Download Report

Physics Including Human Applications
440
Chapter 20
HUMAN VISION
GOALS
When you have mastered the contents of this chapter, you will be able to achieve the
following goals:
Human Vision
Characterize the physical parameters that are significant in human vision.
Visual Defects
Explain the causes and corrections for the visual defects of myopia, hypermetropia,
presbyopia and astigmatism.
Characteristics of Vision
Define the following terms:
visual acuity
accommodation
scotopic vision
dichromat
photopic vision
Corrective Lenses
Solve problems involving visual defects and their corrections using lenses.
PREREQUISITES
Before beginning this chapter you should have achieved the goals of Chapter 18,
Optical Elements, and Chapter 19, Wave Properties of Light.
Chapter 20- Human Vision
Physics Including Human Applications
441
Chapter 20
HUMAN VISION
20.1 Introduction
From your own experience, what descriptive statements can you make concerning the
following aspects of your vision: sensitivity for low light levels, response to different
colors, ability to discriminate between two objects at a distance, and the reliability of
visual depth perception?
The human eye is our main source of information about our external environment. It
responds to light and supplies messages to the brain. In this chapter we will explore the
human visual system and the physical principles behind its operation.
20.2 The Powerful Eye
In this chapter we will examine one of the most amazing optical systems you will ever
encounter. The human eye is an optical system of unsurpassed versatility. Its sensitivity
is so great that it apparently detects single packets of light, yet visual information is
processed only when sufficient numbers of these packets are detected coincidently to
produce a visual pattern. This same system is capable of handling intensities more than
109 times the minimum energy threshold. Let us look at the physics behind the
performance of the human eye.
The eye is analogous to a camera in its basic operation Figure 20.1. There are three
basic parts of the eye: a light focusing system, an automatic aperture system, and a light
sensitive detection system. In the light focusing system, the cornea and the lens account
for the refractive power of the eye. Most of this power is due to the curvature of the
cornea, but the muscles controlling the curvature of the lens provide the flexible
focusing power of the eye. These eye, or ciliary, muscles permit the normal eye to focus
on the retina light from objects very distant, essentially parallel rays of light, as well as
light from objects as close as the normal near point of 25 cm from the eye. Measure your
near point.
Figure 20.1 Diagram of the human eye. Chapter 20- Human Vision
Physics Including Human Applications
442
Move this page toward your eyes until the images of the letters begin to blur. The
distance from your eye to the page at the nearest point of clear vision is your near-point
vision. You have greater eye accommodation than normal if your near-point distance is
less than 25 cm. The power of a lens is a measure of its ability to bend rays of light. A
lens of high power has a small focal length. The numerical value for the power of a lens
in diopters is computed by taking the reciprocal of the focal length of a lens when the
focal length is measured in meters. For example, a lens of 5-cm focal length has a power
of 20 diopters. The human eye has a variable focal length so the eye can change its
power to focus on objects at various distances from the eye. Let us calculate the
accommodation ability of a typical human eye. If the distance from the eye lens to the
retina is given by l (about 3 cm for a human eye), the power of the eye lens for distant
objects, symbolized by P infinity P∞, can be calculated by using the thin lens formula
from Chapter 18,
1/p + 1/q = 1/ f
(18.13)
where p is the distance to the object, q is the distance to the image, and f is the focal
length of the lens. For distant objects the reciprocal of the object distance 1/p is
approximately zero, so the distance from the eye lens to the image on the retina is equal to the
focal length of the eye. Therefore, the power for the eye lens for distant objects P∞,
P∞ = 1/f = 1/l ≅ 30 diopters
(20.1)
where l is the distance from the eye lens to the retina.
The power of the eye for near objects, Pn, can be similarly calculated,
Pn = 1/f = 1/p + 1/q
(20.2)
where p is the distance to an object at your near point (say, 25 cm) and q is the distance
from the eye lens to the retinal,
Pn = 1/near point + 1/l = 1/0.25 + 1/l = 4 + P∞ ≅ 34 diopters
(20.3)
Hence, the change in the power of a human eye lens to accommodate objects at various
distances from the eye is about 4 diopters. Design a simple experiment using a meter
stick to measure the ability of your eyes to accommodate objects at various distances
from your eye. If you wear eyeglasses or contact lenses, perform this experiment both
with and without your glasses or lenses.
The automatic aperture system involves a negative feedback system through the
brain to the iris Figure 20.1. The system automatically adjusts the opening size of the iris
in response to the ambient level of light intensity. In response to high intensity light, the
iris stops down the opening to minimize damage to the detector system and to provide
the proper intensity levels to the detecting system. The response to low light levels,
referred to as dark adaption, takes up to 45 minutes. It provides the eye with its
maximum entrance pupil size and allows the eye to assume its most sensitive light
gathering configuration.
The light sensitive detecting system of the eye consists of the light sensitive
receptors distributed over the back of the eyeball. This light sensitive membrane is
called the retina. There are two kinds of photosensitive cells present in the retina. These
cells are rods and cones, so named because of their shape. The rods are more sensitive to
light and distinguish light from dark when the light level is very low. Vision at low
levels of light intensity is called scotopic vision. The cones are less sensitive to light, but
Chapter 20- Human Vision
Physics Including Human Applications
443
are capable of resolving light into components giving details of images and information
involving the color of the received light. The rods and cones are distributed with
densities greater than 100,000/mm2 on the retina, the greatest concentration of cones
occurring in the region called the fovea.
20.3 The Defective Eye
The most common defects of human vision involve the light refracting system of the
eye. Normal vision provides clear vision for letters that subtendthree minutes of arc at a
distance of 6.10 m (20 ft) from the observer. At this distance, defective eyes may produce
images that are too blurred to read. If a person requires letters at 6.10 m (20 ft) that
normal vision reads at 12.2m (40 ft) the person is said to have 20/40 vision. The
numerator expresses the distance between observer and the letters; the denominator is
the distance at which the normal eye can read these letters. Normal vision is expressed
as 20/20 vision.
EXAMPLE
Find the size of the letters the normal eye can read at 6.10 m (20 ft), if the criterion is that
they subtend 3 minutes of arc at the eye. Using this information what size object is
required at 10 m?
angle (rad) = object size/distance
1 min = 2.91 x 10-4 rad
3 min = 8.73 x 10-4 = x/6.10 m
x = 0.533 cm
At 10 m,
x = 10 m x 8.73 x 10-4
= 8.73 x 10-3 = 0.87 cm.
20.4 Correction of Defective Sight
Many of the defects of the human eye are correctable. Myopia is the term for
nearsightedness. Myopia results when parallel light is focused in front of the retina.
This means that there exists a farthest point of clear vision, usually only a few meters
from the eye. Only objects closer to the eye than the farthest distance of clear vision can
be properly focused on the retina. This defect is most frequently associated with an
elongated eyeball. The correction for myopia is the use of a diverging lens with a focal
length equal to the far point distance, the distance of farthest clear vision. This negative
spectacle lens makes incoming parallel light appear to diverge from the far point (
Figure 20.2).
Chapter 20- Human Vision
Physics Including Human Applications
444
Farsightedness (hypermetropia) due to a short eyeball, results in parallel light being
focused behind the retina Figure 20.3. The same result caused by the hardening of the
lens with aging is called presbyopia. These two conditions mean that the near-point
distance for a person is increased beyond 25 cm. The correction for each of these defects
is the use of a converging lens, which produces for an object at the normal near point
(25 cm) a virtual image at the actual near point of the eye. This means that the focal
length of the correcting lens is given by the equation 1/f = 1/25 - 1/x, where x is the
actual nearpoint of the eye (all distances in cm).
The defect of astigmatism is the formation of a distorted image on the retina. The
usual cause of this defect is the nonuniform curvature of the cornea. This defect can be
corrected by using a lens that is ground to compensate for the cornea defect. Often this
correction is obtained through the use of an appropriately shaped cylindrical lens.
EXAMPLES
1. A man has a far point of 2 m. Find the focal length of the corrective lens for seeing
distant objects, and find the near point of this eye with corrective lens if the man has the
normal near point without glasses.
The lens needed must have a focal length of -2 m so that an object at 2 meters will
have an image at infinity. The near point with this lens is x, where 1/x - 1/25 = 1/f (in
cm). Thus x equals 200/7 cm.
2. A person has a near point of 50 cm. Find the corrective lens required to enable this
person to read at the normal distance of 25 cm.
Using the equation 1/f = 1/25 - 1/50, we find that the corrective lens should have a
focal length of +50 cm.
20.5 The Receptive Eye
As stated earlier research has provided evidence that individual receptor cells on the
retina can be activated by single packets (photons) of light. Fortunately, such
Chapter 20- Human Vision
Physics Including Human Applications
445
phenomena are not processed by the brain; the noise signals by such random activity
would be most distracting. The research shows that of the order of 100 photons per
second will excite a localized set of rods of about 100 in number causing visual
perception to occur. The basis of this coincident detection system is still not clear, and it
continues to be a current area of research.
The wavelength, or color, sensitivity of the human eye is also a subject of presentday research. It has been established that the cone cells in the retina play the central role
in color, or photopic, vision. The cones are most closely packed in the fovea region of the
retina; so when you direct your visual attention to an object your eye muscles rotate
your eyeballs so as to place the focused image of that object on the fovea of the retina.
It has been found that there are four colors whose apparent color does not change as
the intensity of the light focused upon the retina is increased above the threshold
required to stimulate the cone cells. These "pure" colors are red, yellow, green, and blue,
and for a typical observer these pure colors will have dominant wavelengths of 630 nm,
570 nm, 510 nm, and 470 nm, respectively. On the other hand, it is possible for a person
to perceive yellow from a proper mixture of blue and red light, without any 570-nm
light present. So a human being is capable of synthesizing colors in a way that is not
clearly understood at the present time.
In an attempt to refine our understanding of human vision, direct measurements of
the response of the retinal cones to various exciting lights have been made. Three types
of cone cells have been identified, each particularly sensitive to a different range of the
spectrum. However, it is clear that these cones do not send nerve signals directly to the
brain. The cone signals pass through a complex coding system before transmission to
the brain where the visual messages are translated into a person's visual perception.
It has long been known that humans see
objects differently in dim light and in bright light.
We now know that the rod cells in the retina are
most sensitive to low intensity of light, and these
rod cells are most closely packed in the periphery
of the retina, rather than in the fovea. To see
objects in dim lighting you should look slightly to
one side of the object, not directly at it, so that the
light entering the eye falls not on the fovea but on
the periphery of the retina where the rods are
more concentrated. The rods and the cones also
differ in their wavelength response to light. The
rods are more sensitive to blue light than are the
cones. The relative luminosity curves for scotopic
(dim light) and photopic vision are shown in
Figure 20.4. Relative luminosity is defined as the
reciprocal of the normalized threshold intensities,
obtained by dividing each wavelength threshold
by the minimum threshold for the entire system.
(Can you suggest an analogous analysis for the
human ear?)
Chapter 20- Human Vision
Physics Including Human Applications
446
The range of wavelengths that make up the visible spectrum (380 - 770 nm) is
determined by the absorption of ultraviolet by the lens and by the insensitivity of the
receptors in the eye to wavelengths greater than 770 nm.
There are people who cannot discriminate colors. The degree of color- blindness
varies among the population. There is evidence that some color blind persons have only
two of the necessary pigments (thus they are called dichromates) for full color vision.
Those with no color vision are called monochromates.
20.6 The Perceptive Eye
The visual acuity of the eye is defined as the minimum angular separation between two
objects that can be resolved by the eye. Several factors can set this limit, the spacing of
receptor cells with independent nerve paths, diffraction effects (Rayleigh criterion), and
lens system defects. The closest spacing of receptors in the human eye is that of cones in
the fovea, where the center to center distance of adjacent cones is from 2 to 5 microns.
For an eye with 2 cm between cornea and retina this corresponds to about 10-4 rad for
the angle subtended by two stimulated cells separated by one unstimulated cell. This
criterion determines the resolving power of the human eye.
EXAMPLE
Find the distance from the eye for two objects 1 cm apart that subtend an angle of 10- 4
rad
s/r = θ
1 cm/r = 10-4
r = 100 m
Using the Rayleigh criterion and appropriate values for wavelength and for the
aperture of the eye, find the limit for visual acuity, and compare it with the 10-4 rad
value calculated from receptor separation. Does this comparison support the idea that
the visual acuity of the eye is essentially independent of color?
20.7 Binocular Vision
Depth perception in human vision is primarily a result of binocular vision. Each eye
produces a slightly different image on their respective retina due to the different angle
the object subtends at each eyeball. The brain learns to interpret these slightly different
images in terms of distance from the eye. The experimental data suggest that depth
perception can be improved with practice. Experience is used to correlate the actual size
of the object with the size of the image on the retina and the actual distance to the
object. The following cues are important in depth perception:
1. viewing angle differences for each eyeball (Figure 20.5)
2. muscle tension for convergence upon a point of fixation
3. angle subtended by known object at a known distance
4. color contrast interpreted in terms of light scattering which relates to distance
Chapter 20- Human Vision
Physics Including Human Applications
447
5. relation of light and shadows
6. overlapping contours
7. parallax (the apparent movement of objects in the visual field with observer motion)
8. context of object and environment
ENRICHMENT
20.8 Derivation of the Power of Lenses Used Close to the Eye
To derive the power of a lens to be used close to the eye, let the eye lens be replaced by
a thin convex lens of fixed focal length f e in meters. Then the power of the eye will be
given by Pe where
Pe = 1/fe
(20.4)
Next take another lens of focal length f1 in meters with a power of P1 = 1/f1 and place it
close to the first lens. If an object is placed several meters away from the lens we can
find the image distance q1 by applying the thin-lens equation (Equation 18.13),
1/q1 = 1/f1 - 1/p =(p - f1)/pf1
Then
q1 = pf1 /(p - f1 )
(20.5)
This image distance q1 becomes the virtual (negative) object distance for the eye lens so
that the final location of the image in the eye is given by qe:
1/qe = 1/fe -pe = 1/fe +q1
where pe(=-q1) is the object distance to the eye lens.
But the power of the two lens system,
Ps = 1/fs = 1/p + 1/qe = 1/f1 -1/q1 + 1/fe + 1/q1
Ps = 1/f1 + 1/fe =P1 + Pe
(20.6)
Hence the power of a lens-eye system is the sum of the glass lens power plus the eye lens power.
Chapter 20- Human Vision
Physics Including Human Applications
448
EXAMPLE
A man has a near point of 50 cm from his eyes and a far point at infinity. What is the
useful accommodation power of his eyes? Consider the lens of the eye as a simple lens.
1/fe = 1/p + 1/q
For the far point
1/f∞ = 1/∞ + 1/q∞ = 1/q∞
Now 1/f = P∞ = 1/q∞
Pn = 1/0.50 + P∞ = 2.0 + 1/q∞ = 1/fn
accommodation power = P - P∞ = 2.0 diopters
What lens should be prescribed for this individual?
For the normal eye the near point should be 25 cm. This means that the image formed
by the lens must be at least 50 cm from the eye lens.
1/p + 1/q = 1/f
1/25 - 1/50 = 1/f
1/f - (2 - 1)/50 = 1 / 50 cm
P = 1/0.50 m = 2.0 diopters
A general rule for an auxiliary lens for the eye is that the image formed by this lens
must be between the near and far point of the unaided eye. One then uses the principles
of a multiple lens system to prescribe the corrective lens.
SUMMARY
Use these questions to evaluate how well you have achieved the goals of this chapter.
The answers to these questions are given at the end of this summary with the number of
the section where you can find related content material.
Human Vision
1. For each of the following optical properties write a statement to relate it to human
vision: light intensity, light wavelength, lens power, and resolving power.
Visual Defect
2. A person who suffers from hypermetropia is unable to clearly see objects that are
______ the eye lens, has the ______ point distance ______ in size, and can wear
corrective eyeglasses of _______ power.
3. A person who suffers from myopia is unable to clearly see objects that are ______ the
eye lens, has the ______ point distance in size, and can wear corrective eyeglasses of
______ power.
Chapter 20- Human Vision
Physics Including Human Applications
449
Characteristics of Vision
Match each term below with the lettered items.
4. visual acuity
5. scotopic vision
6. photopic vision
7. visual accommodation
8. dichromat
9. binocular vision
a. looking at photographs
b. two-colored rug
m. 0.01 rad
n. reciprocal of normalized intensities
c. contact-lens holder
o. lens power
d. perspicacity
p. iris aperture size
e. depth perception
q. diopters
f. color perception
r. cones
g. low intensity
h. carrots
s. rods
t. 20/20
i. color blindness
u. red, yellow, blue, green
j. blue sensitive
v. Rayleigh criterion
k. red sensitive
w. eye muscle tension
l. fovea region
x. perspiration
Corrective Lenses
10. A young person with a near point of 20 cm, is not able to see an object clearly if it is
more than 5 m from his eye. What kind of lens should you prescribe to correct his
vision?
a. +5 diopters
d. -0.2 diopters
b. -5 diopters
c. +0.2 diopters
e. -2.0 diopters
Answers
1. Section 20.5, Section 20.2, Section 20.6
2. near, near, increased, positive (Section 20.4)
3. far from far, decreased, negative (Section 20.4)
4. d, l, m, p, t, v (Section 20.6)
5. g, j, s (Section 20.5)
Chapter 20- Human Vision
6. f, i, k, l, r, u (Section 20.5)
7. o, q (Section 20.2)
8. f, i (Section 20.5)
9. e, w (Section 20,7)
10. d (Section 20.2)
Physics Including Human Applications
450
ALGORITHMIC PROBLEMS
Listed below are the important equations from this chapter. The problems following the
equations will help you learn to translate words into equations and to solve single
concept problems.
Equations
1/p + 1/q = 1/ f
(18.13)
P = 1/f (m)
(20.1)
P∞ = 1/l (m)
Pn = 1/near point + 1/l
(20.1)
(20.3)
Problems
1. At what distance should a person with 20/20 vision be able to read a letter 1 cm high?
2. What is the power of the lens for each example in Section 20.4, Correction of
Defective Sight?
Answers
1. 11.5 m
2. -0.5 diopters, + 2.0 diopters
EXERCISES
These exercises are designed to help you apply the ideas of a section to physical
situations. When appropriate, the numerical answer is given in brackets at the end of
the exercise.
Section 20.2
1. A girl can adjust the power of her eye's lens-cornea system between limits of +60.0
diopters and +65.0 diopters. With the lens relaxed she can see distant objects such as
stars clearly.
a. Find the girl's near point.
b. How far is her retina from her eye lens? [20.0 cm, 1.67 cm]
Section 20.4
2. A myopic student wears contact lenses of -8.00 diopter power in order to see distant
objects. The distance from the eye lens to the retina is 2.00 cm.
a. Find the power of the student's eye when relaxed.
b. If the student can give an extra +4.00 diopters to his eye lens with ciliary muscles,
find the near point of his eyes without glasses. [58.0 diopters, 8.33 cm]
Chapter 20- Human Vision
Physics Including Human Applications
451
3. A nearsighted person wears glasses of f = -50.0 cm. Through these glasses he has
perfect distant vision, but he cannot focus clearly on the objects closer than 25.0 cm.
How close can he focus if he removes his glasses? What is his far point? [16.7 cm,
50.0 cm]
4. A pair of bifocal lenses have components with focal lengths of 50 cm and -250 cm.
What are the near and far points for the wearer of these bifocal lenses? [50 cm and
250 cm]
PROBLEMS
Each of the following problems may involve more than one physical concept. When
appropriate, the answer is given in brackets at the end of the problem.
5. Typical vision data for a female subject is as follows:
Near Point
Far Point
without glasses
13 cm
39 cm
with glasses
20 cm
∞
What is the subject's power of accommodation without glasses? With glasses, what
is the power of accommodation of the optical system? Explain these results. What is
the power of the glasses lens? Does this person suffer from myopia, hyperopia, or
presbyopia? [5.1 diopters, 5 diopters -2.6 diopters, myopia]
6. An automobile is approaching you on a straight road on a pitch black night. The
lights on the car are 1.5 m apart. Assume the aperture of your eye is 1 cm and the
effective wavelength of the light is 550 nm.
a. Using the Rayleigh criterion, how far will the automobile be from you when you
can just be sure that you are seeing an automobile and not a motorcycle?
b. Using the spacing of receptor cells in the eye, how far will the automobile be from
you when you can just be sure that you are seeing an automobile and not a
motorcycle?
c. According to NASA sources the minimum separation acuity for the average
astronaut is 0.4 minutes of arc. How far will the automobile be from an astronaut
before he can be sure that it is not a motorcycle? [2.2 x 104 m, 1.5 x 104 m, 1.3 x 104 m]
7. A photon of visible light has energy of about 3.6 x 10-19 J. The absolute luminance
threshold for the dark-adapted human eye is 10-5 mL (milli-Lambert= 1.47 x 10-6
watts/cm2). How many photons per second per cm2 are required for minimum
human vision? [4.1 x 107 photons/cm2-sec]
8. In the fovea portion of the eye there are 136 thousand cones per square millimeter,
and the lower absolute threshold of illumination is 10-3 mL. Using the average
energy of a visible photon of light as 3.6 x 10-19 J, how many photons per second are
required to provoke a visual response from a cone? [300 photons/cone - sec]
Chapter 20- Human Vision
Physics Including Human Applications
452
9. On a horizontal angle of 20ø from the fovea there are 158 thousand rods per square
millimeter, and the lower absolute threshold of illumination is 10-5 mL. Using an
average energy of visible light photons as 3.6 x 10-19 J, how many photons per second
are required to provoke a visual response from a rod? [2.58 photons/rod - sec]
10. Given the cone and rod average distribution per unit area in problems 8 and 9, what
is the minimum acuity angle for vision using only cones? Using only rods? The
distance from the cornea to the retina is 25 mm. [2.16 x 10-4 rad, 2 x 10-4 rad]
11. Design an experiment to measure relative luminosity for the eye as shown in Figure
20.4.
12. An older person has a near point of 75 cm. Find the correct lenses to prescribe for
this person. [bifocals, clear glass and +2.67 diopters]
Chapter 20- Human Vision