4228
OPTICS LETTERS / Vol. 38, No. 20 / October 15, 2013
Tuning of surface plasmon polaritons beat length in graphene directional couplers Aldo Auditore,1 Costantino de Angelis,1,* Andrea Locatelli,1 and Alejandro B. Aceves2 1
Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Brescia, Brescia 25123, Italy 2 Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, USA *Corresponding author: [email protected] Received July 22, 2013; revised September 6, 2013; accepted September 6, 2013; posted September 9, 2013 (Doc. ID 194331); published October 14, 2013
We investigate the tuning of the coupling of surface plasmon polaritons between two spatially separated graphene layers. We demonstrate that by slightly changing the chemical potential, a graphene coupler can switch from the bar to the cross state; as a consequence, the coupling coefficient in such structures can be easily controlled by means of an applied electrical signal. © 2013 Optical Society of America OCIS codes: (130.1750) Components; (130.2790) Guided waves; (130.3120) Integrated optics devices. http://dx.doi.org/10.1364/OL.38.004228
After its experimental isolation from bulk graphite [1], graphene has attracted increasing interest from the scientific community. Indeed, this single layer of carbon atoms packed into a honeycomb lattice reveals amazing and very useful properties that have already inspired a huge variety of devices embracing different areas ranging from electronics to optics [2]. As research continues to progress, more graphene-based devices are theoretically conceived and experimentally demonstrated [3–10]. In this framework a very interesting role is now played by the overlapping between two originally separated research areas: plasmonics and graphene photonics. The merging of these two areas has become so important as to create a new research line, graphene plasmonics [11]; one of the main assets graphene has to offer in this framework is the tunability of its electromagnetic properties. As is well known, graphene can sustain surface plasmon polaritons (SPPs) having unique properties as compared to what we are used to with noble metals. In fact a single layer of graphene can support either TE or TM polarized plasmons without suffering from huge loss [12,13]; moreover, as far as TM polarization is concerned, the extremely high confinement factor is particularly favorable to exploring the huge χ 3 nonlinearity of graphene [14–16]. Experimental endeavors have demonstrated evidences of the existence of graphene plasmons by measuring the plasmon resonance of graphene nanoribbon arrays [17], and by acquiring their near-field images [18]. The coupling of SPP between separated graphene layers has been recently analyzed in [19]; however, the very interesting properties arising from the easily tunable optical properties of graphene have not been exploited yet in this framework. Here we show that by slightly changing the chemical potential, a graphene coupler can switch from the bar to the cross state; as a consequence, the output state in such structures can be easily controlled by means of an applied electrical signal. In Fig. 1 we report the basic geometry we will consider in this Letter; two graphene layers are embedded in a dielectric structure: regions 2 and 3 are the cladding and substrate, while region 1 (of width 2s) is the dielectric between the two graphene layers. At the graphene 0146-9592/13/204228-04$15.00/0
boundary we set the following conditions on the tangential components of the electromagnetic field: E⃗ 2;3 − E⃗ 1 × xˆ 0 H⃗ 2;3 − H⃗ 1 × xˆ iωϵ0 ϵrS1–2;3 E⃗ ∥ x s;
(1)
where E⃗ ∥ is the electric field tangent to the graphene layer and ϵrS1−2 (ϵrS1–3 ) is the relative surface permittivity of the graphene layer between regions 1 and 2 (3). As far as the electromagnetic constants of graphene are concerned, we write the linear contribution to the relative complex permittivity as [3] ϵrC
σ 1 σ 1 ϵrS 1 Σ;I − i Σ;R ϵrC;R iϵrC;I ; dg dg ωϵ0 dg ωϵ0
(2)
where dg is the graphene thickness and the surface com1 1 plex conductivity σ 1 Σ σ Σ;R iσ Σ;I (in siemens) is obtained from theoretical models now well established and experimentally validated [20], which give the following dependence of the real and imaginary parts of the conductivity on frequency (ω), temperature (T), and chemical potential (μ): σ0 ℏω 2μ ℏω − 2μ tanh tanh ≃η 2 4kB T 4kB T 3 σ0 4 2μ ℏω 2μ μ − ; ω ≃ − log σ 1 Σ;I π ℏω ℏω − 2μ 9t2
σ 1 Σ;R ω
(3)
where t 2.7 eV is the hopping parameter, ℏ and kB are the reduced Planck’s and Boltzmann’s constants, respectively, σ 0 e2 ∕4ℏ ≃ 6.0853 · 10−5 S with e the electron charge, and η ≥ 1 is a parameter we use here to increase the loss level to better match with recently reported experimental data. Although the physical origin of these extra losses is not yet completely understood (they might originate from enhanced relaxation times at infrared frequency or they could be associated with extrinsic factors such as surface irregularities), they have been measured in recent experiments and data are thus available [21,22]. © 2013 Optical Society of America
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
4229
where g1–2 and g1–3 take into account the contribution of the two graphene layers in the continuity conditions g1–2 iωϵ0 ϵrS;1–2
Note that when graphene is modeled as a brick with a finite thickness (dg 0.34 nm), the results reported in Eq. (3) (where we have conductivities with the dimensions of siemens) need to be normalized as σ → σ∕dg to get conductivities in siemens/m. Note also that this model can be easily extended into the nonlinear regime by adding a nonlinear correction to the surface conduc3 ⃗ 2 tivity as σ Σ σ 1 Σ σ Σ jEj E y;z [23]. Moreover, thanks to the extremely small thickness of the graphene layer, nonlinearity can be analyzed by a parameter embedded into the coefficients describing the continuity of the tangential components of the electromagnetic field [15,16]. However, this issue is well outside the scope of our Letter and deserves to be tackled in a different work suited to address the influence of graphene nonlinearities. To describe SPP propagation along z, we first note that, at first order, the y dependence of the electromagnetic field can be neglected; we then look for guided modes with harmonic temporal dependence expiωt and spatial variation E⃗ 1;2;3 x; z, H⃗ 1;2;3 x; z ∼ exp−iβz Γ1;2;3 x with Γ21;2;3 β2 − ϵr1;2;3 k20 . Obviously the complex wavenumber β, through its real and imaginary parts, describes the evolution of both the phase and the amplitude of the guided modes. We can obviously apply the above modeling to derive the dispersion relation of both TE and TM modes. In the following we describe in detail the TM polarization, which allows for a stronger confinement of the electromagnetic field. We first consider a very general situation where, either using different electrical contacts on the two graphene layers or using a nonsymmetric cladding (ϵ2 ≠ ϵ3 ), the two graphene sheets can be biased in a different way to give rise to an asymmetric coupler (i.e., a coupler where the two guiding layers have different electromagnetic properties). After straightforward algebra we find that coupled SPPs in the system are determined by setting to zero the determinant of the following matrix: 2 6 6 6 M6 6 4
Γ1 s
−Γ1 s
−Γ2 s
−e
e
e
e−Γ1 s
eΓ1 s
0
iωϵ1 Γ1 s Γ1 e
1 −Γ1 s − iωϵ Γ1 e
g1–2 e−Γ2 s
1 −Γ1 s − iωϵ Γ1 e
iωϵ1 Γ1 s Γ1 e
0
0
3
7 −eΓ3 s 7 7 7; (4) 0 7 5 g1–3 e−Γ3 s
g1–3 iωϵ0 ϵrS;1–3
iωϵ3 ; Γ3
and ϵrS;1–2 and ϵrS;1–3 refer to the relative dielectric constant of the two graphene layers, which can in general get different values due to different carriers concentrations. This asymmetric coupler offers a wide variety of possible settings which certainly deserve to be investigated both in the linear and nonlinear regimes. Here we want to give a first prototype example into the possibilities offered by the tunability of the graphene parameters in this framework; we thus focus our attention on a very particular situation corresponding to a linear and symmetric case (ϵ2 ϵ3 and ϵrS;1–2 ϵrS;1–3 ); moreover we use T 300 K, λ 10 μm, and for the sake of simplicity we also set ϵr1 ϵr2 ϵr3 2.25. In this regime the graphene directional coupler has two different eigenstates: the even (odd) supermode corresponding to the out-ofphase (in-phase) hybridization of the SPP guided by the single graphene layers. Note also that the even mode here has always the higher value of the propagation constant. In Fig. 2 we report the solution of the dispersion relation as a function of s for two different situations: continuous lines here refer to the even and odd supermodes corresponding to a chemical potential μ1 0.1 eV in Eq. (3), while the dashed lines refer to a choice of the chemical potential μ2 0.15 eV in Eq. (3). Here we have chosen η 1 in Eq. (3); however, we have verified that the qualitative features of the reported results do not depend on η, at least for η ≤ 5. Obviously for large enough s the two supermodes of the coupler tend to degeneracy and their propagation constant goes into the propagation constant of the SPP of the single graphene layer, with an effective wavelength roughly 318 (154) times smaller than the vacuum wavelength and a plasmon propagation length of roughly 15 nm for a chemical potential μ 0.1 eV (μ 0.15 eV). The main message we can read from Fig. 2 is that a very small change in the chemical potential (i.e., a very small change of an electrical signal applied to the graphene layers) can induce a very big change in the behavior of the system. 500
even
400
neff
Fig. 1. Schematic of the graphene directional coupler: the separation between the layers is equal to 2s.
iωϵ2 ; Γ2
300
even
odd 200
100
odd 5
10
15
20
s [nm]
Fig. 2. Effective index neff Reβ∕k0 of even and odd supermodes of the coupled graphene layers as a function of the separation among the layers. Continuous (dashed) lines refer to a chemical potential of μ1 0.1 eV (μ2 0.15 eV).
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OPTICS LETTERS / Vol. 38, No. 20 / October 15, 2013 120 110 100
Lb [nm]
90 80 70 60 50 40 0.05
0.1
0.15
0.2
0.25
µ [eV]
0.3
0.35
0.4
Fig. 3. Beat length versus chemical potential for a graphene plasmon coupler. Here 2s 10 nm.
In the following we focus our attention at s 5 nm. For this value of the separation between the layers we have computed the beat length of the directional coupler, i.e., the propagation distance needed to observe complete switching from one layer to the other; as is well known, the beat length can be computed from the real part of the propagation constants of the two supermodes of the directional coupler as follows: LB
π Reβeven − Reβodd
5
where βeven (βodd ) are the propagation constants of the even (odd) supermodes. A summary of the obtained results is reported in Fig. 3, where the beat length has been plotted as a function of the chemical potential. We can clearly see there that a very small change of the chemical potential can be used to induce huge changes of the beat length of the coupler. The two particular points (open square and open circle) enlightened in Fig. 3 are the initial conditions in Fig. 4, where we describe the propagation of the electromagnetic signal in the graphene coupler. We note here that the results reported in Fig. 3 are not strongly dependent on the value of η in Eq. (3). We have in fact verified that changing η from 1 (corresponding to the theoretical expected loss level) to 5 (corresponding to the experimental data reported in [21,22, 24]) does not translate into big changes in Fig. 3. This very same observation applies also to the results reported in the rest of the Letter and in particular on the switching behavior described in Fig. 4. In both panels in Fig. 4 total propagation length is set to L ≃ 90 nm. On the upper panel in Fig. 4 the input condition corresponds to the square in Fig. 3 and the coupler is in the cross state; on the lower panel the input condition corresponds to the circle in Fig. 3 and the coupler is in the bar state. We also note here that the results reported in Fig. 3 and in Fig. 4 have been obtained from the analytical approach we have described in this Letter, that is, by solving the equation detM 0, with M as given in Eq. (4); however, using a finite-element-based mode solver [25], we have carefully verified the soundness of our approach by a thorough comparison with numerical results.
Fig. 4. Field evolution in a graphene plasmon directional coupler: upper (lower) panel refers to a chemical potential of μ1 0.1367 eV (μ2 0.0908 eV). Here 2s 10 nm.
In conclusion, we have described the tuning of the coupling of SPPs between two spatially separated graphene layers. We were able to prove that by slightly changing the chemical potential, a graphene coupler can switch from the bar to the cross state; as a consequence, the coupling coefficient in such structures can be easily controlled by means of an applied electrical signal, thus opening interesting opportunities for signal processing. A. A., C. D. A., and A. L. acknowledge financial support from US ARMY under grant no. W911NF-12-1-0202. References 1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 2. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010). 3. A. Vakil and N. Engheta, Science 332, 1291 (2011). 4. M. Liu, X. Yin, and X. Zhang, Nano Lett. 12, 1482 (2012). 5. M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, and A.-D. Capobianco, Opt. Express 20, 23144 (2012). 6. A. Locatelli, A.-D. Capobianco, M. Midrio, S. Boscolo, and C. de Angelis, Opt. Express 20, 28479 (2012). 7. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, Nat. Photonics 5, 411 (2011). 8. J. T. Kim and C. G. Choi, Opt. Express 20, 3556 (2012). 9. R. Alaee, C. Menzel, C. Rockstuhl, and F. Lederer, Opt. Express 20, 18370 (2012). 10. K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, Nature 490, 192 (2012). 11. Q. Bao and K. P. Loh, ACS Nano 6, 3677 (2012). 12. S. A. Mikhailov and K. Ziegler, Phys. Rev. Lett. 99, 016803 (2007).
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS 13. M. Jablan, H. Buljan, and M. Soljacic, Phys. Rev. B 80, 245435 (2009). 14. H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, Opt. Lett. 37, 1856 (2012). 15. A. Auditore, C. De Angelis, A. Locatelli, S. Boscolo, M. Midrio, M. Romagnoli, A.-D. Capobianco, and G. Nalesso, Opt. Lett. 38, 631 (2013). 16. A. V. Gorbach, Phys. Rev. A 87, 013830 (2013). 17. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, Nat. Nanotechnol. 6, 630 (2011). 18. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, Nature 487, 77 (2012).
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19. B. Wang, X. Zhang, X. Yuan, and J. Teng, Appl. Phys. Lett. 100, 131111 (2012). 20. T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008). 21. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, Nature 487, 82 (2012). 22. H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, Nat. Photonics 7, 394 (2013). 23. S. A. Mikhailov and K. Ziegler, J. Phys. Condens. Matter 20, 384204 (2008). 24. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. Garcia de Abajo, ACS Nano 6, 431 (2012). 25. COMSOL Multiphysics 4.2, http://www.comsol.com/.
OPTICS LETTERS / Vol. 38, No. 20 / October 15, 2013
Tuning of surface plasmon polaritons beat length in graphene directional couplers Aldo Auditore,1 Costantino de Angelis,1,* Andrea Locatelli,1 and Alejandro B. Aceves2 1
Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Brescia, Brescia 25123, Italy 2 Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, USA *Corresponding author: [email protected] Received July 22, 2013; revised September 6, 2013; accepted September 6, 2013; posted September 9, 2013 (Doc. ID 194331); published October 14, 2013
We investigate the tuning of the coupling of surface plasmon polaritons between two spatially separated graphene layers. We demonstrate that by slightly changing the chemical potential, a graphene coupler can switch from the bar to the cross state; as a consequence, the coupling coefficient in such structures can be easily controlled by means of an applied electrical signal. © 2013 Optical Society of America OCIS codes: (130.1750) Components; (130.2790) Guided waves; (130.3120) Integrated optics devices. http://dx.doi.org/10.1364/OL.38.004228
After its experimental isolation from bulk graphite [1], graphene has attracted increasing interest from the scientific community. Indeed, this single layer of carbon atoms packed into a honeycomb lattice reveals amazing and very useful properties that have already inspired a huge variety of devices embracing different areas ranging from electronics to optics [2]. As research continues to progress, more graphene-based devices are theoretically conceived and experimentally demonstrated [3–10]. In this framework a very interesting role is now played by the overlapping between two originally separated research areas: plasmonics and graphene photonics. The merging of these two areas has become so important as to create a new research line, graphene plasmonics [11]; one of the main assets graphene has to offer in this framework is the tunability of its electromagnetic properties. As is well known, graphene can sustain surface plasmon polaritons (SPPs) having unique properties as compared to what we are used to with noble metals. In fact a single layer of graphene can support either TE or TM polarized plasmons without suffering from huge loss [12,13]; moreover, as far as TM polarization is concerned, the extremely high confinement factor is particularly favorable to exploring the huge χ 3 nonlinearity of graphene [14–16]. Experimental endeavors have demonstrated evidences of the existence of graphene plasmons by measuring the plasmon resonance of graphene nanoribbon arrays [17], and by acquiring their near-field images [18]. The coupling of SPP between separated graphene layers has been recently analyzed in [19]; however, the very interesting properties arising from the easily tunable optical properties of graphene have not been exploited yet in this framework. Here we show that by slightly changing the chemical potential, a graphene coupler can switch from the bar to the cross state; as a consequence, the output state in such structures can be easily controlled by means of an applied electrical signal. In Fig. 1 we report the basic geometry we will consider in this Letter; two graphene layers are embedded in a dielectric structure: regions 2 and 3 are the cladding and substrate, while region 1 (of width 2s) is the dielectric between the two graphene layers. At the graphene 0146-9592/13/204228-04$15.00/0
boundary we set the following conditions on the tangential components of the electromagnetic field: E⃗ 2;3 − E⃗ 1 × xˆ 0 H⃗ 2;3 − H⃗ 1 × xˆ iωϵ0 ϵrS1–2;3 E⃗ ∥ x s;
(1)
where E⃗ ∥ is the electric field tangent to the graphene layer and ϵrS1−2 (ϵrS1–3 ) is the relative surface permittivity of the graphene layer between regions 1 and 2 (3). As far as the electromagnetic constants of graphene are concerned, we write the linear contribution to the relative complex permittivity as [3] ϵrC
σ 1 σ 1 ϵrS 1 Σ;I − i Σ;R ϵrC;R iϵrC;I ; dg dg ωϵ0 dg ωϵ0
(2)
where dg is the graphene thickness and the surface com1 1 plex conductivity σ 1 Σ σ Σ;R iσ Σ;I (in siemens) is obtained from theoretical models now well established and experimentally validated [20], which give the following dependence of the real and imaginary parts of the conductivity on frequency (ω), temperature (T), and chemical potential (μ): σ0 ℏω 2μ ℏω − 2μ tanh tanh ≃η 2 4kB T 4kB T 3 σ0 4 2μ ℏω 2μ μ − ; ω ≃ − log σ 1 Σ;I π ℏω ℏω − 2μ 9t2
σ 1 Σ;R ω
(3)
where t 2.7 eV is the hopping parameter, ℏ and kB are the reduced Planck’s and Boltzmann’s constants, respectively, σ 0 e2 ∕4ℏ ≃ 6.0853 · 10−5 S with e the electron charge, and η ≥ 1 is a parameter we use here to increase the loss level to better match with recently reported experimental data. Although the physical origin of these extra losses is not yet completely understood (they might originate from enhanced relaxation times at infrared frequency or they could be associated with extrinsic factors such as surface irregularities), they have been measured in recent experiments and data are thus available [21,22]. © 2013 Optical Society of America
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS
4229
where g1–2 and g1–3 take into account the contribution of the two graphene layers in the continuity conditions g1–2 iωϵ0 ϵrS;1–2
Note that when graphene is modeled as a brick with a finite thickness (dg 0.34 nm), the results reported in Eq. (3) (where we have conductivities with the dimensions of siemens) need to be normalized as σ → σ∕dg to get conductivities in siemens/m. Note also that this model can be easily extended into the nonlinear regime by adding a nonlinear correction to the surface conduc3 ⃗ 2 tivity as σ Σ σ 1 Σ σ Σ jEj E y;z [23]. Moreover, thanks to the extremely small thickness of the graphene layer, nonlinearity can be analyzed by a parameter embedded into the coefficients describing the continuity of the tangential components of the electromagnetic field [15,16]. However, this issue is well outside the scope of our Letter and deserves to be tackled in a different work suited to address the influence of graphene nonlinearities. To describe SPP propagation along z, we first note that, at first order, the y dependence of the electromagnetic field can be neglected; we then look for guided modes with harmonic temporal dependence expiωt and spatial variation E⃗ 1;2;3 x; z, H⃗ 1;2;3 x; z ∼ exp−iβz Γ1;2;3 x with Γ21;2;3 β2 − ϵr1;2;3 k20 . Obviously the complex wavenumber β, through its real and imaginary parts, describes the evolution of both the phase and the amplitude of the guided modes. We can obviously apply the above modeling to derive the dispersion relation of both TE and TM modes. In the following we describe in detail the TM polarization, which allows for a stronger confinement of the electromagnetic field. We first consider a very general situation where, either using different electrical contacts on the two graphene layers or using a nonsymmetric cladding (ϵ2 ≠ ϵ3 ), the two graphene sheets can be biased in a different way to give rise to an asymmetric coupler (i.e., a coupler where the two guiding layers have different electromagnetic properties). After straightforward algebra we find that coupled SPPs in the system are determined by setting to zero the determinant of the following matrix: 2 6 6 6 M6 6 4
Γ1 s
−Γ1 s
−Γ2 s
−e
e
e
e−Γ1 s
eΓ1 s
0
iωϵ1 Γ1 s Γ1 e
1 −Γ1 s − iωϵ Γ1 e
g1–2 e−Γ2 s
1 −Γ1 s − iωϵ Γ1 e
iωϵ1 Γ1 s Γ1 e
0
0
3
7 −eΓ3 s 7 7 7; (4) 0 7 5 g1–3 e−Γ3 s
g1–3 iωϵ0 ϵrS;1–3
iωϵ3 ; Γ3
and ϵrS;1–2 and ϵrS;1–3 refer to the relative dielectric constant of the two graphene layers, which can in general get different values due to different carriers concentrations. This asymmetric coupler offers a wide variety of possible settings which certainly deserve to be investigated both in the linear and nonlinear regimes. Here we want to give a first prototype example into the possibilities offered by the tunability of the graphene parameters in this framework; we thus focus our attention on a very particular situation corresponding to a linear and symmetric case (ϵ2 ϵ3 and ϵrS;1–2 ϵrS;1–3 ); moreover we use T 300 K, λ 10 μm, and for the sake of simplicity we also set ϵr1 ϵr2 ϵr3 2.25. In this regime the graphene directional coupler has two different eigenstates: the even (odd) supermode corresponding to the out-ofphase (in-phase) hybridization of the SPP guided by the single graphene layers. Note also that the even mode here has always the higher value of the propagation constant. In Fig. 2 we report the solution of the dispersion relation as a function of s for two different situations: continuous lines here refer to the even and odd supermodes corresponding to a chemical potential μ1 0.1 eV in Eq. (3), while the dashed lines refer to a choice of the chemical potential μ2 0.15 eV in Eq. (3). Here we have chosen η 1 in Eq. (3); however, we have verified that the qualitative features of the reported results do not depend on η, at least for η ≤ 5. Obviously for large enough s the two supermodes of the coupler tend to degeneracy and their propagation constant goes into the propagation constant of the SPP of the single graphene layer, with an effective wavelength roughly 318 (154) times smaller than the vacuum wavelength and a plasmon propagation length of roughly 15 nm for a chemical potential μ 0.1 eV (μ 0.15 eV). The main message we can read from Fig. 2 is that a very small change in the chemical potential (i.e., a very small change of an electrical signal applied to the graphene layers) can induce a very big change in the behavior of the system. 500
even
400
neff
Fig. 1. Schematic of the graphene directional coupler: the separation between the layers is equal to 2s.
iωϵ2 ; Γ2
300
even
odd 200
100
odd 5
10
15
20
s [nm]
Fig. 2. Effective index neff Reβ∕k0 of even and odd supermodes of the coupled graphene layers as a function of the separation among the layers. Continuous (dashed) lines refer to a chemical potential of μ1 0.1 eV (μ2 0.15 eV).
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OPTICS LETTERS / Vol. 38, No. 20 / October 15, 2013 120 110 100
Lb [nm]
90 80 70 60 50 40 0.05
0.1
0.15
0.2
0.25
µ [eV]
0.3
0.35
0.4
Fig. 3. Beat length versus chemical potential for a graphene plasmon coupler. Here 2s 10 nm.
In the following we focus our attention at s 5 nm. For this value of the separation between the layers we have computed the beat length of the directional coupler, i.e., the propagation distance needed to observe complete switching from one layer to the other; as is well known, the beat length can be computed from the real part of the propagation constants of the two supermodes of the directional coupler as follows: LB
π Reβeven − Reβodd
5
where βeven (βodd ) are the propagation constants of the even (odd) supermodes. A summary of the obtained results is reported in Fig. 3, where the beat length has been plotted as a function of the chemical potential. We can clearly see there that a very small change of the chemical potential can be used to induce huge changes of the beat length of the coupler. The two particular points (open square and open circle) enlightened in Fig. 3 are the initial conditions in Fig. 4, where we describe the propagation of the electromagnetic signal in the graphene coupler. We note here that the results reported in Fig. 3 are not strongly dependent on the value of η in Eq. (3). We have in fact verified that changing η from 1 (corresponding to the theoretical expected loss level) to 5 (corresponding to the experimental data reported in [21,22, 24]) does not translate into big changes in Fig. 3. This very same observation applies also to the results reported in the rest of the Letter and in particular on the switching behavior described in Fig. 4. In both panels in Fig. 4 total propagation length is set to L ≃ 90 nm. On the upper panel in Fig. 4 the input condition corresponds to the square in Fig. 3 and the coupler is in the cross state; on the lower panel the input condition corresponds to the circle in Fig. 3 and the coupler is in the bar state. We also note here that the results reported in Fig. 3 and in Fig. 4 have been obtained from the analytical approach we have described in this Letter, that is, by solving the equation detM 0, with M as given in Eq. (4); however, using a finite-element-based mode solver [25], we have carefully verified the soundness of our approach by a thorough comparison with numerical results.
Fig. 4. Field evolution in a graphene plasmon directional coupler: upper (lower) panel refers to a chemical potential of μ1 0.1367 eV (μ2 0.0908 eV). Here 2s 10 nm.
In conclusion, we have described the tuning of the coupling of SPPs between two spatially separated graphene layers. We were able to prove that by slightly changing the chemical potential, a graphene coupler can switch from the bar to the cross state; as a consequence, the coupling coefficient in such structures can be easily controlled by means of an applied electrical signal, thus opening interesting opportunities for signal processing. A. A., C. D. A., and A. L. acknowledge financial support from US ARMY under grant no. W911NF-12-1-0202. References 1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 2. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010). 3. A. Vakil and N. Engheta, Science 332, 1291 (2011). 4. M. Liu, X. Yin, and X. Zhang, Nano Lett. 12, 1482 (2012). 5. M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, and A.-D. Capobianco, Opt. Express 20, 23144 (2012). 6. A. Locatelli, A.-D. Capobianco, M. Midrio, S. Boscolo, and C. de Angelis, Opt. Express 20, 28479 (2012). 7. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, Nat. Photonics 5, 411 (2011). 8. J. T. Kim and C. G. Choi, Opt. Express 20, 3556 (2012). 9. R. Alaee, C. Menzel, C. Rockstuhl, and F. Lederer, Opt. Express 20, 18370 (2012). 10. K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, Nature 490, 192 (2012). 11. Q. Bao and K. P. Loh, ACS Nano 6, 3677 (2012). 12. S. A. Mikhailov and K. Ziegler, Phys. Rev. Lett. 99, 016803 (2007).
October 15, 2013 / Vol. 38, No. 20 / OPTICS LETTERS 13. M. Jablan, H. Buljan, and M. Soljacic, Phys. Rev. B 80, 245435 (2009). 14. H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, Opt. Lett. 37, 1856 (2012). 15. A. Auditore, C. De Angelis, A. Locatelli, S. Boscolo, M. Midrio, M. Romagnoli, A.-D. Capobianco, and G. Nalesso, Opt. Lett. 38, 631 (2013). 16. A. V. Gorbach, Phys. Rev. A 87, 013830 (2013). 17. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, Nat. Nanotechnol. 6, 630 (2011). 18. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, Nature 487, 77 (2012).
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19. B. Wang, X. Zhang, X. Yuan, and J. Teng, Appl. Phys. Lett. 100, 131111 (2012). 20. T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78, 085432 (2008). 21. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, Nature 487, 82 (2012). 22. H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, Nat. Photonics 7, 394 (2013). 23. S. A. Mikhailov and K. Ziegler, J. Phys. Condens. Matter 20, 384204 (2008). 24. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. Garcia de Abajo, ACS Nano 6, 431 (2012). 25. COMSOL Multiphysics 4.2, http://www.comsol.com/.
