Microsoft Word - PHYS070107.doc 67-74 SQU Journal For Science, 12 (1) (2007) © 2007 Sultan Qaboos University 67 Limits of the Efficiency of Imaging with Obstructing Apertures A.T. Mohammed Department of Astronomy, College of Science, University of Baghdad, Jadiria, Iraq, Email: arobie33@yahoo.com. حدود الكفاءة عند التصوير باستخدام فتحات معرقلة للضوء علي طالب محمد استخدام حلول عددية ذات بعدين لتقييم كفاءة الفتحات المعرقلة للضوء من خالل دراسة تأثيرها على حدود قدرة :خالصة تضمن البحث حساب معدل شدة وعرض دالة االنتشار النقطية ، معدل مركبة الترددات لدالة التضمين االنتقالية . التحليل النتائج أوضحت ان الفصل بين الخطين يصبح مميزاً عند .من الخطوطوكذلك معدل الحلقات الجانبية لصورة زوج . مرة من نصف قطر الفتحة االصلية0.6استخدام نصف قطر اإلعاقة المركزية أكبر او يساوي تقريباً 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. KEYWORDS: Obstructing apertures, optical systems, resolution, fourier transform, image quality. 1. 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. According to the Rayleigh criterion, the theoretical resolving power of 5 m optical telescope ( λ=400 nm) are 0.02′′≅ . This value shows the relationship between the resolving power and the telescope apertures. In addition to that criterion, measurements of MTF, strehl ratio, and diffraction limited resolution are also very well A A.T. MOHAMMED 68 considered in quantifying optical systems (William and Bucklund 1989, Harvey and Ftaclas 1995, Brummelaar and Bagnuolo 1995, Brammelaar et al. 1994, Baldwin et al. 2001, Granieri et al. 1998, Jean-Marc et al. 2004, Mohammed et al. 1990, Milanfar and Shakouri, 2002). 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. 2. Theory The fundamental equation to be used for the formation of an image by an ideal optical system is given by: ydxdyyxxfpsyxoyxi ′′′−′−′′=∫∫ ∞ ∞− ),(),(),( (1) ),(),(),( yxpsfyxoyxi ⊗= (2) 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: v)T(u, v).O(u, v)I(u, = (3) 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, ),(),( γηφγη ieU = (4) where ),( γηφ is the random phase of the incident wavefront and the variables ),( γη represent distances in the pupil function. The instantaneous psf of the entire telescope atmosphere system is given by: 2)],(),([),( γηγη UHFTyxpsf = (5) LIMITS OF THE EFFICIENCY OF IMAGING WITH OBSTRUCTING APERTURES 69 Figure 1. Telescope apertures:(a) Uniform ( 0ε = ), (b) 0.6ε = where H ),( γη represents the pupil function and FT denotes Fourier transform operator. The corresponding OTF is the Fourier transform of the psf, thus y)]FT[psf(x,v)T(u, = (6) Equation (6) can also be written in terms of the pupil function and the field distribution incident on the lens as, γηγγηηγη γγηηγη ′′′−′− ′−′−=∫∫ ∞ ∞− ddUU HHvuT ),(*),( ).,(*),(),( (7) where * denotes complex conjugate. The variables η and γ are related to the Fourier space variables u and v by υλγλη f,fu == , where λ is the wavelength and f is the focal length. 3. 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 )NM( × is of size 512512 × 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., Rr=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 ( ), 0 and ( , ) 1Uφ η γ η γ= =⎡ ⎤⎣ ⎦ . 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. (a) (b) A.T. MOHAMMED 70 Figure 2. Perspective plots of the central regions of psf’s, (a) Uniform aperture,(b) ε = 0.2,(c) ε = 0.4, (d) ε = 0.6 and (e) ε = 0.8 (f) ε = 0.9. 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. Figure 3. Plots through the central regions of the psfs. (a) Uniform aperture,(b) ε = 0.2,(c) ε = 0.4,(d) ε = 0.6,(e) ε = 0.8 and (f) ε = 0.9. 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. (a) (b) (c) (d) (e) (f) (a) (c) (d) (e) (f) (b) N or m al iz e in te ns ity N or m al iz e in te ns ity No. of Pixels No. of Pixels 2 LIMITS OF THE EFFICIENCY OF IMAGING WITH OBSTRUCTING APERTURES 71 a M TF N No. of Pixels b c d e f b- ε =0.2 a- ε =0 c- ε =0.4 d- ε =0.6 e- ε =0.8 f- ε =0.9 Figure 4. MTF’s (a) Uniform aperture,(b) ε = 0.2,(c) ε = 0.4,(d) ε = 0.6 (e) ε = 0.8 and (f) ε = 0.9 The above images are then normalized to their maximum values according to the following equation: )0,0( ),( ),( MTF yxMTF yxMTFN = (8) Figure 5. Central plots through Figure 4. where MTF(0,0) represents the value of the central spike. The central lines of the normalized MTFs (i.e. MTFN ) are plotted in Figure 5. (a) (b) (c) (d) (e) (f) A.T. MOHAMMED 72 (a) (b) (c) (d) (e) (f) (g) 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. Figure 6. Line Convolutions:(a) Uniform aperture, (b) ε = 0.2, (c) ε = 0.4, (d) ε = 0.6, (e) ε = 0.8 and (f) ε = 0.9. The images in Figure 6 are then normalized to one at their maximum values and the central lines through these images are plotted. The results are presented in Figure 7. Figure 7. Normalized Intensities through the center of the images in Figure 6. (a) Uniform aperture,(b)ε = 0.2,(c) ε = 0.4,(d) ε = 0.6,(e) ε = 0.8 and (f) ε = 0.9. N or m al iz e In te ns ity No. of Pixels a b d e f c LIMITS OF THE EFFICIENCY OF IMAGING WITH OBSTRUCTING APERTURES 73 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 . Table 1. The variation of MI, AI, FW, AFC and SL at different values of ε. SL AFC FW AI MI ε 1 1 3 1 1 0 1.0808 0.9604 3 1.0412 0.9224 0.2 1.2373 0.8406 3 1.19 0.707 0.4 1.5328 0.6410 1 1.56 0.411 0.6 2.3403 0.3607 1 2.77 0.13 0.8 3.0122 0.1902 1 5.26 0.0362 0.9 The average intensity values (AI) of the psf at different ε is computed by: ∑∑ ∑∑ = = = = = M 1y N 1x 1 1 y)(x,),(A M y N x UpsfyxpsfI (9) where psfU (x,y) is the psf at ε =0. The average frequency components (AFC) of MTF (u,v) at different ε are computed by: ∑∑ ∑∑ = = = = = M 1 N 1 1 1 ),(),MTF(AFC u v M u N v U vuMTFvu (10) where MTFU (u,v) is the MTF at ε =0. The full width of the psf (FW) at different ε are also calculated and presented in this table. These values indicate the width of the central spike of the psfs. The summation of all values that located outside the double lines of the psfs at different ε (see Figure.6) are computed and the values are then divided by the corresponding value of the uniform aperture. The results are describing the actual weight of the side loops (SL). 4. Conclusions The following conclusions could be drawn : 1. The MI value decreases by a factor of 30≅ as ε goes from ε = 0 to ε = 0.9. This great reduction in the height of central spike will enhance the probability of detecting faint companion (Figure 7 describes normalized intensity not actual intensity). 2. As ε increases, AI increases and at ε = 0.9, AI increases by a factor of 5≅ . This is of course will create artifacts in the observed images (see Figure. 2). 3. FW reduced dramatically as ε increases. For ε = 0.6, FW reduces by a factor of 3. This will enhance the resolution by a factor of 3. A.T. MOHAMMED 74 4. AFC reduces by nearly a factor of 5≅ at ε = 0.9. This is because the build up of the side loops in the psfs. AFC is inversely proportional to ε. 5. The SL increases by a factor of 3 at ε = 0.9. SL is linearly proportional to ε. 6. References BALDWIN, J.E., TUBBS, R.N., COX, G.C., MACKAY, C.D., WILSON, R.W. and ANDERSON, M.I. 2001. Diffraction limited 800 nm imaging with 2.56 m Nordic optical telescope. Astronomy and Astrophysics 368: L1-L4. BRUMMELAAR, T.A. and BAGNUOLO, W.G. 1995. Strehl ratio and visibility in long baseline stellar interferometry. Optics Letters 20: 521-23. BRUMMELAAR, T.A., BAGNUOLO, W.G. and RIDGWAY, S.T. 1994. 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