In this article, we demonstrate how to simulate Volume Holographic Gratings (VHG) with the RCWA solver. The modulation of the index of refraction presents a periodicity in the direction of propagation. This periodicity is exploited by the solver to accelerate the simulation of thick holograms.
In this workflow, we use Ansys Lumerical to construct a block of holographic material corresponding to one period of the material. The solver is then configured to consider the repetition of this layer to the desired thickness. Once the RCWA simulation is completed, the result is exported into the Lumerical Sub-Wavelength Model (LSWM) .json format for modelling this grating in a system-level simulation in Ansys OpticStudio Zemax or Ansys Speos.
Overview
Understand the simulation workflow and key results
The volume hologram consists of a slab of photopolymer material in which a modulation of the refraction index was induced within its volume. The modulation corresponds to the recording of an interference pattern between two plane waves illuminating the hologram at a given angle.
This article is divided in 3 main steps as follows:
Step 1: Design of a volume holographic grating
In this section, we show how to use Ansys Lumerical to setup a holographic material with a desired modulation of the refractive index.
Step 2: RCWA simulation with layer repetition
The efficiency of the diffraction orders is computed with the RCWA solver. The response of the VHG is characterized as a function of incident angle and wavelength.
Step 3: Export diffraction characteristics toward Zemax and Speos
The results of the RCWA simulation are saved in .json format and imported into Zemax and Speos so that the hologram can be integrated in a ray-tracing system.
Run and Results
Instructions for running the model and discussion of key results
Step 1: Design of a volume holographic grating
Setting up the modulation of refractive index and the periodicity
- Open the file [[vhg_R_6um_model.fsp]]
- Check the model Setup properties. The parameters related to the recording angle, and the holographic materials are defined. These parameters are used to generate the rectangle object corresponding to the VHG with the correct material properties and dimensions.
- Check how the model was used to set the properties of the VHG rectangle object:
- In the material tab, the index is set with a formula corresponding to the index modulation of the VHG
- In the Geometry tab, the dimension in z corresponds to the period of the VHG in the direction of propagation
- Check how the model was used to adjust the RCWA settings of the hologram structure:
- In the General tab, the index of the background material is set to 1.5
- In the Geometry tab, the x dimension is the period of the VHG in the x-direction
- In the Interface tab, the layer repetition is set to extend the VHG to a thickness of 6um.
The parameters for the recordings can be input in the model properties, and the system is setup by a script with these inputs. Below are the details of how the system is made.
The VHG is modeled as a material of index \(n_0=1.5\), with a modulation index \(\Delta n=0.04\). The hologram recording is performed with the interference of 2 collimated beams described by their k-vector \(\overrightarrow{k_1}\) and \(\overrightarrow{k_1}\). The fringes induce a modulation of the refractive index within the volume of the material that can be described by the following equation:
$$ n=n_0+\Delta n \times cos(\overrightarrow{K}.\overrightarrow{r}) $$
Where \(\overrightarrow{K} = \overrightarrow{k_1}-\overrightarrow{k_2}\) is the grating vector.
In this article, the two recording beams considered are incident on the hologram at angles of 0° and 28° respectively. Note that these angles are given within the recording medium of index 1.5, it corresponds to angles of 0° and 44.7° in air. The beams are incident from opposite directions, so that the hologram is a reflection type, and the recording wavelength is set to 550nm.
As a result, we have:
$$ \overrightarrow{K} = \overrightarrow{k_1}-\overrightarrow{k_2} = \frac{2\pi n_0}{\lambda}(\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}-\begin{bmatrix} cos(28°) \\ 0 \\ -sin(28°) \end{bmatrix}) = \frac{2\pi n_0}{\lambda} \begin{bmatrix} -0.4695 \\ 0 \\ 1.8829 \end{bmatrix} $$
and the modulation index of the holographic material is described by the formula:
$$ n(x,y)=1.5+0.04 \times cos(\frac{2\pi \times 1.5}{0.55}\times (-0.4695x+1.8829z)) $$
Where x and z are the spatial coordinate within the volume of the material in µm. Since the recording beams are considered in the (x,z) plane, there is no dependence in y.
From the equation, we can deduce the periodicity of the structure in the x and z direction to be respectively:
$$ \begin{align*} \left\{ \begin{array}{ll} p_x=\frac{\lambda}{n_0 \times K_x}=0.781 µm \\ p_z=\frac{\lambda}{n_0 \times K_z}=0.1947 µm \end{array} \right. \end{align*} $$
Visualize the modulation of refractive index
- In the RCWA properties, deactivate the layer repetition by setting it to ‘None’ in the “interface tab”
- In the RCWA results tab, select “index report”
- In the Object tree, select the RCWA solver and visualize the index preview
When the layer repetition is active (“multiple” or “thickness”), the index preview tool of the RCWA solver is not supported. Once it is deactivated the index can be visualized.
The slab of material defined correspond to a single period of the VHG, we see that the modulation is sinusoidal around \(n_0=1.5\) with an amplitude of \(\pm 0.02\) (since \(\Delta n=0.04\) ), as set in the model properties.
Step 2: RCWA simulation with layer repetition
- In the interface tab of the RCWA properties, set the layer repetition for a thickness of 6µm.
- In the result tab, make sure that the index report is NOT selected, and select the grating characterization results.
- Run the RCWA solver and visualize the grating characterization results.
- Right click on the grating characterization results of the RCWA solver and export the data in .json format.
The fast method to design VHG is to use the Kogelnik method, that is an analytical solution providing the response of a hologram. It relies on a few approximations, but it is accurate around the Bragg condition.
Another reference we can use to evaluate the result is to run the RCWA simulation through a complete structure without using the repetition layer feature. With the repetition layer, the structure is defined over a thickness of 1 period (0.1947µm), and is slices into 10 layers. The equivalent structure without the repetition is a 6µm thick material with \(10 \times \frac{6\mu m}{0.1947 \mu m} = 308\) layers. For more information on the layer repetition feature, see Layer Repetitions in RCWA
Below are the results obtained with the different methods (for S polarized light).
A VHG is expected to produce a strong order of diffraction when the playback wave is one of the recording wave (Bragg condition), and we can see almost 80% going into the diffraction order \(R_{+1}\) when the playback angle is matching the 0° and 28° incidence angles used for the recording, at the designed wavelength.
All results show good agreement. The RCWA results obtained with the repetition layer feature are almost identical to the data obtained by running the simulation on the full structure, while the running time was improved greatly (by a factor 15 in this case).
With a finer sampling in wavelength and angle, the hologram response can be examined under different playback conditions. Below is the diffraction efficiency in the \(R_{+1}\) order as a function of wavelength and incident angle.
Step 3: Export diffraction characteristics toward Zemax and Speos
Export the results to Zemax OpticStudio
- Copy the .json file in the folder \(Zemax >DLL >Diffractive\)
- Open [[VHG_test_R.zar]]
- In the diffraction properties of the Diffraction Grating object, select the .json file of the VHG
- Run the Ray trace and observe the diffraction behavior on the monitor
In Zemax OpticStudio, the hologram is modeled with the diffraction object, set as a slab of material with refractive index n=1.5. The .json diffraction data are set in the object proprties and load the RCWA diffraction data into the first surface of the object.
When running the ray tracing, we see the energy going into the \(R_{+1}\) order. Since the reflected medium is air, the diffraction angle is not the 28° angle of the medium, but 44.7°. By varying the incident angle of the source, one can observe that out of the Bragg conditions (0° and 44.7° in air), most of the light passes through into the \(T_0\) order.
The shift in Bragg condition can also be examined when varying the wavelength of the playblack source, as illustrated below with a broadband source.
Export the results to Speos
- Repeat the step 2 for a Transmission hologram by setting the holo type to "T" in the model settings
- Copy the .json file to the folder 'SPEOS input files'.
- Open [[LSWM_test_VHG.scdocx]].
- In the properties of the “surface_json” object, select the .json file for the surface layer
- Run the simulation “Direct_Normal_source”
To demonstrate a slightly different case, the same process than above was performed to generate a .json file, but for the case of a transmission hologram. The only setting to change for this case is to change the sign of \(k_{2z}\) (+sin(28°)) and to adjust the thickness of the VHG object to represent the new periodicity value in the z direction (3.1325µm). This is done automatically in the model script by simply selecting T instead of R in the parameters of the Lumerical model.
When running the simulation in Speos, we can see this time that at normal incidence most of the energy is sent to the \(T_{+1}\) order.
Important model settings
Description of important objects and settings used in this model
- To provide an easy comparison with what can be obtained with the Kogelnik model, the VHG calculation was performed within the index background model, with n=1.5. All angles direction should be adjusted according to the Snell law when considered in air.
- The computation time varies greatly depending on the sampling selected for the RCWA solver for the angular and spectral range.
Updating the Model with Your Parameters
Instructions for updating the model based on your device parameters
- The response of the hologram can be customized by adjusting the material parameters (\(n_0\) and \(\Delta n\)) and the recorded angles. Depending on the input parameters, both reflection and transmission holograms can be simulated. The modulation index and the thickness of the material have a strong impact on the bandwidth and the maximum diffraction efficiency of the hologram.
- The RCWA settings should be adjusted to cover the angular and spectral range of interest.
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.
- By combining this workflow with the spatial variation feature of the Lumerical-subwavelength model ( Lumerical Sub-Wavelength Model: How to Simulate a Grating with Spatial Variations ), it is possible to simulate holograms with optical power.
- One common effect in holography is the shrinkage of the material during curing, that induces a compression of the refractive index profile. The modulation formula for the holographic material can be adjusted to take into account this kind of effect
- Another effect that could be considered for the simulation is the saturation of the modulation in the refractive index profile. In the current model, only sinusoidal modulation is considered.
Additional Resources
- Kogelnik, H., "Coupled wave theory for thick hologram gratings, " Bell Syst. Tech. J. 48, 2909-2947 (1969).
- Simulating diffraction efficiency of a volume holographic grating using Kogelnik’s method