It is important to realize that there is not physical difference between interference and diffraction. However, it is traditional to consider a phenomenon as interference when it involves the superposition of only a few waves, and as diffraction when a large number of waves are involved. Another aspect that is important to understand is the fact that every optical instrument only uses a portion of the full incident wavefront. Because of this, diffraction plays a significant role in the detailed understanding of the light train through the device. Even in all of the potential defects in the lens system were eliminated, the ultimate sharpness of the image would be limited by diffraction.
In order to begin to understand diffraction, let's return to Huygen's principle. Recall that this told us that each point on a wavefront can be viewed as a source of secondary spherical wavelets. From this, the progress of the wavefront as it moves through space can theoretically be determined. At any particular time, the shape of the wavefront is made up from the envelope of the secondary wavelets. There is a problem with this approach. In only considering the envelope of the secondary wavelets, Huygen's principle ignores most of the secondary wavelet and retains only the portion which is common to the envelope. As a result of this, Huygen's principle is unable to account for the details of the diffraction process. An example of this can be seen by comparing radio and visible light waves. Radio waves are seen to "bend" around large objects, such as buildings and telephone poles, but visible light creates a fairly distinct shadow. Huygen's principle is independent of any wavelength consideration and predicts the same wavefront configuration in both situations.
This problem was resolved when Fresnel added to Huygen's principle with the idea of interference. The resulting principle, known as the HuygensFresnel principle, states that every unobstructed point of a wavefront, at a given instant in time, 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, taking into consideration their amplitudes and relative phases. As an example of this, consider the following drawing
Define the maximum optical path length difference
as 
.
Assume that 
.
Then when 
,
we also have that 
.
Since the waves were initially in phase, they must all interfere constructively,
no matter where P happens to be. On the other hand, when 
,
the area where 
is limited to a small region extending out directly in from of the aperture,
and it is only there that all of the wavelets interfere constructively.
Beyond this region, some of the wavelets can interfere destructively. This
is the geometric shadow. Remember that the idealized geometric shadow corresponds
to 
.
,
(14.1)
where ri is the distance from the ith oscillator to the observation point P. The sum of the interfering spherical wavelets yields a composite electric field at P that is the real part of
(14.2)
This can be rearranged to be
.The phase difference between adjacent sources
is obtained from the expression 
,
where the maximum optical-path length difference is 
in a medium with an index of refraction n. But, since d is
the distance between two adjacent oscillators, it can be easily seen that
d sin
= r2 - r1. Thus, the field at P becomes
,
(14.3)
where 
is the distance from the center of the line of oscillators to the point
P.
Once we know what the electric field is from the line of oscillators, the flux density can be determined. Using (14.3), the flux density is
(14.4)
where I0 is the flux density
that comes from any single source arriving at P. The principal maxima
occur when 
,
where m = 0, 
1, 2,
…, or using the definition 
,
.
(14.5)
As an example of this, consider an idealized
line source of oscillators, i.e. one for which the width of the slit is
much less than 
.
Each point source emits a spherical wavelet,
(14.6)
where 
0
is the source strength. If we consider a infinitesimal segment of the array
dyi, there are 
sources, where D is the entire length of the array and N
is the total number of sources in the array. Letting N approach
infinity, energy conservation requires that 0
goes to zero in order for the total energy output to remain finite. This
allows us to define the source strength per unit length of the array
as
.
(14.7)
Using this, the electric field at some point P is found from
(14.8)
where r(y) is the distance from the element under consideration to P.
(14.9)
If we write r as an explicit function of y, we get
(14.10)
where
is measured from the x-z plane. The non-linear terms in y
can be ignored when their contribution to the phase is insignificant. This
is true whenever 
is negligible; a condition that is satisfied for all values of
whenever R is large. This is known as the Fraunhofer condition,
where the distance r is linear in y. In turn, this leads
to the fact that the distance to the point of observation, and thus the
phase, can be written as a linear function of the aperture variables.
Returning to Eq. (14.10), we see that Eq. (14.9) becomes
which can be integrated to yield
.
(14.11)
In order to simplify (14.11), define
(14.12)
Then
.From E, the irradiance can be determined to be
.
(14.13)
Notice that when
= 0, 
and I()
is a maximum. This maximum is known as the principal maximum, and
the irrandiance resulting from an idealized coherent line source in
the Fraunhofer approximation becomes
.
(14.14)
Since 
,
when D >> ,
the irradiance drops extremely rapidly as
deviates from zero. Also, when D >> ,
the source, which is a relatively long coherent line source, can be viewed
as a single point emitter radiating predominately in the forward direction.
When the opposite is true, namely that
>> D, then 
is small and the irradiance remains essentially constant for all values
of .
This means that the line source more closely resembles a point source emitting
spherical waves.
We wish to find the far-field flux density at
some arbitrary point P. According to the Huygens-Fresnel principle,
a differential area dS within the aperture may be envisioned as
being covered with coherent secondary point sources. Since dS is
much smaller in extent than ,
all the contributions at P will remain in phase and interfere constructively.
This is true regardless of .
Let A
be the source strength per unit area, and assume that it is constant over
the entire aperture. Then the disturbance at P due to dS
is either the real or imaginary part of
.
(14.15)
The distance from dS to P is
.
(14.16)
The Fraunhofer condition allows us to replace r by R provided that the aperture is relatively small. However, we also need to have a constant phase. This causes Eq. (14.16) to become
.
(14.17)
When R is very large compared to the aperture, this simplifies to
The total electrical field at P is then
.
(14.18)
If the incident wave is originally propagating in the x direction and the aperture is oriented in the y-z plane, then the integral becomes
(14.19)
where A = ab is the area of the
aperture, 
,
and 
.
Thus, the flux density is
(14.20)
where I(0) is the irradiance at the center of the aperture.
,
(14.21)
where
.From symmetry, the result must not depend on 
,
so we can set it to zero in Eq. (14.21). Consider the azimuthal integral
first. The quantity
(14.22)
is known as a Bessel function of the first kind. Comparing it to the azimuthal integral in (14.21), we see that
(14.23)
Using the recurrence relationship for Bessel functions,
where
,this can be evaluated as
(14.24)
where the relationship 
was used. The irradiance becomes
.
(14.25)
At the center of the aperture, the irradiance is
(14.26)
and so Eq. (14.25) becomes
.
(14.27)
.
(14.28)
If 
is the corresponding angular measure, then, using the fact that
,we find that
The Airy disk for each source will be spread
out over a half width ,
centered on the geometric image point. If the angular separation of the
two points is 
,
and if 
the images will be distinct and easily resolved. As the two sources approach
each other, their respective images would also approach each other, overlap,
and blend into a single set of fringes. We can use Lord Rayleigh's criterion
to determine when the two objects are just resolved. This criterion states
that the resolution of two fringes of equal flux density requires that
the principal maximum of one coincide with the first minimum of the other.
Using this criterion, the two objects are just resolved when the
center of one Airy disk falls on the first minimum of the Airy pattern
of the other object. Thus, the angular limit of resolution is
.
(14.29)
We must reconsider the Huygens-Fresnel principle
more closely to understand what is happening in this region. Recall that
we can envision every point on the primary wavefront as a continuous emitter
of spherical secondary wavelets. However, if each wavelet is radiating
uniformly in all directions, then there would be a wave traveling back
towards the source in addition to the normal outgoing wave. Since no such
wave is found experimentally, we must somehow modify the radiation pattern
of the secondary emitters. This can be done by introducing the obliquity,
or inclination factor, K().
The obliquity is used to describe the directionality of the secondary emission.
Kirchoff was the first person to analytically define the obliquity as
(14.30)
where
is the angle made with the normal, k, to the primary wavefront.
Consider a spherical wave emitted from a point
S at a time t = 0. A time t' later, the wave has a
radius of 
and is described by
.
(14.31)
We can divide the wavefront into a series of annular regions. The boundaries of the various regions correspond to the intersections of the wavefront with a series of spheres centered at some observation point, P, with radii given by
where r0 is the minimum distance
from P to the wavefront. These spheres are known as the Fresnel,
or half period, zones. Since each zone is finite in extent,
we can define a ring shaped differential area element dS associated
with the zone. All of the point sources within dS are coherent,
and we can assume that each one radiates in phase with the primary wave.
Thus, in any zone each of the secondary wavelets travel a distance r
to reach P at a time t, and all of the wavelets arrive there
with the same phase, 
.
We can assume that the source strength per unit area A
of the secondary emitters on dS is proportional to the amplitude
of the primary wave,
.The contribution to the optical disturbance at P from the secondary sources on dS is therefore
.
(14.32)
The obliquity factor must vary slowly and thus can be assumed to be constant over a single Fresnel zone. Consider the following drawing
The area element dS is seen to be
which, combined with the law of cosines
yields
(14.33)
Substituting this into Eq. (14.32) and integrating yields
(14.34)
If there are a total of m zones on the wavefront, then the sum of the optical disturbances from all m zones at P is
(14.35)
If m is odd, the series can be written in one of two ways. The first way is
(14.36)
while the second is
(14.37)
This means that either 
or 
.
Using Eqs. (14.36) and (14.37), these conditions become
(14.38)
and
(14.39)
from which is can be concluded that
(14.40)
If m is even, similar arguments leads to a result of
(14.41)
Fresnel showed that the last contributing zone satisfied
so that (14.40) and (14.41) both reduces to
(14.42)
Thus, we see that the optical disturbance generated by the entire unobstructed wavefront is approximately equal to half the contribution from the first zone.
Last updated: July 24, 1997
Comments to: D-Suson@tamuk.edu