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Related Concept Videos

Polar Coordinates01:24

Polar Coordinates

The polar coordinate system offers an alternative to the Cartesian coordinate system for specifying points in a plane, using a distance and an angle instead of x and y coordinates. This system is particularly advantageous in situations involving circular or rotational symmetry, such as in physics or engineering problems involving waves, oscillations, or orbital paths.Defining Polar CoordinatesIn polar coordinates, a point is represented as P(r, ��), where r is the radial distance from a fixed...
Vector Transformation in Rotating Coordinate Systems01:16

Vector Transformation in Rotating Coordinate Systems

Consider a vector rotating about an axis with an angular velocity, such that its tip sweeps a circular path.
Polar and Cylindrical Coordinates01:22

Polar and Cylindrical Coordinates

The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
Transformations of Functions I01:29

Transformations of Functions I

A function's graph can be modified by changing its position or size without altering its overall shape. These transformations allow the graph to be moved across the coordinate plane while preserving its pattern and structure. One of the most common transformations is shifting, which repositions the graph without distorting it.When the output of a function is adjusted by adding or subtracting a constant, the graph shifts vertically. A positive value moves the graph upward, while a negative value...
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Coordinates and Map Projections

Coordinates and map projections are essential tools in accurately representing the Earth's surface for various applications, ranging from navigation to spatial analysis. The latitude and longitude coordinate system is a universally recognized framework for defining locations. Latitude specifies the distance of a point north or south of the equator, measured in degrees from 0° at the equator to 90° at the poles. Longitude indicates a location's position east or west of the prime meridian,...
Spherical Coordinates01:23

Spherical Coordinates

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...

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Optical coordinate transformations.

N Davidson, A A Friesem, E Hasman

    Applied Optics
    |August 20, 2010
    PubMed
    Summary
    This summary is machine-generated.

    A new method designs holographic optical elements for coordinate transformations using analytic ray tracing. This technique enables the creation of computer-generated or computer-originated holograms for advanced optical applications.

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    Area of Science:

    • Optics and Photonics
    • Holography
    • Optical Design

    Background:

    • Holographic optical elements (HOEs) are crucial for manipulating light.
    • General coordinate transformations with HOEs present design challenges.
    • Existing methods may lack versatility for complex transformations.

    Purpose of the Study:

    • To present a novel analytic technique for designing holographic optical elements.
    • To enable HOEs capable of performing general coordinate transformations.
    • To demonstrate the design and realization of a logarithmic coordinate transforming HOE.

    Main Methods:

    • Analytic ray-tracing techniques are employed to determine the grating vector.
    • A Poisson-like equation is solved to obtain the holographic grating function.
    • The grating function is implemented as a computer-generated or computer-originated hologram.

    Main Results:

    • A design methodology for HOEs performing arbitrary coordinate transformations is established.
    • The method successfully designs a holographic element for logarithmic coordinate transformation.
    • The feasibility of analytic design for complex holographic functions is demonstrated.

    Conclusions:

    • The presented technique offers a powerful tool for designing versatile holographic optical elements.
    • This approach facilitates the creation of HOEs for specialized optical functions, such as logarithmic transformations.
    • The analytic design method advances the field of computer-generated holography and optical element fabrication.