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Related Concept Videos

Hyperbolic Functions01:25

Hyperbolic Functions

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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...
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Hyperbolic and Inverse Hyperbolic Functions: Problem Solving01:30

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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...
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Inverse Hyperbolic Functions and Their Derivatives01:25

Inverse Hyperbolic Functions and Their Derivatives

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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...
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Focusing of Light in the Eye01:16

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Light rays enter the eye through the cornea, a transparent dome-shaped tissue that is the eye's outermost layer. The cornea bends or refracts, light rays traveling to the pupil. The shape of the cornea determines how much of the light is bent and whether the image will be focused correctly on the retina at the back of the eye. Once the light has passed through both refraction layers, it converges into a single focal point onto a small area. This is where photoreceptors start transforming...
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Generation of Three-Phase Voltage01:21

Generation of Three-Phase Voltage

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A three-phase AC generator has a rotor with a rotating magnet placed within the stator mounted with the stationary three-phase winding to generate three-phase voltages via mutual induction. These windings are evenly distributed around the inner circumference of the stator and are arranged 120 electrical degrees apart. Three-phase stator windings consist of three separate coils or groups of coils, known as phases, each connected in Y (star) configuration or Delta configuration.
As the rotor...
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Flat Belts: Problem Solving01:28

Flat Belts: Problem Solving

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Flat belts are crucial in many industrial applications as they help transmit power from one pulley to another. The concept of forces and moments is used to determine the maximum moment on a pulley. For instance, consider a flat belt that wraps around two pulleys, A and B, with radii of 30 cm and 10 cm, respectively. The angle between the belt and the horizontal is 20 degrees at the pulleys. As pulley B rotates clockwise and drives pulley A, tension T2 is caused at one end of the belt, while...
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Related Experiment Video

Updated: Jan 25, 2026

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
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Hyperbolic phase function used in a spatial light modulator for flat top focus generation.

Dirk Nodop, Jan Ruecker, Sebastian Waechter

    Optics Letters
    |May 2, 2019
    PubMed
    Summary

    We developed a new method using a liquid crystal on silicon spatial light modulator to transform Gaussian beams into flat top profiles, offering an alternative to Powell lenses for precise beam shaping.

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    Area of Science:

    • Optics and Photonics
    • Beam Shaping Technology

    Background:

    • Gaussian beams are fundamental but often require modification for applications.
    • Flat top beam profiles are crucial for uniform illumination and laser processing.
    • Existing methods like Powell lenses have limitations.

    Purpose of the Study:

    • To introduce a novel transversal beam shaping approach for Gaussian to flat top focus conversion.
    • To compare the proposed hyperbolic phase function method with the Powell lens technique.
    • To derive a strategy for generating squared and circular top hat foci.

    Main Methods:

    • Utilizing a liquid crystal on silicon (LCOS) spatial light modulator (SLM).
    • Implementing a hyperbolic phase function for beam transformation.
    • Theoretical analysis and experimental validation of the phase function.
    • Comparison with the established Powell lens method.

    Main Results:

    • Successfully converted a Gaussian beam to a flat top focus profile.
    • Demonstrated the efficacy of the hyperbolic phase function approach.
    • Provided theoretical and experimental comparison with Powell lenses.
    • Developed a strategy for generating squared and circular top hat profiles.

    Conclusions:

    • The LCOS-based hyperbolic phase function offers a flexible and effective method for beam shaping.
    • This approach provides a viable alternative to traditional beam shaping optics.
    • The derived strategy enables precise control over focus profile generation.