unit - iii diffraction of light

8/9/2019 Unit - III Diffraction of Light

1/10

Mr. A.CHARIS ISRAEL. M.Sc., B.Ed., (Ph.D.)Sr. Lecturer of PHYSICS

Mobile No: +91-9269680853


2/10

PHYSICS-I UNIT III DIFFRACTION OF LIGHT 2009-2010

www.charis-ancha.blogspot.com. These files are created keeping in mind my Students who trust me.Hope you drop comments and suggestions at [email protected]. All the Best.

PHYSICS I for R T U

UNIT III DIFFRACTION OF LIGHT

Diffraction: When light falls on obstacles or small apertures whose size is comparable with the wavelength of light, there is a departure from straight line propagation, the light bends round the corners of the obstacles orapertures and enters in the geometrical shadow. This bending of light is called diffraction.

The amount of bending depends upon the size of the obstacle and the wavelength of wave. The diffraction effects are observed only when a portion of the wavefront is cut off by some obstacle.

Difference between Interference and Diffraction:INTERFERENCE DIFFRACTION

1. The interference takes place between twoseparate wavefronts originating from twocoherent sources.

2. In interference pattern the regions of minimumintensity are usually almost perfectly dark.

3. The width of the fringes may or may not beequal or uniform.

4. In an interference pattern all the maxima are of same intensity.

1. The interaction takes place between thesecondary wavelets originating from differentpoints of the same wavefront.

2. In diffraction pattern the regions of minimumintensity are not perfectly dark.

3. The width of the fringes is never equal.

4. In diffraction pattern all the maxima are of varying intensity.

Two kinds of Diffraction:Diffraction phenomenon can be divided into following two general classes:1. Fraunhofers Diffraction: In this class of diffraction source and the screen are placed at infinity. In this

case the wavefront which is incident on the aperture or obstacle is plane. 2. Fresnels Diffraction: In this class of diffraction source and the screen are placed at finite distances from

the aperture or obstacle having sharp edges. The incident wavefront is either spherical or cylindrical.

FRAUNHOFERS DIFFRACTION AT SINGLE SLIT: (NORMAL INCIDENCE)

Description: The adjacent figure represents a narrow slit AB of width e . Let a plane wavefront WW of

monochromatic light of wavelength ' ' is incident on the slit. Let the diffractedlight be focused by means of a convexlens on a screen. According to Huygen-Fresnel, every point of the wavefront inthe plane of the slit is a source of secondary wavelets. The secondarywavelets travelling normally to the sliti.e., along OP o are brought to focus atP o by the lens. Thus P o is a bright central image. The secondary wavelets travelling at an angle ' ' are focused at

a point P 1 on the screen.The intensity at the point P 1 is either minimum or maximum and depends upon the path difference

between the secondary waves originating from the corresponding points of the wavefront. Mathematical theory:In order to find out the intensity at P1, draw a perpendicular AC on BR.The path difference between secondary wavelets from A and B in direction is BC

. ., sin sini e BC AB e = =

So, the phase difference,2

sine

=

W

'W

P

R

1P

0P

C

B

A

Oe

Screenslit


3/10



PHYSICS I for R T U

Let us consider that the width of the slit is divided into n equal partsand the amplitude of the wave from each part is a . Then the resultant amplitude R is calculated by the method of vector addition as follows.

Consider a polygon of amplitudes as shown in figure. Let OP is theresultant amplitude R .The components along OA is ( cos ) R andcomponents perpendicular to OA is ( sin ) R .

( )cos cos cos 2 ... cos( 1)1 cos cos 2 ... cos( 1) (1) R a a d a d a n d a d d n d = + + + + = + + + +

( )sin sin sin 2 ... sin( 1)

sin sin 2 ... sin( 1) (2) R a d a d a n d

a d d n d = + + +

= + + +

Multiplying eqn(1) by "2sin 2"d , we get

{ }

2 cos sin 2sin 2cos sin ... 2cos( 1) sin2 2 2 2

By using, sin sin 2cos sin , we have2 2

1 132 cos sin 2sin sin sin ... sin sin2 2 2 2 2 2

d d d d R a d n d

C D C DC D

d d d d R a n d n d

= + + + + =

= + + +

1sin sin2 2

again using, sin sin 2sin cos , we have2 2

( 1)2 cos sin 2 sin cos2 2 2

sin ( 1)2cos cos2sin 2

d a n d

C D C DC D

nd n d d R a

nd n d R a

d

= + + + =

=

= (3)

Similarly multiplying eqn(2) by "2sin 2"d , we get

sin ( 1)2sin sin (4)2sin 2

nd n d R a

d

=

Squaring eqn(3) and (4) and then adding, we get2

2 2 2 2 2 2 2

2

2

2 2

2

sin ( 1) ( 1)2cos sin cos sin2 2sin 2

sin sin2 2 (5)sin sin

2 2

nd n d n d R R a

d

nd nd R a or R a

d d

+ = +

= =

Now, the phase difference between any two consecutive waves from the divided parts would be

( )1 1 2 sinTotalphase e d n n

= =

Therefore, the Resultant amplitude R is given by,( )( )

sin sin sin2sin sinsin 2

nd e R a a

d e n

= =

d

R

P a

d

d

d

d

d a

a

a

aaaO

A

BC

D

F

E


4/10



PHYSICS I for R T U

( )sin

where sinsin

for is small sin

R a en

n n n

= =

=

( )sin sin

sin(6)

R a nan

R A

= =

=

Therefore, the Intensity is given by2 2

2 2 sin sin (7)o I R A I I

= = =

Intensity Distribution:Case(i): Principal Maximum: Eqn(7) takes maximum value for 0 =

sin 0

sin 0 ( ) 0 (8)

e

or

= =

= =

The condition 0 =

means that this maximum is formed by the secondary wavelets which travel normally to theslit along OP o and focus at P o. This maximum is known as Principal maximum . Case(ii): Minimum Intensity positions: Eqn(7) takes minimum values for sin 0 = . The values of ' '

which satisfy sin 0 = are, 2 , 3 ...

sin. .,

sin where 1, 2,3,... (9)

n

ei e n

e n n

= =

=

= =

in the above eqn(9) n = 0 is not applicable because corresponds to principal maximum.Therefore, the positions according to eqn(9) are on either side of the principal maximum.Case(iii): Secondary maximum: In addition to principal maximum at 0 = , there are weak secondary maxima between minima positions. Thepositions of these weak secondary maxima can be obtained with the rule of finding maxima and minima of agiven function in calculus.So, differentiating eqn(7) and equating to zero, we have

22 2

2

2

sin sin ( cos sin )0 2 0

0, sin 0 ( sin 0 corresponds to minimum positions)

cos sin 0 tan

dI d A A

d d

A

= = = =

= =Q Q

(10)

The values of ' ' satisfying the eqn(10) are obtainedgraphically by plotting the curves y = and tan y = on the same

graph. The points of intersection of the two curves gives the valuesof ' ' which satisfy eqn(10).The points of intersections are

3 50, , , .2 2 etc =

more exactly 0, 1.430 , 2.462 , 3.471 , .

But 0, gives principal maximum

etc

= =

Substituting the values of ' ' in eqn(7), we get

y =tan y =

02

32

52

2 3

2 5

2


5/10



PHYSICS I for R T U

2 22 2

2 2 20 1 2

3 5sin sin2 2(principal maxima), ., ., ... .3 522 62

2 2

A A I A I A appr I A appr and so on

= = = = =

From the above expressions, I 0 , I 1 , I 2 , it is evident that most of the incident light is concentrated atthe principal maximum.

Intensity distribution graph:A graph showing the variation of intensity with ' ' is as shown

in the adjacent figure.

Width of the central maximum:It is the separation between first minimum on both sides of principal maximum.If the lens L 2 is very near to slit AB or the screen is far awayfrom the lens L 2, then

sin x f

=

where x is the distance of the first minimum P 1 from P 0 and f is the focal length of L 2.But the condition for the first minimum is

sin sin

e e x f

x f e e

= =

= =

Therefore, the width of the central maximum is2

2 . f

W xe = =

Features of width of the central maximum:(i) The width of the central maximum is proportional to ' ' (ii) The width of the central maximum is inversely proportional to e , the width of the slit.(iii) The width of the central maximum is twice that of any other maxima.

FRAUNHOFERS DIFFRACTION BY PLANE DIFFRACTION GRATING(NORMAL INCIDENCE)(Diffraction at N parallel Slits)

Construction: An arrangement consisting of large number of parallel slits of the same width and separated byequal opaque spaces is known as Diffraction grating.

Gratings are constructed by ruling equidistantparallel lines on a transparent material such as glass, with afine diamond point. The ruled lines are opaque to light whilethe space between any two lines is transparent to light and actsas a slit. This is known as plane transmission grating.

When the spacing between the lines is of the order of

the wavelength of light, then an appreciable deviation of thelight is produced.Theory: A section of a plane transmission grating AB placedperpendicular to the plane of the paper is as shown in thefigure.

Let e be the width of each slit and d the width of each opaque space. Then (e+d) is known as gratingelement and XY is the screen. Suppose a parallel beam of monochromatic light of wavelength ' ' be incidentnormally on the grating. By Huygens principle, each of the slit sends secondary wavelets in all directions. Now,the secondary wavelets travelling in the direction of incident light will focus at a point Po on the screen. Thispoint Po will be a central maximum.

I

0 2 3 3 2

W

'W

1P

0P

B

A

ScreenGrating

X

Y


6/10



PHYSICS I for R T U

Now consider the secondary waves travelling in a direction inclined at an angle ' ' with the incidentlight will reach point P 1 in different phases. As a result dark and bright bands on both sides of central maximumare obtained.

The intensity at point P 1 may be considered by applying the theory of Fraunhofer diffraction at a singleslit.

The wavelets proceeding from all points in a slit along their direction are equivalent to a single wave of

amplitudesin

A

starting from the middle point of the slit, where sine =

If there are N slits, then we have N diffracted waves. The path difference between two consecutive slitsis (e+d)sin . Therefore, the phase difference

( )2 sin 2 (1)e d

= + =

Hence the intensity in a direction ' ' can be found by finding the resultant amplitude of N vibrations

each of amplitudesin

A

and a phase difference of ' 2 '

sin sin(2)

sin N

R A

=

Since in the present casesin

, 2a A n N and d

= = =

Ssubstituting these in equationsin 2sin 2

nd a R

d =

and2 2

22

2 2

2

sin sin

sin

sin sin(3)

sino

N I R A

N I I

= = =

The factor2

sin A

gives the distribution of Intensity due to a single slit while the factor

2

2sinsin

N

gives the

distribution of intensity as a combined effect of all the slits.

Intensity Distribution:

Case (i): Principal maxima: The eqn(3) will take a maximum value if sin 0 =

0,1, 2,3...n where n = =

( ) ( )

( )

. ., sin (1)

sin 0,1, 2,... (4)

i e e d n from eqn

or e d n where n

+ =

+ = =

n = 0 corresponds to zero order maximum. For n = 1,2,3, we obtain first, second, third, principal maximarespectively. The sign indicates that there are two principal maxima of the same order lying on either side of zero order maximum.Case(ii): Minima Positions: The eqn(3) takes minimum value if sin 0 N = but sin 0

N m =

( )

( )

. ., sin

N sin

i e N e d m

or e d m

+ =

+ =


7/10



PHYSICS I for R T U

Where m has all integral values except m = 0, N, 2N, , nN, because for these values sin becomes zero andwe get principal maxima. Thus, m = 1, 2, 3, , (N-1). Hence

( )sin 1, 2, ..., ( -1) (5) N e d m where m N + = = gives the minima positions which are adjacent to the principal maxima.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. These are known as secondary maxima.To find their positions

2

2sin sin cos sin sin cos0 2 0

sin sindI A N N N N d

= =

[ ]sin 0, sin 0 cos sin sin cos 0

tan tan (6)

A N only N N N

N N

=

=

Q

The roots of the above equation other than those for which n = give the positions of secondary maxima.

The eqn(6) can be written as tancot N

N

=

From the triangle we have

( ) ( )

( )

2 2

2 2 2

2 2 2 2 2 2 2

2 2

2 2 2

sin cot

sinsin cot sin sin 1 sin

sinsin 1 1 sin

N

N N

N N N

N N

N N

N

=+

= =+ +

=+

Thus

( )

2 2

2 2

sin(7)

1 1 sino

N I I

N

= +

Since intensity of principal maxima is proportional to N 2,

( )( )

2

2 2

2 2 2

1 1 sinIntensity of secondary maxima 1Intensity of principal maxima 1 1 sin

N N

N N

+ = =

+

Hence if the value of N is larger, then the secondary maxima will be weaker and becomes negligible when N becomes infinity.

FORMATION OF SPECTRA WITH GRATING

The principle maxima in a grating are formed in

direction given by ( )sin (1)e d n + =

where ( )e d is the grating element, n the order of themaxima and the wavelength of the incident light.From eqn(1), we conclude that1) For a given wavelength , the angle of diffraction is

different for principal maxima of different orders.2) For white light and for a particular order n, the light of

different wavelengths will be diffracted in differentdirections.

The longer the wavelength, greater is the angle of

2 2cot N +

N

N

cot

2 R

2V 1 R

1V

2 R

2V 1 R

1V

Second Order

First Order First Order

Second Order

ZeroOrder

Grating


8/10



PHYSICS I for R T U

diffraction. So in each order, we will get the spectra having as many lines as the wavelength in the light source.At centre (n = 0, zero order) 0 = gives the maxima of all wavelengths. So here different wavelengths

coincide to form the central image of the same colour as that the light source.Similarly the principal maxima of all wavelengths corresponding to n = 1 will form the first order

spectrum, the principal maxima of all wavelengths corresponding to n = 2 , will form the second order spectrumand so on.

Important characteristics of grating spectra: 1) Spectra of different orders are situated symmetrically on both sides of zero order. 2) Spectral lines are almost straight and quite sharp.3) Spectral colours are in the order from Violet to Red.4) Spectral lines are more dispersed as we go to higher orders.5) Most of the incident intensity goes to zero order and rest is distributed among the other orders.

Maximum number of orders formed by a Grating:The principal maxima in a grating satisfy the condition

( ) ( )sinsin e d e d n Or n ++ = =

The maximum angle of diffraction is 90 o, hence the maximum possible order is given by( ) ( )

max

sin90 oe d e d n

+ += =

Ex: Consider a grating having grating element which is less than twice the wavelength of the incident light, then( )

max max

2

22

e d

n n

+

unit - iii diffraction of light

Documents