Phase-only Pupil Research Articles

Inverse-designed achromatic flat lens enabling imaging across the visible and near-infrared with diameter > 3 mm and NA = 0.3

It is generally thought that correcting chromatic aberrations in imaging requires multiple surfaces. Here, we show that by allowing the phase in the image plane of a flat lens to be a free parameter, it is possible to correct chromatic aberrations over a large continuous bandwidth with a single diffractive surface. In contrast to conventional lens design, we utilize inverse design, where the phase in the focal plane is treated as a free parameter. This approach attains a phase-only (lossless) pupil function, which can be implemented as a multi-level diffractive flat lens that achieves achromatic focusing and imaging. In particular, we experimentally demonstrate imaging using a single flat lens of diameter > 3 mm and focal length = 5 mm (NA = 0.3, f/1.59) that is achromatic from λ = 450 nm (blue) to 1 μm (NIR). This simultaneous achievement of large size, NA, and broad operating bandwidth has not been demonstrated in a flat lens before. We experimentally characterized the point-spread functions, off-axis aberrations, and broadband imaging performance of the lens.

Optimized pupil-plane phase masks for high-contrast imaging.

A differential evolution algorithm (DEA) is shown to improve the design of phase-only pupil plane masks for imaging in the presence of a high-intensity point source. Strehl and suppression ratio metrics for various phase basis functions were experimentally and numerically calculated. Experiments performed with a spatial light modulator demonstrated 100 times suppression of the peak intensity. The Strehl ratio was found to be higher for the DEA masks compared to the individual phase basis function.

Numerical modeling of two-photon focal modulation microscopy with a sinusoidal phase filter.

A spatiotemporal phase modulator (STPM) is theoretically investigated using the vectorial diffraction theory. The STPM is equivalent to a time-dependent phase-only pupil filter that alternates between a homogeneous filter and a stripe-shaped filter with a sinusoidal phase distribution. It is found that two-photon focal modulation microscopy (TPFMM) using this STPM can significantly suppress the background contribution from out-of-focus ballistic excitation and achieve almost the same resolution as two-photon microscopy. The modulation depth is also evaluated and a compromise exists between the signal-to-background ratio and signal-to-noise ratio. The theoretical investigations provide important insights into future implementations of TPFMM and its potential to further extend the penetration depth of nonlinear microscopy in imaging multiple-scattering biological tissues.

Pareto optimality between width of central lobe and peak sidelobe intensity in the far-field pattern of lossless phase-only filters for enhancement of transverse resolution.

Resolution capability of an optical imaging system can be enhanced by reducing the width of the central lobe of the point spread function. Attempts to achieve the same by pupil plane filtering give rise to a concomitant increase in sidelobe intensity. The mutual exclusivity between these two objectives may be considered as a multiobjective optimization problem that does not have a unique solution; rather, a class of trade-off solutions called Pareto optimal solutions may be generated. Pareto fronts in the synthesis of lossless phase-only pupil plane filters to achieve superresolution with prespecified lower limits for the Strehl ratio are explored by using the particle swarm optimization technique.

Generation of a high depth of focus with constant transversal spot size using a phase-only pupil filter

A novel approach is proposed to obtain an extended depth of focus (DoF) in white light with constant transversal spot size within the DoF. It combines a phase-only pupil filter based on multiplexed radial zones with alternating quartic phase functions. The design is first tested via numerical simulations of the point spread function (PSF) based on the scalar diffraction theory. The results for a fourfold gain of the depth of focus are experimentally verified with a phase-only spatial light modulator liquid crystal device combined with a 3D PSF measurement system. A close conformity between the experimental and simulation results proves the effectiveness of the proposed approach.

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.

Designing superresolution optical pupil filter with constrained global optimization algorithm

In this paper, a new method for designing three-zone optical pupil filter is presented. The phase-only optical pupil filter and the amplitude-only optical pupil filters were designed. The first kind of pupil for optical data storage can increase the transverse resolution. The second kind of pupil filter can increase the axial and transverse resolution at the same time, which is applicable in three-dimension imaging in confocal microscopy.

Multiple coaxial foci generation by phase-only pupil filters

A new procedure for creating simultaneous multiple foci along the same optical axis with phase-only pupil-plane filters is proposed. Pupil-plane filters consist in phase-only masks described by simple circularly symmetric analytical functions. The axial separation between foci and their energy distribution are independently controlled by two different parameters of the filter. The main advantage of this new design is that, unlike other methods that required slow iterative algorithms, the proposed phase masks are straightforwardly calculated from analytical expressions. Moreover, real-time dynamic foci control could be achieved by using a computer addressed spatial light modulator that allowed the generation of variable phase patterns. The phase profiles we propose can be combined with multiple transversal foci phase patterns to actually form axial replicas of the transversal foci. Thus, it should be expected that three-dimensional structures were dynamically driven fast enough to perform real-time interactive multiple foci manipulation. Experimental verification of the multiple coaxial foci creation by static phase filters is shown.

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.

Tunable transverse superresolution with phase-only pupil filters

A new set of pure phase filters for realizing transverse superresolution is presented in this paper. The filters, whose significant features are their ability to tune and their simplicity, consist of one half-wave plate between two quarter-wave plates; the half-wave plate is made of two zones that can rotate with respect to each other. By rotating any zone of the half-wave plate, the central lobe width of the irradiance point spread function (PSF) in the transverse direction can be tunably reduced. At the same time, the axial intensity distribution is analysed in detail.

A lateral super-resolution differential confocal technology with phase-only pupil filter

In order to achieve a higher lateral resolution required for ultraprecision measurement of microstructural workpieces, phase-only pupil filtering differential confocal microscopy (PFDCM), a new approach is proposed based on the differential confocal microscopy (DCM), which uses a three-zone phase-only pupil filter with lateral super-resolution capability obtained through optimized design to change the distribution of DCM three-dimensional point spread function, so that the DCM lateral resolution is therefore significantly improved while its axial resolution is slightly improved. Preliminary experimental comparison and analyses indicate that, the lateral and axial resolutions of PFDCM are better than 0.2 μm and 2 nm, respectively, when wavelength of incidence laser beam λ = 632.8 nm , numerical aperture of measuring lens NA = 0.85 , and lateral spot size with a three-zone phase-only pupil filter G T = 0.65 . It is therefore concluded that PFDCM is a new approach to further improvement of lateral resolution in laser probe measurement systems.

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.

Confocal measurement approach for enhancing lateral resolution using a phase-only pupil

Confocal measurement approach is recently widely used as important tools for measurement of three-dimensional microstructures and surface contours because of its good 3D chromatographic imaging capability. But its lateral resolution is limited by the diffraction effect to about 0.4 µm only while the axial resolution has reached the nanometer level. A phase-only confocal measurement approach is therefore proposed to enhance the lateral resolution. In this paper, an optimized superresolution phase-only filter is used in a confocal microscopy system to improve the lateral resolution, and the pinhole in the confocal microscopy system effectively eliminates the disturbance of ambient lighting and improves the imaging quality. Simulation results indicate that, an optimized three-zone pupil with GL = 0.8375 and GA = 0.99 can make the lateral resolution of CMS 1.18 times better while the axial resolution is almost invariable. So it therefore can improve the applicability of confocal microscopy system.

Analytical design of superresolving phase filters

A procedure for designing annular pupil-plane masks to control the light intensity distribution near focus is presented. The method is based on binary phase-only pupil filters that reach 0 and π values. The parameters of the filters are analytically obtained from the figures of merit that characterize the focal light intensity distribution, as the Strehl ratio, the transverse gain or the axial gain. The procedure allows a simple and complete analysis of the binary filters behavior for any application.

Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings

We demonstrate theoretically the feasibility of single-photon 4Pi-confocal microscopy. By inserting a pair of properly designed multi-ring phase-only pupil filters in the illumination path of a 4Pi microscope the height of the sidelobes of the point spread function substantially reduced, so that there is no ambiguity in the 3D image. Then, an axial resolution up to four times higher than that of single-photon confocal microscope can be effectively achieved.

Binary phase-only pupil filters application to high-aperture confocal microscopy

According to the vector diffraction theory, we study a new method to design binary phase-only filters applied to high-aperture objectives, and obtain the approximate relationship between superresolution parameters and filter parameters. The validity of this approximate relationship is dicussed. Then we can design superresolving filters in terms of this relationship. In the end, those filters are applied to confocal microscopy to reduce the strength of secondary lobes. Numerical results show that the pupil filters designed by us can achieve axial superresolution.

Tailoring the axial shape of the point spread function using the Toraldo concept

A novel procedure for shaping the axial component of the point spread function of nonparaxial focusing systems by use of phase-only pupil filters is presented. The procedure is based on the Toraldo technique for tailoring focused fields. The resulting pupil filters consist of a number of concentric annular zones with constant real transmittance. The number of zones and their widths can be adapted according to the shape requirements. Our method is applied to design filters that produce axial superresolution in confocal scanning systems.

Axial superresolution with phase-only pupil filters

A novel filter design for axial superresolution based on a phase-only pupil function is introduced. Solutions for confocal scanning microscopy indicate that the phase-only design reduces the axial spot size by a factor of 2.

Diffractive superresolution elements

Superresolution phase-only pupil filters designed to utilize the degrees of freedom made available by diffractive optics technology are investigated theoretically. These so-called diffractive superresolution elements improve the quality of the superresolved diffraction pattern from the point of view of Strehl ratio, reduction of the spot size, control of the sidelobe effects, optimization procedures, and fabrication tolerances. The performance of these elements is studied, and the nature of the solutions obtainable with binary and multiple-phase structures is analyzed. Design considerations and solutions for applications such as confocal scanning microscopy and optical data storage are presented. Optimization of the degrees of freedom to satisfy desired constraints is discussed and compared with other methods.