Nodal Aberration Theory Research Articles

Alignment algorithm of nonsymmetric off-axis reflective astronomical telescopes based on the modified third-order nodal aberration theory.

For pupil-offset off-axis reflective astronomical telescopes with designed tilts and decenters, owing to the absence of symmetry, axial and lateral misalignments exhibit strong coupling. The astigmatic and coma aberration fields of the misaligned optical systems are not only effected by lateral misalignments but also closely related to axial misalignments. However, the traditional misalignment algorithm based on nodal aberration theory (NAT) usually ignore the effect of axial misalignments on the aberration fields of optical systems when constructing calculation models. As a result, the presence of axial misalignments in pupil-offset off-axis telescopes with designed tilts and decenters will invalidate the traditional NAT-based lateral misalignment algorithm, which makes it difficult to be applied to actual computer-aided alignment experiments. In order to solve this issue, on the framework of modified NAT, third-order astigmatic, third-order coma, and third-order spherical net aberration fields of pupil-offset off-axis systems with designed tilts and decenters induced by axial and lateral misalignments are separated from the total aberration fields, and their inherent relations are analytically expressed. On this basis, in order to construct a solution model that can simultaneously and quantitatively calculate the axial and lateral misalignments, a method is proposed to fit the partial derivative coefficient matrix of misalignments according to field dependence of the net aberrations induced by misalignments. The simulation and actual alignment experiments were performed on a real wide-field off-axis three-mirror telescope using the constructed solution model, which proved the feasibility of the proposed method. Simulation experiments show that for different misalignment conditions generated randomly, both axial and lateral misalignments have achieved convergent solution results. In the actual alignment experiment, the average RMS wavefront errors of the nine field of views is corrected from 1.9 λ to 0.12 λ (λ = 632.8nm) through 3-5 iterations.

Experimental investigation in nodal aberration theory (NAT): separation of astigmatic figure error from misalignments in a Cassegrain telescope.

We present simulations and experimental validations for separating astigmatic figure error from misalignments in Nodal Aberration Theory (NAT) with a high-precision Cassegrain telescope. Both the primary mirror figure error and the secondary mirror misalignments induce binodal astigmatism for the telescope systems. The separation of these two aberration factors plays a crucial role in the telescope alignment process. In this study, the figure error of the aspheric primary mirror of the Cassegrain telescope induced by the mirror mounts was measured interferometrically utilizing a computer-generated hologram (CGH). According to the primary mirror figure error, the astigmatic node locations in the image plane were simulated using real raytracing. The center of the nodes was located on the field center, and the nodes were placed symmetrically with respect to the field center in the image plane. The telescope's alignment was performed using the simulation results, and the node locations were measured on a grid of field points interferometrically. Thereafter, secondary mirror misalignments around the coma-free pivot point were introduced into the optical system, and the node's center was shifted from the field center in the image plane as predicted by NAT. The simulations and interferometric field measurements were performed and compared on a grid of field points for the misaligned state in the presence of primary mirror figure error. The experimental results confirm the predictions from NAT. Statistical analysis was also performed to confirm the accuracy and stability of the measurements.

A simulation and experimental validation of third-order coma in nodal aberration theory with a Cassegrain telescope

A simulation and experimental validation of third-order coma in nodal aberration theory with a Cassegrain telescope

Efficient misalignment calculation for two-mirror optical systems based on wavefront aberrations in a single field of view

Computer-aided alignment (CAA) is a key technology for improving the assembly efficiency and performance of optical systems. However, because of the coma-free pivot point in two-mirror optical systems (TMOSs), the wavefront aberrations from multiple fields of view (FoVs) are required to compute misalignments, and this is not practical. Therefore, misalignment-caused wavefront aberrations were investigated based on the nodal aberration theory, and an analytical aberration computation model for TMOSs was derived. Based on this, an efficient misalignment calculation method based on single-FoV aberrations that introduces known misalignments was proposed and tested by conducting Monte Carlo simulations using the Hilbert telescope. The results showed that micron-level precision can be achieved for the aberration and misalignment calculation without any measurement noise and that a greater number of introduced misalignments can decrease the effect of measurement noise, thereby verifying the effectiveness of the proposed model and method. The findings of this research will open a new direction in CAA by introducing the misalignment rather than the FoV.

Misalignment algorithm of a wide-field survey telescope based on third-order quadratic nodal aberration theory

Off-axis three-mirror anastigmatic (TMA) telescopes with designed decenters and tilts, have many misalignment degrees of freedom and strong coupling between each misalignment degree of freedom. Therefore, it is difficult to establish misalignment equations only using A222 and A131 in nodal aberration theory (NAT). In addition, for off-axis TMA optical systems with designed decenters and tilts, the robustness of the existing fifth-order NAT misalignment calculation algorithm based on high-order Zernike coefficients and boresight errors decreases, so it is difficult to realize its engineering application. To solve the issue of insufficient practicality of the existing misalignment algorithm based on fifth-order NAT, a third-order NAT calculation algorithm based on quadratic aberration field decenter vectors is derived and established. Two concepts of inherent aberration field decenter vector and misalignment aberration field decenter vector are proposed. Taking an off-axis TMA optical system with 6-m focal length as the research object, simulations, and alignment verification experiments were carried out. Compared with the existing fifth-order NAT misalignment algorithm, the results show that when measurement noise is not considered, the two methods can both obtain convergent calculation results, and the average RMS wavefront errors (WFE) of the optical system are both corrected to be below 0.0574 waves. When different levels of measurement noise are introduced, the robustness of the fifth-order NAT misalignment algorithm decreases, and there are even cases where the optical system completely fails to be corrected. However, the algorithm based on quadratic aberration field decenter vectors shows better robustness. Under different levels of measurement noise, this algorithm could correct the average RMS WFE of the optical system to around 0.0574 waves.

Active alignment of space astronomical telescopes by matching arbitrary multi-field stellar image features.

Space-based optical astronomical telescopes are susceptible to mirror misalignments due to space disturbance in mechanics and temperature. Therefore, it is of great importance to actively align the telescope in orbit to continuously maintain imaging quality. Traditional active alignment methods usually need additional delicate wavefront sensors and complicated operations (such as instrument calibration and pointing adjustment). This paper proposes a novel active alignment approach by matching the geometrical features of several stellar images at arbitrary multiple field positions. Based on nodal aberration theory and Fourier optics, the relationship between stellar image intensity distribution and misalignments of the system can be modeled for an arbitrary field position. On this basis, an objective function is established by matching the geometrical features of the collected multi-field stellar images and modeled multi-field stellar images, and misalignments can then be solved through nonlinear optimization. Detailed simulations and a real experiment are performed to demonstrate the effectiveness and practicality of the proposed approach. This approach eliminates the need for delicate wavefront sensors and pointing adjustment, which greatly facilitates the maintainance of imaging quality.

Aberration compensation strategy for the radius of curvature error of the primary mirror in off-axis three-mirror anastigmatic telescopes.

This paper discusses compensation strategies for the aberration fields caused by the error in the radius of curvature (ROC) of the primary mirror (PM) in pupil-offset off-axis three-mirror anastigmatic (TMA) astronomical telescopes. Based on the nodal aberration theory framework, the specific astigmatic and coma aberrations of the off-axis three-mirror system in the presence of the ROC error of the PM are derived. It is demonstrated that some field-dependent aberration components can be induced by ROC error in the off-axis TMA telescopes, apart from the dominating field-constant aberration terms. To reduce the influence of the ROC error on the aberration fields, we propose two aberration compensation strategies: adjusting the position of the PM and introducing axial misalignment of the secondary mirror (SM). Through theoretical analysis and simulations, we conclude that the compensation strategy of changing the axial position of the PM can make the aberration distribution close to the nominal state; the compensation strategy of axially adjusting the SM can make the aberration distribution meet the observation requirements, which is more suitable for space applications. We also discuss compensating the effect of the ROC error using lateral misalignments.

Experimental investigation of binodal astigmatism in nodal aberration theory (NAT) with a Cassegrain telescope system.

We present simulations and an experimental investigation of binodal astigmatism in nodal aberration theory (NAT) for a customized, high-precision Cassegrain telescope system. The telescope system utilizes a five-axis, piezo-actuated flexural mechanism to introduce secondary mirror misalignments and generate aberrations intentionally. The induced aberrations are measured interferometrically and quantified for a grid of field points on the telescope system's image plane. For this purpose, a coma-free pivot point of the secondary mirror was simulated for isolating the binodal astigmatism field response. The separation of the nodes is proportional to the introduced misalignments. A simulation of Fringe Zernike coma and binodal astigmatism was generated using a real ray trace model of the optical system and analyzed to compare to the experimental results. A statistical analysis of the measurements was performed to show the experimental results' accuracy and stability. The experimental results were consistent with the simulations, hence experimentally validating NAT for binodal astigmatism for the first time.

Active alignment of complex perturbed pupil-offset off-axis telescopes using the extension of nodal aberration theory

This paper presents an optical alignment strategy for complex perturbed pupil-offset off-axis reflective telescopes, based on the extension of nodal aberration theory (NAT). First, the direct expansion of the wave aberration function in the vector form for perturbed off-axis systems is given, which is especially convenient for the expansion of the corresponding higher-order terms. The inherent vector relationships between the contributions generated by the aberrations of the on-axis parent systems through pupil transformation are disclosed in detail, which is helpful to understand the aberration behavior of off-axis systems. Then, according to the inherent vector relationships, an analytical alignment model based on NAT for complex cases of perturbed off-axis telescopes is established. It can quantitatively separate the effects of misalignments and surface figure errors from the total aberration fields. The alignment model is solved by using particle swarm optimization algorithm. Then, an optical alignment example of the off-axis three-mirror anastigmatic telescope with misalignments and complex surface figure errors based on the proposed method is demonstrated. After correction, the perturbed telescope can be nearly restored to the nominal states. Finally, Monte Carlo simulations are carried out to show the effectiveness and accuracy of the proposed method.

Synthetic system design method for off-axis stabilized zoom systems with a high zoom ratio.

Stabilized zoom systems possess the advantages such as the simplified system layout, improved system stability, enhanced imaging performance, and a high zoom speed. The complex system design to achieve high performances requires calculations or investigations of the initial system for optimization and improvements, and thus, specific design techniques are pursued. In this study, we propose an automatic optical design scheme of synthetic characteristics for the off-axis stabilized zoom systems, which using focal length variable (FLV) opto-electronic elements and with a high zoom ratio. The study aims at evaluating and synthetically achieving the zooming properties and the image quality balance of entire focus imaging. The multi-element stabilized zoom systems are characterized using the Gaussian brackets expressions and their optimal solution ranges for high zoom ratios are deduced to achieve non-defocusing imaging in specific stroke ranges of FLV elements. Then considering the analytical characterization of the off-axis-induced primary aberrations at multi-conjugate positions, we use a conic surface to deduce the basic expression of the nodal aberration. Thereby the nonlinear global merit function is established with a semi-empirical mathematical model based on nodal aberration theory and nonlinear zoom equation for maintaining the stability of focal length and image plane drift. And the theory of Pareto Optimality is employed in the process of verifying the superiority of the solutions. Finally, a series of solutions for a high zoom ratio and aberration compensation are implemented and the optimal configurations with conical surfaces for an off-axis stabilized zoom system are obtained.

Correction of resonant optical scanner dynamic aberrations using nodal aberration theory.

The rapid oscillation of galvanometric resonant optical scanners introduces linear astigmatism that degrades transverse resolution, and in confocal systems, also reduces signal [V. Akondi et al., Optica 7, 1506, 2020]. Here, we demonstrate correction of this aberration by tilting reflective or refractive optical elements for a single vergence or a vergence range, with and without the use of an adaptive wavefront corrector such as a deformable mirror. The approach, based on nodal aberration theory, can generate any desired third order aberration that results from tilting or decentering optical surfaces.

Optimization method using nodal aberration theory for coaxial imaging systems with radial basis functions based on surface slope

The radial basis functions based on the surface slope (RBF-Slope) freeform surfaces model has demonstrated stronger fitting ability and better optical performance than the conventional RBF model. However, the large number of basis functions and optimization variables of the RBF-Slope model may result in convergence problems during optimization for optical systems consisting of freeform surfaces characterized by RBF-Slope. To overcome these drawbacks, we use Zernike polynomials to link the RBF-Slope model to aberration correction and propose a new optimization method for coaxial imaging systems using the RBF-Slope model based on nodal aberration theory (NAT). The aberrations generated by the conic parameter and Zernike terms up to Z 17 / 18 of the Zernike freeform surface at the non-stop surface in the coaxial imaging system are analyzed, and the gradient descent is implemented to obtain the optimal coefficients of the Zernike surface, which is then fitted by the RBF-Slope surface for further optimization. The method is applied to the optimization of a secondary mirror using the RBF-Slope model in a two-mirror telescope and proved to have better results than traditional commonly used direct optimization. This research offers an important reference for optimization using NAT and provides valuable insight into the optimization method for RBF-Slope freeform surfaces.

Active compensation for optimal RMS wavefront error in perturbed off-axis optical telescopes using nodal aberration theory.

This paper presents an active compensation strategy for RMS wavefront error of perturbed off-axis telescopes in the framework of nodal aberration theory. First, the orthogonalized expression of the wave aberration function in the vector form for perturbed off-axis telescopes is derived by using RMS normalization. The orthogonalized aberration function is applied to analytically describe the RMS wavefront error in perturbed off-axis telescopes with circular apertures. Then, the system compensation model for perturbed off-axis telescopes is established. The compensation model takes the weighted square sum of the RMS wavefront errors at representative field points as the objective function, which is minimized to obtain the optimal compensation solution of off-axis systems with perturbation constraints. The compensation model is solved by using a particle swarm optimization algorithm. Then, the off-axis three-mirror anastigmatic telescope is taken as an example, and the system compensations for the misaligned tertiary mirror and deformed primary mirror are discussed. After compensation, the average RMS wavefront errors in the perturbed off-axis systems are greatly reduced, which can well meet the system requirements. Finally, Monte Carlo simulations of the optimal compensation method and sensitivity table method are carried out to demonstrate the correctness and accuracy of the proposed method.

Field aberrations in terms of the Q-polynomial basis and its relationship to the Zernike basis

The aberrations generated at the image plane of an optical system that includes freeform surfaces described through Q-polynomials can be calculated using nodal aberration theory. By analyzing the definition of each Q-polynomial, they can be compared with Zernike polynomials allowing a relationship between the two bases. This relationship is neither simple nor direct, so a fitting must be made. Once established, the contribution to the aberration field map generated by each surface described through the Q-polynomial can be calculated for any surface that is not at the stop of the system. The Q-polynomials are characterized by their orthogonality in the gradient instead of the surface, which represents an opportunity to restrict the changes in the slope in a simple way and facilitate the manufacturing process. The knowledge of the field aberrations generated by each Q-polynomial allows selecting that which of them are necessary to be introduced as variables in the optimization process for an efficient optimization.

A simulation and experimental validation of third-order coma in nodal aberrationtheory with a Cassegrain telescope

This study presents a simulation and experimental validation of third-order coma in Nodal Aberration Theory (NAT) with a custom-designed Cassegrain system. The Cassegrain system uses a piezo-driven flexure tool to align the telescope's secondary mirror with respect to the primary mirror. This alignment mechanism is also used to misalign the telescope's secondary mirror to induce the intentional aberrations. The third-order Fringe Zernike coma (Z$_{7/8}$) was simulated utilizing the Cassegrain telescope's real ray-trace model (e.g., in Code V) and analyzed for both the telescope's nominal aligned and misaligned states. The simulated aberration of third-order coma was induced utilizing the secondary mirror misalignments and measured interferometrically on a $5\times 5 $ mesh of field points on the telescope's focal plane. The experimental results were compared with simulation results, and we showed that the results are consistent with each other. Hence, we validated the shift of aberration field center (zero coma, or coma node) from the optical axis, where the third-order coma is linear with the field as predicted by NAT for the Cassegrain telescope. For demonstrating the stability and accuracy of the experimental results, a statistical analysis was conducted.

Aberration fields of pupil-offset off-axis two-mirror astronomical telescopes induced by ROC error.

This paper presents a systematic and deep discussion on the aberration field characteristics of pupil-offset off-axis two-mirror astronomical telescopes induced by the radius of curvature (ROC) error based on the framework of the nodal aberration theory (NAT). The expressions of the third-order aberrations in off-axis two-mirror astronomical telescopes with ROC error are derived first. Then the astigmatic and coma aberration fields are discussed, and it is shown in a field constant astigmatism and coma will be induced by ROC error. The aberration compensation between axial misalignments and ROC error are further discussed, and it is shown that the net astigmatic and coma aberration field induced by ROC error can well be compensated by axial misalignments. Importantly, it is also demonstrated that the focal plane shift induced by ROC error can also be compensated at the same time. Also, this paper briefly analyzes the aberration field characteristics when there is the error of conic constant in optical system. Some other discussions are also presented concerning the ROC inconsistency in astronomical telescopes with a segmented primary mirror. This work will lead to a deep understanding of the influence of ROC error in pupil-offset off-axis astronomical telescopes.

Scalar analytical expressions for the field dependence of Zernike polynomials in asymmetric optical systems with circular symmetric surfaces

In this paper we derive scalar analytical expressions describing the full field dependence of Zernike polynomials in optical systems without symmetries. We consider the general case of optical systems constituted by arbitrarily tilted and decentered circular symmetric surfaces. The resulting analytical formulae are inferred from a modified version of the full field dependent wavefront aberration function proposed in the Nodal Aberration Theory (NAT). Such formula is modified with the scope of solving few critical points arising when primary and higher order aberrations are both present in an optical system. It is shown that when secondary aberrations are taken into account in the wavefront aberration function, the final effect is a perturbation to the symmetry of the field dependence of the Zernike polynomials. In particular, the centers of symmetry of the Zernike polynomial field dependences are shifted with respect to the locations predicted using the NAT equations as a consequence of the presence of higher order aberrations. The retrieved analytical expressions are verified through surface fitting to real ray-trace data obtained for a simple optical system.

Zernike monomials in wide field of view optical designs.

Zernike polynomials are universal in optical modeling and testing of wavefronts; however, their polynomial behavior can cause a misinterpretation of individual aberrations. Wavefront profiles described by Zernike polynomials contain multiple terms with different orders of pupil radius (ρ). Zernike polynomials are a sum of high and low orders of ρ to minimize the RMS wavefront error and to preserve orthogonality. Since the low-order polynomials are still contained in the net Zernike sum, there is redundancy in individual monomials. Monomial aberrations, also known as Seidel or primary aberrations, are useful in studying an optical design's complexity, alignment, and field behavior. Zernike polynomial aberrations reported by optical design software are not indicative of individual (monomial) aberrations in wide field of view designs since the low-order polynomials are contaminated by higher order terms. An aberration node is the field location where an individual (monomial) aberration is zero. In this paper, a matrix method is shown to calculate the individual monomial aberrations given the set of Zernike polynomials. Monomial aberrations plotted as a function of field angle (H) indicate the field order (Hn) and the location of true aberration nodes. Contrarily, Zernike polynomial versus field (ZvF) plots can indicate false aberration nodes, due to the polynomial mixing of high- and low-order terms. Accurate knowledge of the monomial aberration nodes, converted from Zernike polynomials, provides the link between a ray-trace model or lab wavefront measurement and nodal aberration theory (NAT). This method is applied to two different optical designs: (1) 120° circular FOV fish-eye lens and (2) 120∘×4∘ rectangular FOV, off-axis, freeform four-mirror design.

Optical system optimization method for as-built performance based on nodal aberration theory.

The traditional optical design process isolates the two steps of system performance optimization and tolerance allocation, making it difficult to achieve optimal design of as-built performance. To solve this problem, this paper proposes an analytical method for optimizing the as-built performance of optical systems. The method uses the nodal aberration theory to derive the wavefront aberration estimated value under the given surface decenter and tilt tolerance, and establishes the optical system as-built performance evaluation model. The as-built performance evaluation does not require a large amount of ray tracing, which can be completed only by tracking the paraxial marginal ray and the principal ray, and the calculation amount is small. The as-built performance evaluation model can be directly used as error function in optical design software for optical system optimization. A Cooke triplet system is taken as an example to compare the as-built performance optimization method, Code V and Zemax OpticStudio's built-in optimization methods and the traditional method which optimizes only nominal performance by Monte-Carlo tolerance analysis.

Design and alignment of an all-spherical unobscured four-mirror image relay for an ultra-broadband sub-petawatt laser.

We describe the design and alignment of an unobscured reflective image relay for use in an ultra-broadband half-petawatt laser system in development at the University of Rochester's Laboratory for Laser Energetics. The design consists of four spherical mirrors in an unobscured configuration. We show the theoretical basis for such a four-mirror design using first-order optical matrix methods and nodal aberration theory. We also describe the process of aligning this design in a test bed and show the results of this alignment method. We were able to achieve a diffraction-limited nominal design. Upon aligning the design in our test bed, we were able to successfully align the design to achieve our target wavefront error of less than 0.3 waves peak to valley and 0.07 waves rms at the central laser-operating wavelength of 910 nm.