Optical Instruments
Scientists study the world as it is, engineers create the world that never has been.
Theodore von Karman (1881-1963)
OBJECTIVES
To examine the operation of some simple optical instruments: eye glasses, magnifier, and
compound microscope.
THEORY
A normal human eye forms sharp images of objects between about 25 cm and infinity.
Many individuals, however, have visual defects which hinder their ability to see clearly over the
full range, and with advancing years essentially everyone will experience a reduction in focusing
ability. Eye glasses, first known from about 1300, are a great aid in such cases. Of course, even
the best eyes have limited resolution, and cannot adequately image very small or very distant
objects. For that people invented microscopes and telescopes, with revolutionary impact on
science and technology. In this lab session you will briefly examine the operation of some of
these devices, starting with the eye itself.
The eye of a typical vertebrate consists of one or two lenses in front of a sheet of neural
material that converts the optical image to electrical signals for further processing elsewhere. In
humans most of the refracting power is due to the front surface (cornea) of the eyeball. The
"crystalline lens" (which is not crystalline) sits directly behind the cornea, inside the eye. It is
equipped with muscles that can change its effective focal length, allowing the eye to focus at the
desired plane. Unlike a camera, the distance between the lens and the image plane is fixed, so all
the accommodation is the result of changes in the crystalline lens.
The optical system of the eye is not particularly good, so the resulting image is somewhat
distorted. Many types of distortion are corrected when the image is processed in the brain, but
some defects are so severe as to hinder vision. The most common problem is nearsightedness or
farsightedness: The inability to focus on far away or nearby objects, respectively. Figure 1
illustrates the problem for a near-sighted (myopic) eye.
If we approximate the optics of the eye by a thin lens, we can understand these refractive
problems with the aid of the Gaussian lens formula
1 1 1
+ =
fe
so si
(1)
25 cm
range of clear vision
normal
eye
near
point
50 cm
far
point
8 cm
near
point
myopic
eye
corrected
myopic
eye
Fig. 1 Comparison of a normal, myopic and corrected myopic eye. The far point and near point
represent the limits of clear vision.
where s o is the distance from object to lens, si is the distance from the lens to the focused image
and fe is the focal length of the lens. The image distance si must be the same as the length, d, of
the eye for a sharp image to be produced on the retina. In a nearsighted person, the lens is too
strong (fe is too small) to form an image at d when so is large. Conversely, in a farsighted person
the lens is too weak (fe is too large) to form an image when s o is small. Since the size of the eye
is fixed, we need to change f. This can be done by surgery or with a lens placed in front of the
eye.
To correct a nearsighted eye, we use a diverging lens to partially counteract the focusing
of the eye lens, as suggested in Fig. 1. Within the thin lens approximation, the effective focal
length f' of two lenses separated by much less than their individual focal lengths is given by
1
1
1
=
+
f fe f c
(2)
where fc refers to the corrective lens. We want to pick f' so that
1 1 1
1
1
+ =
=
+
d f fe fc
(3)
Of course we do not know d (patients usually object to a direct measurement), but we can
measure the farthest distance at which an object is in clear focus for the bare eye. Calling this
distance sf, we obtain
Physics 231 Optical Instruments
2
50 cm
150 cm
presbyopic
eye
near
point
far
point
75 cm
presbyopic
eye with
bifocals
38 cm
25 cm
Fig. 2 Presbyopia (limited accommodation) and correction with a bifocal lens.
1
1 1
+ =
sf d fe
(4)
Solving (3) and (4), we find fc = -s f. As expected, we need a negative lens. A similar argument
will give the correction for a farsighted eye in terms of the closest distance at which the bare eye
can focus.
If the variation in focal length of the crystalline lens is normal, a single lens is adequate to
restore normal vision. With age, however, the crystalline lens gradually becomes rigid and
accommodation is lost, a defect known as presbyopia. The solution is to use different lenses for
near and far vision, often in the form of bifocals as shown in Fig. 2.
Another common defect is astigmatism, in which the eye is unable to sharply focus an
entire image at any distance. Astigmatism is usually produced by an asymmetry of the cornea
which causes the effective focal length of the eye to be different for groups of rays in the
horizontal and vertical planes. A lens in the form of a cylinder section (rather than a sphere
section) will focus only in one plane, and can sometimes be used to correct astigmatism. The
correction is found in the same way as for spherical lenses.
A single-lens magnifier is the logical extension of eye glasses. In trying to examine fine
details we bring an object closer to our eyes so that it subtends a larger angle, but eventually we
reach the point of closest focus and there is no further gain. A short focal length lens placed in
front of the eye allows us to continue this process so that the object appears closer and hence
larger. Referring to Fig. 3, we make s o smaller than the focal length so that the lens forms a
virtual image at the near point of the eye. The eye forms a real image on the retina subtending an
angle ' which is larger than the angle without the lens. A reasonable definition of
magnification would be
Physics 231 Optical Instruments
3
object
y
θ
25 cm
so
virtual y'
image
θ'
y
object
25 cm
-si
Fig. 3 A simple lens used as a magnifier.
M=
y / si
y / 25cm
(5)
Using the simple lens formula and the fact that s i is negative for a virtual image finally yields
M =1+
25cm
f
(6)
where f is the focal length of the magnifying lens.
The magnification available with a single lens is limited because the lens becomes too
small to handle. The compound microscope, shown in simplified form in Fig. 4, allows much
higher magnification with lenses of manageable dimensions. The object to be examined is placed
just beyond the focal point of the objective lens, so that a real magnified image is formed as
shown. The eyepiece lens is positioned so as to form a magnified virtual image of that real
image. The overall magnification is the product of the objective and eyepiece magnifications,
M = Mo Me =
si 25cm 1+
so fe (7)
where the minus sign indicates that the image is inverted. The tube length of the microscope
body, D in Fig. 4, is standardized so objective and eye piece lenses are often marked with their
magnification. Other features of practical importance are discussed in standard textbooks.
Physics 231 Optical Instruments
4
D
si
so
object
fo
fe
fo
objective
eyepiece
virtual
image
Fig. 4 A simplified compound microscope. The focal points of the lenses are marked.
EXPERIMENTAL PROCEDURE
We will mount most of the components (lenses, screens, etc.) we need on an optical
bench. Lenses and other components are installed in sliding holders that fit on the bench. Holders
can be attached to the optical bench with screws when stability is needed. Best results will be
obtained if everything is centered at the same height on a line parallel to the optical bench.
Lenses are specified by focal length or by strength in diopters, the inverse of the focal
length in meters. Please be careful with the lenses, as some of them chip easily.
1. The eye
The eye model is sketched in Fig. 5. Lens C is fixed at 5.75 diopters, to simulate the
cornea. Lens L is either 2.25 diopters, for far vision, or 5.75 diopters, for near vision. This
change simulates the accommodation of the real eye. As in the real eye, lenses C and L should be
very close together, conveniently done with a double lens holder on one slide. Provision is made
to place circular apertures between C and L to simulate the iris. The "retina" is a cardboard
screen, and the object is one of two lighted boxes, a small one on the optical bench for 'near'
vision, or a larger one across the room for 'distant' vision. Our model is a good deal bigger than a
real eye because we chose weaker lenses. The lens and cornea on an intact human eye have a
total power of about 58 diopters, compared to the maximum of about 11 diopters used here.
You should begin by setting the model for distant vision. Install the 5.75 diopter lens
at C and the 2.25 diopter lens at L. Focus the image of the large light box, located as far away
as possible, onto the screen and make note of the distance d. This is the size of the “normal”
model eye. You will probably find the image is improved if you insert the large aperture and
Physics 231 Optical Instruments
5
light box with
object slide
lens C
"cornea"
lens L
"lens"
corrective
lens
screen
"retina"
d
Fig. 5 Optical bench arrangement for the model eye. Only the “near” light box is shown. The
“far” light box should be placed as far away as possible, across the room.
leave it in place during the remainder of the experiment. Is the retinal image upright or
inverted? Reversed? Since these terms are somewhat ambiguous, be sure to explain what you
mean.
Next we will study the range of distances over which the "normal" eye we have
constructed can focus an image. With the 2.25 diopter lens at L, the eye is focused at "infinity".
Without changing d, replace the 2.25 diopter lens at L with a 5.75 diopter lens. Using the small
light box on the slider and the 'crossed-line' object slide, find the "near point", the closest
distance of sharp focus. Is the accommodation range of the model eye reasonably close to
normal? For later use, you should also note the ratio of image size to object size at the near point.
Now make the eye nearsighted by increasing d to a value of 16-17 cm. Find the new near
point, and the ratio of image size to object size. Why might a nearsighted person prefer to
examine a small object without glasses on?
Next, change back to the 2.25 diopter lens, find the far point for the nearsighted eye (s f in
Eq. 4), and compute the necessary correction. Using a separate holder, place the corrective lens
close to the front of the eye, to simulate eyeglasses or a contact lens. Is the image of the far
object now reasonably focused? Does the image change if the correction lens is close to the eye
or a few centimeters away? Install the 5.75 diopter lens at L for near vision, and find the
corrected near point. Is the corrected accommodation roughly the same as in the "normal"
model?
Our study of astigmatism will be qualitative. Remove the corrective lens, and set d to the
"normal" value you found at the beginning of the experiment. Obtain a sharp image of the near
object on the screen. Now place the cylinder lens in front of the eye, where the corrective lens
was. Describe qualitatively what happens to the image when you rotate the cylinder lens. With
the lenses fixed, what happens to the image when you move the retina? You may also find it
interesting to place the cylinder lens in front of you own eye, to better understand why
astigmatism must be corrected for adequate vision.
Physics 231 Optical Instruments
6
screen
and ruler
25 cm
desk lamp
glass
plate
light box
eye
Fig. 6 Optical arrangement for measuring magnification of a lens used as a magnifier.
2. Simple magnifier
Magnifiers are probably familiar, so our study will consist only of a measurement of
the magnification. Dismantle the eye model, install the 5 cm focal length lens on a slider, and
put the multi-line slide on the light box. Look through the lens, with your eye close to it, and
position the light box close to the opposite side of the lens so you see a clearly focused,
magnified virtual image of the slide.
Measuring the magnification is trickier. The classical method is to look through the
lens with one eye, look around the lens at a ruler with the other eye, and position the ruler so
that both images are superimposed and in focus at the observer’s near point. Most people find
this difficult, so we will instead use the arrangement of Fig. 6. A glass plate positioned near
the lens and at 45º to the optical axis acts as a beam combiner. Some of the light coming
from the screen passes through the glass plate, so it is visible to an observer positioned as
shown. Some of the light coming through the lens is reflected from the glass, so the image is
also visible to the same observer.
Looking through the plate as indicated you will see the lens image superimposed on the
screen. Make small adjustments to the position of the light box so that the grid pattern on the
screen is in focus at the same time as the image seen through the lens. You can then use the ruler
to measure the apparent height of the image. Compare the observed ratio of image height to
object height to the magnification you calculate from Eq. 6.
3. Compound microscope model
The arrangement for a compound microscope is sketched in Fig. 7. The 15 cm focal
length lens is the eyepiece and the 5 cm lens is the objective. Following the approximate
dimensions shown, focus an image of the fine grid on the ground glass and then focus the
Physics 231 Optical Instruments
7
light box with
grid slide
objective
ground
glass
eyepiece
30-35 cm
Fig. 7 Optical bench arrangement for the compound microscope.
eyepiece on that image. Remove the ground glass and make small adjustments to the position of
the light box to obtain the best image. Notice the change in your field of view as you move your
eye forward and back relative to the eyepiece. The optimum distance is called the "eye relief",
and is an important design consideration for real instruments. You will probably also notice
some interesting distortions of the image, which you should describe briefly. In a real
microscope considerable effort goes into minimizing these distortions, for obvious reasons.
REPORT
Your report should consist primarily of a qualitative description of your observations,
along with appropriate sketches of your optical setups. Quantitative results are requested at a few
points in the text.
Physics 231 Optical Instruments
8