LEC+9+GEOMETRIC OPTICS

Aims
From this chapter you should gain an understanding of the process of diffraction and its role in
producing distinctive interference patterns. You should also aim to understand how the effects of
diffraction can affect and limit the formation of images. As the classical example of diffraction, you
should be able to describe and explain the Rayleigh criterion and apply it to simple examples.
Minimum learning goals
1. Explain, interpret and use the terms:
diffraction, diffraction pattern, Fresnel diffraction, Fraunhofer diffraction, angular
resolution, Rayleigh criterion, diffraction envelope, principal maximum, secondary maxima,
double slit, diffraction grating.
2. Describe qualitatively the diffraction patterns produced in monochromatic light by single slits,
rectangular apertures, circular apertures, double slits and diffraction gratings.
3. Describe the interference pattern produced by monochromatic light and a grating in terms of
the interference pattern produced by a set of line sources modulated by a diffraction envelope.
4. State and apply the formulas for the angular widths of the central maxima in the Fraunhofer
diffraction patterns of a single slit and a circular hole.
5. State the Rayleigh criterion, explain its purpose and apply it to simple examples.
6. Describe how wavelength, slit width, slit spacing and number of slits affect the Fraunhofer
diffraction patterns produced by multiple slits and gratings.
7. State and apply the formula for the angular positions of maxima in the Fraunhofer diffraction
patterns produced by multiple slits and gratings.
PRE-LECTURE
5-1 SHADOWS
In the ray model we suppose that when light travels through a homogeneous medium it moves along
straight lines. That observation is often called the law of rectilinear propagation. The existence of
shadows is good evidence for the ray model of light. When light from a small (or point ) source
goes past the edges of an opaque object it keeps going in a straight line, leaving the space behind the
object dark (figure 5.1).
Figure 5.1. Straight line propagation of light
A small source of light produces sharp shadows.
L5: Diffraction 70
When the light reaches some other surface the boundary between light and dark is quite sharp.
(On the other hand if the light comes from an extended source the shadow is not so sharp - there is a
region of partial shadow surrounding the total shadow.)
It was not until the about the beginning of the nineteenth century that scientists noticed that
shadows are not really perfectly sharp. Looked at on a small enough scale the edge of a shadow is
not just fuzzy, as you might expect for an extended source of light, but there are also light and dark
striations or fringes around the edge of the shadow. Even more remarkable is the slightly later
discovery that there is always a tiny bright spot right in the middle of the shadow cast by a circular
object (figure 5.2). The fringes and the bright spot cannot be understood in terms of the ray model -
the explanation lies in the wave model. According to the wave theory, the fringes are formed by the
diffraction or bending of light waves around the edges of objects and the subsequent interference
of the diffracted waves. The diffraction of water waves at a hole in a barrier was shown in the video
lecture on interference (L4) and it is sketched in figure 4.4.
One of the effects of diffraction is the production of interference or fringe patterns. Although
these patterns are actually interference patterns in the same sense as those we have already discussed,
when they are produced by the bending of light around obstacles or apertures (holes) they are called
diffraction patterns.
Figure 5.2. Shadow of a small circular object
The edges of the shadow contain fringes and there is a small bright spot in the middle.
Young s twin slits experiment shows that light does not always travel along straight lines. You
can see that it must bend somewhere by looking at figure 5.3. Since there is light at the middle of
the screen but no straight through path from the source to the screen, the light which gets there must
somehow be going around corners. What must be happening is that after the light reaches the first
slit it then spreads out so that some of it reaches the other two slits. Then, having passed through
those slits it spreads out again in many directions until it reaches the screen. This behaviour is
typical of waves, but not of particles.
Light
source
First
slit Pair of
slits
Screen
Figure 5.3. Arrangement for Young s experiment
Light cannot be travelling in a straight line all the way from the source to the screen.
L5: Diffraction 71
The connection between waves and diffraction is much more noticeable for sound than it is for
light. The observation that sound easily travels around corners is strong evidence for the wave
nature of sound.
TEXT & LECTURE
5-2 HUYGENS CONSTRUCTION
A way of describing how diffraction occurs was invented by Christian Huygens (1629-1695) in
about 1679 and was modified much later into the form we now use by Augustin Fresnel (1788 -
1827). Huygens construction is a method for locating the new position of a wave front. Starting
from a known wavefront, we imagine each point on the wavefront to be a new source of secondary
wavelets. The wavelet from each point then spreads out as a sphere (which appears as a circle in
two-dimensional diagrams). After a certain time the new position of the original wavefront is
defined by the boundary or envelope of all the secondary wavelets. Huygens construction for a
plane wave going through a slit is shown in figure 5.4; after it has passed through the hole, the
wavefront is no longer plane, but has curved edges. The result of Huygens construction is
significantly different from the ray model in that it shows light waves spreading into the region of
the geometrical shadow. You should notice that opposite the slit the wavefront is still plane; it bends
only at the edges. This bending effect is noticeable only on a scale comparable with the wavelength
- for a very wide hole comparatively little of the wavefront bends around the edges.
1 2 3 4 5
Figure 5.4. Huygens construction
Five stages in the progress of a wavefront through a slit. The diagrams show the construction for three equal
time intervals after a plane wave reaches the slit, travelling left to right. At any stage the new wavefront is the
boundary of all the wavelets used in the construction.
Although the Huygens construction explains how new wavefronts are formed, it does not
predict the