Chapter 24
Geometrical Optics
Optics
 The study of light is called optics
 Some highlights in the history of optics
 Study of optics dates to at least third century BC
 Eyeglasses invented around 1300
 Microscopes and telescopes invented around 1600
 Applications depend on the ability of lenses and
mirrors to focus light
 Light is an electromagnetic wave and its wave nature
needs to be accounted for
Geometrical Optics
 Applies to the regime where light travels in straight-
line paths
 Effects involving wave interference are not important
 Describes cases in which the wavelength of the light
is much smaller than the size of the objects in the
light’s path
 Wavelength of visible light is less than 1µm
 Describes many everyday applications
 Including the behavior of mirrors and lenses
Rays
 Rays indicate the path and
direction of propagation of
the light wave
 In A, the waves pass
through a large opening and,
to a very good
approximation, follow
straight lines that pass
through the opening
 In B, the opening is about
the same size or smaller
than the wavelength of the
light and needs wave optics
to explain
Section 24.1
Wave Fronts
 Wave front surfaces are determined by the crests and
troughs of the wave
 They are always perpendicular to the associated rays
 The shape of a wave front depends on how the wave is
generated and the distance from the source
Section 24.1
Two Properties of Light
 The motion of light along a light ray is reversible
 If light can travel in one direction along a ray that
connects point A to point B, light can also propagate in
the reverse direction, from B to A
 The perpendicular distance between two wave fronts
is proportional to the speed of light
 Because of the way wave fronts are related to crest
and troughs of a wave
Section 24.1
Ray Tracing
 Light from an object is used by your eye to form an
image of the object
 When your eye combines the rays to form an image,
your brain extrapolates the rays back to their origin
 The method of following the individual rays as they
travel from an object to some other point is called
ray tracing
 Ray tracing involves the use of geometry
Section 24.1
Ray Tracing, cont.
 The figure shows a few
rays from the object
 There are an infinite
number of actual rays
 The light waves
associated with all the
rays contribute to the
image formed by your eye
 In most ray diagrams, we
draw just a few rays from
the top and bottom of the
image
Section 24.1
Image Formation
 Two problems must be considered to understand
how images are formed
 What happens to light rays when they reflect from a
surface such as a mirror or a piece of glass
 What happens to light rays when they pass across a
surface from one material to another such as when
they pass from air into a piece of glass
 You must also distinguish between a flat surface and
a curved surface
Section 24.1
Reflection from a Plane Mirror
 Light rays travel in straight lines until they strike
something
 The rays may be reflected
 Rays may be reflected from a plane mirror
 A flat surface that reflects all or nearly all the light that
strikes it
 If the light is a plane wave, all the rays are parallel
and strike a surface at many different points
Section 24.2
Reflection from a Plane Mirror,
cont.
 Characterize the reflection by a single ray
 The normal (vertical dashed line in fig. B) is
perpendicular to the mirror
 The direction of the incoming and outgoing rays are
measured relative to the normal
Section 24.2
Law Of Reflection - Definitions
 The incoming ray is called the incident ray
 The angle it makes with the normal is called the angle
of incidence, θi
 The outgoing ray is called the reflected ray
 The angle it makes with the normal is called the angle
of reflection, θr
 The Law of Reflection says θi = θr
 Reflection from a perfectly flat mirror is called
specular reflection
Section 24.2
Diffuse Reflection
 If the reflecting surface is
rough, the reflections from
each individual piece of
the surface must be
analyzed
 An incident plane wave
will give rise to many
reflected rays propagating
outward in many different
directions
 This is called diffuse
reflection
Section 24.2
Image Formation – Plane Mirror
 An image formed by a
plane mirror is shown
 Two representative rays
are shown coming from
the object
 There is an infinite
number of rays
emanating from each
point on the object
 The rays that reflected
from the mirror and
reached your eyes form
the image
Section 24.2
Image Formation, cont.
 To your eye, the location of the image is the point
from which these rays appear to emanate
 This point can be found by ray tracing
 Each ray obeys the law of reflection
 Applying geometry will allow the location of the
image to be found
Section 24.2
Image Formation, final
 Characteristics of the image
 The distance from the object to the mirror is the same
as the distance from the image to the mirror
 The size (height) of the image, hi, is the same as the
size (height) of the object, ho
 The image is virtual


The image point is located behind the mirror
The light does not actually pass through the image
 The same analysis can be applied to multiple mirrors
Section 24.2
Refraction
 When a light ray strikes
a transparent material,
some of the light is
reflected and some is
refracted
 The reflected ray obeys
the Law of Reflection
 The refracted ray
passes into the material
 The incident angle is
now denoted as θ1
Section 24.3
Angle of Refraction
 The direction of the refracted ray is measured by
using θ2 (refer to fig. 24.10)
 The value of this angle depends on the incident
angle and the speed of light in the material
 The speed of light in a vacuum is 3 x 108 m/s
 When the light travels through a material substance,
its interactions with the atoms of the material slows
down the wave
Section 24.3
Snell’s Law
v=λf
f constant
v , λ change
 The change in the speed of light from the vacuum
to the material changes the direction of the wave
 From the geometry of the waves in the material,
c
sin 1  sin 2
v
Section 24.3
Snell’s Law, cont.
Snell’s Law, cont.
 The ratio c/v is called the index of refraction and is
denoted by n
 n=c/v
 n is unitless
 Then, sin θ1 = n sin θ2
 This assumes the wave is incident in a vacuum
 A more general statement can be applied to any two
materials with indices of refraction n1 and n2
n1 sin θ1 = n2 sin θ2
 This relationship is called Snell’s Law
Section 24.3
Speeds and n’s for Various
Materials
Section 24.3
Applying Snell’s Law
 Refraction is also reversible
 Snell’s Law applies whether light begins in the
material with the larger or smaller index of refraction
 Possible angles of refraction are always between 0°
and 90°
 The side with the larger index of refraction has the
smaller angle
Direction of Refracted Ray
 Light is refracted toward
the normal when
moving into the
substance with the
larger index of refraction
 Light is refracted away
from the normal when
moving into the
substance with the
smaller index of
refraction
Section 24.3
Total Internal Reflection
 When light is incident
from the side with a
higher index of
refraction, it is bent
away from the normal
 As the incident angle
gets larger, the
refracted angle also
increases
 Eventually, θ2 will reach
90°
Section 24.3
Total Internal Reflection, cont.
 The angle of incidence for which the angle of
refraction is 90° is called the critical angle
 If the angle of incidence is increased beyond the
critical angle, Snell’s Law has no solution for θ2
 Physically, there is no refracted ray
 This behavior is called total internal reflection
 This is only possible when the light is incident from the
side with the larger index of refraction
Section 24.3
Critical Angle
 From Snell’s Law, with θ2 = 90°, θ1 = θcrit
crit
 n2 
 sin  
 n1 
1
 When the angle of incidence is equal to or greater
than the critical angle, light is reflected completely at
the interface
Section 24.3
Fiber Optics
 Total internal reflection is
used in fiber optics
 Optical fibers are
composed of specially
made glass and used to
carry telecommunication
signals
 These signals are sent as
light waves
 They are directed along
the fiber using internal
reflection
Section 24.3
Dispersion
 When light travels in a
material, the speed
depends on the color of
the light
 This dependence of wave
speed on color is called
dispersion
 Since the index of
refraction is slightly
different for each color,
the angle of refraction will
be different for each color
Section 24.3
Dispersion and Prisms
 Dispersion is used by a prism to separate a beam of
light into its component colors
 There are two refractions with the prism
 The red and blue show the extremes of the incident
beams
Section 24.3
Curved Mirrors
 A curved mirror can produce an image of an object
that is magnified
 The image can be larger or smaller than the object
 Magnified images are used in many applications
 Telescopes
 Car’s review mirror
 Many others
Section 24.4
Ray Tracing – Curved Mirror
 A spherical mirror in one
in which the surface of the
mirror forms a section of a
spherical shell
 The radius, R, of the
sphere is the radius of
curvature of the mirror
 The mirror’s principal axis
is the line that extends
from the center of
curvature, C, to the center
of the mirror
Section 24.4
Concave Spherical Mirror
 Properties of concave
spherical mirrors
 Incoming rays that are
close to and parallel to
the principal axis reflect
through a single point F


F is the focal point
It is located a distance ƒ,
the focal length, from the
mirror
 Rays that originate at the
focal point reflect from
the mirror parallel to the
principal axis

From reversibility of light
Section 24.4
Image From a Concave Mirror -Examples
Section 24.4
Image from Concave Mirror –
Ray Diagram
 Trace rays emanating from the top of the object
 The rays all intersect at a single point
 This is the top of the image
 A similar result would be found from rays from other
parts of the object
Section 24.4
Drawing A Ray Diagram
 Three rays are particularly easy to draw
 There are an infinite number of actual rays
 The focal ray
 From the tip of the object through the focal point
 Reflects parallel to the principal axis
Section 24.4
Drawing A Ray Diagram, cont.
 The parallel ray
 From the tip of the object parallel to the principal axis
 Reflects through the focal point
 The central ray
 From the tip of the object through the center of
curvature of the mirror
 Reflects back on itself
 The three rays intersect at the tip of the image
Section 24.4
Properties of an Image
 Magnification is the ratio of the height of the image,
hi, to the height of the object, ho
hi
m
ho
 By convention, the image height of an inverted
image is negative
 Therefore, the magnification is also negative
 Images smaller than the object are said to be
reduced
Section 24.4
Real vs. Virtual Images
 If the rays that form the image all pass through a
point on the image, the image is called a real image
 Real images and virtual images differ
 Light rays only appear to emanate from a virtual
image, they do not actually pass through the image

For a real image, the light rays do actually pass through the
image
 An object and its real image are both on the same side
the mirror

A virtual image is located behind the mirror while the object
is in front
Section 24.4
Concave Mirror and Virtual
Images
 Use ray tracing to find the image when the object is close
to the mirror
 Closer than the focal point
 Use the same three rays
 The rays do not intersect at any point in the front of the
mirror
Section 24.4
Virtual Images
 Extrapolate the rays back behind the mirror
 They intersect at a single image point
 The rays appear to emanate from the image point
behind the mirror
 The image is virtual because light does not actually
pass through any point on the image
 The object and its image are on different sides of the
mirror
 The image is upright and enlarged
Section 24.4
Rules for Ray Tracing – Mirrors
 Construct a figure showing the mirror and its
principal axis
 The figure should also show the focal point and the
center of curvature
 Draw the object at the appropriate point
 One end of the object will often lie on the principal axis
 Draw three rays that emanate from the tip of the
object
 The focal ray passes through the focal point and
reflects parallel to the principal axis
Section 24.4
Rules, cont.
 Three rays, cont.
 The parallel ray is parallel to the principal axis and reflects
through the focal point
 The central ray passes through the center of curvature of
the mirror and reflects back through the tip of the object
 The point where the three rays intersect is the image
point
 This point may be in front of the mirror giving a real image
 This point may be in back of the mirror giving a virtual
image

Found by extrapolation of the rays behind the mirror
Section 24.4
Rules, final
 This ray-tracing procedure can be repeated for any
desired point on the object
 This allows you to find other points on the image
 It is usually sufficient to consider just the tip of the
image
 Other points may be used if needed
Section 24.4
Ray Tracing – Convex Spherical
Mirrors
 A mirror that curves away
from the object is called a
convex mirror
 The center of curvature and
the focal point lie behind the
mirror
 After striking the convex
surface, the reflected rays
diverge from the mirror axis
 The parallel rays converge
on an image point behind
the mirror
 This is the focal point, F
Section 24.4
Ray Tracing – Convex Mirrors,
cont.
 The same three rays are
used as were used for
concave mirrors
 The focal ray is directed
toward the focal point but
is reflected at the mirror’s
surface, so doesn’t go
through F
 The three rays extrapolate
to a point behind the
mirror
 Produces virtual image
Section 24.4
Mirror Equation
 Geometry can be used to find the characteristics of
the image quantitatively
 The distance from the object to the mirror is so
 The distance from the image to the mirror is si
 The given rays produce similar triangles
Section 24.4
Mirror Equation and Focal Length
 From the similar triangles,
1
1
2


so si
R
 For an object at (approximately) infinity, 1/so = 0
 But an “infinite” object will produce parallel rays
 Parallel rays all intersect at the focal point
 Therefore, the focal length can be found from the
radius of curvature of the mirror
R
ƒ 
2
Section 24.4
Mirror Equation and Magnification
 The mirror equation can be written in terms of the
focal length
1
1
1


so si
ƒ
 The magnification can also be found from the similar
triangles shown in fig. 24.30
hi
si
m

ho
so
Section 24.4
Sign Conventions
 All diagrams with mirrors should be drawn with the
light ray incident on the mirror from the left
 The object distance is positive when the object is to
the left of the mirror and negative if the object is to
the right (behind) of the mirror
 The image distance is positive when the image is to
the left of the mirror and negative if the image is to
the right (behind) of the mirror
 The image distance is positive for real images and
negative for virtual images
Section 24.4
Sign Conventions, cont.
 The focal length is positive for a concave mirror and
negative for a convex mirror
 For a concave mirror, ƒ = R / 2
 For a convex mirror, ƒ = - R / 2
 The object and image heights are positive if the
object/image is upright and negative if it is inverted
Section 24.4
Sign Convention, Summary
Section 24.4
Lenses
 A lens uses refraction to form an image
 Typical lenses are composed of glass or plastic
 The refraction of the light rays as they pass from the
air into the lens and then back into the air causes the
rays to be redirected
 Although refraction occurs at both surfaces of the lens,
for simplicity the rays are drawn to the center of the
lens
Section 24.5
Lenses, Focal Point
 Parts B and C show the
simplification of the single
deflection of the rays
 Parallel rays close to the
principal axis intersect at
the focal point
 This is true for incident
rays from either side of
the lens
 The focal points are at
equal distances on the
two sides of the lens
Section 24.5
Spherical Lenses
 The simplest lenses
have spherical surfaces
 The radii of curvature of
the lenses are called R1
and R2
 The radii are not
necessarily equal
Section 24.5
Types of Lenses
 Converging lenses
 All the incoming rays parallel to the principal axis intersect
at the focal point on the opposite side
 Diverging lenses
 All the incoming rays parallel to the principal axis intersect at
the focal point on the same side as the incident rays
Section 24.5
Focal Point – Diverging Lens
 The parallel incident
rays from the left are
refracted away from the
axis
 The rays on the right
appear to emanate from
a point F on the left side
of the lens
 This point F is one of
the focal points of the
lens
Section 24.5
Image from a Converging Lens
 An infinite number of
rays emanate from the
object
 For simplicity, choose
three rays that are easy
to draw
 Start at the tip of the
object
Section 24.5
Rays for a Converging Lens
 The parallel ray is initially
parallel to the principal axis
 Refracts and passes
through the focal point on
the right (FR)
 The focal ray passes
through the focal point on
the left (FL)
 Refracts and goes parallel
to the principal axis on the
right
 The center ray passes
through the center of the
lens, C
Section 24.5
Rays, cont.
 If the lens is very thin, the center ray is not deflected
by the lens
 These three rays come together at the tip of the
image on the right of the lens
 In this case, the image is inverted
 The image is real
 The rays pass through the image
Section 24.5
Rules for Ray Tracing – Lenses
 Construct a figure showing the lens and its principal
axis
 The figure should also show the focal points on both
sides of the lens
 Draw the object at the appropriate point
 One end of the object will often lie on the principal axis
 Draw three rays that emanate from the tip of the
object
 The parallel ray is initially parallel to the principal axis
and after refraction passes through one of the focal
points
Section 24.5
Rules, cont.
 Three rays, cont.
 The focal ray is directed at the other focal point and
after refraction the ray is parallel to the principal axis
 The central ray passes through the center of the lens
and is not deflected
 The point where the three rays or their extrapolation
intersect is the image point
 If the rays actually pass through the lens, the image is
real
 If the rays do not pass through the lens, the image is
virtual
Section 24.5
Rules, final
 Real image
 When a lens forms a real image, the object and image
are on opposite sides of the lens
 Virtual image
 When a lens forms a virtual image, the object and
image are on the same side of the lens
 All other rays that pass through the lens will also
pass through the image
Section 24.5
Ray Tracing – Diverging Lens
 Follow the rules for ray
tracing for lenses
 Since the refracted rays
do not intersect on the
right side of the lens,
extrapolate the rays back
to the left side of the lens
 The extrapolations do
intersect
 The point of intersection is
the image point at the tip
of the image
Section 24.5
Sign Conventions – Lenses,
Diagram
Sign Conventions for Lenses
 Assume light travels through the lens from left to
right
 The object will always be located to the left of the lens
 The object distance is positive when the object is to
the left of the lens
 According to the first convention, the object distance
will always be positive
 The image distance is positive when the image is to
the right of the lens and negative if the image is to
the left of the lens
Section 24.5
Sign Conventions, cont.
 The focal length is positive for a converging lens and
negative for a diverging lens
 The object height is positive if the object extends
above the axis and is negative if the object extends
below
 The image height is positive if the image is extends
above the axis and is negative if the image extends
below
Section 24.5
Thin-Lens Equation
 Geometry can be used
to find a mathematical
relation for locating the
image produced by a
converging lens
 The shaded triangles
are pairs of similar
triangles
Section 24.5
Thin-Lens Equation and Magnification
 The thin-lens equation is found from an analysis of
the similar triangles
1
1
1


so si
ƒ
 The magnification can also be found from the similar
triangles shown
hi
si
m

ho
so
 These results are identical to the results found for
mirrors
Section 24.5