Refraction

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

introduction

Willebrord van Roijen Snell (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 result 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).

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
v

where

n is the index of refraction
c is the speed of light in vacuum
v is the 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.


[magnify]

Index of Refraction for Selected Materials (λ ~ 590 nm)
material index   material index
acetone 1.36   helium (gas) 1.000036
air (–15 °C) 1.00030942   helium (liquid) ??
air (–00 °C) 1.00029238   hydrogen (gas) 1.000140
air (+15 °C) 1.00027712   hydrogen (liquid) 1.0974
air (+30 °C) 1.00026337   lucite 1.495
air (+60 °C) 1.00023958   milk 1.35
alcohol, ethyl (grain) 1.361   oil, microscope 1.515
alcohol, methyl (wood) 1.328   oil, vegetable (50 °C) 1.47
amber 1.546   opticon (epoxy) 1.545
amethyst 1.544   perovskite 2.38
benzene 1.501   plexiglas 1.488
butter (40 °C) 1.455   quartz, crystalline 1.544
butter (60 °C) 1.447   quartz, fused 1.45843
calcite 1.486   ruby 1.76
cd/dvd 1.55   salt 1.516
cocoa butter (40 °C) 1.457   sapphire 1.76
diamond 2.418   sphalerite 2.428
eglestonite 2.49   topaz 1.62
emerald 1.576   turpentine 1.472
emerald, synth flux 1.561   ulexite 1.49
emerald, synth hydro 1.568   vacuum 1 exactly
eye, cornea 1.38   water (ice) 1.309
eye, aqueous humor 1.33   water (liquid, 0 °C) 1.33346
eye, lens cover 1.38   water (liquid, 20 °C) 1.33283
eye, lens 1.41   water (liquid, 100 °C) 1.31766
eye, vitreous humor 1.34   water (vapor) 1.000261
fluorite 1.387   zircon, high 1.960
franklinite 2.36   zircon, low 1.920
glass, borosilicate (pyrex) 1.474   zirconia, cubic 2.173
glass, crown (soda-lime) 1.512      
glass, flint (29% lead) 1.569      
glass, flint (55% lead) 1.669      
glass, flint (71% lead) 1.805      
glass, fused silica 1.459      
glycerin 1.473      

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
n1
sin θc  =  1   when the second media is air
n1

inferior mirage


An inferior mirage.

It is sometimes possible to see over the horizon. The superior mirage or Fata Morgana.

Fata Morgana (superior mirage), the Italian name of the enchanted, half sister of King Arthur. Italian writers and poets described these effects as seen over the straights of Messina, between Italy and Sicily. Although the effects occur worldwide, the Italian name sticks. (From Greenler's book?)

Quote from somebody

Under highly stable atmospheric conditions (typically on calm, clear nights), the radar beam can be refracted almost directly into the ground at some distance from the radar, resulting in an area of intense-looking echoes. This "anomalous propagation" phenomenon (commonly known as AP) is much less common than ground clutter. Certain sites situated at low elevations on coastlines regularly detect "sea return", a phenomenon similar to ground clutter except that the echoes come from ocean waves.

dispersion

rainbow

 
A double rainbow seen on a depressingly gray day in Clinton, Missouri.   You don't need rain for a rainbow. This picture was taken at Niagara Falls, New York.

halo

 
A simple halo.   A 22° halo with an upper tangent arc and two sun dogs. The sun dog on the right is partially obscured by clouds. The sun itself is blocked by the author's bicycle gloved fist.

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.

dn
dλ

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

birefringence

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.


Double Refraction 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

Summary

Problems

practice

  1. Waves travel in all directions in the open ocean, but they always approach the land nearly perpendicular to the shore. Why does this happen?
    • Ocean waves normally form when the wind grips the surface of the water and tries to drag it along. The friction at the interface gives the water a little tug and piles it up into a wave. Short burst of wind make little ripples and strong, steady winds make larger waves or swells. Regardless of size, waves generated by this means generally propagate in the direction that the wind is blowing. Since the wind can and does blow in every direction, waves can and do travel in any direction when they are formed.
       
      [magnify]
       
      The speed of an ocean wave is affected by the depth of the water through which it is propagating. As the sea floor approaches the shore it rises and depth decreases. The shallower the water, the slower the wave speed. (The relationship is a complex one, but near shore speed is approximately proportional to the square root of depth.) Waves entering a medium with slower wave speed are refracted towards the normal. Where the sea floor rises suddenly, the refraction is abrupt. Where it rises gradually, the refraction is gradual. The closer a wave gets, the more perpendicular its propagation. The result is that most waves near the shore will eventually wind up heading very nearly perpendicular to the shore no matter what direction they were traveling in initially.
  2. Write something else.
    • Answer it.
  3. Write something different.
    • Answer it.
  4. Write something completely different.
    • Answer it.

conceptual

  1. refraction-drill.pdf
    The diagrams on the accompanying pdf show a ray of light incident upon an interface. In some cases the ray is traveling through air and entering the glass. In other cases it is traveling through glass and entering the air. On each diagram, sketch the approximate path of the light in the second medium. Ignore any reflected light. It is not necessary to calculate any anlges, but do clearly show the change in direction of the rays, if any, at each surface.

geometric

  1. The diagram below (and on the accompanying pdf) shows 11 parallel rays of light incident on a trapezoidal, crown glass prism. Choose one of the numbered rays at random. On the diagram, trace the transmitted portion of your ray until it emerges from the prism. Just as each incident ray was assigned a number, each emergent ray can be assigned a number. Determine this number.
     
     
  2. The diagram below [magnify] shows 18 rays of light from the sun incident on an equilateral, triangular ice crystal. (The width of the crystal is much larger than the wavelength of the light incident on it.) Choose one of the rays at random. On the diagram, trace the transmitted portion of your ray until it emerges from the crystal. Determine the deviation angle between the incident ray and the emergent ray.
     
     
  3. The diagram below [magnify] shows 20 rays of light from the sun incident on a spherical drop of water. (The diameter of the drop is much larger than the wavelength of the light incident on it.) Choose one of the rays at random. On the diagram, trace the portion of your ray that is transmitted at the interface, reflects once off the back of the drop, and then emerges. Determine the angle between the incident ray and the emergent ray.
     
     
  4. Repeat the problem above, but this time trace the portion of your ray that is transmitted at the interface, reflects twice off the inside of the drop and then emerges.

numerical

  1. Identify the material if the speed of light in the material is …
    1. 4/5 the speed of light in a vacuum
    2. 2/3 the speed of light in a vacuum
    3. 3/4 the speed of light in a vacuum

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