Point Spread Function of Symmetrical Optical System Apodised with Gaussian Filter

Transcription

1 International Journal o Pure and Applied Physics. ISSN Volume 4, Number (8), pp Research India Publications Point Spread Function o Symmetrical Optical System Apodised with Gaussian Filter M. Kalpana Devi*, Ch, Srinivas**, T. Venkat Reddy*** * Department o Physics, KU College o Engineering and Technology, Kakatiya University, Warangal-569, T.S., INDIA. ** Department o Physics, College o Technology, Osmania University, Hyderabad- 57, T.S., INDIA. *** Department o Physics, Kakatiya University, Warangal-569, T.S., India. Abstract In this study, symmetrical optical system is considered. Point spread unction is important parameter in the perormance o optical system. Point spread unction in the case o optical system apodised with Gaussian ilter is derived. Analytical and numerical results are computed or various cases o apodised parameter and are presented graphically. Key words: Optical system, Point spread unction, Gaussians ilter, Deocusing parameter..introduction Optical system is equipped with lenses, mirrors, and prisms that constitute the optical part o instruments such as microscope and telescope. Optical imaging is a technique that depends on illumination o light in ultraviolet, visible, and inrared regions o the electromagnetic spectrum. Optical imaging plays important role as Non Destructive Evaluation (NDE) tool in the health care domain. In optical imaging systems, the Point Spread Function (PSF) describes the response o imaging system to a point source and it plays an important role in the resolution studies o an optical system. There are two important metrics or perormance evaluation o optical systems, one is PSF and other is Optical Transer Function (OTF). The PSF and OTF are mathematically interchangeable in the sense, one is Fourier transorm o other. The diraction theory o PSF was irst studied by Airy. Airy developed expression or PSF amplitude and intensity o a perect system, which is ree rom aberrations. An apodisation unction is used to purposely change the input intensity proile o an optical system. Overviews o PSF analysis is available in the review paper [ and reerences therein]. The Leaver and

2 3 M. Kalpana Devi, Ch, Srinivas, T. Venkat Reddy Smith [] have studied radially symmetric pupil unction having properties o producing a point spread unction which decreases monotonically with the increase o radius. The PSF obtained with the above pupil is rom the secondary maxima and minima but is wider than that o a clear aperture. Magiera [] has studied the ilters which minimize the second moment o the PSF, Magiera, and Pluta [3] have used second moment as a measure o the energy scatter and evaluated the eects o optimal appraising ilters on the distribution o energy in the PSF produced by axial object point. Bareket [4] considered the second moment o the PSF to study the image quality criterion. Tschunko [5] has obtained an expression or the PSF by considering geometrical relations o the incoming and diracted wave ronts and established the diraction integral. Rayalu and Mondal [6] have evaluated the deocusing eect on the PSF o an optical system apodised with parabolic ilters and shaded apertures. Tschunko [7] has studied the PSF o optical system with annular apertures. Venkat et.al. [8] have given a general expression or the PSF with an object having a non-central phase annulus coating. Murthy and Mondal [9] have obtained the PSF o optical system in the sinusoidal amplitude ilters. Linsen and Yaghanng [] have studied the restoration o blurred images by convolution o arbitrary PSF. Chung et. al [] have investigated the inluence o apodisation on aberrated point spread unction. Nakomura and Toyoda [] have proposed an apodised annular pupil to suppress the side lobes in PSF. To the best o our knowledge, no one has analyzed the PSF o symmetrical optical system apodised with Gaussian ilter. In this paper, the same is analyzed. The rest o the paper is organized as ollows. In the section II, mathematical expression or PSF o optical system apodised with Gaussian ilter is derived. In the section III, numerical results are presented. Finally, conclusion is given in section IV.. POINT SPREAD FUNCTION (PSF) WITH GAUSSIAN AMPLITUDE FILTER ζ Q(u,v,w) x P(ξ,η,ζ) q R η a C F ξ W Figure. I. Diraction at an apodised circular aperture. The exact nature o the modiications produced in the amplitude spectrum depends on the type o ilter used. The apodisation ilter considered here is Gaussian type. The schematic representation or diraction at an apodised circular aperture o an optical system is shown in Fig. I. Consider a spherical wave ront with surace area 'W ' with

3 Point Spread Function o Symmetrical Optical System Apodised with Gaussian Filter 33 the radius emerging rom a circular aperture and converging towards to the axial ocal point F. Let G(P) be the optical disturbance at a point P (,, ) in the vicinity o the ocal plane F. Position vector o P is R. Magnitude o R is assumed to be small when compared to that o the radius o the wave ront CF. Our aim is to study the diracted image at the point P. Let x be the distance o the point P rom an arbitrary point Q( u, v, w) on the wave ront at a moment when it is incident on the aperture. Let A be the amplitude o the incident wave ront at a point Q. It is assumed that the incident light is quasi monochromatic light and the wavelength ( ) is very small compared to the radius o the aperture i.e., a. A general expression or complex amplitude at the point P is [3] i A dw G( P) exp( ik ) ( r)exp( ikx). () x W In the above expression k stands or propagation constant, and (r) is the pupil unction, where r is radial co-ordinate o P. The usual inclination actor has been omitted here, since only small angles are involved. I q denotes a unit vector in the direction QF, then rom the igure, we have The surace element dw can be expressed as x q. R () dw d, (3) where dw is the surace element which subtends solid angle d at the point F, then we have dw a rdrd d. (4) Here x can be replaced by as it does not amount any considerable error in Eq. (). Thus Eq. () is reduced to ia exp( ik ) exp( ikx) ( r) dw. Further using Eqs, () and (3), above equation can be written as W ia G ( P) ( r)exp ik q. Rd. (5) The integration extends over the solid angle subtended by the aperture at point P. For a clear aperture ( r), and the Eq. (5) reduces to the Debye integral o an Airy case.

4 34 M. Kalpana Devi, Ch, Srinivas, T. Venkat Reddy ia exp ikqr d. In the aperture plane, let (r, ) and (, ) be the polar coordinates o P and Q, respectively, then we have u a r sin, v a r cos. (7) sin, cos. (8) Since Q is on the spherical wave ront W, we have w a r, a r =, a r =.... (9) Neglecting the terms o higher powers o r, the above equation becomes Then, we have a r. () u v w q. R. From Eqs. (7), (8), and (), we have sin. ar sin cos. ar cos a r R q. Let us now introduce two dimensionless variables y and z to speciy the position o the P. That is and. (6) () a y, () a z, (3) In Eqs. () and (3), y is deocusing parameter. I y, it represents the Gaussian ocal plane. I z / y, the point P lies in the direct light beam, and i z / y, it lies

5 Point Spread Function o Symmetrical Optical System Apodised with Gaussian Filter 35 in the geometrical shadow. From Eqs, () and (3), and using k, we can have Using Eqs. (4) and (4) in Eq. (5), we get ia G( P) k( q. R) zr cos( ) y yr. (4) a ( r)exp izr cos( ) i a y iyr a d where J ( zr ) is the Bessel unction o irst kind and zero order. From the above equation, we get (5) A a iyr i exp i y ( r)exp J ( zr) a (6) A Putting exp i, y a and ater simpliication, we get iyr i a ( r)exp J( zr) (7) The term i a outside the sign o integration does not have any eect on the diraction pattern, hence it can be omitted. Now the diracted light amplitude at a point in the Gaussian ocal plane and away rom the ocusing point F is given by iyr ( r)exp J ( zr) (8) Point spread unction o the optical system can be evaluated by knowing the explicit expression o the pupil unction (r) and then taking the squared modulus o the Eq. (8) at the ocused plane o observation corresponding to y (which represents Gaussian ocal plane). In this case, we have G, Z) ( r) J ( zr) (9) ( In the case, r is normalization distance o Gaussian amplitude ilter, r ( r) e. ()

6 36 M. Kalpana Devi, Ch, Srinivas, T. Venkat Reddy In the Eq. (), is the apodisation parameter, which determines the degree o uniorm transmission with in the apodised region. Substituting Eq. () in Eq. (9), we get The intensity B(,Z) o point spread unction is given by G(, z) e J ( zr) () r B(, z) G(, z). () 3. NUMERICAL RESULTS In this section, the intensity o point spread unction is computed or a Gaussian amplitude ilter using Eqs. () and () against z in the range to ±5 in the stepping.5, and the anodization parameter is taken to be arbitrarily in the range. to. in the step o., and 3.86,,, 3. The results are depicted in Fig. II and Fig. III. From the Fig.II, we can iner that peak o the central maxima o diraction pattern increases while irst minima position decreases in the cases o =. to.4. There is no second order maxima position in this case, that is negative amplitude is zero. For the cases =.5 and.6, peak o the central maxima increases while irst order minima position decreases as in the earlier cases. But the irst order negative peak values increase. For the cases =.7 to., and other higher values, central peak values increase. and irst and second order negative peak values also increase, while minima positions decrease. From these results, it is clear that aperture shading is aecting central maxima. As increases central amplitude and intensity increase and also the radius o the irst dark rings decreases. It can be stated that, the use o ully apodised circular aperture reduces the eects o the anodization. Thus, the aperture shading eiciently sharpens the central maxima as increases. From the Fig. III., it is clearly evident that the pupil unction (r) smoothly decreases as r increases and approaches zero in the cases o. to.4, For other higher values o rom.5 to., the pupil unction (r) decreases r increases smoothly but does not touch zero. In the cases =3.86,,, 3, pupil unction (r) decreases as r increases gradually and latter three curves coincide with the line (r) =. The value o (r) increases as increases. Thus, the Gaussian pupil unction typically suppresses the side lobes but broaden the main lobe o the point spread unction (PSF). The Gaussian amplitude transmittance decreases exponentially or. to.4, and then ater decreases monotonically rom the center towards the edges o Gaussian ilters

7 Point Spread Function o Symmetrical Optical System Apodised with Gaussian Filter G(,z).. = z values Figure.II Variation o amplitude o PSF or various values o [. to., 3.867, =. =.3 =. =.5 =.4 =.6 =3.867 =.9 =. =.7 = Figure. III Pupil Transimission Curves or various values o 4. CONCLUSION First, complex amplitude in the case o Gaussian ilter is derived thereby the point spread unction is derived. The numerical values are computed or the various values o apodisation parameter. This kind o analysis is useul in the design o optical systems in domains o health care, Astronomy, and Communication Engineering, REFERENCES [] Leaver and Smith, 975, Radially symmetric pupil unction having properties o producing a point spread unction which decreases monotonically with the increase o radius, Opt. Commun, 5, 374. [] Magiera, 98, The ilters which minimize the second moment o the PSF,

8 38 M. Kalpana Devi, Ch, Srinivas, T. Venkat Reddy Optik., 56. [3] Magiera and Pluta, 98, Energy scatter and evaluated the eects o optimal appraising ilters on the distribution o energy in the Produced by an actual object point, Optik, 56, 43 [4] Bareket, 979, The second moment o the PSF has studied the image quality criterion. Journal Opt. Soc., AM., 69, 3. [5] Tschunko, 983, Geometrical relations o the incoming and diracted wave ronts and established the diraction integral, Appl. Opt.,, 33. [6] Rayalu and Mondal, 979, The deocusing eect on the PSF o an optical system apodised with parabolic ilters and shaded apertures. [7] Tschunko, 983, PSF S o optical system with annular apertures., Appl. Opt.,, 33,. [8] Venkat Reddy, Bhupal Reddy, Mondal and Clavo, 989, A general expression o the PSF with an objective having a noncentral phase annulus coating, Optica Pura Appl.,, 83. [9] Murthy and Mondal, 989, PSF o optical system in the sinusoidal amplitude ilters, Acts Ciencia Indica, (). [] Linsen and Yaghanng, 989, Restoration o blurred images by convolution o arbitrary PSF, China, Journal o Lasers, 6, 54. [] Chung, Sim, Moen, 99, Inluence o apodisation on aberrated point spread unction Price. SPIE,Intha., Soc. Opt. Eng (USA), 39, 638. [] Nakomura and Toyoda, 99, An apodised annular pupil to suppress the side lobes in point spread unction, Apply. Opt.3,34. [3] Born M., Wol E.,98, Principles o Optics. Pergamon Press.