Diffraction of Light Notes | EduRev

Oscillations, Waves & Optics

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Introduction

The spreading out of a wave when it passes through a narrow opening is usually referred to as diffraction, and the intensity distribution on the screen is known as the diffraction pattern.

The diffraction phenomena are usually divided into two categories: Fresnel diffraction

and Fraunhofer diffraction.

In the Fresnel class of diffraction the source of light and the screen are, in general, at a finite distance from the diffracting aperture In the Fraunhofer class of diffraction, the source and the screen are at infinite distances from the aperture; this is easily achieved by placing the source on the focal plane of a convex lens and placing the screen on the focal plane of another convex lens. The two lenses effectively moved the source and the screen to infinity because the first lens makes the light beam parallel and the second lens effectively makes the screen receive a parallel beam of light.

Diffraction of Light Notes | EduRev

Fraunhofer’s diffraction at a single slit

Let a parallel beam of monochromatic light of wavelength λ be incident normally upon a

narrow slit of width AB = b placed perpendicular to the plane of the paper. Let the diffracted light be focused by a convex lens L on a screen XY placed in the focal plane of the lens. The diffraction pattern obtained on the screen consists of a central bright band,

having alternative dark and weak bright bands of decreasing intensity on both sides.

Diffraction of Light Notes | EduRev

Explanation

In terms of wave theory, a plane wave front is incident normally on the slit AB .

According to the Huygen’s principle, each point on AB sends out secondary wavelets in

all directions. The rays proceeding in the same direction as the incident rays are focused

at O, while those diffracted through an angleθ are focused at P . Let us find out the

resultant intensity at P .

Let AK be the perpendicular to BK . As the optical path from the plane AK to P are

equal, the path difference between the wavelets from A to B in the direction

Path difference = BK = ABsinθ = bsinθ

The corresponding phase difference = Diffraction of Light Notes | EduRev

Let the width AB of the slit is divided into n equal parts. The amplitude of vibration at P . Due to the waves from each part will be the same, say equal to a . The phase difference between the waves from any two consecutive parts is Diffraction of Light Notes | EduRev= δ(say).

Hence the resultant amplitude at P is R 

= Diffraction of Light Notes | EduRev

Let Diffraction of Light Notes | EduRev= α

⇒ R = Diffraction of Light Notes | EduRev

∵ α/n is small

R = Diffraction of Light Notes | EduRev

As n→∞, a→0 but the product na remains finite

Thus Resultant Amplitude at P due is R = Diffraction of Light Notes | EduRev let na = A

Thus resultant intensity at P ; I = R2 = Diffraction of Light Notes | EduRev

The constant of proportionality being taken as unity for simplicity

Condition for Maximum and Minimum intensity Direction for Minimum Intensity

For minimum intensity I = 0 ⇒ Diffraction of Light Notes | EduRev ⇒ sinα = 0 but α is not equal to zero.
Thus α = ±mπ where, m has an integral value1,2,3 except zero

Diffraction of Light Notes | EduRev

⇒bsinθ = ±mλ, where m =1,2,3

Direction for Maximum Intensity

For direction of maximum intensity Diffraction of Light Notes | EduRev

Diffraction of Light Notes | EduRev=0 ⇒ Diffraction of Light Notes | EduRev= 0

Diffraction of Light Notes | EduRev = 0 

⇒ 2cosα − sinα = 0 ⇒ α = tanα

The equation is solved graphically by plotting the curve

y =α and y = tanα

the points of intersection gives (approximately)

α = Diffraction of Light Notes | EduRev

Substituting the value of α into the expression of I

The intensity of the central maxima I0 = Diffraction of Light Notes | EduRev = A2

The intensity of the first maxima I1 = Diffraction of Light Notes | EduRev

The intensity of the second maxima

I2 = Diffraction of Light Notes | EduRev

Thus the ratio of intensities of the successive maxima

Diffraction of Light Notes | EduRev

Diffraction of Light Notes | EduRev


Clearly most of the incident light is concentrated in the central maxima which occurs in the direction α = 0 or Diffraction of Light Notes | EduRev = 0 

or θ = 0 i.e. in the same direction as the incident light

Effect of Slit Width on Diffraction Pattern

If slit is made narrower

Since first minimum on either side of the central maximum occurs in the directionθ , given by equation bsinθ = ±λ

When the slit is narrowed (b is reduced), the angle θ increases which means the central maximum becomes wider. When the slit-width is as small as wavelength(b = λ ) , the first

minimum occurs atθ = 900 , which means central maxima fills the whole space i.e.

condition of uniform illumination.


Example 1: A monochromatic light with a wavelength of λ = 600nm passes through a single slit which has a width of 0.800mm.
(a) What is the distance between the slit and the screen be located if the first minimum in the diffraction pattern is at a distance 1.00mm from the center of the screen?
(b) Calculate the width of the central maximum.

(a) The general condition for destructive interference is

Diffraction of Light Notes | EduRev

For smallθ , we employ the approximation sinθ ≈ tanθ = y/D which yields Diffraction of Light Notes | EduRev

The first minimum corresponds tom =1. If 1 y =1.00mm, then

L = Diffraction of Light Notes | EduRev = 1.33m

(b) The width of the central maximum is W = 2y1 = 2 (1.00 x 10-3m) = 2.00mm


Example 2: A monochromatic light is incident on a single slit of width 0.800mm and a diffraction pattern is formed at a screen which is 0.800 m away from the slit. The second order bright fringe is at a distance 1.60mm from the centre of the central maximum. What is the wavelength of the incident light?

The general condition for destructive interference is

sinθ = Diffraction of Light Notes | EduRev where small-angle approximation has been made.

Thus the position of the mth order dark fringe measured from the central axis is

ym = Diffraction of Light Notes | EduRev

Let the second bright fringe be located halfway between the second and the third dark

fringe. That is y2b =  Diffraction of Light Notes | EduRev

The approximate wavelength of the incident light is then

λ = Diffraction of Light Notes | EduRev = 6.40 x 10-7m


Example 3: Light of wavelength 580 nm is incident on a slit having a width of 0.30 mm. The viewing screen is 2 m from the slit. Find the positions of the first dark fringes and the width of the central bright fringe. What if the slit width is increased by an order of magnitude of 3.0 mm ? What happens to the diffraction pattern?

To analyze the problem, note that the two dark fringes that flank the central bright fringe corresponds to

n = ±1.

Diffraction of Light Notes | EduRev

Diffraction of Light Notes | EduRev

since, y1 = D tanθdark

For small angle θdark ≅ tanθdark ≅ sinθdark ≅ y1 / D

∴ y1 = D sinθdark = Diffraction of Light Notes | EduRev

The width of the central bright fringe is W = 2 |y1| = 2 × 3.87×10−3 m = 7.74 mm

Note that this value is much greater than the width of the slit.

We expect that angles at which the dark bands appear will decrease as b increases

Thus the diffraction pattern narrows.

For b = 3.0 mm, the sine of the angle θdark for the n = ±1 dark fringes are

Diffraction of Light Notes | EduRev

Diffraction of Light Notes | EduRev

The width of the central bright fringe is

W = 2y1 = 2×3.87×10−4 m

⇒ W = 0.774 mm

Notice that this is smaller than the width of the slit


Fraunhofer’s Diffraction at a Double Slit

In Double Slit diffraction experiment, the pattern obtained on the screen is the diffraction

pattern due to a single slit on which a system of interference fringes is superposed.

By Huygen’s principle every point in the slits AB and CD sends out secondary wavelets

in all directions.

Diffraction of Light Notes | EduRev

From the theory of diffraction at a single slit, the resultant amplitude due to wavelets diffracted from each slit in a direction θ is R = Diffraction of Light Notes | EduRev, where α = Diffraction of Light Notes | EduRev

Consider the two slits as equivalent to two coherent sources placed at the middle points

S1 and S2 of the slits and each sending wavelets of amplitude Diffraction of Light Notes | EduRev in a directionθ .

Consequently, the resultant amplitude at a point P on the screen will be the result of interference between two waves of same amplitude Diffraction of Light Notes | EduRev, and having a phase difference δ (say) and path difference S2K = (b + e) sinθ .

Phase difference δ = Diffraction of Light Notes | EduRev

The resultant amplitude R

Diffraction of Light Notes | EduRev

Diffraction of Light Notes | EduRev

Let β = δ/2 ⇒ β = Diffraction of Light Notes | EduRev

Diffraction of Light Notes | EduRev

Therefore the resultant intensity at P is

Diffraction of Light Notes | EduRev

Thus the intensity in the resultant pattern depends on two factors

(i)Diffraction of Light Notes | EduRev, which gives diffraction pattern due to each individual slit and

(ii) cos2 β, which gives interference pattern due to diffracted light waves from the two

slits.

The diffraction term Diffraction of Light Notes | EduRevgives central maximum in the directionθ = 0 , having alternate minima and subsidiary maxima of decreasing intensity on either side the minima are obtained in the direction given by

sinα = 0

⇒ α = ±mπ

Diffraction of Light Notes | EduRev = ±mπ,  where m =1,2,3,.... (but not zero)

Thus the condition of diffraction minimum is

bsinθ = ±mλ       (m =1,2,3,...)

The interference term cos2 β gives a set of equidistant dark and a bright fringe as in Young’s double slit interference experiment. The bright fringe are obtained in the

direction

cos2 β =1  ⇒ β = ±nπ   ⇒Diffraction of Light Notes | EduRev

Thus the condition of maximum is

(b + e)sinθ = ±nλ, where n = 0,1,2,....

The entire pattern may be considered as consisting of interference fringes due to light from both slits, the intensities of these fringes being governed by the diffraction occurring

at the individual slits.

Diffraction of Light Notes | EduRev

Diffraction of Light Notes | EduRev

Diffraction of Light Notes | EduRev

Effect of Slit Width on Diffraction Pattern

Effect of increasing the slit width

If we increase the slit width b , the envelope of the fringe pattern changes so that it central peak is sharper. The fringe spacing, which depends on slit separation, does not change.

Hence less interference maxima now fall within the central diffraction maximum.

Effect of increasing the distance between slits

If b is kept constant and the separation d between them is increased, the fringes becomes closer together, the envelope of the pattern remaining unchanged. Thus more interference maxima fall within the central envelope.

Absent Orders

For certain values of d certain interference maxima become absent from the pattern.

Suppose for some value of θ the following conditions are simultaneously satisfied.

(b + e)sinθ = ±nλ Interference maxima

bsinθ = ±mλ Diffraction minima

Thus according to the first condition there should be an interference maximum in the directionθ , but according to the second condition there is no diffracted light in this direction. Therefore, the interference maximum will be absent in this direction.

Diffraction of Light Notes | EduRev

  • If e = b , then n = 2m = 2,4,6,.... Since m =1,2,3,....Thus 2nd, 4th and 6th order interference maxima will be absent, i.e. they will coincide with1st, 2nd, 3rd order diffraction minima. Thus the central diffraction maximum will have three interference (zero order and two first orders) maxima.
  • If e = 2b , then n = 3m = 3,6,9,.... Thus the 3rd, 6th, 9th order interference maxima will coincide with 1st, 2nd, 3rd order diffraction minima. Thus the central diffraction maximum will have five interference maxima.
  • If e = 3b , then n = 4m = 4,8,12,.... that is 4th, 8th, 12th ……. order interference maxima will coincide with 1st, 2nd and 3rd order diffraction minima. Thus the central diffraction maximum will have seven inference maxima as shown below.

Diffraction of Light Notes | EduRev

Diffraction Grating

A diffraction grating is an arrangement equivalent to a large number of parallel slits of equal widths and separated from one-another by equal opaque spaces. It is made by ruling a large number of fine, equidistant and parallel lines on an optically plane glass

plate with a diamond point. The rulings scatter the light and are effectively opaque while

the un-ruled parts transmit light and act as slits.

Diffraction of Light Notes | EduRev

Let AB be the section of a plane transmission grating, the lengths of the slits being perpendicular to the plane of the paper. Let b be the width of each slit and e the width of

each opaque space between the slits, then d = b + e is called the grating element.

Let a parallel beam of monochromatic light of wavelength λ be incident normally on the

grating. By the theory of Fraunhofer diffraction at a single slit, the wavelets from all points in a slit diffracted in a direction θ are equivalent to a single wave of amplitude Diffraction of Light Notes | EduRev, starting from the middle point of the slit where α = Diffraction of Light Notes | EduRev

Thus if N be total number of slits in the grating, the diffracted rays from all the slits are equivalent to N parallel rays, one each from the middle points S1 , S2 , S3 ... of the slits.

Path difference between the rays from the slits Sand S2 is

S2K = S1S2 sinθ = (b + e) sinθ

The corresponding phase difference = Diffraction of Light Notes | EduRev

Hence the resultant amplitude in the direction θ is R = Diffraction of Light Notes | EduRev

The resultant intensity is I = R2 = Diffraction of Light Notes | EduRev

The first term represents the diffraction pattern produced by a single slit whereas the second term represents the Interference pattern produced by N equally spaced slits. For

N =1 the above equation reduces to a single slit diffraction pattern and for N = 2 , to the

double-slit diffraction pattern.

Following figure shows the plot of Diffraction of Light Notes | EduRev as a function of β for N = 5 . As the value of N becomes very large, the above function becomes narrow and very sharply peaked at β = 0,π ,2π ....

Between the two peaks, the function vanishes when
Diffraction of Light Notes | EduRev m' 1, 2, 3,.....   but m′ ≠ 0,+N,+2N.....

which are referred to as interference minima.

Diffraction of Light Notes | EduRev


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