Image scanning microscopy with a doughnut beam: signal strength and integrated intensity

We discuss the effects of image scanning microscopy using doughnut beam illumination on the properties of signal strength and integrated intensity. Doughnut beam illumination can give better optical sectioning and background rejection than Airy disk illumination. The outer pixels of a detector array give a signal from defocused regions, so digital processing of these (e.g., by simple subtraction) can further improve optical sectioning and background rejection from a single in-focus scan.

Advancing full-field metrology: rapid 3D imaging with geometric phase ferroelectric liquid crystal technology in full-field optical coherence microscopy

Optical coherence microscopy (OCM) is a variant of OCT in which a high-numerical aperture lens is used. Full-field OCM (FF-OCM) is an emerging non-invasive, label-free, interferometric technique for imaging of surface structures or semi-transparent biomedical subjects with micron-scale resolutions. Different approaches to three dimensional full-field optical metrology are reviewed. The usual method for the phase-shifting technique in FF-OCM involves mechanically moving a mirror to change the optical path difference for obtaining en-face OCM images. However, with the use of a broadband source in FF-OCM, the phase shifts of different spectral components are not the same, resulting in the ambiguities in 3D image reconstruction. In this study, we demonstrate, by imaging tissues and cells, a unique geometric phase-shifter based on ferroelectric liquid crystal technology, to realize achromatic phase-shifting for rapid three-dimensional imaging in a FF-OCM system.

Signal strength and integrated intensity in confocal and image scanning microscopy

The properties of signal strength and integrated intensity in a scanned imaging system are reviewed. These properties are especially applied to confocal imaging systems, including image scanning microscopy. The integrated intensity, equal to the image of a uniform planar (sheet) object, rather than the peak of the point spread function, is a measure of the flux in an image. Analytic expressions are presented for the intensity in the detector plane for a uniform volume object, and for the resulting background. The variation in the integrated intensity with defocus for an offset point detector is presented. This axial fingerprint is independent of any pixel reassignment. The intensity in the detector plane is shown to contain the defocus information, and simple processing of the recorded data can improve optical sectioning and background rejection.

Purity of 3D polarization

Measures of purity for 3D partially polarized fields, and in particular, the separation into circularly and linearly polarized contributions, are reexamined, and a new degree of total linear polarization introduced. Explicit expressions for the characteristic decomposition in terms of coherency matrix elements are presented, including the special case of an intrinsic coherency matrix. Parameterization of the coherency matrix in terms of ellipticity, and the directions of the ellipse normal and major axis are investigated. Phase consistency is discussed. A comprehensive collection of results regarding intrinsic polarization properties is presented.

Pixel reassignment in image scanning microscopy with a doughnut beam: example of maximum likelihood restoration

In image scanning microscopy, the pinhole of a confocal microscope is replaced by a detector array. The point spread function for each detector element can be interpreted as the probability density function of the signal, the peak giving the most likely origin. This thus allows a form of maximum likelihood restoration, and compensation for aberrations, with similarities to adaptive optics. As an example of an aberration, we investigate theoretically and experimentally illumination with a vortex doughnut beam. After reassignment and summation over the detector array, the point spread function is compact, and the resolution and signal level higher than in a conventional microscope.

Structured illumination microscopy and image scanning microscopy: a review and comparison of imaging properties

Structured illumination microscopy and image scanning microscopy are two microscopical tech- niques, rapidly increasing in practical application, that can result in improvement in transverse spatial resolution, and/or improvement in axial imaging performance. The history and principles of these techniques are reviewed, and the imaging properties of the two methods compared. This article is part of the Theo Murphy meeting issue 'Super-resolution structured illumination microscopy (part 1)'.

Polarimetric optical scanning microscopy of zebrafish embryonic development using the coherency matrix

Many of the most important resolution improvements in optical microscopy techniques are based on the reduction of scattering effects. The main benefit of polarimetry-based imaging to this end is the discrimination between scattering phenomena originating from complex systems and the experimental noise. The determination of the coherency matrix elements from the experimental Mueller matrix can take advantage of scattering measurements to obtain additional information on the structural organization of a sample. We analyze the contrast mechanisms extracted from (a) the coherency matrix elements, (b) its eigenvalues and (c) the indices of polarimetric purity at different stages of zebrafish embryos, based on previous work using Mueller matrix optical scanning microscopy. We show that the use of the coherency matrix and related decompositions leads to an improvement in the imaging contrast, without requiring any complicated algebraic operations or any a priori knowledge of the sample, in contrast to standard polarimetric methods.

Private versus Shared, Automated Electric Vehicles for U.S. Personal Mobility: Energy Use, Greenhouse Gas Emissions, Grid Integration, and Cost Impacts

Transportation is the fastest-growing source of greenhouse gas (GHG) emissions and energy consumption globally. While the convergence of shared mobility, vehicle automation, and electrification has the potential to drastically reduce transportation impacts, it requires careful integration with rapidly evolving electricity systems. Here, we examine these interactions using a U.S.-wide simulation framework encompassing private electric vehicles (EVs), shared automated EVs (SAEVs), charging infrastructure, controlled EV charging, and a grid economic dispatch model to simulate personal mobility exclusively using EVs. We find that private EVs with uncontrolled charging would reduce GHG emissions by 46% compared to gasoline vehicles. Private EVs with fleetwide controlled charging would achieve a 49% reduction in emissions from baseline and reduce peak charging demand by 53% from the uncontrolled scenario. We also find that an SAEV fleet 9% the size of today's active vehicle fleet can satisfy trip demand with only 2.6 million chargers (0.2 per EV). Such an SAEV fleet would achieve a 70% reduction in GHG emissions at 41% of the lifecycle cost as a private EV fleet with controlled charging. The emissions and cost advantage of SAEVs is primarily due to reduced vehicle manufacturing compared with private EVs.

Analysis of numerical diffraction calculation methods: from the perspective of phase space optics and the sampling theorem

Diffraction calculations are widely used in applications that require numerical simulation of optical wave propagation. Different numerical diffraction calculation methods have their own transform and sampling properties. In this study, we provide a unified analysis where five popular fast diffraction calculation methods are analyzed from the perspective of phase space optics and the sampling theorem: single fast Fourier transform-based Fresnel transform, Fresnel transfer function approach, Fresnel impulse response approach, angular spectrum method, and Rayleigh-Sommerfeld convolution. The evolutions of an input signal's space-bandwidth product (SBP) during wave propagation are illustrated with the help of a phase space diagram (PSD) and an ABCD matrix. It is demonstrated that all of the above methods cannot make full use of the SBP of the input signal after diffraction; and some transform properties have been ignored. Each method has its own restrictions and applicable range. The reason why different methods have different applicable ranges is explained with physical models. After comprehensively studying and comparing the effect on the SBP and sampling properties of these methods, suggestions are given for choosing the proper method for different applications and overcoming the restrictions of corresponding methods. The PSD and ABCD matrix are used to illustrate the properties of these methods intuitively. Numerical results are presented to verify the analysis, and potential ways to develop new diffraction calculation methods are also discussed.

Image scanning microscopy with multiphoton excitation or Bessel beam illumination

Image scanning microscopy is a technique of confocal microscopy in which the confocal pinhole is replaced by a detector array, and the image is reconstructed most straightforwardly by pixel reassignment. In the fluorescence mode, the detector array collects most of the fluorescent light, so the signal-to-noise ratio is much improved compared with confocal microscopy with a small pinhole, while the resolution is improved compared with conventional fluorescence microscopy. Here we consider two cases in which the illumination and detection point spread functions are dissimilar: illumination with a Bessel beam and multiphoton microscopy. It has been shown previously that for Bessel beam illumination in image scanning microscopy with a large array, the imaging performance is degraded. On the other hand, it is also known that the resolution of confocal microscopy is improved by Bessel beam illumination. Here we analyze image scanning microscopy with Bessel beam illumination together with a small array and show that an improvement in transverse resolution (width of the point spread function) by a factor of 1.78 compared with a conventional fluorescence microscope can be obtained. We also examine the behavior of image scanning microscopy in two- or three-photon fluorescence and for two-photon excitation also with Bessel beam illumination. The combination of the optical sectioning effect of image scanning microscopy with multiphoton microscopy reduces background from the sample surface, which can increase penetration depth. For a detector array size of two Airy units, the resolution of two-photon image scanning microscopy is a factor 1.85 better and the peak of the point spread function 2.84 times higher than in nonconfocal two-photon fluorescence. The resolution of three-photon image scanning microscopy is a factor 2.10 better, and the peak of the point spread function is 3.77 times higher than in nonconfocal three-photon fluorescence. The resolution of two-photon image scanning microscopy with Bessel beam illumination is a factor 2.13 better than in standard two-photon fluorescence. Axial resolution and optical sectioning in two-photon or three-photon fluorescence are also improved by using the image scanning modality.

BISTRO

The current trend toward urbanization and adoption of flexible and innovative mobility technologies will have complex and difficult-to-predict effects on urban transportation systems. Comprehensive methodological frameworks that account for the increasingly uncertain future state of the urban mobility landscape do not yet exist. Furthermore, few approaches have enabled the massive ingestion of urban data in planning tools capable of offering the flexibility of scenario-based design.

This article introduces Berkeley Integrated System for Transportation Optimization (BISTRO), a new open source transportation planning decision support system that uses an agent-based simulation and optimization approach to anticipate and develop adaptive plans for possible technological disruptions and growth scenarios. The new framework was evaluated in the context of a machine learning competition hosted within Uber Technologies, Inc., in which over 400 engineers and data scientists participated. For the purposes of this competition, a benchmark model, based on the city of Sioux Falls, South Dakota, was adapted to the BISTRO framework. An important finding of this study was that in spite of rigorous analysis and testing done prior to the competition, the two top-scoring teams discovered an unbounded region of the search space, rendering the solutions largely uninterpretable for the purposes of decision-support. On the other hand, a follow-on study aimed to fix the objective function. It served to demonstrate BISTRO’s utility as a human-in-the-loop cyberphysical system: one that uses scenario-based optimization algorithms as a feedback mechanism to assist urban planners with iteratively refining objective function and constraints specification on intervention strategies. The portfolio of transportation intervention strategy alternatives eventually chosen achieves high-level regional planning goals developed through participatory stakeholder engagement practices.

Eigenvectors of polarization coherency matrices

Calculation of the eigenvectors of two- and three-dimensional coherency matrices, and the four-dimensional coherency matrix associated with a Mueller matrix, is considered, especially for algebraic cases, in the light of recently published algorithms. The preferred approach is based on a combination of an evaluation of the characteristic polynomial and an adjugate matrix. The diagonal terms of the coherency matrix are given in terms of the characteristic polynomial of reduced matrices as functions of the eigenvalues of the coherency matrix. The analogous polynomial form for the off-diagonal elements of the coherency matrix is also presented. Simple expressions are given for the pure component in the characteristic decomposition.

Two-photon image-scanning microscopy with SPAD array and blind image reconstruction

Two-photon excitation (2PE) laser scanning microscopy is the imaging modality of choice when one desires to work with thick biological samples. However, its spatial resolution is poor, below confocal laser scanning microscopy. Here, we propose a straightforward implementation of 2PE image scanning microscopy (2PE-ISM) that, by leveraging our recently introduced single-photon avalanche diode (SPAD) array detector and a novel blind image reconstruction method, is shown to enhance the effective resolution, as well as the overall image quality of 2PE microscopy. With our adaptive pixel reassignment procedure ∼1.6 times resolution increase is maintained deep into thick semi-transparent samples. The integration of Fourier ring correlation based semi-blind deconvolution is shown to further enhance the effective resolution by a factor of ∼2 - and automatic background correction is shown to boost the image quality especially in noisy images. Most importantly, our 2PE-ISM implementation requires no calibration measurements or other input from the user, which is an important aspect in terms of day-to-day usability of the technique.

Two-photon image-scanning microscopy with SPAD array and blind image reconstruction

Two-photon excitation (2PE) laser scanning microscopy is the imaging modality of choice when one desires to work with thick biological samples. However, its spatial resolution is poor, below confocal laser scanning microscopy. Here, we propose a straightforward implementation of 2PE image scanning microscopy (2PE-ISM) that, by leveraging our recently introduced single-photon avalanche diode (SPAD) array detector and a novel blind image reconstruction method, is shown to enhance the effective resolution, as well as the overall image quality of 2PE microscopy. With our adaptive pixel reassignment procedure ∼1.6 times resolution increase is maintained deep into thick semi-transparent samples. The integration of Fourier ring correlation based semi-blind deconvolution is shown to further enhance the effective resolution by a factor of ∼2 - and automatic background correction is shown to boost the image quality especially in noisy images. Most importantly, our 2PE-ISM implementation requires no calibration measurements or other input from the user, which is an important aspect in terms of day-to-day usability of the technique.

Polarization in reflectance imaging

The Sinclair and Kennaugh matrices are widely used in the remote sensing discipline for signals detected in the backward direction. The connections between the Jones matrix and the Sinclair matrix, and between the Mueller matrix and the Kennaugh matrix, are explored. Different operations on the Jones matrix and their corresponding effects on the Mueller matrix, coherency matrix, and coherence vector are derived. As an example, the Sinclair matrix leads to a Mueller-Sinclair matrix, and a transformed coherence vector. The Kennaugh matrix is not, however, a Mueller matrix, but can be determined from the Mueller or Mueller-Sinclair matrices. We consider backscattering through a medium on a perfect mirror. We propose that backscattering from a uniform medium can be modeled as an effective uniform medium situated on a perfectly reflective substrate, and the elementary polarization properties derived. In this way, the concept of a uniform polarizing medium can be extended to the reflectance geometry. An experimental Mueller matrix from the literature is considered as an example.

Pixel reassignment in image scanning microscopy: a re-evaluation

Image scanning microscopy is a technique based on confocal microscopy, in which the confocal pinhole is replaced by a detector array, and the resulting image is reconstructed, usually by the process of pixel reassignment. The detector array collects most of the fluorescent light, so the signal-to-noise ratio is much improved compared with confocal microscopy with a small pinhole, while the resolution is improved compared with conventional (wide-field) microscopy. In previous studies, it has usually been assumed that pixels should be reassigned by a constant factor, to a point midway between the illumination and detection spots. Here it is shown that the peak intensity of the effective point spread function (PSF) can be further increased by 4% by a new choice of the pixel reassignment factor. For an array of two Airy units, the peak of the effective PSF is 1.90 times that of a conventional microscope, and the transverse resolution is 1.53 times better. It is confirmed that image scanning microscopy gives optical sectioning strength identical to that of a confocal microscope with a pinhole equal to the size of the detector array. However, it is shown that image scanning microscopy exhibits axial resolution superior to a confocal microscope with a pinhole the same size as the detector array. For a two-Airy-unit array, the axial resolution is 1.34 times better than in a conventional microscope for the standard reassignment factor, and 1.28 times better for the new reassignment factor. The axial resolution of a confocal microscope with a two-Airy-unit pinhole is only 1.04 times better than conventional microscopy. We also examine the signal-to-noise ratio of a point object in a uniform background (called the detectability), and show that it is 1.6 times higher than in a confocal microscope.

Multiphoton microscopy: a personal historical review, with some future predictions

The historical development of multiphoton microscopy is described, starting with a review of two-photon absorption, and including two- and three-photon fluorescence microscopies, and second- and third-harmonic generation microscopies. The effects of pulse length on signal strength and breakdown are considered. Different contrast mechanisms, including use of nanoparticles, are discussed. Two new promising techniques that can be applied to multiphoton microscopy are described.

Eigenvalues of the coherency matrix for exact backscattering

An important approach to interpretation of the Mueller matrix is based on the eigenvalues of the coherency matrix, given by the roots of a quartic characteristic equation. For the case of backscattering, one eigenvalue is zero from reciprocity arguments, and the characteristic equation reduces to a cubic. These two approaches (quartic and cubic) to calculation of the eigenvalues for exact backscattering are analytically considered and compared. As expected, the cubic approach is usually simpler, but for the special case of two zero eigenvalues, either approach reduces to the predictions of the simple quadratic characteristic equation. Either approach can be used for numerical calculation of the eigenvalues. The variation in different purity measures with the values of the Mueller matrix elements is presented. An experimental Mueller matrix for backscattering from a turbid chiral medium is investigated.

Two-photon focal modulation microscopy for high-resolution imaging in deep tissue

Two-photon microscopy (2PM) is one of the most widely used tools for in vivo deep tissue imaging. However, the spatial resolution and penetration depth are still limited due to the strong scattering background. Here we demonstrate a two-photon focal modulation microscopy. By utilizing the modulation and demodulation techniques, background rejection capability is enhanced, thus spatial resolution and imaging penetration depth are improved. Compared with 2PM, the transverse resolution is increased by 70%, while the axial resolution is increased to 2-fold. Furthermore, when applied in conventional 2PM mode, it can achieve inertial-free scanning in either transverse or axial direction with in principle unlimited scanning speed. Finally, we applied 2PFMM in thick scattering samples to further examine the imaging performance. The results show that the signal-to-background ratio of 2PFMM can be improved up to five times of 2PM at the depth of 500 μm. Fluorescent imaging in the mouse brain tissue. 3D Thy1-GFP hippocampal neurons imaged by (A) 2PM compared with (B) 2PFMM; (C-H) xy maximum-intensity projection imaged by 2PM compared with 2PFMM. Scale bar 50 μm.

Coherency and differential Mueller matrices for polarizing media

The elements of the coherency matrix give the strength of the components of a Mueller matrix in the coherency basis. The Z-matrix (called the polarization-coupling matrix or state-generating matrix) represents a partial sum of the coherency expansion. For transmission through a deterministic medium, the coherency elements can be used directly as generators to calculate the development of polarization upon propagation. The commutation properties of the coherency elements are investigated. New matrices that we call the W-matrix and the X-matrix, both different representations of the Z-matrix in a Jones basis, are introduced. The W-matrix controls the transformation of the Jones vector and also the covariance matrix. The product of the X-matrix with its complex conjugate gives the matrix representation of the Mueller matrix in the Jones basis. The development of Mueller matrix and coherency matrix elements upon propagation through some examples of a uniform medium is investigated. It is shown that the coherency matrix is more easily interpreted than the Mueller matrix. Analytic expressions are presented to calculate the elementary polarization properties from coherency matrix elements or Mueller matrix parameters.

Partially coherent microscope imaging system in phase space: effect of defocus and phase reconstruction

Explicit relationships between the defocused partially coherent cross-coefficient and phase space representations in the image plane are derived. Measurement of a phase space representation in the image plane in principle allows the complex image to be extracted. Implications for phase retrieval using the weak object transfer function or the transport of intensity equation are considered. The phase gradient transfer function, which determines the image for an object exhibiting a slowly varying phase gradient for a partially coherent microscope system, is derived. The effect of the effective source size and geometry on phase imaging with the transport of intensity equation is investigated. The primary consequence of source shape is a rescaling of the phase reconstruction. An annular source is found to give close to a linear response, while at the same time providing improved transverse resolution and an improved response to low spatial frequencies.

Pixellated circle

For applications in optical systems it is often necessary to represent a circular aperture in a pixellated form. An objective parameter is introduced that is a measure of how well an approximate circle can be generated from a small array of square pixels. Both filled circles (disks) and rings are considered. Arrays with a width given by an even number of pixels can also be used to generate quadrants of a circle. Rings with outer and inner profiles given by optimum circles or quadrants can be summed to fill a complete circle or quadrant.

Partially coherent microscope in phase space

Explicit expressions are presented for different phase-space representations (mutual intensity, Wigner distribution function, and ambiguity function) of the partially coherent image wave field in a microscope system. These are separated into system- and object-dependent parts. The partially coherent image in phase space can be described in terms of different 6D system-dependent kernels, all Fourier transforms of the system mutual spectrum, the region of overlap of two displaced objective pupils and the effective source. The image intensity can be expressed in terms of a 4D kernel, the convolution in spatial frequency of the source, and the Wigner distribution function of the objective pupil, given by a marginal of, or a section through, the respective phase-space kernels.

Label-free ultra-sensitive visualization of structure below the diffraction resolution limit

For both fundamental study of biological processes and early diagnosis of diseases, information about nanoscale changes in tissue and cell structure is crucial. Nowadays, almost all currently known nanoscopy methods rely upon the contrast created by fluorescent stains attached to the object or molecule of interest. This causes limitations due to the impact of the label on the object and its environment, as well as its applicability in vivo, particularly in humans. In this paper, a new label-free approach to visualize small structure with nano-sensitivity to structural alterations is introduced. Numerically synthesized profiles of the axial spatial frequencies are used to probe the structure within areas whose size can be beyond the diffraction resolution limit. Thereafter, nanoscale structural alterations within such areas can be visualized and objects, including biological ones, can be investigated with sub-wavelength resolution, in vivo, in their natural environment. Some preliminary results, including numerical simulations and experiments, which demonstrate the nano-sensitivity and super-resolution ability of our approach, are presented.

Factorization of the coherency matrix of polarization optics

We show that the coherency matrix associated with a general depolarizing Mueller matrix can be factorized into the product of a matrix, the coherency matrix factor, and its conjugate transpose. The coherency matrix factor contains all the information in the Mueller matrix, and directly shows useful properties in an illustrative fashion. Propagation through a nondeterministic uniform medium is analyzed. Some examples for simple systems are shown, and an experimental Mueller matrix is considered. The coherency matrix and the coherency matrix factor can be diagonalized, even if the Mueller matrix cannot.

Image formation in image scanning microscopy, including the case of two-photon excitation

The effect of combining the image scanning microscopy (ISM) technique with two-photon fluorescence microscopy is analyzed. The effective spatial frequency cutoff can be doubled, as compared with conventional two-photon fluorescence microscopy, and the magnitude of the optical transfer function near the cutoff of conventional two-photon microscopy is increased by orders of magnitude. For the two-photon case, it is found that the optimum pixel reassignment factor in ISM is not equal to one half, as is often assumed in single-photon fluoresence image scanning microscopy, because the excitation and detection point spread functions are different. The optimum reassignment factor depends on the noise level, and in general the useful cutoff spatial frequency is about 1.8 times that for conventional two-photon microscopy. The effect of altering the reassignment factor in single-photon fluorescence ISM with a Stokes shift is also investigated. Illumination using pupil filters, such as by a Bessel beam, is considered. Using a ring detector array is found to result in good imaging behavior, exhibiting a sharpening of the point spread function by a factor of 1.7 compared with conventional fluorescence. Image formation in ISM can be considered in a four-dimensional spatial frequency space, giving new insight into the imaging properties. This approach is related to phase space representations such as the Wigner distribution function and the ambiguity function. A noniterative algorithm for image restoration is proposed.

Expressions for parallel decomposition of the Mueller matrix: erratum

An error in our paper [J. Opt. Soc. Am. A33, 741 (2016)JOAOD60740-323210.1364/JOSAA.33.000741] is pointed out.

Parameterization of the deterministic Mueller matrix: application to a uniform medium

The extraction of the elementary polarization properties of a uniform medium from a deterministic Mueller matrix has been considered by several researchers. The relationship between a parameterization of the deterministic Mueller matrix that we described recently and the elementary polarization properties for a uniform medium is investigated. The elementary polarization properties can be calculated exactly from the Mueller matrix parameters. Simplified forms for the Mueller matrix parameters in terms of the elementary polarization properties are presented. The effect on the Mueller matrix of varying the ratio of the total diattenuation to the total retardance is investigated. Approximate forms for the elementary polarization properties in terms of the Mueller matrix parameters, valid for weak optical media, are given.

Parameterization of the Mueller matrix

A deterministic Mueller matrix (Mueller-Jones matrix) contains seven independent parameters. By writing the so-called coherence vector in parametric form, the Mueller matrix can also be written in parametric form, where the matrix elements automatically satisfy the known relationships between each other. Three of these parameters are also related to the so-called anisotropy coefficients. The approach is generalized to express all 16 elements of a general Mueller matrix in terms of a scalar and five three-dimensional vectors. Many properties of a Mueller matrix can be written simply in terms of these vectors. Published experimental matrices are considered by this procedure.

Interpretation of the optical transfer function: Significance for image scanning microscopy

The optical transfer function (OTF) is widely used to compare the performance of different optical systems. Conventionally, the OTF is normalized to unity for zero spatial frequency, but in some cases it is better to consider the unnormalized OTF, which gives the absolute value of the image signal. Examples are in confocal microscopy and image scanning microscopy, where the signal level increases with pinhole or array size. Comparison of the respective unnormalized OTFs gives useful insight into their relative performance. The significance of other properties of the general OTF is discussed.

Three-dimensional polarization algebra

If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large (81×81) but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar-Stokes parameters.

Geometry of the Mueller matrix spectral decomposition

An arbitrary Mueller matrix can be decomposed into a sum of up to four deterministic Mueller-Jones matrices, with strengths given by the eigenvalues of an associated Hermitian matrix. A geometrical representation of the eigenvalues in terms of the matrix invariants, using a barycentric (quaternary) plot, is presented. Different polarization purity measures can be expressed in terms of the barycentric coordinates.

Superconcentration of light: circumventing the classical limit to achievable irradiance

Concentration of light is limited by a fundamental physical principle, which ensures that étendue, the product of area and solid angle, can never decrease in an optical system. In microscopy, many superresolving methods, which can overcome the classical resolution limit, have recently emerged. We propose, and demonstrate experimentally, that it is also possible to circumvent the classical light concentration limit. Actually, most superresolution methods exhibit a common drawback: with respect to the total number of emitted photons, they are less efficient than standard widefield microscopy. Most methods "shave"' the point spread function (PSF) by discarding the disturbing signal from its edge. We show, that in contrast to PSF-shaving, methods related to reassignment microscopy (image scanning microscopy, optical photon reassignment, rescan confocal, instant structured illumination microscopy) concentrate all detected photons in their superresolving images and thereby increase the detected signal per sample area compared to widefield microsopy. We term this behavior superconcentration, as it breaks the classical light concentration limit.

Expressions for parallel decomposition of the Mueller matrix

It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system. We introduce forms for the matrices for transforming between the column vector forms of these different matrices. A review of matrix algebra is presented. We find that there is no great advantage, from the point of view of matrix manipulation, in using quantum mechanics ordering rather than the optical ordering of the Stokes parameters, as has been claimed elsewhere.

Image scanning microscopy with a quadrant detector

Confocal scanning microscopy (CSM) is the most widely used modern optical microscopy technique. Theoretically, it allows the diffraction barrier to be surpassed by a factor of 2, but practically this improvement is sacrificed to obtain a good signal-to-noise ratio (SNR). Image scanning microscopy (ISM) solves this limitation but, in the current implementations, the system complexity is increased and the versatility of CSM is reduced. Here we show that ISM can be straightforwardly implemented by substituting the single point detector of a confocal microscope with a quadrant detector of the same size, thus using a small number of detector elements. This implementation offers resolution close to the CSM theoretical value and improves the SNR by a factor of 1.5 with respect to the CSM counterpart without losing the optical sectioning capability and the system versatility.

Realistic wave-optics simulation of X-ray phase-contrast imaging at a human scale

X-ray phase-contrast imaging (XPCI) can dramatically improve soft tissue contrast in X-ray medical imaging. Despite worldwide efforts to develop novel XPCI systems, a numerical framework to rigorously predict the performance of a clinical XPCI system at a human scale is not yet available. We have developed such a tool by combining a numerical anthropomorphic phantom defined with non-uniform rational B-splines (NURBS) and a wave optics-based simulator that can accurately capture the phase-contrast signal from a human-scaled numerical phantom. Using a synchrotron-based, high-performance XPCI system, we provide qualitative comparison between simulated and experimental images. Our tool can be used to simulate the performance of XPCI on various disease entities and compare proposed XPCI systems in an unbiased manner.

Zernike expansion of pupil filters: optimization of the signal concentration factor

Amplitude pupil filters for optimizing the signal concentration factor for a point spread function of given transverse and/or axial widths are derived. The pupil is expanded in a basis of Zernike polynomials. It is shown that the pupil that maximizes the signal concentration factor for a given transverse gain has a quadratically varying amplitude profile, as was shown in a previous paper, while the pupil that maximizes the signal concentration factor for a given axial gain has a quartic amplitude profile.

Diffraction of a focused wave by an aperture: a new perspective

A new approach for calculating the field in the focal region along lines through the focal point of a lens is presented. In particular, the method is applied to a circular aperture. It is also applied to other shaped apertures, including circular sectors or segments, such as a semicircular aperture or Hilbert mask, and to polygonal shapes. The diffracted field is calculated by a one-dimensional Fourier transform, and can be used for accurate calculation at observation points distant from the focus. The approach gives new insight to appreciating the asymptotic behavior of the diffracted field, and the existence of intensity zeros, for different aperture shapes.

Optimization of pupil filters for maximal signal concentration factor

A new optimization for a continuously varying amplitude pupil filter that maximizes the signal concentration factor for a given transverse gain is derived. The filter has a simple parabolic amplitude transmittance, and is an example of a Sonine filter. The connection between different definitions of gain factor is discussed.

Temporal reshaping of two-dimensional pulses

An analytic study of complete cylindrical focusing of pulses in two dimensions is presented, and compared with the analogous three-dimensional case of focusing over a complete sphere. Such behavior is relevant for understanding the limiting performance of ultrafast, planar photonic and plasmonic devices. A particular spectral distribution is assumed that contains finite energy. Separate ingoing and outgoing pulsed waves are considered, along with the combination that would be generated in free space by an ingoing wave. It is shown that for the two dimensional case, in order to produce a temporally symmetrical pulse at the focus, an asymmetric pulse must be launched. A symmetrical outgoing pulse is generated from a source with asymmetric time behavior, or an anti-symmetric input pulse. These results are very different from the corresponding three-dimensional case, and imply fundamental limitations on the performance of ultrafast, tightly focused, two-dimensional devices.

Two-dimensional complex source point solutions: application to propagationally invariant beams, optical fiber modes, planar waveguides, and plasmonic devices

Highly convergent beam modes in two dimensions are considered based on rigorous solutions of the scalar wave (Helmholtz) equation, using the complex source point formalism. The modes are applicable to planar waveguide or surface plasmonic structures and nearly concentric microcavity resonator modes in two dimensions. A novel solution is that of a vortex beam, where the direction of propagation is in the plane of the vortex. The modes also can be used as a basis for the cross section of propagationally invariant beams in three dimensions and bow-tie-shaped optical fiber modes.

Polarized focused vortex beams: half-order phase vortices

A theoretical treatment is presented for the focusing of polarized vortex beams, including the generation of Bessel beams. A combination of a phase vortex with arbitrary topological charge, and a polarization vortex of arbitrary order is considered. Results are given for both paraxial and high NA systems. Conditions for the presence of non-zero on-axis intensity are given. An interesting observation is that half-order phase vortices can exist, without the existence of any phase discontinuity. The behavior of Bessel beams with half-order phase vortices is investigated.

Recent advances in optical microscopy methods for subcellular imaging of thick biological tissues

Optical microscopy has been widely applied in cellular and subcellular imaging. Conventional light microscopes, however, have rather limited imaging depth and are limited to imaging only mechanically sectioned thin samples. Multiphoton microscopy and optical coherence microscopy are common techniques for diffraction-limited imaging beyond an imaging depth of 0.5 mm. Focal modulation microscopy is a novel method that combines confocal spatial filtering with focal modulation to reject out-of-focus backgrounds. Focal modulation microscopy has demonstrated an imaging depth comparable to those of multiphoton microscopy and optical coherence microscopy, near-real-time image acquisition, and capability with a multiple contrast mechanism.

Creation of a 50,000λ long needle-like field with 0.36λ width: comment

In a recent paper, a method for the generation of a long, narrow needle of light was proposed [J. Opt. Soc. Am. A 31, 500 (2014)]. The implications of this on our appreciation of the properties of Bessel beams are discussed.

Multipole and plane wave expansions of diverging and converging fields

This paper presents and compares two basis systems, spherical harmonics and plane waves, for studying diverging and converging beams in an optical system. We show a similarity between a converging field and the time reversed field of a radiation field. We present and analyze the differences between the Debye-Wolf diffraction integral and the multipole theory for focusing of polarized light. The Debye-Wolf diffraction integral gives a well-known anomalous behavior on the optical axis and at the edge of the focused beam that can be avoided by using the multipole theory.

Maréchal condition and the effect of aberrations on Strehl intensity

The effect of aberrations on the Strehl intensity is analyzed. The aberrations are assumed to be random and normally distributed. A variety of different correlation coefficients for the aberration variation are discussed, including Gaussian correlation and Kolmogorov turbulence. For weak aberrations, the Strehl ratio is independent of the correlation function. The Strehl ratio for a given root-mean-square aberration is greater for a smaller number of correlation areas. For Kolmogorov turbulence, the Strehl ratio is lower than for Gaussian correlation.

Analytic method to optimize aperture design in focal modulation microscopy

Focal modulation microscopy (FMM) has been demonstrated more effective than confocal microscopy for imaging of thick biological tissues. To improve its penetration depth further, we propose a simple analytical method to enlarge the modulation depth, the unique property of FMM directly linked to its signal-to-noise ratio. The modulation depth increases as the excitation intensity of the binary phase aperture status is pushed further away from the focal region of the detection optics, thereby creating a dark region in the focal volume, which we call maximally flat crater (MFC). By direct algebraic manipulation, MFCs are achieved for both scalar and vector diffraction optics. Numerical results show that the modulation depth from MFC is very close to the maximum values, with a small difference less than 3% for the same number of subapertures. Applications of bifocus produced by MFC apertures are also discussed.

Focusing of vortex beams: Lommel treatment

Focusing of vortex beams by a lens with circular aperture in the paraxial scalar Debye regime is analyzed. The amplitude in the focal region can be expressed naturally in terms of higher order Lommel functions of two variables. Using recurrence relationships, these can then be expressed in terms of low-order Lommel functions. The phase variation in the focal region is investigated, showing some interesting behavior of the Gouy phase anomaly.

Calculation of the volumetric diffracted field with a three-dimensional convolution: the three-dimensional angular spectrum method

The first Rayleigh-Sommerfeld diffraction formula is treated in an exact form as a three-dimensional (3D) convolution in the spatial domain. Therefore, a 3D Fourier transform can be employed to convert the 3D diffracted electromagnetic field to the reciprocal space without approximations, which we call the 3D angular spectrum (3D-AS) method. It is also demonstrated that if evanescent waves are neglected, the 3D-AS method can be readily implemented numerically, with the results in good agreement with theoretical predictions.

Balanced diffraction aberrations, independent of the observation point: application to a tilted dielectric plate

Balancing of Zernike aberrations breaks down if the defocus term is large enough that the condition (z/λ)≪2/[π(NA)⁴] is not satisfied. A modified Zernike aberration expansion, based on the Zernike aberrations, is developed that accurately includes axial displacement as a low-order term, even for large displacements. This expansion can be used to analyze aberrations for on-axis illumination of a high numerical aperture system. But more importantly, for systems of moderate numerical aperture it allows balanced aberration coefficients to be determined independent of the assumption of a particular reference point. The approach is applied to the case of a tilted dielectric plate. An exact expression is given for the wave front aberration, valid for both large angles of tilt and high beam convergence angles, that is independent of observation distance. Analytical expressions for the third- and fifth-order aberration coefficients are derived. Expressions are given for expansion of multiple-angle power series terms into Zernike polynomials.