Frequency-tunable circular polarization beam splitter using a graphene-dielectric sub-wavelength film

Tuo Chen; Sailing He

doi:10.1364/OE.22.019748

1. Introduction

The capability to manipulate the circular polarization of light is one of the essential optical applications of molecular biology, medical science and analytical chemistry. Most biomolecules, including the basic block of life, DNA, are chiral, which are sensitive to optical stimuli and thus behave differently when exposed to left-handed circular (LCP) waves and right-handed circular (RCP) waves. The most commonly used method to manipulate circular polarization of light is to use half- or quarter-wave plates with a retardation effect in an anisotropic crystal. However, the small differences in permittivity between the corresponding crystallographic directions require a thick retardation distance, resulting in dozens of wavelength thicknesses. On the other hand, progress with chiral heterostructrues [1–3] provides an unprecedented opportunity to manipulate the circular polarization of light by tailoring the refractive index of the material. Nevertheless, most chiral metastructures are based on Bragg reflection and localized plasmons, which lead to a narrow frequency band. Although the spectrum can be broadened by stacking multiple polarizers or introducing a gradient helical pitch [4], the relative strength of different circularly polarized waves and the working spectrum cannot be tuned without changing the physical structure of the chiral material.

Graphene nanophotonics has attracted considerable interest due to its unique optical properties [5–9] and many related applications, ranging from graphene-based optical modulation [10], transformation optics [11], field-effect transistors [12] to numerous other devices [13–16]. Apart from being the thinnest material, graphene is attractive for its physical flexibility, high electron mobility and the possibility of controlling its carrier concentration via external gate voltages or chemical doping. Another promising application of graphene is in magneto-optics. The ability to create a giant Faraday rotation angle has been indentified experimentally in both single-layer and multilayered continuous graphene [17]. The large Hall conductivity [18], which produces extremely significant Faraday rotation originating from the cyclotron effect in the classical region and the inter-laudau-level transitions in the quantum regime, inevitably affects RCP and LCP waves differently, both in amplitude and phase. Epsilon-near-zero (ENZ) material also exhibits highly unusual and intriguing optical properties. One of these properties is the near-zero phase delays while a wave propagates though such an ENZ anisotropic material, which has been proposed for extraordinary transmission in the bend channel [19]. ENZ materials were also employed for radiation pattern control [20, 21], and perfect absorption [22]. Combined with magneto-optical materials, ENZ materials may provide a mechanism for enhancing the effects of nonreciprocity and time-reversal symmetry breaking [23, 24].

In this paper, by employing the properties of ENZ materials and the high Hall conductivity of graphene, we propose a fast tunable magneto-optical sub-wavelength structure that demonstrates efficient manipulation of circular polarization properties of light. The magnetically active structure is transparent for one circularly polarized wave, and opaque for another. In a fairly broad frequency range and at a large oblique incidence angle, such a polarizing effect can be realized and tuned by changing the magnitude of the applied magnetic field or the chemical potential of graphene.

2. Approach for the design of graphene-based circular polarization beam splitter

To model one-atom-thick graphene in a macroscopic electromagnetic description, the graphene can be treated as an ultrathin film with thickness $Δ (Δ \to 0)$ [11]. Asumming exp(-iωt) time harmonic variation, the fourth Maxwell equation can be rearranged as $\nabla \times H = (σ - i ω ξ_{0}) E .$ Denoting the equivalent complex permittivity of Δ thick graphene, we obtain ξ_g = iσ/(ξ₀ωΔ) + 1. As $Δ \to 0$ , due to the large value of the first part, the second part could be neglected. The equivalent parallel complex permittivity of the Δ-thick graphene layer is ξ_g = iσ/(ξ₀ωΔ). In the presence of a static magnetic field B applied perpendicular to its surface, graphene is gyrotropic and its anisotropic permittivity can be described by a tensor:

{\bar{\bar{ξ}}}_{g} = (\begin{matrix} ξ_{g d} & ξ_{g o d} & 0 \\ - ξ_{g o d} & ξ_{g d} & 0 \\ 0 & 0 & ξ_{z} \end{matrix})

where ξ_gd = iσ_xx/(ξ₀ωΔ) and ξ_god = iσ_yx/(ξ₀ωΔ). σ_xx and σ_yx are the longitudinal conductivity (along the E-direction) and Hall conductivity (along the direction perpendicular to E), respectively.

Consider a graphene layer which is biased by a static magnetic field perpendicular to the surface, and its electronic property can be characterized by the surface conductivity, which can be modeled by the Kubo formula [25] through a quantum mechanical analysis, as

\begin{array}{l} σ_{x x} (ω, B) = \frac{e^{2} v_{f}^{2} | e B | ℏ (ω + i 2 Γ)}{i π} \sum_{n = 0}^{\infty} {\frac{1}{M_{n + 1} - M_{n}} \times [f_{d} (M_{n}) - f_{d} (M_{n + 1}) + f_{d} (- M_{n + 1}) - f_{d} (- M_{n})] \\ \times \frac{1}{{(M_{n + 1} - M_{n})}^{2} - ℏ^{2} {(ω + i 2 Γ)}^{2}} + (M_{n} \to - M_{n})} \end{array}

\begin{array}{l} σ_{y x} (ω, B) = - \frac{e^{2} v_{f}^{2} e B}{π} \sum_{n = 0}^{\infty} [f_{d} (M_{n}) - f_{d} (M_{n + 1}) - f_{d} (- M_{n + 1}) + f_{d} (- M_{n})] \\ \times [\frac{1}{{(M_{n + 1} - M_{n})}^{2} - ℏ^{2} {(ω + i 2 Γ)}^{2}} + (M_{n} \to - M_{n})] \end{array}

where ν_f ≈10⁶ m/s is the Fermi velocity, M_n =

\sqrt{n} L

are the Laudau energy levels, and

L^{2} = 2 ћ e B v_{f}^{2}

is the Laudau energy scale. f_d(ξ) = [exp(ξ-µ_c)/(k_BT) + 1]⁻¹ is the Fermi-Dirac distribution, µ_c is the chemical potential, k_B is the Boltzmann constant, T is the environment temperature, ћ = h/(2π), h is the Planck constant, and Γ is the phenomenological scattering rate. The meaning of arrow in Eq. (2) is “replacing M_n with -M_n in the expression of items which have the same priority in calculation”.

The total conductivity depends on two parts of electron transition, namely, the intraband transition and interband transition [7]. The former transition involves energy levels in the same band (both in the conduction band or the valence band), and the latter transition involve levels in different bands. As a result, interband transitions occur essentially at frequencies ћω $\geq$ 2µ_c. The spectral range we study in this paper is consistent with the restriction ћω<2µ_c. Therefore the results of this paper are derived by including the intraband term in (1). Combining this with the fact that the transitions between the levels around μ_c are the strongest ones, it shows that the conductivities follow the Drude model form [26]

σ_{x x} = σ_{0} \frac{1 - i ω τ}{{(ω_{c} τ)}^{2} + {(1 - i ω τ)}^{2}}

and

σ_{y x} = σ_{0} \frac{ω_{c} τ}{{(ω_{c} τ)}^{2} + {(1 - i ω τ)}^{2}}

where

σ_{0} = \frac{2 e^{2} τ}{π ℏ^{2}} k_{B} T \ln (2 \cosh \frac{μ_{c}}{2 k_{B} T})

In this semiclassical expression for the conductivity, τ = 1/(2Γ) is the scattering time and ω_c is the cyclotron frequency, which corresponds to the difference between the neighbor Laudau energy around the Fermi level.

ω_{c} = \frac{M_{n + 1} - M_{n}}{ℏ} \approx \frac{L^{2}}{2 ℏ μ_{c}} = \frac{e B v_{f}^{2}}{μ_{c}}

A DC mobility of high-quality suspended graphene is expected to be 200,000 cm²V⁻¹s⁻¹ [27, 28], and μ > 60,000 cm²V⁻¹s⁻¹ in graphene on a hexagonal boron nitride (h-BN) substrate was experimentally achieved [29]. In a graphene rinbbon, the relaxation time τ = 20 fs was reported [30]. For a multilayered epitaxial graphene on 4H-SiC substrate, the relaxation time τ = 1 ps was determined through carrier mobility measurements [31]. As graphene sheets with a representative value

μ_{c}

= 0.2 eV in our suggested structure, it is possible to consider two values for the relaxation time, namely, τ = 2 ps, which is expected to have high mobility 100,000 cm²V⁻¹s⁻¹ by using

μ = e τ v_{F}^{2} / μ_{c}

[32] and result in larger resonance q-factor in grpahene magnetoplasmons, and τ = 0.2 ps, corresponding to lower mobility 10,000 cm²V⁻¹s⁻¹, which can be possibly achieved in some experimental study of graphene.

The schematic diagram of the suggested structure is shown in Fig. 1, which consists of periodically alternating graphene and dielectric layers. The dielectric layer has permittivity ξ_d = 3. A static magnetic field B is applied perpendicularly to the interface of such two materials. The graphene/dielectric stack can be described as an anisotropic metamaterial with effective constitutive parameters, since the thickness of the dielectric layer t_d is much smaller than the wavelength. There are three effective permittivity parameters, the parallel diagonal and off-diagonal elements ξ_p and g for the electric fields along the graphene/dielectric interface, and ξ_t for the electric field perpendicular to the surface.

{\bar{\bar{ξ}}}_{a v e} = (\begin{matrix} ξ_{p} & - i g & 0 \\ i g & ξ_{p} & 0 \\ 0 & 0 & ξ_{t} \end{matrix})

According to the effective medium theory, we have ξ_p = f∙ξ_gd + (1-f)∙ξ_d, g = f∙i∙ξ_god and 1/ξ_t = f/ξ_z + (1-f)/ξ_d, where f = ∆/(∆ + t_d) is the filling factor of graphene. When ∆ is infinitely small, the three effective permittivity parameters can be reduced to ξ_p = f∙ξ_gd + ξ_d, g = f∙i∙ξ_god and ξ_t = ξ_d. Since ξ_g = iσ/(ξ₀ω∆) and f = ∆/t_d, we can easily get ξ_p = ξ_d + iσ_xx/(ξ₀ωt_d) and g = -σ_yx/(ξ₀ωt_d). Given a frequency in the THz range, without properly tuning the magnitude of the applied static magnetic field B or by properly setting the chemical potential µ_c, which relates to electrostatic biasing and chemical doping, we can achieve near-zero permittivity for ξ_p. The dependence of the frequency of ENZ with both magnetic field B and chemical potential µ_c is rendered in Fig. 2(a) and 2(b). Throughout the paper, the two possible values for the dielectric layer thickness are assumed to be 50 nm and 300 nm, which are far below the wavelength corresponding to dozens of THz (the frequency of ENZ is shown in Fig. 2(a), 2(b) and 3(a)), and the effective medium method used here is valid.

Fig. 1 (a) Schematic view of the graphene/dielectric-stacked structure for circular polarization control in reflection and transmission. A static magnetic field B is applied perpendicularly to the surface. (b) Illustration of our suggested structure.

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Fig. 2 (a) and (b) The dependence of the frequency of ENZ with respect to both magnetic field B and chemical potential µ_c. (c) and (d) The real part of the off-diagonal element g as a function of magnetic field B and the chemical potential µ_c. In (a)-(d), the points a₁-a₄ correspond to the situations where the value of µ_c is fixed to 0.2 eV and increase the magnitude of magnetic field from 1 T to 7 T with a step of 2 T. The points b₁-b₄ correspond to the situations where the magnitude of B is fixed to 3 T and the chemical potential µ_c increases from 0.15 eV to 0.3 eV with a step of 0.05 eV. (e) and (f) Transmittance of the forward RCP and LCP waves through ENZ graphene/dielectric-stack slab as a function of the slab thickness. The lines correspond to the points a_i(i = 1,2,3,4) in (a) and (b) with different applied magnetic fields. (g) and (h) Transmittance of the forward RCP and LCP waves as a function of the slab thickness, with a different chemical potential (corresponding to points b_i, i = 1,2,3,4). Panels to the left correspond to τ = 2 ps while right panels are for τ = 0.2 ps. The dielectric thickness t_d is 50 nm and the environmental temperature is 300 K.

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Fig. 3 (a) The dependence of the frequency of ENZ with respect to both magnetic field B and chemical potential µ_c. (b) The real part of the off-diagonal element g as a function of magnetic field B and the chemical potential µ_c. (c) The transmission and reflection spectrum for a forward propagating RCP wave with different applied magnetic fields. (d) The transmission and reflection spectrum for a forward propagating LCP wave with different applied magnetic fields. (e) The efficiency of the polarizer is characterized by the parameter P_t, which is defined as the multiplication of the degree of polarization and the strength of the transmission of RCP component. (f) The FWHM of the lines in (e) with a dependence on magnetic field B at different levels of chemical potential, which represent the possible bandwidth of the effective operating frequency of the graphene-based polarizer. From (c) to (f), parameters of the structure are the following: the relaxation time 2 ps, the chemical potential 0.2 eV, the thickness of each dielectric layer 300 nm and the slab 15 $μ m$ .

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Optical properties of such a graphene/dielectric stack magnetized along the z-axis are characterized by the permittivity matrix (5). Let us consider the incident electromagnetic wave with the form:

E = (A {\vec{v}}_{R H} + B {\vec{v}}_{L H}) \exp (i [γ x + β z - i ω])

where ${\vec{v}}_{R H}$ and ${\vec{v}}_{L H}$ correspond to the right-handed and left-handed polarization waves, respectively. θ is the angle of incidence, and the transverse wave vector γ = k₀sin(θ).

In the magnetized graphene/dielectric layer:

E = (κ_{1} {\vec{v}}_{1} \exp (i β_{+} z) + κ_{2} {\vec{v}}_{2} \exp (- i β_{+} z) + κ_{3} {\vec{v}}_{3} \exp (i β_{-} z) + κ_{4} {\vec{v}}_{4} \exp (- i β_{-} z)) \exp (i [γ x - i ω])

Note that the electromagnetic field in the magnetic layer consists of four plane waves with different polarization states, which could contain both LH and RH polarization components, especially in oblique incidence. With the Fresnel equation,

β_{\pm}

and

{\vec{v}}_{i}

in the magnetic media can be calculated as:

β_{\pm}^{2} = \frac{γ^{2} (ξ_{t} - ξ_{p}) \pm \sqrt{4 ξ_{t} g^{2} k_{0}^{2} (ξ_{t} k_{0}^{2} - γ^{2}) + γ^{4} (ξ_{p} - ξ_{t})}}{2 ξ_{t}} + ξ_{p} k_{0}^{2} - γ^{2}

{\vec{v}}_{i} = (\begin{matrix} 1 \\ \frac{i g k_{0}^{2}}{β_{i}^{2} + γ^{2} - ξ_{p} k_{0}^{2}} \\ \frac{β_{i} γ}{γ^{2} - ξ_{t} k_{0}^{2}} \end{matrix})

For the plane wave propagating along the magnetization direction, we have γ = 0, and in the forward direction Eq. (8a) reduces to a well known relation for the corresponding wavevectors

β_{\pm} = K_{0} \sqrt{ε_{p} \pm g}

. The plus sign is for a forward propagating RCP wave, and the minus sign is for a forward propagating LCP wave. As the convention for the polarization handedness, the backward propagating waves have the wave vectors,

β_{\pm} = K_{0} \sqrt{ε_{p} \pm g}

. The plus sign is for a LCP wave, and the minus sign is for a RCP wave. The eigenmodes in such a multi-layered structure are two circularly polarized wave

v_{\pm} = (1, \pm i, 0)

, corresponding to wavevector

β_{\pm}

. Assuming a lossless condition, the value of

ε_{p}

goes to zero, and the wave vectors

_{\pm} β_{+}

are real. Therefore the medium is transparent to forward RCP and backward LCP. The other wave vectors

_{\pm} β_{-}

are purely imaginary. This means that the medium is opaque for the forward LCP and backward RCP.

In the graphene/dielectric alternately stacked material, one can achieve a real part of the diagonal element ξ_p in matrix (5) near zero, while its imaginary part remains quite small. As magnetically biased graphene possessing giant gyrotropic properties at THz frequencies, we can also achieve a large off-diagonal element g compared to the conventional material. In Fig. 2(c) and 2(d) we show off-diagonal element g as a function of both the applied static magnetic field B and the chemical potential μ_c. The value of g has a large real part in the frequency range of dozens of THz, which becomes more significant when it is applied by a stronger magnetic field. According to Eq. (4), we can see that the cyclotron frequency has an inverse relationship with the chemical potential, and in Eq. (3) the most significant magneto-optic effect occurs around the cyclotron frequency. If we increase the chemical potential, the frequency of the ENZ will be further away from the cyclotron frequency, resulting in a decrease in the value of g. In Fig. 2(a)-2(d), here are some variations for both the frequency of ENZ and the real value of g, when the relaxation time decreases from 2 ps to 0.2 ps by an order. That is because the introduction of smaller relaxation time will inevitably suppress the strength of gyrotropic resonance.

3. Reflection and transmission for different handed polarization waves by the graphene/dielectric-stacked material

With different values of applied magnetic field and chemical potential, the transmission amplitudes for both circularly polarized waves are shown in Fig. 2(e)-2(h). In Fig. 2(e) and 2(g) (the left panel), as the distance increases, we observe that the RCP forward propagating wave almost preserves its amplitude and has only a little attenuation due to the introduction of an imaginary part in the permittivity. In contrast, the LCP propagating wave is almost completely reduced (most of the energy is reflected) and only a small fraction of it tunnels trough the stack. Such a phenomenon of selected circular polarization transmission will become more significant when we increase the applied static magnetic field and decrease the chemical potential. In the case τ = 0.2 ps (the right panel), both circular polarization waves decay faster than those in the case τ = 2 ps. This is because the imaginary part of the diagonal element ξ_p in Eq. (5) has a reverse relationship with relaxation time. With the premise $ξ_{p, i} ≪ g$ at ENZ frequency, the reduced form of Eq. (8a), $β_{\pm} = k_{0} \sqrt{ε_{p} \pm g}$ can be further simplified to $β_{+} = k_{0} \sqrt{g} + i ξ_{p, i} / (2 \sqrt{g}))$ and $β_{-} = i k_{0} \sqrt{g} + ξ_{p, i} / (2 \sqrt{g}))$ . It can be clearly seen that the loss introduced by reducing the relaxation time will finally bring down the transmission strength for both circular polarization waves. The physics behind such an effect is magnetoplasmons damping, which is related to the relaxation time τ, caused by inter- or intra-band Laudau damping as well as inelastic scattering with phonons and charged impurities. The latter process could be a dominant factor in the situation when the graphene has small a relaxation time.

An incident electromagnetic wave has a linear polarization, which can be decomposed into two opposite circular polarizations, namely, the RCP and LCP propagating waves. Here we assume both circularly polarized waves have unit amplitude and impinge on a graphene/dielectric-stacked layer 15 $μ m$ In order to achieve large value of the off-diagnol element g at the frequency of ENZ, we increase the thickness of dielectric layer to 300 nm. The relationships are shown in Fig. 3(a) and 3(b). With the same strength of magnetic field and chemical potential, the frequency of ENZ has a red shift and g is almost twice as large as the one in Fig. 2. The increase in value of g is due to the approaching of the frequency of ENZ to the cyclotron frequency. The transmission and reflection spectrum of the RCP component are shown in Fig. 3(c). In the lower frequency, most of the wave is reflected. When it comes to the ENZ region, the structure starts to become almost transparent. When we increase the magnitude of the applied magnetic field, the curves have a red shift due to the stronger magneto-optic effect. The spectrum of the LCP component is shown in Fig. 3(d), which is similar to the one for the RCP component: at a lower frequency most of the wave is reflected, and the medium becomes transparent when the frequency increases. However, the positions of the turning points are located at higher frequencies as compared with the RCP components. When we increase the magnitude of the applied magnetic field, the spectrum has a blue shift, which has an opposite moving direction to the RCP component. It is obvious that there is an overlapping region between such two groups of spectra, where the RCP component of the wave can transmit through the structure while the LCP component is reflected. Now we define P_t as the product of the transmission intensity of the RCP component and the quality degree of circular polarization, which is defined as $t_{R C P}^{2} \times (t_{R C P}^{2} - t_{L C P}^{2}) / (t_{R C P}^{2} + t_{L C P}^{2})$ . In the lower frequency, the transmission light has a high quality of circular polarization but a low intensity, while in the higher frequency, the transmission light has a high intensity but low quality of circular polarization, resulting in a near-zero value of P_t for both sides. In the region of ENZ, the suggested structure has both high transmission intensity and high quality of circular polarization. Such a phenomenon is shown in Fig. 3(e). The full width at half maximum (FWHM) of such a peak represents the tolerance of the polarizing effect near the frequency point of ENZ, and has a positive correlation to the magnitude of the applied magnetic field. The relationship is shown in Fig. 3(f). The width of the polarizing band has an almost linear relationship with the magnitude of the magnetic field. However, when we decrease the chemical potential while increasing the magnetic field, the polarizing band will become too close to the cyclotron frequency, where the permittivity of the medium shows some dramatic fluctuation, which results in the disorder at the end of the red line in Fig. 3(f).

Figure 4(a) and 4(b) illustrate the situation of oblique incidence. Here we consider point in Fig. 3(a), with following parameters: the applied magnetic field 3 T, graphene relaxation time 2 ps, the chemical potential 0.2 eV and the corresponding ENZ frequency 9.43 THz. The thickness of the composite layer is 15 μm. The RCP wave and LCP wave incidence are represented in Fig. 4(a) and 4(b), respectively. According to Eq. (8), there are four waves mixing in the magneto-optic material. When we increase the incident angle, it is more likely to excite all four waves in the medium, and thus produce its opposite circular polarization in the transmission spectrum. Both figures show that the graphene/dielectric-stacking polarizer maintains its efficiency at the ENZ region in quite a large incident angle. As the figures show, when the incident angle is less than 45 degrees, the amplitudes of light remain above 80% for both the transmission of RCP incident wave and the reflection of LCP incident wave.

Fig. 4 The transmission and reflection of both the RCP and LCP waves with the oblique incidence of the RCP wave (a) and LCP wave (b). T_rr is the conversion coefficient from the incident RCP wave to the transmitted RCP wave. T_ij and R_ij are the conversion coefficients of the circularly polarized waves, e.g., T_rr is the conversion coefficient from the incident RCP wave to the transmitted RCP wave, and R_lr is the conversion coefficient from incident RCP wave to reflected LCP wave.

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4. Conclusion

We theoretically analyzed the transmission and reflection coefficients of both the RCP and LCP waves at the ENZ frequency range of the graphene/dielectric-stacked material, and these results have been further validated by FDTD calculations, which are much more time-consuming. The composite material shows efficient manipulation of the polarization of the electromagnetic wave in a fairly large range of the incidence angle. Both the transmission ratio of the two circularly polarized waves and the working frequency range can be tuned by changing the magnitude of the applied magnetic field or the chemical potential. The suggested structure with the possibility of easy and fast tuning, together with a strong magneto-optic effect (which is not present in other known materials), may have an impact in a variety of novel devices and applications, beyond circular polarization manipulation and control.

Acknowledgment

The authors are grateful to the partial supports from NSFCs 61178062, 60990322 and 91233208, the National High Technology Research and Development Program (863 Program) of China (No. 2012AA030402), Swedish VRgrant (# 621-2011-4620) and SOARD.

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Frequency-tunable circular polarization beam splitter using a graphene-dielectric sub-wavelength film

Abstract

1. Introduction

2. Approach for the design of graphene-based circular polarization beam splitter

3. Reflection and transmission for different handed polarization waves by the graphene/dielectric-stacked material

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (4)

Equations (12)

Optics Express