## Abstract

We show that a planar waveguide excited by a three-dimensional linear electric dipole exhibits unusual physical properties if it is filled with two equal-thickness layers, where one layer is air and the other layer is a specialized left-handed material (LHM) with the relative permittivity -1/(1 + δ) + *iγ*_{ε}
and permeability -(1 + δ) + *iγ*_{μ}
(δ, γ_{ε} and *γ*_{μ}
are all small parameters). In such a LHM waveguide, extremely high power densities are generated and transmitted under the excitation of the electric dipole no matter whether the LHM layer is lossless or lossy. We also show that a dominant mode always exists in the waveguide, which is not restricted by the guidance condition and the waveguide geometry. The guidance condition only determines the existence of higher guided modes. Numerical experiments verify the above conclusions.

© 2005 Optical Society of America

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### Equations (24)

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(1)
$${\epsilon}_{r1}=-1/\left(1+\delta \right)+i{\gamma}_{\epsilon},\phantom{\rule{1em}{0ex}}{\mu}_{r1}=-\left(1+\delta \right)+i{\gamma}_{\mu}$$
(2)
$${E}_{\mathit{iz}}=-\frac{\mathit{Il}}{8\mathit{\pi \omega}{\epsilon}_{0}}{\int}_{-\infty}^{\infty}\frac{{k}_{\rho}^{3}}{{k}_{0z}}{\tilde{E}}_{\mathit{iz}}{H}_{0}^{\left(1\right)}\left({k}_{\rho}\rho \right)d{k}_{\rho},$$
(3)
$${E}_{\mathit{i\rho}}=-\frac{\mathit{iIl}}{8\mathit{\pi \omega}{\epsilon}_{0}}{\int}_{-\infty}^{\infty}{k}_{\rho}^{2}{\tilde{E}}_{\mathit{i\rho}}{H}_{1}^{\left(1\right)}\left({k}_{\rho}\rho \right)d{k}_{\rho},$$
(4)
$${H}_{\mathit{i\varphi}}=-\frac{\mathit{iIl}}{8\mathit{\pi}}{\int}_{-\infty}^{\infty}\frac{{k}_{\rho}^{2}}{{k}_{0z}}{\tilde{E}}_{\mathit{iz}}{H}_{1}^{\left(1\right)}\left({k}_{\rho}\rho \right)d{k}_{\rho},$$
(5)
$${\tilde{E}}_{0z}={e}^{i{k}_{0z}\mid z\mid}+{R}^{+}{e}^{i{k}_{0z}z}+{R}^{-}{e}^{-i{k}_{0z}z},$$
(6)
$${\tilde{E}}_{1z}={C}^{-}{e}^{-i{k}_{1z}z}+{C}^{+}{e}^{i{k}_{1z}z},$$
(7)
$${\tilde{E}}_{0\rho}=\frac{z{e}^{i{k}_{0z}\mid z\mid}}{\mid z\mid}+{R}^{+}{e}^{i{k}_{0z}z}-{R}^{-}{e}^{-i{k}_{0z}z},$$
(8)
$${\tilde{E}}_{1\rho}=-{C}^{-}{e}^{i{k}_{1z}z}+{C}^{+}{e}^{i{k}_{1z}z}.$$
(9)
$${R}^{+}=-\frac{\left({a}^{-}{c}^{-}p+{a}^{+}{c}^{+}\right)}{D},$$
(10)
$${R}^{-}=-1+{R}^{+}{e}^{i2{k}_{0z}{d}_{0}},$$
(11)
$${C}^{-}=\frac{\left({a}^{-}{b}^{-}-{a}^{+}{b}^{+}\right){\epsilon}_{r0}}{\left({\epsilon}_{r1}D\right)},$$
(12)
$${C}^{+}={C}^{-}{e}^{-i2{k}_{1z}{d}_{2}},$$
(13)
$$D=\left(p{b}^{+}{c}^{-}+{b}^{-}{c}^{+}\right){k}_{0z},$$
(14)
$${a}^{\pm}={e}^{i{k}_{0z}{d}_{1}}\pm {e}^{-i{k}_{0z}{d}_{1}},$$
(15)
$${b}^{\pm}={e}^{i{k}_{0z}{d}_{1}}\pm {e}^{-i{k}_{0z}\left({d}_{1}-2{d}_{0}\right)},$$
(16)
$${c}^{\pm}={e}^{-i{k}_{1z}{d}_{1}}\pm {e}^{i{k}_{1z}\left({d}_{1}-2{d}_{2}\right)},$$
(17)
$$p=\frac{\left({\epsilon}_{r0}{k}_{1z}\right)}{\left({\epsilon}_{r1}{k}_{0z}\right)}.$$
(18)
$$D\prime =\left(2-2{e}^{i2{k}_{0z}\left({d}_{0}-{d}_{2}\right)}\right){k}_{0z}.$$
(19)
$${k}_{0z}=\frac{\mathit{n\pi}}{\left({d}_{2}-{d}_{0}\right)},\phantom{\rule{.5em}{0ex}}\left(n=\mathrm{1,2\dots}\right)$$
(20)
$${E}_{z}=A{e}^{i{k}_{0z}z}{H}_{0}^{\left(1\right)}\left({k}_{\rho}\rho \right),$$
(21)
$${E}_{0z}=A{H}_{0}^{\left(1\right)}\left({k}_{\rho}\rho \right).$$
(22)
$${\epsilon}_{r1}=\frac{-1}{\left(1+\delta \right)},\phantom{\rule{1em}{0ex}}{\mu}_{r1}=-\left(1+\delta \right),$$
(23)
$$D\prime =\frac{\delta {k}_{0z}\left({e}^{i2{k}_{0z}{d}_{2}}-{e}^{i2{k}_{0z}{d}_{0}}\right)}{\left(1+\delta \right)}.$$
(24)
$${E}_{1z}=\left(A\mathrm{cos}{k}_{1z}z+B\mathrm{sin}{k}_{1z}z\right){H}_{0}^{\left(1\right)}\left({k}_{\rho}\rho \right),$$