When two light waves pass through the same point in space simultaneously, interference occurs. Constructive interference happens when the waves are in phase and add to produce a larger wave, while destructive interference occurs when they are out of phase and cancel each other out. The intensity of the resulting interference pattern depends on the phase difference between the waves. In a double slit experiment, the phase difference and resulting interference is determined by the path length difference between waves passing through each slit.

1. Interference
When two light waves pass through a certain position in
space at the same time, superposition happens.
The principle of superposition states:
When two or more waves move simultaneously through
a region of space, each wave proceeds independently
as if the other were not present.
The resulting wave "displacement" at any point and time
is the vector sum of the "displacements" of the individual
waves.
In other word, the resultant displacement is the sum of the
separate displacements of the constituent waves: y = y1 +
y2
1
Superposition principle

2. Interference
Two traveling waves, y1 and y2, whose phases differ only by
the constant φ, have amplitude A, angular wave number k
and angular frequency ω.
They move in the same direction and have the same
frequency and velocity.
To derive an equation for the combined wave, we use the
equations for the two waves, the principle of superposition,
and a trigonometric identity.
2

3. Interference
To simplify the analysis, the first wave is assumed to have a
phase constant of zero, so the phase difference φ is the phase
constant of the second wave.
The principle of superposition says we can add these
equations to determine the equation for the combined wave,
yc. We use the trigonometric identity
to derive the equation for the combined wave shown in Eq. 1.





 





 
=
2
sin
2
cos2sinsin
baba
ba
3

4. Interference
4
Eq. 1
y = transverse displacement
A = amplitude
k = wave number
ω = angular frequency
φ = phase constant

5. Interference
5
The amplitude is the constant factor 2A cos(φ/2).
The sine function describes a wave with the same angular
wave number k and angular frequency ω as the two waves
that combined to create it.
If there is no phase difference between the initial two waves,
φ/2 = 0 and cos(φ/2) = 1, its maximum value. In this case, the
combined wave has double the amplitude of the initial waves.
In other words, there is fully constructive interference.
In addition, any integer multiple of 2π radians (360°) as the
phase difference of the initial waves will cause perfect
constructive interference, because the argument of the
cosine will be a multiple of π rad (180°), giving the cosine a
value of ±1.

6. Interference
6
Conversely, if the phase difference φ is π rad, then φ/2 = π/2
and cos(φ/2) = 0. The function describing the combined
“wave” is constantly zero, and there is complete destructive
interference.
This is to be expected: Two waves that have a phase
difference of π rad cancel out. Any odd integer multiple of π
rad means the cosine is zero and the result is fully
destructive interference.
We have been focusing on the two extremes, fully
constructive and fully destructive interference.
Intermediate interference describes the situation for any
phase difference in between.
The resulting traveling wave is sinusoidal, with the same
wavelength and frequency as the initial waves, but with an
amplitude between zero and double the amplitude of the
contributing waves.

7. Interference
Interference is the interaction of
two or more waves passing the
same point.
Constructive interference
occurs when the waves add in
phase, producing a larger peak
than any wave alone, whereas
destructive interference occurs
when waves add out of phase,
producing smaller peaks than
one of the waves alone
When enhancement
(constructive interference) and
diminution (destructive
interference) conditions alternate
in spatial display, the
interference is said to produce
pattern of fringes (interference
pattern).
7
 The two sources do not need to
be in phase with each other.
 If there is some constant initial
phase difference between the
two sources, the resulting
interference pattern will be
identical to the original pattern,
although it will be shifted in
terms of the location of the
minima and maxima.

8. Condition for Interference
8
If two beams are to interfere to produce a stable pattern,
they must have very nearly the same frequency
A significant frequency difference would result in a rapidly
varying, time dependent phase difference, which in turn
would cause the interference intensity to average to zero
Clearest patterns exist when the interfering waves have
equal or nearly equal amplitudes.

9. 9
Young’s double slit interference
Interference
 Young’s observation when two slits were uniformly
illuminated.
 Consistent with interference between wavefronts
emerging from each slit.

10. 10
 Interference arises due to the path difference between light
emerging from each slit varying as d sin.
Young’s double slit interference

11. 11
 Overall intensity envelope is due to diffraction from each single slit.
  ,...,,m,msind
,...,,m,msind
210
210
2
1 ==
==
ceinterfereneDestructiv
ceinterferenveConstructi
Young’s double slit interference

12. 12
Line spacing for double-slit interference
 A screen is 1.2m from two slits
0.100mm apart. light of
wavelength 500nm is incident
upon the slit. What is the
separation between bright fringes
on the screen?
L
 Given: d= 0.100mm = 1x10-4 m, =500x10-9 m, L= 1.20m
mm
d
mLx
mmmLx
d
m
mmd
00.12
2
20.1
00.61000.520.1
1000.5
1000.1
10500
sin
1,sin
22
3
11
1
3
4
9
1
1
=

=
==
=


=

=
==





13. 13
Wavelengths from double-slit interference
 White light passes though two slits 0.5mm apart allows the
measurement of wavelengths on a screen 2.5m away. The
estimate of the violet and red wavelengths may be obtained.
nm
nm
L
x
m
d
m
d
mmd
violet
red
400
5.2
100.2
1
100.5
700
5.2
105.3
1
100.5
sin,1,sin
34
34
111
=

=
=

=
=

=
==


m
mm
m
mm

14. 14
Coherence
 Interference is only possible if there is a fixed phase relationship
between the radiation emerging from each of the two slits.
 As the light source for Young’s fringes was a
distant source, the sun, and as both slits
“sampled” the wavefront at the same time,
there was interference between the
emerging wavefronts and these slits are
then said to be coherent sources.
 Most light sources are incoherent. An
incandescent light bulb will emit light along
the length of the filament. Light emitted at
each end of the filament bears no phase
relationship to the light emitted at the other.
 Most lasers are sources of very coherent light. They may have a phase
relationship which extends both across the beam and along the beam.
 This high degree of coherence is required to improve the contrast of
interference fringes and in holography.

15. 15
Intensity in the double-slit interference pattern
 The electric field is given by the sum of the fields from each slit.
These vary with angle in terms of a phase difference .
 The intensity in the double slit interference pattern has periodic
maxima corresponding to d sin =m or, in terms of the phase
difference :
 The intensity in between these maxima may be determined as a
function of angle .
 
)cos(cos 00 21
21


=
=
tEtE
EEE



= sin
2
d

16. Intensity in the double-slit interference
pattern
16
 A phasor diagram illustrates the phase summation of the two
equal fields.
 
 2
sin
2
cos2
2
sin)()(
cos2
2
cos2)(
)
2
cos2(2)(
)cos1(2)(
cos22)(
cos2)(
2
,
0
0
000
22
0
2
0
2
0
2
0
2
0
2
0
2
0
21
2
2
22
0
021
00001
00











=
=
==
=
=
=
=
===
tE
tEE
EEE
EE
EE
EEE
EEEEE
EEE

17. 17
 Observe intensity rather than E field
 Assuming I1 = I2 = Io . The cosinus term is the characteristics
of the interference pattern while sin2/2 is the term for the
diffraction.
where
Intensity in the double-slit interference pattern
)
sin
2)(cos1(cos2)( 2
2
2
2121


 oAIIIII ==
4Io
2
cos4cos2)( 2
2121

 oIIIIII ==






=
2
cos2cos1 2 


18. Multiple beam interference
18
The difference in optical path length between adjacent rays
interference between multiple reflection beams.

19. Interference in thin
films
Suppose that a very thin film of air is trapped between two
pieces of glass, as shown in the diagram below.
If monochromatic light is incident almost normally to the film
then some of the light is reflected from the interface between
the bottom of the upper plate and the air, and some is reflected
from the interface between the air and the top of the upper
plate.
The eye focuses these two parallel light beams at one spot on
the retina. The two beams produce either destructive or
constructive interference, depending on whether their path
difference is equal to an odd or an even number of half-wave-
lengths, respectively.
19

20. Interference in thin films
Let t be the thickness of the air
film.
The difference in path-lengths
between the two light rays shown
in the diagram is clearly D = 2 t.
Naively, we expect that
constructive interference
(brightness)
D=m
destructive interference
(darkness)
D = (m+1/2).
where m is an integer
20

21. Interference in thin films
21
Phase difference is introduced between the two rays on reflection.
The first ray is reflected at an interface between an optically
dense medium (glass), through which the ray travels, and a less
dense medium (air). There is a 180o phase change on reflection
from such an interface.
The second ray is reflected at an interface between an optically
less dense medium (air), through which the ray travels, and a
dense medium (glass). There is no phase change on reflection
from such an interface.
Thus, an additional 180o phase change is introduced between the
two rays, which is equivalent to an additional path difference of
/2.
When this additional phase change is taken into account, the
condition for constructive interference becomes
2 t = (m + 1/2)  ,
Similarly, the condition for destructive interference becomes
2 t = m

22. Interference in thin films
22
If the thin film consists of water, oil, or some other transparent
material of refractive index n then the results are basically the
same as those for an air film, except that the wavelength of the
light in the film is reduced from  (the vacuum wavelength) to
/n.
It follows that the modified criteria for constructive and
destructive interference are
2 n t = (m + 1/2) 
and
2 n t = m
respectively.
For white light, the above criteria yield constructive interference
for some wavelengths, and destructive interference for others.
Thus, the light reflected back from the film exhibits those colours
for which the constructive interference occurs.

23. Interference in thin films
23
If the angle of transmittance t is taken into account, the path
difference is expressed by
D=2ntcos t
Then the condition for
constructive interference: D +Dr = m
and
destructive interference: D +Dr = (m + 1/2) 
where m=integer and Dr is the path difference arising from path
change on reflection.
Fringes formed by a dielectric film with an extended source are
referred to as Haidinger fringes, or fringes of equal inclination
(circular fringes). The fringes are formed by parallel beams from
an extended source.

24. 24
Michelson interferometer
 Extremely precise measurements of distance
are possible using an optical interferometer.
 Varying the position of M1 by as little as
100nm would alter the fringes of 400nm light
from dark to bright.
 By sweeping M1 over a large known distance
from the beam splitter, it is possible to
perform a Fourier analysis on the allowed
frequencies and obtain very high resolution
spectra.
 Constructive interference: 2tcos =(m+ ½)


25. 25
Anti-reflection coating
 The deposition of a “quarter-wave” thickness of a
mechanically hard material possessing an appropriate
refractive index allows the surface reflection from glasses,
camera lens and optical surfaces to be minimised.
 Usually MgF2 is used as it’s refractive index is almost halfway
between air and glass.
 Destructive interference occurs at 2nt =m 

26. 26
Wedges and optical testing
 Fringes may easily be generated by
placing a thin wire or sheet of paper
along one edge of a piece of glass resting
on an optically flat surface and
illuminating with monochromatic light.
 Bright bands occur when 2nt =(m+ ½) .
 Extremely useful in determining the
flatness of optical surfaces.

27. 27
Interference in thin films
 An interference effect is seen in the reflection of light from water
with a layer of oil on the surface. Rainbow coloured fringes are
visible.
 Light passing from material of lower to higher
refractive index undergoes a 180° phase change
upon reflection. (Higher to lower has no
associated change.)
 The additional optical path
from A to B to C allows
constructive interference
when equal to a integer
multiple of /noil. (It is
assumed that nair > noil >
nwater.)
 The film thickness for
normal incidence
corresponding to a bright
fringe is (/noil)/2.