The diffraction theory of optical aberrations: Part II: Diffraction pattern in the presence of small aberrations

doi:10.1016/0031-8914(47)90052-9

Physica

Volume 13, Issue 10, December 1947, Pages 605-620

https://doi.org/10.1016/0031-8914(47)90052-9 Get rights and content

Abstract

In this paper the diffraction theory for arbitrary aberrations of a symmetrical optical system is developed for the case that the amount of aberration is small. The aberration function, which measures the deviation of the actual wave-front from a sphere, is expanded in a series of the so-called circle polynomials, which were introduced by Zernike in a problem closely related to the one treated here.

This new expansion appears to have considerable advantages as compared to the customary one. So, for instance, the problem of the counter-balancing of aberrations of various orders can be solved now completely.

Formulae are given, from which the intensity distribution of the diffraction pattern in a receiving plane perpendicular to the principal ray in the neighbourhood of the Gaussian image can be numerically determined without much labour. The formulae are applied to the diffraction patterns of astigmatism and coma. The results of the numerical computations are given in diagrams showing the lines of equal intensity.

References (10)

B.R.A. Nijboer
Physica
(1943)
G.C. Steward
The symmetrical optical system
Phil. Trans. roy. Soc. A
(1925)
B.R.A. Nijboer
Thesis
(1942)
F. Zernike
Physica
(1934)
K. Strehl
Z. Instrumentenkunde
(1895)
K. Strehl
Z. Instrumentenkunde
(1902)

There are more references available in the full text version of this article.

Cited by (55)

Fractional Zernike functions
2024, Journal of Mathematical Analysis and Applications
We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated β-restricted Zernike functions. Mainly, we give the spectral realization of the latter ones and show that they are orthogonal $L^{2}$ -eigenfunctions for certain perturbed magnetic (hyperbolic) Laplacian. The algebraic and analytic properties for the fractional Zernike functions to be established include the connection to special functions, their zeros, their orthogonality property, as well as the differential equations, recurrence and operational formulas they satisfy. Integral representations are also obtained. Their regularity as poly-meromorphic functions is discussed and their generating functions including a bilinear one of “Hardy–Hille type” are derived. Moreover, we prove that a truncated subclass defines a complete orthogonal system in the underlying Hilbert space giving rise to a specific Hilbertian orthogonal decomposition in terms of a class of generalized Bergman spaces.
Non-iterative aberration retrieval based on the spot shape around focus
2022, Optics and Lasers in Engineering
Citation Excerpt :
The paper concludes with a set of numerical simulations assessing the capability of the approach when multiple primary aberrations are present in the system. As can be seen from Fig. 1 primary aberrations generate characteristic focal spot distortion patterns through focus, as discussed by Nijboer in his thesis [18]. For example, in the presence of astigmatism the out of focus intensity distributions are elongated along one direction, with the direction of elongation rotating by 90 degrees when going through focus.
A non-iterative, robust, aberration retrieval method to determine primary aberrations by utilizing the intensity distribution at and around focus is presented. The primary Zernike aberrations (coma, spherical aberration and astigmatism) are retrieved by fitting a set of orthogonal circle functions within the central region of the intensity distribution recorded at 3 different axial planes, typically taken at best focus and either side of focus. Aberration indicators are derived from these fits for each primary aberration and it is shown that these indicators can be used for aberration retrieval. The selected indicators vary almost linearly with the magnitude of aberration up to 0.13λ rms, corresponding to a Strehl ratio of 0.44. In the presence of multiple primary aberrations, the method is found to be reliable for a total rms wavefront deviation below 0.10λ (Strehl ratio of 0.68). This approach is linear and non-iterative and will therefore be beneficial for applications where speed and limiting photon exposure is important such as wavefront correction in biomedical imaging.
GPU-based processing of Hartmann–Shack images for accurate and high-speed ocular wavefront sensing
2019, Future Generation Computer Systems
Citation Excerpt :
This reduces the execution time but also degrades its accuracy. Once the centroids of all the spots have been calculated, the next step is fitting the Zernike polynomial derivatives to the displacements of the spots to obtain the so-called Zernike coefficients [27–30] by solving an overdetermined equation system. In order to parallelize this step, we have tested two different methods.
Hartmann–Shack aberrometry is a widely used technique in the field of visual optics but, high-speed and accurate processing of Hartmann–Shack images can be a computationally expensive/resource intensive task. While some advancements have been made in achieving high-performance processing units, they have not been specifically designed for processing Hartmann–Shack images of the human eye with Graphics Processing Units. In this work, we present the first full-Graphics Processing Unit implementation of a Hartmann–Shacksensor algorithm aimed at accurately measuring ocular aberrations at a high speed from high-resolution spot pattern images. The proposed algorithm, called PaPyCS (Parallel Pyramidal Centroid Search), is inherently parallel and performs a very robust centroid search to avoid image noise and other artifacts. This is a field where the use of Graphics Processing Units have not been exploited despite the fact that they can boost Adaptive Optics systems and related closed-loop approaches. Our proposed implementation achieves processing speeds of 380 frames per second for high resolution (1280x1280 pixels) images, in addition to showing a high resilience to system and image artifacts that appear in Hartmann–Shack images from human eyes: more than 98% of the Hartmann–Shack images, with aberrations of up to 4 $μ$ m Root Mean Square for a 5.12mm pupil diameter, were measured with less than 0.05 $μ$ m Root Mean Square Error, which is basically negligible for ocular aberrations.
Primary aberrations theory in optical systems composed of Cartesian refracting surfaces
2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Fractional Zernike functions
2023, arXiv
Wave Front Aberration Sensors Based on Optical Expansion by the Zernike Basis
2023, Photonics Elements for Sensing and Optical Conversions

View all citing articles on Scopus

^☆: The contents of this paper (as well as of Part 1¹)) form part of the author's thesis, Groningen 1942. Earlier publication of Part II was prevented due to unforeseen circumstances. The diffraction pattern in the presence of larger aberrations will be considered in another communication.

View full text

The diffraction theory of optical aberrations: Part II: Diffraction pattern in the presence of small aberrations☆

Abstract

Physica

The symmetrical optical system

Phil. Trans. roy. Soc. A

Thesis

Physica

Z. Instrumentenkunde

Z. Instrumentenkunde