Freeform surfaces have important applications in optical design. However, the complex surface figure error of freeform imaging systems affects the performance of as-built systems. In this paper, a method for the representation and tolerance analysis of the freeform surface figure error using a sum of specific-probability-distributed Gaussian radial basis functions is presented. This method is easy-used and practical. In addition, the actual manufacturing difficulty of different areas of the freeform surface is considered. Tolerance analysis of a nonsymmetric freeform off-axis three-mirror system was performed, including considering only the surface figure error, as well as an overall tolerance analysis considering both figure error and assembly error. The results demonstrated the effectiveness and feasibility of the proposed method.

Description and tolerance analysis of freeform surface figure error using specific-probability-distributed Gaussian radial basis functions.

The Fourier-based gradient field integration method can efficiently reconstruct transparent surfaces from the measured gradient data in the Shape-from-polarization method. However, for the differentiation operator having large truncation errors and lacking constraints on adjacent heights, it will increase the reconstructing error of the Fourier-based integration method. This paper presents an accurate Fourier-based integration approach to improve the reconstruction accuracy, in which a new differentiation operator is derived by limiting the truncation error and increasing heights and slopes in the operator. In order to modify the data obtained by the new operator to meet the requirements of both periodicity and size for the use of the discrete Fourier transform, we propose a method to extend the raw gradient data by first performing antisymmetric extension and then performing periodic extension. A series of simulations and experiments have been developed to verify the performance of the proposed method. By comparing the approach of this paper with other Fourier-based approaches, including Frankot-Chellappa, Southwell-FT and Simpson-FT, both of the simulation and experiment results show that the proposed Fourier-based integration method performs a higher accuracy than other approaches. In the reconstruction experiment, the reconstruction error can be reduced from 0.065 ~ 0.081 mm to 0.020 mm for the spherical surface, and from 0.060 ~ 0.12 mm to 0.016 mm for the free-form surface.

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An Accurate Fourier-Based Method for Three-Dimensional Reconstruction of Transparent Surfaces in the Shape-From-Polarization Method

High optical performance systems with wide field-of-view (FOV) have important applications in remote sensing. The radial basis functions, which have a prominent local characteristic in surface description, have attracted much attention in recent years. In this paper, an effective design method for the wide FOV imaging system using Gaussian radial basis function freeform surfaces is proposed. The FOV of the optical system is extended from a relatively small value to a larger one, and the Gaussian radial basis function surfaces are extended stepwise based on certain criteria. A high image quality and small distortion off-axis freeform three-mirror system with a wide FOV (${{60}}^\circ \times {0.6}^\circ$) is designed as an example. Tolerance analysis considering both surface figure error and assembly error is performed. The design results demonstrate the effectiveness of the proposed method.

Design method of wide field-of-view imaging systems using Gaussian radial basis functions freeform surfaces.

The description of deformable mirror (DM) surface, which is usually a complex freeform surface, affects the measurement speed and accuracy in a real-time interferometric measurement system with a DM as the dynamic compensator. We propose an accurate and fast description method with automatically configurable Gaussian radial basis function. The distribution and shape factors of GRBFs are related to the complexity of the surface with sufficient flexibility to improve the accuracy, and the fitting results are automatically obtained using a traversal optimization algorithm, which can improve the fitting speed by reducing the number of time-consuming calculations. The feasibility is verified by numerical and practical experiment.

Description method with automatically configurable Gaussian radial basis function for complex freeform surface.

We propose a design scheme and method of a freeform off-axis reflective imaging system with multiple mirrors integrated into one element. The use of a multiple-surfaces-integrated element, described by the Gaussian basis functions freeform surface with local and nonsymmetric properties, significantly decreases the system complexity, as well as reduces the assembly and fabrication difficulty, and achieves high imaging performance. The design theory and process including the initial system design, surface conversion, and system optimization are demonstrated in detail. Three design examples are demonstrated to validate the effect and feasibility of the proposed method, and good imaging performance is obtained.

Designing reflective imaging systems with multiple-surfaces-integrated elements using a Gaussian function freeform surface.

The model fitting degree of optical freeform surfaces is of utmost design importance. We develop a model with radial basis functions based on the surface slope (RBF-slope) for optical freeform surfaces with asymmetric structures. The RBF-slope model improves the basis-function distribution for circular apertures and establishes a relationship between shape factor and local surface slope, which provides the model with better fitting ability than the conventional RBF model (RBF-direct); fitting experiments for off-axis conic surfaces, “bumpy” paraboloids, and the design of a single mirror magnifier demonstrate the efficacy of our approach. Our method can effectively improve aberration balancing of optical freeform surfaces, resulting in high-quality imaging.

Model of radial basis functions based on surface slope for optical freeform surfaces.

It is of great importance to utilize a model to characterize the surfaces while designing an optical imaging system with freeform surfaces. For this purpose, a model with radial basis function based on surface slope (RBF-slope) for optical freeform surfaces was investigated by establishing the relationship between the shape factor and local surface slope, and improving the distributions of the basis functions for circular apertures. We performed the fitting experiment for “bumpy” paraboloids; the results demonstrated that the RBF-slope model has stronger fitting ability than conventional RBF model. A prototype of single mirror magnifier for head-worn display was designed and fabricated, in which the freeform mirror was described and characterized using the RBF-slope model. It can be proved by the design results that the RBF-slope model for optical freeform surfaces has obvious advantages in aberration balancing over conventional model. The primary experiments showed that the freeform surface was well fabricated and expected image display can be achieved.

A single mirror magnifier with freeform surfaces using radial basis functions based on surface slope

To obtain satisfactory performance in characterizing optical freeform surfaces with local features, this paper proposes a model of a radial basis function with slope-based shape factor and distribution (RBF-SSD). Compared to previous RBF-slope models with only slope-based shape factors, the RBF-SSD model relates both shape factors and distribution with the surface slope, ensuring greater fitting ability can be achieved when fitting a surface with local features. Fitting experiments for two different surfaces demonstrated the fitting ability of the RBF-SSD model. An off-axis three-mirror system with 3  °    ×  3.6  °   field of view was designed as an example to show the optical design efficacy of our model.

Kainian Tong

Papers

Model of radial basis functions based on surface slope for optical freeform surfaces.

A single mirror magnifier with freeform surfaces using radial basis functions based on surface slope

Model of radial basis functions with slope-based shape factor and distribution for optical freeform surface