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 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 MINIMA
If the path difference is equal to odd multiple of , θ gives the
direction of minima due to interference
of the 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
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 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. 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. 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. and interference maxima at
MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN:
In double slit experiment , the diffraction minima occurs at
and interference maxima at

12. MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN:
In the double slit experiment, slit width is taken as ‘a’ and 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. 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 satisfied
simultaneously for the same value of θ, then the position of
interference maxima will correspond to the diffraction minima
at the same position on the screen

14. Thus orders 2, 4, 6 etc. of the interference maxima will be
Let a = b in
Then
and
P=0 central diffraction maxima. N=0 central intference maxima. N=2 is missing but n=1 and -1 are present. Total 3 , n=0,1,-1.
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. 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.

19. So resultant intensity of the two
superimposed waves will be
R’2 = R2 + R2 + 2RRcosΦ

20. If Φ=0 I e central maxima I = 4 times that of single slit diffraction.

21. The intensity pattern is in effect a combination of both the single-slit diffraction pattern and the double slit interference pattern.
The amplitude of the diffraction pattern modulates the interference pattern. 
 

22. PLANE DIFFRACTION GRATING

23. A diffraction grating 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. Such a grating is called Transmission grating.
The wave front from the slits interfere with each other similar to the phenomenon of diffraction through double slit.

24. The gratings are prepared by ruling equidistant lines on a glass surface. 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 , if the lines are drawn on a silvered surface then light is reflected from the position of the mirror in between any two lines and such surface acts as Reflection grating.
The spacing between the lines (slits width) is of the order of
wavelength of light. Grating used to study visible region of spectrum contains lines per cm.

25. THEORY OF TRANSMISSION GRATING:
Let XY be the grating surface and MN be the screen. AB is the slit and BC is an opaque portion. The width of slit is ‘a’ and opaque spacing is ‘b’. a+b will be the spacing between two nearest slits.

26. let a plane wave front is incident on the grating surface
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 will corresponds to the position of the central bright maximum.
Consider the secondary waves traveling in a direction deviated at an angle θ with the direction of incident light.

27. Suppose secondary waves come to focus at P1
Suppose 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.

28. PRINCIPAL MAXIMA: Interference
As AB = a and BC = b. Path difference between the secondary
waves starting from A and C = (a+b) sin θ
(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
(a+b) sinθn = nλ where n = 0,1,2,3…..
is the condition for principal maxima.

29. n=0 corresponds to central maxima.
n= 1,2,3, -----etc correspond to 1st order, 2nd order, 3rd order and so on--- principal maxima.
In general the condition for nth order maxima is
d sinθn = nλ (A)
Then what are the maximum number of orders::
let θ = 90o ie maximum value of sinθ 1
n λ ≤ d or n ≤ d \ λ

30. Between two principal maxima, there exist a number of
secondary maxima and minima due to diffraction.
Now d sinθn = nλ
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.

31. CONDITION FOR SECONDARY MININA:
If the path difference so introduced is ,(where N is the total number of lines on the grating surface) then the total path difference between first and the last strips will be:

32. 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
This is a condition for destructive interference
So 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:

33. The number of secondary maxima is (N-2).
Thus the condition for n’th secondary minimum after the nth principal maximum is

34. WIDTH OF PRINCIPLE 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.

35. Dividing (B) by (A) we get:

36. Expanding the equation, we get
For small value of dθ ;

37. 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 decrease 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(dθ is less for violet region.