Phase Only Pupil Filter Design Using Zernike Polynomials

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    A pupil filter is a useful technique for modifying the light intensity distribution near the focus of an optical system to realize depth of field (DOF) extension and superresolution. In this paper, we proposed a new design of the phase only pupil filter by using Zernike polynomials. The effect of design parameters of the new filters on DOF extension and superresolution are discussed, such as defocus Strehl ratio (S.R.), superresolution factor (G) and relative first side lobe intensity (M). In comparison with the other two types of pupil filters, the proposed filter presents its advantages on controlling both the axial and radial light intensity distribution. Finally, defocused imaging simulations are carried out to further demonstrate the effectiveness and superiority of the proposed pupil filter on DOF extension and superresolution in an optical imaging system.


    Wavefront coding , Non-integer phase mask , Depth of field , Imaging system


    In numerous applications it is necessary to extend the depth of field (DOF) and improve the resolution for an optical system, such as microlithography [1], optical data storage [2], and microscopy imaging [3, 4]. One of the most popular techniques to achieve DOF extension and superresolution includes introducing a pupil filter at the pupil plane of an optical system to modify the radial and axial spot size of the point spread function (PSF).

    Since Toraldo made a seminal study on a radial-symmetric filter [5], a lot of work has been done on the design of a pupil filter. Although an amplitude only filter can provide effective superresolution and DOF extension, it still has two main difficulties. The first one is the reduction of image brightness and the other refers to the fabrication, especially for continuously varying amplitude functions [6, 7]. Therefore, in recent years phase only filter design has achieved a rapid development based on the parabolic approximation of the PSF for the general filters introduced by Sheppard [8, 9]. Typically, the design strategies of pupil filters include circular obstructions [10-12], continuously varying functions, such as Bessel functions [13, 14] and Gaussian functions [15], and special forms, such as a spoke wheel filter [16].

    In this paper, a phase only filter is proposed by using Zernike polynomials. Radial and axial spot sizes of an optical system with and without the designed filter are analyzed relatively based on scalar diffraction theory. Then, compared with the other two typical phase filters (three-zones and Gaussian functions), we show its advantages on DOF extension and superresolution. Finally, we further demonstrate the availability and superiority of our proposed phase only filter through defocus imaging simulation.


    According to the Born and Wolf’s theory [17], at the focal region, the amplitude distribution of a converging monochromatic spherical wavefront passing through the pupil plane can be written as:


    Where ρ is the normalized radial coordinate of the pupil plane, P represents the pupil function, J0 represents Bessel function of the first kind of order zero, u and v are the radial and axial dimensionless optical coordinates, given by:


    Where NA is the numerical aperture of the optical system, λ is the wavelength of the incident light, r and z are the usual radial and axial distances.

    Since most practical imaging systems use incoherent illumination, the light intensity distribution near the focal region can be calculated by its amplitude distribution, written as:


    Where * denotes the complex conjugate.

    After introducing the phase pupil filter into the