The Physics
Opus in profectus


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Willebrord Snel (1580–1626) Netherlands. "Although he discovered the law of refraction, a basis of modern geometric optics, in 1621, he did not publish it and only in 1703 did it become known when Huygens published Snell's law in Dioptrica." Snell also studied navigation and proposed the method of triangulation, which is the foundation of geodesy (the branch of mathematics dealing with the measurement of features on the Earth). For some reason, the law named after the man is spelled incorrectly. Snel the man vs. Snell the law.

sin θi  = constant
sin θr

and then

n1 sin θ1 = n2 sin θ2

"This form of Snell's law was actually published by René Descartes (1596–1650) France in La Dioptrique (1637). Snell did discover the relationship but articulated it in a different way. Today it is the form used by Descartes that is called Snell's law."

The index of refraction.

n = c


n =  index of refraction
c =  speed of light in a vacuum
v =  speed of light in a medium

The index of refraction is somewhat related to density, as one would expect. This graph is for transparent minerals. Someone should make one for liquids and see what happens.

Index of refraction for selected materials (λ = 590 nm)
material index   material index
acetone 1.36   glass, borosilicate (pyrex) 1.475
acrylic (lucite, plexiglas) 1.495   glass, crown (soda-lime) 1.512
air (–15 ℃) 1.00030942   glass, fiber (fiberglas) 1.560
air ( 0 ℃) 1.00029238   glass, flint (29% lead) 1.569
air (+15 ℃) 1.00027712   glass, flint (55% lead) 1.669
air (+30 ℃) 1.00026337   glass, flint (71% lead) 1.805
air (+60 ℃) 1.00023958   glass, fused silica 1.4585
alcohol, ethyl (grain) 1.361   glycerol 1.473
alcohol, methyl (wood) 1.328   helium (gas) 1.000036
amber 1.546   helium (liquid) ?
benzene 1.501   hydrogen (gas) 1.000140
butter (40 ℃) 1.455   hydrogen (liquid) 1.0974
butter (60 ℃) 1.447   milk 1.35
calcite 1.486   oil, microscope 1.515
cd/dvd 1.55   oil, mineral 1.470
cocoa butter (40 ℃) 1.457   oil, vegetable (50 ℃) 1.47
diamond 2.418   perovskite 2.38
eglestonite 2.49   quartz, crystalline 1.544
emerald 1.576   quartz, fused silica 1.459
emerald, synth flux 1.561   ruby 1.77
emerald, synth hydro 1.568   salt 1.516
epoxy (opticon) 1.545   sapphire 1.77
eye, cornea 1.38   sphalerite 2.428
eye, aqueous humor 1.33   topaz 1.62
eye, lens cover 1.38   turpentine 1.472
eye, lens 1.41   ulexite 1.49
eye, vitreous humor 1.34   vacuum 1 exactly
fluorite 1.387   water (ice) 1.309
franklinite 2.36   water (liquid, 0 ℃) 1.33346
      water (liquid, 20 ℃) 1.33283
      water (liquid, 100 ℃) 1.31766
      water (vapor) 1.000261
      zircon, high 1.96
      zircon, low 1.92
      zirconia, cubic 2.173

apparent depth

Don't go in the water.

total internal reflection

light traveling from a slow medium to a fast medium

critical angle

  n1 sin θ1  =  n2 sin θ2
  n1 sin θc  =  n2 sin 90°
sin θc  =  n2   in general
sin θc  =  1   when the second media is air

inferior mirage




Dispersion is generally highest in solids and lowest in gases.

Dispersion is often measured in terms of the coefficient of dispersion, which is defined as the difference between the refractive indices for for two prominent lines in the spectrum of hydrogen — the blue F line at 486.1 nm and the red C line at 656.3 nm.

nf − nc

Another common measure of dispersion is the dispersive power.

nf − nc
nD − 1

where nD is the index of refraction for the yellow D line of sodium at 589.0 nm.

Use of a single number to quantify dispersion is rather misleading. Index and wavelength are not linearly related. Dispersion is best quantified as the rate of change of index of refraction with wavelength.


For most transparent materials, a graph of index versus wavelength is curve with a few general characteristics.


Calcite is a common, transparent mineral. It can be found throughout the world, but some of the best samples were originally found in Iceland. Pieces of this mineral are easily split (or cleaved as the geologists say) into parallelogram-faced prismatic chunks. Nonmetallic minerals that cleave easily were called spar in German and so calcite is sometimes also known as Iceland spar. It is of little economic importance by itself (although it is a component of limestone, which is used to make cement), but is of some scientific importance. It has been known for several centuries that light transmitted through calcite takes two paths. This can best be seen by laying a large crystal of it on a page of text. Every letter can be seen twice. This phenomena is known as double refraction or birefringence.

A ray of light incident on a doubly refractive or birefringent material divides into two rays: an ordinary ray (or o ray or ω [omega] ray) and an extraordinary ray (or e ray or ε [epsilon] ray). As the name implies, the o ray behaves in an "ordinary" way, following Snell's law without a problem. The ratio of the sine of the angle of incidence to the sine of the angle of ordinary refraction is a constant. The e ray gets its name because it does not obey this rule.

falls apart here…

Birefringence is only a property of solids

When the incident angle is just right, the o and e rays will follow the same path and the birefringence cannot be seen geometrically. At all other angles, the the two rays will follow different paths. Thus, the index of refraction for extraordinary rays is also a continuous function of direction. The index of refraction for the ordinary ray is constant and is independent of direction

The index of refraction for the extraordinary ray is a continuous function of direction. The index of refraction for the ordinary ray is independent of direction. The two indices of refraction are equal only in the direction of an optic axis.

The measure of birefringence (δ) [delta] is the difference between the indices of refraction of the two rays.

δ = ne − no

In some materials (like calcite) ne < no and the birefringence is less than zero (that is, the e ray is refracted less than the o ray) and the material is said to be optically negative. In other materials (like quartz) the reverse is true and these materials are said to be optically positive. Materials that do not show birefringence are said to be isotropic (like diamond); that is, they behave the same no mater what the alignment of the crystal is relative to the incident ray.

optical behavior   comment examples
isotropic (linear)  
single refraction gases, liquids, glasses, diamond
uniaxial negative  
double refraction
e ray travels faster
calcite, tourmaline, sodium nitrate
uniaxial positive  
double refraction
o ray travels faster
ice, quartz, rutile
triple refraction mica, perovskite, topaz
Types of Refraction
uniaxial minerals no ne δ
beryl Be3Al2(Si6O18) 1.602 1.557 -0.045
calcite CaCO3 1.658 1.486 −0.172
calomel Hg2Cl2 1.973 2.656 +0.683
cinnabar HgS 2.905 3.256 +0.351
hematite Fe2O3 2.940 3.220 +0.287
ice H2O 1.309 1.313 +0.014
lithium niobate LiNbO3 2.272 2.187 −0.085
magnesium fluoride MgF2 1.380 1.385 +0.006
quartz SiO2 1.544 1.553 +0.009
ruby Al 2O3 1.770 1.762 −0.008
rutile TiO2 2.616 2.903 +0.287
peridot   1.690 1.654 −0.036
sapphire Al2O3 1.768 1.760 −0.008
sodium nitrate NaNO3 1.587 1.336 −0.251
tourmaline   1.669 1.638 −0.031
zircon, high ZrSiO4 1.960 2.015 +0.055
zircon, low ZrSiO4 1.920 1.967 +0.047
biaxial minerals α β γ
borax   1.447 1.469 1.472
epsom salt MgSO4·7(H2O) 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)2SiO 1.640 1.660 1.680
perovskite CaTiO3 2.300 2.340 2.380
topaz   1.618 1.620 1.627
ulexite   1.490 1.510 1.520
Index of refefraction for selected nonlinear minerals (λ ~ 590 nm)