=
,
,
. . . The processes of interference and diffraction that you studied in Experiment 5, using microwaves, also occur when the waves are longer or shorter than the few centimeters scale of that experiment. The case of visible light, where the waves are about a hundred thousand times smaller, is of particular interest to us since here, one good detector of the diffracted, or scattered energy is the eye itself, and in many instances the patterns of light produced by diffraction are not only technically interesting but also beautiful in their own right.
Let us review the essential physical ideas that lead to a quantitative description of diffraction effects:
You should review the derivation given in Exp. 5, prior to Eqn. 2, which shows that when light is sent through a simple slit in a screen, the resulting pattern is a system of light stripes with perfect zeros of intensity between them.
sin =
,
,
. . .
When the slit is many wavelengths wide, i.e., 
< < a, we see that
sin << 1, and we can replace sin by itself (in
radians), getting
= n/a n=1,2,3,...
An important proviso in what has been discussed here, and derived in Exp. 5, is that these results for the distribution of light as a function of angle are valid only when the observation point is at a distance very large compared to the aperture width. Diffraction effects are strong, also, nearer the aperture, but the spatial distribution of the light is different there than in the distant region.
An additional important and interesting diffraction effect is that produced by an obstruction placed in an otherwise unobstructed train of waves (Fig. 2, Exp. 5).
Figure 1 Equivalence of diffraction pattern for a disk (a) and a hole (b), except for the overall forward direction beam intensity.
In Fig. 1, we represent the two complementary (or opposite) cases of:
A. An obstruction of width a placed in a somewhat wider beam of waves, and
B. An aperture of width a in an opaque screen that otherwise blocks the beam
A theorem called Babinet's Principle, which is not too difficult to derive by methods such as those used in Exp. 5, may be stated as follows:
Except for the intensity of the central spot, the diffraction pattern produced by an opaque object in a wave train (or beam) is the same as that produced by an aperture of the same size and shape in an otherwise opaque screen.
Thus, according to Babinet's Principle, cases a and b in Fig. 1 will produce basically the same diffraction pattern; the only difference between them is that the beam passing around the obstruction will generally leave a bright extra spot at the center of the screen (in some cases, the central spot may be too dim to see).
This principle is very useful in a practical way. It enables us to measure the size of very small - often microscopic - opaque objects. Consider the example of a very fine fiber, or hair. Its complementary object is a narrow slit in an opaque screen, but we already know the diffraction pattern of the slit from the discussion in Exp. 5. Recall that the diffracted intensity as a function of the angle measured from the beam axis is given by:
I = I0
where
=
sin
.
Since the first zero occurs at = 
, we get the familiar
sin =
for its angular position.
If we now place a viewing screen downstream of the diffracting aperture, and compute the intensity as a function of the distance x on the screen away from the beam axis, we get
x = L tan ,
Figure 2 Setup for measuring angles of the diffraction pattern by measuring x on a screen.
where L is the aperture-to-screen distance. For all the work we will do here,
the angles will be small, so again, we use the approximation
sin tan
which, again, gives
and
x L
L
.
Look again at the intensity distribution of Fig. 5 in Exp. 5; we reproduce it here:
Figure 3
Since nearly all of the energy in the diffraction pattern is contained in the large central peak, in the interval
- < < ,
or, (since = a/), in the angular interval
-
<
<
,
we can state a simple approximate rule that will be adequate for our needs in this experiment. Waves passing through an aperture of width a, or passing over an obstruction of width a, are diffracted into an angular region given approximately by
-
<
<
,
where is the wavelength.
The detailed shape of the diffraction pattern will depend on the shape of the
aperture, or obstruction; however, the general size of the illuminated region
for any kind of aperture is adequately given by = /a, and the
corresponding size on a viewing screen:
x = L/a .
One final consideration is that actual apertures or obstructions are usually two-dimensional, as seen by a beam of waves.
Consider the rectangular aperture in screen S, of Fig. 4. We can describe its diffraction effects in two ways, depending on whether we are considering the x - z plane, where the aperture width is a, or the y-z plane, where it is b. If a < b, as shown, then the angular diffraction spread will be greater in the x-z plane, or along x on the screen S2 than along y. The result is then that an object or aperture elongated in some direction at the screen S1 will produce a diffraction pattern elongated in the perpendicular direction on S2.
Figure 4
Coherent Light
In this experiment we will study the diffraction of visible light produced by a laser.
First, a word is necessary about the unique character of laser radiation. All other sources of light produce what is called incoherent light, which means that a beam of such light is composed of a large number of short trains of waves randomly arranged in phase with respect to one another, and usually having wavelengths that are not all identical. Indeed, ordinary white light contains all the wavelengths in the visible spectrum.
Figure 5 Packets of waves which are incoherent have various wavelengths, directions and phases with respect to each other; The coherent packets are much more homogeneous.
Recall that the mathematical derivation of the formula for diffraction from a slit assumed that a uniform straight wave front existed all across the slit, and radiated Huygens wavelets. No such condition will exist for incoherent light, and as a result, a regular diffraction pattern will not be created. There may be momentarily uniform waves (uniform phase) over the aperture, but it will be erratic; as a result, the diffraction pattern will be smeared out, showing none of the sharp structure we have derived.
Laser light, however, is coherent, which is to say that the beam is a single train of waves that extend uniformly over the whole cross-section. Furthermore, it is monochromatic, meaning that only one single, constant wavelength is present. Thus, in laser light, the idealized diffraction effects discussed above should actually occur.