Interference
In physics, interference is the addition (superposition) of two or more
waves that result in a new wave pattern.
As most commonly used, the term interference usually refers to the
interaction of waves which are correlated or coherent with each other, either
because they come from the same source or because they have the same or
nearly the same frequency.
Two non-monochromatic waves are only fully coherent with each other if
they both have exactly the same range of wavelengths and the same phase
differences at each of the constituent wavelengths.
The total phase difference is derived from the sum of both the path
difference and the initial phase difference (if the waves are generated from 2
or more different sources). It can then be concluded whether the waves
reaching a point are in phase (constructive interference) or out of phase .
Coherent sources
Two sources of light are said to be coherent if the waves emitted
from them have the same frequency and are 'phase-linked'; that is,
they have a zero or constant phase difference.
Consider two waves that are in phase, with amplitudes A1 and A2.
Their troughs and peaks line up and the resultant wave will have
amplitude A = A1 + A2. This is known as constructive
interference.
If the two waves are π radians, or 180°, out of phase, then one
wave's crests will coincide with another wave's troughs and so will
tend to cancel out. The resultant amplitude is A = | A1 − A2 | . If A1
= A2, the resultant amplitude will be zero. This is known as
destructive interference.
Constructive and destructive interference
When two sinusoidal waves superimpose, the resulting waveform
depends on the frequency (or wavelength) amplitude and relative
phase of the two waves. If the two waves have the same amplitude
A and wavelength the resultant waveform will have amplitude between 0 and 2A depending on whether the two waves are in phase or out of phase.
INTERFERENCE BY DIVISION OF WAVEFRONT
The incident wave front is divided in to two parts either by
reflection or by refraction. These two parts of the same wave front
travel unequal distances and reunite at small angles to produce
interference bands.
Fresnel’s biprism, Fresnel’s double mirror and Lloyd’s single
mirror are used to obtain interference by division of wave front.
Fresnel’s biprism
Biprism is a combination of two prisms with their bases joined
and their faces making an angle of about 179 and the other angles
were each of 30’. Light from the monochromatic source S falls on the biprism
P whose refracting edges are vertical. Each half of the biprism
produces a virtual image (i.e S1 and S2) of the source S by
reflection. Let the distance between the two virtual sources be d. D
is the distance of the point C on the screen from the source S i.e.
D=SC. The biprism P refracts light from the source S into two
overlapping beams AF and BE. The closely spaced interference
fringes are produced only in the region AB.
Theory
The point C on the screen is equidistant from S1 and S2 hence
the two waves reaching C reinforce each other and the point C will
be the centre of a bright fringe, while the illumination at any other
point from the screen will depend upon the path difference of the
point from S1 and S2 . On both sides of C, alternately bright and
dark fringes are produced. For any point on the screen to be at the
centre of a bright fringe
( d/D ) x = nλ
(D/d ) λ = x/n = β , the fringe width .
The point will be at the centre of a dark fringe if its distance
from C is (2n+1) λD/ 2d , where n=0,1,2,3……..etc.
INTERFERENCE BY DIVISION OF AMPLITUDE
The amplitude of the in coming beam is divided into two, either by
reflection or refraction. These divided parts reunite after traversing
different paths and interfere constructively or destructively.
Newton’s rings and Michelson’s interferometer are the examples
of these class.
Interference by wedge shaped film
A thin film interference pattern will be a series of alternating bright
and dark fringes of the same color when it is created by a wedge
illuminated by monochromatic light. For example: (1) an air wedge
formed between two pieces of glass, or (2) a sliver of glass
surrounded by air or water.
As the wedge becomes thinner (θ → 0), the fringes are less
numerous as they spread further apart and get wider. Note that the
point of contact is a dark fringe. This results not from the thickness
of the wedge, but from the net phase inversion between the two
reflected rays equals ½ λ. If the plates are placed flat, directly on
top of one another, the plates would be entirely dark. If the plates
remained parallel, but were slowly raised apart, the pattern would
alternate between completely dark and completely bright as the
distance between the plates changed by ¼ wavelengths. [2t = ½ λ].
Newton’s rings
When a plano-convex lens with its convex surface is placed on a
plane glass sheet, an air film of gradually increasing thickness
outward is formed between the lens and the sheet. The thickness of
film at the point of contact is zero. If monochromatic light is
allowed to fall normally on the lens, and the film is viewed in
reflected light, alternate bright and dark concentric rings are seen
around the point of contact. These rings were first discovered by
Newton, that's why they are called Newton’s rings.
Newton's rings are formed due to interference between the light
waves reflected from the top and bottom surfaces of the air film
formed between the lens and glass sheet
Explanation
The phenomenon of the formation of the Newton's rings can be
explained on the basis of wave theory of light. An air film of
varying thickness is formed between the lens and the glass sheet.
When a light ray is incident on the upper surface of the lens, it is
reflected as well as refracted. When the refracted ray strikes the
glass sheet, it undergoes a phase change of 180on reflection.
Interference occurs between the two waves which interfere
constructively if path difference between them is (m+1/2) λ and
destructively if path difference between them is mλ producing
alternate bright and dark rings.
Radius of the rings
Let the radius of curvature of the convex lens is R and the radius of ring is 'r'. Consider light of wave length 'λ' falls on the lens. After
refraction and reflection two rays 1 and 2 are obtained. These rays
interfere each other producing alternate bright and dark rings. At
the point of contact the thickness of air film is zero and the path
difference is also zero and as a 180
O
path difference occurs, so they
cancel each other and a dark ring is obtained at the centre.
Let us suppose that the thickness of air film is 't'.
By using the theorem of geometry,
In thin films, path difference for constructive interference is:
2nt = (m+1/2)λ
Where n= refractive index
For air n = 1 Therefore, 2t = (m+1/2)λ …………….. (2)
For first bright ring m = 0
for second bright ring m = 1
for third bright ring m = 2
Similarly
for N
th
bright ring m = N-1
Putting the value of m in equation (2)
2t = (N-1+1/2)λ
2t = (N-1/2)λ
t =1/2 (N-1/2)λ ………………. (3)
Putting the value of 't' in equation (1)
r
2
= 2Rt
r
2
= 2R . 1/2 (N-1/2)λ
r
2
= R (N-1/2)λ
This is the expression for the radius of N
th bright ring where
rn = radius of N
th bright ring
N = Ring number
R = radius of curvature of lens
λ = Wave length of light
Similarly for dark ring the radius can also be obtained.
EXPERIMENTAL ARRANGEMENT OF NEWTON’S RING APPARATUS
Michelson interferometer
The Michelson interferometer is the most common configuration
for optical interferometry and was invented by Albert Abraham
Michelson. An interference pattern is produced by splitting a beam
of light into two paths, bouncing the beams back and recombining
them. The different paths may be of different lengths or be
composed of different materials to create alternating interference
fringes on a back detector. Fig; Michelson interferometer
There are two paths from the (light) source to the detector. One
reflects off the semi transparent mirror, goes to the top mirror and
then reflects back, goes through the semi-transparent mirror, to the
detector. The other first goes through the semi-transparent mirror,
to the mirror on the right, reflects back to the semi-transparent
mirror, then reflects from the semi-transparent mirror into the
detector.
If these two paths differ by a whole number (including 0) of
wavelengths, there is constructive interference and a strong signal
at the detector. If they differ by a whole number and half
wavelengths (e.g., 0.5, 1.5, 2.5 ...) there is destructive interference
and a weak signal.
Applications of Michelson Interferometer
1. Determination of wavelength of monochromatic
light
The interferometer is adjusted to obtain the fringes of the monochromatic light. The position of the movable mirror is
adjusted till bright fringes appear and the crosswire of the
telescope is adjusted on a particular bright fringe. When the
movable mirror is moved slowly, each fringe gets displaced
parallel to it self in the field of view and the number of fringes
which cross the centre of the field of view gives the measure of the
distance the mirror has moved in terms of wavelength. When the
mirror moves through a distance λ/2 , one fringe shifts and there is
a corresponding change of λ in the path difference.
If m fringes shift across the cross-wire, when the mirror is
moved through a known distance d, then the path difference is 2d.
2d = mλ
or λ = 2d/m
Knowing d and m , the wave length λ can be determined.
2. Resolution of spectral lines.
Let a source S gives out a light consisting of two wavelengths
λ1 and λ2 (λ1> λ2) which differ slightly. Each wavelength
gives rise to its own system of fringes. Due to their
superposition the fringes become distinct and give rise to
positions of maximum and minimum intensity in the field of
view. At the position of maximum a bright fringe of one
system falls over a bright fringe of the other system and at
the position of minimum a bright fringe of one system falls
over the dark fringe of the other system. The position of the
movable mirror is recorded when the centre of maximum is
on the cross-wires. On moving the mirror the visibility
decreases, become almost zero and again improves till the
centre of next maximum is brought on the cross-wires. The
position of the mirror is recorded again. If ‘d’ be the distance through which the mirror is moved for two successive
positions of the maximum distinctness (or indistinctness), the
corresponding path difference is 2d. Suppose m is the
number of fringes of wavelength λ1, lying in this path, then
(m+1) would be the number of fringes of wavelength λ2
covering the same path. Thus
2d = m λ1
also 2d = (m+1) λ2
which gives m= 2d / λ1 and m+1 = 2d/ λ2
therefore (2d/ λ1) - (2d/ λ2) =1
or 2d {(1/ λ1) – (1/ λ2)} =1
or λ1 - λ2 = λ1 λ2 /2d
or δ λ = λ1 λ2 /2d = λ
2
/2d
where λ is the mean of λ1 of λ2 i.e. λ = (λ1+ λ2 )/2
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