Diffraction-Fraunhofer Diffraction

Diffraction
Fraunhofer Diffraction

Contents
1.Introduction
2.Examples
3.History
4.Mechanism-Ancient View
5.Mechanism-Modern Quantum Mechanical View
6.Fresnel Diffraction

7.Distinction between fresnel and fraunhofer diffraction
8.Fraunhofer Diffraction
9.Far Field
10.Focal planeof a positive lens as the far field plane
11.Diffraction due to a Single Slit
12.Circular Aperture

13.Limit of Resolution
14.Resolving Power of Grating
15.Determination of wavelength using Diffraction Grating
16.Fraunhofer Diffraction due to double slit
17.Fraunhofer diffraction due to n slits

1.Introduction
Diffraction refers to various phenomena that occur when a wave
encounters an obstacle or a slit. It is defined as the bending of waves
around the corners of an obstacle or through an aperture into the
region of geometrical shadow of the obstacle/aperture. The
diffracting object or aperture effectively becomes a secondary source
of the propagating wave. Italian scientist Francesco Maria Grimaldi
coined the word "diffraction" and was the first to record accurate
observations of the phenomenon in 1660.

2.Examples
The effects of diffraction are often seen in everyday life. The most
striking examples of diffraction are those that involve light; for
example, the closely spaced tracks on a CD or DVD act as a
diffraction grating to form the familiar rainbow pattern seen when
looking at a disc. This principle can be extended to engineer a
grating with a structure such that it will produce any diffraction
pattern desired; the hologram on a credit card is an example.

a).Diffraction in water c).Diffraction
b).Infinitely many points (3 shown) form at one point

3.History
Thomas Young performed a celebrated experiment in 1803
demonstrating interference from two closely spaced slits.Explaining
his results by interference of the waves emanating from the two
different slits, he deduced that light must propagate as waves.
Augustin-Jean Fresnel did more definitive studies and calculations of
diffraction, made public in 1815 and 1818 and thereby gave great
support to the wave theory of light that had been advanced by
Christiaan Huygens and reinvigorated by Young, against Newton's
particle theory.

4.Mechanism-Ancient View
In traditional classical physics diffraction arises because of the way in which waves propagate;
this is described by the Huygens–Fresnel principle and the principle of superposition of waves.
The propagation of a wave can be visualized by considering every particle of the transmitted
medium on a wavefront as a point source for a secondary spherical wave. The wave
displacement at any subsequent point is the sum of these secondary waves. When waves are
added together, their sum is determined by the relative phases as well as the amplitudes of the
individual waves so that the summed amplitude of the waves can have any value between zero
and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of
maxima and minima.

5.Mechanism-Modern Quantum Mechanical View
In the modern quantum mechanical understanding of light propagation through a slit (or
slits) every photon has what is known as a wavefunction which describes its path from the
emitter through the slit to the screen. The wavefunction — the path the photon will take —
is determined by the physical surroundings such as slit geometry, screen distance and
initial conditions when the photon is created. In important experimentsthe existence of the
photon's wavefunction was demonstrated. In the quantum approach the diffraction pattern
is created by the distribution of paths, the observation of light and dark bands is the
presence or absence of photons in these areas (no interference!). The quantum approach
has some striking similarities to the Huygens-Fresnel principle; in that principle the light
becomes a series of individually distributed light sources across the slit which is similar to
the limited number of paths (or wave functions) available for the photons to travel through
the slit.

6.Fresnel Diffraction
In optics, the Fresnel diffraction equation for near-field diffraction
is an approximation of the Kirchhoff–Fresnel diffraction that can be
applied to the propagation of waves in the near field. It is used to
calculate the diffraction pattern created by waves passing through
an aperture or around an object, when viewed from relatively close
to the object. In contrast the diffraction pattern in the far field
region is given by the Fraunhofer diffraction equation.

The Fresnel Diffraction Integral

Comparison between the diffraction pattern obtained with the Rayleigh-
Sommerfeld equation, the (paraxial) Fresnel approximation, and the (far-field)
Fraunhofer approximation.

7.Distinction between Fresnel and Fraunhofer
Diffraction

8.Fraunhofer diffraction
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves
when the diffraction pattern is viewed at a long distance from the diffracting object, and
also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction
pattern created near the object, in the near field region, is given by the Fresnel diffraction
equation.
The equation was named in honor of Joseph von Fraunhofer although he was not actually
involved in the development of the theory
A detailed mathematical treatment of Fraunhofer diffraction is given in Fraunhofer
diffraction equation.

9.Far Field
When the distance between the aperture and the plane of observation (on which the
diffracted pattern is observed) is large enough so that the optical path lengths from edges
of the aperture to a point of observation differ much less than the wavelength of the light,
then propagation paths for individual wavelets from every point on the aperture to the
point of observation can be treated as parallel. This is often known as the far field

10.Focal plane of a positive lens as the far field
plane
In the far field, propagation paths for individual wavelets from every point on the aperture
to a point of observation are approximately parallel, and the positive lens (focusing lens)
focuses parallel rays toward the lens to a point on the focal plane (the focus point position
depends on the angle of the parallel rays with respect to the optical axis). So, if the focal
length of the lens is sufficiently large such that differences between electric field
orientations for wavelets can be ignored at the focus, then the lens practically makes the
Fraunhofer diffraction pattern on its focal plane as the parallel rays meet each other at the
focus

11.Diffraction due to a Single Slit
In the single slit diffraction experiment, we can observe the bending
phenomenon of light or diffraction that causes light from a coherent
source interfere with itself and produce a distinctive pattern on the
screen called the diffraction pattern. Diffraction is evident when
the sources are small enough that they are relatively the size of the
wavelength of light. You can see this effect in the diagram below.
For large slits, the spreading out is small and generally
unnoticeable.

Case 2:Minimum Intensity Positions

When light from a point source passes through a small circular aperture, it does not
produce a bright dot as an image, but rather a diffuse circular disc known as Airy's disc
surrounded by much fainter concentric circular rings. This example of diffraction is of
great importance because the eye and many optical instruments have circular apertures. If
this smearing of the image of the point source is larger that that produced by the
aberrations of the system, the imaging process is said to be diffraction-limited, and that is
the best that can be done with that size aperture. This limitation on the resolution of
images is quantified in terms of the Rayleigh criterion so that the limiting resolution of a
system can be calculated.

The aperture diffraction pattern above was photographed with Fuji
Sensia 100ASA slide film and then digitized. With the time
exposure necessary to show the side lobes, the central peak was
washed out nearly white. The only retouching of the digital image
was to paint in the washed out part of the central maximum (Airy's
disc). The pinhole was made by placing aluminum foil on a glass
plate, sticking a straight pin into the aluminum foil, and then
rotating the foil. Several pinholes were made, and this one was the
closest to being round.

13.Limit of Resolution
The limit of resolution (or resolving power) is a measure of the ability of the objective lens
to separate in the image adjacent details that are present in the object. It is the distance
between two points in the object that are just resolved in the image. The resolving power
of an optical system is ultimately limited by diffraction by the aperture. Thus an optical
system cannot form a perfect image of a point.
For resolution to occur, at least the direct beam and the first-order diffracted beam must be
collected by the objective. If the lens aperture is too small, only the direct beam is
collected and the resolution is lost.

Consider a grating of spacing d illuminated by light of wavelength λ, at an angle of
incidence i.

Airy Discs
When light from the various points of a specimen passes through the objective and an
image is created, the various points in the specimen appear as small patterns in the image.
These are known as Airy discs. The phenomenon is caused by diffraction of light as it
passes through the circular aperture of the objective.
Airy discs consist of small, concentric light and dark circles. The smaller the Airy discs
projected by an objective in forming the image, the more detail of the specimen is
discernible. Objective lenses of higher numerical aperture are capable of producing smaller
Airy discs, and therefore can distinguish finer detail in the specimen.
The limit at which two Airy discs can be resolved into separate entities is often called the
Rayleigh criterion. This is when the first diffraction minimum of the image of one source
point coincides with the maximum of another.

From the equation it can be seen that the radius of the central
maximum is directly proportional to λ/d. So, the maximum is more
spread out for longer wavelengths and/or smaller apertures.
The primary minimum sets a limit to the useful magnification of the
objective lens. A point source of light produced by the lens is
always seen as a central spot, and second and higher order maxima,
which is only avoided if the lens is of infinite diameter. Two objects
separated by a distance less than θR cannot be resolved.

14.Resolving Power of Grating
The capacity of an optical instrument to show separate images of
very closely placed two objects is called resolving power. The
resolving power of a diffraction grating is defined as its ability to
form separate diffraction maxima of two closely separated wave
lengths.It is defined as the capacity of a grating to form separate
diffraction maxima of two wavelengths which are very close to
each other.

15.Determination of wavelength of light using
Diffraction Grating
Young's Double-Slit Experiment verifies that light is a wave simply
because of the bright and dark fringes that appear on a screen. It is
the constructive and destructive interference of light waves that
cause such fringes.
Constructive Interference:The following two waves (Fig. 1) that
have the same wavelength and go to maximum and minimum
together are called coherent waves. Coherent waves help each
other's effect, add constructively, and cause constructive
interference. They form a bright fringe.

Destructive Interference of Waves
In Fig. 2 however, the situation is different. When the wave with
amplitude A1 is at its maximum, the wave with amplitude A2 is at its
minimum and they work completely against each other resulting in a
wave with amplitude A2 - A1. These two completely out of phase waves
interfere destructively. If A2 = A1, they form a dark fringe.
The bright and dark fringes in Young's experiment follow these
formulas:
Bright Fringes: d sinθk = k λ where k = 0, 1, 2, 3, ...
Dark Fringes: d sinθk = (k - 1/2 ) λ where k = 1, 2, 3, ...

The above formulas are based on the following figures:

Check the following statement for correctness based on the above
figure.
Light rays going to D2 from S1 and S2 are 3(0.5λ) out of phase
(same as being 0.5λ out of phase) and therefore form a dark fringe.
Light rays going to B1 from S1 and S2 are 2(0.5λ) out of phase
(same as being in phase) and therefore form a bright fringe.
Note that SBo is the centerline.
Going from a dark or bright fringe to its next fringe changes the
distance difference by 0.5λ.

Diffraction grating is a thin film of clear glass or plastic that has a large number of lines
per (mm) drawn on it. A typical grating has density of 250 lines/mm. Using more
expensive laser techniques, it is possible to create line densities of 3000 lines/mm or
higher. When light from a bright and small source passes through a diffraction grating, it
generates a large number of sources at the grating. The very thin space between every two
adjacent lines of the grating becomes an independent source. These sources are coherent
sources meaning that they emit in phase waves with the same wavelength. These sources
act independently such that each source sends out waves in all directions. On a screen a
distance D away, points can be found whose distance differences from these sources are
different multiples of λ causing bright fringes. One difference between the interference of
many slits (diffraction grating) and double-slit (Young's Experiment) is that a diffraction
grating makes a number of principle maxima along with with lower intensity maxima in
between. The principal maxima occur on both sides of the central maximum for which a
formula similar to Young's formula holds true.

D = the distance from the grating to the screen
d = the spacing between every two lines (same as every two
sources)
If there are N lines per mm of the grating, then d, the space
between every two adjacent lines or (every two adjacent sources) is
d=1/N or N=1/d
The diffraction grating formula for the principal maxima
is:
d sin θk = k λ where k = 1, 2, 3, ...

A.Determination of (Lines/mm) of the Diffraction Grating:
a)Fix a laser pointer and the diffraction grating (placed in a target holder) on an optical bench as
shown. Try to make a distance D (grating to wall) of about 1.5m.

b)Make sure that the direction of the optical bench is normal (at right angle) to the wall and that
you are measuring the perpendicular distance D from the grating to the wall.
c)Measure y1 , y2 , and D with the precision of mm and record the values.
d)Angles θ1 and θ2 may now be calculated from the measured values as follows:

e)Use the tan-1 function (built-in in your calculator) to calculate θ1 and θ2 .
f )Use angles θ1 and θ2 along with the wavelength given on your laser pointer (in meters)
and the diffraction grating formula to calculate d, the distance between adjacent spaces
(sources) on the grating. Find d once on the basis of k = 1 and once on the basis of k =2 .
Theoretically, the two values you obtain for d must be equal; however, due to
measurement errors, they might be slightly different. Find an average value for d in
meters.
g)From d, determine N, the number of lines per mm of the grating.

2.Red and Violet Wavelengths:
a)Hold a diffraction grating close to your eye and look at the objects around you.You will
see a continuous spectrum of rainbow colors around bright objects. The diffraction grating
separates the colors of white light similar to what a prism does. White light coming from a
bright object separates into its constituent colors as it passes thru the grating and reaches
your eyes. If you are looking through a grating at a bright spot such as the filament of a lit
light bulb, you will be able to direct another person to move to the left or right and mark
the ends of the spectrum you are observing. By measuring the distance between each end
of the spectrum and the bright filament Yviolet or Yred and D the distance from the filament
to the grating (held by you), it is possible to calculate the angles θviolet and θred. Then, by
using the formula d sin θk = k λ , the corresponding wavelengths for violet and red light
can be determined.
Note that through the grating you will see more than one rainbow band. You will see two
or three bands on each side of the center. If you use the 1st band to one side of the center,
then k = 1. For the 2nd band k = 2, and for the 3rd band k = 3.

b)Place the optical bench near the board in your lab or class on a
somewhat high table.
c)Make sure that the optical bench stays at right angle to the board
and mount a light-bulb so that it almost touches the board. Turn the
light bulb on.
d)Hold a diffraction grating at a fixed distance D from the lit bulb.
When you look into the grating, your line of sight must be normal to
the board. A diagram of the set-up is shown below:

where V (in the diagram) is the Violet End of the spectrum, and R the Red end of it. Also
BV is the same as Y1V , the distance from the bulb to the violet end of the first fringe.
Similarly, BR is the same as Y1R, the distance from the bulb to the red end of the first
fringe.

a)While looking into the grating and observing the spectrum, guide
your partner to the extreme ends of the spectrum so that he/she can
mark those points on the board. Your partner must have previously
observed the same spectrum and have a good understanding of the
experimental procedure.
b)When those points are marked, double-check their precision and
measure distances BV and BR to the nearest cm as shown in the
figure. Also measure D.
c)From the data collected, calculate angles θviolet and θred and use
each in the above-mentioned formula separately to find the
corresponding wavelengths.

16.Fraunhofer Diffraction due to Double Slit
In the double-slit experiment, the two slits are illuminated by a single light beam. If the
width of the slits is small enough (less than the wavelength of the light), the slits diffract
the light into cylindrical waves. These two cylindrical wavefronts are superimposed, and
the amplitude, and therefore the intensity, at any point in the combined wavefronts
depends on both the magnitude and the phase of the two wavefronts.These fringes are
often known as Young's fringes.
The angular spacing of the fringes is given by:

The spacing of the fringes at a distance z from the slits is given by
where d is the separation of the slits.
The fringes in the picture were obtained using the yellow light from a sodium light
(wavelength = 589 nm), with slits separated by 0.25 mm, and projected directly onto the
image plane of a digital camera.
Double-slit interference fringes can be observed by cutting two slits in a piece of card,
illuminating with a laser pointer, and observing the diffracted light at a distance of 1 m. If
the slit separation is 0.5 mm, and the wavelength of the laser is 600 nm, then the spacing
of the fringes viewed at a distance of 1 m would be 1.2 mm.

Semi-Quantitative Explanation of Double-Slit
Fringes

Diffraction by a Grating
A grating is defined in Born and Wolf as "any arrangement which
imposes on an incident wave a periodic variation of amplitude or
phase, or both".
A grating whose elements are separated by S diffracts a normally
incident beam of light into a set of beams, at angles θn given by:
This is known as the grating equation. The finer the grating spacing, the greater the
angular separation of the diffracted beams.

If the light is incident at an angle θ0, the grating equation is:
The detailed structure of the repeating pattern determines the form of the
individual diffracted beams, as well as their relative intensity while the
grating spacing always determines the angles of the diffracted beams.
The image on the right shows a laser beam diffracted by a grating into n
= 0, and ±1 beams. The angles of the first order beams are about 20°; if
we assume the wavelength of the laser beam is 600 nm, we can infer that
the grating spacing is about 1.8 μm.

17.Fraunhofer Diffraction due to n slits(Grating)
An arrangement consisting of large number of parallel slits of the same width and
separated by equal opaque spaces is known as Diffraction grating.
Gratings are constructed by ruling equidistant parallel lines on a transparent material such
as glass, with a fine diamond point. The ruled lines are opaque to light while the space
between any two lines is transparent to light and acts as 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 the light is produced.
Theory: A section of a plane transmission grating AB placed perpendicular to the plane of
the paper is as shown in the figure.

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

BY:
P.JOHN ISAAC
BSc(MPCs)
121418468027
St.JOSEPH’S DEGREE AND PG COLLEGE, KING KOTHI,
HYDERABAD.

Diffraction-Fraunhofer Diffraction

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