Explain the phenomenon of total internal reflection
Describe the workings and uses of optical fibers
Analyze the reason for the sparkle of diamonds
A good-quality mirror may reflect more than 90% of the light
that falls on it, absorbing the rest. But it would be useful to
have a mirror that reflects all of the light that falls on it.
Interestingly, we can produce total reflection using an aspect of
refraction.
Consider what happens when a ray of light strikes the surface
between two materials, as shown in Figure \(\PageIndex{1a}\). Part
of the light crosses the boundary and is refracted; the rest is
reflected. If, as shown in the figure, the index of refraction for
the second medium is less than for the first, the ray bends away
from the perpendicular. (Since \(n_1>n_2\), the angle of
refraction is greater than the angle of incidence—that is,
\(θ_1>θ_2\).) Now imagine what happens as the incident angle
increases. This causes \(θ_2\) to increase also. The largest the
angle of refraction \(θ_2\) can be is \(90°\), as shown in Figure
\(\PageIndex{1b}\).
Figure \(\PageIndex{1}\): (a) A ray of light crosses a
boundary where the index of refraction decreases. That is,
\(n_2<n_1\). The ray bends away from the perpendicular. (b) The
critical angle θc is the angle of incidence for which
the angle of refraction is 90°. (c) Total internal reflection
occurs when the incident angle is greater than the critical
angle.
The critical angle \(θ_c\) for a combination of
materials is defined to be the incident angle \(θ_1\) that produces
an angle of refraction of \(90°\). That is, \(θ_c\) is the incident
angle for which \(θ_2=90°\). If the incident angle \(θ_1\) is
greater than the critical angle, as shown in Figure
\(\PageIndex{1c}\), then all of the light is reflected back into
medium 1, a condition called total internal reflection.
(As Figure \(\PageIndex{1}\) shows, the reflected rays obey the
law of reflection so that the angle of reflection is equal
to the angle of incidence in all three cases.)
Snell’s law states the relationship between angles and indices
of refraction. It is given by
\[n_1\sin θ_1=n_2 \sin θ_2. \nonumber \]
When the incident angle equals the critical angle (\(θ_1=θ_c\)),
the angle of refraction is \(90°\) (\(θ_2=90°\)). Noting that
\(\sin 90°=1\), Snell’s law in this case becomes
\[n_1 \, \sin \, θ_1 = n_2. \nonumber \]
The critical angle \(θ_c\) for a given combination of materials
is thus
Total internal reflection occurs for any incident angle greater
than the critical angle \(θ_c\), and it can only occur when the
second medium has an index of refraction less than the first. Note
that this equation is written for a light ray that travels in
medium 1 and reflects from medium 2, as shown in Figure
\(\PageIndex{1}\).
Example \(\PageIndex{1}\): Determining a
Critical Angle
What is the critical angle for light traveling in a polystyrene
(a type of plastic) pipe surrounded by air? The index of refraction
for polystyrene is 1.49.
Strategy
The index of refraction of air can be taken to be 1.00, as
before. Thus, the condition that the second medium (air) has an
index of refraction less than the first (plastic) is satisfied, and
we can use the equation
This result means that any ray of light inside the plastic that
strikes the surface at an angle greater than 42.2° is totally
reflected. This makes the inside surface of the clear plastic a
perfect mirror for such rays, without any need for the silvering
used on common mirrors. Different combinations of materials have
different critical angles, but any combination with \(n_1>n_2\)
can produce total internal reflection. The same calculation as made
here shows that the critical angle for a ray going from water to
air is 48.6°, whereas that from diamond to air is 24.4°, and that
from flint glass to crown glass is 66.3°.
Exercise \(\PageIndex{1}\)
At the surface between air and water, light rays can go from air
to water and from water to air. For which ray is there no
possibility of total internal reflection?
Answer
air to water, because the condition that the second medium must
have a smaller index of refraction is not satisfied
In the photo that opens this chapter, the image of a swimmer
underwater is captured by a camera that is also underwater. The
swimmer in the upper half of the photograph, apparently facing
upward, is, in fact, a reflected image of the swimmer below. The
circular ripple near the photograph’s center is actually on the
water surface. The undisturbed water surrounding it makes a good
reflecting surface when viewed from below, thanks to total internal
reflection. However, at the very top edge of this photograph, rays
from below strike the surface with incident angles less than the
critical angle, allowing the camera to capture a view of activities
on the pool deck above water.
Fiber Optics: Endoscopes to Telephones
Fiber optics is one application of total internal reflection
that is in wide use. In communications, it is used to transmit
telephone, internet, and cable TV signals. Fiber optics employs the
transmission of light down fibers of plastic or glass. Because the
fibers are thin, light entering one is likely to strike the inside
surface at an angle greater than the critical angle and, thus, be
totally reflected (Figure \(\PageIndex{2}\)). The index of
refraction outside the fiber must be smaller than inside. In fact,
most fibers have a varying refractive index to allow more light to
be guided along the fiber through total internal refraction. Rays
are reflected around corners as shown, making the fibers into tiny
light pipes.
Figure \(\PageIndex{2}\): Light entering a thin optic
fiber may strike the inside surface at large or grazing angles and
is completely reflected if these angles exceed the critical angle.
Such rays continue down the fiber, even following it around
corners, since the angles of reflection and incidence remain
large.
Bundles of fibers can be used to transmit an image without a
lens, as illustrated in Figure \(\PageIndex{3}\). The output of a
device called an endoscope is shown in Figure \(\PageIndex{1b}\).
Endoscopes are used to explore the interior of the body through its
natural orifices or minor incisions. Light is transmitted down one
fiber bundle to illuminate internal parts, and the reflected light
is transmitted back out through another bundle to be observed.
Figure \(\PageIndex{3}\): (a) An image “A” is
transmitted by a bundle of optical fibers. (b) An endoscope is used
to probe the body, both transmitting light to the interior and
returning an image such as the one shown of a human epiglottis (a
structure at the base of the tongue). (credit b: modification of
work by “Med_Chaos”/Wikimedia Commons)
Fiber optics has revolutionized surgical techniques and
observations within the body, with a host of medical diagnostic and
therapeutic uses. Surgery can be performed, such as arthroscopic
surgery on a knee or shoulder joint, employing cutting tools
attached to and observed with the endoscope. Samples can also be
obtained, such as by lassoing an intestinal polyp for external
examination. The flexibility of the fiber optic bundle allows
doctors to navigate it around small and difficult-to-reach regions
in the body, such as the intestines, the heart, blood vessels, and
joints. Transmission of an intense laser beam to burn away
obstructing plaques in major arteries, as well as delivering light
to activate chemotherapy drugs, are becoming commonplace. Optical
fibers have in fact enabled microsurgery and remote surgery where
the incisions are small and the surgeon’s fingers do not need to
touch the diseased tissue.
Optical fibers in bundles are surrounded by a
cladding material that has a lower index of
refraction than the core (Figure \(\PageIndex{4}\)). The cladding
prevents light from being transmitted between fibers in a bundle.
Without cladding, light could pass between fibers in contact, since
their indices of refraction are identical. Since no light gets into
the cladding (there is total internal reflection back into the
core), none can be transmitted between clad fibers that are in
contact with one another. Instead, the light is propagated along
the length of the fiber, minimizing the loss of signal and ensuring
that a quality image is formed at the other end. The cladding and
an additional protective layer make optical fibers durable as well
as flexible.
Figure \(\PageIndex{4}\): Fibers in bundles are clad by
a material that has a lower index of refraction than the core to
ensure total internal reflection, even when fibers are in contact
with one another.
Special tiny lenses that can be attached to the ends of bundles
of fibers have been designed and fabricated. Light emerging from a
fiber bundle can be focused through such a lens, imaging a tiny
spot. In some cases, the spot can be scanned, allowing quality
imaging of a region inside the body. Special minute optical filters
inserted at the end of the fiber bundle have the capacity to image
the interior of organs located tens of microns below the surface
without cutting the surface—an area known as nonintrusive
diagnostics. This is particularly useful for determining the extent
of cancers in the stomach and bowel.
In another type of application, optical fibers are commonly used
to carry signals for telephone conversations and internet
communications. Extensive optical fiber cables have been placed on
the ocean floor and underground to enable optical communications.
Optical fiber communication systems offer several advantages over
electrical (copper)-based systems, particularly for long distances.
The fibers can be made so transparent that light can travel many
kilometers before it becomes dim enough to require
amplification—much superior to copper conductors. This property of
optical fibers is called low loss. Lasers emit light with
characteristics that allow far more conversations in one fiber than
are possible with electric signals on a single conductor. This
property of optical fibers is called high bandwidth. Optical
signals in one fiber do not produce undesirable effects in other
adjacent fibers. This property of optical fibers is called reduced
crosstalk. We shall explore the unique characteristics of laser
radiation in a later chapter.
Corner Reflectors and Diamonds
Corner reflectors are perfectly efficient when the conditions
for total internal reflection are satisfied. With common materials,
it is easy to obtain a critical angle that is less than 45°. One
use of these perfect mirrors is in binoculars, as shown in Figure
\(\PageIndex{5}\). Another use is in periscopes found in
submarines.
Figure \(\PageIndex{5}\): These binoculars employ
corner reflectors (prisms) with total internal reflection to get
light to the observer’s eyes.
Total internal reflection, coupled with a large index of
refraction, explains why diamonds sparkle more
than other materials. The critical angle for a diamond-to-air
surface is only 24.4°, so when light enters a
diamond, it has trouble getting back out (Figure
\(\PageIndex{6}\)). Although light freely enters the diamond, it
can exit only if it makes an angle less than 24.4°. Facets on
diamonds are specifically intended to make this unlikely. Good
diamonds are very clear, so that the light makes many internal
reflections and is concentrated before exiting—hence the bright
sparkle. (Zircon is a natural gemstone that has an exceptionally
large index of refraction, but it is not as large as diamond, so it
is not as highly prized. Cubic zirconia is manufactured and has an
even higher index of refraction (≈2.17), but it is
still less than that of diamond.) The colors you see emerging from
a clear diamond are not due to the diamond’s color, which is
usually nearly colorless, but result from
dispersion. Colored diamonds get their color from structural
defects of the crystal lattice and the inclusion of minute
quantities of graphite and other materials. The Argyle Mine in
Western Australia produces around 90% of the world’s pink, red,
champagne, and cognac diamonds, whereas around 50% of the world’s
clear diamonds come from central and southern Africa.
Figure \(\PageIndex{6}\): Light cannot easily escape a
diamond, because its critical angle with air is so small. Most
reflections are total, and the facets are placed so that light can
exit only in particular ways—thus concentrating the light and
making the diamond sparkle brightly.
Explore refraction
and reflection of light between two media with different
indices of refraction. Try to make the refracted ray disappear with
total internal reflection. Use the protractor tool to measure the
critical angle and compare with the prediction from Equation
\ref{critical}.