Module 3 optics 2023 (1)

Physic 1 Module 3: Optics and waves 2Contents 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 Math

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

Physic 1 Module 3: Optics and waves 2

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 3

3.1 Wave review

3.1.1 Description of a propagating wave

• 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

Figure 23 Representation of a typical wave, showing its

direction of motion (direction of travel), wavelength, crests,

troughs and amplitude

Physic 1 Module 3: Optics and waves 4

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

direction of travel; examples include sound

waves in the air

• 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 = uMax

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

Figure 24 When an object bobs up and down on a

ripple in a pond, it experiences an elliptical trajectory because ripples are not simple transverse sinusoidal waves

Physic 1 Module 3: Optics and waves 5

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

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)

Physic 1 Module 3: Optics and waves 6

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

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

• Electromagnetic (EM) waves including

light behave in the same way as other

waves, although it is harder to see

Electromagnetic waves are also

dimensional transverse waves This

two-dimensional 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

• 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

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.

Physic 1 Module 3: Optics and waves 7

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

Figure 28 A representation of a plane wave

• 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

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 8

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

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

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

• 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

• 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 9

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

Figure 31 Depicting the snapshots of the medium for

two pulses of the same amplitude (one upward and one downward) before and during interference;

the interference is destructive

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

the instant of complete overlap,

there is no resulting

displacement of the particles of

the medium When two pulses

with opposite displacements (i.e., one pulse displaced up and the other down) meet at a given location, the upward pull of one pulse is balanced (canceled or destroyed) by the downward pull

of the other pulse Destructive

interference leads to only a momentary

condition in which the medium's

displacement is less than the

displacement of the larger-amplitude

wave

The two interfering waves do

not need to have equal amplitudes in

opposite directions for destructive

interference to occur For example, a

pulse with a maximum displacement of

+1 unit could meet a pulse with a

maximum displacement of -2 units The resulting displacement of the medium during complete overlap is -1 unit, as shown in Figure 32

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

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

When two waves 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 10

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

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) (55)

Physic 1 Module 3: Optics and waves 11

where x1 = O1P and x2 = O2P are the wave paths (distances traveled) from O 1 and O 2 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 A P of the resultant vibration (resultant

amplitude) at the point P is given by

AP = |2Acos {k(x2 − x1)/2}| (58)

• According to Equation 58, A P is time independent and depends only on the path difference,

∆x, of the two wave components:

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 = x 2 − x 1 = (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 180 o 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

Physic 1 Module 3: Optics and waves 12

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 r 2 can be varied by sliding the upper section

• When the path difference is either zero or some integer multiple of the wavelengths λ (that is

r 2 – r 1 = 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 r 2 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

• 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

Physic 1 Module 3: Optics and waves 13

Figure 35 A typical pattern from a two-slit

experiment of interference

Figure 34 Schematic diagram of Young’s double-slit experiment

Two slits behave as coherent sources of light waves that produce an interference pattern on the viewing screen (drawing not to scale)

• 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

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 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

When the light from S 1 and that from S 2

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 14

Figure 36 Geometric construction for describing Young’s

double-slit experiment (drawing not to scale)

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)

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 (62)where n = 0, ±1, ±2,

Physic 1 Module 3: Optics and waves 15

Figure 37 Light intensity versus δ = dsin θ for a double-slit interference pattern when the viewing screen is far from the slits (L >> d)

• 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

where n = 0, ± 1, ±2, ±3, and |n| is the order number

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

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

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 and waves 16

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 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

Solution

(a) The second-order bright fringe Use (64) with n = 2, giving λ = 560 nm

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

d

λ

;

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 d

( y)L

πλ

(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

= 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-to-screen distance L is much

Physic 1 Module 3: Optics and waves 17

greater than the slit separation d (L >> d), and only for small values of θ

3.3 Diffraction and spectroscopy

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