Coherent Sources
Wavefront splitting Interferometer
Young’s Double Slit Experiment
Young’s double slit
© SPK
Path difference:
SP  S P
 D   x  d 2  D   x  d 2
2
2
D  x, d
2
2
  x  d 2
 D 1 
2
D

For 1  y
1  y 
n
2



12
  x  d 2
 D 1 
2
D

 1  nx
x  d 2   x  d 2


2
2D
  2 x  d  2d  xd D
2
2



12
For a bright fringe,
SP  S P  m
For a dark fringe,
SP  S P   2m  1  2
m: any integer
For two beams of equal irradiance (I0)
 xd
I  4 I 0 cos
D
2
Visibility of the fringes (V)
I max  I min
V
I max  I min
Maximum and adjacent minimum of the fringe system
Photograph of real fringe pattern for Young’s double slit
The two waves travel the same distance
– Therefore, they arrive in phase
S'
S
•The upper wave travels one wavelength farther
–Therefore, the waves arrive in phase
S'
S
•The upper wave travels one-half of a
wavelength farther than the lower wave.
This is destructive interference
S'
S
Uses for Young’s Double Slit
Experiment
•Young’s Double Slit Experiment provides a
method for measuring wavelength of the light
•This experiment gave the wave model of light a
great deal of credibility.
Wavefront splitting interferometers
•Young’s double slit
•Fresnel double mirror
•Fresnel double prism
•Lloyd’s mirror
Confocal hyperboloids of revolution in 3D
Path difference

SP  S P 
  m
2
S
m  0, 1, 2, 3
S
x2
y2

1
1 2 1 2

(d   2 )
4
4
=m
-confocal hyperbolae
with S and S as common
foci
Transverse section –Straight fringes
P
S
x
O
d
S
D
The distance of mth bright fringe from central maxima
mD
xm  D sin  m
d
Fringe separation/ Fringe width
D
x 
d
Longitudinal section –Circular fringes
P
rn
N

S
O
S
d
D
For central bright fringe
Path difference = d
m0 
d

For small m
SP  SP  SN  d cos  m  m
2(m  m0 ) 2n
 

d
d
2
m
( n  m  m0 )
Radius of nth bright ring
D 2n
r D 
d
2
2
n
2
2
m
Wavefront splitting interferometers
•Young’s double slit
•Fresnel double mirror
•Fresnel double prism
•Lloyd’s mirror
Interference fringes
Real
Virtual
Localized
Non-localized
Localized fringe
Observed over particular surface
Result of extended source
Non-localized fringe
Exists everywhere
Result of point/line source
Concordance
Discordance
= (q+1/2)
Division of Amplitude
Phase Changes Due To Reflection
• An electromagnetic wave undergoes a phase change of 180°
upon reflection from a medium of higher index of refraction
than the one in which it was traveling
– Analogous to a reflected pulse on a string
μ2
μ1   
1
2
Phase shift
  k0   
C
n1
nf
n2
t
t
D
A
C
D
i
t
A
d
B
B
Optical path difference for the first two reflected beams
  n f [AB  BC]  n1 (AD)
AB  BC  d /cos t
nf
AD  AC sin  i  2d tan  t
sin  t
n1
  2n f dcost
Condition for maxima
dn f cos t  (2m  1)
f
4
m  0, 1, 2,...
Condition for minima
dn f cos t  2m
f
4
m  0, 1, 2,...
Fringes of equal thickness
Constant height contour of a topographial map
Wedge between two plates
1 2
glass
glass
t
x
Path difference
= 2t
Phase difference  = 2kt - 
Maxima 2t = (m + ½) o/n
Minima 2t = mo/n
D
air
(phase change for 2, but not for 1)
Newton’s Ring
• Ray 1 undergoes a phase change of 180 on
reflection, whereas ray 2 undergoes no phase
change
R= radius of curvature of lens
r=radius of Newton’s ring
d  R  R2  r 2
 1  r 2

 R  R 1     ...
 2  R 

1 r2

2 R
For bright ring
1
2 d   ( n  )
2
2
1r
1
2
  ( n  )
2 R
2
1 
1
rbright  (n  ) R  (n  ) R , n  0,1, 2...
2 
2
For dark ring
2d  n
rdark  nR , n  0,1, 2 ...
Reflected Newton’s Ring
Newton’s Ring
1. Optics
Author: Eugene Hecht
Class no. 535 HEC/O
Central library
IIT KGP