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Effect of Defocusing on the Rectangular Wave Response of an Annular Aperture.

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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
With 8 Figures
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.
The performance of a n optical system can be studied with the help of
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.
The irradiance distribution in the images of a rectangular wave object formed by a n optical system is given as 181,
J. N. MAGGOand K. SINCH:Effect of Defocusing on the Rectangular Wave
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
Annalen der Physik
0.4, W,,,= 0.751
7. Folge
E =
Band 27, Heft 1
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-
and K.SINCH:Effect of Defocusing on the Rectangular Wave
a-a 7
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
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.
The authors wish t o thank Prof. P.K.KATTIfor his interest in the work.
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
[l] ROHLER,R., Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft
mbH, Stuttgart, 1967).
[2] HoPKINs,H.H.,Proc. Roy. SOC.A 221 (1955) 91.
R., and D.LEv, J. opt. SOC.Amer. 63 (1963) 324.
R., and A. HOUSTON,
J. opt. SOC.Amer. 63 (1963) 1371.
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.
J. opt. SOC.Amer. 69 (1969) 936.
[9] SINGH,K.,and K.N.CHOPRA,
Appl. Optics 8 (1969) 1695.
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.
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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rectangular, effect, response, defocusing, wave, annular, aperture
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