Freeform Imaging Systems Research Articles

Iterative critical ray tracing for local tolerance analysis of freeform imaging systems

Tolerance analysis of the freeform surfaces serves as a critical bridge between design and manufacturing, offering essential guidance for the desensitization design of optical systems and playing a crucial role in the development of advanced imaging systems. Recently, Deng et al. proposed a direct method [Deng et al. Optica 9, 1039 (2022)] for solving tolerances of freeform surfaces, which reveals the local characteristics of the tolerances of freeform surfaces. However, the method requires dense sampling of the fields of view (FOVs) and the entrance pupil (EP) to cover as many optical surface points as possible, thereby achieving more accurate tolerance envelope solutions. Here, we propose an iterative algorithm called "critical ray tracing" to calculate the critical rays at points on the optical surface and utilize this information for surface tolerance analysis. This method involves fitting the coordinate space of the optical system into a 4D polynomial and employing numerical iteration to determine the rays closest to the preset wave aberration boundary at each surface point. Converting the FOVs and EP sampling into optical surface sampling significantly reduces the number of samples required, achieving computational efficiency without compromising accuracy in determining tolerable surface errors. We demonstrate the effectiveness of our method through tolerance analysis of two different freeform imaging systems. Furthermore, a tolerance analysis example of a complete off-axis three-mirror optical system demonstrates the universality of the process.

Initial design with confocal conic mirrors for freeform optical imaging systems

Freeform surfaces that can offer high degrees of freedom for aberration correction have been widely used in various imaging applications. Initial design is particularly critical in the optical design of freeform imaging systems due to the significantly expanded solution space. Here, we present a method to find an initial point for the optical design of four-mirror freeform imaging systems. The method is computationally simple, easily accessible, and theoretically supported. The effectiveness of the method is demonstrated by designing a four-mirror confocal system without field-constant aberration and linear astigmatism. We also generalize the proposed method to four-mirror confocal systems with “double-pass surface.” The proposed method can yield a promising starting point for the design of freeform off-axis imaging systems.

High-precision analysis of aberration contribution of Zernike freeform surface terms for non-zero field of view.

Clarifying the aberrations arising from freeform surfaces is of great significance for maximizing the potential of freeform surfaces in the design of optical systems. However, the current precision in calculating aberration contribution of freeform surface terms for non-zero field of view is insufficient, impeding the development of freeform imaging systems with larger field of view. This paper proposes a high-precision analysis of aberration contribution of freeform surface terms based on nodal aberration theory, particularly for non-zero field points. Accurate calculation formulas of aberrations generated by Zernike terms on freeform surface are presented. Design examples illustrate that the calculation error of the provided formulas is 78% less than that of conventional theoretical values. Building upon high-precision analysis, we propose an optimization method for off-axis freeform surface systems and illustrate its effectiveness through the optimization of an off-axis three-mirror system. This research extends the applicability of nodal aberration theory in aberration analysis, offering valuable insights for the optimal design and alignment of optical freeform systems.

Optical design mode based on fast automatic design process for freeform reflective imaging systems with modest FOV.

Traditional optical design methods require designer intervention in the system's evolution from the starting point to the final design. Trial-and-error during design optimization improves system performance step by step but requires much time and effort. A new optical design framework, end-to-end fast automatic design, is proposed and achieved for the freeform reflective optics in this paper, which promotes a new optical design mode. Compared with the traditional mode through improving performance after each trial, an optical system with good image quality can be directly obtained in the end-to-end design process with simple input and no human involvement within a short time. If there is still the possibility for performance improvement of the obtained system, the designer can vary the input parameters repeatedly to obtain multiple systems with good image quality. Finally, the desired system is selected from these systems. Compared with the step-by-step trials in traditional optimization, this new optical design mode involves high-speed trials of the end-to-end automatic design process, reducing the dependence on experience and skill. In this paper, an end-to-end fast automatic design method for freeform imaging systems is developed based on a new design route. Using an initial plane system as an input, a freeform system with excellent image quality can be designed automatically within 1-2 min. After several trials of the end-to-end fast design process, three high-performance freeform systems are designed successfully that consider volume control, beam obscuration, and mirror interference.

FreeformNet: fast and automatic generation of multiple-solution freeform imaging systems enabled by deep learning

Using freeform optical surfaces in lens design can lead to much higher system specifications and performance while significantly reducing volume and weight. However, because of the complexity of freeform surfaces, freeform optical design using traditional methods requires extensive human effort and sufficient design experience, while other design methods have limitations in design efficiency, simplicity, and versatility. Deep learning can solve these issues by summarizing design knowledge and applying it to design tasks with different system and structure parameters. We propose a deep-learning framework for designing freeform imaging systems. We generate the data set automatically using a combined sequential and random system evolution method. We combine supervised learning and unsupervised learning to train the network so that it has good generalization ability for a wide range of system and structure parameter values. The generated network FreeformNet enables fast generation (less than 0.003 s per system) of multiple-solution systems after we input the design requirements, including the system and structure parameters. We can filter and sort solutions based on a given criterion and use them as good starting points for quick final optimization (several seconds for systems with small or moderate field-of-view in general). The proposed framework presents a revolutionary approach to the lens design of freeform or generalized imaging systems, thus significantly reducing the time and effort expended on optical design.

Design of compact off-axis freeform imaging systems based on optical-digital joint optimization.

Using a freeform optical surface can effectively reduce the imaging system weight and volume while maintaining good performance and advanced system specifications. But it is still very difficult for traditional freeform surface design when ultra-small system volume or ultra-few elements are required. Considering the images generated by the system can be recovered by digital image processing, in this paper, we proposed a design method of compact and simplified off-axis freeform imaging systems using optical-digital joint design process, which fully integrates the design of a geometric freeform system and the image recovery neural network. This design method works for off-axis nonsymmetric system structure and multiple freeform surfaces with complicated surface expression. The overall design framework, ray tracing, image simulation and recovery, and loss function establishment are demonstrated. We use two design examples to show the feasibility and effect of the framework. One is a freeform three-mirror system with a much smaller volume than a traditional freeform three-mirror reference design. The other is a freeform two-mirror system whose element number is reduced compared with the three-mirror system. Ultra-compact and/or simplified freeform system structure as well as good output recovered images can be realized.

Structural aberration coefficients for freeform imaging systems

Freeform imaging systems are a subject of current interest. To aid in the design of such systems, we introduce structural aberration coefficients for plane symmetric imaging with XY-polynomial freeform surfaces that may also be tilted. Further, we extend the coefficients to systems with an arbitrary number of components and specifically provide analytical equations for two-mirror systems. Distinguishing between system configuration and structural parameters enhances clarity, which we highlight by deriving new conditions for minimum root-mean-square (RMS) spot size and minimum RMS wavefront error in aplanatic two-mirror telescopes.

Automated design of freeform imaging systems for automotive heads-up display applications.

The freeform imaging system is playing a significant role in developing an optical system for the automotive heads-up display (HUD), which is a typical application of augmented reality (AR) technology. There exists a strong necessity to develop automated design algorithms for automotive HUDs due to its high complexity of multi-configuration caused by movable eyeballs as well as various drivers' heights, correcting additional aberrations introduced by the windshield, variable structure constraints originated from automobile types, which, however, is lacking in current research community. In this paper, we propose an automated design method for the automotive AR-HUD optical systems with two freeform surfaces as well as an arbitrary type of windshield. With optical specifications of sagittal and tangential focal lengths, and required structure constraints, our given design method can generate initial structures with different optical structures with high image quality automatically for adjusting the mechanical constructions of different types of cars. And then the final system can be realized by our proposed iterative optimization algorithms with superior performances due to the extraordinary starting point. We first present the design of a common two-mirror HUD system with longitudinal and lateral structures with high optical performances. Moreover, several typical double mirror off-axis layouts for HUDs were analyzed from the aspects of imaging performances and volumes. The most suitable layout scheme for a future two-mirror HUD is selected. The optical performance of all the proposed AR-HUD designs for an eye-box of 130 mm × 50 mm and a field of view of 13° × 5° is superior, demonstrating the feasibility and superiority of the proposed design framework. The flexibility of the proposed work for generating different optical configurations can largely reduce the efforts for the HUD design of different automotive types.

Calculation of aberration fields for freeform imaging systems using field-dependent footprints on local tangent planes.

Aberration theory is a fundamental understanding of the optical aberrations and remains the best way to guide optical system design. The nodal aberration theory, which can be used to describe the aberration fields of freeform imaging systems, is limited by the small field of view (FOV) of the imaging system. In this paper, we propose a method to predict the induced aberration of Fringe Zernike terms with field-dependent footprints. The footprint of each field point is calculated in its corresponding local tangent plane of the optical surface; therefore, a more accurate prediction of the induced aberrations of Fringe Zernike terms can be achieved. Both the FOV and highly tilted architecture of freeform imaging systems are considered when calculating the footprints. Two examples are presented to verify the effectiveness of the proposed method, which we believe can provide good guidance for the design of freeform imaging systems with a relatively large FOV.

Using deep learning to automatically generate design starting points for free-form imaging optical systems.

In this paper, we propose a method to automatically generate design starting points for free-form three-mirror imaging systems with different folding configurations using deep neural networks. For a given range of system parameters, a large number of datasets are automatically generated using the double seed extended curve algorithm and coded optimization. Deep neural networks are then trained using a supervised learning approach and can be used to generate good design starting points directly. The feasibility of the method is verified by designing a free-form three-mirror system with three different folding configurations. This method can significantly reduce the design time and effort for free-form imaging systems, and can be extended to complex optical systems with more optical surfaces.

Susceptibility synthesis of arbitrary shaped metasurfaces

Visual perception relies on light scattering at the object's surface in the direction of observation. By engineering the surface scattering properties, it is possible to realize arbitrary visual percepts. Here, we address theoretically this problem of electromagnetic field transition conditions at conformal interfaces to achieve surface-topography-dependent transmitted and reflected fields. Our analysis, supported by two- and three-dimensional finite-element simulations, provides a solid theoretical framework to design metasurfaces for cloaking, wearable optics, and next-generation freeform imaging systems.

Freeform imaging system with resolution that varies with the field angle in two dimensions.

The human eye's resolution varies with the field angle and has high center resolution and low edge resolution characteristics. In this paper, a freeform imaging system is presented that has resolution distribution characteristics similar to those of the human eye. Field-dependent parameters are used to describe the system's optical properties and a direct design method is proposed to realize the novel functionality. An off-axis reflective freeform imaging system with high center resolution and low edge resolution within a square 30°×30° field of view (FOV) is designed using this method. The maximum instantaneous field of view (IFOV) ratio of center field resolution to edge field resolution is 0.47. Only three freeform surfaces are used to attain good image quality. Simultaneous improvements are observed in both resolution and FOV while the detector remains fixed.

Automated freeform imaging system design with generalized ray tracing and simultaneous multi-surface analytic calculation.

Recently, freeform optics has been widely used due to its unprecedented compactness and high performance, especially in the reflective designs for broad-wavelength imaging applications. Here, we present a generalized differentiable ray tracing approach suitable for most optical surfaces. The established automated freeform design framework simultaneously calculates multi-surface coefficients with merely the system geometry known, very fast for generating abundant feasible starting points. In addition, we provide a "double-pass surface" strategy with desired overlap (not mutually centered) that enables a component reduction for very compact yet high-performing designs. The effectiveness of the method is firstly demonstrated by designing a wide field-of-view, fast f-number, four-mirror freeform telescope. Another example shows a two-freeform, three-mirror, four-reflection design with high compactness and cost-friendly considerations with a double-pass spherical mirror. The present work provides a robust design scheme for reflective freeform imaging systems in general, and it unlocks a series of new 'double-pass surface' designs for very compact, high-performing freeform imaging systems.

Automated freeform imaging system design with generalized ray tracing and simultaneous multi-surface analytic calculation

Automated freeform imaging system design with generalized ray tracing and simultaneous multi-surface analytic calculation

Freeform imaging systems: Fermat\u2019s principle unlocks \u201cfirst time right\u201d design

For more than 150 years, scientists have advanced aberration theory to describe, analyze and eliminate imperfections that disturb the imaging quality of optical components and systems. Simultaneously, they have developed optical design methods for and manufacturing techniques of imaging systems with ever-increasing complexity and performance up to the point where they are now including optical elements that are unrestricted in their surface shape. These so-called optical freeform elements offer degrees of freedom that can greatly extend the functionalities and further boost the specifications of state-of-the-art imaging systems. However, the drastically increased number of surface coefficients of these freeform surfaces poses severe challenges for the optical design process, such that the deployment of freeform optics remained limited until today. In this paper, we present a deterministic direct optical design method for freeform imaging systems based on differential equations derived from Fermat’s principle and solved using power series. The method allows calculating the optical surface coefficients that ensure minimal image blurring for each individual order of aberrations. We demonstrate the systematic, deterministic, scalable, and holistic character of our method with catoptric and catadioptric design examples. As such we introduce a disruptive methodology to design optical imaging systems from scratch, we largely reduce the “trial-and-error” approach in present-day optical design, and we pave the way to a fast-track uptake of freeform elements to create the next-generation high-end optics. We include a user application that allows users to experience this unique design method hands-on.

"First time right" - calculating imaging systems from scratch -INVITED

Freeform optics can be used to greatly extend the functionalities, improve performance, and reduce the volume and weight of optical systems. Today, the design of imaging systems largely relies on efficient ray tracing and optimization algorithms. Such a "step-and-repeat" approach to optical design typically requires considerable experience, intuition, and eventually "trial-and-error" guesswork. This time-consuming process applies especially to freeform optical systems. In this work, we present a deterministic direct optical design method for freeform imaging systems based on differential equations derived from Fermat’s principle and solved using power series. The method allows calculating all optical surface coefficients that ensure minimal image blurring for each individual order of aberrations. We demonstrate the systematic, deterministic, scalable, and holistic character of our method with several catoptric and catadioptric design examples.

Designing freeform imaging systems based on reinforcement learning.

The design of complex freeform imaging systems with advanced system specification is often a tedious task that requires extensive human effort. In addition, the lack of design experience or expertise that result from the complex and uncertain nature of freeform optics, in addition to the limited history of usage, also contributes to the design difficulty. In this paper, we propose a design framework of freeform imaging systems using reinforcement learning. A trial-and-error method employing different design routes that use a successive optimization process is applied in different episodes under an ε-greedy policy. An "exploitation-exploration, evaluation and back-up" approach is used to interact with the environment and discover optimal policies. Design results with good imaging performance and related design routes can be found automatically. The design experience can be further summarized using the obtained data directly or through other methods such as clustering-based machine learning. The experience offers valuable insight for completing other related design tasks. Human effort can be significantly reduced in both the design process and the tedious process of summarizing experience. This design framework can be integrated into optical design software and runs nonstop in the background or on servers to complete design tasks and acquire experience automatically for various types of systems.

Fast automatic design method for freeform imaging systems through system construction and correction.

In traditional optical design, a starting point is selected and coefficients optimization is then performed using software. The process requires considerable time and the involvement of a human with design skills and experience. In this Letter, a fast automatic method for freeform imaging systems design is proposed. Using a plane system as the input, a freeform optical system with high image quality can be designed automatically at high speed. The method consists of system construction and system correction, combining the advantages of the direct design method and the methods based on aberration analysis. After system construction generates a system with fundamental optical parameters, system correction is an iterative process that alternates between image plane correction and surfaces correction to improve the image quality to a high level. Two examples required 5 min 56 s and 6 min 10 s to design freeform systems with near-diffraction-limit image quality.

大视场大相对孔径自由曲面成像系统设计

大视场大相对孔径自由曲面成像系统设计

大视场大相对孔径自由曲面成像系统设计

大视场大相对孔径自由曲面成像系统设计