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

3.5 Layered Particles

229

	z	r
	r1
		L
L 1		x

			Fig. 3.45. Geometry of a layered cylinder
	2
	10
	101				TLAY - parallel
					TLAY - perpendicular
					TINHOMSPH - parallel
	0				TINHOMSPH - perpendicular
	10				TINHOM - parallel
DSCS	10−1				TINHOM - perpendicular
	−2
	10
	10−3
	10−4
	10−5	0	60	120	180	240	300	360
				Scattering Angle (deg)

Fig. 3.46. Normalized di erential scattering cross-sections of a layered particle consisting of a host sphere and a spheroidal inclusion

In Fig. 3.48, we show results for a concentrically layered sphere consisting of three layers of radii ksr1 = 10, ksr2 = 7 and ksr3 = 4. The relative refractive indices with respect to the ambient medium are mr1 = 1.2 + 0.2j, mr2 = 1.5 + 0.1j and mr3 = 1.8 + 0.3j. The scattering curves obtained with the TSPHERE and TLAY routines are close to each other.

230 3 Simulation Results

<DSCS>

2
10
	TLAY - parallel
101	TLAY - perpendicular
	TINHOMSPH - parallel
	TINHOMSPH - perpendicular
0	TINHOM - parallel
10	TINHOM - perpendicular

10−1

10−2

10−3

10−4
	0		30		60		90		120		150		180
					Scattering Angle (deg)

Fig. 3.47. Averaged di erential scattering cross-sections of a layered particle consisting of a host sphere and a spheroidal inclusion

	1
	10
					TLAY - parallel
	0				TLAY - perpendicular
	10				TSPHERE - parallel
					TSPHERE - perpendicular
	−1
DSCS	10
	−2
	10
	−3
	10
	−4
	10
	−5
	10	0	60	120	180	240	300	360
				Scattering Angle (deg)

Fig. 3.48. Normalized di erential scattering cross-sections of a layered sphere

3.6 Multiple Particles

In the following analysis we investigate electromagnetic scattering by a system of particles. The basic routine for analyzing this type of scattering problem is the TMULT routine. As for inhomogeneous scatterers, the individual T -matrices of the particles are input parameters of the code and they may correspond to homogeneous, axisymmetric or nonaxisymmetric particles

3.6 Multiple Particles

231

Z	z2		b2
z1		β2
z20	O2		x2
	n2
a1		a2
O1	x1	x20	X

n1

b1

Fig. 3.49. Geometry of a system of two prolate spheroids

and inhomogeneous, composite or layered particles. The flow diagram of the TMULT routine is as in Fig. 3.34.

Figure 3.49 illustrates a system of two spheroids of semi-axes ksa1 = ksa2 = 4 and ksb1 = ksb2 = 2, and relative refractive indices mr1 = mr2 = 1.5.

The Cartesian coordinates of the center of the second spheroid are ksx20 =

√

ksz20 = 4 2 and y20 = 0, while the Euler angles specifying the orientation of the spheroids are chosen as αp1 = βp1 = 0◦ and αp2 = βp2 = 45◦. The first computational step involves the calculation of the individual T -matrix of a spheroid by using the TAXSYM code, and for this calculation, the maximum expansion and azimuthal orders are Nrank = 10 and Mrank = 4, respectively. The individual T -matrices are then used to compute the T -matrix of the spheroids with respect to the origin O1, and the dimension of the system T - matrix is given by Nrank = 20 and Mrank = 18. In Fig. 3.50, we show the di erential scattering cross-sections computed with the TMULT routine and the multiple multipole method. The scattering curves are in good agreement.

As a second example, we consider a system of five spherical particles illuminated by a plane wave propagating along the Z-axis of the global coordinate system. The spheres are identical and have a radius of ksr = 2 and a relative refractive index of mr = 1.5, while the length specifying the position of the spheres is ksl = 6 (Fig. 3.51). The system T -matrix is computed with respect to the origin O and is characterized by Nrank = 18 and Mrank = 16. The scattering characteristics are computed with the TMULTSPH routine, and numerical results are again presented in the form of the di erential scattering cross-sections. The curves plotted in Fig. 3.52 correspond to a fixed orientation

232 3 Simulation Results

DSCS

101
	TMULT - parallel
100	TMULT - perpendicular
	MMP - parallel
	MMP - perpendicular

10−1

10−2

10−3

10−4

10	−5	60	120	180	240	300	360
	0

Scattering Angle (deg)

Fig. 3.50. Normalized di erential scattering cross-sections of a system of two prolate spheroids computed with the TMULT routine and the multiple multipole method (MMP)

Z

r

l

Y

l

X

l	l

Fig. 3.51. Geometry of a system of five spheres

of the system of spheres (αp = βp = γp = 0◦) and are similar for the T -matrix and the multiple multipole solutions. For a random orientation, we compute the elements of the scattering matrix, and the results are shown in Figs. 3.53 and 3.54 together with the results computed with the SCSMTM code devel-

3.6 Multiple Particles

233

DSCS

101
	TMULTSPH - parallel
100	TMULSPH - perpendicular
	MMP - parallel
	MMP - perpendicular

10−1

10−2

10−3

10−4

10	−5	60	120	180	240	300	360
	0

Scattering Angle (deg)

Fig. 3.52. Normalized di erential scattering cross-sections of a system of five spheres computed with the TMULT routine and the multiple multipole method (MMP)

	1.0
	10
Elements						F11	- TMULTSPH
	0.0					F11	- SCSMTM
						F22	- TMULTSPH
	10
						F22	- SCSMTM
Matrix	−1.0
	10
Scattering	−2.0
	10
	−3.0
	10	0	30	60	90	120	150	180

Scattering Angle (deg)

Fig. 3.53. Scattering matrix elements F11 and F22 of a system of five spheres computed with the TMULT routine and the code SCSMTM of Mackowski and Mishchenko [153]

oped by Mackowski and Mishchenko [153]. The curves are generally close to each other.

Figure 3.55 illustrates a system of four identical spheres that can be used to compare di erent routines. The radius of each spherical particle is ksr = 3 and the relative refractive index is mr = 1.5. In the global coordinate system

234 3 Simulation Results

	0.01
MatrixElements	0.00
	−0.01
Scattering	−0.02				F21	- TMULTSPH
					F21	- SCSMTM
					F43	- TMULTSPH
					F43	- SCSMTM
	−0.03	30	60	90	120	150	180
	0

Scattering Angle (deg)

Fig. 3.54. Scattering matrix elements F21 and F43 of a system of five spheres computed with the TMULT routine and the code SCSMTM of Mackowski and Mishchenko [153]

OXY Z, two local coordinate systems O1x1y1z1 and O2x2y2z2 are defined, and the positions of the origins O1 and O2 are characterized by x01 = y01 = z01 = L and x02 = y02 = z02 = −L, respectively, where ksL = 6.5. In each local coordinate system, a system of two spheres is considered, and the distance between the sphere centers is 2ksl = 3.5, while the Euler angles specifying the orientation of the two-spheres system are αp1 = βp1 = 45◦ and αp2 = βp2 = 30◦. The Cartesian coordinates of the spheres are computed with respect to the global coordinate system and the TMULTSPH routine is used to compute the scattering characteristics. The dimension of the T - matrix of the four-sphere system is given by Nrank = 21 and Mrank = 19. A second technique for analyzing this scattering geometry involves the computation of the T -matrix of the two-sphere system by using the TMULT2SPH routine. In this case, Nrank = 16 and Mrank = 5, and the T -matrix of the twosphere system serves as input parameter for the TMULT routine. Figures 3.56 and 3.57 illustrate the di erential scattering cross-sections for a fixed (αp = βp = γp = 0◦) and a random orientation of the system of spheres. The far-field patterns are reproduced very accurately by both methods.

To demonstrate the capabilities of the TMULTSPH routine we present some exemplary results for polydisperse aggregates. Monodisperse aggregates of small spherical particles are characterized by the number of primary spherule N , the fractal dimension Df , the fractal pre-factor kf , the radius of gyration Rg, and the radius of the primary spherules rp. These morphological parameters are related in the form of a scaling law

3.6 Multiple Particles

235

Z

		z1
	L	x1
		O1
	−L	X
	z2 O	L
	x2	−L
	O2
	z1
		r
	2l	β1
		x1
	O1
	y1	α 1

Fig. 3.55. Geometry of a system of four spheres

DSCS

100
	TMATMULT - parallel
10−1	TMATMULT - perpendicular
	TMATMULTSPH - parallel
	TMATMULTSPH - perpendicular

10−2

10−3

10−4

10−5

10−6	60	120	180	240	300	360
0

Scattering Angle (deg)

Fig. 3.56. Normalized di erential scattering cross-sections of a system of four spheres

236 3 Simulation Results

<DSCS>

100

	TMATMULT - parallel
	TMATMULT - perpendicular
10−1	TMATMULTSPH - parallel
	TMATMULTSPH - perpendicular

10−2

10−3

10−4
	0		30		60		90		120		150		180
					Scattering Angle (deg)

Fig. 3.57. Averaged di erential scattering cross-sections of a system of four spheres

N = kf	Rg		Df	(3.14)
			,
		rp

where the radius of gyration is determined from the position of each primary spherule ri to the geometrical center of the cluster r0,

N

Rg2 = N1 |ri − r0|2 i=1

,N

and r0 = (1/N ) i=1 ri. Equation (3.14) is important because the values of parameters are linked to real physical processes. If Df ≈ 1.8 the process belongs to a di usion limited aggregation and if Df ≈ 2.1 it belongs to a reaction limited aggregation. For simulating di usion limited aggregation, we developed a Fortran program and used the Cartesian coordinates of the spherule as input parameters for the TMULTSPH routine. This program is based on the fast algorithm described by Filippov et al. [66] and generates aggregates by using two di erent methods. In the first method, each new primary spherule will be stuck on the mother aggregate after touching one of the existing spherules. This method is known as the di usion limited algorithm (DLA) and is used for aggregates not larger than about N = 150 spherules. With the second method, complete small aggregates are stuck on the mother aggregate. The small aggregates are generated with the DLA algorithm and consists of about 20 or more primary spherules. This method is called di usion limited cluster cluster aggregation (DLCCA) and gives more realistic aggregates. An example of a monodisperse aggregate representing

3.6 Multiple Particles

237

Fig. 3.58. Monodisperse aggregate with Df = 1.8, N = 130 and rp = 10 nm

Fig. 3.59. Polydisperse aggregate with Df = 1.8, N = 203 and rp between 10 and 20 nm

a soot particle from a combustion processes is shown in Fig. 3.58, while a polydisperse aggregate is shown in Fig. 3.59. For the monodisperse aggregate, the scattering characteristics are shown in Fig. 3.60, and the essential parameters controlling the convergence process are Nrank = 8 and Mrank = 6 for the system T -matrix, and Nrank = 4 and Mrank = 3 for the primary spherules. The di erential scattering cross-sections of the polydisperse aggregate are plotted in Fig. 3.61, and the parameters of calculation for the system T -matrix increase to Nrank = 12 and Mrank = 10. The program described above has been used by Riefler et al. [204] to characterize soot from a flame by analyzing the measured scattering patterns.

238 3 Simulation Results

10−1
10−2
10−3		parallel
		perpendicular
DSCS
10−4
10−5
10−6	0	30	60	90	120	150	180
			Scattering Angle (deg)

Fig. 3.60. Normalized di erential scattering cross-sections of a monodisperse aggregate

	10−3
					parallel
	10−4				perpendicular
	10−5
DSCS	10−6
	10−7
	10−8
	10−9	0	60	120	180	240	300	360
				Scattering Angle (deg)

Fig. 3.61. Normalized di erential scattering cross-sections of a polydisperse aggregate

3.7 Composite Particles

Electromagnetic scattering by axisymmetric, composite particles can be computed with the TCOMP routine. An axisymmetric, composite particle consists of several nonenclosing, rotationally symmetric regions with a common axis of symmetry. In contrast to the TMULT routine, TCOMP is based on a formalism which avoid the use of any local origin translation. The scattering