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Diffractive Effects

Diffraction occurs whenever a portion of a wavefront is obstructed by some opaque object. Close
examination of a shadow under a bright source will reveal that it is made up of finely spaced bright and dark
regions. In this case, light appears not to be propagating in a straight line; the obstacle alters the amplitude
or phase of the light waves such that the regions of the wavefront that propagate beyond the obstacle will
interfere with each other. It is crucial to remember that there is no physical difference between
interference and diffraction. In general interference concerns situations where only a few
waves are interfering, while diffraction concerns a large number of interfering waves. This
distinction is arbitrary. Diffraction occurs with other sorts of waves too. Water waves or sound
waves, for example spread out after they go through a narrow aperture, or bend into the 'shadow' region
behind an obstacle.

The Huygens-Fresnel Principle

In 1609 Christian Huygens proposed that in order to predict how a wavefront will propagate:

Every point on a propagating wavefront serves as the source of spherical secondary wavelets, such that
the wavefront at a later time is the envelope of these wavefronts.

This principle was extended by Fresnel to take into account the wavelength of the waves. The Huygens-
Fresnel principle states:

Every unobstructed point of a wavefront is a source of spherical secondary wavelets with the same
frequency as that of the primary wave. The amplitude of the resultant wave at any forward point is the
superposition of these wavelets (considering their amplitudes and relative phases). The application of this
principle is to waves near an aperture is shown in .

As can be seen from this diagram, if the wavelength of the wave is large compared to the width
of the aperture, the waves will spread out at large angles into the 'shadow' region behind the obstruction.
For larger apertures, the amount of diffraction diminishes. This occurs because light from the spherical
point sources in the aperture interferes constructively in the shadow region. This is always the case when
the λ is greater than the maximum path difference between point sources (| AP - BP| is this
maximum -- then there must always be some sources out-of-phase by λ that interfere
constructively), which occurs when lambda is greater than the width of the slit.

What all this means is that an analysis of diffraction effects can be performed by treating the aperture as
filled with an infinite number of point-sized oscillators each producing spherical wavelets that interfere
with each other to produce a diffraction pattern. We can imagine this in the case of light as the electric
field vibrating the electrons in the obstruction. Using Maxwell's equations we find that the field due to the
oscillation of the electrons exactly cancels the field due to the light wave in the region beyond the
obstruction. If an aperture is removed from the obstruction, however, the electron-oscillators are removed
along with it and hence light will propagate beyond the screen. Up to a sign, it is as if the source and screen
had been removed leaving on the oscillators distributed over the aperture. Thus, up to a sign, we can
consider the pattern created by the aperture as the same as that created by point-source oscillators
distributed over the aperture.

Single Slit Diffraction

When a pattern created by an aperture is viewed on a nearby screen we see a clearly recognizable image of
the aperture with accompanying fringes. This is called Frenel or near-field diffraction. At larger
distances the pattern spreads out much more, such that the image of the aperture is likely to be
unrecognizable; in this region, moving the screen changes only the size of the pattern and not the shape.
This is called the far-field or Fraunhoffer diffraction. We will only treat the latter in the case of a single
slit. Fraunhoffer diffraction is the (linear) limit in which the incoming and outgoing wavefronts are
essentially planar. This usually occurs when L, the distance between the aperture and the screen is L > d^{2}/λ, where d is the width of the aperture. The Fraunhoffer condition can be achieved in practice
by placing a lens with its focus at the source between the source and the aperture
and another lens between the aperture and the screen, with its focus at the screen.

Consider a single slit of width d. Assume that plane, monochromatic waves fall on the slit. Because the
screen on which the diffraction pattern is to be observed is far away (compared to the width of the slit),
light rays heading for any point can be considered essentially parallel. Clearly, all rays heading towards the
center of the screen will arrive in phase, and produce a maximum. Consider waves heading off at some
angle θ_{m} such that the path difference between the point source at A and the point source at B is
λ.

A ray passing through the center of the slit will have a path length exactly λ/2 greater than that
from the source at A, and hence these two waves will interfere destructively. Now consider a point adjacent
to A; the light emitted with have a path difference exactly λ/2 different to light from a point just above
the central point, and will cancel it out. Similarly, for every point between A and the center, there will be a
corresponding point λ/2 away between the center and B that will destructively interfere with it. Hence there
is effectively no light emitted it the direction θ_{m} and it corresponds to a minimum. A similar situation arises
when the path difference between A and B is any whole number of wavelengths--such a situation is shown in , iv). On the other hand, when the path difference between A and B is a half-integer multiple of the
wavelength, such as in iii), there will only be partial cancellation. All the emitters between A and a
point one-third of the way to B will cancel with emitters in the middle third of the slit, leaving a third of the
emitters with nothing to cancel with (that is, however you try to pair off point sources λ/2 different in
path length, one-third will always be left over). This situation corresponds to a maximum. In between the maxima and
minima 0 to 1/3 of the emitters will be unpaired. The mth minimum occurs where the path
difference is mλ. From ii) it is clear that sinθ_{m} = , where d is the
width of the slit. Thus the minima occur at angles:

sinθ_{m} =

At m = 0 there is a maximum. A full analysis of the situation reveals that the irradiance as a function of
the angular displacement, θ is:

I(θ) = I_{0}

where I_{0} is the irradiance of the central maximum. This result is obtained by computing the contribution of
the electric field to a point P on a distance screen and integrating over the slit. When L the distance to the
screen is much greater than d the distance between the slits, the distance to any point P is essentially
the same (L). Thus if each emitter (of unit width) has a source strength ε, ε/L is constant
for all points on the slit. If the slit is oriented in the x direction a differential segment dx contributes to
the field at P an amount:

dE = sin(σt - kr)dx

r is the distance from a point on the slit to P and can be approximated by rL - x sinθ. We can now
perform the integration to find the electric field at P:

E = sin[σt - k(L - x sinθ)]dx

So:

E = sin(σt - kL)

The irradiance is the time average of the electric field squared so recalling that the average of sin^{2}(σt) we
find that:

I =

which is the same as the result above if we recall k = . It also follows from this calculation that
the maxima occur not half way between the minima as might be expected but at the points correponding to the
solution of the transcendental equation:

tan(Πd /λ) = Πd /λ

These are at ±1.4303Π, ±2.4590Π, ±3.4707Π, etc.
A plot of irradiance versus position is shown in .
Clearly the central maximum is much brighter than any of the surrounding maxima, due to the partial cancellation that
occurs.

The analysis by integration over the slits can easily be extended to double or multiple slit systems, or even rectangular or
circular apertures. In the double slit case, the diffraction pattern resembles the double slit interference pattern we found for
Young's experiment, modulated by the envelope of a diffraction pattern similar to the single slit.

As the size of the slits goes to zero, the pattern assumes the form shown in .