The Wave Theory of Light

Waves and Light

Light is not only energy! it carries information from distant

stars and other celestial objects Consists of other electromagnetic

radiation

Waves in 2-D!

Transverse Waves – periodic disturbance where particles in the medium oscillate at right angles to the direction in which the wave travel

Transmission

A wave originating from a point source is circular whereas a wave originating from a linear source is straight

As a wave moves away from its constant-frequency source, the spacing between successive crests or successive troughs (wavelength) remains the same provided the speed of the wave does not change

Wave Structure

Wave front is the leading edge of a continuous crest or tough

Wave ray is a straight line, drawn perpendicular to a wave front, indicating the direction of transmission

Wave Equation

This equation holds true for all types of wave (1D, 2D, and 3D)

A wave slows as it reaches shallow water therefore we have two separate situation:

the wave travelling in deep water has a speed v1 = f1λ1

the wave travelling in shallow water has a speed v2 = f2λ2

v f

Frequency of Waves

If we use the universal wave equation, we can compare how a wave changes in deep water as opposed to shallow water

Frequency is constant!

v1

v2

12

Example #1

The speed and the wavelength of a water wave in deep water are 18.0 cm/s and 2.0 cm, respectively. The speed in shallow water is 10.0 cm/s. Find the corresponding wavelength.

Reflection from a Straight Barrier

When a straight wave front runs into a straight reflective barrier, head on, it is reflected back along its original path

If a wave encounters a straight barrier obliquely (at an angle other than 90°), the wave front will be reflected obliquely as well

The angle formed by the incident wave front and the normal is equal to the angle formed by the reflected wave front and the normal

Angles of Incidence and Reflection

Angle of Incidence (θi) is the angle between the incident wave front and the barrier, or the angle between the incident ray and the normal

Angle of Reflection (θr) is the angle between the reflected wave front and the barrier, or the angle between the reflected ray and the normal

Refraction

Refraction is the bending effect on a wave’s direction that occurs when the wave enters a different medium at an angle

The Normal is a straight line drawn perpendicular to a barrier struck by a wave

Angle of Refraction (θR) is the angle between the normal and the refracted ray, or between the refracted wave front and the boundary

Refraction

For a specific change in medium has a constant value

iR

12 1

2

i n sinR Recall:

Snell’s Law the constant of proportionality (n) and

the index of refraction (n) are the same

When light passes from a vacuum into a substance, n is called the absolute index of refraction

Absolute index of refraction is the index of refraction for light passing from air or a vacuum into a substance

n 1 sin1 n 2 sin2

sin1sin2

v1v2

12

n2n1

Example #2

Partial Reflection-Partial Refraction

When refraction occurs, some of the energy usually reflects as well as refracts

The amount of reflection is more noticeable when a wave travels from shallow to deep water, where the speed increases and again becomes more pronounced as the angle of incidence increases

Total Internal Reflection

Total internal reflection is the reflection of light in an optically denser medium it occurs when the angle of incidence in

the denser medium is greater than a certain critical angle

Diffraction of Water Waves

Wave fronts travel in straight lines as long as the depth of the water is constant and the water is free of obstacles

When waves travel by an edge, longer wavelengths are diffracted more than shorter wavelength

As the wavelength increases, the amount of diffraction increases

Diffraction

When we keep the wavelength (λ) fixed and change the width (w), we find that the amount of diffraction increases as the size of the aperture decreases If waves are to be strongly diffracted they

must pass through an opening of width comparable to their wavelength or smaller

Interference of Waves in Two Dimensions

Constructive Interference occurs when waves build each other up, producing a resultant wave of greater amplitude than the given waves

When a crest from on source meets a crest from another source (or when a trough meets a trough), their energies combine to displace the medium – the energies add together

Destructive Interference occurs when waves diminish one another, producing a resultant wave of lower amplitude than the given waves

When a crest and trough meet, the wave energies displace the medium in opposite directions from each other – they cancel out

To produce a fixed pattern, the interfering waves must have the same frequency (and thus the same wavelength) and also similar amplitudes

Patterns of interference also occurs between two identical waves when they interfere in a 2-D medium such as the water in a ripple tank

Nodal line a line of destructive interference Between the nodal lines are areas of

constructive interference that appear as alternating bright (double-crest) and dark (double-trough) lines of destructive interference

The interference pattern between two identical sources Lines of destructive interference (nodal lines) and areas of

constructive interference

When the frequency of the sources is increased, the wavelength decreases, bringing the nodal lines closer together and increasing their number If the distance between the two sources

is increased, the number of nodal lines will increase

Every dark and bright spot represents constructive interference – either a crest and crest or trough and trough

Every point along the line represents destructive interference because the crests and troughs meet at this line

Mathematical Analysis of the Two-Point-Source Interference Pattern

Two-point-source interference pattern is useful because it allows direct measurement of the wavelength

S1 S2

Path Length

If we take a point P1 on one of the first nodal lines and connect it to each of the two sources by the lines P1S1 and P1S2

Difference in path length of an interference pattern, the absolute value of the difference between the distance for any point P from one source and the distance of the same point P from the other source

Two Point Source Interference Pattern

- Mathematics

n represents the number of nodal lines

θ is the angle of the nodal lined is the distance separating the two charges

P1S1 P1S2 n 1

2

sinn n 1

2

d

xn is the horizontal distance to point Pn from the right bisector and L is the distance from the midpoint of the two sources and point Pn

xnL

n 1

2

d

Example #1

An interference pattern is set up by two point sources of the same frequency, vibrating in phase. A point on the second nodal line is 25.0 cm from one source and 29.5 cm from the other. The speed of the waves is 7.5 cm/s. Calculate the wavelength and the frequency of the source.

Example

A page in a student’s notebook lists the following information, obtained from a ripple tank experiment with two point sources operating in phase: n = 3, x3 = 35 cm, L = 77 cm, d = 6.0 cm, θ3 = 25°, and 5 crests = 4.2 cm.

Determine the wavelength of the waves, using various methods.

Light Paradox

Light: Wave or Particle?

The Nature of Light

Energy can move from one place to another as the energy of moving objects or as the energy of waves

How does light travel?

How does light from a distant source, such as the Sun, bring us energy?

Newton’s Particle Theory:

Light appears to travel in a straight line Also known as “rectilinear

propagation of light”

Light does not bend around corners

When reflected off a flat surface, the angle of incidence equals the angle of reflection Perfectly elastic collision

Newton’s Particle Theory

He believed that light refracts in water because the speed of light increases in water (refracts towards the normal) This is the opposite of what actually occurs

Dispersion was explained by assuming that different light particles have different masses

Huygen’s Wave Model: Light appears to follow the laws of

optics Angle of incidence equals the angle of

reflection

This model adequately predicts that light does bend at an object as other types of waves do The particle theory does not predict or explain this

This theory better explained dispersion of light Same for diffraction

The wave theory proved superior in most observations of light

Huygens’ Principle

Every point on a wave front can be considered as a point source of tiny secondary wavelets that spread out in front of the wave at the same speed as the wave itself. The surface envelops, tangent to all the wavelets, constitutes the new wave front

Hugyen’s Principle

Hugyen’s principle is a method used to construct a succeeding wave based on the previous wave front. The procedure can be used in the same way for all waves, including sound and water waves.

Behaviors of Light Page 468 #1-5 ii) Refraction

i) Reflection

iii) Partial reflection-partial refraction

iv) dispersion

Experimental Evidence that Indicates Light could be a Wave

Both light and waves obey the laws of reflection

Both light and waves obey Snell’s Law The speed of a wave slows down in a more

dense medium, as does light Both waves and light can exhibit partial

reflection-partial refraction and total internal reflection

Waves exhibit dispersion, as does light

Young’s Double Slit Experiment

Central Maximum

1st Order Maximum

1st Order Maximum

2nd order minimum

2nd order minimum

1st order minimum

1st order minimum

2nd Order Maximum

2nd Order Maximum

Any time the path difference is a whole number of wavelengths, constructive interference (a maximum) can be observed

To obtain the nth order maximum, the

path difference would be nλ (n is a whole number)

For Constructive Interference

When using the Maxima as your measurement point:

Central Maximum

1st Order Maximum

1st Order Maximum

2nd order minimum

2nd order minimum

1st order minimum

1st order minimum

2nd Order Maximum

2nd Order Maximum

If the path difference is ,

destructive interference (a minimum) can be observed.

n 1

2

Destructive Interference

When using a minima as your measurement point

Young’s Double – Slit Experiment

From this experiment, it was determined that light makes the same type of interference pattern as water waves. The perpendicular distance between successive nodal lines is given by:

Where L is the perpendicular distance from the slits to the screen

x

L

d

Example #2

Single colour light falling on two slits 0.042 mm apart produces the fifth-order dark fringe at a 3.8° angle. Calculate the wavelength of the light.

You are measuring the wavelength of light from a certain single-colour source. You direct the light through two slits with a separation of 0.15 mm, and an interference pattern is created on a screen 3.0 m away. You find the distance between the first and fourth consecutive dark lines to be 8.0 cm. At what wavelength is your source radiating?

Review

Page 488 #1-3, 7, 8-13, 16, 20, 29*

Diffraction of Light Through a Single Slit

Light passing through a single, narrow slit is diffracted The extent of diffraction increases as

narrower and narrower slits are used The pattern consists of a bright central

region, with dark regions of destructive interference alternating with progressively less intense bright areas

Central maximum is the bright central region in the interference pattern of light and dark lines produced in diffraction

Secondary maxima is the progressively less intense bright areas, outside the central region, in the interference pattern

If you look carefully at the diffraction of water waves through a single opening, you see nodal lines To see how an interference pattern arises

for light, we can analyze the wave behaviour of monochromatic light passing through a single slit

In the case of light diffracted downward through an angle, waves from the top of the slit travel farther than waves from the bottom the smaller the slit width, the larger the

distance between the maxima and minima The longer the wavelength, the greater

the distance between maxima

Minima, or dark fringes

Maxima, or bright fringes occur at the centre of the pattern

sinn nw

sinn w,2w'3w,...

sinm m

1

2

w

sinm 0,32w'52w,...

The Intensity of Light through a Single Slit

For each successive bright area more and more light from the source interferes destructively, in pairs, and therefore the intensity of the light decreases

The separation Δy of adjacent maxima or minima is given by the relationship and the central maximum width is 2Δy

y Lw

Example

Calculate the angle at which 7.50 x 102 nm light produces a second minimum if the single-slit width is 2.0 μm (μm = x10-6)

Example

The first dark fringe in a certain single-slit diffraction pattern occurs at an angle of 15° for light with a wavelength of 580nm. Calculate the width of the slit

Wave Effects of Light

Polarization of Light

Does a light travel like transverse or longitudinal waves?

Plane polarization is a wave that can vibrate in one plane only

Unpolarized a wave that vibrates in all directions perpendicular to the direction of travel

Polarization

Polarization is the confining of vibrations of a wave to one direction

Polarizer is a natural (e.g. clouds) or artificial (e.g. filters) means to achieve polarization

Questions to Answer

How can the intensity of light be diminished or even appear to be cancelled?

What type of wave is light, an how is light polarized by reflection and scattering?

When light passes through a polarizing filter, the light waves are polarized in one plane If the filter is oriented in such a way that

the vibrations are horizontal, then we call the light horizontally polarizedPolarization occurs when the axes of

the polarizing filters are at right angle to each other

Achieving Polarization Polarization can be achieved four ways:

1st – Double Refraction

2nd – Reflectionwhen some absorption takes place at the point at

which light is reflected off a smooth surfaceLight waves reflected from a flat surface are partially

polarized in the horizontal plane

Achieving Polarization

3rd – Scattering Light from the Sun passes through our atmosphere and

encounters small particles that scatter the light Scattering causes the sky to appear blue, since the

shorter wavelengths are scattered more than the longer wavelengths – polarizing filters are used in photography to enhance photographs of the sky and clouds

Polarization occurs when you are looking at the sky through polarized sunglasses

It is the greatest when you are looking at right angles in the direction of the light from the Sun

Achieving Polarization

4th – Uses of Polarizing Filters

They have many uses, including glare reduction, stress analysis, and photography

Many of the practical applications of polarization make use of the phenomenon of optical activity, the ability of some substances, such as sugar, turpentine, insulin, to rotate the plane of polarization of a beam of light

Page 498 #1-3

Diffraction Gratings:

Device whose surface is ruled with close, equally spaced , parallel lines for the purpose of resolving light into spectra:

Very Similar to Double-Slit.

Diffraction Gratings:

Two Types of Diffraction Gratings:

Transmission Grating: Light passes through a series of parallel slits. Refection Grating:

Series of gratings on a mirrored surface: Example CD

Diffraction Gratings:

Diffraction Gratings follow the equation:

Diffraction Gratings can have Slit Patterns exceeding 10 000 lines/cm.

To solve for d: d = 1_____

# of lines/cm

d

mm

sin

How do Diffraction Gratings Differ:

Multiple Slit Source delivers more light Energy, resulting in a brighter interference pattern.

Bright maxima’s are sharper and narrower – less light dispersion.

Slit separation is less – which results in greater separation between maxima’s and increased resolution.

Good – Leads to more accurate measurements = greater accuracy for found wavelength.

Interference in Thin Films:

When Light strikes a Thin Film

Interference of Thin Films: Reflection

When t<<λ: The two waves will be

180 degrees out of phase. Results in a Dark Area at

the top of the vertical soap film.

When t = λ/4: The path difference is

λ/ 2. This Results in the

reats being in phase and constructive Interference produces a bright spot.

When t = λ/2: The reflected rays are

again back out of phase;

Results in Destructive Interference, or a Dark spot on the top of the soap film.

Summary:

Dark areas due to Destructive Interference when the thickness of the thin film is; 0, λ/2, λ, 3 λ/2, etc.

Bright Areas due to Constructive Interference when the thickness of the thin film is; λ/4, 3 λ/4, 5 λ/4, etc.

Interference of Thin Films : Transmission

The Opposite Happens: Dark areas due to Destructive Interference when the thickness

of the thin film is; λ/4, 3 λ/4, 5 λ/4, etc.

Bright Areas due to Constructive Interference when the thickness of the thin film is; 0, λ/2, λ, 3 λ/2, etc.

Page 516 Q 1 - 4

Air Wedge

Air Wedge – The air between two pieces of optically flat glass angled to form a wedge:

X = the distance between Dark Fringes.

L = The length of the wedge

t = is the thickness between the base of the wedge

λ = The Wavelength.

Page 518 Q 5 - 7

Michelson Inferometer

http://video.mit.edu/watch/michelson-interferometer-6561/

http://www.youtube.com/watch?v=uMaFB3jM2qs

Electromagnetic Waves and Light

James Clark Maxwell – 1864: Summarized the relationship between

Electromagnetic Fields: The distribution of electric charges, in space, produces

an electric field. Magnetic field lines are continuous loops without

beginning or end. Electric field lines begin and end on electric charges.

A changing electric field produces a magnetic field. A changing magnetic field produces a magnetic field.

Maxwell’s Essential characteristics of Electromagnetic Waves:

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