In this article, we demonstrate how to build a model for the simulation of a curved waveguide with holographic couplers. Optical power is added to the holographic functions to compensate the power introduced by the curvature of the waveguide.
In this workflow, we build the model for a curved waveguide with a 1D expansion. The responses of the volume holographic grating (VHG) couplers are computed with Ansys Lumerical with the RCWA solver, leveraging the periodicity of the structure within the volume to accelerate the calculation with the layer repetition feature. Optical power is added to the function by sampling spatially the holographic function, and the result with its spatial variation information is then exported to Ansys Zemax OpticStudio through the Lumerical Sub-wavelength Model (LSWM).
Overview
Understand the simulation workflow and key results
The model is based on the system described in the reference paper from Craig T. Draper and Pierre-Alexandre Blanche ( Holographic curved waveguide combiner for HUD/AR with 1-D pupil expansion ). It consists of a curved waveguide, a VHG in-coupler element, and 3 VHGs used for the light extraction with 1D pupil expansion. Each of the VHG presents a spatial variation in their response to compensate for the curvature of the interfaces.
The article is divided into 3 main steps as follows:
Step 1: System model in Ansys Zemax OpticStudio
In this section, we introduce the optical system using only built-in Zemax holographic objects. We show how additional optical power on the holographic couplers can be used to compensate for the waveguide curvature.
Step 2: VHG simulation in Ansys Lumerical
The response of the desired VHG objects are computed in Ansys Lumerical with the RCWA solver. The simulation is accelerated by exploiting the periodicity of the index modulation within the volume. In order to add optical power, a loop is performed on the RCWA runs to compute the response over the surface of the VHG element. At each spatial position, the recording k-vector are adjusted to match the function that we want to induce.
Step 3: Ray Simulation in Ansys Zemax OpticStudio
The VHGs responses are exported and loaded in OpticStudio to replace the ideal objects. We can then perform the ray tracing simulation within the curved waveguide under more realistic conditions.
Run and Results
Instructions for running the model and discussion of key results
Step 1: System model in Ansys Zemax OpticStudio
Setting up the optical model with ideal objects.
- Open the file [[curved_waveguide.zar]]. For this step, we are looking at the first configuration where the holographic couplers are modelled with ideal objects “Toroidal holograms”.
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Check the parameter “Rotation R1” of the object 2 labelled “Hin Zemax”.
- Set it to 171.45mm, the value of the outer curvature of the waveguide, and observe that the rays propagating through the waveguide are not collimated.
- Set it to 171.45*3=514.35mm, and observe that the curvature enables to get the rays coming from the same point of the object to propagate within the waveguide at the same angle.
- Run Ray trace and observe the image in the detector viewer.
For the waveguide design, an image is projected onto a first VHG element by a lens, and the light is sent at an angle such that it meets the total internal reflection angle within the waveguide. Then, the light is extracted when it interacts with one of the outcoupling VHG. Each point of the image corresponds to an angle of propagation within the waveguide.
In the case of a curved waveguide, optical power is introduced for the rays as they bounce off the curved surfaces of the waveguide. Since the curvature of the two surfaces of the waveguide introduces opposite power, there is a sort of compensation, but the initial angle of diffraction induced by the holographic couplers must be adjusted to take the optical power into account.
In Ansys Zemax OpticStudio, the holographic objects are defined with construction points. Then, we can define only a collimated recording beam by setting the construction point far away, or use a close construction point that will define a spherical recording beam.
Then, in the initial optical setup, we are using the curvature of the “Toroidal hologram” object to simulate optical power in the direction of the waveguide curvature (sagittal direction). With this method, we can design the system to propagate correctly through the waveguide and give a clear image. For more details on the design, see the reference .
Step 2: VHG simulation in Ansys Lumerical
In the previous step, the optical power of the hologram was adjusted by varying artificially the curvature of the VHG. However, in practice, the holograms are laminated on top of the waveguide and present the same curvature as the inner or outer surface respectively. For a better simulation, the optical power should be induced by adjusting the recording conditions. This is what we are going to demonstrate in this step, with the RCWA solver.
Design of In-Coupler
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Open the file [[Generate_JSON_VHG_Hin.lsf]].
- Check the Section with the definition of the “Parameters for the VHG”, where the initial construction points are set.
- Check out the loop over the spatial sampling of the VHG. For each location, the k-vectors are adjusted so that the RCWA conditions corresponds to the desired function.
- Run the script and generate the .json file for the in-coupler
The RCWA solver can be used to simulate the response of holographic couplers. The solver includes a layer repetition feature that can be used to accelerate the calculation by exploiting the periodicity of the index modulation along the propagation axis. For more details on how to set up the simulation of a simple VHG, see Volume Holographic Grating – Ansys Optics .
In this case, a loop is performed with the RCWA solver to simulate the response of the VHG at different positions, and the recording k-vectors are adjusted to produce the desired responses considering the curvature of the holographic object.
In the case of the in-coupler, we want an input at normal incidence to be diffracted for a propagation into the waveguide at an angle \(\theta\). For a position (x,y) on the VHG, the recording angles are adjusted as follows
$$ k_1 = \begin{bmatrix} sin(\alpha) \\ 0 \\ cos(\alpha) \end{bmatrix} \& \ k_2 =\begin{bmatrix} sin(\beta) \\ 0 \\ cos(\beta) \end{bmatrix} $$
Where \( \alpha = \frac{x}{R}\), \(\theta=\alpha+\beta\), and R is the radius of curvature at the interface.
The set of RCWA responses computed with the loop is exported with the Lumerical Sub-Wavelength Model (LSWM) in JSON format. This format supports spatial information, as described in this article: Lumerical Sub-Wavelength Model: How to Simulate a Grating with Spatial Variations – Ansys Optics . For this example, all the VHGs have a size of 9mm x 9mm, and we sampled them with a grid of 5x5points. Linear interpolation is used by the LSWM to estimate the response at all locations falling between this grid of points. It provides acceptable results, especially since we are working with mostly collimated beams and slow curvature. For higher accuracy, or to represent a faster variation of the profile, a finer grid would be required.
Note that the points with the same y-coordinates on the grid sampling the VHG are providing the same function because we are adding power only in the sagittal direction. Then, these sets of points are grouped together and labelled with the “index map” going from 1 to 5, and we run only 5 different RCWA simulations instead of 25 independent runs.
Design of Out-Couplers
The same strategy is applied for the design of the out-couplers, but for transmission VHGs. The curvature of the inner part of the waveguide is also of opposite sign than for the in-coupler
- Open the file [[Generate_JSON_VHG_Hout.lsf]].
- Run the script. It performs a loop corresponding to the 3 out-coupler VHG of the 1D expansion and generate 3 separate .json files.
Note that we want the 3 out-couplers to provide the same output direction. However, the normal at the center of each of the 3 VHGs are not parallel. They have a difference of +/- 3.1°, which makes a different output angle of 2.07° in glass. Then, the design output angle \(\theta_{out}\) for each of the 3 VHGs are set to -2.07°, 0°, and 2.07° respectively.
In the case of the in-coupler, we want diffraction efficiency to be as high as possible. For the out-couplers, the diffraction efficiency should be gradually increased over the pupil expansion so that the light is extracted in a balanced manner between the 3 VHGs. Ideally, the efficiency between the 3 out-couplers should be 33%, 50%, and 100% (and that’s how it is set in the Configuration 1 with the ideal elements). In order to achieve this variation of efficiency, the index modulation for the 3 output VHGs are tuned differently.
Higher diffraction efficiency can be obtained by increasing the thickness of the VHGs, but it also has the effect of narrowing the angular bandwidth. It is important to find a good compromise with the level of efficiency and the uniformity of the response over the expected angular bandwidth of the system.
Step 3: Ray Simulation in Ansys Zemax OpticStudio
In this step, we are replacing the ideal Zemax object with the .json files containing the results computed with the RCWA solver at the previous step.
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Open the file [[curved_waveguide.zar]], and switch to the configuration 2.
- In the settings of the “Lenslet Array” objects, check that the correct .json files are selected in the diffraction tab.
- Check that the curvature of the “Lenslet Array” objects are set to match respectively the outer and inner curvature of the waveguide.
- Run a Ray Trace and observe the resulting image in the detector viewer
There is an option with the Zemax objects to compute the efficiency of the VHGs with the Kogelnik method by entering the Volume Hologram parameters. However, since the curvature of the elements is artificially wrong, the results provided would not give an accurate description of the actual behavior of the holographic elements. Then, the diffraction order were set to ideal values manually (100% for the in coupler, and 33%, 50%, and 100% for the out-couplers).
The .json files contain the data related to the VHGs responses computed with the RCWA solver. Compared to the case with the ideal elements, it provides a more realistic representation of the system behavior in terms of geometry (correct curvature of the elements), angular bandwidth, and straylight.
Of course the ideal image looks better, even though we can see some trace of astigmatism left from the cylindrical nature of the system. However, we can see in the 2nd configuration that the correct optical function with compensation of the waveguide curvature can be implemented directly into the optical functions of the VHGs thanks to the Lumerical Sub-Wavelength Model.
For the design of the curved waveguide system, the simulation with the VHG computed with RCWA provides a better representation and visualization of the uniformity and stray-light that are important for the overall system optimization.
Important model settings
Description of important objects and settings used in this model
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To be imported in Zemax, the .json file should be placed in the folder Zemax DLL Diffractive
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The RCWA scripts for the different VHG elements are set with the specific recording condition and expected curvature for each VHGs.
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The RCWA simulation is set with a background material of 1.5, so all angles are in glass.
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In the .json files, the information regarding the period of the structures is passed in the variable “reciprocal_lattice_vector”, that is a 2D vector representing the x and y periodicity. In this example we ignore the periodicity in the y direction.
Updating the Model with Your Parameters
Instructions for updating the model based on your device parameters
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The hologram parameters were set to provide an acceptable angular bandwidth and efficiency. The thickness, index and modulation index of the holographic material should be updated in the Lumerical script to match the material considered.
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A different medium layer could be added on top and/or below of the VHG layer to consider different input and output media.
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.
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A linear interpolation is performed between the sampling grid defined on the VHG elements. For functions with small optical power, a low sampling may be sufficient, but a convergence test should ideally be performed to determine the optimal spatial sampling required to describe accurately the optical function of the hologram.
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Only a 1D expansion is considered in this example, but a 2D system could be considered as well.
Additional Resources
Related Publications
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Craig T. Draper and Pierre-Alexandre Blanche "Holographic curved waveguide combiner for HUD/AR with 1-D pupil expansion", Optics Express, Volume 30, Issue 2 (2022); https://doi.org/10.1364/OE.445091