Singularity Theory and Its Application

Izumiya, Shyūichi; Ishikawa, Goo; Shimada, Ichio; Tokunaga, Hitoo; Sano, Takasi

doi:10.1142/e004

Cited by 5 publications

(3 citation statements)

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“…Singularities represent a pervasive concept in different areas of mathematics and physics, constituting the fundamental skeletons and frameworks for many related concepts and their applications [1][2][3] . The field of singular optics originates from direct observations of singular points in intensity (caustics), phase (vortices), or polarization [4][5][6][7][8] .…”

Section: Introductionmentioning

confidence: 99%

Topological Polarization Singularities in Metaphotonics

Liu,

Shi

et al. 2020

Preprint

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Polarization singularities of vectorial electromagnetic fields locate at the positions (such as points, lines, or surfaces) where properties of polarization ellipses are not defined. They are manifested as circular and linear polarizations, for which respectively the semi-major axes and normal vectors of polarization ellipses become indefinite. First observed in conical diffraction in 1830's, the field of polarization singularities has been systematically reshaped and deepened by many pioneers of wave optics. Together with other exotic phenomena such as non-Hermiticity and topology, polarization singularities have been introduced into the vibrant field of nanophotonics, rendering unprecedented flexibilities for manipulations of light-matter interactions at the nanoscale. Here we review the recent results on the generation and observation of polarization singularities in metaphotonics. We start with the discussion of polarization singularities in the Mie theory, where both electric and magnetic multipoles are explored from perspectives of local and global polarization properties. We then proceed with the discussion of various photonic-crystal structures, for which both near-and far-field patterns manifest diverse polarization singularities characterized by the integer Poincaré or more general half-integer Hopf indices (topological charges). Next, we review the most recent studies of conversions from polarization to phase singularities in scalar wave optics, demonstrating how bound states in the continuum can be exploited to generate directly optical vortices of various charges. Throughout our paper, we discuss and highlight several fundamental concepts and demonstrate their close connections and special links to metaphotonics. We believe polarization singularities can provide novel perspectives for light-matter manipulation for both fundamental studies and their practical applications.

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Section: Introductionmentioning

confidence: 99%

Topological Polarization Singularities in Metaphotonics

Liu,

Shi

et al. 2020

Preprint

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“…Phase transitions, critical phenomena, and the corresponding critical exponents are fundamental topics in statistical mechanics. Though they should have a close relation with critical points of maps [1], catastrophe theory [2], and singularity theory [3,4] in mathematics, there is little application to these simple physical problems. In these geometrical standpoints, it is natural to expect critical points to be singularities on a certain surface or a curve.…”

Section: Introductionmentioning

confidence: 99%

Cusp singularity in mean field Ising model

et al. 2017

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An entropy of the Ising model in the mean field approximation is derived by the Hamilton-Jacobi formalism. We consider a grand canonical ensemble with respect to the temperature and the external magnetic field. A cusp arises at the critical point, which shows a simple and new geometrical aspect of this model. In educational sense, this curve with a cusp helps students acquire a more intuitive view on statistical phase transitions.

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“…The case that a system contains degenerate critical points appears frequently in an [6], the local structure is given as the differentiable function germ:…”

Section: Introductionmentioning

confidence: 99%

Analysis of system structure around degenerate critical points

Nishida

Nakamura

Luo

2006

2006 SICE-ICASE International Joint Conference

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In this paper, we will state the new problem of system structures around degenerate critical points in the viewpoint of control engineering. Then, two fundamental schemes are presented to construct a weak Lyapunov function systematically. The procedures are based on the following deformations. One of them is a collapsing residual singularity defined by the formal local coordinate of Thom's splitting lemma. The other is a blowing-up for the resolution of singularity.

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Singularity Theory and Its Application

Cited by 5 publications

References 0 publications

Topological Polarization Singularities in Metaphotonics

Topological Polarization Singularities in Metaphotonics

Cusp singularity in mean field Ising model

Analysis of system structure around degenerate critical points

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