Interference occurs when two waves superimpose. This can result in constructive or destructive interference depending on the path difference between the waves. Interference is observed with light, sound, and other wave phenomena. Thin film interference is commonly seen in daily life in phenomena like oil slicks and soap bubbles. It is also used in applications like anti-reflective coatings. To study interference, coherent light sources are required where the waves have a constant phase relationship. This can be obtained by dividing a single light source, such as in Young's double slit experiment.

1. WAVE OPTICS
TOPIC-INTERFERENCE

2. Interference
• Superposition of two similar frequency waves
results in different intensities at different points.
• Observed not only for light, other waves e.g.
sound waves, radio waves etc. also show
interference

3. • Why an aspiring Engineer should study the
underlying concepts of interference?
Interference

4. • Interference of sound- Taking use of interference effects of sound,
the auditoriums are designed so as to enhance the voice quality.
• Destructive Interference of Sound (Active noise control)- used in
Automobile Muffler which is to reduce the noise produced by the
exhaust system of car. Muffler senses and produces similar sound
wave but out of phase so that the two interfere destructively.
Why to study Interference?

5. • Interference of light- Interference of light from thin films is used in making
Reflective Coatings-Used to increase reflectivity ~99.9% (e.g. in optical
cavity of laser)
Anti-reflective Coatings- Used in eyeglasses, cameras, binoculars.
Why to study Optical Interference?
• Interference of light- used in interferometery i.e. to measure the wavelength of light

6. Colours of soap bubbles Colours of thin layer of oil spilled on road
Interference of light in daily life?

7. As the pigeon moves about in the
sunshine, the green and purple
colours on its neck feathers can
change suddenly, as viewing
angle shifts. It occurs due to
interference.
Both the experimental and theoretical results
suggest that structural colors in green and
purple neck feathers should originate from the
interference in the top keratin cortex layer
Interference of light in daily life?

8. Interference of light from thin films is something we come
across frequently in daily life and this is the phenomena which
is applied in anti-reflective and reflective coatings.
Therefore, the syllabus is focused on studying interference of
light from thin films.
Interference from Thin Films

9. So let’s recap the basics

10. When two or more light waves of same frequency
superimposes, the resultant intensity in the region of
superposition is in general, different from the sum of
intensities due to individual waves. This redistribution of
intensity of light in the region of superposition of light
beams is called interference.
Optical Interference
Light + Light = ?

11. •At some points, the resultant intensity is greater than the
sum of the intensities due to separate waves (called as
constructive interference) while at some other, lesser than
it(called as destructive interference).
•Interference is a result of the superposition of the waves, its
effect can only be observed in the region of superposition.
Constructive and Destructive Interference

12. If two waves from a source travel two different paths to arrive at a
common point, their phase relationship at that point depends on the
difference in lengths between their paths.
∆= path difference , Ф= phase difference,





2
Important Relation between phase difference and path difference

13. • The wave disturbance at a point P due to one wave at any
instant ‘t’
Y1 = a1 sint
The wave disturbance at the same point at the same instant due
to the other wave
Y2 = a2 sin (t +Ф)
The resultant wave disturbance at P , Y = Y1 + Y2
Y= a1 sint + a2 sin (t + Ф)
Superimposition of two waves

14. • Y = sin t (a1+a2 cos Ф) +a2 sin Ф cost
• Substituting,
a1+a2cos Ф = A cosѲ (1)
and a2sin Ф = A sinѲ (2)
• Y = A cosѲsint + A sinѲcost
or, Y = A sin(t + Ѳ)
• The resultant amplitude at P,

cos
2
1
2
2
2
1 a
a
2
a
a
A 


Superimposition of two waves

15. • resultant intensity at a point after the superposition of two
waves),
I  A2
or, I = kA2=k
where k is a proportionality constant.
]
cos
[ 
2
1
2
2
2
1 a
a
2
a
a 

Conditions of Constructive and Destructive Interference

16. For constructive interference/maxima
Phase difference, Ф= 2nπ n=0,1,2…
or path difference Δ=nλ
Imax =I1+I2+2√(I1I2)= (a1+a2)2
For destructive interference/minima
Phase difference, Ф =(2n-1)π or (2n+1) π
n=1,2… n=0,1,2
or path difference Δ=(2n-1)λ/2 n=1,2…
or, Δ=(2n+1)λ/2 n=0,1,2…
Imin=I1+I2-2√(I1I2) =(a1-a2)2
Conditions for maxima and minima

17. Most Important Conditions For obtaining
Sustained Interference:
◦ The light beams must be coherent.
◦ The light beams must have same frequency.
Note If the phase difference between two
waves is constant with time then the two waves
are coherent. Otherwise, they are incoherent
(eg: two light bulbs).
Conditions for Sustained Interference

18. Methods for Obtaining Coherent light beams
Light beams emitted by two independent light sources are not coherent
as the phase difference between them is not constant with time.
Coherent light beams can be obtained by dividing light coming from a
single source of light in two parts. There are two ways of doing it-
1. Division of wavefront (examples:Young’s double slit experiment
YDSE, Fresnel Biprism etc) : The whole wavefront is divided in two
parts for example by using two slits in YDSE
2. Division of Amplitude (Thin film Interference, Michelson
Interferometer): The amplitude of incident light is divided in two parts
i.e. reflected light and refracted light in thin film interference
How to obtain coherent waves?

19. D
Division of wavefront (YDSE)

20. Division of Amplitude (Thin Film
Interference)

21. Interference from thin film of Uniform
Thickness in reflected light :-

22. Interference from thin film of uniform
thickness (in reflected light) :-
The Path difference between AD and CE is given by:
Path difference = (AB+BC)in film – (AN)in air
= 2µ AB– AN –––––– (1) (as AB=BC)
Now to find AB & AN
Now in rt angle ΔABM,
cos r = BM / AB
or AB = BM / cos r = t / cos r –––––– (2)
To find the value of AN, use rt. Angle ΔANC
Sin i = AN / AC
AN = AC Sin i
AN = 2AM Sin i –––––– (3) (AC = 2AM)
Now to find AM, use rt. Angle ΔAMB
So in ΔAMB,
tan r = AM / BM
or AM = BM tan r
AM = t tan r –––––– (4)

24. Where n = 1,2,3……..

25. Thin film interference for transmitted light :-
i

26. Thin film interference in transmitted system :-
The Path difference between two interfering beams is given by:
Path difference = (BC+CD)in film – (BN)in air
= 2µ BC– BN –––––– (1) (as AB=BC)
Now to find BC & BN
Now in rt angle Δ BMC,
cos r = CM / CB
or CB = CM / cos r = t / cos r –––––– (2)
To find the value of BN, use rt. Angle Δ BND
Sin i = BN / BD
BN = BD Sin i
BN = 2BM Sin i –––––– (3) (BD = 2BM)
Now to find BM, use rt. Angle Δ BMC,
tan r = BM / MC
or BM = MC tan r
BM = t tan r –––––– (4)
Use all corresponding values in Eq. (1)

28. Condition for minima :
P.d. = (2n+1) λ/2
2μt cos r = (2n+1) λ/2
Thus, conditions of maxima and minima in the reflected light are just the
reverse of conditions of maxima and minima for transmitted light. Hence,
reflected and transmitted interference pattern are complimentary to each
other.
Condition for maxima :
P.d. = nλ
2μt cos r = nλ

29. With extended source of light, the rays from the different points of the extended
source incident at different angles at all the portion of the film and reach eye after
reflection. Hence, interference due to whole portion of the film will be visible.
Whereas interference from only a small portion of the film can be observed with a
narrow source of light.
Need of Extended source of light to view interference from
Thin films

30. So now we can understand why thin film/layer appears to be
coloured on illuminating with white light
Yes that is due to interference. If the film is illuminated with white light for a
particular value of ‘r’ (i.e. for a position of eye) only those wavelengths will be
seen in reflected light for which condition of constructive interference is
satisfied and so the film appears coloured. If the position of eye is changed
that is ‘r’ is changed some other colours will satisfy the condition of
constructive interference and so the colours of the film change as the position
of eye is changed.

31. What is Coherence and types of Coherence?
• It is the property of waves that helps in getting sustained interference. Concept of coherence is related
to predictability of phase.
Or we may say that coherence represents fixed phase relationship between the field of wave at different
locations or at different times.
Types of Coherence
1. Temporal Coherence (Longitudinal Coherence): It is the correlation between the phases of field of
the wave at a time t and at a later time t+ ∆t.
• Radiations from a light source consists of finite size wavetrains. The average duration of wavetrains
is represented by τc i.e. the time for which the field of the wave remains sinusoidal.
• At a given point the fields at time t and t+∆t will have definite phase relationship if ∆t << τc
• Therefore, the time duration τc is called coherence time. The wave remains coherent for times of the
order of τc.
• Coherence length lc
: It is the average length of the wavetrains emitted by the source. It is equal to
the distance travelled by light in coherence time lc = c τc
For ordinary sources of light coherence length is of the order of few cm while for laser it may be in
km.
* Coherence time /coherence length is a measure of degree of Temporal coherence.
* Relation between coherence time and line width: Coherence time τc = 1/∆ν and as ν=c/λ ,
∆ν= c ∆ λ/ λ2
• Temporal Coherence is related to the line width ∆ λ of the light source. Narrower is the line width
more will be the extent of temporal coherence.
• Why finite size wavetrains ? May be due to following reasons (i) Due to collision of radiating atom
with other atom (ii) Due to random motion of atoms

32. What is Coherence and types of Coherence?
Temporal Coherence
The interference pattern at P at time t is due to superposition of waves
emanating from S1 and S2 at (t-r1)/c and (t-r2)/c respectively. These two
waves have definite phase relationship if
(r2 – r1)/ c << τc
Therefore, central fringes have good contrast while contrast gets poorer on
moving towards higher order fringes.
Young’s double slit experiment

33. What is Coherence and types of Coherence?
2. Spatial Coherence (Lateral Coherence):
*It is coherence property of the field which is associated with the finite
dimension of the light source. Lateral coherence width =λ/θ
Young’s double slit experiment with finite size source