Freeform System Research Articles

Compact freeform near-eye display system design enabled by optical-digital joint optimization

The near-eye display (NED) systems, designed to project content into the human eye, are pivotal in the realms of augmented reality (AR) and virtual reality (VR), offering users immersive experiences. A small volume is the key for a fashionable, easy-to-wear, comfortable NED system for industrial and consumer use. Freeform surfaces can significantly reduce the system volume and weight while improving the system specifications. However, great challenges still exist in further reducing the volume of near-eye display systems as there is also a limit when using only freeform optics. This paper introduces a novel method for designing compact freeform NED systems through a powerful optical–digital joint design. The method integrates a geometrical freeform optical design with deep learning of an image compensation neural network, addressing off-axis nonsymmetric structures with complex freeform surfaces. A design example is presented to demonstrate the effectiveness of the proposed method. Specifically, the volume of a freeform NED system is reduced by approximately 63% compared to the system designed by the traditional method, while still maintaining high-quality display performance. The proposed method opens a new pathway for the design of a next-generation ultra-compact NED system.

Design Method of Freeform Surface Optical Systems with Low Coupling Position Error Sensitivity.

Freeform off-axis reflective systems are significantly more difficult to align and assemble owing to their asymmetric surface shapes and system structures. In this study, a freeform surface system design method with low coupling position error sensitivity (FCPESM) was proposed. First, we established a mathematical model of a reflective system when it was perturbed by coupling position errors and used the clustering-microelement method to establish the coupling error sensitivity evaluation function. The evaluation function was then applied to the design process of a freeform surface off-axis three-mirror optical system. The results showed that the FCPESM optical design method can significantly relax the assembly tolerance requirements of optical systems on the basis of ensuring image performance. In this study, the reflective system was perturbed by tilt and decenter simultaneously, and the disturbance mechanism of position errors on optical systems was further improved. Through this research, freeform surface systems with both image performance and error sensitivity can be obtained, which makes freeform off-axis reflective systems with better engineering realizability.

Three-dimensional freeform reflector design with a microfacet surface roughness model

This paper proposes a methodology for the design of freeform reflectors with scattering surfaces. We use microfacets, which are small, tilted mirrors superimposed on a smooth surface. We form a simple model of surface roughness and light scattering based on the orientations of the microfacets. Using a least-squares solver to compute the smooth reflector as a starting point, we can subsequently alter the surface using an optimization procedure to account for the scattering. After optimization, the resulting reflector surface produces the desired scattered light distribution. We verify the resulting reflector using raytracing. This study focuses on freeform systems with a collimated incident beam and a far-field target intensity.

Automatic generation method for long-focal-length unobscured freeform optical systems with small volume

The existing design methods for long-focal-length unobscured freeform systems rarely consider the imaging quality requirements and volume constraints simultaneously, causing most of the final designs to not fulfill the requirement of light weight. This study proposes a method to automatically design a long-focal-length unobscured reflective system that satisfies volume constraints while maintaining high imaging quality. First, a method to adaptively set the structural parameter range is proposed, and multiple parameters for different systemic specifications can be effectively calculated within it. Subsequently, the systemic volume and area functions are constructed using the ray tracing method, where the tilt angles, distances between mirrors, and radii of curvature of the mirrors are chosen as the optimization parameters. Third, a comprehensive objective function is jointly established combining ray obscuration and convergence as performance evaluation factors. Then, the structural parameters of a long-focal-length unobscured system with small volume are easily obtained via the simulated annealing method. Finally, the improved W-W method is used to further enhance the imaging quality of the system, and an unobscured freeform reflective optical system with three mirrors is automatically generated. Experimental results demonstrate that our method can automatically calculate the parameter ranges to facilitate the search for structural parameters, and effectively design the long-focal-length unobscured freeform systems with small volume and high imaging quality.

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.

Freeform Liquid–Crystal Polarization Imaging Optics: A Practical Approach from Design to Fabrication

AbstractFreeform liquid–crystal polarization imaging optics (LPIOs) have vast potential for application in next‐generation interactive displays like augmented reality and virtual reality. However, tailoring the phase profile of a freeform LPIO for flexible aberration correction and diffraction efficiency control is still a big challenge due to the difficulties arising from the non‐zero étendue of an imaging system and the finite bandwidth of LPIOs. Here, a practical approach to creating freeform LPIOs from design to fabrication is proposed. A freeform polarization exposure is developed to produce a flexible, precise, and ultra‐high‐resolution photoalignment pattern for a freeform LPIO. A cooperative optimization is performed between the freeform LPIO and the freeform polarization exposure system to find the optimal phase of the freeform LPIO and the optimal surface profiles of the freeform lenses employed in the exposure system to produce freeform wavefronts. The optimal phase of the freeform LPIO is then achieved by superposition interference of the engineered freeform wavefronts via freeform polarization exposure. The effectiveness of the approach and the advantages of freeform LPIOs is experimentally demonstrated. The proposed approach will significantly promote the widespread applications of freeform LPIOs in the field of optics and photonics.

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.

Generating function approach for freeform two-reflector two-target system

We discuss an inverse method to compute a freeform two-reflector two-target system. The optical path length constitutes an integral component and can be expressed in terms of position coordinates at the first target. The system is expressed in terms of a generating function, closed with energy balance and requires a sophisticated least-squares solver to compute the shapes of the reflectors. In a numerical example, we illustrate the algorithm’s capabilities to tackle even the most intricate light distributions.

Differential ray tracing for an efficient computation of the Jacobian in the damped least-squares algorithm

We present a framework for the optimization of optical systems based up on the damped least-squares method and differential ray tracing. First, a mathematical description for the differentiation of the ray-surface intersection is introduced. It is observed that the structure of the optimization variables and operands can be exploited such that the Jacobian for the damped least-squares algorithm can be computed in the same computational complexity as the primal ray tracing for both forward and reverse mode algorithmic differentiation. To demonstrate the advantages of the method a freeform mirror system is analyzed.

Broadband dual-comb hyperspectral imaging and adaptable spectroscopy with programmable frequency combs

We explore the advantages of a free-form dual-comb spectroscopy (DCS) platform based on time-programmable frequency combs for real-time, penalty-free apodized scanning. In traditional DCS, the fundamental spectral point spacing, which equals the comb repetition rate, can be excessively fine for many applications. While fine point spacing is not itself problematic, it comes with the penalty of excess acquisition time. Post-processing apodization (windowing) can be applied to tailor the resolution to the sample, but only with a deadtime penalty proportional to the degree of apodization. The excess acquisition time remains. With free-form DCS, this deadtime is avoided by programming a real-time apodization pattern that dynamically reverses the pulse periods between the dual frequency combs. In this way, one can tailor the spectrometer’s resolution and update rate to different applications without penalty. We show the operation of a free-form DCS system where the spectral resolution is varied from the intrinsic fine 160 MHz up to 822 GHz by applying tailored real-time apodization. Because there is no deadtime penalty, the spectral signal-to-noise ratio increases linearly with resolution by 5000× over this range, as opposed to the square root increase observed for post-processing apodization in traditional DCS. We explore the flexibility to change resolution and update rate to perform hyperspectral imaging at slow camera frame rates, where the penalty-free apodization allows for optimal use of each frame. We obtain dual-comb hyperspectral movies at a 20 Hz spectrum update rate with broad optical spectral coverage of over 10 THz.

Multi-field cosine condition in the design of wide-field freeform microscope objectives.

To break through the limitations of the classical sine condition, a multi-field cosine condition (MCC) is proposed in this paper. Building on this condition, a design method for wide field-of-view (FOV) freeform aplanatic systems is also proposed. This design method ensures that the optical system meets the MCC by correcting the mapping relationships of rays that originate from different apertures on both the object and image sides of the optical system. Using this method, we have successfully designed a chromatic-aberration-free and unobstructed freeform off-axis reflective microscope objective that is characterized by a wide FOV, an extensive spectral working range, and an extended working distance. The objective, which comprises only three freeform mirrors, has a numerical aperture (NA) of 0.4, magnification of 20×, an FOV of 2mm × 2 mm, and an extended working distance of up to 60 mm. The objective demonstrates imaging quality close to the diffraction limit across the visible band to the short-wave infrared band (0.4-3 µm). When compared with a transmission-type wide-field microscope objective with identical NA and magnification properties, the reflective microscope objective nearly triples the FOV.

Geometric characteristics, deployment mechanisms, and digital fabrication methods of a free-form deployable membrane system based on CNC-knitted fabrics and CNC-bent frames

ABSTRACT Deployable membrane structures have played a vital role in creating flexible and adaptable urban and architectural spaces. However, the geometries of deployable membrane structures are often confined to symmetrical and regular shapes, due to the complex interdependencies between structure, material, motion, and geometry. To overcome this challenge, this research proposes a new category of deployable membrane structures that enables a broad range of deployment geometries, including free forms. The proposed system facilitates the transformation of arbitrary free-form surfaces into structural models, simplifies kinematic analysis for free-form deployment, and reduces structural calculations for free-form deployable structures. This study systematically investigated the proposed deployable membrane system in terms of geometric characteristics, deployment mechanisms, and digital manufacturing methods, including investigations of the generation method and geometric categories, summarizations of the folding mode and foldability conditions, and explorations of the CNC manufacturing logic and the G-code generation method. Finally, a full-scale physical prototype is studied to validate the proposed system and related methods.

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.

Interferometric metrology probe for highly sloped freeform optics.

A wide variety of systems are employed to measure the surface profile of aspheric and freeform optical surfaces. Freeform metrology systems must accurately characterize the full surface under test, which can be difficult with steep surface slopes. Here we present an interferometric surface metrology probe for highly sloped aspheric and freeform optical surfaces. The optical design of this probe allows the measurement of surface slopes up to 50deg without tilting the probe, which simplifies stage design and increases the accuracy of the system. A spectrally controlled light source is used to create a virtual ball in front of the probe tip to measure the surface distance and angle. This system produces a cloud of points, to which Zernike polynomials are fit and used to reconstruct the surface. We will show sensitivity tests and accuracy results.

Design method of freeform off-axis three-mirror reflective imaging systems.

In this paper, a novel, to the best of our knowledge, method is proposed to design a freeform off-axis three-mirror reflective imaging system. A special algorithm is demonstrated to calculate the data points on the unknown freeform surface using the rays from the pupil of the center field. Then the three-dimensional shape of the freeform surface is solved by these characteristic points, which serves as a good starting point for further optimization. The benefit of this design method is demonstrated by designing a freeform off-axis three mirror imaging system with high performance. The final system operates at F/2 with an entrance pupil diameter of 400mm and a field of view of 2.4∘×2.4∘. The modulation transfer function (MTF) value of the system reaches 0.6 at 100lp/mm or higher at all fields of view.

Three-mirror free-form off-axis antenna design of beam shaping on a circular substrate

This work presents a design of a three-mirror free-form surface beam shaping system with small dimensions, easy assembly, and good collimation. To facilitate integration and assembly, the three mirrors are integrated into a circular substrate with a radius of 90 mm. Based on the law of energy conservation, Fermat’s principle, and the vector theory of reflection, the input–output mapping is derived, and the three free-form mirrors are constructed point by point simultaneously. Ray tracing results demonstrate that the maximum divergence angle is compressed to 0.467 mrad and the standard deviation of the edge beam radius samples (SDRS) is reduced from 5.42mm to 0.985mm on the receiving plane. The light intensity has a relative root mean square deviation (RRMSD) of only 0.049. Furthermore, the results show that the minimum value of SDRS is 0.940mm at a propagation distance of L=1.72 m, while the minimum value of RRMSD is 0.0431 at L=7.03 m. In addition, the effects of astigmatism and light source offsets on the collimated and shaping effect are analyzed. The design of the free-form surfaces antenna system in this work will provide a new design method for practical engineering and application, and will greatly improve the quality and propagation distance of the Gaussian beam.

Optical freeform reflective imaging system design method with manufacturing constraints.

With the development of space optics, optical freeform surfaces have gradually been utilized in reflective optical imaging systems in recent years. Freeform surfaces not only bring many benefits to the optical imaging system, but also present many challenges to their manufacture. Regardless of the machining method used, machining errors during the fabrication of freeform surfaces will exist, which limits the accuracy of freeform surface machining. In this paper, the deviation root mean square (RMS) of a freeform surface from the reference aspheric surface is proposed to evaluate the manufacturability of the freeform surface by using single-point diamond turning. Then the deviation RMS of freeform surfaces is added to the design process of the optical system as a manufacturing constraint. Subsequently, an off-axis three-mirror system and an off-axis two-mirror system with and without manufacturing constraints are designed, respectively. Then the imaging quality of these optical systems and the linear interpolation error RMS of freeform mirror are analyzed. It can be concluded that, on the basis of reaching the imaging quality requirements, the machining difficulty of a freeform mirror can be reduced when adding manufacturing constraints to the design process.

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.

Automatic Method of Exploring the Landscape of Freeform Dioptric Optical Problems, Working in the Infrared Region

We present an automated method of finding different freeform dioptric starting systems, working in the infrared region, for further optimization in commercial optical design software. Our developed method couples the simultaneous multiple surface (SMS) method, introduced by Benítez and Miñano, with automatic optimization in Zemax OpticStudio. The method allows an optical designer to explore the merit function (MF) landscape of freeform optical problems. In this article, we apply our method to a size, weight, and power (SWaP) problem, and we compare our designed system with a system found in the literature that has the same aperture of F/1.2. Then, we increase the aperture of the system up to F/0.9, taking advantage of the use of freeform surfaces.

Exit pupil quality analysis and optimization in freeform afocal telescope systems.

Afocal telescopes are often used as foreoptics to existing imaging systems to allow for application flexibility. To properly combine an afocal telescope with an existing imaging system, the exit pupil of the afocal telescope and the entrance pupil of the imaging system must be coincident. Additionally, the exit pupil of the afocal telescope must be well-formed; that is, it must be the correct size and shape to mitigate pupil-matching challenges. This work introduces processes for designing freeform afocal telescopes with an emphasis on understanding how to analyze and control the exit pupil quality of such systems. The included 3-mirror design examples demonstrate the advantages of using freeform surfaces in afocal systems and quantify the tradeoffs required to improve the exit pupil quality.