Experiment 4 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,
compound microscope and ophthalmoscope.
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.
A. Optics of the eye and eyeglasses
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 electrochemical 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 4-1
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
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25 cm
range of clear vision
normal
eye
near
point
50 cm
far
point
8 cm
near
point
myopic
eye
corrected
myopic
eye
Fig. 4-1 Comparison of a normal, a myopic and a corrected myopic eye. Shaded bars represent the
range of focus. The far point and near point lines mark the limits of clear vision.
1 1 1
+ =
so si f e
(4-1)
where so 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 combined lenses of the eye. 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 so is small. Since the size of the
eye is fixed, we need to change fe. 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. 4-1. 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
(4-2)
where fc refers to the corrective lens. We want to pick f' so that
1 1 1
1
1
+ =
=
+
d f fe fc
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(4-3)
20
50 cm
150 cm
presbyopic
eye
near
point
far
point
75 cm
presbyopic
eye with
bifocals
38 cm
25 cm
Fig. 4-2 Presbyopia (limited accommodation) and correction with a bifocal lens.
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
1 1 1
+ =
sf d fe
(4-4)
Solving (4-3) and (4-4), we find fc = -sf. 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. 4-2.
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.
B. Magnifying instruments
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. 4-3, we make so smaller than the focal length so that the lens forms a virtual image
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object
ho
θ
25 cm
so
virtual h
i
image
θ'
ho
object
25 cm
si
Fig. 4-3 A simple lens used as a magnifier.
at the near point of the eye. The eye forms a real image on the retina subtending an angle i which is
larger than the angle o without the lens. The magnification would then be
M=
i
ho /so
o ho /25cm
(4-5)
Using the simple lens formula and the fact that si is negative for a virtual image finally yields
M =1+
25cm
f
(4-6)
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. 4-4, allows much higher
D
si
so
object
fo
fe
fo
objective
eyepiece
virtual
image
Fig. 4-4 A simplified compound microscope. The focal points of the lenses are marked.
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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+
fe so (4-7)
where the minus sign indicates that the image is inverted. The tube length of the microscope body,
D in Fig. 4-4, is standardized so objective and eye piece lenses are often marked with their
magnification. Other features of practical importance are discussed in standard textbooks.
C. The ophthalmoscope
The pupil of a normal eye appears black because almost all the light that enters the eye is
absorbed by visual receptors or by a dark inner lining. This avoids scattered light that would
degrade visual images, but it means that the retina and blood vessels inside the eye are difficult to
check for signs of damage or disease. Fortunately, a small amount of the light that enters the pupil
is reflected directly backward from the retina, emerging again from the pupil. This is the cause of
“red-eye” in photos, but the reflected light can also be used to view the retinal surface. The obvious
problem is that a simple light source shining directly into the eye would block the back-reflected
light. The cure is to use a small partially-reflecting surface with the light source, as shown in Fig.
4–5, to illuminate the retinal surface and view the reflected light.
If the subject has normal vision and is looking at a distant point, the lens of the subject’s
eye acts as a simple magnifier to form a virtual image of the illuminated retina with about 14x
magnification. Of course, if the patient is near- or far-sighted, or not accommodated to infinity, the
image will not be in focus for an examiner with normal vision. Standard medical instruments
therefore include additional lenses that can be moved into the viewing path to compensate for these
problems.
beam splitter
examiner
subject
light bulb
Fig. 4-5 Operating principle of the ophthalmoscope
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EXPERIMENTAL PROCEDURE
We will mount most of the components (lenses, screens, etc.) we need on an optical bench.
Best results will be obtained if everything is centered 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 or scratch easily.
A. The model eye
The eye model is sketched in Fig. 4-6. 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. 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 a lighted box on the optical bench or across the
room. 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 leave it in
place during the remainder of the experiment.
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. Install the arrow and crossline slide on the small light box on the slider and 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.
light box with
object slide
lens C
"cornea"
lens L
"lens"
corrective
lens
screen
"retina"
d
Fig. 4-6 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.
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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 (sf in
Eq. 4-4), and compute the necessary correction. Using a separate holder, place the corrective lens in
front of the eye. Is the image of the far object (large light box) 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. 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.
B. Magnifying instruments
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.
To measure the magnification, use a glass plate beam splitter, as you did when locating
virtual images and determining their magnification. In this situation the screen needs to remain
fixed, while you adjust the light box position to obtain a good focus at the screen. You can then
screen
and ruler
25 cm
desk lamp
glass
plate
light box
eye
Fig. 4-7 Use of a beam splitter to measure magnification.
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light box with
grid slide
objective
ground
glass
eyepiece
30-35 cm
Fig. 4-8 Optical bench arrangement for the compound microscope.
measure the height of the virtual image with a ruler. The general layout is shown in Fig. 4-7. Is the
magnification in reasonable agreement with the value calculated from the assumed 5 cm focal
length?
The arrangement for a compound microscope is sketched in Fig. 4-8. 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 multi-line slide on the ground glass. Remove the ground glass and
look through the eyepiece lens to focus on the real image from the objective. 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.
In addition to your qualitative analysis, you should measure the magnification of the
microscope as you did for the simple magnifier. Is your measurement in reasonable agreement with
what you calculate from Eq. 4-7? Since it is difficult to measure the distance from slide to objective
sufficiently accurately, you may prefer to measure Mo from the image on the ground glass.
C. The ophthalmoscope
You can build a model ophthalmoscope to use with the model eye as suggested in Fig. 4-9.
Set up the eye model with the 2.25 and 5.75 diopter lenses and position the screen for normal
light box
no slide
eye
screen
with pattern
glass
plate
Fig. 4-9 Optical layout for a model ophthalmoscope used with the eye model.
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vision, using the patterned side. The light box without any slides becomes the source, and the glass
plate is the beam splitter. If you look into the model you should be able to adjust the position of the
light and beam splitter to illuminate and view a portion of the screen surface. Can you see an image?
Does it appear magnified? Now increase the spacing between lenses and screen to make the model
near-sighted. Does the ophthalmoscope image become fuzzy? Does looking through a negative lens
restore the image?
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.
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