Birefringence is formally defined as the double refraction of light in a transparent, molecularly ordered material, which is a manifestation of the existence of orientation-dependent differences in refractive index. Many transparent solids are optically isotropic, meaning that the index of refraction is equal in all directions throughout the crystalline lattice. Examples of isotropic solids are glass, table salt, many polymers, and a wide variety of both organic and inorganic compounds.

Crystals are classified as being either isotropic or anisotropic depending upon their optical behavior and whether or not their crystallographic axes are equivalent. All isotropic crystals have equivalent axes that interact with light in a similar manner, regardless of the crystal orientation with respect to incident light waves. Light entering an isotropic crystal is refracted at a constant angle and passes through the crystal at a single velocity without being polarized by interaction with the electronic components of the crystalline lattice.

The term anisotropy refers to a non-uniform spatial distribution of properties, which result in different values being obtained when specimens are probed from several directions within the same material. Observed properties are often dependent on the particular probe being employed and often vary depending upon the whether the observed phenomena are based on optical, acoustical, thermal, magnetic, or electrical events. On the other hand, isotropic properties remain symmetrical, regardless of the direction of measurement with each type of probe reporting identical results.

Introduction to Birefringence - Anisotropic crystals, such as quartz, calcite, and tourmaline, have crystallographically distinct axes and interact with light by a mechanism that is dependent upon the orientation of the crystalline lattice with respect to the incident light angle. When light enters the optical axis of anisotropic crystals, it behaves in a manner similar to the interaction with isotropic crystals, and passes through at a single velocity. However, when light enters a non-equivalent axis, it is refracted into two rays each polarized with the vibration directions oriented at right angles to one another, and traveling at different velocities. This phenomenon is termed double refraction or birefringence and is exhibited to a greater or lesser degree in all anisotropic crystals.

Acoustical Model of Anisotropy - The anisotropic character of materials relates to those properties that have different values when measurements are made in different directions within the same material. This interactive tutorial explores how sound waves exhibit anisotropic character as a function of grain structure when traveling through a wooden block, which serves as an excellent model for the behavior of light passing through anisotropic crystals.

Double Refraction (Birefringence) - Calcite is a form of calcium carbonate, commonly referred to as Iceland spar, which has a rhombohedral crystalline shape. Light passing through a crystal of calcite is refracted into two rays, which are separated by a wide margin due to the strong birefringence of the crystal. This interactive tutorial simulates viewing of a ball-point pen and a line of text through a crystal of Iceland spar, producing a double image from the refracted light rays.

Birefringence in Calcite Crystals - As light travels through an anisotropic material, the electromagnetic waves become split into two principal vibrations, which are oriented mutually perpendicular to each other and perpendicular to the direction that the waves propagate. The wave whose electric vector vibrates along the major axis of the index ellipse is termed the slow wave, because the refractive index for this wave is greater than the refractive index for the other wave. The wave vibrating perpendicular to the slow wave is termed the fast wave. This tutorial explores double refraction or birefringence in calcite (calcium carbonate), a colorless, transparent, rhombohedral crystalline salt that is the most common such material found naturally.

The Fresnel or Refractive Index Ellipsoid - The Fresnel, or refractive index, ellipsoid describes the dielectric properties measured in all directions through a material. Measurements through the radius 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 ellipsoid radius. This tutorial explores variations in the shape and dimensions of the ellipsoid as a function of refractive index.

Birefringence Variations with Crystal Orientation - When a beam of light is incident on a birefringent crystal, the waves are split upon entry into orthogonal polarized components (termed ordinary and extraordinary) that travel through the molecular lattice along different pathways, depending on their orientation with respect to the crystalline optical axis. If the incident beam is oblique to the optical axis, the waves diverge during their journey through the crystal. In contrast, the orthogonal wave components follow a co-linear pathway when the incident light beam enters the crystal either parallel or perpendicular to the optical axis. This interactive tutorial explores variations in birefringence that result from orientational variations between the crystal optical axis and the incident light beam.

Birefringent Crystals in Polarized Light - In order to examine how birefringent anisotropic crystals interact with polarized light in an optical microscope, the properties of an individual, isolated crystal can be considered. The specimen material in this tutorial is a hypothetical tetragonal birefringent crystal having an optical axis oriented in a direction that is parallel to the long axis of the crystal. Light entering the crystal from the polarizer will be traveling perpendicular to the optical axis of the crystal, regardless of the crystal orientation with respect to the polarizer and analyzer transmission axes. The virtual microscope viewport presents the crystal as it would appear in the eyepieces of a microscope under crossed-polarized illumination as it is rotated around the microscope optical axis.

Interactive Michel-Levy Birefringence Chart - Quantitative analysis of the interference colors observed in birefringent samples is usually accomplished by consulting a Michel-Levy chart similar to the one illustrated in the tutorial window below. As is evident from this graph, the polarization colors visualized in the microscope and recorded onto film or captured digitally can be correlated with the actual retardation value, thickness, and birefringence of the specimen. The chart is relatively easy to use with birefringent samples if two of the three required variables are known. This interactive tutorial enables visitors to determine the interference color associated with all three values by clicking on selected regions of the interactive chart. A large version of the tutorial is also available.

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