Lec. 7: Ch. 3 - Geometrical Optics
We are
here
1.
2.
3.
4.
Virtual images (review)
Spherical mirrors
Spherical lenses
Aberrations of lenses
http://en.wikipedia.org/wiki/Lens_%28optics%29
Skip 3.3c anamorphic art.
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Review: plane mirrors, specular reflection
• Equal angle rule
• Similar triangles are
useful
θi
θr
Normal
Mirror
• Ray tracing a mirror
Xobject = Ximage
Image point is on the normal
(mirror might need an extension)
Xobject
Extension
Ximage
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1
Basics: What is an object? What is an image?
Xobject
Extension
Ximage
=
3
Basics: What is an object? What is an image?
Xobject
Extension
Ximage
=
In this context, an object is a point that emits light rays
in a range of directions:
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2
Why do we make this definition?
and not here?
How does the “eye”
know the object is here?
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The eye can sense (to some extent) the angle of
incoming rays
The eye effectively traces nearby rays
back toward their intersection; it “sees”
an image at that intersection.
and not here?
How does the “eye”
know the object is here?
(Caution: I’m not really talking about depth perception here (wait for Ch. 8), but this is related to one form of depth
perception, c.f. accommodation.)
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3
How does the eye make sense of all those light rays?
You are not aware of all those rays, but rather of 3 points of light:
It’s often easier to think in terms of objects than individual light rays.
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What is an image?
Xobject
Extension
Ximage
=
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4
What is an image?
Xobject
Extension
Ximage
=
In this context, an image is a point from which light rays
emerge in a range of directions: (the image doesn’t
necessarily emit or produce the light)
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Real vs. Virtual Images
magic,
invisible,
ray
machine
magic,
invisible,
ray
machine
This is a real image. The light rays
actually intersect at (or pass
through) the image.
This is a virtual image. The light rays didn’t
actually intersect at (or pass through) the
image, but the eye sees the image all the
same (the eye is rather amazing).
The (stationary) eye can’t distinguish between the above cases; the rays
entering the eye are exactly the same. But a person could run over and
put a piece of paper where the image is and determine whether it’s real or
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virtual.
5
Curved Mirrors
First, a little geometry review:
This line segment (from center of circle)...
...is perpendicular
(or normal) to this
tangent.
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Rays reflecting from a convex (spherical) mirror
• Ray aimed toward
center of sphere comes
straight back (specular
reflection with normal
incidence)
• What about other rays?
C
All rays aimed at the center C come straight back out.12
6
What happens to all rays that come in parallel?
• Specular reflection
• Find where incoming ray
hits mirror surface
• Find surface normal at
that point (along line from
center--remember
geometry review?)
• Angle of incidence = angle
of reflection
• Reflection (of parallel ray)
looks like it’s coming from
F -- turns out this is true
for all parallel rays.
F
C
θi
θr
The focus is halfway to the center
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What happens to all rays that come in parallel?
Focal point = focus
is behind the surface
• Easy rule for parallel
incoming rays (parallel to
the line between F and C):
they are reflected as if
they came from F.
F
C
The focus is halfway to the center
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7
What about rays aimed at the focus?
(This is the previous rule, backwards)
Focal point = focus is
behind the surface
• Easy rule for rays aimed
at focus:
• An incoming ray aimed
at F gets reflected back
parallel (to the C-F axis).
F
C
The focus is halfway to the center
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Three easy rules for convex, spherical mirrors
1. All rays incident parallel to the CF axis are reflected so that they
appear to be coming from the
focal point F
2. All rays that (when extended)
pass through the center C are
reflected back on themselves.
3. All rays that (when extended)
pass through the focal point F
are reflected back parallel to the
axis
3
F
1
C
2
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8
Ray tracing: convex mirror
Questions:
• Is the image real or virtual?
• Is the image larger or smaller than the object?
• Is the image right-side-up or upside-down?
C
F
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Ray tracing: convex mirror
Questions:
• Is the image real or virtual?
• Is the image larger or smaller than the object?
• Is the image right-side-up or upside-down?
• How could a mirror be useful when used like this?
C
F
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9
Rays reflecting from concave (cavity) mirrors
Ray through the center
reflects straight back at
its source
C
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Incoming parallel rays reflect through focus
• All (incoming parallel) rays
reflect and go through the
focus, about half way from
center to mirror
C
F
As usual, this rule works backwards: incoming rays that go
through the focus reflect back parallel (to the C-F axis).
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10
Rays through focus reflect back parallel to C-F axis.
C
F
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Concave mirrors are very useful
C
light beam emitter
(flashlight)
F
light collector
or solar oven
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11
Three easy rules for concave, spherical mirrors
1. All rays incident parallel to the
C-F axis are reflected through
the focal point F
2. All rays that pass through the
center C are reflected back on
themselves.
3. All rays that pass through the
focal point F are reflected back
parallel to the axis
1
C
F
2
3
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Ray tracing: concave mirror
object outside center
C
F
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12
Ray tracing: concave mirror
object outside center
C
F
Questions:
• Is the image real or virtual?
• Is the image larger or smaller than the object?
• Is the image right-side-up or upside-down?
• How could a mirror be useful when used like this?
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Ray tracing: concave mirror
object between center and focus
C
F
Questions:
• Is the image real or virtual?
• Is the image larger or smaller than the object?
• Is the image right-side-up or upside-down?
• How could a mirror be useful when used like this?
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13
Ray tracing: concave mirror
object between focus and mirror
C
F
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Ray tracing: concave mirror
object between focus and mirror
C
F
Questions:
• Is the image real or virtual?
• Is the image larger or smaller than the object?
• Is the image right-side-up or upside-down?
• How could a mirror be useful when used like this?
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14
ray tracing applet
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We now have many distinct cases
• Convex spherical mirror
• Concave spherical mirror, object outside
center
• Concave spherical mirror, object between
center and focus
• Concave spherical mirror, object between
focus and mirror
For each case, you can now answer: Image
larger? Virtual? Where? What good is it?
AND you can answer these question by ray tracing with
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three simple rules
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On to lenses:
first, review refraction
air
n=1 (nearly)
v = c (nearly)
glass, e.g.
n=1.5
v = c/n < c
Rays bend toward normal when entering slower medium (larger n),
away from normal when entering faster medium (smaller n).
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A trick to remember which way rays bend
Soldiers in mud analogy (challenge: where does this analogy break down?)
As soldiers slow down, space
between them narrows
“rays” (perpendicular
to fronts)
“fronts”
pavement
(soldiers
go fast)
deep mud
(soldiers march
slower through
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deep mud)
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Ray tracing with lenses
Brute force ray tracing:
1. As long as ray stays in
same medium, it goes
straight.
2. At each interface to a
different medium, use
Snell’s law to calculate
how it will bend. Go
back to 1.
This gets tedious!
n>1
n=1
Rays entering
“slower” material
bend toward
normal
Rays entering
“faster” material
bend away from
normal
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Thin convex (converging) lens
focal length
F
F
foci
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17
Thin convex lens: three easy rules for ray tracing
focal length
1) A ray parallel to the axis
is deflected through the
focus on the other side
2) A ray through the center
of the lens continues
undeviated
3) A ray come from the
focus on one side goes
out parallel to the axis
on the other
1
2
3
F
F’
3
foci
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Ray might have to be extended
Note light-focusing property of convex (converging) lens
a good light collector or solar
oven; can also fry ants with
sunlight, but please don’t do that
unless you’re going to eat them
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18
Note light-dispersing property of convex lens
The “backwards” light collector:
create a collimated light beam
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Ray Tracing
Where will this ray go?
foci (focuses?)
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19
Ray Tracing
Where will this ray go?
Suppose it’s emitted
from this object
foci (focuses?)
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Ray Tracing
Where will this ray go?
Suppose it’s emitted
from this object
We know where these 3 rays
go, using the simple ray rules
foci (focuses?)
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20
Ray Tracing
Amazing property of
this lens: all rays from
the object will
converge to the same
point
Where will this ray go?
Suppose it’s emitted
from this object
We know where these 3 rays
go, using the simple ray rules
foci (focuses?)
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Ray Tracing
Where will this ray go?
Suppose it’s emitted
from this object
Amazing property of
this lens: all rays from
the object will
converge to the same
point
We know where these 3 rays
go, using the simple ray rules
foci (focuses?)
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21
Ray Tracing: thin lens, object outside focus
Amazing property of
this lens: all rays from
the object will
converge to the same
point (the image)
See how the rays emerge
from this point (the image)?
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Thin concave (diverging) lens
Guess how this ray will be bent:
F’
F
For diverging lens focal length defined to be
negative (of the distance between focus and lens)
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22
Thin concave (diverging) lens
F
F’
For diverging lens focal length defined to be negative45
Thin concave (diverging) lens: three easy ray rules
1) A ray parallel to the axis
is deflected as if it came
from the focus
2) A ray through the center
of the lens continues
undeviated
3) A ray aimed at the focus
on the other side comes
out parallel
1
2
3
F
F’
Ray might have to be extended
For diverging lens focal length defined to be negative46
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Difference between convex (converging) and concave
(diverging) lenses
1
F
F’
(Rule 3, the backwards
version of rule 1, also differs)
1
F
F’
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Ray tracing a convex lens: object inside focus
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Ray tracing a convex lens: object inside focus
The image appears larger (and farther away) than the object.
This is a magnifying glass.
(Remember: a magnifying glass is a convex lens.)
Aside: near-sighted people need concave/diverging lenses; can a marooned
myopic start a fire with his eye-glasses?
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Convex lens ray tracing: 3 cases
Like concave mirrors, convex lenses have 3 kinds of
cases for ray tracing:
1. object inside focal length
2. object outside focal length, inside twice focal length
3. object outside twice focal length
You can do the ray tracing and answer the following
questions:
Is the image real/virtual?
Is the image larger/smaller than the object?
Is the image erect/inverted?
How can the lens be used?
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