1. Chapter 24
Geometrical Optics

2. Optics The study of light is called optics
Some highlights in the history of optics
Study of optics dates to at least third century BC
Eyeglasses invented around 1300
Microscopes and telescopes invented around 1600
Applications depend on the ability of lenses and mirrors to focus light
Light is an electromagnetic wave and its wave nature needs to be accounted for

3. Geometrical Optics
Applies to the regime where light travels in straight-line paths
Effects involving wave interference are not important
Describes cases in which the wavelength of the light is much smaller than the size of the objects in the light’s path
Wavelength of visible light is less than 1µm
Describes many everyday applications
Including the behavior of mirrors and lenses

4. Rays
Rays indicate the path and direction of propagation of the light wave
In A, the waves pass through a large opening and, to a very good approximation, follow straight lines that pass through the opening
In B, the opening is about the same size or smaller than the wavelength of the light and needs wave optics to explain
Section 24.1

5. Wave Fronts
Wave front surfaces are determined by the crests and troughs of the wave
They are always perpendicular to the associated rays
The shape of a wave front depends on how the wave is generated and the distance from the source
Section 24.1

6. Two Properties of Light
The motion of light along a light ray is reversible
If light can travel in one direction along a ray that connects point A to point B, light can also propagate in the reverse direction, from B to A
The perpendicular distance between two wave fronts is proportional to the speed of light
Because of the way wave fronts are related to crest and troughs of a wave
Section 24.1

7. Ray Tracing
Light from an object is used by your eye to form an image of the object
When your eye combines the rays to form an image, your brain extrapolates the rays back to their origin
The method of following the individual rays as they travel from an object to some other point is called ray tracing
Ray tracing involves the use of geometry
Section 24.1

8. Ray Tracing, cont. The figure shows a few rays from the object
There are an infinite number of actual rays
The light waves associated with all the rays contribute to the image formed by your eye
In most ray diagrams, we draw just a few rays from the top and bottom of the image
Section 24.1

9. Image Formation
Two problems must be considered to understand how images are formed
What happens to light rays when they reflect from a surface such as a mirror or a piece of glass
What happens to light rays when they pass across a surface from one material to another such as when they pass from air into a piece of glass
You must also distinguish between a flat surface and a curved surface
Section 24.1

10. Reflection from a Plane Mirror
Light rays travel in straight lines until they strike something
The rays may be reflected
Rays may be reflected from a plane mirror
A flat surface that reflects all or nearly all the light that strikes it
If the light is a plane wave, all the rays are parallel and strike a surface at many different points
Section 24.2

11. Reflection from a Plane Mirror, cont.
Characterize the reflection by a single ray
The normal (vertical dashed line in fig. B) is perpendicular to the mirror
The direction of the incoming and outgoing rays are measured relative to the normal
Section 24.2

12. Law Of Reflection - Definitions
The incoming ray is called the incident ray
The angle it makes with the normal is called the angle of incidence, θi
The outgoing ray is called the reflected ray
The angle it makes with the normal is called the angle of reflection, θr
The Law of Reflection says θi = θr
Reflection from a perfectly flat mirror is called specular reflection
Section 24.2

13. Diffuse Reflection
If the reflecting surface is rough, the reflections from each individual piece of the surface must be analyzed
An incident plane wave will give rise to many reflected rays propagating outward in many different directions
This is called diffuse reflection
Section 24.2

14. Image Formation – Plane Mirror
An image formed by a plane mirror is shown
Two representative rays are shown coming from the object
There is an infinite number of rays emanating from each point on the object
The rays that reflected from the mirror and reached your eyes form the image
Section 24.2

15. Image Formation, cont.
To your eye, the location of the image is the point from which these rays appear to emanate
This point can be found by ray tracing
Each ray obeys the law of reflection
Applying geometry will allow the location of the image to be found
Section 24.2

16. Image Formation, final Characteristics of the image
The distance from the object to the mirror is the same as the distance from the image to the mirror
The size (height) of the image, hi, is the same as the size (height) of the object, ho
The image is virtual
The image point is located behind the mirror
The light does not actually pass through the image
The same analysis can be applied to multiple mirrors
Section 24.2

17. Refraction
When a light ray strikes a transparent material, some of the light is reflected and some is refracted
The reflected ray obeys the Law of Reflection
The refracted ray passes into the material
The incident angle is now denoted as θ1
Section 24.3

18. Angle of Refraction
The direction of the refracted ray is measured by using θ2 (refer to fig )
The value of this angle depends on the incident angle and the speed of light in the material
The speed of light in a vacuum is 3 x 108 m/s
When the light travels through a material substance, its interactions with the atoms of the material slows down the wave
Section 24.3

19. Snell’s Law
v = λ f
f constant
v , λ change
The change in the speed of light from the vacuum to the material changes the direction of the wave
From the geometry of the waves in the material,
Section 24.3

20. Snell’s Law, cont.

21. Snell’s Law, cont.
The ratio c/v is called the index of refraction and is denoted by n
n = c / v
n is unitless
Then, sin θ1 = n sin θ2
This assumes the wave is incident in a vacuum
A more general statement can be applied to any two materials with indices of refraction n1 and n2
n1 sin θ1 = n2 sin θ2
This relationship is called Snell’s Law
Section 24.3

22. Speeds and n’s for Various Materials
Section 24.3

23. Applying Snell’s Law Refraction is also reversible
Snell’s Law applies whether light begins in the material with the larger or smaller index of refraction
Possible angles of refraction are always between 0° and 90°
The side with the larger index of refraction has the smaller angle

24. Direction of Refracted Ray
Light is refracted toward the normal when moving into the substance with the larger index of refraction
Light is refracted away from the normal when moving into the substance with the smaller index of refraction
Section 24.3

25. Total Internal Reflection
When light is incident from the side with a higher index of refraction, it is bent away from the normal
As the incident angle gets larger, the refracted angle also increases
Eventually, θ2 will reach 90°
Section 24.3

26. Total Internal Reflection, cont.
The angle of incidence for which the angle of refraction is 90° is called the critical angle
If the angle of incidence is increased beyond the critical angle, Snell’s Law has no solution for θ2
Physically, there is no refracted ray
This behavior is called total internal reflection
This is only possible when the light is incident from the side with the larger index of refraction
Section 24.3

27. Critical Angle From Snell’s Law, with θ2 = 90°, θ1 = θcrit
When the angle of incidence is equal to or greater than the critical angle, light is reflected completely at the interface
Section 24.3

28. Fiber Optics Total internal reflection is used in fiber optics
Optical fibers are composed of specially made glass and used to carry telecommunication signals
These signals are sent as light waves
They are directed along the fiber using internal reflection
Section 24.3

29. Dispersion
When light travels in a material, the speed depends on the color of the light
This dependence of wave speed on color is called dispersion
Since the index of refraction is slightly different for each color, the angle of refraction will be different for each color
Section 24.3

30. Dispersion and Prisms
Dispersion is used by a prism to separate a beam of light into its component colors
There are two refractions with the prism
The red and blue show the extremes of the incident beams
Section 24.3

31. Curved Mirrors
A curved mirror can produce an image of an object that is magnified
The image can be larger or smaller than the object
Magnified images are used in many applications
Telescopes
Car’s review mirror
Many others
Section 24.4

32. Ray Tracing – Curved Mirror
A spherical mirror in one in which the surface of the mirror forms a section of a spherical shell
The radius, R, of the sphere is the radius of curvature of the mirror
The mirror’s principal axis is the line that extends from the center of curvature, C, to the center of the mirror
Section 24.4

33. Concave Spherical Mirror
Properties of concave spherical mirrors
Incoming rays that are close to and parallel to the principal axis reflect through a single point F
F is the focal point
It is located a distance ƒ, the focal length, from the mirror
Rays that originate at the focal point reflect from the mirror parallel to the principal axis
From reversibility of light
Section 24.4

34. Image From a Concave Mirror -- Examples
Section 24.4

35. Image from Concave Mirror – Ray Diagram
Trace rays emanating from the top of the object
The rays all intersect at a single point
This is the top of the image
A similar result would be found from rays from other parts of the object
Section 24.4

36. Drawing A Ray Diagram Three rays are particularly easy to draw
There are an infinite number of actual rays
The focal ray
From the tip of the object through the focal point
Reflects parallel to the principal axis
Section 24.4

37. Drawing A Ray Diagram, cont.
The parallel ray
From the tip of the object parallel to the principal axis
Reflects through the focal point
The central ray
From the tip of the object through the center of curvature of the mirror
Reflects back on itself
The three rays intersect at the tip of the image
Section 24.4

38. Properties of an Image
Magnification is the ratio of the height of the image, hi, to the height of the object, ho
By convention, the image height of an inverted image is negative
Therefore, the magnification is also negative
Images smaller than the object are said to be reduced
Section 24.4

39. Real vs. Virtual Images
If the rays that form the image all pass through a point on the image, the image is called a real image
Real images and virtual images differ
Light rays only appear to emanate from a virtual image, they do not actually pass through the image
For a real image, the light rays do actually pass through the image
An object and its real image are both on the same side the mirror
A virtual image is located behind the mirror while the object is in front
Section 24.4

40. Concave Mirror and Virtual Images
Use ray tracing to find the image when the object is close to the mirror
Closer than the focal point
Use the same three rays
The rays do not intersect at any point in the front of the mirror
Section 24.4

41. Virtual Images Extrapolate the rays back behind the mirror
They intersect at a single image point
The rays appear to emanate from the image point behind the mirror
The image is virtual because light does not actually pass through any point on the image
The object and its image are on different sides of the mirror
The image is upright and enlarged
Section 24.4

42. Rules for Ray Tracing – Mirrors
Construct a figure showing the mirror and its principal axis
The figure should also show the focal point and the center of curvature
Draw the object at the appropriate point
One end of the object will often lie on the principal axis
Draw three rays that emanate from the tip of the object
The focal ray passes through the focal point and reflects parallel to the principal axis
Section 24.4

43. Rules, cont. Three rays, cont.
The parallel ray is parallel to the principal axis and reflects through the focal point
The central ray passes through the center of curvature of the mirror and reflects back through the tip of the object
The point where the three rays intersect is the image point
This point may be in front of the mirror giving a real image
This point may be in back of the mirror giving a virtual image
Found by extrapolation of the rays behind the mirror
Section 24.4

44. Rules, final
This ray-tracing procedure can be repeated for any desired point on the object
This allows you to find other points on the image
It is usually sufficient to consider just the tip of the image
Other points may be used if needed
Section 24.4

45. Ray Tracing – Convex Spherical Mirrors
A mirror that curves away from the object is called a convex mirror
The center of curvature and the focal point lie behind the mirror
After striking the convex surface, the reflected rays diverge from the mirror axis
The parallel rays converge on an image point behind the mirror
This is the focal point, F
Section 24.4

46. Ray Tracing – Convex Mirrors, cont.
The same three rays are used as were used for concave mirrors
The focal ray is directed toward the focal point but is reflected at the mirror’s surface, so doesn’t go through F
The three rays extrapolate to a point behind the mirror
Produces virtual image
Section 24.4

47. Mirror Equation
Geometry can be used to find the characteristics of the image quantitatively
The distance from the object to the mirror is so
The distance from the image to the mirror is si
The given rays produce similar triangles
Section 24.4

48. Mirror Equation and Focal Length
From the similar triangles,
For an object at (approximately) infinity, 1/so = 0
But an “infinite” object will produce parallel rays
Parallel rays all intersect at the focal point
Therefore, the focal length can be found from the radius of curvature of the mirror
Section 24.4

49. Mirror Equation and Magnification
The mirror equation can be written in terms of the focal length
The magnification can also be found from the similar triangles shown in fig
Section 24.4

50. Sign Conventions
All diagrams with mirrors should be drawn with the light ray incident on the mirror from the left
The object distance is positive when the object is to the left of the mirror and negative if the object is to the right (behind) of the mirror
The image distance is positive when the image is to the left of the mirror and negative if the image is to the right (behind) of the mirror
The image distance is positive for real images and negative for virtual images
Section 24.4

51. Sign Conventions, cont.
The focal length is positive for a concave mirror and negative for a convex mirror
For a concave mirror, ƒ = R / 2
For a convex mirror, ƒ = - R / 2
The object and image heights are positive if the object/image is upright and negative if it is inverted
Section 24.4

52. Sign Convention, Summary
Section 24.4

53. Lenses A lens uses refraction to form an image
Typical lenses are composed of glass or plastic
The refraction of the light rays as they pass from the air into the lens and then back into the air causes the rays to be redirected
Although refraction occurs at both surfaces of the lens, for simplicity the rays are drawn to the center of the lens
Section 24.5

54. Lenses, Focal Point
Parts B and C show the simplification of the single deflection of the rays
Parallel rays close to the principal axis intersect at the focal point
This is true for incident rays from either side of the lens
The focal points are at equal distances on the two sides of the lens
Section 24.5

55. Spherical Lenses The simplest lenses have spherical surfaces
The radii of curvature of the lenses are called R1 and R2
The radii are not necessarily equal
Section 24.5

56. Types of Lenses Converging lenses Diverging lenses
All the incoming rays parallel to the principal axis intersect at the focal point on the opposite side
Diverging lenses
All the incoming rays parallel to the principal axis intersect at the focal point on the same side as the incident rays
Section 24.5

57. Focal Point – Diverging Lens
The parallel incident rays from the left are refracted away from the axis
The rays on the right appear to emanate from a point F on the left side of the lens
This point F is one of the focal points of the lens
Section 24.5

58. Image from a Converging Lens
An infinite number of rays emanate from the object
For simplicity, choose three rays that are easy to draw
Start at the tip of the object
Section 24.5

59. Rays for a Converging Lens
The parallel ray is initially parallel to the principal axis
Refracts and passes through the focal point on the right (FR)
The focal ray passes through the focal point on the left (FL)
Refracts and goes parallel to the principal axis on the right
The center ray passes through the center of the lens, C
Section 24.5

60. Rays, cont.
If the lens is very thin, the center ray is not deflected by the lens
These three rays come together at the tip of the image on the right of the lens
In this case, the image is inverted
The image is real
The rays pass through the image
Section 24.5

61. Rules for Ray Tracing – Lenses
Construct a figure showing the lens and its principal axis
The figure should also show the focal points on both sides of the lens
Draw the object at the appropriate point
One end of the object will often lie on the principal axis
Draw three rays that emanate from the tip of the object
The parallel ray is initially parallel to the principal axis and after refraction passes through one of the focal points
Section 24.5

62. Rules, cont. Three rays, cont.
The focal ray is directed at the other focal point and after refraction the ray is parallel to the principal axis
The central ray passes through the center of the lens and is not deflected
The point where the three rays or their extrapolation intersect is the image point
If the rays actually pass through the lens, the image is real
If the rays do not pass through the lens, the image is virtual
Section 24.5

63. Rules, final Real image Virtual image
When a lens forms a real image, the object and image are on opposite sides of the lens
Virtual image
When a lens forms a virtual image, the object and image are on the same side of the lens
All other rays that pass through the lens will also pass through the image
Section 24.5

64. Ray Tracing – Diverging Lens
Follow the rules for ray tracing for lenses
Since the refracted rays do not intersect on the right side of the lens, extrapolate the rays back to the left side of the lens
The extrapolations do intersect
The point of intersection is the image point at the tip of the image
Section 24.5

65. Sign Conventions – Lenses, Diagram

66. Sign Conventions for Lenses
Assume light travels through the lens from left to right
The object will always be located to the left of the lens
The object distance is positive when the object is to the left of the lens
According to the first convention, the object distance will always be positive
The image distance is positive when the image is to the right of the lens and negative if the image is to the left of the lens
Section 24.5

67. Sign Conventions, cont.
The focal length is positive for a converging lens and negative for a diverging lens
The object height is positive if the object extends above the axis and is negative if the object extends below
The image height is positive if the image is extends above the axis and is negative if the image extends below
Section 24.5

68. Thin-Lens Equation
Geometry can be used to find a mathematical relation for locating the image produced by a converging lens
The shaded triangles are pairs of similar triangles
Section 24.5

69. Thin-Lens Equation and Magnification
The thin-lens equation is found from an analysis of the similar triangles
The magnification can also be found from the similar triangles shown
These results are identical to the results found for mirrors
Section 24.5