Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function

An analysis of the Strehl ratio and the optical transfer function as imaging quality parameters of optical ele­ ments with enhanced focal length is carried out by employing the Wigner distribution function. To this end, we use four different pupil functions: a full circular aperture, a hyper-Gaussian aperture, a quartic phase plate, and a logarithmic phase mask. A comparison is performed between the quality parameters and test images formed by these pupil functions at different defocus distances. © 2000 Optical Society of America [S0740-3232(00)00605-0] OCIS codes: 100.5070, 110.4850, 110.5100.


INTRODUCTION
In recent years several types o f optical elements that give rise to well-focused energy distributions along three dimensional paths have been extensively reported in the literature (see, e.g., Ref. 1).Annular-type apertures act ing as apodizers to improve the quality of the pointspread function (PSF) and to increase the focad depth have been studied.2-8However, the main drawback of all these methods arises from the fact that both the spa tial resolution and the optical power decrease at the im age plane.An alternative way to achieve an extended depth of field without using apodizers was reported by Hàusler9 for the case in which focusing can be varied through the im a g e -fo x in g process.In this way, the im age field is adequately scanned to produce a well-focused PSF at each image point.Another type o f optical system that allows the concentration of energy in a segment o f the optical axis is the so-called axilens; such lenses have an associated focal length that varies with the radial co ordinate, so their phase retardation functions differ from nential.10,11Therefore, if these phase masks are em ployed as pupil functions o f an imaging optical system, al though there is no decrease in the image intensity the re sulting PSF becomes relatively broad.In a previous paper27 we proposed a method for obtain ing phase retardation functions that give rise to an in crease in the image focal depth.To this end, the W D F of a certain aperture with sm all depth of focus in the image space is sheared in the phase-space domain to originate a new W D F from which its related phase pupil gives rise to a more uniform on-axis image irradiance.In this way, a new phase pupil function with a good performance with respect to defocus, is obtained.A lens axicon with a simi lar phase function was also proposed by Jaroszewicz and Morales,28 with use of geometrical optics.

To
In the present paper we first briefly describe the phasespace formalism and its relationships to different imagequality parameters.In particular, an expression of the SR as a function o f the W D F of a bidimensional radially symmetric aperture is obtained for any out-of-focus plane.Then we extend the approach to analyze the relationship between the OTF of this kind of pupil function and the re lated W D F. W e apply this analysis to study the behavior o f different phase and amplitude pupil functions7,11 at various defocus distances.-------, fJ L -------- -------v , ----------a * \f f \f f

WIGNER DISTRIBUTION FUNCTION: DEFINITIONS AND BASIC RELATIONSHIPS
From Eq. ( 7 Hopkins16,17 has shown that it is pos sible to extend Marechal's treatment by employing OTF 0740-3232/2000/050867-07$15.00 theory to give a tolerance criterion.This method is quite suitable as a merit function in automatic optical design.The relationships between these image quality criteria and the phasedistribution function18,19 (W DF) and the ambiguity function,20 (AF) were employed in several studies to ana lyze the performance of an optical system with respect to :26

For
dvdfiWt X -\{f + z)v, y -\ ( f respectively.From Eq. (1) it can be deduced that this for m alism emphasizes equally the role o f the spatial and the spatial-frequency coordinates.This feature makes these distributions especially suitable for describing the behav ior of optical imaging systems.Am ong several properties o f these distributions, those that are relevant for analyz ing image-quality parameters are the following: the constant incident amplitude, f is the focal length, and i(£ , 77) is the com plex^m plitude transm it tance associated with the exit pupil.By taking into ac-

xFigFig. 3 .
Fig. 1.Normalized intensity of the hyper-Gaussian pupil aper ture, the logarithmic phase mask, the circular pupil function, and the quartic pupil function for extended depth of focus, Az =