Geometric Optics
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Chapter 32
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Geometric Optics
!  The study of light divides itself into three fields
•  Geometric optics
•  Wave optics
•  Quantum optics
!  In the previous chapter we learned that light is an
electromagnetic wave
!  In the next chapter will study the wave properties of light
!  In this chapter we will deal with geometric optics in which
we will treat light that travels in straight lines called light
rays
!  Quantum optics makes use of the fact that light is quantized,
in the form of “photons”
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Spherical Waves
!  Electromagnetic waves spread spherically from a source
!  The concentric yellow spheres
shown below represent the
spreading spherical wave fronts
of the light emitted from the
light bulb
!  The black arrows are the
light rays, which are
perpendicular to the wave
fronts at every point in space
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Plane Waves
!  We can treat light waves from far away sources as plane
waves whose wave front is traveling in a straight line
!  We can further abstract these traveling planes by vectors or
arrows perpendicular to the surface of these planes
!  These planes can then be represented by a series of
parallel rays or just one ray
!  In this chapter we will treat
light as a ray traveling in a
straight line
!  We can solve many problems
geometrically
!  For the remainder of this
chapter we will ignore the
structure of light
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Reflection Law
!  Rays incident on a plane mirror will be
reflected such that the reflected rays have
the same angle relative to the surface
normal vector as the incoming ray
!  The law of reflection is given by
θr = θi
!  Parallel rays incident on a plane mirror will
be reflected such that the reflected rays are
also parallel
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Image Construction: Quantitative
!  Start with an object of height ho located a distance do from
the mirror
!  Represent the object as an arrow
!  Put the tail of the arrow on the optic axis
•  Follow a light ray along the optic axis
!  Follow two rays from the point of the arrow
•  One parallel to the optic axis
•  One intersecting the mirror where the optic axis intersects the
mirror
!  These three rays will be reflected by the mirror
!  An image of the object will be formed with height hi
located a distance di behind the mirror
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Image Construction: Quantitative
!  The bottom of the image will lie on the extrapolation
of the optic axis
!  The top of the image will be located at the intersection of
the extrapolation of the two rays from the point of the
arrow
!  For a plane mirror ho = hi and |do|= |di|
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Monkey mirror: left-right?
!  Again we construct the virtual image with two rays
•  all rays behave the same way
!  The watch is on the top
for the person and the image
(no left-right flip)
!  But: The images is facing in the
opposite direction!
!  Thus, a mirror image is flipped only front-back
•  And the apparent left-right flip is only an indirect consequence
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Example: Full-length Mirror
!  Question:
!  A 184 cm (6 ft 1/2 inch) tall person wants to buy a mirror so that he
can see himself full length. His eyes are 8 cm from the top of his head
!  What is the minimum height of the mirror?
!  Let us abstract the person as a 184 cm tall pole with eyes 8 cm from
the top of the pole
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Example: Full-length Mirror
!  The distance from the floor to the bottom of the mirror is
•  (184 cm – 8cm)/2 = 88 cm
•  the angle of incidence equals the angle of reflection so the point on the mirror
where the person can see the bottom of his feet will be half way between his eyes
and the bottom of his feet
!  Similarly the difference between the top of
the mirror and 184 cm is
•  (8 cm)/2 = 4 cm
!  So the minimum length of the mirror is
•  184 cm – 88 cm – 4 cm = 92 cm
!  The required length of the “Full-length
mirror” is just half the height of the
person wanting to see himself full-length
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Curved Mirrors
!  When light is reflected from the surface of a curved mirror,
the light rays follow the law of reflection at each point on
the surface
!  Light rays that are parallel before they strike the mirror are
reflected in different directions depending on the part of the
mirror that they strike
!  The light rays can be focused or made to diverge
!  An example is a spherical mirror where the reflecting
surface is on the inside of the sphere which causes the
reflected rays to converge
!  This is a concave mirror or converging mirror
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Concave Spherical Mirrors
!  We can abstract this sphere as a two
dimensional semicircle
!  The optical axis of the mirror is a line
through the center of the sphere,
represented by a horizontal dashed line
!  A horizontal light ray above the optical
axis is incident on the surface of the mirror
!  At the point the light ray strikes the mirror,
the law of reflection applies
θi = θr
!  The normal to the surface is a radius line that points to the
center of the sphere marked as C
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Concave Spherical Mirror
!  Now let us suppose that we have many horizontal light rays
incident on this spherical mirror
!  Each light ray obeys the law of
reflection at each point
!  All the reflected rays cross
the optical axis at the
same point called the
focal point F
!  The focal point is half way between point C
and the surface of the mirror
!  C is located at the center of the sphere so the
distance of C from the surface of the mirror is just the
radius of the mirror, R
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Concave Spherical Mirror
!  Thus the focal length f of a spherical mirror is
f=
R
2
!  Here is a photo of light rays incident on an actual
converging mirror
!  We can draw the incident light rays and the reflected light
rays on top of the photo
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Images with a Concave Spherical
Mirror do > f
!  Choose a case where an object with height ho is placed at a
distance do > f from the mirror
!  First ray: from the bottom of the arrow along the optical axis
•  Tells us that the bottom of the image will be located on the optical
axis
•  We typically don’t bother drawing this ray
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Images with a Concave Spherical
Mirror do > f
!  Second ray: starts from the top of the arrow parallel to the
optical axis and is reflected through the focal point
!  Third ray: begins at the top of the arrow, passes through the
center of the sphere, and is reflected back on itself
!  Last two rays intersect at the point where the image of the
top of the object will be located
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Images with a Concave Spherical
Mirror do > f
!  Fourth ray: begins at the top of the arrow, passes through
the focal point, and is reflected parallel to the optical axis
!  This ray intersects rays 2 and 3 at the point where the image
of the top of the object is located
!  Two intersecting rays are sufficient and any two of the three
rays can be used
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Image with a Concave Spherical Mirror
do > f
!  The reconstruction of this special case produces a real image
defined by the fact that the image is on the same side of the
mirror as the object and not behind the mirror
!  The image has a height hi located a distance di from the
surface of the mirror on the same side of the mirror as the
object with height ho and distance do from the mirror
!  The image is inverted and reduced in size
!  The image is called real when you are able to place a screen
at the image location and obtain a sharp projection of the
image on the screen at that point
!  For a virtual image, it is impossible to place a screen at the
location of the image
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Concave Mirror, do < f
!  Now let’s reconstruct another case for a
concave mirror where do < f
!  Again we place the object standing on the optical axis
!  We use three light rays
!  The first establishes that the tail of the image lies on the
optical axis
!  The second ray leaves the top of the object along a radius
and reflected back on itself through the center of the sphere
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Concave Mirror, do < f
!  The third ray starts from the top of the
object parallel to the optical axis and is
reflected through the focal point
!  The reflected rays are clearly diverging
!  To determine the location of the image, we must extrapolate
the reflected rays to the other side of the mirror
!  These two rays intersect a distance di from the surface of the
mirror producing an image with height hi
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Conventions for Mirror Distances
!  In this case, we have the image formed on the opposite side
of the mirror from the object, a virtual image
!  To an observer, the image appears to be behind the mirror
!  The image is upright and larger than the object
!  These results are very different than those for do > f
!  To treat all possible cases for concave mirrors, we must
define some conventions for distances and heights
!  We define distances on the same side of the mirror as the
object to be positive
!  We define distances on the far side of the mirror from the
object to be negative
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Mirror Equation
!
!
!
!
Thus f and do are positive for concave mirrors
In the case of real images, di is positive
In cases where the image is virtual, di is negative
If the image is upright, then hi is positive and if the image is
inverted, hi is negative
!  Using these sign conventions we can express the mirror
equation in terms of do, di, and the focal length f
1 1 1
+ =
do di f
!  The magnification m of the mirror is defined to be
di hi
m=− =
do ho
!  You can find the derivation of the mirror equation in the
book (Derivation 32.1)
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Possible Cases for the Mirror Equation
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