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Crystal Optics
Crystal optics is the branch of optics that describes the behaviour of light in ''anisotropic media'', that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone–Dale relation. Crystals are often naturally anisotropic, and in some media (such as liquid crystals) it is possible to induce anisotropy by applying an external electric field. Isotropic media Typical transparent media such as glasses are '' isotropic'', which means that light behaves the same way no matter which direction it is travelling in the medium. In terms of Maxwell's equations in a dielectric, this gives a relationship between the electric displacement field D and the electric field E: : \mathbf = \varepsilon_0 \mathbf + \mathbf where ε0 is the permittivity of free space and P is the electric polarization (the vector field c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties. Most optical phenomena can be accounted for by using the classical electromagnetic description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice. Practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces. Physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear relationship of voltage and current in an electrical conductor ( Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are ''nonlinear''. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. The word linear comes from Latin ''linearis'', "pertaining to or resembling a line". In mathematics In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties: * Additivity: . * Homogeneity of degree 1: for all α. These properties are known as the superposition principle. In this definition, ''x'' is not necessarily ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Ellipsoid
In crystal optics, the index ellipsoid (also known as the ''optical indicatrix'' or sometimes as the ''dielectric ellipsoid'') is a geometric construction which concisely represents the refractive indices and associated polarizations of light, as functions of the orientation of the wavefront, in a doubly-refractive crystal (provided that the crystal does not exhibit optical rotation). When this ellipsoid is cut through its center by a plane parallel to the wavefront, the resulting intersection (called a ''central section'' or ''diametral section'') is an ellipse whose major and minor semiaxes have lengths equal to the two refractive indices for that orientation of the wavefront, and have the directions of the respective polarizations as expressed by the electric displacement vector . The principal semiaxes of the index ellipsoid are called the ''principal refractive indices''. It follows from the sectioning procedure that each principal semiaxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optic Axis Of A Crystal
An optic axis of a crystal is a direction in which a ray of transmitted light suffers no birefringence (double refraction). An optic axis is a direction rather than a single line: all rays that are parallel to that direction exhibit the same lack of birefringence. Crystals may have a single optic axis, in which case they are ''uniaxial'', or two different optic axes, in which case they are ''biaxial''. Non-crystalline materials generally have no birefringence and thus, no optic axis. A uniaxial crystal (e.g. calcite, quartz) is isotropic within the plane orthogonal to the optic axis of the crystal. Explanation The internal structure of crystals (the specific structure of the crystal lattice, and the specific atoms or molecules of which it is composed) causes the speed of light in the material, and therefore the material's refractive index, to depend on both the light's direction of propagation and its polarization. The dependence on polarization causes birefringence, in which two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartz
Quartz is a hard, crystalline mineral composed of silica ( silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall chemical formula of SiO2. Quartz is the second most abundant mineral in Earth's continental crust, behind feldspar. Quartz exists in two forms, the normal α-quartz and the high-temperature β-quartz, both of which are chiral. The transformation from α-quartz to β-quartz takes place abruptly at . Since the transformation is accompanied by a significant change in volume, it can easily induce microfracturing of ceramics or rocks passing through this temperature threshold. There are many different varieties of quartz, several of which are classified as gemstones. Since antiquity, varieties of quartz have been the most commonly used minerals in the making of jewelry and hardstone carvings, especially in Eurasia. Quartz is the mineral definin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calcite
Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on the Mohs scale of mineral hardness, based on scratch hardness comparison. Large calcite crystals are used in optical equipment, and limestone composed mostly of calcite has numerous uses. Other polymorphs of calcium carbonate are the minerals aragonite and vaterite. Aragonite will change to calcite over timescales of days or less at temperatures exceeding 300 °C, and vaterite is even less stable. Etymology Calcite is derived from the German ''Calcit'', a term from the 19th century that came from the Latin word for lime, ''calx'' (genitive calcis) with the suffix "-ite" used to name minerals. It is thus etymologically related to chalk. When applied by archaeologists and stone trade professionals, the term alabaster is used not just as in geology and mineralogy, where it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birefringence
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress. Birefringence is responsible for the phenomenon of double refraction whereby a ray of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. This effect was first described by Danish scientist Rasmus Bartholin in 1669, who observed it in calcite, a crystal having one of the strongest birefringences. In the 19th century Augustin-Jean Fresnel described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse pola ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polarization (waves)
Polarization ( also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string ''(see image)''; for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves ( shear waves) in solids. An electromagneti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Diagonalization
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) For a finite-dimensional vector space a linear map T:V\to V is called diagonalizable if there exists an ordered basis of V consisting of eigenvectors of T. These definitions are equivalent: if T has a matrix representation T = PDP^ as above, then the column vectors of P form a basis consisting of eigenvectors of and the diagonal entries of D are the corresponding eigenvalues of with respect to this eigenvector basis, A is represented by Diagonalization is the process of finding the above P and Diagonalizable matrices and maps are especially easy for computations, once their eigenvalues and eigenvectors are known. One can raise a diagonal matrix D to a power by simply raising the diagonal entries to that power, and the determi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Tensor
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of order ''r'' represented in coordinates as a quantity with ''r'' indices satisfies :T_ = T_. The space of symmetric tensors of order ''r'' on a finite-dimensional vector space ''V'' is naturally isomorphic to the dual of the space of homogeneous polynomials of degree ''r'' on ''V''. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on ''V''. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics. Definition Let ''V'' be a vector space and :T\in V^ a tensor of order ''k''. Then ''T'' is a symmetric tensor if :\tau_\sigma T = T\, for the braiding maps assoc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
