Conditions For Total Internal Reflection

Total Internal Reflection

Lumen Learning

Learning Objectives

Past the stop of this section, you will exist able to:

Explicate the miracle of total internal reflection.
Describe the workings and uses of cobweb optics.
Analyze the reason for the sparkle of diamonds.

A adept-quality mirror may reflect more than 90% of the light that falls on information technology, absorbing the rest. But it would be useful to have a mirror that reflects all of the light that falls on information technology. Interestingly, we tin produce total reflection using an aspect of refraction.

Consider what happens when a ray of calorie-free strikes the surface between ii materials, such as is shown in Figure 1a. Role of the light crosses the boundary and is refracted; the remainder is reflected. If, every bit shown in the effigy, the index of refraction for the second medium is less than for the first, the ray bends away from the perpendicular. (Since n _ane >due north ₂, the bending of refraction is greater than the angle of incidence—that is, θ _one >θ ₂.) Now imagine what happens every bit the incident angle is increased. This causes θ ₂ to increment also. The largest the angle of refraction θ ₂ can be is 90º, as shown in Figure 1b.The critical angle θ _c for a combination of materials is defined to exist the incident angle θ ₁ that produces an angle of refraction of ninetyº. That is, θ _c is the incident angle for which θ ₂ = 90º. If the incident angle θ ₁ is greater than the critical angle, as shown in Figure 1c, then all of the light is reflected back into medium 1, a status called full internal reflection.

Critical Angle

The incident angle θ ₁ that produces an angle of refraction of 90º is chosen the critical angle, θ _c.

In the first figure, an incident ray at an angle theta 1 with a perpendicular line drawn at the point of incidence travels from n1 to n2. The incident ray suffers both refraction and reflection. The angle of refraction is theta 2. In the second figure, as theta 1 is increased, the angle of refraction theta 2 becomes 90 degrees and the angle of reflection corresponding to 90 degrees is theta c. In the third figure, theta c greater than theta i, total internal reflection takes place and instead of refraction, reflection takes place and the light ray travels back into medium n1. — Figure 1. (a) A ray of light crosses a purlieus where the speed of light increases and the alphabetize of refraction decreases. That is, n ₂ < *due north* _i . The ray bends abroad from the perpendicular. (b) The critical angle θc is the i for which the angle of refraction is. (c) Total internal reflection occurs when the incident angle is greater than the disquisitional angle.

Snell'southward constabulary states the human relationship betwixt angles and indices of refraction. Information technology is given past

n _one sinθ ₁ =n ₂ sinθ ₂.

When the incident angle equals the critical angle (θ _i =θ_c ), the angle of refraction is 90º (θ ₂ = 90º). Noting that sin 90º = 1, Snell'due south law in this case becomes

north ₁ sinθ _ane =n ₂.

The critical angle θ _c for a given combination of materials is thus

$\theta_{c}=\sin^{-1}\left(\frac{n_2}{n_1}\right)\\$ fornorth ₁ >n ₂.

Full internal reflection occurs for whatever incident bending greater than the critical angle θ _c, and it can but occur when the second medium has an index of refraction less than the first. Note the in a higher place equation is written for a light ray that travels in medium 1 and reflects from medium two, as shown in the figure.

Example 1. How Big is the Critical Angle Here?

What is the disquisitional angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air?

Strategy

The index of refraction for polystyrene is found to be 1.49 in Figure two, and the alphabetize 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 the equation $\theta_{c}=\sin^{-1}\left(\frac{n_2}{n_1}\right)\\$ can be used to find the critical bending θ _c. Hither, then, n _two = 1.00 and due north ₁ = 1.49.

Solution

The critical bending is given by

$\theta_{c}=\sin^{-1}\left(\frac{n_2}{n_1}\right)\\$

Substituting the identified values gives

$\begin{array}{lll}\theta_{c}&=&\sin^{-1}\left(\frac{1.00}{1.49}\right)\\\text{ }&=&\sin^{-1}\left(0.671\right)\\\text{ }&=&42.2^{\circ}\end{array}\\$

Discussion

This means that any ray of lite inside the plastic that strikes the surface at an bending greater than 42.2º will be totally reflected. This will make the inside surface of the articulate plastic a perfect mirror for such rays without whatever need for the silvering used on common mirrors. Different combinations of materials accept unlike critical angles, but whatever combination with n ₁ >n ₂ can produce full internal reflection. The same calculation equally made hither shows that the disquisitional angle for a ray going from h2o to air is 48.6º, while that from diamond to air is 24.4º, and that from flint glass to crown drinking glass is 66.3º. There is no full reflection for rays going in the other direction—for case, from air to water—since the condition that the second medium must accept a smaller index of refraction is not satisfied. A number of interesting applications of full internal reflection follow.

Fiber Optics: Endoscopes to Telephones

Light ray enters an S-shaped tube and undergoes multiple reflections, finally emerging through the other end. — Figure 2. Light inbound a thin fiber may strike the inside surface at large or grazing angles and is completely reflected if these angles exceed the critical angle. Such rays go along down the fiber, even following it around corners, since the angles of reflection and incidence remain large.

Fiber optics is one application of total internal reflection that is in broad employ. In communications, it is used to transmit telephone, cyberspace, and cable TV signals. Cobweb eyes employs the transmission of calorie-free down fibers of plastic or drinking glass. Because the fibers are thin, calorie-free inbound i is likely to strike the within surface at an bending greater than the critical angle and, thus, be totally reflected (See Figure ii.) The index of refraction exterior the fiber must exist smaller than within, a condition that is easily satisfied by blanket the outside of the fiber with a material having an appropriate refractive index. In fact, most fibers have a varying refractive alphabetize to permit 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.

Picture (a) shows how an image A is transmitted through a bundle of parallel fibers. Picture (b) shows an endoscope image. — Figure 3. (a) An epitome is transmitted by a packet of fibers that have fixed neighbors. (b) An endoscope is used to probe the body, both transmitting low-cal to the interior and returning an prototype such equally the 1 shown. (credit: Med_Chaos, Wikimedia Eatables)

Bundles of fibers can be used to transmit an image without a lens, as illustrated in Effigy 3. The output of a device chosen an endoscope is shown in Effigy 3b. Endoscopes are used to explore the torso through various orifices or pocket-size incisions. Light is transmitted down ane fiber bundle to illuminate internal parts, and the reflected calorie-free is transmitted back out through some other to exist observed. Surgery tin be performed, such as arthroscopic surgery on the articulatio genus joint, employing cutting tools attached to and observed with the endoscope. Samples can too exist obtained, such as by lassoing an abdominal polyp for external examination.

The image shows a bundle fiber with a medium of refractive index n sub 1 inside surrounded by a medium n sub 2. Medium n sub 2 is made up of cladding material and n sub 1 is the core. — Effigy four. Fibers in bundles are clad by a material that has a lower alphabetize of refraction than the cadre to ensure full internal reflection, fifty-fifty when fibers are in contact with one another. This shows a unmarried fiber with its cladding.

Fiber optics has revolutionized surgical techniques and observations within the body. There are a host of medical diagnostic and therapeutic uses. The flexibility of the fiber optic bundle allows it to navigate around hard and small-scale regions in the body, such equally the intestines, the heart, blood vessels, and joints. Transmission of an intense laser beam to fire abroad obstructing plaques in major arteries every bit well as delivering light to actuate chemotherapy drugs are becoming commonplace. Optical fibers have in fact enabled microsurgery and remote surgery where the incisions are small and the surgeon'due south fingers do not need to impact the diseased tissue.

Fibers in bundles are surrounded by a cladding material that has a lower alphabetize of refraction than the core. (See Figure 4.) The cladding prevents calorie-free from being transmitted between fibers in a parcel. 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 full internal reflection dorsum into the cadre), none can be transmitted between clad fibers that are in contact with one another. The cladding prevents calorie-free from escaping out of the fiber; instead most of the calorie-free is propagated forth the length of the fiber, minimizing the loss of signal and ensuring that a quality image is formed at the other stop. The cladding and an additional protective layer brand optical fibers flexible and durable.

Cladding

The cladding prevents low-cal from being transmitted between fibers in a parcel.

Special tiny lenses that can be attached to the ends of bundles of fibers are being designed and fabricated. Light emerging from a fiber parcel tin can be focused and a tiny spot can exist imaged. In some cases the spot can be scanned, assuasive quality imaging of a region inside the body. Special infinitesimal optical filters inserted at the end of the fiber bundle accept the chapters to image tens of microns below the surface without cutting the surface—not-intrusive diagnostics. This is especially useful for determining the extent of cancers in the tummy and bowel.

Most telephone conversations and Internet communications are now carried by light amplification by stimulated emission of radiation signals along optical fibers. Extensive optical fiber cables take been placed on the bounding main floor and hush-hush to enable optical communications. Optical fiber communication systems offering several advantages over electrical (copper) based systems, particularly for long distances. The fibers can exist made so transparent that light can travel many kilometers earlier it becomes dim plenty to require amplification—much superior to copper conductors. This belongings of optical fibers is chosen low loss. Lasers emit calorie-free with characteristics that permit far more than conversations in one cobweb than are possible with electric signals on a single usher. This holding of optical fibers is called high bandwidth. Optical signals in i cobweb do not produce undesirable furnishings in other adjacent fibers. This property of optical fibers is chosen reduced crosstalk. We shall explore the unique characteristics of laser radiation in a later on chapter.

Corner Reflectors and Diamonds

A light ray that strikes an object consisting of 2 mutually perpendicular reflecting surfaces is reflected dorsum exactly parallel to the management from which it came. This is true whenever the reflecting surfaces are perpendicular, and it is independent of the bending of incidence. Such an object, shown in Figure 5, is called a corner reflector, since the calorie-free bounces from its inside corner. Many inexpensive reflector buttons on bicycles, cars, and alert signs have corner reflectors designed to render light in the management from which it originated. Information technology was more than expensive for astronauts to identify 1 on the moon. Laser signals tin be bounced from that corner reflector to measure out the gradually increasing altitude to the moon with great precision.

Picture (a) shows the lunar expedition with the astronauts and their space shuttle. Picture (b) shows rectangular and round shaped bicycle reflectors. — Effigy 5. (a) Astronauts placed a corner reflector on the moon to measure its gradually increasing orbital distance. (credit: NASA) (b) The bright spots on these bicycle condom reflectors are reflections of the flash of the camera that took this pic on a dark dark. (credit: Julo, Wikimedia Commons)

Corner reflectors are perfectly efficient when the conditions for total internal reflection are satisfied. With common materials, it is easy to obtain a disquisitional bending that is less than 45º. One employ of these perfect mirrors is in binoculars, as shown in Figure 6. Some other use is in periscopes found in submarines.

The picture shows binoculars with prisms inside. The light through one of the object lenses enters through the first prism and suffers total internal reflection and then falls on the second prism and gets total internally reflected and emerges out through one of the eyepiece lenses. — Figure 6. These binoculars employ corner reflectors with full internal reflection to get light to the observer's eyes.

The Sparkle of Diamonds

A light ray falls onto one of the faces of a diamond, gets refracted, falls on another face and gets totally internally reflected, and this reflected ray further undergoes multiple reflections when it falls on other faces. — Figure vii. Light cannot easily escape a diamond, because its critical angle with air is and then small. Most reflections are total, and the facets are placed so that light tin leave merely in item ways—thus concentrating the light and making the diamond sparkle.

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 but 24.4º, and and then when low-cal enters a diamond, it has trouble getting dorsum out. (Run across Figure 7.) Although calorie-free freely enters the diamond, it can get out only if it makes an angle less than 24.4º. Facets on diamonds are specifically intended to make this unlikely, so that the low-cal can exit only in certain places. Good diamonds are very clear, so that the light makes many internal reflections and is concentrated at the few places it tin can exit—hence the sparkle. (Zircon is a natural gemstone that has an uncommonly large index of refraction, but not as big as diamond, then information technology is non as highly prized. Cubic zirconia is manufactured and has an even higher index of refraction (≈ii.17), but still less than that of diamond.) The colors y'all come across emerging from a sparkling diamond are not due to the diamond's color, which is usually nearly colorless. Those colors outcome from dispersion, the topic of Dispersion: The Rainbow and Prisms. 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 pinkish, red, champagne, and cognac diamonds, while effectually 50% of the world'southward clear diamonds come from central and southern Africa.

PhET Explorations: Bending Low-cal

Explore bending of lite between two media with different indices of refraction. Come across how irresolute from air to h2o to drinking glass changes the bending angle. Play with prisms of different shapes and brand rainbows.

Bending Light screenshot. — Click to download the simulation. Run using Java.

Section Summary

The incident angle that produces an angle of refraction of 90º is called critical bending.
Total internal reflection is a phenomenon that occurs at the boundary between 2 mediums, such that if the incident bending in the first medium is greater than the critical bending, then all the light is reflected back into that medium.
Fiber optics involves the transmission of light downward fibers of plastic or drinking glass, applying the principle of total internal reflection.
Endoscopes are used to explore the body through various orifices or minor incisions, based on the transmission of low-cal through optical fibers.
Cladding prevents lite from being transmitted between fibers in a packet.
Diamonds sparkle due to total internal reflection coupled with a large index of refraction.

Conceptual Questions

A ring with a colorless gemstone is dropped into h2o. The gemstone becomes invisible when submerged. Can it be a diamond? Explain.
A loftier-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with petty assimilation. Explicate how it can sparkle with flashes of brilliant color when illuminated by white light.
Is information technology possible that full internal reflection plays a function in rainbows? Explain in terms of indices of refraction and angles, mayhap referring to Figure eight. Some of us take seen the formation of a double rainbow. Is it physically possible to find a triple rainbow?

Figure 8. Double rainbows are not a very common observance. (credit: InvictusOU812, Flickr)
The about common type of delusion is an illusion that light from faraway objects is reflected by a pool of water that is non really there. Mirages are more often than not observed in deserts, when at that place is a hot layer of air near the ground. Given that the refractive index of air is lower for air at higher temperatures, explain how mirages tin can be formed.

Glossary

critical bending: incident angle that produces an angle of refraction of 90º

fiber eyes: transmission of light down fibers of plastic or drinking glass, applying the principle of total internal reflection

corner reflector: an object consisting of ii mutually perpendicular reflecting surfaces, and so that the light that enters is reflected back exactly parallel to the direction from which it came

zircon: natural gemstone with a large index of refraction

Selected Solutions to Problems & Exercises

3. 66.3º

5. > 1.414

7. 1.50, benzene