E E 1
G2 x, y,z 22 zZ -8nR-6rfC ' RniU2E where truncations of G and G2 are done respectively at iVi and iV2. Asymptotically, Thus both series of Gi x,y and G2 x,y have exponential decay G2 x, y - gt exp - nia 2 n2ay 2 2 3.4.95 For the 1-D case, the two series are respectively e v b anc g-Knia 2 2 The two series have the same exponential decay rate if E y 7r ax. For the 2-D case, we choose splitting parameter E such that In the numerical simulations, we use the following parameters A 1, ax 0.95A, ay...
EyPs z fx
where the integration da' is over one period P, and similarly HM - jpda' GP ps,p's h- VsHy p's - Hy j 's h VsGP ps,p's _ f Hy Ps z gt f x 0 lt fix Gp ps,p's eXPiikxn x - x' ikzn z - z' is the periodic Green's function of region 0. In 3.2.11 kzn V s kxn yk2 k2y k n kspn 3.2.13 Making use of periodic Green's function of region 1, we have d lt r' GlP ps, p's h VsEly p's - Ely p's h V'sGlP ps,p's Jpda' G1PiPs, Ps n VsHlyip's - Hlyip's h VsG1PiPs,p's Gipips,p's YE -r ey p ikxnix - x' iklzn z - z'...
REFERENCES AND ADDITIONAL READINGS Xpt
Anastassiu, H. T., M. Smelyanskiy, S. Bindiganavale, and J. L. Volakis 1998 , Scattering from relatively flat surfaces using the adaptive integral method, Radio Sci., 33 1 , 7-16. Axline, R. M. and A. K. Fung 1978 , Numerical computation of scattering from a perfectly conducting random surface, IEEE Trans. Antennas Propagat., 26 3 , 482-488. Belszynski, E., M. Belszynski, and T. Jaroszewicz 1994 , A fast integral-equation solver for electromagnetic scattering problems, IEEE Ant. and Propagat....
Angular Correlation Function And Detection Of Buried Object 1
2 Two-Dimensional Simulations of Angular Memory Effect and Detection of Buried Object 553 2.2 Simple and General Derivation of Memory Effect 553 2.3 ACF of Random Rough Surfaces with Different Averaging Methods 555 2.4 Scattering by a Buried Object Under a Rough Surface 557 3 Angular Correlation Function of Scattering by a Buried Object Under a 2-D Random Rough Surface 3.2 Formulation of Integral Equations 565 3.3 Statistics of Scattered Fields 570 3.4 Numerical Illustrations of ACF and PACF...
pu fV Ef cU sK
11 Computational step we move down the level in this manner. Addition, multiplication and anterpolation. Vim W c m a P m a 5.3.145 5-3.146 The number of computational steps for calculating a and ai 1 is respectively, 2 -2 2 m-2 The same procedure goes on until we reach the first level, which does not need anterpolation. The first level needs QN steps. Thus the total number of steps for dis-aggregation is 12 Computational step Repeat in the same manner for pL. The total number of steps is,...
Vw
Figure 8.1.2 Pair functions for a independent particle position, b gas, c crystalline solid, and d liquid and amorphous solid. h ri2 c n2 n0 J dr3c ri3 h r32 8.1.5a h rl2 c fi2 nQJ df3c f13 g f32 - 1 8.1.56 which is known as the Ornstein-Zernike equation. The physical interpretation of the second term of 8.1.5a , which is the indirect correlation function, is that the indirect influence of particle 1 on particle 2 is a result of particle 1 acting directly on a particle at 7 which in turn exerts...
polar plot dB 1
Figure 7.4.4 Polar plot dB of the phase function of Fig. 7.4.2 of the clustered random case for N 500, lc 0.2A, and L 50A with incident angles at 0 10 and 10 . Upper half-plane shows j s 10 and 0S is 0 180 going clockwise. Lower half-plane shows j s 190 and 0S is 0 gt 180 going counterclockwise. Figure 7.4.4 Polar plot dB of the phase function of Fig. 7.4.2 of the clustered random case for N 500, lc 0.2A, and L 50A with incident angles at 0 10 and 10 . Upper half-plane shows j s 10 and 0S is 0...
rr sr
For the case of vertical polarized incidence, we can perform similar calculations. In the backscattering direction, 6S and lt f s ttks x ksx 9td cos 0i 0t 0s- Hence, the backscattering coefficients are the same for crvv and o gt hh' We have dr r - 1 e ReKd ks gt r 10.2.85 The results of coherent reflection in this section agree with the QCA results in the low frequency approximation. The bistatic intensity agrees with QCA combined with distorted Born approximation. The QCA approximation will be...
fi z Of LL
Then and 67V 2 are real- We first obtain N independent Gaussian random numbers with zero mean and unit variance. We next multiply the numbers by a normalization factor to bo, bN 2, bfn and with n 1,2, , iV 2 1, such that 1.1.17 holds. We then use to get bn with n -1,-2, , N 2 1. The permittivity is then calculated by 1.1.19 . 1.3 Numerical Results and Applications to Antarctica In this section we illustrate the numerical results and application to the Antarctic firn. The Antarctic firn has a...
Jv
jf dfJ i jfc2 J df'g f,f' Jja erj - 1 k2 erj - l vivjfw fufj Jja 10.3.24 where the dyadic Green's function is S r, r f Pjg r, f' 10.3.25 Explicit expressions of G r,r' are given in 2.3.5 - 2.3.7 of Chapter 2. Substituting 10.3.23 and 10.3.24 into 10.3.22 , we have CW 7TT vlflp Einc n J2Y1 c3 k lt 2 erj - l viVjfip G ri,rj fja gt Equation 10.3.26 is the Foldy-Lax multiple scattering equations based on volume integral equation. After the matrix equations are solved, the final scattered field is...
Simulations Of Twodimensional Dense Media
1.1 Extinction as a Function of Concentration 454 1.2 Extinction as a Function of Frequency 456 2 Random Positions of Cylinders 458 2.1 Monte Carlo Simulations of Positions of Hard Cylinders 458 2.2 Simulations of Pair Distribution Functions 460 2.3 Percus-Yevick Approximation of Pair Distribution Functions 461 2.4 Results of Simulations 463 2.5 Monte Carlo Simulations of Sticky Disks 463 3 Monte Carlo Simulations of Scattering by Cylinders 469 3.1 Scattering by a Single Cylinder 469 3.2...
nJ dPa IxM mm
The last term in 12.2.32 is the coherent intensity. Subtracting it from 12.2.32 gives the incoherent intensity, which, however, can contain partial coherent effects. Thus the bistatic intensity of the incoherent field, under the first-order approximation, is nA Ns - 1 j dpaj J exp ikdp paj - pal f L 2 n2AjAdp gp p - As is clear from 12.2.33 , the bistatic intensity of the incoherent field only depends on intrinsic properties of the random media such as quantities like tlA-, L, and gp. The value...
Nf ll jj me [gfj n
The first term in 7.3.18 represents that of conventional radiative transfer theory and the second term represents correlation effects. The last term in 7.3.18 corresponds to the coherent intensity that is in the forward direction. However, from 7.3.11 we have Im F kuk jhnf N f 2 7.3.19 Thus, 7.3.19 only contains the first term of 7.3.18 and does not contain the pair distribution function as in 7.3.18 , nor does it contain the sharply peaked forward scattering. Thus, the first-order solution...
REFERENCES AND ADDITIONAL READINGS Www
Allen, M. P. and D. J. Tildesley 1989 , Computer Simulation of Liquids, Oxford University Press, New York. Baxter, R. J. 1968a , Percus-Yevick equation for hard spheres with surface adhesion, J. Chem. Phys, 49 6 , 2770-2773. Baxter, R. J. 1968b , Ornstein-Zernike relation for a disordered fluid, Aust. J. Phys., 21, 563-569. Baxter, R. J. 1970 , Ornstein-Zernike relation and Percus-Yevick approximation for fluid mixtures, J. Chem. Phys., 52, 4559-4562. Baxter, R. J. 1971 , Distribution...
References And Additional Readings
Fung, A. K. 1994 , Microwave Scattering and Emission Models and Their Applications, Artech House, Norwood, Massachusetts. Gurvich, A. S., V. L. Kalinin, and D. T. Matveyer 1973 , Influence of the internal structure of glaciers on their thermal radio emission, Atm. Oceanic Phys. USSR, 9, 713-717. Hall, D. K. and J. Martinec 1985 , Remote Sensing of Snow and Ice, Chapman and Hall, London. Kong, J. A. 1990 , Electromagnetic Wave Theory, 2nd edition, John Wiley amp Sons, New York. Matzler, C. 1987...
Pi fa vir
Applying boundary conditions 10.3.79 and 10.3.82 gives Vc2 a2. The internal induced electric field is Etn j V i with C2 and 2 as given in 10.3.72 and 10.3.47 , respectively. For the case of 5, let the electric field inside the spheroid be The potentials and are proportional to P21 P21 7 cos and Q iO- i7 cos respectively, where is Legendre function of the 3t 1 T72 cos lt gt 10.3.90 Applying boundary conditions of 10.3.79 and 10.3.80 gives Using 10.3.89 , we get Eind C5 5 where C5 and 5 are as...
k N Ef Eincr GFzF PjAVj erft LEi
Let R unit vector from f' to F, R F r' and R r r'. For F F', by straightforward differentiation, j F, f' G1 R 7 G2 R RR 2.3.5 Gi R -1 ikR k2R2 4 k2R3 2'3-6 G2 R 3 - 3ikR - klR 2-3-7
ikr
-7 5- I dr'r'eikT' --j - 3 A 2 3 lt 5 gt o J g 3 k2 3 k2 in the limit of small a. Putting 2.3.22 in 2.3.19 - l sk2 i - J2 ij Pj 2.3.24 Multiply by AV epi e , and noting that Using a 3 47r 1 3d and AV d3 in 2.3.29 gives The term with imaginary part in the denominator of 2.3.30 is known as radiative correction, which arises for the same reasoning as when scattering by Rayleigh spheres was discussed in Chapter 2, Section 8.2 of Volume I. Let the medium be discretized into rectangular...
Gff
ikz' R ksz h -ksz eik z' kSx k sin 9S cos lt f s ksy k sin 9S sin 4 gt s 13.3.56 ks j_ ksxx ks yy 13.3.5 d Let the incident wave be a plane wave with the following wavevector components kix k sin 6i cos fa 13.3.6a kiy k sin 9i sin fa 13.3.66 The incident wave is downward going in the direction 7t - di, fa while the reflected wave is upward going in the direction 9i,fa . Thus the total field in region 0 is ETE hz RTE kiz ETMR kiz h kiz where Ete and Et m are the amplitudes of the TE and TM...
Particle Positions For Dense Media Characterizations And Simulations
1 Pair Distribution Functions and Structure Factors 404 1.2 Percus-Yevick Equation and Pair Distribution Function for Hard Spheres 406 1.3 Calculation of Structure Factor and Pair Distribution Function 409 2 Percus-Yevick Pair Distribution Functions for Multiple Sizes 411 3 Monte Carlo Simulations of Particle Positions 414 3.1 Metropolis Monte Carlo Technique 415 3.2 Sequential Addition Method 418 4.1 Percus-Yevick Pair Distribution Function for Sticky Spheres 424 4.2 Pair Distribution Function...
REFERENCES AND ADDITIONAL READINGS Vzq
Ail, W. C., J. A. Kong, and L. Tsang 1994 , Absorption enhancement of scattering of electromagnetic waves by dielectric cylinder clusters, Microwave Opt. Technol. Lett., 7 10 , 454-457. Au, W. C., L. Tsang, R. T. Shin, and J. A. Kong 1996 , Collective scattering and absorption in microwave interaction with vegetation canopies, Progress in Electromag. Res., 14, 182-231, EMW Publishers, Cambridge, Massachusetts. Chew, W. C. 1990 , Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold,...
Preface
Electromagnetic wave scattering is an active, interdisciplinary area of research with myriad practical applications in fields ranging from atomic physics to medical imaging to geoscience and remote sensing. In particular, the subject of wave scattering by random discrete scatterers and rough surfaces presents great theoretical challenges due to the large degrees of freedom in these systems and the need to include multiple scattering effects accurately. In the past three decades, considerable...
REFERENCES AND ADDITIONAL READINGS Uip
Agnon, Y. and M. Stiassnie 1991 , Remote sensing of the roughness of a fractal sea surface, J. Geophys. Res., 96 C7 , 12773-12779. Austin, T. R., A. W. England, and G. H. Wakefield 1994 , Special problems in the estimation of power-law spectra as applied to topographical modeling, IEEE Trans. Geosci. Remote Sens., 32, 928-939. Axline, R. M. and A. K. Fung 1978 , Numerical computation of scattering from a perfectly conducting random surface, IEEE Trans. Antennas Propagat., 26 3 , 482-488....
Info Wqe
Figure 6.3.3 Convergence with respect to the neighborhood distance r for co-polarized component. Simulation parameters are Lx Ly 16A, h 0.2A, lx ly 0.6A, er 6.5 1.0 , and 0 10 and - 0 . Figure 6.3.4 Convergence with respect to the neighborhood distance r for cross-polarized component. The simulation parameters are those of Fig. 6.3.3. Next, the matrix solving time dependence on the neighborhood distance r and dielectric constant of the second medium are illustrated. In Fig. 6.3.5, total CPU...
Random Rough Surface Simulations
1 Perfect Electric Conductor Non-Penetrable Surface 114 1.2 Matrix Equation Diriehlet Boundary Condition 1.3 Tapering of Incident Waves and Calculation of Scattered 1.4 Random Rough Surface Generation 124 1.4.1 Gaussian Rough Surface 124 1.4.2 Fractal Rough Surface 132 1.5 Neumann Boundary Condition MFIE for TM Case 134 2.2 Absorptivity, Emissivity and Reflectivity 141 2.3 Impedance Matrix Elements Numerical Integrations 143 2.4 Simulation Results 145 2.4.1 Gaussian Surface and Comparisons with...
Scattering And Emission By A Periodic Rough Surface
1 Dirichlet Boundary Conditions 62 1.1 Surface Integral Equation 62 1.2 Floquet's Theorem and Bloch Condition 63 1.3 2-D Green's Function in 1-D Lattice 64 1.4 Bistatic Scattering Coefficients 67 2 Dielectric Periodic Surface T-Matrix Method 68 2.1 Formulation in Longitudinal Field Components 69 2.2 Surface Field Integral Equations and Coupled Matrix Equations 74 2.3 Emissivity and Comparison with Experiments 81 3 Scattering of Waves Obliquely Incident on Periodic Rough Surfaces Integral...
Threedimensional Wave Scattering From Twodimensional Rough Surfaces
1 Scattering by Non-Penetrable Media 270 1.1 Scalar Wave Scattering 270 1.1.1 Formulation and Numerical Method 270 1.1.2 Results and Discussion 273 1.1.3 Convergence of SMFSIA 277 1.2 Electromagnetic Wave Scattering by Perfectly Conducting Surfaces 278 1.2.1 Surface Integral Equation 278 1.2.2 Surface Integral Equation for Rough Surface Scattering 280 1.2.3 Computation Methods 281 1.2.4 Numerical Simulation Results 286 2 Integral Equations for Dielectric Surfaces 293 2.1 Electromagnetic Fields...
Fast Computational Methods For Solving Rough Surface Scattering Problems 1
1 Banded Matrix Canonical Grid Method for Two-Dimensional Scattering for PEC Case 179 1.2 Formulation and Computational Procedure 180 1.3 Product of a Weak Matrix and a Surface Unknown Column 1.4 Convergence and Neighborhood Distance 188 1.5 Results of Composite Surfaces and Grazing Angle Problems 189 2 Physics-Based Two-Grid Method for Lossy Dielectric Surfaces 196 2.2 Formulation and Single-Grid Implementation 198 2.3 Physics-Based Two-Grid Method Combined with Banded Matrix Iterative...
Info Xan
Figure 6.3.11 Backscattering enhancement of two cases 1-D and 2-D electromagnetic wave incidence on dielectric surface er 6.5 1.0 . Comparison of normalized bistatic scattering coefficients for h 0.5A, I 1.0 , 04 10 , and 0 . with an incident angle of 10 . The peaks near 10 are clearly visible. In the simulation, the co-polarized component requires many realizations for the backscattering peak to converge. This is because in co-polarization, the firstorder scattering obscures the second-order...
Integral Equation Formulations And Basic Numerical Methods
1 Integral Equation Formulation for Scattering Problems 14 1.1 Surface Integral Equations 14 1.2 Volume Integral Equations 17 1.3 Dyadic Green's Function Singularity and Electrostatics 19 3 Discrete Dipole Approximation DDA 27 3.2 Radiative Corrections 29 4 Product of Toeplitz Matrix and Column Vector 37 4.1 Discrete Fourier Transform and Convolutions 38 4.2 FFT for Product of Toeplitz Matrix and Column Vector 42 5 Conjugate Gradient Method 46 5.1 Steepest Descent Method 46 5.2 Real Symmetric...
Is
In Monte Carlo simulations, we calculate the tangential surface fields n x E and h x H . Absorptivity can be calculated by using 6.5.20 . Another formula can be derived as follows. The power absorbed is obtained from the integration of dissipation over region 2 a drV-S2 f dSS2 h Jv2 Js However, since tangential electric and magnetic fields are continuous, we have Using Monte Carlo simulations, the surface fields are calculated. Thus we can use the two formulas derived in this section to...
Info Qor
Figure 2.4.5 Zero padding with period of 2TV. ii g n is needed for n -N 1, N 2, , 0, , TV - 2, N - 1, a total of 2N 1 distinct values. iii x n is defined for n 1, , N. For simplicity, we take M 2N. To illustrate for the case N 4, we need iii g n m , n m 3, 2, 1, 0,1, 2, 3 First we do zero padding of x n Fig. 2.4.5 to get x n . Let x n be the periodic version of x n with period 2N. x f x n for n 1, 2, 3, , TV . for n iV 1, iV 2, , 2iV 2A25 gt y n g n m xi m 2.4.26 Note that the summation has...