1. Physics 1230: Light and Color Chapter #2: Geometric Optics
Lenses, mirages and fiber optics
Lenses in cameras and contact lenses create images by bending light rays (refraction)
Mirages are also due to the bending of light rays.
Another example of bending light is provided by fiber optics used in communications
Rainbows
Rainbows are produced by the spreading of the different wavelengths in white light (also involves bouncing and bending) as white light enters and exits water droplets
Prisms create rainbows by spreading wavelengths
Principles of Geometric Optics
A fancy name for how rays are used to understand shadows, mirrors, lenses and rainbows
Shadows
Shadows and eclipses are produced when light rays are blocked
A pinhole camera also works by blocking rays
Mirrors
Rays bounce off mirrors to produce images (specular reflection)
Why can't you see your reflection in a white piece of paper but you can in a mirror?

2. Which level of physics is needed to explain what properties of light?
Image formation -
ray theory
Wavelength color, polarization and diffraction -
wave theory (electricity and magneticsm)
Interaction of light with atoms -
quantum theory of photons
Constant speed of light no matter how fast the source or observer is moving -
special theory of relativity

3. A point light source emits rays in all directions radially outwards
The rays from two point light sources look like this
The rays only tell us which direction
the light goes in. We know that the light gets dimmer as you move further away from the light source. (Think of the sun. It would be blinding if we were closer to the sun)

4. Shadows appear when rays are blocked
Wall
Wall
Rays that are NOT blocked by the book
unblocked
bright
penumbra
Rays that ARE blocked by the book
Book A
blocked
Book
Shadow
umbra B
2 point light sources
Point light source
The two parts of the penumbra each get light from only one of the two bulbs. The umbra gets no light from either of the two bulbs. The bright region gets light from both of the bulbs.
Extra credit: what happens to the shadow if I move the screen back from the book?

5. Concept question Shadows tell us:
A) What direction the light is shining from
B) That something is blocking the light
C) That light travels in straight lines
D) A, B, & C
E) A & C
Extra Credit Opportunity: how do we prove this?

6. We can extend the definition of the umbra and penumbra to exist in space even without a wall or screen!
Wall
bright
The light from B doesn't reach this penumbra
penumbra A
Book
umbra B
penumbra
The light from A doesn't reach this penumbra
bright

7. We can think about large light sources as being composed of many small light sources

8. Incident rays from a frosted light bulb
An "extended object" consists of many points. Each point on the object emits or reflects rays in all directions (unless the object is a mirror)
MANY reflected rays come from each point on Alex.
This is diffuse reflection
Incident rays from a frosted light bulb
Many rays from different points on a large frosted lightbulb hit Alex's nose

9. The more rays that reach a point the brighter the point
Light source 2
This is why regions outside the penumbra and umbra are brighter
These regions get light rays from both point light sources
The more lights you turn on the brighter the reflected light from objects in the room
See rays at right
Light source 1
Reflected rays
from light 1
Reflected rays
from light 2
Your eye sees
a brighter nose
than with either
light source alone

10. Rays from this part of the sun DO reach the upper penumbra
An extended light source such as the sun (or a large light bulb) also produces an umbra and penumbra in empty space behind the Earth (or another object)
All rays coming from point A on the sun between the two dashed rays are blocked by Earth
All rays coming from point B on the sun between the two dotted rays are blocked by Earth
The umbra gets no light from any portion of the sun
The umbra gets smaller not larger further behind Earth since the Sun is larger than Earth
The penumbra gets light from part of the sun
If you look back from the penumbra you can see part of the sun
When the moon passes completely into the umbra there is a total eclipse of the moon.
When the moon passes into the penumbra there is a partial eclipse of the moon
Rays from this part of the sun DO reach the upper penumbra A
Penumbra
Sun
Umbra
Rays from this part of the sun DON'T reach the upper penumbra because they are blocked by Earth B
Difference between an eclipse of the moon (Earth's shadow crosses the moon) and an eclipse of the sun (when the moon's shadow crosses Earth) and phases of the moon. Phases of the moon have to do with the fact that half the moon always faces the sun and the moon goes around Earth as Earth goes around on its own axis. Phases have to do with viewing angle changes rather than shadow.

11. Solar eclipse Geometry:
moon
NOTE: The umbra is usually about 200km wide

12. Total solar eclipse
During a solar eclipse, the shadow of the Moon passes over the surface of the Earth. From the Earth, we can see the moon blocking the light of the Sun. Looking at the demonstration, you may think that solar eclipses happen very often. The Sun, Earth, and Moon must be lined up just right, in order for a solar eclipse to take place. This happens only two to five times a year. Since the Moon's shadow is so small, compared to the size of the Earth, a solar eclipse can be seen from only small portions of the Earth.

13. Map of the total solar eclipse, Aug 1, 2008

15. Schedule:

16. Lunar eclipse (partial and total)
During a lunar eclipse, the moon passes through the shadow of the Earth. As we look at the moon from the Earth, it looks to us as if the shadow of the Earth is slowly covering the moon. You may think that lunar eclipses happen very often. However, the Sun, Earth, and Moon must be lined up just right, in order for a lunar eclipse to take place. This happens very rarely. In most years there are only two lunar eclipses that can be seen only from certain places on Earth. In a partial lunar eclipse, the moon passes through the penumbra or part of the umbra. In a total lunar eclipse, the moon is completely within the umbra.

17. Extra Credit Opportunity: Based on what we know about eclipses, how do we prove this?

18. A pinhole camera works by blocking rays (demo)
What is an image?
A real image is formed on a screen when one or more rays from each point on the object reach the corresponding points on the screen and no other rays from other points on the object reach those points
Pinhole Camera
blocked rays
Image of light bulb
Light bulb
Notice that this image is upside down and left-right reversed.
Using shadows…

19. previously blocked rays
If a lens is used instead of a pinhole the image is brighter because many of the previously blocked rays are bent so that they arrive at the correct place on the screen image
Camera with lens
Pinhole Camera
blocked rays
previously blocked rays
Image of light bulb
Light bulb
Not just ONE ray from the filament but MANY now arrive at the corresponding image point so the image is BRIGHTER
#1: Why do we need lenses in the modern cameras?
#2: How do we make a photo of an object that does not emit visible light (i.e., ourselves)?

20. One of many rays of light shining on Alex
The object photographed with a pinhole camera does not have to be self-luminous!
One of many rays of light shining on Alex
Pinhole Camera
blocked rays
Image of Alex
Reflected rays off the real Alex go through the hole
and make the image
Alex
Once again this image is upside down and left-right reversed. Early photographs (daguerreotypes) were always left-right reversed;
Note the correspondence between the distances object-camera-screen and image vs. object sizes

21. Finding an image by using rays is called ray tracing
Finding an image by using rays is called ray tracing. Trace rays from the object through the pinhole in the camera to find the image rather than trusting your intuition!
Is the image of Alex smaller or larger than the real Alex?
Smaller
Larger
Same size
Is the image of Alex smaller or larger than the real Alex?
Smaller
Larger
Same size

22. Extra Credit Project (20 points):
Construct a camera on your own (see textbook for details, pages 35 & 36)

23. Materials like metals with many mobile electrons can cancel out the light wave field in the forward direction so there is no transmission but only reflection at certain wavelengths.
Metals reflect all waves below a certain frequency
the plasma frequency - which varies from metal to metal
Silver is particularly interesting because it reflects light waves at all visible frequencies
Its plasma frequency is at the top of the violet so it reflects all of the wavelengths below and appears whitish
Gold and copper have a yellowish-brownish color because they reflect greens, yellows and reds but not blues or violets
Red and green make yellow
Plasma frequency of silver
Plasma frequency of gold
Plasma frequency of copper

24. What is a mirror?
Since silver is such a good reflector a coating of silver on glass makes a good (common) mirror.
If the silver coating is thin enough the mirror can be made to transmit 50% of the light and to reflect the other 50%
This is called a half-silvered mirror
A half-silvered mirror used with proper lighting can show objects on one side or the other of the mirror

25. Law of specular reflection of a ray from a mirror
One of many rays from a light bulb hits Alex's chin.
The ray from the light bulb is diffusely reflected off his chin. We show one of the many rays coming off his chin hitting a mirror.
This is called an incident ray
The normal to the mirror is an imaginary line drawn perpendicular to it from where the incident ray hits the mirror
Normal
This angle
= this angle
The incident ray undergoes specular reflection off the mirror
Note the reflected ray
Mirror
Draw the normal to the mirror
The angle of incidence = the angle of reflection

26. How is an image produced in a mirror? Part 1: Ray-tracing
To find out how Bob "sees" Alex by looking in the mirror we trace rays which obey the law of reflection
Consider an incident ray from Alex's chin which reflects according to the law of reflection at a specific point on the mirror and goes into Bob's eye.
Note - it is not easy to construct this ray! You cannot arbitrarly choose a point on the mirror and expect that the law of reflection will be satisfied
Bob will see only this reflected ray from Alex's chin.
Other refelected rays from Alex's chin will miss his eye (see right)
A ray from Alex's hair will reflect at one point on the mirror into Alex's eye (and satisfies the law of reflection)
Bob looks at Alex's image
Alex
Mirror

27. Bob looks at Alex's image
How is an image produced in a mirror? Part 2: The psychology of ray interpretation
To find the image of Alex we must learn how Bob’s eye (and our eyes) interpret rays
Bob cannot directly know whether the rays entering his eyes have been reflected or not!
We interpret all rays coming into our eye as traveling from a fictitious image in a straight line to our eye even if they are reflected rays!
To find the virtual (fictitious) image of Alex’s chin we extend each reflected ray backwards in a straight line to where there are no real rays
Extend the ray reflected into Bob's eye from Alex's chin backward behind the mirror.
Extend the ray reflected towards Bob's chest (why?) from Alex's chin backward (dashed line) behind the mirror.
The image of Alex's chin will be behind the mirror at the intersection of the two backward-extended reflected rays.
Note all reflected rays from his chin intersect at the same image pt. when extended backwards
Bob looks at Alex's image
Alex
Mirror
• To find the location of his hair in the virtual image we extend any reflected ray from his hair backwards

28. How is an image produced in a mirror
How is an image produced in a mirror? Part 3: The meaning of a virtual image
If we trace rays for every ray from every part of Alex which reflects in the mirror
we get a virtual image of the real Alex behind the mirror. It is virtual because there is no light energy there, no real rays reach it, and it cannot be seen by putting a screen at its position!!
Bob looks at Alex's image
Alex
Bob sees Alex's image in the same place when
he moves his head
When all of the reflected rays from Alex's chin are traced backwards they all appear to come from the virtual image of Alex’s chin
Hence Alex's image is always in the same place regardless of where Bob looks
The image chin is behind the mirror by a distance = to the distance the real chin is in front of the mirror
This is true for all parts of Alex's image
Alex's virtual image is the same size as the real Alex
Alex's image is further away from Bob than the real Alex
Mirror
Virtual image of Alex is behind mirror

29. For simple (flat) mirrors the image location is therefore predictable without knowing where the observer's eye is and without ray-tracing
Mirror
Mirror
Mirror
Mirror

30. Homework HW2 due today after the class
New homework HW3 assigned today: see the course web page;
HW1 – graded (10 points - 100% maximum);

31. Multiple mirrors - a virtual image can act as a real object and have its own virtual image
Question: Where are the images of Alex in the 2 mirrors?
At A only
At B only
At A and B only
At C only
At A, B and C
Mirror
Alex A C B
Mirror
The virtual image at A acts as an object to produce the virtual image of C. It acts as an intermediate image. More precisely it is the red rays which reflect as green rays.

32. A few words about virtual images
Here is the real Alex
Here are some (diffusely reflected) diverging rays coming off his nose
They can be seen by eyes at various locations
We only know his nose is there because our eyes receive the rays
Therefore, we would see an image (virtual) of Alex if those rays reached our eyes even when he wasn't there.
Mirrors can provide those rays!
The (imaginary) extension of (reflected) rays behind the mirror look just like the real rays from the real Alex
Mirror (incident rays not shown)
Use flat mirror to demonstrate difference between real and virtual image of my face after this slide

33. Speed of Light in free space & in materials
Note: the fact that light has a unique speed was not obvious at the beginning . At that time, the real questions were: 1. Is the speed of light infinite or finite? 2. If the speed is finite, is it variable or a constant? Everyday experience suggests that speed is very great and might be infinite. How one can measure speed of light?
General method: speed= distance/time
The difficulty: when speed is very great, distance must be large and/or clock must have lots of resolution;
Echo Method to measure the speed of a sound wave?
Michelson: clock is rotating mirror wheel; result= 299,774 km/s

34. Echo method : measuring the speed of light

36. Reflection of waves occurs where the medium of propagation changes abruptly
Part of the wave can be transmitted into the second medium while part is reflected back
You can hear someone from outside the pool when you are underwater because sound waves are transmitted from the air through the water (with different speed in each).
When light waves are incident on a glass slab they are mostly transmitted but partly reflected (about 4%)!
Glass slab
Is the speed of light in the glass slab the same as in the free space???
No.

37. How can reflection require that the speed of the wave changes
How can reflection require that the speed of the wave changes? We thought the speed of light was always c = 3 x 108 m/s!
The speed of an electromagnetic (EM) wave is constant (for every wavelength) in empty space!
The speed of light is slower than c in glass, water and other transparent media
(Einstein showed that light can never travel faster than c)
The speed of light in a medium is v = c/n, where n is a number larger than one called the index of refraction
n = 1.5 for glass
n = for water
n = for vegetable oil
Light is reflected and transmitted at a boundary because
When a light wave travels in a medium the electric field of the light jiggles the electrons in the medium.
This produces new electric fields which can cancel or add to the original light wave both in the forward and backward directions
These are the transmitted and reflected light waves

38. Refractive indices of different materials
Can we see a glass rod immersed into the oil with the same refractive index?
A. Yes
B. No
Material
Refractive Index
Air
1.0008
Water
1.330
Glass
1.5
Diamond
2.417
Ruby
1.760
Oil

39. Refraction is the bending of a ray after it enters a medium where its speed is different
A ray going from a fast medium to a slow medium bends towards the normal to the surface of the medium
A ray going from a slow medium to a fast medium bends away from the normal to the surface of the medium
The speed of light in a medium is v = c/n, where n is a number larger than one called the index of refraction and c = 3 x 108 m/s
n = 1.3 for glass
n = for water
Hence, a ray going into a medium with a higher index of refraction bends towards the normal and a ray going into a medium with a lower index of refraction bends away from the normal
Air (fast medium)
Normal
Glass or
water (slow)
nair < nwater
< 1.33
Normal
Air (fast medium)
Glass or
water (slow)
How about light going into a medium with exactly the same index of refraction?

40. Ray-bending together with our psychological straight-ray interpretation determine the location of images underwater
The precise amount of bending is determined by the law of refraction (sometimes called Snell's law):
ni sinqi = nt sinqt
Here, qi = angle between incident ray and normal,
and qt = angle between transmitted ray and normal
ni and nt are the indices of refraction in the medium containing the incident ray and in the medium containing the transmitted ray
Fig 2.49 Fisherman and fish
normal
transmitted ray
image of fish for someone out of water
incident ray
fish
In order to observe the fish from outside the water a transmitted ray must enter your eye.
You will think it comes from a point obtained by tracing it backwards,
Extend any 2 of the many many transmitted rays from the fish backwards to find the image of the fish (where they intersect).
The location of that image will be the same for any observer outside of the water.

41. What we see and how different it can be from what it seems to be
The woman will see the underwater part of body being
Smaller than it really is;
Much larger than it really is;
Of natural size;
The boy will see the underwater part of body being
Smaller than it really is;
Much larger than it really is;
Of natural size;

42. For the glass-air interface
Total internal reflection is an extreme case of a ray bending away from the normal as it goes from a higher to a lower index of refraction medium (from a slower to a faster medium)
Just below the critical angle for total internal reflection there is a reflected and a transmitted (refracted) ray
Normal
Air (fast medium)
Critical angle
Glass or
water (slow)
Just above the critical angle for total
internal reflection there is a reflected ray
but no transmitted (refracted) ray
Normal
Glass or
water (slow)
For the glass-air interface
Angles are made with the normal

43. Total internal reflection
The precise amount of bending is determined by the law of refraction (sometimes called Snell's law):
ni sinqi = nt sinqt
Here, qi = angle between incident ray and normal,
and qt = angle between transmitted ray and normal
ni and nt are the indices of refraction in the medium containing the incident ray and in the medium containing the transmitted ray
Show that the internal reflection is a consequence of the Snell’s law

44. What we see and how different it can be from what it seems to be
If the critical angle condition is satisfied, will the boy see the part of body above water:
yes;
No.
Extra Credit:
Refractive index of water is 1.33; What is the critical angle for the case of air-water interface?

45. 0ptical fibers: how they work
Application of total internal reflection phenomenon;
Telecommunications;
Can see inside a body of a living person – applications in medicine;
Total internal reflection

46. Mirages can occur if the index of refraction increases as the ray goes deeper into the slower medium?
Refraction of light
Light propagates along a curved line because of the multiple refraction phenomena
However, in our mind, analyzing what we see, we do not take this into account and assume that light entering our eyes traveled along straight lines
Higher index
Higher index
High index
Incident ray

47. Morris, in Heavens Command, describes a view from Grosse Isle in the Gulf of St. Lawrence : "in the early morning sun the islands are inverted in mirage, and seem to hang there, suspended between sky and water." In the early morning, the air gets warm faster than the water, so there is warmer air above the cooler air that is lying just above the water. How do the light rays bend, resulting in the mirage Morris describes.
Since the index of refraction decreases at higher altitudes the red ray coming from the island continuously bends away from the (vertical) normal.
Think of your car with right wheels on a good road and left wheels on the muddy terrain
At the top of its trajectory critical reflection occurs and the reflected ray then refracts towards the normal on its way down until it reaches the eye.
The brain then interprets this curved ray as a straight ray tangent to the curved one at the point of entry into the eye.
Hence, the island appears to be in the sky.
Hot air
Sky
Grosse Isle
Cool air
Water
Image of Grosse Isle
Hot air, low density, index of refraction closer to 1, FAST medium
Sky
Cool air, higher density,
index of refraction > 1, SLOW medium
Grosse Isle

48. A closer look at the curved ray from Grosse Isle to the eye of the beholder
As the curved ray moves up and bends towards the horizon it is bending away from the normal
Think of marching soldiers leaving deepest mud
Fast
At the top of its trajectory the ray undergoes total internal reflection
Then the curved ray begins to move down and towards the normal.
We can see sun after it is below the geometrical horizon:
Slow
High index
Highest index

49. More common mirages occur when the lower air is hotter than the cooler air
Lower temp means higher density
Higher index
Slower light speed
When the road (or desert) is hot the air is thin close to the road (or desert) so the density is lower.
The density rises slowly above the road to heights which are cooler.
The index of refraction of air increases as the density increases.
A mirage results because we interpret the last ray to enter our eye as having traveled a straight line rather than a curve
Ray bends away from normal
Higher temp means lower density
Lower index
Faster light speed
HOT ROAD

50. Dispersion is responsible for rainbows
Dispersion causes the spreading of all of the colors in white light
This can be done by a prism to create a spectrum or by raindrops to create a rainbow
Dispersion occurs because the speed of short wavelength light (blues) is slightly slower than that of long wavelengths (reds) in glass or water
Hence short wavelengths (blues) bend more towards the normal than long wavelengths (reds) when white light enters glass or water. Blues also bend more away from the normal than reds when leaving glass or water

51. What is the normal to a curved surface and how is it used to find rays?
To find the normal to a curved surface at a point where a ray hits that surface (and will be reflected or refracted)
First draw a tangent line to the curve (or tangent plane to the surface)
The normal is perpendicular to that line or plane and going through the point
Once you have drawn the normal you can draw the reflected or refracted ray

52. How does a single raindrop contribute to a rainbow?
White light enters the waterdrop
Remember white light contains rays of all wavelengths
The blue ray is refracted closer to the normal than the red ray
Remember light travels slower in water than in air
This greater bending of the blue ray than the red ray is called dispersion
We have not shown the green, yellow, and orange rays but they each bend by amounts more than the red and less than the blue
All of the rays reflect off the inside of the raindrop and then undergo another dispersion as they exit the raindrop
The laws of reflection and refraction are always obeyed
Raindrop
Dispersion occurs
here during refraction
white light comes in
Reflections
A spectrum of colors comes out

53. How we see all the colors in the rainbow?

54. Primary & Secondary Rainbows
Note inverted color sequence;
Secondary
Raindrop
Dispersion occurs
here during refraction
white light comes in
Reflections
A spectrum of colors comes out
Primary

55. Question about the Rainbow
The colors in rainbow are caused by:
(A) Dispersion of light;
(B) Absorption of light;
(C) Total internal reflection and dispersion of light;
(D) Light sources of different color;
(E) Scattering;

56. Scattering of light: how we see fog, clouds, milk
Clouds –water droplets, crystals of ice, or both;
Fog – water droplets;
There are tiny droplets in milk too;
Why we see clouds, fog, and milk being of more or less white color?
Small droplets scatter light of all colors reflected in all directions;

57. Question A pen semi-immersed in water appears bent because of light
A. reflection;
B. refraction;
C. dispersion;
D. A,B, &C;
E. total internal reflection;

58. Demonstrations Rainbow;
Dispersion of light after going through a prism;
Retro-reflector;
Half-silvered mirror;

59. Extra credit project: magic box or periscope (p. 44 & p
Extra credit project: magic box or periscope (p.44 & p.52 of the textbook)
Periscope – how could we have skyscrapers & perfect mountain views in Boulder at the same time;
20 points extra credit;
To be submitted before the last exam;
Magic box: can show on Pearl street

60. Read chapter 2 of the textbook
Wave propagation in different media and reflection/refraction at interfaces (1st HW problem);
Mirages;
Sunsets;
Subsuns;
Sun pillars;
Reflections of diamonds;
Dry vs. wet road and diffuse vs. specular reflection;
Sun dogs, halos, & more;
HW #3 is due on Thursday;
Chapter # 3: mirrors and lenses;