Strehl Ratio of Point Spread Function with First-Order Parabolic Filter

Strehl Ratio of Point Spread Function with First-Order Parabolic Filter

P Thirupathi 1 , Sreehari Pagidipally 2

1 Department of Mathematics, University College of Engineering,Osmania University, Hyderabad, Andhra Pradesh, India.

2 Department of Mathematics, Osmania University, Hyderabad, Andhra Pradesh, India.

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 a first-order super-resolving parabolic filter. The results obtained have been discussed graphically.

Keywords Mathematical Optics, Parabolic Filters, Fourier Optics, Strehl Ratio .. etc.

1. 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 first-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.

2. 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 Lunenburgs 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 HOPKINS [5]. WILKINS [6], while solving the modified Lunenburg apodization problems discussed the Strehl ratio.BARAKAT and LEVIN [7] have studied the apodization problem of determining the diffraction pattern to have the largest possible Strehl ratio, for a rotationally symmetric optical system. BARAKAT and HOUSTON [8] computed Strehl ratio for an annular aperture possessing third- order and fifth- order spherical aberration. They have adopted the approach of MARECHAL [9] to minimize the mean square deviation of the wave front and hence maximize the Strehl ratio.

   DEVELIS [10], in his study of comparisons of methods of evaluation, discussed the Strehl ratio and its relation to Marechal tolerance. HOPKINS [11] 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 [12]. This work has been extended by ASAKURA and NAGAI [13] 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 [14] 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. KUSAKAWA and OKUDAIRA [15], in their study of Wiener apodization problems obtained pupil functions for different Strehl ratios. The relation between the minimum obtainable second order-moment and the prespecified Strehl ratio has been discussed by them. HAZRA [16] 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 finding the diffraction pattern has specified Sparrow limit of resolution and the maximum possible Strehl criterion has been

   solved by PENG and WILKINS [17], for both incoherent and coherent illumination, respectively. WILKINS [18], has solved the apodization problem for maximum Strehl ratio and specified Rayleigh limit of resolution. STAMNES [19], while re-examining the Lunenburg apodization problem in the frame-work of non- paraxial optics, concluded that a converging spherical wave with a uniform energy distribution as compared to a converging spherical wave with a uniform energy distribution over the aperture, always gives better results, as far as Strehl ratio is concerned. MAHAJAN [20] calculated the Strehl ratio, quite accurately from the phase aberration variance. KIBE and WILLIAMS [21] have studied Strehl ratio for a specified Rayleigh limit and for maximum central irradiance. McCUTCHENS theorem has been used by LOHMANN and OJEDA CASTANEDA [22], to derive the condition for axial symmetry and periodicity of Strehl ratio, which may serve as a focus criterion.

   Strehl ratio for the Straubel class of apodization filters has been studied by RAO, MONDAL and SESHAGIRI RAO [23], who concluded that the Strehl ratio is the encircled

   I 0, 0

   p

   SR (1)

   I A 0, 0

   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

   G p 0, 0

   SR 2 (2)

   GA 0,

   Where the symbol G and G stand for the point

   energy enclosed within a circle of infinitesimal radius. Strehl P A

   ratio for triangular and associated filters has been investigated by VISWANATHAM [24]. Several others like DEVARAYALU [25], SHANKARAIAH [26], have employed

   different apodisers and studied the effect on the Strehl ratio.URBANCZYK [27], has determined the Strehl ratio for the optical imaging systems changing the state of light polarization. HEROLOSKI [28] has derived closed form solutions for Strehl ratio of an untruncated, aberrated Gaussian beam system.

   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 [29]. RAMNATHAN [30] examined the effect of Kaiser Pupils on the Strehl ratio. MURTY [31], used co- sinusoidal filters and investigated the influence of apodization and defocusing, with both circular and annular apertures on Strehl ratio. SURENDAR [32] has evaluated he Strehl ratio for apodised optical systems, circular and annular, using Lanczos filters and determined that apodisations in combination with obscuration further lowers the Strehl ratio. KARUNASAGAR [33] 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 [34].

   spread function for the actual optical system used and the perfect system respectively, thus,

   1 2

   SR 4 f r rdr ..(3)

   0

   Where f(r)= ( r2 ) Thus finally,

   1

   2

   SR 4 r2 rdr (4)

   0

   =0.75

   =0.50

   =0.25

   =0

   0.7

   0.6

   0.5

   SR

   0.4

   0.3

   0.2

   0.1

   0

   0 0.1 0.2 0.3 0.4 0.5

   0.6 0.7 0.8 0.9 1

3. 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.

   Fig.1 Variotion of SR with for =0,0.25,0.50&0.75

4. RESULTS AND DISCUSSIONS:

In the figure 1, 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 figure that the various SR curves for all the values of are straight lines parallel to each other i.e. having the same slope with respect to the -axis. These curves can

therefore, be mathematically represented by the following straight line equation.

SR= m .(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

the same. 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.

ACKNOWLEDGEMENT

One of the authors Dr. P.Thirupathi thanks his employer Osmania University, Hyderabad where he is working as an Assistant Professor(c), Department of Mathematics, University College of Engineering, Hyderabad, Andhrapradesh, India.

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