Double
Rainbow
1
2
Bar at the Folies Bergères’ by Edouard Manet (1882)
Chapter 35
Interference
The concept of optical interference is critical to understanding many natural
phenomena, ranging from color shifting in butterfly wings (iridescence) to
intensity patterns formed by small apertures. These phenomena cannot be
explained using simple geometrical optics, and are based on the wave
nature of light.
In this chapter we explore the wave nature of light and examine several key
optical interference phenomena.
35- 3
Light as a Wave
Huygen’s Principle: All points on a wavefront serve as point sources of
spherical secondary wavelets. After time t, the new position of the wavefront will
be that of a surface tangent to these secondary wavelets.
Fig. 35-2
35- 4
Law of Refraction
1
2
1 v1
t 


v1 v2
2 v2
1
sin 1 
(for triangle hce)
hc
2
sin  2 
(for triangle hcg)
hc
c
sin 1 1 v1
Index of Refraction: n 


v
sin  2 2 v2
c
c
n1 
and n2 
v1
v2
sin 1 c n1 n2


sin  2 c n2 n1
Fig. 35-3
Law of Refraction:
n1 sin 1  n2 sin2
35- 5
Wavelength and Index of Refraction
n v
v

  n    n 
 c
c
n
v
cn c
fn 

  f The frequency of light in a medium is the same
as it is in vacuum
n  n 
Since wavelengths in n1 and n2 are different,
the two beams may no longer be in phase
L
L
Ln1
Number of wavelengths in n1: N1 


Fig. 35-4
n1  n1

L
L
Ln2
Number of wavelengths in n2 : N 2 


n 2  n2

Ln2 Ln2 L
Assuming n2  n1: N 2  N1 

  n2  n1 



N2  N1  1/2 wavelength  destructive interference
35- 6
Rainbows and Optical Interference
Fig. 35-5
The geometrical explanation of rainbows given in Ch. 34 is incomplete.
Interference, constructive for some colors at certain angles, destructive for
other colors at the same angles is an important component of rainbows
35- 7
Diffraction
For plane waves entering a single slit, the waves emerging from the slit
start spreading out, diffracting.
Fig. 35-7
35- 8
Young’s Experiment
For waves entering a two slit, the emerging waves interfere and form an
interference (diffraction) pattern.
Fig. 35-8
35- 9
Locating Fringes
The phase difference between two waves can change if the waves travel paths of
different lengths.
What appears at each point on the screen is determined by the path length difference
DL of the rays reaching that point.
Path Length Difference: DL  d sin
Fig. 35-10
35-10
Locating Fringes
if DL  d sin    integer     bright fringe
Maxima-bright fringes:
d sin   m for m  0,1, 2,
Fig. 35-10
if DL  d sin    odd number     dark fringe
Minima-dark fringes:
d sin   m  12   for m  0,1,2,
 2 
1  1.5 
m

1
dark
fringe
at:


sin
m  2 bright fringe at:   sin 1 

35-11

d
d




Coherence
Two sources to produce an interference that is stable over
time, if their light has a phase relationship that does not
change with time: E(t)=E0cos(wt+f)
Coherent sources: Phase f must be well defined and
constant. When waves from coherent sources meet, stable
interference can occur. Sunlight is coherent over a short length
and time range. Since laser light is produced by cooperative
behavior of atoms, it is coherent of long length and time
ranges
Incoherent sources: f jitters randomly in time, no stable
interference occurs
35-12
Intensity in Double-Slit Interference
E1  E0 sin wt and E2  E0 sin wt  f 
2 d
I  4 I 0 cos2 12 f
f
sin 

maxima when: 12 f  m for m  0,1, 2,
 d sin   m for m  0,1, 2,
 f  2m 
E
2
2 d

sin 
(maxima)
minima when: 12 f   m  12    d sin    m  12   for m  0,1,2,
(minima)
I avg  2 I 0
Fig. 35-12
E1
35-13
Proof of Eqs. 35-22 and 35-23
Eq. 35-22 E  t   E0 sin w t  E0 sin w t  f   ?
E  2  E0 cos    2 E0 cos 12 f
E 2  4 E02 cos2 12 f
I E2
 2  4cos2 12 f  I  4 I 0 cos2 12 f
I 0 E0
Eq. 35-23 
   f
Fig. 35-13
phase   path length 
 difference   difference 



2

 phase  2  path length 
 difference     difference 




f
2

 d sin  
35-14
Combining More Than Two Waves
In general, we may want to combine more than two waves. For eaxample,
there may be more than two slits.
Prodedure:
1. Construct a series of phasors representing the waves to be combined.
Draw them end to end, maintaining proper phase relationships between
adjacent phasors.
2. Construct the sum of this array. The length of this vector sum gives the
amplitude of the resulting phasor. The angle between the vector sum
and the first phasor is the phase of the resultant with respect to the first.
The projection of this vector sum phasor on the vertical axis gives the
time variation of the resultant wave.
E4
E3
E
E1
E2
35-15
Interference from Thin Films
f12  ?
 0
Fig. 35-15
35-16
hitt
A 5:0-ft woman wishes to see a full length
image of herself in a plane mirror. The
minimum length mirror required is:
A. 5 ft
B. 10 ft
C. 2.5 ft
D. 3.54 ft
E. variable: the farther away she stands the
smaller the required mirror length
17
question
Two thin lenses (focal lengths f1 and f2) are
in contact. Their equivalent focal length is:
A. f1 + f2
B. f1f2/(f1 + f2)
C. 1=f1 + 1=f2
D. f1 /f2
E. f1(f1 f2)=f2
18
hitt
The image of an erect candle, formed using a
convex mirror, is always:
A. virtual, inverted, and smaller than the candle
B. virtual, inverted, and larger than the candle
C. virtual, erect, and larger than the candle
D. virtual, erect, and smaller than the candle
E. real, erect, and smaller than the candle
19