86 Polarization

Many people wear polarized sunglasses to reduce glare while driving, fishing, skiing, or spending time near water. These lenses make it easier to see by blocking much of the reflected light that produces bright glare (Figure 86.1). The ability of polarized sunglasses to reduce reflections is based on an important property of light known as polarization. In this chapter, we explore what polarization is, how it is produced, and why it is useful. Polarization provides strong evidence that light behaves as a transverse electromagnetic wave, and it has numerous practical applications ranging from photography and LCD screens to medical imaging and biochemical analysis.

What Is Polarization?

Light is an electromagnetic (EM) wave. As discussed in earlier chapters, electromagnetic waves are transverse waves, meaning that their electric and magnetic fields oscillate perpendicular to the direction in which the wave travels. The electric field, magnetic field, and direction of propagation are always mutually perpendicular.

Polarization describes the orientation of the oscillations of a transverse wave. For electromagnetic waves, the polarization direction is defined as the direction of the electric field.

Do not confuse this definition with the term electric polarization used earlier in this textbook to describe the separation of positive and negative charges within a material. Although both concepts use the word "polarization," they describe different physical phenomena.

Why Only Transverse Waves Can Be Polarized

A useful analogy is provided by waves traveling along a rope. If the rope is shaken up and down, the wave oscillates vertically. If it is shaken side to side, the wave oscillates horizontally. These waves have well-defined directions of vibration and are therefore polarized. Now imagine placing a narrow vertical slot in the path of each wave. The vertically oscillating wave passes through the slot, while the horizontally oscillating wave is blocked because its motion is perpendicular to the opening. A polarizing filter acts in much the same way for light. Instead of selecting mechanical vibrations, it selects one orientation of the electric field.

Unpolarized and Polarized Light

Most natural light sources, including the Sun, incandescent bulbs, and many LEDs, emit unpolarized light. Although each individual electromagnetic wave has a well-defined direction of polarization, the enormous number of waves emitted by these sources have random orientations. As a result, every possible polarization direction is represented equally. Because the electric field points in many different directions, unpolarized light has no preferred orientation of oscillation.

How a Polarizing Filter Works

A polarizing filter converts unpolarized light into polarized light by transmitting only one orientation of the electric field while absorbing or blocking the others. Commercial polarizing filters are made from long-chain molecules that are aligned in the same direction. These molecules interact much more strongly with electric fields that oscillate parallel to their length than with electric fields that oscillate perpendicular to them. As a result, only the component of the electric field aligned with the filter's transmission direction is allowed to pass through. The direction that a filter transmits is called its polarization axis.

Two Polarizing Filters

The behavior of polarized light is easiest to understand by considering two polarizing filters placed one after another. The first filter converts the incoming unpolarized light into polarized light. After passing through this filter, the electric field oscillates in only one direction. The amount of light transmitted by the second filter depends on the angle between its polarization axis and the direction of polarization of the incoming light.

• If the two filter axes are parallel, all of the polarized light transmitted by the first filter also passes through the second.
• If the second filter is rotated, only the component of the electric field parallel to its axis is transmitted.
• If the two axes are perpendicular, no light is transmitted by an ideal pair of filters.

This simple experiment demonstrates that light possesses a definite direction of polarization.

The Component of the Electric Field

Suppose polarized light strikes a polarizing filter whose transmission axis makes an angle [latex]\theta[/latex] with the direction of polarization. Only the component of the electric field parallel to the filter axis passes through. Using vector components, the transmitted electric-field amplitude is

Because the intensity of an electromagnetic wave is proportional to the square of the electric-field amplitude, the transmitted intensity is reduced even more than the electric field itself. This relationship leads directly to one of the most important equations in polarization, known as Malus's law, which relates the transmitted intensity to the angle between the light's polarization direction and the filter axis.

Malus's Law

The intensity of light transmitted through a polarizing filter depends on the angle between the light's direction of polarization and the transmission axis of the filter. Suppose polarized light with intensity [latex]I_0[/latex] strikes a polarizing filter whose axis makes an angle [latex]\theta[/latex] with the electric field. Only the component of the electric field parallel to the transmission axis passes through the filter:

Because the intensity of an electromagnetic wave is proportional to the square of the electric-field amplitude, the transmitted intensity is

This relationship is known as Malus's law, named after the French physicist Étienne-Louis Malus, who discovered it in 1809.

Malus's law predicts several important limiting cases.

• [latex]\theta=0^\circ[/latex]: The polarization direction is parallel to the filter axis, so

All of the polarized light passes through.

• [latex]\theta=45^\circ[/latex]:

Exactly one-half of the incident intensity is transmitted.

• [latex]\theta=90^\circ[/latex]:

An ideal polarizing filter blocks all of the incident polarized light.

Polarization by Reflection

In the previous section, we learned that polarizing filters can produce polarized light by transmitting only one orientation of the electric field. Surprisingly, polarization can also occur naturally when light is reflected from a surface. You can observe this effect yourself by wearing polarized sunglasses while looking at sunlight reflected from water, a wet road, or a glass window. As you rotate the sunglasses, the reflected light becomes alternately brighter and dimmer. The glare never disappears completely under most viewing conditions because the reflected light is usually only partially polarized. This phenomenon explains why polarized sunglasses are so effective at reducing glare while driving, boating, skiing, or fishing.

Why Reflection Produces Polarization

When unpolarized light strikes the boundary between two transparent materials, such as air and water or air and glass, part of the light is reflected while the remainder is transmitted (refracted) into the second material. The reflected and refracted waves do not contain the same mixture of polarization directions. Light whose electric field oscillates perpendicular to the plane of incidence is reflected more efficiently than light oscillating parallel to that plane. Consequently, the reflected beam becomes partially polarized. For everyday horizontal surfaces such as lakes, roads, or windows, this means that the reflected light is predominantly horizontally polarized. Polarized sunglasses are therefore manufactured with a vertical transmission axis, allowing most vertically polarized light from the environment to pass while blocking much of the horizontally polarized glare.

Brewster's Angle

At one special angle of incidence, the reflected light becomes completely polarized. This angle is called Brewster's angle, named after the Scottish physicist Sir David Brewster. Brewster's angle depends only on the refractive indices of the two media and is given by

where

• [latex]n_1[/latex] is the refractive index of the medium in which the incident light travels,
• [latex]n_2[/latex] is the refractive index of the second medium, and
• [latex]\theta_b[/latex] is Brewster's angle.

An interesting geometric property accompanies Brewster's angle. At this angle, the reflected and refracted rays are always perpendicular to one another.

Atomic Explanation: How Polarizing Filters Work

The operation of a polarizing filter can also be understood from the behavior of electrons inside the material. A polarizing filter consists of millions of long-chain molecules that are aligned in nearly the same direction during manufacturing. These molecules contain electrons that can move relatively freely along the length of each molecule but are much less able to move perpendicular to it.

When an electromagnetic wave encounters the filter, its oscillating electric field exerts forces on these electrons. If the electric field oscillates parallel to the molecules, the electrons are driven back and forth along the molecular chains. As they oscillate, they absorb energy from the incoming light, greatly reducing that polarization component. If the electric field oscillates perpendicular to the molecules, the electrons cannot move very far. Because little energy is absorbed, this component of the wave passes through the filter with relatively little attenuation.

Why This Happens

Electrons behave somewhat like tiny masses attached to springs.

• If they are free to move in the direction of the electric field, they absorb energy efficiently.
• If their motion is restricted, very little energy is absorbed.

This microscopic behavior explains why polarizing filters transmit only one orientation of the electric field.

Polarization by Scattering

Reflection is not the only process that can polarize light. Scattering—the redirection of light after it interacts with small particles or molecules—can also produce polarized light. One of the most familiar examples is the blue sky. If you wear polarized sunglasses and slowly rotate them while looking at a clear blue sky, you will notice that the brightness of the sky changes. This occurs because sunlight scattered by molecules in Earth's atmosphere is partially polarized. This observation provides another demonstration that light is a transverse electromagnetic wave.

How Scattering Produces Polarization

When unpolarized sunlight enters Earth's atmosphere, it interacts with molecules of nitrogen, oxygen, and other gases. The oscillating electric field of the incoming light causes electrons within these molecules to oscillate as well. Oscillating electrons behave like tiny radio antennas, re-emitting electromagnetic waves in many directions. Because the electrons oscillate only in directions perpendicular to the incoming light, the scattered light acquires a preferred polarization direction. The strongest polarization is observed when looking at scattered light at an angle of approximately 90° relative to the incoming sunlight.

Why Isn't the Sky Completely Polarized?

If a single air molecule scattered sunlight only once, the scattered light viewed at the appropriate angle could be nearly completely polarized. In reality, sunlight undergoes multiple scattering events before reaching your eyes. Light arriving from different directions has different polarization states. When these scattered waves combine, the result is only partially polarized light. This explains why rotating polarized sunglasses changes the brightness of the sky but never makes it completely black.

Applications of Polarization by Scattering

The polarization produced by scattering has many scientific and technological applications.

• Photographers use polarizing filters to darken blue skies and reduce atmospheric haze.
• Astronomers measure polarized light to study dust clouds, magnetic fields, and planetary atmospheres.
• Environmental scientists analyze polarized scattered light to detect aerosols, smoke, and air pollution.
• Remote-sensing satellites use polarization measurements to estimate cloud properties and ocean surface conditions.

Modern Polarized Sunglasses

Most high-quality sunglasses combine several optical technologies to improve vision and protect the eyes.

• Polarizing filters reduce glare from reflective surfaces such as water, roads, and snow.
• Tinted lenses reduce the overall intensity of visible light.
• Anti-reflection coatings minimize reflections from the lens surfaces themselves.
• Photochromic lenses automatically darken in sunlight because molecules within the lens change structure when exposed to ultraviolet (UV) radiation and return to their transparent state indoors.

Liquid Crystals and Other Polarization Effects in Materials

Many modern technologies rely on materials that change the polarization of light. One of the most familiar examples is the liquid crystal display (LCD), which is found in smartphones, laptop computers, digital watches, medical monitors, and televisions. Although liquid crystals flow like liquids, their elongated molecules can maintain an ordered alignment similar to that of a crystal. This unique combination of fluidity and molecular order gives them unusual optical properties, including the ability to rotate the plane of polarization of light.

Liquid Crystal Displays (LCDs)

A liquid crystal can rotate the polarization of light passing through it by approximately 90°. When an electric voltage is applied, the molecules realign and no longer rotate the polarization. This simple principle allows each pixel of an LCD to switch between bright and dark states in a fraction of a second.

How an LCD Pixel Works

Each LCD pixel is sandwiched between two polarizing filters whose transmission axes are perpendicular to one another.

1. Unpolarized light first passes through the front polarizer, becoming linearly polarized.
2. If no voltage is applied, the liquid crystal rotates the polarization by about 90°, allowing the light to pass through the second polarizer.
3. When a voltage is applied, the liquid crystal molecules align with the electric field and no longer rotate the polarization.
4. Because the light now reaches the second polarizer with the wrong polarization, it is blocked and the pixel appears dark.

Modern color LCDs contain millions of pixels. Each pixel consists of three independently controlled subpixels with red, green, and blue filters. By adjusting the amount of light passing through each subpixel, the display produces millions of different colors.

Optical Activity

Some transparent materials have the remarkable ability to rotate the plane of polarization of linearly polarized light. Such substances are said to be optically active. Examples include sugar solutions, glucose, insulin, collagen, and many biological molecules whose structures are not superimposable on their mirror images. These molecules are described as chiral.

The angle through which the polarization rotates depends on several factors:

• the chemical composition of the material,
• the concentration of the substance,
• the distance the light travels through the material, and
• the wavelength of the light.

Stress-Induced Optical Activity

Normally, materials such as glass and many plastics do not significantly alter the polarization of light. However, mechanical stress changes their internal molecular arrangement, making them temporarily optically active. This phenomenon, called photoelasticity, causes different parts of a stressed object to rotate the polarization of light by different amounts. When viewed between crossed polarizers, colorful interference patterns appear because different wavelengths undergo slightly different polarization changes.

Engineers frequently use photoelasticity to identify regions of high stress in mechanical components before manufacturing full-scale parts.

Birefringence

Some crystals possess another fascinating polarization property called birefringence, or double refraction. When unpolarized light enters a birefringent crystal, it splits into two separate rays that travel at different speeds and often in different directions.

The two rays are known as:

• Ordinary ray (o-ray): follows Snell's law using the crystal's ordinary refractive index.
• Extraordinary ray (e-ray): experiences a different refractive index and therefore does not obey Snell's law in the usual way.

Each ray is polarized in a direction perpendicular to the other.

Problems

84. 84. What angle is needed between the direction of polarized light and the transmission axis of a polarizing filter to reduce the transmitted intensity by one-half?
    85. The transmission axes of two polarizing filters are separated by an angle of [latex]45.0^\circ[/latex]. By what factor does the second filter reduce the intensity of the light emerging from the first filter?
    86. A beam of completely polarized light has an intensity of [latex]150\ \text{W/m}^2[/latex]. What is its intensity after passing through a polarizing filter whose transmission axis is oriented at an angle of [latex]89.0^\circ[/latex] relative to the light's polarization direction?
    87. A beam of polarized light has an intensity of [latex]1.00\ \text{kW/m}^2[/latex]. At what angle should a polarizing filter be oriented to reduce the transmitted intensity to [latex]10.0\ \text{W/m}^2[/latex]?
    88. Example 86.1 states that a polarizing filter oriented at an angle of [latex]18.4^\circ[/latex] reduces the intensity of polarized light by only [latex]10.0\%[/latex], meaning that [latex]90.0\%[/latex] of the original intensity is transmitted. Verify this statement using Malus's law.
    89. Show that if three ideal polarizing filters are arranged so that the second is oriented at [latex]45^\circ[/latex] relative to the first and the third is oriented at [latex]90.0^\circ[/latex] relative to the first, the transmitted intensity is [latex]25.0\%[/latex] of the intensity leaving the first filter. Compare this result with the case in which only the first and third filters are used.
    90. Suppose the transmitted intensity through two polarizing filters separated by an angle [latex]\theta[/latex] is [latex]I[/latex], while the transmitted intensity for an angle [latex]90.0^\circ-\theta[/latex] is [latex]I'[/latex]. Prove that
        [latex]I+I'=I_0.[/latex]

        Use the following trigonometric identities:

        • [latex]\cos(90^\circ-\theta)=\sin\theta[/latex]
        • [latex]\cos^2\theta+\sin^2\theta=1[/latex]
    91. At what angle will light reflected from diamond be completely polarized?
    92. Determine Brewster's angle for light traveling in water that is reflected from crown glass.
    93. A scuba diver observes light reflected from the water's surface. At what angle will the reflected light be completely polarized?
    94. Light inside a sheet of crown glass is reflected from a water interface, as in an aquarium. At what angle is the reflected light completely polarized?
    95. Light reflected from a window at an angle of [latex]55.6^\circ[/latex] is completely polarized. Determine the index of refraction of the window material and identify the most likely type of glass.
        1. Light reflected from a gemstone at an angle of [latex]62.5^\circ[/latex] is completely polarized. Could the gemstone be diamond?
        2. If the gemstone were immersed in water instead of air, at what angle would complete polarization occur?
    96. If [latex]\theta_b[/latex] is Brewster's angle for light reflected from the top of an interface between two materials, and [latex]\theta_b'[/latex] is Brewster's angle for light reflected from below the interface, prove that
        [latex]\theta_b+\theta_b'=90.0^\circ.[/latex]
    97. Integrated Concept. A polarizing filter reduces the intensity of polarized light to [latex]50.0\%[/latex] of its original value. By what factor are the amplitudes of the electric and magnetic fields reduced?
    98. Integrated Concepts. Suppose you wear two pairs of polarized sunglasses whose transmission axes differ by [latex]15.0^\circ[/latex]. Compared with wearing only one pair, by what factor is the time required for the light entering your eye to deliver the same amount of energy increased? Assume the lenses are otherwise perfectly transparent.
    99. Integrated Concepts.
        1. On a sunny day, the solar intensity is [latex]1.00\ \text{kW/m}^2[/latex]. A circular lens of diameter [latex]0.200\ \text{m}[/latex] focuses sunlight onto water contained in a black aluminum beaker. Two ideal polarizing sheets are placed in front of the lens with their transmission axes separated by [latex]20.0^\circ[/latex]. Assuming the sunlight is unpolarized and that 80.0% of the transmitted energy is absorbed, determine the initial rate of temperature increase (in [latex]^\circ\text{C/s}[/latex]) of the water. The aluminum beaker has a mass of [latex]30.0\ \text{g}[/latex] and contains [latex]250\ \text{g}[/latex] of water.
        2. Do the polarizing sheets become warm during this process? Explain your reasoning.
    100. Critical Thinking.A semiconductor laser emits light at a wavelength of [latex]400.0\ \text{nm}[/latex]. The light is directed normally (perpendicular) onto a diffraction grating that has 1,000 lines etched over a distance of [latex]10.00\ \text{mm}[/latex].
         1. What is the diffraction angle for the first-order maximum?
         2. What is the diffraction angle for the second-order maximum?
         3. The laser is replaced with another laser. The first-order diffraction angle is now [latex]23.9^\circ[/latex]. What wavelength of light does the new laser produce?
         4. If the diffraction grating were replaced with another grating having twice the spacing between adjacent lines, what would the first-order diffraction angle be?
         5. Would the magnitude of the diffraction angle be different for light diffracted to the left compared with light diffracted to the right? Explain.