Intensity Point Spread Function Research Articles

Universal linear intensity transformations using spatially incoherent diffractive processors

Under spatially coherent light, a diffractive optical network composed of structured surfaces can be designed to perform any arbitrary complex-valued linear transformation between its input and output fields-of-view (FOVs) if the total number (N) of optimizable phase-only diffractive features is ≥~2NiNo, where Ni and No refer to the number of useful pixels at the input and the output FOVs, respectively. Here we report the design of a spatially incoherent diffractive optical processor that can approximate any arbitrary linear transformation in time-averaged intensity between its input and output FOVs. Under spatially incoherent monochromatic light, the spatially varying intensity point spread function (H) of a diffractive network, corresponding to a given, arbitrarily-selected linear intensity transformation, can be written as H(m, n; m′, n′) = |h(m, n; m′, n′)|2, where h is the spatially coherent point spread function of the same diffractive network, and (m, n) and (m′, n′) define the coordinates of the output and input FOVs, respectively. Using numerical simulations and deep learning, supervised through examples of input-output profiles, we demonstrate that a spatially incoherent diffractive network can be trained to all-optically perform any arbitrary linear intensity transformation between its input and output if N ≥ ~2NiNo. We also report the design of spatially incoherent diffractive networks for linear processing of intensity information at multiple illumination wavelengths, operating simultaneously. Finally, we numerically demonstrate a diffractive network design that performs all-optical classification of handwritten digits under spatially incoherent illumination, achieving a test accuracy of >95%. Spatially incoherent diffractive networks will be broadly useful for designing all-optical visual processors that can work under natural light.

Influence of Apodization Pupil Function on the Intensity Point Spread Function of Optical System

Apodization is the significant modification of the impulse response of the optical system. This modification plays a vital role in improving the imaging and focusing characteristics foe various optical systems. We present our studies on apodization of pupils with amplitude filter respect to images of point objects . The central intensity distribution, the Full width at half maximum (FWHM)of point spread function, first maxima and first minima are evaluated and compared with and without apodisation .our results shown that the resolution of the intensity of the PSF has been modified as the degree of apodization increases .

Deep learning estimation of modified Zernike coefficients and recovery of point spread functions in turbulence.

Recovering the turbulence-degraded point spread function from a single intensity image is important for a variety of imaging applications. Here, a deep learning model based on a convolutional neural network is applied to intensity images to predict a modified set of Zernike polynomial coefficients corresponding to wavefront aberrations in the pupil due to turbulence. The modified set assigns an absolute value to coefficients of even radial orders due to a sign ambiguity associated with this problem and is shown to be sufficient for specifying the intensity point spread function. Simulated image data of a point object and simple extended objects over a range of turbulence and detection noise levels are created for the learning model. The MSE results for the learning model show that the best prediction is found when observing a point object, but it is possible to recover a useful set of modified Zernike coefficients from an extended object image that is subject to detection noise and turbulence.

Engineering of Aberrated PSF by Asymmetric Apodization with the Complex Shaded Aperture

The point spread function (PSF) produced by a coherent optical system under the influence of defocus, coma, and primary spherical aberration (PSA) is examined in this work. This paper deals with asymmetric apodization and pupil engineering to control monochromatic aberrations. To reduce the influence of monochromatic aberrations on the diffracted PSF, this approach uses amplitude and phase apodization. Analytical investigations on intensity PSF are carried out with varying amounts of aberrations and degrees of amplitude and phase apodization. Computed central peak intensity and full width at half maxima (FWHM) and analyzed. The resolution of a diffraction-limited optical imaging system is improved by using an asymmetric optical filter that minimizes the effect of defocus.

Lateral resolution enhanced interference microscopy using virtual annular apertures

The lateral resolution in microscopic imaging generally depends on both, the wavelength of light and the numerical aperture of the microscope objective lens. To quantify the lateral resolution Ernst Abbe considered an optical grating illuminated by plane waves. In contrast, the Rayleigh criterion holds for two point sources or point scatterers separated by a lateral distance, which are supposed to emit spherical waves. A portion of each spherical wave is collected by the objective lens and results in an Airy disc corresponding to a diffraction limited intensity point spread function (PSF). If incoherent illumination is employed the intensity PSFs related to different scatterers on an object are added resulting in the well-known Rayleigh resolution criterion. In interference microscopy instead of the intensity the electric field scattered or diffracted by an object will be affected by the transfer function of the optical imaging system. For a reflective object the lateral resolution of an interference microscope can be again characterized by the Abbe limit if the object under investigation is a grating. However, if two irregularities on a flat surface are being imaged the resolution no longer obeys the Rayleigh criterion. Instead, it corresponds to an optical system with an annular aperture and thus surpasses the prediction given by the Rayleigh criterion. This holds true for both, amplitude as well as phase objects, as it will be elucidated in this study by theoretical considerations, simulation results and an experimental proof of principle.

Three-dimensional transfer function of optical microscopes in reflection mode.

Three-dimensional (3D) transfer functions build the basis for a comprehensive characterization of optical imaging systems in the spatial frequency domain. Utilizing the projection-slice theorem, the 2D modulation transfer function of an incoherent imaging system can be derived from a 3D transfer function by integration with respect to the axial spatial frequency. For a diffraction limited microscope with homogeneous incoherent pupil illumination, the modulation transfer function equals the 2D autocorrelation function of a circular disc. However, until now to the best of our knowledge no 3D transfer function has been published, which exactly leads to the 2D modulation transfer function of a diffraction limited microscope in reflection mode. In this article, we derive a formula, which after integration with respect to the axial spatial frequency coordinate perfectly fits to the diffraction limited 2D modulation transfer function. The inverse three-dimensional Fourier transform of the 3D transfer function results in a complex-valued 3D point spread function, from which the depth of field, the lateral resolution and, in addition, the corresponding 3D point spread function of both, a conventional and an interference microscope, can beobtained.

Aspherical microlenses enabled by two-photon direct laser writing for fiber-optical microendoscopy

Fiber-optical microendoscopy has made significant improvements to in vivo neural imaging, minimally invasive diagnostics, and microsurgery. However, high resolution, miniaturization, and low complexity cannot be simultaneously achieved together in the lens system for fiber-optical microendoscopy because current lens systems are in shape and dimensions restricted by limitations of manufacturing. Recently, two-photon direct laser writing (DLW) has been implemented in the fabrication of low-resolution micro-optics structures. Here, we demonstrate a high-resolution miniaturized singlet aspherical microlens fabricated on the fiber facet using DLW. The microlens has a high numerical aperture (NA), of 0.9, in air with only one aspherical surface, and is 10–20 times smaller in diameter than a typical gradient refractive index (GRIN) microlens. The designs of aspherical microlenses with NAs of 0.3, 0.6, and 0.9 in air are aberration-free at three wavelengths (561, 590, and 630 nm). The full width at half maximum of the effective intensity point spread function of a 0.9 NA aspherical microlens is 0.85 μm. We demonstrate fiber-optical microendoscopy imaging with a 0.6 NA aspherical microlens. The proposed aspherical microlens can potentially be applied to the development of a high-resolution, extremely miniaturized fiber-optical microendoscope.

Machine learning based adaptive optics for doughnut-shaped beam.

The doughnut-shaped beam has been widely applied in the field of super-resolution microscopic imaging, micro-nanostructure lithography, ultra-high-density storage, and laser trapping. However, how to maintain the doughnut-shaped focus inside the scattering medium becomes a challenge, due to the wavefront aberrations. Here we demonstrate a machine learning based adaptive optics method to recover the doughnut-shaped focus with high speed. In our method, the relationship between the distorted doughnut-shaped intensity point spread function and the coefficients of the first 15 Zernike modes for phase correction is established. Experimental results show that the wavefront aberration with 101,784 optical control elements can be predicted within ~17 ms even using a personal computer, and 97.5% correction accuracy can be achieved in 200 repeated tests. Besides, we successfully apply this method in the scanning microscopy theoretically. With a large number of optical control elements and fast operation speed, our method may pave the way for many important applications in bioimaging, such as deep tissue stimulated emission depletion (STED) microscopy.

Optically-Monitored Nanopore Fabrication Using a Focused Laser Beam

Solid-state nanopores (ssNPs) are extremely versatile single-molecule sensors and their potential have been established in numerous biomedical applications. However, the fabrication of ssNPs remains the main bottleneck to their widespread use. Herein, we introduce a rapid and localizable ssNPs fabrication method based on feedback-controlled optical etching. We show that a focused blue laser beam irreversibly etches silicon nitride (SiNx) membranes in solution. Furthermore, photoluminescence (PL) emitted from the SiNx is used to monitor the etching process in real-time, hence permitting rate adjustment. Transmission electron microscopy (TEM) images of the etched area reveal an inverted Gaussian thickness profile, corresponding to the intensity point spread function of the laser beam. Continued laser exposure leads to the opening of a nanopore, which can be controlled to reproducibly fabricate nanopores of different sizes. The optically-formed ssNPs exhibit electrical noise on par with TEM-drilled pores, and translocate DNA and proteins readily. Notably, due to the localized thinning, the laser-drilled ssNPs exhibit highly suppressed background PL and improved spatial resolution. Given the total control over the nanopore position, this easily implemented method is ideally suited for electro-optical sensing and opens up the possibility of fabricating large nanopore arrays in situ.

Adaptive optics scanning system with two-zone phase filters

ABSTRACTAn adaptive confocal system employing pupil filters and two light sources has been designed, providing improved transverse resolution. Coherent transfer functions are studied to show the transfer capability at different frequencies and the limitations of the filters in obtaining clear improvements in the focal plane. Then, intensity point spread functions are analysed to reveal the dependence of the resolution on the size of the pinhole and the inner radius of the filter. An experiment has been carried out using a liquid crystal phase modulator and an artificial eye to validate the simulation results. It proves that the system with two light sources can correct the aberrations, with the basic effect of the phase modulation being to decrease the low spatial frequencies and increase the high spatial frequencies. Improvement of the transverse resolution can be achieved by sacrificing light intensity and contrast.

A simple method for quality evaluation of micro-optical components based on 3D IPSF measurement.

This paper presents a simple method based on the measurement of the 3D intensity point spread function for the quality evaluation of high numerical aperture micro-optical components. The different slices of the focal volume are imaged thanks to a microscope objective and a standard camera. Depending on the optical architecture, it allows characterizing both transmissive and reflective components, for which either the imaging part or the component itself are moved along the optical axis, respectively. This method can be used to measure focal length, Strehl ratio, resolution and overall wavefront RMS and to estimate optical aberrations. The measurement setup and its implementation are detailed and its advantages are demonstrated with micro-ball lenses and micro-mirrors. This intuitive method is adapted for optimization of micro-optical components fabrication processes, especially because heavy equipments and/or data analysis are not required.

Phase‐retrieved pupil function and coherent transfer function in confocal microscopy

This work reports on the retrieval of the pupil function and coherent transfer function of a coherent reflection type confocal microscope from simulated measurements of the intensity point spread function. Two phase retrieval algorithms are presented in this vein, which incorporate the multiple pupil dependence of image formation in confocal microscopy. Verification of the algorithms follows by numerical simulations.

AXIAL SUPERRESOLUTION BY PHASE FILTER IN OPTICAL COHERENCE TOMOGRAPHY

Axial superresolution in optical coherence tomography (OCT) by a three-zone annular phase filter is demonstrated. In the proposed probe of a spectral domain OCT system, the width of the central lobe of the axial intensity point spread function is apodized by the filter to be within the coherence gate determined by the light source, while its sidelobes are lying outside the coherence gate without contributing to the coherence imaging. By measurement of the depth response of the OCT system before and after inserting the filter, an improvement of about 20% in axial resolution is confirmed. OCT imaging on biological sample of orange fresh is also conducted, demonstrating increased depth discrimination without the negative contribution from sidelobes realized by the phase filter in combination with the coherence gate intrinsic to OCT. It comes to a conclusion that we can obtain axial superresolution by filter in OCT system without the degrading influence of large sidelobes.

Polychromatic intensity point spread function of a birefringent lens

We study the intensity point spread function (IPSF) of a uniaxial birefringent lens sandwiched between two linear polarizers under broadband illumination having flat-top spectral profile. The optic axis of the birefringent crystal is perpendicular to the lens axis. For monochromatic input illumination, a low-power birefringent lens offers enhanced resolution under parallel-polarizers configuration and apodization under crossed-polarizers configuration with a single focus as well. At a relatively high value of birefringence parameter, the proposed system behaves as a double-focus lens. The present communication uses a novel technique to study the effect of polychromatic illumination on the imaging characteristics of birefringent lenses. The studies reveal that the imaging characteristics of the low-power birefrigent lens does not deteriorate considerably under polychromatic illumination. However, for a polychromatic beam with large bandwidth, the imaging behavior of the system having high birefringence parameter deviates appreciably from that under strictly monochromatic illumination. The said system even loses its inherent bi-focal nature due to depolarization of polarized polychromatic light upon passage through the birefringent crystal.

Hybrid pupil filter design using Bessel series

We propose a simple method of designing pupil filters for transverse super-resolution. For this end we represent the amplitude Point Spread Function (PSF) as a series expansion, constructed from the Fourier transform of a basis of Bessel functions. With this representation we optimize the intensity PSF according to certain desired characteristics, such as a smaller disk diameter than the corresponding, clear aperture, Airy disk. It is proved that by using few basis functions, it is possible to design pupils with similar or better PSF characteristics than previously reported.

Method of obtaining wavefront slope data from through-focus point spread function measurements

A method is described for analyzing point source imagery collected with various amounts of defocus to obtain wavefront slope data at the exit pupil of an imaging system. Integration of this slope data yields the system wavefront aberration function. The method is based on a geometric optics interpretation of intensity point spread function measurements in the caustic region near focus. Algorithm performance is examined through Monte Carlo simulations. Application of the method to segmented-aperture systems is also explored. The proposed method is used to generate initial wavefront estimates for an iterative phase retrieval algorithm, significantly improving the capture range over a blind phase retrieval approach when segment tilt errors are large.

Simulation of a polarization pupil filter using magneto-optic lens under elliptically polarized illumination

An active lens made of magneto-optic glass element and placed before a linear polarizer is shown to simulate the pupil plane amplitude and phase filter due to thickness dependent Faraday rotation variation in radial direction. Changing the applied magnetic flux density can dynamically change the pupil characteristics. The variation of intensity point spread function (IPSF) and optical transfer function (OTF) with magnetic flux density for elliptically polarized input beam is studied.

Resolution analysis of a gradient-index rod and a gradient-index lens array

The physical optics analysis of a gradient-index (GRIN) rod and a GRIN lens array with aberrations is presented. We investigated the optical path length and aberration of a GRIN rod without definition of the stop plane. We also defined an effective aberration transmission function to include aberrations into physical optics analysis. Our theoretical results of impulse responses agree excellently with experiments. For a single GRIN rod, we obtained a theoretical value of 10.3 microm and an experimental value of 10.4 microm for the full width at half-maximum of the intensity point-spread function. For a GRIN array, the theoretical value of 19.2 microm and the experimental measurement of 19.9 microm agree to within 4%. This physical optics methodology with aberrations included can be applied to optical design software. The resolution difference in xerographic process for test material along the "parallel-to-perpendicular" directions is observed. It agrees with the theoretical result for the intensity impulse response of the GRIN array calculated with a second-order correction.

Measured double-pass intensity point-spread function after adaptive optics correction of ocular aberrations

The double-pass intensity point-spread function was recorded in four subjects using a monochromatic source emitting at 543 nm, through a 6.7-mm diameter pupil i) at the fovea after adaptive optics correction of the ocular aberrations, ii) at the fovea without adaptive optics correction, and iii) at 2 degrees of eccentricity with adaptive optics correction. The half-width at half-maximum of the double-pass point-spread function was narrower after correction of the ocular aberrations. At 2 degrees of eccentricity this width was larger than at the fovea. The minimum widths were about 1.1 arcmin in dark pigmented eyes and 1.6 arcmin in light pigmented eyes. These values are 6 to 9 times larger than the width expected from diffraction alone.

Zernike representation and Strehl ratio of optical systems with variable numerical aperture

We consider optical systems with variable numerical aperture (NA) on the level of the Zernike coefficients of the correspondingly scalable pupil function. We thus present formulas for the Zernike coefficients and their first two derivatives as a function of the scaling factor ε ≤ 1, and we apply this to the Strehl ratio and its derivatives of NA-reduced optical systems. The formulas for the Zernike coefficients of NA-reduced optical systems are also useful for the forward calculation of point-spread functions and aberration retrieval within the Extended Nijboer–Zernike (ENZ) formalism for optical systems with reduced NA or systems that have a central obstruction. Thus, we retrieve a Gaussian, comatic pupil function on an annular set from the intensity point-spread function in the focal region under high-NA conditions.