Strehl Ratio with Higher-Order Parabolic Filter

International
OPEN ACCESS Journal
Of Modern Engineering Research (IJMER)
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 80|
Strehl Ratio with Higher-Order Parabolic Filter
P Thirupathi1, T Ramakrishna Goud2
1Department of Mathematics, University College of Engineering, Osmania University, Hyderabad, Telangana,
India
2Department of Mathematics, University College of Science Saifabad, Osmania University, Hyderabad,
Telangana, India
I. Introduction
It is well-known that the image of a point object obtained even with a diffraction limited system is not a
point. There is a spread of light flux over a considerable region of space in the focus of the image plane, the
actual nature of the spread, known as the Point Spread Function is controlled by the size and shape of the
aperture and the type of the non-uniformity of transmission. The importance of this was first realized by
LOMMEL [1] and he developed the theory of the distribution of light at and near the focus of an optical
system with a circular aperture. In the present paper, we shall present the results of our studies on one of the
most important image-quality assessment parameters, the Strehl ratio which is based on the point-spread
function of the optical system and is apodised with higher-order parabolic filters. Initially, the Strehl ratio was
introduced as “Definitionshelligkeit” by its originator Strehl himself. In its original nomenclature, the term
“definition” was used to mean “distinctness” of an outline or detail in the image.
II. Previous Studies on Strehl-Ratio
Strehl ratio[2,3] is an important quality assessment parameter for imaging systems and its
maximization by the use of amplitude filters has been attempted by several workers. BARAKAT [4], in his
study on solutions to Lunenburg’s apodization problems, investigated the Strehl ratio for both circular and slit
apertures. It is not a physically measurable quantity in the strict sense of the word but nevertheless is a common
measure of theoretical performance of the system. WILKINS [5], while solving the modified Lunenburg
apodization problems discussed the Strehl ratio. BARAKAT and HOUSTON [6] computed Strehl ratio for
an annular aperture possessing third-order and fifth- order spherical aberration. They have adopted the approach
of MARECHAL [7] to minimize the mean square deviation of the wave front and hence maximize the Strehl
ratio.
DEVELIS [8], in his study of comparisons of methods of evaluation, discussed the Strehl ratio and its
relation to Marechal tolerance. HOPKINS [9] stated that for highly corrected optical systems, that is those
substantially satisfying the Rayleigh quarter-wave criterion, the Strehl ratio may be used as diffraction based
criterion of image quality. Strehl ratio, for circular apertures with a ring- shaped  - phase change, has been
investigated by ASAKURA and MISHINA [10]. This work has been extended by ASAKURA and NAGAI
[11] to modify annular and annulus apertures. It has been found that the Strehl ratio is always reduced in
comparison with that of a clear aperture as long as the semi-transparent and phase annulus aperture is used.
KUSAKAWA [12] has studied the problem of finding the pupil function which minimizes the
dispersion factor (Excluded energy), subject to the condition that the Strehl ratio, must have a certain
prespecified value. The relation between the minimum obtainable second order-moment and the prespecified
Strehl ratio has been discussed by them. HAZRA [13] studied the problem of maximization of Strehl ratio for
the more general case of partially space-coherent illumination. Hazra restated the criterion of “maximization of
Strehl ratios” as the criterion of “maximization of effective central illumination within a circle of
infinitesimally small radius around the centre of the diffraction pattern”. The apodization problem of
Abstract: In all the branches of science, engineering and technology, it is known that the output due to
an input impulse function, spatial or temporal, is never an impulse. There is a spread of the input impulse
function in the output due to the noise introduced by the physical device. It was Strehl who first
introduced the important image-quality assessment parameter “Definitionshelligkeit” or simply known
as the Strehl Ratio (SR) after his name. In this paper, we have studied this parameter for an optical
system apodised with the higher-order super-resolving parabolic filters. The results obtained have been
discussed graphically.
Key-words: Mathematical Optics, Higher order Parabolic Filters, Fourier Optics, Strehl Ratio ….. etc.

finding the diffraction pattern has specified Sparrow limit of resolution and the maximum possible Strehl
criterion has been solved by PENG and WILKINS [14], for both incoherent and coherent illumination,
respectively. MAHAJAN [15] calculated the Strehl ratio, quite accurately from the phase aberration variance.
KIBE and WILLIAMS [16] have studied Strehl ratio for a specified Rayleigh limit and for maximum central
irradiance. McCUTCHEN’S theorem has been used by LOHMANN and OJEDA CASTANEDA [17], to derive
the condition for axial symmetry and periodicity of Strehl ratio, which may serve as a focus criterion.
Formulae for estimating the Strehl, coefficient in the presence of third and Fifth-order aberrations as
well as defocusing have been obtained by GRAMMATIN and OKISHEVA [18]. RAMNATHAN [19]
examined the effect of Kaiser Pupils on the Strehl ratio. MURTY [20], used co-sinusoidal filters and
investigated the influence of apodization and defocusing, with both circular and annular apertures on Strehl
ratio. SURENDAR [21] has evaluated the Strehl ratio for apodised optical systems, circular and annular,
using Lanczo’s filters and determined that apodisations in combination with obscuration further lowers the
Strehl ratio. KARUNASAGAR [22] has evaluated the Strehl ratio for both circular and annular apertures
apodised with generalized Hanning filters for the first, second, third and the fourth orders of the filter
considered. A good account and a comprehensive review on Strehl ratio can be found in the reference [23].
III. Definition of Strehl ratio
STREHL suggested the use of the relative intensity of the diffraction as a measure of the image
quality. The strehl ratio (SR) is defined as the ratio of the central intensity of the PSF of the system and that of
the uniform pupil function for diffraction limited system.
 
 
0,0
0,0
I
p
SR
I
A
 …………… (1)
Where the subscripts P and A referred to the parabolic and Airy pupils respectively. I
p
(0,0) represent the
intensity point spread function at centre (0,0) of the diffraction pattern due to the optical system used and
I
A
(0,0) represent the same for the diffraction-limited perfect system. According to the above expression for
SR can be written in terms of respective pupil function as follows. Therefore,
 
 
2
2
0,0
0,0
G
p
SR
G
A
 ……………(2)
Where the symbol P G and A G stand for the point spread function for the actual optical system used and the
perfect system respectively, thus,
 
2
1
0
SR 4 f r rdr
 
  
 
 ……………..(3)
Where f(r)=
N ( r ) 2   where N=2,3,4&5.
Thus finally,
SR=
2
1
0
2 ) ( 4 



  r rdr N   ……………(4)

Fig:1 Second order: Variation of SR with β for α=0, 0.25, 0.5& 0.75 Fig:2 Third order: Variation of SR with β for α=0, 0.25, 0.5& 0.75 Fig:3 Fourth order: Variation of SR with β for α=0, 0.25, 0.5& 0.75

Fig:4 Fifth order: Variation of SR with β for α=0, 0.25, 0.5& 0.75
IV. Results and Discussions
Fig 1 to 4 represents the SR curves for values of   0,0.25.0.5&0.75 with β=0 to 1 with
deference 0.1. It is observed from the figures that for all the values of , as the values of  are increased,
over all curves increases, maintaining of course, their super-resolving parabolic shape. The most important
feature to be observed in this figure is that the order increasing then all the curves starts from origin i.e. for
second order the curves start from deferent values and the fourth and fifth order the curves start from origin. If
the order increasing the  =0 curve coincide with the  -axis.
In the figure 1,,2,3&4 we have shown variation of Strehl ratio with various values of apodisations
parameter and for various values of the D.C.bias  =0,0.25,0.5,0.75 it is observed from the figures that the
various SR curves for all the values of  are parabola curves. These curves can therefore, be mathematically
represented by the following equation.
SR=
2   r ……….(5)
Where m is the slope of SR curve and  is its intercept on the SR-axis. The important point to be
mentioned here is that the effect of  on the SR values is quantitatively different. Quantitatively, however, the
SR values depend prominently on the  values. Because, higher is the value of  the Strehl ratio values are
quantitatively higher than those for lower values of . However, it must be pointed out that we can not increase
the value of  indefinitely in order to keep the over-all value of f(r)  1, in order to satisfy the fundamental
passivity condition of an optical system.
REFERENCES
[1]. LOMMEL, E., Abh.Bayer Akad., vol.15, Abth.2, 1885
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[3]. STREHL, K., (1902), Z., f Instrumkde, 22, 213
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[6]. BARAKAT, R. and HOUSTON, A., (1963), J. Opt. Soc. Am., 53, 1244.
[7]. MARECHAL, A., (1947), Rev, Opt., 26, 257
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[10]. ASAKURA, T., MISHINA, H., (1970), Jap. J. Appl. Phys., 9,195
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[16]. KIBE, J.N., WILKINS, J.E., Jr., (1984), J, Opt, Soc, Am, A., 1,337. [17]. LOHMANN, A.W., OJEDA-CASTNEDA, J, (1984), Opt, Acta, 31, 603 [18]. GRAMMATIN, A.P., OKISHEVA, E.V., (1986), Sov, J, Opt, Techno (USA), 53,460 [19]. RAMANATHAN, S., (1986), Ph.D, Thesis, p, 41. Osmania Univ, Hyderabad. [20]. MURTY, P.V.V.S., (1992), p.32, in Thesis entitled ” Studies on Diffracted Field and Imaging Characteristics of Optical Systems with Co sinusoidal Apodisations Filters”, presented to Osmania University, for Ph.D. [21]. SURENDAR, K., (1993), P, 69, Ph.D Thesis, “Studies on Diffracted Field and Image Characteristics of Optical Systems Apodised with Lanczo Filters” SURENDAR, K., GOUD, S, L., MONDAL, P.K., (1992), Acta ,Cienca Indica 18, P, 6134 [22]. KARUNASAGAR, D., (2003), p, 62, Ph.D Thesis: “Studies on the Performance of Optical Systems Apodised with Generalized Hanning amplitude Filters” [23]. RATHNAM, C., “Fourier Analytical Investigations on the performance of multiple-Annuli coded Aperture in multiplexed Tomography”, Ph.D thesis, Osmania University, Hyderabad, A.P, India, 2005. [24] P Thirupathi 1 , Sreehari Pagidipally 2 “Strehl Ratio of Point Spread Function with First-Order Parabolic Filter” International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 3 Issue 3, March – 2014.

Strehl Ratio with Higher-Order Parabolic Filter

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Strehl Ratio with Higher-Order Parabolic Filter