1. Light Through a Single Slit
Light passes through a slit or opening and then illuminates a screen
As the width of the slit becomes closer to the wavelength of the light, the intensity pattern on the screen and additional maxima become noticeable

2. Single-Slit Diffraction
Water wave example of single-slit diffraction
All types of waves undergo single-slit diffraction
Water waves have a wavelength easily visible
Diffraction is the bending or spreading of a wave when it passes through an opening

3. Huygen’s Principle
It is useful to draw the wave fronts and rays for the incident and diffracting waves
Huygen’s Principle can be stated as all points on a wave front can be thought of as new sources of spherical waves

4. Double-Slit Interference
Light passes through two very narrow slits
When the two slits are both very narrow, each slit acts as a simple point source of new waves
The outgoing waves from each slit are like simple spherical waves
The double slit experiment showed conclusively that light is a wave
Experiment was first carried out by Thomas Young around 1800

5. Young’s Double-Slit Experiment
Light is incident onto two slits and after passing through them strikes a screen
The light intensity on the screen shows an interference pattern

6. Young’s Experiment, cont.
The experiment satisfies the general requirements for interference
The interfering waves travel through different regions of space as they travel through different slits
The waves come together at a common point on the screen where they interfere
The waves are coherent because they come from the same source
Interference will determine how the intensity of light on the screen varies with position

7. Young’s Experiment Assume the slits are very narrow
According the Huygen’s principle, each slit acts as a simple source with circular wave fronts as viewed from above
The light intensity on the screen alternates between bright and dark as you move along the screen
These areas correspond to regions of constructive interference and destructive interference

8. Double Slit Analysis
Need to determine the path length between each slit and the screen
Assume W is very large
If the slits are separated by a distance d, then the difference in length between the paths of the two rays is
ΔL = d sin θ

9. Double Slit Analysis
If ΔL is equal to an integral number of complete wavelengths, then the waves will be in phase when they strike the screen
The interference will be constructive
The light intensity will be large
If ΔL is equal to a half number of complete wavelengths, then the waves will not be in phase when they strike the screen
The interference will be destructive
The light intensity will be zero

10. Conditions for Interference
For constructive interference, d sin θ = m λ
m = 0, ±1, ±2, …
Will observe a bright fringe
For destructive interference, d sin θ = (m + ½) λ
Will observe a dark fringe

11. Double-Slit Intensity Pattern
The angle θ varies as you move along the screen
Each bright fringe corresponds to a different value of m
Negative values of m indicate that the path to those points on the screen from the lower slit is shorter than the path from the upper slit

12. Spacing Between Slits Notation: For m = 1,
d is the distance between the slits
W is the distance between the slits and the screen
h is the distance between the adjacent bright fringes
For m = 1,
Since the angle is very small, sin θ ~ θ and θ ~ λ/d
Between m = 0 and m = 1, h = W tan θ

13. Approximations
Since the wavelength of light is small, the angles involved in the double-slit analysis are also small
For small angles, tan θ ~ θ and sin θ ~ θ
Using the approximations, h = W θ = W λ / d

14. Interference with Monochromatic Light
The conditions for interference state the interfering waves must have the same frequency
This means they must have the same wavelength
Light with a single frequency is called monochromatic (one color)
Light sources with a variety of wavelengths are generally not useful for double-slit interference experiments
The bright and dark fringes may overlap or the total pattern may be a “washed out” sum of bright and dark regions
No bright or dark fringes will be visible

15. Single-Slit Interference
Slits may be narrow enough to exhibit diffraction but not so narrow that they can be treated as a single point source of waves
Assume the single slit has a width, w
Light is diffracted as it passes through the slit and then propagates to the screen

16. Single-Slit Analysis
The key to the calculation of where the fringes occur is Huygen’s principle
All points across the slit act as wave sources
These different waves interfere at the screen
For analysis, divide the slit into two parts

17. Single-Slit Fringe Locations
If one point in each part of the slit satisfies the conditions for destructive interference, the waves from all similar sets of points will also interfere destructively
Destructive interference will produce a dark fringe

18. Single-Slit Analysis: Dark Fringers
Conditions for destructive interference are
w sin θ = ±m λ
m = 1, 2, 3, …
The negative sign will correspond to a fringe below the center of the screen

19. Single-Slit Analysis: Bright Fringes
There is no simple formula for the angles at which the bright fringes occur
The intensity on the screen can be calculated by adding up all the Huygens waves
There is a central bright fringe with other bright fringes that are lower in intensity
The central fringe is called the central maximum
The central fringe is about 20 times more intense than the bright fringes on either side
The width of the central bright fringe is approximately the angular separation of the first dark fringes on either side

20. Single-Slit – Central Maximum
The full angular width of the central bright fringe = 2 λ / w
If the slit is much wider than the wavelength, the light beam essentially passes straight through the slit with almost no effect from diffraction

21. Double-Slit Interference with Wide Slits
When the slits of a double-slit experiment are not extremely narrow, the single-slit diffraction pattern produced by each sit is combined with the sinusoidal double-slit interference pattern
A full calculation of the intensity pattern is very complicated

22. Diffraction Grating
An arrangement of many slits is called a diffraction grating
Assumptions
The slits are narrow
Each one produces a single outgoing wave
The screen is very far away

23. Diffraction Grating
Since the screen is far away, the rays striking the screen are approximately parallel
All make an angle θ with the horizontal axis
If the slit-to-slit spacing is d, then the path length difference for the rays from two adjacent slits is
ΔL = d sin θ
If ΔL is equal to an integral number of wavelengths, constructive interference occurs
For a bright fringe, d sin θ = m λ
m = 0, ±1, ±2, …

24. Diffraction Grating, final
The condition for bright fringes from a diffraction grating is identical to the condition for constructive interference from a double slit
The overall intensity pattern depends on the number of slits
The larger the number of slits, the narrower the peaks

25. Grating and Color Separation
A diffraction grating will produce an intensity pattern on the screen for each color
The different colors will have different angles and different places on the screen
Diffraction gratings are widely used to analyze the colors in a beam of light

26. Diffraction and CDs
Light reflected from the arcs in a CD acts as sources of Huygens waves
The reflected waves exhibit constructive interference at certain angles
Light reflected from a CD has the colors “separated”

27. Crystal Diffraction of X-rays
Diffraction effects occur with other types of waves
The atoms of a crystal are arranged in a periodic way, forming planes
These planes reflect em radiation
Leads to interference of the reflected rays

28. X-Ray Diffraction, cont.
The effective slit spacing is the distance between atomic planes
Typically 3 x m
Compared to 10-4 m or 10-5 m for a grating
X-rays have the appropriate wavelength to diffract
The planes give dots instead of fringes
By measuring the angles that give constructive interference, the distance between the planes can be measured

29. Optical Resolution
For a circular opening of diameter D, the angle between the central bright maximum and the first minimum is
The circular geometry leads to the additional numerical factor of 1.22

30. Telescope Example Assume you are looking at a star through a telescope
Diffraction at the opening produces a circular diffraction spot
Assume there are actually two stars
The two waves are incoherent and do not interfere
Each source produces its own different pattern

31. Rayleigh Criterion
If the two sources are sufficiently far apart, they can be seen as two separate diffraction spots (A)
If the sources are too close together, their diffraction spots will overlap so much that they appear as a single spot (C)

32. Rayleigh Criterion, cont.
Two sources will be resolved as two distinct sources of light if their angular separation is greater than the angular spread of a single diffraction spot
This result is called the Rayleigh criterion
For a circular opening, the Rayleigh criterion for the angular resolution is
Two objects will be resolved when viewed through an opening of diameter D if the light rays from the two objects are separated by an angle at least as large as θmin

33. Limits on Focusing
A perfect lens will focus a narrow parallel beam of light to a precise point at the focal point of the lens
The ray optics approximation ignores diffraction
The real focus is spread over a disc

34. Limits on Focusing, cont.
If the lens has a diameter D, it acts like an opening and according to the Rayleigh criterion produces a diffracted beam spread over a range of angles
Diffraction spreads the focal point over a disk of radius r
The focal length is limited to

35. Limits on Focusing, final
The wave nature of light limits the focusing qualities of even a perfect lens
It is not possible to focus a beam of light to a spot smaller than approximately the wavelength
The ray approximation of geometrical optics can be applied at size scale much greater than the wavelength
When a slit or a focused beam of light is made so small that its dimensions are comparable to the wavelength, diffraction effects become important

36. Scattering
When the wavelength is larger than the reflecting object, the reflected waves travel away in all direction and are called scattered waves
The amplitude of the scattered wave depends on the size of the scattering object compared to the wavelength
Blue light is scattered more than red
Called Rayleigh scattering

37. Blue Sky
The light we see from the sky is sunlight scattered by the molecules in the atmosphere
The molecules are much smaller than the wavelength of visible light
They scatter blue light more strongly than red
This gives the atmosphere its blue color

38. Scattering, Sky, and Sun Blue sky Sun near horizon
Although violet scatters more than blue, the sky appears blue
The Sun emits more strongly in blue than violet
Our eyes are more sensitive to blue
The sky appears blue even though the violet light is scattered more
Sun near horizon
There are more molecules to scatter the light
Most of the blue is scattered away, leaving the red

39. Nature of Light
Interference and diffraction show convincingly that light has wave properties
Certain properties of light can only be explained with a particle theory of light
Color vision is one effect that can be correctly explained by the particle theory
Have strong evidence that light is both a particle and a wave
Called wave-particle duality
Quantum theory tries to reconcile these ideas

40. Color Vision
Color vision is due to light detectors in the eye called cones
The three types of cones are sensitive to light from different regions of the visible spectrum
Particles of light, photons, carry energy that depends on the frequency of the light