<p>Surface plasmon polariton (SPP) waves are electromagnetic (EM) waves that travel over the interface between a metal and a dielectric. Typically, the conventional Drude model (DM), which uses integer order derivatives, has been employed to investigate SPP behavior. Since the classical DM does not exactly match experimental values for metal dielectric constants, it has been modified with the fractional calculus method to improve its modeling capabilities. The fractional method allows us to analyze system behaviors in non-integer domains that classic integer-order models cannot capture. In this paper, we use a fractional Drude model (FDM) with non-integer order derivatives to study SPP wave characteristics. A planar silver-glass interface is unitized to investigate the effect of the fractional-order parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11468_2025_3059_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>, which defines the order of the FDM. The impact of small variations of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11468_2025_3059_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> from unity on dielectric constant and refractive index of metal is explored. Furthermore, important SPP properties such as effective propagation length, dispersion curve, and bandgaps are examined for various values of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11468_2025_3059_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>. Notably, setting <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11468_2025_3059_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$ \gamma = 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> recovers results in agreement with the standard DM, validating the proposed mechanism. This theoretical study indicates that the optical characteristics are influenced by both the applied electric field frequency <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11468_2025_3059_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$ \omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> and the non-integer order <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11468_2025_3059_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> of FDM. This fractional order of FDM is tuned to ensure a good fit with the experimental data. It is hoped that these findings would be useful to plasmonic researchers.</p>

Incorporation of Fractional Drude Model to Study the SPP Waves

期刊论文
发表日期：

2025年6月9日

查看全文

Sher Ali,
Qaisar Abbas Naqvi,
Aqeel Abbas Syed

摘要

Surface plasmon polariton (SPP) waves are electromagnetic (EM) waves that travel over the interface between a metal and a dielectric. Typically, the conventional Drude model (DM), which uses integer order derivatives, has been employed to investigate SPP behavior. Since the classical DM does not exactly match experimental values for metal dielectric constants, it has been modified with the fractional calculus method to improve its modeling capabilities. The fractional method allows us to analyze system behaviors in non-integer domains that classic integer-order models cannot capture. In this paper, we use a fractional Drude model (FDM) with non-integer order derivatives to study SPP wave characteristics. A planar silver-glass interface is unitized to investigate the effect of the fractional-order parameter $ \gamma $ $γ$ , which defines the order of the FDM. The impact of small variations of $ \gamma $ $γ$ from unity on dielectric constant and refractive index of metal is explored. Furthermore, important SPP properties such as effective propagation length, dispersion curve, and bandgaps are examined for various values of $ \gamma $ $γ$ . Notably, setting $ \gamma = 1 $ $γ = 1$ recovers results in agreement with the standard DM, validating the proposed mechanism. This theoretical study indicates that the optical characteristics are influenced by both the applied electric field frequency $ \omega $ $ω$ and the non-integer order $ \gamma $ $γ$ of FDM. This fractional order of FDM is tuned to ensure a good fit with the experimental data. It is hoped that these findings would be useful to plasmonic researchers.