Chapter 27
Lenses and Optical
Instruments
Lenses
Converging lens
Diverging lens
Thin Lenses


A thin lens consists of a piece of
glass or plastic, ground so that each
of its two refracting surfaces is a
segment of either a sphere or a
plane
Lenses are commonly used to form
images by refraction in optical
instruments
Thin Lens Shapes



These are
examples of
converging lenses
They have positive
focal lengths
They are thickest
in the middle
More Thin Lens Shapes



These are
examples of
diverging lenses
They have
negative focal
lengths
They are thickest
at the edges
Glass lens (nG = 1.52)
The focal length of a lens is determined by the shape
and material of the lens.
Same shape lenses: the higher n, the shorter f
Lenses with same n: the shorter radius of curvature,
the shorter f
Typical glass, n = 1.52
Polycarbonate, n = 1.59 (high index lens)
Higher density plastic, n ≈ 1.7 (ultra-high index lens)
Q. A parallel beam of light is sent through an aquarium.
If a convex lens is held in the water, it focuses the beam (……..
……………………. ) than outside the water
nair = 1, nwater = 1.33
(a) closer to the lens
(b) at the same position as
(c) farther from the lens
Rules for Images
• Trace principle beams considering one end of an object
•
•
•
•
off the optical axis as a point light source.
A beam passing through the focal point runs parallel to
the optical axis after a lens.
A beam coming through a lens in parallel to the optical
axis passes through the focal point.
A beam running on the optical axis remains on the optical
axis.
A beam that pass through the geometrical center of
a lens will not be bent.
Find a point where the principle beams or their imaginary
extensions converge. That’s where the image of the point source.
two focal points: f1 and f2
Parallel beams: image at infinit
Virtual image
Magnifying glass
Virtual image
1/p + 1/q = 1/f
Magnification, M = -q/p
Negative M means that the image is upside-down.
For real images, q > 0 and M < 0 (upside-down).
Lens equation and magnification
1/p + 1/q = 1/f
M = -q/p
This eq. is exactly the same as the mirror eq.
Now let’s think about the sign.
positive
negative
p
real object
imaginary object
(multiple lenses)
q
real image
(opposite side of object)
imaginary image
(same side of object)
f
M
for converging lens
for diverging lens
erect image
inverted image
1/p + 1/q = 1/f
1/2f + 1/q = 1/f
1/q = 1/2f
M = -q/p = -1
two focal points: f1 and f2
1/p + 1/q = 1/f
1/f + 1/q = 1/f
1/q = 0  q = infinite
Parallel beams: image at infinit
Virtual image
Magnifying glass
1/p + 1/q = 1/f
2/f + 1/q = 1/f
1/q = -1/f
M = -(-f)/(f/2) = 2
Virtual image
positive f
Ex. 27.1 A thin converging lens has a focal length of 20 cm.
An object is placed 30 cm from the lens. Find the image
Distance, the character of image, and magnification.
f = 20, p = 30
1/q = 1/f – 1/p
= 1/20 – 1/30
= 1/60
q = 60
real image (opposite side)
M = -q/p
= -60/30
= -2 < 0 inverted
Magnifier


Consider small object held in front of eye
• Height y
• Makes an angle  at given distance from the
eye
Goal is to make object “appear bigger”: ' > 
y

Magnifier

Single converging lens
• Simple analysis: put eye right behind lens
• Put object at focal point and image at infinity
• Angular size of object is , bigger!
Outgoing
rays
Rays seen coming
from here

f
y

f
Image at
Infinity
1 1 1
 
q f p
Angular Magnification
(Standard)




Without magnifier: 25 cm is closest distance to view
• Defined by average near point. Younger people do better
•   tan  = y / 25
With magnifier: put object at distance p = f
• '  tan ' = y / f
Define “angular magnification” m = ' / 
Note that magnifiers work better for older people because
near point is actually > 25cm
~y/25
’~y/f
M= ’/  = 25/f
Example

Find angular magnification of lens
with f = 5 cm
25
m 
5
5
25
m 
1  6
5
Standard
Maximum
Optical Instruments
Eye Glasses
Perfect Eye
Nearsighted
Nearsighted can be corrected with a diverging lens.
 A far object can be focused on retina.
Farsighted
A
Power of lens: diopter = 1/f (in m)
(+) diopter  converging lens
(-) diopter  diverging lens
Larger diopter
 Stronger lens (shorter f)
Material
n
Cornea
1.38
Aqueous
Humor
Lens
1.331.34
1.411.45
1.34
Vitreous
Humor
Air
Water
1.00
1.33
Combinations of Thin Lenses
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The image produced by the first lens is
calculated as though the second lens
were not present
The light then approaches the second
lens as if it had come from the image of
the first lens
The image of the first lens is treated as
the object of the second lens
The image formed by the second lens is
the final image of the system
Combination of Thin Lenses, 2

If the image formed by the first lens
lies on the back side of the second
lens, then the image is treated at a
virtual object for the second lens
• p will be negative

The overall magnification is the
product of the magnification of the
separate lenses