Lecture 26-1
Lens Equation
(<0)
1 1 1
 
p q f
 True for thin lens and paraxial rays.
 magnification m = h’/h = - q/p
Lecture 26-2
Signs in the Lens Equation for Thin Lenses
1 1 1
 
p q f
q
m
p
• p is positive for real object
• p is negative for virtual object
• q is positive for real image
yi
f q
q


yo
f
p
• q is negative for virtual image
• m is positive if image is upright
• m is negative if image is inverted
• f is positive if converging lens
• f is negative if diverging lens
Lecture 26-3
Aberrations
Cameras, …
Chromatic aberration
correct
nblue > nred
Spherical aberration
Parabolic
mirrror
Large
telescopes, …
Lecture 26-4
Geometric Optics vs Wave Optics
• Geometric optics is a limit of the general optics where wave
effects such as interference and diffraction are negligible.
 Geometric optics applies when objects and apertures
involved are much larger than the wavelength of light.
 In geometric optics, the propagation of light can be
analyzed using rays alone.
• Wave optics (sometimes also called physical
optics) - wave effects play important roles.
 Wave optics applies when objects and
apertures are comparable to or smaller than the
wavelength of light.
 In wave optics, we must use the concepts
relevant to waves such as phases, coherence,
and interference.
Lecture 26-5
Coherence
 When the difference in phase between two (or more) waves
remains constant (i.e., time-independent), the waves are said
to be perfectly coherent.
- laser light and light transmitted through a small aperture are coherent.
- light from a light bulb and sun light over some area are incoherent.
 A single light wave is said to be coherent if any two points
along the propagation path maintains a constant phase difference.
Only coherent waves can
produce interference
fringes!
Coherence length: the spatial
extent over which light
waves remain coherent.
Lecture 26-6
Interference of Two Coherent Waves
Snapshot of
wave fronts at
a given instant
I  c 0 E 2
 ( E1  E2 ) 2
Constructive interference (in phase)
B,C
Destructive interference (completely
out of phase)
A
Lecture 26-7
Intensity of Interference Fringes
Let the electric field components of the two
coherent electromagnetic waves be
E1  E0 sin t
E2  E0 sin(t   )
The resulting electric field component point P
is then
E E E
1
2
 E0 sin t  sin(t   ) 
 

 2 E0 cos sin  t  
2 
2
Intensity is proportional to E2
I Em2
2
 2 I  4 I 0 cos
I 0 E0
2
I=0 when  = (2m+1)p , i.e. half cycle + any number of cycle.
Lecture 26-8
Thin Film
n1  n2 , n3  n2
• Thin here means that the thickness is
comparable to the wavelength of the light.
• The reflected light waves from the two
sides of a thin film interfere.
Destructive interference eliminates
(or minimizes) the reflected light!
e.g., non-reflecting lens coating
Phase change by p
• Phase difference could come from:
reflection, path length difference,
different indices of refraction
• If the incident light propagates from a
medium of lower index of refraction toward
one of higher index of refraction, the phase
of the reflected wave shifts by p.
(neither highlow index nor transmitted light)
Lecture 26-9
Thin-Film Interference-Cont’d
(Assume near-normal incidence.)
Path length difference:
l  2t
destructive
m


(m  1/ 2) constructive
where

0
n
• ray-one got a phase change of 180o due to reflection from air to
glass.
• the phase difference due to path length is:  ' 
•then total phase difference:  = ’+180.
l

 2p 
l
0
n
 2p
Lecture 26-10
Thin-Film Continued
The previous discussion was for the situation in which n2 > n1 and
n2 > n3 , i.e., the index of refraction of the film is larger than those
of the surrounding media, but they are also valid if the index of
refraction of the film is smaller than those of the surrounding
media (n2 < n1 and n2 < n3 ).
The equations fail for some of the following situations. Which one(s)?
If the film has an intermediate index of refraction
Conditions for maxima/minima will reverse!
Lecture 26-11
Warm-up quiz
Monochrome light of wavelength  in air is incident normal
to a thin layer of oil film floating on water as shown. If the
film thickness is 5/(4n1). Which of the following statement
is true?
a) The reflection is dark by destructive
interference of rays 1 and 2
b) The reflection is bright by constructive
interference of rays 1 and 2
c) The reflection is colorful by
interference of rays 1 and 2.
1
2
oil
(n1>1.33)
water
(n=1.33)
Lecture 26-12
Newton’s Ring
The air between the glass plates acts like a thin film.
• Since the thickness of the film changes over the radius of the
plates, alternating bright and dark fringes form, when the
plates are illuminated. Because of the curvature of the upper
piece, the film thickness varies more rapidly at larger radius.
Thus the fringe separation is smaller toward the outside.
Lecture 26-13
Two (narrow) slit Interference
Young’s double-slit experiment
• According to Huygens’s principle,
each slit acts like a wavelet. The
the secondary wave fronts are
cylindrical surfaces.
• Upon reaching the screen C, the
two wave interact to produce an
interference pattern consisting of
alternating bright and dark bands
(or fringes), depending on their
phase difference.
Constructive vs. destructive
interference
Lecture 26-14
Interference Fringes
For D >> d, the difference in path lengths
between the two waves is
L  d sin 
• A bright fringe is produced if the path
lengths differ by an integer number of
wavelengths,
d sin   m , m  0, 1,
• A dark fringe is produced if the path
lengths differ by an odd multiple of
half a wavelength,
d sin   (m  1/ 2) , m  0, 1,
y ~ D*tan(θ)
Lecture 26-15
Intensity of Interference Fringes
Let the electric field components of the two
coherent electromagnetic waves be
E1  E0 sin t
E2  E0 sin(t   )
The resulting electric field component point P
is then
E E E
1
2
 E0 sin t  sin(t   ) 
 

 2 E0 cos sin  t  
2 
2
Intensity is proportional to E2
I Em2
2
 2 I  4 I 0 cos
I 0 E0
2
I=0 when  = (2m+1)p , i.e. half cycle + any number of cycle.
Lecture 26-16
Intensity of Interference Fringes-Cont’d
For Young’s double-slit experiment, the phase difference is
  2p 
d sin 


maxima
 22pp (mm, for
1/2), for mimina
4 I 0 cos
2

2
Lecture 26-17
Physics 241 –Quiz A
Light of wavelength  in air is incident normal to a
thin layer of glass held in air as shown. If the
reflection is suppressed (dark) by interference of
rays 1 and 2, what is a possible thickness d of the
glass layer?
a) /(4n1)
b) /4
c) /(2n1)
d) /2
e) 3/4
1
2
glass
(n1>1)
Lecture 26-18
Physics 241 –Quiz B
Light of wavelength  in air is incident normal to a
thin layer of oil film floating on water as shown. If
the reflection is suppressed (dark) by interference
of rays 1 and 2, what is a possible thickness d of
the oil film?
a) /(4n1)
b) /4
c) /(2n1)
d) /2
e) 3/4
1
2
oil
(n1>1.33)
water
(n=1.33)
Lecture 26-19
Physics 241 –Quiz C
Light of wavelength  in air is incident normal to a
thin layer of oil film floating on water as shown. If
the reflection is bright (constructive interference)
by interference of rays 1 and 2, what is a possible
thickness d of the oil film?
a) /(4n1)
b) /4
c) /(2n1)
d) /2
e) 3/4
1
2
oil
(n1>1.33)
water
(n=1.33)