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.