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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.

The properties of signal strength and integrated intensity in a scanned imaging system are reviewed. These prop-erties 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.(c) 2022 Optica Publishing Group

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)’.

Surface topography measuring interference microscopy is a three-dimensional (3D) imaging technique that provides quantitative analysis of industrial and biomedical specimens. Many different instrument modalities and configurations exist, but they all share the same theoretical foundation. In this paper, we discuss a unified theoretical framework for 3D image (interferogram) formation in interference microscopy. We show how the scattered amplitude is linearly related to the surface topography according to the Born and the Kirchhoff approximations and highlight the main differences and similarities of each. With reference to the Ewald and McCutchen spheres, the relationship between the spatial frequencies that characterize the illuminating and scattered waves, and those that characterize the object, are defined and formulated as a 3D linear filtering process. It is shown that for the case of near planar surfaces, the 3D filtering process can be reduced to two dimensions under the small height approximation. However, the unified 3D framework provides significant additional insight into the scanning methods used in interference microscopy, effects such as interferometric defocus and ways to mitigate errors introduced by aberrations of the optical system. Furthermore, it is possible to include the nonlinear effects of multiple scattering into the generalized framework. Finally, we consider the inherent nonlinearities introduced when estimating surface topography from the recorded interferogram.

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.

Improved expressions are given for the performance parameters for transverse and axial gains for complex pupil filters. These expressions can be used to predict the behavior of filters that give a small axial shift in the focal intensity maximum and also predict the changes in gain for different observation planes.

If the pinhole in a confocal microscope is replaced by a detector array, each detector pixel simultaneously gives a confocal image. The resultant four-dimensional data set can be reconstructed to give improved resolution and signal to noise ratio.

Image Scanning Microscopy (ISM) [1] successfully overcomes the trade-off between resolution and signal-to-noise ratio of traditional confocal microscopes by considering the spatial distribution of the fluorescence emission light and by reassigning the detected photons accordingly (i.e., pixel-reassignment). A recent implementation [2] upgrades a confocal to an image scanning microscope by substituting the traditional single-element detector with a SPAD array detector. Notably, the SPAD array samples the fluorescence signal from the detection/probing volume both in space (such as cameras) and in time (such as single element detectors, e.g., SPADs), providing potentially significant extra information for a variety of experimental contexts. To fully exploit this advantage, we present a versatile FPGA-based time-resolved microscopy platform that parallelly acquires all the SPAD array signals with a sub-nanosecond temporal resolution, thanks to a Digital Frequency Domain architecture. In the context of stimulated emission depletion (STED) microscopy, we leverage the platform to decrease the STED power needed to achieve a target spatial resolution [3] . In particular, we show how to synergically exploit both the spatial and temporal extra information to implement a new and dedicated photon-reassignment method for STED microscopy. This method not only takes advantage of the pixel-reassignment principle but also compensates for all the different sources of background which typically reduce resolution and imaging quality in STED microscopy, i.e., incomplete depletion [4] , [5] , direct-excitation from the STED beam and out-of-focus signal. Additionally, the platform allows for fluorescence lifetime imaging and a straightforward pulsed interleaved excitation (PIE) implementation, enabling dual-color STED microscopy ( Fig. 1 ).

Mueller matrix microscopy is an advanced imaging technique providing a full characterization of the optical polarization fingerprint of a sample. The Lu-Chipman (LC) decomposition, a method based on the modeling of elementary polarimetric arrangements and matrix inversions, is the gold standard to extract each polarimetric component separately. However, this models the optical system as a small number of discrete optical elements and requires a priori knowledge of the order in which these elements occur. In stratified media or when the ordering is not known, the interpretation of the LC decomposition becomes difficult. In this work, we propose a new, to our knowledge, representation dedicated to the study of biological tissues that combines Mueller matrix microscopy with a phasor approach. We demonstrate that this method provides an easier and direct interpretation of the retardance images in any birefringent material without the use of mathematical assumptions regarding the structure of the sample and yields comparable contrast to the LC decomposition. By validating this approach through numerical simulations, we demonstrate that it is able to give access to localized structural information, resulting in a simple determination of the birefringent parameters at the microscopic level. We apply our novel, to our knowledge, method to typical biological tissues that are of interest in the field of biomedical diagnosis.

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.

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.

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 similar to 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 similar to 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. (C) 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement H2020 Marie Sklodowska-Curie Actions (AdaptiveSTED, No. 794531); FondsWetenschappelijk Onderzoek (G092915, V429717N); European Research Council (Bright Eyes, No. 818699). Vicidomini, G (corresponding author), Ist Italiano Tecnol, Mol Microscopy & Spect, Genoa, Italy giuseppe.vicidomini@iit.it

We propose a straightforward implementation of two-photon image scanning microscopy (2PE-ISM) that, by leveraging our recently introduced single-photon avalanche diode (SPAD) array detector and a novel blind image reconstruction algorithm is shown to dramatically improve the optical resolution of two-photon imaging, in various test samples. We show how our computational ISM approach is able to adapt to changing imaging conditions, thus ensuring optimal image quality. We also show how our recently introduced blind deconvolution approaches can be integrated into the image reconstruction workflow to further improve the image quality.

Microscopy using a scanning approach is reviewed, with numerous references to previous work, as at 1987, before the confocal laser scanning microscope was widely available commercially. The advantages of microscope imaging using a scanning technique are described. The principles of confocal microscopy are introduced. Use of Bessel beam illumination in a confocal microscope is discussed. This combination can increase spatial resolution while retaining some confocal optical sectioning. The advantages of imaging using a scanning method with a point source are detailed. For example, differential phase contrast may be performed to obtain phase information. Use of a point source allows different types of spectroscopy to be performed in a spatially resolved fashion. An advantage of point-source systems for such applications is that imaging, performed with the incident radiation, is separated from wavelength selection and analysis of the emitted radiation, thus simplifying system design and resulting in superior performance. Use of a point source to excite carriers in a semiconducting sample allows the electronic properties of materials and devices to be investigated. Advantages of object scanning, rather than beam scanning, are discussed. Advantages of using a laser source are described, including the use of the heterodyne technique as a synthetic confocal pinhole, and nonlinear microscopy such as two-photon fluorescence and CARS (coherent anti-Stokes Raman spectroscopy). It should be noted that 1987, as well as being before commercial confocal microscopes were widely available, was before the special properties of Bessel beams were appreciated, and before two photon fluorescence microscopy was first experimentally demonstrated.

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.

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 ISM platform – based on a new single-photon avalanche diode (SPAD) array detector – coupled with a novel blind image reconstruction method, is shown to improve the effective resolution, as well as the overall image quality of 2PE microscopy. Indeed, in stark contrast to conventional single-point detectors, SPAD array detectors give access to the images of any excited scanning region, from which it is possible to decode information about the aberrations/distortions – occurring during imaging – able to substantially improve the reconstruction. Most importantly, our 2PE-ISM implementation requires no calibration or other input from the user; it works like any familiar two-photon system, but produces higher resolution images deep into thick samples. In our view, this novel implementation is the key for making 2PE-ISM mainstream.

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.

Almost all known nanoscopy methods rely upon the contrast created by fluorescent labels attached to the object of interest. This causes limitations on their applicability to in vivo imaging. A new label-free spectral encoding of spatial frequency (SESF) approach to nanoscale probing of three-dimensional structures has been developed. It has been demonstrated that spatial frequencies, encoded with optical wavelengths, can be passed though the optical system independent of the resolution of the imaging system. As a result information about small size structures can be detected even using a low resolution imaging system. Different versions of the SESF imaging have been published [1-7], including a novel contrast mechanism for high resolution imaging [1], real time nano-sensitive imaging [2], reconstruction the axial (along depth) spatial frequency profiles for each point with nano-sensitivity to structural changes [3], and the adaptation of the SESF approach to depth resolving imaging [4,5]. Recently the SESF approach has been applied to break the diffraction limit and dramatically improve resolution [6,7]. Here we present further development of the SESF approach including correlation mapping SESF imaging. Both results of numerical simulation and preliminary experimental results, including biological objects, will be presented. [1] Alexandrov, et.al., Opt. Lett. 36 3323 (2011). [2] Alexandrov, et.al., Opt. Express 20 (8) 9203 (2012). [3] Alexandrov, et.al., Appl. Phys. Let., 101 033702 (2012). [4] Uttam, et.al., Opt. Express, 21, 7488 (2013). [5] Alexandrov, et.al., Nanoscale, 6, 3545 (2014). [6] Alexandrov, et.al., Sci. Rep., 5, doi: 10.1038/srep13274 (2015). [7] Alexandrov, et.al., J. Biophotonics, https://doi.org/10.1002/jbio.201700385 (2018).

Image scanning microscopy (ISM) can improve the effective spatial resolution of confocal microscopy to its theoretical limit. However, current implementations are not robust or versatile, and are incompatible with fluorescence lifetime imaging (FLIM). We describe an implementation of ISM based on a single-photon detector array that enables super-resolution FLIM and improves multicolor, live-cell and in-depth imaging, thereby paving the way for a massive transition from confocal microscopy to ISM. A single-photon detector array enables robust and versatile image scanning microscopy (ISM) on any confocal microscope. This implementation makes super-resolution FLIM possible and eases a transition from confocal microscopy to ISM.

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.

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.

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.

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.

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.

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.

Conventional optical microscopes, such as brightfield, darkfield, phase contrast or differential interference contrast microscopes are partially coherent imaging systems. Imaging in a partially coherent system was first analyzed by Hopkins only in 1953. He propagated the mutual intensity through the optical system, but did not give an expression for the mutual intensity of the image itself. The mutual intensity is a four dimensional (4D) quantity that contains information about the modulus and phase of the image wave field, which depends on the object’s complex refractive index in 3D. The mutual intensity is related to other representations such as the Wigner distribution function (WDF) and ambiguity function. Explicit expressions for different phase space representations of the image wave field are given. The expressions separate into system and object dependent parts. In addition, explicit relationships between the defocused partially coherent cross-coefficient and phase space representations in the image plane are derived.

If a scanning illumination spot is combined with a detector array, we acquire a 4 dimensional signal. Unlike confocal microscopy with a small pinhole, we detect all the light from the object, which is particularly important for fluorescence microscopy, when the signal is weak. The image signal is basically a cross-correlation, and is highly redundant. It has more than sufficient information to reconstruct an improved resolution image. A 2D image can be generated from the measured signal by pixel reassignment. The result is improved resolution and signal strength, the system being called image scanning microscopy. A variety of different signal processing techniques can be used to predict the reassignment and deconvolve the partial images. We use an innovative single-photon avalanche diode (SPAD) array detector of 25 detectors (arranged into a 5× 5 matrix). We can simultaneously acquire 25 partial images and process to calculate the final reconstruction online.

We demonstrate a fast, switchable achromatic phase shifter operating on the geometric phase principle, using ferroelectric liquid crystal technology for rapid 3D biological imaging in a full field optical coherence tomography.

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.

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

The use of an array of detectors can help overcoming the traditional limitation of confocal microscopy: the compromise between signal and theoretical resolution. Each element independently records a view of the sample and the final image can be reconstructed by pixel reassignment or by inverse filtering (e.g. deconvolution). In this work, we used a SPAD array of 25 detectors specifically designed for this goal and our scanning microscopy control system (Carma) to acquire the partial images and to perform online image processing. Further work will be devoted to optimize the image reconstruction step and to improve the fill-factor of the detector.

Three-dimensional imaging at high-spatiotemporal resolutions and over large penetration depths is key for unmasking the dynamics and structural organization of complex biological systems. However, the need to axially shift the focus, with consequent limitations in imaging speed, and signal degradation at large depths due to scattering effects, makes this task challenging. Here, we present a novel approach in 2-photon excitation microscopy that allows fast volumetric imaging and enhanced signal-to-background (S/B) in thick tissue. Our technique is based on ultrafast beam shaping at each pixel by means of an acoustic optofluidic lens. Shaping the excitation beam with different phase profiles enables both high-speed axial focus shifting, for continuous volumetric imaging, and controlled aberrated imaging, advantageous for out-of-focus background removal. We provide a theoretical description of our approach, and demonstrate volumetric imaging of neuronal cells from a mouse brain slice with enhancements in S/B up to a factor of 10 over a depth of 600 μm.

Many papers have claimed the attainment of super-resolution, i.e. resolution beyond that achieved classically, by measurement of the profile of a feature in the image. We argue that measurement of the contrast of the image of a dark bar on a bright background does not give a measure of resolution, but of detection sensitivity. The width of a bar that gives an intensity at the center of the bar of 0.735 that in the bright region (the same ratio as in the Rayleigh resolution criterion) is λ/(13.9×numerical aperture) for the coherent case with central illumination. This figure, which compares with λ/(numerical aperture) for the Abbe resolution limit with central illumination, holds for the classical case, and so is no indication of super-resolution. Theoretical images for two points, two lines, arrays of lines, arrays of bars, and grating objects are compared. These results can be used a reference for experimental results, to determine if super-resolution has indeed been attained. The history of the development of the theory of microscope resolution is outlined.

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.

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.

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.

Concentration of light is limited by a fundamental physical principle, which ensures that etendue, 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.