Zero retardation point

Two additional features of the mirror system are necessary for successful operation. The first is a means of sampling the interferogram at precisely spaced retardation intervals. The second is a method for determining exactly the zero retardation point to permit signal averaging. If this point is not known precisely. 

If we make use of the fact that the interferogram as represented in Eq. 2.12 is symmetrical about 8 = 0 but the first data point is actually sampled before the zero retardation point, at 8 = —e the interferogram takes the form... 

Let us now consider the classical Fourier transform from a more mathematical basis by considering the case of a symmetrical interferogram that has been measured from the zero retardation point [1]. The integral... 

The optical path difference (OPD) between the beams that travel to the fixed and movable mirror and back to the beamsplitter is called retardation, 8. When the path length on both arms of the interferometer are equal, the position of the moving mirrors is referred to as the position of zero retardation or zero path difference (ZPD). The two beams are perfectly in phase on recombination at the beamsplitter, where the beams interfere constructively and the intensity of the beam passing to the detector is the sum of the intensities of the beams passing to the fixed and movable mirrors. Therefore, all the light from the source reaches the detector at this point and none returns to the source. To understand why no radiation returns to the source at ZPD one has to consider the phases on the beam splitter. 

This last representation is completely equivalent to the analytidty of t(ai) in Im 0 and the statement that a,t(a>) go to zero as u - oo. The analyticity property in turn is a direct consequence of the retarded or causal character of T(t), namely that it vanishes for t > 0. If t(ai) is analytic in the upper half plane, but instead of having the requisite asymptotic properties to allow the neglect of the contribution from the semicircle at infinity, behaves like a constant as o> — oo, we can apply Cauchy s integral to t(a,)j(o, — w0) where a>0 is some fixed point in the upper half plane within the contour. The result in this case, valid if t( - oo is... 

An evaluation of the retardation effects of surfactants on the steady velocity of a single drop (or bubble) under the influence of gravity has been made by Levich (L3) and extended recently by Newman (Nl). A further generalization to the domain of flow around an ensemble of many drops or bubbles in the presence of surfactants has been completed most recently by Waslo and Gal-Or (Wl). The terminal velocity of the ensemble is expressed in terms of the dispersed-phase holdup fraction and reduces to Levich s solution for a single particle when approaches zero. The basic theoretical principles governing these retardation effects will be demonstrated here for the case of a single drop or bubble. Thermodynamically, this is a case where coupling effects between the diffusion of surfactants (first-order tensorial transfer) and viscous flow (second-order tensorial transfer) takes place. Subject to the Curie principle, it demonstrates that this retardation effect occurs on a nonisotropic interface. Therefore, it is necessary to express the concentration of surfactants T, as it varies from point to point on the interface, in terms of the coordinates of the interface, i.e.,... 

In liquids the static kinetics precedes the diffusion accelerated quenching, which ends by stationary quenching. The rate of the latter k = AkRqD has a few general properties. In the fast diffusion (kinetic control) limit Rq — 0 while k —> ko. In the opposite diffusion control limit Rq essentially exceeds a and increases further with subsequent retardation of diffusion. As the major quenching in this limit occurs far from contact, the size of the molecules plays no role and can be set to zero. This is the popular point particle approximation (ct = 0), which simplifies the analytic investigation of diffusional quenching. For the dipole-dipole mechanism the result has been known for a very long time [70] ... 

As mentioned earlier, it is also impractical to collect data at infinitesimally spaced values of 8. Data in Fourier transform spectroscopy is usually collected at discrete steps determined by a coincident interferometer with a mode-locked helium-neon (He-Ne) laser at X = l/o = 632.8 nm. The fringes of this laser are detected with a diode and sampling at every zero crossing of this line will provide data for retardation values spaced at 316.4 nm. In addition, one must insure that the mirror is moved slowly enough that the detector can relax between each sample so the data for each point is independent. 

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