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Optical Indicatrix
GLY 4200
Fall, 2012
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Geometrical Representation
• The idea of a geometrical representation of the variance of the index of refraction goes back to Sir Lazarus Fletcher (1854-1921) in The Optical Indicatrix and the Transmission of Light in Crystals (London, 1892)
• The geometrical representation is called an indicatrix
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Types of Indicatrix
• There are three general types of indicatrix
• These are Isotropic Uniaxial Biaxial
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Constructing the Indicatrix
• A vector, equal in magnitude to the index of refraction, can be assumed to originate at some origin
• The direction is arbitrary• The heads of an infinite collection of such
vectors, all pointing outward from the origin, would describe the geometrical figure called the indicatrix
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Isotropic Indicatrix
• The index of refraction is equal in all directions
• In the case of an isotropic substance, the indicatrix is a perfect sphere
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Dependence on Wavelength
• The magnitude of the vector, representing the index of refraction, will change depending on the wavelength of the light
• Thus, the size of the sphere will change, but the isotropic indicatrix remains a sphere
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Viewing Isotropic Minerals
• An isotropic crystal viewed in crossed nicols always remains in extinction
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High-Order Axes
• Crystals of the hexagonal and tetragonal systems have a unique high order axis (6 or 3–fold, hexagonal; 4-fold, tetragonal).
• It was discovered that rhombohedral calcite crystals give rise to not one but two refracted rays when light is incident on the surface
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Double Refraction
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Explanation of Double Refraction
• Can be explained using the indicatrix theory if it is assumed that the index of refraction varies in different directions
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Uniaxial Crystals
• For a uniaxial crystal it is found that the indicatrix is an ellipsoid, rather than a sphere
• Light propagating perpendicular to the c-axis direction (light vibrates parallel to the c-axis) will experience an index of refraction called ε (epsilon)
• Light which vibrates perpendicular to the c-axis will experience, in any plane, an index of refraction called ω (omega)
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Uniaxial Indicatrix
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Atomic Environment
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Uniaxial Positive
• Ellipsoid is prolate, like a football
• EGO is POSITIVE (Epsilon Greater than Omega)
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Uniaxial Negative
• Ellipsoid is oblate, like a hamburger
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Optic Axis
• Optic axis of a uniaxial crystal is the high-order symmetry axis
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Circular and Principle Sections
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Random Sections
• The principle axis will be ω and ε’, where ε’ is intermediate to ω and ε in value
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Calculation of ε’
• θ is the angle between the plane which cuts the ellipsoid and the c-axis
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Wave Normal
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Wave Front
• Green lines show the wave front
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Quartz Uniaxial Figures
• Diagram shows the variation of the indices of refraction with viewing direction in quartz
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Lack of Symmetry Restrictions
• Crystals which belong to the orthorhombic, monoclinic, or triclinic systems possess no axis higher than 2-fold
• There is nothing in the crystal symmetry which requires the index of refraction in the a-b plane to have a single value A2 fold takes a into –a, b into –b Whereas A4 fold takes a into b, etc.
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Biaxial Crystals
• Crystals which belong to the orthorhombic, monoclinic and triclinic systems are known as biaxial crystals
• The index of refraction is triaxial ellipsoid
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Triaxial Ellipsoid
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Optical Axes in Biaxial Crystals
• The direction of the fastest ray is called X, and the slowest is called Z
• These directions are perpendicular
• A third direction, perpendicular to XZ plane, is called Y
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Biaxial Indices of Refraction
• α is the lowest (X - direction)
• β is intermediate (Y - direction),
• γ is highest (Z - direction)
• These correspond to crystal directions but not in a particular order
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Birefringence
• Numerical difference γ - α is called the birefringence
• This value will be experienced by a ray traveling perpendicular to the X-Z plane
• Note that β is not the average of α & γ but merely a value in between these two
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Optic Axial Plane
• This plane, often abbreviated OAP, shows the maximum birefringence
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Biaxial Principal Sections
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Indices of Principle Axes
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Location of nβ
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Circular Sections
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Biaxial Positive
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Biaxial Negative
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Extreme Cases
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Vz and α, β, and γ
co s( )( )( )( )V z
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2V and 2E