**Last Update: 2023-08-29**

In this article, an example is demonstrated to set up an exit pupil expander (EPE) using the RCWA tool for an augmented reality (AR) system in OpticStudio. The planning of gratings in k-space (optical momentum) is first explained, and the details of setting up each grating is discussed.

**Authored By Michael Cheng**

## Introduction

This article is part 1 of 4 articles, which introduces the concept of k-space and discusses how to plan for an exit pupil expander design based on this concept. Links to other parts are listed below for reader’s convenience.

The system introduced in this article includes gratings. The diffraction grating efficiency is modeled by the RCWA DLL. The details of the RCWA tool are not discussed in this article. Readers can check the following knowledgebase article for more information.

Simulating diffraction efficiency of surface-relief grating using the RCWA method

## Exit Pupil Expander

Exit pupil expansion (EPE) is one of the common techniques used in waveguide-based AR systems. In Figure 1, an ideal system is shown where light is coupled into the waveguide with a surface-relief grating (SRG) and coupled out from the waveguide with another SRG. Ideally, the design should let beams from each field overlap at the exit pupil so that that the eye can receive the full image better.

Figure 1 An ideal structure of a waveguide-based AR system where output light from each field can better fill the eye’s pupil.

However, without adequate design, the light from each field can never overlap at the pupil of the eye as shown in Figure 2.

Figure 2 Without adequate design, light coming out from the waveguide usually diverge and can never overlap at the eye’s pupil.

This is where pupil expansion can be useful. As shown in Figure 3, by properly arranging all the elements in the system, when light hits the out-coupling grating, beams can partially continue propagating inside the waveguide and partially be coupled out. This results in the exit pupil is expanded, and therefore the structure is often called an exit pupil expander (EPE). With an EPE, light from each field can overlap at an area where the eye pupil is supposed to be placed to see the whole image. This area is also called an eye box.

Figure 3 A waveguide system with pupil expansion in one dimension.

Note that Figure 3 only shows an example of EPE in one dimension, which is practically not useful because light at the other dimension still diverges and thus limits the available field of view (FOV). In this article, we demonstrate how a system with 2-dimensional pupil expansion, like in Figure 4, can be set up.

Figure 4 A waveguide system with 2-dimensional pupil expansion.

## Plan the grating period and direction

In this section, the concept of k-space is introduced. The k-space is a very useful tool for planning the arrangement of gratings in a waveguide.

For a ray, one can define its wave vector , where (L,M,N) is a unit vector, λ0 is the wavelength in vacuum, and n is the refractive index of the material where the ray is propagating in. To analyze and plan the grating parameters for the EPE, it’s more useful to consider the normalized wave vector , where is the wave vector of the same ray in vacuum. Note this normalized wave vector n*(L,M,N), is called optical momentum in Hamiltonian optics. Since the vector component N can always be calculated from L and M with N=sqrt(1-L^2+M^2), any single ray can be fully represented by the x and y components of its normalized wave vector (nL, nM) without any loss of information, as shown at the right side of Figure 5. For convenience, the space formed by (nL, nM) is called k-space in this article.

Figure 5 Concept of k-space.

There are many interesting and useful characteristics in k-space, including:

- All the possible ray propagation direction in a medium form a circular area in k-space. The radius of this circle is equal to the refractive index of the material.
- When a ray refracts from one media to the other, this ray’s position in k-space is unchanged. This follows the Interface conditions for electromagnetic fields.
- If a ray refracts from a region with higher index and its position in k-space is larger than that in the next region, total internal reflection (TIR) happens because there is no possible propagation direction for the next media, as shown in lower right side in Figure 6.

Figure 6 All available ray propagations in a material can be represent as a circular area with radius equal to refractive index in the k-space. Refraction doesn’t move the ray’s position in k-space. If a ray’s position in k-space is larger than the available circle in next media, the ray cannot transmit and undergoes TIR.

- What a grating does to a ray is to move the ray’s position in k-space by a vector , where λ0 is the wavelength in vacuum, m is diffraction order, Λ is period of the grating, and (fx,fy,fz) is a unit vector represent the direction of periodicity of the grating. Note that while the diffraction order can be any integer number as shown in Figure 7, in most of cases, the grating is simply designed for -1 or +1 order.

Figure 7 If a ray is diffracted by a grating, its position in the k-space is moved by a vector of .

In this article, 3 gratings are used to build the EPE. These 3 gratings will couple the rays from air to the waveguide, turn the rays’ direction inside the waveguide, and then couple the rays from the waveguide back into air.

As shown in Figure 8, the light source emits from air and the whole FOV is represented as an area in k-space. The function of first grating is to move the whole FOV to the TIR zone, which is the area between the first and second k-space circles.

In this example, the grating value is designed as below.

- The index of the waveguide is 1.8, which is also the radius of the outer circle in k-space.
- Assume the EPE is designed for a wavelength of 0.55 µm (in vacuum).
- The first grating will move the ray by an amount of +1.4 in x direction in k-space. This means a ray normally incident on the grating, which is at (0,0) in the k-space, will be moved to the central place between the inner (radius = 1.0) and outer circle (radius = 1.8) in k-space. Since the amount of movement by a grating in k-space is , the period of the grating is then 0.55/1.4=0.393 µm, assuming the diffraction order m is +1.
- The function of the second grating is turning the rays’ propagation by 90 degrees. In k-space, this means the grating moves the ray by a vector of 1.4*sqrt(2)*(-1/sqrt(2),-1/sqrt(2)), as shown by the “2nd” arrow in the Figure 8. Similarly, this means the period of the second grating is 0.55/1.4/sqrt(2)=0.278 µm. And the grating direction should be rotated by 45 degrees.
- The third grating is similar to first one. Its period is same as the first grating, but the orientation should be rotated by 90 degrees relative to the first grating. This grating can restore the ray position in k-space, for example, from (0,-1.4) in k-space back to (0,0), as when it’s coupled in. As a result, the ray can exit the grating with exactly the same angle as when it first reached the waveguide.

Note that k-space can only describe how the ray’s propagation direction is changed by each grating. It doesn’t describe how the grating should be placed on the waveguide. In this article, the gratings will be placed as shown in the right side of Figure 8.

Figure 8 At the left side, it can be seen how an incident ray’s propagation direction is rotated by each grating. On the right side, it shows how the ray propagates and how each grating is arranged in the waveguide.

**Other design**

In this article, we mainly introduce an EPE deisgn with 3 one dimensinoal gratings. It is also possible to design the EPE waveguide and only 2 gratings (1D + 2D). In this case, the 2D grating will be hexagonal periodic and it moves the rays in a hexagonal way as below. The principle of how a grating changes the ray propagation in this design is exactly same. The difference of the following design compared to the design talked above is that we can use a single 2D grating to replace two 1D gratings. We will not talk more about this design in this series of articles. If users are interested in this type design, more details are introduced int Exit Pupil Expander with 1D-2D Gratings.

**Next Article:** How to simulate exit pupil expander (EPE) with diffractive optics for augmented reality (AR) system in OpticStudio: part 2

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