A light ray which is reflected from a surface will be reflected in the
plane formed by the normal to the surface and the incident ray. It
will be reflected on the opposite side of the normal as the incident
ray, and the angle it forms with the normal will be equal to that
formed by the incident ray.
Diagram 3.1 Reflection of a ray of light
If the surface on which the light falls is irregular and opaque, the
light will be reflected back into the incident medium, but in a random
manner. In
other words, the law of reflection will be satisfied, but the irregularities
in the surface will keep the rays from forming an image. The reflection
is
DIFFUSED. SPECULAR reflection, also called REGULAR reflection,
occurs when the surface is polished, and results in all rays of light
reflecting back in a non-random manner with respect to each other.
The polished surface must have its irregularities considerably smaller
than the
wavelength of the light falling on it in order to result in specular
reflection and image formation. For the balance of this course, REFLECTION
will
refer to specular reflection.
Light can be modeled with either the wave theory or the photon theory.
Diffuse reflection is easily diagrammed with either theory. When
we see a
non-shiny surface, we are looking at diffuse reflections.
Diagram 3.2 Specular and diffuse reflection of rays of light,
using the photon model.
Diagram 3.3 Specular and diffuse reflection of light waves, using
the wave model.
REFLECTION AT A PLANE SURFACE
When parallel rays of light fall on, or are incident on, a smooth, flat
(or plano) surface, the rays will be reflected. Parallel incident
rays remain
parallel, and converging or diverging rays continue to converge or
diverge at the original angle to each other. The observer will see
an image which
is not the surface itself. Instead, the observer will see images
of the objects which are on the same side of the surface as the observer.
The
images will appear to be erect, will be reversed side-for-side, and
will be the same distance behind the surface as the object is in front
of the
surface.
Diagram 3.4 Image formed by a polished reflecting surface
From diagram 3.4 we can see that the head of the arrow, which is on
the left, is on the left of the image which appears to be behind the surface.
Because the rays emerge from the reflecting surface at the original
angle with respect to each other, the image appears to be the same distance
behind the surface as the object is in front of it. This is the
same as saying that the reflecting surface did not change the vergence
of the rays;
therefore, the apparent distance of the image to the observer did not
change. The lack of change in vergence also results in the image
appearing to be the same size as the object. Our brains use the
law of rectilinear propagation (light travels in a straight line) to determine
how far
away an object is. The brain interprets the image to be where
the rays would be coming from if they had traveled in a straight line to
get to the
eye. So we interpret the object to be where the image is, behind
the mirror, even though intellectually we know none of the light entering
our
eye actually came from there.
The image is considered to be a virtual image because the light rays do not actually go through the point from which they appear to emanate.
Curved mirrors pose a more complex problem.
REFLECTION AT A CURVED SURFACE
A curved surface may be either concave (center bulging away from the
observer) or convex (center bulging toward the observer.) We will
be
concerned only with curved surfaces which are spherical in nature (described
by a circle or sphere).
A surface which is spherical in nature will have a center of curvature.
This is the point that is equidistant from all points on the surface of
the
sphere. We will suppose that the surface is mirrored, or reflects
all incident light.
On diagram 3.5, we are assuming that a curved mirror is made up of many
small plane mirrors. We can then apply the laws of reflection to
the mirror
as a whole. C denotes a point which is equidistant from each
small plane mirror. It is the center of curvature. The point
on an imaginary axis
through C where all of the reflected rays cross is f, the focal point.
The incident rays in this diagram are parallel to each other and to the
axis.
Diagram 3.5: Reflection of rays striking a curved surface.
A light ray traveling from the center of curvature and incident on the
curved surface will meet the surface at a right angle to the tangent to
the
circle at that point. In diagram 3.5, the tangent to the circle
is represented by a small plane mirror. This line from the center
of curvature is the
NORMAL at that point on the curve. Since the angle of reflection
is equal to the angle of incidence, and the angle of incidence is zero,
the ray will be
reflected back on the same path, and it will again go through the center
of curvature. A ray impacting on the surface but not going through
the center
of curvature will be reflected on the opposite side of the normal,
and will leave at an angle equal to the incident angle.
The AXIS of the sphere is any ray traveling through the center of curvature,
and the VERTEX of the sphere is where the axis we choose
meets the shpere. Diagram 3.6 shows a PARAXIAL ray AB impacting
on a spherical reflecting surface. That is, it is a ray traveling
parallel to the
chosen axis, and near but not on the axis. C represents the center
of curvature and V represents the vertex.
Diagram 3.6: Paraxial ray reflection from a concave surface.
The ray is reflected, and will cross the axis. The law of reflection
says that angle ABC = angle CBD. Since AB is parallel
to CD, angle ABC =
angle BCD, the triangle BCD is an equilateral triangle; and side CD
= BD. If ray AB were very close to the axis, BD would become almost
equal to
DV, and thus the point of intersection, D, would be close to 1/2 the
distance from the center of curvature to the vertex. DV represents
the
focal length f. Therefore, as the distance between the paraxial
rays and the axis approaches zero, f approaches one-half C.
As you can see from the diagram, rays further from the axis cross
the axis closer to the vertex V. This is spherical aberration, which
was discussed in the lectures on Aberrations. A surface that is parabolic
in shape, instead of spherical, will correct this aberration. We
use spherical shapes
instead of parabolic because they are easier to make. In optical
systems with a small aperture (the rays of interest are near the axis)
the spherical
and parabolic surfaces are almost identical. For systems where
the aperture is large with respect to the focal length, parabolic curves
become
necessary.
If we accept the premise that the object is 'close' to the axis, we
may now state the laws for constructing the image formed by a curved surface.
In
diagram 3.7, there are rays reflecting off of every point of the object
and diverging in all directions. We choose four principle rays from
the tip of
the object to describe the image which will be formed by all of the
rays that reflect from the concave surface. These particular rays
are described
by the following rules:
- The ray parallel to the axis will reflect through the focal point, at 1/2 the radius of curvature.
- The ray through the focal point will reflect parallel to the axis.
- The ray through the center of curvature will reflect upon it self.
- The ray incident at the vertex of the mirror reflects back at the same angle to the axis but on the other side of the axis.
Diagram 3.7: Formation of an image by a reflecting concave surface.
Other rays from the tip of the object will also be reflected following
the law of reflection, and they will also cross at the point where these
four
cross. Rays emerging from any point along the object will cross
where the corresponding point is on the image. The result for this
diagram is an
image which is inverted, smaller than the object, and real. It
is considered real because a film placed at this position would show an
actual image,
unlike the image shown for the plane mirror. All erect images
are virtual, all inverted images are real. (and vice versa.)
The position and size of the image can be computed algebraically. Consider diagram 3.8, showing principal rays 3 and 4.
Diagram 3.8: Image formation for a concave mirror.
The angles aVc and bVc are equal by definition of reflection, and so
triangles aVo and bVi are similar right triangles. Therefore,
O/p = I/q,
resulting in O/I = p/q. Also, angles aco and bci are equal.
Therefore, the measure of
We can also see from the original equation O/I = p/q, that the magnification
of the image, I/O, will have the same value as q/p. We will
use the following sign convention: real images have negative
length, and virtual images have positive length. Our convention for
p and q will be
that they are both positive if they are real. Thus, for mirrors,
the image distance is positive if the image is to the left of the mirror.
Therefore,
magnification becomes M = -q/p = I/O.
We must also have a sign convention for C and f. The center of
curvature and focal length will be positive for concave mirrors, where
they are on
the left of the reflecting surface, and negative for convex mirrors,
where they are on the right of the reflecting surface.
Note on the drawing that if the object is tall enough in comparison
to the focal length and center of curvature to make the drawing understandable,
then spherical aberration comes into effect, and the rays do not meet
exactly where they should. Rays 3 and 4 do not require the rule that
f is
1/2 the center of curvature, and thus they result in the drawn image
forming at the correct position.
We may now compute the position and size of the images for verification
of our drawings. For example, in diagram 3.7, if we state that p
= 11cm,
C = 4cm, and f = 2cm, then
1/2 = 1/11 + 1/q
1/q = 1/2 - 1/11 = 0.409
q = 1/0.409 = 2.44cm
M = -q/p = -2.44/11 = -0.22
resulting in an image which is about 1/4 the size of the object, inverted
and real, and 2.44cm to the left of the mirror vertex. These distances
and lengths may be verified on the drawing. When you do your drawings,
note that the differences between the drawn and computed image's size and
distance are due to aberrations resulting from the large size of the object
in relation to f and C.
The ray tracings for the convex or diverging mirrors follow the same
four rules, but they require tracing back where the rays would have come
from
in order to be going in their reflected directions. Thus, these
images are all virtual.
Diagram 3.9: Image formation at a diverging mirror.
Reading assignment: Stoner & Perkins, Optical Formulas Tutorial, page 185-192.
EXERCISES
1. In diagrams 3.10-18., trace at least two of the four principal rays, and show the image formed by each concave or convex mirror. Perform the mathematical calculations necessary to verify that the images in the drawings are correct.
Diagram 3.10
Diagram 3.11
Diagram 3.12
Diagram 3.13
Diagram 3.14
2. Describe the changes in the size and placement of the image in a concave mirror as the object approaches the mirror from a distance.
Diagram 3.15
Diagram 3.16
Diagram 3.17
Diagram 3.18
3. Describe the changes in the size and placement of the image
in a convex mirror as the object approaches the mirror from a distance.
The answers are here. Do them yourself first. If you just look at my answers and say "Yep, I would have gotten that" then you will have learned nothing. Does not matter to me. I already know how to do them. May very well matter to you.
Copyright 2001, Ellen Stoner,
MALS, ABOM