Advances in Optical Technologies

Volume 2016, Article ID 1608342, 5 pages

http://dx.doi.org/10.1155/2016/1608342

## Spherical Aberration of Point Spread Function with Asymmetric Pupil Mask

^{1}Science and Research Laboratory of Automated Systems of Science Researches, Samara National Research University, Moskovskoye Shosse 34, Samara 443086, Russia^{2}Department of Physics (H&S), CMR Institute of Technology, Medchal Road, Kandlakoya, Hyderabad, Telangana 501 401, India^{3}Optics Research Group, Department of Physics, University College of Science, Osmania University, No. 49, Hyderabad, Telangana 500007, India

Received 31 July 2016; Accepted 9 November 2016

Academic Editor: Paramasivam Senthilkumaran

Copyright © 2016 Naresh Kumar Reddy Andra and Karuna Sagar Dasari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Point spread function underneath spherical wave aberration with antiphase apodization has been obtained by one-dimensional pupil mask functions. In the presence of spherical aberration, suppression of optical side-lobes has increased on one side of the point spread function with the width of the periphery strips within the pupil mask. On introducing wave aberration effect, there exists dependence of the lateral resolution of central peak of the asymmetric point spread function on the amount of amplitude masking. However, the magnitude of intensity of central peak is originated be to amplified by the highest degree of amplitude and phase masking. Additionally, for aberrated asymmetric PSF, FWHM increases and it further decreases with the control parameters of amplitude and phase mask. The magnitude of this corollary can quantify the super resolution of diffracted structures under spherical aberration.

#### 1. Introduction

Asymmetric apodization is the designed modification of the impulse response of an optical system such that simultaneous elimination of secondary maxima and narrowing the central peak in the field of diffraction. This method can change the performance of an optical imaging system by redistributing the light flux in the image plane. There has been a certain number of investigations involving asymmetric apodization for different considerations, that is, improvement of side-lobe suppression [1, 2], application of asymmetric apodization to linear arrays and circular arrays [3, 4] to improve the axial resolution of the optical system [5, 6], reduction of Sparrow resolution limit with incoherent illumination [7] to achieve axial and lateral resolution in confocal imaging [8] and to detect the extra solar planets [9], and improvements of dispersion of chirped fiber gratings [10]. In these studies the optical system has asymmetrically apodized. It is clear that by employing asymmetric apodization we obtained low side-lobes and sharp main peaks on one side of the PSF at the cost of improved side-lobes and wider main peaks on counter side.

It is well known that diffractive aberrations are caused by deviations from geometrical optics due to the wave nature of light. These effects place the limit on the optical system imaging performance. The third-order Gaussian theory describes that primary monochromatic aberrations like spherical aberrations arise due to geometrical deviations and may also result from the pupil functions. The spherical aberration can also be explained by saying that the deviation of the principal surface from its ideal shape causes the focal length to slightly vary with the radius of zone of the aperture. There has been a significant number of studies involving diffractive monochromatic aberrations with different aperture systems and techniques [11–13].

It is clear that, through suitable apodization technique in the exit pupil of optical system, the diffractive object may achieve high resolution over a confined region in the image plane of an optical system in the presence of Seidel aberration effect. This is the origin for the current investigation. Based on the extensive study made on the system of apodization, we came into conclusion that asymmetric apodization could be the solution for aberrated optical imaging system to achieve the super resolution. This technique is effective in improving the axial and lateral resolution. In this study we investigated cylindrical optical elements or slit functions with asymmetric apodization employed to redistribute the light energy in the focal plane under geometrical aberration. The study has a wide range of potential applications such as confocal imaging, medical optics, laser beam profiling, atmospheric science, and design of antenna arrays in communication engineering. In this paper work, we obtained asymmetric point spread function of one-dimensional asymmetric pupil mask configured analytically in the structure of two periphery fine strips of certain width subjected to phase masking and a central cylindrical region of the aperture is shaded with amplitude mask. Here we obtained first side-lobe intensities, central peak positions, and FWHM values for various amounts of amplitude and phase apodization.

#### 2. Theory and Formulation

Within the diffraction wave theory, the amplitude impulse reaction of spherical aberration strained optical imaging system is the Fourier transform of the asymmetric pupil mask function which consists of three regions having homogeneous amplitude transmittance, namely, two fine edge strips with reverse phase transmittance of the forms and and zero phase transmittance for the central cylindrical amplitude mask of the one-dimensional optical element. Because of exceptionally deep reduction ability and constant working angles throughout the regions of considered edge elements, we consider the asymmetric phase functions. Thus, we consider complex pupil mask with real amplitude transmittance at the central region and complex conjugated outer edge elements. In this case, the resultant complex light amplitude distribution in the focal plane is the sum of the diffraction amplitude contributing to the central cylindrical region of aperture system of width () and diffraction amplitudes contributing to the narrow left and right periphery strips with opposite phase transmittances and given by The diffraction field amplitude contributing to the left periphery fine strip is as follows: On introducing wave aberration, such as primary spherical aberration, the diffraction amplitude contributing to central cylindrical amplitude masking zone of a pupil function of width is as follows:

The diffraction field amplitude contributing to the right periphery fine strip is as follows: Therefore, the total field amplitude on the image plane of the optical imaging system with asymmetric pupil mask is where , is the coordinate in the pupil plane, is the reduced dimensionless diffraction optical coordinate, *λ* is the wavelength of the stimulated light radiation, and is the spherical aberration control parameter of the optical imaging system. “” and *β* are phase (asymmetric) apodization and amplitude apodization control parameters, respectively. Impact of primary spherical aberration on resultant intensity PSF, which is the actual quantifiable quantity, can be obtained by taking square modulus of (5). Accordingly,

#### 3. Results and Discussion

The results of investigations on the effects of spherical aberration on the light flux distributions in the image plane of an optical imaging system with asymmetric pupil functions have been obtained from (6) as a function of dimensionless optical coordinate “” varying from −15 to +25 by employing an iterative method of numerical integration. Gaussian quadrature method [14] holds the most important properties and accuracy in finding the positions and intensities of primary and secondary maxima on either side of PSF under primary spherical aberration, though results are reported on one side, the PSF only, in which side-lobes are suppressed to a great extent and central peak is found to be sharp. This is the right half of the entire pattern. Our main attention has been drawn on central peak position and its resolution, optical side-lobe intensities for various amounts of spherical aberration, and amplitude-phase apodization. These parameters are enough for judging the imaging performance and efficiency of asymmetric pupil mask of spherical aberration strained optical imaging system. The PSF curves are presented in Figure 1 for various degree of asymmetric apodization when the central region of the pupil function is transparent when = *π*/2. Figure 1 shows that on the left side of the pattern the central maxima is transferred, widened, and lowered at the cost of narrow central peak on the right half side. This effect depends on the width () of edge element of the aperture.