Ray Optics
Wave Optics
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We have seen that light is the wigglings of the electromagnetic
field.
Light behaves like a wave.
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Wave Optics
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The wave theory of
light has let us
understand
diffraction.
d sin θ = m λ
d~λ
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Wave Optics
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If the wavelength is much smaller than the holes, then there is
almost no diffraction!
In this case, light passing
through a screen leaves
sharp shadows.
The light travels in rays!
The wave theory of light
isn’t useful here. We would
like to use a new
description in terms of
light rays.
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Light in a Medium
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When light travels through a material medium it appears to travel
at a different speed.
This change comes from the interaction of the light with the
charges in the material.
v = c/n
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n is called the index of refraction, and is a property of the
material.
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Index of Refraction
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The index depends on the
material we’re in.
It depends on the density of
charges, electrostatic
properties, etc…
In general, we just look up the
value.
The index will be important
when we discuss what happens
when light goes from one
material medium to another.
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Ray Optics
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We know that light has a
frequency, but for this
description we’ll ignore it.
We’ll treat the light as though it
was moving in rays.
The easiest case to see is a
laser!
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Reflection
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Using the principles of ray
optics, we can understand
some interesting
properties of light.
When incident on a
different medium, light
can be reflected back.
This is easiest to see in
mirrors.
We can actually
understand this from the
wave model of light, but
it’s easier in the ray
model!
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The Law of Reflection
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Send in a laser beam at incident angle, θi. How does it reflect?
θi = θr
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Reflection
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From the geometry, the light
from the mirror appears to
come from behind the mirror
s = s’
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What about for an object?
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Reflection and Refraction
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Light doesn’t
just bend
when it
encounters a
medium.
Remember
there is
always a
reflected
wave and a
transmitted
wave!
The light
bends!
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Reflection and Refraction
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Let’s look at this from the ray optics point of view.
The light bends
when it goes from
one medium to
another.
There is a weak
reflected wave,
too.
How do we
determine the
amount of
bending?
Let’s go back to
the wave
description!
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The Law of Refraction
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Let’s look at the two
waves.
The frequencies are
the same!
The wavelengths
change!
λ1n1 = λ2n2
l = λ1 sinθ1
l = λ2 sinθ2
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Snell’s Law
n1 sinθ1 = n2 sinθ2
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Refraction
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Refraction is almost as easy to work out as reflection!
Light bends towards the normal in the more dense material, and
away from the normal in the less dense material!
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Total Internal Reflection
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Here’s an interesting
case!
What happened to
the blue light?
(n1 / n2) sinθ1 = sinθ2
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The sine function
can’t be bigger than
1 (for real angles)!
There is NO
refracted wave!
The critical angle
(n1 / n2) sinθ1 = 1
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Dispersion
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The index usually is not constant! The index depends on frequency.
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This gives rise to a familiar phenomenon...
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Dispersion
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Lenses
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We’ve seen that light bends when it travels through a material.
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By curving the material, we can do some very interesting things!
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Where does the light come together?
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Lenses
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The rays focus at the focal
point.
The focal point is located at
the focal length, f.
For a converging lens, the
parallel rays focus at the focal
point.
For a diverging lens, the parallel
rays appear to diverge from the
focal point.
We’ll soon see what happens if
the rays aren’t parallel.
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Converging Lenses
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The focal length works both ways.
Parallel lines from far away focus
at the focal point.
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Rays from a source at the
focal point will be refracted
to give parallel rays.
Now - what happens if we
look at an object?
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Converging Lenses
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Now let’s consider the light coming from an object.
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The light rays are bent different amounts, focusing at the image
point.
The light rays don’t focus at the focal point!
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How do we determine the image point?
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Ray Tracing Diagrams
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There are several steps for a converging lens
1. Draw an optical axis with the lens centered on it.
2. Represent the object with an arrow on the axis at a distance s.
3. Draw the three special rays:
a. A ray parallel to the axis going through the far focal point
b. A ray through the near focal point, parallel to the axis
c. A ray straight through the center of the lens
4. Extend the rays until they converge
5. Measure the image distance s’.
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Ray Tracing Diagrams
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We don’t always have to measure the image point!
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The thin-lens equation.
1/s + 1/s’ = 1/f
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Converging Lenses
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What happens if the object is inside the focal length?
Now, trace the lines
back!
We get a virtual
image.
Notice - the lines
appear to spread out.
We get magnification!
m = - s’/s
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Diverging Lenses
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So far we have been discussing converging lenses. What about
diverging lenses?
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Diverging lenses spread out the light.
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How do we draw the ray tracing diagrams for the diverging lenses?
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Diverging Lenses
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There are slightly different steps for a diverging lens
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1. Draw an optical axis with the lens centered on it.
2. Represent the object with an arrow on the axis at a distance s.
3. Draw the three special rays:
a. A ray parallel to the axis going through the near focal point
b. A ray through the far focal point, parallel to the axis
c. A ray straight through the center of the lens
4. Trace the diverging rays backwards
5. Measure the image distance s’.
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Again, let’s look at a diagram.
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Fermat’s Principle
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There is yet another, deeper,
way of understanding the law
of reflection.
Of all the different ways that
light can go, it takes the path
of least time!
Light could bounce off any of
the spots of a mirror.
Since light is traveling at a
constant speed, it takes the
shortest distance.
θi = θr
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Fermat’s Principle
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The Principle of Least
Time also predicts
refraction!
The speed is not constant,
here!
The path of least distance
is NOT the path of least
time!
The actual path is the one
that minimizes the total
travel time!
n1 sinθ1 = n2 sinθ2
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The Principle of Least
Action…
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