a
b
Figure 1 Waves impinge on a hole. False version is (a): correct one is (b).
a
b
Figure 2 Waves impinge on an obstacle. Once again, the correct version is (b).
The problem of diffraction is simply that of what happens when a train of waves strikes an obstruction. While we are going to discuss the problem specifically for electromagnetic radiation, it is much more general than that; the analysis we shall do here applies, almost without any change, to sound waves and even ocean waves.
The discussion can be stated quite well, in fact, by considering the behavior of ocean waves in a familiar situation. Suppose you are in an airplane watching the waves approach a breakwater which has a gap in it. The waves which come into the gap can certainly get through. But what happens to them then? Do they continue on in the manner of Fig. 1a? It is our experience that they do not; rather, we know that they spread radially away from the gap, as in Fig. 1b.
Similarly, one knows that an object such as a board set into the path of a wave train in water will not cast a long "shadow" as in Fig. 2a, but will cause only a momentary break in the otherwise straight waves (Fig. 2b).
a
b
Figure 1 Waves impinge on a hole. False version is (a): correct one is (b).
a
b
Figure 2 Waves impinge on an obstacle. Once again, the correct version is (b).
The behavior of waves in these circumstances is accurately described by a very simple but profound principle set forth by Christiaan Huygens in the 17th century. Before stating it, we will review briefly the way in which wave motion in general is treated mathematically.
In order to treat travelling waves in a quantitative fashion, a formal mathematical expression must be used to describe them. The most important case is that in which the wave has a sinusoidal shape. For example, an observer in a fixed position at the water level in Fig. 1 would measure the water level rising and falling as a sinusoidal function of time. Thus, is we let A represent the water level referred to its average position at any given time t, we can write
A = A0 cos 2
t = A0 cos 
t
where is the frequency of the wave in cycles per second, =
2 is its angular frequency, and A is the maximum change in the water
level from its average position. (Notice that there is only a trivial
difference between writing cos t and sin t.)
If this same observer were to look out over the waves he would also record a sinusoidal variation of the water level as a function of position, which can be described by
A = A0 cos 2
=
A0 cos kx
where the wavelength, 
is the distance from one crest to the next, k =
2/ is called the propagation constant, and x is the position
relative to the observer measured along a line perpendicular to the waves.
The argument of the cosine function (called the phase angle or, simply, the
phase) at any arbitrary time and position must then be an algebraic sum of the
two phase angles t and kx. Furthermore, if we know that crests per
second pass a given spot and that the distance between them is , then
the velocity at which the wave crests are travelling is v = =
/k. Thus, a surfer coming straight in on the wave and maintaining a
fixed level must be travelling at this velocity; i.e., at the position on x =
vt the water level A must be a constant. This will be true if the phase angle
of the wave is proportional to the quantity x - vt = x - (/k)t.
Therefore, the general expression for the amplitude of a travelling wave is
A = A0 cos (kx - t).
You can now check the above expression at fixed position (say x = 0), at fixed time, and for an observer moving along with the wave to see that it does indeed correctly describe a travelling wave.
We define a plane wave as a wave moving through 3 dimensional space, e.g., sound or light waves, such that all points in a plane perpendicular to the direction of travel are in the same phase (have the same phase angle). A spherical wave emanates from a point, so that all points on a sphere centered at the original point are in the same phase.
Now, Huygens' principle states simply that every point on a wave front (i.e., a surface of equal phase) acts as a point source of spherical waves. To see what results this may have, let us return to our two-dimensional case of surface waves in water. The analogy to Huygens' principle here would be that each point along a wave is a source of circular waves. Let us draw our open breakwater again, and the wave which is just coming through it:
Figure 3 Huygens' wavelet construction.
Now, if we wish to predict the form of this wave a time t later, we simply draw a great number of circles of radius r = vt and having centers all along the original wave. Their envelope will indeed be the wave shape at the new time. (For 3 dimensions, we would superimpose spherical waves originating from every point on some initial wave front.) This clearly describes the behavior shown in Fig. 1b and 2b.
We will next consider the problem of these "diffracted" waves in a little more detail. We ask the following question: If we are at a distance from an aperture (the opening in our breakwater) which is large compared to the aperture width, how does the amplitude of the waves vary with the angle between the observer and the aperture? We assume that the plane (or straight waves are incident upon the opening, and parallel to it (Fig. 4).
Figure 4 Setup for integration of amplitudes from a single slit, out to point P.
The problem will be to calculate the wave amplitude at a point P which is at an
angle 
from the normal to the aperture and at a distance r from it.
Our reasoning here is qualitatively quite simple. We invoke Huygens' principle again, and assume that across the aperture there are, in a straight line, a multitude of little generators of spherical (or circular ) waves, all oscillating in perfect synchronism together.
The amplitude of the wave at point P will be the sum of all the contributions from these little sources. The only complicating fact here is that point P is farther from the sources at the left of the aperture than from those at the right, and consequently, the various contributions will not all have the same phase when they arrive - the waves will cancel each other out in varying degrees depending on the angle at which the observation is made. (Since we have assumed that r>>a, i.e., that P is quite distant from the aperture, we are safe in the assumption that the waves coming to P from all equal increments dy of the opening have the same maximum amplitude, and that they only differ in phase.)
dAp = A0 cos (kr - t) dy.
Sources off the aperture center are closer to or farther from P by distances y
sin (See Fig. 4), and so, the general increment is
dAp= A0
,
and we simply integrate this over the width of the opening to get Ap:
Ap() = A0
The result, which you should evaluate for yourself, is:
Ap() = A0a
cos (kr - t).
The constant A0 contains the dependence of the maximum amplitude upon the distance r. This will not interest us for now, however, since our main concern is with the angular dependence.
In most instances, the diffracted wave intensity is of more interest than the amplitude itself, since our means of detection of most wave phenomena are sensitive only to intensity. As mentioned in Experiment 4, intensity is proportional to the square of the wave amplitude, and therefore, the angular intensity dependence of the diffracted waves has the form
I = I0
where
=
.
The radiation pattern then will look something like Figure 5:
Figure 5
This pattern contains a quite surprising result. It shows us that while the
wave pattern does indeed spread out, as we guessed from a qualitative
application of Huygens' principle, there are particular angles where the
intensity goes exactly to zero. We could certainly not have anticipated such a
result from the general "common-sense" arguments which preceded our
calculation. We see that the "null angle" is related in a simple way to the
ratio between the wavelength and a, the aperture width:
I = 0 for 
= , 2, . . .,
or, since k = 
,
I = 0 for sin =
,
,
. . . .
In this experiment, you will determine for an electromagnetic wave by
measuring the null angle for a given aperture width.
Interference
Interference pheomena in wave mechanics are not different from what we have just described, since, for example, the existence of a null in the diffraction pattern is due to a perfect, cancelling interference between waves from different parts of the aperture.
However, this particular term is usually reserved for interference between waves from discrete, well-separated sources. The most commonly used and studied case is an array of linear or point sources of waves, such as slits in an opaque screen (light) or dipole antennas (radio waves).
Consider the array of narrow slits in a screen in Figure 6. We can suppose
that electromagnetic plane waves are incident on the screen (as they are in
this experiment), and we will simplify the problem by making a<<,
so each slit radiates isotropically. (This will actually not occur, but the
matter will not concern us here.)
Figure 6 The two-slit interference geometry.
Now, for points of observation very far from the screen, strong intensity maxima will occur when all the slit contributions arrive in phase; but this clearly requires that the differences in distance must be an integral number of wavelengths, or that
d sin = n.
The greater the number of slits, the sharper and more closely defined in angle are the maxima. Optical spectrographs employ gratings having as many as 105 apertures, and these allow measurements of wavelength to accuracies of the order of one part in 106. But even two slits allow a determination of wavelength to an accuracy somewhat better than can be accomplished by a measurement of the single-slit diffraction pattern.