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Hyperbolic Functions

Hyperbolic Functions

A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...

Inverse Hyperbolic Functions and Their Derivatives

Inverse Hyperbolic Functions and Their Derivatives

The shape of a suspension bridge cable hanging under its own weight is described by a catenary curve, which is modeled using the hyperbolic cosine function. This mathematical model accurately captures the balance between gravity and tension acting along the cable. When a particular vertical position on the cable is known, the corresponding horizontal position can be determined using the inverse hyperbolic cosine function, allowing for a detailed analysis of the cable's geometry.Inverse...

Crystal Field Theory - Tetrahedral and Square Planar Complexes

Crystal Field Theory - Tetrahedral and Square Planar Complexes

Tetrahedral Complexes
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...

Planar Rigid-Body Motion

Planar Rigid-Body Motion

Understanding the movement of a rigid body in planar motion involves recognizing that every particle within this body is traversing a path that maintains a consistent distance from a specific plane. This concept is fundamental in the study of physics and mechanical engineering, and it allows us to comprehend better how objects move in space.
Planar motion is typically divided into three distinct categories. The first is rectilinear translation, demonstrated by a subway train that moves along...

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

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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Related Experiment Video

Updated: Jan 28, 2026

Characterizing Dissipative Elastic Metamaterials Produced by Additive Manufacturing

Characterizing Dissipative Elastic Metamaterials Produced by Additive Manufacturing

Published on: June 28, 2024

Magnetically controlled planar hyperbolic metamaterials for subwavelength resolution.

Bo Han Cheng¹, Hong Wen Chen², Kai Jiun Chang²

¹Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan.

Scientific Reports

|December 15, 2015

Summary

Researchers developed a magnetically controlled InSb-dielectric structure for subwavelength resolution in THz imaging. This breakthrough overcomes diffraction limits, enabling higher-density optoelectronic components and advanced photolithography applications.

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Area of Science:

Optoelectronics
Metamaterials
Terahertz (THz) technology

Background:

Conventional optics face limitations in resolving subwavelength features due to the evanescent nature of scattered signals.
The demand for high-density optoelectronic components necessitates overcoming the diffraction limit, particularly in photolithography.

Purpose of the Study:

To propose and analyze a novel magnetically controlled InSb-dielectric multi-layered structure.
To achieve subwavelength resolution in the Terahertz (THz) region by dynamically controlling the structure with an external magnetic field.

Main Methods:

Utilized transfer matrix method and effective medium approach to elucidate the super-resolution mechanism.
Performed electromagnetic numerical simulations to validate the proposed design's feasibility.

Main Results:

The InSb-dielectric structure demonstrates the ability to resolve subwavelength structures at varying frequencies by adjusting the magnetic field.
Increased magnetic field strength enhances the resolving power for a fixed incident frequency.

Conclusions:

The magnetically controlled layered structure offers a practical route for multi-functional materials and real-time super-resolution imaging.
This technology has significant implications for advanced photolithography and the development of high-density optoelectronic components.