DIFFRACTION-FRAUNHOFER TYPE

DIFFRACTION-FRAUNHOFER TYPE

UNIVERSITY OF CALCUTTA

UG PHYSICS HONOURS SEMESTER-II

1

Dr. Kamalika Hajra

The idea: Geometrical boundary of the shadow of an obstacle is usually not sharp

• Geometrical boundary of the image of a slit is not sharp

• Some light as if creeps into the area of the shadow bending of light

Huygen’s Principle: Every point on a wavefront acts as a source of secondary wavelets

• This effect is confined to the forward envelope.

• Cannot explain bending of light

Dr. Kamalika Hajra 2

Huygens-Fresnel’s Principle

Every unobstructed point of a wavefront, at a given instant, serves

as a source of spherical secondary wavelets (with the same

frequency as the primary wave); the amplitude of the optical field

at any point beyond is the superposition of all these wavelets


Diffraction pattern may be of two different varieties, depending on the

distance of the source and the screen from the slit

Fresnel diffraction

Fraunhoffer diffraction

Fresnel type of diffraction Fraunhofer kind of diffraction

Source and screen are at finite

distance from the slit

Source and screen are at infinite distance

from the slit

Dr. Kamalika Hajra

5

How Fraunhoffer diffraction is made effective in the laboratory:

keeping the source at the focus of a converging lens, thus rendering the emerging

rays parallel, which will be incident on the slit, hence effectively placing the source at

infinite distance from the source;

the screen is effectively placed at infinite distance from the screen by placing it

at the focal plane of another converging lens which is positioned infront of the diffracted

rays on the other side of the lens

Dr. Kamalika Hajra

6

FRAUNHOFFER DIFFRACTION DUE TO A SINGLE SLIT

Slit width=b

Assumption:

The slit is considered to be made up of “n”

number of discrete points A1, A2,…..,An, such that

consecutive points are ∆ distance apart.

Assuming and

)1(nb

n 0

nb

Phase difference between two rays arising from consecutive imaginary points and

is given by , where is the angle that the diffracted rays make with the

normal to the slit.

Thus the resultant field at any point P on the screen is given by

jA 1jA

sin

2

]}1{cos(.......)cos([cos ntttaE

Dr. Kamalika Hajra

7

Adding the trigonometric terms

Where the amplitude of the resultant field is

Defining and

We get , at any point P, field

And intensity , is the intensity at

])1(2

1cos[ ntEE

2sin

2sin

n

aE

sinb naA

)cos(sin

tAE

2

2

0

sin

II

222

0 anAI 0

Dr. Kamalika Hajra

8

CONDITION OF MAXIMA AND MINIMA: INTENSITY DISTRIBUTION

Intensity I is max or min when

CASE I: If

This gives the position of the Central Maximum where

If

This gives the positions of the

diffraction minima

0d

dI0]

1

sin

cos[

sin22

2

0

I

0sin

2

2

0 0

0II

0 0sin ,......2,1, mm

mb

mb

sin

,sin

Dr. Kamalika Hajra

9

CASE II:

This is a transcendental equation, which is solved graphically and the roots are given by the

points os intersection of the equations and

Intersection points are found at

etc.

Giving the positions of secondary maxima with drastically decreasing intensity.

Intensity distribution graph for fraunhoffer single

slit diffraction.

1

sin

cos tan

y tany

46.2

43.1


FRAUNHOFFER DIFFRACTION DUE TO A DOUBLE SLIT

-

b

-

a

-

b

-

Width of each slit=b

Distance between two slits=a

Also, a+b=d

1st slit consists of imaginary points ,

which are distance apart.

2nd slit consists of imaginary points ,

which are also distance apart.

Phase difference between rays from consecutive

sources, that are diffracted at an angle is

given by

nAAA ....,, 21

nBBB ,....,, 21

sin

2

At any point P` on the screen,

Field due to rays diffracted by 1st slit,

where

Field due to rays diffracted by 2nd slit, , the phase difference

between corresponding

rays from the two slits\

Thus resultant field at the point P` is

)cos(sin

1

tAE

]cos[sin

12

tAE

sin2

sin

1 d

b

21 EEE

Dr. Kamalika Hajra

11

Solving, we get, , where

Thus, Intesity of the pattern is of the form

single slit diffraction

Note that the intensity is a product of two terms

double slit interference

)2

cos(cossin

2 1

tAE

sinsin

2

2

1

2

1 dd

2

0

2

2

2

0 ,cossin

4 AIII

2

2sin

2cos

Dr. Kamalika Hajra

12

MAXIMA AND MINIMA :

Intensity , when either ……..(i) and/or ……(ii)

(A) (B)

,n=0,1,2,3,…. ,m=1,2,3,…

INTERFERENCE MINIMA DIFFRACTION MINIMA

Minima can occur due to interference or diffraction or both

0I 0cos2 0sin

2

2

)2/1(sin

2)12(0cos2

nd

n

mb

m

sin

0sin 2

Dr. Kamalika Hajra

13

Usually variation of the diffraction term is negligible compared to the

interference term

Thus the maxima of the pattern is governed by the interference tern only, i.e., by

Now, for maxima,

, n=0,1,2,3,……..

Condition of maxima for two slit interference is the condition of maxima of

the entire pattern, provided, variation of the diffraction term is not too rapid.

db

2cos

nd

nd

sin

sin

1cos2

Dr. Kamalika Hajra

14

MISSING ORDERS:

Condition of interference maxima

Condition of diffraction minima

Hence all such maxima will be absent, and are called missing orders.

e.g., let p=3

so that 3rd, 6th, 9th, etc. orders of maxima

will be missing

Thus, when, in a double slit set up, the interference

maxima coincides with a diffraction minima,

the corresponding orders of maxima are missing

from the pattern and are called

missing orders .

nd sin

mb sin

When both

conditions are

satisfied for the

same value of

)(saypm

n

b

d pbd

bd 3

Dr. Kamalika Hajra

15

INTENSITY PATTERN FOR DOUBLE SLIT DIFFRACTION

ANGULAR WIDTH

Angular width between Angular width between

consecutive minima (interference) : consecutive maxima (diffraction) :

bbn

bnnn

)1(sinsin 1 d

nn

sinsin 1

Dr. Kamalika Hajra

16

FACTORS AFFECTING THE DOUBLE SLIT INTERFERENCE PATTERN:

Effect of increasing slit width, b:

thus central peak becomes sharper and acomodates lesser number

of interference fringes

Effect of increasing slit separation ,d:

thus more interference fringes will be acomodated within the central

diffraction maxima

Effect of increasing wavelength, : angular width for both patterns increase

bsin

b

d

sin d

Dr. Kamalika Hajra

17

SINGLE SLIT VS DOUBLE SLIT

SINGLE SLIT DOUBLE SLIT

Central bright region followed by secondary

maxima

Equally spaced interference fringes modulated by a

diffraction envelope

Spacing and intensity of the pattern governed

by the factor b

Spacing and intensity of the pattern governed by factors b

and d

No possibility of missing orders Missing orders present

Dr. Kamalika Hajra

18

FRAUNHOFER DIFFRACTION BY MULTIPLE SLITS: PLANE DIFFRACTION GRATING

Width of each slit = b

Separation between consecutive slits = a

Each slit consists of n imaginary sources ,

distance apart

Separation between corresponding pairs of

such sources in adjacent slits :

The resulting field E at any arbitrary point P on

the screen is the sum of N terms, where N is the

total number of slits.

bad

Calculation shows, , where]}2

1{cos[

sin

sinsin1

Nt

NAE

sin

sin

d

b

Dr. Kamalika Hajra

19

Thus intensity

2

2

2

2

0sin

sinsin NII

Single slit diffraction term N slit interference term

Note: For N=1, single slit diffraction intensity

For N =2, double slit diffraction intensity

Thus above expression of intensity is the most general one for any number of slits!

2

2

0

sin

II

2

2

2

0 cossin

II


POSITIONS OF MAXIMA AND MINIMA:

A. Principal maxima:

At , , i.e., undetermined

However by appl ying L`Hospital’s rule, we see that

So that intensity becomes

Thus condition of obtaining principal maxima are:

, m=0,1,2,3,…

Note that , cannot exceed , so that there will only be a

limited number of principal maxima

m0

0

sin

sin2

2

N

NN

Ltm

sin

sin

2

22 sin

INI

md

m

sin

1sin m

d


B. Minima and secondary maxima:

Intensity will be 0 when

(i) Either , n=1,2,3,……, i.e., positions of single slit

diffraction minima

OR

(ii)

These are the secondary minima and there are (N-1) such minima.

Evidently, between two such minima there will lie a maxima,

called secondary maxima.

For secondary maxima, we get

Giving,

nb sin

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

0sin

NppNd

pNN

....3,2,,0 NNNp

cotcot NN

maxmaxsec

22

maxmaxsec

sin1

II

N

II


Note :

The intensity of higher order principal maxima will be modified due to presence of the

diffraction pattern envelope.

If we take for example N=6, intensity of central principal maxima ~

no. of minima between two consecutive principal maxima = N-1=5

no. of secondary maxima between two principal maxima = N-2=4

For a given angle, if a principal maxima coincides with a diffraction minima, it will be absent

and result in ‘missing orders’

In addition to secondary minima, diffraction minima are also there but for large N, they are

hardly contribute at higher orders

The angle between the first minimum on either side of the principal maximum is given by

, for order m.

Thus higher the value of N, sharper is the principal maximum.

362 N

m

mNd

cos


PLANE TRANSMISSION GRATING

Basically consists of an optically transparent

sheet on which slits are made by ruling grooves .

The grooves act as the opaque spaces and the

space in between forms the slit

Typically, N~15000 slits/inch

Condition of obtaining principal maxima

Is also called the grating equation

And d=a+b, is called the grating element

.....3,2,1,0,sin mmd

Grating spectrum:

When incident light is polychromatic, value of will differ

for different values of except for the central maximum

Thus for , the component colours will be separated

resulting in appearance of spectra .

0m

Dr. Kamalika Hajra

24

RESOLVING POWER OF A GRATING: RAYLEIGH’S CRITERION

Resolving power ability to distinguish two very close spectral lines and

Rayleigh’s criterion: Two neighbouring lines will be just resolved when the centre of

one maximum falls on the first minimum of the adjacent line.

this gives, chromatic resolving power of a grating

for order and N rulings to be

Thus, higher the value of N, better is the resolving power of the grating.

d

mN

thm

Dr. Kamalika Hajra

25

GRATING SPECTRUM VS PRISM SPECTRUM

GRATING SPECTRUM PRISM SPECTRUM

Large number of spectra on either side of central

principal maximum

Only a single spectrum

Overlapping occurs between spectra at higher

orders

Not applicable

Central principal spectrum is the brightest with

diminishing intensity on either side

Not applicable

Red is most deviated Violet is most deviated

Ghost lines and missing orders present No such thing

Angular dispersive power does not depend on

material of the grating

Angular dispersive power depends on material of

the prism

Dr. Kamalika Hajra

26

Questions and Answers

Dr. Kamalika Hajra

27

DIFFRACTION-FRAUNHOFER TYPE

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