Review Notes Chapters 5,6,12,13,14 – P106
11/5/05
Chap. 5 - The Human Eye and Vision I: Producing the Image
1. The major parts of the eye are the cornea, iris, eyelens, retina, aqueous humor, and
vitreous humor. The pupil is the opening of the iris.
2. The “simplified eye” consists of a converging lens and a screen (retina) 16.67 mm
away.
3. The equivalent focusing power of the relaxed eye for a lens in air is about 60 diopters.
Focusing power in diopters = 1/focal length in meters. So the relaxed eye focuses
parallel rays (or, equivalently, an object at infinity) on the retina, which is 1/60 m (or
16.67 mm) away from the lens in the simplified eye.
4. Most of focusing power is from corneal refraction (about 42 diopters). The remaining
18 diopters is from the eyelens.
5. The eye focuses on objects at various distances by changing the focal length of the
eyelens over a fairly small range (about 4 diopters ) using the ciliary muscle. This
process is known as accommodation. The normal amount of accommodation yields a
maximum focusing power of 64 diopters and a minimum object distance of 25 cm.
This is the normal adult near point, the closest the object can be to the eye for a sharp
image to be produced. Children may have 10 diopters of accommodation and
therefore a closer near point. You should verify that the thin lens equation of chapter 3
is satisfied with xo = 0.25 m, xi = 0.0167 m, and f = (1/64) m.
1/f = 1/xo + 1/xi
where xo = object distance, xi = image distance, and f = focal length of lens.
6. The retina contains photoreceptors of two types: about 120 million rods and about 7
million cones. Most of the cones are concentrated in a small area of the retina called
the fovea located on the visual axis. The retina has a blind spot with no
photoreceptors where the optic nerve exits the eye. The eye continuously scans to
avoid a blank spot in the image the brain produces.
7. The rods are responsible for vision in dim light (scotopic). This is black and white
vision and is fairly coarse-grained. The cones are responsible for vision in bright
light (photopic). This is color vision and is fine-grained.
8. The overall sensitivity of the retina greatly increases over tens of minutes as we go
from bright light to dim light conditions (e.g. a movie theater). This process is called
adaptation. This adaptation process is much more important in allowing us to see
over a huge range of light intensities than is the action of the iris, which can only
change the area of the pupil opening by about a factor of 20.
9. The delay between a flash of light and the retinal response is known as latency and is
shorter for a high intensity flash. Retinal response continues for a brief time after the
flash ends. This is known as persistence of response with the retinal response lasting
for about 1/25 s at low intensities to 1/50 s at high intensities.
- page 2 Chap 6 - Optical Instruments
1. Normal vision allows us to focus clearly on objects from 25 cm away to infinity.
This requires a range of refractive power from 60 D to 64 D.
2. The three common vision deficiencies that are corrected with spherical lenses are
myopia, hyperopia, and presbyopia. Astigmatism is corrected with a cylindrical
lens.
3. Myopes are said to be “near sighted” and cannot see distant objects clearly. They
have uncorrected far points (the most distant object that can be sharply focused)
closer than infinity, which means they can’t focus parallel rays onto the retina. They
have too much refractive power, which means the minimum power is more than 60 D.
Myopia is corrected with a diverging lens (negative focal length and negative
refractive power), which combines with the eye to produce a minimum power of 60 D.
When one sets the xo equal to the uncorrected far point in the lens equation and solves
for (1/f), one obtains the uncorrected minimum power. The appropriate corrective lens
is the negative value that adds to that power to produce 60 D. Note that (1/xi) in the
equation is always 60 m-1 in this type of problem. The proper image distance xi for the
eye cannot change.
4. Hyperopes are said to be “far sighted” and cannot see nearby objects clearly. Their
uncorrected near points are beyond 25 cm. They have too little refractive power,
which means the maximum power is less than 64 D. Hyperopia is corrected with a
converging lens (positive focal length) which combines with the eye to produce a
maximum power of 64 D. When one sets the xo equal to the uncorrected near point in
the lens equation and solves for (1/f), one obtains the uncorrected maximum power.
The appropriate corrective lens is the positive value that adds to that power to produce
64 D.
5. Presbyopia results when our accommodation drops below 4 diopters and usually
occurs when we are in our 40’s. For people with otherwise normal vision, this causes
the near point to move out beyond 25 cm. It is corrected with “reading glasses”,
which have low power (< 4 diopters) converging lenses. A myope or hyperope, who
also suffers from presbyopia, will require bifocals with separate prescriptions for near
and distant viewing. For example, a myope with a far point of 1 meter and only 2 D of
accommodation has an uncorrected power range of 61-63D. She would need a -1 D
lens for distance viewing, but a +1 D lens for close viewing.
6. A magnifying glass is a converging lens that allows us to look at objects placed closer
to the eye than the normal 25 cm near point and thus to see more detail. The object is
placed just inside the focal point of the lens, and the eye is placed immediately behind
the lens. The rays from the object are then nearly parallel when they leave the lens,
and can be focused on the retina by the relaxed eye. Another way of saying the same
thing is that the lens forms a very distant, very large virtual image that your eyes can
easily focus on. The magnifying power of a magnifying glass is defined as the ratio of
image size on the retina using the magnifier to the image size on the retina without the
magnifier. Assuming the object has to be placed at 25 cm without the magnifier, the
magnification is just the ratio of the two object distances or 25/f with f in cm.
Magnifying power much greater than 5X generally produces unacceptable aberrations.
- page 3 7. The compound microscope achieves greater magnification than a magnifying glass
by using a second lens, the objective, to produce an enlarged real image that can be
viewed by the eyepiece as if with a magnifying glass. Both lenses are converging.
For a microscope of typical size, the magnification = -(160cm/fo)*(25cm/fe). The
first term comes from producing the enlarged image with the objective lens with focal
length fo, and the second term is just the magnifying glass formula for an eyepiece
with focal length fe. Since fo and fe are both in the denominator, short focal length
lenses for both the objective and eyepiece produce the most magnification.
8. Refracting astronomical telescopes also use two converging lenses, one as an
objective and one as an eyepiece. Telescope magnification = -fo/fe so the lens with
the shorter focal length is used as the eyepiece to achieve maximum magnification.
The minus sign indicates the image will be inverted.
Chap. 12 - Wave Optics
1. Diffraction is wave spreading that occurs after a wave passes an obstacle or through
an aperture that is not huge compared to the wavelength of the wave. The amount of
spreading depends on the ratio λ/b where b is the width of the opening or obstacle.
The spreading increases as λ/b increases, i.e., small openings produce more
diffraction. For example, light passing through a window (λ ~ 500 nm, b ~ 1 m, λ/b =
5x10-7) produces no visible diffraction effect. A sound wave passing through the same
open window (λ ~ 1 m, λ/b = 1) spreads a lot. We don’t have to stand in front of the
window to hear what’s happing outside, even if the walls are soundproof. We do have
to stand in front of the window to see what’s happening outside.
2. Two waves are coherent if their relative phase is constant. This is only possible for
waves with the same frequency and wavelength and traveling in the same direction at
the same speed.
3. Phase differences between two coherent waves can be stated in degrees or in number
of wavelengths. In-phase waves have a phase difference of 0°, 360°, 720°, etc. A
phase difference of 360° indicates that one wave has slipped one wavelength relative
to the other and so is back in phase with the first. Out-of-phase waves have a phase
difference of 180°, 540°, 900°, etc. A phase difference of 180° indicates that one
wave has slipped ½ wavelength relative to the other.
4. Interference can only occur between two coherent waves.
5. The interference will be constructive if the two waves are in phase, crest on crest and
trough on trough.
6. The interference will be destructive if the two waves are out of phase, crest of one on
the trough of the other.
7. Diffraction and interference are phenomena that only occur for waves. Since light
shows these effects, it has wave properties.
8. It is easy to produce two coherent sources of sound waves by sending the same signal
to two speakers. At any point in space, the type of interference (constructive or
destructive) between the waves from the two speakers will depend on the path
difference from the two speakers to the point of interest. If the two speakers
themselves are in phase, then the total phase difference between the waves depends
only on the path difference and in degrees is (|d2-d1|/λ)*360°. Here |d2-d1| is the
- page 4 absolute value of the path difference and * means to multiply. If the two speakers are
out of phase (accomplished by reversing the two wires to one of the speakers), then
the total phase difference is 180° + (|d2-d1|/λ)*360°. These numbers should be
compared with those given in item (3) above to determine whether the interference is
constructive or destructive.
9. It is difficult to create two coherent light sources, so two tricks are used. One is wave
splitting in which two sources are created by passing a light beam through two,
narrow, closely placed slits. The two slits then become two in-phase coherent sources
of light. The other trick is amplitude splitting in which part of the amplitude an
incoming wave is reflected at the top of a thin film and part at the bottom of the thin
film. The two reflected waves are coherent and can interfere after they leave the film.
The film must be thin so that the extra path of the wave that makes a round trip in the
film does not exceed the coherence length of the light.
10. In Young’s double slit interference, the interference pattern from the two slits is
displayed on a screen. The phase difference between the two waves is (|d2-d1|/λ)*360°,
just as for sound in item (8). If |d2-d1| is 0, λ, 2λ, or any integral number of wavelengths,
constructive interference will result. If |d2-d1| is, 0.5λ, 1.5λ, or any odd number of half
wavelengths, destructive interference will result. The result is alternating bright and
dark fringes on the screen. There is always a bright fringe on the center line from
between the two slits since d2 – d1 = 0 there. The spacing between two successive bright
fringes or two successive dark fringes is Dλ/d, where D is the perpendicular distance
from the slits to the screen, λ is the wavelength of the light, and d is the separation
between the centers of the two slits. The spacing increases with increasing wavelength,
so the pattern will be more spread out for red than for blue.
11. In thin film interference the path difference |d2-d1| between the two reflected waves is
2t, where t is the thickness of the film. This assumes the incident light arrives nearly
perpendicular to the film surface (ϑi = 0°). Therefore, the phase difference resulting
from the path difference is (2t/λ’)*360°. Here λ’ is the wavelength in the film and is
related to the wavelength in air by λ’ = λ air/nfilm. If the reflections at the top and bottom
of the film are both hard or both soft, the total phase difference is just that due to the
path difference, (2t/λ’)*360°. However if one of the reflections is hard and one is soft,
an additional 180° phase shift is produced. In that case the total phase difference is 180°
+ (2t/λ’)*360°. These values of total phase difference should be compared with the
values given in item (3) to determine if the interference is constructive or destructive. A
hard reflection occurs when a wave reflects from a material with a higher index of
refraction. A soft reflection occurs when it reflects from a material with a lower index
of refraction.
12. Thin film interference produces the colors we see in soap bubbles and oil slicks. If we
observe a bright color in the reflected light, the film thickness must be such that the
interference is constructive for that wavelength. Reflection can be suppressed in camera
lenses by applying a thin film 0.25λ’ thick so that destructive interference results.
13. Even if there is only a single narrow slit, an interference pattern called a single slit
diffraction pattern will appear on the screen. This results from self interference
between portions of the wave from different parts of the slit. The width of the central
bright maximum is 2Dλ/b, where b is the slit width. All the other maxima are half that
- page 5 wide Dλ/b. Note that the width of the central maximum increases as the ratio λ/b
increases. The wave spreading is increasing with λ/b as mentioned in item (1) above.
14. A diffraction grating is an array of many, regularly spaced slits. When light passes
through a diffraction grating, the distance from the center line to the first bright spot is
given by the same formula as Young’s double slit -- Dλ/b. For a grating, b is very small
since the slits are very close, so the pattern is spread out more. Another result of having
many, many slits is that the bright fringes are very narrow. Since the spacing depends
on wavelength, a diffraction grating can be used to spread out white light into all the
colors of the spectrum. The longer wavelengths (red) will be farther from the center
line. This is opposite from the case when a prism produces the color spectrum. In that
case blue is refracted most and appears farthest from the line of the original light ray.
Chap. 13 – Scattering and Polarization
1. Polarization can only exist in transverse waves. In transverse waves the disturbance
is perpendicular to the direction of wave travel. Remember that diffraction and
interference are the other two behaviors unique to waves and apply to both
longitudinal and transverse waves.
2. Linearly polarized waves (also called plane polarized) are characterized by the
wave disturbance (the electric field for electromagnetic waves like light) lying in a
single plane.
3. Ordinarily, light is not polarized. The electric field may be in any direction
perpendicular to the direction the light is traveling and is constantly changing.
4. Polarized (or partially polarized) light can be produced by scattering from small
particles, by reflection from insulators (not metals), and by selective absorption by
special filters called polarizers (an example is the plastic Polaroid).
5. Scattering of light by particles small compared to the wavelength (e.g., by air
molecules) is called Raleigh scattering. The probability for scattering increases
rapidly with increasing frequency in the visible range, which is why blue light (higher
frequency) scatters more than red light (lower frequency) in the air. When light is
scattered sideways, it is also polarized.
6. Specular reflection from a horizontal insulator produces light that is partially
polarized with its plane of polarization horizontal. When the light is incident at
Brewster’s angle (a special angle of incidence that depends on the indices of refraction
of the initial material n1 and the reflecting material n2; tanϑB = n2/n1), the reflected
light is 100% polarized. Brewster’s angle is ~56° when light in air reflects from glass;
it is ~53° when light reflects from water. Diffuse reflection does not produce
polarized light.
7. The specularly reflected light comprises “glare” that can be blocked by a properly
oriented polarizer. The horizontally polarized reflected light from a horizontal surface
is blocked by polarizing sunglasses with a vertical axis.
8. If unpolarized light is incident on a polarizer, only light with the electric field aligned
with the polarizer’s axis is passed. Half the intensity is lost. The light that passes the
polarizer is 100% plane polarized with its plane of polarization aligned with the axis of
the polarizer.
- page 6 9. If a second polarizer is placed after the first with its axis perpendicular to that of the
first (crossed polarizers), no light will pass the second polarizer. A third polarizer
placed between the first two with its axis at an intermediate angle will rotate the plane
of polarization so that some light will pass through all three polarizers. This illustrates
the rather amazing fact that a vector like the electric field can always be divided into
two components that point in any two perpendicular directions. As long a component
of the polarized light is aligned with a polarizer’s axis, some light will pass.
Chap. 14 – Holography
1. A hologram records both the intensity and phase of light reflected from an object.
This is accomplished by recording the interference pattern between the reflected
object beam and a reference beam.
2. A developed hologram looks like a complicated interference pattern; it looks nothing
like the object. In order to view the image, the hologram must be illuminated by a
reconstruction beam that mimics the original reference beam that was present during
the creation of the hologram.
3. In producing any stable interference pattern, the two interfering beams must maintain
their relative phase; i.e., they must be coherent. This is also true in producing
holograms. Therefore, most holograms are exposed on motionless isolation tables.
Relative motion of the object and reference beams of so much as a half wavelength of
light during the exposure would smear out the interference fringes.
4. The extra phase information recorded in a hologram compared to a normal
photograph, allows a three dimensional image to be produced. Moving your head
while viewing a hologram allows different perspectives and produces parallax effects.
5. Another difference between holograms and photographs is that the entire scene is
encoded in all parts of the hologram. Cut a hologram in two, and you still observe the
entire scene, albeit with lower resolution.