What is Diffraction?
Diffraction
is the deviation of a wave from its straight ray propagation. This occurs when part of the wave is
obstructed by a boundary. Diffraction
is a wave phenomenon so light undergoes diffraction because of its wave
nature.
Why does it occur?
In
accordance with Huygen’s
Principle as a wave from a single source propagates through space
each wavefront
can be thought of as consisting of numerous individual secondary wavefronts. Diffraction occurs because part of the wave
(some of the secondary wavelets) that is obstructed changes phase or amplitude
and interferes with the rest of the unaltered wave (the rest of the secondary
wavelets). This causes multiple interference
patterns resulting in the diffraction of the wave.
What affects the diffractive behavior
of a light source at a boundary?
-The
wavelength of the light source
-The
distance from light source to the boundary
-The
distance from the boundary to the screen where the diffraction
pattern is projected
-The
size, shape, and nature of the boundary or the obstruction
What are the conditions for Fresnel
Diffraction?
Fresnel
diffraction occurs when either the distance
from the source to the obstruction or the distance from the obstruction to the
screen is comparable to the size of the obstruction. These comparable distances and sizes lead to
unique diffractive behavior.
Why is Fresnel Diffraction different
than other types of diffraction?
The
approximations made for Fraunhofer Diffraction are no longer
valid. The light source can no longer
be considered a planar wavefront at the aperture because it can longer be
approximated to originate at infinity.
It must be considered a spherical wavefront.
-The
relative phase
difference for a curved wavefront is not constant.
-Amplitudes
of the individual light waves (secondary wavefronts) at the observation point
are not equal because the distances traveled by each element, or wavefront, can
no longer be considered approximately equal.
Therefore, the intensity of light on the screen varies from
point to point.
How is Fresnel Diffraction dealt with
mathematically?
-All
parameters (lengths, distances, widths, etc.) must be considered in the
mathematical interpretation of Fresnel diffraction because of their comparable
sizes.
-One
can determine the diffraction pattern caused by Fresnel diffraction by
determining the intensity of light at each point on the viewing screen.
a b
The intensity at each point
on the screen can be determined by a parameter V
V
= v1 - v2
where
v1
= [s/2 + k][2(a +b)/ab(wavelength))]1/2
v2 = [-s/2 + k][2(a +b)/ab(wavelength))]1/2
and
k depends on q (the distance from the center of the screen
to the point (x,y) on the screen where you wish to determine the intensity)
v1 and v2
depend on k
so for each distance q there is a
different v1 and v2
therefore
there is a unique V for every point on the screen and its value
determines the intensity at that point
How is
Fresnel Diffraction represented graphically?
This parameter V and, therefore the resultant Fresnel diffraction pattern, can be represented graphically as well.
The
resultant amplitude (intensity) of light at an observation point can be
determined by vectorially adding (head to tail) each of the individual
amplitude components.
For Fraunhofer diffraction, where the relative phase of these elements is constant, the vector addition simply
gives an arc length of a circle and the intensity is the cord length of that
arc.
Fresnel diffraction introduces another phase shift due to the curvature of the wave. Therefore, each consecutive amplitude is
longer displaced the same amount (forming a circular arc length) rather these
elements bend into a spiral curve.
What
is a Cornu Spiral?
The
Cornu Spiral is the geometric depiction of the Fresnel diffraction patterns due
to different barriers. Each point on
the spiral has a value that corresponds to certain v1(k) or v2(k). Therefore, the cord length between
any two points v1
and v2 on the spiral, corresponding to
a certain k value gives V for that
value k (and thus q). V determines the intensity at that point q (which has both an x- and y-coordinate) on
the screen.
How do different barriers and apertures
affect the diffraction pattern?
One can see from the graph above that the cord
length between two points determines the intensity of the pattern produced
on the screen. Because the bottom
half of the barrier extends to infinity, v1 always has an infinite value. Since the Cornu spiral spirals inward
infinitely, an infinite value is found at the center of the spiral. The points 1,2,3,4,5 correspond to
different v2
values. The smaller the length of the cord the less intense the light is at that point on the screen. The
zero point on the graph (point 3) represents the
center of the screen (where the barrier
begins in this case). One can see that some light extends past
this barrier into the shadow produced by the barrier.
|
The
diffraction pattern due to an opaque circle is identical to that of a
circular aperture except for the bright spot in the center of the
pattern. The mathematics of
Fresnel's theory and the Cornu Spiral predict this bright spot but it is
often very hard to see. Poisson
actually tried to argue that Fresnel's theory was wrong because it
predicted such a spot that had not been found experimentally. This spot is now often referred to as Poisson's Spot. |
References
Dauger,
Dean. Fresnel Diffraction. http://www.dauger.com/fresnel/
Weisstein,
Eric. Fresnel Diffraction - from Eric's Wonderful World of Physics. http://scienceworld.wolfram.com/physics/FresnelDiffraction.html
Fresnel
Diffraction. http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/fresnelcon.html
Hecht,
Eugene. Optics 4ed. Addison Wesley, San Francisco : 2002.