<p>The parity-time (PT) symmetric behaviour of surface plasmon polaritons (SPPs) propagating at the interface of an atomic lattice and silver-silica nanocomposite material is investigated. Significant two-dimensional parity-time symmetry conditions exist for dispersion relation, group index, group velocity, wavelength, propagation time, rotary plasmon drag and penetration depths of SPPs. The real part of the SPP dispersion relation varies in the range of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(0.8 k_0 \le Re[k_{spp}(x,y)]\le 2.0 k_0\)</EquationSource> </InlineEquation> over the position region <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(-1\ell _0 \le \;x,y\; \le 1\ell _0 \)</EquationSource> </InlineEquation> and satisfies the PT-symmetry condition <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(Re[k_{spp}(x,y)]=Re[k_{spp}(-x,-y)]^\)</EquationSource> </InlineEquation>, representing even symmetry with respect to spatial inversion. In contrast, the imaginary part of the SPP dispersion relation varies in the range of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(0 \le Im[k_{spp}(x,y)]\le 0.08 k_0\)</EquationSource> </InlineEquation> over the position region <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(-1\ell _0 \le \;x,y\; \le 1\ell _0 \)</EquationSource> </InlineEquation> and satisfies the PT-symmetry condition <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(Im[k_{spp}(x,y)]=-Im[k_{spp}(-x,-y)]^\)</EquationSource> </InlineEquation>, representing odd symmetry where the loss spectrum in one region is exactly balanced by the gain spectrum in the opposite region. The group index of SPPs varies in the range of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(0 \le n_g^{spp}(x,y) \le 291\)</EquationSource> </InlineEquation> and the corresponding group velocity of SPPs is in the range of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(0 \le v^{spp}_g(x,y) \le 1.03\times 10^6 m/s\)</EquationSource> </InlineEquation>. The reported wavelength <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\lambda _{spp}(x,y)\)</EquationSource> </InlineEquation> of SPPs is 0.15<i>mm</i> and their propagation time <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(t_{spp}(x,y)\)</EquationSource> </InlineEquation> is <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(0.29\; nanoseconds\)</EquationSource> </InlineEquation>. The group index, group velocity, wavelength, propagation time, penetration depth in the atomic medium and penetration depth in the silver-silica nanocomposite material also indicate even symmetry with respect to spatial inversion. Furthermore the normal rotary plasmon drag of both clockwise and anticlockwise spinning angular frequencies <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(\omega _{s}=\pm 20 rad/s\)</EquationSource> </InlineEquation> is reported in the range of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(-1.30\; \mu rad \le \theta ^{spp}_{drag}(x,y) \le 1.30\; \mu rad\)</EquationSource> </InlineEquation>, and exhabits the even symmetry indicating a spatially balanced angular momentum transfer within the plasmonic system. The reported results are useful for the development of advanced sensing technologies, optical waveguides, optoelectronics, and photovoltaic devices.</p>

Parity-Time Symmetry of Surface Plasmon Polaritons at the Interface Between Atomic Lattice and Ag-\(SiO_2\) Nanocomposite

期刊论文
发表日期：

2026年4月27日

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Shehzad Khan,
Attaur Rahman,
Laila A. AL-Essa,
Mati ur Rahman

摘要

The parity-time (PT) symmetric behaviour of surface plasmon polaritons (SPPs) propagating at the interface of an atomic lattice and silver-silica nanocomposite material is investigated. Significant two-dimensional parity-time symmetry conditions exist for dispersion relation, group index, group velocity, wavelength, propagation time, rotary plasmon drag and penetration depths of SPPs. The real part of the SPP dispersion relation varies in the range of \(0.8 k_0 \le Re[k_{spp}(x,y)]\le 2.0 k_0\) over the position region \(-1\ell _0 \le \;x,y\; \le 1\ell _0 \) and satisfies the PT-symmetry condition \(Re[k_{spp}(x,y)]=Re[k_{spp}(-x,-y)]^*\) , representing even symmetry with respect to spatial inversion. In contrast, the imaginary part of the SPP dispersion relation varies in the range of \(0 \le Im[k_{spp}(x,y)]\le 0.08 k_0\) over the position region \(-1\ell _0 \le \;x,y\; \le 1\ell _0 \) and satisfies the PT-symmetry condition \(Im[k_{spp}(x,y)]=-Im[k_{spp}(-x,-y)]^*\) , representing odd symmetry where the loss spectrum in one region is exactly balanced by the gain spectrum in the opposite region. The group index of SPPs varies in the range of \(0 \le n_g^{spp}(x,y) \le 291\) and the corresponding group velocity of SPPs is in the range of \(0 \le v^{spp}_g(x,y) \le 1.03\times 10^6 m/s\) . The reported wavelength \(\lambda _{spp}(x,y)\) of SPPs is 0.15mm and their propagation time \(t_{spp}(x,y)\) is \(0.29\; nanoseconds\) . The group index, group velocity, wavelength, propagation time, penetration depth in the atomic medium and penetration depth in the silver-silica nanocomposite material also indicate even symmetry with respect to spatial inversion. Furthermore the normal rotary plasmon drag of both clockwise and anticlockwise spinning angular frequencies \(\omega _{s}=\pm 20 rad/s\) is reported in the range of \(-1.30\; \mu rad \le \theta ^{spp}_{drag}(x,y) \le 1.30\; \mu rad\) , and exhabits the even symmetry indicating a spatially balanced angular momentum transfer within the plasmonic system. The reported results are useful for the development of advanced sensing technologies, optical waveguides, optoelectronics, and photovoltaic devices.