Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

To give a justification of the above expansion we consider the integral representation

2π π

epol(β, α)ejke(β,α)·r = epol(β, α)ejk(β ,α )·r

00

×δ(α − α)δ(cos β − cos β) sin β dβ dα , (1.27)

Using the orthogonality relations of vector spherical harmonics we see that the expansion coe cients amn and bmn are given by

2π π

amn = 4jn f (β, α, β , α ) · mmn (β , α ) sin β dβ dα

00

=4jnepol(β, α) · mmn(β, α) ,

2π π

00

=−4jn+1epol(β, α) · nmn(β, α) .

Substituting (1.28) and (1.29) into (1.27) and taking into account the integral representations for the regular vector spherical wave functions (cf. (B.26) and (B.27)) yields (1.25).

The polarization unit vector of a linearly polarized vector plane wave is given by (1.18). If the vector plane wave propagates along the z-axis we have β = α = 0 and for β = 0, the spherical vector harmonics are zero unless m =

±1. Using the special values of the angular functions πn1 and τn1 when β = 0,

we obtain

a±1n = −jn√2n + 1 (±j cos αpol + sin αpol) ,

b±1n = −jn+1√

18 1 Basic Theory of Electromagnetic Scattering

Thus, for a vector plane wave polarized along the x-axis we have = jn−1√2n + 1 ,

b1n = b−1n = jn−1√2n + 1 ,

while for a vector plane wave polarized along the y-axis we have

a1n = a−1n = jn−2√2n + 1 , b1n = −b−1n = jn−2√2n + 1 .

Gaussian Beam

Many optical particle sizing instruments and particle characterization methods are based on scattering by particles illuminated with laser beams. A laser beam has a Gaussian intensity distribution and the often used appellation Gaussian beam appears justified. A mathematical description of a Gaussian beam relies on Davis approximations [45]. An nth Davis beam corresponds to the first n terms in the series expansion of the exact solution to the Maxwell equations in power of the beam parameter s,

s = wl0 ,

where w0 is the waist radius and l is the di raction length, l = ksw02. According to Barton and Alexander [9], the first-order approximation is accurate to s < 0.07, while the fifth-order is accurate to s < 0.02, if the maximum percent error of the solution is less than 1.2%. Each nth Davis beam appears under three versions which are: the mathematical conservative version, the L-version and the symmetrized version [145]. None of these beams are exact solutions to the Maxwell equations, so that each nth Davis beam can be considered as a “pseudo-electromagnetic” field.

In the T -matrix method a Gaussian beam is expanded in terms of vector spherical wave functions by replacing the pseudo-electromagnetic field of an nth Davis beam by an equivalent electromagnetic field, so that both fields have the same values on a spherical surface [81, 83, 85]. As a consequence of the equivalence method, the expansion coe cients (or the beam shape coe cients) are computed by integrating the incident field over the spherical surface. Because these fields are rapidly varying, the evaluation of the coe cients by numerical integration requires dense grids in both the θ- and ϕ-direction and the computer run time is excessively long.

For weakly focused Gaussian beams, the generalized localized approximation to the beam shape coe cients represents a pleasing alternative (see, for instance, [84, 87]). The form of the analytical approximation was found in part by analogy to the propagation of geometrical light rays and in part by numerical experiments. This is not a rigorous method but its use simplifies and significantly speeds up the numerical computations. A justification of

and

u = 2Q ρ0ρn .

w02

The normalization constant Kmn is given by

Kmn = 2jn n(n + 1)

2n + 1

for m = 0, and by

for m = 0. If both coordinate systems coincide (ρ0 = 0), all expansion coe - cients are zero unless m = ±1 and

The Gaussian beam becomes a plane wave if w0 tends to infinity and for this specific case, the expressions of the expansion coe cients reduce to those of a vector plane wave.

We next consider the general situation depicted in Fig. 1.7 and assume that the auxiliary coordinate system ObXbYbZb and the global coordinate system OXY Z have the same spatial orientation. The Gaussian beam propagates in a direction characterized by the zenith and azimuth angles βg and αg, respectively, while the polarization unit vector encloses the angle αpol with the xb-axis of the beam coordinate system Obxbybzb. As before, the particle coordinate system is obtained by rotating the global coordinate system through the Euler angles αp, βp and γp. The expansion of the Gaussian beam in the particle coordinate system is obtained by using the addition theorem for vector spherical wave functions under coordinate rotations (cf. (B.52) and (B.53)), and the result is

∞n

Ee(r) = amnM 1mn (ksr) + bmnN 1mn (ksr) ,

n=1 m=−n