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Multiple Lenses
Draw Rays !
• We determine the effect of a system of lenses by considering
the image of one lens to be the object for the next lens.
+1
0
-1
+2
∴
s1 = +1.5m, f1 = +1m
s = 3m
'
1
For the second lens:
∴
s2' = −0.8 m
+4
+6
+5
f = -4m
f = +1m
For the first lens:
+3
s1'
m1 = − = − 2
s1
s2 = +1m, f2 = -4m
s 2'
4
m2 = −
=+
s2
5
1 1 1
1
1
1
= − =
−
=
'
s1 f1 s1 1 m 1.5 m 3 m
1
1 1
1
1
5
=
−
=
−
=
−
4m
s2'
f 2 s2 − 4 m 1 m
m = m 1m 2 = −
8
5
Multiple Lenses
• Objects of the second lens can be virtual. Let’s move the
second lens closer to the first lens (in fact, to its focus):
+1
0
-1
+2
+3
+4
+5
+6
f = +1m
For the first lens:
∴
f = -4m
s1 = +1.5m, f1 = +1m
s = 3m
'
1
For the second lens:
∴ s2' = +4 m
s1'
m1 = − = − 2
s1
s2 = -2m, f2 = -4m
1 1 1
1
1
1
=
−
=
−
=
s1'
f1 s1 1 m 1.5 m 3 m
1
1 1
1
1
1
=
−
=
−
=
s2'
f 2 s2 − 4 m − 2 m 4 m
s 2'
m2 = −
= +2
s2
Note the negative object distance for the 2nd lens.
m = m1m2 = −4
The eye
• The “Normal Eye”
The Eye
– Far Point ≡ distance that relaxed eye can focus onto retina = ∞
– Near Point ≡ closest distance that can be focused on to the retina
~ 25 cm
1 1 1
1
= + ' = 0+
f s s
2.5 cm
f = 2.5 cm
1 1 1 1
1
= + ' =
+
f s s 25 2.5
f = 2.3 cm
2.5cm
25cm Therefore the normal eye acts as a lens with a focal length
which can vary from 2.5 cm (the eye diameter) to 2.3 cm which
allows objects from 25 cm → ∞ to be focused on the retina!
This is called “accommodation”
Diopter: 1/f Eye = 40 diopters, accommodates by about 10%, or 4 diopters
Eye corrections (glasses, contacts)
Near-sighted eye is elongated, image of distant object forms in front of retina
Add diverging lens, image forms on retina
Far-sighted eye is short, image of close object forms behind retina
power = 1/f; f in meters
Add converging lens, image forms on retina
The farsighted eye
Near point changes with age:
7 cm
→ 200 cm
7 years
60 years
Most distinct vision is at near point. Image is largest.
Example: Reading glasses. The near point of a person’s
eye is 75 cm. What power reading glasses should be used
to bring the near point to 25 cm?
s = 25 cm
s ′ = −75 cm
1 1 1
1
1
+ = =
+
s s ′ f 25 cm − 75 cm
1
1
=
= 2.67 diopters
f 0.375 m
i
virtual
F
o
s’
F
s
Assumes eye very close. Results are slightly different when distance
between them is taken ino consideration.
Old Preflight
A person with normal vision (near point 28cm) is standing in front of a
plane mirror. What is the closest distance to the mirror the person can
stand and still see himself in focus?
a) 14 cm
b) 28 cm
c) 56 cm
s
s’
Object for eye = Image of self
• Distance from eye to object = s+s’
• Set s+s’ = near point
Old Preflight
Two people who wear glasses are camping. One of them is
nearsighted and the other is farsighted. Which person's glasses will
be useful in starting a fire with the sun's rays?
a) the farsighted person's glasses
b) the nearsighted person's glasses
What do you need to start a fire?
• REAL IMAGE ! (light is focused to a point)
• Converging lens gives REAL IMAGE
• Far-sighted people need converging lenses!
Special Lens Combinations
If two thin lenses are close together, they act
effectively as a single lens. The focal length of
the “doublet” is given by
1
f doublet
1
1
=
+
f1 f2
f1
f2
fdoublet
Note the power (=1/f) of the
combination is just Pdoublet = P1 + P2
Lecture 27, ACT 2
• Hildegard’s retina is 2.5 cm behind
the lens, which has a minimum
focal length of 2.6 cm.
2.5 cm
feye = 2.6 cm
1. What does the focal length fcl of her contact
lens need to be?
(a) 65 cm
(b) -65 cm
(c) -0.1 cm
2. What is the power Pcl of the contact lens?
(a) 1.5 D
(b) -1.5 D
(c) 1000 D
Lecture 27, ACT 2
• Hildegard’s retina is 2.5 cm behind
the lens, which has a minimum
focal length of 2.6 cm.
2.5 cm
feye = 2.6 cm
1. What does the focal length fcl of her contact
lens need to be?
(a) 65 cm
1
1
1
=
+
f
f eye
f cl
(b) -65 cm
(c) -0.1 cm
1
1
1
=
−
f cl 2.5 cm
f eye
f eye (2.5 cm )
2.6 × 2.5
f cl =
=
= 65 cm
f eye − 2.5 cm
2.6 − 2.5
Lecture 27, ACT 2
• Hildegard’s retina is 2.5 cm behind
2.5 cm
the lens, which has a minimum
focal length of 2.6 cm.
f = 2.6 cm
2. What is the power Pcl of the contact lens?
eye
(a) 1.5 D
(b) -1.5 D
(c) 1000 D
1
1
Pcl ≡
=
= 1.5 D
f cl 0.65 m
Note: We could have solved for the
power directly:
Pcl ≡ Pneed − Peye = 40 D − 38.5 D
Angular Magnification
• Our sense of the size of an object (in the absence of other
clues) is determined by the size of image on the retina.
• This is proportional to the angle subtended by the object:
h α1
Bigger image
h’1 ~ α1
h α2
h’2 ~ α2
Smaller image
• The magnification of an optical system can then be
equivalently defined as the ratio of the output angular spread
to the input angular spread:
M ≡
θ out
θ in
Magnifying Glass
• Our sense of the size of an object is determined by the size
of image on the retina.
– If the object were closer to our eye, it would subtend a larger
angle.
– However, we can only focus on an object if it is no closer than
the near-point distance Lnp (~25 cm).
– We can use a simple magnifier to create an enlarged virtual
image outside Lnp.
Positive “f” lens
Lnp
h
α
h
β
~f
Object at Near Point - can’t get nearer
h
α ≈
L np
Define Angular Magnification:
β Lnp
M ≡ ≈
α
f
Object just inside Focal Point
of simple magnifier
h
β≈
f
⇒ Choose f < Lnp
Ma gnifier
Ma
At normal distances a small
object (small letter print)
subtends a small angle θ.
with a lens or magnifier we
can magnify the subtended
angle to θ′.
The ratio is call angular
magnification
θ′
M =
θ
Which is NOT the same as
lateral magnification.
Telescopes
• The purpose of a telescope is to gather light from distant
objects and produce a magnified image.
– Refracting telescopes use lenses so that the objects can be viewed
directly.
θ1
f1
θ1 ≈
f2
θ2
θ1
M =
h’
h'
f1
θ2 ≈
h'
f2
θ2
h '/ f 2
f
=
= 1
h '/ f 1
f2
θ1
– Reflecting telescopes use mirrors to create the image
» Most astronomical telescopes are reflectors, since the most
important feature for these telescopes is the light gathering
ability, and it is easier to make a large mirror than it is to make
large lenses.
detector
Telescope
Angular magnification
y′ /
θ′
M = =−
θ
y/
f2
=−
f1
f1
f2
Note, light gathering power depends on objective lens
Diameter.
Hubble Space Telescope
The HST was launched in 1990; it was
discovered that a lens had been ground
incorrectly, so all images were blurry!
A replacement “contact lens”, COSTAR,
was installed in 1993.
(Now, Hubble’s instruments have
built-in corrective optics, so
COSTAR is no longer needed.)
Before COSTAR
After COSTAR
Aperture of primary mirror: 2.4 m (~8 ft.)
Mass of primary mirror: 828 kg (~1800 lbs)