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.

Scatter plot of n vs. ρ

Index of refraction for selected materials (λ = 590 nm)
material index
acetone 1.36
acrylic (lucite, plexiglas) 1.495
air (−15 °C) 1.00030942
air (0 °C) 1.00029238
air (+15 °C) 1.00027712
air (+30 °C) 1.00026337
air (+60 °C) 1.00023958
alcohol, ethyl (grain) 1.361
alcohol, methyl (wood) 1.328
amber 1.546
barium fluoride 1.475
benzene 1.501
butter (40 °C) 1.455
butter (60 °C) 1.447
calcite 1.486
cd/dvd 1.55
cocoa butter (40 °C) 1.457
diamond 2.418
eglestonite 2.49
emerald 1.576
emerald, synth flux 1.561
emerald, synth hydro 1.568
epoxy (opticon) 1.545
eye, cornea 1.38
eye, aqueous humor 1.33
eye, lens cover 1.38
eye, lens 1.41
eye, vitreous humor 1.34
fluorite 1.387
franklinite 2.36
material index
glass, borosilicate (pyrex) 1.475
glass, crown (soda-lime) 1.523
glass, fiber (fiberglas) 1.560
glass, flint (45% lead) 1.618
glass, fused silica 1.4585
glycerol 1.473
helium (gas) 1.000036
helium (liquid) ?
hydrogen (gas) 1.000140
hydrogen (liquid) 1.0974
lithium fluoride 1.392
milk 1.346
oil, microscope 1.515
oil, mineral 1.470
oil, vegetable (50 °C) 1.47
perovskite 2.38
quartz, crystalline 1.544
quartz, fused silica 1.459
ruby 1.77
salt 1.516
sapphire 1.77
sphalerite 2.428
topaz 1.62
turpentine 1.472
ulexite 1.49
vacuum 1 exactly
water (ice) 1.309
water (liquid, 0 °C) 1.33346
water (liquid, 20 °C) 1.33283
water (liquid, 100 °C) 1.31766
water (vapor) 1.000261
zinc selenide 2.630
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 of a beam of light through a triangular glss prism

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 phenomenon is known as double refraction or birefringence.

Double refraction (birefringence) through a piece of calcite

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.

Types of refraction
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
biaxial triple refraction mica, perovskite, topaz
Index of refefraction for selected nonlinear minerals (λ ~ 590 nm)
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