There are two common methods for calculating the diffraction efficiency of Volume Holographic Grating (VHG): the Kogelnik method and the Rigorous Coupled-Wave Analysis (RCWA) method. The Kogelnik method is an approximate approach that relies on several assumptions and may not be accurate under certain conditions. On the other hand, RCWA is a precise computational method that can accurately calculate situations where the Kogelnik method fails. In the past, we have provided workflows for using the Kogelnik method to calculate the diffraction efficiency of VHGs. In this article, we will provide a simulation approach using the RCWA method to calculate the diffraction efficiency of VHGs.

**Authored By Yihua Hsiao, Michael Cheng**

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## Introduction

Diffraction elements are indispensable components in many optical systems, such as the AR's exit pupil expander (EPE) system, for controlling the direction of light. In the other article, Simulating diffraction efficiency of surface-relief grating using the RCWA method, we have extensively discussed that SRG and VHG are both gratings, but they differ in the way their periodic variations are achieved. VHG can also be considered a type of grating due to the periodic variation in its material's refractive index. The calculation of its diffraction efficiency generally involves two methods: the Kogelnik method and the RCWA method. The first part of this article will explore the theoretical differences between these two methods.

In the second part, we will introduce how to use the User-defined diffraction DLL and provide detailed explanations of the setup parameters. Users can utilize this DLL to design their own VHG optical systems.

Finally, an example in an AR EPE system will be presented to demonstrate the integration of the RCWA method's calculations for VHG elements with the ray tracing engine.

## Theoretical Comparison of Kogelnik Method and RCWA Method

The Kogelnik method, introduced in 1969, is based on several assumptions：

1. The hologram thickness must be sufficiently thick.

2. Diffraction exists only in the 0th or 1st order.

3. The refractive index of the hologram is the same as that of the surrounding environment.

The third assumption has a significant impact on many applications, such as in EPE systems, where accurate simulation becomes challenging when the refractive index of the hologram differs from that of the environment.

Furthermore, this method requires the definition of modulation coefficients for refractive index and absorption, as indicated in the following equations and figures. Under these assumptions, it is possible to calculate the diffracted electric field and diffraction efficiency for both transmission and reflection holograms. For further details, please refer to Simulating diffraction efficiency of a volume holographic grating using Kogelnik’s method.

The RCWA method assumes homogeneity in the Z-direction and periodicity in the XY-direction for gratings. If the Z-direction is not homogeneous, it can be divided into multiple homogeneous layers to satisfy the assumption of Z-direction homogeneity for each layer.

In essence, the RCWA method consists of three main components:

1. Solving eigenvalues in the hologram. Initially, the electromagnetic field and dielectric tensor are expanded using Fourier series, and this result is input into the Maxwell equations to determine the eigenvalues.

2. Continuity of the tangential electric field and wave vector at two interfaces. This boundary condition is used to solve for the electric field.

3. Calculation of the electromagnetic field in the multiple layers segmented in the Z-direction and their connection using the scattering matrix method.

For further details, please refer to Simulating diffraction efficiency of surface-relief grating using the RCWA method.

For a comparison of the two methods, please refer to the table below. The Kogelnik method, due to its numerous assumptions, comes with several limitations; however, its primary advantage lies in its computational speed. Nevertheless, for scenarios where the refractive index of the surrounding material differs from that of the hologram, the RCWA method must be employed to accurately calculate the results.

Kogelnik | RCWA | |

Refractive index modulation (∆n/n) | Reflective holograms should <0.16 and transmissive holograms should <0.06. |
No limit |

Thickness |
Holograms should be thick enough |
No limit |

Calculation of polarization |
Approximation |
No limit |

Speed of calculation |
Fast (analytical solution) |
Slow (solution of eigenvalue and simultaneous equations) |

Limitations of environmental materials |
Only the same refractive index as the hologram |
No limit |

## Usage Guide for the RCWA Hologram DLL

A DLL (Dynamic Link Library) has been developed for simulating holograms using the RCWA method, and users can download it from the attachment. In this section, we will explain the setup procedure using the example file 'test_example1.zmx' provided in the attachment. First, you can open the 'Diffraction Property' using objects like 'Hologram lens.' Within the NSC editor, this object allows you to configure parameters such as position, shape, and material, while other parameters related to holograms are adjusted within the 'Diffraction Property' section. It is important to note that hologram parameters configured within the NSC editor are ineffective. The following section will provide a detailed explanation of each parameter.

- DLL: Please select 'vhg_constant_hologramXXXXXX.dll,' where XXXXXX represents the version name.
- Start Order and Stop Order refers to the beginning and ending orders of diffracted rays. Orders outside this range will not be traced.
- Holo type: 1 signifies both construction beams as either diverging or converging rays simultaneously. 2 denotes one construction beam as diverging and the other as converging. This definition aligns with Hologram 1 and Hologram 2 in this article. Currently, this DLL only supports the scenario where the construction beams are parallel rays. For simplicity, it is recommended to set this parameter to 2.
- For the definitions and detailed explanations of Max Order, Use Coat file, Rotate Grating, Interpolation, Only these orders, and Stochastic mode parameters, please refer to this article.
- x1, y1, z1, x2, y2, z2: These represent the starting coordinates of the two generated rays. For example, in the figure below, the starting point of Construct Ray 1 is (1, 0, 1), generating green parallel rays. Construct Ray 2 starts at (0, 0, 1), generating pink parallel rays. The interference result is shown by the brown lines, and a detailed distribution can be viewed using the 'Programming … User Extensions … RCWAvisualization.exe' as explained in the Visualization tool section of this article.
- n1, n2: The refractive indices of the materials where construction beam 1 and construction beam 2 originate.
- Holo index, Index modulation (dn): Refractive index modulation follows the below formula where n0 represents the Holo index, and ∆n denotes the Index modulation (dn).
- Layer per period: This parameter determines how many layers are sliced in the Z-direction for each period. A higher value signifies more accurate calculations but may also require a longer computational time.

## Example Simulations of EPE Systems

In the following articles, we demonstrate the construction of optical systems for EPE simulations using different optical elements:

- Using SRG-based components to create an EPE optical system within an AR system: How to simulate exit pupil expander (EPE) with diffractive optics for augmented reality (AR) system in OpticStudio
- Demonstrating the construction of an EPE optical system using VHG calculated based on the Kogelnik method: Building Exit Pupil Expansion (EPE) based volume holographic gratings (VHG)

In this article, we utilize the RCWA method introduced in these articles to calculate VHG components and construct an EPE optical system. For detailed information about EPE, please refer to the preceding two articles. The example in this article employs the same setup, with a two-dimensional QR code pattern slide object serving as the light source, along with lenses. Rays pass through three VHG components calculated using the RCWA method DLL introduced in this article. Finally, a lens and a detector are used to simulate the human eye."

The simulation results are shown in the right image of the figure below. Simultaneously, the left image in the figure represents the results obtained using the Kogelnik method. The simulation assumptions of the Kogelnik method assume that the VHG has the same refractive index as the surrounding environment. However, in the EPE system, the refractive index of the waveguide plate differs from that of the VHG. Therefore, in theory, the RCWA method provides a more accurate representation. In the simulation results obtained with the RCWA method, oscillations in intensity distribution can be observed, which are absent in the results from the Kogelnik method. This difference illustrates the necessity of employing the RCWA method when simulating scenarios where the refractive index of the VHG differs from that of the surrounding environment.

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