Interference due to transmitted light in
thin films
P
P
Ss
rr
ii
A
AIR AIR
C
C
C’
A
tt
r
r

B
B
r r
r r
i
r
r
i
i

M
M
DD
AIR AIR
i NN
Q
R
Q
Here at B reflection takes place at the rarer
Medium .so no phase change change occurs.
BM normal to CD and DN normal to BR.
So optical path difference between DQ and BR
is
x = (BC+CD) -BN
n1sini = n2sinr
Sinr = sini
=sini/sinr
In BND,
In BMD,
sini=BN/BD
sinr=MD/BD
=sini/sinr =BN/MD
BN = MD
Since <CPB=<CBP=r
and
BC=CP=CD
x = (BC+CD) –BN
= (DP)- (MD)
= (DP-MD)
= (MP)
In BPM cosr = MP/BP
So MP = BP cosr
= 2t cos r since BP=2t
SO,
x =  PM = 2 t cos r
Conditions for constructive interference will be
2 t cos r = n
And conditions for destructive interference will
(2n  1)
2 t cos r 
2
One of the important applications of the thin
film interference is reducing the reflectivity of
lens surface.
ar n1 n > n
1
2
ai
at
n2
n1  n 2
ar 
a i  a ir
n1  n 2
2 n1
at 
ai  ait
n1  n 2
ai, ar, at are amplitudes of incident,reflected and
transmitted waves.
n2>n1 , ar is negative showing that reflection
occurs at a denser medium a phase change  comes
n1  n 2
2n1
r
,t 
n1  n 2
n1  n 2
r and t are reflection and transmission
coefficients
a
r’a
t’a
n2
n1
n 2  n1
2n 2
r' 
 r, t ' 
n 2  n1
n1  n 2
4n1n 2
1  tt'  1 
2
( n1  n 2 )
2
 n1  n 2 
2
  r
 
 n1  n 2 
These are the Stokes’ relations.
Non reflecting films
• Reflectivity is the fraction of incident light reflected
by a surface for normal incidence.
• Reflectivity depends upon the refractive index  of
the material. It is given by
  1


  1
2
• For glass  = 1.5.
• Reflectivity = 0.04
• 4% of incident light is reflected for normal
incidence. Remaining 96% is transmitted.
• The loss of energy due to reflection is one major reason of
clarity reduction. There is also a reduction in the intensity of
the images since less light is transmitted through the lenses.
• When films are coated on lens of prism surface the
reflectivity of these surfaces is appreciably reduced.
• Initially the coating were made by depositing several
monomolecular layers of an organic substance on glass
plates.
• Now it is done by either evaporating calcium or magnesium
fluoride on the surface in vacuum or by chemical treatment
of the surface with acids which leave a thin layer of silica
on the surface.
• No light is destroyed by non reflecting film, but there is
redistribution means decrease in reflection results increase
in transmission.
Thickness of nonreflecting thin film
1
a
r
2
air(a)
t
film(f)
glass(g)
g f> a
Two interfering beams will interfere constuctively if
2ftcosr = n
Rays will interfere destructively if
2ftcosr = (2n+1)/2
For normal incidence <r=900
2ft = (2n+1)/2
So 2ft = /2
for min
thickness, n = 0

t 
4 f
If a film having thickness of /4f and having
refractive index less than that of the glass is
coated on glass, then waves reflected from the
upper surface of the film destructively
interfere with the waves reflected from the
lower surface of the film. Such a film known as
a non reflecting film.
ar
r, t
1
a
2
ar’tt’
na
nf
r’, t’
at
ar’t
ng
Amplitude of ray 1 = ar
Amplitude of ray 2 = ar’tt’
na  nf
2n a
r
,t 
na  nf
na  nf
nf  ng
2n f
r' 
, t ' 
nf  ng
na  nf
na  nf
2n a
ar 
a, at 
a
na  nf
na  nf
nf  ng
2n a
2n f
ar' tt' 


a
nf  ng na  nf na  nf
nf  ng
2n a n f


a
2
(n a  n f ) n f  n g
For complete destructive interference ray 1 and 2 must
Have the same amplitude, i.e.
nf  ng
na  nf
4n f n a
a

a
2
na  nf
na  nf  nf  ng
4n f n a
is verynearlyqeual to unity
2
n a  n f 
nf  ng
na  nf
a
a
na  nf
nf  ng
 nanf  nang  n  nf ng
2
f
 nanf  nang  n  nf ng
2
f
 n  na ng
2
f
 nf 
na ng
This equation gives the estimate of refractive index of
This film which should be coated on a surface to reduce
Its reflectivity. If na= 1 (for air) and ng = refractive index
of glass then
n f  refractiveindex of thinfilm
 ng
 nf  ng