Geometric Optics
Mirrors, light, and image
formation
Geometric Optics
• Understanding images and image
formation, ray model of light,
laws of reflection and refraction,
and some simple geometry and
trigonometry
• The study of how light rays form
images with optical instruments
Reflection and refraction on plane mirrors
REFLECTION AND REFRACTION AT A
PLANE SURFACE
Key terms
• Anything from which light rays radiate
– Object
• Anything from which light rays radiate
that has no physical extent
– Point object
• Real objects with length, width, and
height
– Extended objects
Key terms
Specular
reflection
Reflection on a
plane surface where
reflected rays are in
the same directions
Diffused
reflection
Relfection on
a rough
surface
Key terms
Virtual
image
Image formed if
the outgoing rays
don’t actually
pass through the
image point
Real image
Image formed if
the outgoing
rays actually
pass through
the image point
Image formation by a Plane mirror
Image formation by a Plane mirror
Ray
Diagrams
Line of Sight
Principle
• a diagram that traces the path
that light takes in order for a
person to view a point on the
image of an object
• suggests that in order to view
an image of an object in a
mirror, a person must sight
along a line at the image of the
object.
Reflection at a Plane Surface
Image formation by a Plane mirror
θ
θ
M
θ
s
V
θ
s’
M’
Image formation by a Plane mirror
• M is the object and M’ is the virtual
image
• Ray MV is incident normally to the plane
mirror and it returns along its original
path
• s= object distance
• s’= image distance
• s=-s’
Image formation by a Plane mirror
• Sign rules
For the object distance:
– When the object is on the same side of the
reflecting or the refracting surface as the
incoming light, s is positive
For the image distance:
– When the image is on the same side of the
reflecting or the refracting surface as the
outgoing light, s’ is positive
Image of an extended object
Q
Q’
V’
θ
y
M
θ
θ
sθ
θ
V
s’
y’
M’
Image of an extended object
• Lateral magnification
–Ratio of image height to object height
–M=y’/y
• Image is erect
• m for a plane mirror is always +1
• Reversed means front-back dimension
is reversed
Reflection on Concave and Convex mirrors
REFLECTION AT A SPHERICAL
SURFACE
Reflection at a Concave Mirror
P
C
V
P’
Reflection at a Concave Mirror
R
•Radius of curvature
C
• Center of curvature
• The center of the sphere of
which the surface is a part
V
• Vertex
• The point of the mirror surface
CV
•Optic axis
Graphical Methods for Mirrors
IMAGE FORMATION ON
SPHERICAL MIRRORS
Graphical Method
• Consists of finding the point of
intersection of a few particular rays
that diverge from a point of the object
and are reflected by the mirror
• Neglecting aberrations, all rays from
this object point that strike the mirror
will intersect at the same point
Graphical Method
• For this construction, we always
choose an object point that is not
on the optic axis
• Consists of four rays we can usually
easily draw, called the principal
rays
Graphical Method
A ray parallel to the axis, after
reflection passes through F of
a concave mirror or appears
to come from the (virtual) F
of a convex mirror
A ray to V is reflected
forming equal angles
with the optic axis
A ray through (or
proceeding toward)
F is reflected
parallel to the axis
A ray along the radius
through or away from C
intersects the surface
normally and is reflected
back along its original path
Object is at F
Object is between F and Vertex
Object is at C
Object is between C and F
Positions of objects for concave
mirrors
Image formation by concave mirrors
Position of object
Position of image
Character of image
Between F and C
Real, inverted,
reduced
At C
Real, inverted, same
size
Between C and F
Beyond C
Real, inverted,
enlarged
At F
At infinity
No image
Beyond the vertex
Virtual, upright,
enlarged
At V
Virtual, upright, same
size
Beyond C
At C
Between F and
Vertex
At V
Reflection at a Concave Mirror
If α dec, θi
is nearly
parallel
Rays nearly
parallel or
close to R
Paraxial
rays
Reflection at a Concave Mirror
If α inc, P’
is close to
V
Image is
smeared
out
Spherical
Aberration
Reflection at a Concave Mirror
C
F
s at infinity s’= R/2
V
Reflection at a Concave Mirror
• All reflected rays converge on the
image point
• Converging mirror
• If R is infinite, the mirror
becomes plane
Reflection at a Concave Mirror
The incident parallel
rays converge after
reflecting from the
mirror
They converge at a F
at a distance R/2
from V
F is Focal point,
where the rays are
brought to focus
f is the focal length,
distance from the
vertex to the focal
point
f= R/2
Reflection at a Concave Mirror
C
F
s’ at infinity s= R/2
V
Reflection at a Concave Mirror
The object
is at the
s=f=R/2
focal
point
1/s+
1/s’= 1/f
Object image relation, spherical
mirror
1/s +1/s’= 2/R
1/s’=0; s’ at
infinity
Image of an Extended Object
m= y’/y
Lateral
magnification
m= y’/y= -s’/s
Lateral
magnification for
spherical mirrors
Reflection at a Convex Mirror
F
s or s’ at infinity
C
s’ or s= R/2
Image formation on spherical mirrors
• Sign rules
For the object distance:
–When the object is on the same side
of the reflecting or the refracting
surface as the incoming light, s is
positive; otherwise, it is negative
Image formation on spherical mirrors
• Sign rules
For the image distance:
–When the image is on the same side
of the reflecting or the refracting
surface as the outgoing light, s’ is
positive; otherwise, it is negative
Image formation on spherical mirrors
• Sign rules:
For the radius of curvature of a spherical
surface:
–When the center of curvature C is on
the same side as the outgoing light, the
radius of curvature is positive,
otherwise negative
Reflection at a Convex Mirror
• The convex side of the spherical mirror
faces the incident light
• C is at the opposite side of the outgoing
rays, so R is neg.
• All reflected rays diverge from the same
point
• Diverging mirror
Reflection at a Convex Mirror
Incoming rays are
parallel to the optic
axis and are not
reflected through F
s is positive, s’ is
negative
Incoming rays diverge,
as though they had
come from point F
behind the mirror
F is a virtual focal
point
Refraction at spherical interface
REFRACTION AT A SPHERICAL
SURFACE
Refraction at a Spherical Surface
V
C
Refraction at a Spherical Surface
na/s + nb/s’= (nb-na)/R
Object-image
relation, spherical
refracting surface
na/s + nb/s’=0
At a plane refracting
Lateral
surface
magnification,
m=y’/y= -(n s’/n s)
a
b
spherical
refracting surface
Biconcave and biconvex thin lenses
GRAPHICAL METHOD FOR
LENSES
Lenses
Lenses
Biconvex
lens;
converging
Biconcave
lens;
diverging
Lenses
Only F is
needed
for the
ray
diagram
Chief ray
through
the center
is
undeviated
For concave lens, the
rays appear to have
passed through F on
the object’s side of the
lens
Ray parallel is
refracted in such a
way that it goes
through F on
transmission
through the lens
Focal ray is parallel
to the axis of
transmission
Lens maker's equation
ANALYTICAL METHOD FOR THIN
LENSES
Equations for thin lenses
1/s
+
1/s’=
1/f
1/f=(n-1) [(1/R
Object-image
relation, thin
lenses
2
m=y’/y= -s’/s
Lateral
magnification,
thin lenses
)(1/R
)]
1
Lensmaker’s equation