8/10/2019 Point Spread Functions Of Aberrated Optical Systems With Annular Apertures

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8/10/2019 Point Spread Functions Of Aberrated Optical Systems With Annular Apertures

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International Journal of Scientific Research and Engineering Studies (IJSRES)Volume 1 Issue 4, October 2014

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III. RESULTS AND DISCUSSION

Expressions (2) (3)and (5) have been used to compute the point spread functions of the apodised apertures for variousvalues of central obscuration parameter . Figure 1 depicts theintensity distribution curves for various degrees of apodisation

parameter for circular aperture (=0) for the first orderHanning amplitude filter ( =1). Form the profile of theintensity distribution curves it is evident that for = 0.50, i.e.,for partial apodisation there appears to be a total eliminationof the optical side-lobes thus shaping the point spread functionto the desired profile. Table-1 provides the computed values ofthe radius of the first, second and third dark rings for circularaperture in the case of first order filter for different values of.

-10 -5 0 5 10

0.0

0.2

0.4

0.6

0.8

1.0

Figure 1: Intensity Distribution Curves For = 1

FirstMinimaPosition

SecondMinimaPosition

ThirdMinimaPosition

0.000.10

2.66652.6674

6.12056.1226

9.59499.5976

0.200.300.400.500.60

0.700.800.90

2.67052.67722.69342.77072.5314

2.61762.63692.6477

6.12986.14556.18356.36235.8117

6.00656.05146.0765

9.60929.63499.69699.98089.1137

9.41029.48219.5227

1.00 2.6565 6.0971 9.5565Table 1

FirstMinimaPosition

SecondMinimaPosition

ThirdMinimaPosition

0.000.10

3.83173,8581

7.01547.0300

10.172810.1833

0.200.300.400.500.600.700.800.90

3.94634.13104.51245.33046.19773.34431.54842.1656

7.07977.19287.48368.53789.60787.97456.71726.8507

10.217910.298010.521811.707312.880111.30379.9824

10.07351.00 2.4560 6.9826 10.1631

Table 2 lists the tabulated values of the radius of the first,second and third dark rings for annular aperture ( = 0.8) inthe case of first order filter for varied values of . Figure 2illustrated the variation in the size of the central maxima as the is varied from 0 to 1. There is a clear reduction in the widthof the central lobe for higher values of apodisation. Thisfeature of contraction in the diffraction pattern is also reflectedwhen is varied from 0 to 0.8.

0.0 0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

6

7

v/s RADIUS OF FIRST DARK RING

R A D I U S

Figure 2: Variation Of The Radius Of First Dark RingFigure 3 shows the reduction of the optical side- lobes for

= 0.50 when = 1 for various values of central obscuration parameter. The PSF thus produced by the optical system for = 0.4 exhibits super-resolving nature of the impulse responseof the system. Figure 4. Show the variations in the width ofthe central lobe for second order filter. The width of thecentral lobe decreases for higher values of apodisation.However, there is an enlargement in the central maximum for

apodisation values ranging from = 0.6 to = 0.8. In thisrange of apodisation there is a degradation of the resolution ofthe optical system.

8/10/2019 Point Spread Functions Of Aberrated Optical Systems With Annular Apertures

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International Journal of Scientific Research and Engineering Studies (IJSRES)Volume 1 Issue 4, October 2014

ISSN: 2349-8862

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-10 -5 0 5 10

0.000

0.005

0.010

0.015

0.020

0.025

0.030

B ( Z )

Z

Figure 3: Intensity Distribution Curves For = 0.5.

-10 -5 0 5 10

0.00

0.02

0.04

0.06

0.08

0.10

- - - - - - - - - - - - - - - - - - - -

B ( Z ) - - - - - - - - - - - - - - - - -

>

=0.0

Y1=2

Y=2

Figure 4: Intensity Distribution Curves For Annular

A perture With =0.1

-10 -5 0 5 10

0.00

0.02

0.04

0.06

0.08

0.10

- - - - - - - - - - - - - B

( Z ) - - - - - - - - - - - - - - - - - - >

Y1=2

Y=2

Figure 5: Intensity Distribution Curves For Annular Aperture

With =0.3

Figs. 4 and 5 depict the intensity distribution profiles forannular apertures for central obscuration parameter for =0.1and =0.3 when the optical system is under extreme influenceof primary spherical aberration and defect of focus. Thechosen amplitude filter is effective in suppressing the opticalside-lobes so that the impulse response (PSF) of the opticalsystem results in a well tailored distribution of the intensity.

IV. CONCLUSIONS

The process of apodising the optical system withgeneralized Hanning amplitude filter, suppresses fully or

partially the optical side- lobes. For = 0.5 these side -lobes aretotally eliminated for all the four orders of the optical filterconsidered. Central obstruction reduces the radius of the Airydisc below the classic limit of 3.8317. The radius of the firstdark ring for clear annular apertures gets reduced by about onethird for large obstructions ( = 0.8). Energy is transferredfrom the Airy disc to surrounding rings, resulting in theincreases in the strength of the optical side-lobes. Theemployed Hanning amplitude filters are effective in shapingthe point spread function of the given optical systems.

REFERENCES

[1] Jacquint P and Roizen-Dossier B 1964 Progress in Opticsed. E. Wolf (North-Holland Publishing Company,Amsterdam) Vol III, Chapter 2.

[2] T.R.Corle and G.S.Kino, Confocal scanning OpticalMicroscopy and Related Imaging Systems, AcademicPress, San Diego, 1996.

[3] A.Boi ivin, Theorie et calcul des Figures de Diffraction deRevolution, Gauthier villars, Paris, 1964.

[4] D. Karuna Sagar, Ph.D. Thesis, Osmania University,Hyderabad, India, 2003.

[5] Abdul Muhith and D. Karuna Sagar, Defocused pointspread functions of two annulus apertures apodised withHanning amplitude filters, J. P. A. P. Vol.21, No.3, July-Sept., 2009, pp 493-495.