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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:
Page 534 Q 1 - 4