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(64) Coherence - Coherence A practical monochromatic light...

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Coherence A practical monochromatic light source consists of a collection of similar atoms that are continually excited by collisions, and then spontaneously decay back to their electronic ground states, in the process emitting photons of characteristic angular frequency , where is the difference in energy between the excited state and the ground state, and is Planck's constant divided by (Hecht and Zajac 1974). An excited electronic state of an atom has a characteristic lifetime, , which can be calculated from quantum mechanics, and is typically (ibid.). It follows that when an atom in an excited state decays back to its ground state it emits a burst of electromagnetic radiation of duration and angular frequency . However, according to the bandwidth theorem (see Section 9.3 ), a sinusoidal wave of finite duration has the finite bandwidth (1013) In other words, if the emitted wave is Fourier transformed in time then it will be found to consist of a linear superposition of sinusoidal waves of infinite duration whose frequencies lie in the approximate range to . We conclude that there is no such thing as a truly monochromatic light source. In reality, all such sources have a small, but finite, bandwidths that are inversely proportional to the lifetimes, , of the associated excited atomic states. How do we take the finite bandwidth of a practical ``monochromatic'' light source into account in our analysis? In fact, all we need to do is to assume that the phase angle, , appearing in Equations ( 993 ) and ( 1007 ), is only constant on timescales much less that the lifetime, , of the associated excited atomic state, and is subject to abrupt random changes on timescales much greater than . We can understand this phenomenon as being due to the fact that the radiation emitted by a single atom has a fixed phase angle, , but only lasts a finite time period, , combined with the fact that there is generally no correlation between the phase angles of the radiation emitted by different atoms. Alternatively, we can account for the variation in the phase angle in terms of the finite bandwidth of the light source. To be more exact, because the light emitted by the source consists of a superposition of sinusoidal waves of frequencies extending over the range to , even if all the component waves start off in phase, the phases will be completely scrambled after a time period
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has elapsed. In effect, what we are saying is that a practical monochromatic light source is
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