Graphene perfect absorber of ultra-wide bandwidth based on wavelength-insensitive phase matching in prism coupling

We proposed perfect absorbers of ultra-wide bandwidths based on prism coupling with wavelength-insensitive phase matching, which consists of three dielectric layers (Prism-Cavity-Air) with monolayer graphene embedded in the cavity layer. Due to inherent material dispersion of the dielectric layers, with the proper choice of the incidence angle and the cavity thickness, the proposed perfect absorbers can satisfy the phase matching condition over a wide wavelength range, inducing enormous enhancement of the absorption bandwidth. The requirement on the material dispersions of the prism and the cavity layer for the wavelength-insensitive phase matching over a wavelength range of the interest has been derived, and it has been demonstrated that the various kinds of materials can meet the requirement. Our theoretical investigation with the transfer matrix method (TMM) has revealed that a 99% absorption bandwidth of ~300 nm with perfect absorption at λ  = 1.51 μm can be achieved when BK7 and PDMS are used as the prism and the cavity layer, respectively, which is ~7 times wider than the conceptual design based on the non-dispersive materials. The full width at half maximum of our designed perfect absorber is larger than 1.5 μm.

. Theoretical model of the proposed graphene perfect absorber composed of Prism(n1)-Cavity(n2)-Air(n3), where monolayer graphene is embedded in the cavity and n1 > n2 > n3 = 1. The red thin layer represents the monolayer graphene of 0.34 nm thickness as an absorbing medium. 12 is Fresnel coefficients of the reflection at the prism-cavity interface, and scatt is the scattered waves via the multiple reflections from the other interface, assuming that the amplitude of the incidence wave (black dashed line) is unity.
We provide additional theoretical derivations to show that the reflection phase is approximately  at the resonance condition (or wavelength) in the proposed graphene perfect absorber based on prism coupling ( Fig. S1). Because the graphene is too thin (0.34 nm), the graphene does not almost affect the resonance condition. So, the reflection phase of the dielectric multilayer structures (prism-cavity-air) without graphene is assumed to be the same as that of the perfect absorption condition [1,2]. The perfect absorption occurs when the waves reflected from the prism-cavity interface (12) destructively interfere with the scattered waves via the multiple reflection from the other interface (scatt), assuming that the amplitude of the incidence wave is 1. This means that the reflection coefficient can be given by 12 0 scatt r   = + = (S1) and thus, the amplitude and phase information is as follows.
12 12 If the graphene layer is removed, | r | = -1 due to total internal reflection (TIR), and 12 12 So, we obtain As a result, considering that |r| = -1, and |12| is small enough compared to |scatt|, For a given material combination of the prism and the cavity layer, the resonance conditions of r = -1 are determined by the cavity thickness and incidence angle. Figure S2. Schematic of the dielectric multilayer structures (prism-cavity-air) without a graphene layer,

Reflection coefficient based on TMM
When the graphene layer is removed, the reflection coefficient of the proposed absorber structure (Fig. S2) can be derived by the transfer matrix method (TMM) [3,4]. We considered transverse electric (TE, or spolarized) wave illumination from the prism with an incidence angle of , whose electric field is perpendicular to the incidence plane. In general, the reflection or transmission coefficients can be calculated with the TMM, in which electric fields in one position can be related to those in other positions through a transfer matrix, M. For the considered multilayer structure, total transfer matrix relating the incoming (E1+) and outgoing (E1-) components at the prism-cavity interface, and the incoming (E3-) and outgoing (E3+) components at the cavity-air interface is written as In particular, at the resonance condition of r = -1, Eq. (S11) is rearranged as follows. When the relative position of graphene layer embedded in the cavity changes, the optimal (perfect) absorption wavelength changes so as to satisfy critical coupling condition, considering that wavelengthinsensitive phase matching is not almost affected by the graphene layer. Actually, analysis for dA = 0.50d2 and dA = 0.25d2 are already discussed in the manuscript: in detail, the three curves in Fig 5(b) correspond to cross-section at optimal d2 in Fig. S3(a) and Fig. S3(c). Fig. S3(b) shows the absorption map for dA = 0.35d2. The perfect absorption occurs at not  = 1.51m but  = 1.38m (indicated by optimal point P3).
Obviously, for TM polarization, the condition of wavelength-insensitive phase matching by the inherent material dispersion of dielectric layers is identical to that of TE polarization [1,5], ignoring birefringence of materials. However, perfect (or ultra-broadband) absorption cannot be obtained for both polarizations simultaneously because a loss rate is sensitive to polarization. Boundary condition of electromagnetic fields, owing to the very large absolute permittivity of graphene (for example, above 16 at  = 1.51m) compared to cavity permittivity, for TM polarization, the dominant surface-normal electric field intensity in graphene and resultant loss rate are very low, as shown in Fig. S4(a). In material combination considered here, the multi-layer graphene should be necessarily applied to achieve graphene perfect absorber of ultra-wide bandwidth. As seen in Fig. S4(b-d), regardless of the number of graphene layer, absorption peak branches follow the loci of the resonance conditions with the reflection phase of .