# Tunable Plasmonic Talbot Effect Based on Graphene Monolayer
## 3 Calculation Results and Discussion
### The Talbot Effect Distance \(Z_{t}\) as a Function of the Multiple Parameters  
In order to study the Talbot effect in more detail, we investigate the properties of the Talbot effect distance \(Z_{T}\) as a function of the structural and material parameters. According to the Equation (9), the change of \(Z_{T}\) is directly related to the period \(D\) and effective wavelength \(\lambda_{n}\). The effective wavelength \(\lambda_{n}\) is in connection with the chemical potential of graphene and incident wavelength \(\lambda_{0}\). Therefore, in this paper, \(Z_{T}\) is investigated by changing the period \(D\), chemical potential of graphene and the incident wavelength. Firstly, the Talbot effect distance \(Z_{T}\) is investigated as a function of the period \(D\). The chemical potential of graphene is set as 0.3 eV, and the incident wavelength \(\lambda_{0}\) is fixed on 7000 nm, which means that \(\lambda_{n}\) is 64.3 nm. The period D starts at 80 nm and increases at a step of 20 nm and up to 260 nm. As demonstrated in Figure 3, \(Z_{T}\) grows with the period D in a parabolic way, which agrees well with Equation (9). Note that there are some small deviations between the numerical values and the predicted data obtained from Equation (9), which is due to the fact that the imaginary parts of the surface conductivity are ignored in the numerical calculation of Equations (3)-(9). Consequently, the analytical result of wavelength \(\lambda_{n}\) is slightly larger than the numerical results. Figure 3: The Talbot distance \(Z_{T}\) as a function of period D, where the input wavelength is 7 \(\mu\) m and chemical potential is 0.3 eV. Figure 2: \(|E|^{2}\) field distribution of graphene plasmonic Talbot effect with different incident wavelengths. The chemical potential is 0.3 eV, period is 160 nm, and the input wavelength is (**a**) \(\lambda_{0}\) =7 \(\mu\)m. (**b**) \(\lambda_{0}\) = 8 \(\mu\)m. (**c**) \(\lambda_{0}\) = 9 \(\mu\)m. (**d**) \(\lambda_{0}\) = 10 \(\mu\)m. (**e**) \(\lambda_{0}\) = 11 \(\mu\)m. (**f**) The imaginary part (described as \(\text{Im}\Big{(}n_{eff}\Big{)}=\text{Im}\big{(}\beta/k_{0}\big{)}\) ) and the real part (denoted by \(\text{Re}\Big{(}n_{eff}\Big{)}=\text{Real}(\beta/k_{0})\) ) of the effective refractive index of graphene supported plasmon as a function of the incident wavelength which is obtained from MATLAB. On the other hand, each Talbot image can be represented by the brightest interference fringe point, and the phase difference between the adjacent beams of light satisfies \(\Delta\varphi=2k\pi\), which is originally given by  
\[\Delta\varphi=\Delta\varphi_{0}+m\frac{2\pi}{\lambda_{n}}D\sin\theta\approx \Delta\varphi_{0}+m\frac{2\pi}{\lambda_{n}}D\tan\theta=\Delta\varphi_{0}+m \frac{2\pi}{\lambda_{n}}D\frac{x}{z} \tag{10}\]  
where \(\Delta\varphi_{0}\) is the phase different between the adjacent beams of light at \(z=0\). The distance \(z\) between the adjacent Talbot images is given by  
\[z=x\frac{nD}{\lambda_{n}} \tag{11}\]  
so, according to Equation (10) as the period \(D\) gets larger, the distance \(z\) between the two adjacent images gets larger which results from the increase in the Talbot distance \(Z_{T}\). Next, the Talbot distance \(Z_{T}\) as a function of the incident wavelength \(\lambda_{0}\) and the chemical potential \(\mu_{c}\) of graphene sheet is studied. In Figure 4a, the Talbot distance \(Z_{T}\) is recorded in the wavelength range from 5 to 9 \(\upmu\)m with the chemical potential ranging from 0.3 to 0.5 eV. For a given chemical potential, the Talbot distance decreases with the increase in incident wavelength. In addition, the Talbot distance decreases with increased chemical potential for a fixed wavelength. The rigorous data of \(Z_{T}\) obtained by Equation (9) are also plotted in Figure 4a in solid line. In general, the theoretical predictions in Equation (9) match well with the value of numerical calculation, which is illustrated the accuracy of research methods.