1. FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT
While dealing with interference, we considered the interference
of the secondary waves originating from the two slits. Each slit
also produces a diffraction pattern of its own.
So, the resultant pattern at any pattern on the screen will
consists of:
1) Interference pattern due to two slits
2) Two diffraction patterns, one due to each slit

2. FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd.)
Consider two rectangular slits AB and CD parallel to each other
and perpendicular to the plane of paper. The width of each slit is
‘a’ and are separated from each other by opaque portion whose
width is ‘b’. Lens L is placed between slits and the screen,
focuses the interference-cum-diffraction pattern on the
screen.

3. FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd.)
Let a plane wave front be incident on the surface XY. All the
secondary waves travelling in a direction parallel to OP come
to focus at P. Therefore point P corresponds to the position of
the central bright maximum.
INTERFERENCE MAXIMA AND MINIMA:
Let us consider the secondary waves travelling in a direction
inclined at an angle θ with initial direction. From the triangle
CAN

4. INTERFERENCE MAXIMA AND MINIMA (contd.):
If the path difference is equal to odd multiple of , θ gives the
direction of minima due to interference pattern
of the secondary waves from the two slits.
Hence
Putting n = 1, 2, 3, ….. the values of θ1, θ2, θ3, θ4 etc.
corresponding to the directions of minima can be obtained.

5. INTERFERENCE MAXIMA AND MINIMA (contd.):
From equation (I)
On the other hand if the secondary waves travel in a
direction such that the path difference is even multiple
of , gives the direction of the maxima due to
interference of light waves emanating from the
two slits.

6. INTERFERENCE MAXIMA AND MINIMA (contd.):
Putting n = 1, 2, 3, ….. the values of etc.
corresponding to the directions of the maxima can
be obtained. From equation (II) we get
and

7. INTERFERENCE MAXIMA AND MINIMA (contd.):or
As θ is small
Thus, the angular separation between any two consecutive
minima (or maxima) is equal to
The angular separation is inversely proportional to (a+b), the distance
between two slits.

8. FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd.)
DIFFRACTION MAXIMA AND MINIMA :
Let us consider the secondary waves travelling in a direction
inclined at an angle φ with the initial direction of the incident
light. The path difference then will be given by
If the path difference BM is equal to λ the wavelength of light
used, then we can divide the slit into two equal halves each of
width a/2, corresponding to each point in the upper half of the
slit there will be a point in lower half of the slit, such that the
path difference between the waves originating from them and
diffracted at an angle φ will be λ/2.

9. FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd.)
DIFFRACTION MAXIMA AND MINIMA (CONTD.):
So the angle φ in this case will give the direction of
minimum. If φn be the angle of diffraction for nth diffraction
minimum, then
Putting n = 1, 2, 3, ….. the values of etc.
corresponding to the direction of diffraction minima can be
obtained.

10. FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd.)

11. FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd.)

12. FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd.)
MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN:
In the diffraction pattern due to a double slit, the slit width is
taken as ‘a’ and the separation between the slits as ‘b’. If the slit
width ‘a’ is kept constant, the diffraction pattern remains the
same.
Keeping ‘a’ constant, if the spacing ‘b’ is altered the spacing
between the interference maxima changes. Depending on the
relative values of ‘a’ and ‘b’ certain orders of interference
maxima will be absent in the resultant pattern.

13. MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN (CONTD.):
The directions of interference maxima are given by
The directions of diffraction minima are given by
In equations ‘(A) and (B) ‘n’ and ‘p’ are integers. If the values
of ‘a’ and ‘b’ are such that the equation are satisfies
simultaneously for the same value of θ, then the position of
interference maxima corresponds to the diffraction minima
at the same position on the screen

14. MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN (CONTD.):
Let a = b
Then
and
If p = 1, 2, 3, 4 etc. then n = 2, 4, 6 etc.
Thus orders 2, 4, 6 etc. of the interference maxima will be
missing in the diffraction pattern. There will be three
interference maxima in the central diffraction maxima.

15. MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN (CONTD.):
If 2a = b
Then
and
If p = 1, 2, 3, 4 etc. then n = 3, 6, 9 etc. Thus orders 3, 6, 9 etc.
of the interference maxima will be missing in the diffraction pattern.
On the both sides of central maximum, the number of interference
maxima is 2 and hence there will be five interference maxima
in the central diffraction maximum. The position of the third
interference maximum will corresponds to the first diffraction
minimum.

16. If MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN (CONTD.):
The two slits join and all the orders of interference maxima
will be missing. The diffraction pattern observed on the
screen is similar to that due to a single slit of width equal
to 2a.

17. DIFFERENCE BETWEEN SINGLE SLIT AND DOUBLE SLIT DIFFRACTION PATTERN:
The main difference between the two diffraction patterns is
that double slit pattern is the superposition of single slit
diffraction pattern and a double slit interference pattern.
The principal maximum of double slit pattern, envelops a
number of interference maxima and minima.
The intensity of the principal maximum of double slit diffraction
pattern is about four times as compared to that of single slit

18. DIFFERENCE BETWEEN SINGLE SLIT AND DOUBLE SLIT DIFFRACTION PATTERN (CONTD.):
The intensity of principal maximum of single slit diffraction pattern decreases gradually to zero but that of double slit diffraction pattern varies between certain maximum and minimum values, before attaining zero value. This happens because of superposition of interference maxima and minima on principle maximum.
DIFFRACTION GRATING

19. PLANE DIFFRACTION GRATING

20. PLANE DIFFRACTION GRATING
A diffraction grating is an extremely useful device and in one of
its forms it consists of a very large number of narrow slits side
by side. The slits are separated by opaque spaces.
When a wave front is incident on a grating surface, light is
transmitted through the slits and obstructed by the opaque
portion. Such a grating is called Transmission grating.
The secondary wave front from the position of the slits interfere
with each other similar to the interference of waves in Young’s
double slit experiment.

21. PLANE DIFFRACTION GRATING
Frauhoffer used the first grating which consisted of large number
of parallel fine wires stretched on a frame. Now a days, gratings
are prepared by ruling equidistant lines on a glass surface. The
lines are drawn with a fine diamond point. The space between
any two lines is transparent to light and the line portion is opaque
to light. Such surface acts as transmission grating.
On the other hand , lines are drawn on a silvered
surface (plane or concave) then light is reflected from the
position of the mirror in between any two lines and such
surface acts as Reflection grating.

22. PLANE DIFFRACTION GRATING
If the spacing between the lines is of the order of the
wavelength of light, then an appreciable deviation of light is
produced. Grating used to study visible region of spectrum
contains lines per cm.
Gratings, with originally ruled surface are only few. For practical
purpose replicas of the original grating are prepared. On the
original grating surface a thin layer of collodion solution is
poured and the solution is allowed to harden. Then the film of
collodion is removed from the grating surface and fixed
between two glass plates. This arrangement serves as
Transmission grating.

23. PLANE DIFFRACTION GRATING
THEORY OF PLANE TRANSMISSION GRATING:
Let XY be the grating surface and MN be the screen, both are
perpendicular to the plane of paper. Here AB is the slit and BC is
an opaque portion. The width of slit is ‘a’ and opaque spacing
between two slits is ‘b’.

24. THEORY OF PLANE TRANSMISSION GRATING(contd.):
let a plane wave front is incident on the grating surface. All the
secondary wave fronts traveling in the same direction as that of
incident light will come to focus at point P on the screen. The
point P where all the secondary waves reinforce one another
corresponds to the position of the central bright maximum.

25. THEORY OF PLANE TRANSMISSION GRATING(contd.):
Consider the secondary waves traveling in a direction inclined
at an angle θ with the direction of incident light. The collecting
lens also suitably rotated such that the axis of the lens is
parallel to the direction of secondary waves.
The secondary waves come to focus at P1 .The intensity at P1
will depend on path difference between the secondary waves
originating from the corresponding points A and C of two
neighboring slits.

26. THEORY OF PLANE TRANSMISSION GRATING(contd.):
PRINCIPAL MAXIMA:
As AB = a and BC = b. Path difference between the secondary
waves starting A and C is equal to AC Sin θ. But
AC = AB + BC = a + b
So path difference= δ = AC Sin θ = (a+b) Sin θ
If δ = nλ
Then
δ = (a+b) Sin θn= nλ (A)
where n = 1, 2, 3, 4, ………
(a+b) is called Grating element/constant.
For a grating with 15,000 lines per inch,
The value of grating constant is
(a+b) =2.54/15,000 cm
gives P1 as a point of maximum intensity.

27. THEORY OF PLANE TRANSMISSION GRATING(contd.):
PRINCIPAL MAXIMA:
For such points, where relation (A) is satisfied the
waves from all slits are exactly in phase. Hence they are
points of maximum possible intensity and are termed as
Principal maxima.
δ = (a+b) Sin θn= nλ (A)
Equation(A) gives the condition for nth order Principal maximum.
The position of Principal maxima is independent of the number
of slits. Since amplitude of disturbance from each slit is
independent of θ, so all principal maxima have almost same
intensity.
Between the two principal maxima, there exist a number of
secondary maxima and minima.

28. THEORY OF PLANE TRANSMISSION GRATING(contd.):
CONDITION FOR SECONDARY MININA:
If the angle of diffraction changes from θn to θn+dθ, there will
be corresponding change in the path difference between the
waves originating from A and C.
If the path difference so introduced be ,(where N is the
total number of lines on the grating surface) then the total
path difference between the rays originating from the
corresponding points of the first and the last strips will be:

29. THEORY OF PLANE TRANSMISSION GRATING(contd.):
CONDITION FOR SECONDARY MININA:
If the diffraction grating be divided into two halves, then the
corresponding points in the upper and lower halves will have
a path difference of
The superposition of the waves from the corresponding points
in the upper and lower halves will lead to destructive
interference and the direction θn+dθ will correspond to first
minimum after the nth Principal Maximum
Between nth and (n+1)th principal maxima, there will be (N-1)
secondary minima corresponding to a path difference:

30. THEORY OF PLANE TRANSMISSION GRATING(contd.):
CONDITION FOR SECONDARY MININA:
The number of secondary maxima is (N-2). Thus the condition
for n’th secondary minimum after the nth principal maximum is

31. THEORY OF PLANE TRANSMISSION GRATING(contd.):
WIDTH OF CENTRAL MAXIMA:
The direction of nth principal maximum is given by:
Let and give the directions of the first secondary
minima on the two sides of the nth order primary maxima, then
Where N is the total number of lines on the grating surface.

32. THEORY OF PLANE TRANSMISSION GRATING(contd.):
WIDTH OF CENTRAL MAXIMA:
Dividing (B) by (A) we get:

33. THEORY OF PLANE TRANSMISSION GRATING(contd.):
WIDTH OF CENTRAL MAXIMA:
Expanding the equation, we get
For small value of dθ ;

34. WIDTH OF CENTRAL MAXIMA (conclusions):
In equation (C), dθ refers to half the angular width of the
principal maximum. The half width dθ is
Inversely proportional to N, the total number of lines and
The value of is more for higher orders because
the increase in the value of is less than the
increase in the order n
half width of the principal maximum is less for higher orders.
the larger the number of lines on the grating surface
the smaller is the value of
is higher for longer wavelengths and hence spectral lines are sharper towards violet than the red end of the spectrum.

35. OBLIQUE INCIDENCE Let a parallel beam of light be
incident obliquely on the grating
surface at an angle of incidence i.
Then the path difference between
the secondary waves passing
through the points A and C
= FC + CE
From triangle AFC it is seen that
Above equation holds good if the beam is diffracted upwards.

36. OBLIQUE INCIDENCE(contd.)
If the beam is diffracted downwards then the path
difference becomes:
For nth primary maximum

37. OBLIQUE INCIDENCE(contd.)
The deviation of the diffraction beam =
For deviation to be minimum, must be minimum.
This is possible if the value of is maximum:
Thus, the deviation produced in the diffracted beam is a minimum when the angle of incidence is equal to the angle of diffraction.
Let Dm be the angle of minimum deviation. Then
NEXT
SLIDE

38. OBLIQUE INCIDENCE(contd.)
Condition of the principal maximum of the nth order for a wavelength λ.

39. ABSENT SPECTRA WITH A DIFFRACTION GRATING
Then
not possible
Hence the first order spectrum
will be absent.
Similarly second, third etc. orders will be absent if
In general if
nth order spectrum will be absent

40. ABSENT SPECTRA WITH A DIFFRACTION GRATING(contd..)
The condition for absent spectra can be obtained from following
considerations. For the nth order principal maximum
Further if the value of a and are such that:
Then the effect of the wave front from any slit will be zero.
Considering each slit to be made of two halves, the path
difference between the secondary waves from the
corresponding points will be
they cancel one
another’s effect

41. ABSENT SPECTRA WITH A DIFFRACTION GRATING(contd..)
If two conditions given by equations (A) and (B) are satisfied
simultaneously then
The values of n = 1, 2, 3, 4……. Refers to the orders of principal maxima that are absent in the diffraction pattern

42. HIGHEST POSSIBLE ORDER OF PRINCIPAL MAXUMUM
The maximum value for
Substituting this value in equation (A)
Here n is the highest order for principal maximum visible. If n is not an integer, then the highest possible order is given by the integer lower than n.

43. INTENSITY DISTRIBUTION OF DIFFRACTION GRATING