Double Refraction by a Uniaxial Crystal

Crystalline materials can have different indices of refraction in different crystallographic directions. Crystals belonging to the hexagonal, tetragonal, or rhombohedral classes are uniaxial, in that they possess a unique optical axis, most often coincident with the crystallographic axis. Light traveling through such an anisotropic medium can exhibit double refraction or birefringence, in which an unpolarized incident light ray splits into two polarized rays with mutually perpendicular planes of vibration. The ray with its electric field vibrating perpendicular to the optical axis is called the ordinary ray, and is characterized by an index of refraction . The ray that vibrates parallel to the optical axis is called the extraordinary ray, with its index of refraction designated . The birefringence is the difference Δ, which can be positive or negative.
The best-known birefringent crystal is the mineral calcite (Iceland spar), the colorless, transparent rhombohedral salt calcium carbonate, . The optical axis coincides with the three-fold axes of the equilateral groups, along which the ions are also situated. For 590 nm light, the yellow sodium-D lines, the indices for calcite are =1.658, =1.486, =0.172. Some other common uniaxial minerals are: tourmaline, beryl, quartz, ruby, sapphire and zircon. Birefringent materials find several applications in optics, for example Nicol prisms and quarter-wave plates.
In this Demonstration, the angle of incidence of a light ray entering a crystal can be varied, as can the two refractive indices (the default values are those of calcite). The direction of the optical axis is marked with a gray arrow. The ordinary ray is shown in orange, the extraordinary ray in blue. After refraction by the crystal, the light emerges in two parallel rays. You also have the option to display the polarizations of the various rays.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Snapshot 1: conventional refraction when and are equal
Snapshot 2: double refraction by a hypothetical crystal with large birefringence
Snapshot 3: same crystal with polarizations shown


    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+