1

Diffraction

1

Diffraction“Any deviation of light rays from rectilinear path which is neither reflection nor refraction is knownas diffraction.” (Sommerfeld)

Types or kinds of diffraction:

1. Fraunhofer (1787-1826)

2. Fresnel (1788-1827)

•In addition to interference, waves also exhibit another property – diffraction.

• It is the bending of the waves as they pass by some objects or through an aperture.

• The phenomenon of diffraction can be understood using Huygen’s principle

Diffraction of ocean water wavesOcean waves passing through slits in Tel Aviv, Israel

Diffraction occurs for all waves, whatever the phenomenon.

Huygens Principle : when a part of the wave front is cut off by an obstacle, and the rest admitted through apertures, the wave on the other side is just the result of superposition of the Huygens wavelets emanating from each point of the aperture, ignoring the portions obscured by the opaque regions.

Secondary wavelets from apertures

μv > μRRefraction

Deviation for blue is larger than that for red

Deviation for red is larger than that for blue

Diffraction

Fraunhofer diffraction

Single slit diffractionPrincipal maximum

First minimum

Second minimum

Young’s Two Slit Experiment and Spatial Coherence

If the spatial coherence length is less than the slit separation, then the relative phase of the light transmitted through each slit will vary randomly, washing out the fine-scale fringes, and a one-slit pattern will be observed.

Fraunhofer diffraction patterns

Good spatial coherence

Poor spatial coherence

Diffraction from one- and two-slits

Fraunhofer diffraction patterns

One slit

Two slits

Diffraction from small and large circular apertures

Far-field intensity pattern

from a small aperture

Far-field intensity

pattern from a large aperture

Diffraction from multiple slits

Slit DiffractionPattern Pattern

Superposition of large number of phasorsof equal amplitude a and equal successivephase difference δ. We have to find the resultant phasor.

GP series

2/2/2/

2/2/2/

1

1

iii

ininin

i

in

eee

eeae

e

eaA

2/1

2/sin

2/sin

nie

na

25

δ

δ

8δ

7δ

6δ

5δ

4δ

3δ

2δ

δδ/2

δ/2

δ/2

2/12/sin2 nienrA

2/1

2/sin

2/sin

nie

naA

2

Problem: Obtain intensity formula by integrationf

x sin θ is the path difference at a point x where the slit element dx is placed.

Integrate: a exp(iδ)

For single slit path differencebetween the two ends of the slit

Δ = b Sin θ

Phase difference = 2π Δ/λ = nδ β = nδ/2 = πb sin θ/λ

Intensity for single slit

I

β

β = π b sin θ / λ Minima at

β = + m π _

Maxima at

tan β = β

m = 1, 2, 3…

2

For Principal maxima

0

sin1

( ) (0)I I

β = π b Sin θ / λ

2

3

2sin ( cos sin )(0) 0

dII

d

For extrema of I()

tan β = β

tan β = βMaxima

y=tan β

y= β

β

For subsidiary maxima

1.4303 , 2.4590 , 3.470 ,.....

Full width of central maximum

= λ / b

Δ x0 = f λ / bDiffraction envelope size

Point source

Smearing effectof diffraction

Δ x0

f

(First minima at sin ~ = /b)

Double slit

Phase=(2π/) * x sin

2

Double slit intensity pattern for d=5b

Single slit diffraction pattern X double slit interference pattern

b sin θ = m λ

d sin θ = n λ

Diffraction Minima at

Interference Maxima at

/m = 0

Condition for Missing orders

When the above two equations are satisfied at the same point in the pattern (same θ), dividing one equation by the other gives the condition for missing orders.

bm

nd

Missing orders 5, 10, 15, 20….

d/b = 5

When we use the double-source equation to find locations of

bright spots, we find that there are some places where we

expect to see bright spots, but we see no light. This is known

as a missing order, and it happens because at that location

there's a zero in the single slit pattern.

Remember!

If the zero in the single slit pattern, and a zero in the double slit

pattern coincides, it is not called a missing order, as there is no

order to be missing!

Also, if there is a local peak in the single slit pattern, and a zero in

the double source pattern, there will still be a zero (remember, we

multiply the functions!) - this is also not a missing order.

N slit grating

Normal incidence

Transmissiongrating

= Number of slits

We know that the amplitude due to each single slit:

= δ /2

)2/sin(

)2/sin(

N

aA

sin

a

Intensity pattern

for small b

InterferenceDiffraction

sin2(x) sin2(5*x)

Behavior of

sin2(5*x) / sin2(x)

sin2(10*x) / sin2(x)

When = 0, or mπ (m = 0, ±1, ±2,…)

md sin (m = 0, ±1, ±2,…)

The intensity is maximum when this condition is satisfied. These are called the Primary maxima.m gives the order of the maximum.The intensity drops away from the primary maxima.The intensity becomes zero (N-1) times between two successive primary maxima.There are (N-2) secondary maxima in between.As N increases the number of secondary maxima increases and the primary maxima becomes sharper.

NN

sin

sin

Intensity

I

γ

N=15

The intensity becomes zero (N-1) times between two successive primary maxima.There are (N-2) secondary maxima in between.

For two wavelengths

d/b = 4

5th Order6th Order

7th Order

1st Order

Principal maxima

Minima

When = 0, or mπ (m = 0, ±1, ±2,…)

md sin (m = 0, ±1, ±2,…)

Because at minima we should have, sin 0

If maximum is at m and Δw is the width, then the intensity should be zero at (m + Δw )

Nmd wm

))(sin(

π + π/Nπ - π/N

I

γ

Oblique incidence

Maxima

m

Nm

d

Nm

d

wm

wm

1)sin(

)sin(

Let us estimate the width of the mth order Principal maximum

dNmw

wm

cos

,1with )sin( Expand w

sin (A+B) = sin A cos B + cos A sin Bd sin θm = mλ

Dispersive power of grating is defined as

Width of principal maxima

When does the principal maxima get sharper?

md sinDifferentiate:

Chromatic resolving power of a grating

= m N

Chromatic resolving power of a prism

-

Reflection grating

Transmission grating

Bragg’s law X-ray diffraction from crystals:2d Sin θ = n λ

30 40 50 60 70 80 90 1000

2000

4000

6000

8000

650oC

550oC

450oC

250oC

as-quenched

bulk Cu (222

)(3

11)

(220

)

(200

)

(111

)

Inte

nsity

(cps

)

2 (degree)

X-ray diffraction from (95% Cu 5% Co) crystals:

68

Visibility of the fringes (V)

max min

max min

I IV

I I

Maximum and adjacent minimum of the fringe system

s

wacV sin

Reference: Eugene Hecht, Optics, 4th Ed. Chapter 12

s

YaII o

2cos4

o

w

w

o dYYYs

aAYI

2/

2/

2 )(cos)(

)(cos2

oo YYs

aAdYdI

s

wacV sin