Diffraction Ibh Uem Kolkata 2015 (1)

DIFFRACTION OF LIGHT

Indrani Bhattacharya

MODULE 2 : OPTICS 1

UEM, B. Tech., 1ST YEAR, MODULE 2, OPTICS 1, 2015

Diffraction of light: Fresnel and Fraunhofer class. Fraunhofer diffraction for single slit and double

slits. Intensity distribution of N-slits and plane

transmission grating (No deduction of theintensity distributionsfor N-slits is necessary), Missing orders.

Rayleigh criterion, Resolving power of grating andmicroscope(Definition and formulae)

TOPICS TO BE COVERED


DEFINITION

The slight bending of light round the edge of an object , whose size is comparable with the wavelength of light, and spreading the same into the regions of the geometric shadow is called the Diffraction of light.

The light waves are diffracted only when the size of the obstacle is com--parable to the wavelength of light.

If the opening is much larger than the light's wavelength, the bending will be almost unnoticeable.


What is the Difference between INTERFERENCE & DIFFRACTION ?



DIFFERENT TYPES OF DIFFRACTION PHENOMENA

FRESNEL TYPE

FRAUNHOFFER TYPE


FRESNEL DIFFRACTION PHENOMENON

The Fresnel diffraction deals with the near-field diffraction, wherethe Source , the Screen or both are at finite distances from the obstacle and the distances are important.

In Fresnel diffraction, the incident wavefront is either Spherical orCylindrical.

In Fresnel Diffraction, the centre of diffraction may be bright or darkdepending upon the number of Fresnels Zone.


FRAUNHOFER DIFFRACTION PHENOMENON

The Fraunhofer diffraction deals with the far-field diffraction, wherethe Source , the Screen or both are at infinite distances from theobstacle; it is viewed at the focal plane of an imaging lens.here, the inclination is important.

In Fraunhofer diffraction, the incident wavefront is Plane.

In Fraunhofer Diffraction, the centre of diffraction pattern is always bright.


Computer simulation of Fraunhofer diffraction by a rectangular aperture


Computer simulation of the Airy diffraction pattern, of Circular Aperture


FRAUNHOFER DIFFRACTION DUE TO SINGLE SLIT

Concept of a Single Slit :

A slit is a rectangular aperture whose length is as large compared to itsbreadth.

The Width of the Slit is comparable to the Wavelength of light used.

Theory / Principle

The study of Diffraction pattern is based upon the superposition of Huygens secondary wavelet which are supposed to be generated atevery point on the wavefront of the slit.


i

A

B

dx

O

C

E F P

Single Slit

Lens

Plane ofObservation

Figure 3 : Showing experimental arrangement of Fraunhofer DiffractionPattern through Single Slit.


AB is a long narrow slit of width = a, placed perpendicular to theplane of paper and is illuminated by a parallel beam of monochro--matic light of wavelength .

The figure shows the cross-section of the slit.

O is the centre of the slit.

Let i be the angle of incidence with the normal to the plane of theslit.

Due to Diffraction, the rays will generate secondary wavelets in all possible directions and are focussed on the Focal Plane of theConverging Lens L.

Parallel Diffracted Rays will be focussed by the lens on the screenat P to form the image of the Slit.

SALIENT POINTS ABOUT THE EXPERIMENTAL SET UP



The Point of Observation is at INFINITY, and obviously, it is a case of Fraunhofer Class of Diffraction.

Let us consider a diffrcating element of width, dx at C at a distance xfrom O.

OE and OF are two Perpendiculars drawn on the Incident and the Diffracted rays ; hence, OE and OF will represent Incident and Diffracted Wavefronts.

We will be finding out the expression for the Intensity of Light at P.


DERIVATION OF THE EXPRESSION FOR INTENSITY AT P

The equation of Vibration on the Incident Wavefront OE can be represented by :

where c is the velocity of Light.

The equation of Vibration on the Diffracted Wavefront OF can be represented by :

)1...(2

cti

Aey

)2....()(

2ECFcti

Aey



But we can write :

which means :

Hence, eqn (2) reduces to :

sinsin xixCFECECF

)3.....(sinsin xixECF

)4....()(

2

xcti

Aey



Neglecting the effects of disturbance due to inclination and others, the disturbance at P due to element dx at C is given by :

Hence, the Total Disturbance at P due to the whole Slit is given by :

)5.....()(

2

dxAeydxdsxcti

2/

2/

22

2/

2/

)(2

a

a

xicti

a

a

xcti

dxeAeS

dxAeydxdsS



)6.....(2

2

2

2

i

ee

a

AaeS

i

eeAeS

ai

aicti

ai

aicti



Substituting

we have from previous expression :

Xa

X

XAaeS

i

ee

X

AaeS

cti

iXiXcti

sin

2

2

2

)7....(sin

2cti

eX

XAaS



)7....(sin

2cti

eX

XAaS

Examining expression (7) , it appears that the Vibration is truly Simple Harmonic.

The Intensity at P is proportional to the Square of Amplitude, i.e.,

Let us choose the Constant of Proportionality as Unity, i.e.,

)8....(22

SKISI

1K



Hence, the Intensity at P will be :

)9...(sinsin

2

222

2

X

XaA

X

XAaI

)10...(

sinsin

sinsinsin

2

2

22

ia

ia

aAI

2

2

22

2

2

22

sinsin

sinsinsinsin

ia

ia

aAa

a

aAI



CONDITIONS FOR MAXIMA AND MINIMA

)9...(sinsin

2

222

2

X

XaA

X

XAaI

)11...(sin2cossin2

2.sincossin2.

3

222

4

2222

X

XXXXaA

dX

dI

X

XXXXXaA

dX

dI




From eqn.(10), we can explore the condition of Maxima and Minima by putting :

which implies :

)12...(0dX

dI

0sin

2

222

X

XaA

dX

d

)13...(0

sincossin2

3

22

X

XXXXaA




)14...(0

sincossin2

3

22

X

XXXXaA

We have as conditions :

)...(aX

)...(0sin bX

)...(0sincos cXXX




From condition (a), we have :

which must be true, when ;

But since , this condition is invalid and we reject this one.

From condition (b), we have :

which means :

aX

0

0

0sin X

)15....(

sin0sin

a

nn

a

na




If we find and put the condition in eqn. (15), we can find that :

This gives the Condition for Minimum and the other successive minima are given by :

From eqn. (16) it is apparent that, the Minima are Equidistant.

2

2

dX

Id

02

2

dX

Id

)16,....(3

,2

, 321aaa




From condition (c) we have :

Condition (c) gives the condition for maxima.

The value of X satisfying the eqn. (c) is obtained by drawing two curves :

(i) Y =X & (ii) Y = tan X on the same scale.

)'....(tancos

sin

)...(0sincos

cXXX

X

cXXX




The Point of Intersection of the two curves will give the values of X, satisfying the equation tan X = X, the maximum value of intensity will occur at :

which is Central Band.

gives the next value of the Intensity :

with

)17...(sin

0 222

222

000

0

aAX

XaALtIX

X

43.11 X

22

1 0469.0 aAI )18....(43.11 a




gives

with

gives

with

46.22 X22

2 0168.0 aAI

)19....(46.22a

47.33 X22

3 0083.0 aAI

)20....(47.33a

XY

Y = X

X

Intensity

0 2 3--2-3

Y= t

anX

1.4

3

2.4

6

3.4

7

-1.4

3

-2.4

6

-3.4

7

Plotting of Intensity distributionsDue to single slit




Expressions (17 ), (18 ), (19 ) and (20) represent that :

The diffraction pattern consists of a Bright Central Maximum with Intensity represented by .

The central maximum is followed by minimum of Zero Intensity.

The minimum of Zero Intensity is followed by secondary maximum of intensity at at both sides of the centralmaxima.

It is also apparent from the graph is that the maxima are NOT equidistant at lower orders.

0I

43.11 X1I




The angle of Diffraction, for the first minimum on either side of the Central Maximum is given by :

The Condition for Minima is given by :

For Normal Incidence : i = 0;

Therefore : ;

For First Order Minima, we have, n=1 for .

1

)15....(a

n

a

ni

sinsin

a

n sin

1




a

1sin

For First Order Minima, we have, n=1 for .

Since is small,

Hence, the Angular Width of the Principal Maxima, i.e., is given by :

i.e., inversely proportional to the Width of the Slit.

1

1 11sin

a

1

12

a

22 1


FRAUNHOFER DIFFRACTION DUE TO DOUBLE SLIT

Concept of Double Slit :

Double slit is an arrangement where two single slits are placed in parallelon the same plane

The Width of each Slit is generally identical and much smaller than their lengths.

The slits have an opaque space between them.

The widths of both the slits and Opaque Space should be of the order ofwavelength used.


Figure shows Same double-slit assembly (0.7 mm between slits); in topimage, one slit is closed. In the single-slit image, a diffractionpattern (the faint spots on either side of the main band) forms due tothe nonzero width of the slit. A diffraction pattern is also seen in thedouble-slit image, but at twice the intensity and with the addition ofmany smaller interference fringes.


FRAUNHOFER DIFFRACTION DUE TO DOUBLE SLIT

Theory / Principle

The study of Diffraction pattern due to Double Slits consists of diffractionfringes caused by rays diffracted from both slits superposed on the

interference fringes caused by rays coming from each pair of correspondingpoints on the two slits.


Figure showing double-slit diffraction pattern by a plane wave


a

b

aP

d

1S

2S

O

1O

i A P B

x

dx

Figure showing Fraunhofer diffraction thru. Double slits MN and RS

Lens

Plane ofObservation

N

M

S

R

MN

RS

Slit 1

Slit 2

21SS Plane of theSlita/2

a/2



represents the section of the slit.

a is the width of each slit and b is the width of the opaque space.

and are the mid-points of the slits MN and RS respectively.

The distance between and is d which is equal to (a+b).

is the angle of incidence.

The rays will be diffracted at all possible directions. Let us considera beam of rays for which the angle of diffraction is .

L is the Converging lens which focuses the diffracted rays on its focalplane.

As the Source and Point of Observation both are effectively at Infinity,obviously, it is a case of Fraunhofer class of diffraction.

21SS

O

1OO

i

'O


ANALYTICAL STUDY OF INTENSITY OF DOUBLE SLIT DIFFRACTION PATTERN

2S

1O

A P B

x

dx

a

i

To find the Intensity on the screen, let us consider a diffracting elementof width dx at P where .

From let us draw two perpendiculars and on the incident and diffracted rays.

xPO 1

1O BO1AO1





2S

1O

A P B

x

dx

a

i

Let the equation of vibration of the incident wavefront is representedby :

where A is the amplitude of the wave and is the wavelength. BO1

AO1

)1...(2

cti

Aey






2S

1O

A P B

x

dx

a

i

Let the equation of vibration of the diffracted wavefront is representedby :

BO1

)2...(

2APBcti

Aey







2S

1O

A P B

x

dx

a

i

Now,

where :

)3...()sin(sin

sinsin

xixAPB

xixPBAPAPB

)4...(sinsin i



Expression (2) reduces to:

)5...(

2

xcti

Aey

2S

1O

A P B

x

dx

a

i

Lens

Plane ofObservation

1P




2S

1O

A P B

x

dx

a

i

Lens

Plane ofObservation

1P

The disturbance at due to element dx at P is given by :

)6.....()(

2

dxAeydxdsxcti

1P

S

R


)7...(

2/

2/

22

1

2/

2/

)(2

111

a

a

xicti

a

a

xcti

dxeAeS

dxAedxydsS

The disturbance at due to the entire slit is given by :

The disturbance at due to the other slit is :

1P


RS

MN1P

)8...(

2/

2/

22

2

ad

ad

xicti

dxeAeS



Hence, the disturbance due to both the slits at will be : 1P

dia

a

xicti

ad

ad

xia

a

xicti

edxeAeS

dxedxeAeSSS

22/

2/

22

2/

2/

22/

2/

22

21

1

di

a

a

xicti

e

i

eAeS

2

2/

2/

22

12



Rearranging we can write :

diaiai

cti

ei

ee

a

AeS

2//

2

12

Let us put : Xa

)10...(2

sin2

cos1sin

12

2

22

didX

XAaeS

ei

ee

X

AaeS

cti

diiXiX

cti



Hence, the Effective amplitude of Vibration at due to two slits is :

The resultant Intensity is Proportional to the square modulus of amplitude.

Let the Constant of Proportionality is equal to 1.

Hence, the Intensity at is given by ::

1P

)11...(2

sin2

cos1sin

.

did

X

XAaAmp

1P



dX

XaAI

dX

XaAI

ddX

XaAI

2cos1

sin2

2cos22

sin

2sin

2cos1

sin

2

222

2

222

2

2

2

222

d

X

XaAI 2

2

222 cos2.

sin2

)12...(cossin

4 22

222

d

X

XaAI



)12...(cossin

4 22

222

d

X

XaAI

Substituting , we can write :

dY

)13...(cossin

4 22

2

0 YX

XII

Where : )14...(220 aAI


)13...(cossin

4 22

2

0 YX

XII


Upon examining eqn. (13), it is predominant that the factor arises due to the Single Slit Pattern.

The next factor arises due to the Disturbance coming from

the two slits having a Phase Difference .

Hence, the Minimum Intensity of Interference occurs when :

d2cos

2

2sin

X

X

d

2

2)12cos(0cos2

n

d



CONDITION FOR MINIMA

2

)12(sinsin

sinsin,

2)12(

2)12cos(0cos2

nbai

ibad

nd

nd

For Normal Incidence, i=0; hence ::

)14...(2

)12(sin

2)12(sin

ba

nnba


CONDITION FOR MINIMA

)14...(

2

)12(sin

ba

n

Successive minima are obtained for different values of n e.g.,

;..

2

7,3;

2

5,2

;2

3,1;

2,0

ban

ban

ban

ban

It is apparent the minima are equidistant; the distances between the successive minima are equal and is

ba


CONDITION FOR MAXIMA

For maxima, the condition will be :

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

cos1cos2

nnba

nd

nd

,sinsinsin&

i

bad

for normal incidence, where i=0

)15....()(

sinba

n


CONDITION FOR MAXIMA

)15....()(

sinba

n

The Successive maxima are obtained for different values of n e.g.,

,.....)(

3,

)(

2,

)( bababa

This shows that the maxima are equidistant and is ba



)13...(cossin

4 22

2

0

d

X

XII

GRAPHICAL STUDY OF INTENSITY OF DOUBLE SLIT DIFFRACTION PATTERN

The term is called Interference term which gives a set of

equidistant dark and bright fringes as in the Youngs double slit interferenceexperiment.

The term is called Diffraction Term having a central maxima

at X=0, i.e., in the direction = 0.

The Maxima also occurs at values of X at : which

are called Secondary Diffraction Maxima.

d2cos

2

2sin

X

X

,....2

5,

2

3



)13...(cossin

4 22

2

0

d

X

XII

The minima are obtained when :

Which means :

That is :

for normal incidence, i=0.

except 0.

)15...(0sin

2

2

X

X

mX

mX

sin0sin

,...3,2,1,sin

,sin

mma

ma



)16,......(3,2,1,sin mma

This Minima are known as Diffraction Minima.

The Diffraction Pattern consists of a Central Maxima in the direction =0with alternate minima and secondary maxima of diminishing intensity oneither side.

The resultant Intensity distribution against angular position, is shown in figure.


Figure showing DOUBLE SLIT Interference Pattern


Figure showing SINGLE SLIT Diffraction Pattern


FIGURE SHOWING DOUBLE SLIT DIFFRACTION PATTERN

(In Radians)

INTENSITY

Documents

Diffraction Ibh Uem Kolkata 2015 (1)