DIFFRACTION

Interference

Two or more waves traveling in the same medium travel independently and can pass
through each other.  In regions where they overlap we only observe a single disturbance.
We observe  interference.  When two or more waves interfere,  the resulting
displacement is equal to the vector sum of the individual displacements.  If two
waves with equal amplitudes overlap in phase, i.e. if crest meets crest and trough meets
trough, then we observe a resultant wave with twice the amplitude.  We have
constructive interference.  If the two overlapping waves, however, are completely  out
of phase, i.e. if crest meets trough, then the two waves cancel each other out completely.
We have destructive interference.

Fraunhofer diffraction through a single slit 

The single slit
Assume light from a distant source passes through a narrow slit.  
According to the Huygen-Fresnel principle, the total field at a point y
on the screen is the superposition of wave fields from an infinite
number of point sources in the aperture region.  Each point s on the
wave front inside the aperture ( –a/2  ≤ s  ≤ a/2) is the source of a
spherical wave.  A distance r from the point s the electric field is due
to this point sources is
dE = (Asds/r)cos(kr-ωt).
If r0 is the distance from the point s = 0 on the optical axis to a point y
on the screen, then the contribution dE to the total amplitude on the
screen from the point at s = 0 is
dE(y) = (Asds/r0)cos(kr0-ωt).
Here As/r0 is the amplitude per unit width and ds is the infinitely small
width of a point source.  For off-axis points for which s  ≠ 0, the
distance is longer or shorter than r0 by an amount .
The contribution dE(y) to the total amplitude on the screen from an
off-axis point (s ≠ 0) is
dE(y) = (Asds/(r0+ (s)))cos(k(r0+ (s))-ωt).
To find the total amplitude E(y) we have to add up  the contributions
from all points on the aperture.  Because there are an infinite number
of points, the sum becomes an integral.  .
We define sinθ =   /s.  Since r0  >> , we approximate 1/(r0+ ) with
1/r0.  However we cannot drop the inside the cosine function, since
k (s) is not necessarily much smaller than 2π.

The time-averaged intensity has a peak in the center with smaller
fringes on the sides.
For small angles we may approximate sinθ ≅ θ.  Then the first zeros on
the sides of the central peak occur when
πasinθ/λ ≅ πaθ/λ = π, or θ = λ/a.
On the screen we see a pattern similar to that shown in the figure
below. The positions of all maxima and minima in the diffraction pattern from a single slit can
also be found from the following simple arguments.
When light passes through a single slit whose width w is on the order of the wavelength
of the light, then we observe a single slit diffraction pattern.  Huygen's principle tells us
that each part of the slit can be thought of as an  emitter of waves.  All these waves
interfere to produce the diffraction pattern. Consider a slit of width w as shown in the
diagram below.
For light leaving the slit in a particular direction, we may have destructive interference
between the ray at the top edge (ray 1)and the middle ray (ray 5).  If these two rays
interfere destructively, so do rays 2 and 6, 3 and 7, and 4 and 8.  In effect, light from one
half of the opening interferes destructively and cancels out light from the other half.  Ray
1 and ray 5 are half a wavelength out of phase if ray 5 must travel 1/2 wavelength further
than ray 1.  We need
(w/2)sinθ = λ/2 or wsinθ = λ
for destructive interference to produce the first dark fringe.  Other dark fringes in the
diffraction pattern produced by a single slit are found at angles q for which
wsinθ = mλ.
If the interference pattern is viewed on a screen a distance L from the slits, then the
wavelength can be found from the spacing of the fringes.  We approximately have
λ = zw/(mL),
where z is the distance from the center of the interference pattern to the mth dark line in
the pattern.  That applies as long as the angle q is small, i.e. as long as z is small
compared to L


 The progression to a larger number of slits shows a pattern of narrowing the high
intensity peaks and a relative increase in their peak intensity. This progresses toward the
diffraction grating, with a large number of extremely narrow slits. This gives very narrow
and very high intensity peaks that are separated widely. Since the positions of the peaks
depends upon the wavelength of the light, this gives high resolution in the separation of
wavelengths. This makes the diffraction grating like a "super prism".

Plane transmission diffraction grating 


When there is a need to separate light of different wavelengths with high resolution, then
a diffraction grating is most often the tool of choice. This "super prism" aspect of the
diffraction grating leads to application for measuring atomic spectra in both laboratory
instruments and telescopes. A large number of parallel, closely spaced slits constitutes a
diffraction grating. The condition for maximum intensity is the same as that for the
double slit or multiple slits, but with a large number of slits the intensity maximum is
very sharp and narrow, providing the high resolution for spectroscopic applications. The
peak intensities are also much higher for the grating than for the double slit.

Mercury spectrum

The two aspects of the grating intensity relationship can be illustrated by the diffraction
from five slits. The intensity is given by the interference intensity expression
modulated by the single slit diffraction envelope for the slits which make up the grating:
Total intensity expression:
Grating Intensity Comparison A diffraction grating is the tool of choice for separating the colors in incident
light.Diispersiive Power of a Diiffractiion Gratiing
The dispersive power of a diffraction grating is defined as the rate of
change of the angle (θ) of diffraction with the wave length(λ) of light.
(dθ/dλ)
For planer transmission grating, we know
Diff. eq(1) w.r.t λ, we get
Linear dispersive power

Resolving power

The resolving power of an optical instrument is its ability to separate the images of two
objects, which are close together.  Some binary stars in the sky look like one single star
when viewed with the naked eye, but the images of the two stars are clearly resolved
when viewed with a telescope.
Why?
The merging of the images in the eye is caused by diffraction.
If you look at a far-away object, then the image of the object will form a diffraction
pattern on your retina.  For two far-away objects,  separated by a small angle q, the
diffraction patterns will overlap.  You are able to resolve the two objects as long as the
central maxima of the two diffraction patterns do not overlap.  The two images are just
resolved when one central maximum falls onto the first minimum of the other diffraction
pattern.  This is known as the Rayleigh criterion.  If the two central maxima overlap the
two objects look like one
(a + b) sinθ = nλ...................( )1

The width of the central maximum in a diffraction pattern depends on the size of the
aperture, (i.e. the size of the slit).  The aperture of your eye is your pupil.  A telescope has
a much larger aperture, and therefore has a greater resolving power.  The minimum
angular separation of two objects which can just be resolved is given by qmin = 1.22l/D,
where D is the diameter of the aperture.  The factor of 1.22 applies to circular apertures
like the pupil of your eye or the apertures in telescopes and cameras.
The closer you are to two objects, the greater is the angular separation between them.  Up
close, two objects are easily resolved.  As your distance from the objects increases, their
images become less well resolved and eventually merge into one image.
Rayleigh’s Criterion for Resolution
“The two point sources or spectral lines of equal intensity are just resolved by an optical
instrument when the central maximum of diffraction pattern due to one falls on the first minimum
of the diffraction pattern of the other”