Nodal Aberration Theory Research Articles

Computation of astigmatic and trefoil figure errors and misalignments for two-mirror telescopes using nodal-aberration theory.

In active optics systems, one concern is how to quantitatively separate the effects of astigmatic and trefoil figure errors and misalignments that couple together in determining the total aberration fields when wavefront measurements are available at only a few field points. In this paper, we first quantitatively describe the impact of mount-induced trefoil deformation on the net aberration fields by proposing a modified theoretical formulation for the field-dependent aberration behavior of freeform surfaces based on the framework of nodal aberration theory. This formulation explicitly expresses the quantitative relationships between the magnitude of freeform surfaces and the induced aberration components where the freeform surfaces can be located away from the aperture stop and decentered from the optical axis. On this basis, and in combination with the mathematical presentation of nodal aberration theory for the effects of misalignments, we present the analytic expressions for the aberration fields of two-mirror telescopes in the presence of astigmatic primary mirror figure errors, mount-induced trefoil deformations on both mirrors, and misalignments. We quantitatively separate these effects using the analytical expressions with wavefront measurements at a few field points and pointing errors. Valuable insights are provided on how to separate these coupled effects in the computation process. Monte Carlo simulations are conducted to demonstrate the correctness and accuracy of the analytic method presented in this paper.

Ray-based optical design tool for freeform optics: coma full-field display.

The field of optical fabrication has progressed to a point where manufacturing optical quality freeform surfaces is no longer prohibitive. However, to stimulate the development of freeform systems, optical designers must be provided with the necessary tools. Full-field displays are an example of such a tool. Identifying the field dependence of the dominant aberrations of a freeform system is critical for a controlled optimization and with the help of full-field displays, this can be accomplished. Of specific interest is coma, an often system-limiting aberration and an aberration that has recently been directly addressed with freeform surfaces. In this research, we utilize nodal aberration theory to develop a ray-based method to generate a coma full-field display that circumvents wavefront fitting errors that can affect Zernike polynomial-based full-field displays for highly aberrated freeform starting designs.

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.

Nodal aberration properties of coaxial imaging systems using Zernike polynomial surfaces.

As a combination of the nodal aberration theory (NAT) and the freeform surface optical design, the nodal aberration properties of Zernike polynomial surfaces in a coaxial imaging system are analyzed in detail in this paper. The aberrations induced by the Zernike polynomial terms on either the reflective or the refractive surfaces in the system can be quantitatively calculated, which can be used to predict the surface figure or mount-induced errors, or as guidance for aberration correction and optimization during the freeform optical design. Specially, some new kinds of aberrations with complex field dependence can be generated by the Zernike polynomial terms, which further lead to the discovery of linear or multilinear nodes (cruciate nodes) in the aberration field. This linear nodal property will benefit the design of some kinds of freeform optical systems with a linear field of view. Furthermore, an iterative method based on the NAT is proposed to correct the field-constant aberrations.

Theory of aberration fields for general optical systems with freeform surfaces.

This paper utilizes the framework of nodal aberration theory to describe the aberration field behavior that emerges in optical systems with freeform optical surfaces, particularly φ-polynomial surfaces, including Zernike polynomial surfaces, that lie anywhere in the optical system. If the freeform surface is located at the stop or pupil, the net aberration contribution of the freeform surface is field constant. As the freeform optical surface is displaced longitudinally away from the stop or pupil of the optical system, the net aberration contribution becomes field dependent. It is demonstrated that there are no new aberration types when describing the aberration fields that arise with the introduction of freeform optical surfaces. Significantly it is shown that the aberration fields that emerge with the inclusion of freeform surfaces in an optical system are exactly those that have been described by nodal aberration theory for tilted and decentered optical systems. The key contribution here lies in establishing the field dependence and nodal behavior of each freeform term that is essential knowledge for effective application to optical system design. With this development, the nodes that are distributed throughout the field of view for each aberration type can be anticipated and targeted during optimization for the correction or control of the aberrations in an optical system with freeform surfaces. This work does not place any symmetry constraints on the optical system, which could be packaged in a fully three dimensional geometry, without fold mirrors.

Extending nodal aberration theory to freeform surfaces

Extending nodal aberration theory to freeform surfaces

The astigmatic aberration field in active primary mirror astronomical telescopes

Abstract Stemming from the exceptional results obtained with the pathfinding NTT telescope commissioned in 1989, the current generation of large astronomical telescopes, including the VLT, Gemini, Subaru, LBT, VST, and many more, use an active primary mirror in conjunction with the position of the secondary mirror to optimize performance with a cycle time typically measured in minutes over the course of acquiring a long exposure image. Recent discoveries, based in research enabled by the nodal aberration theory (NAT), have created a complete, linear model for the effect of, and the interaction of, the alignment compensators used during an exposure when an active primary mirror has a role in the process of maintaining performance in combination with the rotation of the secondary mirror around a specific external pivot point. This scenario, presented in the context of NAT, clearly illustrates important, limiting interactions between the secondary mirror position and the primary mirror astigmatic figure residual created through active control. It also points to the need for the wavefront sensor algorithm to anticipate that the astigmatic field will be binodal. The science of weak galactic lensing is exceptionally sensitive to binodal astigmatism, making this aberration field an urgent area of research.

Third-order aberrations of a plane symmetric optical system

The wave aberration theory of non-symmetrical optical systems is useful for understanding the misalignment in symmetrical systems and designing of off-axis mirror systems. A theory about the third-order aberrations for sub-aperture plane symmetric optical system is developed by using the aberration theory for full-aperture axially symmetric spherical systems. This paper proves the nodal aberration theory, namely, the points in the field with zero third-order aberration will deviate from the field center except for spherical aberration. It also reveals that the nodal aberrations arise from the transformation of the aberrations in full-aperture systems. This theory can be efficiently used in non-symmetric optical system designing process.

Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces

This paper introduces the path forward for the integration of freeform optical surfaces, particularly those related to φ-polynomial surfaces, including Zernike polynomial surfaces, with nodal aberration theory. With this formalism, the performance of an optical system throughout the field of view can be anticipated analytically accounting for figure error, mount-induced errors, and misalignment. Previously, only misalignments had been described by nodal aberration theory, with the exception of one special case for figure error. As an example of these new results, three point mounting error that results in a Zernike trefoil deformation is studied for the secondary mirror of a two mirror and three mirror telescope. It is demonstrated that for the case of trefoil deformation applied to a surface not at the stop, there is the anticipated field constant contribution to elliptical coma (also called trefoil) as well as a newly identified field dependent contribution to astigmatism: field linear, field conjugate astigmatism. The magnitude of this astigmatic contribution varies linearly with the field of view; however, it has a unique variation in orientation with field that is described mathematically by a concept that is unique to nodal aberration theory known as the field conjugate vector.

LSST Telescope Alignment Plan Based on Nodal Aberration Theory

ABSTRACTThe optical alignment of the Large Synoptic Survey Telescope (LSST) is potentially challenging, due to its fast three-mirror optical design and its large 3.5° field of view (FOV). It is highly advantageous to align the three-mirror optical system prior to the integration of the complex science camera on the telescope, which corrects the FOV via three refractive elements and includes the operational wavefront sensors. A telescope alignment method based on nodal aberration theory (NAT) is presented here to address this challenge. Without the science camera installed on the telescope, the on-axis imaging performance of the telescope is diffraction-limited, but the field of view is not corrected. The nodal properties of the three-mirror telescope design have been analyzed and an alignment approach has been developed using the intrinsically linear nodal behavior, which is linked via sensitivities to the misalignment parameters. Since mirror figure errors will exist in any real application, a methodology to introduce primary-mirror figure errors into the analysis has been developed and is also presented.

A new family of optical systems employing φ-polynomial surfaces

Unobscured optical systems have been in production since the 1960s. In each case, the unobscured system is an intrinsically rotationally symmetric optical system with an offset aperture stop, a biased input field, or both. This paper presents a new family of truly nonsymmetric optical systems that exploit a new fabrication degree of freedom enabled by the introduction of slow-servos to diamond machining; surfaces whose departure from a sphere varies both radially and azimuthally in the aperture. The benefit of this surface representation is demonstrated by designing a compact, long wave infrared (LWIR) reflective imager using nodal aberration theory. The resulting optical system operates at F/1.9 with a thirty millimeter pupil and a ten degree diagonal full field of view representing an order of magnitude increase in both speed and field area coverage when compared to the same design form with only conic mirror surfaces.

Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations

Building on earlier work on the nodal aberration theory of third-order aberrations and a subset of fifth-order terms, this paper presents the multinodal field dependence of the family of aberrations describing the shape of the medial focal surface (the focal surface upon which the minimum RMS wavefront error is measured) and the astigmatic aberrations with respect to this surface through the fifth order. Specifically, the multinodal field dependence for W(420M) and W₄₂₂ (the field-quartic medial surface and field-quartic astigmatism) are derived and presented as well as their influence on the magnitude and nodal field dependence of the companion lower-order terms, W(220M) and W₂₂₂. This paper provides the first derivations of field-quartic aberrations presented by the author in the refereed literature.

Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT)

We present the nodal aberration field response of Ritchey-Chrétien telescopes to a combination of optical component misalignments and astigmatic figure error on the primary mirror. It is shown that both astigmatic figure error and secondary mirror misalignments lead to binodal astigmatism, but that each type has unique, characteristic locations for the astigmatic nodes. Specifically, the characteristic node locations in the presence of astigmatic figure error (at the pupil) in an otherwise aligned telescope exhibit symmetry with respect to the field center, i.e. the midpoint between the astigmatic nodes remains at the field center. For the case of secondary mirror misalignments, one of the astigmatic nodes remains nearly at the field center (in a coma compensated state) as presented in Optics Express 18, 5282-5288 (2010), while the second astigmatic node moves away from the field center. This distinction leads directly to alignment methods that preserve the dynamic range of the active wavefront compensation component.

Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations

Building on an earlier work on the nodal aberration theory of the 3rd-order aberrations [J. Opt. Soc. Am. A22, 1389 (2005)] and the first paper in this series on the nodal aberration theory of higher-order aberrations [J. Opt. Soc. Am. A26, 1090 (2009)], this paper continues the derivation and presentation of the intrinsic, characteristic, often multinodal geometry for each type/family of the 3rd- and 5th-order optical aberrations as categorized by parallel developments for rotationally symmetric optics. The first paper in this series on the higher-order terms developed the nodal properties of the spherical aberration family, including W(060), W(240M), and W(242), and for completeness 7th-order spherical aberration W(080). This second paper in the series develops and presents the intrinsic, characteristic, often multinodal properties of the family of comatic aberrations through 5th order, specifically W(151), W(331M), and W(333) [field-linear, 5th-order aperture coma; field-cubed, 3rd-order aperture coma; and field-cubed, elliptical coma (a 3rd-order in aperture 5th-order vector aberration)]. This paper will present the first derivations of trinodal aberrations by the author.

Misalignment-induced nodal aberration fields in two-mirror astronomical telescopes

We present the effects of misalignments on the field dependence of the third-order aberration fields of traditional, two-mirror astronomical telescopes in the context of nodal aberration theory, which we believe is the most general and extensible framework for describing and improving on-station performance. While many of the advantages of nodal aberration theory, compared to other, often power series expansion-based descriptions of misalignment effects on aberrations, become particularly important when analyzing telescopes with more than two mirrors, or in the presence of figure errors; this paper aims to provide and demonstrate the fundamental concepts needed to fully describe the state of correction of misaligned two-mirror telescopes. Importantly, it is shown that the assumption that perfect performance on axis ensures a fully aligned telescope is false, and we demonstrate that if Ritchey-Chrétien telescopes are aligned for zero coma on axis as the sole criterion, formidable misalignments will likely remain, leading to image quality degradation, particularly beyond midfield caused by astigmatism with binodal field dependence (i.e., astigmatism goes to zero at two points in the field).

Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration

Building off an earlier work on multinodal third-order aberrations [J. Opt. Soc. Am. A22, 1389 (2005)], this is the first in a series of papers that derives and illustrates the characteristic multinodal geometry for each of the fifth-order aberrations. Part I (as this paper will be referred to) will present the spherical aberration family: specifically, W(060), W(240M) and W(242), and W(080) (fifth-order spherical, oblique spherical, and seventh-order spherical). Nodal aberration theory is proving to be very effective as both an optical design tool for fully unobscured off-axis telescopes and as an analysis method, particularly in the context of the response of any imaging optical systems to misalignment. It is important to recognize that this multinodal approach to aberration theory is not restricted to small perturbations. The remaining papers in this series will result in a complete presentation of the intrinsic characteristic multinodal properties of each of the fifth-order aberrations. As such, this series provides a definitive theory of the optical aberrations of (nonanamorphic) imaging systems with a circular aperture stop.

The misalignment induced aberrations of TMA telescopes

The next major space-borne observatory, the James Webb Space Telescope, will be a 6.6M field-biased, obscured, three-mirror anastigmat (TMA). Over the used field of view, the performance of TMA telescopes is dominated by 3(rd) order misalignment aberrations. Here it is shown that two dominant 3(rd) order misalignment aberrations arise for any TMA telescope. One aberration, field constant 3(rd) order coma is a well known misalignment aberration commonly seen in two-mirror Ritchey Chretien telescopes. The second aberration, field-asymmetric, field-linear, 3(rd) order astigmatism is a new and unique image orientation dependence with field derived here for the first time using nodal aberration theory.

Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry

Many authors, dating back to at least the 1950s, have presented mathematical expansions of the wave-front aberration function for optical systems without symmetry, typically based on limiting assumptions and simplifications, with some of the most recent work being done by Howard and Stone [Appl. Opt. 39, 3232 (2000)]. This paper reveals that in fact there are no new aberrations in imaging optical systems with near-circular aperture stops but otherwise without symmetry. What does occur is that the field dependence of an aberration often changes when symmetry is abandoned. Each aberration type develops a characteristic field behavior in a system without symmetry. Specifically, for example, astigmatism, develops a binodal field dependence; e.g., there are typically two points in the field with zero astigmatism, and typically neither point is on axis. This construct, nodal aberration theory, for understanding the aberrations in systems without symmetry becomes a direct extension of an optical designer's knowledge base. Through the use of real ray-based analysis methods, such as Zernike coefficients, it is possible to understand completely the aberrations of optical systems without symmetry in terms of rotationally symmetric aberration theory with the simple addition of the concept of field nodes.