module 3 optics

8/3/2019 Module 3 Optics

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MINISTRY OF EDUCATION AND TRAINING

NONG LAM UNIVERSITY

FACULTY OF FOOD SCIENCE AND TECHNOLOGY

Course: Physics 1

Module 3: Optics and WavePhenomena

Instructor: Dr. Son Thanh Nguyen

Academic year: 2008-2009


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Physic 1 Module 3: Optics 2

Contents

Module 3: Optics and wave phenomena

3.1. Wave review

1) Description of a wave

2) Transerve waves and longitudinal waves

3) Mathematical description of a traveling (propagating) wave with constant amplitude4) Electromagnetic waves

5) Spherical and plane waves

3.2. Interference of sound waves and light waves

1) Interference of sinusoidal waves Coherent sources

2) Interference of sound waves

3) Interference of light waves

3.3. Diffraction and spectroscopy

1) Introduction to diffraction

2) Diffraction by a single narrow slit - Diffraction gratings

3) Spectroscopy: Dispersion Spectroscope Spectra

3.4. Applications of interference and diffraction

1) Applications of interference2) Applications of diffraction

3.5. Wave-particle duality of light and particles

1) Photoelectric effect Einsteins photon concept

2) Electromagnetic waves and photons

3) Wave-particle duality De Broglies postulate


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3.1. Wave review

1) Description of a propagating wave

Simply stated, a wave is a way in which energy is transferred from place to place withoutphysical movement of material from one location to another. In wave motion, the energy is

carried by a disturbance of some sort. This disturbance, whatever its nature, occurs in adistinctive repeating pattern. Ripples on the surface of a pond, sound waves in air, and

electromagnetic waves in space, despite their many obvious differences, all share this basic

defining property.

In other words, wave is a periodic disturbance that travels from one place to another without

actually transporting any matter. The source of all waves is something that is vibrating, moving

back and forth at a regular, and usually fast rate.

We must distinguish between the motion of particles of the medium through which the wave is

propagating and the motion of the wave pattern through the medium, or wave motion. Theparticles of the medium vibrate at fixed positions; the wave progresses through the medium.

Familiar examples of waves are waves on a surface of water; waves on a stretched string;

sound waves; light and other forms of electromagnetic radiation.

While a mechanical wave such as a sound wave exists in a medium, waves of electromagnetic

radiation including light can travel through vacuum, that is, without any medium.

Periodic waves are characterized by crests (highs) and troughs (lows), as shown in Figure 23.

Within a wave, the phase of a vibration of the mediums particle (that is, its position within the

vibration cycle) is different for adjacent points in space because the wave reaches these points at

different times.

Waves travel and transfer energy from one point to another, often with little or no permanent

displacement of the particles of the medium (that is, with little or no associated mass transport);

instead there are oscillations (vibrations) around almost fixed locations.

Figure 23: Representation of a typical wave, showing itsdirection of motion, wavelength, and amplitude.


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2)Transverse and longitudinal waves

In terms of the direction of particless vibrations and that of the wave propagation, there

are two major kinds of waves: transverse waves and longitudinal waves.

Transverse waves are those with

particless vibrations perpendicular to thewave's direction of travel; examples include

waves on a stretched string and

electromagnetic waves.

Longitudinal waves are those with

particless vibrations along the wave's

direction of travel; examples include sound

waves.

Apart from transverse waves and

longitudinal waves, ripples on the surface of a pond are actually a combination of transverse and

longitudinal waves; therefore, the points on the water surface follow elliptical paths, as shown in

Figure 24.

3)Mathematical description of a traveling (propagating) wave with constant

amplitude

Transverse waves are probably the most important waves to understand in this module;

light is also a transverse wave. We will therefore start by studying transverse waves in a simple

context: waves on a stretched string.

As mentioned earlier, a transverse, propagating wave is a wave that consists of oscillations of

the mediums particles perpendicular to the direction of wave propagation or energy transfer. Ifa transverse wave is propagating in the positive x-direction, the oscillations are in up and down

directions that lie in the yz-plane.

From a mathematical point of view, the most primitive or fundamental wave is harmonic

(sinusoidal) wave which is described by the wave function

u(x, t) = Asin(kx t) (47)

where u is the displacement of a particular particle of the medium from its midpoint, A

the amplitude of the wave, k the wave number, the angular frequency, and t the time.

In the illustration given by Figure 23, the amplitude is the maximum vertical distance betweenthe baseline and the wave or the maximum departure of the wave from the undisturbed state.

The units of the amplitude depend on the type of wave - waves on a string have an amplitude

expressed as a distance (meters), sound waves as pressure (pascals), and electromagnetic waves

as an magnitude of the electric field (volts/meter). The amplitude may be constant or may vary

with time and/or position. The form of the variation of amplitude is called the envelope of the

wave.

Figure 24: When an object bobs up and down on aripple in a pond, it experiences an ellipticaltrajectory because ripples are not simpletransverse sinusoidal waves.


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The period T is the time for one complete cycle for an oscillation. The frequency f (also

frequently denoted as ) is the number of periods per unit time (one second) and is measured in

hertz. T and f are related by

f=1

T (48)

In other words, the frequency and period of a wave are reciprocals of each other. The

frequency is equal to the number of crests or cycles passing any given point per unit time (a

second).

The angular frequency represents the frequency in terms of radians per second. It is related

to the frequency f by

= 2f (49)

There are two velocities that are associated with waves. The first is the phase velocity, vp or v,

which gives the rate at which the wave propagates, is given by

v =k

(50)

The second is the group velocity, vg, which gives the velocity at which variations in the

shape of the wave's amplitude propagate through space. This is the rate at which information can

be transmitted by the wave. It is given by

vg =k

(51)

The wavelength (denoted as ) is the distance between two successive crests (or troughs) of awave, as shown in Figure 22. This is generally measured in meters; it is also commonly

measured in nanometers for the optical part of the electromagnetic spectrum. The wavelength is

related to the period (or frequency) and speed of a wave (phase velocity) by the equation

= vT = v/f (52)

For example, a radio wave of wavelength 300 m traveling at 300 million m/s (the speed

of light) has a frequency of 1 MHz.

The wavenumber k is associated with the wavelength by the relation

k=2

(53)

Example: Thomas attaches a stretched string to a mass that oscillates up and down once

every half second, sending waves out across the string. He notices that each time the mass

reaches the maximum positive displacement of its oscillation, the last wave crest has just

reached a bead attached to the string 1.25 m away. What are the frequency, wavelength, and

speed of the waves? (Ans. f = 2 Hz, = 1.25 m, v = 2.5 m/s)


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Figure 26: Spherical waves emitted by a

point source. The circular arcs represent thespherical wave fronts that are concentric with

the source. The rays are radial lines pointing

outward from the source, perpendicular to

the wave fronts.

4)Electromagnetic waves

As described earlier, a transverse, moving wave is a wave that consists of oscillations

perpendicular to the direction of energy transfer.

If a transverse wave is moving in the positive x-direction, the oscillations are in up and down

directions that lie in the yz-plane.

Electromagnetic (EM) waves including

light behave in the same way as other

waves, although it is harder to see.

Electromagnetic waves are also two-

dimensional transverse waves. This two-

dimensional nature should not be confused

with the two components of anelectromagnetic wave, the electric and

magnetic field components, which are

shown in shown in Figure 25. Each of these

fields, the electric and the magnetic, exhibits

two-dimensional transverse wave behavior,

just like the waves on a string, as shown in

Figure 25.

A light wave is an example of an electromagnetic wave which is shown in Figure 25. In

vacuum light propagate with phase speed: v = c = 3 x 108 m/s.

The term electromagnetic just means that the energy is carried in the form of rapidlyfluctuating electric and magnetic fields. Visible light is the particular type of electromagnetic

wave (radiation) to which our human eyes happen to be sensitive. But there is also invisible

electromagnetic radiation, which goes completely undetected by our eyes. Radio, infrared, and

ultraviolet waves, as well as x rays and gamma rays, all fall into this category.

Figure 25:Electric and magnetic fieldsvibrate perpendicular to each other.

Together they form an electromagnetic wave

that moves through space at the speed oflight c.


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Figure 27: Far away from a point source, thewave fronts are nearly parallel planes, and therays are nearly parallel lines perpendicular tothese planes. Hence, a small segment of a

spherical wave is approximately a plane wave.

Figure 28: A representation of a plane wave

moving in the positive x direction with a speed v.The wave fronts are planes parallel to

the yz plane.

5) Spherical and plane waves

If a small spherical body, considered as a point, oscillates so that its radius varies sinusoidally

with time, a spherical wave is produced, as shown in Figure 26. The wave moves outward from

the source in all directions, at a constant speed if the medium is uniform. Due to the mediums

uniformity, the energy in a spherical wave propagates equally in all directions. That is, no one

direction is preferred over any other.

It is useful to represent spherical waves with a series of circular arcs concentric with the

source, as shown in Figure 26. Each arc represents a surface over which the phase of the wave is

constant. We call such a surface of constant phase a wave front. The distance between adjacent

wave fronts equals the wavelength . The radial lines pointing outward from the source and

perpendicular to the wave fronts are called rays.

Now consider a small portion of a wave front

far from the source, as shown in Figure 27. In

this case, the rays passing through the wave

front are nearly parallel to one another, and the

wave front is very close to being planar.

Therefore, at distances from the source that are

great compared with the wavelength, we can

approximate a wave front with a plane. Any

small portion of a spherical wave front far from

its source can be considered a plane wave front.

Figure 28 illustrates a plane wave propagating

along the x axis, which means that the wave

fronts are parallel to the yz plane. In this case, the

wave function depends only on x and t and has the

form

u(x, t) = Asin(kx - t) (54)

That is, the wave function for a plane wave isidentical in form to that for a one-dimensional

traveling wave (equation 47). The intensity is the

same at all points on a given wave front of a plane

wave.

In other words, a plane wave have wave fronts

that are planes parallel to each other, rather than


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Figure 29: Depicting the snapshots of the medium fortwo pulses of the same amplitude (both upward) beforeand during interference; the interference isconstructive.

Figure 30: Depicting the snapshots of the medium for twopulses of the same amplitude (both downward) before andduring interference; the interference isconstructive.

spheres of increasing radius.

3.2. Interference of sound waves and light waves

Interference of waves

What happens when two waves meet while they travel through the same medium? What affect

will the meeting of the waves have upon the appearance of the medium? These questionsinvolving the meeting of two or more waves in the same medium pertain to the topic of wave

interference.

Wave interference is a phenomenon which occurs when two waves of the same frequency and

of the same type (both are transverse or longitudinal) meet while traveling along the same

medium. The interference of waves

causes the medium to take on a

shape which results from the net

effect of the two individual waves

upon the particles of the medium.

In other words, interference is the

ability of two or more waves to

reinforce or partially cancel each

other.

To begin our exploration of wave

interference, consider two sine

pulses of the same amplitude traveling in different directions in the same medium.

Suppose that each is displaced upward 1 unit at its crest and has the shape of a sine wave.

As the sine pulses move toward each other, there will eventually be a moment in time when they

are completely overlapped. At that moment, the resulting shape of the medium would be an

upward displaced sine pulse with an amplitude of 2 units. The diagrams shown in Figure 29depict the snapshots of the medium for two such pulses before and during interference. The

individual sine pulses are drawn in red and blue, and the resulting displacement of the medium is

drawn in green.

This type of interference is called constructive interference. Constructive interference

is a type of interference which occurs at any location in the medium where the two interfering

waves have a displacement in the same direction and their crests or troughs exactly coincide.

The net effect is that the two wave motions reinforce each other, resulting in a wave of greater

amplitude. In the case mentioned

above, both waves have an upward

displacement; consequently, the

medium has an upwarddisplacement which is greater than

the displacement of either

interfering pulse. Constructive

interference is observed at any location

where the two interfering waves are

displaced upward. But it is also

observed when both interfering


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Figure 32: Depicting the before and duringinterference snapshots of the medium for two pulses ofdifferent amplitudes (one upward, +1 unit and onedownward, -2 unit); the interference isdestructive.

Figure 31: Depicting the snapshots of the medium fortwo pulses of the same amplitude (one upward and onedownward) before and during interference;the interference isdestructive.

waves are displaced downward. This is shown in Figure 30 for two downward displaced pulses.

In this case, a sine pulse with a maximum displacement of -1 unit (negative means a

downward displacement) interferes with a sine pulse with a maximum displacement of -1 unit.

These two pulses are again drawn in red and blue. The resulting shape of the medium is a sine

pulse with a maximum displacement of -2 units.

Destructive interference is a type of interference which occurs at any location in the mediumwhere the two interfering waves have displacements in the opposite direction. For instance,

when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum

displacement of -1 unit, destructive interference occurs. This is depicted in the diagram shown in

Figure 31.

In Figure 31, the interfering

pulses have the same maximum

displacement but in opposite

directions. The result is that the two

pulses completely destroy each other

when they are completely overlapped.

At the instant of complete overlap,there is no resulting displacement of

the particles of the medium. When two

pulses with opposite displacements

(i.e., one pulse displaced up and the

other down) meet at a given location, the upward pull of one pulse is balanced (canceled or

destroyed) by the downward pull of the other pulse. Destructive interference leads to only a

momentary condition in which the medium's

displacement is less than the displacement of

the largest-amplitude wave.

The two interfering waves do not

need to have equal amplitudes in opposite

directions for destructive interference to

occur. For example, a pulse with a maximum

displacement of +1 unit could meet a pulse

with a maximum displacement of -2 units.

The resulting displacement of the medium

during complete overlap is -1 unit, as shown

in Figure 32.

The task of determining the shape of the resultant wave demands that the principle of

superposition is applied. The principle of superposition is stated as follows:

When two waves interfere, the resulting displacement of the medium at any location is

the algebraic sum of the displacements of the individual waves at that same location.

In the cases mentioned above, the summing of the individual displacements for locations of

complete overlap was easy and given in the below table.


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Maximum displacement of

Pulse 1

Maximum displacement of

Pulse 2

Maximum resulting

displacement

+1 +1 +2

-1 -1 -2

+1 -1 0

+1 -2 -1

1) Interference of sinusoidal waves Coherent sources

Mathematics of two-point source interference

We already found that the adding together of two mechanical waves can be constructive or

destructive. In constructive interference, the amplitude of the resultant wave is greater than that

of either individual wave, whereas in destructive interference, the resultant amplitude is less than

that of either individual wave. Light waves also interfere with each other. Fundamentally, all

interference associated with light waves arises when the electromagnetic fields that constitute

the individual waves combine.

Conditions for interference

For sustained interference in waves to be observed, the following conditions must be met:

The sources must maintain a constant phase with respect to each other. The sources should of a single wavelength (or frequency).

Such wave sources are termed coherent sources.

We now describe the characteristics of coherent sources. As we saw when we studied

mechanical waves, two sources (producing two traveling waves) are needed to create

interference. In order to produce a stable interference pattern, the individual waves must

maintain a constant phase relationship with one another.

As an example, the sound waves emitted by two side-by-side loudspeakers driven by asingle amplifier can interfere with each other because the two speakers are coherent - that is,

they respond to the amplifier in the same way at the same time.

A common method for producing two coherent sources is to use one monochromatic

source to generate two secondary sources. For example, a popular method for producing two

coherent light sources is to use one monochromatic source to illuminate a barrier containing two

small openings (usually in the shape of slits). The light emerging from the two slits is coherent

because a single source produces the original light beam and the two slits serve only to separate

the original beam into two parts (which, after all, is what was done to the sound signal from the

side-by-side loudspeakers).

Consider two separate waves propagating from two coherent sources located at O1 and O2. Thewaves meet at P, and according to the principle of superposition, the resultant vibration at P is

given by

uP = u1 + u2 = Asin(kx1t) + Asin(kx2t) (55)

where x1 = O1P and x2 = O2P are the wave paths (distances traveled) from O1 and O2 to

P, respectively.


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Figure 33: An acoustical system for demonstratinginterference of sound waves. A sound wave from the

speaker (S) propagates in the tube and splits intotwo parts at point P. The two waves, which

superimpose at the opposite side, are detected atthe receiver (R). The upper path length r2 can bevaried by sliding the upper section.

For the sake of simplicity, we have assumed A1 = A2 = A.

Using the trigonometric identity: sin + sin = 2sin{(+)/2}cos{(-)/2} (56), from

equation (55) we have

uP = 2Acos{k(x2 x1)/2}sin{k(x1 + x2)/2 - t} (57)

From equation (57), we see that the amplitude AM of the resultant vibration (resultant

amplitude) at the point P is given by

AP = |2Acos {k(x2 x1)/2}| (58)

According to equation (58), AP is time independent and depends only on thepath difference,

x, of the two wave components

x = x2 x1 (59)

From equations (53), (58) and (59), we can easily see the following cases:

Case 1: x = x2 x1 = n2/k = n where n = 0, 1, 2, or the path difference is

zero or some integer multiple of wavelength.

We have AP = 2A. The amplitude of the resultant wave is 2A - twice the amplitude of

either individual wave. In this case, the

interfereing waves are said to be

everywhere in phase and thus interfere

constructively. There is a constructive

interference at P.

Case 2: x = x2 x1 = (n +0,5)/k = (2n + 1)/2 where n = 0, 1,

2, orthe path difference is odd

multiple of half wavelength.

We have AP = 0. The resultant

wave has zero amplitude. In this case,

the interfereing waves are exactly 180o

out of phase and thus interfere

destructively. There is a destructive

interference at P.

2) Interference of sound waves

One simple device for demonstrating

interference of sound waves is illustrated

in Figure 33. Sound from a loudspeaker S

is sent into a tube at point P, where there is

a T-shaped junction.


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Figure 34: Schematic diagram of Youngs double-slit experiment.Two slits behave as coherent sources of light waves that produce aninterference pattern on the viewing screen (drawing not to scale).

Half of the sound power travels in one direction, and half travels in the opposite direction.

Thus, the sound waves that reach the receiver R can travel along either of the two paths. The

distance along any path from speaker to receiver is called the path length r.The lower path

length r1is fixed, but the upper path length r2can be varied by sliding a U-shaped tube, which issimilar to that on a slide trombone.

When the path difference is either zero or some integer multiple of the wavelength (that is

r2 r1 = n, where n = 0, 1, 2, . . .), the two waves reaching the receiver at any instant are inphase and interfere constructively. For this case, a maximum in the sound intensity is detected at

the receiver.

If the path length r2 is adjusted such r2 r1 = (n + 1/2), where n = 0, 1, 2, . . ., the two

waves are exactly rad, or 180, out of phase at the receiver and hence cancel each other. In this

case of destructive interference, no sound is detected at the receiver.

3) Interference of light waves

Two-point source light interference patterns

Any type of wave, whether it is a water wave or a sound wave should produce a two-point

source interference pattern if the two sources periodically disturb the medium at the same

frequency. Such a pattern is always characterized by a pattern of alternating nodal and antinodal

lines. Let's discuss what one might observe if light were to undergo two-point source

interference. What will happen if a "crest" of one light wave interferes with a "crest" of a second

light wave? And what will happen if a "trough" of one light wave interferes with a "trough" of a

second light wave? And finally, what will happen if a "crest" of one light wave interfered with a

"trough" of a second light wave?

Whenever light waves constructively interfere (such as when a crest meeting a crest or a

trough meeting a trough), the two waves act to reinforce one another and to produce an

enhanced lightwave. On the other

hand, whenever

light waves

destructively

interfere (such as

when a crest meets

a trough), the two

waves act to destroy

each other and

produce no light

wave. Thus, the

two-point sourceinterference pattern

would still consist

of an alternating

pattern of antinodal

lines and nodal

lines. For light

waves, the antinodal


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Figure 35: A typical pattern from a two-slitexperiment of interference.

lines are equivalent to bright lines, and the nodal lines are equivalent to dark lines. If such an

interference pattern could be created by two light sources and projected onto a screen, then there

ought to be an alternating pattern of dark and bright bands on the screen. And since the central

line in such a pattern is an antinodal line, the central band on the screen ought to be a bright

band.

YOUNGS DOUBLE-SLIT EXPERIMENT

In 1801, Thomas Young successfully showed that light does produce a two-point source

interference pattern. In order to produce such a pattern, monochromatic light must be used.

Monochromatic light is light of a single color; by use of such light, the two sources will vibrate

with the same frequency.

It is also important that the two light waves be vibrating in phase with each other; that is, the

crest of one wave must be produced at the same precise time as the crest of the second wave.

(These waves are often referred to as coherent light waves.)

As expected, the use of a monochromatic light source and pinholes to generate in-phase light

waves resulted in a pattern of alternating bright and dark bands on the screen. A typical

appearance of the pattern is shown in Figure 35.

To accomplish this, Young used a single light source (primary source) and projected the light

onto two very narrow slits, as shown in Figure 34. The light from the source will then diffract

through the slits, and the pattern can be

projected onto a screen. Since there is only

one source of light, the set of two waves

which emanate from the slits will be in

phase with each other.

As a result, these two slits, denoted as S1

and S2, serve as a pair of coherent lightsources. The light waves from S1 and S2

produce on a viewing screen a visible

pattern of bright and dark parallel bands called fringes, as shown in Figure 35. When the light

from S1 and that from S2 both arrive at a point on the screen such that constructive interference

occurs at that location, a bright fringe appears. When the light from the two slits combines

destructively at any location on the screen, a dark fringe results.


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Figure 36: Geometric construction for describing Youngs double-slit experiment (not to scale).

We can describe Youngs experiment quantitatively with the help of Figure 36. The viewing

screen is located a

perpendicular distance

L from the double-

slitted barrier. S1 and S2

are separated by a

distance d, and the

source ismonochromatic. To

reach any arbitrary

point P, a wave from the

lower slit travels farther

than a wave from the

upper slit by a distance

d sin . This distance is

called the path

difference (lowercase

Greek delta).

If we assume

that two rays, S1P and S2P,

are parallel, which is

approximately true because

L is much greater than d,

then is given by

= S2P S1P = r2 r1 = d sin (60)

where d = S1S2 is the distances between the two coherent light sources (i.e., the two slits).

If is either zero or some integer multiple of the wavelength, then thetwo waves are in phase at point P and constructive interference results. Therefore,

the condition for bright fringes, or constructive interference, at point P is

= r2 r1 = n (61)

where n = 0, 1, 2, .

The number n in equation (61) is called the order number. The central bright fringe at = 0(n = 0) is called the zeroth-order maximum. The first maximum on either side, where n = 1, is

called the first-order maximum, and so forth.

When is an odd multiple of/2, the two waves arriving at point P are 180 out of phase andgive rise to destructive interference. Therefore, the condition for dark fringes, or destructive

interference, at point P is

= r2 r1 = (n + 1/2) (62)

where n = 0, 1, 2, ....


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It is useful to obtain expressions for the positions of the bright and dark fringes measured

vertically from O to P. In addition to our assumption that L >> d, we assume that d >> . These

can be valid assumptions because in practice L is often of the order of 1 m, d a fraction of a

millimeter, and a fraction of a micrometer for visible light. Under these conditions, is small;

thus, we can use the approximation sin tan . Then, from triangle OPQ in Figure 36, we see

that

y = OP = L tan L sin (63)

From equations (60), (61) and (63), we can prove that the positions of the bright fringes

measured from O are given by the expression

ybright = nL

d

(64)

Similarly, using equations (60), (62) and (63), we find that the dark fringes are located at

ydark = (n + 1/2) Ld (65)

As we demonstrate in the following example, Youngs double-slit experiment provides a

method for measuring the wavelength of light. In fact, Young used this technique to do just that.

Additionally, the experiment gave the wave model of light a great deal of credibility. It was

inconceivable that particles of light coming through the slits could cancel each other in a way

that would explain the dark fringes. As a result, the light interference show that light is of wave

nature.

Example: A viewing screen is separated from a double-slit source by 1.2 m. The distance

between the two slits is 0.030 mm. The second-order bright fringe is 4.5 cm from the center line.

(a)Determine the wavelength of the light. (Ans. = 560 nm)(b) Calculate the distance between two successive bright fringes. (Ans. 2.25 cm)


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Figure 37: Light intensity versus = d sin for a double-

slit interference pattern when the viewing screen is far fromthe slits (L >> d).

Intensitydistribution of the

double-slit

interference pattern

So far we have

discussed the

locations of only thecenters of the bright

and dark fringes on a

distant screen. We

now direct our

attention to the

intensity of the light

at other points

between the positions

of constructive and

destructive

interference.

In other words, wenow calculate the

distribution of light

intensity associated

with the double-slit

interference pattern.

Again, suppose that the two

slits represent coherent sources of sinusoidal waves such that the two waves from the slits have

the same frequency f and a constant phase difference.

Recall that the intensity of a light wave, I, is proportional to the square of the resultant electricfield magnitude at the point of interest, we can show that (see pages 1191 and 1192, Hallidays

book).

I = Imaxcos2d

( y)L

(66)

where Imax is the maximum intensity on the screen, and the expression represents the time

average.

Constructive interference, which produces light intensity maxima, occurs when the quantity

y/L is an integral multiple of, corresponding to y = (L/d)n. This is consistent with equation

(64).


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Figure 38: Diffraction of a light wave: (a) If radiationwere composed of rays or particles moving in perfectly

straight lines, no bending would occur as a beam oflight passed through a circular hole in a barrier, andthe outline of the hole, projected onto a screen, wouldhave perfectly sharp edges. (b) In fact, light is diffractedthrough an angle that depends on the ratio of thewavelength of the wave to the size of the gap. The resultis that the outline of the hole becomes "fuzzy," as shownin this actual photograph of the diffraction pattern.

A plot of light intensity versus

= d sin is given in Figure 37.

Note that the interference pattern

consists of equally spaced

fringes of equal intensity.

Remember, however, that this

result is valid only if the slit-to-

screen distance is much greaterthan the slit separation (L >> d),

and only for small values of.

3.3. Diffraction and

spectroscopy

1) Introduction to

diffraction

Diffraction is the deflection, or

"bending," of a wave as it passes

a corner or moves through a

narrow gap. For any wave, the

amount of diffraction is

proportional to the ratio of the

wavelength to the width of the

gap. The longer the wavelength

and/or the smaller the gap, the

greater the angle through which

the wave is diffracted. Thus,

visible light, with its extremely

short wavelengths, shows

perceptible diffraction onlywhen passing through very

narrow openings. (The effect is

much more noticeable for sound

waves, however - no one thinks twice about our ability to hear people even when they are

around a corner and out of our line of sight.)

Diffraction is normally taken to refer to various phenomena which occur when a wave

encounters an obstacle whose size is comparable to the wavelength. It is described as the

apparent bending of waves around small obstacles and the spreading out of waves past small

openings. Diffraction occurs with all waves, including sound waves, water waves, and

electromagnetic waves such as visible light, x-rays, and radio waves. Diffraction is a property

that distinguishesbetween wave-like and particle-like behaviors.

A slit of infinitesimal width which is illuminated by light diffracts the light into a series of

circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform

intensity. The light at a given angle is made up of contributions from each of these point sources,

and if the relative phases of these contributions vary by more than 2, we expect to find minima

and maxima in the diffracted light.


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The effects of diffraction can be readily seen in everyday life. The most colorful examples ofdiffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act

as a diffraction grating to form the familiar rainbow pattern we see when looking at a disk. All

these effects are a consequence of the fact that light is a wave.

Diffraction arises because of the way in which waves propagate; this is described by theHuygensFresnel principle. This principle states that

Eeach point of an advancing wave front is in fact the center of a fresh disturbance and

the source of a new train of waves; and that the advancing wave as a whole may be regarded

as the sum of all the secondary waves arising from points in the medium already traversed.

The propagation of a wave can be visualized by considering every point on a wavefront as a

point source for a secondary radial wave. The subsequent propagation and addition of all these

radial waves form the new wavefront, as shown in Figure 38. When waves are added together,

their sum is determined by the relative phases as well as the amplitudes of the individual waves,

an effect which is often known as wave interference. The resultant amplitude of the waves can

have any value between zero and the sum of the individual amplitudes. Hence, diffraction

patterns usually have a series of maxima and minima (see Figure 38b).

To determine the form of a diffraction pattern, we must determine the phase and amplitude of

each of the Huygens wavelets at each point in space and then find the sum of these waves. There

are various analytical models which can be used to do this including the Fraunhoffer diffraction

equation for the far field and the Fresnel diffraction equation for the near field.

As a result, diffraction effects are classified into either Fresnel or Fraunhofer types.Fresneldiffraction is concerned mainly with what happens to light in the immediate neighborhood of adiffracting object or aperture, so is only of concern when the illumination source is close by.

Fraunhofer diffraction is the light-spreading effect of an aperture when the aperture (or object)is lit by plane waves, i.e., waves that effectively come from a source that is infinitely far away.

Because of Fraunhofer diffraction, a telescope can never form a perfect image. A point-likesource, for example, will be seen as a small disk surrounded by a series of rings; a thin line on a

planet will become widened into a band, which decreases in intensity on both sides. The only

way to overcome the limitations of diffraction is to use a telescope of larger aperture.

Diffraction is set to work in diffraction gratings. Here, light passed through a series of very

accurately ruled slits. Gratings are ruled from 70 lines/mm (for infrared work) to 1800 lines/mm

(for ultraviolet work).

2) Diffraction by a single narrow slit

Single-slit diffraction

This is an attempt to more clearly visualize the nature of single-slit diffraction. The

phenomenon of diffraction involves the spreading out of waves past openings which are on the

order of the wavelength of the wave. The spreading of the waves into the area of the geometrical

shadow can be modeled by considering small elements of the wavefront in the slit and treating

them like point sources.


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In general, diffraction occurs when waves pass through small openings, around obstacles, or

past sharp edges, as shown in Figure 39. When an opaque object is placed between a point

source of light and a screen, no sharp boundary exists on the screen between a shadowed region

and an illuminated region. The illuminated region above the shadow of the object contains

alternating light and dark fringes. Such a display is called a diffraction pattern (see Figure 38.3,

Hallidays book, page 1213). Figure 38.3 shows a diffraction pattern associated with the shadowof a penny.

In this module we restrict our attention to Fraunhofer diffraction, which occurs, for example,

when all the rays passing through a narrow slit are approximately parallel to one another (a plane

wave). This can be achieved experimentally either by placing the screen far from the opening

used to create the diffraction or by using a converging lens to focus the rays once they pass

through the opening, as shown in Figure 39a.

A bright fringe is observed along the axis at = 0, with alternating dark and bright fringes

occurring on either side of the central bright one. Figure 39b is a photograph of a single-slit

Fraunhofer diffraction pattern.

We can find the angle at which a first minimum is obtained in the diffracted light by the

following reasoning. The light from a source located at the top edge of the slit interferes

destructively with a source located at the middle of the slit, when the path difference between

them is equal to /2. Similarly, the source just below the top of the slit will interfere

destructively with the source located just below the middle of the slit at the same angle. We can

continue this reasoning along the entire height of the slit to conclude that the condition for

destructive interference for the entire slit is the same as the condition for destructive interference

Figure 39: (a) Fraunhofer diffraction pattern of a singleslit. The pattern consists of a central bright fringeflanked by much weaker maxima alternating with darkfringes (drawing not to scale). (b) Photograph of asingle-slit Fraunhofer diffraction pattern.


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Figure 40: Intensity distribution for aFraunhofer diffraction pattern from a

single slit of width a. The positions of twominima on each side of the centralmaximum are labeled (drawing not to

scale).

Figure 41: Diffraction grating is an opticaldevice used to disperse light into a spectrum.

between two narrow slits a distance apart that is half the width of the slit (see section 3.2.3). The

path difference is given by (asin)/2 so that the minimum intensity occurs at an angle min givenby

a sin min = (67)

where a is the width of the slit.

A similar argument can be used to show that if

we imagine the slit to be divided into four, six

eight parts, etc, minima are obtained at angles n

given by

a sin n = n (68)

where n is an integer greater than zero.

The intensity distribution for a Fraunhofer

diffraction pattern from a single slit of width ashown in Figure 40.

It should be noted that this analysis applies only

to the far field, that is at a distance much larger than the width of the slit.

Diffraction gratings

Diffraction grating is an optical device used to

disperse light into a spectrum. It is ruled with

closely-spaced, fine, parallel grooves, typically

several thousand per cm, that produce interference

patterns in a way that separates all the components

of the incoming light, as shown in Figure 41. A

diffraction grating can be used as the main

dispersing element in a spectroscope (see the next

section).

In other words, a diffraction grating is the tool

of choice for separating the colors in incident

light.

The diffraction grating, a useful device for analyzing light sources, consists of a large number

of equally spaced parallel slits. A transmission grating can be made by cutting parallel lines on aglass plate with a precision ruling machine. The spaces between the lines are transparent to the

light, and hence act as separate slits.

A plane wave is incident from the left, normal to the plane of the grating. The pattern

observed on the screen is the result of the combined effects of interference and diffraction. Each

slit produces diffraction, and the diffracted beams interfere with one another to produce the final

pattern.


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Figure 42: Intensity versus

sin for a diffraction grating.The zeroth-, first-, and

second-order maxima areshown.

The waves from all slits are in phase as they leave the slits. However, for some arbitrary

direction measured from the horizontal, the waves must travel different path lengths before

reaching a particular point on the viewing screen.

The condition for maximum intensity is the same as that for a double slit (see section 3.2.3).

However, angular separation of the maxima is generally much greater because the slit spacing is

so small for a diffraction grating. The diffraction pattern produced by the grating is thereforedescribed by the equation

d sin = m (69)

where m = 0, 1, 2, 3 and |m| is the order number; is a selected wavelength; d is the spacing

of the grooves; and is the angle of incidence of light. Equation (69)states the condition for

maximum intensity.

The diffraction grating is thus an immensely useful tool for the separation of the spectral lines

associated with atomic transitions. It acts as a "super prism", separating the different colors of

light much more than the dispersion effect in a prism.

We can use equation (69) to calculate the wavelength if

we know the grating spacing d and the angle . If the

incident radiation contains several wavelengths, the mth-

order maximum for each wavelength occurs at a specific

angle. All wavelengths are seen at = 0, corresponding to

the zeroth-order maximum (m = 0).

The first-order maximum (m = 1) is observed at an angle

that satisfies the relationship sin = /d; the second-order

maximum (m = 2) is observed at a larger angle , and so

on.

The intensity distribution for a diffraction grating

obtained with the use of a monochromatic source is shown

in Figure 42. Note the sharpness of the principal maxima

and the broadness of the dark areas. This is in contrast to

the broad bright fringes characteristic of the two-slit

interference pattern (see section 3.2.3).

Diffraction gratings are most useful for measuring wavelengths accurately. Like prisms,

diffraction gratings can be used to disperse a spectrum into its wavelength components (see the

next section). Of the two devices, the grating is the more precise if one wants to distinguish two

closely spaced wavelengths.

Example: Light of wavelength 580 nm is incident on a slit having a width of 0.300 mm.

The viewing screen is 2.00 m from the slit. Find the positions of the first dark fringes and the

width of the central bright fringe. (Ans. 3.87 mm, 7.74 mm)


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3) Spectroscopy: Dispersion Spectroscope - Spectra

Spectroscopy

Spectroscopy is the study of the way in which atoms absorb and emit electromagneticradiation. Spectroscopy pertains to the dispersion of an object's light into its component colors

(or energies). By performing the analysis of an object's light, scientists can infer the physical

properties of that object (such as temperature, mass, luminosity, and chemical composition).

We first realize that light acts like a wave. Light has particle-like properties too (see section3.5).

The wave speed of a light wave is simply the speed of light, and different wavelengths of light

manifest themselves as different colors. The energy of a light wave is inversely-proportional to

its wavelength; in other words, low-energy light waves have long wavelengths, and high-

energy light waves have short wavelengths.

Electromagnetic spectrum

Physicists classify light waves by their energies (wavelengths). Labeled in increasing energy

or decreasing wavelength, we might draw the entire electromagnetic spectrum, as shown in

Figure 43.

Notice that radio, TV, and microwave signals are all light waves, they simply lie at

wavelengths (energies) that our eyes do not respond to. On the other end of the scale, beware the

high energy UV, x-ray, and gamma-ray photons. Each one carries a lot of energy compared to

their visible-and radio-wave counterparts.

Figure 43: The electromagnetic spectrum. Notice how small the visible region of thespectrum is, compared to the entire range of wavelengths.


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Figure 44: In a prism, material dispersion

(a wavelength-dependent refractive index)causes different colors to refract at

different angles, splitting white light into arainbow.

Dispersion

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its

frequency. Media having such a property are termed dispersive media.

The most familiar example of dispersion is probably a rainbow, in which dispersion causes the

spatial separation of a white light into

components of different colors (differentwavelengths), see Figure 44. Dispersion is

most often described for light waves, but it

may occur for any kind of wave that interacts

with a medium or passes through an

inhomogeneous geometry. In optics,

dispersion is sometimes calledchromatic

dispersion to emphasize its wavelength-

dependent nature.

The dispersion of light by glass prisms is

used to construct spectrometers. Diffraction

gratings are also used, as they allow more

accurate discrimination of wavelengths.

The most commonly seen consequence of

dispersion in optics is the separation of white light into a color spectrum by a prism. From

Snell's law, it can be seen that the angle of refraction of light in a prism depends on the

refractive index of the prism material. Since that refractive index varies with wavelength, it

follows that the angle that the light is refracted by will also vary with wavelength, causing an

angular separation of the colors known as angular dispersion.

A white light consists of a collection of component colors. These colors are often observed as

white light passes through a triangular prism. Upon passage through the prism, the white light isseparated into its component colors - red, orange, yellow, green, blue, and violet.


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Spectroscope

A spectroscope is a device used for splitting a beam of radiation (light) into its component

frequencies (or wavelengths) and delivering them to a screen or detector for detailed study (see

Figure 45). In other words, spectroscope is an optical system used to observe luminous spectra

of light sources.

In its most basic form, this device consists of an opaque barrier with a slit in it (to define a

beam of light), a prism or a diffraction grating (to split the beam into its component colors), and

an eyepiece or screen (to allow the user to view the resulting spectrum). Figure 44 shows such

an arrangement.

In many large instruments the prism is replaced by a device called a diffraction grating,consisting of a sheet of transparent material with many closely spaced parallel lines ruled on it.

The spaces between the lines act as many tiny openings, and light is diffracted as it passes

through these openings. Because different wavelengths of electromagnetic radiation are

diffracted by different amounts as they pass through a narrow gap, the effect of the grating is to

split a beam of light into its component colors.

Principle of operation of a spectroscope

We use the source of interest to light a narrow slit. A first collimating lens is placed on the

path of light to send a parallel beam on a prism or a grating. After the dispersion of light, a

second lens projects on a screen the image of the slit, resulting many color lines. Each linecorrespond to a wavelength. This series of lines constitutes the spectrum of the light source.

Examples are shown in Figure 46, including:

i. White light is broken up into a continuous spectrum, from red to blue (visible light).

ii. A incandescent gas gives bright lines of specific wavelengths; it is an emission

spectrum and the position of the lines are characteristic of this gas.

Figure 45:Diagram of a simple spectroscope. A small slit in the opaque barrier

on the left allows a narrow beam of light to pass. The light passes through a prismand is split up into its component colors. The resulting spectrum can be viewed

through an eyepiece or simply projected onto a screen.


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iii. The same cold gas is placed between the source of white light and the spectroscope. It

absorbs some of the radiations emitted by this source. Dark lines are observed at the same

positions as the bright lines of the previous spectrum. It is an absorption spectrum.

SPECTRA

In the domain of electromagnetic radiation, a spectrum is a series of radiant energies arranged

in order of wavelength or frequency. The entire range of frequencies is subdivided into wideintervals in which the waves have some common characteristic of generation or detection, such

as the radio-frequency spectrum, infrared spectrum, visible spectrum, ultraviolet spectrum, and

x-ray spectrum.

Spectra are also classified according to their origin or mechanism of excitation, as emission,

absorption, continuous, line, and band spectra. An emission spectrum is produced whenever the

radiation from an excited light source are dispersed. An absorption spectrum is produced against

a background of continuous radiation by interposing matter that reduces the intensity of

radiation at certain wavelengths or spectral regions. The energies removed from the continuous

spectrum by the interposed absorbing medium are precisely those that would be emitted by the

medium if properly excited. A continuous spectrum contains an unbroken sequence of

wavelengths or frequencies over a long range. Line spectra are discontinuous spectra

characteristic of excited atoms and ions, whereas band spectra are characteristic of molecular

gases or chemical compounds.

Within the visible spectrum, various light wavelengths are perceived as colors ranging from

red to blue, depending upon the wavelength of the wave. White light is a combination of all

visible colors mixed in equal proportions. This characteristic of light, which enables it to be

combined, so that the resultant light is equal to the sum of its constituent wavelengths, is called

additive color mixing.

The term spectrum (plural form, spectra) is applied to any class of similar entities or properties

strictly arrayed in order of increasing or decreasing magnitude. In general, a spectrum is adisplay or plot of intensity of radiation (particles, photons, or acoustic radiation) as a function of

mass, momentum, wavelength, frequency, or some other related quantity.


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3.4. Applications of interference and diffraction

1) Applications of interference

Interference can be used to measure the wavelength of a monochromatic light (see the example

in section 3.2.3).

A common example of the applications of interference involves the interference of radio wave

signals which occur at the antenna of a home when radio waves from a very distant transmitting

station take two different paths from the station to the home. This is relatively common forhomes located near mountain cliffs. In such an instance, waves which travel directly from the

transmitting station to the antenna interfere with other waves which reflect off the mountain

cliffs behind the home and travel back to the antenna, as shown in Figure 47.

Figure 46: Examples of continuous spectrum, line spectrum and absorptionspectrum.


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In this case, waves are taking two different paths from the source to the antenna - a direct

path and a reflected path. Clearly, each path is represented by a different distance traveled from

the source to the home, with the reflected pathway corresponding to the longer distance of the

two. If the home is located some distance d from the mountain cliffs, then the waves which take

the reflected path to the home will be traveling an extra distance given by the expression 2d. The

2 in this expression is due to the fact that the waves taking the reflected path must travel past the

antenna to the cliffs (a distance d) and then back to the antenna from the cliff (a second distance

d).

Thus, the path difference of 2d results in destructive interference whenever it is equal toa half number of wavelength.

Since radio stations transmit their signals at a specific and known frequencies, the

wavelengths of these light waves can be determined by relating them to the transmitted

frequencies and the light speed in vacuum (3 x 108

m/s).

Creating holography

Holography is a method (technique) of producing a three-dimensional image of an object by

recording on a photographic plate or film the pattern of interference formed by a split laser beam

and then illuminating the pattern either with a laser or with ordinary light.

The technique is widely used as a method for optical image formation and in addition has been

successfully used with acoustical (sound) and radio waves.

The technique is accomplished by recording the pattern of interference between the wave

emanating from the object of interest and a known reference wave, as shown in Figure 48a. In

general, the object wave is generated by illuminating the (possibly three-dimensional) subject of

interest with a highly coherent beam of light, such as one supplied by a laser source. The waves

reflected from the object strike a light-sensitive recording medium, such as photographic film or

plate.

Simultaneously a portion of the light is allowed to bypass the object and is sent directly

to the recording plate, typically by means of a mirror placed next to the object. Thus incident onthe recording medium is the sum of the light from the object and a mutually coherent reference

wave.

Figure 47: An example of radio wave interference.


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Figure 49: Depicting thephotoeffect.

The light diffraction thus imposes a limiting resolution of any optical instrument.

3.5. Duality of lights and particles

1) Photoelectric effect Einsteins photon concept

Photoelectric effect

Photoelectric effect is a process whereby light falling on a

surface knocks electrons out of the surface. The photoelectric

effect refers to the emission, or ejection, of electrons from the

surface of, generally, a metal in response to incident light, as

illustrated in Figure 49.

According to Figure 49, when shining a violet light on a

clean sodium (Na) metal in a vacuum, electrons were ejected

from the surface. It means the photoelectric effect occurred.

The remarkable aspects of the photoelectric effect are:

1. The electrons are emitted immediately. It means there is

no time lag.

2. Increasing the intensity of the light increases the number

of photoelectrons ejected, but not their maximum kinetic

energy.

3. No electron is emitted until the light has a threshold

frequency, no matter how intense the light is.

4. A weak violet light will eject only a few electrons, but their maximum kinetic energies are

greater than those for an intense light of longer wavelengths. It means that the maximum

kinetic energies of ejected electrons increase when the wavelength of the shining light is

shorter.

5. The maximum kinetic energies of the emitted electrons is independent of the intensity of

the incident radiation.

These observations baffled physicists for many decades, since they cannot be explained if light

is thought of only as a wave. If light were to be a wave, both the maximum kinetic energy and

the number of the electrons emitted from the metal should increase with an increase in the

intensity of light. Observations contradicted this prediction; only the number, and not the

maximum kinetic energy, of the electrons increases with the increase of the intensity of the

shining light.

Einsteins photon concept

Einstein (1905) successfully resolved this paradox by proposing that

*The incident light consists of individual quanta, called photons, that interact with the

electrons in the metal like discrete particles, rather than as continuous waves.


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* For a given frequency, or 'color,' of the incident light, each photon carries an energy

E = hf (70)

where h is Planck's constant (h = 6.626069 x 10-34

Joule seconds) and f the frequency of the

light.

*Increasing the intensity of the light corresponds, in Einstein's model, to increasing thenumber of incident photons per unit time (flux), while the energy of each photon remains the

same (as long as the frequency of the radiation was held constant).

Clearly, in Einstein's model, increasing the intensity of the incident radiation would cause

greater numbers of electrons to be ejected, but each electron would carry the same average

energy because each incident photon carries the same energy. This assumes that the dominant

process consists of individual photons being absorbed by electrons and resulting in the ejection

of a single electron for one photon absorbed. Likewise, in Einstein's model, increasing the

frequency f, rather than the intensity, of the incident light would increase the maximum kinetic

energy of the emitted electrons.

Both of these predictions were confirmed experimentally.

The photoelectric effect is perhaps the most direct and convincing evidence of the existence of

photons and the 'corpuscular' or particle nature of light and electromagnetic radiation. That is, it

provides undeniable evidence of the quantization of the electromagnetic field and the limitations

of the classical field equations of Maxwell.

Albert Einstein received the Nobel prize in physics in 1921 for explaining the photoelectric

effect and for his contributions to theoretical physics.

Energy contained within the incident light is absorbed by electrons within the metal, giving the

electrons sufficient energy to be knocked out of, that is, emitted from, the surface of the metal.

According to the classical Maxwell wave theory of light, the more intense the incident light is

the greater the energy with which the electrons should be ejected from the metal. That is, the

maximum kinetic energy of ejected (photoelectric) electrons should increase with the intensity

of the incident light. This is, however, not the case.

The minimum energy required to eject an electron from the surface of a metal is called the

photoelectric work function of the metal, often denoted as . Thus the condition for the

photoeffect to occur is

hf (71)

Let = hf0 (72)

The condition for the photoeffect to occur becomes

f f0 (73)

fo is called the threshold frequency of the metal.


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Using f = c/ and letting f0 = c/0, equation (73) becomes

0 (74)

0 is called the threshold wavelength of the metal. , f0, and 0 depend on the nature of

the metal of interest.

(71), (73), and (74) set the condition for the photoelectric effect to occur.

The maximum kinetic energy of the emitted electrons, EkinMax, is thus given by the energy of

the photon minus the photoelectric work function

EkinMax = hf - (75)

EkinMax thus depends on the frequency of light falling on the surface, but not on the

intensity of the shining light.

From equation (75) we see that the emitted electrons move with greater speed if the appliedlight has a higher frequency provided that (71) is satisfied.

Example: Lithium, beryllium, and mercury have work functions of 2.3 eV, 3.90 eV, and

4.50 eV, respectively. If 400-nm light is incident on each of these metals, determine (a) which

metal exhibits the photoelectric effect and (b) the maximum kinetic energy of the emitted

electrons in each case. (Ans. (b) 0.81 eV)

2) Electromagnetic waves and photons

Light as a wave

In the early days of physics (say, before the nineteenth century), very little was known aboutthe nature of light, and one of the great debates about light was over the question of whether

light is made of a bunch of "light particles," or whether light is a wave. Around 1800, a man

named Thomas Young apparently settled the question by performing an experiment in which he

shone light through very narrow slits and observed the result (see section 3.2.3). Here's the idea

behind it.

Suppose you have a whole bunch of ping-pong balls. You stand back about fifteen feet

from a doorway, and one by one you dip the balls in paint and throw them through a door, at a

wall about 5 feet behind the door. You will get a bunch of colored dots on the wall, scattered

throughout an area the same shape as the door you are throwing them through. This is how

particles (such as ping-pong balls)

behave.

On the other hand, waves do not

behave this way. Think of water

waves. When a wave encounters an

obstacle, it goes around it and

closes in behind it. When a wave

passes through an opening, it


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spreads out when it reaches the other side (diffraction, see section 3.3.1). And under the right

conditions, a wave passing through an opening can form interesting diffraction patterns on the

other side, which can be deduced mathematically.

Young shone momochromatic light through two very narrow slits, very close together. He then

observed the result on a screen. Now if light is made up of particles, then the particles should

pass straight through the slits and produce two light stripes on the screen, approximately the

same size as the slits. (Just like the ping-pong balls in the picture above.). On the other hand, iflight is a wave, then the two waves emerging from the two slits will interfere with each other

and produce a pattern of many stripes, not just two.

Young found the interference pattern with many stripes, indicating that light is a wave.

Later in the nineteenth century, James Clerk Maxwell determined that light is an electromagnetic

wave: a transverse wave of oscillating electric and magnetic fields. When Heinrich Hertz

experimentally confirmed Maxwell's result, the struggle to understand light was finished.

Light as particles

As mentioned earlier, when light is shone on a metal surface, electrons can be ejected from

that surface. This is called the photoelectric effect. Without going into detail, if one assumes that

light is a wave, as Young showed, then there are certain features of the photoelectric effect that

simply seem impossible. What Einstein showed is that if one assumes that light is made up of

particles (now called "photons"), the photoelectric effect can be explained successfully, as

discussed in the previous section.

3) Wave-particle duality - De Broglies postulate

Wave-Particle Duality

Is light a wave, or is light a flow of particles? Under certain conditions, such as when we shine

it through narrow slits and look at the result, it behaves as only a wave can. Under other

conditions, such as when we shine it on a metal and examine the electrons that comes off, light

behaves as only particles can. This multiple personality of light is referred to as wave-particle

duality.


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Light behaves as a wave, or as particles, depending on what we do with it, and what we try to

observe.

A wave-particle dual nature was soon found to be characteristic of electrons as well. The

evidence for the description of light as waves was well established before the time when the

photoelectric effect first introduced firm evidence of the particle nature of light. On the otherhand, the particle properties of electrons was well documented when the De Broglies postulate

and the subsequent experiment by Davisson and Germer established the wave nature of

electrons, as shown in Figure 50.

De Broglies postulate

In 1924 Louis de Broglie proposed the idea that all matter displays the wave-particle duality as

photons do. According to De Broglies postulate, for all matter and for electromagnetic radiation

alike, the energy E of the particle is related to the frequency f of its associated wave, by the

Planck relation.

E = hf (76)

and that the momentum p of the particle is related to its wavelength by what is known as the

De Broglie relation.

p =h

(77)

where h is Planck's constant.

The DavissonGermer experiment was a physics experiment conducted in 1927 which

confirmed De Broglies hypothesis, which says that particles of matter (such as electrons) havewave properties. This is a demonstration of wave-particle duality of electrons.

Description of the DavissonGermer experiment

The experiment consisted of firing an electron beam from an electron gun on a nickel

crystal at normal incidence (i.e., perpendicular to the surface of the crystal), as shown in Figure

50. The angular dependence of the reflected electron intensity was measured by an electron

Figure 50: Left: photoeffect showing particle nature of light;Right: Davisson-Germer experiment showing wave nature of electrons.


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detector and was determined to have the same diffraction pattern as those predicted by Bragg for

x-rays.

n = 2d sin (78)

where

n is an integer determined by the order given, is the wavelength of x-rays, moving electrons, protons, and neutrons,

dis the spacing between the planes in the atomic lattice, and

is the angle between the incident ray and the scattering planes.

Before the acceptance of De Broglies hypothesis, diffraction was a property that was

thought to be only exhibited by waves. Therefore, the presence of any diffraction effects by

matter demonstrated the wave-like nature of matter. When De Broglie wavelength was inserted

into the Bragg condition, the observed diffraction pattern was found as predicted, thereby

experimentally confirming De Broglies hypothesis for electrons.


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REFERENCES

1) Halliday, David; Resnick, Robert; Walker, Jearl. (1999) Fundamentals of Physics 7th ed.

John Wiley & Sons, Inc.

2) Feynman, Richard; Leighton, Robert; Sands, Matthew. (1989) Feynman Lectures on Physics.

Addison-Wesley Publishing Company.

3) Serway, Raymond; Faughn, Jerry. (2003) College Physics 7th ed. Thompson, Brooks/Cole.

4) Sears, Francis; Zemansky Mark; Young, Hugh. (1991) College Physics 7th ed. Addison-

Wesley Publishing Company.

5) Beiser, Arthur. (1992) Physics 5th ed. Addison-Wesley Publishing Company.

6) Jones, Edwin; Childers, Richard. (1992) Contemporary College Physics 7th ed. Addison-

Wesley Publishing Company.

7) Alonso, Marcelo; Finn, Edward. (1972) Physics 7th ed. Addison-Wesley PublishingCompany.

8) Michels, Walter; Correll, Malcom; Patterson, A. L. (1968) Foundations of Physics 7th ed.

Addison-Wesley Publishing Company.

9) WEBSITES

http://en.wikipedia.org/wiki/Bragg%27s_law

http://www.daviddarling.info/encyclopedia/D/diffraction.html

http://www.daviddarling.info/encyclopedia/O/optics_and_optical_phenomena.html

http://www.scienceclarified.com/He-In/Interference.html

http://www.phy.duke.edu/~rgb/Class/phy51/phy51/node51.html

http://www.spokaneschools.org/Science/Second/physics/gr7sound.htm

http://www.physicsforums.com/showthread.php?t=96788

http://library.thinkquest.org/10796/ch11/ch11.htm


http://phet.colorado.edu/simulations/sims.php?sim=Wave_Interference


http://van.physics.illinois.edu/qa/listing.php?id=1918

http://www.shsu.edu/~chm_tgc/sounds/flashfiles/pee.swf

http://hypertextbook.com/physics/

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

http://www.britannica.com/EBchecked/topic/555255/sound

http://www.sparknotes.com/testprep/books/sat2/physics/chapter17section2.rhtmlhttp://en.wikipedia.org/wiki/Spectrometer

http://spie.org/x32350.xml

http://www.factmonster.com/ce6/sci/A0846216.html

http://www.shsu.edu/~chm_tgc/sounds/flashfiles/pee.swf

http://hypertextbook.com/physics/

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

module 3 optics

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