On-the-fly surface manufacturability constraints for freeform optical design enabled by orthogonal polynomials

Automatic control method for freeform surface shapes

Designing freeform optics with high degrees of freedom can improve their optical performances; however, there are high requirements for controlling the surface shapes of such optics. Optical designers need to add constraints to the optimization process and make repeated adjustments to ensure the manufacturability of these shapes; this process is cumbersome and relies heavily on the experience of the designer. In this study, an automatic control method for freeform surface shapes is proposed. By adding an outer loop to the optimization process, the principal curvature and sag departure of the sampling points are gradually controlled during the optimization cycle based on the system requirements and surface evaluation results. The method was implemented in CODE V and successfully applied to a design example in freeform prism optics.

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.

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.

Integrated analysis of industrial limitations and image quality: an end-to-end design approach

There is a trend in optical system design toward explicitly considering real-world industrial demands in the metrics to be optimized, from which emerges a cost-performance trade-off. Another relevant recent tendency is the so-called end-to-end design, where the design metric is an expected quality index of the final image, after digital restoration. We propose an integrated approach for analyzing the cost-performance trade-off in end-to-end designs. We exemplify it with a simple optical model where the cost is determined by the inclusion of an aspherical surface. We show that the resulting optimal trade-off configurations when applying an end-to-end design are substantially different from a conventional design. Such differences, as well as the increase in performance, are especially significant for lower-cost configurations.

Grating waveguides by machine learning for augmented reality

We propose a machine-learning-based method for grating waveguides and augmented reality, significantly reducing the computation time compared with existing finite-element-based numerical simulation methods. Among the slanted, coated, interlayer, twin-pillar, U-shaped, and hybrid structure gratings, we exploit structural parameters such as grating slanted angle, grating depth, duty cycle, coating ratio, and interlayer thickness to construct the gratings. The multi-layer perceptron algorithm based on the Keras framework was used with a dataset comprised of 3000-14,000 samples. The training accuracy reached a coefficient of determination of more than 99.9% and an average absolute percentage error of 0.5%-2%. At the same time, the hybrid structure grating we built achieved a diffraction efficiency of 94.21% and a uniformity of 93.99%. This hybrid structure grating also achieved the best results in tolerance analysis. The high-efficiency artificial intelligence waveguide method proposed in this paper realizes the optimal design of a high-efficiency grating waveguide structure. It can provide theoretical guidance and technical reference for optical design based on artificial intelligence.

Design of freeform mirrors using the concentric rings method

Through the methodology of optical surface design based on concentric rings, this paper proposes the design of freeform mirrors, initially by employing segmented rings, each of them with different spherical radii of curvature, and then by employing segmented conic rings with different conic constants in each of the segments. These surfaces will then produce the desired images. For the case of segmented spherical rings, mathematical expressions were deduced to obtain the image points as a function of the radii of curvature. Furthermore, it is shown that in the case where conic rings were used, there is a decrease in spherical aberration, which allows the manipulation of the generated image. Finally, several proposals are presented for the design of mirrors to generate both the desired size of the image and the desired distribution of energy, together with their analyses.

Freeform hyperspectral imager design in a CubeSat format

A freeform pushbroom hyperspectral imager design was investigated as a combination of a freeform reflective triplet imager and a freeform reflective triplet spectrometer used in double-pass. The design operates at about F/2 with a 15-degree cross-track field-of-view and a 30 mm entrance pupil diameter. The design process led to achieving a small volume of less than 2 liters that fits comfortably within a 3U CubeSat geometry, exemplifying the compactness of this hyperspectral imager. We report the freeform sag departures and maximum slopes of the freeform surfaces, as well as the manufacturing tolerances together with an evaluation of the system stray light, all of which highlight the feasibility of a design in this class to be manufactured. This design uniquely positions itself on the landscape of compact hyperspectral imagers.

Roadmap for the unobscured three-mirror freeform design space

In rotationally symmetric lens design, there are rule-of-thumb boundaries on field-of-view and aperture for well-known design forms that provide valuable information to the designer prior to starting a design. In the design space of unobscured three-mirror imagers, freeform optics have been shown to provide a significant benefit over conventional surface shapes, but the degree to which they improve the performance for any given combination of field-of-view, entrance pupil diameter, and F-number remains unknown. Thus, designers of these systems are not afforded any pre-design information to inform their specification decisions. Here, we designed over 200 systems to establish a first-of-its-kind roadmap of specification ranges over which an unobscured three-mirror imager using freeform surfaces can achieve diffraction-limited performance in the visible spectrum. The scalability of the findings to the infrared regions of the spectrum is also addressed.

Advances in reconfigurable optical design, metrology, characterization, and data analysis

Reconfigurable freeform optical systems greatly enhance imaging performance within non-symmetric, compact, and ergonomic form factors. In this paper, several advances improve design, testing, and monitoring of these systems. Specific enhancements include definition of polynomials for fast and efficient parameterizations of vector distributions in non-circular apertures and merit based function optimization. Deflectometry system improvements enable metrology for almost any conceivable optic shape and guide deterministic optical figuring process during the coarse grinding phase by including modulated infrared sources. As a demonstration of these improvements, parametric optimization is tested with the tomographic ionized-carbon mapping experiment, a reconfigurable optical system. Other case studies and demonstrations include metrology of a fast, f/1.26 convex optic, an Alvarez lens, and real-time monitoring of an array of independently-steerable hexagonal mirror segments as well as an induction formed surface and inflatable Mylar mirror.

Efficient representation of freeform gradient-index profiles for non-rotationally symmetric optical design

Conventional optical designs with gradient index (GRIN) use rotationally-invariant GRIN profiles described by polynomials with no orthogonality. These GRIN profiles have limited effectiveness at correcting aberrations from tilted/decentered or freeform systems. In this paper, a three-dimensional orthogonal polynomial basis set (the FGRIN basis) is proposed, which enables the design of GRIN profiles with both rotational and axial variations. The FGRIN basis is then demonstrated via the design of a 3D GRIN corrector plate targeted to correct the rotationally-variant aberrations induced from a tilted spherical mirror. A sample corrector is manufactured and tested, showing significant correction of astigmatism. The FGRIN basis opens a new design space of 3D rotational variant GRIN profiles, which has the potential of replacing multiple freeform surfaces and simplifying complex systems.

Off-axis conics as base surfaces for freeform optics enable null testability

When conducting interferometric tests of freeform optical surfaces, additional optical components, such as computer-generated holograms or deformable mirrors, are often necessary to achieve a null or quasi-null. These additional optical components increase both the cost and the difficulty of interferometric tests of freeform optical surfaces. In this paper, designs using off-axis segments of conics as base surfaces for freeforms are explored. These off-axis conics are more complex base surfaces than typically-used base spheres but remain null-testable. By leveraging off-axis conics in conjunction with additional orthogonal polynomial departures, designs were found with up to an order-of-magnitude of improvement in testability estimates relative to designs that use base spheres. Two design studies, a three-mirror telescope and a wide field-of-view four-mirror telescope, demonstrate the impact of using off-axis conics as the base surface.

Estimating field-dependent nodal aberration theory coefficients from Zernike full-field displays by utilizing eighth-order astigmatism

When using freeform surfaces in optical design, the field dependence of the aberrations can become quite complex, and understanding these aberrations facilitates the design process. Here we calculate the field dependence of low-order Zernike astigmatism (Z5/6) up to the eighth order in nodal aberration theory (NAT). Expansion of NAT astigmatism terms to the eighth order facilitates a more accurate fit to the Zernike astigmatism data. We then show how this estimated field dependence can be used to quantitatively analyze a freeform telescope design. This analysis tool adds to the optical designer's arsenal when up against the challenge of designing with freeform optics.

Freeform geometrical optics II: from parametric representation to CAD/CAM

Freeform optical surfaces are of great importance because of two main properties. The first is their ability to enhance the image quality of image-forming optical systems; the second is their inherent reduction in the number of surfaces in image and nonimage-forming optical systems. However, the main characteristic of freeform surfaces is that they lack symmetry about any spatial axis. This attribute allows describing freeform surfaces with a mathematical parametric representation. Unfortunately, parametric representation can be extremely extended. On the other hand, when describing freeform surfaces, the explicit representation is commonly preferred because of its compactness and CAD-format exportable easiness. Parametrically represented freeform surfaces can nonetheless be exported to a CAD format, with no significant departure of surface shape, as shown here. The vector method presented here guarantees that the surface's sampling density be proportional to the irradiance on the surface.

Effect of freeform surfaces on the volume and performance of unobscured three mirror imagers in comparison with off-axis rotationally symmetric polynomials

The invention of new design techniques for unobscured reflective systems using freeform surfaces has expanded the optical design space for these system types. We illustrate how the use of freeform surfaces can expand the design space of the Three Mirror Compact design type to allow both better performance at a given system volume and smaller volumes for a given performance target. By evolving designs using conventional off-axis asphere type surfaces to ever smaller volumes and then converting these off-axis asphere descriptions to centered Zernike descriptions, we show that the wavefront error improves by up to 69% in this case by allowing the surfaces to break rotational symmetry. In addition, we show that evolving designs from the same starting point as the off-axis asphere designs but instead using a centered Zernike description can produce a design with a 39% smaller volume in this case while maintaining the same diffraction-limited performance.