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Optical Diffraction
Parag Bhattacharya
Department of Basic Sciences
School of Engineering and Technology
Assam Don Bosco University
Concept of Diffraction
Optical Diffraction is a general characteristic of wave phenomena
that occurs whenever a portion of a wavefront of light is obstructed in
some way, leading to the deviation of light from rectilinear
propagation.
The Huygens-Fresnel Principle:
Every unobstructed point on a wavefront, at a given instant,
serves as a source of spherical secondary wavelets (with the same
frequency as that of the primary wave). The amplitude of the optical
field at any point beyond is the superposition of all these wavelets
(considering their amplitudes and relative phases).
In interference phenomena, the superposing beams originate from a
discrete number of sources.
In diffraction phenomena, the superposing beams originate from a
continuous distribution of sources.
The longer the wavelength compared to the aperture, the greater is
the effect of diffraction.
Two broad categories of diffraction:
Fresnel Diffraction
 Also called near-field diffraction.
 Either the source or the screen
or both are at finite distance from
the diffraction aperture.
 Spherical wavefronts involved.
Fraunhofer Diffraction
 Also called far-field diffraction.



The shape of the aperture is
recognisable with slight fringing
around its periphery.


As separation between the
aperture and the screen
increases, the image of the
aperture becomes more
structured and fringes become
more prominent.

Both the source and the screen
are at infinite distance from the
diffraction aperture.
Planar wavefronts only involved.
The projected pattern is
comprised of intensity variations,
and bears little or no
resemblance to the aperture.
As separation between the
aperture and the screen
increases, the size of the pattern
changes but its shape remains
the same.
Two broad categories of diffraction:
Fresnel Diffraction
 Example:
Fraunhofer Diffraction
 Example:
Fraunhofer Diffraction due to a
single slit
Setup for single-slit diffraction
Δ
ds
θ
s
b
θ
θ
y
P
f
Single-slit
Focussing
Lens
Screen
Strategy for applying the superposition principle

Portion of wavefront at the slit acts as an extended
source.

Intensity at point P on the screen due to parallel
rays from different portions of wavefront at the slit.

We consider each infinitessimal portion ds of the
wavefront at the slit as a source.

Then we find the electric field due to ds.

The resultant field is calculated by integrating over
the entire slit width b.
Single-slit Diffraction:
We define the field strength, i.e., field per unit length E0, to be
uniform over the wavefront (since the wavefront is planar).
Then the source field at length element ds is directly proportional to
ds, i.e., E0ds
Thus, the field at P due to ds is
(1)
where r is the radial distance from a source
For the wave from the centre of the slit, i.e., s = 0, we set r = r0
Then for a wave originating from a source ds at a height s, we
have, r = r0 + Δ.
Therefore,
(2)
Single-slit Diffraction:
(2
)
Since Δ is very small compared to r0, it can be ignored in the
amplitude term.
But cannot be ignored in the phase (Why???)
From the geometry, we have Δ = s sin θ
Then (2) becomes:
(3)
Then the total field at P is:
(4
)
Single-slit Diffraction:
(4)
This integration gives:
Substituting the limits of integration, we obtain,
We define a new quantity:
(5)
Single-slit Diffraction:
(5)
This gives:
Therefore, we finally obtain,
(6)
Single-slit Diffraction:
Consider 2 parallel waves emitted from the slit:
Δ
b/2
θ
θ

1st wave from the centre

2nd wave from either endpoint
The separation between these 2 points on the
wavefront portion is b/2.
Say, these 2 waves make an angle θ with the
optic axis.
Then path difference between these waves is:
And the corresponding phase difference is:
Therefore, β can be interpreted as the phase
difference (at point P) between 2 waves emitted
from points on the slit separated by b/2.
Single-slit Diffraction:
(6)
The encircled quantity gives the amplitude of the resultant field at
the point P.
Thus, the amplitude at P:
and
Therefore the intensity I is:
Single-slit Diffraction:
Hence,
(7)
where,
We define a new function called the sinc function as:
Then (7) becomes:
(8)
In (8), I0 is a constant.
So the intensity variation due to Fraunhofer diffraction follows the
square of the sinc function.
Hence, we now briefly have a look at the sinc function.
The sinc function


The sinc function has the property that its value approaches 1 as
its argument approaches 0
The zeroes of the sinc function occur when sin β = 0, i.e.,
sinc β
1
-4π
-3π
-2π
-π
0
π
2π
3π
4π
β
sinc β
The sinc function
1
-4π
-3π
-2π
-π
π
0
2π
3π
4π
β
The sinc squared function
sinc2 β
1
-4π
-3π
-2π
-π
0
π
2π
3π
4π
β
Intensity distribution
The intensity varies as:
The maximum value of the intensity is I0. This occurs at β = 0. This
position is known as the central maximum.
The dark regions occur at the zeros of the sinc function. These
occur at β = mπ.
These positions are known as points of minima.
(9)
From the geometry of the set-up, y is small compared to f.
Hence θ is also small.
Thus, using the small-angle approximation:
Intensity distribution
(9)
Thus, (9) becomes,
Central
Maximum
I
Points of
Minima
-y
-4λf/b
-3λf/b -2λf/b
Secondary
Maxima
I0
-λf/b
0
λf/b
2λf/b
3λf/b
4λf/b
y
Fraunhofer Diffraction due to a
plane transmission grating
Plane Transmission Grating
An arrangement of a large number of close, parallel, straight,
transparent and equidistant slits, each having equal width a, with
neighboring slits separated by an opaque region of width b.
a
b
d
d
The grating element (or grating period) is d = a + b
Typical there can be 15,000 lines (slits) per inch on a grating.
Plane Transmission Grating
Consider light incident on N slits of a grating.
The resultant diffraction field at any point on the screen is obtained
by integrating the field due to all the elements over N slits.
O'1
O
O1
Transmission
Grating
Focussing
Lens
Screen
Plane Transmission Grating
The integration over N slits gives the resultant field at a point P on
the screen as:
(1)
(Derivation not necessary)
The quantity in square brackets is the amplitude of the resultant
field.
Here,
and
(2)
where, d: grating element and a: slit size
Then the intensity variation on the screen is:
(3)
Diffraction
Term
Interference
Term
Intensity variation
(3)
I
Central
Maximum
2π
3rd Order
π
2nd Order
1st Order
0th Order
α:
β:
sin θ :
Principal
Maxima
3π
Intensity variation

1
α:
β:
sin θ :
For a grating with N slits, between 2 successive
principal maxima there are:

N – 1 points of minima

N – 2 secondary maxima
π
2π
3π
Positions for the principal maxima
(3)
Diffraction
Term
Interference
Term
where,
The positions of all the minima and principal maxima are governed
by the interference term.
These positions occur when we have
(4)
principal maxima occur for
secondary minima occur for p = all other integer values
(5)
Positions for the principal maxima
(4)
principal maxima occur for
As per (5), for principal maxima, we must have
Thus, the condition for principal maxima becomes:
(5)
Positions for the principal maxima
Thus, the condition for principal maxima becomes:
Therefore,
(6)
Equation (6) is known as the diffraction grating equation.
The quantity m is called the order of the diffraction.
Measurement of wavelength of light
Experimental set-up in the lab:
m=2
m=1
m=0
Laser
m=–1
Transmission
Grating
m=–2
Screen
Measurement of wavelength of light
Experimental set-up in the lab:
m=2
S2
m=1
θ2
S1
θ1
m=0
L
For any order m, we have
We have,
(6)
From (6) and (7), we have,
(7)
Fresnel Zone Plates

A zone plate is a device used to focus light.

A zone plate uses diffraction in order to focus light (unlike lenses
and spherical mirrors that use reflection or refraction).

Based on the analysis by Augustin Jean-Fresnel.

A zone plate consists of a set of radially symmetric rings, known
as Fresnel zones, which alternate between opaque and
transparent.

Light hitting the zone plate will
diffract around the opaque
zones.

The zones can be spaced so
that the diffracted light
constructively interferes at the
desired focus, creating an image
there.
The spacings between the opaque and transparent zones are such
that light diffracted through the transparent zones constructively
interfere at the desired focus.
Fresnel Zones
Suppose we have a zone plate with the desired focal length f.
We consider a source point located at a distance r from the centre
of the zone plate.
Then the optical path length between S
and F is:
(1)
S
r
O
l
f
F
If S is at the centre (r = 0), then
(2)
Fresnel Zones
Thus wave from any source point would have a path difference with
that from the central source point given by l – l0
All the source points for which the path difference with l0 is less
than λ/2, would constructively interfere at the focus, i.e.,
(3)
S
r
O
l
f
F
All points on the plate that satisfy (3)
form a circular region, called the 1st
zone (shaded in red)
Fresnel Zones
Source points outside the 1st zone that have path difference
greater than λ/2, but less than λ, would destructively interfere with
the sources in the 1st zone, at the focus, i.e.,
(4)
S
r
O
l
f
F
All points on the plate that satisfy (4)
form an annular region, called the 2nd
zone (shaded in blue)
Fresnel Zones
Thus, we have:
1st Zone:
2nd Zone:
Proceeding in this manner, the nth zone may be defined as the
collection of those points on the plate whose path difference satisfy
the following relation:
(5)
where, n is a positive integer such that 1 ≤ n ≤ N, and N is the total
number of zones in the plate.
Point sources from odd zones (n = 1, 3, 5,...)
-> interfere constructively with the
1st
zone
Point
sources from even zones (n = 2, 4, 6,...)
-> interfere destructively with the
1st zone
Fresnel Zones
Point sources from odd zones (n = 1, 3, 5,...)
-> interfere constructively with the
1st
zone
Point
sources from even zones (n = 2, 4, 6,...)
-> interfere destructively with the
1st zone
In order to maximise the constructive interference at the focus, the
zone plate is constructed as:

Odd-numbered zones are made transparent (to enable light to
pass through)

Even-numbered zones are made opaque (to block the light)
Note: The choice of the 1st zone as the reference is arbitrary. If the
zone plates are inverted, i.e., opaque odd zones and transparent
even zones, the effect would be the same.
(Can you figure out why?)
Fresnel Zones
Zone Plate with transparent
odd zones
Zone Plate with transparent
even zones
(Positive Zone Plate)
(Negative Zone Plate)
Determination of the zone radii
(1)
(2)
The outer zone boundary for the nth zone is given as:
(6)
Using (1) and (2) in (6), we obtain,
Solving for rn, we obtain the zone radius for the nth zone as:
(7)
Determination of the zone radii
(7)
Generally,
Hence,
Thus, the 2nd term in (7) is negligible compared to the 1st term.
Therefore, in the 1st approximation,
(8)
(8) shows that
Thus,
Home-work
Q1. Arrive at the expression for the surface area of
the nth zone.
Q2. Show that, in the 1st approximation, the area of
any zone is constant.
References:
1)
2)
3)
“Optics”, 4th Ed., Eugene Hecht & A. R. Ganesan,
Pearson
“Introduction to Optics”, 3rd Ed., Frank L. Pedrotti,
Leno M. Pedrotti & Leno S. Pedrotti, Pearson
“Physics for Engineers and Technologists”, Samrat
Dey, EBH Publishers
Thank You!
Assam Don Bosco University