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| Principles of Light Modulation |
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| Let us consider the case of electromagnetic radiation in an ionised medium. Suppose we increase/decrease the amplitude of an electromagnetic wave passing through an ionised gas. These modulations of the electromagnetic wave could contain coded information (Morse code for example). One would expect that these signals would be transmitted faster than the speed of light. Our answer to this is that the modulated sine wave is not a pure sine wave. It can be Fourier analyzed into a group of pure sine waves, all with somewhat different frequencies. Each pure sine wave is expected to be traveling with speed u (>c). However, in sum a case, the modulation pattern travels with what is called the group velocity ng. It may be quite different from the velocities of the component pure sine waves and in this case ng < c. The group velocity is given by : |
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| Previously, we were adding sine waves of exactly the same frequency. Now, we shall add sine waves of different frequencies, but close to each other. In this case, as time goes on, the sine waves would get increasingly out of phase with each other. The simplest case would be the sum of two equal sine waves of frequencies w1 and w2, as shown in the below figure. |
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| Two sine waves of slightly different frequency are in phase at the origin and successively get out of and in phase moving away from the origin. |
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The sum of the two sine waves |
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| This can be put in a more useful form by using the relation for the sum of two cosines from trigonometry: |
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| is the modulation functions or "envelope" (the outer curve in the above figure). In this case, the modulation function happens to be a sine wave of lower frequency. |
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| It is possible to obtain modulation functions of other shapes by adding more sine waves of slightly different frequencies. An important example is that of a single pulse of oscillation: Such a single pulse would look like the central hump in the above figure and is called a wave packet. We shall now show how a wave packet can be built up out of a group of neighboring sine waves. Starting with the result in below figure (b), we could "kill off" the neighboring bunches of oscillation by adding a third sine wave of frequency r with amplitude equal to the height of the bunches. [Note that A(t) changes sign for alternate bunches]. This new sine wave will add to the central bunch, and will be 1800 out of phase with its nearest neighbors as shown in below figure. The plot G(w) shows the relative strengths of the three sine waves that are being added together. In order to kill off the next set of neighboring bunches, we could add the two sine waves |
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| as shown in the below figure. These next two sine waves will hardly affect the central bunch, but their sum will be 180o out of phase with the next set of neighboring bunches. |
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| The sum of three sine waves |
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| Plot of the relative amplitudes |
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| The sum of sine waves |
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| Plot of the relative amplitudes |
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| An infinite number of neighboring sine waves must be added together to form a single wave packet with no neighboring bunches. This situation is shown in the above figure, where the function G(w) displays the relative amplitudes of the component sine waves. The shape of G(w) is called Gaussian function. |
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| The sum of infinite number of sine waves |
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| Plot of the relative amplitudes. G(w) is a Gaussian function with mean equal to v and standard deviation Dw |
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