IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (4) 2009 Studying of Optical System Includes Elliptical Aperture Point Spread Function A. B.Hassan, H. M. Ali Department of Physics, College of Education, Ibn Al-Haitham, University of Baghdad Abstract In this work, optical system with elliptical aperture using point spread function was studied. This is due to its comparison with an optical system with a circular aperture. The present work deals with the theoretical study of intensity distribution within the image. In this work, a special formula was derived which is called the point spread function (PSF) by using a pupil function technique. The work deals with the limited optical system diffraction only (ideal system), and the system with focal shift. Also a graphic relation was founded between eccentricity and the best of focal depth given to at least (80%) of intensity. Theory The complex amplitude at any point in the image plane [1] is given by: )1...(),( 1 ),( )(2   y x vyuxi dxdyeyxf A vuF  Where (u,v)= image plane coordinates (x,y)= xit pupil coordinates A= exit pupil area f(x,y) is the pupil function [2], which has the form: ),().,(),( yxikweyxyxf  ),( yx Is the real amplitude distribution across wave front which is equal to unity in most cases. (k) The wave number and equals to  2 . ),( yxw is wave aberration function [3]: )2..()(),( 22 1 2 n N n n yxwyxw   Point spread function (distribution of illuminance in image plane due to point source) G(u,v) is given by the squared modulus of complex amplitude [4]whichis: 2 ),(),( vuFvuG  )3..().,(),( 2 ),(2   y x vyuvi dxdyeyxfvuG  We have asymmetrical intensity distribution in image plane, so we can cancel one of the image plane coordinates (v = 0): IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (4) 2009 (z) is a disp lacement coordinate which equals to (z=2πu) N is a normalizing constant, where G (0) =1 The pupil function condition using elliptical aperture equation [shown in fig (1)] )5..( 10 1 ),( 2 2 2 2 2 2 2 2 ),(                b y a x b y a x e yxf yxikw Where A= π, and ellipse area (A=abπ), so (ab=1) Substitute eq (5) into eq (4): for a diffraction – limited system (aberration free system ), where w(x,y)=0 )7...(1)0( 2 1 1 22 22        by by yaax yaax dxdyNG When we solve equation [7] by (Gauss quadrrature method): 2 1  N We substitute N into eq (6): 2 1 1 2 22 22 1 )(        by by yaax yaax izx dxdyezG  )8...()]sin()[cos( 1 )( 2 1 1 2 22 22                by by yaax yaax dxdyzxizxzG  The term (isin (zx)) into eq (8) was canceled because (sin) is odd function. )9...()cos( 1 )( 2 1 1 2 22 22                by by yaax yaax dxdyzxzG  )4..().,()( 2   y x izxdxdyeyxfNzG 2 1 1 22 22 ).,()(        by by yaax yaax izx dxdyeyxfNzG )6..()( 2 1 1 22 22        by by yaax yaax izx dxdyeNzG IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (4) 2009 The equation (9) refers to ideal system (free aberration system). Now consider having a longitudinal focal shift (aberrated system): Following the same procedures; the intensity is given by: )9..( 2 221 221 )]22(22sin [ 2 221 221 )]22(22cos[ 2 1 )(                                  by by yaax yaax dxdyyxwzx by by yaax yaax dxdyyxwzxzG   Results and Discussion Twenty point Gauss quadrature were used [5] to evaluate the integral in eq (8). The results thus obtained for the normalized intensity are given in table (1), it shows the available values of parameter (a) [consider area of aperture (A) which always equals to (π)] and asymmetric intensity distribution for it, when (a=1) (circular aperture) and (1.5 , 2 , 2.5) for elliptical aperture. Fig (2) shows how the diffraction pattern for a circular aperture varies with a pattern of elliptical aperture, so that the radius of the intensity distribution for elliptical aperture is smaller than for a circular aperture. Thus the central peak of the intensity is sharper for elliptical aperture, so the resolution of the elliptical aperture should be better than of the circular aperture. Fig. (3) shows the focal shift for a circular aperture which varies with an elliptical aperture, that the depth of focus for a circular aperture is better than for the elliptical aperture. Finally; fig (4) shows the relationship between eccentricity (e) for ellipse aperture and the best of focal depth which gives (80%) of intensity , that the tolerance of focal shift was better when decrease aperture flatting (approaches to a circular aperture), that means the eccentricity approaches to zero References 1 .Horitz, P. (1976).Appl. Opt. 15:167-171 2.Barakat, R. (1998). Opt. Commun. 156 (4):235-239 3.Kinter, E.C. J. Modern. (1999). Opt. 46:1031-1042 4.Refrgier, A. ; Mcmahon, R.G. and Helfand, D.J. (2000) J.Opt. Soc. Am. 17:1185-1191 5.Al-Jizany, A.B. (2001) Msc. Thesis (Baghdad University). )(),( 22 2 yxwyxw  2009) 4 (22مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد دراسة منظمة بصریة تحتوي فتحة بیضویة باستخدام دالة االنتشار النقطیة هبه ممتاز علي،عالء بدر حسن الجیزاني جامعة بغداد ، تربیة ابن الهیثم كلیة ال، قسم الفیزیاء ةصالخال ة بـصریة ة ، وتمــت مقارنـة النتـائج مــع تمـت دراسـة منظومــ تحتـوي فتحــة بیـضویة باسـتخدام دالــة االنتـشار النقطیـ فـي مـستوى الـصورة ، واشـتقاق صـیغة هذا البحث مع دراسة نظریـة لتوزیـع الـشدة یتعامل . فتحة دائریة يمنظومة بصریة ذ ـام (البحــث مــع منظومــة بــصریة محــددة بــالحیود فقــط تعامــل باســتخدام تقنیــة دالــة البؤبــؤ ،ة للفتحــة البیــضویة خاصــ النظـ . خطأ بؤري و، ومنظومة ذ) المثالي عالقـة بیانیـة تـربط عامـل االخـتالف وجـدتإن االخـتالف المركـزي للفتحـة البیـضویة هـو دالـة لمقـدار تفلطـح الفتحـة ، ومنهـا .األقلمن الشدة على ) %80(وسماحیة الخطأ البؤري التي تعطي ) e(المركزي IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (4) 2009 Table (1): Intensity distribution in a circular and elliptical aperture (PSF) z a=1(circle) a=1.5 a=2 a=2.5 -10 0.0000768 0.000752 0.000046 0.000097 -9 0.0029791 0.000032 0.000432 0.000015 -8 0.0034486 0.001381 0.00013 0.000046 -7 0.0000016 0.000223 0.000366 0.000345 -6 0.0084941 0.002979 0.001381 0.000752 -5 0.0171527 0.001306 0.000076 0.000697 -4 0.0010863 0.008494 0.003449 0.000076 -3 0.0511216 0.010533 0.008494 0.001306 -2 0.3326821 0.051122 0.001086 0.017153 -1 0.7746855 0.553501 0.332682 0.158195 0 1.0001215 1.000122 1.000122 1.000122 1 0.7746855 0.553501 0.332682 0.158195 2 0.3326821 0.051122 0.001086 0.017153 3 0.0511216 0.010533 0.008494 0.001306 4 0.0010863 0.008494 0.003449 0.000076 5 0.0171527 0.001306 0.000076 0.000697 6 0.0084941 0.002979 0.001381 0.000752 7 0.0000016 0.000223 0.000366 0.000345 8 0.0034486 0.001381 0.00013 0.000046 9 0.0029791 0.000032 0.000432 0.000015 10 0.0000768 0.000752 0.000046 0.000097 Table (2): Axial intensity for circular and elliptical aperture w20 a=1(circle) a=1.5 a=2 a=2.5 0 1.000122 1.000122 1.000122 1.000122 0.2 0.875232 0.636018 0.255324 0.161672 0.4 0.572814 0.200478 0.137527 0.085907 0.6 0.254552 0.125858 0.083003 0.05337 0.8 0.054673 0.080299 0.054111 0.036835 1 0 0.042107 0.039839 0.030017 1.2 0.024324 0.029903 0.030451 0.059813 1.4 0.046765 0.013408 0.032086 0.064329 1.6 0.035791 0.009179 0.021195 0.051075 1.8 0.010794 0.004254 0.009275 0.002354 2 0 0.00365 0.032252 0.001014 IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (4) 2009 Fig.(1): Elliptical aperture -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -10 -8 -6 -4 -2 0 2 4 6 8 10 z G(z) a=1 a=1.5 a=2 a=2.5 Fig. (2): Intensity distribution in a circular and elliptical aperture (PSF) A=π b a IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (4) 2009 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 w20 G(w) a=1 a=1.5 a=2 a=2.5 Fig. (3): Axial intensity for a circular and elliptical aperture 0 0.2 0.4 0.6 0.8 1 1.2 0.040.070.10.110.130.190.240.250.260.27 w20 e Fig. (4): Relationship between eccentricity (e) and the best focal depth which gives (80%) of intensity