## Abstract

We present a new optical system for a combiner-type head-up display (HUD) without any asymmetrical elements by employing a confocal, off-axis, two-hyperboloid mirror as an aberration corrector. From an off-axial aberration analysis, we initially obtain an off-axis two-mirror system corrected for linear astigmatism and spherical aberration by configuring its parameters to satisfy the confocal condition. In addition, to compensate the down angle in the HUD, the image display plane is tilted to satisfy the Scheimpflug condition. This design approach enables us to easily balance the residual aberrations without asymmetrical components, which results in an excellent starting design. The final optical system for an HUD has a virtual image of 7.3 in at 2 m away from an eye box having an area of $130\times 50\text{\hspace{0.17em}}{\mathrm{mm}}^{2}$.

© 2019 Optical Society of America

## 1. INTRODUCTION

An optical system for a head-up display (HUD) is generally composed of an off-axis two-mirror system at finite conjugates. Because of the complexity of aberrations caused by the off-axis configuration, it is not easy to design a well-corrected off-axis reflective imaging system. To balance these non-symmetrical aberrations, most optical systems for an HUD have used asymmetric optical components such as a freeform surface or cylinder [1–3]. However, since freeform surfaces have a complex geometry, much effort and high cost are required to align and produce them. In addition, adding another asymmetric optical component may reduce the contrast, owing to the ray reflection on that element, and lead to increased costs.

Even though much research has been reported on aberration analyses and design methods for mirror systems, these studies have dealt with on-axis systems or infinite conjugate systems that are mainly used for telescopes [4–10]. Therefore, they do not provide a proper solution for an off-axis system working at finite conjugates, such as an HUD. To solve these problems, in this study, we present the expressions for zero-spherical-aberration, linear astigmatism–free, and Scheimpflug conditions by introducing an optical axis ray [11–13]. These formulas are applicable to an off-axis system working at finite conjugates.

In this study, we propose a new optical system for an HUD without any asymmetrical elements by employing a confocal, off-axis, two-hyperboloid mirror as an aberration corrector. To initially achieve a reasonable and compact mirror system corrected for spherical aberration and linear astigmatism at the given conjugates, we analytically examine the optical parameters for an HUD. Next, the image plane of this off-axis mirror system is tilted to compensate the down angle of an HUD by satisfying the Scheimpflug condition. This design approach enables us to obtain a compact HUD with better performance even though there are no asymmetrical components. Thus, the key point of this study is an optical system for an HUD without any asymmetrical elements.

## 2. ABERRATIONS OF SINGLE OFF-AXIS MIRROR SYSTEM AT FINITE CONJUGATES

#### A. Single Off-Axis Hyperboloidal Mirror Corrected for Spherical Aberration at Finite Conjugates

This section introduces a single convex hyperboloidal mirror with the conic constant $K$, which is designed to correct spherical aberration at finite conjugates, as shown in Fig. 1.

The standard equation for a hyperbola with center (0, 0) is given by [13]

where $a$ is the half length of the transverse axis, $b$ is the half length of the conjugate axis, and $F$ and ${F}^{\prime}$ are the first and second foci of a hyperbola, respectively. In addition, from the definition of a hyperbola, we obtain In Fig. 1, $l$ is the object distance $(l<0)$ from a point $P$ on the mirror to the first focus, and ${l}^{\prime}$ is the image distance $({l}^{\prime}>0)$ from a point on the mirror to the second focus. They are measured along the optical axis ray (OAR) that passes through the two foci and the aperture-stop center [11]. It can be shown from these geometries of a hyperbola that In Eq. (4), the transverse magnification $(m)$ of a single convex hyperboloidal mirror is defined by $m={l}^{\prime}/l$. In addition, applying the law of Sines to the $\mathrm{\Delta}FP{F}^{\prime}$ of Fig. 1, we obtain the distance $(d)$ from the center of the hyperbola to one of the foci as#### B. Scheimpflug Condition of an Off-Axis System

Figure 2 shows the tilted image plane owing to the tilted object plane in an on-axis system. For a tilted system at the finite conjugates, it is apparent from Fig. 2 that the object $({\theta}_{o})$ and image $({\theta}_{i})$ plane tilt angles are related by [7,8]

Substituting the magnification of Eq. (9) for the zero spherical aberration in Eq. (10), we obtain

Figure 3 shows spot diagrams of a single off-axis hyperboloidal mirror designed to satisfy Eq. (11). Its object is assumed to be located on the tilted plane by the angle ${\theta}_{o}=30\xb0$ from the plane normal to a line segment of the OAR. Zero spherical aberration yields the focused spot at center field (0°, 30°). In addition, since the image plane is tilted to satisfy the Scheimpflug condition, it can be shown from Fig. 3 that the aberrations over all fields are symmetrically balanced.

## 3. ELIMINATION CONDITIONS OF SPHERICAL ABERRATION AND LINEAR ASTIGMATISM, AND SCHEIMPFLUG CONDITION OF AN OFF-AXIS N-MIRROR SYSTEM AT FINITE CONJUGATES

Unlike axially symmetric systems, an off-axis mirror system introduces linear astigmatism in the image, which is the dominant aberration along with spherical aberration. In addition to these aberrations, considering the Scheimpflug condition for the tilted conjugate planes yields a good starting point for the design of this type of imaging system.

Figure 4 shows a confocal off-axis reflective imaging system consisting of $N$ mirrors. Every consecutive pair of mirrors shares a common focus, although they do not share a common axis. Owing to the property of conic mirrors, this reflecting system provides a perfect image at the center of the field, which is the focus of the final mirror. In this confocal off-axis $N$-mirror system, from Chang’s study [12], the linear astigmatism–free condition is given by

Meanwhile, although the mirror system satisfies the linear astigmatism–free condition of Eq. (12), the object plane might be not perpendicular to the OAR owing to the down angle in the HUD. To compensate that as shown in Fig. 4, the image plane should be tilted to satisfy the Scheimpflug condition as follows [12]:

## 4. DESIGN EXAMPLE

#### A. Initial Optical Design for HUD

In designing an off-axis mirror system for an HUD, the locations of the virtual image and LCD image display are set to be the object and image surfaces, respectively. This configuration simplifies the ray path in designing an off-axis mirror system. However, because all rays actually emanate from the LCD display, they are reversed conjugates.

To correct the dominant aberrations and build a suitable optical system for an HUD, a confocal, off-axis, two-hyperboloid mirror imaging system is suggested, as shown in Fig. 5. Since the primary mirror has a virtual object, a hyperboloid is good solution for this mirror. Similarly, to have a compact HUD, the distance between two mirrors should be short while maintaining the magnification of the system. This can be realized by introducing a hyperboloid to the secondary mirror to have a virtual object ${F}_{2}$. For a two-mirror system for an HUD, by substituting $N=2$ in Eqs. (14) and (15), we obtain

From Eqs. (16) and (17), an initial optical system for our HUD is configured as follows: the virtual image is set to be projected at a down angle of 5° and placed 2 m away from the eye box. Hence, in Fig. 5, the object distance $({l}_{1})$, OAR incidence angle $({i}_{1})$ for the primary mirror, and the angle $({u}_{1})$ between the primary mirror axis and the OAR are determined to meet the specifications of an HUD. In addition, to locate the second focal point $({F}_{1}^{\prime})$ of the primary mirror on its axis, the image distance $({l}_{1}^{\prime})$ is properly set for a given ${l}_{1}$, ${i}_{1}$, and ${u}_{1}$. Inserting these parameters into Eq. (9) gives the conic constant $({K}_{1})$ and magnification $({m}_{1})$ of the primary mirror, in which the spherical aberration is eliminated.

Next, considering the magnification of the system, the object distance $({l}_{2})$ and the incidence angle $({i}_{2})$ of the OAR for secondary mirror ${M}_{2}$, and the angle $({u}_{2})$ between the secondary mirror axis and the OAR should be determined to reasonably locate the LCD display plane. At this point, to effectively image the virtual object $({F}_{2})$ of ${M}_{2}$ on the plane of the LCD display by the secondary mirror, a hyperboloid is a good choice for the secondary mirror. By substituting these parameters and magnification $({m}_{1})$ of the primary mirror in Eq. (16), we obtain the conic constant $({K}_{2})$ and image distance $({l}_{2}^{\prime})$ of the secondary mirror, which yield the solutions for zero spherical aberration and a linear astigmatism–free system.

Finally, considering the line of sight of the driver, the object plane (actually the virtual image) in an HUD is tilted at a down angle ${\theta}_{o}=5\xb0$ from the plane normal to a line segment of the OAR. To compensate this tilted object, the image plane (actually the LCD display) should be tilted by the angle ${\theta}_{i}=1.163\xb0$, which is calculated from the Scheimpflug condition of Eq. (17) for ${m}_{1}=-0.425$, ${m}_{2}=-0.546$, and ${\theta}_{o}=5\xb0$.

Using the design process just outlined, an initial design corrected for spherical aberration, linear astigmatism, and Scheimpflug condition is analytically derived. The detailed parameters for an initially designed confocal, off-axis, two-hyperboloid mirror system for an HUD are summarized in Table 1, and its layout is shown in Fig. 5. In addition, spot diagrams of this system are illustrated in Fig. 6, which indicate that the dominant aberration is a third-order coma, as expected.

Meanwhile, since this system satisfies the Scheimpflug condition, it is clear from the spot diagrams that the aberrations in the system are symmetrically balanced with respect to the center field, similar to an on-axis mirror system. Thus, since the dominant aberrations are corrected, this off-axis mirror system provides a good starting design for an HUD system. This analytical design approach for a confocal, off-axis, two-hyperboloid mirror system is an important point of this study.

#### B. Optimized Design for an HUD

The confocal, off-axis, two-hyperboloid mirror system shown in Fig. 5 and Table 1 is used as a starting point to design a final optical system for an HUD. Namely, the mirror system of Fig. 5 is optimized to meet the current specifications given in Table 2 [2,14–16].

Because an HUD system has an off-axis optical system, its aberration properties are not rotationally symmetric. To balance these non-symmetrical aberrations, most imaging systems are designed using freeform surfaces such as $xy$ polynomial, Zernike polynomial, or anamorphic asphere, or by adding another asymmetric optical component such as a cylinder [1–4]. Since these freeform surfaces have different aspheric coefficients on the $x$ and $y$ axes, they are not axially symmetric so that a high cost and much effort are required to produce and measure them. In addition, adding an asymmetric optical component makes an HUD system larger and reduces the contrast owing to the ray reflection on that element.

In this study, however, we obtained a confocal, off-axis, two-hyperboloid mirror system for an HUD through an initial design process executed analytically. Since this system is initially corrected for spherical aberration and linear astigmatism, and its image plane is already tilted to satisfy the Scheimpflug condition, we do not need optical components such as a freeform surface or a cylinder as an asymmetrical aberration corrector. This configuration leads to a simple and cheap optical system for an HUD, and is a significant merit of this study.

In the initial design of Fig. 5, the eye box and field size should be increased in order to meet the current specifications for an HUD. The area where a driver can easily recognize a virtual image is known as the eye box. Considering the driver’s eyes in a combiner-type HUD, the eye box is extended to $130\text{\hspace{0.17em}}\mathrm{mm}\times 50\text{\hspace{0.17em}}\mathrm{mm}$, and the field of view (FOV) for a virtual image is enlarged to $4.6\text{\hspace{0.17em}}(\mathrm{H})\xb0\times 2.6\text{\hspace{0.17em}}(\mathrm{V})\xb0$ to have a virtual image size of 7.3 in at 2 m away from the eye box [2,14–16].

In order to improve the overall performance in an extended aperture and field system, we balance the aberrations of the starting data by aspherizing two mirrors. The equation for the aspheric surface used is given as

Additionally, an HUD system has a large virtual image along with asymmetrical properties, which generally leads to large non-symmetrical distortion. While designing the optical system for an HUD, the changes of distorted images owing to the head motion should be considered. The rays emanating from the object fully pass through the eye box, and then are imaged on the image plane. In spot diagrams, different ray positions at a given field denote the shifts of image points formed by the changed chief rays, which are induced by eye movement within the eye box. Thus, to reduce the changes in distortion owing to the head motion of the driver, we should design the optical system to have small spot sizes over all fields.

From this design process, an optical system for an HUD having good performance is finally obtained, and Fig. 7 denotes its layout. Table 3 lists the detailed mirror design data with the aspheric coefficients in a finally designed HUD system. Aberrations in all fields are sufficiently balanced so that the RMS spot sizes in most fields are reduced to less than 200 μm, as shown in Fig. 8.

Finally, an HUD is a binocular system in which the virtual images are recognized by both eyes. Hence, the binocular disparities are an important aberration in that system, and they are analyzed [15,16]. Assuming that the average distance between both eyes is 65 mm, the disparities are evaluated over all positions within the eye box. The maximum convergence disparity (in horizontal) and divergence disparity (in vertical) are 0.75 and 0.56 mrad, respectively. They are sufficiently reduced and meet the requirements, as listed in Table 2.

#### C. Evaluation of Distortion Changes Within the Eye Box

An optical system for an HUD generally has large and non-symmetrical distortion owing to its off-axis configuration. This distorted image may reduce the driver’s recognition and attention, so we need to examine the distortion in detail.

In this study, the distortion is first evaluated at the center position of the eye box. To examine the changes in distortion owing to the head motion of the driver, we evaluated the distorted images at four other eye positions located at the corner of the eye box, as shown in Fig. 9(a). Figure 9(b) shows how the distortion is evaluated at each eye position. In this figure, ${x}_{o}$ and ${y}_{o}$ are the paraxial image heights of the horizontal and vertical directions, respectively. In addition, $x$ and $y$ are the real image heights of the corresponding directions. To calculate the distortions in the horizontal and vertical directions, we define the following two quantities of distortion as

In Table 4, $\mathrm{\Delta}{x}_{D}(\%)$ and $\mathrm{\Delta}{y}_{D}(\%)$ denote the changed amounts of distortion at each eye position from that at the center eye position, and it is shown they are less than 2.28% over all fields. Even though this system is an off-axis mirror system, the changed amounts of distortion are so small that the distorted image owing to the head motion may not be noticeable. Consequently, this system has sufficient performance to fulfill the requirements of a head-up display system [15].

## 5. CONCLUSION

By use of an aberration analysis for a confocal, off-axis, two-hyperboloid mirror system, this study suggests a new off-axis reflective imaging system for a combiner-type HUD. In order to have a good initial design, the spherical aberration and linear astigmatism were corrected using a confocal, off-axis, two-hyperboloid-mirror system. In addition, to compensate the tilted object plane owing to the down angle of an HUD, we tilted the image plane to satisfy the Scheimpflug condition. These serial processes provided a better starting point for the optical design of an HUD.

Since the asymmetric aberrations of this system were initially corrected well, we did not use asymmetrical optical components such as a freeform surface or cylinder as an aberration corrector. This configuration was the key point of this study. After balancing the residual aberrations, an optical system for an HUD having good performance was finally obtained.

The changed amounts of distortions within the eye box are evaluated and found to be less than 2.28% over all fields. The binocular disparities within the eye box are at most 0.75 mrad, and thus meet the requirements. The size of the virtual image was 7.3 in at 2 m distance from the eye box, and the area of the eye box was $130\times 50\text{\hspace{0.17em}}{\mathrm{mm}}^{2}$. As a result, this design approach for an off-axis mirror system is expected to be useful in finding design solutions for a compact HUD.

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