Introduction

Optical isolators, the photonic analogue of diodes, allow nonreciprocal unidirectional light transmission1,2,3,4,5,6,7,8. They have become indispensable components in optical telecommunications and laser applications for stabilizing propagating photons and preventing undesired feedback1,2,3,4,5,6,7,8. However, the main component of the optical isolators, Faraday rotator, is usually composed of magneto-optical materials with weak Faraday effect. It requires a long propagation distance in the order of operating wavelength to achieve good isolation and therefore, optical isolators are generally bulky1. In attempt to reduce costs and realize on-chip integration, there is a keen interest to miniaturize the nonreciprocal devices through designing novel structures2. Recent advance in fabrication of ultrathin two-dimensional (2D) graphene down to monolayer may provide the possibility to further push the optical isolation to atomic scale and has attracted more and more people's interests. This is due to the fact that graphene can become nonreciprocal and gyrotropic under an external magnetic field9,10,11,12,13,14,15,16. However, because of its ultrathin thickness, graphene is almost transparent for light17,18 and can only turn the polarization of incident light by several degrees under a modest magnetic field11,12. Hence, external bulky resonating structure and polarizer are still needed when incorporating graphene into the traditional isolators13,14. Achieving optical isolation in the atomic scale is still very challenging.

In this paper, we theoretically proposed the idea of atomically thin nonreciprocal optical isolation for circularly polarized waves by using graphene monolayer. We show that under an external magnetic field, electrons in graphene would experience distinct magnitudes of Lorentz force for left-handed or right-handed circularly polarized wave (LCP or RCP) incidence at cyclotron frequency. Therefore, for the same circularly polarized wave, graphene can be tailored to be transparent to the forward propagation direction, whereas to be reflective to the backward propagation direction. Moreover, for the same propagation direction, graphene can be tailored to be transparent to one of the circularly polarized waves, whereas to be reflective to the other polarization handedness. As a result, the polarization-selective isolator can isolate and filter circular polarizations simultaneously. Our work provides an analytical study on the extreme limit of thickness for optical isolation, which might be useful in the quest to miniaturize nonreciprocal photonics in the future.

Results

Fig. 1 schematically shows the atomically thin and polarization-selective optical isolation based on graphene monolayer. Under an external magnetic field perpendicular to graphene plane, graphene (in the xy plane) becomes gyrotropic and can be described by an asymmetric permittivity tensor: . For plane wave propagating along magnetization direction, the eigenmodes in gyrotropic graphene are LCP and RCP. Without loss of generality, we analytically calculate the transmissivity of circularly polarized waves normally incident from a dielectric medium with relative permittivity ε1 onto the gyrotropic graphene layer and transmitted into a second dielectric medium with relative permittivity ε3. Detailed calculation can be found in Methods. The calculated wavevectors for LCP and RCP in gyrotropic graphene are as follows:

from which we see that for the same circularly polarized wave they have different phase velocities (vph = ω/k) in the forward and backward propagations. Moreover, since and , the transmissivity of LCP in one propagation direction is the same with that of RCP in the opposite propagation direction. With this symmetry, we only need to analyze the forward propagating transmissivity.

Figure 1
figure 1

Schematics of the atomically thin and polarization-selective optical isolation for circularly polarized waves.

An external magnetic field is applied perpendicular to graphene plane. Only one circularly polarized wave can propagate through the gyrotropic graphene in one propagation direction. The arrows in the helixes show the instantaneous electric field distributions of circularly polarized waves.

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The ideal case of graphene monolayer without loss is firstly discussed in Fig. 2(a). The parameters in Fig. 2(a) are as follows: the chemical potential μc = 0.5 eV, the cyclotron frequency ωc/2π = 10 THz, the damping constant γ = 0 and ε1 = ε3 = 1. From the analytical derivations (details are in Methods), we can get forward propagating transmissivity t+LCP = 0 and t+RCP = 0.99 at cyclotron frequency, indicating good isolation performance. The underlying mechanism of the remarkable contrast in transmissivity at cyclotron frequency is that the electron velocity in graphene under LCP incidence is largely different from that under RCP incidence, making gyrotropic graphene highly conductive and reflective for one circular polarization incidence while transparent for another circular polarization incidence. From a microscopic point of view, by using free electron model19, , where meff is the effective electron mass and is the electric field of incident wave being either RCP or LCP. Under time harmonic excitation, we can derive

where . In this paper, we set the external magnetic field along direction and the scalar cyclotron angular frequency is defined as . For the forward propagation, we have for LCP incidence with and for RCP incidence with . Hence, the magnitude of Lorentz force acting on graphene electrons under LCP incidence is much larger than that under RCP incidence at ω = ωc. As a result, can be theoretically very large and is reduced to half at cyclotron frequency. Since the electric current density and , gyrotropic graphene becomes highly conductive under LCP incidence, indicating gyrotropic graphene behaves like a perfect electric conductor and LCP is thus totally reflected. In contrast, for RCP incidence, graphene has a finite negative permittivity and wave is evanescent inside of the graphene. As its skin depth (1/Im(k+RCP) > 100 nm) is far larger than graphene's atomic thickness, graphene is highly transparent to RCP incident wave.

Figure 2
figure 2

Transmissivity of RCP (t+RCP) and LCP (t+LCP) through gyrotropic graphene monolayer in the forward propagation.

(a) Comparison of t+RCP and t+LCP under the lossless (2πγ/ωc = 0) and loss (2πγ/ωc = 1) conditions with ωc/2π = 10 THz. (b) Loss influence on the isolation performance with different cyclotron frequencies. t+RCP/t+LCP increases as 2πγ/ωc or ωc decreases. Other parameters are as follows: μc = 0.5 eV and ε1 = ε3 = 1.

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Because of the collision between electrons, loss is inevitable in practical graphene samples. In the following part, we studied the loss influence on the isolation performance. When loss is considered, the electron velocity in equation (2) becomes . For the forward propagation and by setting along direction, we have for LCP incidence and for RCP incidence. Therefore, as long as 2πγ/ωc is sufficiently small, can still be theoretically very large and at cyclotron frequency can still be achieved. Fig. 2(a) shows the analytical transmissivity of LCP and RCP under the lossless (2πγ/ωc = 0) and loss (2πγ/ωc = 1) conditions. One can see that the loss degrades the performance of isolation. Fig. 2(b) shows the transmissivity ratio between RCP and LCP at cyclotron frequency (t+RCP/t+LCP) as a function of 2πγ/ωc with different cyclotron frequencies, from which one can see that, on one hand, t+RCP/t+LCP increases as 2πγ/ωc decreases, on the other hand, with the same value of 2πγ/ωc, t+RCP/t+LCP increases as the cyclotron frequency decreases. Hence, better isolation performance can be obtained by keeping 2πγ/ωc sufficiently small or setting smaller cyclotron frequency.

To verify the analytical prediction of the atomically thin isolation, Fig. 3 shows the numerical simulations of the lossless case at 10 THz with the use of Finite Element Method (COMSOL Multiphysics). In the simulation, the following parameters are used: μc = 0.5 eV, ωc/2π = 10 THz, γ = 0 and ε1 = ε3 = 1. We set the incident field with unit magnitude. When propagating in the forward direction, RCP is almost totally transmitted through graphene, as shown in Fig. 3(a), featured with unit magnitude of maximum Ey at both sides of graphene. Meanwhile, LCP is totally reflected, as shown in Fig. 3(b), characterized with twice unit magnitude of maximum Ey at the left graphene side and near zero at the right side. Moreover, from Fig. 3(b), one can see that the total electric field at the graphene boundary is always zero, indicating that graphene exhibits zero impedance, verifying our previous theoretical analysis that in this case graphene behaves like a highly conductive material. When propagating in the backward direction, RCP is totally reflected in Fig. 3(c) and LCP is almost totally transmitted in Fig. 3(d), behaving inversely to the forward propagation.

Figure 3
figure 3

Simulation demonstration of the atomically thin optical isolation for circularly polarized waves under the lossless condition.

Ey field distribution of RCP and LCP waves propagating in (a–b) the forward and (c–d) backward directions at 10 THz. The black dashed lines denote the location of graphene and the green arrows represent the wave incident direction. The parameters are: μc = 0.5 eV, ωc/2π = 10 THz, γ = 0 and ε1 = ε3 = 1. Under the lossless condition, graphene monolayer can isolate and filter the circular polarizations simultaneously.

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The numerical verifications for the loss case are shown in Fig. 4. Based on previous practical graphene parameters, we set ωc/2π = 0.3 THz, μc = 0.5 eV and γ = 0.63 THz in the demonstration. These correspond to B = −1 Tesla, the 2D carrier density ns = 2 × 1013 cm−2 and the electron mobility μ = 28000 cm2V−1s−1. These parameter settings are achievable in experiments, because that the 2D carrier density ns in graphene is experimentally reported to be gated efficiently from very low (1010 cm−2) to very high values (1014 cm−2)20,21,22 and the electron mobility up to μ = 60000 cm2V−1s−1 has been experimentally obtained in high-quality suspended23 or hexagonal boron nitride substrate supported24 graphene samples. We also consider the graphene is mounted on a substrate. The influence of surrounding dielectric substrate on the isolating performance is shown in Fig. 4(a,b), where symmetric structure is assumed with ε1 = ε3 = εd to avoid additional background reflection. Smaller εd is found favorable to achieve better isolation performance. To ensure the high electron mobility in graphene and therefore a smaller damping constant γ, hexagonal boron nitride with εd = 4.825 is chosen. From Fig. 4(b), one can see that high t+RCP/t+LCP can be obtained in a wide frequency range with the maxima of 25.96 at 0.311 THz (3.7% higher than ωc = 0.3 THz). When t+RCP/t+LCP > 10 is defined as a reference, a 0.26 THz bandwidth can be obtained with frequency ranging from 0.2 to 0.46 THz. Fig. 4(c–f) show the numerical simulations of the nonreciprocal isolation at 0.3 THz. In the forward propagation, RCP wave passes graphene with a transmissivity of 31.5% in Fig. 4(c), while LCP is efficiently blocked with a trivial transmissivity of 1.23% in Fig. 4(d). When propagating in the backward direction, it behaves inversely in Fig. 4(e,f). From these demonstration, one see that an atomically thin wide-band THz nonreciprocal optical isolation is possible by using practical graphene samples, showing the appealing potential applications of graphene in ultrathin nonreciprocal device design.

Figure 4
figure 4

Atomically thin optical isolation under the loss condition.

(a) Influence of εd on the forward propagating transmissivity of LCP and RCP waves, where ε1 = ε3 = εd. (b) The isolation performance represented by t+RCP/t+LCP for different εd. (c–f) Ey field distribution of circularly polarized waves propagating in the forward and backward directions at 0.3 THz. The Parameters are: μc = 0.5 eV, ωc/2π = 0.3 THz, γ = 0.63 THz and εd = 4.8. The black dashed lines denote the location of graphene and the green arrows represent the wave incident direction.

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Discussion

Previous graphene based traditional Faraday isolators14,16 are generally composed by a graphene sheet, a resonating structure and two wire grids. In that Faraday isolator structure, the gyrotropic graphene only serves as a Faraday rotator, the thickness of the resonating structure is around half of the operating wavelength14 and the two wire grids serving as the polarizers are indispensable to realize isolation. When considering the thickness of the resonating structure and wire grids, the total thickness of previous graphene based Faraday isolators14,16 is usually in the order of operating wavelength, far larger than our atomic thickness. Hence, previous graphene based Faraday isolators14,16 are different from the demonstration in this work where no external wire grids or other resonant structures are needed. Flexible tunability is a potential advantage of graphene based devices26,27. Since the best performance of the ultrathin graphene isolation exists near cyclotron frequency, the isolation working frequency range should also be tunable by external magnetic field.

In conclusion, we have theoretically proposed the atomically thin and polarization-selective nonreciprocal optical isolation for circular polarizations. The mechanism underlying the atomically thin optical isolator is revealed from a microscopic viewpoint. Our work shows a feasible solution to control the propagation of circularly polarized waves in graphene and to manipulate light nonreciprocally at atomic scale, which opens up new possibilities for further innovations in functional ultrathin nonreciprocal optics.

Methods

Graphene monolayer is treated as a homogeneous film with thickness d = 1 nm20,28 and characterized by an anisotropic permittivity tensor diag[εeq εeq εz]. Based on Kubo formula, εeq can be cast to the Drude model9,10,20,28,29,

where and Ginter are optical conductivity attributed to intra-band and inter-band transitions respectively, εinter = 1 + iGinter/ωε0d, is the plasma frequency, γ is the inverse of relaxation time and ε0, e, and kB are the dielectric permittivity in vacuum, electron charge, reduced Planck constant and Boltzmann constant, respectively. When setting μc = 0.5 eV and temperature T = 300 K, we have the effective electron mass meff = 0.1 × 0.91 × 10−30 kg and ωp/2π = 410 THz. At the operation frequency below 1 THz, εinter is equal to 3.9.

Gyrotropic graphene can be described by an asymmetric permittivity tensor , where and . Hence, and . The scalar cyclotron angular frequency ωc is defined as , when the external magnetic field is along direction.

Wave polarizations can be viewed by either the temporal or spatial view points. In the case of a circularly polarized wave, from the temporal view point, the tip of the electric field vector describes a circle at a fixed point in space as time progresses. If the right-hand (left-hand) thumb points in the direction of propagation while the fingers point in the direction of the tip motion, the wave is defined as right-hand (left-hand) circularly polarized. From the spatial view point, the instantaneous electric field vector of the RCP (LCP) wave describes a left-hand (right-hand) helix along the direction of propagation, as shown in Fig. 1.

To achieve the condition of good isolation for circular polarizations, the transmissivity of the forward propagation in Fig. 1 are analytically calculated in the following. The calculation of backward propagation can be done by the same methodology. Setting in the lth region with l = 2 standing for gyrotropic graphene and l = 1,3 for isotropic dielectric. After calculation, we have:

where , , , , , , , dz1 = 0 and dz2 = d. The reflection and transmission coefficients are and , respectively. After some algebra, we have

The reflectivity and transmissivity are and , respectively. From equation (5) and equation (6), r+L/RCP and t+L/RCP are strongly dependent on ωc, ωp and γ, when the operating frequency is around ωc. In particular, for LCP incidence in the lossless case, when ω → ωc, , , , , , , and thus r+LCP = 1 and t+LCP = 0. For RCP incidence in the lossless case at ω = ωc, is negatively large but finite and we get t+RCP = 0.99 as shown in Fig. 2(a) through numerical computation. When considering the loss case and setting ωp as a constant, t+L/RCP is strongly dependent on zπγ/ωc and ωc, which can be seen from Fig. 2(b).

When μc and γ are given, the corresponding ns and μ can be theoretically calculated by using 10,29 and 10,29, respectively, where Fermi velocity υF ≈ 0.95 × 106 m/s. When Tesla and μc = 0.5 eV, we can neglect Landau quantization influence and use the Drude model directly for graphene according to Ref. [10], because the following conditions are satisfied: the Landau energy scale and the operating frequency (set below 1 THz) energy .