Parity-time-symmetric plasmonic metamaterials

Hadiseh Alaeian and Jennifer A. Dionne

Phys. Rev. A 89, 033829 – Published 14 March 2014

Abstract

We theoretically investigate the optical properties of parity-time ( $PT$ )-symmetric three-dimensional metamaterials composed of strongly coupled planar plasmonic waveguides. By tuning the loss-gain balance, we show how the initially isotropic material becomes both asymmetric and unidirectional. Investigation of the band structure near the material's exceptional point reveals several interesting optical properties, including double negative refraction, Bloch power oscillations, unidirectional invisibility, and reflection and transmission coefficients that are simultaneously equal to or greater than unity. The highly tunable optical dispersion of $PT$ -symmetric metamaterials provides a foundation for designing an unconventional class of three-dimensional bulk synthetic media, with applications ranging from lossless subdiffraction-limited optical lenses to nonreciprocal nanophotonic devices.

Received 31 May 2013

DOI:https://doi.org/10.1103/PhysRevA.89.033829

Authors & Affiliations

Hadiseh Alaeian^1,2 and Jennifer A. Dionne^2,3

¹Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA
²Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA
³Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

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Issue

Vol. 89, Iss. 3 — March 2014

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Images

Figure 1
(a) Schematic of the plasmonic metamaterial, and the associated band diagrams for transverse magnetic modes (TM). The metamaterial is composed of coupled plasmonic waveguides, with deeply subwavelength metal and dielectric thicknesses of $t_{m} = t_{d} = 30$ nm. The unit-cell period is $Λ = 150$ nm. The refractive index of the dielectric is $n \pm i k$ , with $n = 3.2$ and $k$ corresponding to the non-Hermiticity parameter. Band diagrams are included for (b) $k = 0$ , (c) $k = 0.2$ , (d) $k = 0.3$ , and (e) $k = 0.5$ . Bands $B_{1}$ and $B_{2}$ are positive refractive index bands, while bands $B_{3}$ and $B_{4}$ correspond to negative refractive index bands. The horizontal dotted black lines in (b)–(e) indicate the surface plasmon resonance frequency $ω_{sp}$ . The black circles in panels in the same panels indicate the exceptional points of the dispersion, beyond which the purely real modes evolve to the complex modes characterized by either loss or gain (dashed black lines).
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Figure 2
Equifrequency contours of the metamaterial, plotted for both positive and negative index bands at wavelengths of $λ = 954$ nm ( $B_{1}$ , $B_{2}$ ), 604 nm ( $B_{3}$ ), and 445 nm ( $B_{4}$ ). Band $B_{4}$ is circular for zero and small $k$ , corresponding to an isotropic metamaterial. For all $k$ , bands $B_{1}$ and $B_{4}$ remain elliptical, while band $B_{2}$ is hyperbolic. However, band $B_{3}$ undergoes a hyperbolic to elliptical transition with increasing $k$ .
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Figure 3
Real ( $H_{y}$ ) for a 10-unit-cell metamaterial upon illumination by a plane wave in air of wavelength $λ = 445$ nm and angle $θ = 45^{\circ}$ . (a) The results for $k = 0$ while (b) and (c) show the results for $k = 0.445$ when illuminated from the loss ( $+ z$ ) and gain ( $- z$ ) side, respectively. The overlaid arrows indicate the direction of the averaged Poynting vectors in the corresponding regions. I in each panel indicates the incident wave vector, while T shows the transmitted plane wave. The lower graphs in each panel show the distribution of the power in each layer of the stack. The metallic layer is represented in gray, while the gain and loss regions are represented in red and yellow, respectively.
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Figure 4
Real ( $H_{y}$ ) when a semi-infinite metamaterial is illuminated with a plane wave at $λ = 445$ nm along $x$ direction (endfire illumination). The illumination angle $θ$ equals 45 $^{\circ}$ in (a) and (b) and $- 45^{\circ}$ in (c). The side graphs in each panel show the distribution of the magnetic field 1 $μ$ m away from the interface.
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Figure 5
Reflected and transmitted powers of the metamaterial with increasing $k$ and varying numbers of periods. (a) Reflected power upon illumination from the left (loss) side of the metamaterial; (b) reflected power upon illumination from the gain (right) right side; (c) quotient of reflected powers; and (d) transmitted power. In all panels the incident angle is $θ = 45^{\circ}$ and the illumination wavelength is $λ = 445$ nm. The inset in (b) and (d) show the behavior of the reflected and transmitted powers around $k = 0.035$ , where $R_{R}$ vanished and $T = 1$ independent of the number of periods.
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Figure 6
Variation of the scattering parameters as a function of incident angle $θ$ for (a) $k = 0.035$ and (b) $k = 0.445$ . Comparison between the phase of the transmitted beam and a free-space beam as a function of incident angle for (b) $k = 0.035$ and (d) $k = 0.445$ .
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