Super-resolving Pupil Filters Research Articles

Programmable super-resolution pupil phase filter with a deformable mirror

Compared with binary diffractive super-resolving elements, programmable super-resolution pupil filters permit the analysis of various filter designs and allow the filters to be changed rapidly to modify the response of an optical system. In this Letter, a deformable mirror is employed as a programmable super-resolution pupil phase filter. Continuous phase-only filters based on the Zernike polynomial series are designed by the genetic algorithm and fitted through closed-loop adaptive optics with a piezoelectric deformable mirror. Experimental super-resolution results are in agreement with the theoretical predictions. This method has no polarization light requirement and is convenient for application.

Performance improvement of super-resolving pupil filters via combination with nonlinear saturable absorption films

With the continuous development of the field of information technology, there has been a demand for recording mark size of optical data storage, optical imaging resolving power, and characteristic linewidth of photolithography to reach nanoscale. However, it is very difficult to realize the goal due to the optical diffraction limit restrictions. Much interest has focused on the study of optical far-field super-resolution spot by using pupil filters. However, common concerns have continued to plague super-resolving pupil filters based on either scalar diffraction theory or vector diffraction theory. These concerns include the fact that the side lobe becomes non-negligible when the central lobe is squeezed to a certain extent. Moreover, it is difficult to reduce the super-resolving spot to nanoscale. In this work, we proposed a novel method to combine the super-resolving pupil filters with nonlinear saturable absorption thin films to reduce the central spot size to nanoscale, lower the intensity ratio of side lobe to central lobe, and elongate the depth of focus or tunable tolerance distance between the super-resolving spot and sample. The simulated results indicate that by using the three-zone annular binary phase filter as the super-resolving pupil filter and Sb2Te3 as the nonlinear saturable absorption thin films, the central spot size can be reduced to nanoscale, the side lobe intensity is squeezed to about 10% of the central lobe intensity, and the tunable tolerance distance between the super-resolving spot and the sample is about two times that of the depth of focus of the diffraction limited spot at the incident laser wavelength of 405nm and the numerical aperture of focusing lens of 0.95. The combination of the super-resolving pupil filters with the nonlinear saturable absorption thin films is very useful for nano-optical data storage, maskless nanolithography, and nano-optical imaging. It is also easy to use in actual applications because of the operation in the far-field range.

Quasi-achromatic superresolving phase only pupil filters

Reported in this paper is a new design of the quasi-achromatic phase-only superresolving pupil filters (QAPSFs) relatively independent of wavelength based on the zero order Bessel function of the first kind. The effect of design parameters of the new filters on the superresolving performance parameters such as the normalized spot size (G) and the Strehl ratio (S) is discussed and analyzed in detail. As an example, the performance parameters of QAPSFs made of optical quartz glass are simulated and compared with phase only pupil filters, which indicate that this kind of filters is less independent of wavelength in the range from 300nm to 589nm for superresolution, and a high Strehl ratio is also obtained. The equivalent performance parameters are derived to characterize the superresolution performance of the quasi-achromatic filter.

Limitations of superoscillation filters in microscopy applications

The idea of superresolving pupil filters comes from the concept of superoscillations that may occur in regions of a band-limited signal with small amplitude having oscillations faster than the fastest Fourier component of the signal. In optical microscopy, superresolution can be achieved by appropriate design of pupil functions where the angular aperture determines the ultimate focal spot smaller than the Abbe diffraction limit outside the evanescent field region. The angular aperture cannot be increased indefinitely and the huge sidelobes cannot be avoided that are present in superresolving filters. The limitations of using such kind of filters in microscopy applications are discussed through computational examples.

Non-iterative method for designing super-resolving pupil filters

We propose a method of designing pupil filters for transverse super-resolution without making use of recursive algorithms or the parabolic approximation for the point spread function (PSF). We represent the amplitude of the PSF as an expansion of orthogonal functions from the Fourier-Bessel transform of a Dini series. Their coefficients are related with desired features of the PSF, such as the transversal super-resolution gain and the intensity of the secondary maxima. We show the possibility to derive closed formulas to obtain large super-resolution gains with tolerable side-lobe intensities, at the expense of increasing the intensity of a chosen secondary lobe.

Modified Shack–Hartmann wavefront sensor using an array of superresolution pupil filters

A conventional Shack-Hartmann wavefront sensor works with an array of lenslets which produces an array of focal spots in the back focal plane of the lenses. The displacements of the focal spots from a reference position give a measure of the mean local wavefront slopes. To determine the positions of the focal spots, centroiding algorithms have to be used. In this work, the use of superresolution pupil filters to reduce the size of the focal spots is analyzed, as well as its effect on the variance of the centroid estimate, seeking for an enhancing of the sensor accuracy.

Convex objective function-based design method developed for minimizing side lobe

The existence of multiple local solutions makes it very difficult to search for filter parameters to achieve a desired side lobe level during the design of superresolution pupil filters. To deal with the difficult issue of side lobe control in the designing process, a convex objective function-based design method is developed through phase rotation and variable replacement to transform the complicated solving process with multiextreme subintervals into a simple optimization process with a convex interval. A group of constant annular complex superresolving filters are designed using the developed method. The comparison of the superresolving filters designed in this way with the well-known continuous phase filter and 3-zone multiphase diffractive superresolution filters proves the validity of the developed method.

Synthetic complex superresolving pupil filter based on double-beam phase modulation

To take advantage of phase-only filters to reduce reflected stray light, two phase plates are used in a synthetic complex superresolving pupil filter to get a small spot for laser direct writing. The amplitude modulation in a conventional complex superresolving pupil filter (CCSPF) is substituted by an equivalent phase modulation. The equivalence between the proposed filter and a CCSPF is established for a general design. The effectiveness of the proposed filter was proven through experiments to obtain a small main lobe width of 0.86 times the size of the airy spot.

Effect of fabrication errors on lateral super-resolution property of a three-zone pupil filter

Two models have been established using the basic definitions of super-resolution characteristic parameters for normalized spot size G T and Strehl ratio S T of a three-zone lateral super-resolution pupil filter with fabrication errors to quantitatively analyze the effect of these fabrication errors on the lateral super-resolution property. These new models established to describe the analytic relationship of G Te and S Te of a pupil filter with its fabrication errors and its transmission function A( ρ), phase function φ( ρ) and structural parameters directly relate the super-resolution parameters of a three-zone lateral super-resolution pupil filter to its fabrication errors to make the quantitative analyses of the effect of fabrication errors easier, thereby providing a theoretical basis for the analysis, design and fabrication of a three-zone lateral super-resolution pupil filter. The models established for G Te and S Te have been used to analyze the effect of the fabrication errors of a pupil filter on its super-resolution property with a three-zone phase-only pupil filter as an example.

Approach to higher spatial resolutions in a laser probe measurement system using a phase-only pupil filter

Phase-only pupil-filtering differential confocal measurement, a new approach, is proposed to improve the spatial resolution of a laser probe measurement system (LPMS) for ultraprecise measurement of microstructural workpieces. The proposed approach uses a lateral superresolution pupil filter for sharpening the main lobe of the Airy spot to improve the LPMS lateral resolution and uses the differential confocal measurement method to improve the LPMS axial resolution, thereby improving the LPMS spatial resolution. In addition to improving the spatial resolution, linearity, and antiinterference capability of a confocal measurement system, the proposed approach can be used for bipolar absolute measurement and improvement of the measurement range. Experimental comparison and analyses indicate that the lateral resolution of a phase-only pupil-filtering differential confocal system can be improved by 50% over that of an LPMS with the same parameters, and a lateral resolution better than 0.27 µm and an axial resolution better than 3 nm can be achieved when the wavelength of the incident beam is =632.8 nm, the numerical aperture of the measuring lens is NA=0.65, and uM=4. It is therefore concluded that the phase-only pupil filtering differential confocal measurement method is a new approach to a higher spatial resolution of LPMSs and can be used for ultraprecise measurement of surface microcontours and microdimensions.

Image Restoration Phase-Filtering Lateral SuperresolutionConfocal Microscopy

Image restoration phase-filtering lateral superresolution confocalmicroscopy, a new approach, is proposed to achieve lateralsuperresolution using a confocal microscope. This approach uses alateral superresolution pupil filter to preliminarily improve itslateral resolution and uses a single-image superresolution restorationtechnique based on a maximum likelihood estimate to further improve itslateral resolution. The new approach has the advantages of a low costand the remarkable superresolution effect without excessive systemcomplexity. Experiments indicate that the proposed approach can improvethe lateral resolution of a confocal microscope from 0.3 μm toless than 0.1 μm when λ = 632.8 nm and NA=0.85.

Effect of fabrication errors on axial super-resolution property of a three-zone pupil filter

Two models have been established using the basic definitions of super-resolution characteristic parameters for normalized spot size G A and Strehl ratio S A, of a three-zone axial super-resolution pupil filter with fabrication errors, to quantitatively analyze the effect of these fabrication errors on the axial super-resolution property. These new models established to describe the analytic relationship of G Ae and S Ae of a pupil filter with its fabrication errors and its transmission function A( ρ), phase function φ( ρ) and structural parameters, directly relate the super-resolution parameters of a three-zone axial super-resolution pupil filter to its fabrication errors to make the quantitative analyses of the effect of fabrication errors easier, thereby providing a theoretical basis for the analysis, design and fabrication of a three-zone axial super-resolution pupil filter. The models established for G Ae and S Ae have been used to analyze the effect of the fabrication errors of a pupil filter on its super-resolution property, with a three-zone phase-only pupil filters as example.

Effect of fabrication errors on superresolution property of a pupil filter

Three analytical models have been established for superresolution parameters G(Ae), G(Te), and S(e) related to transmission function A(rho), phase function of phi(rho), and the structural parameters with fabrication errors of an N-zone circular-symmetrical superresolution pupil filter. These new models established, directly relate the superresolution parameters of an N-zone super-resolution pupil filter to its fabrication errors to make the quantitative analyses of the effect of fabrication errors easier, thereby providing a theoretical basis for the analysis, design, and fabrication of an N-zone super-resolution pupil filter. The models established for G(Ae), G(Te), and S(e) have been used to analyze the effect of the fabrication errors of a three-zone phase-only pupil filter on its superresolution property, to verify their validities.

Superspeckles: a new application of optical superresolution

The effect of optical superresolution on speckle correlations is studied. Simulations reveal that using a lateral superresolution pupil filter more than twice the out of plane correlation length of the clear pupil can be achieved. This means that the measurement range in speckle correlation measurements doubles. To verify the correlation length an experiment is performed using a liquid crystal (LCD) spatial light modulator as a programmable superresolution filter. The results corroborate the simulation.

Discrete continuous-phase superresolving filters

A new type of phase-only superresolving pupil filter with a discrete continuous-phase profile is presented that is a combination of discrete multilevel-phase modulation and continuous-phase modulation. This type of filter can achieve better superresolution performance than the continuous-phase filters reported in Opt. Lett. 28, 607 (2003). Therefore, with regard to the superresolution effect, this type of filter deserves study for practical applications. More importantly, the diffraction performance of this type of filter can explain the effect of a discrete-phase filter illuminated with a continuous wave front, whose superresolving performance cannot be analyzed with previous superresolution methods.

Design and comparison of amplitude-type and phase-only transverse super-resolving pupil filters

An optimization procedure for designing super-resolving filters is presented, which is to design amplitude-type and phase-only transverse super-resolving Toraldo filters by nonlinear programming at the special Strehl ratio (the ratio of the maximum intensity of the central core of the super-resolving pattern to that of the Airy pattern). Results of comparisons between the performance of our filters and those of the references indicate that the former is superior to the latter. The super-resolving characters of the two types of filters are also discussed and compared.

Transverse or axial superresolution in a 4Pi-confocal microscope by phase-only filters.

A novel procedure to design axial and transverse superresolving pupil filters for the 4Pi-confocal microscope is presented. The method is based on the use of a series of figures of merit developed to describe the effect of inserting two identical filters in the two arms of the illumination path of the microscope. As a practical implementation, we have applied our method to obtain superresolving continuous phase-only filters. Different resolution-improving phase functions are shown for the transverse and the axial direction. These filters provided axial gain up to 1.3 and transverse gain up to 1.4 without an increase in sidelobes.

Three-dimensional behavior of apodized nontelecentric focusing systems.

The scalar field in the focal volume of nontelecentric apodized focusing systems cannot be accurately described by the Debye integral representation. By use of the Fresnel-Kirchhoff diffraction formula it is found that, if the aperture stop is axially displaced, the focal-volume structure is tuned. We analyze the influence of the apodizing function and find that, whereas axially superresolving pupil filters are highly sensitive to the focal-volume reshaping effect, axially apodizing filters are more inclined to the focal-shift effect.

Annular binary filters for controlling the axial behaviour of optical systems

The one-dimensional (1D) version of the iterative Fourier transform algorithm (IFTA) and a modified error diffusion algorithm are proposed for binarizing rotationally symmetric pupil filters designed to shape the axial impulse response of optical system. The resulting binary masks consist of a set of transparent and opaque annular zones of equal area or equal width. A numerical experiment in which we examine the performance of the binarization methods is carried out. In this experiment the resemblance between the axial diffractive behaviour of the binary version of an axially superresolving pupil filter, and that of the original continuous-tone filter is evaluated. It is shown that the performance of the binary mask obtained with 1D IFTA is much better than that of the annular binary filters obtained by other digital half-toning techniques which preserve the rotational symmetry of binarized pupils.

Binarization of continuous-tone pupil filters: a comparison of several digital halftoning procedures

Nine digital halftoning procedures, including two algorithms proposed by us, are compared to find out which one is best suited for binarization of continuous-tone pupil filters. The procedures we deal with include iterative, error diffusion, error convergence and one-pixel algorithms. In our numerical experiment the performance of binary filters is examined in terms of two parameters: the resemblance of their amplitude impulse response (AIR) to the AIR of the original continuous filter, as well as their light efficiency. The examination is carried on for super-resolving, apodizing and sine-wave pupil filters. The results are to a certain extent object dependent, nevertheless some general conclusions are presented.