# Tunable Plasmonic Talbot Effect Based on Graphene Monolayer
## 2 Model of the Plasmonic Talbot Effect and Calculation Methods  
The monolayer graphene sheet lies on the substrate (plane XOZ) with a relative dielectric constant of \(\underline{\varepsilon}\), and is covered by a media with dielectric constant of \(\overline{\varepsilon}\), as schematically shown in Figure 1a. The incident side is divided into \(n\) identical segments, as shown in Figure 1b. The unit amplitude plane waves in transverse magnetic (TM) mode with phase difference are alternately incident upon each segment. In order to simplify the structure of the model in simulation and reduce the computation, the distribution of diffraction waves after the grating is considered solely and the process of incident wave passing through the grating is omitted. Instead adjacent incident waves with a fixed phase difference are used to describe the wave function through the grating. All the incident waves in each segment interfere with each other and form multiple intensity enhancements in both X and Z directions. One period of the grating includes two neighboring segments. Consequently, image arrays appear periodically along the propagation direction. This is the basic mechanism of the Talbot effect. Each period of the grating is composed of the two adjacent segments. Figure 1: (**a**) The graphene sheet is surrounded by air with a dielectric constant \(\varepsilon_{0}\) and placed on silicon dioxide substrate with a dielectric constant \(\underline{\varepsilon}=3.9\). (**b**) The calculation window of the model structure of the plasmonic Talbot effect. The periodic function of the grating at \(z=0\) is given by the Fourier series [12]:  
\[T(x,0)=\sum_{n=-\infty}^{+\infty}C_{n}\exp\left(\frac{i2\pi nx}{D}\right) \tag{1}\]  
when wavelength is less than period \(D\), or the propagation distance is greater than the wavelength, according to the superposition of the multiple slit interferences in each segment, the amplitude at point \((x,z)\) is given by [17]:  
\[T(x,z)=\exp(-i\beta z)\sum_{n=-\infty}^{+\infty}C_{n}\exp\left(\frac{i2\pi nx}{ D}\right) \tag{2}\]  
where \(\beta\) is the propagation constant of graphene along the z-axis. For the plasmon supported by graphene, \(\beta\) is described by [28]:  
\[\beta=\frac{i2\omega\varepsilon_{0}}{\sigma_{s}}\frac{\overline{\varepsilon}+ \underline{\varepsilon}}{2} \tag{3}\]  
where \(\underline{\varepsilon}\) and \(\overline{\varepsilon}\) are the dielectric constants of the material on both sides of graphene and \(\varepsilon_{0}\) is the dielectric constant of vacuum. \(\omega\) is the angular frequency, and \(\sigma_{s}\) is the surface conductivity of the graphene sheet. According to [PERSON]'s formula [29; 30], graphene's surface conductivity is composed of the contributions from interband electron-electron transition and the intraband electron-photon scattering. \[\sigma_{s}=\sigma_{intra}+\sigma_{inter} \tag{4}\]  
where the intraband electron-photon scattering \(\sigma_{intra}\) is described by  
\[\sigma_{intra}=\frac{ie^{2}k_{B}T}{\pi\hbar(\omega+i\tau^{-1})}[\frac{\mu_{c }}{k_{B}T}+2\ln(1+\exp(-\frac{\mu_{c}}{k_{B}T}))] \tag{5}\]  
and the interband electron-electron transition \(\sigma_{inter}\) is given by  
\[\sigma_{inter}=\frac{ie^{2}}{4\pi\hbar}\ln(\frac{2\left|\mu_{c}\right|-(\omega+ i\tau^{-1})}{2\left|\mu_{c}\right|+(\omega+i\tau^{-1})}) \tag{6}\]  
where \(T\) is the temperature which is set as 300 K, \(\tau\) is the electron momentum relaxation time, and which is set as 1 ps in this article. \(\hbar\) is the reduced Planck constant. \(\mu_{c}\) is the chemical potential, which could be modified by either chemical doping or the external gate voltage. The effective refractive index of graphene indicated as \(n_{eff}\) is described by Equation (7), where the real part of \(n_{eff}\) represents the dispersion characteristic and imaginary part implies the propagation loss of the plasmon.