Diffraction and Interference Patterns

1. Diffraction and Huygens' Principle

Diffraction is the bending of waves around corners that occurs when a portion of a wavefront is cut off by a barrier or a obstacle. To explain diffraction, we need to first introduce Huygensˇ Principle.

In 1678, Christian Huygens discovered a geometric method that can be used to describe the propagation of any wave through space. Huygensˇ principle (or Huygensˇ construction) states that each point of a primary wavefront serves as the source of spherical secondary wavelets that advance with a speed of frequency equal to those of the primary wave. The primary wavefront at some later time is the envelop of these wavelets. Figure 1 shows the application of Huygensˇ principle to the propagation of a plane wave and of a spherical wave. Kirchhoff later showed that Huygensˇ principle is a consequence of the wave equation, and the intensity of the wavelets is zero in the backward direction.

If we treat each point of the wavefront as a point source of spherical waves, then it is easy to see why waves bend around edges. (see Figure 2)

2. Interference

Where two or more waves overlap we only observe a single disturbance. It is called interference. The resulting intensity ( or amplitude) is equal to the vector sum of each individual intensity. If two waves with idenitical amplitude and wavelength overlap in phase, i.e. if crest meets crest and trough meets trough, then we observe a resultant wave with twice the amplitude, it is called constructive interference. If the two overlapping waves, are completely out of phase, i.e. if crest meets trough, then we have destructive interference. (ie. the two waves cancel each other out completely. ) (see Figure 3.)

3. Double-Slit interference pattern

Suppose we have two slits that are separated by a distance d. At very large distances from the slits, the lines from the two slits to some point P on the screen are approximately parallel, and the path difference is approximately dsin(x) as illustrated in Figure 4.

If the path difference is exactly a multiple of the wavelength l, then crest meets crest and we have an interference maximum (when constructive interference occurs). The formula for all interference maxima is given by:

When the path difference is (m+1/2)l, we have interference minima and the formula is given by:

The distance y measured along the screen from the central point to the mth bright fringe is related to the angle x by

where L is the distance from the slits to the screen ( see Figure 4). For small angle l, we have

so dsin(x) is given approximately by

Substituting this into the formula for interference maxima, we get

4. Single-slit interference pattern

In the previous section we assumed the slits were very narrow so that we could consider them to be point source of circular waves. We could therefore assume that the intensity due to one slit acting alone was the same at any point P on the screen. When the slit is not so narrow, the intensity on a screen far away is dependent on the angle x between the ray to point P and the normal line between the slit and the screen.(see Figure 5)

Consider each point on a wavefront to be a point source of light in accordance with Huygensˇ principle. Suppose we have a slit of width a and there are 100 point sources, each represented by a dot. We divide the slit into two regions, with the upper half of dots in one region and the lower half of dots in the other. The path difference between source 1 and source 51 is (1/2)a sin(x). If this path difference is equal to one half of the wavelength l, then the waves from these two sources cancel each other. Similarly, we can see that each pair of sources separated by (a/2) will also cancel, and the resulting light intensity will be zero. The expression for points of zero intensity is thus given by:

Since sin(x) is at most 1, if a, the width of the slit is smaller than the wavelength l then there is no diffraction pattern. In Figure 5, the distance y from the centre to point P is related to the angle x and distance L from the slit to the screen by:

For small angle x, we get:

written by Chin-Hung,Richard,Hsu
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