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This example characterizes a diffraction grating in response to a broadband planewave at normal incidence. Lumerical provides a set of grating scripts as well as “grating order transmission” analysis group, making it easy to calculate common results such as number of grating orders, diffraction angles and grating efficiencies at different wavelengths. The grating analysis group is also useful to obtain the fraction of power for a specific grating order.

### Files and Required Products

FDTD

Minimum product version: 2019a r1

### Contents

Overview

Run and results

Updating the model with your parameters

Taking the model further

## 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. The “grating order transmission” analysis group uses various grating-related commands and returns a comprehensive list of results useful for general characterization of grating:

Number of grating orders

Grating efficiency for each grating order

Grating efficiency into S- or P-polarized light for each grating order

Direction cosine of each grating order (Equivalently, theta and phi values in the farfield 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

1.        Open and run the simulation file (diffraction_grating_FDTD.fsp.)

2.        Open and run the script file (diffraction_grating_FDTD.lsf.)

Note: The provided script collects some representative results from the two analysis groups ("grating_T" and "grating_R") and visualize them. Alternatively, you can manually run the analysis (Right click on the analysis group and select "Run analysis") and explore the results in the Results View window.

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 larger number of diffraction orders at 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 $$\mu m$$, below which new grating orders kick in.

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 higher order.

There appears to be some discontinuities near 0.9 $$\mu m$$. These are to do with the 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 farfield 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 degree.

## Important model settings

Description of important objects and settings used in this model

The grating analysis group forms the key part of this simulation and its setting requires special attention:

Spans and surface normal: For correct grating analysis, the span of this object should be extended beyond the periodic boundaries. The "surface normal" parameter in setup properties determines the direction of the grating projection. The frequency monitor will be automatically set up based on these two parameters.

Plot option: The analysis group has an option for plotting some representative results. To enable the automatic plotting, set the "make plots" to "1" in the "Analysis – Variables" tab of the object. Alternatively, you can directly access the full list of the grating analysis results using a separate script (as was done in the diffraction_grating_FDTD.lsf) and have more freedom in the visualization of the results.

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.

Boundaries: Current simulation uses a source with a $$30^{\text{o}}$$ polarization. To consider the phase changes at x and y boundaries, we used Bloch boundaries in x and y instead of Periodic boundaries.

Arrayed structures: To better represent an infinitely periodic structures, it is recommended that the unit cell is repeated on each side of the periodic boundaries rather than drawing a single unit cell. Details about how this works can be found here.

PML profile: Diffraction gratings can have certain orders propagating at steep angles to the PML, making the absorption by the default PML poor. To alleviate such problem, this example uses the "steep angle" PML profile. For further information about how to choose the PML profile, please visit here.

## 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” is 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. The grating analysis group also works in 2D.

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 change the plane wave type from "Bloch/periodic" to "BFAST" to allow a fix-angle injection for broadband plane wave source.

## 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 trianglular lattice.