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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

Colors  &  Dispersion
Rainbows, Blue Skies, Red Sunsets

[ This ] may be repeated with great ease,
whenever the sun shines, and without any
other apparatus than is at hand to everyone
Thomas Young (1773-1829)   Nov. 24, 1803

Related articles on this site:

Related Links (Outside this Site)

The Rainbow  by  Dr. James B. Calvert  (University of Denver).
Rainbows  by  Jerry L. Stanbrough  (Batesville High School).
Angles in a Rainbow  by  Thayer Watkins  (Photography)
Why is the sky blue?  by  Philip Gibbs  (May 1997)
Atmospheric optics:   Fogbow & glory, holy light, ice halos, sundogs, etc.  Primary & secondary rainbows,  Alexander's dark band, supernumeraries. Dewbow, glassbow.

Wikipedia :   Dispersion  |  Rainbow  |  CIE chromaticity diagram (1931)  |  Fraunhofer lines  |  Low-dispersion glass

Video:  MIT OpenCourseWare   Vibrations & Waves  (8.03)  by  Walter Lewin.
Explanation  of  Walter Lewin's Glassbow  (2004-06-20 / APOD 2004-09-13).
Colour mixing: The mystery of Magenta  by  Steve Mould  (RI, 2013-02-13).
 Visible spectrum accurately rendered in RGB on grey background
 Double Rainbow in Yellowstone
Double rainbow in Yellowstone (Wyoming)

Wave Dispersion  &  Colors of the Rainbow

 PWP @ AT&T Worldnet. In Memoriam 2000-2010.
(2005-09-29)   Dispersion Relation
The celerity of a wave as a function of its frequency.

The  dispersion relation  of a propagation medium is what gives the celerity of a wave in terms of either its frequency  (n )  or its wavelength  (l).

The simplest dispersion relation is that of a nondispersive medium, for which the celerity (u) is constant.  For example, the celerity of electromagnetic waves in a vacuum is equal to Einstein's constant  (u = c).

One common way to specify the  dispersion relation  is by giving the  pulsatance  w = 2pn  as a function of the  wave number  k = 2p/l

w   =   w(k)

More generally, this relation has a vectorial counterpart involving the  wave vector  (k)  which is appropriate for a transmission medium which isn't necessarily  isotropic.  (Recall that we use  bold type  to denote a vector.)

w   =   w(k)

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 still working on this one...

(2015-07-26)   Dispersion Models  &  Empirical Dispersion Equations
Approximate relations between wavelength and frequency.

Historically, spectral colors were characterized by their  vacuum wavelength.  Now that we use a system of units where the celerity of light in a vacuum is a defined constant, that's no longer more accurate than specifying the frequency.  The great advantage of the latter is that it doesn't depend on the properties of a perfectly transparent medium.

Wavelength, on the other hand, does depend on the celerity of light in the medium of propagation.  By definition, the  dispersion equation  is the relation between wavelength and frequency.  The product of those two is the  phase celerity  which is equal, by definition, to the  speed of light in a vacuum  (c = Einstein's constant)  multiplied into the medium's  index of refraction (n):

l n   =   n c

Propagation in a dispersive medium can be described by complex quantities, according to the Question.

Causality  implies the subtle  Kramers-Kronig relations.  However, the index n need not be a real number less than 1  (in the presence of absorption, celerity can exceed the speed of light, as is often the case in the X-ray domain).

1836:  Augustin Cauchy  (1789-1857; X1805)

1864:  Charles Briot  (1817-1772; ENS 1838)

Also known as the Schott equation because the Schott optical glass company used it until 1992  (when they switched to the Sellmeier formulation, presented next).

1871:  Wilhelm Sellmeier

18??:  Hartman

Not consistent with the Kramers-Kronig relations.

18??:  Hendrick A. Lorentz  (1853-1928) :

1900:  Paul Drude (1863-1906) :

1960:  Conrady

1986:  Forouhi-Bloomer Amorphous Dispersion:

For thin films,  A. Ramin Forouhi  and  I. Bloomer  deduced dispersion equations for the refractive index, n, and extinction coefficient, k, which were published in 1986 and 1988.

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Overview   |   Cauchy's empirical equation (1836)   |   Sellmeier equation (1871)   |   Wilhelm Sellmeier ()

(2005-09-29)   Group Speed
The speed at which a wave may carry information.

A wave where a single frequency is present is unable to carry any information. 

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v   =   dw / dk   =   -l dn / dl

 Arms of Lord Rayleigh
 (John W. Strutt) 1842-1919; Nobel 1904 (2008-01-24)   Rayleigh scattering   (Tyndall effect, 1859)
(2007-07-24)   What makes the sky blue and sunsets red?
(2007-07-13)   Why do we perceive the Sun as yellow?

In 1859, John Tyndall (1820-1893) observed that small particles suspended in a fluid scatter bluish light  (short wavelength)  more strongly than reddish light  (long wavelength).  This scattering of light by tiny particles is known either as the  Tyndall effect  or  (more commonly)  Rayleigh scattering.  The intensity of the effect varies inversely as the  fourth power  of the wavelength involved.

One crude way to explain the main part of effect is to consider that an incoming electromagnetic wave produces induced dipoles which radiate energy away at the same frequency as the driving wave.

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 still working on this one...

(2008-01-24)   Index of refraction of water
Different colors travel at different speeds in water.

For visible light in water, the index of refraction  (n)  goes from  1.331  for red light  to about  1.343  for violet light.  More precisely:

Absolute Index of Refraction of Water  (n)
n  (20°C)l  (vacuum)Fraunhofer Line
1.3312656.281 nmC   ( Ha ) Red
 627.661 nma   ( O)Orange
1.3330589.3 nmD   ( Na ) Yellow
 527.039 nmE   ( Fe ) Green
1.3372486.134 nmF   ( Hb ) Blue
1.3404434.047 nmG'   ( Hg ) Indigo
1.3435396.847 nmH   ( Ca) Violet

Data gleaned for the relative index of water with respect to either air or vacuum:

  • Sodium light  (yellow, 589.3 nm)  in water at  t °C  (accuracy 0.00002):
    nvacuum   =   1.33401 - 10-7 (66 t + 26.2 t2 - 0.1817 t3 + 0.000755 t4 )

Index of Refraction of Water

 Main mode of reflection 
 off a spherical raindrop (2008-01-24)   Reflection by a raindrop
Several types of reflections are possible.

Let  n  be the index of refraction of the water inside a spherical raindrop  (relative  to the surrounding air).  The dominant mode of reflection is pictured at right.

Elementary geometry gives the angle  q  between the incident and emergent rays as a function of the angles of incidence  (i)  and refraction  (r)  which the rays make with the [centripetal] normal lines at each of the three relevant diopters:

q   =   4 r  -  2 i

As  i  increases  (starting from 0)   so does  q,  until a maximum is reached where the relation  2 dr = di  makes  dq  vanish.  At that point, Snell's law and the vanishing of its derivative provide two simultaneous equations:

n  sin r   =     sin i
  n  cos r   =   2  cos i

Putting   sin i  =  x ,  we first relation gives   sin r  =  x/n. 
Squaring the second one, we obtain:

n 2 ( 1 - x 2 / n 2 )   =   4 ( 1 - x 2 )

Therefore,   x =  (4-n 2 ) / 3 .   Using   cos 2i  =  1-2x 2   we obtain:

i   =   ½ arccos (2n2/3 - 5/3)

Similarly,  cos 2r  =  1-2x2 / n2   gives   r   =   ½ arccos (5/3 - 8/3n2 ) .  So:

    qmax   =   2 arccos (5/3 - 8/3n2 )  -  arccos (2n2/3 - 5/3)    

With  n = 1.3312  (red light in water at 20°C)  we obtain  qmax = 42.34°.  On the other hand,  n = 1.3435  (violet light)  yields  qmax = 40.58°. 

For graphics, we used  q = 42.4°,  i = 59.4°,  r = 40.4°  (n = 1.3308).  As  i  is near the Brewster angle of 53.08°, strong polarization occurs.

What the main reflection mode produces is the familiar sight of a beautiful  42°  rainbow (the primary rainbow) around the direction  opposite  to the Sun, as explained in the next article.

(2008-01-27)   Primary and Secondary Rainbows
The spectacular show put on by water droplets.

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 still working on this one...

Hyperphysics   |   Alexander's dark band   |   Alexander of Aphrodisias (fl. AD 200)

(2008-01-27)   The  22° Halo
From ice crystals in high-altitude cirrus clouds.

Under the same conditions, a halo also exists around the Sun but it's much harder to detect because of the blinding effect of direct sunlight.

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 still working on this one...

(2009-12-22)   What's the "length" of a rainbow ?
A spherical circle of angular radius  q  has a circumference  2 p sin q.

At the beginning of his celebrated lecture on rainbows  (part of the 8.03 freshman course on the physics of waves at MIT)  Walter Lewin  asks several questions.  Those are mostly about physical properties but the one pertaining to  "the length of a rainbow"  requires a mathematical digression related to  spherical geometry :

The  angular circumference  of a circle of  angular radius  q  is equal to:

2 p sin q   =   360° sin q

This translates into about  242.47°  for the  entire  circle of a rainbow  (whose angular radius on the red side is  42.34°).  The actual curvilinear length of a rainbow depends on what percentage of the whole circle is visible...

For example, if the tangents to the extremities of the visible arc of a rainbow make an angle of  45°,  then  1/8  of the whole circle is visible and the curvilinear length of the actual arch on the celestial sphere is  45° sin q  (or about  30.3°).

In spherical geometry, the length of a curve is obtained by adding the angular widths of all the infinitesimal line elements it is composed of.

Clearly, Professor Lewin did not  mean  to involve spherical geometry in that simple-minded question.  Yet, a thorough answer requires such a viewpoint.

(2011-02-07)   Decomposition of white light by a prism
The colors produced by a prism are  not  the colors of the rainbow.

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(2011-02-07)   Diffraction Grating
How the reflection of white light is spread at orders  m > 0.

A diffraction grating can be considered to be a mirror that reflects light only on strips separated by a distance  d.  When light of wavelength  l  falls on such a grating at normal incidence it is reflected at any angle  q  which allows constructive interference,  which is whenever there's an integer  m  such that:

m l   =   d sin qm

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Wikipedia :   Diffraction grating

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