INTERFERENCE OF LIGHT

1. Interference of Light
Akash Shukla

2. Contents
• Introduction
• Condition for Interference
• Path Difference & Phase Difference
• Young’s double Slit Exp.
• Condition for good interference pattern
• How to get two coherent source?
• Fresnel's Bi-Prism
• Interference due to parallel thin films
• Reflected light
• Refracted light
• Newton’s Ring Exp.
• Michelson's Interferometer
• Application of Michelson's Interferometer

3. Introduction
A phenomenon that
S1 occurs when two
light beams meet.
S2
Constructive Interference Destructive Interference

4. Condition for interference
• The sources must be coherent i.e. they must maintain a constant
phase with respect to each other.
• The sources should be monochromatic i.e. of a single wavelength.

5. Path difference & Phase Difference
P
r1
Y
• The difference between optical r2
path of two rays, which are in S1
cosntent phase difference with
each other meet at a same point d O
is called Path difference.
S
S2
x
D
2
Phase Difference x
2
Phase Difference Path Difference

6. Young’s Double Experiment
P
r1
Y
r2
S1
d O
S
S2
x
D

7. Superposition principle
Two waves of different amplitude and phase
Y1 a1 sin t
Y2 a2 sin t
a1 , a2 is the amplitude of Y1 & Y2 and is the phase Difference
Superimpose at a point P due to superposition principle
Y = Y1 + Y2
Y a1 sin t a2 sin t a1 a2 cos R cos ......(1)
Y sin t a1 a2 cos a2 cos t sin a2 sin R sin .................(2)
Y R cos sin t R sin cos t
Y R sin t

8. Cont….
Squaring and adding equation 1 and 2
R2 a12 2
a2 2a1a2 cos
Intensity is directly proportional to square of amplitude.
The resultant intensity is not just directly sum of individual
intensities due to the separated waves I R2
Suppose that amplitudes of both the waves are same a1, a2 = a
I 4a 2 cos 2
2

9. Cont….
Condition for maxima: intensity is maxima when cos φ is maximum i.e.
cos 1
i.e. Phase Difference 2n
Path Difference( x) 2n n n 0,1, 2,3, 4....
2
Condition for minima: intensity is minimum when cos φ is minimum i.e.
cos 1
i.e. Phase Difference 2n 1
Path Difference( x) 2n 1 2n 1 n 0,1, 2,3, 4....
2 2

10. Conclusion
Constructive interference
Path difference natural multiple of wavelength (λ)
Phase Difference even multiple of pi (π)
Destructive interference
Path difference odd multiple of wavelength (λ/2)
Phase Difference odd multiple of pi (π)

11. Fringe Width (β)
P
By the S1PQ
2 2 2 Y
S1 P S1Q PQ
2 S1 Q
2 d
S1 P D2 Y
2
d O
S x
R
S2
By the S2 PR
2 2 2
S2 P S2 R PR
2 D
2 d
S2 P D2 Y
2

12. Cont….
2 2
2 2 d d
S2 P S1P Y Y
2 2
2 2
d d
x Y2 Yd Y 2 Yd
2 2
x 2Yd
S 2 P S1 P S 2 P S1 P 2Yd
x 2D 2Yd
Yd
x
D
Yd
i.e. Path Difference( x)
D

13. Cont….
Consider two cases
1. if the Pth fringe is a bright fringe 2. if the Pth fringe is a Dark fringe
Path Difference( x) n Path Difference( x ) 2n 1
2
Yd
n Yd
D 2n 1
D 2
n D
Y D
d Y 2n 1
Distance between two 2d
consecutive bright fringes Y1 Distance between two consecutive
and Y2 Dark fringes Y1 and Y2
D 2 D 3D 5 D
Y1 and Y2 Y1 and Y2
d d 2d 2d
2 D D D 5 D 3D D
Fringe Width Fringe Width
d d d 2d 2d d
Spacing between any two consecutive maxima and minima is the same

14. Conclusion
Fringe width is :-
• Directly proportional to wavelength (λ)
• Directly proportional to distance between screen from two
source (D)
• Inversely proportional to separation between two source (d)

15. Condition for good Interference
• light waves are in same phase.
• light waves are in constant path difference.
• light waves are same wavelength(i.e. monochromatic).
• Light waves have same amplitude.
• Distance between source and screen is large.
• Distance between two coherent source is less.

16. How to get two Coherent Source?
• Division of Wave front
Incident wave front is divided in to two parts by phenomenon of
reflection or refraction. e.g. Lloyd’s Mirror, Fresnel bi-prism
• Division of Amplitude
Incident light amplitude is divided in to two parts by phenomenon of
reflection or refraction. e.g. Newton’s Rings, Michelson’s
interferometer, beam splitter

17. Fresnel’s Bi-Prism
• It consists of two thin acute angled prisms joined at the bases. It is
constructed as a single prism of obtuse angle of 1790. The acute
angle on both side is about 30 . A portion of the incident light is
refracted downward and a portion upward.
30’
1790
30’

18. Experimental Setup
A Fringes of large width
S1 E
Fringes of equal width
d S C
c
S2 F Fringe Width
D
B d
a b d
Wavelength
D
D d
Wavelength
a b
Determination of wavelength of light by Fresnel’s Bi-prism

19. d by Angle of Deviation Method
By the formula of Prism
A sin 2
S1 ( + ) and are very
sin 2 less quantity so
d S C
1
Total deviation produce is 2 then
S2
B SS1 SS 2 a tan a
a d d
2a
2 2
d 2a
d 2a 1

20. d by displacement method
By the magnification formula
b a d1 a d b
and 2
A d b d a
S1
d1 d 2
d d1
1
C d d
d d1d 2
d2
S2
B
a b

21. Interference by Reflected Light
Optical Path difference between two reflected light rays
BF and DR N
F Q
i R
A i
Path BC CD in film Path in Air
i
BC CD BF B r D
From the Snell’s Law
t r
sin i BF BD BF E
sin r ED BD ED
BF ED C
r
BC CD ED t
BC CE ED ED
PC CE
P
PF

22. Cont….
PF
cos r
BP N
PF cos r BP F Q
i R
A i
PF 2t cos r
i
PF , PF 2t cos r B r D
Path Difference 2t cos r
t r
A ray reflected at a surface backed by a E
denser medium surface abrupt phase
change of π or path difference of λ/2
C
r
Total P. D. 2t cos r t
2
P

23. Conclusion
Bright Fringe Path difference odd
P. D. 2t cos r n multiple of wavelength
2 (λ/2)
Phase Difference odd
2 t cos r 2n 1 multiple of pi (π)
2
n 0,1, 2,3,...
Dark Fringe
P. D. 2t cos r 2n 1
2 2 Path difference natural
2 t cos r n multiple of wavelength (λ)
Phase Difference even
n 0,1, 2,3,..... multiple of pi (π)

24. Interference by Transmitted light
R
Inside the film, reflection at different N t
points takes place at the surface backed r
A i
by rarer medium , thus no abrupt
change of phase and path taken place. D
B
Bright Fringe Q
2 t cos r n n 0,1, 2,3,... r t
Dark Fringe r
2 t cos r 2n 1 n 0,1, 2,3,..... C i E
2
P
T1
T

25. Newton’s Rings
Path Difference 2 t cos r
2
(for air Film), r=0 (normal incident)
Telescope
2t
2
Point of contact t=0
Dark Fringe
2
Central spot is
Dark
nth maxima
S
2t n
2
In this system t is constant along a circle so that the fringes
are In form of circle.

26. Cont….
2t n
2
2t 2n 1 Bright Fringe
2
Dark Fringe C
2t n
By the property of the circle
EB ED AE EC
O 2R
r r t 2R t
r2 2 Rt t 2 t2 2 Rt B
2
E D
2 Rt r t
r2 A
2t
R

27. Cont….
For Bright Fringe
R r = D/2
r2 2n 1
2
D2 2n 1 2 R
D 2 R 2n 1
Dn 2n 1
Diameter of nth Bright ring is
For Dark Fringe proportional
to root of odd natural number.
r2 n R Diameter of nth Dark Ring is
D2 proportional
n R to root of natural numbers
4 Dn n

28. Application of Newton’s Rings
• Determination of Wavelength of light source

29. Michelson’s Interferometer
M1
G1
L G2
M2

30. Application of Interferometer
• Determination of wavelength of light
• Determination of two neighborhood wavelength
• Determination of refractive index and thickness of glass plate