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

3.3 Homogeneous, Axisymmetric and Nonaxisymmetric Particles

209

z

O	x
2a

	2b
	ReZ
	O	ImZ
zN		z2 z1

Fig. 3.13. Geometry of an oblate cylinder and the distribution of the discrete sources in the complex plane

Table 3.3. Surface parameters of particles with extreme geometries

Particle type	a (µm) b (µm) αp (◦)			βp (◦)
Prolate spheroid	8.5	0.85	0	0
Fibre	3	0.06	0	0
Oblate cylinder	0.03	3	0	0
Cassini particle	1.1	1.125	0	45

Table 3.4. Maximum expansion and azimuthal orders for particles with extreme geometries

Particle type	Nrank	Nint
Prolate spheroid	100	1000
Fibre	50	3000
Oblate cylinder	36	5000
Cassini particle	28	1000

is mr = 1.5 and the scattering characteristics are computed in the azimuthal plane ϕ = 0◦. The parameters describing the geometry and orientation of the particles are given in Table 3.3, while the parameters controlling the convergence process are listed in Table 3.4. Note that the Cassini particle has a diameter of about 3.15 µm and an aspect ratio of about 1/4.

210 3 Simulation Results

In Figs. 3.14–3.17 we plot the normalized di erential scattering crosssections together with the results computed with the discrete sources method for parallel and perpendicular polarizations, and for the case of normal incidence. It is apparent that the agreement between the curves is acceptable.

We conclude this section with an extensive validation test for an oblate cylinder of radius ksb = 15, length 2ksa = 7.5 and relative refractive index

102
101			TAXSYM - parallel
			TAXSYM - perpendicular
100			DSM - parallel
			DSM - perpendicular
10 1
DSCS
10 2
10 3
10 4
10 5
6
10 180	120	60	0	60	120	180

Scattering Angle (deg)

Fig. 3.14. Normalized di erential scattering cross-sections of a prolate spheroid with a = 8.5 µm and b = 0.85 µm. The curves are computed with the TAXSYM routine and the discrete sources method (DSM)

DSCS

10 1
10 2		TAXSYM - parallel
		TAXSYM - perpendicular
10 3		DSM - parallel
		DSM - perpendicular
10 4
10 5
10 6
10 7
10 8
9	60	0	60	120	180
10 180 120

Scattering Angle (deg)

Fig. 3.15. Normalized di erential scattering cross-sections of a fibre particle with a = 3.0 µm and b = 0.06 µm. The curves are computed with the TAXSYM routine and the discrete sources method (DSM)

3.3 Homogeneous, Axisymmetric and Nonaxisymmetric Particles

211

103				TAXSYm - parallel
102
				TAXSYM - perpendicular
101				DSM - parallel
100				DSM - perpendicular
10−1
DSCS 10−2
10−3
10−4
10−5
10−6
−7	−120	−60	0	60	120	180
10−180

Scattering Angle (deg)

Fig. 3.16. Normalized di erential scattering cross-sections of an oblate cylinder with a = 0.03 µm and b = 3.0 µm. The curves are computed with the TAXSYM routine and the discrete sources method (DSM)

102
	TAXSYM - parallel
101	TAXSYM - perpendicular
	DSM - parallel
100	DSM - perpendicular
10−1
DSCS 10−2

10−3

10−4
10−5	−120	−60	0	60	120	180
−180

Scattering Angle (deg)

Fig. 3.17. Normalized di erential scattering cross-sections of a Cassini particle with a = 1.1 and b = 1.125. The curves are computed with the TAXSYM routine and the discrete sources method (DSM)

mr = 1.5. The T -matrix calculations are performed with Nrank = 30 sources distributed in the complex plane and by using Nint = 300 integration points on the generatrix curve. The scattering characteristics plotted in Fig. 3.18 are calculated for the Euler angles of rotation αp = βp = 0◦ and for the case of normal incidence. Also shown are the results computed with the discrete

212	3	Simulation Results
		104				TAXSYM - parallel
		103
						TAXSYM - perpendicular
		102				DSM - parallel
		101				DSM - perpendicular
						MMP - parallel
		100				MMP - perpendicular
		DSCS 10−2				DDA - parallel
						CST - parallel
		10−1				DDA - perpendicular
		10−3				CST - perpendicular
		10−4
		10−5
		10−6
		10−7	0	30	60	90	120	150	180

Scattering Angle (deg)

Fig. 3.18. Normalized di erential scattering cross-sections of an oblate cylinder with ksb = 15 and 2ksa = 7.5. The curves are computed with the TAXSYM routine, discrete sources method (DSM), multiple multipole method (MMP), discrete dipole approximation (DDA) and finite integration technique (CST)

DSCS

100
	TNONAXSYM - parallel
10−1	TNONAXSYM - perpendicular
	DDA - parallel
	DDA - perpendicular

10−2

10−3

10−4

10−5	30	60	90	120	150	180
0

Scattering Angle (deg)

Fig. 3.19. Normalized di erential scattering cross-sections of a dielectric cube computed with the TNONAXSYM routine and the discrete dipole approximation (DDA)

sources method, multiple multipole method, discrete dipole approximation and finite integration technique. No substantial di erences between the curves are noted. Further scattering patterns for finite fibres having a large aspect ratio have been published by Pulbere and Wriedt [196].

3.3.2 Nonaxisymmetric Particles

In Fig. 3.19, we compare the results obtained with the TNONAXSYM routine to the discrete dipole approximation solutions. Calculations are performed for

3.3 Homogeneous, Axisymmetric and Nonaxisymmetric Particles

213

z

2l	O	y
	x

a

Fig. 3.20. Geometry of a hexagonal prism

a dielectric cube of length 2ksa = 10 and relative refractive index mr = 1.5. The incident wave propagates along the Z-axis of the global coordinate system and the Euler orientation angles are αp = βp = γp = 0◦. Convergence is achieved for Nrank = 14 and Mrank = 12, while the numbers of integration

points on each square surface are Nint1 = Nint2 = 24.
	As a second example, we consider a hexagonal prism (Fig. 3.20) of length
2l	=	1.154 µm,	hexagon side	a =	1.154 µm and relative refractive	index
mr	=	1.5. The	wavelength of	the	incident radiation is taken to be	λ =

0.628 µm, and the orientation of the hexagonal prism is specified by the Euler angles αp = βp = 0◦ and γp = 90◦. For this application, the maximum expansion and azimuthal orders are Nrank = 22 and Mrank = 20, respectively. Figure 3.21 compares results obtained with the TNONAXSYM routine and the finite integration technique. The agreement between the curves is acceptable.

Figure 3.22 illustrates the normalized di erential scattering cross-sections for a positive uniaxial anisotropic cube of length 2a = 0.3 µm, and relative refractive indices mrx = mry = 1.5 and mrz = 1.7. The principal coordinate system coincides with the particle coordinate system, i.e., αpr = βpr = 0◦. Calculations are performed at a wavelength of λ = 0.3 µm and for the case of normal incidence. The parameters controlling the T -matrix calculations are Nrank = 18 and Mrank = 18, while the numbers of integration points on each cube face are Nint1 = Nint2 = 50. The behavior of the far-field patterns obtained with the TNONAXSYM routine, discrete dipole approximation and finite integration technique is quite similar.

Figure 3.23 shows plots of the di erential scattering cross-sections versus the scattering angle for a negative uniaxial anisotropic ellipsoid of semi-axis lengths a = 0.3 µm, b = 0.2 µm and c = 0.1 µm, and relative refractive indices mrx = mry = 1.5 and mrz = 1.3. The orientation of the principal coordinate system is given by the Euler angles αpr = βpr = 0◦ and the wavelength of

214	3	Simulation Results
			10	1
							TNONAXSYM - parallel
			0				TNONAXSYM - perpendicular
			10				CST - parallel
							CST - perpendicular
			−1
		DSCS	10
			−2
			10
			−3
			10
			−4
			10
			−5
			10	0	30	60	90	120	150	180

Scattering Angle (deg)

Fig. 3.21. Normalized di erential scattering cross-sections of a dielectric hexagonal prism computed with the TNONAXSYM routine and the finite integration technique (CST)

100

10−1

	DSCS	10−2
		10−3

10−4

10−50

TNONAXSYM - parallel

TNONAXSYM - perpendicular

DDA - parallel

DDA - perpendicular

CST - parallel

CST - perpendicular

30

60

90

120

150

180

Scattering Angle (deg)

Fig. 3.22. Normalized di erential scattering cross-sections of a uniaxial anisotropic cube computed with the TNONAXSYM routine, discrete dipole approximation (DDA) and finite integration technique (CST)

the incident radiation is λ = 0.3 µm. The maximum expansion and azimuthal orders are Nrank = 11 and Mrank = 11, respectively, and the numbers of integration points in the zenith and azimuthal directions are Nint1 = 100 and

3.3 Homogeneous, Axisymmetric and Nonaxisymmetric Particles

215

	100				TNONAXSYM - phi = 0˚
					TNONAXSYM - phi = 90˚
	10−1				CST - phi = 0˚
					CST - phi = 90˚
DSCS	10−2
	10−3
	10−4	30	60	90	120	150	180
	0

Scattering Angle (deg)

Fig. 3.23. Normalized di erential scattering cross-sections of a uniaxial anisotropic ellipsoid computed with the TNONAXSYM routine and the finite integration technique (CST)

Nint2 = 100, respectively. The curves show that the di erential scattering cross-sections are reasonably well reproduced by the TNONAXSYM routine.

Next we present calculations for a dielectric, a perfectly conducting and a chiral cube of length 2ksa = 10. The refractive index of the dielectric cube is mr = 1.5 and the chirality parameter is βki = 0.1. The scattering characteristics are computed by using the symmetry properties of the transition matrix. For particles with a plane of symmetry perpendicular to the axis of rotation (mirror symmetric particles with the surface parameterization r(θ, ϕ) = r(π − θ, ϕ)) the T matrix can be computed by integrating θ over the interval [0, π/2], while for particles with azimuthal symmetry (particles with the surface parameterization r(θ, ϕ) = r(θ, ϕ+2π/N ), where N ≥ 2) the T matrix can be computed by integrating ϕ over the interval [0, 2π/N ]. For T -matrix calculations without mirror and azimuthal symmetries, the numbers of integration points on each square surface are Nint1 = Nint2 = 30. For calculations using azimuthal symmetry, the numbers of integration points on the top and bottom quarter-square surfaces are Nint1 = Nint2 = 20, while for calculations using mirror symmetry, the numbers of integration points on the lateral half-square surface are Nint1 = Nint2 = 20. The integration surfaces are shown in Fig. 3.24, and the di erential scattering cross-sections are plotted in Figs. 3.25–3.27. The results obtained using di erent techniques are generally close to each other.

216 3 Simulation Results

z

O

y

2a

x

Fig. 3.24. Integration surfaces for a cube

		101
		100
	DSCS	10−1
		10−2

10−3

10−40

parallel perpendicular

parallel - azimuthal symmetry perpendicular - azimuthal symmetry parallel - azimuthal and mirror symmetry

perpendicular - azimuthal and mirror symmetry

60

120

180

240

300

360

Scattering Angle (deg)

Fig. 3.25. Normalized di erential scattering cross-sections of a dielectric cube using the symmetry properties of the transition matrix

3.3.3 Triangular Surface Patch Model

Some nonaxisymmetric particle shapes such as ellipsoids, quadratic prisms and regular polyhedral prisms are directly included in the Fortran program

3.3 Homogeneous, Axisymmetric and Nonaxisymmetric Particles

217

DSCS

101	parallel - azimuthal symmetry
	perpendicular - azimuthal symmetry
100	parallel
	perpendicular
10−1

10−2

10−3

10	−4	60	120	180	240	300	360
	0

Scattering Angle (deg)

Fig. 3.26. Normalized di erential scattering cross-sections of a chiral cube using the symmetry properties of the transition matrix

	1
	10
	0
	10
	−1
DSCS	10
	−2
	10
	−3
	10
	−4
	10
	−5
	10	0

parallel - azimuthal and mirror symmetry perpendicular - azimuthal and mirror symmetry parallel perpendicular

60

120

180

240

300

360

Scattering Angle (deg)

Fig. 3.27. Normalized di erential scattering cross-sections of a perfectly conducting cube using the symmetry properties of the transition matrix

as it is provided on the CD-ROM with the book. To handle arbitrary particle geometries, the program can also read particle shape data from an input file.

The particle shape data is based on a surface description using a triangular surface patch model. There are various 3D object file formats suitable for a meshed particle shape. We decided for the Wavefront .obj file format but the

218 3 Simulation Results

program will only support the polygonal format subset and not the free-form geometry (also included in the .obj file format). In this case, a triangular surface patch model of the particle surface has to be generated by an adequate software. For shapes given by an implicit equation, the HyperFun polygonizer [106, 183] supporting high-level language functional representations is a possible candidate for surface mesh generation. Function representation is a generalization of traditional implicit surfaces and constructive solid geometry, which allows the construction of complex shapes such as isosurfaces of realvalued functions [184]. The HyperFun polygonizer generates a VRML output of a su ciently regular triangular patch model, which is then converted to the

.obj file format by using the 3D Exploration program.

To compute the T -matrix elements by surface integrals we employ a modified midpoint or centroid quadrature. The integral over each surface patch is approximated by multiplying the value of the integrand at the centroid by the patch area [79]

f dS ≈
	f (vi,c)area[vi,1, vi,2, vi,3] ,	(3.12)

Si

where, vi,1, vi,2, vi,3 are the vertices spanning a triangle and vi,c denotes the mass center of the triangle [vi,1, vi,2, vi,3]

		1	3
	vi,c =		vi,j .	(3.13)
		3

j=1

For a su ciently large number of surface patches, the use of this centroid integration is satisfactorily accurate. In convergence checks versus the number of triangular faces, we found that this centroid quadrature is quite stable and the computational results are not much influenced by the number of integration elements.

As an example, we consider a sphere which has been cut at a quarter of its diameter on the z-axis as shown in Fig. 3.28. The cut sphere has been meshed

Fig. 3.28. Geometry of a cut sphere with 10,132 faces