Physical Optics

Chapter 21

Image Credit: NASA

The mystery of light has been a great curiosity for scientists, and has led to some of the greatest discoveries. Understanding light has led to practical applications like lasers and fiber optics for communication as shown in the image above. Understanding light has led to answers about why is the sky blue and sunset red. The image on the right shows both of these aspects at the same time with the photograph of sunrise from the International Space Station by astronaut Ron Garan. The lessons of this chapter will explore our understanding of the mysteries of light from the point of view of it being a wave in what is called Physical optics. These lessons set up even more applications for the next chapter using lenses and mirrors to recreate reality through images from the point of view of light as a particle.

Light as a Wave – Lesson 1

Lesson Objectives

• Be able to state the historical perspective of light

• Know the evidence that supports light as a wave

• Describe the law of reflection

• Relate to the magnitude of the speed of light

• Understand Raleigh’s scattering

Vocabulary

Huygen’s principle

Photon

Diffraction

Ray

Check Your Understanding

1. James Clerk Maxwell made a great discovery when he divided the constants from the electric field and magnetic field equations. What did Maxwell discover?

2. What do X-rays and radio waves have in common?

Introduction

In the quest to understand nature, light has been a favorite theme throughout history. Isaac Newton was made famous by his investigations of light and the presentation of the reflecting telescope which is still called a Newtonian reflector today. The phenomenon of light has been investigated and explained with theories that describe light as a particle and also as a wave. This lesson will focus on the wave theory of light popularized by Isaac Newton.

Lesson Content

Historical Perspective on Light as a Wave

Early models of light proposed by Christian Huygens described light as a wave. Huygens lived in the middle of the Renaissance period a generation before Isaac Newton. It was Huygens

who developed the pendulum approximation equation, 2 LTg

π= , which he used to design the

first pendulum clock that could be used on ships to aid in navigation. Huygens also used a telescope to discover the largest moon of Saturn, Titan, and also rightly claimed that the “ears” of Saturn were flat thin rings. To explain light, Huygens imagined that light traveled as waves where the fronts of the plane waves were really combinations of many wave crests. Figure 1 shows Huygen’s principle with the many wave crests making up the wave fronts where the slowing of wavelets leading to a change in direction of the light wave and its apparent bending. Notice that the crests of waves are shifted but still maintain their wave front.

Figure 1. Huygen’s view of light as wave

Isaac Newton observed that white light would pass through a glass prism and separate into a spectrum of colors. He would conclude that light was corpuscular, or like a particle. The corpuscular theory of light held that model for a century strongly based on Newton’s reputation. However, experiments performed by Young and Fresnel would show that light indeed acted like a wave. The lessons of this chapter will explore some wave phenomenon that finally overturned Newton’s particle premise about light.

As science is an interpretation of men about how nature works, the rules change with the evidence. A young scientist named Albert Einstein would make both sides of history happy by helping justify that light is both a particle and a wave. In fact, the only Nobel Prize awarded to Einstein was for his work with the Photoelectric effect which supported the idea that light is made of wave-particles called photons. Figure 2 includes a link to a TED talk that shows a short burst of photons acting like a particle but also bouncing around and bending like waves.

Evidence for Light as a Wave

Work done before and after Newton helped justify the theory that light was a wave. The specific phenomena that support the wave theory of light include reflection, refraction, diffraction, and the Doppler effect. The Doppler effect was described in lesson 4 of chapter 11, with sound as the primary example. However, it has been shown that moving light sources will show a shift in their wavelengths depending on their speed and instead of hearing a change in pitch as with sound, there is a change in color. Object moving away from an observer to have longer wavelengths and thus red shifted. Object approaching an observer will have compressed wavelengths that shift toward blue, or blue shifted. Doppler radar is used to determine the dynamics of an approaching storm, speed of aircraft, and even police radar guns.

https://www.youtube.com/watch?v=SoHeWgLvlXI

Figure 2. Multimedia Resource: Watching Light Waves, TED Talks

Evidence for the wave theory of light was strong before the time of Newton as is evident with Francesco Grimaldi’s diffraction experiments. However, Grimaldi did not offer a compelling argument for the cause and Newton’s reputation was too great so wave theory won the day. It is interesting to note that Newton repeated Grimaldi’s experiments and did others, but Newton’s eyesight probably kept him from seeing the dark absorption lines as further evidence of light as a wave in his spectra projections.

Grimaldi set up a dark room and let light pass through a circular opening. A rod was held in the column of light in front of a screen as shown in figure 3. Instead of casting a circular shadow that was the same size of the rod, Grimaldi noticed that the shadow was larger than the actual rod. Even more astounding was the appearance of light rings and a central light inside of the shadow. If light is a wave, it will bend around edges so that the edges become new sources of light waves. Those new bent waves of light will interfere with each other such that some of them may add up to make light appear inside of the shadow and to also make the edges fuzzy. These shadow phenomena were called diffraction by Grimaldi.

Figure 3. Grimaldi’s Diffraction Experiment

Refraction is the bending of light as it passes from one medium to another where the optical density is different. Optical density is a fuzzy conceptual description of the ability of a transparent material to interact with light. Figure 1 shows a wave front passing into a optically denser medium and the edge of the waves that enter first slow down so by the time the rest of wave enters the medium, the overall direction has changed.

Law of Reflection

Reflection is also a wave behavior and is shown in figure 4. A line drawn perpendicular to a set of parallel wave fronts is called a ray, so that figure 4 shows an incoming ray of light and a reflected ray of light after hitting a mirror. The incoming ray is called the incident ray and the outgoing ray is called the reflected ray. The law of reflection states that the angle of the incident ray with will be equal to the angle of the reflected ray with all angles measured with respect to the perpendicular normal direction: i rθ θ= .

Speed of a Light Wave

James Clerk Maxwell found that the speed of a disturbance in an electric and magnetic field was 83.0 10 m

s× . This speed was immediately recognized as significant to Maxwell. Clever timing

experiments with the moons of Jupiter had been completed in the late 1600’s to find the speed of light, and better values were available and matched Maxwell’s electromagnetic wave speed. Maxwell proved that light is an electromagnetic wave. This is yet another piece of evidence that light is a wave. This speed is difficult to imagine. If mirrors were set up around the circumference of the Earth, light would travel 7 ½ times around the Earth in one second. The

speed of light is written as 83.0 10 msc = × , where the c stands for celeritas, Latin for swiftness.

Figure 4. The law of reflection

Rayleigh Scattering

Why is the sky blue? This classic question has an answer that provides further evidence of the wave theory of light. The atmosphere of the Earth is mostly diatomic nitrogen, N2. The size of an N2 molecule is smaller than the wavelengths of visible light. Blue light has a wavelength close to the violet end of the spectrum around 400nm, and red light is around 700nm. The diameter of a N2 molecule is 0.11nm which is much smaller. When the waves of light travel through the Nitrogen atmosphere, the smaller waves will scatter in random directions compared to the longer waves. The end product is that the sky appears blue in all directions due to this scattered light due to the wave properties of light. As the sun sets, the light must pass through a thicker cross section of atmosphere so that a high percentage of the blue light has been removed leaving a reddish orange sunset. A final implication of Rayleigh scattering can be seen with a laser pointer at night. The beam is visible because the light scatters around the beam making it visible and very useful to point out constellations at night to your friends. Figure 5 shows the blue sky and a red sunset together, and scattered light from a green laser pointer.

Image credit: NASA

Figure 5. Examples of Rayleigh Scattering

Lesson Summary

• Light was considered a particle for a century based almost entirely on the reputation of Isaac Newton.

• Experiments in reflection, refraction, diffraction, and the Doppler effect all show that light has wave properties.

• The law of reflection applies to rays of light that reflect off of a surface with the same angle on the other side of the perpendicular normal direction.

• Further proof that light is a wave comes from the explanation of why the sky is blue through the phenomenon of Rayleigh scattering.

Review Questions

Review Problems

Further Reading / Supplemental Links

•

Points to Consider

1. How does the incident angle compare to a refracted angle? Are they equal or at a constant proportion to each other?

2. If a light wave goes into an optically denser medium, air to water for instance, does its wavelength change or does its frequency change? What about speed?

Refraction – Lesson 2

Lesson Objectives

• Describe optical density with the index of refraction

• Understand how Fermat’s principle explains refraction

• Know how to apply Snell’s law

• Explain rainbows with dispersion

Vocabulary

Index of refraction

Critical angle

Dispersion

Check Your Understanding

1. How does the speed of light provide evidence that light is a wave?

2. Movies often show a person being shot by a bullet fly through the air in response to being hit. Why is this unrealistic?

Introduction

Light as a wave will interact with matter. These interactions are the source for the phenomena investigated in this lesson. Light as a wave has a wave speed related to a frequency and wavelength. Waves can bend around corners and combine with other waves to make smaller or larger waves. These happen with light waves as well. As much evidence as there was to support the wave theory of light, it would still take the brilliant work of many minds to interpret something like the fuzziness of a shadow to be proof that light is a wave. The lessons presented here are the end result of many great long investigations.

Lesson Content

Effect of Transparent Material on Light Waves

Light that travels through water will move slower than light traveling through air. For transparent materials, the atoms interact with the light waves by absorbing and emitting them in a cycle that slows down the progress of the light. The actual interactions are described by the seemingly strange rules for the physics of atoms called quantum mechanics to be covered

in another chapter’s lessons. Some materials will have more interactions with light than other materials. The measure of the material interaction with light is called the index of refraction. This index, written as the letter n, is expressed as the ratio of the speed of light in a vacuum to the speed of light in the material as shown in equation 1.

(1) cnv

=

Table 1 shows a list of indices of refraction for various substances.

Substance Index of Refraction

Air 1.00029

Ice 1.31

Water 1.333

Crown glass 1.52

Zircon 1.923

Diamond 2.417

Gallium phosphide 3.50

Table 1. Index of Refraction for Various Substances

The speed of the light wave in the medium can be written as c f λ= in a vacuum. The speed

of light in a medium is v f λ= . It is important to note that the number of waves entering a

medium must be conserved due to conservation of energy, so that the frequency is the same.

Using equation 1 with these two wave velocities yields: 0

medium

fnf

λλ

= , which can be expressed

as the wavelength of the medium in terms of the vacuum wavelength in equation 2.

(2) 0medium n

λλ =

Refraction

Refraction is the bending of the light path as it goes from one medium to another. There are several perspectives to take to explain this bending phenomenon, but the most convincing may be that first performed by 17th century French lawyer turned mathematician Pierre de Fermat. Fermat said that light would take the path of least resistance to get from point P to point Q shown in figure 6. Light will not follow the straight line between P and Q since light slows down in the second medium. The red line is the best path that minimizes the time traveled. The

practical analogy is a person trying to cross a long swimming pool to get to some diagonal point. The quickest way to cross would be the red line not the black line since you run faster than you swim.

Figure 6. The bent path of light passing between two mediums

This is a max/min problem in calculus where the function minimized is time with respect to the distance. The total time it takes light to travel from point P and Q can be written as:

1 2

1 2TOTAL

r rtv v

= + , where the r’s represent the red line distance from the points to the bending

point at the yellow-blue interface. The velocity of light in each medium can be expressed in

terms of the index of refraction. 11

cvn

= and 22

cvn

= . The Pythagorean theorem can be used

to express the distances in terms of x’s and the y value. 2 21 1r x y= + and ( )22

2 2r x L y= + −

. Combining all of these creates a new equation for total time.

( )222 221

1 2

TOTAL

x L yx yt c c

n n

+ −+= + . To minimize the time, the derivative of the function for

time in terms of y, which is the variable distance to the interface point, can be set equal to zero.

( )( )2 2 221 21 2

1 1 0TOTAL L ydt ydy v vx y x L y

− −= + =

+ + −

Look carefully at the fractions next to the velocity terms. They are the sines of the θ1 and θ2

angles. This max/min expression will become: 1 21 2

1 1sin sin 0v v

θ θ− = . Using the definition of

the index of refraction, the velocities can be substituted: 1 21 2sin sin 0n n

c cθ θ− = . The final

expression has the speed of light cancelled out and is called Snell’s law, as shown in equation 3.

(3) 1 1 2 2sin sinn nθ θ=

Using Snell’s Law

As light waves enter a thicker medium, they will shift their direction as the wave reorients its direction according to Snell’s law. Figure 7 shows the apparent bending of a pole as it is goes into water, since the light from each part of the pole bends to make it look closer to the surface.

Figure 7. Appearance of bending due to refraction

Example

Consider the image from figure 7. If the angle of incidence is 30 degrees, predict the angle of refraction.

Solution:

Use Snell’s Law. 1 1 2 2sin sinn nθ θ=

2 1sin sin 0.38air

water

nn

θ θ

= =

22O

Something curious happens when the angle of incidence continues to increase. It eventually reaches the point where the sine is equal to 1 which corresponds to a refracted angle of 90 degrees. The light wave would bend so much that it never actually penetrates the medium. At every angle greater than this critical angle, light will reflect instead of refract. Snell’s law can be used to express the critical angle by setting one of the angles equal to 90 degrees as shown in equation 4. Light can travel through bending optical fiber if the angle is not too much, and is used to send information (fiber optics).

(4) 1

2

sin Cnn

θ =

Rainbows

Isaac Newton observed that white light would separate into a spectra of colors when passing through a prism. This observation was Snell’s Law in action with the curious fact that blue light bends more than red light. This separation effect of different wavelengths of light as it refracts is called dispersion and it causes rainbows. The mostly spherical rain drops provide a kind of round prism which creates the arc shaped spectra suspended against the backdrop of the sky.

Lesson Summary

• The index of refraction describes the extent of interaction of a medium with light and

can be expressed as cnv

= .

• Light waves will take the path of least resistance through multiple mediums as described with Snell’s law: 1 1 2 2sin sinn nθ θ=

• Light will reflect instead of refract beyond a critical angle.

• Rainbows are caused by blue light bending more than red light in droplets of rain due to an effect called dispersion.

Review Questions

Review Problems

Further Reading / Supplemental Links

Points to Consider

1. What will happen if the light wave from a reflection and the original source coincide?

2. Light entering a dark room from a slightly open door will mostly shine on a sliver of light on a wall in the room, but the rest of the room is partially illuminated. How does this happen?

Young’s Double Slit Experiment & Interference – Lesson 3

Lesson Objectives

• Explain how Young’s experiment describes the wave nature of light

• Make predictions about the patterns produced by double slit diffraction

• Analyze the diffraction patterns caused by a single slit

• Describe the effect of multiple slits on diffraction patterns

• Understand thin film interference

Vocabulary

Interference

diffraction grating

Check Your Understanding

1. What is Snell’s Law?

2. How does the wave behavior of light cause rainbows?

Introduction

Probably the most powerful proof that light has wave behavior is interference effects. However, these effects take careful observation and even more careful explanation. Lesson 1 of this chapter introduced the first subtle evidence from interference but called it diffraction. The great CalTech physics professor Richard Feynman said in his Lecture in Physics that “no-one has ever been able to define the difference between interference and diffraction satisfactorily.” He goes on to say “it is just a question of usage, and there is no specific, important physical difference between them.” The focus of this lesson is Young’s experiment with light waves passing through narrow slits and creating very distinct interference patterns.

Lesson Content

Young’s Experiment

If plane waves run into a barrier with a single opening, a series of circular point source waves are created. The animated image in Figure 1 shows this scenario. The scenario becomes more interesting if there are two such slits in the wall where each slit becomes a point source wave. The two new waves will alternately constructively and destructively interfere and cause patterns

of bright and dark spots on a screen for light waves in a phenomenon called interference or sometimes diffraction. Thomas Young discovered and explained these patterns in 1803. Figure 2 shows Young’s original sketch of the combination of the two waves to create the pattern also shown in figure 2.

Figure 1. Animated image of a plane wave encountering a narrow slit

Figure 2. Young’s sketch of the double slit experiment and a diffraction pattern

The alternating bright and dark spots are due to constructive and destructive interference, and the locations of these spots is a geometry problem addressed in chapter 11, lesson 3 with sound waves. When a single source of light passes through a double slit as shown above, the light “waves” create a regular pattern of bright lines (or spots for lasers). This is due to the light waves constructively and then destructively interfering with each other where each slit opening is like a new source of light waves. To predict where the lines or spots will appear, some simple geometry can be used where the path length of the bottom slit is one wavelength longer than the path length of the top slit for the distance above the central maximum. The 2nd maximum will be 2 wavelengths different for the bottom path versus the top. The exact same

lines or spots will appear below the maximum as the situation is symmetric and the same rules apply by reversing the rule for top and bottom slits.

Analyzing Double-Slit interference patterns

Figure 4. Path length difference in double-slit diffraction

An approximation is made to make the math even simpler. Since L is usually much greater than d, the three lines from the spot to the centers of the two slits plus the midpoint are all assumed to be parallel. This approximation works well in experiments to several decimal places when dealing with visible light in lab situations.

Notice that the bottom diagonal line in figure 4 has a bold portion. This bold portion must be some whole number of waves so that the bottom diagonal arrives at the screen at the maximum so that constructive interference occurs and a bright line/spot is seen. A triangle is formed using the width d as the hypotenuse and the relationship in equation 5 is true.

(5) sind mθ λ= where the integer m = 1 is for the 1st maximum, and so on.

The approximation comes from the fact that sin tanθ θ θ≈ ≈ for small angles. The definition of tangent turns the above definition into equation 6.

(6) maxyd mL

λ =

Solving for ymax gives us the equation seen in many physics books shown in equation 7.

(7) maxm Ly

dλ

=

Equation 8 is a more precise equation does not assume the rays are parallel.

(8)

( )

( )

22 2

max 22

2mm L

yd m

λλ

λ

− =−

When the path length is different by a half wavelength for the double-slit, the light will destructively interfere so that equations 5 & 7 can be rewritten for minima in equation 9 & 10.

(9) (10)

Single-slit interference

A single slit can also show an interference pattern with the entire slit acting as a source of Huygen wavelets. The geometric relationships look similar to the double slit results, except they are 180 degrees out of phase. Imagine two light rays where one starts at the edge of the slit (point A on figure 5) and the other starts in the middle of the slit (point B on figure 5). If the path length is off by a half wavelength, the two rays of light will destructively interfere and create a minima on the screen (dark spot). The relationship between the two rays is expressed in equation 11. To better imagine the single slit pattern, imagine the spherical wave created just below the upper edge. You can see that there will be another spherical wave just below the midpoint that is exactly ½ wavelength off in path length so that all of the waves described by Huygens will cancel out to make a minima. The small angle approximation leads to the expression for the position of the minima, and the width of the central maximum in equation 12.

(11) sin2 2w m λθ =

sinw mθ λ=

*for single-slit minima*

(12) minm Ly

aλ

=

(13) 2 Lw

aλ

=

The width of the slit is defined as the distance between the 1st two minima, thus the value of 2 in equation 13. Figure 5 shows a plot of intensity and also an actual single slit pattern. The maxima values will be found where the path length is shifted by a half wavelength so the integer m is replaced with m+½.

( )12sind mθ λ= + ( )1

2min

m Ly

dλ+

=

Figure 5. Single slit interference

Multiple slit interference

A double slit interference pattern is the result of two single slits. The distance between the two slits is typically much larger than the single slit width so the equations 5 and 7 apply. Adding more slits will extend the pattern with the central maxima of each slit contributing to a brighter pattern. A transparent glass or plastic can be etched with an incredible number of grooves or lines that will act as slits for light to create a device called a diffraction grating. A laser light shown through a diffraction grating will produce a series of bright spots whose separation can be described with equation 7. If light made of many colors passes through a diffraction grating, the red light will bend less than blue light through each line so that a spectrum of light will be built by the total grating. These gratings are very useful to look at sources of light to find the dark and bright spectral lines to determine chemical composition of samples here on Earth and samples throughout the universe. A CD and DVD are basically diffraction gratings where the reflected laser patterns are read as music, movies, or data.

Thin-film interference

A layer of oil on water is very thin and can be a single layer of molecules thick in some cases. Figure 6 shows a ray of light reflecting off of a thin layer combining with a ray of light that penetrate the layer and reflected. Constructive interference can occur at thicknesses of ½ or ¼ wavelength depending on whether the layer behind the film has a higher index of refraction. Figure 6 shows the case when n3 is larger than n2 so that a 180 degree phase change occurs at the bottom reflection. This applies to the case of an oil layer on top of water. Equation 14 shows the relationship for the minimum thickness of the film that will result in constructive interference in terms of the wavelength and index of refraction.

(14) 4 4

mediumtn

λ λ= = due to equation 2 when there is a medium with a higher n behind

In the case of a soap bubble with air on the other side, the film will cause a reflection on the back side with no phase change making the path length difference equal to a ½ wavelength.

(15) 2 2

mediumtn

λ λ= = when there is a medium with a lower n behind

Figure 6. Thin-film interference for n3>n2>n1

Lesson Summary

• Young’s experiment with a single wavelength of plane light waves passing through two narrow slits showed the effects of constructive and destructive interference with an alternating light and dark patter.

• The geometry of double slit interference predicts the positions of the maxima with

maxm Ly

dλ

= and the minima with ( )1

2min

m Ly

dλ+

= .

• Single slit interference displays Huygen’s principle with the entire slit acting as a source

of spherical waves the positions of the maxima with ( )1

2max

m Ly

dλ+

= and the minima

with minm Ly

dλ

= .

• A diffraction grating is a transparent device with multiple slits etched into it that can show a wide equally bright diffraction pattern, and can also display a spectrum of light due to the different indices of refraction for different wavelengths of light.

• A thin film like oil on water or a soap bubble will show very colorful patterns which are due to constructive and destructive interference, and the thickness of the layers can be predicted.

Review Questions

Review Problems

Further Reading / Supplemental Links

Waves & Interference Patterns visualization: https://www.youtube.com/watch?v=dNx70orCPnA

Points to Consider

1. If light is an electric and magnetic wave, is the orientation of those waves important?

2. How does a light filter work for both colors and intensity?

Polarization – Lesson 4

Lesson Objectives

• Understand the concept of polarization by filter

• Predict the angle of incidence for maximum polarization by reflection

• Describe phenomenon of light polarized by scattering

• Know the types of polarization

Vocabulary

Polarizer

Brewster’s angle

Linear polarization

Circular polarization

Check Your Understanding

1. What is the difference between interference and diffraction?

2. In terms of Huygen’s principle, what is the difference between single and double slit interference?

Introduction

Polarization is a word used to describe physical phenomenon and in societal terms. A polarized society may only have one of a couple very different points of view. In circuits, polarity may refer to state of being positive or negatively charged. The physical optics definition of polarization refers to the orientation of the changing electric and magnetic fields that make up light waves. This abstract definition will lead to some very real applications and explanations of nature all around us.

Lesson Content

Polarization by Filtering

The presence of light is due to a stream of electromagnetic waves. Direction of the propagation of the transverse electromagnetic wave is always perpendicular to both the electric and magnetic fields, which are also perpendicular to each other. For a particular unpolarized wave particle of light, the orientation of the electric and magnetic field will be a random direction. It

is possible to filter unpolarized light with a polarizer so that it only contains waves of light that have the same orientation, thus making it polarized light. This filtering can be done by reflection of some dielectric surfaces, by scattering, and by passing the light through a transparent filter that has its molecules aligned to absorb all light that is not aligned in a specific orientation. The polarized filter will seem to filter the light to reduced its energy so that the angle between the preferred orientation and the actual orientation results in a change in energy of the light: 0 cospolarizedE E θ= . The actual energy of the individual photons of light does not

change, but the total energy of the light source drops since some light is absorbed. The energy intensity of the light can be expressed by the Poynting vector as described in chapter 19:

20

02ES

cµ= . Since unpolarized light is randomly oriented, the average electric field will be the

average value of the square of cosine which is 2102 E . If a second filter is placed in series with

the first, light will be reduced by a further cosine term squared using the angle between the preferred orientations of the two filters. The resulting power density, I, is derived with the Poynting vector is expressed with Malus’ Law (equation 16).

(16) 2

0 cos2

II θ= for light passing through two polarized filters

Polarized filters are used in sunglasses and 3D movie theatres.

UCLA Physics Demo of Polarization

Paul Hewitt Explains Polarization

Figure 8. Video demonstration and explanations of polarization

Polarized Light Through Reflection

Figure 7 has a ray of unpolarized light reflecting and refracting off of a surface interface between two indices of refraction. An example could be air and water. If an unpolarized ray of light reflects off of a surface interface, some of the light will reflect. For a given intensity of

electric field parallel to the direction of propagation, there is a relationship to the intensity of the reflected electric field not derived here but shown in equation 17.

(17) ( )( )

1 21 2 2 1|| || ||

1 2 2 1 1 2

tancos coscos cos tanreflected incident incident

n nE E En n

θ θθ θθ θ θ θ

−−= =

+ +

Figure 7. Polarization by Reflection

The implication becomes more obvious when the incident angle is perpendicular to the

refracted angle, or 01 2 90θ θ+ = . Equation 17 shows that the tangent term on the bottom

becomes infinite and the parallel component of the reflected electric field goes to zero. This means that all of the reflected electric field is perpendicular to the direction of the reflected light ray which is by definition 100% polarized. The angle at which this occurs is called the Polarization angle or Brewster’s angle. Snell’s law can be combined with the trig identity,

2 1sin cosθ θ= , in the case that 01 2 90θ θ+ = to produce Brewster’s law (equation 18)

(18) 21

1

tan nn

θ =

Example

Sunlight reflects off a flat water surface and glass surface. What is the difference in the angle where maximum polarization occurs for the two surfaces?

Solution:

Using 1.00airn = , the incident angle for maximum polarization for water and glass can be found

using (18): 11.33tan1.00

θ = 01 53.1Bθ θ= = & 1

1.50tan1.00

θ = 01 56.3Bθ θ= =

03.3θ∆ =

This polarization by reflection phenomenon does not work for all reflective surfaces, in particular metals.

Polarization by Scattering

Scattering is a process of absorption and re-emission of light by small particles. Scattering seems to imply that the light is merely redirected, but the process involves the quantum mechanics of the small particles in how they interact with the light. If unpolarized light from a source runs into a collection of small particles, light will be absorbed and re-emitted in random directions. The light that is re-emitted in a direction perpendicular to the original light source will be aligned the same way. Since the emitted frequency will equal the original frequency, the perpendicular scattered light will have the same component of electric field. As a rule, all light scattered 90 degrees to the original source will be completely polarized. Mix this with the fact that blue light is 10 times more likely to scatter than red light yields a further explanation for the blue sky and red sunsets. It also explains that the part of the sky 90 degrees from the Sun will be more polarized than the rest of the blue sky. Since rainbows are the result of reflected sun light through water droplets, rainbows are strongly polarized.

Types of Polarization

Through the discussions thus far, the orientation of the light waves considered have been linear. Linear polarization has constant orientations for the electric and magnetic field so that a set angle could be used to describe the light waves orientation. On the left in figure 8 is shown a constant orientation for the red electric field and blue magnetic field resulting in the black linearly polarized light wave. Figure 8 on the right shows a circularly polarized light wave where the orientation of both the electric and magnetic field change direction constantly making a regular spiral orientation that changes at a set frequency. Any electromagnetic wave can be polarized, so the types mentioned have applications in other realms of the spectrum. A combination of vertical and horizontally polarized radio waves can better penetrate buildings, which makes them ideal for your local radio stations to use. A rotating receiver would receive circularly polarized radio waves better than horizontally polarized radio waves from a stationary source.

Linear polarized electromagnetic wave

Circular polarized electromagnetic wave

Figure 8. Types of Polarized waves

Lesson Summary

• Polarization is the extent to which the orientation of light waves are the same.

• Special transparent filters called polarizers can only let light waves pass that have a particular orientation.

• There is an incident angle for non-metallic reflective surfaces that will have a maximized polarization effect, where the angle is called Brewster’s angle and can be found using

21

1

tan nn

θ = .

• Light that is absorbed and re-emitted by small particles is called scattered light, and scattered light that is emitted 90 degrees away from the source will be completely polarized.

Review Questions

Review Problems

Further Reading / Supplemental Links

Points to Consider

1. What are some applications of bending light?

2. What are some simple experiments that could be done with a polarizer?