Three-mirror Anastigmatic Telescope Research Articles

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 a Focusing Mechanism Actuated by Piezoelectric Ceramics for TMA Telescope.

With the development of space telescopes towards high-resolution and intelligent imaging, the scale and complexity of the focal plane components of large-aperture, off-axis, three-mirror anastigmatic (TMA) optical systems are increasing. Traditional focal plane focusing technology reduces the system reliability and increases the scale and complexity. This paper proposes a three-degrees-of-freedom focusing system based on a folding mirror reflector, with a piezoelectric ceramic actuator as the driver. An environment-resistant flexible support was designed for the piezoelectric ceramic actuator through an integrated optimization analysis. The fundamental frequency of the large-aspect-ratio rectangular folding mirror reflector focusing mechanism was around 121.5 Hz. After testing, it was found to meet the requirements of the space mechanics environment. This system shows promise for application to other optical systems as an open-shelf product in the future.

Convex optical freeforms using fringe Zernike overlay approach for two-mirror and three-mirror telescopes for space applications

The challenge in manufacturing and testing of convex optical parts is well known in comparison to its concave counterpart and an optimized imaging system invariably features a combination of both. Of late, freeform mirrors find lot of interest in space applications as the optical systems remain compact within the allocated envelope. As is well known, these freeform mirrors are difficult to manufacture and test due to their lack of rotational or translational symmetry and poses additional challenges if it has convex profile. In the current investigations, two new convex freeform optical surfaces are proposed which is based on fringe Zernike overlay technique. The first convex freeform envisaged is for a two-mirror off-axis telescope where the secondary mirror is an off-axis freeform and the second convex freeform for a three-mirror anastigmat telescope where the secondary mirror is an on-axis convex freeform. Substitution of these convex freeform surfaces in the two-mirror and three-mirror telescope systems aid in the correction of asymmetrical aberrations to a wider view field, thereby enhancing the performances of the de-centred and tilted systems in the respective configurations. The procedure adopted for the manufacturing of these convex freeform surfaces using bonnet polishing methods and the interferometric tests using modified-Hindle null corrector and subaperture-stitching techniques for evaluation are described in detail.

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.

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.

Comparisons between an on-axis three-mirror anastigmat telescope and an off-axis one: polarization aberrations.

Due to the ability to achieve a wide-diffraction-limited field of view, three-mirror anastigmat (TMA) telescopes are widely used in many applications that demand high imaging quality and detection accuracy, in which polarization aberrations play increasingly important roles. In this paper, polarization properties of on-axis elements and those of off-axis elements are analyzed and compared via polarization aberration theory. Then, an on-axis TMA (OTMA) telescope and an off-axis TMA (FTMA) telescope are designed and optimized to be diffraction-limited systems in parallel. Their constructional parameters and first-order parameters are set to be identical. Via polarization ray tracing, polarization aberrations of the 2 telescopes are obtained and compared in terms of diattenuation map, retardance map, Jones pupil, Pauli pupil, and amplitude response matrix. Results demonstrate that polarization aberrations are closely related to the structural features of mirrors and telescopes. The diattenuation and retardance maps of power mirrors in the OTMA telescope are rotationally symmetric, while those in the FTMA telescope are not. A Maltese cross appears in off-diagonal elements of the Jones pupil of the OTMA telescope and disappears in the FTMA telescope. Amplitude response matrices of polarization crosstalk components in the OTMA telescope are with a four-peak structure, while those in the FTMA telescope are with a two-peak structure. After quantitative comparisons, it is found that diattenuation in the OTMA telescope is smaller than that in the FTMA telescope as well as in the cases of retardance and polarization crosstalk. There are also similarities between the 2 TMA telescopes. Polarization aberrations of secondary mirrors (SM) are greater than those of both primary mirrors (PM) and tertiary mirrors (TM). What is more, polarization aberrations of TM are so small that they can be ignored safely. The numerical sum of Pauli coefficients of PM, SM, and TM is nearly identical to the cumulative Pauli coefficients. Hence, the TMA telescopes can be treated as weak polarization elements. This work compares polarization aberration performance of on- and off-axis TMA telescopes and can provide important references for designing and optimizing telescopes in which polarization aberrations matter.

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.

Stray light control and analysis for an off-axis three-mirror anastigmat telescope

Off-axis three-mirror anastigmat (TMA) telescopes provide excellent correction of aberrations over a large field-of-view in one direction. In a push broom configuration, this lightweight and compact optical configuration enables high-performing imaging. Moreover, passive multi-spectral acquisition can be achieved using a filter stack positioned at the detector vicinity. Stray light is a typical limiting factor for instrument performance and its control is specific to the type of optical configuration. We describe the stray light control and analysis methods in an off-axis TMA. The design intends to control first-order scattering from non-optical surfaces and to block straight shots. This is achieved through usage of elements such as apertures and baffles, both internal and external to the instrument. It will be demonstrated that the aperture stop is a critical element, whose stray light contribution can be controlled with V-groove vanes. Impact of mirrors roughness, a manufacturing physical limitation, is evaluated by modeling the bidirectional scattering distribution function and computing the stray light distribution at the detector for point-like source illumination. We show that the scattering on the different mirrors broaden differently the spot size, as the successive mirrors focus or expand the scattered rays differently. Finally, the ghost reflections inside the filter stack are evaluated and we show that there is no cross-talk between the different channels.

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.

Improvement of a computer-aided alignment algorithm for the nonsymmetric off-axis reflective telescope

Due to complications of the off-axis three-mirror anastigmat (TMA) telescope, each optical element in the off-axis TMA telescope is introduced with theoretical eccentricity and tilt. Moreover, the introduction of freeform surfaces and other optical elements with complex surface features generally causes the initial alignment accuracy of the optical path to be low. A large initial alignment error amplifies the sensitivity of the misalignment calculation accuracy to the measurement error of the Zernike coefficient, resulting in difficulty obtaining convergence results for a computer-aided alignment algorithm. Considering the above issues, the alignment sensitivity of each component in the optical path is analyzed in this. The large conditional number of the sensitivity matrix results in poor algorithm robustness. Thus, an adaptive damping factor least-squares algorithm model is proposed and derived to improve the efficiency of the classical least-squares algorithm. A method for piecewise optimization of the damping factor is also deduced. Experiments based on a 0.6 m off-axis TMA telescope verify the effectiveness of the algorithm. Simulation and integration experiments show that the proposed method can reduce the accuracy requirements of the initial alignment and improve the adaptability of Zernike coefficient measurement noise. The alignment procedure is carried out for three iterations, and the average of the five field-of-view wave aberration values is enhanced from 2.1λ (RMS; λ=632.8nm) to 0.09λ (average). The improved algorithm can solve the large initial alignment error of a nonsymmetrical off-axis reflective optical system with a freeform surface as well as the problem of the low success rate of the misalignment value due to low Zernike coefficient measurement accuracy.

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.

Analytical design method of three-mirror anastigmatic telescope with mirror spacings as free design parameters

We present an analytical design method for the three-mirror anastigmatic (TMA) telescope with mirror spacings as the free design parameter. After the optical designer determines, the system focal length and the mirror spacings according to the design requirements, the design solutions of all TMA telescopes that meet the conditions can be obtained directly according to the formulas for the mirror radius and conic constants derived. The method here can predetermine the mirror position and system envelope size before design, and can quickly give all design solutions. We give a design example using the method and compare and discuss all design solutions of TMA.

Aberrational interactions between axial and lateral misalignments in pupil-offset off-axis two-mirror astronomical telescopes.

Due to the absence of rotational symmetry, the effects of axial and lateral misalignments couple tightly together, which leads to special aberration field characteristics. This paper will present an in-depth and systematic discussion on the interactions between the effects of axial and lateral misalignments in pupil-offset off-axis two-mirror astronomical telescopes. The aberration function of this class of telescopes in the presence of axial and lateral misalignments is derived. The specific expressions of two dominant non-rotationally symmetric aberrations, i.e., astigmatism and coma, are obtained and the aberration field characteristics are discussed. Importantly, it is shown that under certain conditions, a node will arise in the field of view for these two kinds of aberrations. Then the aberrational compensation mechanisms between axial and lateral misalignments are quantitatively explicated, and it is shown that the non-rotationally symmetric aberrations induced by axial misalignments can well be compensated by lateral misalignments. However, we also find that in this process, the defocus aberration induced by these two kinds of misalignments will accumulate (rather than cancel out). Therefore, in practice, it is better to separate these two kinds of misalignment. Finally, we propose a simple method to decouple axial misalignments from lateral misalignments with wavefront measurement at one field position. Most of this work can be extended to other kind of pupil-offset off-axis astronomical telescopes, such as off-axis three-mirror anastigmatic telescopes.

Optical design for CETUS: a wide-field 1.5-m aperture UV payload being studied for a NASA probe class mission study

As part of a study funded by NASA headquarters, we are developing a probe-class mission concept called the Cosmic Evolution through UV Spectroscopy (CETUS). CETUS includes a 1.5-m aperture diameter telescope with a large field of view (FOV). CETUS includes three scientific instruments: a far ultraviolet (FUV) and near ultraviolet (NUV) imaging camera (CAM); a NUV multiobject spectrograph (MOS); and a dual-channel point/slit spectrograph (PSS) in the Lyman ultraviolet (LUV), FUV, and NUV spectral regions. The large FOV three-mirror anastigmatic (TMA) optical telescope assembly (OTA) simultaneously feeds the three separate scientific instruments. That is, the instruments view separate portions of the TMA image plane, enabling parallel operation by the three instruments. The field viewed by the MOS, whose design is based on an Offner-type spectrographic configuration to provide wide FOV correction, is actively configured to select and isolate numerous field sources using a next-generation micro-shutter array. The two-channel CAM design is also based on an Offner-like configuration. The PSS performs high spectral resolution spectroscopy on unresolved objects over the NUV region with spectral resolving power, R ∼ 40,000, in an echelle mode. The PSS also performs long-slit imaging spectroscopy at R ∼ 20,000 in the LUV and FUV spectral regions with two aberration-corrected, blazed, holographic gratings used in a Rowland-like configuration. The optical system also includes two fine guidance sensors, and wavefront sensors that sample numerous locations over the full OTA FOV. In-flight wavelength calibration is performed by a wavelength calibration system, and flat-fielding is also performed, both using in-flight calibration sources. We describe the current optical design of CETUS and the major trade studies leading to the design.

Experimental study on the extension of nodal aberration theory to pupil-offset off-axis three-mirror anastigmatic telescopes

The aberration fields of misaligned on-axis telescopes can be described by nodal aberration theory. However, traditional nodal aberration theory cannot directly apply to pupil-offset off-axis systems. In our previous work, the net aberration fields of pupil-offset off-axis two-mirror astronomical telescopes induced by lateral misalignments were investigated by extending nodal aberration theory to include pupil-offset off-axis telescopes with a system-level pupil coordinate transformation through simulation. An experimental study on the net aberration fields of pupil-offset off-axis three-mirror anastigmatic (TMA) telescopes induced by lateral misalignments is further presented. Specifically, the astigmatism and coma aberration fields as well as their inherent relations are analytically expressed, simulated, and quantitatively validated with a real pupil-offset off-axis TMA telescope. Meanwhile, the differences between the aberration fields of misaligned off-axis and on-axis TMA telescopes are revealed and explicated. Our work not only contributes to a deep understanding of the net aberration fields of pupil-offset off-axis TMA telescopes induced by lateral misalignments but also represent an important validation for the extension of nodal aberration theory to pupil-offset off-axis telescopes.

Alignment algorithm for three-mirror anastigmatic telescopes based on nodal aberration theory

ABSTRACTFor most of the alignment algorithms, alignment errors and figure errors cannot be separated due to the coupling effect among optical elements. We present an alignment algorithm for TMA telescopes on the framework of nodal aberration theory (NAT). Based on NAT, we firstly determine the axial misalignments of secondary (SM) and tertiary (TM) mirrors. Then we decouple the lateral misalignments and angular misalignments of SM and TM. Finally, we decouple the figure errors of primary mirror (PM) from the alignment errors of SM and TM. To validate the algorithm, a TMA telescope is designed for simulation. In the simulation, modulation transfer function (MTF) is chosen to evaluate the telescope before and after correction. It is found that the algorithm presented in this paper maintains high precision. In the end, a Monte Carlo simulation is conducted to further demonstrate the accuracy of the presented alignment algorithm even in poor conditions.

Compensation factors for 3rd order coma in three mirror anastigmatic (TMA) telescopes.

Misalignment induced third-order coma with respect to misaligned parameters in TMA optical systems is derived by using Nodal Aberration Theory, which yields the compensation factors that can be used to accomplish coma compensation in both coaxial and off-axis misaligned TMA telescopes. By using the compensation factors, coma free point for the tertiary mirror in TMA telescopes is derived and proved to be the negative form of the one for the secondary mirror in the Cassegrain telescope. The compensation factors can also be used to design the off-axis TMAs due to their capability of eliminating the coma over the field of view.

Elimination of the field-dependent aberrations of the JWST-like space telescopes in the multi-field fine-phasing process.

This paper investigates the alignment strategies for eliminating the field-dependent aberrations of the class of large three-mirror anastigmatic (TMA) space telescopes with a segmented primary mirror (PM) like the James Webb Space Telescope (JWST) in the multi-field fine-phasing process based on the framework of nodal aberration theory. During the single-field (on-axis field) fine-phasing process, the individual segment tip, tilt, and piston errors, as well as the de-space of the secondary mirror, are well corrected, and the PM is also adjusted to compensate for those aberrations induced by the misalignments of other mirrors at the center of the science field of view. However, interrogating off-axis field points can reveal the presence of large wavefront errors due to mirror misalignments. Eliminating these field-dependent aberrations is the main goal of the multi-field fine-phasing process. This paper first presents an analytic study on an established alignment strategy used for eliminating the field-dependent aberrations. While it is demonstrated that this alignment strategy has the ability to reduce the field dependency of the wavefront errors, it will, however, also be revealed that this strategy still exhibits some problems, and its alignment efficiency is low. Then, a new alignment strategy with higher alignment efficiency is further proposed. Detailed simulations with a TMA telescope that has similar parameters with the JWST are performed to illustrate the efficiency and rationality of the proposed strategy. This work can not only contribute to an in-depth understanding of the multi-field fine-phasing process, but also present a possibility to improve the efficiency of this process.

Alignment of a three-mirror anastigmatic telescope using nodal aberration theory.

Most computer-aided alignment methods for optical systems are based on numerical algorithms at present, which omit aberration theory. This paper presents a novel alignment algorithm for three-mirror anastigmatic (TMA) telescopes using Nodal Aberration Theory (NAT). The aberration field decenter vectors and boresight error of misaligned TMA telescopes are derived. Two alignment models based on 3rd and 5th order NAT are established successively and compared in the same alignment example. It is found that the average and the maximum RMS wavefront errors in the whole field of view of 0.3° × 0.15° are 0.063 λ (λ = 1 μm) and 0.068 λ respectively after the 4th alignment action with the 3rd order model, and 0.011 λ and 0.025 λ (nominal values) respectively after the 3rd alignment action with the 5th order model. Monte-Carlo alignment simulations are carried out with the 5th order model. It shows that the 5th order model still has good performance even when the misalignment variables are large (-1 mm≤linear misalignment≤1 mm, -0.1°≤angular misalignment≤0.1°), and multiple iterative alignments are needed when the misalignment variables increase.

Alignment metrology for the Antarctica Kunlun Dark Universe Survey Telescope

Dome A is the highest point on the Antarctic Plateau, and the Chinese expedition team was the first to arrive there in 2005 January. It is an excellent site for astronomical observations. The Kunlun Dark Universe Survey Telescope (KDUST), a proposed next-generation Chinese Antarctic optical telescope, is currently being planned. KDUST is a coaxial three-mirror anastigmatic telescope with an entrance pupil diameter of 2.5 m and a field of view of 1|${^{\circ}_{.}}$|5. The telescope should be fully aligned in order to achieve diffraction-limited image quality. Based on the current optical model for KDUST, in this study we describe a modified alignment metrology and give the preliminary results of numerical simulations. Using the acquisition of field-dependent aberrations, the algorithm, which is based on double Zernike polynomial decomposition and the least-squares fitting method, delivers a suitable set of corrections for resolving misalignments including the multi-axis motion of the secondary and tertiary mirrors. Moreover, the algorithm can also be used to calculate the primary mirror's low-order figure errors mixing with the misalignments of the secondary mirror.