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CHAPTER 24 The Wave Nature of Light

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Units

• Waves Versus Particles; Huygens’ Principle and Diffraction • Huygens’ Principle and the Law of Refraction • Interference – Young’s Double Slit Experiment • The Visible Spectrum and Dispersion • Diffraction by a Single Slit or Disk • Diffraction Grating • The Spectrometer and Spectroscopy • Interference by Thin Films • Michelson Interferometer • Polarization • Liquid Crystal Displays • Scattering of Light by the Atmosphere

What can we say about the nature of light? Does it travel as a stream of particles away from its source, or does light travel in the form of waves that spread outward from the source? In this section we will investigate the wave nature of light.

Huygens, Christiaan (1629-1695)

Dutch physicist who was the leading proponent of the wave theory of light. He developed the concept of the wavefront, but could not explain color. In 1656, Christiaan Huygens built the world's first pendulum clock.

Waves Versus Particles; Huygens’ Principle and Diffraction

Huygens’ Principle: Every point on a wave front acts as a point source; the wavefront as it develops is tangent to their envelope

Huygens principle is useful for analyzing what happens when waves impinge on an obstacle and the wave fronts are partially interrupted.

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The bending of waves behind obstacles into the “shadow region” is known as diffraction. Since diffraction occurs for waves, but not for particles, it is one way for distinguishing the nature of light.

Waves Versus Particles; Huygens’ Principle and Diffraction

Huygens’ Principle is consistent with diffraction:

The laws of reflection and refraction were well known in Newton’s time. The law of reflection could not distinguish between the two theories. The law of refraction is another matter. Consider a ray of light entering a medium where it is bent toward the normal, as when traveling from air into water.

Huygens’ Principle and the Law of Refraction

This bending can be constructed using Huygens principle if we assume the speed of light is less in the second medium. Newton favored a particle theory of light which predicted the opposite result, that the speed of light would be greater in the second medium. In 1859 the French physicist Jean Foucault confirmed the wave-theory.

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Jean Bernard Léon Foucault 1819 -1868) was a French physicist best known for the invention of the Foucault pendulum, a device demonstrating the effect of the Earth's rotation. He also made an early measurement of the speed of light, invented the gyroscope, and discovered eddy currents.

Huygens’ Principle and the Law of Refraction

Huygens’ Principle can also explain the law of refraction. As the wavelets propagate from each point, they propagate more slowly in the medium of higher index of refraction. This leads to a bend in the wavefront and therefore in the ray. The frequency of the light does not change, but the wavelength does as it travels into a new medium.

Highway mirages are due to a gradually changing index of refraction in heated air.

Wave Optics

• The wave nature of light is needed to explain various phenomena – Interference – Diffraction – Polarization

• The particle nature of light was the basis for ray (geometric) optics

Interference

• Light waves interfere with each other much like mechanical waves do • All interference associated with light waves arises when the electromagnetic fields that

constitute the individual waves combine

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Conditions for Interference

• For sustained interference between two sources of light to be observed, there are two conditions which must be met

– The sources must be coherent • They must maintain a constant phase with respect to each other

– The waves must have identical wavelengths

Producing Coherent Sources

• Light from a monochromatic source is allowed to pass through a narrow slit • The light from the single slit is allowed to fall on a screen containing two narrow slits • The first slit is needed to insure the light comes from a tiny region of the source which is

coherent • Old method • Currently, it is much more common to use a laser as a coherent source • The laser produces an intense, coherent, monochromatic beam over a width of several

millimeters • The laser light can be used to illuminate multiple slits directly

Thomas Young (1773-1829) Young is perhaps best known for his work in physical optics, as the author of series of research which did much to establish the wave theory of light, and as the discoverer of the interference of light. In Young's double-slit experiment, c. 1801, he passed a beam of light through two parallel slits in an opaque screen, forming a pattern of alternating light and dark bands on a white surface beyond. This led Young to reason that light was composed of waves.

Young’s Double Slit Experiment

• Thomas Young first demonstrated interference in light

waves from two sources in 1801 • Light is incident on a screen with a narrow slit, So

• The light waves emerging from this slit arrive at a second screen that contains two narrow, parallel slits, S1 and S2

• The narrow slits, S1 and S2 act as sources of waves

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• The waves emerging from the slits originate from the same wave front and therefore are always in phase

Young’s Two-Slit Experiment

In this experiment, the original light source need not be coherent; it becomes so after passing through the very narrow slits. If light consists of particles, the final screen should show two thin stripes, one corresponding to each slit. However, if light is a wave, each slit serves as a new source of “wavelets,” as shown, and the final screen will show the effects of interference. This is called Huygens’s principle.

Resulting Interference Pattern

• The light from the two slits form a visible pattern on a screen • The pattern consists of a series of bright and dark parallel bands called fringes • Constructive interference occurs where a bright fringe appears • Destructive interference results in a dark fringe

Fringe Pattern

• The fringe pattern formed from a Young’s Double Slit Experiment would look like this

• The bright areas represent constructive interference • The dark areas represent destructive interference

Interference Patterns

• Constructive interference occurs at the center

point • The two waves travel the same distance

– Therefore, they arrive in phase

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• The upper wave has to travel farther than the lower

wave • The upper wave travels one wavelength farther

– Therefore, the waves arrive in phase • A bright fringe occurs

• The upper wave travels one-half of a wavelength farther than the lower wave

• The trough of the bottom wave overlaps the crest of the upper wave

• This is destructive interference – A dark fringe occurs

Young’s Two-Slit Experiment This diagram illustrates the numbering of the fringes.

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Between the maxima and the minima, the interference varies smoothly.

Example 1: Line spacing for double-slit interference. A screen containing two slits 0.100 mm apart is 1.20 m from the viewing screen. Light of

wavelength 500nm falls on the slits from a distant source. Approximately how far will adjacent bright interference fringes be on the screen?

Given 4 90.100 1 10 , 500 10d mm x m x m and L = 1.20m, the first order fringe(m = 1)

occurs at an angle given by This is a very small angle, so we can take sin ,

with in radians. The first order fringe will occur a

distance 1x above the center of the screen given by

So The second order fringe (m = 2) will occur at Above the center, and so on. Thus the lower order fringes are 6.00 mm apart. Example 2: Wavelengths from double slit interference. White light passes through two slits 0.50mm apart, and an interference pattern is observed on a screen 2.5m away. The first-order fringe resembles a rainbow with violet and red light at opposite ends. The violet light falls about 2.0mm and the red 3.5mm from the center of the central white fringe. Estimate the wavelengths for the violet and red light. For the violet light: or 400nm. For red light: or 700nm

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4

(1)(500 10 )sin 5.00 10

1.00 10

m x mx

d x m

1 1 1/ tanx L

3

1 1 (1.20 )(5.00 10 ) 6.00x L m x mm

2 2

212.0x L L mm

d

4 37sin 5.0 10 2.0 10

4.0 101 2.5

d d d x x m x mx m

m m m L m

4 37sin 5.0 10 3.5 10

7.0 101 2.5

d d d x x m x mx m

m m m L m

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Interference – Young’s Double-Slit Experiment Since the position of the maxima (except the central one) depends on wavelength, the first- and higher-order fringes contain a spectrum of colors.

The Visible Spectrum and Dispersion Wavelengths of visible light: 400 nm to 750 nm Shorter wavelengths are ultraviolet (UV); longer are infrared (IR).

The index of refraction of a material varies somewhat with the wavelength of the light.

This variation in refractive index is why a prism will split visible light into a rainbow of colors.

Actual rainbows are created by dispersion in tiny drops of water.

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Diffraction by a Single Slit or Disk

Light will also diffract around a single slit or obstacle.

The resulting pattern of light and dark stripes is called a diffraction pattern. This pattern arises because different points along a slit create wavelets that interfere with each other just as a double slit would.

Example 3: Single-slit diffraction maximum.

Light of wavelength 750nm passes through a slit 31.0 10x mm wide. How wide is the central maximum (a) in degrees, and (b) in centimeters, on a screen 20 cm away? (a) The first minimum occurs at

So 49o is the angle between the center and the first minimum. The angle subtended by the whole central maximum, between the minima above and below the

center, is twice this or 98o (b) The width of the central maximum is 2x, where tan / 20x cm . So

Diffraction Grating

• The diffracting grating consists of many equally spaced parallel slits

– A typical grating contains several thousand lines per centimeter • The intensity of the pattern on the screen is the result of the combined effects of

interference and diffraction

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7.5 10sin 0.75

1 10

x m

D x m

2 2(20 )(tan 49 ) 46ox cm cm

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• The condition for maxima is

– d sin θbright = m λ

• m = 0, 1, 2, … • The integer m is the order number of the

diffraction pattern • If the incident radiation contains several

wavelengths, each wavelength deviates through a specific angle

• • • All the wavelengths are focused at m = 0

– This is called the zeroth order maximum

• The first order maximum corresponds to m = 1 • Note the sharpness of the principle maxima

and the broad range of the dark area – This is in contrast to the broad, bright

fringes characteristic of the two-slit interference pattern

The maxima of the diffraction pattern are defined by

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X-ray diffraction is used to determine crystal structure – the spacing between crystal planes is close enough to the wavelength of the X-rays to allow diffraction patterns to be seen. A grating spectroscope allows precise determination of wavelength:

Diffraction can also be observed upon reflection from narrowly-spaced reflective grooves; the most familiar example is the recorded side of a CD. Some insect wings also display reflective diffraction, especially butterfly wings.

• A diffraction grating can be used in a three-beam method to keep the beam on a CD on track

• The central maximum of the diffraction pattern is used to read the information on the CD

• The two first-order maxima are used for steering

Interference by Thin Films Another way path lengths can differ, and waves interfere, is if the travel through different media. If there is a very thin film of material – a few wavelengths thick – light will reflect from both the bottom and the top of the layer, causing interference. This can be seen in soap bubbles and oil slicks, for example.

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The wavelength of the light will be different in the oil and the air, and the reflections at points A and B may or may not involve reflection.

A similar effect takes place when a shallowly curved piece of glass is placed on a flat one. When viewed from above, concentric circles appear that are called Newton’s rings.

One can also create a thin film of air by creating a wedge-shaped gap between two pieces of glass.

Interference in Reflected Waves Constructive interference: Destructive interference:

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Interference can also occur when light refracts and reflects from both surfaces of a thin film. This accounts for the colors we see in oil slicks and soap bubbles. Now, we have not only path differences and phase changes on reflection; we also must account for the change in wavelength as the light travels through the film.

• Facts to remember – An electromagnetic wave traveling from a medium of index of refraction n1 toward a

medium of index of refraction n2 undergoes a 180° phase change on reflection when n2 >

n1

• There is no phase change in the reflected wave if n2 < n1

– The wavelength of light λn in a medium with index of refraction n is λn = λ/n where λ is

the wavelength of light in vacuum

• Ray 1 undergoes a phase change of 180° with respect to the incident ray

• Ray 2, which is reflected from the lower surface, undergoes no phase change with respect to the incident wave

• Ray 2 also travels an additional distance of 2t before the waves recombine

• For constructive interference – 2nt = (m + ½ ) λ m = 0, 1, 2 …

• This takes into account both the difference in optical path length for the two rays and the 180° phase change

• For destruction interference – 2 n t = m λ m = 0, 1, 2 …

• Two factors influence interference – Possible phase reversals on reflection – Differences in travel distance

• The conditions are valid if the medium above the top surface is the same as the medium below the bottom surface

• If the thin film is between two different media, one of lower index than the film and one of higher index, the conditions for constructive and destructive interference are reversed

• Be sure to include two effects when analyzing the interference pattern from a thin film – Path length – Phase change

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Newton’s Rings

• Another method for viewing interference is to place a planoconvex lens on top of a flat glass surface

• The air film between the glass surfaces varies in thickness from zero at the point of contact to some thickness t

• A pattern of light and dark rings is observed – This rings are called Newton’s Rings – The particle model of light could not explain the origin of the rings

• Newton’s Rings can be used to test optical lenses

Problem Solving Strategy with Thin Films

• Identify the thin film causing the interference • Determine the indices of refraction in the film and the media on either side of it • Determine the number of phase reversals: zero, one or two • The interference is constructive if the path difference is an integral multiple of λ and

destructive if the path difference is an odd half multiple of λ – The conditions are reversed if one of the waves undergoes a phase change on

reflection Example 4: Thickness of soap bubble skin. A soap bubble appears green ( 540 )nm at the point on

its front surface nearest the viewer. What is the smallest thickness the soap bubble film could have? Assume

1.35n .

(540 )100

4 (4)(1.35)

nmt nm

n

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Interference in Thin Films, Example

• An example of different indices of refraction

• A coating on a solar cell • There are two phase changes

CD’s and Thin Film Interference

• A CD has multiple tracks – The tracks consist of a sequence of pits of varying length formed in a reflecting

information layer • The pits appear as bumps to the laser beam

– The laser beam shines on the metallic layer through a clear plastic coating

Reading a CD

• As the disk rotates, the laser reflects off the

sequence of bumps and lower areas into a photodector

– The photodector converts the fluctuating reflected light intensity into an electrical string of zeros and ones

• The pit depth is made equal to one-quarter of the wavelength of the light

• When the laser beam hits a rising or falling bump edge, part of the beam reflects from the top of the bump and part from the lower adjacent area

– This ensures destructive interference and very low intensity when the reflected beams combine at the detector

• The bump edges are read as ones • The flat bump tops and intervening flat plains are read as zeros

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DVD’s

• DVD’s use shorter wavelength lasers – The track separation, pit depth and minimum pit length are all smaller – Therefore, the DVD can store about 30 times more information than a CD

Albert A. Michelson1852 -1931 Michelson touched on many departments of physics but, perhaps due to a special instinct which he appeared to possess, he excelled in optics. He performed early measurements of the velocity of light with amazing delicacy and in 1881 he invented his interferometer for the purpose of discovering the effect of the Earth's motion on the observed velocity.

Michelson Interferometer The Michelson interferometer is centered around a beam splitter, which transmits about half the light hitting it and reflects the rest. It can be a very sensitive measure of length.

Polarization

Light is polarized when its electric fields oscillate in a single plane, rather than in any direction perpendicular to the direction of propagation. Polarized light will not be transmitted through a polarized film whose axis is perpendicular to the polarization direction.

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When light passes through a polarizer, only the component parallel to the polarization axis is transmitted. If the incoming light is plane-polarized, the outgoing intensity is:

This means that if initially unpolarized light passes through crossed polarizer, no light will get through the second one.

Light is also partially polarized after reflecting from a nonmetallic surface. At a special angle, called the polarizing angle or Brewster’s angle, the polarization is 100%.

Liquid Crystal Displays (LCD)

Liquid crystals are unpolarized in the absence of an external voltage, and will easily transmit light. When an external voltage is applied, the crystals become polarized and no longer transmit; they appear dark. Liquid crystals can be found in many familiar applications, such as calculators and digital watches.

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Color LCD displays are more complicated; each pixel has three subpixels to provide the different colors. A source of light is behind the display (unlike calculators and watches, which use ambient light). The pixels must be able to make finer adjustments than just on and off to provide a clear image.

• Rotation of a

polarized light beam by a liquid crystal when the applied voltage is zero

• Light passes through the polarizer on the right and is reflected back to the observer, who sees the segment as being

bright

• When a voltage is

applied, the liquid crystal does not rotate the plane of polarization

• The light is absorbed by the polarizer on the right and none is reflected back to the observer

• The segment is dark

• Changing the applied voltage in a precise pattern can – Tick off the seconds on a watch – Display a letter on a computer display

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CHAPTER 24

WAVE NATURE OF LIGHT

CONCEPTS

1. For a thin spherical lens, the focal length of red light compared to the focal length of blue light is greater.

2. The wave theory of light is attributed to Christian Huygens.

3. The particle theory of light is attributed to Isaac Newton.

4. The principle responsible for light spreading as it passes through a narrow slit is diffraction.

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5. The principle responsible for alternting light and dark bands when light passes through two or more narrow slits is called interference.

6. The principle responsible for the fact that certain sunglasses can reduce glare from

refected surfaces is polarization.

7. The principle on which fiber optics is based is total internal reflection.

8. The principle which allows a rainbow to form is dispersion.

9. The principle on which lenses work is refraction.

10. The principle which explains why a prism separates white light into different colors is dispersion.

11. Objects under water appear closer than they are because of refraction.

12. All points on a given wave front have the same phase.

13. Rays are always perpendicular to wave fronts.

14. The spacing between adjacent wave fronts is one-half waavelength.

15. When a light wave enters into a medium of different optical density its speed and wavelength change.

16. When a beam of light (wavelength = 590 nm), originally traveling in air, enters a piece of glass (index of refraction 1.50), its frequency is unaffected.

17. When a beam of light (wavelength = 590 nm), originally traveling in air, enters a piece of glass (index of refraction 1.50), its wavelength is reduced to 2/3 its original value.

18. The index of refraction of diamond is 2.42. This means that a given frequency of light travels 2.42 times faster in vacuum than it does in diamond.

19. When a ray of light passes obliquely from one medium to another, the direction of travel, wavelength and speed changes.

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20. White light is a mixture of all frequencies.

21. You can only see a rainbow if the sun is behind you.

22. The color on the outer edge of the primary rainbow is red.

23. Two light sources are said to be coherent if they are of the same frequency, and maintain a constant phase difference.

24. Ina double slit experiment, it is observed that the distance between adjacent maxima on a remote screen is 1 cm. The distance between adjacent maxima is increased to 2 cm when the slit separation is cut in half.

25. At the first maxima on either side of the central bright spot in a double slit experiment, light from each opening arrives in phase.

26. At the first maxima on either side of the central bright spot in a double slit experiment, light from one opening travels two wavelengths of light farther than light from the other opening.

27. The colors on an oil slick are caused by reflection and interference.

28. When a beam of light, which is traveling in glass, strikes an air boundary, there is no phase change in the reflected beam.

29. When a beam of light, which is traveling in air, is reflected by a glass surface, there is a 180o phase change in the reflected beam.

30. A soap film is being viewed in white light. As the film becomes very much thinner than the wavelength of blue light, the film appears black because it reflects no visible light.

31. In terms of the wavelength of light in soapy water, the minimum thickness of soap film that will reflect a given wavelength of light is one-fourth wavelength.

32. After a rain, one sometimes sees brightly colored oil slicks on the road. These are due to interference effects.

33. Consider two different gratings; one has 4000 lines per cm and the other one has 6000 lines per cm. The 6000-line grating produces the greater dispersion.

34. For a beam of light, the direction of polarization is defined as the direction of the electric field’s vibration.

35. The polarization of sunlight is greatest at both sunrise and sunset.

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