Simple expressions for performance parameters of complex filters, with applications to super-Gaussian phase filters

Abstract

To study the three-dimensional (3-D) behavior produced by complex filters, we have extended the expressions for the axial and the transverse gain to the case in which the best image plane is not near the paraxial focus. Super-Gaussian phase filters are proposed to control the 3-D image response of an optical system. Super-Gaussian phase filters depend on several parameters that modify the shape of the phase filter, producing tunable control of the 3-D response of the optical system. The filters are capable of producing a wide range of optical effects: transverse superresolution with high depth of focus, 3-D superresolution, and transverse apodization with different axial responses. © 2004 Optical Society of America OCIS codes: 110.0110, 220.0220, 350.5730, 220.1230. The three-dimensional (3-D) response of an optical system must be very different depending on its application. A common goal in many optical systems, for instance, in optical storage, 1 is to improve transverse resolution. Nevertheless, in most cases, a specific axial behavior is needed. For instance, in photolithography a high depth of focus (DOF) is also required. To achieve transverse superresolution, both in photolithography and in optical storage, very short wavelengths and high numerical apertures (NAs) are used. This has two effects: on one side the optical system must be well corrected (which is diff icult), and on the other high NA reduces the DOF (DOF is related to l͞NA 2 ). For this reason different superresolution techniques 2,3 have been proposed as ways to relax the optical system requirements or simply to improve the resolution of an optical system by providing enough DOF. In other systems, such as scanning microscopy, transverse and axial superresolution is desirable to obtain 3-D images. 4,5 There are different methods that have been used to modify the impulse response of an optical system. Annular transmission pupils have been widely used to improve transverse and (or) axial resolution. 4 -6 Continuously varying amplitude f ilters have also been investigated to produce transverse superresolution and (or) high DOF. 3,6,7 However, in recent years, phase-only filters have been studied to check whether they can improve the performance of amplitude-only f ilters. 1In one approach, binary phase filters are used 1 to obtain elements that are easy to produce. In other works continuous phase filters have been investigated, 8,9 but in some cases although transverse superresolution is obtained it is produced by very complex phase masks. 9In a recent study 8 a simple continuous phase filter was used to obtain transverse superresolution.In this Letter we propose super-Gaussian phase filters that have a shape that depends on four parameters. These parameters give us enough degrees of freedom to modify the whole 3-D image response in a controlled way. The f ilters are capable of producing a wide range of optical effects: they can produce transverse superresolution that increases the DOF, 3-D superresolution, and even transverse apodization ...

Programmable apodizer to compensate chromatic aberration effects using a liquid crystal spatial light modulator

“…The analysis of Eqs. (4)(5)(6) shows that when polychromatic illumination is used the PSF can change with the wavelength due to three factors. First, the complex amplitude function in the exit pupil f λ (r) may vary for the different wavelengths because the support media of the applied filter may exhibit a wavelength dependent transmission.…”

“…Sheppard and Hegedus introduced the transverse and axial gains to study the influence of transmission filters in planes near the paraxial focus [2]. These parameters have been generalized for phase filters when the focalization is produced near the paraxial focus by de Juana et al [4] and by Ledesma et al for phase filters in any defocused plane [5]. We have also analysed some filters that give an equal axial response while providing a transverse hyperresolving or a transverse apodizing response [6].…”

Phase Only Pupil Filter Design Using Zernike Polynomials

“…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][11][12], continuously varying functions, such as Bessel functions [13,14] and Gaussian functions [15], and special forms, such as a spoke wheel filter [16].…”

Improved expressions for performance parameters for complex filters

“…De Juana et al [2] extended the gain parameters to the case of general-phase filters, for the case when the intensity maximum is shifted only a small distance from the geometrical focus. Ledesma et al [3] introduced an alternative approach for any complex filter, in which the plane of best focus is calculated first, and generalized gain parameters in the surroundings of the shifted focus are then calculated. This approach is much more flexible and is preferable for many phase filters, as the intensity peaks on the axis can be situated far from the geometrical focal plane, with the filter acting like a zone plate.…”

“…These expressions have been validated for the particular case of a phase mask that produces a small axial shift of the focal intensity. The method fails completely, however, if an inflection point in axial intensity occurs between the intensity peak and the geometrical focus ͉͑u 0 ͉ Ͼ 5͒, so that if the intensity peak is distant from the geometrical focal plane, the approach of [3] is necessary.…”