Limits of the Efficiency of Imaging with Obstructing Apertures

ةصلاخ :   ةردق دودح ىلع اهريثأت ةسارد للاخ نم ءوضلل ةلقرعملا تاحتفلا ةءافك مييقتل نيدعب تاذ ةيددع لولح مادختسا ليلحتلا . ةيلاقتنلاا نيمضتلا ةلادل تاددرتلا ةبكرم لدعم ، ةيطقنلا راشتنلاا ةلاد ضرعو ةدش لدعم باسح ثحبلا نمضت جوز ةروصل ةيبناجلا تاقلحلا لدعم كلذكو طوطخلا نم .  دنع ً ازيمم حبصي نيطخلا نيب لصفلا نا تحضوأ جئاتنلا ً ابيرقت يواسي وا ربكأ ةيزكرملا ةقاعلإا رطق فصن مادختسا 0.6 ةيلصلاا ةحتفلا رطق فصن نم ةرم .   ABSTRACT: Two-dimensional numerical solutions are carried out to asses the quality of obstructing apertures in terms of the diffraction limited resolution. This include the quality of the point spread function (psf), the modulation transfer function (MTF), and an image of double lines. These are average intensity of the psf (AI), maximum intensity of the psf,(MI), full width at half maximum of the psf (FW) average frequency components of MTF (AFC), and average side loops of an image of a double lines. The results indicate that the separation of the two lines becomes recognizable using central obstruction of radius equal to or greater than approximately 0.6 times the radius of the primary aperture.


Introduction
ll telescopes have an inherent limitation to their angular resolution due to the diffraction of light at the telescope aperture.The diffraction causes an optical system to behave as a low-pass filter in the formation of an image.The cut-off frequency is directly determined by the shape and size of the limiting pupil in the optical system.The incoming light is approximately a plane wave since the source of the light is so far away.
There are several criteria for analyzing the performance of an optical imaging systems.The Rayleigh criterion is generally regarded as a fundamental limit in predicting the performance of optical imaging systems.
Many studies have been presented in the literatures concerning imaging with obstructing aperture (Fienup 2000, Mohammed 2006, Chakraborty and Thompson 2005).
The aim of this paper is to present the quantitative assessment of the limitations imposed by obstructing apertures on the psf and MTF in order to determine the constraints on the efficiency of imaging with such apertures.

Theory
The fundamental equation to be used for the formation of an image by an ideal optical system is given by: Equations ( 1) and ( 2) are equivalent and representing a convolution equation.Where i(x,y) is the observed image intensity, o(x,y) is the object intensity, psf (x,y) represents the image blurring function caused by the imaging system and ⊗ denotes convolution operator.
The Fourier transform of (2) is given by: where I(u,v) and O(u,v) are, respectively, the complex Fourier transforms of the image intensity i(x,y), and the object intensity o(x,y); T(u,v) which represents the Fourier transform of the psf, is an important function known as the optical transfer function (OTF).The modulation or amplitude of the complex function T(u,v) is called MTF.In general, the resolution of an imaging system is limited only by the luck of large optical elements that are free from inherent distortions.Now consider an extremely distant quasimonochromatic point source located on the optical axis of a simple imaging system.In the absence of atmospheric turbulence, this source would generate a plane wave normally incident on the lens.In the presence of the atmosphere, the plane wave incident on the inhomogeneous medium propagates into the medium, and ultimately a perturbed wave falls on the lens.The field distribution incident on the lens can be expressed as, Equation ( 6) can also be written in terms of the pupil function and the field distribution incident on the lens as, where * denotes complex conjugate.The variables η and γ are related to the Fourier space variables u and v by , where λ is the wavelength and f is the focal length.

Simulations
The size of the pupil function H(u,y) is taken to be a two dimensional circular function of radius R and of unity magnitude; this array pixels.This size is taken as large as possible in order to keep the theoretical diffraction limiting resolution vanishing to zero inside this array.This aperture is said to be a uniform aperture.
The central obstructing aperture is simulated by calculating the parameter ε.This parameter represents the ratio of the radius of obstructing circle (r) to the radius of the uniform aperture, R, i.e., R r = ε .Telescope apertures of ε = 0 and ε = 0.6 are demonstrated in Figure 1.
We consider the object to be imaged is an extremely distant quasimonochromatic point source located on the axis of an optical telescope.In the absence of atmospheric turbulence, this source would generates a plane wave ( ) The psfs are computed via Equation ( 5).The perspective plots of the central regions of these psfs (14 pixels by 14 pixels ) are shown in Figure 2. It should be pointed out here that the central spikes of the psfs are very sharp.This is because R is taken to be very large, R=120 pixels.The line plots through the centre of these regions are shown in Figure 3.The MTFs are also computed according to Equation ( 6) or ( 7) respectively and absolute values are taken for T (u,v).The results are shown in Figure 4.The psfs at different values of ε are convolved with the double line object presented in Figure 6(a).The lines are one pixel wide separated by a distance of 2 pixels.This value is chosen because the full width of the psf of the uniform aperture is 3 pixels.The results are shown in Figure 6.For ε = 0, the line plot shows a little peak at the center.As ε increases, the separation of the double lines becomes recognizable and the side loops become severe.
The normalized maximum intensity values (MI) of the psfs are calculated and presented in the Table 1 given below as follows.First, the maximum value of the psf of the uniform aperture (ε = 0) is calculated.Secondly, the maximum intensity values at different ε are divided by the maximum value at ε = 0.This measure is taken in order to examine the dropness in the value of the central spike of the psf as a function of ε .
. The variation of MI, AI, FW, AFC and SL at different values of ε.The average intensity values (AI) of the psf at different ε is computed by: