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

Optics Express
Vol. 17,
Issue 22,
pp. 19791-19797
(2009)
•https://doi.org/10.1364/OE.17.019791

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

Hongyi Chen, Shixiang Xu, and Jingzhen Li

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Hongyi Chen, Shixiang Xu, and Jingzhen Li, "Omnidirectional constant transmission and negative Brewster angle at planar interfaces associated with a uniaxial medium," Opt. Express 17, 19791-19797 (2009)

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Abstract

This paper presents our detailedly theoretical analyses on omnidirectional constant transmission and negative Brewster angle at planar interfaces associated with a uniaxial medium. The amplitude reflection and transmission coefficients at planar interfaces associated with uniaxial media are derived by examining the boundary condition and the dispersion relation. It is found that under certain conditions, the coefficients are constants independent of the incident angle. Another interesting phenomenon is that an interface between isotropic and uniaxial media can exhibit negative refraction under Brewster condition. Our results offer considerable potential device applications of uniaxial media.

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

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

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}),

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

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

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

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 .

k_{z \pm}^{'} = - β B / 2 γ \pm A

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

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} .

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}

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}

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

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}) .

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}] .

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} .

γ = ε_{1}

ε_{z} ε_{x} = ε_{1}^{2} .

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

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}] .

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

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

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

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

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

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Equations (27)

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