1. The Interactions Between Light and Matter The Geometric Optics of Image Formation Also called "Ray Tracing"

2. (I) Clear Materials Bend Rays Light bending is called "refraction".
Air
Water
where a straight ray
would come from.
where the ray
really comes from.

3. Snell’s Law: Describes the binding of light
Snell’s Law: Describes the binding of light. Bending of Light is called "Refraction"
The amount of bending depends on a property
of the material called "index of refraction", n.
So we find that the light going from fast medium (like air) to slow medium (like glass) is bent toward the normal, and light going from slow medium to fast bends away from the normal. This statement is the law of refraction, also known as Snell’s law.
The quantity of the bend is governed by an equation. But before we show you the equation, we need to go over some basic trigonometry.
Water
n is low
Glass
n is high

4. Snell’s Law: The Angles and the Normal
Refraction (bending) is described
by the change in angle
measured from the normal. 1
The normal is an imaginary line
perpendicular to the surface.
Now that we’ve gone over the trig stuff, the rest is simple. Snell’s law states that as light enters medium 2 with index of refraction n2 at an angle q1 from the normal, from medium 1 with index of refraction n1, then the light exits the interface with angle q2 from the normal. This example assumes that the n1 is less than n2. 2

5. All other values of n are >1.
Snell’s Law: The equations 1 2 n1 n2
Material #2
Material #1
Snell’s Law: n1sin1= n2sin2
Define n = 1 for a vacuum
All other values of n are >1.

6. Refraction for Different Materials
light
45
AIR
WATER
GLASS
DIAMOND
32
28
16

7. Snell’s Law: n1sin1= n2sin2 Examples
Material #2
Material #1
Material Index of Refraction, n
Vacuum 1 (exactly)
Air (approximately 1.000)
Water
Glass
Diamond 2.4

8. Snell’s Law: n1sin1= n2sin2 It works exactly the same in reverse.
Material #1 1 n1 n2 2
Material #2
Material Index of Refraction, n
Vacuum 1 (exactly)
Air (approximately 1.000)
Water
Glass
Diamond 2.4

9. Into and out of a flat plate of glass.
Air, n1 = 1.00
Air, n3 = 1.00
n1sin1= n2sin2 q4 q3 q2 q1
n3sin3= n4sin4

10. and the input and output rays are parallel.
It can be shown that
q1= q4
q2= q3
and the input and output rays are parallel.
Glass
n2 = 1.5
Air, n1 = 1.00
Air, n3 = 1.00 q4 q3 q2 q1

11. Trigonometry Review RULES THAT DEFINE SIN, COS, TAN of an ANGLE:
Hypotenuse (r)
Opposite
Side (y)
sin() = y/r (opp/hyp)
cos() = x/r (adj/hyp)
tan() = y/x (opp/adj) 
Adjacent
Side (x)
We need to go over the three basic trigonometric functions. To do this, we need to look at a right triangle, where one of the angles subtended by two sides is 90 degrees. The two sides are known as legs, and the side opposite to the 90 degree angle is the hypotenuse. The hypotenuse is the longest of the three sides.
Now, say we have an angle theta () on the top. The leg that touches theta is the adjacent leg, and the other is the opposite leg.
Sine of theta is the ratio between the opposite leg and the hypotenuse,
Cosine of theta is the ratio between the adjacent leg and the hypotenuse,
And the tangent of theta is the ratio between the opposite leg and the adjacent leg.
An easy mnemonic device is SOH-CAH-TOA, or “Sine Opposite over Hypotenuse, Cosine Adjacent over Hypotenuse, Tangent Opposite over Adjacent.”

12. (II) Refraction Is How Lenses Work

13. Using Refraction to Focus Light.
Glass Lens in Air
n1=1
Parallel Rays
n2=1.5
Up to now, the interface between the mediums has always been flat. Now consider a case where there are a whole bunch of flat surfaces but at a different angle. The light would bend according to Snell’s law, and we can calculate where all the rays would go. By increasing the number of flat surfaces and making each surface smaller and smaller, all of a sudden the interface becomes a curved surface. This is a description of the front surface of a lens.
Focal point of lens
Optical Axis
Focal length of lens, f

14. Parallel rays come to focus at one point on the image plane.
Glass Lens in Air
n1=1
n2=1.5
Optical
Axis
Parallel Rays
different direction
Image Plane
Focal length of lens, f

15. A Chief Ray is a ray heading toward or away from the center of the lens.
Glass Lens in Air
n1=1
Examples of Chief Rays
n2=1.5
Optical
Axis
Focal length of lens, f

16. Thin Lens Approximation: Chief Rays pass through the lens without deviation.
Glass Lens in Air
n1=1
Examples of Chief Rays
n2=1.5
Optical
Axis
Focal length of lens, f

17. (III) The Rules of Ray Tracing

18. We identify two types of very important rays:
(A) Collimated Rays:
These are the rays that enter or exit the lens parallel to the optical axis. These rays pass through a focal point.
(B) Chief Rays:
These are the rays that go through
the center of the lens on the optical axis. These rays are un-deviated.

19. Light from the object passes through the lens.
Follow the Ray Tracing Rule
Thin Lens
Ray Tracing Rule:
Select ONLY two tracing
rays, e.g. one of type (A),
and one of type (B), each
from the tip of the object.
(A)
(B)
Object
Ray (A) passes through
the focal point.
Ray (B) is
not deviated.
Optical
Axis f

20. Light from the object passes through the lens.
Thin Lens
Optical
Axis f
Object
Light from the object passes through the lens.
Follow the Ray Tracing Rule
The point of intersection
is where the tip of
the object comes
to a clear focus.
(A)
(B)
All other rays from this
point must come to
focus at the same point.

21. This is how a projector works.
Object is a slide h
Light
Source f f h'
M = h' h

22. Magnification: The ratio of the size of
the image to the size of the object.
Magnification < 1 in this example,
so the image is smaller than
the object.
Object h' h
M =
Optical
Axis f

23. Traditional Ray Tracing Terms
Focal lengths for a thin
lens in air: f = f'
Object
(collimated ray)
(chief ray) h h' f f' L L'
Object distance and height (L, h)
Image distance and height (L, h)

24. Try some different object locations, L. We observe 6 special cases.
Object distance = L Image distance = L' h f f h'
M = h' h

25. Case (I) Object distance L =  f
h' = 0 h
M = = 0 h'
Case (I) Object distance L = 
Image formed at the focal point, and magnification = 0
NOTE: this (chief) ray’s origin is the TIP of the object (same as for the collimated ray)

26. Case (II) L between  and 2f. As object moves to the right,
the image size increases.
eye
looks small
& inverted h h' f f
M = h' h
Image is real
and inverted.

27. eye
looks small
& inverted h h' f f
M = h' h

28. Case (III) At L = 2f, h = h', and M = 1.
eye
looks same
size & inverted h h' f f 2f 2f
M = h' h

29. Case (IV): L between 2f and f, (a) the image is still inverted, and
(b) h' > h, and M > 1.
h' still increases as the object moves toward the lens.
looks larger
& inverted h
eye h' f f 2f 2f
M = h' h

30. (a) the image is still inverted, and (b) h' > h, and M > 1.
For L between 2f and f,
(a) the image is still inverted, and
(b) h' > h, and M > 1.
h' still increases as the object moves toward the lens.
looks much
larger & inverted h
eye f f 2f 2f
M = h' h h'

31. Case(V): L = f. The rays are parallel. They
cross at infinity, so h' =  = M.
This is the point of maximum confusion!
looks very
confusing h
eye f f 2f 2f
M = h' h

32. Case (VI): L between f and the lens. The rays diverge
and look AS IF they come from an image that
(a) is erect and (b) enlarged, h'>h, m > 1.
This is called a "virtual image".
We look through the lens,
and it is a
magnifying glass! h'
M = > 1 h h
eye f f
The rays diverge!!

33. This is how a magnifying glass works!
This is called a "virtual image".
We look through the lens,
and it is a
magnifying glass! h'
M = > 1 h h
eye f f
The rays diverge!!

34. eye Move the object from the focal point toward the lens, and
the virtual image gets smaller, but M > 1. f
M = > 1 h' h
Looks smaller and smaller
as it nears the lens.
eye

35. Case (VII): The image is at the lens, so L = 0.
The image is also at the lens, it is erect,
and it is the same size.
M = = 1 h' h
Object
Image f f

36. Know and be able to ray trace the seven cases
of the thin, concave lens.
Object location
(1) at infinity
(2) > 2f
(3) at 2f
(4) between 2f and f
(5) at f
(6) between f and the lens
(7) at the lens
Know how h' and M change as the object moves
toward the lens and away from the lens.

37. (IV) Magnifying Power of a Lens

38. The ability of a lens to magnify an object
is determined by the focal length, f.
Lens #1 h
eye f f

39. As f increases, the lens magnifies less.
Long f = low magnifying power.
Short f = high magnifying power.
Lens #2 h
eye f f

40. Define the lens power, D = 1/f If f is in meters, then D is
called the "diopter" of the lens.
Lens #2 h
eye f f

41. A practical measure of magnifying power.
The "X" of the lens:
A practical measure of magnifying power.
Without a lens, most people can focus comfortably
no closer than 23 cm from the object.
In focus for the average eye
eye h
do = 23 cm

42. The "X" of the lens. If we want to make the object look larger,
we move it closer to the eye.
However, this makes it out of focus to the eye.
eye h
distance d

43. The "X" of the lens.
In order to make the close object appear in focus,
we place a magnifying lens very close to the eye.
By trial and error, we find a lens that will do this.
Most people find that a lens with a focal length, f,
that is approximately equal to the distance, x, works best.
eye h
distance d  f

44. The "X" of the lens. eye do = 23 cm distance d  f
The object started out at distance do = 23 cm.
The lens of focal length f allowed it to be in focus at a
distance of d = f. In other words, the lens brings the
object closer by a factor of do/d = do/f = 23/f.
So, define the magnification of the lens as X=23/f.
eye h h
do = 23 cm
distance d  f

45. The "X" of the lens. eye do = 23 cm distance d  f
For example, a lens of f = 2.3cm
allows most people to move an object from
23 cm to 2.3 cm. This makes the object appear 10 times
as close. Thus, we call the lens a 10X lens.
eye h h
do = 23 cm
distance d  f

46. (V) Another Type of Lens

47. This kind of lens is called a "Convex Lens".
(B)
Object
The image focal point, denoted by F prime, is the point where the parallel rays entering the positive lens converge to. It is actually half the distance to the center of the curvature of the lens. [The center of curvature may be represented by the letter C and is explained as the center of a circle for which the lens edge is also the edge of the circle.]
The object focal point, denoted by F, is exactly the same distance from the lens on the other side. f

48. This kind of lens is called a "Concave Lens".
It is possible to define focal points
and ray tracing for this kind of lens also,
but that is beyond the scope of this course.
Lens
In a concave lens, the image focal point is located on the object side of the lens. Therefore, the focal length is negative. As we learned before, the object focal point and the object focal length is exactly opposite of the image focal point and focal length.

49. (VI) When Things Go Wrong!
Aberrations

50. The ideal lens focuses all rays from a single object point onto a single image point.

51. The failure of a lens to focus all the rays at the same point.
Aberration
The failure of a lens to focus all the rays at the same point.
Object
(A)
(B)
circle of confusion

52. The failure of a lens to focus all the rays at the same point.
Aberration
The failure of a lens to focus all the rays at the same point.
Object
(A)
(B)
circle of confusion
and a blurred image

53. Dispersion: The cause of one kind of aberration called chromatic aberration.
Dispersion - Index of refraction, n, depends on the frequency (wavelength) of light.
So far, we have treated every material as though it has one index of refraction. The truth is that the index of refraction also depends on the frequency of the photon. This is called dispersion and is seen here in a glass prism. Different materials have different degrees of dispersion.
Dispersion is responsible for the colors produced by a prism:
red light “bends” less within the prism, while blue light “bends” more.

54. Chromatic Aberration
Dispersion results in a lens having different focal points for different wavelengths - this effect is called chromatic aberration.
Results in a “halo” of colors.
Solution: Use 2 lenses of different shape and material (“achromatic doublet”).
White light
F’Blue
F’Red
Object (small dot)
Since dispersion causes a different degree of bending for different wavelengths of photons (or colors), it may cause deviations from a perfect image. This is called chromatic aberration, where the lens has different focal lengths for different wavelengths. This in turn causes the image to be surrounded by a halo of colors. A point source is imaged as a halo of colors.
A solution for this aberration is to use a pair of lenses made of two different materials so that all the colors focus at the same point. Such a lens is called an achromatic doublet, and is used in precision optical systems such as photographic lenses.
Image with chromatic aberration .

55. Spherical Aberration: The shape of the lens has to be ideal
Spherical Aberration: The shape of the lens has to be ideal. However, it is easier and cheeper to make lenses that are shaped like the surface of a sphere.
A sphere is almost, but not
quite exactly the ideal shape. F’
Object (small dot)
Another common aberration is spherical aberration. As mentioned before, the surface of the lens is ground as a sphere with a fixed radius. Such a surface performs well as long as the surface remains relatively flat. Otherwise, the rays that hit the near edge of the sphere begin to be refracted too much so that they don’t bend toward the focal point. The resulting image is a blurred image; a point source would be imaged as a blurred spot. Solutions for spherical aberration include use of a lens where the surface is not quite spherical (called aspherical) or, use a compound lens where more than one lens is used.
Image with spherical aberration .