Ripple structure

We discussed in Section 4.3 the electromagnetic normal modes, or virtual modes, of a sphere, which are resonant when the denominators of the scattering coefficients an and bn are minima (strictly speaking, when they vanish, but they only do so for complex frequencies or, equivalently, complex size parameters). But ext is an infinite series in an and bn, so ripple structure in extinction must be associated with these modes. The coefficient cn (dn) of the internal field has the same denominator as bn(an). Therefore, the energy density, and hence energy absorption, inside the sphere peaks at each resonance there is ripple structure in absorption as well as scattering. 

For each index n there is a sequence of values of x for which the mode associated with an or bn is excited. We may therefore label each (nonoverlapping) extinction peak with the type of mode [electric (an) or magnetic (/ )], the index n, and the sequential order of x (Chylek, 1976) for example, a 0, a 0, and so on, where the superscript indicates the order of x the extinction peaks between x = 50 and x = 52 for a water droplet are so labeled in Fig. 1 1.7. 

Three peaks, labeled a 0, aj0, a Q, in the real part of a60(x) for a water droplet are shown in Fig. 11.9 the real part of the refractive index is fixed but the imaginary part is varied. These optical constants could be obtained by, for example, adding a little dye (food coloring) to water this would increase k without appreciably changing n. Note that the horizontal scale is different for 

The peaks appear to be Lorentzian, and if we were to plot Im a60(x)) this would be even more obvious. That this is indeed so, to good approximation, was shown by Fuchs and Kliewer (1968), who discussed in detail the normal modes of an ionic sphere. 

For given k, the peaks broaden with increasing order. As absorption is increased for a fixed order each peak is broadened at the expense of its height, particularly the a 0 peak the broader, higher-order peaks are not nearly as severely damped by absorption. This accounts for the rapid loss of the sharpest extinction peaks with increased absorption (Fig. 11.12). 

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Examples of Extinction Interference and Ripple Structure Reddening... 

We shall defer detailed discussion of the ripple structure, which is considerably more complicated both mathematically and physically than the interference structure, until Chapter 11. Suffice it to say for the moment that the ripple structure has its origins in the roots of the transcendental equations (4.54) and (4.55), the conditions under which the denominators of the scattering coefficients vanish. 

Both the interference structure and the ripple structure are strongly damped when absorption becomes large, as it does in water if 1 /X is greater than about 6 pm x this is analogous to damping of interference bands in the transmission spectrum of a slab (see Fig. 2.8). If the droplet is small compared with the wavelength, then peaks in the bulk absorption spectrum are seen in the particle extinction spectrum for example, the extinction peaks in Fig. 4.6 at about 6 jam-1 for a 0.05-jum-radius droplet and at about 0.3 jum-1 for a 1.0-jam droplet are neither interference nor ripple structure but bulk absorption peaks. This illustrates the fact that absorption dominates over scattering for small a/X if there is any appreciable bulk absorption. 

The effect of averaging over one or more particle parameters—size, shape, orientation—is to efface details extinction fine structure, particularly ripple structure, to a lesser extent interference structure (Chapter 11) and undulations in scattering diagrams. If the details disappear upon averaging over an ensemble perhaps the best strategy in this instance would be to avoid the details of individual-particle scattering altogether and reformulate the problem statistically. 

Volume-normalized extinction is plotted in Fig. 11.2 as a function of photon energy for several polydispersions of MgO spheres both scales are logarithmic. For comparison of bulk and small-particle properties the bulk absorption coefficient a = Airk/X is included. Some single-particle features, such as ripple structure, are effaced by the distribution of radii. The information contained in these curves is not assimilated at a glance they require careful study. 

Ripple structure, beginning with the sharpest at large size parameters, is the first to disappear as a is increased. As the distribution is further widened, the interference structure fades away. For the widest distribution the only remaining features are reddening at small size parameters, and, at the other extreme, an asymptotic approach to the limiting value 2. < if ( K /... 

Figure 11.7 High-resolution (Ax = 10 4) calculation of the ripple structure in extinction by a water droplet (m = 1.33 + 110-8). After Chylek et al. (1978a).

Ripple structure was observed in scattering at 90° by water droplets as they nucleated and grew in a cloud chamber (Dobbins and Eklund, 1977). We shall show in Section 11.7 that ripple structure is easily observed in extinction by... 

The power required to levitate an oil drop as its size parameter is varied by tuning the dye laser wavelength is shown in the lower curves of Fig. 11.11. The calculated radiation pressure efficiency (plotted as 1 /QpT) is shown in the middle curve and Qext in the upper curve the refractive index m = 1.47 + 110—6 is approximately constant over the small wavelength interval. This figure is taken from Chylek et al. (1978b), who identified the peaks in the upper curve. Curve a of the experimental results is for values of x calculated from the drop size determined microscopically with an accuracy of 5%. The ripple structure... 

Figure 11.12 The effect of increasing absorption on interference and ripple structure.

A glance at the curves in Fig. 11.15 reveals extinction characteristics similar to those for spheres at small size parameters there is a Rayleigh-like increase of Q a with x followed by an approximately linear region broad-scale interference structure is evident as is finer ripple structure, particularly in the curves for the oblate spheroids. The interference structure can be explained... 

Perhaps the greatest difference between the extinction calculations for prolate and oblate spheroids is in the ripple structure, which is much more obvious for the latter and even persists to the largest a/b ratios shown, although with reduced amplitude. 

Mie theory does an admirable job of predicting extinction by spherical particles with known optical constants even the finest details it predicts—ripple structure—have been observed in extinction by single spheres. Several different causes—a distribution of sizes or shapes, and absorption—have the same effect of effacing the ripple structure or even the broader interference structure. 

Asymptotic expressions for extinction and absorption efficiencies of spheres averaged over a size parameter interval Ax it (i.e., with no ripple structure) have been derived by Nussenzveig and Wiscombe (1980). 

Chylek, P., 1976. Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section, J. Opt. Soc. Am., 66, 285-287. 

Fig. 1 A. shows the absorption and fluorescence spectra of 9AAHH in a PVA film. It can be seen that there exists an overlap between the two spectra. Fig. IB. shows the fluorescence spectra of 9AAHH doped single microparticles. Here the spectra contain ripple structures...

Besides differential scanning calorimetry, electron microscopy can also serve for characterizing the mixing behavior of multicomponent vesicular systems. The ripple structure of phospholipids with saturated alkyl chains (also referred to as smectic Bca phase, Fig. 35) is taken to indicate patch formation (immiscibility) in mixed phos-close enough (1-2 nm) lipid molecules are able to diffuse from one membrane to the between the pre- and main-transition of the corresponding phospholipid, electron... 

Fig. 35. Scheme of the "ripple structure of bi layers of saturated phospholipids found between the pre-transition and main-transition of the membrane 86)... 

Fig. 36. Electron micrograph of ripple structure and patch formation in 1 1 mixed bilayers of (18, n = 12) and dimyristoylphosphatidylcholine. Bar represents 250 nm...

The third view of the same matrix in Figure 3 is from a point underneath the surface which reveals that the valleys show a ripple structure like that of the ridges seen from above. It is clear from this view that the sharp minima are very close to the backward direction and are at very large particle size. 

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