FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT While dealing with interference, we considered the...

FRAUNHOFFER DIFFRACTION AT DOUBLE SLITWhile 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

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 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.


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.


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


or

As θ is small

Thus, the angular separation between any two consecutive

minima (or maxima) is equal toThe angular separation is inversely proportional to (a+b), the distance between two slits.

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.

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.



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.

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


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.


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

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.

DIFFRACTION GRATING

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.


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.


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 10000 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.


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’.

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.

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.


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

(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

If δ = nλThenδ = (a+b) Sin θn= nλ (A)where n = 1, 2, 3, 4, ………

gives P1 as a point of maximum intensity.


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.

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.

δ = (a+b) Sin θn= nλ (A)

PRINCIPAL MAXIMA:


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:



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:



The number of secondary maxima is (N-2). Thus the condition

for n’th secondary minimum after the nth principal maximum is


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.


WIDTH OF CENTRAL MAXIMA:

Dividing (B) by (A) we get:

THEORY OF PLANE TRANSMISSION GRATING(contd.):WIDTH OF CENTRAL MAXIMA:

Expanding the equation, we get

For small value of dθ ;

WIDTH OF CENTRAL MAXIMA (conclusions):

In equation (C), dθ refers to half the angular width of the

principal maximum. The half width dθ isInversely 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.

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.

OBLIQUE INCIDENCE(contd.)If the beam is diffracted downwards then the

path difference becomes:

For nth primary maximum

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

OBLIQUE INCIDENCE(contd.)

Condition of the principal maximum of the nth

order for a wavelength λ.

ABSENT SPECTRA WITH A DIFFRACTION GRATING

If Then not possible

Hence the first order spectrumwill be absent.

Similarly second, third etc. orders will be absent if

In general if

nth order spectrum will be absent

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

If two conditions given by equations (A) and (B) are satisfied

simultaneously then

ABSENT SPECTRA WITH A DIFFRACTION GRATING(contd..)

The values of n = 1, 2, 3, 4……. Refers to the orders of principal maxima that are absent in the diffraction pattern

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.

INTENSITY DISTRIBUTION OF DIFFRACTION GRATING

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT While dealing with interference, we considered the...

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