Birefringence 174412 219985759 2008-06-17T19:32:37Z FiriBot 6990130 robot Adding: [[ro:Refracţie dublă]] [[Image:Calcite.jpg|right|thumb|400px|A calcite crystal laid upon a paper with some letters showing the double refraction]] '''Birefringence''', or '''double refraction''', is the decomposition of a [[Ray (optics)|ray]] of [[light]] into two rays (the '''ordinary ray''' and the '''extraordinary ray''') when it passes through certain types of material, such as [[calcite]] [[crystal]]s or [[boron nitride]], depending on the [[polarization]] of the light. This effect can occur only if the structure of the material is [[anisotropic]] (directionally dependent). If the material has a single [[anisotropy#Physics|axis of anisotropy]] or [[optical axis]], (i.e. it is [[Index ellipsoid#Uniaxial indicatrix|uniaxial]]) birefringence can be formalized by assigning two different [[refractive index|refractive indices]] to the material for different polarizations. The birefringence magnitude is then defined by :<math>\Delta n=n_e-n_o\,</math> where ''n''<sub>o</sub> and ''n''<sub>e</sub> are the refractive indices for polarizations perpendicular ('''ordinary''') and parallel ('''extraordinary''') to the axis of anisotropy respectively. The reason for birefringence is the fact that in anisotropic media the electric field vector <math>\vec E</math> and the dielectric displacement <math>\vec D</math> can be nonparallel (namely for the extraordinary polarisation), although being linearly related. Birefringence can also arise in [[Magnetism|magnetic]], not [[dielectric]], materials, but substantial variations in magnetic [[Permeability (electromagnetism)|permeability]] of materials are rare at optical frequencies. == Creating birefringence == While birefringence is often found naturally (especially in crystals), there are several ways to create it in [[optical isotropy|optically isotropic]] materials. *Birefringence results when isotropic materials are deformed such that the isotropy is lost in one direction (ie, stretched or bent). [http://www.oberlin.edu/physics/catalog/demonstrations/optics/birefringence.html Example] *Applying an electric field can induce molecules to line up or behave asymmetrically, introducing anisotropy and resulting in birefringence. (''see'' [[Pockels effect]]) *Applying a magnetic field can cause a material to be '''circularly birefringent''', with different indices of refraction for [[polarization#Theory|oppositely-handed circular polarizations]] (''see'' [[Faraday effect]]). == Examples of uniaxial birefringent materials == {| class="wikitable sortable" style="float:right; margin: 0em 0em 1em 1em;" |+ Uniaxial materials, at 590 nm<ref name=hypertextbook>{{cite web|last=Elert|first=Glenn|title=Refraction|work=The Physics Hypertextbook|url=http://hypertextbook.com/physics/waves/refraction/}}</ref> |- ! Material || n<sub>o</sub> || n<sub>e</sub> || Δn |- | [[beryl]] Be<sub>3</sub>Al<sub>2</sub>(SiO<sub>3</sub>)<sub>6</sub>||1.602 ||1.557 ||-0.045 |- | [[calcite]] CaCO<sub>3</sub> || 1.658 || 1.486 || -0.172 |- | [[calomel]] Hg<sub>2</sub>Cl<sub>2</sub> || 1.973 || 2.656 || +0.683 |- | [[ice]] H<sub>2</sub>O || 1.309 || 1.313 || +0.014 |- | [[lithium niobate]] LiNbO<sub>3</sub>|| 2.272|| 2.187|| -0.085 |- | [[magnesium fluoride]] MgF<sub>2</sub>|| 1.380|| 1.385|| +0.006 |- | [[quartz]] SiO<sub>2</sub>|| 1.544|| 1.553|| +0.009 |- | [[ruby]] Al<sub>2</sub>O<sub>3</sub>|| 1.770|| 1.762|| -0.008 |- | [[rutile]] TiO<sub>2</sub>|| 2.616|| 2.903|| +0.287 |- | [[peridot]] (Mg, Fe)<sub>2</sub>SiO<sub>4</sub> || 1.690|| 1.654|| -0.036 |- | [[sapphire]] Al<sub>2</sub>O<sub>3</sub>|| 1.768|| 1.760|| -0.008 |- | [[sodium nitrate]] NaNO<sub>3</sub>|| 1.587|| 1.336|| -0.251 |- | [[tourmaline]] (complex silicate )|| 1.669|| 1.638|| -0.031 |- | [[zircon]], high ZrSiO<sub>4</sub>|| 1.960|| 2.015|| +0.055 |- | zircon, low ZrSiO<sub>4</sub>|| 1.920|| 1.967|| +0.047 |} Many [[plastic]]s are birefringent, because their molecules are 'frozen' in a stretched conformation when the plastic is moulded or extruded.<ref>[http://www.dep.uminho.pt/home/rec_humanos/mostra_curriculum.php3?pessoa=12&&menu=5&&idcategoria=1 The Use of Birefringence for Predicting the Stiffness of Injection Moulded Polycarbonate Discs] </ref> For example, [[cellophane]] is a cheap birefringent material, and [[Polaroid]] sheets are commonly used to examine for orientation in birefringent plastics like [[polystyrene]] and [[polycarbonate]]. Birefringent materials are used in many devices which manipulate the polarization of light, such as [[wave plate]]s, [[polarizer|polarizing]] [[prism (optics)|prisms]], and [[Lyot filter]]s. There are many birefringent crystals: birefringence was first described in calcite crystals by the [[Denmark|Danish]] scientist [[Rasmus Bartholin]] in [[1669]]. Birefringence can be observed in [[amyloid]] plaque deposits such as are found in the brains of [[Alzheimer's disease|Alzheimer's]] victims. Modified proteins such as [[immunoglobulin]] light chains abnormally accumulate between cells, forming fibrils. Multiple folds of these fibers line up and take on a beta-pleated sheet [[conformation]]. [[Congo red]] dye [[Intercalation (chemistry)|intercalates]] between the folds and, when observed under polarized light, causes birefringence. Cotton (Gossypium hirsutum) fiber is birefringent because of high levels of cellulosic material in the fiber's secondary cell wall. Slight imperfections in [[optical fiber]] can cause birefringence, which can cause distortion in [[fiber-optic communication]]; see [[polarization mode dispersion]]. [[Silicon carbide]], also known as Moissanite, is strongly birefringent. The refractive indices of several (uniaxial) birefringent materials are listed below (at wavelength ~ 590 nm)<ref name=hypertextbook/> == Biaxial birefringence == {| class="wikitable sortable" style="float:right; clear:both; margin: 0em 0em 1em 1em;" |+ Biaxial materials, at 590 nm<ref name=hypertextbook/> |- ! Material || ''n''<sub>α</sub> || ''n''<sub>β</sub> || ''n''<sub>γ</sub> |- |[[borax]] ||1.447 ||1.469 ||1.472 |- |[[Magnesium sulfate|epsom salt]] MgSO<sub>4</sub>·7(H<sub>2</sub>O) ||1.433 ||1.455 ||1.461 |- |[[mica]], [[biotite]] ||1.595 ||1.640 ||1.640 |- |mica, [[muscovite]] ||1.563 ||1.596 ||1.601 |- |[[olivine]] (Mg, Fe)<sub>2</sub>SiO<sub>4</sub> ||1.640 ||1.660 ||1.680 |- |[[perovskite]] CaTiO<sub>3</sub> ||2.300 ||2.340 ||2.380 |- |[[topaz]] ||1.618 ||1.620 ||1.627 |- |[[ulexite]] ||1.490 ||1.510 ||1.520 |} '''Biaxial birefringence''', also known as '''trirefringence''', describes an anisotropic material that has more than one axis of anisotropy. For such a material, the refractive index tensor '''n''', will in general have three distinct [[eigenvalues]] that can be labeled ''n''<sub>α</sub>, ''n''<sub>β</sub> and ''n''<sub>γ</sub>. == Measuring birefringence == Birefringence and related optical effects (such as [[optical rotation]] and linear or [[circular dichroism]]) can be measured by measuring the changes in the polarization of light passing through the material. These measurements are known as [[polarimetry]]. A common feature of optical microscopes is a pair of crossed [[polarizer|polarizing]] filters. Between the crossed polarizers, a birefringent sample will appear bright against a dark (isotropic) background. == Applications of birefringence == Birefringence is widely used in optical devices, such as [[liquid crystal display]]s, [[electro-optic modulator|light modulators]], [[Lyot filter|color filters]], [[wave plate]]s, [[optical axis gratings]], etc. It also plays an important role in [[second harmonic generation]] and many other [[Nonlinear optics|nonlinear processes]]. It is also utilized in medical diagnostics: needle aspiration of fluid from a [[gout]]y joint will reveal negatively birefringent [[urate]] crystals. Some artists also work with birefringence, the most notable being contemporary American artist [http://www.austine.com/aboutaustine.shtml Austine Wood Comarow] who coined the term "Polage" to describe her polarized light collages. The artist works by cutting hundreds of small pieces of [[cellophane]] and other birefringent films and laminating them between plane polarizing filters. Comarow's Polage art is exhibited at the [[Museum of Science, Boston]], the [[New Mexico Museum of Natural History and Science]] in [[Albuquerque]], NM, and la [[Cité des Sciences et de l'Industrie]] (the City of Science and Industry) in [[Paris]]. It is also used as a spatial low-pass filter in electronic cameras, where the thickness of the crystal is controlled to spread the image in one direction, thus having the effect of a spatial low-pass filter (by increasing the spot-size). This is essential to the proper working of all television and electronic film cameras, to avoid spatial aliasing, the folding back of frequencies higher than can be sustained by the pixel matrix of the camera. In [[ophthalmology]], scanning laser polarimetry utilises the birefringence of the retinal nerve fibre layer to quantitate its thickness indirectly, which is of use in the assessment and monitoring of [[glaucoma]]. ==Elastic birefringence== Another form of birefringence is observed in anisotropic [[elastic deformation|elastic]] materials. In these materials, [[S-wave|shear wave]]s split according to similar principles as the light waves discussed above. The study of birefringent shear waves in the earth is a part of [[seismology]]. Birefringence is also used in optical mineralogy to determine the chemical composition, and history of minerals and rocks. == Electromagnetic waves in an anisotropic material == {| class="wikitable" style="float:right; clear:both; margin: 0em 0em 1em 1em;" |+ Effective refractive indices in uniaxial materials ! rowspan="2" | Propagation<br>direction ! colspan="2" | Ordinary ray ! colspan="2" | Extraordinary ray |- ! Polarization ! n<sub>eff</sub> ! Polarization ! n<sub>eff</sub> |- | ''z'' | ''xy''-plane | <math>n_o</math> | n/a | n/a |- | ''xy''-plane | ''xy''-plane | <math>n_o</math> | ''z'' | <math>n_e</math> |- | ''xz''-plane | ''y'' | <math>n_o</math> | ''xz''-plane | <math>n_e < n < n_o</math> |- | other | colspan="4" | analogous to ''xz''-plane |} The behavior of a light ray that propagates through an anisotropic material is dependent on its polarization. For a given propagation direction, there are generally two perpendicular polarizations for which the medium behaves as if it had a single effective refractive index. In a uniaxial material, rays with these polarizations are called the extraordinary and the ordinary ray (''e'' and ''o'' rays), corresponding to the extraordinary and ordinary refractive indices. In a biaxial material, there are three refractive indices ''α'', ''β'', and ''γ'', yet only two rays, which are called the fast and the slow ray. The slow ray is the ray that has the highest effective refractive index. For a uniaxial material with the ''z'' axis defined to be the optical axis, the effective refractive indices are as in the table on the right. For rays propagating in the ''xz'' plane, the effective refractive index of the ''e'' polarization varies continuously between <math>n_o</math> and <math>n_e</math>, depending on the angle with the ''z'' axis. The effective refractive index can be constructed from the [[Index ellipsoid]]. ===Mathematical description=== More generally, birefringence can be defined by considering a dielectric [[permittivity]] and a refractive index that are [[tensor]]s. Consider a [[plane wave]] propagating in an anisotropic medium, with a relative permittivity tensor '''ε''', where the refractive index '''n''', is defined by <math>n\cdot n = \epsilon</math>. If the wave has an electric [[vector (spatial)|vector]] of the form: {{Equation|1=\mathbf{E=E_0}\exp i(\mathbf{k \cdot r}-\omega t) \,|2=2}} where '''r''' is the position vector and ''t'' is time, then the [[wave vector]] '''k''' and the angular frequency ω must satisfy [[Maxwell's equations]] in the medium, leading to the equations: {{Equation|1=-\nabla \times \nabla \times \mathbf{E}=\frac{1}{c^2}(\mathbf{\epsilon} \cdot \frac{\part^2 \mathbf{E} }{\partial t^2})|2=3a}} {{Equation|1= \nabla \cdot (\mathbf{\epsilon} \cdot \mathbf{E}) =0 |2=3b}} where ''c'' is the [[speed of light]] in a vacuum. Substituting eqn. 2 in eqns. 3a-b leads to the conditions: {{Equation|1={{!}}\mathbf{k}{{!}}^2\mathbf{E_0}-\mathbf{(k \cdot E_0) k}= \frac{\omega^2}{c^2} (\mathbf{\epsilon} \cdot \mathbf{E_0}) |2=4a}} {{Equation|1=\mathbf{k} \cdot (\mathbf{\epsilon} \cdot \mathbf{E_0}) =0 |2=4b}} For the matrix product <math>(\epsilon\cdot\mathbf E)</math> often a separate name is used, the ''dielectric displacement vector'' <math>\mathbf D</math>. So essentially birefringence concerns the general theory of linear relationships between these two vectors in anisotropic media. To find the allowed values of '''k''', '''E'''<sub>0</sub> can be eliminated from eq 4a. One way to do this is to write eqn 4a in [[Cartesian coordinates]], where the ''x'', ''y'' and ''z'' axes are chosen in the directions of the [[eigenvector]]s of '''ε''', so that {{Equation|1=\mathbf{\epsilon}=\begin{bmatrix} n_x^2 & 0 & 0 \\ 0& n_y^2 & 0 \\ 0& 0& n_z^2 \end{bmatrix} \,|2=4c}} Hence eqn 4a becomes {{Equation|1=(-k_y^2-k_z^2+\frac{\omega^2n_x^2}{c^2})E_x + k_xk_yE_y + k_xk_zE_z =0|2=5a}} {{Equation|1=k_xk_yE_x + (-k_x^2-k_z^2+\frac{\omega^2n_y^2}{c^2})E_y + k_yk_zE_z =0|2=5b}} {{Equation|1=k_xk_zE_x + k_yk_zE_y + (-k_x^2-k_y^2+\frac{\omega^2n_z^2}{c^2})E_z =0|2=5c}} where ''E''<sub>x</sub>, ''E''<sub>y</sub>, ''E''<sub>z</sub>, ''k''<sub>x</sub>, ''k''<sub>y</sub> and ''k''<sub>z</sub> are the components of '''E'''<sub>0</sub> and '''k'''. This is a set of linear equations in ''E''<sub>x</sub>, ''E''<sub>y</sub>, ''E''<sub>z</sub>, and they have a non-trivial solution if their [[determinant]] is zero: {{Equation|1=\det\begin{bmatrix} (-k_y^2-k_z^2+\frac{\omega^2n_x^2}{c^2}) & k_xk_y & k_xk_z \\ k_xk_y & (-k_x^2-k_z^2+\frac{\omega^2n_y^2}{c^2}) & k_yk_z \\ k_xk_z & k_yk_z & (-k_x^2-k_y^2+\frac{\omega^2n_z^2}{c^2}) \end{bmatrix} =0\,|2=6}} Multiplying out eqn (6), and rearranging the terms, we obtain {{Equation|1=\frac{\omega^4}{c^4} - \frac{\omega^2}{c^2}\left(\frac{k_x^2+k_y^2}{n_z^2}+\frac{k_x^2+k_z^2}{n_y^2}+\frac{k_y^2+k_z^2}{n_x^2}\right) + \left(\frac{k_x^2}{n_y^2n_z^2}+\frac{k_y^2}{n_x^2n_z^2}+\frac{k_z^2}{n_x^2n_y^2}\right)(k_x^2+k_y^2+k_z^2)=0\, |2=7}} In the case of a uniaxial material, where ''n''<sub>x</sub>=''n''<sub>y</sub>=''n<sub>o</sub>'' and ''n<sub>z</sub>''=''n<sub>e</sub>'' say, eqn 7 can be factorised into {{Equation|1=\left(\frac{k_x^2}{n_o^2}+\frac{k_y^2}{n_o^2}+\frac{k_z^2}{n_o^2} -\frac{\omega^2}{c^2}\right)\left(\frac{k_x^2}{n_e^2}+\frac{k_y^2}{n_e^2}+\frac{k_z^2}{n_o^2} -\frac{\omega^2}{c^2}\right)=0\,.|2=8}} Each of the factors in eqn 8 defines a surface in the space of vectors '''k''' — the '''surface of wave normals'''. The first factor defines a [[sphere]] and the second defines an [[ellipsoid]]. Therefore, for each direction of the wave normal, two wavevectors '''k''' are allowed. Values of '''k''' on the sphere correspond to the '''ordinary rays''' while values on the ellipsoid correspond to the '''extraordinary rays'''. For a biaxial material, eqn (7) cannot be factorized in the same way, and describes a more complicated pair of wave-normal surfaces.<ref name="bornwolf">Born M, and Wolf E, ''Principles of Optics'', 7th Ed. 1999 (Cambridge University Press), §15.3.3</ref> Birefringence is often measured for rays propagating along one of the optical axes (or measured in a two-dimensional material). In this case, '''n''' has two eigenvalues which can be labeled ''n''<sub>1</sub> and ''n''<sub>2</sub>. '''n''' can be diagonalized by: {{Equation|1=\mathbf{n} = \mathbf{R(\chi)} \cdot \begin{bmatrix} n_1 & 0 \\ 0 & n_2 \end{bmatrix} \cdot \mathbf{R(\chi)}^\textrm{T} |2=9}} where '''R'''(χ) is the rotation matrix through an angle χ. Rather than specifying the complete tensor '''n''', we may now simply specify the ''magnitude'' of the birefringence Δ''n'', and ''extinction angle'' χ, where Δ''n'' = ''n''<sub>1</sub>&nbsp;−&nbsp;''n''<sub>2</sub>. == See also == {{Commons|Birefringence}} * [[Cotton-Mouton effect]] * [[Crystal optics]] * [[John Kerr (physicist)|John Kerr]] * [[Periodic poling]] * [[Dichroism]] ==References== {{reflist}} ==External links== * http://www.olympusmicro.com/primer/lightandcolor/birefringence.html * [http://www.youtube.com/watch?v=BEClYQbuG7U] Video of stress birefringence in Polymethylmethacrylate (PMMA or Plexiglas). * [http://www.campoly.com/application_notes.html Application note on the theory of birefringence] *<cite>Austine Wood Comarow: Paintings in Polarized Light</cite>, James Mann, Wasabi Publishing, 2005, ISBN 0-9768198-0-5. 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