Time-domain surface plasmon polaritons on a graphene sheet

A time-domain analysis is presented of the transient field excited by a vertical electric dipole on a graphene sheet described through a local scalar conductivity model valid in the intraband regime. The analysis is carried out by means of the double-deformation method of Tsang and Kong [J. Math. Phys. 20, 1170 (1979)], a modified modal technique which allows for defining a causal surface-plasmon-polariton (SPP) contribution to the total transient field that is also computationally very efficient for a graphene sheet. Numerical results are presented and validated against independent numerical full-wave simulations. The proposed approach provides a viable route to the direct time-domain investigation of the role of SPPs in graphene-based structures employed in ultra-wide-band nanosystems.

Figure 1

An infinite graphene sheet placed in vacuum on the $z=0$ plane, excited by a transient vertical electric dipole (VED) with electric-dipole moment $p_{\mathrm{e}}\left(t\right){\mathbf{u}}_z$ placed at the origin of the Cartesian reference system. The graphene sheet is described in the frequency domain through a local, scalar surface impedance $Z_{\mathrm{g}}=R_{\mathrm{g}}+j\omega L_{\mathrm{g}}$. Polar coordinates $(\rho ,\varphi )$ are also shown for an observation point P located on the sheet.

Figure 2

Sommerfeld integration path (SIP) for the calculation of the FD field $B_{\varphi }$ as an inverse Hankel transform of order 1. The singularities of the spectral field are also indicated: branch points at $k_{\rho }=\pm k_0$ (circles), associated with the determination of the vertical wave number $k_z$, and the relevant Sommerfeld branch cuts (red wiggly lines); branch point at $k_{\rho }=0$ (circle), associated with the Hankel function $H_1^{\left(2\right)}(·)$, and the relevant branch cut (orange wiggly line); and poles at $k_{\rho }=\pm k_{\rho \mathrm{SPP}}\left(\omega \right)$ (crosses), associated with FD SPP modes.

Figure 3

Path deformation in the $k_{\rho }$ plane: SDP through the branch point $k_{\rho }=k_0$ and evolution of the SPP pole $k_{\rho \mathrm{SPP}}$ as a function of the radian frequency $\omega$.

Figure 4

Path deformation in the $\omega$ plane: SDPs through the origin for $t>s_0\rho$ (positive imaginary axis) and $t<s_0\rho$ (negative imaginary axis) and evolution of the SPP pole ${\omega }_{\rho \mathrm{SPP}}$ as a function of the imaginary part $q$ of the radial wave number along the $k_{\rho }$-SDP ($k_{\rho }=k_0-jq$). Two branch-point singularities are also shown with the relevant branch cuts, arising from the square-root function in the definition of the vertical wave number $k_z$ and from the Hankel function $H_1^{\left(2\right)}\left(·\right)$.

Figure 5

Normalized time-domain magnetic induction ${\widehat{b}}_{\varphi }\left(\rho ,t\right)$ as a function of the normalized time $t/\left(s_0\rho \right)$: comparison between our formulation (blue solid line) and a brute-force numerical inversion of the relevant spectral induction (cyan dashed curve). The incident induction is also reported for reference (gray dashed curve). Parameters: $\rho =0.1$ mm, $T_{\mathrm{c}}=5$ ps.

Figure 6

Normalized time-domain magnetic induction ${\widehat{b}}_{\varphi }\left(\rho ,t\right)$ as a function of the normalized time $t/\left(s_0\rho \right)$: comparison between our formulation (blue solid line), a brute-force numerical inversion of the relevant spectral induction (cyan dashed curve), and results obtained with cst (black circles). Parameters: As in Fig. 5, with a source wave form multiplied by $t$.

Figure 7

Normalized time-domain magnetic induction and its constituent parts as functions of the normalized time $t/\left(s_0\rho \right)$ for $\rho =0.1$ mm and current profile as in (20) with $T_{\mathrm{c}}=0.5$ ps: (a) $0<t/\left(s_0\rho \right)<3$ and (b) $t/\left(s_0\rho \right)>3$.

Figure 8

Same as in Fig. 7 for $T_{\mathrm{c}}=2$ ps (a) and $T_{\mathrm{c}}=5$ ps (b).

Figure 9

Normalized time-domain magnetic induction and its constituent parts as functions of the normalized time $t/\left(s_0\rho \right)$ for $\rho =1$ mm and current profile as in (20) with $T_{\mathrm{c}}=1$ ps.

Figure 10

Absolute value of the normalized time-domain magnetic induction and its constituent parts in a logarithmic scale as functions of $\rho$ with and current profile as in (20) with $T_{\mathrm{c}}=1$ ps for $t=3s_0\rho$ (a) and $t=7s_0\rho$ (b).

Figure 11

“Phase diagrams” showing the main character of the transient field as a function of the normalized time $t/\left(s_0\rho \right)$ (horizontal axis) and normalized radial distance $\rho /{\lambda }_{\mathrm{s}}$, with ${\lambda }_{\mathrm{s}}=c{T}_{\mathrm{c}}$ (vertical axis, logarithmic scale): (a) case of $T_{\mathrm{c}}=1$ ps and (b) case of $T_{\mathrm{c}}=5$ ps.

Figure 12

SPP pole singularity $k_{\rho \mathrm{SPP}}={\beta }_{\rho \mathrm{SPP}}-j{\alpha }_{\rho \mathrm{SPP}}$ of the spectral field ${\tilde{B}}_{\varphi }\left(k_{\rho },\omega \right)$: (a) pole locus (blue line) in the complex plane of the normalized radial wave number ${\widehat{k}}_{\rho }=k_{\rho }/k_0$, parameterized by the radian frequency $\omega$; (b) the relevant dispersion curves for the normalized phase constant ${\widehat{\beta }}_{\rho \mathrm{SPP}}$ (black line) and attenuation constant ${\widehat{\alpha }}_{\rho \mathrm{SPP}}$ (red line). Parameters: ${\tau }_{\mathrm{s}}=0.5$ ps, ${\mu }_{\mathrm{c}}=0$ eV, $T=300$ K.

Figure 13

Pole singularities in the complex plane of the normalized radian frequency $\widehat{\omega }$: pole loci ${\widehat{\omega }}_{\mathrm{p},i}$ ($i=1,2,3,4$), parameterized by the imaginary part $q$ of the radial wave number on the SDP $k_{\rho }=\omega s_0-jq$, and poles of the source spectrum $\pm {\omega }_{\mathrm{ps}}$ (black and gray diamonds), independent of $q$. The graphene conductivity has parameters as in Fig. 12. Source parameters: $T_{\mathrm{c}}=20$ ps, $a=4/T_{\mathrm{c}}$.

Figure 14

Pole singularities in the complex plane of the normalized radian frequency $\widehat{\omega }$: real and imaginary parts of ${\widehat{\omega }}_{\mathrm{p},i}$ as functions of the normalized imaginary part $\widehat{q}$ of the radial wave number on the SDP $k_{\rho }=\omega s_0-jq$: (a) $i=1,2$ and (b) $i=3,4$.

Figure 15

Path deformation in the complex plane of the normalized radian frequency $\widehat{\omega }=\omega /\mathrm{\Gamma }$ from the positive real axis (dashed line) to the vertical SDP through the origin (solid line), for $t>s_0\rho$. The poles (crosses) and the BPs (circles) with the relevant BCs (wiggly lines) are also indicated.

Figure 16

Same as in Fig. 15, for $t<s_0\rho$.