Refraction on Spherical Surface
h
θ a = α + φ ; φ = β + θb ⎫
na sin θ a = nb sin θ b
⎬ naα + nb β = (nb − na )φ
naθ a ≅ nbθ b
⎭
s’
s
f
h’
Refraction on Spherical Surface
Magnification
na
n
n − na
+ b = b
s
s'
R
R positive if C on
transmission side;
negative otherwise
h
h
h
; tan β =
; tan φ =
s +δ
s '−δ
R −δ
n
n
n −n
h
h
h
⇒ a + b = b a
α≅ ; β≅ ; φ≅
s
s'
R
s
s'
R
tan α =
Refraction on Spherical Surface
Example: Fish bowl. A fish is 7.5
cm from the front of the bowl.
Find the location and magnification of the
fish as seen by the cat. Ignore effect of bowl.
Radius of bowl = 15 cm. na = 1.33
s
For small angles
tan θ ≅ sin θ
na y
n y′
=− b
s
s′
n s′
y′
=− a
m=
y
nb s
n s′
y′
=− a
y
nb s
b
I s’
− y′
y
tan θ b =
s
s′
na sin θ a = nb sin θ b
m=
R negative
O
tan θ a =
na
n
n − na
+ b = b
s
s'
R
a
m=−
na
n
n − na
nb
n − na na
−
+ b = b
= b
s
s'
R
s'
R
s
nb
1
s′ =
=
= −6.44 cm
nb − n a n a
− 0.33
1.33
−
−
− 15 cm 7.5 cm
R
s
The fish appears closer and larger.
na s ′
1.33 (−6.44 cm)
=−
∗
= 1.14
nb s
1
7.5 cm
Lens Equation – thin lens
Lenses
• A lens is a piece of transparent material shaped such that
parallel light rays are refracted towards a point, a focus:
– Convergent Lens
» light moving from air into glass
will move toward the normal
» light moving from glass back into
air will move away from the normal
» real focus
– Divergent Lens
» light moving from air into glass
will move toward the normal
» light moving from glass back into
air will move away from the normal
» virtual focus
Positive f
Negative f
na
n
n − na
+ b = b
s1
s1 '
R1
nb
n
n − nb
+ a = a
s2
s2 '
R2
For air, na=1 and glass, nb=n, and s2=-s1’. Neglects l.
1
n
n −1
+
=
s1
s1 '
R1
−
n
1
1− n
+
=
s1 '
s2 '
R2
Eliminate, s1’, s1 → s, s2’ → s’
⎛ 1
1
1
1 ⎞ 1
⎟⎟ =
+
= (n − 1)⎜⎜ −
s
s'
⎝ R1 R2 ⎠ f
⇐ Lensmaker eqn.
f > 0, converging
f < 0, diverging
1
Lensmaker’s example
Lensmaker’s Formula
n = 1.5
The thin lens equation for a lens with two curved sides:
⎛ 1
1
1 ⎞
⎟
= (n − 1 )⎜⎜
−
f
R 2 ⎟⎠
⎝ R1
⎛ 1
1
1 ⎞
⎟
= (n − 1 )⎜⎜
−
f
R 2 ⎟⎠
⎝ R1
R2 = -15 cm
R1 = +10 cm
Incident Light
R1
1
1 ⎤
⎡1
= (1.5 − 1) ⎢ −
⎥
f
⎣10 − 15 ⎦
f = 12 cm
R2
R1 is first surface; R2 is second.
R1 is first surface; R2 is second.
R is positive if center of curvature on transmission side.
R is positive if center of curvature on transmission side.
f positive if converging lens; negative if diverging.
f positive if converging lens; negative if diverging.
Sign conventions for lenses
(same as mirrors)
Convergent Lens example
• Draw some ray diagrams:
h
1 1
1
+ =
s s'
f
Parallel ray – goes through F2
Focal ray (F1 ) – becomes parallel
Central ray - undeviated
F1
s
fs
s' =
s− f
s’
f
Light coming from ∞,
focuses at focal point.
f
F2
• IF
object & incoming ray on same side, positive s,
otherwise negative.
• IF image & outgoing ray on same side, positive s′,
otherwise negative.
• If converging lens, then f is positive; for diverging lens,
f negative.
h’
1 1 1
+ =
s s' f
s' h'
M =− =
s
h
13. A parallel laser beam of width w1 is incident on a two lens
system as shown below.
w2
s' =
fs
s− f
M =−
s ' h'
=
s h
6. The lens in your eye
a) is always a positive (i.e., converging) lens
b) is always a negative (i.e., diverging) lens
c) is sometimes positive, and sometimes negative, depending on
whether you are looking at an object near or far away
d) is positive if you are near-sighted and negative if you are farsighted
7. The image on the back of your retina is
Each lens is converging. The second lens has a larger focal
length than the first (f2> f1).
What does the beam look like when it emerges from the
second lens?
a) The beam is converging
b) The beam is diverging
c) The beam is parallel to the axis with a width < w1
d) The beam is parallel to the axis with a width = w1
e) The beam is parallel to the axis with a width > w1
a) inverted
b) noninverted
8. The image on the back of your retina is
a) real
b) virtual
9. The image on the back of your retina is
a) enlarged
b) reduced
2
Lecture 26, ACT 1
• A lens is used to image an object on a
screen. The right half of the lens is covered.
– What is the nature of the image on the
screen?
Lecture 26, ACT 1
object
lens
(a) left half of image disappears
(b) right half of image disappears
(c) entire image reduced in intensity
screen
• A lens is used to image an object on a
screen. The right half of the lens is covered.
– What is the nature of the image on the
screen?
object
lens
(a) left half of image disappears
(b) right half of image disappears
(c) entire image reduced in intensity
screen
• All rays from the object are brought to a focus at the screen by the lens.
• The covering simply blocks half of the rays.
• Therefore the intensity is reduced but the image is of the entire object!
•( one more thing… In our examples of image formation, we only needed
the top half of the lens to form the image)
s’
h
s
f
h’
Sign conventions for lenses
(same as mirrors)
Divergent lens example
• Because the lens is divergent, f is negative:
• IF
h
F2 h’
s
f
F1
s’
f
Parallel ray – goes through F2
Focal ray (F1 ) – becomes parallel
Central ray - undeviated
1 1
1
+ =
s s'
f
fs
s =
s− f
'
s' h'
M =− =
s
h
Graphical images of Thin Lens
f is negative.
object & incoming ray on same side, positive s,
otherwise negative.
• IF image & outgoing ray on same side, positive s′,
otherwise negative.
• If converging lens, then f is positive; for diverging lens,
f negative.
1 1 1
+ =
s s' f
s' =
fs
s− f
Compound Lens Example
with ray tracing
M =−
s ' h'
=
s h
Image of 1st lens becomes
object of 2nd lens Resulting
image is upright
Work on board
3
Summary of Lenses and Mirrors
• We have derived, in the paraxial (and thin lens)
approximation, the same equations for mirrors and lenses:
1
1
1
+
=
s
s′
f
M = −
s′
s
when the following sign conventions are used:
Variable
Mirror
Lens
f>0
f<0
concave
convex
converging
diverging
s>0
s<0
real (front)
virtual (back)
real (front)
virtual (back)
s’ > 0
s’ < 0
real (front)
virtual (back)
real (back)
virtual (front)
Principal rays “connect”
object and image
one goes through the
center of the lens or
mirror
the other goes parallel to
the optic axis and then is
refracted or reflected
through a focal point
the third one is like the
second one….
Optical Aberrations
Why really good lenses cost a lot
(But spherical
Spherical aberration
Chromatic aberration
Due to dispersion (index of
refraction depends on
frequency), focal length can be
different for different colors.
Astigmatism
Curvature of the lens not
symmetric in transverse
directions → slightly cylindrical
→ different focal lengths
Multiple lenses can be used
to improve aberrations
Spherical Aberration
lenses are
much cheaper
to grind &
polish!)
Outside the “paraxial limit”,
optimal focusing occurs only for a
parabolic lens. Spherical lenses
look ~parabolic for narrow field of
view.
Summary
Chromatic Aberration
• Lens Equation
– same as mirror equation
1
1
1
+
=
s
s′
f
• Lensmaker’s Formula
1 ⎛ n 2 − n1
= ⎜⎜
f
⎝ n1
⎞⎛ 1
1 ⎞
⎟⎟ ⎜⎜
⎟⎟
−
⎠ ⎝ R1 R 2 ⎠
• Aberrations
– spherical – ideal lens shape is parabolic
– chromatic – index of refraction depends on frequency
– multiple lenses can reduce aberrations
www-optics.unine.ch/education/optics_tutorials/achromat.html
Next time: Multiple lenses, Optical instruments, Lasers
The Lens Equation
• We now derive the lens equation which determines the image
distance in terms of the object distance and the focal length.
– Convergent Lens:
h
s’
s
f
h’
Ray Trace:
• Ray through the center of the lens (light blue) passes through undeflected.
• Ray parallel to axis (white) passes through focal point f. These are principal
h′
h
rays!!!
two sets of similar triangles: h ′ = s ′
=
s′ − f
f
s
h
same as mirror eqn
1
1
1
′
′
s
s
−
f
eliminating h’/h:
+
=
=
if we define
s
s′
f
s
f
s’ > 0
f>0
magnification: also same as mirror eqn!!
M < 0 for inverted image.
M
= −
s′
s
4