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Chapter 23: Geometric Optics
Review Basic Geometry!
Ray Approximation
• The rays are straight
lines perpendicular to
the wave fronts
• With the ray
approximation, we
assume that a wave
moving through a
medium travels in a
straight line in the
direction of its rays
Light Rays: Ignore Diffraction and
Interference of waves!
Diffraction depends on SLIT WIDTH: the smaller the width,
relative to wavelength, the more bending and diffraction.
We will assume that λ<<d , where d is the diameter of the opening.
This approximation is good for the study of mirrors, lenses, prisms, etc.
Reflection & Refraction
i   r
n1 sin 1  n2 sin  2
Following the Reflected and
Refracted Rays
•Ray  is the
incident ray.
•Ray  is the
reflected ray.
•Ray  is refracted
into the lucite.
•Ray  is internally
reflected in the lucite.
•Ray  is refracted as
it enters the air from
the lucite.
Section 35.5
Law of Reflection
• The normal is a line
perpendicular to the
surface
– It is at the point where the
incident ray strikes the
surface
• The incident ray makes an
angle of θ1 with the
normal
• The reflected ray makes
an angle of θ1’ with the
normal
Specular
Reflection
• Specular reflection is
reflection from a
smooth surface
• The reflected rays are
parallel to each other
• All reflection in this
text is assumed to be
specular
Diffuse
Reflection
• Diffuse reflection is
reflection from a rough
surface
• The reflected rays travel in
a variety of directions
• A surface behaves as a
smooth surface as long as
the surface variations are
much smaller than the
wavelength of the light
Law of Reflection
i   r
Why are most materials Opaque?
(Opaque – Can’t see through)
They absorb light without re-emitting it. Vibrations given
by the light to their atoms and molecules are turned into
random kinetic energy – they become slightly warmer.
Opacity: Mirrors
Free electrons in opaque reflective surfaces
can vibrate, absorb & re-emit at any frequency.
Mirror Mirror
Transparency
Selective Absorption
Glass resonates strongly with UV and
infrared, absorbing those frequencies
while transmitting visible frequencies.
Refraction:
Bending Light into Focus
Refraction: Bending of Light
Transmitted through Materials
Light Bends because it Slows Down.
Atoms are Optical Tuning Forks
Light slows down as it travels through
glass because it takes time to be
absorbed and re-emitted.
Light Slows Down in Materials
Light Bends Toward the Normal when going from a
medium of lower refractive index to one that has a
higher refractive index and visa versa.
lower n
higher n
Index of Refraction
c
n
v
n 1
Vacuum: 1
Water: 1.33
Glass: 1.46
Diamond: 2.4
The Index of Refraction
• Refraction: Light Bends in
Transmission
• The speed of light in any
material is less than its speed
in vacuum
• The index of refraction, n,
of a medium can be defined
as
• For a vacuum, n = 1
– We assume n = 1 for air
speed of light in a vacuum c λ
n
 
also
speed of light in a medium v λn
• For other media, n > 1
λ  λ in vacuum  • n is a dimensionless number
greater than unity, not
n


λn  λ in a medium 
necessarily an integer
Some Indices of Refraction
Frequency Doesn’t Change!
• As light travels from one
medium to another, its
frequency does not
change
– Both the wave speed and
the wavelength do change
– The wavefronts do not
pile up, nor are created or
destroyed at the
boundary, so ƒ must stay
the same
Snell’s Law of Refraction
Angles are always measured from the normal.
n1 sin 1  n2 sin  2
Snell’s Law – Example
Light is refracted into a crown glass
slab. n1 = 1.00 and n2 = 1.52
If θ1 = 30.0o, θ2 = ?
θ2 = sin-1(n1 / n2) sin θ1 = 19.2o
The ray bends toward the normal,
as expected because n2 > n1
Snell’s Law of Refraction
In general:
n1 sin 1  n2 sin  2
n11  n22
n1
 2  1
n2
If n2  n1 , then 1  2
measured from the normal!
Prelab
Emerging Beam is Parallel to Incident Beam but
offset distance d, called the Lateral Shift and is the
subject of this week’s lab!
Fig. 35-15, p. 989
Following the Reflected and
Refracted Rays
•Ray  is the
incident ray.
•Ray  is the
reflected ray.
•Ray  is refracted
into the lucite.
•Ray  is internally
reflected in the lucite.
•Ray  is refracted as
it enters the air from
the lucite.
Section 35.5
Beam & Refraction Directions
• Possible directions of the
beam are indicated by rays
numbered 1 through 5
• The refracted rays are bent
away from the normal since
n 1 > n2
Total Internal Reflection
 2  90
n1 sin 1  n2 sin  2
n2
sin  C 
n1
The Critical Angle
Critical Angle
• There is a particular angle
of incidence that will
result in an angle of
refraction of 90°
– This angle of incidence is
called the critical angle, θC
n2
sin θC 
(for n1  n2 )
n1
• An application of internal
reflection
• Plastic or glass rods are used to
“pipe” light from one place to
another
• Applications include:
– medical use of fiber optic
cables for diagnosis and
correction of medical problems
– Telecommunications
• A flexible light pipe is called an
optical fiber
• A bundle of parallel fibers
(shown) can be used to construct
an optical transmission line
Fiber Optics
Critical Angle Sample Problem
A ray of light, emitted by a laser located beneath the surface of an
unknown liquid with air above it, undergoes total internal
refection as shown. What is the index of refraction for the liquid?
What is its likely identification?
If you pass white light through a prism,
it separates into its component colors.
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
The index of refraction depends on
WAVELENGTH.
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
The speed and wavelength change but
the FREQUENCY does NOT.
Fr
Frequency depends on the oscillating source!
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
Why does Violet Light bend more
than Red Light?
Violet light slows down more because the atoms in the material
are tuned to higher frequencies. As the violet light travels
through glass it takes more time to be absorbed and re-emitted.
Variation of Index of Refraction with Wavelength
speed of light in a vacuum c λ
n
 
speed of light in a medium v λn
• This dependence of n on λ
is called dispersion
• The index of refraction for
a material generally
decreases with increasing
wavelength
• Violet light bends more
than red light when passing
into a refracting material
Refraction in a Prism
•Since all the colors have
different angles of deviation,
white light will spread out
into a spectrum.
– Violet deviates the
most.
– Red deviates the least.
– The remaining colors
are in between.
Section 35.7
Dispersion via Diffraction
constructive : d sin   m, m  0,1, 2,3
If you pass white light through a prism,
it separates into its component colors.
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
The index of refraction depends on
WAVELENGTH.
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
The speed and wavelength change but
the FREQUENCY does NOT.
Fr
Frequency depends on the oscillating source!
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
Why does Violet Light bend more
than Red Light?
Violet light slows down more because the atoms in the material
are tuned to higher frequencies. As the violet light travels
through glass it takes more time to be absorbed and re-emitted.
Variation of Index of Refraction with Wavelength
speed of light in a vacuum c λ
n
 
speed of light in a medium v λn
• This dependence of n on λ
is called dispersion
• The index of refraction for
a material generally
decreases with increasing
wavelength
• Violet light bends more
than red light when passing
into a refracting material
Angle of Deviation
• Since all the colors
have different angles
of deviation, white
light will spread out
into a spectrum
– Violet deviates the most
– Red deviates the least
– The remaining colors are
in between
Dispersion Sample Problem
The index of refraction for
violet light in silica flint glass
is 1.66, and that for red light is
1.62. What is the angular
dispersion of visible light
passing through a prism of
apex angle 60.0° if the angle of
incidence is 50.0°? red (660
nm) violet (410 nm)
Use Snell’s Law twice and some
geometry! Angles are always
measured from the normal.
n1 sin 1  n2 sin  2
Thin Film
Interference
Interference in Thin Films
When reflecting off a medium of greater refractive index, a
light wave undergoes a phase shift of ½ a wavelength.
Wave 1 undergoes a phase shift of 180 degrees.
From Low to High,
a phase change of
pi!
From High to Low,
a phase change?
NO!
Interference in Thin Films
• The wavelength of ray 1 in the
film is /n
• For constructive interference
2t = (m + ½) /n (m = 0, 1, 2 …)
This takes into account both the
difference in optical path length for
the two rays and the 180° phase
change
• For destructive interference
2t = m/n
(m = 0, 1, 2 …)
Problem: Thin Films
A thin film of gasoline floats on a puddle
of water. Sunlight falls almost
perpendicularly on the film and reflects
into your eyes a yellow hue. Interference
in the the thin gasoline film has eliminated
blue (469nm in vacuum) from the
reflected light. The refractive indices of
the blue light in gasoline and water are
1.40 and 1.33 respectively.
Determine the minimum nonzero
thickness of the film.
What color do you see?
Thin Film Interference
The light reflected from a soap bubble
(n = 1.40) appears red ( = 640 nm). What is
the minimum thickness (in nm)?
a. 124
b.104
c. 114
d.134
e. 234
How are Rainbows Formed?
Dispersion: Raindrops Act like Prisms
• A ray of light strikes a drop
of water in the atmosphere
• It undergoes both reflection
and refraction
– First refraction at the
front of the drop
• Violet light will
deviate the most
• Red light will deviate
the least
The Rainbow
• At the back surface the light is
reflected
• It is refracted again as it
returns to the front surface and
moves into the air
• The rays leave the drop at
various angles
– The angle between the white
light and the most intense violet
ray is 40°
– The angle between the white
light and the most intense red ray
is 42°
Observing the Rainbow
• If a raindrop high in the sky is observed, the red ray is seen
• A drop lower in the sky would direct violet light to the
observer
• The other colors of the spectra lie in between the red and
the violet
The droplets form a circular arc, with each droplet within the arc
dispersing light and reflecting it back towards the observer with
the greatest concentration of outgoing rays found at these 40-42
degree angles of deviation. Every droplet within the arc is
refracting and dispersing the entire visible light spectrum
(ROYGBIV).
Rainbow facts
• an observer is in a position to see only a single color from any
one droplet of water.
• your rainbow is slightly different from the rainbow seen by
others
• your rainbow moves with you
• disk within the bow is brighter because of overlapping of
multiple refractions (which don’t occur outside the disk)
A line drawn from your eye to the top of the rainbow forms a 42degree angle with the imaginary line from the sun through your
eye. (If there is a secondary rainbow, it forms an angle of 51degrees). Because these angles determine the position of the
rainbow in the sky, it will sink as the sun rises and rise as the sun
sinks. At some points, the entire rainbow, not just the bottom half,
will be below the horizon where you can't see it. That's why you'll
never see a summer rainbow at midday.
Double Rainbow
• The secondary rainbow is
fainter than the primary
• The secondary rainbow
arises from light that
makes two reflections
from the interior surface
before exiting the raindrop
• Higher-order rainbows are
possible, but their
intensity is low
•
•
Halos are caused by the light of the sun or moon passing through a very thin layer of
cirruform (ice-crystal) clouds in the upper atmosphere. The ice crystals refract the light
of the moon, similar to the way water droplets in the lower atmosphere can refract
sunlight to produce a rainbow. Just like a rainbow, strong halos can have bands of color
in them, due to slightly different refractive properties of the ice crystals for different
colors. Essentially, halos ARE rainbows caused by primary refraction in ice crystals.
Some interesting facts about halos: Halos always occur exactly 22 degrees away from
the sun or moon. Occasionally, intense halos can be double halos, just as intense
rainbows can be doubled. Intense halos can also produce "moondogs" or "sundogs,"
very bright regions on the halo evenly spaced at 90 degree intervals around the halo.
Physics Fun on an Airplane
Always sit on the side opposite the
sun when traveling north-south!!
Why is the Sky Blue?
Galileo
In the early 17th century, many scientists believed that there was no
such thing as the "speed of light"; they thought light could travel any
distance in no time at all. Galileo disagreed, and he came up with an
experiment to measure light's velocity: he and his assistant each took
a shuttered lantern, and they stood on hilltops one mile apart. Galileo
flashed his lantern, and the assistant was supposed to open the shutter
to his own lantern as soon as he saw Galileo's light. Galileo would
then time how long it took before he saw the light from the other
hilltop. The problem was that the speed of light is simply too fast to
be measured this way; light takes such a short time (about 0.000005
seconds, in fact) to travel one mile that there's no way the interval
could have been measured using the tools Galileo had.
The Speed of Light?
•
•
•
•
186,000 miles per second
300,000 kilometers per second
3 x 108 m/s
first successfully determined by
Danish astronomer Ole Roemer in
1675: 2.3 x 108 m/s
• First Terrestrial Measurement by
Fizeau in 1849: 2.9979 x 108 m/s
• In 1926, Michelson used a rotating
prism to measure the time it took
light to make a round trip from
Mount Wilson to Mount San
Antonio in California, a distance
of about 22 miles (36 km). The
precise measurements yielded a
speed of 186,285 miles per second
(299,796 kilometres per second).
Huygens’s Principle
Construction for a Plane Wave
• Huygens assumed that light is a form
of wave motion rather than a stream of
particles
• All points on a given wave front are
taken as point sources for the
production of spherical secondary
waves, called wavelets, which
propagate outward through a medium
with speeds characteristic of waves in
that medium
• After some time has passed, the new
position of the wave front is the
surface tangent to the wavelets
Huygens’s Construction for a
Spherical Wave
• The inner arc represents
part of the spherical wave
• The points are
representative points
where wavelets are
propagated
• The new wavefront is
tangent at each point to
the wavelet
Huygens’s Principle Prove the
Laws of Reflection & Refraction
Huygens’s Principle and the Law of
Reflection
• Triangle ABC is congruent
to triangle ADC
• cos g = BC / AC
• cos g’ = AD / AC
• Therefore, cos g = cos g’
and g  g’
• This gives θ1 = θ1’
• This is the law of
reflection
Huygens’s Principle and the Law
of Refraction
• Ray 1 strikes the
surface and at a time
interval Δt later, ray 2
strikes the surface
• During this time
interval, the wave at A
sends out a wavelet,
centered at A, toward
D
Huygens’s Principle and the Law of
Refraction
• The wave at B sends out a
wavelet, centered at B, toward C
• The two wavelets travel in
different media, therefore their
radii are different
• From triangles ABC and ADC,
we find
BC v1t
sin θ1 

AC AC
sin 1 v1 c n1 n2



sin 2 v 2 c n2 n1
AD v 2t
and sin θ2 

AC AC
n1 sin θ1  n2 sin θ2
Why aren’t images of objects
produced on the wall without a
lens or hole?
Why aren’t images of objects
produced on the wall without a
lens or hole?
Law of Reflection
Construction of an Optical Fiber
• The transparent core is
surrounded by
cladding
– The cladding has a lower n
than the core
– This allows the light in the
core to experience total
internal reflection
• The combination is
surrounded by the
jacket
The heating effect of a medium such as glass or the Earth’s
atmosphere that is transparent to short wavelengths but opaque
to longer wavelengths: Short get in, longer are trapped!
Lenses and Mirrors
Image Formation
Real and Virtual Images
Real images can be
displayed on screens
Virtual Images can not be
displayed onto screens.
Notation for Mirrors and Lenses



The object distance is the distance
from the object to the mirror or lens:
Denoted by p
The image distance is the distance
from the image to the mirror or lens:
Denoted by q
The lateral magnification of the
mirror or lens is the ratio of the
image height to the object height

Denoted by M
The focal point: f

The radius of curvature: R
M
Image height h'

Object height h
1 1 2 1
  
p q R f
FRONT is on the same side as the object and BACK is the other side!
The Principal axis goes through the focal point and the center of
curvature of the lens or mirror!!
Flat Mirrors Make Virtual Images
Virtual Image: An image that cannot be projected onto a surface.
A virtual image only appears like light rays came from the
location of the image, they are not really there.
Flat mirrors make Virtual Images.
Images Formed by
Flat Mirrors






Light rays leave the source and
are reflected from the mirror
Images are always located by
extending diverging rays back to
a point at which they intersect
One ray starts at point P, travels
to Q and reflects back on itself
Another ray follows the path PR
and reflects according to the law
of reflection
h’ = h for all images
Flat mirrors make virtual images
Reversals in a Flat Mirror

A flat mirror produces
an image that has an
apparent left-right
reversal

For example, if you
raise your right hand
the image you see
raises its left hand
Properties of the Image Formed
by a Flat Mirror – Summary

The image is as far behind the mirror as the
object is in front


|p| = |q|
The image is unmagnified

The image height is the same as the object height



The image is virtual
The image is upright


h’ = h and M = 1
It has the same orientation as the object
There is a front-back reversal in the image
Mirror Reflection
Convex & Concave
“Object” on the left, image on the right.
Convex Mirror
Convave Mirror
Lateral Magnification
Image height h'
M

Object height h
Magnification does not always mean bigger, the
size can either increase or decrease.
M>1: Increase
Positive: Upright
M<1: Decrease
Negative: Inverted
Focal Length Shown by
Parallel Rays
Focal Length& Radius of
Curvature



When the object is very far
away, then p → ∞ and the
incoming rays are essentially
parallel
In this special case, the image
point is called the focal point
The distance from the mirror to
the focal point is called the
focal length


The focal length is ½ the radius of
curvature
R = 2f
Ray Diagrams:Concave Mirrors




Ray 1 is drawn from the top of the object parallel to the principal axis and is
reflected through the focal point, F
Ray 2 is drawn from the top of the object through the focal point and is reflected
parallel to the principal axis
Ray 3 is drawn through the center of curvature, C, and is reflected back on itself
The intersection of any two of the rays at a point locates the image. The third ray
serves as a check of the construction
Concave Mirror, p > R




The center of curvature is between the object
and the concave mirror surface (f >0)
The image is real (q>0)
The image is inverted (M<0)
The image is smaller than the object (absM<1)
1 1 2 1
  
p q R f
Concave Mirror, p < f




The object is between the mirror surface and the
focal point (p>0)
The image is virtual (q<0)
The image is upright (M>0)
The image is larger than the object (M>1)
Ray Diagrams:Convex Mirrors



Ray 1 is drawn from the top of the object parallel to the principal
axis and is reflected away from the focal point, F
Ray 2 is drawn from the top of the object toward the focal point
and is reflected parallel to the principal axis
Ray 3 is drawn through the center of curvature, C, on the back
side of the mirror and is reflected back on itself
Convex Mirror





The object is in front of a convex mirror (p>0)
The focal point distance q is negative (q <0)
The image is always virtual and upright (M>0)
As the object distance decreases, the virtual image size increases
The image is smaller than the object (0<M<1)
Sign Conventions: Mirrors
1 1 2 1
  
p q R f
Optics Activity Fun with Mirrors!!!
Lenses

Image formation is a consequence of
light traveling in straight lines

The first camera—the pinhole camera—
illustrates this fact.
Lenses
A lens nicely bends the straight-line paths
of light.
Lenses
A converging lens can project an image.
Lenses
Key features of lenses

principal axis


focal point


line joining the centers of curvature of the two lens
surfaces
point at which all the light rays come together
focal length

distance between the center of the lens and either
focal point
Lens Refraction
Converging & Diverging
Converging Lens
Diverging Lens
Lenses
Lenses
 two common types

converging (convex) lens



thicker at the center than edges
converges light
diverging (concave) lens


thinner at the center than edges
diverges light
Focal Length:Converging Lens
Focal Length:Diverging Lens
Converging Thin Lens Shapes



These are examples
of converging lenses
They have positive
focal lengths
They are thickest in
the middle
Diverging Thin Lens Shapes



These are examples
of diverging lenses
They have negative
focal lengths
They are thickest at
the edges
Signs for Thin Lenses
1 1 2 1
  
p q R f
h'
q
M

h
p
Compare Signs for Mirrors
and Thin Lenses
Thin Lenses
Ray Diagram for Converging
Lens, p > f



The image is real (q>0)
The image is inverted (M<0)
The image is on the back side of the lens (q>0)
Ray Diagram for Converging
Lens, p < f




The image is virtual (q < 0)
The image is upright (M>0)
The image is larger than the object (M>1)
The image is on the front side of the lens (q<0)
Ray Diagram for Diverging
Lens

For a diverging lens, the image is always virtual
and upright (M>0)


This is regardless of where the object is placed
The image is on the front side of the lens (q<0)
Optics Activity Fun with Lenses!!!!
Combination of Thin Lenses,
example