IBN AL- HAITHAM J . FO R PURE & APPL. SC I VO L.22 (4) 2009 Studying o f Optical System Includes Elliptical Aperture Point Spread Function A. B.Hassan, H. M. Ali Departme nt of Physics, College of Education, Ibn Al-Haitham, Unive rsity of Baghdad Abstract In this work, op tical sy st em with ellip tical aperture using p oint sp read function was st udied. This is due to its comparison with an op tical sy st em with a circular aperture. The present work deals with the theoretical study of intensity distribution within the image. In this work, a sp ecial formula was derived which is called the p oint sp read function (PSF) by using a p up il function technique. The work deals with the limited op tical sy st em diffraction only (ideal sy st em), and the sy st em 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 p oint in the image plane [1] is given by: )1...(),( 1 ),( )(2  y x vyuxi dxdyeyxf A vuF  Where (u,v)= image plane coord inates (x,y )= xit p up il coordinates A= exit p up il area f(x,y ) is the pup il function [2], which has t he form: ),().,(),( yxikweyxyxf  ),( yx Is t he real amp litude dist ribution across wave front which is equal to unity in most cases. (k) T he wave number and equals to  2 . ),( yxw is wave aberration function [3]: )2..()(),( 22 1 2 n N n n yxwyxw   Point sp read function (distribution of illuminance in image plane due to p oint 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 asy mmetrical 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 equ als t o (z=2πu) N is a normalizing constant, where G (0) =1 The pup il 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) Subst itut e eq (5) into eq (4): for a diffraction – limited sy st em (aberration free sy stem ), where w(x,y )=0 )7...(1)0( 2 1 1 22 22        by by yaax yaax dxdyNG When we solve equ ation [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 t o ideal sy st em (free aberration sy stem). Now consider havin g a lon gitudinal focal shift (aberrated sy st em): Following the same p rocedures; 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 p oint 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 p arameter (a) [consider area of aperture (A) which alway s equals to (π)] and asy mmetric intensity distribution for it, when (a=1) (circular aperture) and (1.5 , 2 , 2.5) for ellip tical aperture. Fig (2) shows how the diffraction p att ern for a circular aperture varies with a p att ern of ellip tical aperture, so that the radius of the intensity distribution for ellip tical aperture is smaller than for a circular aperture. Thus the central p eak of the intensity is sharp er for ellip tical aperture, so the resolution of the ellip tical aperture should be bett er than of the circular ap erture. Fig. (3) shows the focal shift for a circular aperture which varies with an ellip tical aperture, that t he dep th of focus for a circular aperture is bett er than for the elliptical aperture. Finally; fig (4) shows t he relationship between eccentricity (e) for ellipse aperture and the best of focal dep th which gives (80%) of intensity , that t he tolerance of focal shift was bett er when decrease aperture flatting (app roaches to a circular aperture), that means the eccentricity app roaches to zero Re ferences 1 .Horitz, P. (1976).Appl. Opt. 15:167-171 2.Barakat, R. (1998). Opt. Commun. 156 (4):235-239 3.Kinter, E.C. J. M odern. (1999). Op t. 46:1031-1042 4.Refrgier, A. ; M cmahon, R.G. and Helfand, D.J. (2000) J.Opt . Soc. Am. 17:1185-1191 5.Al-Jizany, A.B. (2001) M sc. Thesis (Baghdad University ). )(),( 222 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 تحتـوي فتحــة بیـضویة باسـتخدام دالــة االنتـشار النقطیـة ، وتمــت مقارنـة النتـائج مــع تمـت دراسـة منظومــة بـصریة فـي مـستوى الـصورة ، واشـتقاق صـیغة هذا البحث مع دراسة نظریـة لتوزیـع الـشدة یتعامل . فتحة دائریة يمنظومة بصریة ذ النظــام (البحــث مــع منظومــة بــصریة محــددة بــالحیود فقــط تعامــل باســتخدام تقنیــة دالــة البؤبــؤ ،ة للفتحــة البیــضویة خاصــ . 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): Inte nsity distributi on in a circul ar and elliptical aperture (PS F) 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 circul ar 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 distributi on in a circul ar and elliptical aperture (PS F) 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): Rel ationship between eccentricity (e) and the best focal depth which gives (80%) of intensity