Image Formation
Image Formation
We will use geometrical optics: light propagates in straight lines
until its direction is changed by reflection or refraction.
When we see an object directly, light comes to us straight from the
object.
When we use mirrors and lenses, we see light that seems to come
straight from the object but actually doesn’t.
Thus we see an image (of the object), which may have a different
position, size, or shape than the actual object.
Images Formed by Plane Mirrors
When we use mirrors and lenses, we see light that seems to come
straight from the object but actually doesn’t. Thus we see an
image, which may have a different position, size, or shape than
the actual object.
object
image
virtual image
(the ray reaching your
eye doesn’t really come
from the image)
But…. the brain thinks the ray came from the image.
Images Formed by Plane Mirrors
You can locate each point on the image with two rays:
1. A ray normal to the mirror
object
image
Image is reversed
front to back
Images Formed by Plane Mirrors
You can locate each point on the image with two rays:
1. A ray normal to the mirror
2. The ray that reaches the observer’s eye
object
image
Image is reversed
front to back
Images Formed by Plane Mirrors
You can locate each point on the image with two rays.
object
image
Images Formed by Plane Mirrors
The distance from the image to the mirror equals
the distance from the object to the mirror: d = d
d
object
d
image
Also, the height of
the image equals the
height of the object
Images Formed by Plane Mirrors
Properties of Mirror Images Produced by Plane Mirrors
• The distance from the image to the mirror equals
the distance from the object to the mirror: d = d
• The mirror image is upright but reversed right to left
• The mirror image is reversed front to back
• The mirror image is the same size as the object
A flower has a height h, and is at a distance d, from a plane mirror.
An observer is at table level, a distance d behind the flower.
Find y as a function of h
Parabolic Mirrors
• Shape the mirror into a
parabola of rotation (In one
plane it has cross section given
by y = x2).
• All light going into such a
mirror, parallel to the parabola’s axis of rotation, is
reflected to pass through a
common point - the focus.
• What about the reverse?
•
Parabolic Mirrors
• These present the concept of a focal point - the point to which
the optic brings a set of parallel rays together.
• Parallel rays come from objects that are very far away (and,
after reflection in the parabolic mirror, converge at the focal
point or focus).
• Parabolas are hard to make. It’s much easier to make
spherical optics, so that’s what we’ll examine next.
Spherical Mirrors
Concave
Convex
Spherical Mirrors
Convex Mirror
Concave Mirror
A ray incident parallel to
the axis reflects as if
coming from the focus.
A ray incident parallel to
the axis reflects passing
through the focus.
The focal distance f
is given by f = - R/2
The focal distance f
is given by f = R/2
Spherical Mirrors
To analyze how a spherical mirror works we draw
some special rays, apply the law of reflection where
they strike the spherical surface, and find out where
they intersect.
f
c
A ray parallel to the mirror axis reflects through the focal point f
A ray passing through the focus reflects parallel to the axis
A ray that strikes the center of the mirror reflects symmetrically
A ray passing through the center of curvature c, returns on itself
Image Formation
Trace one ray incident parallel to the axis
Trace a second ray incident through the focus
The image is at the intersection of the two rays.
Repeat for every point in the image.
In practice only ‘head’ and ‘tail’ are needed.
f
c
Spherical Mirrors - Concave
When the object is beyond c, the image is:
real (on the same side as the object),
reduced, and inverted.
f
c
Spherical Mirrors - Concave
Object between c and f.
f
c
Image is real, inverted, magnified
Spherical Mirrors
Object between f and the mirror.
f
c
Image is virtual, upright, magnified
The Mirror Equation
h0 d0
h0 d0  R

and

hi di
hi R  di
Next, 1 
1 R
R
di
d0 d0  R
Then,

di R  di
R R
1 1 1

 2 
 
d 0 di
d 0 di f
1
d0
The Mirror Equation
c
f=R/2
f
l’
l
1 1 1
  '
f l l
Magnification
h
f
h’
c
The magnification is given by the ratio M = h’ / h = - l’/ l
Curved Mirrors
mirror equation
focal length
magnification
1 1
1


l l'
f
f  R/ 2
h'
l'
M 

h
l
Curved Mirrors
Sign conventions:
Distance in front of the mirror  positive
Distance behind the mirror  negative
Height above center line  positive
Height below center line  negative
Concave
Convex
Image With a Convex Mirror
c
f
Here the image is virtual (apparently positioned behind
the mirror), upright, and reduced. Can still use the mirror
equations (with negative distances for f, c=R, and l’).