Interference light waves 1

14.1 Superposition of Waves
Consider a region in space where two or more waves pass through at the same time. According to the superposition principle, the net displacement is simply given by the vector or the algebraic sum of the individual displacements. Interference is the combination of two or more waves to form a composite wave, based on such principle. The idea of the superposition principle is illustrated in Figure 14.1.1.
(a)
(b)
(c)
(d)
Figure 14.1.1 Superposition of waves. (b) Constructive interference, and (c) destructive interference.
Suppose we are given two waves,
11011122022(,)sin(), (,)sin()xtkxtxtkxt???? 2 ???±+== ± +? (14.1.1)
the resulting wave is simply
1011122220(,)sin()sin()xtkxtkxt???????+=±+± + (14.1.2)
The interference is constructive if the amplitude of (,)xt?is greater than the individual ones (Figure 14.1.1b), and destructive if smaller (Figure 14.1.1c).
As an example, consider the superposition of the following two waves at : 0t=
12()sin, ()2sin4xxxx????==???
+ ?? (14.1.3)
The resultant wave is given by
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()12()()()sin2sin12sin2cos4xxxxxx?
? =? +? = + ?? + ?? = + + x
?? (14.1.4)
where we have used
sin()sincoscossin????? ? +=+ (14.1.5)
and sin(/4)cos(/4)2/2??==. Further use of the identity
[ ]
2222222222sincossincoscossinsincossin()abaxbxabxxabababxxabx?????+=++??++??=++=++ (14.1.6)
with
1tanba????=???? (14.1.7)
then leads to
()522sin()x x ??=+ + (14.1.8)
where 1tan(2/(12))30.40.53 rad.??=+=°= The superposition of the waves is depicted in Figure 14.1.2.
Figure 14.1.2 Superposition of two sinusoidal waves.
We see that the wave has a maximum amplitude when sin()1x?+=, or /2x??=?. The interference there is constructive. On the other hand, destructive interference occurs at 2.61radx ??=?=, wheresin()0?=.
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In order to form an interference pattern, the incident light must satisfy two conditions:
(i) The light sources must be coherent. This means that the plane waves from the sources must maintain a constant phase relation. For example, if two waves are completely out of phase with ??=, this phase difference must not change with time.
(ii) The light must be monochromatic. This means that the light consists of just one wavelength 2/k??=.
Light emitted from an incandescent lightbulb is incoherent because the light consists o waves of different wavelengths and they do not maintain a constant phase relationship. Thus, no interference pattern is observed.
Figure 14.1.3 Incoherent light source
14.2 Young’s Double-Slit Experiment
In 1801 Thomas Young carried out an experiment in which the wave nature of light was demonstrated. The schematic diagram of the double-slit experiment is shown in Figure 14.2.1.
Figure 14.2.1 Young’s double-slit experiment.
A monochromatic light source is incident on the first screen which contains a slit . The emerging light then arrives at the second screen which has two parallel slits S0S1 and S2. which serve as the sources of coherent light. The light waves emerging from the two slits then interfere and form an interference pattern on the viewing screen. The bright bands (fringes) correspond to interference maxima, and the dark band interference minima.
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