Enhanced reflective dichroism from periodic graphene ribbons via total internal reflection

Guang Chen; Xiao Lin; Zuojia Wang

doi:10.1364/OE.27.022508

1. Introduction

Hyperbolic metamaterials, as artificial subwavelength-structured media, are featured by the exotic hyperbolic dispersion [1], and is a special type of uniaxial crystals with the different components of permittivity/permeability tensors having opposite signs. An electromagnetic field in such a medium could find itself in a new class of material that combines both metal and dielectric properties [2]. As a result, the wave vectors span an unlimited range in these media (if the nonlocal response is neglected), enabling their wide use in the control of light flow, including negative refraction [3–5], high-resolution imaging [6,7], and enhancement of spontaneous emission [8,9]. Hyperbolic dispersion has also been demonstrated in metasurfaces with optically thin thicknesses [10,11]. The polaritons on hyperbolic metasurfaces have an extremely anisotropic in-plane propagation and lead to intriguing applications, such as planar hyperlens [12], 2D topological transitions [13], and super-Coulombic optical interactions [14].

Hyperbolic dispersions also exist in natural 2D materials with in-plane anisotropy, such as black phosphorous and patterned graphene [15–17]. Polaritons in 2D material shows high spatial confinement and enable light manipulation at the extreme nanoscale. Combined with the advantages of low loss and flexible tunability (e.g., via gating), 2D materials are appealing in nanophotonic devices. For instance, topological transitions from elliptical to hyperbolic dispersions can be implemented in an array of graphene ribbons [18]. Broadband all-angle negative refraction of highly squeezed hyperbolic polaritons can be supported by graphene nanoribbon arrays [19]. Highly spatial confinements in 2D nanoribbons make them attractive for enhancing light-matter interactions and mid-IR biosensing [20]. A graphene monolayer modulated by periodic gate bias can generate an anisotropic surface conductivity tensor and offers a canalization regime required for planar hyperlens [21]. Polarization control and nonlinearity enhancement can be achieved in periodic patterned graphene [22,23]. Stacked graphene ribbons are capable of covering the reflective phase over 2π range and thus promise efficient implementations of focusing mirrors [24,25]. In most works [13,18,26–28], periodic 2D ribbons are usually modeled by an effective anisotropic conductivity. However, the effective medium theory can only work well under the condition that the pitch of periodic structures is much smaller than the wavelength of supported polaritons. If the pitch approaches the polaritonic wavelength, the effective medium theory becomes less accurate and even invalid. This way, a more accurate and general homogenization approach is highly wanted to describe the effective surface conductivity of graphene ribbon arrays.

In this paper, a rigorous mode-expansion theory is utilized to calculate the effective conductivity of periodic graphene ribbons. We find that the high-order Floquet modes play an important role in determining the electromagnetic property of the periodic graphene ribbons especially at the resonant frequency. Importantly, the revealed phenomenon cannot be predicted by the previous electrostatic approximation. Moreover, the obtained effective conductivity is adopted to study the scattering behaviors of graphene ribbons. We find that the reflective dichroism of graphene ribbons can be enhanced via the total internal reflection (TIR). To be specific, by rotating its principle axis, a graphene ribbon array can act as an atomically thin chiral absorber, showing large absorption for one circular polarization and large reflection for the other.

2. Effective medium theory for periodic graphene ribbons

Figure 1 schematically depicts the geometry of our system, in which graphene ribbons are periodically arranged at the boundary of two dielectrics (with their relative permittivity as $ε_{1}$ and $ε_{2}$ respectively). Each ribbon has a width of W and a period L in the u direction. For such a periodic structure, the scattering of light from the ribbon array can be rigorously analyzed through the mode-expansion theory (MET). In the following, we derive the effective medium equation and compare the effectiveness with previous results.

Fig. 1 Schematic illustration of the periodic graphene ribbons under study and its effective surface conductivity tensor.

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2.1 scattering of p-polarized waves from periodic graphene ribbons

We assume that light propagates from region 1 to region 2. The total field of p-polarized light in the reflection side can be expressed by [29]

{\bar{E}}_{1}^{p} = \frac{1}{k_{i}} (\hat{u} k_{z} - \hat{z} k_{u}) e^{i {\bar{k}}_{i} \cdot \bar{r}} + \frac{1}{k_{i}} \sum_{n} R_{n}^{p p} (\hat{u} k_{z n}^{r} - \hat{z} k_{u n}) e^{i {\bar{k}}_{r n} \cdot \bar{r}},

{\bar{H}}_{1}^{p} = \frac{1}{η_{i}} \hat{v} e^{i {\bar{k}}_{i} \cdot \bar{r}} + \frac{1}{η_{i}} \sum_{n} R_{n}^{p p} \hat{v} e^{i {\bar{k}}_{r n} \cdot \bar{r}} .

The transmitted field is thus written as

{\bar{E}}_{2}^{p} = \frac{1}{k_{t}} \sum_{n} T_{n}^{p p} (\hat{u} k_{z n}^{t} - \hat{z} k_{u n}) e^{i {\bar{k}}_{t n} \cdot \bar{r}},

{\bar{H}}_{2}^{p} = \frac{1}{η_{t}} \sum_{n} T_{n}^{p p} \hat{v} e^{i {\bar{k}}_{t n} \cdot \bar{r}} .

Here,

{\bar{k}}_{i} = \hat{u} k_{u} + \hat{z} k_{z}

is the incident wave vector,

{\bar{k}}_{r n} = \hat{u} k_{u n} + \hat{z} k_{z n}^{r}

and

{\bar{k}}_{t n} = \hat{u} k_{u n} + \hat{z} k_{z n}^{t}

are the reflected and transmitted wave vectors of the n-th Floquet mode, respectively.

k_{u n} = k_{u} + n k_{G}

,

k_{G} = 2 π / L

. The z-component wave vectors of the diffracted light are

k_{z n}^{r} = - \sqrt{k_{i}^{2} - k_{u n}^{2}}

and

k_{z n}^{t} = \sqrt{k_{t}^{2} - k_{u n}^{2}}

.

η_{i} = \sqrt{μ_{0} / ε_{1}}

and

η_{t} = \sqrt{μ_{0} / ε_{2}}

are the characteristic impedances of two media.

R_{n}^{p p}

and

T_{n}^{p p}

correspond to the reflection and transmission coefficients of the n-th diffraction order,

n = 0, \pm 1, \pm 2 \dots

. In our case, the discontinuity at the boundary attributes to the anisotropic surface currents induced in periodic graphene ribbons. In a unit cell, the tangential magnetic fields in two regions can be related by

{\begin{matrix} \hat{z} \times ({\bar{H}}_{2}^{p} - {\bar{H}}_{1}^{p}) = \frac{1}{2} \hat{u} σ_{s} (E_{1 u}^{p} + E_{2 u}^{p}), | u | \leq W / 2 \\ \hat{z} \times ({\bar{H}}_{2}^{p} - {\bar{H}}_{1}^{p}) = 0, W / 2 < | u | \leq L / 2 \end{matrix},

and the continuity of tangential electric fields due to the absence of magnetic surface current:

E_{1 u}^{p} = E_{2 u}^{p}

. Here the center of graphene ribbon is assumed at the origin of coordinates. Substitute Eqs. (1)-(4) into the boundary condition of Eq. (5), multiply two sides by

e^{- i m k_{G} u}

and integrate over

- L / 2 \leq u \leq L / 2

, then we can obtain

δ_{m 0} + R_{m}^{p p} \frac{k_{z m}^{r}}{k_{z}} - T_{m}^{p p} \frac{k_{z m}^{t} k_{i}}{k_{z} k_{t}} = 0,

δ_{m 0} + R_{m}^{p p} - \frac{η_{i}}{η_{t}} T_{m}^{p p} = \frac{W}{L} η_{i} σ_{s} \sum_{n} T_{n}^{p p} \frac{k_{z n}^{t}}{k_{t}} sinc [\frac{(- m + n) W}{L}] .

Based on Eqs. (6) and (7), we can strictly calculate all scattering coefficients

R_{m}^{p p}

and

T_{m}^{p p}

for given geometric dimensions, fixed material parameters and incident angles. In particular, if we let m = 0 and W = L, Eq. (7) transforms into a normal case that relates the boundary fields at a homogeneous conductive surface

δ_{00} + R_{0}^{p p} - \frac{η_{i}}{η_{t}} T_{0}^{p p} = η_{i} σ_{u}^{e f f} T_{0}^{p p} \frac{k_{z 0}^{t}}{k_{t}},

with

σ_{u}^{e f f}

representing the effective conductivity of the homogeneous surface in the u direction. Combining Eqs. (7) and (8), we can get the formula of the effective surface conductivity for p-polarized light

σ_{u}^{e f f} = \frac{W}{L} σ_{s} \sum_{n} \frac{T_{n}^{p p} k_{z n}^{t}}{T_{0}^{p p} k_{z 0}^{t}} \sin c [\frac{n W}{L}] .

2.2 scattering of s-polarized waves from periodic graphene ribbons

The effective medium theory for s-polarized light is almost the same as that for p-polarized one. The total fields in the reflected (with the subscript ‘1’) and transmitted (with the subscript ‘2’) regions are written by

{\bar{E}}_{1}^{s} = - \hat{v} e^{i {\bar{k}}_{i} \cdot \bar{r}} - \sum_{n} R_{n}^{s s} \hat{v} e^{i {\bar{k}}_{r n} \cdot \bar{r}},

{\bar{H}}_{1}^{s} = \frac{1}{η_{i} k_{i}} (\hat{u} k_{z} - \hat{z} k_{u}) e^{i {\bar{k}}_{i} \cdot \bar{r}} + \frac{1}{η_{i} k_{i}} \sum_{n} R_{n}^{s s} (\hat{u} k_{z n}^{r} - \hat{z} k_{u n}) e^{i {\bar{k}}_{r n} \cdot \bar{r}},

{\bar{E}}_{2}^{s} = - \sum_{n} R_{n}^{s s} \hat{v} e^{i {\bar{k}}_{t n} \cdot \bar{r}},

{\bar{H}}_{2}^{s} = \frac{1}{η_{t} k_{t}} \sum_{n} T_{n}^{s s} (\hat{u} k_{z n}^{t} - \hat{z} k_{u n}) e^{i {\bar{k}}_{t n} \cdot \bar{r}} .

Here

R_{n}^{s s}

and

T_{n}^{s s}

are the reflection and transmission coefficients for the n-th diffraction order. By applying the boundary conditions and integration over the unit cell, we can similarly obtain the relation among scattering coefficients

δ_{m 0} + R_{m}^{s s} - T_{m}^{s s} = 0,

δ_{m 0} + R_{m}^{s s} \frac{k_{z m}^{r}}{k_{z}} - T_{m}^{s s} \frac{η_{i} k_{z m}^{t} k_{i}}{η_{t} k_{z} k_{t}} = \frac{W}{L} η_{i} σ_{s} \sum_{n} T_{n}^{s s} \sin c [\frac{(- m + n) W}{L}] .

The effective surface conductivity for s-polarized light is then derived as

σ_{v}^{e f f} = \frac{W}{L} σ_{s} \sum_{n} \frac{T_{n}^{s s}}{T_{0}^{s s}} \sin c [\frac{n W}{L}] .

2.3 Verification of the effective medium theory

We next verify the effectiveness of the Eqs. (9) and (16) by comparing with simulation results. The surface conductivity of graphene can be calculated from the random-phase approximation in the local limit [30,31]

\begin{array}{l} σ_{s} (ω) = \frac{i 2 e^{2} k_{B} T}{π ℏ^{2} (ω + i τ^{- 1})} \ln [2 \cos h (\frac{E_{F}}{2 k_{B} T})] \\ + \frac{e^{2}}{4 ℏ} [\frac{1}{2} + \frac{1}{π} arc \tan (\frac{ℏ ω - 2 E_{F}}{2 k_{B} T}) - \frac{i}{2 π} \ln \frac{{(ℏ ω + 2 E_{F})}^{2}}{{(ℏ ω - 2 E_{F})}^{2} + 4 {(k_{B} T)}^{2}}], \end{array}

where k_B is the Boltzmann constant, T = 300 K is the temperature, ω is frequency,

τ = μ E_{F} / e V_{F}^{2}

is carrier relaxation lifetime, E_F is the Fermi level, μ = 10000 cm²/(V s) is the mobility, and V_F = 10⁶ m/s is the Fermi velocity.

The geometric dimension of graphene ribbons is set to W = 0.25 μm and L = 0.5 μm. The permittivities of two dielectric are selected as $ε_{1} = 2.25 ε_{0}$ and $ε_{2} = ε_{0} .$ From the effective medium theory with the electrostatic approximation [18,19,27], the effective surface conductivity of such periodic 2D ribbons is estimated by

σ_{u}^{e f f} = \frac{L σ_{s} σ_{C}}{W σ_{C} + (L - W) σ_{s}}, σ_{v}^{e f f} = σ_{s} \frac{W}{L},

where

σ_{C} = - i (ω ε_{0} ε_{e f f} L / π) \ln [\csc (π G / 2 L)]

is equivalent conductivity associated with the near-field coupling between adjacent ribbons,

ε_{e f f} = 0.5 (ε_{1} + ε_{2}) / ε_{0}

is the relative effective permittivity of the medium that embed the ribbons [32]. This theory is valid when the plasmon wavelength in the u direction is much smaller than the period L. Since

σ_{C}

is derived by an electrostatic condition, we term this method as an electrostatic effective approach (EEA).

The difference between two effective medium approaches are illustrated in Fig. 2. Under p-polarized incidence (Fig. 2(a)), MET approach indicates a resonant mode at 18.2 THz and thus the effective conductivity in the u direction (σ_u) has a Lorentzian resonance (black and red curves for the real and imaginary parts, respectively). As a comparison, the effective conductivity calculated by EEA, corresponding to yellow and blue curves, indicates no resonance. In this scenario, the electrostatic condition is no longer applicable because high-order Floquet modes grow where the inverse of lattice constant becomes comparable to the wave numbers. For s-polarized mode, both effective medium approaches provide identical results, as shown in Fig. 2(b). Full wave simulation (e.g., COMSOL simulation) allows us to identify the accuracy of the MET. As depicted in Figs. 2(c) and 2(d), the reflection coefficients of the graphene ribbons calculated from effective conductivity are fully consistent with those obtained from numerical simulations. Here, a unit cell of graphene ribbon has been simulated by both 2D and 3D frequency domain solvers, with Floquet periodic boundaries applied in the u and the v directions. Plane wave modes are excited by the periodic ports. The maximum element size of mesh is 20 nm. Transition boundary condition is used to model the electromagnetic property of graphene.

Fig. 2 Effective medium analysis of graphene ribbons. The effective conductivity for (a) the p polarization and (b) the s polarization. MET indicates a resonant mode around 18.2 THz. Comparison between the calculated and simulated reflection spectra for (c) p polarization and (d) the s polarization. E_F = 0.4 eV.

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Through the rigorous mode-expansion calculation, we can investigate the weight of different Floquet modes at resonant and off-resonant frequencies. The reflection ( $R_{n}^{p p}$ ) and transmission coefficients ( $T_{n}^{p p}$ ) are plotted in Fig. 3. As the working frequency blue shifts, the second-order reflection coefficients (Figs. 3(a)–3(d)) for p-polarized light increases gradually, exceeding the zero-order one at the resonance (18.2 THz), and then fall again. Similarly, high-order fields are enhanced as well in the transmitted region at the resonance frequency, as depicted in Figs. 3(e)–3(h). The enhancement of high-order fields attributes to the invalidation of EEA at specific conditions. On the contrary, the MET approach is in principle more accurate and suitable to analyze the macroscopic electromagnetic behaviors of periodic 2D ribbons, without losing the high-order details.

Fig. 3 The reflection coefficients of Floquet modes at (a) 14 THz, (b) 16 THz, (c) 18.2 THz, and (d) 20 THz. (e-h) The corresponding transmission coefficients. E_F = 0.4 eV.

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2.4 General effective conductivity tensor and plane wave propagation

The subwavelength of both the 2D ribbons and the period allows one to describe the optical properties of an anisotropic/hyperbolic metasurface by a general effective conductivity tensor [33]

\bar{\bar{σ}} = [\begin{matrix} σ_{x x} & σ_{x y} \\ σ_{y x} & σ_{y y} \end{matrix}] = [\begin{matrix} σ_{u}^{e f f} \cos^{2} φ + σ_{v}^{e f f} \sin^{2} φ & (σ_{v}^{e f f} - σ_{u}^{e f f}) \sin φ \cos φ \\ (σ_{v}^{e f f} - σ_{u}^{e f f}) \sin φ \cos φ & σ_{u}^{e f f} \sin^{2} φ + σ_{v}^{e f f} \cos^{2} φ \end{matrix}] .

Here, φ is the angle between the x direction of the global coordinates and the u direction of the local coordinates of the 2D ribbons. With this general conductivity tensor, the scattering of the periodic 2D ribbons for arbitrary plane wave incidence can be obtained from Maxwell’s equations using boundary conditions. We assume the plane of incidence is the xz plane (i.e., the wave vector has the form of

{\bar{k}}_{i} = \hat{x} k_{x} + \hat{z} k_{z}),

θ_{i}

and

θ_{t}

refer to the angles of incidence and refraction, respectively. We then obtain the reflection coefficients

r_{p p} = - \frac{(σ_{x x} + Y_{2}^{p} - Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}}{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}},

r_{s p} = - \frac{\sqrt{Y_{1}^{s} Y_{1}^{p}} σ_{y x}}{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}},

r_{p s} = - \frac{\sqrt{Y_{1}^{p} Y_{1}^{s}} σ_{x y}}{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}},

r_{s s} = - \frac{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} - Y_{1}^{s}) - σ_{x y} σ_{y x}}{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}},

and the transmission coefficients

t_{p p} = \frac{2 \sqrt{Y_{2}^{p} Y_{1}^{p}} (σ_{y y} + Y_{2}^{s} + Y_{1}^{s})}{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}},

t_{s p} = - \frac{2 \sqrt{Y_{2}^{s} Y_{1}^{p}} σ_{y x}}{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}},

t_{p s} = - \frac{2 \sqrt{Y_{2}^{p} Y_{1}^{s}} σ_{y x}}{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}},

t_{s s} = \frac{2 \sqrt{Y_{2}^{s} Y_{1}^{s}} (σ_{x x} + Y_{2}^{p} + Y_{1}^{p})}{(σ_{x x} + Y_{2}^{p} + Y_{1}^{p}) (σ_{y y} + Y_{2}^{s} + Y_{1}^{s}) - σ_{x y} σ_{y x}} .

Here,

Y_{1}^{p} = Y_{1} / \cos θ_{i}

,

Y_{1}^{s} = Y_{1} \cos θ_{i}

,

Y_{2}^{p} = Y_{2} / \cos θ_{t}

and

Y_{2}^{s} = Y_{2} \cos θ_{t}

are the wave admittances at oblique incidence,

Y_{i} = \sqrt{ε_{i} / μ_{0}}

(i = 1, 2) is the characteristic admittance of medium. Noting that the phase responses of both reflection and transmission coefficients are determined by the electric fields based on the global Cartesian coordinate, that is, E_x for p polarization and E_y for s polarization, respectively.

We can combine above results and use two Jones matrices (R for reflection and T for transmission) to describe the scattering properties of the metasurface

R = [\begin{matrix} r_{p p} & r_{p s} \\ r_{s p} & r_{s s} \end{matrix}], T = [\begin{matrix} t_{p p} & t_{p s} \\ t_{s p} & t_{s s} \end{matrix}] .

A coordinate transformation from the Cartesian base to the circular base gives us the Jones matrices for circular polarizations [34]

R_{c i r c} = [\begin{matrix} r_{R R} & r_{R L} \\ r_{L R} & r_{L L} \end{matrix}] = \frac{1}{2} [\begin{matrix} r_{p p} - r_{s s} + i (r_{p s} + r_{s p}) & r_{p p} + r_{s s} - i (r_{p s} - r_{s p}) \\ r_{p p} + r_{s s} + i (r_{p s} - r_{s p}) & r_{p p} - r_{s s} - i (r_{p s} + r_{s p}) \end{matrix}],

T_{c i r c} = [\begin{matrix} t_{R R} & t_{R L} \\ t_{L R} & t_{L L} \end{matrix}] = \frac{1}{2} [\begin{matrix} t_{p p} + t_{s s} + i (t_{p s} - t_{s p}) & t_{p p} - t_{s s} - i (t_{p s} + t_{s p}) \\ t_{p p} - t_{s s} + i (t_{p s} + t_{s p}) & t_{p p} + t_{s s} - i (t_{p s} - t_{s p}) \end{matrix}] .

The subscripts ‘R’ and ‘L’ correspond to right-handed and left-handed circularly polarized light, respectively. Noting that reflection and transmission matrices for circular polarization have slight difference in their form, due to the flipping of wave vector in the z direction upon reflection.

3. Enhanced reflective dichroism by total internal reflection

Dichroism refers to the absorption difference between two eigenstates, including linear dichroism for linear polarizations and circular dichroism (CD) for spin photons. For an infinitely thin film standing in free space, the theoretical maximum absorption is 50% due to the symmetric radiation property of surface electric currents [35]. In the following section, we will discuss how total internal reflection boost the absorbing efficiency in anisotropic 2D metasurfaces, both for linear and circular polarizations.

3.1 Enhanced linear absorption

The schematic illustration of TIR enhanced absorption is shown in Fig. 4(a). When the angle of incidence exceeds the critical angle ( $θ_{i} > θ_{c}$ ), the transmitted waves become pure evanescent fields and no energy is capable of propagating in the second region. As a consequence, the second dielectric acts as a reflective mirror and the incident waves can dissipate dramatically at the 2D ribbon interface. Figures 4(b) and 4(c) show the reflection spectra of a 2D ribbon metasurface at different incident angles. The Fermi level is 0.4 eV for the 2D material with other parameters identical to those in Section 2.3, and the critical angle here is $θ_{c} = \arcsin (1 / 1.5) = 41.8 °$ . Under p-polarized illuminations, the minimum reflection (0.12) occurs at 18.3 THz with the incident angle of 55°, corresponding to an absorption of 98.6%. For s-polarized illuminations, however, the reflection keeps higher than 0.995 over the whole frequency range from 15 to 25 THz. Specifically, the linear dichroism defined by the absorption difference can achieve as high as 0.98 at resonance.

Fig. 4 Enhanced reflective linear dichroism via total internal reflection. (a) The schematic illustration of total internal reflection. Reflection spectra for (b) the p polarization and (c) s polarization for graphene ribbons with E_F = 0.4 eV. The reflection spectra for (d) p polarization and (e) s polarization at different Fermi levels with a fixed incident angle of 55°.

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The working frequency of the TIR enhanced linear dichroism can be flexibly tuned by adjusting the Fermi energy of the 2D material. As shown in Fig. 4(d), the reflection valley blue shifts from 15.8 to 20.5 THz while increasing E_F from 0.3 to 0.5 eV. The reflection spectra for the s polarization (Fig. 4(e)), on the contrary, vary extremely little and still kept at high levels. Such large reflective dichroism is achieved at an infinitely thin interface rather than bulky multilayer slabs. Compared with metamaterial absorbers [35,36], TIR enhanced absorption only utilizes a single 2D film and has no requirement on back mirrors. Furthermore, a transparent window still exists below the critical angle, thus providing a dual functional platform that can either absorb or transmit incident waves with reduced reflection.

3.2 Enhanced chiral absorption

In this section, we will present a switching between linear absorption and chiral absorption in a single 2D material interface. Generally, strong chiroptical responses requires the breaking of mirror symmetry, i.e., construction of three dimensional chiral structures [37]. Chiral metamirrors are capable of absorbing spin photons by design of planar or bulk chiral meta-atoms standing on metallic backgrounds [38–41]. Anisotropic plasmonic antennas made by gold nanoparticles provide spin selectivity at the boundary of two dielectrics [42]. All these approaches required three dimensional structures and accurate periodic alignments in both directions.

To demonstrate the spin-selective capability of the 2D ribbons, we calculate the absorption performance of a thin sheet with the general conductivity tenser described in Eq. (19). Since only resistances contribute to the energy dissipation, we only investigate the dependence of chiral absorption on real parts of the conductivities. Figures 5(a) and 5(b) show the calculated absorption as conductivity tensor varies. The angle φ is set to 45° so as to breaking the mirror symmetry with respect to the plane of incidence. Maximum absorption for LCP light happens around the point with the conductivity combination of $σ_{u}^{e f f} Z_{0} = 0.7$ and $σ_{v}^{e f f} Z_{0} = 2.4,$ Z₀ is the characteristic impedance of vacuum. Similarly, full absorption for RCP light occurs while exchange the values of two conductivity components, equivalent to the reversal of the sign of φ. The CD is defined as A_LCP - A_RCP, where A_LCP and A_RCP represent the absorption for LCP and RCP light, respectively. The calculated CD performance is plotted in Fig. 5(c) and it is demonstrated that the 2D surface switches its handedness around the isotropic condition ( $(σ_{u}^{e f f} = σ_{v}^{e f f}) .$ It is worth noting that high CD occurs at the anisotropy region with suitable resistive values. Compared with the effective conductivity obtained from MET approach in Fig. 2, we find an appropriate group of parameters of graphene ribbons at 18.2 THz, where $σ_{u}^{e f f} Z_{0} = 2.7 + 0.1 i$ , and $σ_{v}^{e f f} Z_{0} = 0.07,$ in which high-efficient spin selectivity could happen. It is worth noting that CD must be exactly zero at normal incidence, due to the intrinsic mirror symmetry of the structure. At oblique incidence, however, the mirror symmetry with respect to the plane of incidence can be broken and results in the enhancement of CD. This is the phenomenon of extrinsic chirality.

Fig. 5 Chiral absorption performance of a general anisotropic conductive layer. The dependence of the absorption on conductivities for (a) RCP and (b) LCP light. (c) The corresponding CD performance. Both the angle of incidence and the rotation angle φ are set to 45°.

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The performance of chiral absorption and reflective CD is illustrated in Fig. 6(a). When graphene ribbons stand on the dielectric surface with its optical axis rotated by φ from the x axis, it can act as an atomically thin chiral mirror [38,43]. RCP light can be highly reflected without reversing its handedness, while LCP one is efficiently absorbed. Figures 6(b) and 6(c) plots the absorption spectra for RCP and LCP light, respectively. In the case of φ = 0°, off-diagonal conductivities vanish $(σ_{x y} = σ_{y x} = 0)$ and two absorption curves overlap with each other (blue curves), indicating no chiroptical response. As φ increases, CD appears and reaches its maximum at resonance frequencies. Specifically, CD raises up to 0.93 at 18.25 THz when φ = 30°, as the green curve shows in Fig. 6(d). It should be noted that the reflection for RCP light is distinctive from that in ordinary metallic mirror. As depicted in Fig. 6(e), the co-polarization reflection coefficient (r_RR) is much higher than the cross-polarization one (r_LR). It indicates a handedness-preserving behavior upon reflection, which is not achievable in conventional isotropic mirrors. Furthermore, the cross-polarization reflection coefficients for two polarizations are exactly equal (r_RL = r_LR). The underlying mechanism can be understood by the reciprocity in combination with the structural symmetry. Reciprocity theorem requires $r_{R L}^{f o r w a r d} = r_{L R}^{b a c k w a r d} .$ Mirror symmetry about the optical axis of graphene ribbon arrays ensures no difference between forward and backward input light. Consequently, the cross-polarization reflection coefficients must be identical, namely, $r_{R L}^{f o r w a r d} = r_{L R}^{b a c k w a r d} = r_{L R}^{f o r w a r d} .$

Fig. 6 Enhanced chiral absorption in graphene ribbons. (a) Schematic illustration of the chiral absorption via total internal reflection and rotation of optical axis. Absorption spectra for (b) RCP and (c) LCP light for various values of φ. (d) The corresponding CD spectra. (e) The reflection spectra for φ = 30°. E_F = 0.4 eV and the angle of incidence is 45°.

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As the graphene is a tunable material, the CD spectra become tunable by increment of the Fermi level. The dependence of chiral performance on the Fermi level is plotted in Fig. 7. As E_F increases from 0.3 to 0.5 eV, the absorption spectra for both LCP and RCP experience blue shifts. The resonant frequency increases from 15.8 THz, 18.25 THz to 20.45 THz, with the maximum LCP absorption reaching 0.86, 0.96 and 0.95, respectively. As a comparison, the maximum RCP absorption changes little. The corresponding maximum CDs are 0.83, 0.93 and 0.92, respectively. Therefore, the 2D ribbon array can act as a tunable chiral film that changes the chiroptical responses of the total internal refection.

Fig. 7 Tunability of the enhanced chiral absorption at different Fermi levels. Absorption spectra for (a) LCP and (b) RCP light. (c) CD spectra. The Fermi level E_F changes from 0.3 eV to 0.5 eV, φ = 30° and the angle of incidence is 45°.

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4. Conclusion

In summary, we have applied a rigorous mode expansion theory to study the scattering properties of graphene ribbons. A generalized effective conductive tensor is developed with much higher accuracy than the electrostatic effective approach. At plasmonic resonance, mode expansion calculation reveals that high order Froquet modes are significantly enhanced and therefore the electrostatic effective approach no longer holds. We have also discussed the absorption enhancement in graphene ribbon arrays on top of a dielectric, with the efficiency reaching 98.6%. Switching between linear and circular dichroism in reflection occurs as the orientation of the principle axis varies, indicating an ultrathin coating film with versatile chiroptical responses. Combined with the tunability and high confinement properties in 2D materials, we expect our work could find potential applications in polarization filters, integrated optics and biosensing devices.

Funding

National Natural Science Foundation of China (NSFC) (61801268); Natural Science Foundation of Shandong Province (NSFSP) (ZR2018QF001); the Young Scholars Program of Shandong University.

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Enhanced reflective dichroism from periodic graphene ribbons via total internal reflection

Abstract

1. Introduction

2. Effective medium theory for periodic graphene ribbons

2.1 scattering of p-polarized waves from periodic graphene ribbons

2.2 scattering of s-polarized waves from periodic graphene ribbons

2.3 Verification of the effective medium theory

2.4 General effective conductivity tensor and plane wave propagation

3. Enhanced reflective dichroism by total internal reflection

3.1 Enhanced linear absorption

3.2 Enhanced chiral absorption

4. Conclusion

Funding

References

Cited By

Figures (7)

Equations (30)

Optics Express