both sides, which is the essence of omnidirectional total transmission in this case. However, refraction is still in existence for the directions of wave vector and ray vector change after transmission in most of the cases. For uniaxial media on both sides, omnidirectional total transmission can happen when γ ₁ A ₁/ε_z ₁ ε_x ₁=γ ₂ A ₂/ε_z ₂ ε_x ₂, which means γ ₁=γ ₂ and ε_z ₁ ε_x ₁ = ε_z ₂ ε_x ₂. From Eq. (6) one can immediately conclude that θ ₂=-θ ₁ at the surface between the same uniaxial media with arbitrary θ ₁, omnidirectional total transmission occurs.

From above discussion, both Eqs. (17a) and (17b) must be fulfilled for omnidirectional total transmission. A question arised naturally is what happens if only one of the equations is fulfilled. From Eq. (14), the condition for omnidirectional constant transmission (amplitude transmission coefficient is a constant irrespective of the value of B) is Eq. (17a), which gives match of wave vectors so that the amplitude reflection and transmission coefficients keep constantly in spite of the variation of the incident angle. In addition, Eq. (17b) can be considered as giving match of refractive index so that reflected wave disappears and omnidirectional total transmission occurs under condition of wave vectors matching.

When ε ₁ is between ε_z and ε_x, θ can be chosen to be ±arcsin[(ε ₁ -ε_z)/(ε_x-ε_z)]^1/2 to fulfill Eq. (17)a), and the amplitude reflection coefficient is a constant for arbitrary incident angle:

r = [1 / {(ε_{z} ε_{x})}^{1 / 2} - 1 / ε_{1}] / [1 / {(ε_{z} ε_{x})}^{1 / 2} + 1 / ε_{1}] .

It is obvious that omnidirectional total transmission, where both Eqs. (17a) and (17b) are fulfilled and r=0, is a special case of omnidirectional constant transmission. However, the condition for omnidirectional constant transmission is much looser than for omnidirectional total transmission and is easy to meet. At the wavelength of 632.8 nm, if using calcite as the uniaxial medium with ε _x=1.486² and ε_z=1.658² one can find there are a lot of isotropic media with ε ₁ between ε _z and ε _x, e.g. Ba(NO₃)₂, CdF₂, CsCl and NaCl, which can be chosen [16] to realize omnidirectional constant transmission but not omnidirectional total transmission.

It should be pointed out that the amplitude reflection and transmission coefficients are still constants even at incident angle of near 90° and total reflection never occurs under condition of Eq. (17a). Otherwise, total reflection may occur at graze incidence from one side. For γ≈ε ₁, the amplitude reflection coefficient changes dramatically near the critical angle. For example, if the incident angle θ of the 632.8 nm beam is 84° from isotropic side, and calcite is used in the other side, the refractive index of an isotropic medium changes from γ ^1/2 to γ ^1/2+0.01, absolute value of the amplitude reflection coefficient can change from 0.0011 to 1. Such property can be used for a filter with highly sensitive intensity control and a new kind of optical bistable switch. However, the mechanism and an easy way to change the refractive index still need to be found out.

The discussion above shows that the amplitude reflection and transmission coefficients can be constant. However, for a beam splitter the reflectivity and transmissivity [1] (the ratios of power) are of much interest. From Eqs. (10) and (11), it can be shown that

S_{i}^{'} = Re [{(α B + β k_{i z}^{'} / 2, 0, γ A)}^{'} H_{0}^{2} / 2 ω ε_{0} ε_{z} ε_{x}],

S_{r}^{'} = Re [{(α B + β k_{r z}^{'} / 2, 0, - γ A)}^{'} r^{2} H_{0}^{2} / 2 ω ε_{0} ε_{z} ε_{x}] .

The intensities of incident and reflected wave are I_i = S’_i and I_r = S’_r. It is easy to prove that I_i/I_r is not equal to ׀r׀² but has a more complex expression since k^’_iz≠k^’_rz in Eqs. (19) and (20) because the reflected angle is not equal to the incident angle in most of the case in an anisotropic medium and the cross section of the wave changes after reflection. So constant transmission of amplitude indicates constant transmission of power is not straight forward for anisotropic case. Here we further examine how the incident field is divided into the reflected and transmitted fields and calculate the reflectivity and transmissivity. The amount of energy in the incident wave on a unit area of the boundary per second is therefore J_i=S’_icosθ^’_si=S’_iS’_iz/S’_i=S’_iz, and the energy of the reflected wave leaving a unit area of the boundary per second is given by similar expressions J_r = -S’_rz. So using Eqs. (19) and (20), the reflectivity can be given as

R = J_{r} / J_{i} = {| r |}^{2} .

And the transmissivity can be given as

T = 1 - {| r |}^{2}

from energy conservation. Equations (21) and (22) indicate that omnidirectional constant transmission gives a result of omnidirectional constant reflectivity and transmissivity. Such property can be used for a beam splitter with splitting ratios independent with incident angle. It is quite useful for a beam splitter as well as an attenuator under wide incidence. What is more, the adjusting precision can be greatly reduced with great improvement of insensitivity on excursion. For a fixed anisotropic medium, such beam splitter can be designed by choosing isotropic media with refractive indexes between ε_x ^1/2 and ε_z ^1/2 and rearrange the optical axis to tune the omnidirectional constant reflectivity from 0 to (ε_x ^1/2 -ε_z ^1/2)²/(ε_x ^1/2 -ε_z ^1/2)², which is from 0 to 0.003 for calcite crystal.

Now we turn our attention to the Brewster angle. The definition of Brewster angle can be extended to the surfaces of transparent uniaxial crystals [9]. In isotropic cases, the Brewster angles are given as tanθ_B ₁ =n ₂/n ₁ and tanθ_B ₂ =n ₁/n ₂ with θ_B ₁ and θ_B ₂ are the Brewster angles of medium 1 (with refractive index n ₁) and medium 2 (with refractive index n ₂) at an interface between them. In optically anisotropic surfaces, from Eq. (13), the expression of B for r=0 is

B^{2} = (ε_{x} ε_{z} - ε_{1} γ) ε_{1} k_{0}^{2} / (ε_{x} ε_{z} - ε_{1}^{2}),

and the Brewster angle θ_B ₁ is given by

\tan^{2} θ_{B 1} = (ε_{x} ε_{z} - ε_{1} γ) / (ε_{1} γ - ε_{1}^{2}) .

So the existence and size of θ_B ₁ depend on γ for fixed ε_x, ε_z, and ε ₁. Brewster angle exists under condition that γ is between ε ₁ and ε_xε_z/ε ₁. For ε ₁ is between ε_x and ε_z, θ_B ₁ can be tuned from 0 to 90° by changing γ [9]. It should be pointed out that under condition of Eq. (17a) θ_B ₁ is equal to 90° and loses its meaning. However, the numerator and the denominator of the right side of Eq. (24) are both 0 under conditions of Eqs. (17a) and (17b). So θ_B ₁ can be considered as uncertain when omnidirectional total transmission happens.

We can get the relation of θ_B ₁ and θ_B ₂ from Eqs. (16), (23) and (24):

\tan^{2} θ_{B 1} / {(\tan θ_{B 2 \pm} \pm | β | / 2 γ)}^{2} = ε_{1}^{2} / γ^{2} .

So for Brewster angle θ_B ₁ of isotropic medium 1, there are two Brewster angles of anisotropic medium 2. The smaller angle θ_B ₂₊ corresponding to “+” and the larger angle θ_B _2- corresponding to “_” in Eq. (25), which are different for wave propagating from right to left or the inverse. Under the condition γ≈ε_xε_z/ε ₁, Brewster angle θ_B ₁ can be sufficiently small so that θ_B ₂₊ is negative thus negative refraction happens under Brewster condition. From Eqs. (24) and (25), numerical calculation shows that the Brewster angle θ_B ₂₊ can be negative while θ_B ₁ is between 0 and 6.1°. When γ≈ε ₁, all the Brewster angles are near 90°. The relations of Brewster angles and γ at the surface of Ba(NO₃)₂ (ε ₁=1.5714²) and calcite are shown in Fig. 2(a) with γ changing from ε_xε_z/ε ₁ to ε ₁. The Brewster angles are sensitive in dependence on γ as well as refractive indexes thus may provide a new technique for optical axis orientation and refractive indexes measurement. A frequency-selective device may also be derived by selecting a dispersive uniaxial medium such that Brewster angles are different for different frequencies.

Fig. 2 (a)Brewster angles θ_B ₁ and θ_B _2± as functions of γ at the surface of Ba(NO₃)₂ (ε ₁=1.5714²) and calcite (ε_x=1.486², ε_z=1.658²); (b)Brewster angle θ_B ₁ and critical angle θ_c ₁ as functions of γ at the same kind of surface.

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If γ<ε ₁, total reflection can occur for wave incident from medium 1, and the corresponding critical angle θ_c ₁=arcsin(γ/ε ₁). The Brewster angle θ_B ₁ and the critical angle θ_c ₁ are shown in Fig. 2(b) with γ changing from ε_xε_z/ε ₁ to ε ₁. It can be seen that θ_B ₁ and θ_c ₁ become closer and closer to 90° when γ≈ε ₁. Such feature is also very different from that in isotropic case.

4. Conclusion

To summarize, we have presented a detailedly theoretical analysis on omnidirectional constant transmission and the features of Brewster angles at planar interfaces associate with a uniaxial medium. Omnidirectional constant transmission, which is useful for a beam splitter and an attenuator under the situation of wide incidence, can occur at the interfaces associated with a uniaxial medium under certain conditions. The condition for omnidirectional constant transmission is much looser than for omnidirectional total transmission and quite easy to fulfill practically. The tuning character of Brewster angle at planar interfaces associated with uniaxial media can be used to steer light and also provides new methods for refracting light at any special incident angle without loss. Especially, omnidirectional constant transmission and negative Brewster angle offer mechanisms of amplitude control for negative refraction in anisotropic media. Moreover, these properties of uniaxial media give new ways of bending, polarizing, angular dispersion, energy filtering for light with extraordinary characters and offer considerable potential device applications. Such properties may also provide new experimental opportunities to investigate the physical phenomena associated with uniaxial media.

Acknowledgement

This work is supported by the National Natural Science Foundations of China (Grant Nos. 60477042 and 60878017) and Shenzhen Key Laboratory of Micro-nano Photonic Information Technology.

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Omnidirectional constant transmission and negative Brewster angle at planar interfaces associated with a uniaxial medium

Abstract

1. Introduction

2. Theory

3. Results and discussions

4. Conclusion

Acknowledgement

References and links

Cited By

Figures (2)

Equations (27)

Optics Express