The augmented reality head-up display (AR-HUD) can integrate with real-world road conditions, projecting rich driving information into the driver’s field of vision as images or text, significantly improving driving safety and user experience. To better incorporate AR, the AR-HUD system has evolved from a single-focal-plane display to a multi-focal-plane display, which has increased the complexity of optimization design. The traditional trial-and-error method requires substantial man power and time, and the final optimization direction and design results often depend heavily on the designer’s experience. Therefore, it is necessary to develop a simple and effective method for the automatic construction and optimization of bifocal or multi-focal plane AR-HUD systems, ensuring the ability to simultaneously display near-field basic information and far-field interactive information.

The initial structure of an off-axis three-mirror system is typically derived from the coaxial three-mirror system, but the final structure often shows significant deviations, and the intermediate construction process is time-consuming. In this paper, the windshield point cloud is first represented by XY polynomials. The full field of view is then tracked, and characteristic rays within the pupil are sampled. Point cloud envelope information is computed using Malus’ law and the law of refraction. The field of view expansion is solved step-by-step, ensuring contiguity across adjacent fields. Data fitting is constructed using smooth spline polynomial (SSP) characterization to construct the initial architecture. The edge field of view data is further optimized using iterative optical feedback, with iterative coefficients adjusted during the process. The root-mean-square error is evaluated based on the imaging performance quality function, and the optimal fitting coefficients are determined. The desired free-form mirror is then iteratively generated, resulting in the final structure of the bifocal-plane AR-HUD optical system.

As an example, a bifocal-plane AR-HUD system is designed with a far-field virtual image distance (VID) of 10 m for a 12°×3.5° field of view and a near-field VID of 3.8 m for a 6°×2° field of view. The eye box size is 130 mm×50 mm. The real automotive windshield point cloud is extracted and represented by XY polynomial fitting. At the same time, an initial system with two unshielded inclined and eccentric planes is established. During construction, the center field of view and the edge field of view are first defined, and point cloud locations are calculated using Malus’ law. The coordinate envelope information of the point cloud in the full field of view is derived using a field-of-view extension method and fitted to a third-order XY polynomial form (Fig. 11). The validity of the optical path feedback iteration method is verified, and after 15 iterations, the average spot radius is reduced by 61.73% compared to the initial configuration (Fig. 11). Finally, after ensuring that the boundary conditions for continuity are met, a sixth-order free-form surface characterized by SSP is compared with a sixth-order XY polynomial primary mirror for image quality optimization. At a reference wavelength of 0.587 μm, the wavefront RMS of the far-field surface is reduced by 26% (Fig. 12). The edge field-of-view performance also improved, with the modulation transfer function (MTF) of the SSP design approaching diffraction limits. At 6.5 lp/mm, the MTF is greater than 0.5, representing a 10% improvement compared to the traditional XY polynomial expression. The distortion in the SSP design is also well-controlled (Fig. 12).

The algorithm framework proposed in this paper enables the rapid generation of a stable initial bifocal-plane AR-HUD architecture for a single optical machine in limited space. It further optimizes system image quality to meet AR-HUD performance requirements at varying projection distances. This method effectively improves the iteration efficiency of freeform surface systems, reaching the threshold value after only a few iterations. It also considers the fusion of adjacent freeform surface. The comprehensive characterization using SSP enhances the optimization efficiency for designers. The bifocal-plane AR-HUD design example verifies the feasibility and practicality of this algorithm. The resulting image quality meets application requirements and provides a valuable reference for the future development of multi-focal-plane AR-HUD systems. This method can also be effectively applied to other freeform surface imaging systems, such as ultra-short focus projection systems, and head-mounted display system.