[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) The celerity of a wave as a function of its frequency.
The dispersion relation of a propagation mediumis what gives thecelerity of a wave in termsof either its frequency ( ) or its wavelength ().
The simplest dispersion relation is that of anondispersive medium, for which thecelerity (u) is constant. For example, the celerity ofelectromagnetic waves in a vacuum isequal toEinstein's constant (u = c).
One common way to specify the dispersion relation is by givingthe pulsatance = 2 as a function of the wave number k =
= (k)
More generally, this relation has a vectorial counterpart involving the wave vector (k) which is appropriate for a transmission medium whichisn't necessarily isotropic. (Recall that we use bold type to denote avector.)
= (k)
(2015-07-26) 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 adefinedconstant, that's no longer more accurate than specifying the frequency. The great advantage of the latter is that it doesn't depend on the propertiesof 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 wavelengthand 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 intothe medium's index of refraction (n):
= n c
Propagation in a dispersive medium can be described bycomplex quantities,according to theQuestion.
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 theSchott optical glass companyused 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) The speed at which a wave may carry information.
A wave where a single frequency is present is unable to carry any information.
v = d / dk = 2 d / d
(2008-01-24) (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 thatan incoming electromagnetic wave produces induceddipoles which radiate energy away at thesame frequency as the driving wave.
(2008-01-24) Different colors travel at different speeds in water.
For visible light in water, the index of refraction (n) goes from for red light to about 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): nvacuum = 1.33401 10-7(66 t + 26.2 t2- 0.1817 t3+ 0.000755 t4)
(2008-01-24) Several types of reflections are possible.
Let n be the index of refraction of the water inside aspherical raindrop (relative to the surrounding air). The dominant mode of reflection is pictured at right.
Elementary geometry gives the angle between the incident and emergent rays as a function of the angles ofincidence (i) and refraction (r) which the rays makewith the [centripetal] normal lines at each of the three relevantdiopters:
= 4 r 2 i
As i increases (starting from 0) so does , until a maximum is reachedwhere the relation 2 dr = di makes d vanish. At that point,Snell's lawand 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 2 = (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:
max = 2 arccos (5/3 - 8/3n2) arccos (2n2/3 - 5/3)
With n = 1.3312 (red light in water at 20°C) we obtain max = 42.34°. On the other hand, n = 1.3435 (violet light) yields max = 40.58°.
= 42.4°, i = 59.4°, r = 40.4° (n = ). As i is near theBrewster 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 thenext article.
(2008-01-27) The spectacular show put on by water droplets.
(2008-01-27) From ice crystals in high-altitude cirrus clouds.
Under the same conditions, a halo also exists around the Sun but it's muchharder to detect because of the blinding effect of direct sunlight.
(2020-02-10)
(2009-12-22) A spherical circle of angular radius has a circumference 2 sin .
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 asksseveral 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 is equal to:
2 sin = 360° sin
This translates into about 242.47° for the entire circle of a rainbow (whose angularradius on the red side is 42.34°). The actual curvilinear lengthof 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 makean angle of 45°, then 1/8 of the whole circle is visible andthe curvilinear length of the actual arch on the celestial sphere is 45° sin (or about 30.3°).
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) The colors produced by a prism are not the colors of the rainbow.
(2011-02-07) How the reflection of white light is spread at orders m > 0.
A diffraction grating can be considered to be a mirror that reflects lightonly on strips separated by a distance d. When light of wavelength falls on such a grating at normal incidence it is reflected at any angle which allows constructive interference, which is whenever there's an integer m such that:
