Annalen der Physik. 7. Folge, Band 27, Heft 1,1971, S. 12-16 J. A. Barth, Leipzig Effect of Defocusing on the Rectangular Wave Response of an Annular Aperture By J.S.MAGGOand K. SIXGII With 8 Figures Abstract Modulation of a general rectangular wave object by a defocused annular aperture illuminated by incoherent light has been studied. Results for the modulation transfer function and image irradiance distribution have been plotted and discussed. Similarity between the system sine wave response and rectangular wave response has been shown for a particular object function. Introduction The performance of a n optical system can be studied with the help of FOCEIER methods [l I which enable us t o measure a very important image assessment parameter namely the optical transfer function. The importance of the system performance analysis in terms of its sine wave response is well known. The spatial frequency response of clear and annular aperture in the absence or presence of various aberrations using sine wave targets has been reported by many workers [2-7J. Due t o the inherent difficulties in manufacturing a sinewave grating with a uniform contrast all over, much interest is being shown in using rectangular and triangular wave objects. Investigations using these types of objects have been reported by SISGHand KAVATIIERAR, [8], SINGHand CIIOPRA [Y] and KATTIe t al. [lo, 111. The purpose of the present paper is t o present the results of investigations on the rectangular wave responsc of a dcfocused annular aperture. An annular aperture is of fundamental importance when reflecting components are used in the image forming system. Importancc of such a n apcrturc has been considerably cmphasised by SINXI and DHILLON[12] and need not be repeated. Theory The irradiance distribution in the images of a rectangular wave object formed by a n optical system is given as 181, 13 J. N. MAGGOand K. SINCH:Effect of Defocusing on the Rectangular Wave 04 08 12 16 2.0 E"ig.1-4. Rectangular wave response of a n annular aperture for Fig.1. E = 0.4, W,,, = 0.251 Fig.2. E = 0.4, W,,, = 0.61 where a and b are the mean intensity and the modulation respectively and p is the period. For n = 1, the quantity T ( n w )reduces t o the sine wave response of the system. The maximum frequency that the system passes is 2.0 so that nw 5 2.0 represents the allowable harmonic components of the rectangular wave. The number of allowable harmonics goes on decreasing with increasing spatial frequency w . I n order t o calculate the image irradiance distribution, the factor l ' ( o )or the transfer function of the system should be explicitly known. STEEL[13] and O'NEILL [14] have independently calculated the transfer function of a n annular [15] has made use of sampling theorem techniques t o get aperture. BARAKAT the transfer function of a n annular aperture as where on are positive zeroes of J, and t(v,/2) is the point spread function evaluated a t v J 2 as where W ( r )is the aberration function, which can be expanded as 14 Fig.3. Annalen der Physik E = 0.4, W,,,= 0.751 Fig.4. * 7. Folge E = * Band 27, Heft 1 * 1971 0.6, W,,,= 0.261 Here W,, is the coefficient of defocusing, W,, the coefficient of third order spherical aberration and so on. I n order to evaluate the point spread function, 64 - point GAUSS - quadrature is used. The computations were done on an ICL 1909 computer. The interval ( E , 1) is changed to the standard GAUSS interval (- 1, 1) by a linear transformation + where u is one of the GAUSSpoints. The value of the point spread function is then automatically substituted into the series (4). The transfer function thus evaluated is further used by the computer to evaluate the image irradiance distribution. Assuming the object contrast to be unity, the image contrast can then be found as Results and Discussions The contrast is computed by making use of equations (1)through (6). Different values of IX (a = 0.1, 0.25, 0.5, 0.75) have been used for two values of obscuration ratio E = 0.4 and 0.6. The amounts of defocusing considered are 0.252, 0.52 and 0.752 respectively. Figs. (1-6) represent the results of investi- J.hT.MAGGo and K.SINCH:Effect of Defocusing on the Rectangular Wave \\ 1.5 a-a 7 wu Pig.l5 and 6. Rectangular wave response of an annular aperture for Fig.6. F = 0.6, W,,, = 0.766 Fig.6. E = 0.6, WO2,= 0.51 gations. It is seen from the figures that the contrast falls rapidly for low frequencies for increasing amounts of defocusing. A slight improvement in contrast is obtained in the intermediate frequency region for oc = 0.10. The curves and HOUSTON marked S depict the sine wave response as calculated by BARAKAT [5]. The contrast in this case falls rapidly for low frequencies and becomes constant in the medium frequency range before gradually falling to zero in the high frequency region. KO spurious resolution has been observed in the cases considered but it is clear from fig. 6 ( W,,, = 0.752) that if we increase the amount of defocusing spurious resolution will step in. The irradiance distribution in the images has been calculated by making use of equation (1)for different values of wz' and a t three spatial frequencies w = 0.1, 1.0 and 1.5. Parameters a and b are both equal t o 0.5. Fig. 7-8 shows the plot of irradiance. It is observed from the figures that the diffraction figures get broadened for all values of o( for increasing amounts of defocusing. The broadening also increases as the obscuration ratio increases. Acknowledgements The authors wish t o thank Prof. P.K.KATTIfor his interest in the work. 16 Annalen der Physik * 7. Folge * Band 27, Heft 1 * 1971 Fig. 7 and 8. Distribution of irradiance in the images of a rectangular wave object Fig.8. E = 0.4, W,,, = 0.51 Fig.7. E = 0.4, W,,, = 0.261 References [l] ROHLER,R., Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft mbH, Stuttgart, 1967). [2] HoPKINs,H.H.,Proc. Roy. SOC.A 221 (1955) 91. 131 BARAKAT, R., and D.LEv, J. opt. SOC.Amer. 63 (1963) 324. [4] BARAKAT, R., and A. HOUSTON, J. opt. SOC.Amer. 63 (1963) 1371. [5] BARAKAT,R., and A.HOUSTON, J. opt. SOC.Amer. 56 (1965) 538. [6] LEVI,L., and R.H.AUSTING,Appl. Optics 7 (1968) 967. 171 LENSIKII,A.V.,Sov. J. Opt. Tech., 35 (1968) 230. [8] SINGH,K.,and A.K.KAVATHEKAR, J. opt. SOC.Amer. 69 (1969) 936. [9] SINGH,K.,and K.N.CHOPRA, Appl. Optics 8 (1969) 1695. -101 KATTI,P.K., K.SINGHand A.K.KAVATHEKAR, Opt. Acta 16 (1969) 629. 313 KATTI,P.K.,J.N.MAaao and K.Singh, Atti Fond. Ronchi 26 (1970). -121 SINGH,K.,and H.S.DHILLON,J. opt. SOC.Amer. 59 (1969) 395. [13] STEEL,W.H., Rev. Opt. 32, 4, 143 (1953) 269. r14] O’NEILL, E.L., J. opt. SOC.Amer. 46 (1956) 285. [I51 BARAKAT,R., J. opt. SOC.Amer. 64 (1964) 920. Ne wDelhi/India, Indian Institute of Technology Departments of Physics. Bei der Redaktion eingegangen am 10. August 1970. Anschr. d. Verf.: Dr. J.N.MAGGoand Dr. KSINQH Department of Physics, Applied Optics Section, Indian Institute of Technology, Hauz Khas New Delhi 29,Hndia

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