Recent research in augmented reality (AR) eyewear has prompted interest in using volume holographic optical elements for this application. However, many sensing operations in AR systems require the use of wavelengths in the near-infrared (NIR) (750 to 900 nm). These wavelengths typically exceed the sensitivity range of available commercial holographic recording materials (450 to 650 nm), which complicates the design of optical elements with power since significant aberrations result when the reconstruction wavelength differs from the construction wavelength. Several methodologies for designing a waveguide hologram imaging system in NIR are reviewed and evaluated. The design approach presented in our work integrates the most effective practices such as fabrication point source location optimization and aberration analysis to realize effective holographic waveguide couplers formed with visible wavelength light and reconstructed in the NIR. The technique is demonstrated by designing and fabricating an input waveguide hologram in conjunction with a multiplexed output coupling hologram. The resulting input/output waveguide holograms can achieve an image resolution of (∼3 lp / mm) with a 0.6-mm-thick glass substrate that has a refractive index of 1.8.

## 1.

## Introduction

Recently, there has been an increase in the interest of eye-tracking methods for augmented reality (AR) and mixed reality systems.^{1}^{–}^{8} Eye-tracking is important for AR eyewear as it improves high image fidelity and contrast across the field of view (FOV) without excessive demands on the power of the projection system. This in turn leads to longer battery lifetimes and greater utility of the AR system. The human eye most sensitive to the light falling onto the fovea region of the retina, where a high density of cone photoreceptors is located.^{7}^{,}^{9} Since the eye is constantly moving while viewing a scene, an eye-tracking system is required to monitor eye movement in real time to adjust the high-resolution image rendering around the fovea region.^{3}^{,}^{9} Different eye-tracking systems have been investigated, among them the video-oculography that functions by monitoring the eye position with a camera system and is a good candidate with relatively high resolution and accuracy.^{10}^{–}^{13}

Compared to the traditional bulky free-space video-oculography-based tracking system, the waveguide eye-tracking system has a more compact format, but it has a much lower resolution and efficiency.^{1}^{–}^{8} Volume holographic optical elements (HOEs) are well suited to AR eyewear applications.^{14} This results from the ability to fabricate components that realize complex optical functions (such as high efficiency) in relatively thin (2 to $50\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$) films that can be deposited on either flat or curved surfaces.^{15} Two promising materials for AR eyewear are dichromated gelatin (DCG)^{16}^{–}^{18} and dry-processed Covestro photopolymers (CPP).^{19}^{,}^{20} However, many sensing operations in AR systems require the use of near-infrared (NIR) wavelengths in the 750- to 900-nm range. These wavelengths typically exceed the normal sensitivity range of DCG (350 to 550 nm) and PP materials (450 to 650 nm).^{16}^{,}^{17}^{,}^{19}^{–}^{21} This complicates the design of optical elements that have focusing power since significant aberrations result when the reconstruction wavelength differs from the construction wavelength. Several solutions have been proposed to resolve this problem, including point source optimization, recursive aberration correction, and preaberration of the construction beams.^{22}^{–}^{27} For mass production of substrate-mode holograms that couple light into waveguides with optical power, the simplest construction method is desirable and suggests the use of point source optimization with a single exposure.

The purpose of this work is to present and verify a point source location optimization design methodology for realizing substrate-mode volume holographic lenses that couple light into AR eyewear substrates and are suitable for eye motion-sensing applications. The method uses nonsequential raytracing to control image aberration combined with coupled-wave analysis^{28}^{,}^{29} describing hologram diffraction efficiency. The design method is experimentally verified in a commercially available photopolymer with a substrate-mode holographic lens formed with light at 532 nm and reconstructed at 850 nm.

## 2.

## System Design

## 2.1.

### Design Configuration

The basic geometrical configuration of the eye-tracking system is illustrated in Fig. 1. A 1951 USAF target is used as the object. ${n}_{w}$ and ${n}_{H}$ are, respectively, the refractive index values of the waveguide and HOE. ${d}_{H}$, ${d}_{w}$ are the thickness values of the waveguide and HOE. ${w}_{H}$ is the width of the holographic lens and ${d}_{0}$ is the distance from the eye to the HOE. For convenience, the reconstruction wavelength is called ${\lambda}_{2}$ and the construction wavelength is ${\lambda}_{1}$.

The first step of the design process is to assume operation at the reconstruction wavelength in the NIR and unfold the rays propagating through the waveguide substrate. The in-coupling HOE is designed to have a focusing power that converges light at a particular wavelength from a collimating state to a point with coordinates $({x}_{f},{z}_{f})$, as shown in Fig. 2. The light emanating from the object with an FOV angle of $\alpha $ is diffracted by a transmission type waveguide HOE at an angle larger than the critical angle of the waveguide.

In this type of holographic lens, the grating vector varies across the aperture and significantly reduces the recoupling of diffracted light by the hologram. This in turn leads to improved efficiency and contrast of the desired diffracted beam. The diffracted light that propagates through the waveguide at position ${x}_{f}$ is then diffracted out of the waveguide with a reflection or transmission type planar waveguide HOE and captured by an imaging system. An alternative way to couple the light out from the waveguide is by placing a prism that is index-matched to the waveguide at the location of the exit pupil. The focusing point of the in-coupling waveguide HOE with optical power (at ${\lambda}_{2}$) should satisfy the condition given by

## Eq. (1)

$${\mathrm{tan}}^{-1}\left(\frac{{x}_{f}-{w}_{H}/2}{-{z}_{f}}\right)\ge {\mathrm{sin}}^{-1}\left(\frac{1}{{n}_{H}}\right),$$## Eq. (2)

$${W}_{\text{exit}}=\sqrt{{x}_{f}^{2}+{z}_{f}^{2}}\left(\frac{\alpha \text{\hspace{0.17em}}\mathrm{cos}(\alpha )}{\mathrm{cos}({\theta}_{l})}\right),$$When the holographic lens has a focal lens of 30 mm, an FOV $\alpha =3.0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$, and a diffraction angle of 60 deg, then the resulting exit pupil size is $\sim 1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$.

## 2.2.

### Bragg Condition and Point-Source Optimization

Two factors must be simultaneously optimized in the design of a holographic lens—the diffraction efficiency and image quality. The diffraction efficiency is determined by how closely the volumetric Bragg condition is satisfied during the reconstruction process.^{28} The image quality is primarily dependent on the lateral component of the grating period within the holographic lens. The Bragg condition in this design is given by

## Eq. (4)

$${\overrightarrow{\mathbf{k}}}_{o}-{\overrightarrow{\mathbf{k}}}_{r}=\overrightarrow{\mathbf{K}}={\overrightarrow{\mathbf{k}}}_{i}-{\overrightarrow{\mathbf{k}}}_{p},$$## Eq. (5)

$$\overrightarrow{\mathit{K}}=\frac{2\pi}{\mathrm{\Lambda}}[\mathrm{sin}\text{\hspace{0.17em}}\phi \hat{\mathbf{x}}+\mathrm{cos}\text{\hspace{0.17em}}\phi \hat{\mathbf{z}}].$$${\overrightarrow{\mathbf{k}}}_{o}$ and ${\overrightarrow{\mathbf{k}}}_{r}$ are, respectively, the propagation vectors of the object and reference beams at ${\lambda}_{1}$, ${\overrightarrow{\mathbf{k}}}_{I}$ and ${\overrightarrow{\mathbf{k}}}_{p}$ are the propagation vectors of diffracted and reconstruction beams at ${\lambda}_{2}$. $\phi $ is the angle of the grating vector relative to the $z$ axis. The Bragg matching condition for this case is depicted in Fig. 3. When the volumetric Bragg condition is satisfied during the reconstruction, the diffraction efficiency of the diffracted light is maximized.

If the object and reference beams are two plane waves, the grating vector $\overrightarrow{\mathbf{K}}$ is a constant across the hologram aperture. In this case, while the HOE is formed and reconstructed at different wavelengths, the Bragg condition can be satisfied across the entire aperture by matching the construction and reconstruction conditions as depicted in Fig. 3. However, when a holographic lens is formed, the grating vector $\overrightarrow{\mathbf{K}}$ varies across the aperture. In this case, if the construction and reconstruction share the same wavelength, the holographic lens can be formed using one plane wave and a spherical wave (like a traditional holographic lens) without aberration. Another case is that if the wavelengths used in the construction and reconstruction are different, then the Bragg matching between the construction and reconstruction should be made for every point across the entire aperture of the HOE to ensure high diffraction efficiency during the reconstruction. For this condition, the construction beams are no longer straightforward tilted plane waves or spherical waves, and they are aberrated waves that cannot be generated easily with traditional optical systems.

Since the grating vector varies across the aperture, the ideal construction angles at ${\lambda}_{1}$ must also be varied. Several methods for optimizing this procedure have been proposed.^{29}^{–}^{32} The reconstruction wavefronts at ${\lambda}_{2}$ can be expressed as ${\varphi}_{p}(x,y,0)$ and ${\varphi}_{i}(x,y,0)$ corresponding to ${\overrightarrow{\mathbf{k}}}_{p}(x,y,0)$ and ${\overrightarrow{\mathbf{k}}}_{i}(x,y,0)$, where ${\varphi}_{p}(x,y,0)$ is a planar wavefront and ${\varphi}_{i}(x,y,0)$ is a spherical wavefront. The phase function required to form the holographic lens can be described as

This is the phase function introduced by the HOE. During reconstruction, if the incident wavefront is ${\varphi}_{p}(x,y,0)$, a diffraction wavefront with the phase ${\varphi}_{p}(x,y,0)$ is formed. Following the Bragg condition given in Eq. (4), ${\overrightarrow{\mathbf{k}}}_{r}(x,y,0)$ and ${\overrightarrow{\mathbf{k}}}_{\mathrm{o}}(x,y,0)$ at ${\lambda}_{1}$ can be determined for different $(x,y)$ coordinates within the hologram aperture. The two aberrated wavefronts at ${\lambda}_{1}$ required to record the HOE are ${\varphi}_{r}(x,y,0)$ and ${\varphi}_{\mathrm{o}}(x,y,0)$ and correspond to the required propagation vectors ${\overrightarrow{\mathbf{k}}}_{r}(x,y,0)$ and ${\overrightarrow{\mathbf{k}}}_{o}(x,y,0)$ for the construction process. In this step, the phase introduced by the holographic lens at ${\lambda}_{1}$ can be expressed as

Ideally, $\mathrm{\Delta}\varphi ={\varphi}_{\mathrm{HOE}}^{\prime}-{\varphi}_{\mathrm{HOE}}=0$, and this results in an aberration-free reconstruction with high diffraction efficiency. Also this indicates that in order to form an aberration-free holographic lens working at a wavelength different from the construction wavelength, the two construction beams must be preaberrated. Some methods, including recursive holographic recording with a set of point sources^{27}^{,}^{29}^{,}^{33} and generating preaberrated construction beams with a computer-generated hologram,^{22}^{,}^{29} have been proposed and explored. However, preaberrating the wavefronts for construction at ${\lambda}_{1}$ cannot be easily implemented due to the complexity of the phase functions. Therefore, instead of generating the sophisticated aberrated wavefronts, a point-source approximation method that accomplishes both Bragg matching and sufficient image resolution for the eye-tracking application is used to calculate the approximate wavefronts for the construction process.

After completing the wavefront functions for the construction beams at ${\lambda}_{1}$, spherical wavefronts generated from two point-sources $P1({x}_{P1},{z}_{P1})$, $P2({x}_{P2},{z}_{P2})$ are used to approximate the two aberrated wavefronts. The phase difference between the aberrated construction waves and the approximate construction waves can be specified as

## Eq. (8)

$${W}_{\mathrm{WFEr}}({x}_{P1},{z}_{P1})={\sum}_{j=1}^{M}{\sum}_{i=1}^{N}[{\varphi}_{r}({x}_{i},{y}_{j},0)-{\varphi}_{P1}({x}_{i},{y}_{j},0)],$$## Eq. (9)

$${W}_{\mathrm{WFEo}}({x}_{P2},{z}_{P2})={\sum}_{j=1}^{M}{\sum}_{i=1}^{N}[{\varphi}_{o}({x}_{i},{y}_{j},0)-{\varphi}_{P2}({x}_{i},{y}_{j},0)],$$Once the waveguide parameters are known, the wavefronts for the two construction beams can be computed using Eqs. (8) and (9) to determine the approximate construction point locations that result in the smallest reconstruction wavefront error. A representative set of parameters for an eye-tracking waveguide hologram are given in Table 1. Figure 5 shows two contour maps of ${W}_{\mathrm{WFE}}$ as a function of the locations of the point sources $P1({x}_{P1},{z}_{P1})$ and $P2({x}_{P2},{z}_{P2})$ for this design. Figure 5(a) gives ${W}_{\mathrm{WFEo}}({x}_{P2},{z}_{P2})$ of the object beam and Fig. 5(b) illustrates ${W}_{\mathrm{WFEr}}({x}_{P1},{z}_{P1})$ of the reference beam. The two points corresponding to the construction waves at ${\lambda}_{1}$ with the minimum wavefront error are found to be $P1(33.0,185.0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm})$ and $P2(30.6,33.6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm})$.

## Table 1

The parameter values used in the optimization.

Parameter | Symbol | Value |
---|---|---|

Reconstruction wavelength | ${\lambda}_{2}$ | 850 nm |

Construction wavelength | ${\lambda}_{1}$ | 532 nm |

Reconstruction angle | ${\theta}_{r}$ | 0° |

Designed focal point | $({f}_{x},{f}_{z})$ | (20.3, 26.6 mm) |

Hologram width | ${w}_{H}$ | 16 mm |

Hologram refractive index | ${n}_{H}$ | 1.50 |

Based on the optimized two-point source locations, the first 20 Zernike coefficients describing the wavefront errors between the approximate wavefronts and the object/reference wavefronts are calculated and plotted in Fig. 6. The wavefront errors for both the reference and object beams are dominated by the sixth and eighth Zernike polynomials corresponding to astigmatism and coma. The third Zernike term refers to the large tilt of the reference beam in the $x$ direction. In addition, it is apparent that the object beam wavefront error is much larger than the reference beam since it has a larger numerical aperture (NA). The wavefront aberration compensation with optical elements such as a spatial light modulator can base on these calculated Zernike polynomials.

The resulting optimized construction point source locations $P1(33,185\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm})$ and $P2(30.6,33.6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm})$ formed at ${\lambda}_{1}$ are shown in Fig. 7(a). A plot of the ideal grating vectors across the aperture of the HOE and the grating vectors formed with the optimized point source locations are shown in Fig. 7(b). In this case, the difference between the two grating vectors increases at the edge of the aperture.

The diffraction efficiency for this hologram as a function of incident angle at a wavelength of 850 nm is shown in Fig. 8(a) and the hologram parameters for the plots are given in Table 2. Each curve indicates the diffraction efficiency at a different position across the hologram aperture and shows that significant diffraction efficiency (20% to 80%) can be maintained across the entire hologram aperture. The diffraction efficiency profiles of the reference beam shown in Fig. 8(b) indicate significant overlappings that can introduce ghost images.

## Table 2

Parameters used in the DE calculation.

Parameter | Symbol | Value |
---|---|---|

Reconstruction wavelength | ${\lambda}_{2}$ | 850 nm |

Construction wavelength | ${\lambda}_{1}$ | 532 nm |

Construction point 1 | $P1$ | (33, 185 mm) |

Construction point 1 | $P2$ | (30.6, 33.6 mm) |

Width of HOE | ${w}_{H}$ | 16 mm |

Refractive index of HOE | ${n}_{H}$ | 1.50 |

Index modulation | ${n}_{1}$ | 0.022 |

HOE material thickness | ${d}_{H}$ | $16\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ |

## 2.3.

### Output HOE Design

The first step in the design of the holographic waveguide output coupler is to determine the angular range of the rays diffracted by the incident hologram coupler and propagating through the waveguide. Using the parameters for the input coupling hologram given in Table 2, the angular range of rays within the waveguide varies from 45 deg to 58 deg. This indicates that an angular bandwidth of 13 deg is required within the waveguide material. For a transmission HOE with a thickness of $16\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, a refractive index modulation of 0.02, two construction angles of {0 deg, 50 deg}, a wavelength at 850 nm for construction and reconstruction, the FWHM of the DE is only 2.2 deg, and the null-to-null angular bandwidth is only $\sim 3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ within the HOE material. This range is insufficient for eye-tracking applications.

There are several ways to increase the angular bandwidth of the hologram. The simplest method is to decrease the hologram thickness while increasing the refractive index modulation amplitude. However, this is not always possible with commercially available materials such as the CPP, which has a limited range of thickness and index modulation values. Another method is to multiplex several holograms with the same lateral grating period at a specific location along with the hologram aperture. In order to increase the angular bandwidth to 13 deg, at least five holograms must be multiplexed with the $16\text{-}\mu \mathrm{m}$ thick Covestro material. In addition, since this material has a maximum refractive index modulation of 0.03, each grating can only share a refractive index modulation of 0.006. Figure 9(a) shows the DE profiles of each multiplexed grating with the construction angles given in Table 3, where angles are measured relative to the normal of the HOE surface and assumed to be within the same medium as the HOE. The HOE has a thickness of $16\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, and each grating has a refractive index modulation of 0.006. The step angle between two adjacent reconstruction angles is 2.5 deg, which was found to provide high diffraction efficiency and low cross-coupling between gratings.

## Table 3

Angles used in the multiplexing (within the HOE).

Terms | Reconstruction angles | Construction angles |
---|---|---|

Grating #1 | $\{-3.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg},47.0\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}\}$ | {6.5 deg, 37.3 deg} |

Grating #2 | $\{-1.6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg},49.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}\}$ | {8.3 deg, 39.6 deg} |

Grating #3 | {0.0 deg, 52.0 deg} | {10.1 deg, 41.9 deg} |

Grating #4 | {1.5 deg, 54.5 deg} | {11.8 deg, 44.2 deg} |

Grating #5 | {2.9 deg, 57 deg} | {13.4 deg, 46.5 deg} |

Simulation of the resulting image with the multiplexed output coupling hologram is shown in Fig. 9(b) when used with the in-coupling HOE fabricated with the optimized two point-sources and parameter values given in Table 4. As indicated, the output coupling hologram with five multiplexed gratings covers the angular bandwidth of the rays diffracted by the input coupler. However, the image intensity is not uniform across the aperture in the $x$ direction due to the variation of the DE profiles [Fig. 9(a)] and recoupling effects. An image resolution of $\sim 3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{lp}/\mathrm{mm}$ with degraded contrast (ghost images) can be achieved, which matched well with the modeling. The ghost images are partly caused by the sidelobes of the in-coupling HOE, the insufficient holographic focusing power, and noise introduced by the out-coupling HOE.

## Table 4

Parameters used in the waveguide system with an output HOE.

Parameter | Symbol | Value |
---|---|---|

Wavelength of illumination | ${\lambda}_{2}$ | 850 nm |

Half field of view | HFOV | 2.5 deg |

RI of HOE | ${n}_{H}$ | 1.50 |

RI of waveguide | ${n}_{w}$ | 1.80 |

Thickness of waveguide | ${d}_{w}$ | 0.6 mm |

Thickness of HOE | ${d}_{H}$ | $16\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ |

Position of USAF target | ${d}_{o}$ | 20 mm |

Position of the out-coupling HOE | ${d}_{\mathrm{OH}}$ | 29 mm |

Grating periods of output HOE | ${\mathrm{\Lambda}}_{x}$ | 709.7 nm |

Width of the out-coupling HOE | ${w}_{\mathrm{OH}}$ | 2 mm |

Width of the in-coupling HOE | ${w}_{H}$ | 16 mm |

Focal length of imaging lens | ${f}_{i}$ | 5 mm |

## 3.

## Experimental Results

## 3.1.

### In-Coupling HOE Fabrication

In order to demonstrate the effectiveness of the hologram design process, experimental holographic waveguide couplers were fabricated and compared to the model results. The holograms were recorded in the CPP with a thickness of $16\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ and a maximum index modulation of 0.03. The point source locations were optimized for minimum aberration in the reconstructed image at a wavelength of 532 nm. The geometrical configuration of the recording setup is illustrated in Fig. 10(a), where two right-angle prisms with side lengths of 50 and 33 mm are used to couple the construction beams into the CPP film. Prism #1 is used to couple the object beam onto the hologram, and prism #2 is used to in-couple the reference beam. The point source locations are computed for construction in the air with a waveguide refractive index of 1.50. A specialized prism mount was fabricated to hold the two prisms using a 3D printer, as shown in Fig. 10(b). The locations of the point-source locations are marked on two rods extending from the mount. The experimental recording setup is shown in Fig. 10(c), where two aspherical lenses with NAs 0.25 and 0.50 are used to generate the two construction wavefronts. In addition, an ND filter with variable density values is placed between prism #1 and lens 2 to uniformly adjust the beam ratio across the aperture since the object beam has different irradiance values across its aperture.

Figure 11 shows the spectral transmittance of the simulation and experiment computed and measured at the center of the HOE. The experimental parameters for the HOE have a thickness of $16\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, a refractive index modulation of 0.022, a reconstruction angle of 0 deg. The measured spectral transmittance shown in Fig. 11(a) agrees well with the simulation results when the HOE is reconstructed at the construction beam angle. Note that in order to maximize the diffraction efficiency at 850 nm that the hologram is overmodulated at the construction wavelength of 532 nm. Likewise, when the HOE is reconstructed at the designed reconstruction angle (0 deg), the measured and modeled spectral transmittances shown in Fig. 11(b) indicate a decrease in the peak DE of the experimental hologram compared to the simulated result. This difference is mainly due to the absorption of the holographic material and the resolution limitation of the transmittance measurement. The refractive index modulation used in the data fitting is 0.022. This indicates that the overall system efficiency can approach 70% if there is no loss during the out-coupling step.

## 3.2.

### Out-Coupling HOE Fabrication

To experimentally record the output coupling HOE, a recording system consisting of two high-precision translation stages and two rotation stages is used, as shown in Fig. 12(a). The construction angles in the air are given in Table 3 and are set using precision stages that have 0.1-deg resolution capability. The intensities of the two recording beams are adjusted to be nearly equal using ND filters. During the recording process, grating #3 with the central DE profile shown in Fig. 9(a) is formed first with the designed angles, then mirror 1 and the 3D printed platform holding the prism are rotated to the desired angles to form grating #2. After each recording, translation stages 1 and 2 are shifted to make the two beams overlap on the same recording area. The third through fifth gratings are recorded in a similar manner sequentially. The two recording beams are blocked by an optical shutter when the stages are repositioned; backlash issues in the mounts are avoided by only moving stages in one direction.

With construction angles given in Table 3 and the recording setup illustrated in Fig. 12(a), an out-coupling HOE with five multiplexed gratings within the same volume of hologram material is formed, and the resulting transmittance profiles as a function of wavelength are shown in Fig. 12(b). The incident angle selected during measurement is chosen to be the reference beam angle (10.1 deg) for the construction (grating #3), and the DE profile indicates a significant broadening in the spectral bandwidth of the HOE compared to that of a single grating depicted with the dashed curve (in yellow). With the construction angles given in Table 3 and a refractive index modulation of 0.006, a $16\text{-}\mu \mathrm{m}$-thick HOE has a peak DE of $\sim 40\%$ at 532 nm and $\sim 20\%$ at 850 nm. The simulated DE in Fig. 12(b) utilizes five different refractive index modulation values from left to right of 0.004, 0.0047, 0.0054, 0.0059, and 0.006. Hence, the experimental data agree well with the calculated results using approximate coupled-wave analysis 28.

With the designed and fabricated in-coupling and out-coupling HOEs, the measured images are shown in Fig. 14 (with the testing setup shown in Fig. 13). For this measurement, the entire FOV is scanned using a CCD camera from left to right across the output hologram aperture. The resolution in the $x$ direction is lower than the $y$ direction since diffraction occurs in the $x$ direction. The scanned images shown in Fig. 14 indicate a resolution of $\sim 3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{lp}/\mathrm{mm}$, which agree well with the modeled image results. The aberration dominating the image degradation is the astigmatism, which has been explored and discussed in our previous work.^{8}

## 3.3.

### Discussion

The image quality from the experiment consisting of the in-coupling HOE fabricated with the two point-sources and the out-coupling HOE recorded with five multiplexed holograms agrees well with the simulated result that has a resolution of $\sim 3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{lp}/\mathrm{mm}$. In addition, the system has an average efficiency of $\sim 15\%$ operating at 850 nm. The diffraction efficiency could be improved to $>80\%$ if the out-coupling HOE is designed to have a DE of 100%. (It is only 20% at 850 nm in this work due to the limited index modulation range.) Notice that in this design, a thicker or lower refractive index of the waveguide can also improve image quality by increasing the critical angle in the waveguide, decreasing the number of ray reflections, and decreasing recoupling with the input coupling hologram.

However, several issues still exist, including ghost images, waveguide wedge effects, and distortion. Ghost images are partly introduced by the sidelobes of the HOE and the insufficient focusing power. The sidelobes can be decreased by recording the HOE with a refractive index modulation value less than the value corresponding to the peak DE. Furthermore, the focusing power can be increased with a more significant rotation of the grating vector across the hologram aperture. Moreover, the tilt and distortion issues can be corrected by introducing aberrated wavefronts for construction predicted by the Zernike polynomials decomposition in Sec. 2.2.

## 4.

## Conclusions

In this work, a design methodology for eye-tracking waveguide holograms is presented that includes, wavefront/diffraction efficiency optimization using point-source location optimization, wavefront aberration analysis, and nonsequential raytracing. The design method is verified by forming an experimental holographic input coupling lens in a $16\text{-}\mu \mathrm{m}$-thick photopolymer deposited on a 0.6-mm-thick glass substrate with a refractive index of 1.80. The tested result shows an average of 70% in-coupling efficiency. In addition, an out-coupling waveguide HOE multiplexed with five gratings is designed and fabricated that has an average DE of 40% at 532 nm and 20% at 850 nm. The waveguide eye-tracking system consisting of the in-coupling and out-coupling HOEs is modeled with nonsequential raytracing and experimentally tested. The resulting image has a resolution of $\sim 3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{lp}/\mathrm{mm}$ with an average efficiency of 15%. Furthermore, the primary aberrations of the holographic lens are astigmatism and coma, which can be partly corrected with preaberrated wavefronts for construction. The $3\text{-}\mathrm{lp}/\mathrm{mm}$-resolution is a good representation of the potentials of the holographic waveguide eye-tracking system, based on what the improved system with higher image quality, contrast, and efficiency can be pursued further in future work. Overall, the straightforward waveguide HOE design methodology has the potentials to fabricate holographic waveguide couplers and systems operating at a wavelength that differs from the construction wavelength.

## References

## Biography

**Jianbo Zhao** received his BS degree in physics from Nankai University and his MS degree in optics from James C. Wyant College of Optical Sciences in 2015 and 2018. He is a PhD student at the University of Arizona. He is working on improving solar and AR systems’ performance by designing volume holographic optical elements under the instruction of Dr. Raymond K. Kostuk.