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

(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)

(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).

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).

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.

(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.

v =
dw / dk
=
-l^{2 }
dn / dl

(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.

(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:

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): n_{vacuum}
= 1.33401 -
10^{-7 }(66 t + 26.2 t^{2 }- 0.1817 t^{3 }+ 0.000755 t^{4 })

(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 its derivative translate into 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^{ 2 } = (4-n^{ 2 }) / 3 .
Using cos 2i = 1-2x^{ 2 } we obtain:

i = ½ arccos (2n^{2}/3 - 5/3)

Similarly, cos 2r = 1-2x^{2 }/ n^{2}
gives
r = ½ arccos (5/3 - 8/3n^{2 }) . So:

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

(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.

(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.

(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: