Characterize a diffraction grating in response to a broadband planewave at normal incidence. Lumerical provides a set of grating scripts for the DGTD solver, making it easy to calculate common results such as the number of grating orders, diffraction angles and grating efficiencies at different wavelengths.
Overview
Understand the simulation workflow and key results
The diffraction grating in this example is a 2D array of half-ellipsoids on a planar surface. A broadband (0.85-1 \(\mu m\)) planewave is normally incident on the surface grating from the substrate, resulting in multiple diffraction orders in the transmission and the reflection regions. A script command called gratingprojection returns a comprehensive list of results necessary for general characterization of grating:
- Number of grating orders
- Grating efficiency for each grating order
- Direction cosine of each grating order (Equivalently, theta and phi values in the far-field half-sphere)
The above results are returned as a function of wavelength and can be directly used in your grating design or further processed to yield the figure of merit of your interest.
Run and results
Instructions for running the model and discussion of key results
- Open and run the simulation file ([[diffraction_grating_DGTD.ldev]]).
- Open and run the script file ([[diffraction_grating_DGTD.lsf]]).
Number of grating orders vs. wavelength
The following plot shows the number of transmitted/reflected orders the grating supports in terms of the wavelength. It can be noted that
- The grating supports a larger number of diffraction orders at a shorter wavelength.
- The reflection shows larger number grating order than the transmission. This is because the index of the substrate (1.45) is larger than that of the air, meaning a shorter effective wavelength in the substrate. This is consistent with the above observation.
- Both the transmission and reflection show abrupt changes in the number of grating orders at 0.9 um, below which new grating orders start to appear.
Fractional power into a specific diffraction order vs. wavelength
In many cases, it might be necessary to calculate how much of the transmitted/reflected power is converted into a specific diffraction order:
$$T(n,m) = \frac{\text{Transmitted power to (n,m) order}}{\text{Total transmitted power}}$$
It can be observed from the following plot that
- The transmission to (0,0) order, T(0,0), is the same as the total transmission for wavelengths over 0.9 \(\mu m\). This is because the grating supports only a single transmission order at this wavelength range as shown in the previous plot. The differences between the T_Total and the T(0,0) can be attributed to the transmission into higher diffraction orders since there is no absorption in the material used.
- For the reflection, the transmission to (0,0) order is negligible over the whole wavelength range, meaning most of the reflected power is converted to a higher order.
- There appear to be some discontinuities near 0.9 \(\mu m\). These are to do with Wood’s anomaly and can be noticeable at the wavelengths where the number of grating orders changes.
Diffraction angle for a specific diffraction order vs. wavelength
The diffraction angle of the grating is also dependent on the operating wavelength and exhibits different values for different orders. The only exception is the (0,0) order, which is fixed by the angle of the incident beam (theta=0 and phi=0 in this example). The following plot shows the diffraction angle of the transmitted (0,1) order in terms of the wavelength. This specific order starts to appear at 0.9 \(\mu m\) and propagates almost parallel to the substrate. As the wavelength gets shorter, its propagation direction moves towards the polar axis (z-axis in this example.)
Diffraction efficiencies and angles at a specific wavelength
The focus so far has been on how the number of diffraction orders, diffraction efficiencies and diffraction angles change in terms of the wavelength. It is also insightful to learn about the behavior of the whole grating orders for a specific wavelength. This can be best visualized by representing each supported order as a point in the far-field semi-sphere. The following images show the transmitted and reflected orders at 0.85 \(\mu m\). The results are consistent with those presented above. For example,
- There are 3 and 11 diffraction orders for the transmission and the reflection, respectively.
- The diffraction angle for the transmitted (0,1) order is about 70 degrees.
Important model settings
Description of important objects and settings used in this model
PML performance : Diffraction gratings can have multiple diffraction orders, resulting in some orders propagating at steep angle. To improve the absorption properties of the pml, you might need to modify the settings in the pml boundary conditions and/or “thickness” of the pml in the “Simulation region” object by specifying the “shell thickness”.
Homogeneous environment : The grating analysis assumes that the medium at the location of the monitor and beyond (towards the propagation direction) is homogeneous. If there are any index changes on or beyond the monitor, the grating analysis will give incorrect results.
Mesh override : There are mesh override objects over the transmission and reflection monitors. This is to allow more spatial data points for the near-field monitors, hence improving the accuracy of the grating projection results.
Updating the model with your parameters
Instructions for updating the model based on your device parameters
Different geometry : When replacing the geometry with your own, make sure that the spans of the “FDTD” are updated to match the period of your structures. If your grating has an identical cross-section in one direction, you can run 2d simulations instead.
Non-normal incidence : The current example deals with a normal incidence. If you want to simulate the response of the grating to a broadband angled injection, you need to run single frequency simulations and sweep the frequency over the frequency range of interest.
Taking the model further
Information and tips for users that want to further customize the model
Non-rectangular lattice: The grating projection in Lumerical assumes a rectangular array of unit cells. However, you can also use it for gratings with non-rectangular lattices or mixed periodicities. In the triangular-lattice gratings shown below, you can form a larger rectangular unit cell (red) composed of two smaller unit cells(yellow) of the triangular lattice.
Additional resources
Additional documentation, examples and training material
See also
Related Ansys Innovation Courses