Module 3 optics 2023 (1)

MINISTRY OF EDUCATION AND TRAINING
NONG LAM UNIVERSITY
FACULTY OF CHEMICAL ENGINEERING AND FOOD TECHNOLOGY

Course: Physics 1
Module 3:
Optics and wave phenomena

Instructor: Dr. Nguyen Thanh Son

Academic year: 2021-2022


Contents

Module 3: Optics and wave phenomena
3.1 Wave review
3.1.1 Description of a wave
3.1.2 Transverse waves and longitudinal waves
3.1.3 Mathematical description of a traveling (propagating) wave with constant amplitude
3.1.4 Electromagnetic waves
3.1.5 Spherical and plane waves
3.2 Interference of sound waves and light waves
3.2.1 Interference of sinusoidal waves – Coherent sources
3.2.2 Interference of sound waves
3.2.3 Interference of light waves
3.3 Diffraction and spectroscopy
3.3.1 Introduction to diffraction
3.3.2 Diffraction by a single narrow slit - Diffraction gratings
3.3.3 Spectroscopy: Dispersion – Spectroscope – Spectra
3.4 Applications of interference and diffraction

3.4.1 Applications of interference
3.4.2 Applications of diffraction
3.5 Wave-particle duality of matter
3.5.1 Photoelectric effect – Einstein’s photon concept
3.5.2 Electromagnetic waves and photons
3.5.3 Wave-particle duality – De Broglie’s postulate

Physic 1 Module 3: Optics and waves

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3.1 Wave review
3.1.1 Description of a propagating wave

Figure 23 Representation of a typical wave, showing its
direction of motion (direction of travel), wavelength, crests,
troughs and amplitude.
• 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.
• In wave motion, 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.
• 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. While 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 and waves

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3.1.2 Transverse waves and longitudinal waves
In terms of the direction of particles’ 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’ 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’ 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 in the air.
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.
3.1.3 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 = u Max
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 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 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.
• 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

Physic 1 Module 3: Optics and waves

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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 (one
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 pattern propagate through space. This is also 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 23. 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 wave number 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)

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Example: A sinusoidal wave traveling in the positive x-direction has an amplitude of
15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. Find the angular wave number k,
period T, angular frequency ω, and speed v of the wave.
Ans. k = 0.157 rad/cm; T = 0.125 s; ω = 50.3 rad/s; v = 3.2 m/s

3.1.4 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 are 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 electromagnetic waves, as shown in Figure 25. In vacuum, light
propagates with phase speed: v = c = 3 x 108 m/s.

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• 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.
3.1.5 Spherical and plane waves

• If a small spherical body, considered 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 to 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 radial 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

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.
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 – ωt)

(54)

Physic 1 Module 3: Optics and waves

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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.


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 has wave fronts
that are planes parallel to each other, rather than spheres of increasing radius (Figure 28).
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 effect

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 a
phenomenon in which 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 opposite 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
Physic 1 Module 3: Optics and waves

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Figure 30 Depicting the snapshots of the medium for two
pulses of the same amplitude (both downward) before and
during interference; the interference is constructive.


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
displaced upward. But it is also observed when both interfering 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 directions. 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 diagrams 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
Figure 31 Depicting the snapshots of the medium for
the instant of complete overlap,
two pulses of the same amplitude (one upward and one
there is no resulting
downward) before and during interference;
displacement of the particles of
the interference is destructive.
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 larger-amplitude
wave.
The two interfering waves do
not need to have equal amplitudes in
Figure 32 Depicting the before and during
opposite directions for destructive
interference snapshots of the medium for two pulses of
interference to occur. For example, a
different amplitudes (one upward, +1 unit and one

pulse with a maximum displacement of downward, -2 unit); the interference is destructive.
+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 meet, the resulting displacement of the medium at any location is the
algebraic sum of the displacements of the individual waves at that location.
Physic 1 Module 3: Optics and waves

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• In the cases mentioned above, the summing of the individual displacements for locations of
complete overlap was easy and given in the below table.
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

3.2.1 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 the larger amplitude of the individual waves. Light waves also interfere with each other.
Fundamentally, 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 of waves have the same frequency.
• The sources of waves must maintain a constant phase with respect to each other.
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 of the same frequency (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 sources of
waves - 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 waves emerging from the two slits are
coherent because a single source produces the original light beam and the two slits only serve 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 point 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)
Physic 1 Module 3: Optics and waves

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(55)


where x1 = O1P and x2 = O2P are the wave paths (distances traveled) from O1 and O2 to P,
respectively.
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 AP 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,
∆x, of the two wave components:
∆x = x 2 − x1

(59)

From Equations 53, 58 and 59, we can easily see the following cases:
Case 1:

∆x = x2 − x1 = n2π/k = nλ

where n = 0, ±1, ±2, … or the path difference is zero or some integer multiple of
wavelengths.
We have AP = 2A. The amplitude of the resultant wave is 2A - twice the amplitude of
either individual wave. In this case, the interfering (component) waves are said to be everywhere
in phase and thus interfere constructively. There is a constructive interference at P.
Case 2:

∆x = x2 − x1 = (2n + 1)π/k = (2n + 1)λ/2 = (n + 1/2)λ

where n = 0, ±1, ±2, … or the path difference is odd multiple of half wavelengths.
We have AP = 0. The resultant wave has zero amplitude. In this case, the interfering
(component) waves are exactly 180o out of phase and thus interfere destructively. There is a
destructive interference at P.
3.2.2 Interference of sound waves

• One simple device for demonstrating interference of sound waves is illustrated by Figure 33.

Sound from a loudspeaker S is sent into a tube at point P, where there is a T-shaped junction.
• 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.
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• When the path difference is either zero or some integer multiple of the wavelengths λ (that is
r2 – r1 = nλ, where n = 0, ±1, ±2, . . .), the two waves reaching the receiver at any instant are in
phase and reinforce each other. For this case, a maximum in the sound intensity is detected at
the receiver. We have constructive sound wave interference 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. For
this case, a minimum in the
sound intensity is detected at
the receiver. We have
destructive sound wave
interference at the receiver.

3.2.3 Interference of light
waves

♦ Two-point source light
interference patterns

Figure 33 An acoustical system for demonstrating
interference of sound waves. A sound wave from the
speaker S propagates into 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.

• 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 interferes 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
lines and nodal lines. For light waves, the antinodal 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
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• 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
Figure 34 Schematic diagram of Young’s double-slit experiment.
the light onto two very
Two slits behave as coherent sources of light waves that produce an
narrow slits, as shown
interference pattern on the viewing screen (drawing not to scale).
in Figure 34. The light
from the source will then diffract through the slits, and the interference 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
sources. 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.
Figure 35 A typical pattern from a two-slit
When the light from S 1 and that from S2
experiment of interference.
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.
• We can describe Young’s experiment quantitatively with the help of Figure 36. The viewing
screen is located at a perpendicular distance L from the double-slit 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 dsin θ. This
Physic 1 Module 3: Optics and waves

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distance is called the path difference δ (lowercase Greek delta). Note θ is the angle between the
ray to point P and the normal line between the slit and the screen.
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

δ = S2P – S1P = r2 – r1 = dsin 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
wavelengths,
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
Figure 36 Geometric construction for describing Young’s doubleslit experiment (drawing not to scale).

δ = r2 – r1 = nλ
where n = 0, ±1, ±2,

(61)

.

• The absolute value of n or |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 and waves

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(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 meter, 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 = Ltan θ ≈ Lsin 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
λL
d
where n = 0, ± 1, ±2, ±3, ... and |n| is the order number.
ybright = n

(64)

• Similarly, using equations 60, 62 and 63, we find that the dark fringes are located at
ydark = (n + 1/2)

λL
d

• 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
Figure 37 Light intensity versus δ = dsin θ for a
double-slit source by 1.2 m.
double-slit interference pattern when the viewing
The distance between the two
screen is far from the slits (L >> d).
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)
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(65)


(a) The second-order bright fringe

Solution
Use (64) with n = 2, giving λ = 560 nm.

(b) Using (64), the distance between two successive bright fringes = yn-1 - yn-1 =

λL
;
d

Plugging numbers gives the value of 2.25 cm.

♦ 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.
• 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 = Imaxcos2 (

πd
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 π y/λL is an integral
multiple of π, corresponding to y
= n(λL/d). This is consistent
with Equation 64.
• A plot of light intensity versus
δ = dsinθ 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 L is much

Figure 38 Diffraction of a light wave: (a) If radiation
were composed of rays or particles moving in perfectly
straight lines, no bending would occur as a beam of
light passed through a circular hole in a barrier, and
the outline of the hole, projected onto a screen, would
Physic 1 Module 3: Optics and waves
16
have perfectly sharp edges. (b) In fact, light is diffracted
through an angle that depends on the ratio of the
wavelength of the wave to the size of the gap. The result

is that the outline of the hole becomes "fuzzy," as shown
in this actual photograph of the diffraction pattern.


greater than the slit separation d (L >> d), and only for small values of θ.
3.3 Diffraction and spectroscopy
3.3.1 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 only when 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 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 of uniform intensity, thus serving as a point source. The light at a given angle is a
combination of the 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.
• 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
Each 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 wave front as a
point source for a secondary radial wave. The subsequent propagation and addition of all these
radial waves form the new wave front, 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
Physic 1 Module 3: Optics and waves

17


are various analytical models which can be used to do this, including the Fraunhofer diffraction
equation for the far field and the Fresnel diffraction equation for the near field.

• As a result, ddiffraction effects are classified into either Fresnel or Fraunhofer type. 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).
3.3.2 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 wave front in the slit and treating
them like point sources.

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.
Physic 1 Module 3: Optics and waves

18


• 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
appearing 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 between two narrow slits a
distance apart that is half the width of the
Figure 40 Intensity distribution for a
slit (see Section 3.2). The path difference is
Fraunhofer diffraction pattern from a
given by (asinθ)/2 so that the first minimum
single slit of width a. The positions of two

intensity occurs at an angle θmin given by
minima on each side of the central
asin θmin = λ

(67)

maximum are labeled (drawing not to
scale).

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 θmin/n given by
asin θmin/n = nλ
where n = ± 1, ±2, ±3, ... and |n| is the order number.

Physic 1 Module 3: Optics and waves

19

(68)


• The intensity distribution for a Fraunhofer diffraction pattern from a single slit of width a is
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.
• The width of the central maximum of the diffraction pattern observed on the screen, denoted
by ∆, is given by
∆ = 2y1

(68’)

where y1 is the distance between the central line and the first minimum (see Figure 40).
Example: A single slit 0.10 mm wide which is 5.0 m from the screen is illuminated by light of
wavelength 580 nm. Find
(a) The position y1 of the first minimum;
(b) the width of the central maximum of the diffraction pattern observed on the screen.
Solution
(a) Because L = 5 m >> a = 0.10 mm 0.10 mm, θ is small; thus, we can use the
n =1
y1
approximation sin θ ≈ tan θ, then yn = Ltanθmin/n = Lsinθmin/n. The first minimum
= Lsinθmin/1 (*). Using (68) gives sinθmin/1 = λ/a (**). Combining (*) and (**) leads to y1 = Lλ/a.
Plugging numbers gives y1 = 1.45 cm.
(b) Using (68’) gives ∆ = 2y1 = 2.9 cm.

♦ 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 grooves per centimeter, that
produce interference patterns in a way that separates all the color 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 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
Physic 1 Module 3: Optics and waves
20

Figure 41 Diffraction grating is an optical
device used to disperse light into a spectrum.


interference and diffraction. Each slit produces diffraction, and the diffracted beams interfere
with one another to produce the final
pattern.

• 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).
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
dsin θmax/m = 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

(grating spacing). 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.

Figure 42 Intensity versus
sin θ for a diffraction grating.
The zeroth-, first-, and
second-order maxima are
shown.

• 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 mth-order 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 θmax/1 = 1λ/d; the second-order maximum (m = 2) is observed at a larger angle θ, and so on
(Figure 42).
• 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).
• 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: The wavelengths of the hydrogen alpha line and the hydrogen beta line are 653.4 nm
and 580.8 nm, respectively; using a grating with 2.00 x 10 5 lines (grooves/slits) per meter, what
is the angular separation for these two spectral lines in the first order?
Physic 1 Module 3: Optics and waves

21


Solution
In this problem, d = 1 divided by the number of lines per meter = 1/(2.00 x 105/m) = 5 x 10-6 m.
For λ1 = 653.4 x 10 -9 m and m = 1, using (69)
sin θ1 = (1)(653.4 x 10-9 m)/(5 x 10-6 m) =
o
θ1 = 7.51 .
0.131
θ2 =
For λ2 = 580.8 x 10 -9 m and m = 1
sin θ2 = (1)(580.8 x 10 -9 m)/(5 x 10-6 m) = 0.116
o
6.67 .
As a result, in the first order, the angular separation for the α and β lines is (7.51 - 6.67)o = 0.84 o.

3.3.3 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.
• The 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 or wavelengths. Labeling 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.

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

♦ 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.
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23


• A white light consists of a collection of component colors. These colors are often observed as
white light passes through a triangular prism. Upon passing through the prism, the white light is

separated into its component colors - red, orange, yellow, green, blue, and violet.

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 onto 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 45 shows such
an arrangement.
• In many large instruments, the prism is replaced by 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 collimating lens is placed on the path of
light to send a parallel beam on a prism or a diffraction grating. After the dispersion of the light,
a second lens projects on a screen the image of the slit, resulting many color lines. Each line
corresponds to a wavelength. This series of lines constitutes the spectrum of the light source.
Examples are shown in Figure 46, including:
Physic 1 Module 3: Optics and waves

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i. White light is broken up into a continuous spectrum, from red to blue (visible light).
ii. An incandescent gas gives bright lines of specific wavelengths; it is an emission
spectrum and the positions of the lines are characteristic of the gas.
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
• 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.
• 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
is dispersed.
- 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.
- 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.
• 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.

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