# Photonic Stopband Filters Based on Graphene-Pair Arrays
## 1 Introduction  
Bragg gratings, which are formed by modulating the refractive indices of materials or spatial structures of systems periodically, are extensively utilized for optical filters [1, 2], fiber lasers [3] and optical reflectors [4, 5]. Two identical Bragg gratings are assigned on both sides of a bulk dielectric to form the distributed feedback Bragg gratings, which are usually applied to realize wavelength selectors [6]. The periodical refractive index in Bragg gratings can be formed with a photosensory mask of which the host material in Bragg gratings, such as photosensitive glass, is covered by the photosensory mask, which has periodical transmittable fringes distributed in space. Then, by illuminating with an ultraviolet light, the periodical refractive index is formed [7]. Furthermore, the gratings of refractive index can also be achieved through irradiating directly the photosensitive glass with an ultraviolet light. Two lights interfere and further form interferometric fringes, which are utilized to inscribe the Bragg gratings. Such gratings are generally called fiber Bragg gratings [8]. Moreover, two materials with different refractive indexes are arranged alternately along the horizontal coordinate to form a periodic structure, which can be called general Bragg gratings, also known as the one-dimensional periodic photonic crystal [9]. Photonic crystals can induce a photonic band structure, being similar to the energy band of electrons in semiconductors. There are plenty of discrete photonic bandgaps in wave-vector space and each of them locates between two adjacent transmission photonic bands [10]. Lights with a wavelength in these photonic bandgaps cannot transmit the photonic crystals, and the light may be totally reflected back. So, the photonic bandgaps act as the stopbands of light waves and can be utilized for photonic filters immediately [11, 12]. However, once the filters based on photonic crystals are synthesized, the stopbands widths and the center frequencies of stopbands are determined permanently and cannot be adjusted conveniently in some specific application scenarios. Graphene is a two-dimensional ultra-thin material that has emerged in recent years, and it has excellent electrical conductivity and many fascinating optical properties [13]. ABragg grating can be formed by inserting a series of graphene monolayers into a bulk dielectric to form periodic graphene arrays in space. The photonic crystals based on graphene arrays have been extensively studied so far [14], such as solitons [15; 16; 17], supermodes [18], optical bistability [19], Rabi oscillations [20; 21], etc. The works in the literature about the research progress on graphene arrays are listed in Table 1. In particular, graphene has unique advantages, as it is utilized for exciting surface plasmon polariton [22; 23; 24]. Therefore, these unique periodic structures constructed by graphene arrays can be viewed as photonic crystals as well. In addition, the optical characteristic quantities are a function of the graphene surface conductivity, which can be modulated through the chemical potential of graphene. In practice, one can change the graphene chemical potential by loading an external gate voltage on graphene or by chemical doping [13], so the characteristics induced in the graphene-based photonic crystals can be flexibly regulated by the chemical potential of graphene. Photonic crystals based on graphene arrays are formed by arranging a monolayer of graphene periodically in the horizontal direction [25]. Of them, the photonic bandgap structure exists in the wave vector space as well. Compared with monolayer graphene, the photonic crystals composed of graphene pairs have a stronger resonance of light waves [21], that is, a more powerful reflection light may be induced in the stopbands, so the falling and rising edges of the bandgaps become steeper, and the extinction effect at the bandgap edge is better.