Interference and Diffraction

Interference

Principle of Superposition of Waves
• When two or more waves arrive at a point in the
medium simultaneously then each wave
produces its own displacement independent of
the other waves. The resultant displacement at
that point is given by the vector sum of
individual displacements produced by each
wave.

Constructive and distructive
Interference

Condition for Constructive Interference
(brightness):-
• Overlapping (crest-crest, Trough-Trough)
• Path difference = n λ [n = 0, 1, 2, ...........]
• Phase difference=2nπ [n = 0, 1, 2, ...........]
• Intensity Iα A2

Condition for Destructive Interference
(darkness):-
• Overlapping (crest-Trough, crest -Trough)
• Path difference = (2n – 1) λ/2 [n = 1, 2, 3, 4 ……..]
• Phase difference = nπ
• Intensity Iα A2

Path Length and Path Difference

Young’s Experiments

Multiple Choice Questions
• 1. In interference of light
(a) Light energy is created
(b) Light energy is destroyed
(c) Light energy is redistributed
(d) Light energy is doubled
• 2. Select the correct statement
(a) Only longitudinal waves produce interference.
(b) Only transverse waves produce interference.
(c) Only standing waves produce interference.
(d) Both longitudinal and transverse waves produce
interference.

• 3. Two sources of light are said to be coherent if
they emit light waves of the same
(a) Frequency and speed
(b) Wavelength and constant phase difference
(c) Frequency and amplitude
(d) Intensity and frequency
• 4. Which one of the following quantities is
conserved in the interference of light waves?
(a) Phase difference (b) Amplitude
(c) Intensity (d) Path difference

Solutions
• 1. (c) Light energy is redistributed
In interference of light, light energy is redistributed.
• 2.(d) Both longitudinal and transverse waves produce
interference.
Both longitudinal and transverse waves produce interference.
• 3. (b) Wavelength and constant phase difference
Two sources of light are said to be coherent, if they emit
light waves of the same wavelength and of constant phase
difference.
• 4. (c) Intensity
In interference of light waves, intensity of light is conserved.

Theory of Interference Band

• Expression for path difference S2P – S1P :-
• In ∆PNS2 and ∆PMS. By Pythagoras theorem.
S2P2 = S2N2 + PN2 = D2 + (x + d/2)2............. (1)
S1P2 = S1M2 + PM2 = D2 + (x - d/2)2.............(2)
• S2P2 – S1P2 = (x + d/2)2 – (x – d/2)2
(S2P– S1P)(S2P+ S1P) = x2 + 2x d/2 + d2/4 – x2 + 2x d/2 –d2/4
(S2P– S1P)(S2P+ S1P) = 2xd ..................... (3)
If x, d < < D, then S1P ≈ S2P ≈ D
Equation (3) becomes.
2D (S2P – S1P) = 2xd
S2P – S1P = xd/D.................. (4)

• For a point P to be bright.
S2P – S1P = n λ/2 .................... (5)
From (3) and (4),
xd/D = n λ
x = n λ D/d
Let xn, xn + 1 = distances of nth and n + 1th bright
bands from O.
• xn = nλ D/d and xn - 1 = (n + 1)λ D/d
X = Band width of bright band
= xn+1 – xn
= (nλ D/d) + (λ D/d) – (n λ D/d)
X = λD/d

• For a point P to be dark
S2P – S1P = (2n – 1) λ /2 ..................... (8)
From (3) and (8),
xd/D = (2n – 1) λ/2
x = (2n – 1) λD/2d ...................... (9)
Let xn, xn +1 = distances of nth and n+ 1th dark bands from O.
Using these in (9).
Xn = (2n – 1) λ D/2d
Xn – 1 = [2(n + 1) – 1] λD/2d = (2n + 1) λD/2d
X = Band width of dark band
= Xn-1 – Xn
= (2nλD/2d) + (λ D/2d) λ – (2n D/2d) + (λ D/2d)
X = λ D/d ...................... (10)

Conditions for Obtaining Well Defined
Steady Interference Pattern
• Two sources must be
i. equally bright :- This will not give a well-defined interference
pattern.
ii. Monochromatic :- This will not give a well defined
interference pattern.
iii. Coherent:- This produces unstable interference pattern.

• iv. as narrow as possible:- These will not a well defined
interference pattern.
• v. as closed as possible :-
Since X = λ D/d. For λ, D = constant
X α 1/d. Therefore smaller the distance between the
sources higher the bond width. This gives a well
defined interference pattern.
• Vi:- Equally Polarized
• Vii:- distance between sources and screen should be as
large as possible

Biprism
• Two thin prisms of extremely small refracting
angle are connected bases to base. Such a
prism formed is called as “Biprism”. Biprism is
used for producing two coherent sources from
a single source.

Biprism Experiment (to find unknown
wavelength )

Measurement of D
• Distance between slit and the eyepiece is
measured directly from the scale on the
optical bench.

b. Measurement of X
• For this purpose vertical wire of the cross wire is
made coincide with one edge of the white band and
corresponding reading (xn) on the micrometer scale
is recorded using slow motion screw. The cross wire
is moved through known number of bands (n).
Vertical wire is again made coincide with one edge of
the bright band and corresponding heading (xn) on
the micrometer is recorded. Mean band width X is
obtained by the formula.
• X = Xn – X0/n

Measurement of d (method of
conjugate foci)