13. optics.pdf

Optics
• Reflection
• Diffuse reflection
• Refraction
• Index of refraction
• Speed of light
• Snell’s 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

Reflection
Most things we see are thanks to reflections, since most objects
don’t 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.

Law of Reflection
i r
i = r
Normal line
(perpendicular to surface)
i
n
c
i
d
e
n
t
r
a
y
s
r
e
f
l
e
c
t
e
d
r
a
y
s

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
direction 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.

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).

Reflection & Refraction
Reflected Ray
Incident Ray
R
e
f
r
a
c
t
e
d
R
a
y
θr
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, θr
, will be greater than the angle of incidence, as depicted on
the left. If the light slows down in the new medium, θr
will be less than
the angle of incidence, as shown on the right.
normal
Reflected Ray
Incident Ray
Refracted Ray
θr
normal

Axle Analogy
θr
Imagine you’re 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 wheel 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
grass
sidewalk
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.

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:
n = Index of Refraction
c = Speed of light in vacuum
v = Speed of light in medium
n =
c
v
Note that a large index of refraction
corresponds to a relatively slow
light speed in that medium.
Medium
Vacuum
Air (STP)
Water (20º C)
Ethanol
Glass
Diamond
n
1
1.00029
1.33
1.36
~1.5
2.42

Snell’s Law
Snell’s 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
sinθi
= nr
sinθr
Here ni
is the index of refraction in the original
medium and nr
is the index in the medium the
light enters. θi
and θr
are the angles of
incidence and refraction, respectively.
θi
θr
ni
nr
Willebrord
Snell

Snell’s 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.
sinθ1
= x / d, so x = d sinθ1
sinθ2
= y / d, so y = d sinθ2
Speed of A: v1
= x / t
Speed of B: v2
= y / t
Continued…
•
•
•
•
A
A B
B
θ1
θ2
x
y
d
n1
n2

Snell’s Law Derivation
(cont.)
v1
/ c sinθ1
1/n1
sinθ1
n2
v2
/c sinθ2
1/n2
sinθ2
n1
= ⇒ = =
⇒ n1
sinθ1
= n2
sinθ2
v1
x/ t x sinθ1
=
v2
y/ t y sinθ2
= = So,
•
•
•
•
A
A B
B
θ1
θ2
x
y
d
n1
n2

Refraction Problem #1
1. Find the first angle of refraction
using Snell’s law.
2. Find angle ø. (Hint: Use
Geometry skills.)
3. Find the second angle of
incidence.
4. Find the second angle of
refraction, θ, using Snell’s Law
19.4712º
θ
Glass, n2
= 1.5
Air, n1
= 1 30°
ø
79.4712º
10.5288º
Horiz. ray,
parallel to
base 15.9º
Goal: Find the angular displacement of the ray after having passed
through the prism. Hints:

1. Find θ1
(just for fun).
2. To show incoming & outgoing
rays are parallel, find θ.
3. Find d.
4. Find the time the light spends in
the glass.
Extra practice: Find θ if the bottom
medium is replaced with air.
θ1
20º
θ
d
glass
H2
0
H2
0
10m
20º
20º
0.504 m
5.2 ·10-8
s
26.4º
n1
= 1.3
n2
= 1.5
Goal: Find the distance the light ray is displaced due to the thick
window and how much time it spends in the glass. Some hints are
given.

θ = ?
36°
Goal: Find the exit angle relative to the horizontal.
19.8°
glass
air
The triangle is isosceles.
Incident ray is horizontal, parallel to the base.
answer: θ =
air

Reflection Problem
θ
50º
answer: θ = 10º
center of
semicircular mirror
with horizontal base
Goal: Find the angle 𝜃 of the incident ray relative to horizontal
so that reflected ray will be vertical.

The Brewster angle is the angle of incidence the produces reflected
and refracted rays that are perpendicular. At this angle reflected light is
perfectly polarized.
Brewster Angle
From Snell, n1
sinθb
= n2
sinθ.
α = θb
since α + β = 90º,
and θb
+ β = 90º.
β = θ since α + β = 90º,
and θ + α = 90º. Thus,
n1
sinθb
= n2
sinθ = n2
sinβ = n2
cosθb
tanθb
= n2
n1
θb
θb
θ
α
β
n2
n1
Sir David
Brewster

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, θc
. 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
lower optical density than it is currently
traveling in (so that it speeds up).
Otherwise, the light would bend toward
the normal. Moreover, the sine inverse of
a ratio bigger than one is not a real
number.
θc
= sin-1 nr
ni
ni
nr
θc
From Snell,
n1
sinθc
= n2
sin90°
Since sin 90° = 1, we
have n1
sinθc
= n2
and
the critical angle is

Critical Angle Sample Problem
Calculate the critical angle for the diamond-air boundary.
θc
= sin-1
(nr
/ ni
)
= sin-1
(1 / 2.42)
= 24.4°
Any light shone at the interface
beyond this angle will be
reflected back into the diamond.
θc
air
diamond
Refer to the Index of Refraction chart for the information.

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
n2
Total internal reflection can be used for practical applications
like fiber optics.
θ > θc
θ
n1
n2
>

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.
Optical fibers have many advantages over
copper wires. They are less expensive,
thinner, lightweight, and more flexible. They
don’t heat up since they use light signals
instead of electricity. 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
spool of optical fiber
Continued…

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…

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).

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 as well as the
lingering daylight after the sun is below the horizon.
More Mirages
Continued…

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 many layers,
each slightly hotter than the layer above it. The cooler air is denser and 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 air is above the hot air.

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.

Observer
Apparent
position
of sun
Actual
position
of sun
Atmosphere
Lingering daylight after the sun
is below the horizon is another
effect of refraction. Light
travels at a slightly slower
speed in earth’s 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 sunlight is
Sunlight after Sunset
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.
Earth

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 smaller. Thus, red light refracts less
than violet. (The pic is exaggerated.) This effect causes white light to
split into its 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.

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.
Atmospheric Optics

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 (depending 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
sun’s rays; the droplets reflecting violet light make an angle of 40o
.
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

Secondary Rainbow
The secondary rainbow is a rainbow of radius
51°, occasionally visible outside the primary
rainbow. It is produced when the light entering
a cloud droplet is reflected twice internally
and then exits the droplet. The color spectrum
is reversed with respect to the primary
rainbow, with red appearing on its inner edge.
Primary
Secondary
Alexander’s
dark region

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
taken 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.

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 can’t be projected on paper,
screens, or film since the light rays do not really converge there.
Examples are forthcoming.

Plane Mirror
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 converge there . For
that reason, this image at P’ is a virtual image.
Object
Virtual
Image
P P’
O I
do
di
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
.
Continued…

Object Image
P B
M
P’
do
di
h h’
Mirror
Two rays from object P strike the mirror at points B and M. Each ray is
reflected such that i = r.
Triangles BPM and BP’M are
congruent by ASA (try to 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.
With plane mirrors, the image is reversed left to right. When you raise
your left hand in front of a mirror, your image raises its right hand. But
why aren’t top and bottom reversed?
object image
Plane Mirror (cont.)

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.

Concave Mirrors
•Concave mirrors are approximately spherical and have a principal
axis that goes through the center, C, of the imagined sphere from 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. Its length is 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).

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 = β. By
alt. int. angles, the angle at C is also β, so α = 2β. s is the
arc length from the principal axis to the pt. of incidence.
Now imagine a circle centered at F with radius f.
If the incident ray is close to the
principal axis, an arc length of
angle ⍺ centered at F also has length
of about s. From s = rθ, we have
s = rβ and s ≈ f α = 2fβ. Thus,
r β ≈ 2 f β, and r = 2 f.
r = 2f
• • α
β
β
β
C F
r
f
s
t
a
n
g
e
n
t
l
i
n
e

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.

•
• C
F • •
C
F
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 don’t 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…
Spherical Aberration

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 Parabolic

Concave Mirrors: Object beyond C
• •
C F
object
image
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.

Concave Mirrors: Object between C and F
• •
C F
object
image
An incoming parallel ray
reflects through the focus, and a
ray coming in through the focus
reflects parallel to the principal
axis. So, 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.

Concave Mirrors: Object in front of F
• •
C F
object
image
A ray coming from the focus
toward the object tip will reflect
parallel to the principal axis. A
ray coming from the tip parallel
to the axis will reflect through the
focus. These diverging rays
appear to be emanating from
behind the mirror (dotted lines).
Thus when an object is placed in
front of F, its image is located
behind the mirror. It is a virtual,
upright image that is larger in
size than the object. It is virtual
since no light rays actually
intersect at the image.

Concave Mirrors: Object at C or F
What happens when an object is placed at C?
What happens when an object is placed at F?
The image will be formed at C also, but it
will be inverted. It will be real and the
same size as the object.
No image will be formed. All rays will
reflect parallel to the principal axis and will
never converge. The image is “at infinity.”

Other Rays in Ray Diagrams
There are two other convenient rays to use for many of these diagrams. One
that goes straight through the center and reflects back on itself. Another hits
the mirror right on the axis, where it’s easy to use i = r. Any two of the four
red rays would be sufficient to locate the image. Doing at least three is a
good way to check your work.
object
image
• •
C F
In order to produce a clear image,
ALL rays emanating from the tip of
the object that hit the mirror need to
reflect through the tip of the image.
A few other rays are shown in
green. These rays aren’t
convenient, though, for ray
diagrams, as the red ones are. Note
that the angle of incidence always
equals the angle of reflection. (The
normal is perpendicular to the
tangent line for any ray.)

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.
light rays
•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.

Convex Mirror Diagram
• C
F
object
image
A parallel, incident ray reflects
as if it had come from the focus.
Conversely, a ray heading
toward the focus bounces off
parallel. A ray toward the center
reflects back on itself. The
image formed by a convex
mirror, no matter where the
object is placed, will be virtual,
upright, and smaller than the
object. Therefore, there is only
one ray diagram for this mirror.
As the object is moved closer to
the mirror, the image will
approach the size of the object.
•

+ for real images
- for virtual images
+ for concave mirrors
- for convex mirrors
1
f =
1
do
1
di
+
f = focal length, di
= image distance, do
= object distance
di
f
Mirror/Lens Equation
Mirror Sign Convention:

Mirror/Lens Equation Derivation
From ΔPCO, β = θ + α, so 2β = 2θ + 2α.
From ΔPTO, γ = 2θ + α , so -γ = -2θ - α.
Adding equations yields 2β - γ = α.
β =
s
r
s
γ ≈
di
α ≈
s
do
(cont.)
•
C
θ
θ
α
β
γ
s object
image
di
O
P
T
From s = r × central
angle, we have
s = r β, s ≈ do
α, and
s ≈ di
γ (for rays close
to the principal axis).
Thus:
do
• •
•

Mirror/Lens Equation Derivation (cont.)
2s
r
- s
di
=
s
do
1
do
2
r =
1
di
+
2
2f =
1
do
1
di
+
1
f =
1
do
1
di
+
From the last slide, β = s/r, α ≈ s/d0
, γ ≈ s/di
, and 2β - γ = α.
Substituting into the last equation yields:
•
C
θ
θ
α
β
γ
s
object
image
di
do
O
P
T
The last equation applies to convex and concave mirrors, as well as to
lenses, provided a sign convention is adhered to.

Magnification
m = magnification
hi
= image height (negative means inverted)
ho
= object height
m =
hi
ho
By definition,
Magnification is simply the ratio of image height to
object height. A positive magnification means the
image is upright, while negative means it’s inverted.

Magnification Identity: m =
-di
do
hi
ho
=
•C
object
image,
height = hi
di do
To derive this let’s look at two rays. One hits the mirror on the axis.
The incident and reflected rays each make angle θ 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
θ ho
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 being a
consequence of our sign
convention. (In this picture
hi
is negative and di
is
positive.) m < 0 means that
the image is inverted.

Mirror Equation Sample Problem
Suppose All Star, who is 3.5 feet
tall, stands 27 feet in front of a
concave mirror with a radius of
curvature of 20 feet. For practice,
draw a ray diagram with all four
“convenient rays.” Where will his
image be reflected and what will its
size be?
di
=
hi
=
• •
C F
15.88 feet
-2.06 feet

Mirror Equation Sample Problem 2
• •
C
F
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. Draw a ray
diagram again. Where is she
standing and how tall is she?
do
=
ho
=
19.25 feet
2.75 feet
image
reflective
side

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

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.

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
light 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.
•
• • •
F F 2F
2F
Reality Approximation
•
• • •
F F 2F
2F

Convex Lenses
Rays traveling parallel to the principal
axis of a convex lens will refract
through the focus.
Rays traveling directly through the
center of a thin convex lens will leave
the lens traveling in the exact same
direction.
•
• • •
F F 2F
2F
•
• • •
F F 2F
2F
Rays traveling from the focus will
refract parallel to the principal axis.
•
• • •
F F 2F
2F

Convex Lens: Object Beyond 2F
•
• • •
F F 2F
2F
object
image
As before, coming in
parallel means exiting
through the focus and vice
versa. 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.
This diagram is analogous to the concave mirror with the object located beyond C.
As was the case with mirrors, all other “inconvenient rays” coming from
the tip of the object will intersect at the tip of the image (green ray).

Convex Lens: Object Between 2F and F
•
• • •
F F 2F
2F
object
image
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.
This diagram is analogous to the concave mirror with the object between C and F.

Convex Lens: Object within F
•
• • •
F F 2F
2F
object
image
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.
convex lens used
as a magnifier
This diagram is analogous to the concave mirror with the object placed within F.

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 thin concave lens will leave
the lens traveling in the exact same
direction, just as with a convex lens.
Concave Lenses
•
• • •
F F 2F
2F
•
• • •
F F 2F
2F
•
• • •
F F 2F
2F
Rays traveling toward the
focus will refract parallel to
the principal axis.

Concave Lens Diagram
•
• • •
F F 2F
2F
object
image
A parallel ray diverges as if
from the focus on the near
side of the lens. A ray
heading toward the far side
focus comes out parallel. A
ray straight through the
center is unaffected. 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.
This diagram is analogous to the convex mirror.

Lens Sign Convention
di
+ for real image
- for virtual image
f
+ for convex lenses
- for concave lenses
1
f =
1
do
1
di
+
f = focal length
di
= image distance
do
= object distance

Lens/Mirror Sign Convention
The general rule for lenses and mirrors is this:
di
+ for real image
- for virtual image
Also, 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.

Lens Sample Problem
•
• • •
F F 2F
2F
Tooter, who stands 4 feet
tall (counting his
snorkel), finds himself 24
feet in front of a convex
lens and he sees his
image reflected 35 feet
behind the lens. What is
the focal length of the
lens and how tall is his
image?
f =
hi
=
14.24 feet
-5.83 feet
Tooter
Tooter
image
(height = 5.83 feet)

Convex Lens in Water
H2
O
Glass Glass
Air
Because glass has a higher index of refraction than 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.

Air
Glass
n = 1.5
Air
H2
O
n = 1.33
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 will.
Convex Lens Made of Water

Air & Water Lenses
H2
O
Convex lens made of Air
Concave lens made of H2
O
Air
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.
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.
What would be the situation with a concave lens made of air
submerged in water?

Chromatic Aberration
As in a raindrop or a prism, different
wavelengths 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

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-
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 becomes rounder,
shortening the focal length so that close objects can be focused on the
retina. The retina contains photoreceptors--rods and cones. Rods are
more sensitive to light not to color. Three different cone cells (red,
green, and blue) to detect those colors. Rods and cones relay
information to the brain via the optic nerve.

Hyperopia The first eye shown suffers from
farsightedness, which is also known as
hyperopia. This is due to an eyeball
that is too short or a cornea that is too
flat. This means the focal length 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.
Distant objects can still be seen by a
farsighted person because light rays are
coming in parallel and are easier to
bring to a focus.
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.

Myopia The first eye suffers from
nearsightedness, or myopia. This is
due to an eyeball that is too long or a
cornea that is too curved. This means
the focal length is too short, causing
the images of distant objects to be
focused in front of the retina.
The second eye’s 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.
Nearsighted people are able to see near
objects because light rays diverge from
them as they enter the eye, which
require more distance or focusing
power to bring the rays to a focus.
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.

Refracting telescopes use a convex lens at its aperture. 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, which is given by the ratio of the focal lengths, a good
telescope also needs resolving power, which is its ability to distinguish objects
with very small angular separations.
Refracting Telescopes
F
Air
objective
lens
m = fO
/ fe
eyepiece
fO
fe

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.

Huygens’ Principle
Christiaan
Huygens
•
• •
• •
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.
Animation
original wavefront
next wavefront
wave propagation

a
Diffraction: Single Slit P
Continued…
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, let’s 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, λ/2,
destructive interference will
occur at P and there will
be a dark spot there.
Extra
distance
a/2
screen

Single Slit (cont.)
Let’s zoom in on the small triangle in the last slide. Since a/2 is
extremely small compared to the distance 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 θ. Since AB is the hypotenuse of a
right triangle, the extra distance is given by (a/2)sinθ. Thus, using
E
x
t
r
a
d
i
s
t
a
n
c
e
a/2
T
o
p
o
i
n
t
P
T
o
p
o
i
n
t
P
θ θ
A
C
B
θ
(a/2)sinθ = λ/2, or equivalently,
a sinθ = λ, we can locate the first dark
spot on the screen. Other dark spots can
be located by dividing the slit further.

a
P
screen
Diffraction: Double Slit
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. If
d1
- d2
= n λ (a multiple of the wave-
length), 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
x
d1
- d2
= [ L2
+ (x + a / 2)2
]½
- [ L2
+ (x - a / 2)2
]½
However this is impractical and
unnecessary. There is a simple way to
get a great approximation, which is
always done in practice...

Double Slit (cont.)
a
P
screen
d1 d2 L
In practice, L is far greater than a, meaning
that segments measuring d1
and d2
are
virtually parallel. Thus, both rays make an
angle θ relative to the vertical, and the bottom
right angle of the triangle is also θ (just like
in the single slit case). This means the extra
distance traveled is given by a sinθ.
Therefore, the required condition for a bright
spot at P is that there exists a natural number,
n such that:
θ θ
asinθ = n λ
If white light is shone at the
slits, different colors will be
in phase at different angles.

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 sinθ = nλ, light of
wavelength λ will be reinforced at an angle of θ. 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. A CD stores information in plastic pits coated with aluminum. Destructive
interference of the laser used to read the CD occurs at the edge of the pits. This is
just like what happens with thin films. The tracks of a CD are very close together
(about 625/mm), about the same spacing as a diffraction grating.
n = 0 n = 1
n = 1
n = 2 n = 2

Credits
Snork pics: http://www.geocities.com/EnchantedForest/Cottage/7352/indosnor.html
Snorks icons: http://www.iconarchive.com/icon/cartoon/snorks_by_pino/
Snork seahorse pic: http://members.aol.com/discopanth/private/snork.jpg
Mirror, Lens, and Eye pics:
http://www.physicsclassroom.com/
Refracting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/refracting.htmlRefracting 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
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.htmlWillebrord
Snell and Christiaan Huygens pics: http://micro.magnet.fsu.edu/optics/timeline/people/snell.html Chromatic Aberrations:
http://www.dpreview.com/learn/Glossary/Optical/Chromatic_Aberrations_01.htmWillebrord Snell and Christiaan Huygens
pics: http://micro.magnet.fsu.edu/optics/timeline/people/snell.html Chromatic Aberrations:
http://www.dpreview.com/learn/Glossary/Optical/Chromatic_Aberrations_01.htm
Mirage Diagrams: http://www.islandnet.com/~see/weather/elements/mirage1.htm
Sir David Brewster pic: http://www.brewstersociety.com/brewster_bio.htmlSir David Brewster pic:
http://www.brewstersociety.com/brewster_bio.html Mirage pics:
http://www.polarimage.fi/Sir David Brewster pic: http://www.brewstersociety.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/les%20mirages.html
Diffuse reflection: http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1d.html
Diffraction: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/grating.html

13. optics.pdf

Recommended

Recommended

More Related Content

Similar to 13. optics.pdf

Similar to 13. optics.pdf (20)

More from rajatrokade185

More from rajatrokade185 (11)

Recently uploaded

Recently uploaded (20)

13. optics.pdf