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Related Concept Videos

Toroids01:27

Toroids

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A toroid is a closely wound donut-shaped coil constructed using a single  conducting wire. In general, it is assumed that a toriod consists of  multiple circular loops perpendicular to its axis.
When connected to a supply, the magnetic field generated in the toroid has field lines circular and concentric to its axis. Conventionally, the direction of this magnetic field is expressed using the right-hand rule. If the fingers of the right hand curl in the current direction, the thumb...
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Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half...
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Gauss's Law: Planar Symmetry01:27

Gauss's Law: Planar Symmetry

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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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Centroid for the Paraboloid of Revolution01:16

Centroid for the Paraboloid of Revolution

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The paraboloid of revolution is an axially symmetric surface generated by rotating a parabola around its axis. This shape has several applications in mechanical engineering due to its advantageous structural properties, such as strength against stress concentration points and rotational symmetry.
The centroid for the paraboloid of revolution is the point where all the mass of the paraboloid is concentrated. This centroid is important for engineering applications, as it determines how forces are...
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Torsion of Noncircular Members01:16

Torsion of Noncircular Members

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Circular shafts undergoing torsional stress maintain their cross-sectional integrity due to their axisymmetric nature. This symmetry ensures an even distribution of stress, allowing the shaft to withstand torsion without distorting. In contrast, square bars, lacking this axial symmetry, experience significant distortion across their cross-sections when subjected to torsion, with the exception of along their diagonals and at lines connecting midpoints. A detailed examination of a cubic element...
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Geodesic conformal gradient device based on a torus.

Yixiao Ge, Wen Xiao, Huanyang Chen

    Optics Letters
    |August 29, 2024
    PubMed
    Summary

    This study designs a gradient refractive index device for toroidal surfaces, enabling Gaussian beam focusing. It overcomes topological challenges using inversion transformations for optical field control.

    Area of Science:

    • Optics
    • Non-Euclidean Geometry
    • Gradient Index Optics

    Background:

    • Optical fields on non-Euclidean geometry are a growing research area.
    • Geodesic conformal transformation theory connects curved surfaces with planar gradient refractive indices.
    • Existing theories do not cover surfaces with non-trivial topology (genus > 0).

    Purpose of the Study:

    • To design a gradient planar device for toroidal surfaces using geodesic conformal transformation theory.
    • To achieve Gaussian beam focusing on surfaces with non-trivial topology.
    • To address and rectify challenges posed by non-zero genus in optical devices.

    Main Methods:

    • Applying geodesic conformal transformation theory to toroidal surfaces.
    • Designing a gradient refractive index (GRIN) planar device.

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  • Utilizing inversion transformations to manage discontinuous boundaries.

    • Main Results:

    • Successful design of a gradient planar device for toroidal surfaces.
    • Demonstration of Gaussian beam focusing capabilities.
    • Rectification of one-to-two discontinuous boundaries caused by non-zero genus.

    Conclusions:

    • Geodesic conformal transformation theory can be extended to surfaces with non-trivial topology.
    • Gradient planar devices can effectively control curved optical fields on complex surfaces.
    • Inversion transformations are a viable method for overcoming topological limitations in optical device design.