# EME: Grating Coupler A shallow-etch silicon grating coupler: how much of the guided light it sends up toward a fiber, how much is lost to the substrate, and the fiber angle that couples best. The grating is uniform across its width, so it is solved as a slab, with a window two cells wide and periodic in x. A PML above and below absorbs the light the grating radiates once it has left the window. That light is carried by the lossy modes of each cross-section, so the basis needs many of them (100 here). The junctions between the etched and unetched cross-sections conserve power at any number of modes, so what is transmitted, reflected and radiated adds up to the power launched. A monitor, a plane inside the circuit, samples the field 1 um above the silicon. `monitor_coupling()` gives its overlap with a tilted Gaussian beam (a 10.4 um fiber mode here), and a `sweep()` over the beam’s angle reuses the solved field, so it costs no further solve. By reciprocity the same number is the coupling from the fiber into the waveguide. *This code example is licensed under the* [BSD 3-Clause License](../_static/LICENSE.html). Python ```python import sys import numpy as np import emodeconnection as emc from matplotlib import pyplot as plt ## Set simulation parameters wavelength = 1550 # [nm] wavelength dx, dy = 20, 20 # [nm] resolution h_core = 220 # [nm] silicon thickness etch = 70 # [nm] grating etch depth period = 630 # [nm] grating period fill = 0.5 # [-] unetched fraction of each period num_periods = 20 # [-] h_box = 2000 # [nm] buried oxide h_air = 2000 # [nm] air above the silicon lead = 2000 # [nm] unetched waveguide before and after the grating mfd = 10.4e3 # [nm] fiber mode field diameter num_modes = 100 # [-] enough for the radiated light, see below ## The grating is uniform along x, so it is solved as a slab: a window two ## cells wide and periodic in x. A PML above and below absorbs the light the ## grating radiates once it has left the window, so the lossy modes in the ## basis carry the radiation. em = emc.EMode(emode_cmd=sys.argv[1:], simulation_name='grating_coupler') em.settings( wavelength=wavelength, x_resolution=dx, y_resolution=dy, window_width=2 * dx, window_height=h_box + h_core + h_air, num_modes=num_modes, boundary_condition='0P', pml_NSEW_bool=[1, 1, 0, 0], num_pml_layers=10, background_material='Air') em.shape(name='BOX', material='SiO2', height=h_box) em.shape(name='core', material='Si', height=h_core) em.label_profile(name='unetched') em.shape(name='core', material='Si', height=h_core - etch) em.label_profile(name='etched') ## Draw EME sections: a lead, the periods, and a lead out em.straight_section(name='lead_in', profile='unetched', length=lead) em.straight_section(name='gap', profile='etched', length=period * (1 - fill)) em.straight_section(name='tooth', profile='unetched', length=period * fill) for _ in range(num_periods - 1): em.copy_section(section_name='gap') em.copy_section(section_name='tooth') em.straight_section(name='lead_out', profile='unetched', length=lead) em.EME() ## A monitor 1 um above the silicon, centered on the grating z_center = lead + num_periods * period / 2 em.monitor(name='fiber_plane', w=z_center, y=h_box + h_core + 1000) ## Where the power goes, for 1 W in the fundamental mode S = em.get('S_matrix') T = np.abs(S.transmission(0, 1)[0, 0]) ** 2 R = np.abs(S.reflection(0)[0, 0]) ** 2 up = em.monitor_field('fiber_plane').power_through_plane print(f'transmitted: {T * 100:.1f} %') print(f'reflected: {R * 100:.1f} %') print(f'radiated up through the monitor: {up * 100:.1f} %') print(f'the rest, mostly into the substrate: {(1 - T - R - up) * 100:.1f} %') ## Coupling to a fiber at each tilt: the solved field is reused, only the ## overlap with the tilted Gaussian is evaluated again angles = np.arange(0, 25.1, 1) # [deg] data = em.sweep(key='angle', values=angles, result='monitor_coupling', monitor='fiber_plane', beam=dict(mode_field_diameter=mfd)) coupling = np.asarray(data['monitor_coupling']) best = angles[np.argmax(coupling)] print(f'best fiber angle: {best:.0f} deg, coupling {10 * np.log10(coupling.max()):.2f} dB') plt.figure(figsize=(6, 4)) plt.plot(angles, 10 * np.log10(coupling), marker='o', ms=3) plt.xlabel('Fiber angle (degrees)') plt.ylabel('Coupling to the fiber (dB)') plt.grid(True, alpha=0.3) plt.savefig('grating_coupling_vs_angle.png', dpi=300, bbox_inches='tight') ## The field along the grating: the guided mode decays as it radiates, up ## toward the fiber and down into the substrate, at the grating's angle em.plot(component='Ex', plane='z-y', plot_function='real') em.close() ``` Console output: ```default EMode3D 1.0.8 - email Solving S-matrices... Solving section: lead_in... completed in 0.0 sec Solving section: gap... completed in 0.0 sec Solving section: tooth... completed in 0.0 sec Solving section: 4... completed in 0.0 sec Solving section: 5... completed in 0.0 sec Solving section: 6... completed in 0.0 sec Solving section: 7... completed in 0.0 sec Solving section: 8... completed in 0.0 sec Solving section: 9... completed in 0.0 sec Solving section: 10... completed in 0.0 sec Solving section: 11... completed in 0.0 sec Solving section: 12... completed in 0.0 sec Solving section: 13... completed in 0.0 sec Solving section: 14... completed in 0.0 sec Solving section: 15... completed in 0.0 sec Solving section: 16... completed in 0.0 sec Solving section: 17... completed in 0.0 sec Solving section: 18... completed in 0.0 sec Solving section: 19... completed in 0.0 sec Solving section: 20... completed in 0.0 sec Solving section: 21... completed in 0.0 sec Solving section: 22... completed in 0.0 sec Solving section: 23... completed in 0.0 sec Solving section: 24... completed in 0.0 sec Solving section: 25... completed in 0.0 sec Solving section: 26... completed in 0.0 sec Solving section: 27... completed in 0.0 sec Solving section: 28... completed in 0.0 sec Solving section: 29... completed in 0.0 sec Solving section: 30... completed in 0.0 sec Solving section: 31... completed in 0.0 sec Solving section: 32... completed in 0.0 sec Solving section: 33... completed in 0.0 sec Solving section: 34... completed in 0.0 sec Solving section: 35... completed in 0.0 sec Solving section: 36... completed in 0.0 sec Solving section: 37... completed in 0.0 sec Solving section: 38... completed in 0.0 sec Solving section: 39... completed in 0.0 sec Solving section: 40... completed in 0.0 sec Solving section: 41... completed in 0.0 sec Solving section: lead_out... completed in 0.0 sec completed in 0.6 sec transmitted: 5.4 % reflected: 0.4 % radiated up through the monitor: 36.2 % the rest, mostly into the substrate: 58.1 % Sweeping the beam's angle on monitor 'fiber_plane'... completed in 26.7 sec best fiber angle: 12 deg, coupling -5.93 dB Exited EMode ``` Figures: ![image](examples/Figures/EME_Grating_Coupler/grating_coupling_vs_angle.png)![image](examples/Figures/EME_Grating_Coupler/grating_Ex.png)