This example demonstrates how to use the DGTD to compute the diffraction efficiency and deflection angles of a periodic grating consisting of an array of sub-wavelength silicon beams on top of a glass substrate. As explained in the referenced article, this type of grating is used to steer a beam of light by carefully controlling the separation and width of the silicon beams. To get the phase delay required to steer light by a given deflection angle, the grating is operated at a wavelength in which the silicon beams exhibit strong dispersion and introduce noticeable losses. For this reason, the diffraction efficiency is an important figure of merit in addition to the achieved deflection angle.
Lumerical products R2018a or newer
Since the grating is long and uniform along the length of the silicon beams, a 2D simulation is sufficient to provide a complete characterization of the grating. As shown in the above figure, silicon beams (or nanobeams) of two different widths (30 and 55nm) are employed. All the nanobeams have the same height (75nm) and are arranged in pairs separated by a small gap (85nm). The grating period (380nm) was chosen so that a light source incident from above at the proposed operating wavelength (520nm) will produce three propagating grating orders into the substrate (refractive index of 1.78635). The grating was designed so that the first grating order carries most of the transmitted power and has a deflection angle of fifty degrees for a source at normal incidence. The transmitted power is very sensitive to the polarization of the incoming light. If the electric field is linearly polarized along the length of the nanobeams, most of the incoming power will be transmitted, and, if the electric field is linearly polarized in the perpendicular direction most of the incoming power will be reflected back. The project file nanobeams_grating.ldev contains a 2D DGTD simulation of the grating with the appropriate boundary conditions (Bloch and PML) and a plane wave source linearly polarized along the length of the nanobeams. This particular setup is called a transmission mode metasurface in the referenced paper. Two frequency domain monitors (named T and R) record the transmitted and reflected fields. The script file nanobeams_grating.lsf sweeps the angle of incidence of the source and computes the relative power into each of the transmitted and reflected grating orders. When run, the script generates the results shown below.
The first figure shows the relative power into each transmitted and reflected grating order. As expected, there are three transmitted grating orders propagating into the substrate and most of the power is carried by the first one (N = 1). The relative power into each grating order is calculated by calling the gratingprojection command with the fields recorded by each of the T and R monitors. In addition to the monitor fields, the gratingprojection command requires the period of the grating and the wave vector of the source, these are retrieved by using the getperiodicity and getsourcedirection commands. The second figure shows the deflection angle (relative to the normal incidence direction of the source) corresponding to each of the transmitted and reflected grating orders. As expected, the first transmitted order (N = 1) provides a deflection of 50 degrees for a source at normal incidence. Since the deflection angles depend only on the period of the grating, the wave vector of the source and the refractive index of the substrate and source media, the deflection angles can be calculated by using the gratingorders command without performing a full grating projection.
Relative power into each transmitted and reflected grating order.
Deflection angle for each transmitted and reflected grating order.
The above results are in reasonable agreement with the simulation results presented in the referenced paper, however, to get a precise match, it is necessary to know the exact refractive index values used for the silicon nanobeams. Since this information is not provided, representative values for amorphous silicon are employed here. The bottom two figures compare the obtained grating efficiency (relative power into the first grating order) and deflection angle with the simulation results presented in the referenced paper. To compute the relative power into each propagating grating order, the DGTD grating projection commands are used. Since the deflection angle is independent of the refractive index of the silicon nanobeams, it was possible to get an excellent match. However, to get a precise match for the diffraction efficiency, it is necessary to refine the refractive index data for the silicon nanobeams.
Diffraction efficiency comparison for the first transmission order.
Deflection angle comparison for the first transmission order.
Note: Broadband sweep
The simulations in this example are performed for a single wavelength. Although the DGTD solver is capable of simulating a broad wavelength range with a single simulation, the results will be valid for a broad wavelength range only in simulations where the source injects light at normal incidence. If the source injects light at an angle, the application of Bloch boundary conditions limits the validity of the results to the center wavelength. Therefore, to obtain accurate results for multiple wavelengths, one simulation for each wavelength is required.
Lin et al,. "Optical metasurfaces for high angle steering at visible wavelengths", Scientific Reports, 7, 2286 (2017)
Grating projection toolbox, Mie Scattering (DGTD), Chromatic Polarizer, Photothermal Heating in Plasmonic Nanostructures, getperiodicity, getsourcedirection, gratingorders, gratingprojection, Diffraction grating