Research of the effect of aberrations on image quality in optical systems
                                                            A.V. Kozhevnikov1
                                   1
                                    Samara National Research University, 34 Moskovskoe Shosse, 443086, Samara, Russia

Abstract

This work presents a study of the influence and compensation of aberrations in optical imaging systems by superimposing surfaces described
by Zernike polynomials. The aim is to investigate how to compensate for aberrations and their effectiveness when dealing with aberrations of
different types. The study is done in the Zemax packet by modeling the optical imaging system and passing test images through it.


Keywords: wave aberrations; Zernike polynomials; imaging optical system



1. Introduction

   The aberration of the optical system is the deformation of images that occur at the output of the optical system. The name
comes from the lat. Aberratio - evasion, removal. Deformation consist in the fact that the optical images do not completely
correspond to the object. This is manifested in the blurriness of the image and is called monochromatic geometric aberration or
color image - chromatic aberration of the optical system. Most often, both types of aberration appear together.
   In the paraxial region, the optical system works almost perfectly, the dot is represented by a point, and the straight line is a
straight line, etc. However, as the point moves away from the optical axis, the rays from it intersect in the image plane not at one
point. Thus, a circle of dispersion arises, i.e. there are aberrations. The dimension of the aberration can be determined by
calculating the geometric and optical formulas through a comparison of the coordinates of the rays, and also approximately using
the formulas of the theory of aberrations [1].
   There is a description of the phenomenon of aberration in both the ray theory (deviation from identity is described through
geometric aberrations and ray scattering patterns) and in the representations of wave optics (the deformation of a spherical light
wave along the path through an optical system is estimated). Usually, ray theory is used to characterize optical systems with
large aberrations, otherwise the principles of wave optics are applied.
   As a rule, analysis of wave aberrations is performed on the basis of Zernike polynomials [1-4]. In [5-8], it was proposed to
use a multi diffractive optical elements for an optical wavefront decomposition on the basis of Zernike polynomials. Moreover,
the use of optical elements that are consistent not only with Zernike functions, but also their superpositions, makes it possible to
perform optical measurements ensuring the restoration of the shape of the wave front [9-11].
   In this work, we investigate the effect of aberrations described by Zernike polynomials on the imaging properties of an optical
system using the Zemax packet [12]. The aim of the work is to investigate methods of aberration compensation [13-15] and
their effectiveness in dealing with aberrations of different types.

2. Modeling of wave aberrations by varying the coefficients of a polynomial surface

    The first stage will be modeled several surfaces described by Zernike polynomials. For this part, an optical system consisting
of two refractive lenses is used, one of which is the surface under investigation. Light with wavelengths of 450, 550 and 650 nm
is transmitted through such a system. As an estimation of aberration distortions, an image of the Latin letter "F", passed through
the described optical system, is considered. The most interesting are the coefficients - "Zernike standard coefficients".




                                       .

                                                    Fig.1. Two-dimensional scheme of the optical system


    There are even and odd Zernike polynomials. Even polynomials are defined as: 𝑍(𝜌, 𝜑) = 𝑅𝑛𝑚(𝜌) cos(𝑚𝜑), and odd as:
 𝑍𝑛−𝑚(𝜌, 𝜑) = 𝑅𝑛𝑚(𝜌) sin(𝑚𝜑). Where m and n are nonnegative integers, such that n> m, φ - azimuth angle, and ρ - radial
 distance 0 <= ρ <= 1. Zernike polynomials are limited in the range from -1 to +1. Radial polynomials are defined as: 𝑅𝑛𝑚 (𝜌) =
                 (−1)𝑘 (𝑛−𝑘)!
 ∑(𝑛−𝑚)/2
  𝑘=0           𝑛+𝑚      𝑛−𝑚       𝜌𝑛−2𝑘
            𝑘!(     −𝑘)!(   −𝑘)!
                 2        2




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                                                 Computer Optics and Nanophotonics / A.V. Kozhevnikov
            Table.1. The result of simulation of wave aberrations by the Zernike row decomposition.
              The value and order            «Zernike standard           Diagram of point dispersion        Test image after passing
           of the coefficient to be        coefficients»                 «Spot diagram»                 through the simulated system.
                     set




         Zernike 2 = 1.3




         Zernike 3 = 1.0




         Zernike 4 = 0.008




         Zernike 5 = 0.04




         Zernike 6 = 0.05




         Zernike 10 = 0.5




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                                                Computer Optics and Nanophotonics / A.V. Kozhevnikov
   The orthogonality of the Zernike polynomials gives them great advantages when analyzing aberrations in comparison with
the exponentiation basis. The main advantages are:
   1) The absolute values of the coefficients of the decomposition in the Zernike polynomials decrease with increasing degree
       of polynomials, that is, the Zernike row, as a rule, always converges, which cannot be said about the exponentiation
       rows;
   2) Each coefficient of the row gives the contribution of aberrations of a given type and order to the total wave aberration
       from the position of mutual balance of all types of aberrations. This means that the individual types of aberrations
       represented by the Zernike polynomial decomposition affect image quality quite independently of each other.
        Table 2. The result of compensating wave aberrations.
             Values of the coefficients            «Zernike                                            Diagram of point dispersion
                of the basic and                 standard                                                  «Spot diagram»
             compensating surfaces.            coefficients»                 Wavefront map
                       2nd order
                  Zernike 1 = 22
                  Zernike 2 = 22




                     3nd order
                  Zernike 1 = 10
                  Zernike 2 = 10




                     4nd order
                  Zernike 1 = 1.3
                  Zernike 2 = 1.3




                     5nd order
                  Zernike 1 = 1.0
                  Zernike 2 = 1.0




                     6nd order
                  Zernike 1 = 1.0
                  Zernike 2 = 1.0




                    10nd order
                  Zernike 1 = 1.0
                  Zernike 2 = 1.0




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                                                    Computer Optics and Nanophotonics / A.V. Kozhevnikov
     In each case, the coefficients of different orders are taken, from 1st to 10th, they are assigned the maximum values at which
the test image preserves the original contour and sharpness. The parameter "Zernike max term" is set to 16, it is responsible for
the number of polynomial surface parameters.
     As can be seen from Table 1, the variation of the coefficients of the polynomial surface leads to deformations in the point
dispersion diagram of and on the test image passed through the test system.

 3. Investigation of compensation of aberrations in the optical system

   In this section, we study the compensation of aberrations in a test optical imaging system by superimposing surfaces described
by Zernike polynomials. The aim is to search for the maximum efficiency of compensating wave aberrations by adding a second
polynomial surface. The effect is achieved due to the complex conjugacy of polynomials.

    The optical system used consists of two mirrors arranged at a certain angle to each other and an auxiliary paraxial surface used
to parallelize the incoming light beam. In both mirrors there are polynomially described surfaces.

   As can be seen from Table 2, the addition of the second Zernike surface allows us to compensate for the simulated
aberrations in the optical system.




                                                              Fig.2. Optical system of two mirrors.

4. Conclusion

   The decomposition of the wave aberrations in the Zernike row was conducted in this work. Wave aberrations of various types
were modeled and their effect on image quality in optical systems was examined. An optical system of two mirrors with two
polynomial surfaces is simulated. The first surface was used to model the aberrations themselves, the second was used to
compensate them. The principle used is based on the complex conjugacy of polynomials.
   The possibility of compensation of wave aberrations by superposition of surfaces described by Zernike polynomials is
investigated. The maximum values of aberrations that can be compensated for by relatively effective compensation due to the
use of polynomial surfaces were obtained.
   The study was carried out in a Zemax packet by modeling test optical imaging systems, and passing test images through them.

Acknowledgments

    The work was supported by the Ministry of Education and Science of the Russian Federation.

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