Chapter 10

The Interference of Light

10.1 INTRODUCTION

Introduction

Interference and diffraction are important properties of waves that occur under certain conditions

They have no equivalent in classical particle motion

Thomas Young first established that light was a wave after demonstrating the two slit interference pattern

Introduction

When discussing waves in general, we refer to water waves since they are easy to visualise, and use the same terminology

Introduction

The use of crest and trough refers to the points where the electric and magnetic fields are at maximum in either direction

10.2DIFFRACTION

Diffraction in general

Diffraction is the change in direction of propagation of a wave as it passes by an obstacle while remaining in the same medium

This is shown below where the lines represent crests of the wave

Diffraction in general

This is observable in ocean waves, and is the reason sound can travel around the corner of a building

Diffraction through a single slit

When two barriers are used to make a slit, waves passing through will diverge, diffracting around both barriers

The spreading out (divergence) of plane waves as they pass through an opening is an example of diffraction

Diffraction through a single slit

Why isn’t this phenomenon seen everyday with light waves?

Diffraction through a single slit

Diffraction of light is difficult to see due to the short wavelength of light

As seen below, the angle of divergence depends on the ratio of the width of the gap to the wavelength of the wave

Diffraction through a single slit

As the width of the slit increases, the divergence of the wave decreases

As the wavelength of the wave increases, the divergence of the wave increases

Diffraction through a single slit

Light passing through a 1cm slit will diffract approximately 0.00003°

Effectively the light passes through with no observable divergence

To be able to observe diffraction of visible light, it needs to be passed through slits of width between 10-5m to 10-6m

Diffraction through a single slit

http://www.youtube.com/watch?v=BH0NfVUTWG4

Diffraction of light by a narrow slit

When a plane light wave (parallel beam of light) passes through a narrow slit, it diverges and falls on a screen with the following pattern

Diffraction of light by a narrow slit

When diffraction occurs, there is no change in the frequency, velocity or wavelength of the wave

The only changes are in the amplitude and direction

Diffraction of light by a narrow slit

The pattern consists of a wide band of light, with narrower bands on each side

Diffraction of light by a narrow slit

On the right is a graph of intensity, the distance between the centre of the dark regions is approximately equal, and the first dark band is also this distance from the central peak

10.3COHERENT WAVE SOURCES

Phase

Two wave sources are in phase with each other if they are emitting waves that are in phase with each other

This means that if one source is emitting a crest, the other is emitting a crest; if one is emitting a trough, the other is emitting a trough, etc.

Both sources must be producing light of the same frequency, and hence the same wavelength

Coherent Wave Sources

The term monochromatic light refers to light of a single frequency

Two (or more) sources of waves are coherent if they maintain a constant phase relationship with each other

They must, therefore, be emitting light of the same type and frequency

Incandescent Sources of Light

An incandescent solid (or liquid) is one that has been caused to glow through the application of heat

Examples of incandescent solids include:• an iron bar that heated until it glows• the tungsten filament in a light bulb• the glowing coals in a fire Examples of incandescent liquids include:• molten iron• lava

Incandescent Sources of Light

When materials are heated, the molecules and atoms within them vibrate faster

Since atoms consist of charged particles (electrons & protons) we have vibrating (accelerating) charged particles

The particles don’t all vibrate with the same frequency, but with a range of frequencies

Incandescent Sources of Light

The graph below shows the number of particles vibrating at particular frequencies

Incandescent Sources of Light

Since the electromagnetic waves emitted have a range of frequencies, light emitted from an incandescent source is not monochromatic

Since the E-M waves have different frequencies, they cannot be in phase

Thus, light emitted from incandescent sources is not coherent

10.4CONSTRUCTIVE & DESTRUCTIVE INTERFERENCE

The Principle of Superposition

If 2 or more travelling waves are passing through a medium, the resultant wave is found by adding together vectorially the displacements due to all the individual waves at all points in the medium

The Principle of Superposition

In the example to the right, the two waves are travelling in opposite directions

As they meet they interact constructively, producing a wave of greater intensity at that point

The Principle of Superposition

The waves then continue through one another, emerging unaffected by the interaction

The Principle of Superposition

The Principle of Superposition

When two peaks (or troughs) of in phase waves coincide, their displacements are added vectorially to produce constructive interference

+ =

The Principle of Superposition

When the peak of a wave coincides with the trough of another wave out of phase by λ/2, their displacements are added vectorially to produce destructive interference

+ =

The Principle of Superposition

When out of phase or waves of different frequencies waves interact, any combination of constructive or destructive interference can occur

The Principle of Superposition

http://www.acs.psu.edu/drussell/demos/superposition/superposition.html

10.5GENERAL TWO SOURCE WAVE INTERFERENCE

Conditions for Constructive & Destructive Interference

The following are conditions necessary for observing interference of light:

• the light from the different sources must overlap

• the light must be of the same frequency

• the sources must be coherent

• the amplitude of the sources must be approximately equal

• external light must be excluded

Conditions for Constructive & Destructive Interference

Consider two sources of waves emitting identical waves in phase with one another

Conditions for Constructive & Destructive Interference

Point P is equidistant from each source, thus the waves from each source will be in phase and constructively interfere

Conditions for Constructive & Destructive Interference

At point Q, there will be a path difference between the distances travelled by each wave

Conditions for Constructive & Destructive Interference

The path difference between the two waves to Q is = QS2 – QS1

Conditions for Constructive & Destructive Interference

If the path difference is one wavelength, then the crests and troughs will constructively interfere

Conditions for Constructive & Destructive Interference

If the path difference is half a wavelength, then the crests and troughs will destructively interfere

The Two Source Interference Pattern in Two Dimensions

The diagram below shows circles representing crests and troughs of waves emanating from two sources producing waves in phase

The Two Source Interference Pattern in Two Dimensions

The purple lines represent crests and the blue lines represent troughs

The Two Source Interference Pattern in Two Dimensions

The dotted lines are lines of zero amplitude, where destructive interference is occurring; sometimes referred to as nodal lines

The Two Source Interference Pattern in Two Dimensions

Half way between the dotted lines (but not shown) are lines of maximum amplitude; also known as anti-nodal lines

The Two Source Interference Pattern in Two Dimensions

This pattern of interference can be seen using water waves, but applies to light, sound, and other waves also

Class Problems

Conceptual Questions: 1-2, 5-6

Descriptive Questions: 1-2

Computational Questions: 1-2

10.6THE TWO SLIT INTERFERENCE OF LIGHT

The Two Slit Apparatus

The two source interference pattern was first observed by Thomas Young in 1801

Setting up two coherent sources is difficult, Young got around this by setting up a coherent light source that first passed through one slit, which caused diffraction of the wavefront

The Two Slit Apparatus

A second screen behind the first slit had a double slit in it

The wavefront reached both slits simultaneously, effectively creating two coherent sources

The Two Slit Apparatus

The apparatus used is known as Young’s double slit interferometer

The Two Slit Interference Pattern for Light

The pattern produced by constructive and destructive interference of the two wavefronts is shown below

The Two Slit Interference Pattern for Light

At the centre bright fringe there is no path difference between the two sources, thus the waves interact constructively

The Two Slit Interference Pattern for Light

As we move away from the centre, the path difference increases

The Two Slit Interference Pattern for Light

When the path difference becomes λ/2 we get a dark fringe due to destructive interference at these points

The Two Slit Interference Pattern for Light

As we move further out, bright fringes occur when the path difference = Δs = mλ, where m is an integer, and dark fringes when Δs = (m + ½)λ

The Two Slit Interference Pattern for Light

Unlike the single slit pattern where intensity of the bright fringes increases with distance away from the central bright fringe, the double slit bright fringes maintain the same intensity

The Two Slit Interference Pattern for Light

This diagram is exaggerated in the vertical direction to emphasise the effect, the actual total dispersion is less than 3° from the central bright fringe

Formula for Path Difference to Any Point on the Screen

At any point other than the perpendicular bisector EC, the angular displacement is given by θ, as shown below

Formula for Path Difference to Any Point on the Screen

The path difference is given by Δs = |BP – AP|

(AP = DP)

Formula for Path Difference to Any Point on the Screen

Using trigonometry,

Class Problems

Conceptual Questions: 3-4

Descriptive Questions: 3

Computational Questions: 3-4

Formula for the Angular Position of Any Maximum

The maxima of the interference pattern occur when the path difference, Δs = mλ

Thus, d sinθ = mλ, where d is the distance between the two slits

Likewise, the minima occur when d sinθ = (m + ½)λ

Formula for the Angular Position of Any Maximum

If P is the first order maxima, the path difference to P from the slits is one wavelength,

i.e. at P, d sinθ = λ

Formula for the Angular Position of Any Maximum

Since θ is very small, to a very good approximation,

sin θ ≈ tan θ

Formula for the Angular Position of Any Maximum

From the triangle PEC, tanθ =

Formula for the Angular Position of Any Maximum

Therefore, d sinθ = λ, becomes

i.e.

Formula for the Angular Position of Any Maximum

The distance between adjacent maxima or minima on the screen is given by

This is also referred to as the bandwidth

The fringe separation depends on: the wavelength of monochromatic light used (λ), the distance to the screen (L), and the distance between the slits (d)

Determination of the Wavelength of Light

Young’s double slit interferometer can be used to determine the wavelength of monochromatic light

By using slits a given distance apart, at a certain distance from the screen, the distance between fringes can be found and used to determine the wavelength of the light source

The wavelength can also be determined by measuring the angle of the mth maxima and using the formula

d sinθ = mλ

Speckle on reflection of laser light

When a laser is reflected off a rough surface, some sections of the beam appear brighter than others, this is called laser speckle

This is due to interference; since the beam is coherent, the rough surface causes a path difference in the reflections of the beam

Thus when the light falls on our eye, at some points there is constructive, and other points destructive interference

Class Problems

Conceptual Questions: 7, 10

Descriptive Questions: 4

Computational Questions: 5-7, 14-15

10.7TRANSMISSION DIFFRACTION GRATINGS

Principle of operation

Transmission diffraction gratings consist of very many, closely spaced parallel slits.

These slits are much closer than the double slits in Young’s interferometer, d = 10-5-10-6m

This is achieved by scratching the surface of glass or plastic; the scratches do not transmit light

Principle of operation

When light passes through the slits it diffracts

Because of the very small size of the slits, the diffraction is large

Principle of operation

Constructive and destructive interference can be observed from these multiple slits

Path Difference between Adjacent Waves

If monochromatic light is passed through the grating, we effectively have a large number of sources of light in phase

Path Difference between Adjacent Waves

If a lens is used to focus the rays, the path difference between adjacent rays can be found with the triangle ABN

Path Difference between Adjacent Waves

In triangle ABN, , therefore, AN = ABsinθ

Path Difference between Adjacent Waves

This means that the path difference between adjacent rays = d sinθ

This is the same formula for double slits, however we did not need to make the approximations relying on θ being small

Therefore this formula is valid for all angles

The Diffraction Pattern

When we had double slit diffraction, the fringes were quite wide, with the intensity progressively decreasing until the path difference is ½λ

However the diffraction intensity drops away almost immediately when we move away from the straight through position

This is because of the number of interfering waves, each destructively interferes with another wave even at very small angles

The Grating Pattern

This interference causes bright narrow lines of constructive interference when the path difference is a multiple of the wavelength, with very low intensity in between

The Grating Equation

When the path difference is mλ, there is constructive interference

10.ACOMPACT DISCS

Binary Numbers (not examinable)

In the decimal system, there are 10 digits (base 10); each place value is 10x larger than the previous one

e.g. 7931 = 7x103 + 9x102 + 3x101 + 1x100

Different number systems have different bases, e.g. the hexadecimal system used in computing uses base 16 and represents the numbers 11-16 using the letters A-F

Binary Numbers (not examinable)

In the binary system, there are two digits, 0 and 1

The number 110011 would thus be equal to:

25 + 24 + 0 + 0 + 21 + 20 = 51

Digital information is represented with binary because it allows data to be stored or transmitted using only 2 variables or states: on or off

Converting an analogue waveform to digital (not examinable)

Sound waves are converted by microphones into continuous electrical signals called analogue waveforms

Converting an analogue waveform to digital (not examinable)

Sound waves are converted by microphones into continuous electrical signals called analogue waveforms

Converting an analogue waveform to digital (not examinable)

A analogue to digital converter (ADC) measures the intensity of the signal at a rate of 44.1 kHz

The intensity is represented by a number between -32,768 and +32,768, this range is represented as a 16 digit binary number

e.g. an amplitude of 15,670 is recorded as:

0011 1101 0011 0110

The construction of a CD (not examinable)

The cross section of a CD is shown below

During manufacture, a series of pits are put into the polycarbonate layer representing the digital information

The construction of a CD (not examinable)

These pits are coated with aluminium, to become reflective

When viewed from the bottom side, the pits will appear as bumps to the reading laser of the CD player

The construction of a CD (not examinable)

The bumps are very small, only about 0.5 μm wide; there is about 8km of track on a single CD

Relating the Bumps to Binary Digits

Whenever a bump begins or ends, this generates a signal

A signal is interpreted as a 1, no signal is interpreted by 0

How the Laser Reads the Data

The laser used in CD players has a wavelength of 780nm, this is in the near infrared region of the spectrum

However when it is in the polycarbonate (refractive index of 1.55) its speed is reduced, which reduces the wavelength to approximately 500nm (in the green region)

Importantly, the height of the bumps (110nm) is approximately ¼ of this wavelength

How the Laser Reads the Data

How the Laser Reads the Data

The laser is focused on the disc to a spot 1.7μm in diameter

About when the laser is over a bump, about 35% is reflected of the bump and 65% off the flat section (land)

Because the bumps are ¼ of a wavelength, the light reflected will have a path difference of approximately ½ a wavelength

How the Laser Reads the Data

The path difference causes most of the light incident on the detector to undergo destructive interference

This results in a reduced signal from the detector

The change in signal is interpreted by the CD player as a 1, no change in signal indicates a 0

Keeping the Laser on Track

Data from the CD is read from the inside out so that it is possible to have smaller CDs

The CD spins at a rate of 200-500 rpm, the laser remains relatively stationary moving outward over the length of the audio tracks

The CD slows as the laser moves out to ensure that the track speed (and therefore data rate) stays constant

Keeping the Laser on Track

To ensure that the laser doesn’t switch tracks while it is in operation, tracking beams are used

This is achieved using a diffraction grating

Keeping the Laser on Track

Almost all of the light from the reflected tracking beams should return to it’s detector since it should be reflected off of the ‘land’

Keeping the Laser on Track

If the light incident on the tracking beam detector is reduced, it is due destructive interference caused by the tracking beam partially passing over the bumps, and the mechanism ‘knows’ to correct itself

Class Problems

Computational Questions: 23, 25, 28-30, 32