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Geometrical optics
In this section we study optical systems involving lenses and
mirrors, developing an understanding of devices such as
microscopes and telescopes, and biological systems such as the
human eye.
In the previous section we were studying “physical optics”,
focusing on the fundamental description of electromagnetic
waves, and on basic phenomena such as reflection, refraction,
and diffraction. In “geometrical optics”, we apply the principles
and equations of reflection, refraction, and dispersion to
understand optical systems. The first few slides bring forward
the topics we need from the previous section.
Recall…
Definition of light rays
The motion of a point on a wave
front may be indicated by a
direction vector perpendicular to
the front. These direction
vectors may in turn be
connected into single lines or
curves, called “rays”. to show
the overall motion of a wave
front.
Typically, a given problem will
focus on the behavior of either
the wave fronts or the rays, and
often only one of the two will be
shown. One can always be
constructed from the other by
drawing perpendicular segments
and connecting them.
Electromagnetic waves in materials
Recall…
Electromagnetic waves travel more slowly in materials than they do in vacuum.
For visible light, we are most interested in transparent materials such as glasses,
plastics, liquids (especially water), and gases. The ratio of the speed of light in
vacuum to the speed in the given material, is called the “index of refraction”, n:
n
c
Material
Index of
Refraction

Vacuum
1.0000
Air
1.0003
Ice
1.31
Water
1.333
Ethyl Alcohol
1.36
Plexiglas
1.51
Crown Glass
1.52
Light Flint Glass
1.58
Dense Flint Glass
1.66
Zircon
1.923
Diamond
2.417
Rutile
2.907
Gallium phosphide
3.50
The table at right lists the index of refraction
for a number of common (and uncommon)
materials. You can see the trend that the
index of refraction rises with density. If we
want to calculate the speed of light in these
materials, we solve for v above:
c

n
Fundamentally:

1


1
K 0 K m  0

1
K  0 0

c
c

K n
<--lowest optical
density
<--highest optical
density
Recall…
Laws of Reflection
The Laws of Reflection:
(1) The angle of incidence equals the angle of
reflection: qa = qr .
(2) The incoming and outgoing rays, and the
normal, are all in one plane.
Law of Refraction
Recall…
Light traveling from a medium of lower index of refraction to higher bends
toward the normal. From higher to lower (eg. glass to air) the bending is away.
n1 < n2
n1 > n2
n1 n2
n1 n2
The angles are calculated
from the Law of Refraction:
n1 sin q1  n2 sin q2
The critical angle, beyond which total
internal reflection occurs, for n1 > n2 only:
n2
sin q c 
n1
How a lens focuses light
Light rays going through glass plates
emerge at their incident angles. But
overall bending can be achieved by
creating wedges, or “prisms”, of glass.
A stack of
prisms can
bend light
toward
one line.
A symmetric
stack of
prisms can
bend (focus)
to a line on
its symmetry
plane.
Smoothing the front and back surfaces into spherical sections will create a lens
that can focus incoming parallel rays to a sharp point, known as the focal point.
Focal point of a converging lens
As the picture shows at right, rays from
infinity (parallel rays), going through a
converging lens, will pass through a focal
point. This is the way we defined “focal
point” on the previous slide.
The second picture shows that a point
source of light placed at a focal point will
cause parallel rays (plane waves) to
emerge from the other side of the lens.
(1) Every lens has two focal points—one
on each side, the same distance from the
center, and (2) the lenses pictured here
are called “converging” lenses since they
bend light toward their optical axis,
causing the rays to converge. Notice
that, beyond F2 in the top picture, they
are diverging.
The “object-image relation” for a thin lens
As we know, a lens can focus light
to form an image on a screen. In
the diagram at right, the object is
the arrow on the left, and the
image is the inverted red arrow on
the right. If we have been given
the position and size of the object,
the “problem” is to find the position
and size of the image.
The method we are using here is called “ray tracing”. There are light rays leaving
the top of the object in all directions. By “knowing” any two rays, we can find where
they cross, and that is the position where the light from the top of the object focuses.
We know that (1) the ray traveling parallel to the lens axis will pass through the focal
point and, (2) the ray passing through the center of the lens will be straight (for a thin
lens). We can calculate the crossing position, Q’, from similar triangles:
OPQ sim OP’Q’: y   y 
s
s
s s  f s

 1
Equate y’/y:
s
f
f
y
s

y
s
OAF2 sim P’Q’F2: y   y 
1 1 1
 
s f s
f
s  f
1 1 1
 
s s f
then
Object-image relation
y
s  f

y
f
m
y
s

y
s
Magnification
Images created by light rays passing through lenses
From our everyday experience, we know that images we see in mirrors are
inverted left-right (“mirror images”). Look at your left hand in a mirror some
time, and you’ll see a right-handed image of your left hand.
This does not happen with images created by light passing through a lens. The
ray tracing construction on the previous slide showed that a converging lens
inverts an image in the “y” direction. It does the same thing in the “x” direction.
As the picture below shows, the net result is that the image is rotated, but that
its “handedness” is preserved. This is still true in systems with multiple lenses,
since rays pass through the lenses in succession, and each lens preserves
handedness.
Sign rules for object and image distance, and radii of curvature
Sign rule for the object distance: When the object is on the same side of the
reflecting or refracting surface as the incoming light, the object distance s is
positive; otherwise, it is negative.
Sign rule for the image distance: When the image is on the same side of the
reflecting or refracting surface as the outgoing light, the image distance s’ is
positive; otherwise, it is negative.
Sign rule 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, it is negative.
Focal point of a diverging lens
When we introduced the terminology
“converging lens”, you may have suspected
that lurking in the shadows would be a
“diverging lens”. Well, here it is! Any lens
that is thinner at the middle than at the edges
is a diverging lens, which causes incoming
light rays to spread (diverge) instead of
focusing (converging).
Oddly enough, diverging lenses still have a
“focal point”. It is the point from which
parallel light seems to come when passed
through the lens (upper picture). Also, light
focusing to a point can be made parallel if its
focal point coincides with that of the lens
(lower picture). Fortunately, no new
derivations are required. We can use the
object-image relation with a diverging lens
simply by making the focal length, f, negative!
( We’ll discuss this again later.)
Consider a converging lens,
f, followed by a diverging
lens with f = -f.
The lenses at the top are all
examples of converging
lenses, and at the bottom,
diverging lenses.
Consider (1) how this creates
convenience, (2) why certain
combinations of a and b clearly
have no net effect.
“Real” images and “virtual” images
In the upper picture, light from
the object at P is passing
through a converging lens and
brought to a focus at P’, where
it creates a real image on a
screen. Diffuse reflection from
the screen is delivering light to
the eye.
In the lower picture, light from
the object at P is passing
through a diverging lens. If we
look through the lens we see a
virtual image that appears to be
at location P’. This time the
light comes directly from the
lens to the eye.
As we shall see, however,
under the right geometrical
conditions, either type of lens
can lead to either type of
image!
Ray tracing & object-image relation for diverging lenses
Recall the rays we constructed to
analyze the converging lens, then
construct analogous rays for the
diverging lens. Note: we have
drawn three rays in each case, but
any two are enough to specify the
image location.
If we repeated the derivation of the
object-image relation for this lens,
we would see that we get the
same equation as before as long
as we specify the focal length of
the diverging lens as a negative
number. Diverging lenses are
sometimes called “negative
lenses”.
1 1 1
 
s s f
Object-image relation
m
y
s

y
s
Magnification
Images produced with a converging lens
MEPI
The lensmaker’s equation
Imagine that you have a
piece of glass, with index of
refraction n, that you want to
form into a lens. The problem
involves determining the radii
of curvature for the spherical
surfaces needed to achieve a
certain focal length. For this
we use (without derivation)
the lensmaker’s equation:
 1
1
1 

 (n  1) 
f
 R1 R2 
Note that radii with C on the
outgoing side, such as R1, ,
have a positive sign, whereas
those on the incoming side
have a negative sign.
If the two radii are equal in
magnitude, this simplifies
to:
1 2(n  1)

f
R
The pinhole camera
Image from a cylindrical oatmeal box camera
The human eye
Near-sightedness
Far-sightedness
Correcting astigmatism
Operator’s manual
Lens “power”, in “diopters”
For lenses used to correct myopia or hyperopia in the eye, instead of specifying
the focal length of the corresponding diverging or converging lens, it is
conventional to specify the same information as the “power” of the lens in units of
“diopters”. The conversion is simple: the power is the reciprocal of the focal
length in meters. Eg. f = .4 m  2.5 diopters for a converging lens, with
negative numbers for diverging lenses (since f is negative).
A magnifying glass
A normal human eye can focus at a nearest
distance of about 25 cm. Usually we use a
converging lens to create a real image on a
screen. But if that “screen” is our eye,
placed just inside the focal point, we will see
a large virtual image appearing “nearby”.
The virtual image looks much larger than the
original because it covers a larger angular
range. This causes the image projected on
our retinas to be larger by (approximately)
the ratio of these angles, which we see as
magnification, M:
M 
q
y/ f
25 cm


q y / 25 cm
f
A magnifying glass cannot be used for M
greater than about X3 to X4 because the
lens becomes thick and introduces
aberrations. For higher magnifications, we
need an optical microscope. (Below.)
Two lens system
For optical systems with more than one optical element (lens, mirror), the image
position and magnification are found by stepping through the elements one at a
time, finding the intermediate image at each step, until light rays have reached the
final image position. At each step, the previous image becomes the new object,
and for each element in the system one applies the object-image relation for that
particular element: 1 1 1
s

s

f
Optical microscope
An optical microscope, otherwise
known as a “compound microscope”
has two lenses. The one nearest
the object (“objective lens”) forms a
magnified real image at location F1’ ,
then the lens nearest the eye
(“eyepiece lens”) acts as a
magnifying glass to magnify it
further. The total magnification M is
the product of the magnifications of
the objective, m1, and eyepiece, M2.
Once again, the near focus limit of
the eye at 25 cm is the limiting
criterion.
s
s
Magnification of
m1   1   1
the objective lens:
s1
f1
MEPI
Magnification of M  25 cm
2
f2
the eyepiece:
(25 cm) s1
Combined
M  m1 M 2 
magnification:
f1 f 2
Refracting telescope
Like the compound
microscope, the
refracting telescope
uses an objective lens to
create a real image at a
location within the tube,
and this image is then
viewed with an eyepiece
lens to magnify the
image further. Like the
magnifying glass, the
total magnification for a
telescope is best
understood in terms of
“angular magnification”.
 y
F2 angular range of
tan q  q 
object at infinity:
f1
MEPI
Magnification, from
ratio of angles:
Eye angular range
of image at F2:
M
tan q   q  
y / f 2
f
q

 1
q
y  / f1
f2
y
f2
Images in a plane mirror
To analyze lens optics we used the
Law of Refraction, but for mirrors of
any shape in an optical system, we
use the Law of Reflection. Diffuse
light coming from any point on an
object reflects from the mirror
surface with q a  qr . Some of the
light from this point enters our eye,
forming a virtual image. For a plane
mirror, the image distance behind
the mirror surface is equal to the
object distance in front of the mirror.
For an extended object, in a plane
mirror, the virtual image is the same
size as the object. So the
magnification is 1. Discuss the rays
we have constructed to find the
image position.
Chromatic aberration
Dispersion, the variation of the index of refraction with frequency in transparent
materials, causes problems with the focal properties of lenses. As we saw, in
prisms, blue light bends more than red light. So the same effect must happen in
lenses—where one assumes that ray paths are independent of color. The first
picture below shows how lenses will have slightly different focal lengths for
different colors. This effect is called “chromatic aberration” and, when
noticeable, the colors are seen to separate at the “edges” of images. The
second picture shows one way to reduce this effect: an “achromatic doublet”,
made of two lenses with different indices of refraction (often found in cameras).
MEPI
Spherical mirrors, and their focal points
In many optical systems, concave
or convex spherical mirrors may
take the place of lenses. As seen
in the pictures at right, parallel rays
entering a concave spherical
mirror will focus at a point. And
parallel rays shining on a convex
spherical mirror will appear to
come from a focal point.
Notice: f = R/2
So concave spherical mirrors are
similar in their action to converging
lenses, and convex, to diverging
lenses. In fact, following the sign
rules listed in an earlier slide, we
may use the object-image relation
to find images formed by spherical
mirrors.
Mirrors have several advantages over lenses. Among them: (1) they are based
on the Law of Reflection, so there is no chromatic aberration, and (2) it is
practical to make them VERY large. (More later on the latter.)
The images formed by concave spherical mirrors
Discuss
The focal length of a spherical mirror
Two slides back, we saw (without any proof), that the focal length of a
spherical mirror is f = R/2, where R is the radius of curvature. The text goes
through a rather lengthy derivation of this result, using the two pictures below.
We won’t reproduce the proof here.
This simple formula is fine for doing homework problems. But it is based on
the small angle approximation, and is not adequate for real optical systems.
In fact, most large, high quality concave or convex mirrors are not spherical
sections, they are parabolic sections. We’ll see why on the next slide.
Spherical aberration  mirrors should be
parabolic for perfect focus
This demonstration shows that the geometry of
a spherical mirror does not cause all rays to
cross its axis at exactly the same point.
Mathematics shows that the surface giving a
perfect focus must be parabolic. (Try sketching
rays entering the edge of a hemisphere to see
why.) A sphere approximates a parabola at
small angles, the domain where the simple
formula is acceptable.
Spherical
Parabolic
The U of A Mirror Lab
Roger Angel, a Professor of Astronomy at the
University of Arizona, invented the method
used to produce all the gigantic telescope
mirrors being installed in observatories
worldwide. The mirrors are spin-cast in a
rotating oven at the Mirror Lab, below the
football stadium seats. The oven melts the
borosilicate glass as it rotates, causing the
glass to flow into a paraboloid shape that
requires very little grinding and polishing to
create the final mirror.
The U of A Mirror Lab
A mirror, following spin-casting, being readied
for removal from the oven.
The U of A Mirror Lab
After grinding, the mirrors are polished to a
surface accuracy of a fraction of a wavelength.
The U of A Mirror Lab
Roger Angel
8.4 meter diameter mirror in place at the Large
Binocular Telescope on Mount Graham
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