=Paper= {{Paper |id=Vol-2744/short47 |storemode=property |title=Wavefront Parameters Recovering by Using Point Spread Function (short paper) |pdfUrl=https://ceur-ws.org/Vol-2744/short47.pdf |volume=Vol-2744 |authors=Olga Kalinkina,Tatyana Ivanova,Julia Kushtyseva }} ==Wavefront Parameters Recovering by Using Point Spread Function (short paper)== https://ceur-ws.org/Vol-2744/short47.pdf

Wavefront Parameters Recovering by Using Point
Spread Function

Olga Kalinkina[0000-0002-2522-8496], Tatyana Ivanova[0000-0001-8564-243X]
and Julia Kushtyseva[0000-0003-1101-6641]

ITMO University, 197101 Kronverksky pr. 49, bldg. A, Saint-Petersburg, Russia
oskalinkina@itmo.ru, tvivanova@itmo.ru,
julia.kushtyseva@gmail.com

Abstract. At various stages of the life cycle of optical systems, one of the most
important tasks is quality of optical system elements assembly and alignment
control. The different wavefront reconstruction algorithms have already proven
themselves to be excellent assistants in this. Every year increasing technical ca-
pacities opens access to the new algorithms and the possibilities of their appli-
cation. The paper considers an iterative algorithm for recovering the wavefront
parameters. The parameters of the wavefront are the Zernike polynomials coef-
ficients. The method involves using a previously known point spread function
to recover Zernike polynomials coefficients. This work is devoted to the re-
search of the defocusing influence on the convergence of the algorithm. The
method is designed to control the manufacturing quality of optical systems by
point image. A substantial part of the optical systems can use this method with-
out additional equipment. It can help automate the controlled optical system ad-
justment process.

Keywords: Point Spread Function, Wavefront, Zernike Polynomials, Optimiza-
tion, Aberrations.

1 Introduction

At the stage of manufacturing optical systems, one of the most important tasks is
quality of optical system elements assembly and alignment control. There are various
methods for solving this problem, for example, interference methods. However, in
some cases, one of which is the alignment of the telescope during its operation, other
control methods are required [1, 2], for example control by point image (point spread
function) or the image of another known object. Among the advantages of using this
method for telescopic systems control is economic expediency. Firstly, there is no
need for external intervention in a working telescope. Secondly, equipment that al-
lows recording the point spread function has the lowest cost in comparison with inter-
ferometric control methods equipment.
__________
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0).
2 O. Kalinkina, T. Ivanova and J. Kushtyseva

Wavefront reconstruction algorithms have been widely developed in holography,
adaptive optics, microscopy, and other fields [3]. Some algorithms restore the distri-
bution of the wavefront in the form of a sample of values, others allow you to restore
the describing parameters. A necessary condition for the operation of such algorithms
is the assumption that in the process of wavefront propagation through the optical
system, only the phase component changes, while the amplitude component remains
constant.
The Gerchberg – Saxton algorithm, and its modifications are the most frequently
used and found in the literature. It is an iterative reconstruction of the phase infor-
mation from the known amplitudes of the sampled image and diffraction plane inten-
sity pictures measured. By given the amplitude of a signal and its Fourier transform,
the algorithm attempts to recover the phase information for the Fourier transform, and
thereby reconstruct the signal.
The algorithm alternates between Fourier and inverse Fourier transforms, using the
input amplitudes at each iteration to improve the phase estimates. Using this idea,
many algorithms were developed with the next inputs data - the two or more related
images that differ in a certain phase diversity. It is noted that using more images in-
creases the speed and convergence of the algorithms.
In this paper, we solve the problem of reconstructing the wavefront from only one
known point image. The purpose of this work is to develop an algorithm for determin-
ing the wavefront parameters, which are the Zernike polynomials coefficients of the
wave function expansion, by known point spread function.

2 Method

2.1 Zernike polynomials
One of the most common ways to describe the wavefront profile is a Zernike polyno-
mials fitting coefficients (1):
𝑊 (𝜌, 𝜑) = ∑𝑛 ∑𝑚 𝐶𝑛𝑚 𝑍𝑛𝑚 (𝜌, 𝜑) (1)

where n, m are the indices of polynomials with -n ≤ m ≤ n, n + m is even; Cnm is the
fitting coefficient; Znm (ρ, φ) is the Zernike polynomial with the corresponding indices
n, m; φ is the azimuthal angle; ρ is the radial distance (1 ≥ ρ ≥ 0).
Each Zernike fitting coefficient is representing different types of optical system
aberrations. Table 1 shows the correspondence of Zernike polynomials to different
types of 3rd order aberrations. These Zernike coefficients are linearly independent,
thus individual aberration contributions to an overall wavefront may be isolated and
quantified separately.
It should also be noted that another advantage of the Zernike polynomials is the
stability to computational errors in the process of modeling optical systems on a com-
puter or in the mathematical processing of measurement results.
Wavefront Parameters Recovering by Using Point Spread Function 3

Table 1. Zernike polynomials
Zernike polynomials Aberrations
𝑍20 (𝜌, 𝜑) = 2𝜌2 − 1 Defocus
𝑍40 (𝜌, 𝜑) = 6𝜌4 − 6𝜌2 + 1 Spherical
𝑍1−1(𝜌, 𝜑) = 𝜌 sin 𝜑 Y Tilt
𝑍11 (𝜌, 𝜑) = 𝜌 cos 𝜑 X Tilt
𝑍2−2(𝜌, 𝜑) = 𝜌2 sin 2𝜑 45° Astigmatism
𝑍22 (𝜌, 𝜑) = 𝜌2 cos 2𝜑 0° Astigmatism
𝑍3−1(𝜌, 𝜑) = (3𝜌2 − 2) sin 3𝜑 Y Coma
𝑍31 (𝜌, 𝜑) = (3𝜌2 − 2) cos 3𝜑 X Coma

2.2 Point spread function calculation
The point spread function (PSF) is a two-dimensional function describe an image of
point source. PSF can be calculating by inverse Fourier transform of the pupil func-
tion (2):
2
ℎ(𝜂′𝑥 , 𝜂′𝑦 ) = [𝐹−1 (𝑓0 (𝜌𝑥 , 𝜌𝑦 ))] (2)

where 𝑓0 (ρ𝑥 , ρ𝑦 ) is the pupil function, ρ𝑥 , ρ𝑦 are the canonical pupil coordinates,
η′𝑥 , η′𝑦 are the image canonical coordinates.
The pupil function can be described as (3):

𝜏 1/2 (ρ𝑥 , ρ𝑦 ) ∙ 𝑒 2𝜋𝑖𝑊(ρ𝑥 ,ρ𝑦 ) , (ρ𝑥 2 + ρ𝑦 2 ) ≤ 1
𝑓0 (ρ𝑥 , ρ𝑦 ) = { (3)
0, (ρ𝑥 2 + ρ𝑦 2 ) ≥ 1

where 𝜏(ρ𝑥 , ρ𝑦 ) − is the pupil transmission function, 𝑊(ρ𝑥 , ρ𝑦 ) is the wave aberra-
tion function, the expression (ρ𝑥 2 + ρ𝑦 2 ) ≤ 1 defines the region inside the circle
pupil.
The aberration-free PSF of optical systems of diffraction-limited quality is repre-
sented by a diffraction scattering circle, which is called an Airy disk (Figure 1a), with
a central maximum at a point corresponding to an ideal image. Aberrations of the
optical system and alignment errors have a noticeable effect on the image of a point
object. Thus, in real optical systems, the form of PSF differs from the aberration-free
one and have more complex structure. In Figure 1b, one can see the PSF recorded
during the control of a real telescopic system.
4 O. Kalinkina, T. Ivanova and J. Kushtyseva

Fig. 1a. Aberration-free PSF

Fig. 1b. PSF in real telescopic system

2.3 Algorithm
The direct problem in modeling optical systems is to calculate the PSF from a known
set of aberrations represented by the corresponding Zernike polynomials.
The inverse problem is to restore the wavefront in the form of Zernike polynomials
fitting coefficients from the known PSF intensity distribution. The inverse problem
can be solved using parametric optimization. The parameters are the coefficients for
Zernike polynomials. The minimized function is the standard deviation of the refer-
ence PSF from the PSF calculated at each optimization step (4) (Figure 2).
1 2
∑𝑥′ ∑𝑦′ ‖ℎ0 𝑥′𝑦′ − ℎ𝑥′𝑦′ ‖ → min (4)
2
Wavefront Parameters Recovering by Using Point Spread Function 5

Fig. 2. Minimized function

In this work, we use only four polynomials to describe wave front: astigmatism (c22
and s22 coefficients), and coma (c31 and s31 coefficients).
Modeling and solving the optimization problem is carried out using the Ceres
Solver [5] library for C++ with open source code developed by Google.
Solving the optimization problem using Ceres Solver is (5):
1 2
∑𝑖 ‖𝑓𝑖 (𝑥𝑖 1 , 𝑥𝑖 2 … 𝑥𝑖 𝑘 )‖ → min, − ∞ < 𝑥𝑗 < +∞ (5)
2 𝑥

where 𝑓𝑖 (𝑥𝑖 1 , 𝑥𝑖 2 … 𝑥𝑖 𝑘 ) is the objective function, 𝑥𝑖 1 , 𝑥𝑖 2 … 𝑥𝑖 𝑘 are optimization pa-
rameters.
As method efficiency test, we set aberration coefficients (c22, s22, c31, s31), then
calculate PSF. This PSF became input for optimization and obtained during optimiza-
tion coefficients can be compare with initial ones. If result coefficients differ from
initial ones less, then 10-5 – we can say method work correctly.

3 Defocusing influence on the method convergence

The algorithm application to focused PSF, produce in some cases incorrect determina-
tion of the astigmatism coefficients (c22, s22) signs, so quite small number of PSFs
with various combinations of the initial coefficients were successfully restored.
Avoiding this problem is using PSF with defocus, describe by Zernike polynomial
coefficient c20. In the result sign of astigmatism is defining correctly and number of
successfully restored combination of various aberration coefficients is increased.
Numerical experiments to define best defocusing value (c20 coefficient value) for
stable algorithm work and successful restore aberration coefficients were performed.
Thus, when checking algorithm application to a large number of modeled PSFs, the
optimal value of the defocus was determined as range from 0.1 to 0.2 wavelength
(absolute value). In this case aberration coefficients were successfully restored for
aberration coefficients less than 0.5 wavelength (absolute value).
6 O. Kalinkina, T. Ivanova and J. Kushtyseva

Table 2. Optimization result

initial optimization initial optimization
с20 = 0,1λ
values result values result

c31[λ] 0,05689 0,05689 0,31352 0,31352

s31[λ] 0,07813 0,07813 0,35279 0,35279
c22[λ] 0,09347 0,09347 0,32016 0,32016

s22[λ] 0,06019 0,06019 0,39148 0,39148
Number of
5 14
iterations

PSF with aberration more than 0.5 wavelength have a more complex mathematical
form, which in some cases impossible to restore by optimization. But it was finding
out that for every combination of coefficients c22, s22, c31, s31 is possible to find
appropriate defocusing value for successful definition, and this value could be differ-
ent for different coefficients combination. Figure 3 shows the PSF examples corre-
sponding to one set of coefficients (c22 = 0.2 λ, s22 = 0.6 λ, c31 = 0.6 λ, s31 = 0.4 λ)
with different defocus values c20. The successful restoration were only in the one
value c20 = 0.6 λ (highlighted in the figure).

Fig. 3. Example of PSF with manually selected defocus
Wavefront Parameters Recovering by Using Point Spread Function 7

4 Conclusion

The algorithm presented in the work is intended to determine the wavefront parame-
ters from the point spread function. Focused PSF does not provide enough infor-
mation to estimate aberrations, and the presence of defocusing simplifies the determi-
nation of aberration coefficients. To determine small aberrations (less than 0.5 wave-
length absolute value), the optimal defocus is in the range of 0.1-0.2 wavelength abso-
lute value.
PSF with aberration more than 0.5 wavelength is more difficult to restore, but for
every combination of coefficients c22, s22, c31, s31 is possible to find appropriate
defocusing value for successful definition. However, this defocus value must be
searching for each case separately.

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