Spectral statistics of light in one-dimensional graphene-based photonic crystals

The optical properties and spectral statistics of light in one-dimensional photonic crystals in the representative classes of ${\left(\mathrm{AB}\right)}^N$ (composed of dielectric layers) and ${\left(\mathrm{AGBG}\right)}^N$ (composed of periodic stacking of graphene-dielectric layers) have been investigated using the transfer matrix method and random matrix theory. The proposed method provides new predictions to determine the chaos and regularity of the optical systems. In this analysis, the chaoticity parameter with $q=0$ for Poisson distribution and $q\to 1$ for Wigner distribution is determined based on the random matrix theory. It has been shown that two kinds of chaos and regularity modes can be found with Brody distribution. Also, as a part of this work, we found out the regular pattern in both classes of ${\left(\mathrm{AB}\right)}^N$ and ${\left(\mathrm{AGBG}\right)}^N$ when results were fit to a Brody distribution. Moreover, the effects of different parameters such as the number of unit cells, incident angle, state of polarization, and chemical potential of the graphene nanolayers on the structures' regularity are discussed. It is found that the regular patterns are seen in the band gaps. The results show that the structure ${\left(\mathrm{AGBG}\right)}^N$ has an extra photonic band gap compared to ${\left(\mathrm{AB}\right)}^N$, which is tunable by changing the chemical potential of the graphene nanolayers. Therefore, the possibility of external control of the regularity using a gate voltage in the graphene-based photonic crystals is obtained. Finally, comparing of TE and TM waves based on the random matrix theory, which interpolates between regular and chaotic systems, indicates that the Poisson statistics well describes the TE waves.

Figure 1

Schematic representation of 1D-PCs with the structure (a) ${\left(\mathrm{AB}\right)}^N$ and (b) ${\left(\mathrm{AGBG}\right)}^N$.

Figure 2

The transmission (T), reflection (R), and absorption (A) spectra of ${\left(\mathrm{AGBG}\right)}^N$ for (a) TE (left panel) and (b) TM polarization (right panel) at the fixed incident angle ($\theta ={45}^{\circ }$).

Figure 3

The transmission spectra of different structures ${\left(\mathrm{AB}\right)}^N$ (up panel) and ${\left(\mathrm{AGBG}\right)}^N$ (down panel) for TE (left panel) and TM polarization (right panel) at the fixed incident angle ($\theta ={45}^{\circ }$). The i, ii, and iii on top of the figures indicate the lower, median, and upper regions of frequency, respectively.

Figure 4

The transmission spectra of different structures ${\left(\mathrm{AB}\right)}^N$ (top panel) and ${\left(\mathrm{AGBG}\right)}^N$ (bottom panel) in the plane of period number and frequency ($N,f$) for TE (left panel) and TM polarization (right panel).

Figure 5

The transmission spectra of different structures ${\left(\mathrm{AB}\right)}^N$ (top panel) and ${\left(\mathrm{AGBG}\right)}^N$ (bottom panel) in the plane of incident angle and frequency ($\theta ,f$) for TE (left panel) and TM polarization (right panel).

Figure 6

The transmission spectra of ${\left(\mathrm{AGBG}\right)}^N$ for TE (left panel) and TM polarization (right panel) for different chemical potential values ${\mu }_c$ = 0.2 and 0.4 eV.

Figure 7

The spectral statistics of the structure ${\left(\mathrm{AB}\right)}^N$ for different period numbers $N$ = 5, 10, 15, and 20 at the fixed incident angle ($\theta ={45}^{\circ }$) for TE polarization.

Figure 8

The spectral statistics of the structure ${\left(\mathrm{AB}\right)}^N$ for different period numbers $N$ = 5, 10, 15, and 20 at the fixed incident angle ($\theta ={45}^{\circ }$) for TM polarization.

Figure 9

The spectral statistics of the structure ${\left(\mathrm{AGBG}\right)}^N$ for different period numbers $N$ = 5, 10, 15 and 20 at the fixed incident angle ($\theta ={45}^{\circ }$) for TE polarization.

Figure 10

The spectral statistics of the structure ${\left(\mathrm{AGBG}\right)}^N$ for different period numbers $N$ = 5, 10, 15, and 20 at the fixed incident angle ($\theta ={45}^{\circ }$) for TM polarization.

Figure 11

The transmission and spectral statistics of the structure ${\left(\mathrm{AB}\right)}^N$ at the fixed incident angle ($\theta =0^{\circ }$) for TE and TM polarizations.

Figure 12

The transmission and spectral statistics of the structure ${\left(\mathrm{AB}\right)}^N$ at the fixed incident angle ($\theta ={30}^{\circ }$) for TE polarization.

Figure 13

The transmission and spectral statistics of the structure ${\left(\mathrm{AB}\right)}^N$ at the fixed incident angle ($\theta ={60}^{\circ }$) for TE polarization.

Figure 14

The transmission and spectral statistics of the structure ${\left(\mathrm{AB}\right)}^N$ at the fixed incident angle ($\theta ={30}^{\circ }$) for TM polarization.

Figure 15

The transmission and spectral statistics of the structure ${\left(\mathrm{AB}\right)}^N$ at the fixed incident angle ($\theta ={60}^{\circ }$) for TM polarization.

Figure 16

The transmission and spectral statistics of the structure ${\left(\mathrm{AGBG}\right)}^N$ at the fixed incident angle ($\theta =0^{\circ }$) for TE and TM polarizations.

Figure 17

The transmission and spectral statistics of the structure ${\left(\mathrm{AGBG}\right)}^N$ at the fixed incident angle ($\theta ={30}^{\circ }$) for TE polarization.

Figure 18

The transmission and spectral statistics of the ${\left(\mathrm{AGBG}\right)}^N$ periodic structure at the fixed incident angle ($\theta ={60}^{\circ }$) for TE polarization.

Figure 19

The transmission and spectral statistics of the structure ${\left(\mathrm{AGBG}\right)}^N$ at the fixed incident angle ($\theta ={30}^{\circ }$) for TM polarization.

Figure 20

The transmission and spectral statistics of the structure ${\left(\mathrm{AGBG}\right)}^N$ at the fixed incident angle ($\theta ={60}^{\circ }$) for TM polarization.