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A monochromatic fully polarized electromagnetic wave propagating in free
space possesses, in general, two families of singular lines (C lines) on
which the transverse field is circularly polarized. The distribution of
polarization ellipses around a C line shows that it obeys the same
classification scheme as the isotropic points of a two-dimensional
symmetric tensor: that is, a given section of a C line may belong to one
of three different line patterns and it may be elliptic or hyperbolic.
In addition it may be left-or right-handed. However, the way in which
the polarization ellipses are executed in time shows that C lines may
also be regarded as singularities of phase, analogous to line
dislocations or interference fringes in scalar waves. From this point of
view, a given line is of edge or screw type, according to its
orientation, or, more generally, is a curved line of mixed edge-screw
type. The whole field is divided into regions of opposite hand by
surfaces of linear polarization and each family of C lines is confined
to just one of these regions.

Content uploaded by John Frederick Nye

Author content

All content in this area was uploaded by John Frederick Nye on Feb 25, 2016

Content may be subject to copyright.

... However, the topological information about the optical anisotropy of the biological layer appears to be integrally averaged over all coordinates and geometric dimensions of MM map images within a quantitative statistical analysis. For statistical quantification of polarization-detected local variations in optical anisotropy parameters, statistical analysis of scale-selective samples from MM data derived from polarization-singular [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] and scale-selective [35][36][37][38][39][40] wavelet analysis may be most appropriate. ...

... The basic principles of complex vector singular analysis are formulated and described in details elsewhere, 19,[22][23][24][25][26] considered for optical fields, [27][28][29] and practically implemented in biomedical imaging. [30][31][32][33][34] Thus, for the first time, to our knowledge, biological tissues with a presence of linearly birefringence were characterized analytically in terms of the formation of linear and circular polarization singularities. ...

... The approbation of this approach demonstrates a significant improvement in the sensitivity and accuracy of MM polarimetry in the differential diagnosis of inflammatory and oncological conditions. [30][31][32][33][34][35][36]42,43 However, the polarization-singular [19][20][21][22][23][24][25][26][27][28][29]41,44,45 and scale-selective wavelet [37][38][39][40]46,47 approaches in biomedical diagnosis require further developments. ...

Significance:
Mueller-matrix polarimetry is a powerful method allowing for the visualization of malformations in biological tissues and quantitative evaluation of alterations associated with the progression of various diseases. This approach, in fact, is limited in observation of spatial localization and scale-selective changes in the poly-crystalline compound of tissue samples.
Aim:
We aimed to improve the Mueller-matrix polarimetry approach by implementing the wavelet decomposition accompanied with the polarization-singular processing for express differential diagnosis of local changes in the poly-crystalline structure of tissue samples with various pathology.
Approach:
Mueller-matrix maps obtained experimentally in transmitted mode are processed utilizing a combination of a topological singular polarization approach and scale-selective wavelet analysis for quantitative assessment of the adenoma and carcinoma histological sections of the prostate tissues.
Results:
A relationship between the characteristic values of the Mueller-matrix elements and singular states of linear and circular polarization is established within the framework of the phase anisotropy phenomenological model in terms of linear birefringence. A robust method for expedited (up to ∼15 min) polarimetric-based differential diagnosis of local variations in the poly-crystalline structure of tissue samples containing various pathology abnormalities is introduced.
Conclusions:
The benign and malignant states of the prostate tissue are identified and assessed quantitatively with a superior accuracy provided by the developed Mueller-matrix polarimetry approach.

... L lines and C lines are called polarization singularities and are vector versions of scalar vortex lines in wave fields, existing in light [24][25][26] and acoustic and water waves [27] (both acoustic and water waves have a vector nature [28,29]) where some property of the general polarization ellipse is not defined. In 3D fields, polarization singularities are often described as the underlying skeleton that embeds highly complex topologies into the field's polarization texture [30,31]. ...

... (13) near a dual-3D vortex (E(r 0 ) = H(r 0 ) = 0). Note that by normalizing E, the argument to Im{} in Eq. (13) defines the local electric wave vector [25]: ...

We present a study of 3D electromagnetic field zeros, uncovering their remarkable characteristic features and propose a classifying framework. These are a special case of general dark spots in optical fields, which sculpt light’s spatial structure into matter-moving, information-rich vortices, escape the diffraction limit for single-molecule imaging, and can trap particles for nanoscale manipulation. Conventional dark spots are 2D in two aspects: localized in a plane and having a non-zero out-of-plane field component. We focus on non-paraxial fields, where 3D dark spots can exist non-stably at fully localized points, making distinct imprints in the flux of energy and momentum, and in the light’s polarization texture. With this work, we hope to enhance current dark spot applications, or inspire new ones impossible with lower-dimensional zeros.

... depending on two angles (θ 0 , ψ 0 ) relative to the eigenstates of the linearly polarised background field, |x⟩, |y⟩. For general θ 0 and ψ 0 the initial polarisation forms an ellipse in the two-state space with orientation defined by an angle tan 2γ 0 = tan 2θ 0 cos ψ 0 between the ellipse's semi major-axis and |x⟩ [64,65]. When the relative phase angle ψ 0 is an integer multiple of π, the phase becomes real and the polarisation ellipse collapses to a line in the two-state space: the incoming photon is then linearly polarised at an angle θ 0 to |x⟩. ...

When photons propagate in vacuum they may fluctuate into matter pairs thus allowing the vacuum to be polarised. This \emph{linear} effect leads to charge screening and renormalisation. When exposed to an intense background field a \emph{nonlinear} effect can arise when the vacuum is polarised by higher powers of the background. This nonlinearity breaks the superposition principle of classical electrodynamics, allowing for light-by-light scattering of probe and background photons mediated through virtual pairs dressed by the background. Vacuum polarisation is a \emph{strong-field} effect when all orders of interaction between the virtual pair and the background must be taken into account. In this investigation we show that multiple scattering processes of this type may be observed by utilising high-energy laser pulses with long pulse duration, such as are available at facilities like ELI Beamlines. In combination with appropriate sources of high-energy probe photons, multiple probe-background light-by-light scattering allows for testing the genuine nonlinear regime of strong-field QED. This provides access to the uncharted nonperturbative regime beyond the weak-field limit.

... In vector waves, singularities manifest as lines of circular polarization in three-dimensional space, upon which the orientation of the major axis of the polarization ellipse is undefined; the typical form of these singularities is usually referred to as C-lines. 3 In a closed path around a C-line, the orientation changes by a half-integer multiple of 2π, and this multiple is called the topological index. Phase and polarization singularities typically intersect the transverse plane of an optical beam at a point; for polarization, we then refer to C-points. ...

Historically, infinity was long considered a vague concept—boundless, endless, larger than the largest—without any quantifiable mathematical foundation. This view changed in the 1800s through the pioneering work of Georg Cantor, who showed that infinite sets follow their own seemingly paradoxical mathematical rules. In 1924, David Hilbert highlighted the strangeness of infinity through a thought experiment now referred to as the Hilbert Hotel paradox, or simply Hilbert’s Hotel. The paradox describes a “fully” occupied imaginary hotel having an infinite number of single-occupancy rooms. The manager can always find a room for new guests by simply shifting current guests to the next highest room, leaving the first room vacant. The investigation of wavefield singularities has uncovered the existence of a direct optical analogy to Hilbert’s thought experiment. Since then, efforts have been made to investigate the properties of Hilbert’s Hotel by controlling the dynamics of phase singularities in “fractional” order optical vortex beams. Here, we have taken such proposals to the next level and experimentally demonstrated Hilbert’s Hotel using both phase and polarization singularities of optical fields. Using a multi-ramped spiral-phase-plate and a supercontinuum source, we generated and controlled fractional order vortex beams for the practical implementation of Hilbert’s Hotel in scalar and vector vortex beams. Using a multi-ramped spiral-phase-plate, we show the possibility for complicated transitions of the generalized Hilbert’s Hotel. The generic experimental scheme illustrates the usefulness of structured beams in visualizing unusual mathematical concepts and also for fractional vector beams driven by fundamental and applied research.

... Polarization singularities in monochromatic fields, on the other hand, have a multivalent definition in the literature, requiring only one or more parameters of the polarization ellipse (e.g., azimuthal angle and ellipticity angle) to be singular (1,4,8,(14)(15)(16)(17)(18)(19)(20)(21)(22). They cannot be considered as complete polarization singularities as the polarization is either still defined at the singularity {e.g., L lines and bright C points [(22) and the Supplementary Materials]} or singular only in a specific basis and not topologically protected {e.g., V points, dark C points [(8) and the Supplementary Materials]}. ...

Optical singularities play a major role in modern optics and are frequently deployed in structured light, super-resolution microscopy, and holography. While phase singularities are uniquely defined as locations of undefined phase, polarization singularities studied thus far are either partial, i.e., bright points of well-defined polarization , or are unstable for small field perturbations. We demonstrate a complete, topologically protected polarization singularity; it is located in the four-dimensional space spanned by the three spatial dimensions and the wavelength and is created in the focus of a cascaded metasurface-lens system. The field Jacobian plays a key role in the design of such higher-dimensional singularities, which can be extended to multidimen-sional wave phenomena, and pave the way for unconventional applications in topological photonics and precision sensing.

... Figure 1A shows the IR polarization state in the transverse plane when the two driving beams are collinear. On the optical axis, the polarization is purely circular, forming a "C point" (25,26). Around the C point, the field is elliptically polarized, and the orientation of the ellipse varies by π when travelling along a loop about the optical axis (Fig. 1A). ...

Symmetries and conservation laws of energy, linear momentum, and angular momentum play a central role in nonlinear optics. Recently, paraxial light fields with nontrivial topology have been attracting a keen interest. Despite not being eigenstates of the orbital and spin angular momenta (OAM and SAM), they are eigenstates of the generalized angular momentum (GAM) operator-a mixture of the OAM and SAM operators with fractional eigenvalues. By driving high harmonic generation with a polarization Möbius strip carrying a half-integer GAM charge and implementing angular momentum characterization methods in the extreme ultraviolet range, we demonstrate the linear scaling of the GAM with the harmonic order, each harmonic carrying a precise half-integer GAM charge. Our work shows that beyond SAM and OAM, the GAM is, in some situations, an appropriate quantum number. It paves the way for finer manipulations and applications of light beams containing fractional-order polarization singularities.

... In a closed path around an optical vortex, the phase always changes by an integer multiple of 2π; this multiple is referred to as the topological charge. In vector waves, singularities manifest as lines of circular polarization in threedimensional space, upon which the orientation of the major axis of the polarization ellipse is undefined; the typical form of these singularities are usually referred to as C-lines 3 . In a closed path around a C-line, the orientation changes by a halfinteger multiple of 2π, and this multiple is called the topological index. ...

Historically, infinity was long considered a vague concept - boundless, endless, larger than the largest - without any quantifiable mathematical foundation. This view changed in the 1800s through the pioneering work of Georg Cantor showing that infinite sets follow their own seemingly paradoxical mathematical rules. In 1924, David Hilbert highlighted the strangeness of infinity through a thought experiment now referred to as the Hilbert Hotel paradox, or simply Hilbert's Hotel. The paradox describes an "fully" occupied imaginary hotel having infinite number of single-occupancy rooms, the manager can always find a room for new guest by simply shifting current guests to the next highest room, leaving first room vacant. The investigation of wavefield singularities has uncovered the existence of a direct optical analogy to Hilbert's thought experiment. Since then, efforts have been made to investigate the properties of Hilbert's Hotel by controlling the dynamics of phase singularities in``fractional'' order optical vortex beams. Here, we have taken such proposals to the next level and experimentally demonstrated Hilbert's Hotel using both phase and polarization singularities of optical fields. Using a multi-ramped spiral-phase-plate and a supercontinuum source, we generated and controlled fractional order vortex beams for the practical implementation of Hilbert's Hotel in scalar and vector vortex beams. Using a multi-ramped spiral-phase-plate, we show the possibility for complicated transitions of the generalized Hilbert's Hotel. The generic experimental scheme illustrates the usefulness of structured beams in visualizing unusual mathematical concepts and also for fractional vector beams driven fundamental and applied research.

We show that Skyrmion ﬁeld lines, constructed from the local Stokes parameters, trace out lines of constant optical polarisation

Achieving the conversion from surface waves (SWs) to propagating waves has captivated long-standing interest, and various ingenious metasurfaces benefiting from the powerful control capability for electromagnetic waves are able to realize efficient SWs directional radiation. Nevertheless, most existing schemes still suffer from the bottlenecks of single radiation channel, uncontrollable radiation intensity, and immutable radiation pattern, which immensely hinder their practical application in high-integration intelligent devices. Herein, a series of appealing strategies are proposed to achieve the dual-channel SWs directional radiation with customizable radiation intensity and switchable radiation pattern. The dual-channel SWs radiation metadevice based on the phase modulation metasurface is designed to directionally radiate SWs in left-handed circular polarized channel and right-handed circular polarized channel and possesses the broadband frequency scanning characteristic. More strikingly, the intensity-customizable dual-channel SWs radiation metadevice loaded with lumped resistors can control the realized gain of two circular polarized radiation beams, and the pattern-switchable dual-channel SWs radiation metadevice loaded with PIN diodes can dynamically adjust the radiation direction of the radiation beams. Numerous simulations and experiments of the proof-of-concept prototypes with modular design corroborate the theoretical predictions. Our methodology shows unprecedented flexibility in regulating SWs directional radiation and has enormous potential in engineering applications.

Topological properties of optical strips of the vectors determining the polarization ellipse orientation which are constructed on non-planar contours were studied for the electric field reflected from a gold particle of ellipsoidal shape while irradiated by a plane monochromatic wave. The twisting of each optical strip traced near the particle can be characterised by the sum of the intrinsic twist index of the strip and writhe and full geometric torsion coefficients of the bypass contour. It is shown that the intrinsic twist index generally does not exceed half the linking number of the strip with the polarization singularity lines of the scattered near-field. The twists of the strips are localized near three distinct surfaces, converging on the C -line and usually change their directions. It is shown that the intrinsic twist index is the most suitable for the role of the main topological feature of the optical strip.

The properties and significance of disclinations in electromagnetic
diffraction patterns are examined. A disclination is defined in terms of
a hypothetical field pattern, and it is shown that field patterns
containing disclinations can exist as solutions of Maxwell's equations.
Solutions are first constructed for stationary disclinations in various
orientations; later, solutions representing curved and moving
disclinations are constructed. The effect of polarization on a straight
disclination is examined, and the disclination in the magnetic field
associated with an electric field is addressed. The effect of the scalar
perturbation is considered, as well as the influence of its combination
with the polarization effect. The sheath of apparent linear polarization
and the field of a perturbed pure screw disclination are treated.

When an ultrasonic pulse, containing, say, ten quasi-sinusoidal oscillations, is reflected in air from a rough surface, it is observed experimentally that the scattered wave train contains dislocations, which are closely analogous to those found in imperfect crystals. We show theoretically that such dislocations are to be expected whenever limited trains of waves, ultimately derived from the same oscillator, travel in different directions and interfere - for example in a scattering problem. Dispersion is not involved. Equations are given showing the detailed structure of edge, screw and mixed edge-screw dislocations, and also of parallel sets of such dislocations. Edge dislocations can glide relative to the wave train at any velocity; they can also climb, and screw dislocations can glide. Wavefront dislocations may be curved, and they may intersect; they may collide and rebound; they may annihilate each other or be created as loops or pairs. With dislocations in wave trains, unlike crystal dislocations, there is no breakdown of linearity near the centre. Mathematically they are lines along which the phase is indeterminate; this implies that the wave amplitude is zero.