a
b
For analyzing this phenomenon, we use the Huyghen-Fresnel principle.
1.2 Huygen-Fresnel principle:
For analyzing properties of wave processes, Huygen introduced the
concept of wave front, and the rule how to draw a wave front from
known sharp of at some former time
Huygen’s principle (1678):
All points on wavefront are point
sources for spherical secondary
wavelets with speed, frequency
that equal to initial wave. The wave front at a
later time is the envelope of these wavelets.
Wavefront at
t=0
Wavefront at
time t
Basing on Huygen’s principle one can interpret diffraction as
interference of light from secondary sources. For example, every
point of circular aperture becomes a secondary source, and what we
see in the screen is the interference of secondary sources.
But this interpretation is only qualitively, for a quantitive analyze,
we need more. It has not been known from Huygen’s principle:
how can determine the amplitude and the phase of secondary waves ?
Fresnel’s complementary statement:
Observation
point
Wave
front
Fresnel states that for the vibration
at P due to waves from the secondary
source dS we have the following formula:
dS
where
a
0
&
(t +
)
→ the amplitude &
phase of vibration of secondary sources at dS on the wave front S
K
→ a coefficient which depends on the angle
K
decreases
as
increases;
K = 0
when
= / 2
The total vibration
at
P
is
1.3 Analysis of diffraction through circular aperture:
Having Huygen-Fresnel principle we tend to analyze the phenomenon
of diffraction through a circular aperture.
1.3.1. The Frsenel method to devide a spherical wave front into
adjacent zones (Fresnel zones):
1-th zone
2-nd zone
3-rd zone
4-th zone
Calculate the area of the
m-th zone:
where S
m
is the area of
the m-th spherical segment:
(h
m
– the hight of segment)
h
m
is defined from the following equation:
Remark: The area of a zone does not depend on m. It means that the
areas of zones are approximatly the same (for values of m that are not
large).
We have also the formula for the radius of the m-zone:
r
m
is proportional to
From the Fresnel formula for dξand all the described properties of
Fresnel zones we can lead to the following formula for the amplitudes
of vibrations at P which are sent from Fresnel spherical zones :
from the 1-st zone
from the m-th zone
,,,
…
Further, the phases of vibrations from two adjacent zones have the
phase difference → we can write for the total amplitude of
vibrations at P:
It equals a half of amplitude due to
the 1-st zone !
1.3.2 Come back to the experiment of diffraction through an aperture:
→ What happens if there is a screen with circular aperture in the
light propagation line ?
Suppose that the part of wave front based on the aperture
incorporates m zones:
+ for odd m
- for even m
(odd m)
(even m)
Screen with aperture
Screen
y y
Conclusions of diffraction picture on the screen:
Depending on the size of aperture, the number of open zones m is
odd or even:
• If m is odd → at the center point P there is a light spot
• If m is even → at the center point P there is a dark spot
Besides the center point P, at other points in the screen, the light
intensity has maxima or minima, depending on the distance from
the center point P.
Owing to the symmetry, light & dark fringers on the screen are
circles centered at P.
Comparison the Fresnel zone pictures according to
a) the axis SP; b) the axis SP’; c) the axis SP”
a)
b)
c)
To understand why the light intensity is different at P, P’ & P”
we describe the Fresnel zones for three cases, and make a
comparison. Here we suppose that the aperture exposes 3 zones.
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