Nodal Aberration Theory Research Articles

Desensitization design method of an off-axis three-mirror anastigmatic optical system based on the nodal aberration theory

A design method for an off-axis three-mirror anastigmatic optical system with low misalignment sensitivity is proposed. Based on the transformed pupil coordinate and nodal aberration theory, the analytical expressions between the optical parameters and misalignments are derived and an as-built performance evaluation model is established. The effects of mirror spacings and the off-axis magnitude value on the misalignment sensitivity are analyzed. With the desensitization design method proposed in this paper, a low-sensitivity off-axis TMA optical system can be designed. The simulation experiments show that, the misalignment sensitivity of the off-axis TMA optical system obtained by our method is approximately 60% that of the system obtained by nominal performance optimization method, and the as-built performance is about 1.49∼1.54 times that of the nominal performance optimization method.

Design of a zoom head-up display optical system based on a nodal aberration theory

We propose a zoom dual-path augmented reality head-up display optical system. The system uses two image generation units to display the basic and interactive information. Based on the node aberration theory, the aberration evaluation function was constructed. By solving the optimal solution of the aberration evaluation function, the initial structure of the interactive information display optical path was obtained. In order to solve the vergence-accommodation conflict in the process of vehicle speed change, the interactive information optical path was designed as a zoom optical path. The zoom distance is 10–20 m, and the basic information optical path will not be interfered with in the process. In addition, the tolerance analysis of the system was carried out to prove the machinability and stability of the system. This study makes the imaging of the augmented reality head-up display system more in line with the characteristics of the human eye and improves driving comfort.

An analytic expression for the field dependence of three-order Zernike aberrations in decentered and/or tilted optical systems

The double Zernike polynomials can describe the aperture dependence and field dependence of non-coaxial optical system aberrations simultaneously, and characterize the complex optical aberrations, which are of great value in the optical system design and alignment. In this paper, the specific analytic forms of all third-order double Zernike aberrations in decentered and/or tilted optical systems are derived based on the nodal aberration theory. This paper realizes the orthogonalization of the third-order aberrations with different aperture dependence and field dependence. The calculation process of the third-order double Zernike aberrations is presented. The ray tracing of the Cassegrain telescope verifies the analytic derivation of the double Zernike aberrations.

Compact off-axis reflective optical system design combining freeform mirror and freeform detector

Minimizing system size and weight without sacrificing the desired imaging performance is a crucial and challenging goal in the design of off-axis optical systems. In this paper, we propose a design concept that combines freeform mirror and freeform detector to achieve more compact off-axis optical systems. Initially, by applying nodal aberration theory, the optimal focal surface shape for off-axis systems is demonstrated to be an off-axis paraboloid or even a freeform surface, rather than the commonly assumed spherical shape. Subsequently, we show the ability of the design concept to reduce off-axis system volume through a design example of a fast and wide-field off-axis three-mirror system. Results indicate that, under the same design parameters and imaging quality, the design with a freeform detector is 80% smaller than that with a flat detector in terms of volume. Finally, the bending capability of the freeform detector presented in this paper is ensured through finite element analysis, and its surface accuracy is achievable based on the current precision level of curved detector fabrication. The proposed design concept holds promise for application in planetary science, where the imaging performance of the instrument is highly required and the size and weight are strictly limited.

Optimization of optical system for reducing tolerance sensitivity based on the high-order nodal aberration theory

This paper proposes an optimization method aimed at addressing the challenge of balancing system imaging performance and tolerance performance. The method is based on the high-order nodal aberration theory. The calculation method for high-order aberration coefficients is derived, along with the change in aberrations after introducing tolerance perturbation. The evaluation system includes both the imaging performance of the system and its tolerance factors. A comprehensive merit function is created for optimization. Three aspherical system examples were designed for verification, and the optimization results of this method show significantly lower tolerance sensitivity compared to the general optimization results and the SAB function optimization results of CODE V.

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.

Tolerance desensitization method based on principal component analysis and nodal aberration theory

Tolerance desensitization method based on principal component analysis and nodal aberration theory

Sigma vector calculations in nodal aberration theory and experimental validation using a Cassegrain telescope

Nodal Aberration Theory (NAT) was developed to explain the field dependency of aberration field centers in the image plane of nominally rotationally symmetric optical systems that have lost their symmetry through misalignments. A new insight into the theory led to calculating the sigma vectors, which locate the aberration field centers, using the angle between a real-ray trace of the optical axis ray (OAR) and the normal of the local surface where “local” refers to the object and image optical spaces of that surface. Here, we detail the sigma vector calculations for general optical systems and provide an experimental investigation of a misaligned system with a high-precision customized Cassegrain telescope. In the simulations, a Newtonian telescope, a Cassegrain telescope, and a three-mirror anastigmat telescope were misaligned intentionally in ray-tracing software. The sigma vectors were calculated analytically for the third-order aberrations of astigmatism and coma. Experimentally, the same perturbations were implemented for the Cassegrain telescope system, and the aberrations were quantified through interferometric measurements on a grid of field points in the image plane that verified the analytical derivation and simulations.

Design of off-axis aspheric four-mirror non-axial mechanical zoom optical system with large relative aperture.

A large relative aperture is essential to improve the spatial resolution of zoom systems. To overcome the limitations of the existing off-axis reflective mechanical zoom system with a low zoom rate and a small relative aperture, this paper proposes a non-axis movement method for increasing the degrees of freedom. On the basis of nodal aberration theory, passive eccentricity is changed into active eccentricity to achieve wave aberration balance in the multiple structures of the zoom imaging system. An off-axis aspherical four-mirror non-axial mechanical zoom optical system is designed and fabricated. The prototype has been successfully processed and assembled with the help of computer-aided alignment technology. The prototype's F-number is 4 and zoom ratio is 4.57:1. Experimental results verify the feasibility of the proposed method.

Active Alignment of Large-Aperture Space Telescopes for Optimal Ellipticity Performance

Ellipticity performance of space telescopes is important for exploration of dark matter. However, traditional on-orbit active optical alignment of space telescopes often takes “minimum wavefront error across the field of view” as the correction goal, and the ellipticity performance after correcting the wave aberration is not optimal. This paper proposes an active optical alignment strategy to achieve optimal ellipticity performance. Based on the framework of nodal aberration theory (NAT), the aberration field distribution corresponding to the optimal full field-of-view ellipticity is determined using global optimization. The degrees of freedom (DOFs) of the secondary mirror and the folded flat mirror are taken as the compensation DOFs to achieve the optimal ellipticity performance. Some valuable insights into aberration field characteristics corresponding to optimal ellipticity performance are presented. This work lays a basis for the correction of ellipticity for complicated optical systems.

Analysis and correction of misalignment-induced aberrations in two-mirror telescopes with stop on the secondary mirror.

We study the problem of misalignment aberration analysis and correction of the two-mirror telescopes with stop on the secondary mirror. The variation law of the system's aberration field is analyzed with nodal aberration theory when the primary mirror with an astigmatic figure error is misaligned. The analytic expression among the system wave aberration, misalignments, and astigmatism figure error is given, and the correction model of system misalignment aberration is established. The simulation experiment shows that the relative error of the prediction of system misalignment coma and astigmatism based on this model is less than 4.1%.

Two-mirror aerial mapping camera design with a tilted-aplanatic secondary mirror for image motion compensation.

The optical compensation method is a promising method to compensate the image motion by driving the optical element of the system. However, the tilt motion of the optical element could induce the degradation of the image quality. Based on the Nodal Aberration Theory (NAT), this paper proposes a novel optical image motion compensation design with a secondary tilt-aplanatic two-mirror optical system (STATOS), which achieves the insensitivity on the image quality of the optical system. First, the properties of coma-free pivot point depending on the location of stop at primary mirror or secondary mirror have been analyzed respectively. Then the calculating method about construction parameters in terms of STATOS is developed. Further, the maximum tilt angle of secondary mirror could be solved using NAT under ordered specific restriction. Finally, a prototype is designed with SM-tilting result of MTF value maintain above 0.32 at the cut-off frequency of detector, where the principal distance could remain stable and the changed distortion is better than 0.03%. A test flight has been carried out to prove the desired results and feasibility of STATOS used in an aerial mapping camera.

Application of nodal aberration theory in aberration compensation of the imaging system (invited)

Application of nodal aberration theory in aberration compensation of the imaging system (invited)

Analytical correction of pupil-offset off-axis telescopes with complex figure errors and misalignments based on nodal aberration theory

Active optical correction is necessary for space large-aperture optical telescopes, especially for systems with high performance requirements. This paper proposes an analytic correction method for pupil-offset off-axis optical telescopes with misalignments and complex figure errors on the basis of nodal aberration theory (NAT). An analytical description of the wave aberration contribution from the complex surfaces in off-axis telescopes is derived, which gives a unified description for the impacts of the complex surfaces located at the stop and away from the stop. Using the analytical description, a correction model for off-axis telescopes with figure errors and misalignments is established. The off-axis optical telescope with trefoil and astigmatic figure errors and lateral misalignments is taken as a typical example, the presented method is discussed. After the corresponding correction, the perturbed off-axis telescope is almost restored to the system nominal states, which can meet the specification requirements well. Finally, the Monte Carlo simulations are implemented to demonstrate the effectiveness of the presented approach.

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.

Wave Aberration Correction for an Unobscured Off-Axis Three-Mirror Astronomical Telescope Using an Aberration Field Compensation Mechanism

Studying how to use the coupling characteristics of net aberration fields induced by different perturbation parameters to realize the wave aberration compensation correction of perturbed telescopes is of great significance for the development of active optics. Based on nodal aberration theory, this paper studies the wave aberration compensation correction method of an unobscured off-axis three-mirror telescope. Specifically, first of all, it theoretically analyzes the coupling effects and compensation relationships of net aberration fields induced by different perturbation parameters of the telescope. Furthermore, it establishes wave aberration correction models with the secondary mirror as the compensator and the third mirror as the compensator for the telescope, respectively. In the end, it verifies the two compensation correction models by simulations. The results show that the tolerance of the secondary mirror compensation correction mode (SMCM) to the perturbation parameter threshold is significantly better than that of the third mirror compensation correction model (TMCM). When the introduced perturbation parameter threshold is small, the correction accuracy of the two models for the wave aberrations is equivalent, and both reach the order of 10−3λ (RMS, λ = 632.8 nm). When the perturbation parameter threshold is increased, the correction accuracy of SMCM can still be maintained at the order of 10−3λ but the correction accuracy of TMCM would decrease by an order of magnitude.

Optical design method of a three-mirror anastigmatic telescope with low misalignment sensitivity based on the nodal aberration theory.

We propose a design method for a three-mirror anastigmatic telescope with low misalignment sensitivity and deduce the analytic expression between the misalignment aberration and its optical parameters based on the nodal aberration theory. We establish an optical system as-built performance evaluation model. Using this model as the system's as-built performance evaluation indicator, we can get an optical system that could have both low misalignment sensitivity and good image quality after optimization. The design results of a field bias three-mirror anastigmatic telescope show that the misalignment aberration of the system can be reduced by changing the spacing of the mirrors. When the spacing between the primary mirror and the secondary mirror increases and the spacing from the secondary mirror to the third mirror and the third mirror to the image plane decreases, the misalignment sensitivity will drop significantly. If the mirror spacing is changed by 10%, the misalignment sensitivity of the telescope optimized by our method is only about 85% of that of the traditional method.

Optimization Method for Low Tilt Sensitivity of Secondary Mirror Based on the Nodal Aberration Theory

The optical system that combines imaging and image motion compensation is conducive to the miniaturization of aerial mapping cameras, but the movement of optical element for image motion compensation will cause a decrease in image quality. To solve this problem, reducing the sensitivity of moving optical element is one of the effective ways to ensure the imaging quality of aerial mapping cameras. Therefore, this paper proposes an optimization method for the low tilt sensitivity of the secondary mirror based on the Nodal aberration theory. In this method, the analytical expressions of the tilt sensitivity of the secondary mirror in different tilt directions are given in the form of zernike polynomial coefficients, and the influence of the field of view on the sensitivity is expressed in the mathematical model. The desensitization optimization function and desensitization optimization method are proposed. The catadioptric optical system with a focal length of 350 mm is used for desensitization optimization. The results show that the desensitization function proposed in this paper is linearly related to the decrease of sensitivity within a certain range, and the standard deviation of the system after desensitization is 0.020, which is 59% of the system without desensitization. Compared with the traditional method, the method in this paper widens the range of angle reduction sensitivity and has a better desensitization effect. The research results show that the optimization method for low tilt sensitivity of the secondary mirror based on the Nodal aberration theory proposed in this paper reduces the tilt sensitivity of the secondary mirror, revealing that the reduction of the sensitivity depends on the reduction of the aberration coefficient related to the misalignment in the field of view, which is critical for the development of an optical system for aerial mapping cameras that combines imaging and image motion compensation.

Optimization of the performance of hand‐held FFOCT scanner

In this work, we have established an improved handheld full-field optical coherence tomography (FFOCT) system based on the tandem structure of Michelson interferometer and Fizeau interferometer. We have analyzed the influence of wave aberrations of the handheld FFOCT as a function of orientation deviation of mirror M2 on the system point spread function (PSF) by using the nodal aberration theory. It is found that the range of the orientation deviation of mirror M2 must be less than 0.1° to generate stable FFOCT images. The effect of wave aberrations of defocus as a result of the refractive index difference on the system PSF has also been calculated, the measuring depth in the case of the sample with refractive index of 1.5 and the propagation medium is air (n = 1) should be less than 20 μm. Theoretical analyses have been demonstrated by the experimental results of aluminum powder, onion and porcine myocardium. By comparing the defocused FFOCT images of aluminum powder with defocused FFOCT images of onion and porcine myocardium, it can be concluded that reducing the refractive index difference between the medium and sample can increase the imaging depth of handheld FFOCT system.

Automatic initial configuration in off-axis reflective optical system design using combined nodal and Seidel aberration.

Currently, an off-axis reflective optical system is designed always from a coaxial initial system, and good imaging performance depends heavily on the choice of good initial configurations. This paper first establishes an imaging-performance evaluation function of the off-axis initial configuration based on nodal aberration theory and Seidel aberration theory. An automatic optimization method of the off-axis initial configuration using the global simulated annealing algorithm is proposed. Two design examples of off-axis three-mirror reflective optical systems are used to demonstrate the effectiveness and simplicity based on the initial configuration design method. Example 1: a series of optical systems with a large 10∘×3∘ field of view (FOV) is quickly and simply implemented using simple surfaces, and the optical systems exhibit perfect imaging quality. Example 2: starting from the good automatic initial configurations, two larger 20∘×3∘ FOV optical systems with focal lengths of 500 mm and f-numbers of five are designed, resulting in good imaging performance and providing a valuable design reference.