Samuel J. Ling; Jeff Sanny; William Moebs

Learning Objectives

By the end of this section, you will be able to:

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 1.14(a). 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, $θ_{2} > θ_{1} .)$ 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 part (b). 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 1.14(c), then all of the light is reflected back into medium 1, a condition called total internal reflection. (As the figure 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.)

In figure a, an incident ray at an angle theta 1 with a perpendicular line drawn at the point of incidence travels from n 1 to n 2. The incident ray undergoes both refraction and reflection. The angle of refraction o the refracted ray in medium n 2 is theta 2. The angle of reflection of the reflected ray in medium 1 is theta 1. In figure b, the incident angle is theta c which is larger than the angle of incidence in figure a. The angle of refraction theta 2 becomes 90 degrees and the angle of reflection is theta c. In figure c, the angle of incidence theta 1 is greater than theta c, total internal reflection takes place and only reflection takes place. The light ray travels back into medium n 1, with the reflection angle being theta one.

Figure 1.14 (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

θ_{c}

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.

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} .

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} .

The critical angle $θ_{c}$ for a given combination of materials is thus

θ_{c} = {sin}^{−1} (\frac{n_{2}}{n_{1}}) for n_{1} > n_{2} .

1.5

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 1.14.

Example 1.4

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

θ_{c} = {sin}^{−1} (\frac{n_{2}}{n_{1}})

to find the critical angle $θ_{c},$ where $n_{2} = 1.00$ and $n_{1} = 1.49 .$

Solution

Substituting the identified values gives

θ_{c} = {sin}^{−1} (\frac{1.00}{1.49}) = {sin}^{−1} (0.671) = 42.2 ° .

Significance

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 °

.

Check Your Understanding 1.3

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?

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 1.15). 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.

Light ray enters an S-shaped optical fiber and undergoes multiple internal reflections at the fiber walls, finally emerging through the other end.

Figure 1.15 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 1.16. The output of a device called an endoscope is shown in Figure 1.16(b). 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 (a) shows how an image A is transmitted through a bundle of parallel fibers. Figure (b) shows an endoscope image.

Figure 1.16 (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 1.17). 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.

The figure shows a fiber with a medium of refractive index n 1 surrounded by a medium n 2. Medium n sub 2 is made up of cladding material and n sub 1 is the core. The light ray reflects at the interface between the core and the cladding, staying inside the core as it travels along the fiber.

Figure 1.17 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 (The Law of Reflection) 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 1.18. Another use is in periscopes found in submarines.

The figure shows binoculars with prisms inside. The light through one of the object lenses enters through the first prism and undergoes two total internal reflection, exiting parallel to the incident ray but shifted over so it then falls on the second prism. The ray again total internally reflects twice and shifts to emerge out through one of the eyepiece lenses parallel to the incident ray.

Figure 1.18 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 1.19). 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. The colors result from dispersion, which we discuss in 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.

A light ray falls onto one of the faces of a diamond, gets refracted, falls on another face and gets totally internally reflected since the angle of incidence at the diamond air interface is larger than the critical angle. This reflected ray further undergoes multiple reflections when it falls on other faces.

Figure 1.19 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.

Interactive

Explore refraction and reflection of light between two media with different indices of refraction by engaging this Phet simulation. 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 1.5.

1.4 Total Internal Reflection