Diffraction Grating with Anisotropic Materials

In this article, two types of diffraction gratings made with anisotropic liquid crystal (LC) materials are characterized using the RCWA solver. In the first case, the thickness of the cell and the orientation of the LC molecule are tuned to reach 100% efficiency into the 1^st order of diffraction at certain wavelengths. In the 2^nd case, a polarization volume grating (PVG) that transmits or reflects into different diffraction orders based on the handedness of the incident circularly polarized incident light is simulated.

In this workflow, we use Ansys Lumerical’s RCWA solver to construct the gratings and simulate their response. The first grating is formed with liquid crystal molecules having their long axis oriented in the transverse (\(xy\)) plane, providing in-plane anisotropy. A periodic spatial variation on the LC orientation is induced to design the grating. The LCs in each period have a cycloidal orientation pattern. The diffraction characteristics are then exported into the Lumerical Sub-Wavelength Model (LSWM) JSON format for modelling this grating in a system-level simulation in Zemax.

In the next part, we simulate a PVG based on twisted LCs [2]. The LCs exhibit in-plane anisotropy, but their in-plane orientations vary along the propagation (\(z\)) direction. In addition to the periodic spatial variation of the LCs in the  plane, the grating also exhibits periodicity along the propagation (\(z\)) direction. This is more efficiently simulated using the “Layer Repetitions” feature in RCWA. The results of the RCWA simulation are exported into a [[.json]] file and then used in Zemax for raytracing.

[[NOTE:]] The RCWA support for full anisotropy requires Ansys Lumerical version 2025 R1.3 or higher. In plane-anisotropy (covered in the example) requires 2024 R1.3 or higher.

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Overview

Understand the simulation workflow and key results

In the first case, the diffraction grating consists of a single layer of liquid crystal (5CB). The grating structure is obtained by inducing a periodic spatial variation on the orientation of the long axis of the LC molecules within the XY plane. By designing the grating properly with a cycloid diffraction pattern, it is possible to eliminate the 0^th order and to distribute the light entirely to the 1st orders.

This modelling is divided into 3 main steps as follows:

Step 1: Design a grating with a cycloidal director pattern

In this section, we show how to use Ansys Lumerical to setup a LC cell where the orientation of the long axis is varying spatially.

Step 2: RCWA simulation with in-plane anisotropy

The efficiency in the different orders is computed with the RCWA solver. The thickness is tuned to obtain the elimination of the 0 ^th order at the desired wavelength.

Step 3: Export grating characteristics toward Zemax

The results of the RCWA simulation can be saved in [[.json]] format and imported directly into Zemax so that the grating can be included in a ray-tracing system.

In the 2^nd case, we simulate a twisted LC grating. We create a material called “LC 1.7” in the material database, which is used as the base anisotropic material. The ordinary index of the material is 1.5 and the extraordinary index is 1.7. The grating reflects right-handed circularly polarized light into the +1 order and transmits left-handed circularly polarized light into the fundamental order. The 3 steps involved in simulating the PVG are enlisted below:

Step 4: Design a grating with a twisted director pattern

In this step, we set up a twisted LC cell in the RCWA solver of Ansys Lumerical. The LC directors lie in the \(x\)-\(y\) plane, their in-plane orientation also changes along the propagation (\(z\)) direction.

Step 5: RCWA simulation with general anisotropy

A 2D Y-normal RCWA simulation is run to calculate the diffraction efficiencies of different orders at a wavelength of 550 nm.

Step 6: Export grating characterization results to Zemax

The grating characterization results are saved in a [.json]] file and imported into Zemax for raytracing.

Run and Results

Instructions for running the model and discussion of key results

Step 1: Design a grating with a cycloidal director pattern

1. Open the file [[lc_grating_RCWA.fsp]]
2. Right click on the model in the object tree and examine the Setup
   1. In the variable tabs, the main parameters for the systems are defined. It includes the wavelength range of the simulation, the thickness and the material of the LC cell, and the period of the grating.
   2. In the Script tab, examine how the different parameters are used to construct the setup. In particular, the LC rotation attribute is defined with a spatial variation where the vector defining the direction of the LC long axis is set as a function of the spatial dimension X.

During the fabrication process of an LC cell, the default orientation of the molecules is defined by a photo-alignment layer. In the simulation setup, the spatial variation is set according to a cycloidal pattern so that the angle between the long axis of the LC molecules and the x-axis is as follows:

$$\theta (x) = \cos(\frac{\pi x}{\Lambda})$$

Where \(\Lambda\) is the period of the grating. For more information on how to define the spatial variation with an attribute, see LC Rotation - Simulation object , or Matrix Transformation - Simulation object for a more general tool.

The LC pattern can be visualized directly in the viewport, where one period of the cycloidal pattern is shown (not on scale). The overall system is illustrated below:

Step 2: RCWA simulation with in-plane anisotropy

3. Run the script [[lc_grating_RCWA_script.lsf]]

The RCWA is set to run over a spectral range of 0.37µm to 0.9µm, at normal incidence. The script runs the solver and extracts the diffraction efficiency in the first orders (DE _1st order = DE ₊₁ + DE _-1 ) for two different thicknesses (3µm and 5µm).

The results highlight the fact that at certain wavelengths (e.g. 0.44µm) 100% of the energy is sent to the first orders.

Step 3: Export grating characteristics toward Zemax

5. Select the RCWA solver. In the result tab right click on ‘grating_characterization’ and 'Export to JSON'
6. Open ‘[[lc_grating_RCWA_ray_tracing.zprj]]’ with Zemax
7. Trace rays for different wavelengths, 498nm and 600nm, and for different polarization of the source (linear and circular) and observe the impact on the Transmission properties of the grating.

The [[.json]] file is applied to define the diffraction characteristics of the ‘Diffraction Grating’ object. The source sends light at normal incidence on the grating and two detectors capture respectively the reflection and transmission signals.

By modifying the wavelength of the system, we can verify that the transmission behavior computed with Lumerical is well transcribed to the optical system. At some wavelengths, the 0th order is completely eliminated (e.g. @600nm) while for other spectral components (e.g. @498nm) there is no energy diffracted away from the 0 ^th order.

By using linear polarization, the grating distributes the light equally between the +1 and -1 order. By setting the polarization to be circular (\(Jx=Jy=1\), and either \(X\)-Phase or \(Y\)-phase set to 90), all the energy is directed to a single order (+1 or -1 for either left of right circular polarization).

Step 4: Setup a twisted LC cell in RCWA

8. Open the file [[PVG_RCWA.fsp]].
9. The grating geometry is parameterized in the model Setup.
   1. The input parameters can be found in model->Setup->Variables. This includes the main grating parameters such as the periods in \(x\) and \(y\) directions, the thickness of the LC, and the materials/refractive indices.
   2. The script used to set up the geometry using the “Variables” can be found in model->Setup->Script.

A schematic of the PVG is shown below:

In the script, the orientation unit vector of the LC is defined in terms of \(\theta\) and \(\phi\), where \(\theta\) is the out-of-plane rotation angle (90 deg here) and \(\phi\) is the in-plane rotation angle.

The “Layer Repetition” feature is used to handle the periodicity of the grating along the \(z\)-direction (For more information on the layer repetition feature, see Layer Repetitions in RCWA – Ansys Optics). The grating has a total thickness of 4 um, with the period being 0.4672 um along the \(z\)-axis direction. Only one period is set up in the GUI using the script described above. The total thickness of the structure that must be considered for the overall calculations is defined in the “layer repetition” menu in the “Interfaces” tab of the RCWA solver settings.

Step 5: RCWA simulation

10. The RCWA simulation is run, and the grating characterization results are saved into a [[.json]] file using the GUI.

Step 6: Import the grating characteristics to Zemax

11. Open ‘[[PVG_ray_tracing.zprj]]’ with Zemax
12. Trace rays for a wavelength of 550 nm and different circular polarizations of the source and observe the impact on the transmission and reflection of the grating.

The [[.json]] file created in the previous step is applied to the ‘Diffraction Grating’ object to define its diffraction characteristics. The Multi-Configuration capability is used to perform the raytracing for left-handed and right-handed circularly polarized light. The source sends light at normal incidence on to the grating and two detectors capture the transmission and the reflection.

For right-handed circularly polarized light (\(Jx=Jy=1\), \(X\)-phase =0 and \(Y\)-phase = -90), the grating reflects almost all the light into the +1 order as evident from the detector results shown below:

For left-handed circularly polarized light (\(Jx=Jy=1\), \(X\)-phase =0 and \(Y\)-phase = 90), the grating transmits all the light into the fundamental order as shown below:

The grating behavior matches the results reported in [2].

Important model settings

Description of important objects and settings used in this model

• To be imported in Zemax, the [[.json]] file should be placed in the folder Zemax\DLL\Diffractive.

Updating the Model with Your Parameters

Instructions for updating the model based on your device parameters

• In the 1^st case, the simulation was performed with the material 5CB-Li, that is commonly used for nematic liquid crystal, while in the 2^nd case a new material was created. In similar fashion, the desired material can easily be selected from or created in the Material Database.
• The grating parameters can be changed from model->Setup->Variables based on the requirements. For example, the period can be changed to control the angle of diffraction.
• The orientation of the LC within the simulation domain can be changed as per user requirements by modifying the script used to calculate the LC unit vector orientation in model->Setup->Script.

Taking the Model Further

There are some considerations that are not covered in this demonstration but users could pay more attention when they try to follow this process for their systems.

• In this demonstration, the simulations are performed in two different conditions to highlight the diffraction properties of such structures. An optimization run or a simple sweep can be performed to automatically find the optimal thickness of the cell required for the desired wavelength.
• Similar analysis can be also extended to simulate LCs that have out-of-plane orientation (\(\theta \neq 90\)). Starting 2025 R1.3, RCWA supports fully anisotropic materials, please see this.

Additional Resources

Additional documentation, examples and training material

Related Publications

1. Katarzyna A. Rutkowska and Anna Kozanecka-Szmigiel "Design of Tunable Holographic Liquid Crystalline Diffraction Gratings", Sensors 2020, 20(23), 6789 https://doi.org/10.3390/s20236789
2. Yishi Weng, Daming Xu, Yuning Zhang, Xiaohua Li, and Shin-Tson Wu, "Polarization volume grating with high efficiency and large diffraction angle," Opt. Express 24, 17746-17759 (2016)