Spherical wave functions radiating

The surface fields ei i and hi i are the tangential components of the electric and magnetic fields in the domain bounded by the closed surfaces Sx and 52. Taking into account the completeness property of the system of regular and radiating vector spherical wave functions on two enclosing surfaces... 

To compute the T matrix of the two-particles system and to derive a scattered-field expansion centered at the origin O of the global coordinate system we use the Stratton-Chu representation theorem for the scattered field Eg in Dg. In the exterior of a sphere enclosing the particles, the expansion of the approximate scattered field in terms of radiating vector spherical wave functions reads as... 

The scattered field is expanded in terms of radiating vector spherical wave functions... 

Returning to the scattering problem of a particle near an arbitrary infinite surface, we see that our analysis is complete if, according to Step 1, we are able to represent the plane electromagnetic wave and the radiating vector spherical wave functions and as integrals over vector plane waves. 

As in (B.3), the expansion of quasi-plane waves in terms of radiating spherical wave functions is given by... 

The superscript T stands for the regular vector spherical wave functions while the superscript 3 stands for the radiating vector spherical wave functions. It is useful to note that for n = m = 0, we have mI q = = 0. Mi, ,... 

In the far-held region, the asymptotic behavior of the spherical Hankel functions for large value of the argument yields the following representations for the radiating vector spherical wave functions [40] ... 

Passing to the radiating vector spherical wave functions we consider the integral representation (B.28) and the relation r = ro + ri. For ri > ro, this representation can written as... 

For ri < ro, we represent the radiating vector spherical wave functions as... 

Let us consider the non-relativistic limit of the relativistic operators describing radiation. Expressing the small components of the four-component wave functions (bispinors) in terms of the large ones and expanding the spherical Bessel functions in a power series in cor/c, we obtain, in the non-relativistic limit, the following two alternative expressions for the probability of electric multipole radiation ... 

Output resonance is reached (for a droplet with a fixed radius) when specific wavelengths within the inelastic emission profile correspond to MDR s with different n,t values. For those wavelengths, the droplet can be envisioned as an optical cavity with a large Fresnel number and Q-factors which are dependent on the specific n and I values. The portion of the inelastic radiation detected is that allowed to "leak" out of the droplet cavity. However, it is the internal field distributions of the electromagnetic waves at X, and X, which are best described by spherical harmonic functions and not by plane waves as in the case of an extended medium, that affect the nonlinear optical interactions. Such interactions in droplets can be illustrated by several well known examples in nonlinear spectroscopy of liquids in an optical cell. 

The regular and radiating spherical vector wave functions can be expressed as integrals over vector spherical harmonics [26]... 

The following theorems state the completeness and linear independence of the system of regular and radiating spherical vector wave functions on two enclosing surfaces. 

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