Notes on Homework
• Please tear the "ruffle edges" off of homework
done in a notebook and torn out
• If the homework is two pages long, staple the
pages together
• If you have a problem with the grading, you think
something was graded incorrectly, points were
added incorrectly, or something was entered
wrong into D2L, put your homework and a
written explanation in Molly’s mailbox.
Chapter 2: Geometrical Optics
• Reflection
– Half-silvered mirrors
– Locating an image
– Multiple reflections
• Refraction
– Total internal
reflection
– Fiber Optics
– Mirages
• Dispersion
– Prisms
– Rainbows
– Diamonds
Transmission and Reflection
Incident ray
Transmitted ray
Reflected ray
The incident ray causes electrons in the material to oscillate,
which generates a new wave. Some of this new wave travels
backwards as the reflected wave, some travels forward, and
combines with the incident wave as the transmitted wave.
Half-silvered mirrors
Silver is a good mirror because of all the oscillating charges. But
what if the layer is too thin to allow the electrons to move enough
to fully cancel the forward going wave?
This is called a “half-silvered”
mirror, because half the light is
transmitted and half is reflected.
Often called “one way mirrors”,
this is not technically correct.
Light passes both ways through
the mirror, but typical use leads
them to appear to be only one
way mirrors.
Half-Silvered Mirrors: Application
Clicker Question
If someone turns the light
on in the policeman’s room,
what does the suspect see?
A. Only his reflection
B. Only the policeman
C. Both his own reflection
and the policeman
??
Law of Reflection
Ray tracing:
1. Draw the incident ray
2. Draw the “normal”, a line
at right angles to the
surface, at the point
where the incident ray hits
3. The reflected ray will be at
the same angle to the
normal as the incident ray
normal to surface
incident ray
reflected ray
reflective surface (mirror)
This law is independent of the reflective surface material. It only
matters that the surface is reflective.
Notes on Ray Tracing
• There are an infinite number of rays that you can
draw, but some of them are more useful than
others:
Pinhole camera
Object
Image
Notes on Ray Tracing
• There are an infinite number of rays that you can
draw, but some of them are more useful than
others:
Pinhole camera
Object
Image
We only NEED these two rays to find the position and size of the image
Specular and Diffuse Reflections
From a mirror or mirror-like
surface
From a rough or highly
textured surface, like a
sheet of paper, cloth, or fur
Clicker Question
• Where are you most likely to see a good reflected
image of yourself?
A. A pool of water
B. A polished metal sheet
C. A piece of window glass
Reflections from the surface of water
Reflected Image Formation
Virtual Images
• The virtual image is in the same place
regardless of the location of the viewer
• The image is called virtual because no
real rays reach the image, and it cannot
be seen by putting a screen at its position
Image Formation in Flat Mirrors
Multiple Mirrors
C
A
B
A virtual image can act as an intermediate image and generate
its own virtual image
Multiple Mirrors
The mirrors do not have to be at right angles
To avoid ray-tracing at weird angles, you can extend one of the
mirrors and position the image in the appropriate spot
Retroreflectors
• Retroreflectors are a special set of multiple mirrors that reflect
incoming rays back the way they came, parallel to the incoming ray
• They are used in bike reflectors and other safety equipment
Refraction
• When going from a fast to a
slow medium, light bends
towards the normal to the
surface
• When going from a slow to a
fast medium, light bends away
from the normal to the surface
• The speed of the medium is
related to its index of
refraction , n
– n = 1.3 for glass
– n = 1.5 for water
• Remember there is still a
reflection as well
Normal
Air (fast medium)
Glass or
water
(slow)
Normal
Air (fast medium)
Glass or
water
(slow)
Refraction Analogy
Normal to
front edge of
slower terrain
Rows of soldiers after
they hit slow terrain
Rows of soldiers before
they hit rough (muddy)
slow terrain
Incident ray
• The rows of marching soldiers
are analogous to the wave
fronts of light
• When the soldiers hit muddy
terrain they slow down
• This causes the rows or parts of
rows in the mud to move less
far in a given time.
• Another way of saying this is
that the ray perpendicular to
the rows of soldiers outside the
mud bends towards the normal
to the muddy region for rows in
the mud
Refraction Analogy
Normal to
front edge of
slower terrain
Rows of soldiers after
they hit slow terrain
Rows of soldiers before
they hit rough (muddy)
slow terrain
Incident ray
• You can imagine that the
muddier the terrain, the more
the soldiers will slow down, and
the more the “ray” will bend
towards the normal.
• Different optical materials are
differently “muddy”, which we
identify using the index of
refraction.
• Higher index materials are
“muddier” and bend the ray
more than lower index materials
n = 1.3 for glass
n = 1.5 for water
n = 2.4 for diamond
Locating an Image Underwater
• The ray from the fish’s fin is
bent away from the normal
as it exits the water
• We interpret the ray as
having arrived in a straight
line.
• Adding one more ray from
the fish’s fin (the ray normal
to the surface is chosen
because it is not bent)
allows us to locate the
image of the fish
There is still a reflection here
as well
Clicker Question
• Where does the fish
see the fisherman?
A. Where he is
B. Not at all
C. Higher than where
he is
D. Lower than where
he is
Clicker questions Ch. 2
10.The light and fish are
both underwater in the
lake. Where does the
fish see the light and/or
the image of the light?
a)
b)
c)
d)
e)
Higher
Lower
Where the light actually is
(c) and (a)
(c) and (b)
Top of lake
Total Internal Reflection
• Recall that rays moving
from glass (slow) to air
(fast) bend away from the
normal
• The amount by which they
bend away depends on the
material
• For any material, we can
find an angle where there is
no transmitted ray, called
the critical angle
Glass
Glass
Glass
Fiber Optics
Mirages
Incident ray
• Occur when the index of refraction
increases as you travel into the medium
• In our analogy with the marching
soldiers this means they go slower and
slower as they advance.
• Imagine the mud getting deeper and
deeper as they advance.
• The ray keeps bending more and more
towards the normal as the mud
deepens
• In the limit as the muddier transitions
become more gradual the refracted ray
looks curved
Mirages
(color indicates temperature, not a sunset or sunrise)
Increasing
temperature
Hot air is less dense, and is thus a “faster” medium than cold air. This
leads to the ray bending away from the normal. Because we
interpret rays as traveling in straight lines, we see a “reflection” in
the road, and assume it is wet.
“Flying Islands” Mirage
“Flying Islands” Mirage
(color indicates temperature, not a sunset or sunrise)
Decreasing
temperature
Cold Ocean
The temperature near the surface of the ocean is often cooler than
the air above it, we see the upside down version of the road
mirage, as the ray travels from cool to warm air, and is bent away
from normal. Again, we interpret rays as traveling in straight lines,
and thus the island appears to float above the ocean
Dispersion
So far, what we’ve learned about refraction means that
in a prism would refract like this:
But we’ve seen that what really happens is this:
Dispersion
• Dispersion means that different colors
(frequencies/wavelengths) of light have different
speeds through a material (different index of
refraction), and thus refract at different angles.
• In most transparent optical materials, blue (high
frequency) light has a higher index than red (low
frequency), and thus blue light is bent more than
red.
Clicker Question
• Where does blue come out of prism, compared to
red?
A)
red
blue
B)
blue
red
C) Somewhere in the middle
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
Rainbows
Diamonds
• Diamond has a very high
index of refraction leading to
a small critical angle for total
internal reflection
• Diamonds are cut such that
most light entering will hit a
back facet at more than the
critical angle, reflecting many
times before returning out
the front
• This long path through the
diamond leads to a lot of
dispersion, spreading the
colors out
Diamond: n=2.4
24.5°