B.Tech 1st year Physics notes Unit :4 WAVE OPTICS

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OPTICS
INTERFERENCE
The phenomenon in which alternate bright and dark bands are observed due to the
superposition of light waves proceeding from two coherent sources of light is known as
„Interference‟.
Two monochromatic sources of light are said to be coherent if they emit radiation of the
same wavelength with the same amplitude and with a constant or zero phase difference.
Coherent sources can be obtained in two ways;
i) Division of a wave front (Ex. Young‟s Double slit Expt.)
ii) Division of amplitude (Ex. Newton,s Rings, Air-Wedge)
Air Wedge (Wedge-shaped Film):
Expt. to measure the diameter of a thin wire (or) thickness of a very thin object:
An Air Wedge is formed between two optically plane glass plates by inserting a thin wire
(or a very thin object) of uniform cross section at one end. The other ends of the two
plates are in direct contact at O. Monochromatic light of wavelength  from an extended
source is reflected vertically downwards on to the wedge by the inclined glass plate G. A
traveling microscope M is placed above G with its axis vertical. It is focused to observe
clearly the interference fringes in the light reflected from the air wedge (Fig. 1.a). These
interference bands are equidistant and are alternately bright and dark \with a constant
fringe width. By measuring the positions of different orders of dark bands using M, the
fringe width (β) is calculated. The distance of the object from the edge of contact (l) also
is measured. Knowing λ, the thickness (d) of the wire (or object) is calculated using the
l
formula, d .
2

M SOURCE S
G
S
B
E
Q



Q
A D
d
O   t
P
O P
Fig. 1. (a) C

Fig. 1. (b)

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THEORY FOR AIR- WEDGE
Consider a wedge-shaped film of refractive index  enclosed by the two optically plane
surfaces OP and OQ inclined at an angle θ to each other(Fig. 1. (b)). This can be formed
between two optically plane glass plates by inserting a thin wire (or a very thin object) of
uniform cross section at one end. The other ends of the two plates are in direct contact at
O. The thickness t of the film is zero at the edge of contact at O. It gradually increases as
one moves from O towards P. The film OPQ is illuminated by a parallel beam of
monochromatic light SA of wavelength . A part of it is reflected along AB and the
remaining is transmitted along AC in the wedge shaped film. At C, the ray suffers
reflection and travels along CD and DE. The rays AB and DE are coherent since they
originate from the same incident ray SA. Hence, interference occurs at A between the
reflected rays AB and DE showing different orders of alternate dark and bright straight
line fringes. They are parallel to each other. Successive fringes have equal thickness.
Order of interference fringe depends on the thickness t of the wedge shaped film at the
place of interference.
Expression for fringe width:
The geometrical path difference between the two interfering rays is AC + CD =
2t cos r where t is the thickness of airfilm, r is the angle of refraction. For normal
incidence, angle of refraction r = 0 and so cos r  cos 0  1 .
 geometrical path difference = 2t . Besides, there is an additional path difference of
/2 due to phase reversal during reflection at the denser medium at C. So the
Total path difference, TPD = geometrical path difference + additional path difference of

/2 due to phase reversal.  TPD = 2t  …………………….. (1)
2

The condition for mth order bright fringe is TPD = 2  tm  .= m 
2
 
(or) 2tm  m   (2m  1) ………….. (2)
2 2
where t = tm for mth order fringe and. m = 1,2.3 …
 
The condition for mth order dark fringe is TPD = 2  tm  .= (2m-1)
2 2
 
(or) 2tm  (2m  1)  (m  1) ……….. (3)
2 2
Suppose the (m+1)th dark fringe is formed where the thickness of the air film is tm+1.
Then, 2  tm 1   (m  1)  1  [replacing m by (m+1) in eqn (3)
= m ……………… (4)
Subtracting (3) from (4), 2 (tm  1  tm )   ………………….. (5)

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Let Xm+1 and Xm be the distances of the (m+1)th and mth dark fringes from O. d =
diameter of the wire; l = distance between the edge of contact O and the wire and  be
the angle of wedge. Then,
tm  1 tm d
   tan     (6) , since  is very small
xm  1 xm l
d d
Using eqn. (6) we have, tm  1   xm  1 ; and tm  xm
l l
d d d
 tm  1  tm    xm  1  - xm   xm  1  - xm 
l l l
d
But xm  1  xm    fringe width .   tm  1  tm   
l
Substituting the value of  tm  1  tm  in Eq. (5), tm tm+1
d Q
we get 2  
l
l d
  ................... (7)  P
2 d O
Xm Xm+1

, d, λ and l are constants. Therefore, fringe width β is constant. Similarly, if we consider
two consecutive bright fringes, the fringe width β will be the same as givenl in eqn (7).
l
Rearranging eqn (7) we have d  .................(8)
2
l
For air film  = 1 and hence, d  .................(9) .The thickness of the object (diameter
2
of the wire) can be calculated from eqn (9) knowing the value of  and measuring the
values of  and l. If the wedge shaped film between the plates has a Refractive index
(RI)  then eqn (8) is used.
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l d l 1
Consider   ................... (7) . From eqn (6) we find that   and hence 
2 d l d 
 
Substituting this in eqn (7) we get   . For air film,   since  = 1 for air.
2 2
Thus, the angle of the wedge is
 
 ( for a film of RI  ) and  for air film (  = 1 for air)
2 2
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Uses of Air-wedge Experiment
By Air-wedge experiment,
1) Wavelength of a monochromatic source can be determined

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2) Thickness of very thin objects of uniform cross section can be found.
3) Planeness of transparent glass plates can be tested.
4) Refractive index of transparent liquids can be determined.
Testing a surface for planeness:
Suppose, it is desired to test the planeness of surface OQ of a transparent glass
plate. Then, a wedge shaped air film is formed between an optically plane glass plate OP
and the surface under test (OQ). The fringes will be straight if the surface under test is
perfectly plane. If the surface OQ is not perfectly plane, the fringes will be irregular in
shape and are not straight lines . In practice, perfectly plane surfaces are produced by
polishing the surfaces and testing them from time to time, until the fringes are straight. In
testing for planeness, an extended source of light should be used.
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NEWTON‟S RINGS
EXPERIMENT: When a plano-convex lens of radius R is placed on a plane glass
plate, an air film is formed between them in the region surrounding the point of contact,
O. Suppose monochromatic light of wavelength  from a source S falls on a plane glass
plate G inclined at 45 to the horizontal. G reflects a part of the light normally on plano-
convex lens arrangement. A traveling microscope M is placed above G with its axis
vertical. It is focused to observe clearly the interference fringes in the light reflected
from the Newton.s rings arrangement. (Fig. 3.a). These interference bands are concentric
circular fringes which are alternately bright and dark with a dark spot at the center. They
are localized at the spherical surface of the lens.
By measuring the radii rn of different orders of circular fringes, Radius of
curvature and focal length of the lens can be determined accurately. Also, Refractive
index of transparent liquids can be determined.


M
G SOURCE

S




Fig. 3. (a)




O