Graphene has a Kerr nonlinear coefficient which is several order of magnitudes larger than that of some common semiconductors such as Si and GaAs. Due to the large Kerr nonlinearity, graphene supports propagation of quasi one-dimensional spatial solitons. This page demonstrates how to simulate spatial soliton propagation in a dielectric waveguide with an embedded graphene sheet using the volumetric permittivity approach described in Modeling methodology.
In this example, we consider a dielectric slab waveguide with two 300nm thick dielectric plated sandwiching a graphene sheet as shown below. Light can be confined in the waveguide as a vertical slab mode and by self-focusing effect of the Kerr effect horizontally.
To see the self-focusing effect of the propagating wave, we prepare an input beam which has a slab mode profile Eslab_x(y) vertically and a sech(x) profile horizontally. To create the input beam profile, we perform three steps below.
- Using MODE, we calculate the slab mode Eslab_x(y)
- Extract the mode profile Esech_x(y) and then multiply it by sech(x) function.
- Import the profile Einput(x,y)=Eslab_x(y)*sech(x) into an import source object in the FDTD
To take the steps above, open the MODE simulation file graphene_soliton_mode_calculation.lms and run graphene_soliton_mode_calculation.lsf. It will create a text file which stores the slab mode profile Eslab_x(y) is created. The file name is "mode_profile.txt". After creating the file, run graphene_soliton_source.lsf to create the final input beam profile Enput(x,y). The file name is "sourcefile.fld", which can be imported into the source object as described in the source import page. In the simulation file graphene_soliton_propagation.fsp, the input beam profile is already imported.
The figures below show the propagation of light in the waveguide, for weak and strong electric field intensity, recorded by the power monitor "graphene_XZplane", which is located in the middle of the graphene layer. The field intensity is adjusted by modifying the amplitude in the source settings (General tab); for the weak and strong field examples below we use amplitudes of 3.5e5 and 3.5e7, respectively. We can confirm that the beam diverges for weak electric field,
while it keeps almost same profile for strong electric field due to the self-focusing effect by a large Kerr nonlinearity.
Note: Scaling of χ(3)
The current 'Kerr nonlinear' material model uses simple algorithm to incorporate the nonlinear effect P=ε0(χ(1)+χ(3)E2)E. The implementation is primarily designed for, and stable when the condition χ(1)>>χ(3)E2 is met. Using a thin layer of graphene (e.g. 1nm) requires a very large χ(3)E2 to create the spatial soliton because only a small fraction of the modal field is confined in the graphene layer.To make the simulation stable, we use a thicker layer (20nm), which allows us to scale down the size of χ(3)E2 for the same self-focussing effect. In principle, a more sophisticated algorithm (possibly created with a user defined material plugin) could solve this problem without the need to use such a thick layer.
- M.L. Nesterov, J. Bravo-Abad, A.Y. Nikitin, F.J. Garcia-Vidal and L. Martin-Moreno, "Graphene supports the propagation of subwavelength optical solitons," Laser Photonics Rev. Vol. 7 (2), L7–L11 (2013).