Amplitude and Phase Fluctuations of a Light Wave

The electric field of a single-mode laser wave can be represented by Ei t) = Eo t)cos[(OLt + ki r + 

Even a well-stabilized laser, where all technical noise has been eliminated (Vol. 1, Sect. 5.4.5) still shows small fluctuations AEo of its amplitude and A(/ of its phase, because of quantum fluctuations. While technical fluctuations may be at least partly eliminated by difference detection (Fig. 1.9), this is not possible with classical means for photon noise caused by uncorrelated quantum fluctuations. 

These fluctuations are illustrated in Fig. 9.95 in two different ways the time-dependent electric field E t) and its mean fluctuations of the amplitude 0 and phase (p are shown in an E t) diagram and in a polar phase diagram with the axes E and 2- In the latter, amplitude fluctuations cause an uncertainty of the radius r = o, whereas phase fluctuations cause an uncertainty of the phase angle (p (Fig. 9.95b). Because of Heisenberg s uncertainty relation it is not possible that both uncertainties of amplitude and phase become simultaneously zero. 

In order to gain a deeper insight into the nature of these quantum fluctuations, let us regard them from a different point of view the EM field of a well-stabilized single-mode laser can be described by a coherent state (called a Glauber state [1334]) 

For coherent states (9.86) of the radiation field, and also for a thermal-equilibrium radiation field one obtains the symmetric relations 

The electric field of a single-mode laser wave can be represented by 

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The above discussion has centered on wave motion imposed on a surface by, for instance, an oscillating bar. But thermal fluctuations cause wave motion of small amplitude even on interfaces that are not disturbed by external means. With laser light scattering techniques it is possible to measure interfadal tension from analysis of surface fluctuations. This method has been applied to the measurement of ultralow interfacial traisions between liquid phases (Bouchiat and Meunier, 1972 Cazabat et al., 1983 Zollweg et al., 1972). Presumably it could also be used to determine surface compressibility or other rheological properties. 

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