Numerical Illustrations with Finite Dielectric Cylinders in .NET

Writer DataMatrix in .NET Numerical Illustrations with Finite Dielectric Cylinders
1.4.1 Numerical Illustrations with Finite Dielectric Cylinders
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For numerical illustrations, we choose the nonspherical particles to be finite dielectric cylinders, which provide a reasonable scattering model for vegetable canopy. Scattering by the finite cylinders will be calculated numerically using a method of moment body of revolution (MoM-BOR) code. In this code, surface integral equations are solved by using the method of moments [Harrington, 1968]. The variations of the unknown electric and magnetic surface fields are approximated by staggered pulse functions in the t-direction and are expanded in Fourier series in the </J-direction. The detailed discretization procedure can be found in Glisson and Wilton [1980] and Joseph [1990J. For computational efficiency, it is important to note that it is required to calculate the inverse of the impedance matrix or admittance matrix only once for a cylinder of fixed length, radius, and permittivity. The admittance matrix of the cylinder is then stored. Given incident fields of arbitrary polarization, incident angle, and orientation of the cylinder, the surface fields induced on the surface of the cylinder are calculated by multiplying the admittance matrix with the incident field. In this case, the total
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1.4 Active Remote Sensing of a Layer of Nonspherical Particles
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surface fields on the curve sides as well as on the two ends of the cylinder are calculated. The bistatic scattering amplitudes can then be calculated. In our implementation, we further store the bistatic scattering amplitudes in the body frame for each harmonic. For an arbitrary incident field and with arbitrary orientation, the bistatic scattering amplitudes in the principal frame can be calculated from that of the stored scattering amplitudes by rotation of coordinates and by interpolation. Step 1: Calculation of Impedance Matrix and Admittance Matrix Given the length, radius, and complex permittivity of the cylinder, the impedance matrix and the admittance matrix of the cylinder are calculated by solving the surface integral equations only one time with MoM-BOR. The admittance matrix is then stored. Step 2: Calculation of Bistatic Scattering Amplitudes for Each Harmonic in the Body Frame We shall use the same notations as 1, in which the infinite cylinder approximation was used. Consider an incident field given in the body frame by
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i E = Evbi'Ubi + Ehbihbi Si = sin (hd:b + cos (hiZb Vvi = cos (hiXb - sin (hiZb hbi = fib
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(8.1.84) (8.1.85) (8.1.86) (8.1.87)
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Because of rotational symmetry, we have taken <Pbi = 0 without loss of generality. Given the admittance matrix stored in step 1, a product can be taken between the admittance matrix and the incident field to give the surface fields at each discretized point on the surface for each harmonic. These include E~m(Zb), H~m(Zb), E;'m(Zb), and H;'m(Zb) on the curved side,
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E~m(Pb)' H~m(Pb)' E~m(Pb), and H~m(Pb) on the upper side, and E%m(Pb), H%m(Pb), E~m(Pb), and H~m(Pb) on the lower side, where Zb denotes the coordinate on the curved side and Pb denotes the cylindrical radial coordinate
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on the upper side and the lower side. The bistatic scattering amplitudes in the body frame are
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!VbVb ( hs, <Pbs; Obi, <Pbi
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eim</>b.' !vbvbm(Obs, Ob;)
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(8.1.88)
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!vbhb(fhs, <Pbs; (hi, <Pbi
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eim</>bS !vbhbm((hs,fhi) eim</>hs fhhvhm((hs,fhi) eim</>hS fhhhbm ((hs,fhi)
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(8.1.89)
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fhbVh((hs, <Pbs;f)bi, <Pbi = 0) = fhbhb(fhs,<Pbs;(hi,<Pbi =0) =
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(8.1.90)
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The bistatic scattering amplitudes for each harmonic, fVbvbm(fhs, ()bi), !vbhhm(()bs, Obi), !hbvhm(Obs, Obi), and ihhhhm(Obs, Obi), can be expressed in terms of the surface fields as follows
!Vbobm(Obs,fhi) =
(_i)m~ [ -
ik{ sin (hsrJHq,mJm(ws)
+ :: COSObsrJHzmJm(ws)}
( _i)m + _ _ et'k( cos () bi-COS (})L b" "2
[imk Jm(ksin()bsPb) dpbPb. 2 0 k sm ObsPb { - E~m(Pb) + rJHi:m(Pb) cos ()bs}
- kEzmJ:n(ws)] a
+ kJ:n(k sin ObsPb) {Ei:m(Pb)
+ rJH~m(Pb) cos Obs} ]
+ (_i)m e-ik(cos(lhi-COS(}b.,)~ r
dPbPb [_
k sm ObsPb
Jm(k sin ObsPb)
X { -
E*m (Pb)
+ rJHf:m (Pb) cos Obs}
(8.1.92)
kJ:n(ksinObsPb){E~m(Pb) +rJHtm(Pb)COSObS}]
(_i)m~ [ ik{ sinObsEq,mJm(ws)
ihbobm(Obs,Obi) =
+ :: COsObsEzmJm(Ws)} + krJHzmJ:n(ws)]
+ (_i)m eik(cos(lbi-COS(}bS)~
x {rJH~m (Pb)
r dPbPb [ k !mk Jm(k ()bsPb
sin ObsPb)
+ E~m (Pb) cos ()bs}
+ kJ:n(ksin()bsPb)
{-rJH~m(Pb) + E~m(Pb) COS Obs} ]
1.4 Active Remote Sensing of a Layer of Nonspherical Particles
+ (_i)m e-ik(coslhi-cos(lb,,)%