This 2D FDTD example shows how to obtain broadband characteristics of a grating coupler and compares the results to experimental data. Furthermore, this application example demonstrates correct use and benefits of the multi-frequency beam calculation and compares the results with standard single frequency beam calculation. To better understand the differences between single and multi-frequency beam calculation visit this page.

## Setup

### Grating coupler structure

The simulated SOI structure is optimized for maximum coupling efficiency at 1310nm and it consists of a 200nm thick Si waveguide that is placed on 700nm thick SiO2 layer. The waveguide is then covered by a 700nm thick SiO2 cladding as depicted on figure 2. The grating itself has duty cycle of 50%, grating period of 500nm and the etching depth of the groves is 30nm as shown on figure 3.

Figure 2: Index profile of the simulated structure

### Gaussian beam setup

Correct source setup representing the laser beam is as important as optimizing the grating coupler structure in order to achieve the maximum coupling efficiency. In particular, we need to determine the optimal incident angle θ_{in}, laser beam waist radius ω_{0} and the distance d between the center of the input laser beam and the end of the grating coupler(beginning of the waveguide).

The optimal incident angle can be obtained by sweeping a range of angles and recording the power coupled into the waveguide at the optimized single frequency of 1310nm. To do this, open the Broadband Grating Coupler.fsp and launch the "Sweep Injection Angle" sweep. Once the sweep is finished, you can visualize the "Coupled power" sweep result and observe that the maximum coupled power is achieved around 13.9 degrees, which is well aligned with the experimental results from the paper.

The optimal beam waist diameter can be calculated by the following equation:

$$\omega_{0}=1.37 L_{c} \cos \left(\theta_{i n}\right)$$

where Lc is grating characteristic length. Lc calculated in the paper is 13 +/-1um, which gives us optimal beam waist radius between 16 and 18.6 um. Alternative method how to estimate the optimal beam waist is to use MODE, specifically the FDE solver. To do this, simply copy the simulated structure to MODE or use the associated file Broadband Grating Coupler.lms and search for modes around effective index of 2.86 obtained by the following equation:

$$n_{e f f}=\frac{k_{i n} \sin \left(\theta_{i n}\right)+\frac{p 2 \pi}{\Lambda}}{k_{i n}}$$

where

p is the diffraction order

is the grating period

k_{in }is the module of the incident wave vector

The Finite-Difference Eigenmode solver finds and calculates the spatial profile of the grating coupler mode(Figure 4) at 1310nm and gives us better understanding of the optimal beam profile. This information is helpful during the FDTD simulation setup when we need to determine the Gaussian beam properties. Note that since we are using fully vectorial beam profile(thin lens option), the beam waist is determined by the Numerical Aperture. The beam profile related to specific beam NA can be shown by the "Visualize beam data" button(Figure 5). Additionally, the beam options tab allows us to define the distance from focus and position the beam waist near the grating coupler plane rather than at the source injection plane.

Figure 4: Grating coupler mode profile at 1310nm

Figure 5: Gaussian beam source settings

The last missing parameter that is needed to build the FDTD simulation is the optimal distance between the center of the beam and the end of the grating d. This distance is given by a simple relationship d=Lc, which is between 12 and 14um. This property is in our simulation defined by the source position relative to the grating couple structure.

### Monitor setup

In the paper, the coupled power is measured indirectly as:

$$P_{\text {coupled}}=P_{\text {injected}}-\left(P_{\text {reflected}}+P_{\text {transmitted}}\right)$$

We reproduced this approach by placing power transmission monitors below and above the grating coupler to measure the reflection and transmission that is later subtracted from the injected power during post processing. In addition to this approach, we used also a direct measurement method when we placed a power transmission monitor at the end of the waveguide in order to directly measure the power coupled into the waveguide. Note that the modal fields extend beyond the waveguide interface and therefore the monitor must be extended as well in order to measure 100% of the coupled power. The extend of the modal fields can be obtained by the mode expansion monitor.

## Results and Discussion

### Coupled power as a function of wavelength

In order to plot the wavelength dependence of the coupled power it is necessary to run a broadband simulation with angled injection. This type of simulation can benefit from using the multifrequency beam calculation and we demonstrate this by comparing the results of two simulations with multifrequency beam calculation on and off. In addition, we will investigate the difference between the indirect and direct measurement of the coupled power. To run the simulation, first open the Broadband Grating Coupler.fsp and then run script Broadband Grating Coupler.lsf. The following plot will be produced once the simulations are completed:

Figure 6: Simulated coupled power as function of wavelength between 1280 - 1340 nm

The maximum simulated coupled power at θ=13.9 degrees is 55.7%, which is well aligned with the measured maximum of 56%. Moreover, the multifrequency beam calculation delivered better accuracy when compared to the results obtained with beam source calculated at single frequency, which showed 10% narrower bandwidth at FWHM. Notice that the results are nearly identical at 1310nm. This is a result of the single frequency simulation being centered at 1310nm and therefore, the calculated beam is accurate at this specific wavelength.

The comparison of direct measurement of power coupled into the waveguide and the indirect calculation using reflected and transmitted power shows nearly identical results. Hence, using the direct method seems to be more convenient since it requires only one monitor and no post processing.

### Coupled power as function of injection angle

To simulate the coupled power as function of injection angle we use a parameter sweep to collect the values of coupled power at the predefined range of wavelengths. To do this simply run the script Broadband Grating Coupler Injection Angle Sweep.lsf. The script sets the simulation into single frequency mode as we are interested to obtain this characteristics at the optimized frequency of 1310nm. Additionally, the simulation with single frequency calculation is slightly faster, which is convenient as we need to run multiple simulations during the parameter sweep. Finally, the scrip will collect the sweep results and plot the characteristics shown on figure 7. Again, the results show maximum coupling efficiency of 55.7% at 13.9 degrees, which is well aligned with the measured values.

Figure 7: Simulated coupled power as function of injection angle between 11 - 18 degrees at 1310nm

### Related Publications

Vivien et al., Light injection in SOI microwaveguides using high-efficiency grating couplers, Journal of Lightwave Technology, Vol. 24, No. 10, 2006

### See Also

Inverse Design of Grating Coupler