Edge Imaging with Obscured Apertures Apodised by Amplitude Filters

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1 International Journal of Innovative Research in Science, Engineering an Technology Vol., Issue 3, March 13 Ege Imaging with Obscure Apertures Apoise by Amplitue Filters Venkanna Mekala 1, Karuna Sagar Dasari Optics Research Group, Department of Physics, Nizam College, Osmania University, Hyeraba-57 1 Professor, Optics Research Group, Department of Physics, Nizam College, Osmania University, Hyeraba- 57 ISSN: Abstract. The analytical stuies were mae for obscure apertures apoise with amplitue filter uner the influence of efocus. The coherent ege-response of optical systems apoise with shae aperture in the case of shrink aperture has been stuie. The image quality assessment parameters have been evaluate by masking the aperture from the outsie. Results are rawn for various obscure apertures an compare with an without apoisation. It is foun that, this type of shaing an shaping of apertures is foun to be effective in enhancing the image quality an the resolving power of the optical imaging systems. Key wors: Ege Imaging, Coherence, Apoisation, Amplitue filters, Ege-ringing, Ege-graient an Egeshift. I.INTRODUCTION A coherent optical system is linear in the complex fiel amplitue which means, that the resultant amplitue is the sum of the component amplitues an the role of transfer function is playe by the pupil function itself. The transfer function of optical system has a sharp cut-off in coherent illumination. An ege object has strong high frequency components. The cut-off of the coherent optical system is effectively at a low value as compare to Fourier spectrum of the sharp ege [1] an physical result is the unwante ege ringing. The presence of these spurious fringes in the ege response an the apparent shift of the image ege lea to the ifficulty in fining the location an the measurement of the ege. Stuies on this subject inicate the importance of coherent imagery in areas like spatial filtering techniques an microscopy. Literature is rich in the stuies on circular apertures, employing apoisation to reuce the ringing [-1]. In this paper, we propose to stuy the joint effects of apoisation an obscuration on the iffraction images of coherently illuminate straight ege. The ege-ringing, ege-shift an ege-graient of the ege fringes have been evaluate for ifferent values of apoisation using shrink apertures. For this we consiere rotationally symmetric, iffraction-limite an efocuse coherent optical system. These investigations have suggeste the use of certain pupil functions in conjunction with optimal apoizers to assess the ege image quality [11-17]. II.THEORY The mathematical representation of amplitue transmission of an opaque straight ege is given by A (u, v) = 1 for u A (u, v) = for u < (1) Copyright to IJIRSET 645

2 f(r) International Journal of Innovative Research in Science, Engineering an Technology Vol., Issue 3, March 13 ISSN: This inicates that the transmission function is iscontinuous at u =. The Fourier transform for this equation gives the amplitue spectrum of the object an is given by [18] 1 1 a ( x, y) where x is the Dirac-elta function. The moifie object amplitue spectrum at the exit pupil of the optical system is given by ' a where f x, y is the pupil function of the optical system. For the given optical system the complex amplitue istribution in the image plane is given by the inverse Fourier transform of expression (3). Thus The present work constitutes one-imensional ege conition an hence, the general form of amplitue istribution is given by ' ' ' 1 1 A ( u, v ) y x () i x x, y ax, y f x, y (3) ' ' A( u, v ) pupil a ' ' x, y f x, yexp i u x v yxy (4) 1 sin( Zx) f ( x, ) x x where Z=πu ' an f(x, ) is the coherent transfer function of the system. The coherent transfer function f(x, ) in the current stuy is rotationally symmetric an satisfies the conition (5) f x, f x, Pupil function for shae aperture is f r 1 r (6) (7) where r is the normalize istance of an arbitrary point on the pupil from its centre an β is the apoisation parameter. The term β controls the egree of non-uniformity of transmission over the pupil. The value of β=, correspons to iffraction limite Airy system having uniform transmission of unity over the entire aperture... r Fig.1 Pupil transmission curves for various values of β Copyright to IJIRSET 646

3 International Journal of Innovative Research in Science, Engineering an Technology Vol., Issue 3, March 13 ISSN: For this apoizer the amplitue transmittance ecreases monotonically from the center towars the eges of the pupil. The above Fig.1 shows this phenomenon in greater etail. Higher spatial frequency components of the object are iffracte by a larger angle an hence these go preominantly through the ege of the aperture. As the pupil transmittance is ecrease at the eges as compare to that of the center, ue to apoization, the result is reuction in the higher spatial frequency components in the image. This manifests as partial or full suppression of the unesire optical sie lobes or seconary maxima, which consequently enhances image features. On introucing wave aberration such as efect-of-focus expression (5) takes the form ' ' ' 1 1 A ( u, v ) 1 x sin( Zx) f ( r)exp( i ) x x (8) For the given shrink aperture shae with the amplitue filter the expression (8) becomes, ' ' ' 1 1 A ( u, v ) (1 r x sin( Zx) )exp( i ) x x (9) Where < ε 1 is the raial obscuration parameter. Now the intensity istribution of an ege image forme by an apoize optical system is given by the square moulus of expression (9). Thus B( u') B( Z) A'( u') B Z A ' Z III. RESULTS AND INTERPRETATION The investigations on the effects of efocus an obscuration parameter on the images of ege objects forme by coherent optical systems apoise by the amplitue filter in the case of shrink aperture have been evaluate using the expressions (1) by employing Matlab7.8. The intensity istribution B(Z) in the images of straight ege objects has been obtaine for ifferent values of imensionless iffraction variable Z varying from -3 to +. The image quality assessment parameter such as ege-ringing, ege-shift an ege-graient have been stuie for various values of apoisation, efocus an obscuration parameters. The outer obscuration parameter of the aperture consiere are ε=1,.9,,.7 an. However the value ε=1 represents the circular aperture. Fig.a shows the intensity istribution profile of the straight ege for unapoise an aberration free optical system (Airy case) for both circular an shrink apertures. The ege ringing is pronounce an is insensitive to obscuration parameter ε as the system is unapoise. Fig.b illustrates the case where the efocus parameter =π is introuce. For shrink apertures, the negative maximum amplitue increases with obscuration parameter ε an hence the presence of ringing is much more pronounce. From the fig.c it is observe that, at the efocuse plane π, by increasing the shrinking zone of the aperture the unwante ege ringing has been reuce along with ege shift an improving the ege graient. In Figs., e&f, the optical system is apoise by shae apertures. We fin increase in ege ringing with obscuration parameter ε for =, 1 1 x Sin Z x (1 ) exp[ ( ) ] r i x x (1) Copyright to IJIRSET 647

4 International Journal of Innovative Research in Science, Engineering an Technology Vol., Issue 3, March 13 ISSN: π planes, but it ecreases an increase in ege graient at =π in the presence of apoisation. Hence this efocuse plane may be esignate as the optimum receiving image plane. It is evient that the apoise optical systems are more sensitive to aperture obscuration than unapoise ones. In Figs.c & f, the image intensity istribution curves are almost similar an clearly seen that, at certain efocuse planes, aperture shaping lowering the ringing effect even in the absence of shaing, however, it is more effective along with aperture shaing. Fig.3a shows the variation of the ege- ringing with obscuration parameter ε for ifferent efocus planes without apoisation. The magnitue of ege-ringing is foun to be almost constant for all the values of ε as the system is aberration-free an unapoise, i.e., for Airy pupil. For the plane =π ege-ringing is increasing with ε but it is in ecreasing tren an attain the minimum (.13946) value at ε=.7 for =π. The curve for =π cuts the curve for = at the value of ε aroun 75 an cuts the curve for =π at the value of ε aroun. For these values the ege-ringing is almost the same. Fig.3b shows the variation of the ege- ringing with obscuration parameter ε for ifferent efocus planes in the presence of apoisation. The ege-ringing is much more pronounce with ε varies from 1(circular aperture) to (shrunk aperture) for the planes =, π. But it attains much lower value for =π at ε=.7, however it is in increasing tren for ε<.7. Figs.3c an 3 illustrate the variation of ege- shift with raial obscuration ε for various values of without an with apoisation respectively. The minimum shift occurs for the circular aperture (ε=1) an is inepenent of ε for airy pupils. The ege-shift is increasing in the efocuse optical systems for ε=1,.9& even in the presence of apoisation. However it is in ecreasing tren for ε.7 for both unapoise an apoise systems at certain efocuse plane =π. The minimum shift occurs for the circular aperture (ε=1) an is epenent of ε for airy pupil an shae pupils. The ege-shift is increasing in the efocuse optical systems as the obscuration zone increases for both unapoise an apoise optical systems. Fig.3e an 3f show the fall of ege-graient with obscuration (ε=1,.9,,.7&) regarless of the values of apoisation for =, π. But it rises with obstructing zone of the aperture from 1(circular aperture).7(shrink aperture) at =π. This increase in ege graient improves the egrae ege images. IV. CONCLUSIONS The important conclusions of the investigations on the effects of efocus an aperture shaping on the images of straight ege objects forme by the coherent optical systems apoise with the shae aperture filter are summarize as: i. The unapoize optical systems are less sensitive to aperture shaping in the absence of efocus than the apoise ones. ii. The unwante ege ringing is foun to reuce even in the presence of efocus with the aperture obscuration. iii. Apoisation along with aperture shaping is useful in improving the performance of efocuse coherent optical systems. iv. For =π the ege ringing shows a ecreasing tren an there is an improvement in ege graient for both unapoise an apoise optical systems with aperture shaping. Hence at this efocuse plane the optical system may be consiere to be optimum in ege imaging. Hence this type of aperture shaing an shaping is foun to be very effective in improving the image quality an resolution capabilities of the optical imaging systems for coherent ege imaging. Copyright to IJIRSET 648

5 B(Z) B(Z) B(Z) B(Z) B(Z) B(Z) International Journal of Innovative Research in Science, Engineering an Technology Vol., Issue 3, March 13 ISSN: (a) (b) (c) () Fig. The intensity (e) profiles for un-apoise an apoise pupil functions uner the (f) influence of efocus an obscuration Copyright to IJIRSET 649

6 E.G E.G E.S E.S E.R E.R International Journal of Innovative Research in Science, Engineering an Technology Vol., Issue 3, March 13 ISSN: (a) (b) (c).7.9 () (e) (f) Fig.3 Shows the effect of Obscuration on Ege ringing (a & b), Ege shifting (c & ) an Ege graient (e & f) at ifferent efocuse planes Copyright to IJIRSET 65

7 ISSN: International Journal of Innovative Research in Science, Engineering an Technology Vol., Issue 3, March 13 Fig.4 Shows shae Circular aperture of unit raius (first) an various Shrunk apertures [1]. P.S. Consiine, J.O.S.A 58, 11 (1968). []. R. W. Smith, Opt.Commu.4, 157 (1971). [3]. R. W. Smith, Opt.Commu.6, 8 (197). [4]. R. W. Smith, Opt.Commu.9, 61 (1973). [5]. F.G. Leaver an R. W. Smith, Optics 39,158 (1973). [6]. T.Araki an T. Asakura, Opt.Commun., 3, 373 (1977). [7]. T. Asakura an T.Araki, Optik, 46, 365 (1976). REFERENCES [8]. K. P. Rao, P.K. Monal an T.Sheshagiri Rao, Optik, 5, 73 (1977). [9]. J. P. Mills an B.J. Thampson, J. O. S. A(A), 3, 74 (1985). [1]. J. Sethuraman an R.S. Sirohi. Opt. Commun. 8, 1, 11 (1979). [11]. Ramanathan S, Rao K P, Monal & Krishnarayalu G, Inian J Pure & Appl Phys, 17, 91 (1971). [1]. Thompson B J & Krisl M E, Photogr Sci & Engg. 1, 19 (1977) [13]. Rao K P, Monal P K & Seshagiri Rao, Optik, 5, 73 (1978). [14]. Menez J A, Monal P K, Murthy P V V S & Siva Prasa P, Optica Pure Y Applicaa, 4, 57 (1991). [15]. Surener K, Gou S L & Monal P K, J Opt, 1 (3), 75 (199). [16]. Ramanathan S, Rao, K P Monal P K & Krishnarayalu G, Inian J Pure & Appl Phys, 19, 83 (1981). [17]. Das P K, Optical Signal Processing, (Springer-Verlag, Berlin, Heielberg, Germany), 59 (1991). [18]. [18] R. N. Bracewell, The Fourier Transform an its Applications, Revise n e.,(mcgraw-hill Book Co., NewWork,(1986). f Copyright to IJIRSET 651