Degeneracy in freeform surfaces described with orthogonal polynomials

Takaki, Nick; Bauer, Aaron; Rolland, Jannick P.

doi:10.1364/ao.57.010348

Cited by 22 publications

(7 citation statements)

References 16 publications

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“…The constraints for controlling the system distortion, eliminating light obscuration and constraining system size can be established using real ray trace data. The sag difference between the freeform surface and base sphere should be controlled, in order to improve the manufacturability and testability of freeform surface 17 .…”

Section: Design Processmentioning

confidence: 99%

“…For Zernike polynomial surface, the overall sag difference can be controlled by constraining the average sag difference along the edge of the circumcircle of the surface aperture to be zero. This can be done by controlling the surface coefficients of the rotationally symmetric terms directly 17 . For Q-type polynomials, the average sag difference along the edge of the circumcircle of the surface aperture is naturally zero if proper normalizing radius is set.…”

Section: Design Examplesmentioning

confidence: 99%

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Design and exploration of freeform three-mirror imaging systems with integrated primary and tertiary mirrors using same expression

Zhou

Yang

Cheng

et al. 2023

AOPC 2022: Novel Optical Design; And Optics Ultra Precision Manufacturing and Testing

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A typical and widely used configuration of freeform imaging system is freeform off-axis three-mirror systems. However, as the system configuration and surface shape have no rotational symmetry, the assembly process will be very difficult. In this paper, we designed off-axis three-mirror systems in which the primary and tertiary mirrors are integrated into one single element. The two mirrors use different portions of a single mirror, which is represented by one mathematical expression. The design process of the system is demonstrated in detail, and three different kinds of light folding geometry are explored. The system design using different types of freeform surface is also explored and the imaging performance is compared. Finally, tolerance analysis which considers the local and random nature of surface error and actual manufacturing difficulty of the freeform surface is conducted.

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Section: Design Processmentioning

confidence: 99%

Section: Design Examplesmentioning

confidence: 99%

Design and exploration of freeform three-mirror imaging systems with integrated primary and tertiary mirrors using same expression

Zhou

Yang

Cheng

et al. 2023

AOPC 2022: Novel Optical Design; And Optics Ultra Precision Manufacturing and Testing

View full text Add to dashboard Cite

show abstract

“…For orthogonal polynomial departures (e.g., Zernike polynomials), while restrictions on the coefficients can be found by calculating derivatives, designers may find it easier to make restrictions by inspecting the representations of these polynomials. For example, only four of the first 36 Fringe Zernike terms will contribute to local x-astigmatism at the origin (Z5, Z12, Z21, and Z32 in Fringe ordering, or Z 2 ñ for ñ = 0, ..., 3 using the two-index notation 12,13 ), so the requirement can be written as…”

Section: Conical Optics Strict Vs Approximate Confocality and Freefor...mentioning

confidence: 99%

Leveraging confocality and Cartesian reflectors in freeform optical designs

Papa,

Takaki,

Schiesser

2023

International Optical Design Conference 2023

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“…In surface representation, Zernike polynomials have emerged as a means of describing the shape of freeform optical surfaces [127][128][129][130][131]. State-of-the-art lens design programs, such as Zemax and CODE V, empower optical designers to use Zernike polynomials to represent freeform surfaces, which are called Zernike surfaces.…”

Section: Optical Designmentioning

confidence: 99%

Zernike polynomials and their applications

Niu

Tian

2022

J. Opt.

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The Zernike polynomials are a complete set of continuous functions orthogonal over a unit circle. Since first developed by Zernike in 1934, they have been in widespread use in many fields ranging from optics, vision sciences, to image processing. However, due to the lack of a unified definition, many confusing indices have been used in the past decades and mathematical properties are scattered in the literature. This review provides a comprehensive account of Zernike circle polynomials and their noncircular derivatives, including history, definitions, mathematical properties, roles in wavefront fitting, relationships with optical aberrations, and connections with other polynomials. We also survey state-of-the-art applications of Zernike polynomials in a range of fields, including the diffraction theory of aberrations, optical design, optical testing, ophthalmic optics, adaptive optics, and image analysis. Owing to their elegant and rigorous mathematical properties, the range of scientific and industrial applications of Zernike polynomials is likely to expand. This review is expected to clear up the confusion of different indices, provide a self-contained reference guide for beginners as well as specialists, and facilitate further developments and applications of the Zernike polynomials.

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Degeneracy in freeform surfaces described with orthogonal polynomials

Cited by 22 publications

References 16 publications

Design and exploration of freeform three-mirror imaging systems with integrated primary and tertiary mirrors using same expression

Design and exploration of freeform three-mirror imaging systems with integrated primary and tertiary mirrors using same expression

Leveraging confocality and Cartesian reflectors in freeform optical designs

Zernike polynomials and their applications

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