Efficient Modeling Of Graphene Based Optical Devices

In the last years the unique optical properties of graphene have attracted the attention of the scientific community, and novel graphene-based photonic, plasmonic and optoelectronic devices have been proposed for a plethora of applications [1, 2]. In this framework, numerical and analytical modelling play a crucial role both for design and analysis purposes. As a matter of fact, well-established models for the complex bi-dimensional linear conductivity of graphene have been reported in the literature (see e.g. [3, 4]), and a strong third-order optical nonlinear response has also been predicted both theoretically and experimentally [5, 6]. Nevertheless the analysis of graphene-based optical devices remains quite challenging from the numerical point of view. Indeed, it was demonstrated that graphene layers can beaccurately modeled in conventional full-wave solvers by treating them as volumetric media with knownconductivity and proper atomic thickness (< 1 nm) [7, 8], but the required fine discretization of theseultra-thin layers results in a huge computation burden.In order to overcome this limitation, a different and more efficient approach to this kind of problem hasrecently been proposed. In fact, it was demonstrated that all the effects induced by the presence ofgraphene layers embedded in dielectric media can be modeled by discontinuities of the magnetic fieldwhich take into account the surface currents flowing on the graphene layers. In this way, the wholeanalysis is greatly simplified and a more relaxed discretization step can be used. By applying thistechnique, amplitude equations for surface plasmons in graphene have been derived [9], and thepeculiar properties of directional couplers composed of a pair of graphene layers have been thoroughlystudied, both in the linear [10] and in the nonlinear regime [11]. In particular, in Ref. [10] we havecalculated the dispersion relations of the supermodes of a symmetric graphene plasmonic coupler byillustrating a procedure which allows to treat the more general case of asymmetric structures.Graphene layers can also be sandwiched within conventional slab waveguides in order to electricallytune the optical properties of these structures. In this context, we have demonstrated that also thenonlinear phase shift which is induced by the strong third-order nonlinearity of graphene can beincorporated into a boundary condition for the tangential magnetic field, and we discussed the existenceof nonlinear modes sustained by graphene layers in dielectric waveguides [12]. Moreover, we haveshown that the beat length of dielectric couplers can be controlled by inserting graphene layers in themiddle of those structures and then tuning the bias voltage in order to vary the dielectric constant ofgraphene, thus shifting only the effective index of the even supermode [13].Last, but not least, we have recently proposed to exploit the idea of modelling the graphene as a purelybi-dimensional sheet which imposes a boundary condition on the magnetic field to realize a novel ultra-fast field propagator tailored for graphene-based devices [14]. The algorithm is derived from the wellknown Beam Propagation Method (BPM), which has been widely used in the last decades for theanalysis of wave propagation in photonic devices. The key point of the method is the finite-differenceformulation of the second-order derivative, which allows to discretize the discontinuous magnetic fieldby including in the diffractive operator all the effects which stem from the presence of the graphenelayers, thus avoiding to resort to sub-nanometer sampling steps. The novel BPM has been validatedfirst by demonstrating the undistorted propagation of the even and the odd supermodes of the graphenecoupler described in [10, 11], as it is possible to verify in Fig. 1. Then, a single waveguide has beenexcited and the field evolution along the coupler has been evaluated by propagating the input field withthe reformulated BPM technique. In Fig. 2 we demonstrate the high tunability of this kind of structure byreporting results obtained by slightly varying the chemical potential of the two layers. In Fig. 3 asystematic comparison between the beat length calculated by using the BPM and analytical resultsobtained from the solution of the dispersion relations is depicted. The excellent agreement betweennumerical and analytical results constitutes a strong validation of the proposed technique.The results that we illustrate are fundamental to show that the novel BPM algorithm allows ultrafast andaccurate analysis of complex photonic devices wherein graphene layers are introduced in order toexploit the high tunability of their optical parameters. These findings open the way to the realization of abrand new class of field propagators specifically tailored for the analysis of graphene-based structures. References[1] Q. Bao, and K. P. Loh, ACS Nano, 6 (2012) 3677.[2] F. Xia, H. Yan, and P. Avouris, Proc. IEEE, 101 (2013) 1717.[3] G. W. Hanson, IEEE Trans. Antennas Propagat., 56 (2008) 747.[4] T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B, 78 (2008) 085432.[5] S. A. Mikhailov, and K. Ziegler, Journ. Phys.: Condens. Matter, 20 (2008) 384204.[6] E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, Phys. Rev. Lett., 105 (2010)097401. [7] M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, and A.-D.Capobianco, Opt. Express, 20 (2012) 23144.[8] A. Locatelli, A.-D. Capobianco, M. Midrio, S. Boscolo, and C. De Angelis, Opt. Express, 20 (2012) 28479.[9] A. V. Gorbach, Phys. Rev. A, 87 (2013) 013830.[10] A. Auditore, C. De Angelis, A. Locatelli, and A. Aceves, Opt. Lett., 38 (2013) 4228.[11] D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. B, 88(2013) 045443. [12] A. Auditore, C. De Angelis, A. Locatelli, S. Boscolo, M. Midrio, M. Romagnoli, A.-D. Capobianco,and G. Nalesso, Opt. Lett., 38 (2013) 631.[13] A. Locatelli, A.-D. Capobianco, G. Nalesso, S. Boscolo, M. Midrio, and C. De Angelis, Opt.Commun., 318 (2014) 175. [14] A.-D. Capobianco, A. Locatelli, S. Boscolo, C. De Angelis, and M. Midrio, submitted to IEEEPhoton. Technol. Lett., (2014).