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J. Eur. Opt. Society-Rapid Publ. 21, 34( 2025)
Fig. 3.( a) Central wavelength variation with the grating period.( b) Transmission curves at varying etching depth.
Fig. 4.( a) Electric field distribution in a plane parallel to the propagation direction.( b) Electric field amplitude at 1555 nm.
the propagation direction z, given in equation( 4) as a function of the fill factor of the grating.
! 2 ff � b gðff
Þ ¼ a exp �; ð4Þ c
where g( ff) is the efficiency and the variable ff corresponds to the fill factor in the different regions of the grating. a, b, and c are constants determining the parameters of the Gaussian beam shape.
3 Simulation results
2D-FDTD( finite-difference Time-domain) simulations are performed using ANSYS Lumerical commercial software. The frequency domain power transmission is obtained from the monitor placed at the topside of the grating and is calculated through the following formula:
Z
TðxÞ ¼
1 2
RðPðxÞÞdS |
|
|
|
monitor
Source powerðxÞ
|
; |
ð5Þ |
where T( x) is the normalized transmission as a function of frequency, P( x) is the pointing vector and dS is the surface normal.
All parameters of the grating, i. e., the height of the grating layer, etching depth, and period, have been varied
and studied to optimize the efficiency of the grating. The period set the central wavelength of highest efficiency for each section N of the complete grating, see Figure 3a for K = 370 nm, 380 nm, 390 nm, 400 nm corresponding to k c = 1555 nm, 1565 nm, 1570 nm, and 1585 nm, respectively.
Considering K = 370 nm corresponding to a maximal efficiency at k c = 1555.5 nm, the etching depth h and the height of the grating, i. e., SiGe layer thickness t, are investigated to optimize further the transmission above the grating. Transmission curves examples are shown in Figure 3b for three etching depths. h influences the central wavelength and the efficiency. The optimum is found for h = 195 nm. Once the geometry is determined for each grating section, a final FDTD simulation of the entire grating is performed. The height of the SiGe layer is set to t = 911nm and the etching depth is h = 195 nm. The period is set to K = 370nm.
The field distribution in a plane along the propagation direction( xz-plane) is given in Figure 4a. Even though the Bragg condition allows the outcoupling at an angle of 0 °, usually a tilt angle is preferred. This is explained by the presence of strong back reflections due to second order diffraction [ 20 ]. The simulated angle of the output beam at the central wavelength, i. e., 1555 nm, is 34 ° which is close to the calculated angle, i. e., 37 ° through the Bragg condition in equation( 3).
The efficiency along the topside of the grating can then be calculated from equation( 6) using data extracted from the FDTD simulations. Intensity is collected and sum over