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

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.

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

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.

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.

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

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.

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.

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.

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd.)

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

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.

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

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.

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.

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.

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

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.

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

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

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.   

PLANE DIFFRACTION GRATING

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.

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 10000 lines per cm.

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.

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.

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.

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.

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

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.

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:

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:

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

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.

Dividing (B) by (A) we get:

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

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.