Computer Simulation for the Effects of Optical Aberrations on Solar Images Using Karhunen-Loeve polynomials

Numerical simulations were carried out to evaluate the effects of different aberrations modes on the performance of optical system, when observing and imaging the solar surface. Karhunen-Loeve aberrations modes were simulated as a wave front error in the aperture function of the optical system. To identify and apply the appropriate rectification that removes or reduces various types of aberration, their attribute must be firstly determined and quantitatively described. Wave aberration function is well suitable for this purpose because it fully characterizes the progressive effect of the optical system on the wave front passing through the aperture. The Karhunen-Loeve polynomials for circular aperture were used to describe wave front deviations and to predict the initial state of adaptive optics corrections. The results showed that increasing the aberration modes causes an increase in the blurring of the observed image. Also, we conclude that the optical phase error is increased significantly when aperture’s radii are increased. Keyword: Karhunen-Loeve, Zernike Polynomials, Wave Front Aberration, Solar Adaptive Optics, and PSF. دودح ةددعتم مادختساب ةيدمذلا رورلا ىمع ةيرربلا غويزلا تا ريثأتل ةيبوساح ةاكاحم Karhunen-Loeve نادح يفون دئا ر* 1 يمع ركاش ىده , 2 عيدو تمكح ءافو , 3 1 ,قا رعلا,دادغب ,مؽمعمل خركلا ةعماج 2 قا رعلا,دادغب,دادغب ةعماج ,مؽمعلا ةيمك,ءازفلاو غمفلا ػدق 3 سدحتلا ػدق قا رعلا,دادغب,مؽمعمل خركلا ةعماج ,ءايزيفؽيجلاو يئاشلا سدحتلا ةيمك,يئاشلا ةصلاخلا ريثأت ػييقتل ةيبؽساح ةاكاحم ذيفشت ػت دصر للاخ ،ةيرربلا ةمؽعشسلا ىمع ةفمتخسلا ةيشاؽغلا طاسنا ل غيزلا طاسنا .سسذلا حطس ريؽرتو Karhunen-Loeve ةحتف ةلادل ةجؽسلا ةهبج يف أطخلا تمثم لًاوا بجي غيزمل ةفمتخسلا عاؽنلاا ضفخ وا ءازقمل بؽغرسلا حيحرتلا قيبطتو ديدحتل .ةيرربلا ةمؽعشسلا اماست فرت اهنلأ ةجؽسلا ةيشاؽغ ةلاد ؽه ديج لكذب ضرغلا اذه بساشي .ايسك اهتافصو اهتا زيسم صاشتقا ISSN: 0067-2904 Hassan et al. Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2463-2473 2464 ع يرربلا ماعشمل يسكا رتلا رثلاا ةددعتم مادختسا ػت .ةحتفلا يف عقؽم لك للاخ ؼم رسي يذلا ءؽزلا ىم دودح Karhunen-Loeve ةيلولاا ةلاحلاب ؤبشتملو ةجؽسلا ةهبج يف تافا رحنلاا فصؽل ةيرئاد ةحتفل .ةرؽطسلا تايرربلا تاحيحرتل رؽرلا ةيشاؽغ دادزت فؽس ةيشاؽغلا طاسنا ةدايز دشع ةسا ردلا جئاتن ترهظا .ةدؽصرسلا راعشسلا ةحتف ةدايزب دادزي رؽطمل يرربلا أطخلا نا ،ازيا اشجتشتسا . Introduction Adaptive optics (AO) allow ground based telescopes to achieve their theoretical diffraction-limited resolution by compensating for the blurring effects of the Earth’s atmosphere. Over the past two decades, the high angular resolution provided by AO coupled with the largest telescopes and modern science instruments became an important tool for studying and understanding our universe [1]. Using Karhunen-Loeve polynomials for generating atmospheric turbulence was introduced. This method takes into account the temporal and spatial effects of simulating atmospheric turbulence on the observed images [2]. In addition, the use of Karhunen-Loeve polynomials was introduced rather than using Zernike polynomials, as they are a statistically independent set of orthonormal polynomials [3]. The analysis is based on the decomposition of the distorted wavefront over the aperture in an orthonormal series with randomly independent coefficients [4]. Model Approach In the case of a modal representation, the estimate represents a coefficient of an aperture function. The wave front is represented by a set of functions; each function is valid over the whole aperture. They have to be linearly independent; the phase can then be given by [5]:


Introduction
Adaptive optics (AO) allow ground based telescopes to achieve their theoretical diffraction-limited resolution by compensating for the blurring effects of the Earth's atmosphere. Over the past two decades, the high angular resolution provided by AO coupled with the largest telescopes and modern science instruments became an important tool for studying and understanding our universe [1].
Using Karhunen-Loeve polynomials for generating atmospheric turbulence was introduced. This method takes into account the temporal and spatial effects of simulating atmospheric turbulence on the observed images [2]. In addition, the use of Karhunen-Loeve polynomials was introduced rather than using Zernike polynomials, as they are a statistically independent set of orthonormal polynomials [3]. The analysis is based on the decomposition of the distorted wavefront over the aperture in an orthonormal series with randomly independent coefficients [4].

Model Approach
In the case of a modal representation, the estimate represents a coefficient of an aperture function. The wave front is represented by a set of functions; each function is valid over the whole aperture. They have to be linearly independent; the phase can then be given by [5]: where are the coefficients and are the functions. Karhunen-Loeve polynomials will be discussed to approximate the wavefront.

Karhunen-Loeve Polynomials
Karhunen-Loeve (KL) polynomials are each a sum of Zernike polynomials. However, they have statistically independent coefficients [6]. This is important due to the nature of atmospheric turbulence, as described by the Kolmogorov model following Kolmogorov statistics [7]. KL polynomials are given as [8]: where is the constructing matrix of KL polynomials (Table-1) and N is the number of Zernike orders in which the KL order j is represented. To make a realization of a wave front after being distorted by the Earth's atmosphere, Fried derived Zernike-Kolmogorov residual errors. Thus, a wave front can be represented as [10]: The wavefront is now represented as a sum of KL polynomials with the Zernike-Kolmogorov residual error weights in the 's. The 's in Equation (2), which was calculated from the Zernike-Kolmogorov residual errors and ∆j, were measured through many experimental procedures by Fried [3,10], calculated by Noll, and given in Table-2 in terms of D/r 0 . Thus, a realization of atmospheric turbulence can be simulated for different severities of turbulence and different apertures [4 and 9]. Compensation of the atmospheric seeing can be achieved using the technology of Adaptive Optics, a technique which is being pursued by every major ground-based observatory. The techniques of AO are those by which telescope optics are adjusted on a rapid time scale to compensate for distortions in the wavefront entering a telescope. It is also possible to apply postdetection corrections to images and achieve high angular resolution on bright sources with ground-based telescopes [11].
( ⁄ ) ⁄ KL functions are built on the basis of Zernike polynomial expansion. This approach allows to reduce KL equations combined into the system of linear algebraic equations which are expressed for matrix elements analytically in terms of hypergeomatric function [12]. Table 3-Karhunen-Loeve Polynomials up to 4 th order [2] order Aberration mode

Results and Discussion
The first 15 KL polynomials are written and shown graphically in Figure-1     As we know, the ideal shape of the PSF generally assumes perfect axial and rotational symmetry. Figures-(2-4) show the change rate of PSF shape as well as its wings and difference from the ideal state. However, it is quite close to the effect of each mode of the optical aberrations. In other words, PSF image shape is close to the shape of the aberration effect for each aberration mode (like defocus, astigmatism, coma, spherical aberration, and so on). Now, the comparative analysis of various quality metrics will be presented. In order to provide substantive results, evaluation with subjective measures requires careful selection of the test subject to correlate with human perception of image quality. Normalized Root Mean Square Error (NRMSE), Strehl Ratio (SR), Normalized Cross-Correlation (NCC) and Normalized Absolute Error (NAE) are the most commonly used criteria for image visual quality assessment. To investigate a relationship between the fidelity criteria and KL aberration modes (piston, tip, tilt, focus, astigmatism x, astigmatism y, coma x, coma y, trefoil x and trefoil y), Figures-(6, 7, 8, and 9) are used. In the above figures (6, 8, and 9) the fidelity criteria were proportional to the radius of the telescope's aperture, which means that with the increase of the telescope's aperture, the effect of the atmospheric turbulence will be increased. As we know, the SR metric parameter represents a measurement of the  Figure-7 shows the decrease of SR with the increase of the aberration modes for both types of aperture radii, as expected. The difference between SR values for the piston mode KL1 for two aperture radii shows the high sensitivity for increasing the diameter of the telescope.
To study the relationship between a single aberration mode (focus K4) and aperture's radii for each objective fidelity criteria, Figures-(10, 11, 12, and 13) are employed.

Conclusions
The simulations showed that the average PSF values for aberration circular aperture with KL modes from K1 to K10 (in Figures-2, 3 and 4) and their effects on the original image (sun spot) were demonstrated in the convolved images in Figure-5. Clearly, increasing the aberration mode will cause an increase in the blurring of the observed image (sun spot). From Figures-(6, 7, 8, and 9) for both tow aperture's radii, it can be concluded that the optical phase error in terms of KL polynomials was increased with the increase of the aberration modes order. Figures-(10, 11, 12, and 13) showed a relationship between the focus (K4) aberration modes as a function of aperture's radii. Evidently, it can be concluded that the optical phase error (focus) was increased significantly when aperture's radii were increased. When this increase is larger than 128 pixel radius, the behavior of the phase error is almost stable. Finally, the sum of all KL aberration modes can be used as an initial state of the adaptive optics process. The suggested future work to develop our simulation is to use multi-conjugate AO system for more reliable results.