Measuring Birefringence of Anisotropic Crystals By Ben Grober

Outline Goals Background Method Results Discussion Crystals Birefringence Samples Method Senarmont Method Results Discussion

Goals Accurately measure phase retardation of anisotropic crystal Calculate birefringence of anisotropic crystal Identify minerals based on measurements

Crystals Minerals are naturally occurring crystalline solids There are two types of crystals: Isotropic and Anisotropic Isotropic Crystals Single refractive index Totally extinct under cross polarized light Anisotropic Crystals Two or three refractive indices Has birefringence colors under cross polarized light

Anisotropic Crystals Have two or three refractive indices The difference in refractive indices creates birefringence Light is split into two eigenwaves Two waves interfere with each other This birefringence can be seen as colors under crossed polarized light

Types of Anisotropic Crystals There are two types of anisotropic crystals: uniaxial and biaxial Uniaxial Crystals Two refractive indices One optic axis Biaxial Crystals Three refractive indices Two optic axis

Optic Axis

Birefringence ∆𝑛=( 𝑛 𝑒 − 𝑛 𝑜 ) Birefringence is the result of a crystal having two or more refractive indices Birefringence is a natural property of anisotropic crystals All anisotropic minerals have a range of birefringence values Useful for identifying minerals and other applications ∆𝑛=( 𝑛 𝑒 − 𝑛 𝑜 )

Birefringence Colors The product of birefringence can be seen under cross polarized light as colors The color depends on many factors including: Birefringence Sample thickness Crystal orientation Birefringence never changes, even if color does

Birefringence Color Chart

Samples Using plagioclase and hornblende crystals Both are anisotropic and biaxial They have very different birefringence values

Samples Plagioclase Hornblende

Method Senarmont method Pass light from a light source through light diffuser Light then passes through interference filter Light passes through polarizer and then our sample From sample, light passes through Fresnel Rhomb Observe light after passing through analyzer

Method Analyzer Sample Interference filter Fresnel rhomb Polarizer White light source Light diffuser Interference filter Polarizer Sample Fresnel rhomb Analyzer

(or Senarmont compensator) Method Optical element Symbol Jones matrix Output polarization Polarizer (horizontal ) Sample (slow axis at 45º) Fresnel rhomb (or Senarmont compensator) (s =horizontal) Analyzer (oriented at 90º+d/2) x y x y apply to x y g x y x(s) y(p) x y d/2 x y d/2 x y

Method Orient crystal with slow axis at 45 degrees to polarizer Use interference filter (676nm, 630nm, 532nm, 450nm) Rotate analyzer until crystal is fully extinct Note angle on analyzer Repeat 4 times for each wavelength 𝛿 2 = 𝜋 𝜆 Δ𝑛𝑑

Method

Results Plagioclase at 94o Hornblende at 327o Wavelength (nm) d/2 (degrees) 676 44.0, 43.5, 43.5, 43.0 336.5, 336.0, 332.5, 334.5 630 48.0, 46.5, 46.5, 46.0 357.5, 356.0, 357.5, 354.0 532 56.0, 57.5, 58.0, 56.5 428.0, 427.0, 424.0, 432.0 450 68.0, 66.0, 67.5, 65.0 514.0, 515.5, 512.5, 510.0

Results ∆𝑛= 𝑆𝑙𝑜𝑝𝑒 (180 𝑜 )𝑑

Results Measured Accepted Plagioclase 0.006 0.007-0.013 Hornblende 0.045 0.014-0.034

Discussion Measured values were very close to accepted values Plagioclase error most likely due to optic axis Hornblende error most likely due to sample thickness

Summary Birefringence is a natural property of minerals Birefringence is important for identifying minerals and has a variety of other uses Found phase retardation of crystal to within 2nm of accuracy Found birefringence of crystal fairly accurately Error due to crystal thickness and orientation