Abstract

In a dense discrete random medium, the propagation and scattering of waves are affected not only by the individual properties of the particles such as size, shape, and permittivity, but also by group properties such as the statistics of relative particle positions and relative orientations. We use Monte Carlo simulations to investigate the interactions of electromagnetic waves with a dense medium consisting of spheroidal particles for cases of random orientation and for cases of aligned orientation. A shuffling process is used to generate the positions of densely packed spheroids. Multiple-scattering equations are formulated by means of the volume integral equation and are solved numerically. The scattering results are averaged over realizations. Numerical results are presented for the extinction rates and the phase matrices. Salient features of the numerical results indicate that (1) the extinction rates of densely packed small spheroids are smaller than those of independent scattering; (2) for aligned spheroids, the extinction rates are polarization dependent; and (3) the co-polarized part of the phase matrix for densely packed spheroids is smaller than that of independent scattering, while the cross-polarized part is larger than that for independent scattering. This means that the ratio of cross-polarization to co-polarization is significantly higher than that of independent scattering.

© 1998 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.
  2. A. Ishimaru, Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  3. C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of spherical particles: experiment and theory,” Waves Random Media 2, 225–234 (1992).
    [CrossRef]
  4. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).
  5. J. B. Guidt, G. Gouesbet, J. N. Le Toulouzan, “Accurate validation of visible infrared double extinction simultaneous measurements of particle sizes and number density by using densely laden standard media,” Appl. Opt. 29, 1011–1022 (1990).
    [CrossRef] [PubMed]
  6. R. West, D. Gibbs, L. Tsang, A. K. Fung, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
    [CrossRef]
  7. L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations,” Opt. Lett. 17, 314–316 (1992).
    [CrossRef] [PubMed]
  8. L. Zurk, L. Tsang, K. H. Ding, D. P. Winebrenner, “Monte Carlo simulation of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12, 1772–1781 (1995).
    [CrossRef]
  9. K. H. Ding, C. E. Mandt, L. Tsang, J. A. Kong, “Monte Carlo simulations of pair distribution functions of dense discrete random media with multiple sizes of particles,” J. Electromagn. Waves Appl. 6, 1015–1030 (1992).
  10. C. H. Chan, L. Tsang, “A sparse matrix canonical grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
    [CrossRef]
  11. W. C. Chew, J. H. Lin, X. G. Yang, “An FFT T-matrix method for 3D microwave scattering solution from random discrete scatterers,” Microwave Opt. Technol. Lett. 9, 194–196 (1995).
    [CrossRef]
  12. J. W. Perram, M. S. Wertheim, J. L. Lebowitz, G. O. Williams, “Monte Carlo simulations of hard spheroids,” Chem. Phys. Lett. 105, 277–280 (1984).
    [CrossRef]
  13. J. W. Perram, M. S. Wertheim, “Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function,” J. Comput. Phys. 58, 409–416 (1985).
    [CrossRef]
  14. R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).
  15. R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, N.J., 1973).
  16. J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  17. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

1995 (3)

L. Zurk, L. Tsang, K. H. Ding, D. P. Winebrenner, “Monte Carlo simulation of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12, 1772–1781 (1995).
[CrossRef]

C. H. Chan, L. Tsang, “A sparse matrix canonical grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
[CrossRef]

W. C. Chew, J. H. Lin, X. G. Yang, “An FFT T-matrix method for 3D microwave scattering solution from random discrete scatterers,” Microwave Opt. Technol. Lett. 9, 194–196 (1995).
[CrossRef]

1994 (1)

1992 (3)

L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations,” Opt. Lett. 17, 314–316 (1992).
[CrossRef] [PubMed]

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of spherical particles: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

K. H. Ding, C. E. Mandt, L. Tsang, J. A. Kong, “Monte Carlo simulations of pair distribution functions of dense discrete random media with multiple sizes of particles,” J. Electromagn. Waves Appl. 6, 1015–1030 (1992).

1990 (1)

1985 (1)

J. W. Perram, M. S. Wertheim, “Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function,” J. Comput. Phys. 58, 409–416 (1985).
[CrossRef]

1984 (1)

J. W. Perram, M. S. Wertheim, J. L. Lebowitz, G. O. Williams, “Monte Carlo simulations of hard spheroids,” Chem. Phys. Lett. 105, 277–280 (1984).
[CrossRef]

1982 (1)

Brent, R. P.

R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Chan, C. H.

C. H. Chan, L. Tsang, “A sparse matrix canonical grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
[CrossRef]

Chew, W. C.

W. C. Chew, J. H. Lin, X. G. Yang, “An FFT T-matrix method for 3D microwave scattering solution from random discrete scatterers,” Microwave Opt. Technol. Lett. 9, 194–196 (1995).
[CrossRef]

Ding, K. H.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fung, A. K.

Gibbs, D.

Gouesbet, G.

Guidt, J. B.

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

Ishimaru, A.

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of spherical particles: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

Kong, J. A.

K. H. Ding, C. E. Mandt, L. Tsang, J. A. Kong, “Monte Carlo simulations of pair distribution functions of dense discrete random media with multiple sizes of particles,” J. Electromagn. Waves Appl. 6, 1015–1030 (1992).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

Kuga, Y.

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of spherical particles: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

Le Toulouzan, J. N.

Lebowitz, J. L.

J. W. Perram, M. S. Wertheim, J. L. Lebowitz, G. O. Williams, “Monte Carlo simulations of hard spheroids,” Chem. Phys. Lett. 105, 277–280 (1984).
[CrossRef]

Lin, J. H.

W. C. Chew, J. H. Lin, X. G. Yang, “An FFT T-matrix method for 3D microwave scattering solution from random discrete scatterers,” Microwave Opt. Technol. Lett. 9, 194–196 (1995).
[CrossRef]

Mandt, C. E.

L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations,” Opt. Lett. 17, 314–316 (1992).
[CrossRef] [PubMed]

K. H. Ding, C. E. Mandt, L. Tsang, J. A. Kong, “Monte Carlo simulations of pair distribution functions of dense discrete random media with multiple sizes of particles,” J. Electromagn. Waves Appl. 6, 1015–1030 (1992).

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of spherical particles: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Perram, J. W.

J. W. Perram, M. S. Wertheim, “Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function,” J. Comput. Phys. 58, 409–416 (1985).
[CrossRef]

J. W. Perram, M. S. Wertheim, J. L. Lebowitz, G. O. Williams, “Monte Carlo simulations of hard spheroids,” Chem. Phys. Lett. 105, 277–280 (1984).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tsang, L.

C. H. Chan, L. Tsang, “A sparse matrix canonical grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
[CrossRef]

L. Zurk, L. Tsang, K. H. Ding, D. P. Winebrenner, “Monte Carlo simulation of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12, 1772–1781 (1995).
[CrossRef]

R. West, D. Gibbs, L. Tsang, A. K. Fung, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
[CrossRef]

L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations,” Opt. Lett. 17, 314–316 (1992).
[CrossRef] [PubMed]

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of spherical particles: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

K. H. Ding, C. E. Mandt, L. Tsang, J. A. Kong, “Monte Carlo simulations of pair distribution functions of dense discrete random media with multiple sizes of particles,” J. Electromagn. Waves Appl. 6, 1015–1030 (1992).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

Wertheim, M. S.

J. W. Perram, M. S. Wertheim, “Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function,” J. Comput. Phys. 58, 409–416 (1985).
[CrossRef]

J. W. Perram, M. S. Wertheim, J. L. Lebowitz, G. O. Williams, “Monte Carlo simulations of hard spheroids,” Chem. Phys. Lett. 105, 277–280 (1984).
[CrossRef]

West, R.

Williams, G. O.

J. W. Perram, M. S. Wertheim, J. L. Lebowitz, G. O. Williams, “Monte Carlo simulations of hard spheroids,” Chem. Phys. Lett. 105, 277–280 (1984).
[CrossRef]

Winebrenner, D. P.

Yang, X. G.

W. C. Chew, J. H. Lin, X. G. Yang, “An FFT T-matrix method for 3D microwave scattering solution from random discrete scatterers,” Microwave Opt. Technol. Lett. 9, 194–196 (1995).
[CrossRef]

Zurk, L.

Appl. Opt. (1)

Chem. Phys. Lett. (1)

J. W. Perram, M. S. Wertheim, J. L. Lebowitz, G. O. Williams, “Monte Carlo simulations of hard spheroids,” Chem. Phys. Lett. 105, 277–280 (1984).
[CrossRef]

J. Comput. Phys. (1)

J. W. Perram, M. S. Wertheim, “Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function,” J. Comput. Phys. 58, 409–416 (1985).
[CrossRef]

J. Electromagn. Waves Appl. (1)

K. H. Ding, C. E. Mandt, L. Tsang, J. A. Kong, “Monte Carlo simulations of pair distribution functions of dense discrete random media with multiple sizes of particles,” J. Electromagn. Waves Appl. 6, 1015–1030 (1992).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Microwave Opt. Technol. Lett. (2)

C. H. Chan, L. Tsang, “A sparse matrix canonical grid method for scattering by many scatterers,” Microwave Opt. Technol. Lett. 8, 114–118 (1995).
[CrossRef]

W. C. Chew, J. H. Lin, X. G. Yang, “An FFT T-matrix method for 3D microwave scattering solution from random discrete scatterers,” Microwave Opt. Technol. Lett. 9, 194–196 (1995).
[CrossRef]

Opt. Lett. (1)

Waves Random Media (1)

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of spherical particles: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

Other (6)

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

R. P. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, N.J., 1973).

J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

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Figures (5)

Fig. 1
Fig. 1

Electric field Einc(r) incident on N nonoverlapping small spheroids that are randomly positioned and oriented in a volume V.

Fig. 2
Fig. 2

Extinction rate as a function of fractional volume of particles, with relative permittivity of particles r=3.2, for prolate spheroids with ka=0.2 and e=1.8. Dotted curve, medium with spheres; pluses, medium with randomly oriented spheroids; circles and crosses, medium with aligned spheroids but with the incident wave being vertically polarized and horizontally polarized, respectively.

Fig. 3
Fig. 3

Extinction rate as a function of elongation ratio of randomly oriented spheroids with ka=0.2 and fractional volume f=20%.

Fig. 4
Fig. 4

Phase matrix as a function of scattering angle for ka=0.2, fractional volume f=10%, elongation ratio e=1.8, and relative permittivity of particles r=3.2 for randomly oriented spheroids. In the simulations, N=2000 particles are used and the results are averaged over Nr=50 realizations. (a) P11, (b) P21, (c) P12, (d) P22. Circles, dense-medium results; crosses, independent-scattering results.

Fig. 5
Fig. 5

Phase matrix as a function of scattering angle. Parameters and symbols are as for Fig. 4, except that fractional volume f=30%.

Tables (1)

Tables Icon

Table 1 Values of the Phase Matrix for Various Fractional Volumesa

Equations (111)

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p(r)=jforrinVjforrinthebackground.
P(r)=χ(r)E(r),
χ(r)=p(r)-1
E(r)=Einc(r)+k2dr g(r, r)P(r)-dr g(r, r)P(r)
g(r, r)=exp(ik|r-r|)4π|r-r|
E(r)=Einc(r)+k2j=1NVjdr g(r, r)Pj(r)-j=1NVjdr g(r, r)Pj(r).
Ej(r)=α=1Nbαjαfjα(r).
rj=xjx+yjy+zjz
zbj=sin βj cos αjx+sin βj sin αjy+cos βjz.
fj1=zbj 1v0j
fj2=xbj 1v0j
fj3=ybj 1v0j
fj4=1A4j(xbjxbj+ybjybj-2zbjzbj),
fj5=1A5j(zbjxbj+xbjzbj),
fj6=1A5j(zbjybj+ybjzbj),
fj7=1A7j(xbjxbj-ybjybj),
fj8=1A7j(ybjxbj+xbjybj).
E(r)=Einc(r)+j=1Nα=1Nbajαqjα(r),
qjα(r)=k2Vjdrg(r, r)fjα(r)(rj-1)-Vjdrg(r, r)·fjα(r)(rj-1)
qjα(r)Cjαfjα(r),
alβ=Vldrflβ(r)·E(r)=Vldrflβ(r)·α=1Nbalαflα(r)=Vldrflβ(r)·Einc(r)+j=1jlNα=1NbajαVldr×flβ(r)·qjα(r)+α=1NbalαVldrflβ(r)·qlα(r)=Vldrflβ(r)·Einc(r)+j=1jlNα=1NbajαVldrflβ(r)·qjα(r)+alβClβ.
alβ=1(1-Clβ)Vldrflβ(r)·Einc(r)+j=1jlNα=1NbajαVldrflβ(r)·qjα(r).
Vldrflβ(r)·Einc(r)
=v0lflβ·Einc(rl)forβ=1, 2, 30forβ>3.
Es(r)=k2 exp(ikr)4πr(vsvs+hshs)·j=1Nα=1Nbajα(rj-1)×Vjdr exp(-iks·r)fjα(r),
Es(r)k2 exp(ikr)4πr(vsvs+hshs)·j=1Nα=1Nbajα(rj-1)×v0jfjα exp(-iks·rj).
xj=xj+Δηx
yj=yj+Δηy
zj=zj+Δηz,
θj=πηθ,
ϕj=2πηϕ,
F(r, λ)=λFA(r-rA, ΩA)+(1-λ)FB(r-rB, ΩB),
Fi(r, λ)=(r-ri)TAi-1(Ωi)(r-ri),
Ai(Ωi)=ai2aiaiT+bi2bibiT+ci2ciciT.
FAB(rA, ΩA, rB, ΩB)=0λ1max{rmin[F(r, λ)]},
FAB(rA, ΩA, rB, ΩB)
×<1ifAandBoverlap,=1ifAandBareexternallytangent,>1ifAandBarenotoverlapping.
Vldrflβ(r)·qjα(r)
=(rj-1)vojvolk2flβ·G(rl, rj)·fjα,
G(rl, rj)=l+k2g(r, r)
V=Nvf,
Einc(r)=y exp(ikz)
Es=Evsvs+Ehshs.
Es=rNrσ=1NrEsσ,
Esσ=rEsσ-Es,
Esσ=Evsσvs+Ehsσhs.
σvsN=1Nrσ=1Nr|Evsσ|2,
σhsN=1Nrσ=1Nr|Ehsσ|2.
κe=1V0πdθs sin θs02πdϕs(σvsN+σhsN).
p(β, α)=sin β4π
02πdα0πdβp(β, α)=1.
alβ=1(1-Clβ)Vldrflβ(r)·Einc(r).
P11(θs)=σvsNV,
P21(θs)=σhsNV.
P12(θs)=σhsNV,
P22(θs)=σvsNV.
Pα=(p-)fα.
f1=z 1v0,
f2=x 1v0,
f3=y 1v0,
f4=1A4(xx+yy-2zz),
f5=1A5(zx+xz),
f6=1A6(zy+yz),
f7=1A7(xx-yy)
f8=1A8(yx+xy)
Vdrfα·fβ=δαβ,
Eind(α)=Cαfα.
C1=-(p-)ξ02lnξ0+1ξ0-1-1(ξ02-1),
C2=C3=-12p-ξ0ξ0-ξ02-12lnξ0+1ξ0-1,
C4=-3ξ02(ξ02-1)p-×12(3ξ02-1)lnξ0+1ξ0-1-3ξ0,
C5=C6=-(ξ02-1)2p-(2ξ02-1)×3ξ02-2ξ02-1-3ξ02lnξ0+1ξ0-1,
C7=C8=-ξ02-14p-×ξ032(ξ02-1)lnξ0+1ξ0-1-3ξ03-5ξ0ξ02-1,
ξ0=11-a2/c2.
E=f2=x 1v0,
P=(p-) xv0.
Φout=Φin
-ξ·Φout+ξ·Φin=ξ·P.
Φin=A[(ξ2-1)(1-η2)]1/2 cos ϕ,
Φout=BQ11(ξ)P11(η)cos ϕ,
P11(η)=1-η2,
Q11(ξ)=ξ2-1ξξ2-1-12lnξ+1ξ-1.
A=12p- f0ξ0v0ξ0-ξ02-12lnξ0+1ξ0-1,
Eind=-Φin.
Eind(2)=C2f2,
E=f5.
P=(p-)A5(zx+xz).
Φin=-Af02ξη[(ξ2-1)(1-η2)]1/2 cos ϕ,
Φout=Bξ2-13ξ2-2ξ2-1-3ξ2lnξ+1ξ-1×3η1-η2 cos ϕ.
A=1A5-(ξ02-1)2p-(2ξ02-1)×3ξ02-2ξ02-1-3ξ02lnξ0+1ξ0-1.
f0=c2-a2
x=f0[(ξ2-1)(1-η2)]1/2 cos ϕ,
y=f0[(ξ2-1)(1-η2)]1/2 sin ϕ,
z=f0ξη.
P22(ξ)P22(η)exp(2iϕ)
=9(ξ2-1)(1-η2)exp(2iϕ),
=9f02(x2-y2+i2xy),
P21(ξ)P21(η)exp(iϕ)
=-9ξη[(ξ2-1)(1-η2)]1/2exp(iϕ),=-9zf02(x+iy),
P2-1(ξ)P2-1(η)exp(-iϕ)
=-z4f02(x-iy),
P2-2(ξ)P2-2(η)exp(-2iϕ)
=164f02(x2-y2-2ixy),
P20(ξ)P20(η)=14(9ξ2η2-3η2-3ξ2+1),=-34x2+y2f 02+32z2f02-12.
Φ4=-12(x2+y2)+z2-f 023,
Φ5=-zx,
Φ6=-zy,
Φ7=-12(x2-y2),
Φ8=-xy.
P11(indep.)P11(dense)
P11(dense)P21(dense)
P11(indep.)P21(indep.)

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