Describe the effects of dispersion in producing rainbows
Summarize the advantages and disadvantages of dispersion
Everyone enjoys the spectacle of a rainbow glimmering against a
dark stormy sky. How does sunlight falling on clear drops of rain
get broken into the rainbow of colors we see? The same process
causes white light to be broken into colors by a clear glass prism
or a diamond (Figure \(\PageIndex{1}\)).
We see about six colors in a rainbow—red, orange, yellow, green,
blue, and violet; sometimes indigo is listed, too. These colors are
associated with different wavelengths of light, as shown in Figure
\(\PageIndex{2}\). When our eye receives pure-wavelength light, we
tend to see only one of the six colors, depending on wavelength.
The thousands of other hues we can sense in other situations are
our eye’s response to various mixtures of wavelengths. White light,
in particular, is a fairly uniform mixture of all visible
wavelengths. Sunlight, considered to be white, actually appears to
be a bit yellow, because of its mixture of wavelengths, but it does
contain all visible wavelengths. The sequence of colors in rainbows
is the same sequence as the colors shown in the figure. This
implies that white light is spread out in a rainbow according to
wavelength. Dispersion is defined as the spreading of white light
into its full spectrum of wavelengths. More technically, dispersion
occurs whenever the propagation of light depends on wavelength.
Any type of wave can exhibit dispersion. For example, sound
waves, all types of electromagnetic waves, and water waves can be
dispersed according to wavelength. Dispersion may require special
circumstances and can result in spectacular displays such as in the
production of a rainbow. This is also true for sound, since all
frequencies ordinarily travel at the same speed. If you listen to
sound through a long tube, such as a vacuum cleaner hose, you can
easily hear it dispersed by interaction with the tube. Dispersion,
in fact, can reveal a great deal about what the wave has
encountered that disperses its wavelengths. The dispersion of
electromagnetic radiation from outer space, for example, has
revealed much about what exists between the stars—the so-called
interstellar medium.
Nick Moore’s video discusses dispersion of a pulse as he taps a
long spring. Follow his explanation as Moore replays the high-speed
footage showing high frequency waves outrunning the lower frequency
waves. https://www.youtube.com/watch?v=KbmOcT5sX7I
Refraction is responsible for dispersion in rainbows and many
other situations. The angle of refraction depends on the index of
refraction, as we know from Snell’s law. We know that the index of
refraction n depends on the medium. But for a given
medium, n also depends on wavelength (Table
\(\PageIndex{1}\)).
Table \(\PageIndex{1}\): Index of Refraction (\(n\)) in
Selected Media at Various Wavelengths
Medium
Red (660 nm)
Orange (610 nm)
Yellow (580 nm)
Green (550 nm)
Blue (470 nm)
Violet (410 nm)
Water
1.331
1.332
1.333
1.335
1.338
1.342
Diamond
2.410
2.415
2.417
2.426
2.444
2.458
Glass, crown
1.512
1.514
1.518
1.519
1.524
1.530
Glass, flint
1.662
1.665
1.667
1.674
1.684
1.698
Polystyrene
1.488
1.490
1.492
1.493
1.499
1.506
Quartz, fused
1.455
1.456
1.458
1.459
1.462
1.468
Note that for a given medium, n increases as wavelength
decreases and is greatest for violet light. Thus, violet light is
bent more than red light, as shown for a prism in Figure
\(\PageIndex{3b}\). White light is dispersed into the same sequence
of wavelengths as seen in Figures \(\PageIndex{1}\) and
\(\PageIndex{2}\).
Example \(\PageIndex{1}\): Dispersion of White
Light by Flint Glass
A beam of white light goes from air into flint glass at an
incidence angle of 43.2°. What is the angle between the red (660
nm) and violet (410 nm) parts of the refracted light?
Strategy
Values for the indices of refraction for flint glass at various
wavelengths are listed in Table \(\PageIndex{1}\). Use these values
for calculate the angle of refraction for each color and then take
the difference to find the dispersion angle.
Solution
Applying the law of refraction for the red part of the beam
Although 0.6° may seem like a negligibly small angle, if this
beam is allowed to propagate a long enough distance, the dispersion
of colors becomes quite noticeable.
Exercise \(\PageIndex{1}\)
In the preceding example, how much distance inside the block of
flint glass would the red and the violet rays have to progress
before they are separated by 1.0 mm?
Answer
9.3 cm
Rainbows are produced by a combination of refraction and
reflection. You may have noticed that you see a rainbow only when
you look away from the Sun. Light enters a drop of water and is
reflected from the back of the drop (Figure \(\PageIndex{4}\)).
The light is refracted both as it enters and as it leaves the
drop. Since the index of refraction of water varies with
wavelength, the light is dispersed, and a rainbow is observed
(Figure \(\PageIndex{4a}\)). (No dispersion occurs at the back
surface, because the law of reflection does not depend on
wavelength.) The actual rainbow of colors seen by an observer
depends on the myriad rays being refracted and reflected toward the
observer’s eyes from numerous drops of water. The effect is most
spectacular when the background is dark, as in stormy weather, but
can also be observed in waterfalls and lawn sprinklers. The arc of
a rainbow comes from the need to be looking at a specific angle
relative to the direction of the Sun, as illustrated in Figure
\(\PageIndex{4b}\). If two reflections of light occur within the
water drop, another “secondary” rainbow is produced. This rare
event produces an arc that lies above the primary rainbow arc, as
in Figure \(\PageIndex{4c}\), and produces colors in the reverse
order of the primary rainbow, with red at the lowest angle and
violet at the largest angle.
Dispersion may produce beautiful rainbows, but it can cause
problems in optical systems. White light used to transmit messages
in a fiber is dispersed, spreading out in time and eventually
overlapping with other messages. Since a laser produces a nearly
pure wavelength, its light experiences little dispersion, an
advantage over white light for transmission of information. In
contrast, dispersion of electromagnetic waves coming to us from
outer space can be used to determine the amount of matter they pass
through.