Optics

1
Optics

• Reflection
• Diffuse reflection
• Refraction
• Index of refraction
• Speed of light
• Snells law
• Geometry problems
• Critical angle
• Total internal reflection
• Brewster angle
• Fiber optics
• Mirages
• Dispersion

• Prisms
• Rainbows
• Plane mirrors
• Spherical aberration
• Concave and convex mirrors
• Focal length radius of curvature
• Mirror / lens equation
• Convex and concave lenses
• Human eye
• Chromatic aberration
• Telescopes
• Huygens principle
• Diffraction

2
Reflection
Most things we see are thanks to reflections,
since most objects dont produce their own
visible light. Much of the light incident on an
object is absorbed but some is reflected. the
wavelengths of the reflected light determine the
colors we see. When white light hits an apple,
for instance, primarily red wavelengths are
reflected, while much of the others are
absorbed. A ray of light heading towards an
object is called an incident ray. If it reflects
off the object, it is called a reflected ray. A
perpendicular line drawn at any point on a
surface is called a normal (just like with normal
force). The angle between the incident ray and
normal is called the angle of incidence, i, and
the angle between the reflected ray and the
normal ray is called the angle of reflection, r.
The law of reflection states that the angle of
incidence is always equal to the angle of
reflection.
3
Law of Reflection
Normal line (perpendicular to surface)
r
i
reflected rays
incident rays
i r
4
Diffuse Reflection
Diffuse reflection is when light bounces off a
non-smooth surface. Each ray of light still
obeys the law of reflection, but because the
surface is not smooth, the normal can point in a
different for every ray. If many light rays
strike a non-smooth surface, they could be
reflected in many different directions. This
explains how we can see objects even when it
seems the light shining upon it should not
reflect in the direction of our eyes. It also
helps to explain glare on wet roads Water fills
in and smoothes out the rough road surface so
that the road becomes more like a mirror.
5
Speed of Light Refraction
As you have already learned, light is extremely
fast, about 3 ? 108 m/s in a vacuum. Light,
however, is slowed down by the presence of
matter. The extent to which this occurs depends
on what the light is traveling through. Light
travels at about 3/4 of its vacuum speed (0.75 c
) in water and about 2/3 its vacuum speed (0.67 c
) in glass. The reason for this slowing is
because when light strikes an atom it must
interact with its electron cloud. If light
travels from one medium to another, and if the
speeds in these media differ, then light is
subject to refraction (a changing of direction at
the interface).
Refraction of light waves
Refraction of light rays
6
Reflection Refraction
At an interface between two media, both
reflection and refraction can occur. The angles
of incidence, reflection, and refraction are all
measured with respect to the normal. The angles
of incidence and reflection are always the same.
If light speeds up upon entering a new medium,
the angle of refraction, ?r , will be greater
than the angle of incidence, as depicted on the
left. If the light slows down in the new medium,
?r will be less than the angle of incidence, as
shown on the right.
Reflected Ray
Reflected Ray
Incident Ray
Incident Ray
?r
Refracted Ray
normal
normal
Refracted Ray
?r
7
Axle Analogy
Imagine youre on a skateboard heading from the
sidewalk toward some grass at an angle. Your
front axle is depicted before and after entering
the grass. Your right contacts the grass first
and slows, but your left wheel is still moving
quickly on the sidewalk. This causes a turn
toward the normal. If you skated from grass to
sidewalk, the same path would be followed. In
this case your right wheel would reach the
sidewalk first and speed up, but your left wheel
would still be moving more slowly. The result
this time would be turning away from the normal.
Skating from sidewalk to grass is like light
traveling from air to a more
overhead view
optically dense medium like glass or water. The
slower light travels in the new medium, the more
it bends toward the normal. Light traveling from
water to air speeds up and bends away from the
normal. As with a skateboard, light traveling
along the normal will change speed but not
direction.
sidewalk
grass
?r
8
Index of Refraction, n
The index of refraction of a substance is the
ratio of the speed in light in a vacuum to the
speed of light in that substance
Medium Vacuum Air (STP) Water (20º
C) Ethanol Glass Diamond
n 1 1.00029 1.33 1.36 1.5 2.42
n Index of Refraction c Speed of light in
vacuum v Speed of light in medium
Note that a large index of refraction corresponds
to a relatively slow light speed in that medium.
9
Snells Law
?i
ni
nr
?r
Snells law states that a ray of light bends in
such a way that the ratio of the sine of the
angle of incidence to the sine of the angle of
refraction is constant. Mathematically, ni sin?
i nr sin?r Here ni is the index of refraction
in the original medium and nr is the index in the
medium the light enters. ? i and ?r are the
angles of incidence and refraction, respectively.
Willebrord Snell
10
Snells Law Derivation
Two parallel rays are shown. Points A and B are
directly opposite one another. The top pair is at
one point in time, and the bottom pair after time
t. The dashed lines connecting the pairs are
perpendicular to the rays. In time t, point A
travels a distance x, while point B travels a
distance y. sin?1 x / d, so x d
sin?1 sin?2 y / d, so y d sin?2 Speed of
A v1 x / t Speed of B v2 y / t
Continued
11
Snells Law Derivation (cont.)
v1 x / t x sin?1



So,
v2 y / t y sin?2
v1 / c sin?1 1 / n1 sin?1
n2



?
v2 / c sin?2 1 / n2 sin?2
n1
? n1 sin?1 n2 sin?2
12
Refraction Problem 1
Goal Find the angular displacement of the ray
after having passed through the prism. Hints

• Find the first angle of refraction using Snells
  law.
• Find angle ø. (Hint Use Geometry skills.)
• 3. Find the second angle of incidence.
• Find the second angle of refraction, ?, using
  Snells Law

19.4712º
79.4712º
Air, n1 1
30
10.5288º
Horiz. ray, parallel to base
ø
15.9º
?
Glass, n2 1.5
13
Refraction Problem 2
Goal Find the distance the light ray displaced
due to the thick window and how much time it
spends in the glass. Some hints are given.
?1
20º
20º
1. Find ?1 (just for fun). 2. To show incoming
outgoing rays are parallel, find ?. 3. Find
d. 4. Find the time the light spends in
the glass. Extra practice Find ? if bottom
medium is replaced with air.
H20
n1 1.3
20º
0.504 m
glass
10m
n2 1.5
5.2 10-8 s
d
H20
?
26.4º
14
Refraction Problem 3
Goal Find the exit angle relative to the
horizontal.
19.8
?
36
air
? ?
glass
The triangle is isosceles.Incident ray is
horizontal, parallel to the base.
15
Reflection Problem
Goal Find incident angle relative to horizontal
so that reflected ray will be vertical.
?
10º
?
50º
center of semicircular mirror with horizontal base
16
Brewster Angle
The Brewster angle is the angle of incidence the
produces reflected and refracted rays that are
perpendicular.
From Snell, n1 sin?b n2 sin?.
a ?b since ? ? 90º, and ?b
? 90º. ß ? since ? ? 90º, and
? ? 90º. Thus, n1 sin?b n2 sin? n2 sin?
n2 cos?b
tan?b n2 / n1
Sir David Brewster
17
Critical Angle
The incident angle that causes the refracted ray
to skim right along the boundary of a substance
is known as the critical angle, ?c. The critical
angle is the angle of incidence that produces an
angle of refraction of 90º. If the angle of
incidence exceeds the critical angle, the ray is
completely reflected and does not enter the new
medium. A critical angle only exists when light
is attempting to penetrate a medium of higher
optical density than it is currently traveling
in.
nr
ni
?c
From Snell, n1 sin?c n2 sin 90?
Since sin 90? 1, we have n1 sin?c n2 and
the critical angle is
nr
?c sin-1
ni
18
Critical Angle Sample Problem
Calculate the critical angle for the diamond-air
boundary.
Refer to the Index of Refraction chart for the
information.
?c sin-1 (nr / ni) sin-1 (1 / 2.42)
24.4? Any light shone on this boundary beyond
this angle will be reflected back into the
diamond.
air
diamond
?c
19
Total Internal Reflection
Total internal reflection occurs when light
attempts to pass from a more optically dense
medium to a less optically dense medium at an
angle greater than the critical angle. When this
occurs there is no refraction, only reflection.
n1
n1
n2 gt
n2
? gt ?c
?
Total internal reflection can be used for
practical applications like fiber optics.
20
Fiber Optics
Fiber optic lines are strands of glass or
transparent fibers that allows the transmission
of light and digital information over long
distances. They are used for the telephone
system, the cable TV system, the internet,
medical imaging, and mechanical engineering
inspection.
spool of optical fiber
Optical fibers have many advantages over copper
wires. They are less expensive, thinner,
lightweight, and more flexible. They arent
flammable since they use light signals instead of
electric signals. Light signals from one fiber
do not interfere with signals in nearby fibers,
which means clearer TV reception or phone
conversations.
A fiber optic wire
Continued
21
Fiber Optics Cont.
Fiber optics are often long strands of very pure
glass. They are very thin, about the size of a
human hair. Hundreds to thousands of them are
arranged in bundles (optical cables) that can
transmit light great distances. There are three
main parts to an optical fiber

• Core- the thin glass center where light travels.
  
• Cladding- optical material (with a lower index
  of refraction than the core) that surrounds the
  core that reflects light back into the core.
• Buffer Coating- plastic coating on the outside
  of an optical fiber to protect it from damage.

Continued
22
Fiber Optics (cont.)
Light travels through the core of a fiber optic
by continually reflecting off of the cladding.
Due to total internal reflection, the cladding
does not absorb any of the light, allowing the
light to travel over great distances. Some of
the light signal will degrade over time due to
impurities in the glass.
There are two types of optical fibers

• Single-mode fibers- transmit one signal per
  fiber (used in cable TV and telephones).
• Multi-mode fibers- transmit multiple signals per
  fiber (used in computer networks).

23
Mirage Pictures
Mirages
24
Mirages
Mirages are caused by the refracting properties
of a non-uniform atmosphere. Several examples of
mirages include seeing puddles ahead on a hot
highway or in a desert and the lingering daylight
after the sun is below the horizon.
More Mirages
Continued
25
Inferior Mirages
A person sees a puddle ahead on the hot highway
because the road heats the air above it, while
the air farther above the road stays cool.
Instead of just two layers, hot and cool, there
are really
many layers, each slightly hotter than the layer
above it. The cooler air has a slightly higher
index of refraction than the warm air beneath it.
Rays of light coming toward the road gradually
refract further from the normal, more parallel to
the road. (Imagine the wheels and axle on a
light ray coming from the sky, the left wheel is
always in slightly warmer air than the right
wheel, so the left wheel continually moves
faster, bending the axle more and more toward the
observer.) When a ray is bent enough, it
surpasses the critical angle and reflects. The
ray continues to refract as it heads toward the
observer. The puddle is really just an inverted
image of the sky above. This is an example of an
inferior mirage, since the cool are is above the
hot air.
26
Superior Mirages
Superior mirages occur when a layer of cool air
is beneath a layer of warm air. Light rays are
bent downward, which can make an object seem to
be higher in the air and inverted. (Imagine the
wheels and axle on a ray coming from the boat
the right wheel is continually in slightly warmer
air than the left wheel. Thus, the right wheel
moves slightly faster and bends the axle toward
the observer.) When the critical angle is
exceeded the ray reflects. These
mirages usually occur over ice, snow, or cold
water. Sometimes superior images are produced
without reflection. Eric the Red, for example,
was able to see Greenland while it was below the
horizon due to the light gradually refracting and
following the curvature of the Earth.
27
Sunlight after Sunset
Lingering daylight after the sun is below the
horizon is another effect of refraction. Light
travels at a slightly slower speed in Earths
atmosphere than in space. As a result, sunlight
is refracted by the atmosphere. In the morning,
this refraction causes sunlight to reach us
before the sun is actually above the horizon. In
the evening, the
Apparent position of sun
Observer
Actual position of sun
Earth
Atmosphere
sunlight is bent above the horizon after the sun
has actually set. So daylight is extended in the
morning and evening because of the refraction of
light. Note the picture greatly exaggerates this
effect as well as the thickness of the atmosphere.
Different shapes of Sun
28
Dispersion of Light
Dispersion is the separation of light into a
spectrum by refraction. The index of refraction
is actually a function of wavelength. For longer
wavelengths the index is slightly small. Thus,
red light refracts less than violet. (The pic is
exaggerated.) This effect causes white light to
split into it spectrum of colors. Red light
travels the fastest in glass, has a smaller index
of refraction, and bends the least. Violet is
slowed down the most, has the largest index, and
bends the most. In other words the higher the
frequency, the greater the bending.
Animation
29
Atmospheric Optics
There are many natural occurrences of light
optics in our atmosphere.
One of the most common of these is the rainbow,
which is caused by water droplets dispersing
sunlight. Others include arcs, halos, cloud
iridescence, and many more.
Photo gallery of atmospheric optics.
30
Rainbows
A rainbow is a spectrum formed when sunlight is
dispersed by water droplets in the atmosphere.
Sunlight incident on a water droplet is
refracted. Because of dispersion, each color is
refracted at a slightly different angle. At the
back surface of the droplet, the light undergoes
total internal reflection. On the
way out of the droplet, the light is once more
refracted and dispersed. Although each droplet
produces a complete spectrum, an observer will
only see a certain wavelength of light from each
droplet. (The wavelength depends on the relative
positions of the sun, droplet, and observer.)
Because there are millions of droplets in the
sky, a complete spectrum is seen. The droplets
reflecting red light make an angle of 42o with
respect to the direction of the suns rays the
droplets reflecting violet light make an angle of
40o.
Rainbow images
31
Primary Rainbow
32
Secondary Rainbow
Secondary
Primary
Alexanders dark region
33
Supernumerary Arcs
Supernumerary arcs are faint arcs of color just
inside the primary rainbow. They occur when the
drops are of uniform size. If two light rays in a
raindrop are scattered in the same direction but
have take different paths within the drop, then
they could interfere with each other
constructively or destructively. The type of
interference that occurs depends on the
difference in distance traveled by the rays. If
that difference is nearly zero or a multiple of
the wavelength, it is constructive, and that
color is reinforced. If the difference is close
to half a wavelength, there is destructive
interference.
34
Real vs. Virtual Images
Real images are formed by mirrors or lenses when
light rays actually converge and pass through the
image. Real images will be located in front of
the mirror forming them. A real image can be
projected onto a piece of paper or a screen. If
photographic film were placed here, a photo could
be created.
Virtual images occur where light rays only appear
to have originated. For example, sometimes rays
appear to be coming from a point behind the
mirror. Virtual images cant be projected on
paper, screens, or film since the light rays do
not really converge there. Examples are
forthcoming.
35
Plane Mirror
Object
Rays emanating from an object at point P strike
the mirror and are reflected with equal angles of
incidence and reflection. After reflection, the
rays continue to spread. If we extend the rays
backward behind the mirror, they will intersect
at point P, which is the image of point P. To an
observer, the rays appear to come from point P,
but no source is there and no rays actually
converging there . For that reason, this image at
P is a virtual image.
P
P
Virtual Image
do
di
O
I
The image, I, formed by a plane mirror of an
object, O, appears to be a distance di , behind
the mirror, equal to the object distance do.
Animation
Continued
36
Plane Mirror (cont.)
Two rays from object P strike the mirror at
points B and M. Each ray is reflected such that
i r.
P
Triangles BPM and BPM are congruent by ASA (show
this), which implies that do di and h h.
Thus, the image is the same distance behind the
mirror as the object is in front of it, and the
image is the same size as the object.
do
di
P
B
h
M
h
Image
Object
object
image
Mirror
With plane mirrors, the image is reversed left to
right (or the front and back of an image ). When
you raise your left hand in front of a mirror,
your image raises its right hand. Why arent top
and bottom reversed?
37
Concave and Convex Mirrors
Concave and convex mirrors are curved mirrors
similar to portions of a sphere.
light rays
light rays
Concave mirrors reflect light from their inner
surface, like the inside of a spoon.
Convex mirrors reflect light from their outer
surface, like the outside of a spoon.
38
Concave Mirrors

• Concave mirrors are approximately spherical and
  have a principal axis that goes through the
  center, C, of the imagined sphere and ends at the
  point at the center of the mirror, A. The
  principal axis is perpendicular to the surface of
  the mirror at A.

• CA is the radius of the sphere,or the radius of
  curvature of the mirror, R .
• Halfway between C and A is the focal point of
  the mirror, F. This is the point where rays
  parallel to the principal axis will converge when
  reflected off the mirror.
• The length of FA is the focal length, f.
• The focal length is half of the radius of the
  sphere (proven on next slide).

39
r 2 f
To prove that the radius of curvature of a
concave mirror is twice its focal length, first
construct a tangent line at the point of
incidence. The normal is perpendicular to the
tangent and goes through the center, C. Here, i
r ?. By alt. int. angles the angle at C is
also ?, and a 2 ß. s is the arc length from
the principle axis to the pt. of incidence. Now
imagine a sphere centered at F with radius f.
If the incident ray is close to the principle
axis, the arc length of the new sphere is about
the same as s. From s r ?, we have s r ß
and s ? f a 2 f ß. Thus, r ß ? 2 f ß, and
r 2 f.
tangent line
?
?
s
?
?


f
C
F
r
40
Focusing Light with Concave Mirrors
Light rays parallel to the principal axis will be
reflected through the focus (disregarding
spherical aberration, explained on next slide.)


In reverse, light rays passing through the focus
will be reflected parallel to the principal axis,
as in a flood light.
Concave mirrors can form both real and virtual
images, depending on where the object is located,
as will be shown in upcoming slides.
41
Spherical Aberration
F


C
Spherical Mirror
Parabolic Mirror
Only parallel rays close to the principal axis of
a spherical mirror will converge at the focal
point. Rays farther away will converge at a point
closer to the mirror. The image formed by a large
spherical mirror will be a disk, not a point.
This is known as spherical aberration.
Parabolic mirrors dont have spherical
aberration. They are used to focus rays from
stars in a telescope. They can also be used in
flashlights and headlights since a light source
placed at their focal point will reflect light in
parallel beams. However, perfectly parabolic
mirrors are hard to make and slight errors could
lead to spherical aberration.
Continued
42
Spherical vs. Parabolic Mirrors
Parallel rays converge at the focal point of a
spherical mirror only if they are close to the
principal axis. The image formed in a large
spherical mirror is a disk, not a point
(spherical aberration).
Parabolic mirrors have no spherical aberration.
The mirror focuses all parallel rays at the focal
point. That is why they are used in telescopes
and light beams like flashlights and car
headlights.
SPHERICAL vs. PARABOLIC
43
Concave Mirrors Object beyond C
The image formed when an object is placed beyond
C is located between C and F. It is a real,
inverted image that is smaller in size than the
object.
object


C
F
Animation 1
Animation 2
44
Concave Mirrors Object between C and F
The image formed when an object is placed between
C and F is located beyond C. It is a real,
inverted image that is larger in size than the
object.
object


C
F
Animation 1
Animation 2
45
Concave Mirrors Object in front of F
The image formed when an object is placed in
front of F is located behind the mirror. It is a
virtual, upright image that is larger in size
than the object. It is virtual since it is
formed only where light rays seem to be diverging
from.
object


C
F
Animation
46
Concave Mirrors Object at C or F
What happens when an object is placed at C?
Animation


What happens when an object is placed at F?
47
Convex Mirrors

• A convex mirror has the same basic properties as
  a concave mirror but its focus and center are
  located behind the mirror.
• This means a convex mirror has a negative focal
  length (used later in the mirror equation).
• Light rays reflected from convex mirrors always
  diverge, so only virtual images will be formed.

• Rays parallel to the principal axis will reflect
  as if coming from the focus behind the mirror.
• Rays approaching the mirror on a path toward F
  will reflect parallel to the principal axis.

48
Convex Mirror Diagram
The image formed by a convex mirror no matter
where the object is placed will be virtual,
upright, and smaller than the object. As the
object is moved closer to the mirror, the image
will approach the size of the object.
object
image


C
F
49
Mirror/Lens Equation Derivation
From ?PCO, ? ? ?, so 2? 2? 2?. From
?PCO, ? 2? ? , so -? -2? - ?.
Adding equations yields 2? - ? ?.
P
From s r ?, we have s r ß, s ? di a, and
s ? di a (for rays close to the principle
axis). Thus
?
object
s
?
?
?
?

T
O
C
image
di
do
(cont.)
50
Mirror/Lens Equation Derivation (cont.)
From the last slide, ? s / r, ? ? s / d0 , ?
? s / di , and 2 ß - ? ?.
Substituting into the
last equation yields
P
2s
s
s
-
?
object

r
s
?
di
do
?
?
?
1
2
1



r
T
do
di
O
C
image
1
1
2


2f
do
di
di
1
1
1


f
do
di
do
The last equation applies to convex and concave
mirrors, as well as to lenses, provided a sign
convention is adhered to.
51
Mirror Sign Convention
f focal length di image distance do object
distance
1
1
1


f
do
di
for real image - for virtual image
di
for concave mirrors - for convex mirrors
f
52
Magnification
By definition,
m magnification hi image height (negative
means inverted) ho object height
Magnification is simply the ratio of image height
to object height. A positive magnification means
an upright image.
53
Magnification Identity
To derive this lets look at two rays. One hits
the mirror on the axis. The incident and
reflected rays each make angle ? relative to the
axis. A second ray is drawn through the center
and is reflected back on top of itself (since a
radius is always perpendicular to an tangent line
of a
circle). The intersection of the reflected
rays determines the location of the tip of the
image. Our result follows from
similar triangles, with the negative sign a
consequence of our sign convention. (In this
picture hi is negative and di is positive.)
object
?
ho

C
image, height hi
di
do
54
Mirror Equation Sample Problem
Suppose AllStar, who is 3 and a half feet tall,
stands 27 feet in front of a concave mirror with
a radius of curvature of 20 feet. Where will his
image be reflected and what will its size be?
di
15.88 feet
hi
-2.06 feet
55
Mirror Equation Sample Problem 2
Casey decides to join in the fun and she finds a
convex mirror to stand in front of. She sees her
image reflected 7 feet behind the mirror which
has a focal length of 11 feet. Her image is 1
foot tall. Where is she standing and how tall is
she?
do
19.25 feet
ho
2.75 feet
56
Lenses
Lenses are made of transparent materials, like
glass or plastic, that typically have an index of
refraction greater than that of air. Each of a
lens two faces is part of a sphere and can be
convex or concave (or one face may be flat). If
a lens is thicker at the center than the edges,
it is a convex, or converging, lens since
parallel rays will be converged to meet at the
focus. A lens which is thinner in the center
than the edges is a concave, or diverging, lens
since rays going through it will be spread out.
Convex (Converging) Lens
Concave (Diverging) Lens
57
Lenses Focal Length

• Like mirrors, lenses have a principal axis
  perpendicular to their surface and passing
  through their midpoint.
• Lenses also have a vertical axis, or principal
  plane, through their middle.

• They have a focal point, F, and the focal length
  is the distance from the vertical axis to F.
• There is no real center of curvature, so 2F is
  used to denote twice the focal length.

58
Ray Diagrams For Lenses
When light rays travel through a lens, they
refract at both surfaces of the lens, upon
entering and upon leaving the lens. At each
interface the bends toward the normal. (Imagine
the wheels and axle.) To simplify ray diagrams,
we often pretend that all refraction occurs at
the vertical axis. This simplification works well
for thin lenses and provides the same results as
refracting the light rays twice.
Reality
Approximation
59
Convex Lenses

• Rays traveling parallel to the principal axis of
  a convex lens will refract toward the focus.
• Rays traveling directly through the center of a
  convex lens will leave the lens traveling in the
  exact same direction.

F
F
2F
2F
Rays traveling from the focus will refract
parallel to the principal axis.
60
Convex Lens Object Beyond 2F
The image formed when an object is placed beyond
2F is located behind the lens between F and 2F.
It is a real, inverted image which is smaller
than the object itself.
object




F
2F
F
2F
Experiment with this diagram
61
Convex Lens Object Between 2F and F
The image formed when an object is placed between
2F and F is located beyond 2F behind the lens.
It is a real, inverted image, larger than the
object.
object




F
2F
F
2F
62
Convex Lens Object within F
The image formed when an object is placed in
front of F is located somewhere beyond F on the
same side of the lens as the object. It is a
virtual, upright image which is larger than the
object. This is how a magnifying glass works.
When the object is brought close to the lens, it
will be magnified greatly.




F
2F
F
2F
object
convex lens used as a magnifier
63
Concave Lenses

• Rays traveling parallel to the principal axis of
  a concave lens will refract as if coming from the
  focus.
• Rays traveling directly through the center of
  a concave lens will leave the lens traveling in
  the exact same direction, just as with a convex
  lens.

Rays traveling toward the focus will refract
parallel to the principal axis.
64
Concave Lens Diagram
No matter where the object is placed, the image
will be on the same side as the object. The image
is virtual, upright, and smaller than the object
with a concave lens.
object




F
2F
F
2F
image
Experiment with this diagram
65
Lens Sign Convention
f focal length di image distance do object
distance
1
1
1


f
do
di
for real image - for virtual image
di
for convex lenses - for concave lenses
f
66
Lens / Mirror Sign Convention
The general rule for lenses and mirrors is this
for real image - for virtual image
di
and if the lens or mirror has the ability to
converge light, f is positive. Otherwise, f
must be treated as negative for the mirror/lens
equation to work correctly.
67
Lens Sample Problem




F
2F
F
2F
14.24 feet
-5.83 feet
68
Lens and Mirror Applet
This application shows where images will be
formed with concave and convex mirrors and
lenses. You can change between lenses and mirrors
at the top. Changing the focal length to negative
will change between concave and convex lenses and
mirrors. You can also move the object or the
lens/mirror by clicking and dragging on them. If
you click with the right mouse button, the object
will move with the mirror/lens. The focal length
can be changed by clicking and dragging at the
top or bottom of the lens/mirror. Object
distance, image distance, focal length, and
magnification can also be changed by typing in
values at the top.
Lens and Mirror Diagrams
69
Convex Lens in Water
Because glass has a higher index of refraction
that water the convex lens at the left will still
converge light, but it will converge at a greater
distance from the lens that it normally would in
air. This is due to the fact that the difference
in index of refraction between water and glass is
small compared to that of air and glass. A large
difference in index of refraction means a greater
change in speed of light at the interface and,
hence, a more dramatic change of direction.
70
Convex Lens Made of Water
Glass
Since water has a higher index of refraction than
air, a convex lens made of water will converge
light just as a glass lens of the same shape.
However, the glass lens will have a smaller focal
length than the water lens (provided the lenses
are of same shape) because glass has an index of
refraction greater than that of water. Since
there is a bigger difference in refractive index
at the air-glass interface than at the air-water
interface, the glass lens will bend light more
than the water lens.
Air
n 1.5
H2O
Air
n 1.33
71
Air Water Lenses
On the left is depicted a concave lens filled
with water, and light rays entering it from an
air-filled environment. Water has a higher index
than air, so the rays diverge just like they do
with a glass lens.
Air
Concave lens made of H2O
To the right is an air-filled convex lens
submerged in water. Instead of converging the
light, the rays diverge because air has a lower
index than water.
H2O
Convex lens made of Air
What would be the situation with a concave lens
made of air submerged in water?
72
Chromatic Aberration
As in a raindrop or a prism, different
wave-lengths of light are refracted at different
angles (higher frequency ? greater bending). The
light passing through a lens is slightly
dispersed, so objects viewed through lenses will
be ringed with color. This is known as chromatic
aberration and it will always be present when a
single lens is used. Chromatic aberration can be
greatly reduced when a convex lens is combined
with a concave lens with a different index of
refraction. The dispersion caused by the convex
lens will be almost canceled by the dispersion
caused by the concave lens. Lenses such as this
are called achromatic lenses and are used in all
precision optical instruments.
Chromatic Aberration
Achromatic Lens
Examples
73
Human eye
The human eye is a fluid-filled object that
focuses images of objects on the retina. The
cornea, with an index of refraction of about
1.38, is where most of the refraction occurs.
Some of this light will then passes through the
pupil opening into the lens, with an index of
refraction of about 1.44. The lens is flexi-
Human eye w/rays
ble and the ciliary muscles contract or relax to
change its shape and focal length. When the
muscles relax, the lens flattens and the focal
length becomes longer so that distant objects can
be focused on the retina. When the muscles
contract, the lens is pushed into a more convex
shape and the focal length is shortened so that
close objects can be focused on the retina. The
retina contains rods and cones to detect the
intensity and frequency of the light and send
impulses to the brain along the optic nerve.
74
Hyperopia
The first eye shown suffers from farsightedness,
which is also known as hyperopia. This is due to
a focal length that is too long, causing the
image to be focused behind the retina, making it
difficult for the person to see close up things.
The second eye is being helped with a convex
lens. The convex lens helps the eye refract the
light and decrease the image distance so it is
once again focused on the retina. Hyperopia
usually occurs among adults due to weakened
ciliary muscles or decreased lens flexibility.
Formation of image behind the retina in a
hyperopic eye.
Convex lens correction for hyperopic eye.
Farsighted means can see far and the rays focus
too far from the lens.
75
Myopia
The first eye suffers from nearsightedness, or
myopia. This is a result of a focal length that
is too short, causing the images of distant
objects to be focused in front of the retina.
The second eyes vision is being corrected with
a concave lens. The concave lens diverges the
light rays, increasing the image distance so that
it is focused on the retina. Nearsightedness is
common among young people, sometimes the result
of a bulging cornea (which will refract light
more than normal) or an elongated eyeball.
Formation of image in front of the retina in a
myopic eye.
Concave lens correction for myopic eye.
Nearsighted means can see near and the rays
focus too near the lens.
76
Refracting Telescopes
Refracting telescopes are comprised of two convex
lenses. The objective lens collects light from a
distant source, converging it to a focus and
forming a real, inverted image inside the
telescope. The objective lens needs to be fairly
large in order to have enough light-gathering
power so that the final image is bright enough to
see. An eyepiece lens is situated beyond this
focal point by a distance equal to its own focal
length. Thus, each lens has a focal point at F.
The rays exiting the eyepiece are nearly
parallel, resulting in a magnified, inverted,
virtual image. Besides magnification, a good
telescope also needs resolving power, which is
its ability to distinguish objects with very
small angular separations.
F
77
Reflecting Telescopes
Galileo was the first to use a refracting
telescope for astronomy. It is difficult to make
large refracting telescopes, though, because the
objective lens becomes so heavy that it is
distorted by its own weight. In 1668 Newton
invented a reflecting telescope. Instead of an
objective lens, it uses a concave objective
mirror, which focuses incoming parallel rays. A
small plane mirror is placed at this focal point
to shoot the light up to an eyepiece lens
(perpendicular to incoming rays) on the side of
the telescope. The mirror serves to gather as
much light as possible, while the eyepiece lens,
as in the refracting scope, is responsible for
the magnification.
78
Huygens Principle
Christiaan Huygens, a contemporary of Newton, was
an advocate of the wave theory of light. (Newton
favored the particle view.) Huygens principle
states that a wave crest can be thought of as a
series of equally-spaced point sources that
produce wavelets that travel at the same speed as
the original wave. These wavelets superimpose
with one another. Constructive interference
occurs along a line parallel to the original wave
at a distance of one wavelength from it. This
principle explains diffraction well When light
passes through a very small slit, it is as if
only one of these point sources is allowed
through. Since there are no other sources to
interfere with it, circular wavefronts radiate
outwards in all directions.
Christiaan Huygens
Applet showing reflection and refraction Huygens
style
79
Diffraction Single Slit
P
screen
Light enters an opening of width a and is
diffracted onto a distant screen. All points at
the opening act as individual point sources of
light. These point sources interfere with each
other, both constructively and destructively, at
different points on the screen, producing
alternating bands of light and dark. To find the
first dark spot, lets consider two point
sources one at the left edge, and one in the
middle of the slit. Light from the left point
source must travel a greater distance to point P
on the screen than light from the middle point
source. If this extra distance is a half a
wavelength, ?/2, destructive interference will
occur at P and there will be a dark spot there.
Extra distance
a / 2
applet
a
Continued
80
Single Slit (cont.)
Lets zoom in on the small triangle in the last
slide. Since a / 2 is extremely small compared to
the distanced to the screen, the two arrows
pointing to P are essentially parallel. The extra
distance is found by drawing segment AC
perpendicular to BC. This means that angle A in
the triangle is also ?. Since AB is the
hypotenuse of a right triangle, the extra
distance is given by (a / 2) sin?. Thus, using
(a / 2) sin? ?/2, or equivalently, a sin?
?, we can locate the first dark spot on the
screen. Other dark spots can be located by
dividing the slit further.
To point P
C
Extra distance
To point P
?
?
?
A
a / 2
B
81
Diffraction Double Slit
P
screen
Light passes through two openings, each of which
acts as a point source. Here a is the distance
between the openings rather than the width of a
particular opening. As before, if d1 - d2 n ?
(a multiple of the wavelength), light from the
two sources will be in phase and there will a
bright spot at P for that wavelength. By the
Pythagorean theorem, the exact difference in
distance is
d1
d2
L
d1 - d2 L2 (x a / 2)2 ½
- L2 (x - a / 2)2 ½
Approximation on next slide.
a
Link 1 Link 2
x
82
Double Slit (cont.)
P
screen
In practice, L is far greater than a, meaning
that segments measuring d1 and d2 are
virtually parallel. Thus, both rays make an angle
? relative to the vertical, and the bottom right
angle of the triangle is also ? (just like in
the single slit case). This means the extra
distance traveled is given by a sin?. Therefore,
the required condition for a bright spot at P is
that there exists a natural number, n, such that
d1
d2
L
a sin? n ?
?
?
If white light is shone at the slits, different
colors will be in phase at different angles.
a
Electron diffraction
83
Diffraction Gratings
A different grating has numerous tiny slits,
equally spaced. It separates white light into its
component colors just as a double slit would.
When a sin? n ?, light of wavelength ? will
be reinforced at an angle of ?. Since different
colors have different wavelengths, different
colors will be reinforced at different angles,
and a prism-like spectrum can be produced. Note,
though, that prisms separate light via refraction
rather than diffraction. The pic on the left
shows red light shone through a grating. The CD
acts as a diffraction grating since the tracks
are very close together (about 625/mm).
84
Credits
Snork pics http//www.geocities.com/EnchantedFore
st/Cottage/7352/indosnor.html Snorks icons
http//www.iconarchive.com/icon/cartoon/snorks_by_
pino/ Snork seahorse pic http//members.aol.com/d
iscopanth/private/snork.jpg Mirror, Lens, and Eye
pics http//www.physicsclassroom.com/
Refracting
Telescope pic http//csep10.phys.utk.edu/astr162/
lect/light/refracting.html Reflecting
Telescope pic http//csep10.phys.utk.edu/astr162/
lect/light/reflecting.html Fiber
Optics
http//www.howstuffworks.com/fiber-optic.htm
Willebrord Snell and Christiaan Huygens pics
http//micro.magnet.fsu.edu/optics/timeline/people
/snell.html Chromatic Aberrations
http//www.dpreview.com/learn/Glossary/Optical/Chr
omatic_Aberrations_01.htm

Mirage Diagrams http//www.islandnet
.com/see/weather/elements/mirage1.htm
Sir David Brewster pic http//www.brewstersociet
y.com/brewster_bio.html
Mirage pics
http//www.polarimage.fi/
http//www.greatestplaces.org/mirage/desert1.html
http//www.ac-grenoble.fr/college.ugine
/physique/les20mirages.htmlDiffuse reflection
http//www.glenbrook.k12.il.us/gbssci/phys/Class/r
efln/u13l1d.htmlDiffraction http//hyperphysics.
phy-astr.gsu.edu/hbase/phyopt/grating.html