Infinite guided modes in a planar waveguide with a biaxially anisotropic metamaterial

Qiang Cheng and Tie Jun Cui

Qiang Cheng^{1} and Tie Jun Cui^{1}

^{1}Center for Computational Electromagnetics and the State Key Laboratory of Millimeter Waves, Department of Radio Engineering, Southeast University, Nanjing 210096, China

Qiang Cheng and Tie Jun Cui, "Infinite guided modes in a planar waveguide with a biaxially anisotropic metamaterial," J. Opt. Soc. Am. A 23, 1989-1993 (2006)

We investigate in detail the guided modes in a two-layered planar waveguide where one layer is filled with an ordinary right-handed material (RHM) and the other is filled with a biaxially anisotropic metamaterial. We show that the mode properties are closely dependent on the spatial dispersion relation of the anisotropic medium. When the dispersion equation for the anisotropic medium becomes a two-sheet or a one-sheet hyperbola type, an infinite number of guided modes can be supported simultaneously in the waveguide, which is completely different from the cases of RHM and isotropic metamaterial. We also investigate the mode distributions of the planar waveguide in the lossy case, where we discover that the dominant mode in the waveguide is a forward wave while the higher-order modes are backward waves under the two-sheet hyperbolic dispersion. Numerical results validate our theoretical analysis.

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Case I: Comparison of the Locations of Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases^{
a
}

${k}_{yp}$

${k}_{yp}^{\prime}$

66.9992

$67.0037+i0.6661$

175.8001

$175.7997+i0.2142$

318.7120

$318.7120+i0.2933$

395.6254

$395.6254+i0.2667$

Here, ${h}_{0}={h}_{1}=2\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$, $f=10\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}$, ${k}_{yp}$ is the real pole in the lossless case with ${\u03f5}_{1x}=4$, ${\mu}_{1y}=1$, and ${\mu}_{1z}=1$, and ${k}_{yp}^{\prime}$ is the complex pole in the lossy case with ${\u03f5}_{1x}=4+i0.001$, ${\mu}_{1y}=1+i0.001$, and ${\mu}_{1z}=1+i0.001$.

Table 2

Case II: Comparison of the Locations of the First Ten Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases^{
}

${k}_{yp}$

${k}_{yp}^{\prime}$

116.0135

$116.0183+i0.4159$

$-395.5881$

$-395.5879+i0.7194$

$-609.7860$

$-609.7859+i0.8037$

$-795.7551$

$-795.7547+i0.9409$

$-971.0731$

$-971.0725+i1.0886$

$-1140.8946$

$-1140.8940+i1.2403$

$-1307.4398$

$-1307.4390+i1.3937$

$-1471.8531$

$-1471.8523+i1.5482$

$-1634.7935$

$-1634.7926+i1.7034$

$-1796.6701$

$-1796.6692+i1.8589$

Here, ${h}_{0}={h}_{1}=2\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$, $f=10\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}$, ${k}_{yp}$ is the real pole in the lossless case with ${\u03f5}_{1x}=4$, ${\mu}_{1y}=1$, and ${\mu}_{1z}=-1$, and ${k}_{yp}^{\prime}$ is the complex pole in the lossy case with ${\u03f5}_{1x}=4+i0.001$, ${\mu}_{1y}=1+i0.001$, and ${\mu}_{1z}=-1+i0.001$.

Table 3

Case III: Comparison of the Locations of the First Ten Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases^{
a
}

${k}_{yp}$

${k}_{yp}^{\prime}$

110.6722

$110.6723+i0.0217$

331.1530

$331.1529+i0.1656$

455.0563

$455.0562+i0.3115$

542.2226

$542.2225+i0.4222$

656.3832

$656.3831+i0.5572$

785.3671

$785.3669+i0.7024$

922.9745

$922.9740+i0.8523$

1065.8997

$1065.8992+i1.0046$

1212.2841

$1212.2836+i1.1584$

1361.0248

$1361.0241+i1.3130$

Here, ${h}_{0}={h}_{1}=2\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$, $f=10\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}$, ${k}_{yp}$ is the real pole in the lossless case with ${\u03f5}_{1x}=4$, ${\mu}_{1y}=-1$, and ${\mu}_{1z}=1$, and ${k}_{yp}^{\prime}$ is the complex pole in the lossy case with ${\u03f5}_{1x}=4+i0.001$, ${\mu}_{1y}=-1+i0.001$, and ${\mu}_{1z}=1+i0.001$.

Table 4

Case IV: Comparison of the Locations of Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases^{
a
}

${k}_{yp}$

${k}_{yp}^{\prime}$

154.2187

$154.2187+i0.0080$

Here, ${h}_{0}={h}_{1}=2\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$, $f=10\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}$, ${k}_{yp}$ is the real pole in the lossless case with ${\u03f5}_{1x}=-4$, ${\mu}_{1y}=1$, and ${\mu}_{1z}=1$, and ${k}_{yp}^{\prime}$ is the complex pole in the lossy case with ${\u03f5}_{1x}=-4+i0.001$, ${\mu}_{1y}=1+i0.001$, and ${\mu}_{1z}=1+i0.001$.

Tables (4)

Table 1

Case I: Comparison of the Locations of Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases^{
a
}

${k}_{yp}$

${k}_{yp}^{\prime}$

66.9992

$67.0037+i0.6661$

175.8001

$175.7997+i0.2142$

318.7120

$318.7120+i0.2933$

395.6254

$395.6254+i0.2667$

Here, ${h}_{0}={h}_{1}=2\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$, $f=10\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}$, ${k}_{yp}$ is the real pole in the lossless case with ${\u03f5}_{1x}=4$, ${\mu}_{1y}=1$, and ${\mu}_{1z}=1$, and ${k}_{yp}^{\prime}$ is the complex pole in the lossy case with ${\u03f5}_{1x}=4+i0.001$, ${\mu}_{1y}=1+i0.001$, and ${\mu}_{1z}=1+i0.001$.

Table 2

Case II: Comparison of the Locations of the First Ten Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases^{
}

${k}_{yp}$

${k}_{yp}^{\prime}$

116.0135

$116.0183+i0.4159$

$-395.5881$

$-395.5879+i0.7194$

$-609.7860$

$-609.7859+i0.8037$

$-795.7551$

$-795.7547+i0.9409$

$-971.0731$

$-971.0725+i1.0886$

$-1140.8946$

$-1140.8940+i1.2403$

$-1307.4398$

$-1307.4390+i1.3937$

$-1471.8531$

$-1471.8523+i1.5482$

$-1634.7935$

$-1634.7926+i1.7034$

$-1796.6701$

$-1796.6692+i1.8589$

Here, ${h}_{0}={h}_{1}=2\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$, $f=10\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}$, ${k}_{yp}$ is the real pole in the lossless case with ${\u03f5}_{1x}=4$, ${\mu}_{1y}=1$, and ${\mu}_{1z}=-1$, and ${k}_{yp}^{\prime}$ is the complex pole in the lossy case with ${\u03f5}_{1x}=4+i0.001$, ${\mu}_{1y}=1+i0.001$, and ${\mu}_{1z}=-1+i0.001$.

Table 3

Case III: Comparison of the Locations of the First Ten Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases^{
a
}

${k}_{yp}$

${k}_{yp}^{\prime}$

110.6722

$110.6723+i0.0217$

331.1530

$331.1529+i0.1656$

455.0563

$455.0562+i0.3115$

542.2226

$542.2225+i0.4222$

656.3832

$656.3831+i0.5572$

785.3671

$785.3669+i0.7024$

922.9745

$922.9740+i0.8523$

1065.8997

$1065.8992+i1.0046$

1212.2841

$1212.2836+i1.1584$

1361.0248

$1361.0241+i1.3130$

Here, ${h}_{0}={h}_{1}=2\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$, $f=10\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}$, ${k}_{yp}$ is the real pole in the lossless case with ${\u03f5}_{1x}=4$, ${\mu}_{1y}=-1$, and ${\mu}_{1z}=1$, and ${k}_{yp}^{\prime}$ is the complex pole in the lossy case with ${\u03f5}_{1x}=4+i0.001$, ${\mu}_{1y}=-1+i0.001$, and ${\mu}_{1z}=1+i0.001$.

Table 4

Case IV: Comparison of the Locations of Poles of the Integrands in Eqs. (1, 2) (or Zeros of s) for the Lossless and Lossy Cases^{
a
}

${k}_{yp}$

${k}_{yp}^{\prime}$

154.2187

$154.2187+i0.0080$

Here, ${h}_{0}={h}_{1}=2\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$, $f=10\phantom{\rule{0.2em}{0ex}}\mathrm{GHz}$, ${k}_{yp}$ is the real pole in the lossless case with ${\u03f5}_{1x}=-4$, ${\mu}_{1y}=1$, and ${\mu}_{1z}=1$, and ${k}_{yp}^{\prime}$ is the complex pole in the lossy case with ${\u03f5}_{1x}=-4+i0.001$, ${\mu}_{1y}=1+i0.001$, and ${\mu}_{1z}=1+i0.001$.