Download Course: Physics 1 Module 3: Optics and Wave Phenomena

MINISTRY OF EDUCATION AND TRAINING
NONG LAM UNIVERSITY
FACULTY OF FOOD SCIENCE AND TECHNOLOGY
Course: Physics 1
Module 3: Optics and Wave
Phenomena
Instructor: Dr. Son Thanh Nguyen
Academic year: 2008-2009
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 amplitude
4) 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 interference
2) Applications of diffraction
3.5. Wave-particle duality of light and particles
1) Photoelectric effect – Einstein’s photon concept
2) Electromagnetic waves and photons
3) Wave-particle duality – De Broglie’s postulate
Physic 1 Module 3: Optics
2
3.1. Wave review
1) Description of a propagating wave
Figure 23: Representation of a typical wave, showing its
direction of motion, wavelength, and amplitude.
• Simply stated, a wave is a way in which energy is transferred from place to place without
physical 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 a
distinctive 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. The
particles 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 medium’s 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.
Physic 1 Module 3: Optics
3
2) Transverse and longitudinal waves
In terms of the direction of particles’s vibrations and that of the wave propagation, there
are two major kinds of waves: transverse waves and longitudinal waves.
• Transverse waves are those with
particles’s vibrations perpendicular to the
wave's direction of travel; examples include
waves on a stretched string and
electromagnetic waves.
• Longitudinal waves are those with
particles’s vibrations along the wave's
Figure 24: When an object bobs up and down on a
direction of travel; examples include sound ripple in a pond, it experiences an elliptical
waves.
trajectory because ripples are not simple
transverse sinusoidal 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 medium’s particles perpendicular to the direction of wave propagation or energy transfer. If
a 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 − ωt)
(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, ω the angular frequency, and t the time.
• In the illustration given by Figure 23, the amplitude is the maximum vertical distance between
the 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.
Physic 1 Module 3: Optics
4
• The period T is the time for one complete cycle for an oscillation. The frequency f (also
frequently denoted as ν) 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 ω represents the frequency in terms of radians per second. It is related
to the frequency f by
ω = 2πf
(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= ω
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 =
∂ω
∂k
(51)
• The wavelength (denoted as λ) is the distance between two successive crests (or troughs) of a
wave, 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
λ = 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=
2π
λ
(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, λ = 1.25 m, v = 2.5 m/s)
5
Physic 1 Module 3: Optics
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.
Figure 25: Electric and magnetic fields
vibrate perpendicular to each other.
Together they form an electromagnetic wave
that moves through space at the speed of
light c.
• Electromagnetic (EM) waves including
light behave in the same way as other
waves, although it is harder to see.
Electromagnetic waves are also twodimensional transverse waves. This twodimensional nature should not be confused
with the two components of an
electromagnetic 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.
Figure 26: Spherical waves emitted by a
point source. The circular arcs represent the
spherical wave fronts that are concentric with
the source. The rays are radial lines pointing
outward from the source, perpendicular to
the wave fronts.
• 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 rapidly
fluctuating 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.
Physic 1 Module 3: Optics
6
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 medium’s
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 λ. 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
Figure 27: Far away from a point source, the
wave fronts are nearly parallel planes, and the
rays are nearly parallel lines perpendicular to
these planes. Hence, a small segment of a
spherical wave is approximately a plane 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 - ωt)
(54)
That is, the wave function for a plane wave is
identical 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
Physic 1 Module 3: Optics
7
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.
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 questions
involving 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.
Figure 29: Depicting the snapshots of the medium for
two pulses of the same amplitude (both upward) before
and during interference; the interference is constructive.
• 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 29
depict 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 upward
displacement which is greater than
the displacement of either
interfering pulse. Constructive
interference is observed at any location
where the two interfering waves are
Figure 30: Depicting the snapshots of the medium for two
displaced upward. But it is also
pulses of the same amplitude (both downward) before and
observed when both interfering
during interference; the interference is constructive.
Physic 1 Module 3: Optics
8
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 medium
where 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,
Figure 31: Depicting the snapshots of the medium for
there is no resulting displacement of
two pulses of the same amplitude (one upward and one
the particles of the medium. When two
downward) before and during interference;
pulses with opposite displacements
the interference is destructive.
(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.
Figure 32: Depicting the before and during
interference snapshots of the medium for two pulses of
different amplitudes (one upward, +1 unit and one
downward, -2 unit); the interference is destructive.
• 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.
Physic 1 Module 3: Optics
9
Maximum displacement of
Pulse 1
+1
-1
+1
+1
Maximum displacement of
Pulse 2
+1
-1
-1
-2
Maximum resulting
displacement
+2
-2
0
-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 a
single 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. The
waves meet at P, and according to the principle of superposition, the resultant vibration at P is
given by
uP = u1 + u2 = Asin(kx1 − ωt) + Asin(kx2 − ωt)
(55)
where x1 = O1P and x2 = O2P are the wave paths (distances traveled) from O1 and O2 to
P, respectively.
Physic 1 Module 3: Optics
10
For the sake of simplicity, we have assumed A1 = A2 = A.
• Using the trigonometric identity: sinα + sinβ = 2sin{(α+β)/2}cos{(α-β)/2} (56), from
equation (55) we have
uP = 2Acos{k(x2 − x1)/2}sin{k(x1 + x2)/2 - ωt}
(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 the path difference,
Δx, of the two wave components
Δx = x2 − x1
(59)
From equations (53), (58) and (59), we can easily see the following cases:
Case 1: Δx = x2 − x1 = n2π/k = nλ 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: Δx = x2 − x1 = (n +
0,5)π/k = (2n + 1)λ/2 where n = 0, ±1,
±2, … or the 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.
Physic 1 Module 3: Optics
Figure 33: An acoustical system for demonstrating
interference of sound waves. A sound wave from the
speaker (S) propagates in the tube and splits into
two parts at point P. The two waves, which
superimpose at the opposite side, are detected at
the receiver (R). The upper path length r2 can be
varied by sliding the upper section.
11
• 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 r1 is fixed, but the upper path length r2 can be varied by sliding a U-shaped tube, which is
similar to that on a slide trombone.
• When the path difference is either zero or some integer multiple of the wavelength λ (that is
r2 – r1 = nλ, where n = 0, ±1, ±2, . . .), the two waves reaching the receiver at any instant are in
phase 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)λ, where n = 0, ±1, ±2, . . ., the two
waves are exactly π 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 light
wave. 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 source
interference pattern
would still consist
of an alternating
pattern of antinodal
Figure 34: Schematic diagram of Young’s double-slit experiment.
lines and nodal
Two slits behave as coherent sources of light waves that produce an
lines. For light
interference pattern on the viewing screen (drawing not to scale).
waves, the antinodal
Physic 1 Module 3: Optics
12
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.
♦ YOUNG’S 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 light
Figure 35: A typical pattern from a two-slit
sources. The light waves from S1 and S2
experiment of interference.
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.
Physic 1 Module 3: Optics
13
• We can describe Young’s experiment quantitatively with the help of Figure 36. The viewing
screen is located a
perpendicular distance
L from the doubleslitted barrier. S1 and S2
are separated by a
distance d, and the
source is
monochromatic. 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 θ. This distance is
called the path
difference δ (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 δ is given by
Figure 36: Geometric construction for describing Young’s doubleslit experiment (not to scale).
δ = S2P – S1P = r2 – r1 = d sin θ
(60)
where d = S1S2 is the distances between the two coherent light sources (i.e., the two slits).
If δ is either zero or some integer multiple of the wavelength, then the
two waves are in phase at point P and constructive interference results. Therefore,
the condition for bright fringes, or constructive interference, at point P is
δ = r2 – r1 = nλ
(61)
where n = 0, ±1, ±2, ….
• The number n in equation (61) is called the order number. The central bright fringe at θ = 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 δ is an odd multiple of λ/2, the two waves arriving at point P are 180° out of phase and
give rise to destructive interference. Therefore, the condition for dark fringes, or destructive
interference, at point P is
δ = r2 – r1 = (n + 1/2)λ
where n = 0, ±1, ±2, ....
Physic 1 Module 3: Optics
14
(62)
• 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 >> λ. These
can be valid assumptions because in practice L is often of the order of 1 m, d a fraction of a
millimeter, and λ a fraction of a micrometer for visible light. Under these conditions, θ is small;
thus, we can use the approximation sin θ ≈ tan θ. Then, from triangle OPQ in Figure 36, we see
that
y = OP = L tan θ ≈ L sin θ
(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 = n
λL
d
(64)
• Similarly, using equations (60), (62) and (63), we find that the dark fringes are located at
ydark = (n + 1/2)
λL
d
(65)
• As we demonstrate in the following example, Young’s 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. λ = 560 nm)
(b) Calculate the distance between two successive bright fringes. (Ans. 2.25 cm)
Physic 1 Module 3: Optics
15
♦ Intensity
distribution of the
double-slit
interference pattern
• So far we have
discussed the
locations of only the
centers 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, we
now calculate the
distribution of light
intensity associated
with the double-slit
interference pattern.
Figure 37: Light intensity versus δ = d sin θ for a doubleslit interference pattern when the viewing screen is far from
the slits (L >> d).
• 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 electric
field magnitude at the point of interest, we can show that (see pages 1191 and 1192, Halliday’s
book).
I = Imaxcos (
πd 2
y)
λL
(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
πy/λL is an integral multiple of π, corresponding to y = (λL/d)n. This is consistent with equation
(64).
Physic 1 Module 3: Optics
16
• A plot of light intensity versus
δ = d sinθ 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-toscreen distance is much greater
than the slit separation (L >> d),
and only for small values of θ.
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
Figure 38: Diffraction of a light wave: (a) If radiation
gap. The longer the wavelength
were composed of rays or particles moving in perfectly
and/or the smaller the gap, the
straight lines, no bending would occur as a beam of
greater the angle through which
light passed through a circular hole in a barrier, and
the wave is diffracted. Thus,
the outline of the hole, projected onto a screen, would
visible light, with its extremely
have perfectly sharp edges. (b) In fact, light is diffracted
short wavelengths, shows
through an angle that depends on the ratio of the
perceptible diffraction only
wavelength of the wave to the size of the gap. The result
when passing through very
is that the outline of the hole becomes "fuzzy," as shown
narrow openings. (The effect is
in this actual photograph of the diffraction pattern.
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 distinguishes between 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 2π, we expect to find minima
and maxima in the diffracted light.
Physic 1 Module 3: Optics
17
• The effects of diffraction can be readily seen in everyday life. The most colorful examples of
diffraction 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 the
Huygens–Fresnel 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. Fresnel
diffraction is concerned mainly with what happens to light in the immediate neighborhood of a
diffracting 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-like
source, 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.
Physic 1 Module 3: Optics
18
Figure 39: (a) Fraunhofer diffraction pattern of a single
slit. The pattern consists of a central bright fringe
flanked by much weaker maxima alternating with dark
fringes (drawing not to scale). (b) Photograph of a
single-slit Fraunhofer diffraction pattern.
• 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,
Halliday’s book, page 1213). Figure 38.3 shows a diffraction pattern associated with the shadow
of 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 θ = 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 λ/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
19
Physic 1 Module 3: Optics
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 (a sinθ)/2 so that the minimum intensity occurs at an angle θmin given
by
a sin θmin = λ
(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 θn
given by
a sin θn = nλ
(68)
where n is an integer greater than zero.
Figure 40: Intensity distribution for a
Fraunhofer diffraction pattern from a
single slit of width a. The positions of two
minima on each side of the central
maximum are labeled (drawing not to
scale).
• The intensity distribution for a Fraunhofer
diffraction pattern from a single slit of width a
shown 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.
Figure 41: Diffraction grating is an optical
device used to disperse light into a spectrum.
• 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 a
glass 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.
Physic 1 Module 3: Optics
20
• The waves from all slits are in phase as they leave the slits. However, for some arbitrary
direction θ 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 therefore
described by the equation
d sin θ = mλ
(69)
where m = 0, ± 1, ±2, ±3 and |m| is the order number; λ is a selected wavelength; d is the spacing
of the grooves; and θ 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 θ. If the
incident radiation contains several wavelengths, the mthorder maximum for each wavelength occurs at a specific
angle. All wavelengths are seen at θ = 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 θ = λ/d; the second-order
maximum (m = 2) is observed at a larger angle θ, 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).
Figure 42: Intensity versus
sinθ for a diffraction grating.
The zeroth-, first-, and
second-order maxima are
shown.
• 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)
Physic 1 Module 3: Optics
21
3) Spectroscopy: Dispersion – Spectroscope - Spectra
♦ Spectroscopy
• Spectroscopy is the study of the way in which atoms absorb and emit electromagnetic
radiation. 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 section
3.5).
• 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 highenergy 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 the
spectrum is, compared to the entire range of wavelengths.
Physic 1 Module 3: Optics
22
♦ 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 (different
wavelengths), 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 called chromatic
dispersion to emphasize its wavelengthdependent 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.
Figure 44: In a prism, material dispersion
(a wavelength-dependent refractive index)
causes different colors to refract at
different angles, splitting white light into a
rainbow.
• 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 is
separated into its component colors - red, orange, yellow, green, blue, and violet.
Physic 1 Module 3: Optics
23
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 prism
and is split up into its component colors. The resulting spectrum can be viewed
through an eyepiece or simply projected onto a screen.
♦ 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 line
correspond 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.
Physic 1 Module 3: Optics
24
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 wide
intervals 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 a
display or plot of intensity of radiation (particles, photons, or acoustic radiation) as a function of
mass, momentum, wavelength, frequency, or some other related quantity.
Physic 1 Module 3: Optics
25
Figure 46: Examples of continuous spectrum, line spectrum and absorption
spectrum.
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 for
homes 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.
Physic 1 Module 3: Optics
26
Figure 47: An example of radio wave interference.
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 to
a 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 on
the recording medium is the sum of the light from the object and a mutually coherent reference
wave.
Physic 1 Module 3: Optics
27
Figure 48b: Obtaining images from a
hologram.
Figure 48a: Recording a hologram.
.
The photographic recording obtained is known as a hologram (meaning a “total
recording”); this record generally bears no resemblance to the original object, but rather is a
collection of many fine fringes which appear in rather irregular patterns. Nonetheless, when this
photographic transparency is illuminated by coherent light, one of the transmitted wave
components is an exact duplication of the original object wave, as shown in Figure 48b. This
wave component therefore appears to originate from the object (although the object has long
since been removed) and accordingly generates a virtual image of it, which appears to an
observer to exist in three-dimensional space behind the transparency. The image is truly threedimensional in the sense that the observer's eyes must refocus to examine foreground and
background, and indeed can “look behind” objects in the foreground simply by moving his or
her head laterally.
2) Applications of diffraction
♦ Diffraction gratings (see section 3.3.3)
♦ Limiting of resolution of an optical instrument
• The ability of optical instrument such as a microscope to distinguish between closely spaced
objects is limited because of the wave nature of light.
• Consider light waves from different objects far from a narrow slit, and these objects can be
considered as two noncoherent point sources S1 and S2. If no diffraction occurred, two distinct
bright spots (or images) would be observed on the viewing screen. However, because of
diffraction, each source is imaged as a bright central region flanked by weaker bright and dark
fringes. What is observed on the screen is the sum of two diffraction patterns: one from S1 and
the other from S2.
• If the two sources are far enough apart to keep their central maxima from overlapping, their
images can be distinguished and are said to be resolved; as a result the observer can see S1 and
S2 distinguishably.
• If the sources are close together, however, the two central maxima overlap, and the images are
not resolved; as a result the observer can not see S1 and S2 distinguishably.
Physic 1 Module 3: Optics
28
• The light diffraction thus imposes a limiting resolution of any optical instrument.
3.5. Duality of lights and particles
1) Photoelectric effect – Einstein’s 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.
Figure 49: Depicting the
photoeffect.
• 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.
♦ Einstein’s 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.
Physic 1 Module 3: Optics
29
* 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 the
number 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 φ. Thus the condition for the
photoeffect to occur is
Let
hf ≥ φ
(71)
φ = hf0
(72)
The condition for the photoeffect to occur becomes
f ≥ f0
(73)
fo is called the threshold frequency of the metal.
Physic 1 Module 3: Optics
30
Using f = c/λ and letting f0 = c/λ0, equation (73) becomes
λ ≤ λ0
(74)
λ0 is called the threshold wavelength of the metal. φ, f0, and λ0 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 - φ
(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 applied
light 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 about
the 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
Physic 1 Module 3: Optics
31
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, if
light 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 Broglie’s 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.
Physic 1 Module 3: Optics
32
Figure 50: Left: photoeffect showing particle nature of light;
Right: Davisson-Germer experiment showing wave nature of electrons.
• 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 other
hand, the particle properties of electrons was well documented when the De Broglie’s postulate
and the subsequent experiment by Davisson and Germer established the wave nature of
electrons, as shown in Figure 50.
♦ De Broglie’s postulate
• In 1924 Louis de Broglie proposed the idea that all matter displays the wave-particle duality as
photons do. According to De Broglie’s 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 λ by what is known as the
De Broglie relation.
p=
h
λ
(77)
where h is Planck's constant.
• The Davisson–Germer experiment was a physics experiment conducted in 1927 which
confirmed De Broglie’s hypothesis, which says that particles of matter (such as electrons) have
wave properties. This is a demonstration of wave-particle duality of electrons.
• Description of the Davisson–Germer 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
Physic 1 Module 3: Optics
33
detector and was determined to have the same diffraction pattern as those predicted by Bragg for
x-rays.
nλ = 2d sin θ
(78)
where
•
•
•
•
n is an integer determined by the order given,
λ is the wavelength of x-rays, moving electrons, protons, and neutrons,
d is the spacing between the planes in the atomic lattice, and
θ is the angle between the incident ray and the scattering planes.
Before the acceptance of De Broglie’s 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 Broglie’s hypothesis for electrons.
Physic 1 Module 3: Optics
34
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. AddisonWesley Publishing Company.
5) Beiser, Arthur. (1992) Physics 5th ed. Addison-Wesley Publishing Company.
6) Jones, Edwin; Childers, Richard. (1992) Contemporary College Physics 7th ed. AddisonWesley Publishing Company.
7) Alonso, Marcelo; Finn, Edward. (1972) Physics 7th ed. Addison-Wesley Publishing
Company.
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://library.thinkquest.org/10796/ch9/ch9.htm
http://phet.colorado.edu/simulations/sims.php?sim=Wave_Interference
http://library.thinkquest.org/10796/ch9/ch9.htm
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.rhtml
http://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
Physic 1 Module 3: Optics
35