N slit Diffraction.pdf

Grating diffractions

1. Lecture –7
N-Slits diffraction –Diffraction Grating
Dr.Shailesh J.

2. An arrangement consisting of large number of parallel slits of the
same width and separated by equal opaque spaces is known as
Diffraction grating.
❑Ruling equidistant parallel lines on a transparent material such as glass,
with a fine diamond point.
❑The ruled lines are opaque to light while the space between any two lines is
transparent to light and acts as a slit.
❑This is known as plane transmission grating.

3. Consider, plane transmission grating AB(having N slits) placed perpendicular to
the plane of the paper
▪Let‘e’be the width of each slit and ‘d’the width of each opaque space. Then(e+d)is known as grating
element andXYis the screen.
▪a parallel beam of monochromatic light of wavelength ‘λ’ incident on grating.
▪The pointP
owill be a central maximum.
▪Secondary waves travelling in a direction inclined at an angle ‘θ’ with the incident light will reach pointP
1in
different phases.
▪Intensity at P1 will can be considered by applying ‘Fraunhofer diffraction at a single slit, amplitude is given
by where
▪For N slits, path difference (e+d) sinθand Phase difference

4. Hence the intensity in a direction can be found by finding the resultant amplitude of N
vibrations each of amplitude and a phase difference of
R=
??????sin????????????/2
sin??????/2
(For single slit diffraction)
For present case of N slits, d =
then,
Distribution of Intensity due to a single slit
Distributionofintensityasa
combinedeffectofalltheslits

5. Intensity Distribution:
n = 0corresponds to zero order maximum.
Forn = 1,2,3,…we obtain first, second, third,… principal maxima respectively.
Case (i): Principal maxima:
Case(ii): Minima Positions:
But,
then
then
Wheremhas all integral values exceptm = 0, N, 2N, …, nN,because for these values
And we get principal maxima. Thus,m = 1, 2, 3, …, (N-1).Hence
gives the minima positions which are adjacent to the principal maxima.

6. Case(iii): Secondary maxima:As there are(N-1)minima between two adjacent principal maxima there must
be(N-2)other maxima between two principal maxima.
Therootsoftheaboveequationotherthanthoseforwhich givethepositionsofsecondarymaxima
Only

7. Since intensity of principal maxima is proportional toN
2
Hence if the value ofNis larger, then the secondary maxima will be weaker and
becomes negligible whenNbecomes infinity

9. Formation of multiple spectra with grating
The diffraction of the nth order principle maxima for a wavelength ‘λ’
is given by the equation
(e+d) sinθ= ±nλ
•For particular λ, the angle of diffraction θis different for
principle maxima of different orders.
•For larger the λ, the greater the angle of diffraction θ
•For θ= 0 which gives maxima of all wavelength
•For n= 1 will form first order spectrum
•For n= 2 will form second order spectrum and so on
Maximum number of order possible with grating
(e+d) sinθ= ±nλ
n
max= (e+d) / λ
For example, if (e+d) < 2λ Therefore, n
max<
2λ
λ
< 2
Hence, Only first order is possible.

10. Effect of increase in the width of ruled surface
The diffraction of the nth order principle maxima for a wavelength ‘λ’
is given by the equation (e+d) sinθ
n= nλor N(e+d) sinθ
n= Nnλ
Consider the first minimum adjacent to the ‘n
th
’ maximum in increasing direction
of θ. Let its direction be (θ
n+ dθ
n) where dθ
n is the angular half width of the n
th
maximum. Now direction of minima is given by
N(e+d) sinθ= mλwhere m = N n+1
N(e+d) sin(θ
n+ dθ
n)= (Nn+1)λ (dθ
n is very small)
dθ
n =
λ
??????(e+d)cosθ
n
Larger the width of ruled surface (e+d) the smaller is the angular half width and
sharper are the maxima.

11. Dispersive power of grating
The diffraction of the nth order principle maxima for a wavelength ‘λ’
is given by the equation
(e+d) sinθ= nλ
&#3627408517;??????
&#3627408517;??????
=
??????
&#3627408518;+&#3627408517;????????????????????????
The dispersive power of grating is defined as the ratioof variation of
angle of diffraction with wavelength. i.e.
????????????
????????????

12. Facts and Applications:
➢To measure the width of slit.
➢To determine the wavelength of monochromatic source.
➢To measure the Young's modulus of the material of a wire.
➢Wave nature of light.
➢Grating is used in spectrometers, monochromators, spectrographs, and in
many other scientific optical instruments.
➢Analysis of sample in research
1. CD reflecting rainbow colors
2. Holograms
3. Sun appears red during sunset
4. From the shadow of an object
5. Bending of light at the corners of the door
6. Spectrometer
7. X-ray diffraction
8. To separate white light

13. Class Work
Q.1. Rising and setting sun appears to be reddish because
A)Diffraction sends red ray to the earth at these times
B)Scattering due to dust particles and air molecules are responsible for this effect
C)Refraction is responsible for this effect
D)Polarization is responsible for this effect
Q.2. Which of the following phenomena produces the colorof the soap bubble?
A)Diffraction
B)Interference
C)Dispersion
D)Polarisation
Q.3. The first diffraction minima due to a single slit diffraction is at angle 30
o
for alight of wavelength
500 nm. The width of the slit is:
A)5 x 10
-5
cm B) 10 x 10
-5
cm C) 2.5 x 10
-5
cm D) 1.25 x 10
-5
cm
Q. 4. The third order diffracted image, corresponding to the wavelength 440 nm, coincides with the
second order corresponding to an unknown wavelength in the spectrum produced by grating. Find the
unknown wavelength.