# See-Through Near-Eye Display with Built-in Prescription and Two-Dimensional Exit Pupil Expansion

^{1}

^{2}

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## Abstract

**:**

## Featured Application

**The proposed see-through near-eye display is applicable for the augmented/mixed reality devices, e.g., smart glasses or headsets etc. In addition to being a wearable computing device, it can be used as a pair of everyday eyeglasses with a built-in prescription.**

## Abstract

^{2}. Besides, the prescription for correcting the refractive errors can be integrated as well. The design rules are set forth in detail, followed by the results and discussion regarding the efficiency, field of view (FOV), exit pupil, angular resolution (AR), modulation transfer function (MTF), contrast ratio (CR), distortion, and simulated imaging.

## 1. Introduction

## 2. Design Rules

#### 2.1. Proposed Structure

_{1}, W

_{2}, H, T, and γ are the front width, back width, height, thickness and bevel angle of plano-concave lens, respectively.

#### 2.2. Plano-Concave Lens

_{r}the real object distance, and s′ the image distance. For the real image, rays emitted from the real object were diverged by the plano-concave lens so as to offset the over-focusing of the eye. A “real” image―by which we mean that it is the image of a real object, albeit this image is technically virtual―was formed at a closer distance. The object distance s, image distance s′ and diopter or optical power of the lens P shall be correlated via the lens-maker’s equation [18]

^{−1}when s

_{r}= ∞ m and s′ = −0.333 m. It should be mentioned that the “real” image was not observed by the eye. Rather, the eye saw the real object, as the rays derailed by the lens got back on track through the accommodation of the eye [19].

_{1}of the lens rested partly upon the interpupillary distance d

_{ip}, which was on average 64 mm [20]. If the front surface of the lens is center-aligned with the eye, then the front width W

_{1}of lens is

_{b}is the width of the bridge of the smart glasses. Say d

_{b}= 18 mm, W

_{1}= 46 mm. Further, the front width W

_{1}, back width W

_{2}, thickness T, and bevel angle γ shall be correlated via

_{2}= 47.93 mm, and T = 7.47 mm. The height H was a freelance parameter as long as it met the ergonomics. Pursuant to the above design rules, a plano-concave lens can be tentatively designed with the parameters itemized in Table 1.

#### 2.3. Pico Projector

_{v,}and the focal length, f, of the projection lens. It would be straightforward to write the FOV of the pico projector as

_{m}is the size of the microdisplay. Say D

_{m}= 0.165 inch (4.191 mm) for both horizontal and vertical dimensions and s

_{v}= f = 10 mm, then FOV = 24° (horizontal) × 24° (vertical), i.e., 33° (diagonal). Once the FOV was given, the exit pupil measured at the eye relief (ER)―the distance starting from the last surface of the pico projector to the pupil of eye―could be determined with

_{p}is the aperture of the projection lens, and also the entrance pupil of our system. If the ER = 12 mm and A

_{p}= 6 mm, then EP = 1 mm, which is unacceptably small. Since there is not much room to further shorten the eye relief―especially for NEDs without a built-in prescription―the simplest way to expand the exit pupil was to scale up the projection lens. That being said, the wearability was compromised due to the added volume and weight. Table 2 lists the customized specifications necessary for designing the pico projector. Other than the values already mentioned, the resolution of the microdisplay was 640 × 640, the pixel size was 6.5 µm, and the contrast ratio (CR) was 100,000.

#### 2.4. Multiplexed Slanted Grating

_{g}the slant angle. To couple light into and out of the lens, both the ISG and OSG were constructed as the transmission gratings, whose allowable diffraction angles shall satisfy [22]

_{i}is the refractive index of the incident medium, n

_{e}the refractive index of the exit medium, θ

_{i}the incident angle (relative to the grating normal), θ

_{m}the diffraction angle of m

^{th}order, m the diffraction order, and λ the wavelength. As Equation (7) indicates, for a single-period slanted grating, both of its spectral and angular bandwidths were intrinsically narrow. To solve this issue, a multiple of waveguides with different gratings can be stacked together for the full color [15]. As an alternative solution, we resorted to the multiplexing of plural gratings on a single layer [23,24]. The multiplexed grating can be decomposed into three sub-gratings of the same slant angle but of different periods, widths and heights, as shown in Figure 5. Each sub-grating was designed for certain wavelength and incident angle. Say the periods of sub-gratings are p

_{A}, p

_{B}, and p

_{C}, respectively, then the collective period P

_{m}of the multiplexed grating shall be the least common multiple of the above three. To minimize the reflection at the interface between the grating and the lens, it was desirable to etch the slanted gratings out of the lens so that there is no mismatch in their refractive indices. Speaking of the fabrication, photolithography, electron-beam lithography, and focused ion beam are among the feasible lithography techniques [25].

#### 2.5. Exit Pupil Expansion

_{i}that can be written as

_{i}of the ISG. Say A

_{p}= 6 mm and α = 14.48°, A

_{i}= 13.25 mm. Let β symbolize the angle of beam incident to the OSG, as shown in Figure 7, where only the axial rays are depicted. The horizontal length of the elliptical exit pupil will be elongated to A

_{o}

_{i}= 3.8 mm and β = 75.52°, A

_{o}= 15.2 mm. As a result, the ISG is of 3.8 × 13.25 mm

^{2}and the OSG of 15.2 × 13.25 mm

^{2}. In case of misalignment, their actual sizes were supposed to be bigger. If the beam incident to the OSG is perpendicular to the bevel or the ISG, then β is equal to the bevel angle γ, which affects the thickness of the plano-concave lens. Incidentally, for the off-axis rays, problems such as pupil mismatch, vignetting, etc. merit special care.

## 3. Results and Discussion

#### 3.1. Simulation Settings

#### 3.2. Diffraction Efficiency

#### 3.3. Total Efficiency

_{i}the average DE of the ISG, DE

_{o}the average DE of the OSG, and A

_{p}the diameter of the entrance pupil. For our projection lens, f = 10 mm, A

_{p}= 6 mm, thus f# = 1.67. For the entire FOV at the wavelength of 565 nm, DE

_{i}= 14.4% and DE

_{o}= 10.5%, thus η = 0.55%.

#### 3.4. Field of View

_{r}, describes the angular extent of the curved front surface of the plano-concave lens, which is given by

_{r}is limited by the size of the plano-concave lens. As W

_{1}= 46 mm, H = 30 mm (see Table 1), and ER = 12 mm, FOV

_{r}= 133° (125° × 103°). The FOV of the virtual image, FOV

_{v}, hinges on both the pico projector and gratings. As the out-coupling angles of the ISG and OSG span over the input FOV of the pico projector, the output FOV

_{v}is therefore conserved as 33° (24° × 24°), which is comparable to HoloLens 1′s FOV [15].

#### 3.5. Exit Pupil

^{2}. Revisiting Figure 7 and Equation (6), the final exit pupil becomes

_{o/i}= 15.2 or 13.25 mm, and FOV = 24° × 24°, EP = 10 × 8 mm

^{2}. It shall be noted that although both the expanded and duplicated exit pupils can be calculated in the same way, the expanded exit pupil without gaps is undisputedly more solid than the duplicated exit pupils with gaps in between [8,9,10,11,12,13,14,15,16].

#### 3.6. Angular Resolution

_{h}and N

_{v}are the number of pixels along the horizontal and vertical directions, respectively. For FOV

_{v}= 33°, N

_{h}= 640, and N

_{v}= 640, AR = 2.19′.

#### 3.7. Modulation Transfer Function

#### 3.8. Contrast Ratio

_{o}is the CR of the real/virtual object. As the horizontal/vertical resolution is 640 and the FOV = 24°, the corresponding spatial frequency shall be 13.33 cycle/degree. According to Figure 10, at the field of 0° for the real image, CR

_{r}= 5 (CR

_{o}= ∞ and MTF = 0.69), while for the virtual image, CR

_{v}= 3 (CR

_{o}= 100,000 and MTF = 0.44).

#### 3.9. Distortion

#### 3.10. Simulated Imaging

## 4. Conclusions

^{2}. As opposed to the conventional EPEs, in which the exit pupil is duplicated or cloned into many, our solution leverages the oblique intersection of the viewing cone. More importantly, full color can be realized by the multiplexed gratings on a single layer. Other than the EPE, another benefit is the integration of prescriptions for a better user experience. Based on the simulation, its overall performance, including the efficiency, FOV, AR, MTF, CR, distortion, and simulated imaging have been investigated. For the real image, the FOV is 133° (diagonal), MTF is above 0.252 at 30 cycle/degree, CR is 5, and distortion is 0.02%. For the virtual image, the total efficiency is 0.55%, FOV is 33° (diagonal), AR is 2.19′, MTF is above 0.032 at 30 cycle/degree, CR is 3, and distortion is 3.4%. Merits aside, low efficiency, pupil mismatch, vignetting, etc. are identified as the remaining issues and will be dealt with in our future work.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the proposed see-through near-eye display (NED). W

_{1}, W

_{2}, H, T, and γ are the front width, back width, height, thickness and bevel angle of plano-concave lens, respectively.

**Figure 2.**Optical path diagram for imaging the real object. R is the radius of curvature of the front surface of lens, s

_{r}the real object distance, and s′ the image distance.

**Figure 3.**Optical path diagram of the pico projector. D

_{m}is the size of the microdisplay, s

_{v}the virtual object distance, f the focal length of the projection lens, A

_{p}the aperture of the projection lens, FOV the field of view, ER the eye relief, and EP the exit pupil.

**Figure 4.**Cross-sectional profile of slanted grating. p is the grating period, h the grating height, w the grating width, θ

_{g}the slant angle, θ

_{i}the incident angle (relative to the grating normal), and θ

_{m}the diffraction angle of m

^{th}order.

**Figure 5.**Decomposition of multiplexed grating into three sub-gratings. Say the periods of sub-gratings are p

_{A}, p

_{B}, and p

_{C}, respectively, then the collective period P

_{m}of the multiplexed grating shall be the least common multiple of the above three.

**Figure 6.**Illustration of the vertical expansion of the exit pupil by the in-coupling slanted grating (ISG). α is the angle between the optical axis of viewing cone and the plane of the ISG, A

_{p}the aperture of the projection lens, A

_{i}the vertical length of intersected exit pupil.

**Figure 7.**Illustration of the horizontal expansion of the exit pupil by the out-coupling slanted grating (OSG). β is the angle of beam incident to OSG, γ the bevel angle, W

_{i}the width of the ISG, A

_{o}the horizontal length of elongated exit pupil.

**Figure 8.**Diffraction efficiencies (DEs) of all possible diffraction orders combined of the ISG and OSG calculated with respect to the wavelength. The average DEs of the ISG and OSG over the entire spectrum (460 to 660 nm) are 48.0% and 16.7%, respectively.

**Figure 9.**Diffraction efficiencies of the ISG and OSG calculated with respect to the out-coupling angle for the wavelength of 565 nm. The average DEs of the ISG and OSG over the full horizontal/vertical FOV (±12°) are 14.40% and 10.53%, respectively.

**Figure 10.**Calculated MTFs of (

**a**) real and (

**b**) virtual images. At the spatial frequency of 30 cycle/degree, MTFs for all fields of the real and virtual images are above 0.252 and 0.032, respectively.

**Figure 11.**Calculated distortion with respect to the field angle. It can be seen that the distortions are 0.02% and 3.4% for the real and virtual images, respectively.

**Figure 12.**(

**a**) Original (photographer: C. P. Chen, location: Peterhof Grand Palace, St. Petersburg, Russia), (

**b**) real, and (

**c**) virtual images. Compared to the original one, the real image is virtually lossless, while the virtual image has a mild pincushion distortion and slightly decreased brightness.

Object | Parameter | Value |
---|---|---|

Plano-concave lens | W_{1} | 46 mm |

W_{2} | 47.93 mm | |

H | 30 mm | |

T | 7.47 mm | |

γ | 75.52° | |

P_{w} | −3 m^{−1} | |

n@565 nm | 1.5880 ^{1} | |

R | 0.1960 m |

^{1}Polycarbonate is chosen as the lens material.

Object | Parameter | Value |
---|---|---|

Microdisplay | D_{m} (diagonal) | 0.233 inch |

D_{m} (horizontal/vertical) | 0.165 inch | |

Resolution | 640 × 640 | |

Pixel size | 6.5 µm | |

CR | 100,000 | |

Projection lens | f | 10 mm |

FOV (diagonal) | 33° | |

FOV (horizontal/vertical) | 24° | |

Aperture | 6 mm | |

ER | 12 mm | |

EP | 1 mm |

Surface | Surface Type | Radius (mm) | Thickness (mm) | Refractive Index ^{1} | Semi-Aperture (mm) |
---|---|---|---|---|---|

real object | sphere | infinity | infinity | ||

1 | asphere | −196.0000 | 7.4700 | 1.5880 | 1.5000 |

2 | sphere | infinity | −336.2359 | 2.6760 | |

real image | sphere | infinity | 0 | 99.7948 |

^{1}Refractive index is left empty when the medium is air.

Surface | Surface Type | Radius (mm) | Thickness (mm) | Refractive Index ^{1} | Semi-Aperture (mm) |
---|---|---|---|---|---|

virtual image | sphere | infinity | infinity | ||

1 | asphere | 4.1494 | 3.6000 | 1.7258 | 3.0000 |

2 | asphere | 4.1379 | 10.0000 | 2.2399 | |

microdisplay | sphere | infinity | 0.0000 | 3.4613 |

^{1}Refractive index is left empty when the medium is air.

Surface | Y Radius (mm) | Conic Constant (K) | 4th Order Coefficient (A) | 6th Order Coefficient (B) | 8th Order Coefficient (C) |
---|---|---|---|---|---|

1 | −196.0000 | 0.0000 | 0.0003 | −0.0001 | 2.7720 × 10^{−5} |

Surface | Y Radius (mm) | Conic Constant (K) | 4th Order Coefficient (A) | 6th Order Coefficient (B) | 8th Order Coefficient (C) | 10th Order Coefficient (D) | 12th Order Coefficient (E) |
---|---|---|---|---|---|---|---|

1 | 4.1494 | −0.1154 | 0.0003 | 5.7069 × 10^{−7} | 6.9614 × 10^{−6} | −5.4048 × 10^{−7} | 2.1806 × 10^{−8} |

2 | 4.1379 | −0.6547 | 0.0055 | 0.0003 | 0.0003 | −6.7023 × 10^{−5} | 9.9782 × 10^{−6} |

Object | Sub-Object | Parameter | Value |
---|---|---|---|

ISG | Sub-grating A | p | 511 nm |

h | 1015.80 nm | ||

θ_{g} | 29.31° | ||

w | 55.67 nm | ||

Sub-grating B | p | 587 nm | |

h | 637.79 nm | ||

θ_{g} | 29.31° | ||

w | 272.30 nm | ||

Sub-grating C | p | 688 nm | |

h | 697.85 nm | ||

θ_{g} | 29.31° | ||

w | 244.51 nm | ||

Multiplexed grating | P_{m} | 19.95 μm | |

OSG | Sub-grating A | p | 317 nm |

h | 702.98 nm | ||

θ_{g} | 42.413° | ||

w | 128.67 nm | ||

Sub-grating B | p | 364 nm | |

h | 440.53 nm | ||

θ_{g} | 42.413° | ||

w | 237.36 nm | ||

Sub-grating C | p | 427 nm | |

h | 633.81 nm | ||

θ_{g} | 42.413° | ||

w | 310.07 nm | ||

Multiplexed grating | P_{m} | 12.39 μm |

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## Share and Cite

**MDPI and ACS Style**

Zhang, W.; Chen, C.P.; Ding, H.; Mi, L.; Chen, J.; Liu, Y.; Zhu, C.
See-Through Near-Eye Display with Built-in Prescription and Two-Dimensional Exit Pupil Expansion. *Appl. Sci.* **2020**, *10*, 3901.
https://doi.org/10.3390/app10113901

**AMA Style**

Zhang W, Chen CP, Ding H, Mi L, Chen J, Liu Y, Zhu C.
See-Through Near-Eye Display with Built-in Prescription and Two-Dimensional Exit Pupil Expansion. *Applied Sciences*. 2020; 10(11):3901.
https://doi.org/10.3390/app10113901

**Chicago/Turabian Style**

Zhang, Wenbo, Chao Ping Chen, Haifeng Ding, Lantian Mi, Jie Chen, Yuan Liu, and Changzhao Zhu.
2020. "See-Through Near-Eye Display with Built-in Prescription and Two-Dimensional Exit Pupil Expansion" *Applied Sciences* 10, no. 11: 3901.
https://doi.org/10.3390/app10113901