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Encyclopedia > Index ellipsoid
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The index ellipsoid is a diagram of an ellipsoid that depictes the orientation and relative magnitude of refractive indices in a crystal. Jump to: navigation, search Egg-shaped. ... The refractive index of a material is the factor by which electromagnetic radiation is slowed down (relative to vacuum) when it travels inside the material. ... Quartz crystal A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ...


The equation for the ellipsoid is construction used the electric displacement vector, D, and the dielectric constants, The term displacement can have one of several meanings, depending on context: Displacement (distance), a physical quantity in kinematics Particle displacement, acoustics of sound in air Displacement (fluid), a different physical quantity, used in fluid mechanics and navigation; used as a measure of a ships size Engine displacement, a...

Defining the field energy, W, as:
and the reduced displacement as:

then the index ellipsoid is defined by the equation,

.

The semiaxes of this ellipsid are dielectric constants of the crystal. This ellipsoid can be used to determine the polarization of an incoming wave, with wave vector, s, by taking the intersection of the plane with the index ellipsoid. The axes of the resulting ellipse are the resulting polarization directions. The dielectric constant εr (represented as or K in some cases) is defined as the ratio: where εs is the static permittivity of the material in question, and ε0 is the vacuum permittivity. ... This article treats polarization in electrodynamics. ... A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ... In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ...


Uniaxial Indicatrix

An important special case of the index ellipsoid occurs when the ellipsoid is an ellipsoid of revolution, e.g. constructed by rotationing an ellipse around either the minor or major axis. In this case, there is only one optical axis; the axis of rotation. In such a case, the material is uniaxial and has only one principal symmetry axis. In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. ...


References

External Links

  • Java Tutorial of Index Ellipsoids

  Results from FactBites:
 
Olympus Microscopy Resource Center: Specialized Microscopy Techniques - Fresnel or Refractive Index Ellipsoid - ... (910 words)
The radius of the ellipsoid yields the refractive index (n), or the square root of the dielectric constant for waves whose electric displacement vectors lie in the direction of the radius of the ellipsoid within an anisotropic medium.
The value of the refractive index given by the radius of the Fresnel ellipsoid is valid for all waves with an electric vector that vibrates in the direction of that radius, regardless of the wave's direction of propagation.
The index ellipse yields the refractive indices and indicates the vibration direction (of the electric vector) of the light waves, because the waves vibrate with the electric vectors along the major and minor axes of the index ellipse.
Refraction Of Light - LoveToKnow 1911 (3935 words)
This constant is called the relative refractive index of the second medium, and may be denoted by / l ab, the suffix ab signifying that the light passes from medium a to medium b; similarly,u ba denotes the relative refractive index of a with regard to b.
Hence it occurred to him that similar relations with respect to an ellipsoid with three unequal axes would give the speeds and polarizations of the waves in a biaxal crystal, and the results thus deduced he found to be in accordance with all known facts.
the central circular section of the ellipsoid and the normal section of the cylinder touching the ellipsoid along this section: whence by a known proposition the ray describes a cone whose sections parallel to the given planes are circles.
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