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The non-reciprocity of the edge magnetoplasmon modes of a graphene strip is leveraged to design a non-reciprocal magnetoplasmon graphene coupler, coupling only in one direction. The proposed coupler consists of two coplanar parallel magnetically biased graphene strips. In the forward direction, the modes along the adjacent strip edges of the strips have the same wavenumber and therefore couple to each other. In the backward direction, the modes along the adjacent strip edges have different wavenumbers and therefore no coupling occurs.
© 2013 Optical Society of America
1. Introduction
Graphene, a one atom thick carbon layer material, has spurred huge research interest since it was first produced in 2004 [1–3], owing to its unique properties, such as high mobility, ambipolarity and half integer quantum Hall effect [2, 3]. In the area of plasmonics, graphene has been shown to exhibit unique properties, such the capability of supporting both TE and TM plasmons [4–8], gate tunability [9, 10] and has been extensively investigated as a candidate towards the realization of enhanced and novel plasmonic devices [4,11–15]. Moreover, when it is biased by a perpendicular magnetic field, it exhibits gyrotropic and non-reciprocal properties, which have been recently investigated at microwave, terahertz and optical frequencies [16–20].
A magnetically biased graphene strip supports edge and bulk magnetoplasmons with non-reciprocal properties [20, 21]. This non-reciprocity can be exploited in the design of novel non-reciprocal plasmonic devices. In this paper, we propose a non-reciprocal magnetoplasmon graphene coupler, whose operation is based on the non-reciprocity of the edge magneto-plasmons of magnetically biased graphene strips. The proposed structure exhibits coupling in the forward direction, whereas coupling is prohibited in the backward direction. The structure is simulated using the 2D finite difference frequency domain (FDFD) technique [22] where graphene is modeled as a zero-thickness 2D conductive sheet with a conductivity tensor following the Drude model [16]. In Sec. 2, the nonreciprocity of the edge magnetoplasmon modes in a magnetically biased graphene strip is discussed. The coupler structure is introduced and analyzed in Sec. 3.
2. Magnetoplasmons in a graphene strip
A graphene strip supports an infinite number of 2D-bulk modes and two almost degenerate symmetrical and anti-symmetrical edge modes. When magnetically biased, the degeneracy of the two edge modes is lifted and these edge modes exhibit different dispersions. The slow-wave factor and loss of the edge and bulk magnetoplasmons of a magnetically biased graphene strip are plotted in Fig. 1. The edge modes are represented in red and the bulk modes in blue. The dashed curves correspond to the dispersion curves of an infinite graphene sheet. The corresponding electric field patterns for different modes are shown in Fig. 2. The edge modes propagating along the right and left edges of the strip have opposite right and left handed circular polarizations, as shown in Fig. 3. In magnetically biased graphene, which exhibits the conductivity tensor σ̄ = σd(x̂x̂ + ẑẑ) + σo(x̂ẑ − ẑx̂), where σd and σo are the diagonal and off-diagonal conductivities, respectively, the right and left-hand circularly polarized waves see different scalar conductivities, σd + jσo and σd − jσo, respectively. Therefore, the two edge modes exhibit different dispersions, as observed in Fig. 1.
Fig. 1 Slow-wave factor and loss for a graphene strip with parameters w = 100 μm, τ = 0.1 ps, ns = 1013 cm−2 and B0 = 1 T. Edge modes are plotted in red and bulk modes in blue. The dashed curve shows the dispersion for an infinite graphene sheet with the same parameters. The gray area corresponds to the light cone.
Fig. 3 Electric field on the graphene strip for the edge modes propagating on the right and left edges. (a) Point A on the right edge sees a counter clockwise rotating electric field as the wave (mode 2+) propagates along the graphene strip. (b) Point B on the left edge sees a clockwise rotating electric field as the wave (mode 1+) propagates along the graphene strip.
The loss of the edge modes and the first two bulk modes of a magnetically biased graphene strip is shown in Fig. 1(b). At frequencies close to cut-off, the loss becomes maximum, as in all conventional waveguides (the first mode shows a similar trend at lower frequencies). This is the result of the zigzagging propagation of the modes between the edges of the strip and the increase of the deviation angle from the strip axis as frequency decreases [20].
The electric field patterns for the magnetoplasmon modes of the structure in Fig. 1 are shown in the top and bottom rows of Fig. 2 for the forward (+z) and backward (−z) directions, respectively. The edge modes propagating on the same edge [(1+ and 2−) or (2+ and 1−)] have different dispersions, i.e. the edge modes propagating on the same edge have different phase velocities. This non-reciprocity can be used to realize novel non-reciprocal plasmonic devices, like non-reciprocal plasmonic phase shifters [20], isolators and couplers. In the next section, we exploit this non-reciprocity to design a non-reciprocal magnetoplasmon coupler which exhibits coupling only in one direction of propagation, while it prohibits coupling in the opposite direction.
3. Non-reciprocal magnetoplasmon coupler
Figure 4 shows the configuration of the proposed edge-coupled coplanar nonreciprocal magnetoplasmon coupler. The structure is biased by a magnetic static field perpendicular to the plane of the strips. The two strips are chemically doped with different levels of doping. If the conductivity is tuned in a way that the two edge modes propagating along the adjacent edges (the inner edges of the structure) have similar dispersion properties in the forward (+z) direction, these two modes are phase matched and hence couple to each other, as illustrated in Fig. 4(a). In contrast, for propagation in the opposite direction (−z), the corresponding modes have different dispersions due to the non-symmetric dispersion of the edge modes and therefore do not couple, as illustrated in Fig. 4(b).
Fig. 4 Non reciprocal plasmonic coupler, consisting of two parallel graphene plasmonic waveguides. Both waveguides are biased with a magnetostatic field perpendicular to their plane. (a) Feeding through port 1. (b) Feeding through port 2.
Figure 5 shows the dispersion curves for the edge and bulk magnetoplasmon modes of two (separate) graphene strips, that will be later combined to form a non-reciprocal coupler. The solid curves show the dispersion curves for the magnetoplasmons of a graphene strip with carrier density ns = 1013 cm−2. This strip is tuned to be the right-hand strip of the coupler. The dashed curves show the dispersion curves for a graphene strip with carrier density ns = 8 × 1012 cm−2. This strip is tuned to be the left-hand strip of the coupler. The coupler is designed to operate in the 4–6 THz frequency range. The edge modes are shown in red and the bulk modes in blue. In Fig. 5(a), it is seen that the phase velocity of the modes R1+ of the right strip and L2+ of the left strip are matched in the region indicated by the right ellipse. In Fig. 6, different combinations of edge modes are shown. We see that for the forward (+z) propagation, mode R1+ propagates on the left edge of the right strip and mode L2+ propagates on the right edge of the left strip (red box). Therefore, if the strips are placed close enough to each other, these two modes satisfy the proper conditions for coupling, which will be verified in simulation results.
Fig. 5 Dispersion curves for edge (red) and bulk (blue) modes of two isolated graphene strips with different parameters. The solid curves show the slow-wave factor and loss for a graphene strip with parameters w = 100 μm, τ = 0.1 ps, ns = 1013 cm−2 and B0 = 1 T. The dashed curves show the slow-wave factor and loss for a graphene strip with parameters w = 100 μm, τ = 0.1 ps, ns = 8 × 1012 cm−2 and B0 = 1 T. Phase matched regions are emphasized by ellipses.
Fig. 6 Electric field magnitude for the edge modes of the graphene strips of Fig. 5 showing different possible scenarios when they are placed side by side. The right strip has the parameters w = 100 μm, τ = 0.1 ps, ns = 1013 cm−2 and B0 = 1 T. The left strip has the parameters w = 100 μm, τ = 0.1 ps, ns = 8 × 1012 cm−2 and B0 = 1 T.
For the backward direction (−z), the situation is different. In this case, referring to Fig. 5(a), the modes with matching wave numbers (emphasized by the left ellipse) are R1− and L2−. However, referring to Fig. 6 for backward propagation, we see that these modes are propagating on the opposite (far) edges of the two strips (green box) and therefore can not couple. The modes propagating on the near edges of the strips for the backward direction are R2− and L1− (blue box). However, referring to Fig. 5(a), these modes have different dispersions (marked with small circles) and therefore can not couple.
The coupler structure is simulated in Fig. 7, with the two strips having a separation of s = 2 μm. The mode coupling is seen in the field patterns for the forward (+z) and backward (−z) directions shown in Fig. 8. In the forward direction, modes R1+ and L2+ of Fig. 5(a) couple, their dispersion curves (shown in black) change and they form a symmetrical (S+) and an anti-symmetrical (A+) mode (black curves in Fig. 7). The electric field pattern for these symmetrical and anti-symmetrical modes is shown in Fig. 8(a) for the forward propagation. Figure 9 shows the transverse electric vectorial fields for these two modes, whose symmetry and anti-symmetry are clearly apparent.
Fig. 7 Dispersion curves of the edge and bulk modes of the graphene plasmonic coupler of Fig. 4 for the forward and backward propagation with parameters wR = 100 μm, τ = 0.1 ps, ns = 1013 cm−2 and B0 = 1 T for the right strip, and wL = 100 μm, τ = 0.1 ps, ns = 8 × 1012 cm−2 and B0 = 1 T for the left strip and spacing s = 2 μm.
Fig. 8 Electric field magnitude for the edge modes of the coupler of Fig. 4 propagating on the near edges of the strips at the frequency f = 6 THz. (a) Forward direction. The edge modes couple and give two coupled symmetrical and anti-symmetrical modes. (b) backward direction. The modes do not couple.
In the backward (−z) direction, the modes R2− and L1− propagating on the near edges, do not couple, because they are phase mismatched. The electric field patterns for these modes are shown in Fig. 8(b) for the backward propagation, showing two decoupled modes.
The dispersion curves for the bulk modes of the coupler in Fig. 7 are relatively unchanged, compared to the dispersion curves of the bulk modes of each strips, shown in Fig. 5. This is because the bulk modes are propagating inside their respective strips and therefore only weakly couple with the modes of the other strip.
Assume now that the structure is excited at port 1 (Fig. 4) with a transverse electric field E1(x, y) = EL2+(x, y). Neglecting the bulk modes, which do not contribute to coupling, the transverse electric field along the structure is given by
where with all the fields appropriately normalized. The integrals are taken over the transverse plane. The output electric fields at ports 2 and 4 (Fig. 4) for a coupler of length l are then and [see Figs. 5(a) and 6], where Here subscripts A and S refer to anti-symmetrical and symmetrical modes, respectively. The difference in the wave numbers βA and βS causes the energy to be periodically transferred between the two edges [23]. The coherence length, corresponding to the shortest distance of maximal power transfer from port 1 to port 4, is found by plotting and using (3a) and (3b), respectively, versus l.The output powers at ports 2 and 4 of the coupler are plotted in Fig. 10. Figure 10(a) shows the forward coupling, where the coupler is excited at port 1, showing the power at through (port 2) and coupled (port 4) ports for different coupler lengths. Figure 10(b) shows the backward coupling, when the coupler is excited at port 2, plotting the power at through and coupled ports (ports 1 and 3) for different coupler lengths. It is seen in Fig. 10(a) that the power is gradually transferred to port 4 and exceeds the power at port 2 between l = 0.5λ0 and l = 1.2λ0. Although a relatively high carrier density is used in the coupler, loss exceeds 60 dB, which seems prohibitive for practical purposes. However, lower sheet resistances than the 230 Ω/□ used in the simulation have been reported in the literature. Nitric acid doping of graphene can provide a sheet resistance of 150 Ω/□ [24], the layer by layer doping method provides a sheet resistance of 50 Ω/□ [25], a 4-layer nitric acid doped graphene with a sheet resistance of 30 Ω/□ was reported in [24] and a hybrid graphene-metallic nanogrid structure exhibiting a record sheet resistance of 3 Ω/□ was reported in [26]. Figure 11 shows the coupling performance of the coupler for sheet resistances of 80 Ω/□, 30 Ω/□ and 15 Ω/□.
Fig. 10 Output powers at the through and coupled ports when the coupler is excited (a) at port 1, (b) at port 2. λ0 is the free space wavelength.
Figure 11(a) shows the power transferred to the through and coupled ports (ports 2 and 4, respectively) when the coupler is excited at port 1. It is seen that using lower resistance graphene strips in the coupler dramatically improves the coupling performance. For a coupler of length l = 0.7λ0, a 15 Ω/□ sheet resistance can provide a coupling of −3 dB in the forward direction and an isolation of 30 dB in the backward direction.
Fig. 11 Output powers for couplers with low resistance graphene strips at the through and coupled ports when the coupler is excited (a) at port 1, (b) at port 2.
4. Conclusions
A non-reciprocal graphene magnetoplasmon coupler has been proposed and analyzed. The coupler consists of two coplanar parallel magnetically biased graphene strips. Its operation principle is based on the non-reciprocity of the edge magnetoplasmon modes of a graphene strip. For a properly designed coupler, it was shown that the edge modes propagating in the forward direction can be tuned to be be phase matched so as to couple. In the backward propagation direction however, the edge modes have different dispersions and do not couple. The conductivity was shown to be very critical for proper coupling. For a practical non-reciprocal magnetoplasmon coupler, graphene strips with sheet resistances as low as 15 Ω/□ is required.
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