Acoustic and electromagnetic scattering analysis using discrete sources x null field method in electromagnetic theory

Acoustic and electromagnetic scattering analysis using discrete sources x null field method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources x null field method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources x null field method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources x null field method in electromagnetic theory

X NULL-FIELD METHOD IN ELECTROMAGNETIC THEORY This chapter is devoted to the analysis of the electromagnetic scattering using the null-field method (NFM) with discrete sources Our presentation is focussed on the exterior and the transmission boundary-value problems We begin by constructing projection methods for the general null-field equations Next we will present the conventional null-field method with discrete sources for solving the exterior Maxwell problem The efficiency of the conventional null-field method in comparison to other projection methods will be investigated from a computational point of view We will also discuss the use of Tikhonov regularization and will present some numerical results We then consider the conventional null-field method for solving the transmission problem For the sake of completeness we will list the different formulations of the method which serves as input for our computer simulations Next we proceed to establish the expression of the transition matrix and to investigate general constraints of the transition matrix such as symmetry and unitarity We will demonstrate that the use of distributed sources rather than localized sources improves the symmetry of the transition matrix Finally, we will present some computer simulations They include scattering by particles with large size parameters, particles with extreme geometries, cubs, ellipsoids and rough particles 241 242 CHAPTER X NFM IN ELECTROMAGNETICS PROJECTION METHODS We begin our analysis with the exterior Maxwell boundary-value problem Let hs be the unique solution to the general null-field equation (6.103) and let the scattered field E^, H^ be given by (6.107) Replace in (6.107) the surface density h^ by hsNi where h^iv refers to an approximation of hs The resulting expressions define for x £ Ds the approximate scattered field E ^ , H f and for x € A the residual field E ^ , 6H^ Under these circumstances, the estimates l|E.-E^L,G + l|H.-HrL,G < CWK-KMks (10.1) and I I ^ E ^ L G + ll^H^llocG ^ C'WK-K^hs (10.2) hold in any closed sets Gs C Ds and Gi C Di In accordance with (10.1), the approximate solution E ^ , H ^ converge to the exact solution E5,Hs if hsN converge in the L^-norm to h^ In this section, if not stated otherwise, the set {^i''^,^J,''^} represents the sets of localized and distributed spherical vector wave functions and vector Mie potential For the system of magnetic and electric dipoles we take {*if,^;.'|} or {^];l^l:i} instead of {^]^\^l'^} , where ^j^f = •^i'p» ^jip — -^jip ^^^ P = 1» By convention when we refer to the nullfield equations (7.24) we refer implicitly to all equivalent forms of these equations We are now in position to discuss projection methods for the general null-field equations At this end let us consider the null-field equations (7.24) written for convenience as (10.3) (nxh*,nx^^>2^5 = j (n x e5,n x ^^>2 ^ 1/ = 1,2, Let the sequence N ^=1 solves the null-field equations ( h ; ^ , n X *3^2,s = j (n X e5,n X $3^2,s ' (10.5) PROJECTION METHODS 243 V = l,2, ,iV, and let h^ = n x h* Then, from (10.3) and (10.5) we see that h^jy also satisfies the projection relations (10.6) A > (10.19) holds Here A^r id the truncated matrix ^N A\\ Al2 A21 A22 , V,^X= l,2, ,iV, (10.20) with elements ^i//i 411 = ( n x * ; , , n x (nx*^*))^^^, 412 = ( n x ^ ; „ n x (nx*^*))^^^ All = ( n x * ; , , n x (nx$^*))2^, ^22 = ( n x $ ; „ n x (nx4>^*)>2^ (10.21) However, the systems of radiating and regular spherical vector wave functions not form a Riesz basis The purpose of the numerical results presented in Tables 10.1-10.3 is to illustrate the properties of the distributed 247 CONVENTIONAL NULL-FIELD METHOD TABLE 10.1 Maximal and minimal eigenvalues for a spheroidal surface with ka = 10 and kb = and different truncation sizes M?L„ ^ m a x — A^ ^ m a x — ^U f^max — ^ m a x ^ oo the projection schemes PS2 and PS3 correspond to the method with normal equations and for A = we obtain a scheme in which the surface current densities are approximated by radiating spherical vector wave functions The same behavior of the solution can be observed if ^J'^ and $J,'^ are the distributed spherical vector wave functions Ml^i^^ and Af^^- ^^ ^his case rimax denotes the number of discrete sources In Figure 10.3 we plot the differential scattering cross-section for a prolate spheroid with ka = 10 and kb = 2.5, The curves are computed with the conventional method and projection scheme PS2 In order to achieve convergence we use 16 sources in the first case, while 18, 22 and 26 sources corresponding to A = 0, A = 0.1 and A = 1.0, respectively, are necessary in the second case The discrepancy concerning the number of sources used is more pronounced if the aspect ratio increases The results plotted in Figure 10.4 correspond to a prolate spheroid with ka = 10 and kb = 1.25 As before, 16 sources gives accurate results in the conventional method, while 38, 42 and 48 discrete sources corresponding to A = 0, A = 0.1 and A = 1.0, respectively, are required in the projection scheme PS2 The parameter A appearing in projection schemes PS2 and PS3 can be used to control the rate of convergence and therefore gives more flexibility 284 CHAPTER X NFM IN ELECTROMAGNETICS FIGURE 10.20 llustration of the cone-sphere geometry We start by investigating the accuracy of various formulations of the null-field method for the scattering problem of a cone-sphere particle We compare the null-field method with localized, multiple and distributed spherical vector wave functions and the null-field method with sub-domain bases For this purpose we choose the residual of the electric field on spherical surfaces with shifted origins as error criterion For our numerical experiments we consider a cone-sphere particle with a size parameter offc^a= 6, a refractive index of M = 1.5 and a half-cone angle of a = 45^ The geometry of the particle is illustrated in Figure 10.20 The direction of propagation of the incident wave is along the zaxis and the polarization direction encloses an angle of Qp = 45° with the X-axis The residual of the electric field, computed by using the single spherical coordinate-based null-field method is plotted in Figure 10.21 As expected, the null-field condition deteriorates significantly outside the maximal inscribed sphere The instability of the method is illustrated in Figure 10.22, in which the residual electric field is computed in the vicinity of the tip for diflferent numbers of unknowns The residual field increases with increasing nmax, and conversely the scattered field solution oscillates However, the above procedure can be used to obtain the optimal solution which assures a minimal residual field In Figure 10.23 we show the values of the residual of the electric field, computed on the spherical surfaces with centersfc^it;= andfcat/;= , when the origin of the internal field expansion is changed This shifting procedure is temporary and of limited use, for though it does provide better residual field values in the regions around the shifted origin, the null-field condition in other parts of the dielectric object begins to deteriorate significantly as the inscribed sphere is shifted away from the center CONVENTIONAL NULL-FIELD METHOD 0.3 285 Residual electric field 0.2H 0.1 0.0 -0.1 kw FIGURE 10.21 Residua) of the electric field computed with the single spherical coordinate-based null-field method The origin of the spherical surface is located at the point kaw, which varies between and 1.8 Residual electric field 15 FIGURE knowns 10.22 16 17 18 19 20 Number of unknown n „^^ Residual of the electric field as a function on the number of un- 286 CHAPTER X NFM IN ELECTROMAGNETICS f) a - Residual electric field 0.5- ^ -^kw = 8.0 — k w = 0.0 -V,^^ 0.40.30.20.1- 0.1- , r — 1 — - I r — - 1 0.3 0.6 Origin of 0.9 • • • 1.2 • 1.5 expansion FIGURE 10.23 Residual of the electric field at the points kgw = and ksw = when the origin of the expansion is shifted along the z-axis In Figure 10.24 we show the residual electric field computed by using the null-field method with multiple spherical vector wave functions The data correspond to different arrangements of poles In each case the residual field is smaller than x 10'"'* over the entire analysis domain In order to compare the scattered field solutions computed with different theories we use this method as reference Specifically, the differential scattering crosssection normalized by na^ will be evaluated in the azimuthal plane (p = The differential scattering cross-section computed by using the multiple spherical vector wave functions is shown in Figure 10.25 The residual of the electric field computed by the null-field method with distributed spherical vector wave functions is shown in Figure 10.26 The maximal value of the residual field is smaller than x 10~^ In view of Figure 10.26 we may conclude that this method is superior to the single spherical coordinate-based null-field method, but inferior to the null-field method with multiple spherical vector wave functions In Figure 10.27 we show the residual of the electric field computed by using the null-field method with sub-boundary bases In accordance with Wall [151] we found that local basis functions are more suitable in representing surface fields near the tip than global basis functions We mention here, that the evaluation of the residual of the electric field within Di is not relevant, because the surface current densities computed according to (10.66) not guarantee the null-field condition in Ds- In this case an additional criterion, consisting in the evaluation of the residual fields in the exterior of S, must be used However, the relative error of the differential CONVENTIONAL NULL-FIELD METHOD 287 Residual electric field FIGURE 10.24 Residual of the electric field computed with the null-field method with multiple spherical vector wave functions The data correspond to the following arrangements of poles: (a) P = 2, (kszi = 0.0, nj^^x = 16)> i^sZ2 = 7.5, ri^^ = 4), (b) P = 2, (kszi = 0.0, ni,ax = 17), (ksZ2 = 7.5, n^^ = 4) and (c) P = 3, (fc^^i = 0.0, n i , ^ = 15), (ksZ2 = 6.5, nl,^^ = 3), {ksz^ = 7.5, nl,^ = 2) DSCS lE+01 —p-polarized -»-s-polarized lE+00 lE-01 ; lE-02 lE-03 lE-04 lE-05 — — — r — " ^ — • T — , — , — 1 I I — » ' • — • — 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.25 Differential scattering cross-section computed with the null-field method with multiple spherical vector wave functions 288 CHAPTER X NFM IN ELECTROMAGNETICS Residual electric field lE-02 FIGURE 10.26 Residual of the electric field computed with the null-field method with distributed spherical vector wave functions The curves corespond to a number of sources: (a) rimax = 15, (b) nmax = 16 and (c) rimax = 17 f ield electric n "^ Residual -, • • - s p l i n e functions - ^ l i n e a r functions - — c o n v e n t i o n a l NFM i // 0.1- •< 1 — ^ ^ ^ • • 0.1- kw FIGURE 10.27 Residual of the electric field computed with the null-field method with sub-domain bases (linear and spline functions) The number of unknowns is »lmax = 22 CONVENTIONAL NULL-FIELD METHOD 289 Relative error (%) 0.8 —linear functions -^spline functions -B-conventional NFM 0.6 0.44 0,2A 0.0 30 60 90 120 150 180 (a) Scattering angle (deg) n fi n 0.6- Relative error (%) —linear functions -^spline functions -B-conventional NFM 0.4- 0.2- 0.0 J • No 30 60 90 120 150 180 (b) Scattering angle (deg) FIGURE 10.28 Relative error of the differential scattering cross-section using the null-field method with sub-domain bases: (a) p-polarization and (b) s-polarization scattering cross-section, plotted in Figure 10.28, indicates the superiority of the sub-boundary bases for surface current densities representation Conclusions concerning the computing time and the accuracy of the solution can be summarized as follows The single spherical coordinatebased null-field method needs less computing time but has an unstable convergence The null-field method with multiple and distributed spherical vector wave functions need rather lengthy computations (by a factor of 1.2-1.3 versus the standard null-field method), but give fast convergent and higher accurate solutions The null-field method with sub-boundary 290 CHAPTER X NFM IN ELECTROMAGNETICS bases requires lengthy computation (by a factor of 1.5-1.6 versus the standard null-field method), has a slow convergence, but gives a more accurate solution than the single spherical coordinate-based null-field method The null-field method with spherical vector wave functions distributed in the complex plane has been implemented in a computer program In the following we will present computed results for several scatterers that gradually increase in complexity In our examples we consider bodies of revolution We define the laboratory frame OXYZ and choose the direction of propagation of the incident wave to be along the Z-axis The particle coordinate system is denoted by Oxyz^ where the 2:-axis represents the symmetry axis of the particle The orientation of the symmetry axis of the particle is given by the Euler angles a and /? The incident wave is linearly polarized and the polarization direction encloses an angle ap with the X-axis Our first objective is to demonstrate the validity of the null-field method with spherical vector wave functions located in the complex plane We verify the accuracy of the proposed method for a spherical scatterer with a size parameter offc^a= 12 by comparing our numerical results with those obtained by use of the Mie series The differential scattering cross-section will be evaluated in the aaimuthal plane (p = 0° of the laboratory frame for ap = 0° and Qp = 90® The spherical vector wave functions were distributed on the symmetry axis of the particle, i.e the Re z-axis, on the imaginary axis in the complex plane, i.e the Im z-axis, and on a circle contains in D, concentric with the boundary The results are shown in Figure 10.29 Complete agreement between our method and the exact modal expansion serves as an evidence of the accuracy of the proposed method In Figures 10.30 (a) we consider a spheroidal particle with a refractive index of M = 1.5, a size parameter of fc^a = and an aspect ratio of a/b = 10 The symmetry axis of the particle is along the Z-axis For this case we used a collection of spherical vector wave functions located on the symmetry axis of the particle For comparison we have plotted the differential scattering cross-section computed by using the multiple multipole method These results clearly demonstrated that no significant differences exist between the scattering diagrams Numerical experiments with the null-field method have shown fast convergence of the scattered field, which is apparent from the results given in Figure 10.30 (b) for a spheroidal particle with a refractive index of M = 1.5, a size parameter of fc«a = 36 and an aspect ratio of a/b = 18 Now we consider an oblate spheroid with a refractive index of M = 1.5, a size parameter of ksb = and an aspect ratio of a/b = 1/10 For this scatterer we choose the spherical vector wave functions on the imaginary axis in the complex plane Plots of the differential scattering cross-section computed by the null-field method and discrete sources method are shown in Figure 10.31 291 CONVENTIONAL NULL-FIELD METHOD DSCS lE+01 lE+00 p-polarized lE-01 lE-02 • /^WIMVFA/ II lE-03 i l E - • Mie lE-05 • s-polarized —NFM —1—1—1—1—.—1—r—1—1— 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.29 Differential scattering cross-section for a spherical particle with M : 1.5 and fc^a = 12 The data correspond to Mie solution and the null-field method The next problem we analyzed involves a cylinder with kg D — 10 and L/D = 1/10 The orientation of the symmetry axis of the particle is defined by the Euler angles: a = 0° and /3 = 0° and /3 = 90° A comparison was also made by using the multiple multipole method The results are shown in Figure 10.32 Prom our examples we may conclude that the null-field method with distributed spherical vector wave functions can be used to solve the scattering problems of highly deformed particles Our next numerical computations deals with the analysis of nullfield method with distributed dipoles and vector Mie potentials In our examples we consider particles without rotational symmetry We first verify the accuracy of the null-field method with discrete sources distributed on auxiliary surfaces by using the single spherical coordinate-based null-field method as reference For this purpose we consider a dielectric ellipsoid whose surface is described by x'^ 1/^ ^^ (10.96) and choose the size parameters to be fc^a = 1.2, kgb = and kgC = The direction of propagation of the incident wave is along the z-axis The plots in Figure 10.33 represent the diflFerential scattering cross-section evaluated in the azimuthal planes (^ = 0° and (p = 90° for ap = 0° The auxiliary surfaces S~ and 5"^ are chosen to be homothetic to 5, with a homothetic ratio of 0.5 and 2, respectively A total of 110 sources is required to obtain an agreement with the single spherical coordinate-based null-field method 292 CHAPTER X lE-02 NFM IN ELECTROMAGNETICS DSCS lE-03 s-polarized lE-04 lE-05 lE-06 p-polarized \ lE-07 lE-08 j •MMP —NFM lE-09 (a) -T , , , • » 1 r— 30 60 90 120 150 180 Scattering angle (deg) DSCS s-polarized (fc>) 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.30 Differential scattering cross-section for prolate spheroids: (a) M • 1.5, ksa = 6, a/b = 10, (6) M = 1.5, ksa = 36, a/b = 18 293 CONVENTIONAL NULL-FIELD METHOD DSCS lE-Ol s-polarized lE-02 lE-03 i lE-04 i p-polarized V / lE-05 i lE-06 I lE-07 \ • DSM ~NFM — — — — — — — — — — — — — — T—1 r—i 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.31 Differential scattering cross-section for an oblate with M = 1.5, ksb = and a/b = 1/10 Results for the problem of plane wave scattering by a dielectric cube are shown in Figure 10.34 The cube size parameter is 41/X — 5, where / is the side length Evaluation of the differential scattering cross-section is done in the plane 9:? = 0° for ap = 0° and ap = 90° The selected auxiliary surfaces are taken to be spherical surfaces of radii a_ = //4 and a^ = 3//2, respectively The scattering characteristics are also computed by using the discrete dipole approximation, that is by using the public domain DDACode DDSCAT, version 5, by Draine and Flatau A good agreement with the single spherical coordinate-based null-field method solution has been obtained for 144 sources We note here that, in the case of tips or edges on the main surface, the fields should have singularities near the point of geometrical singularities The usual technique is to smooth the geometrical singularities with certain radii of curvature or to incorporate surface densities capable of representing the correct singularity in subdomains of interest In the null-field method a simple technique which consists of computing the surface integrals using Gaussian quadrature is often used For particles consisting of more than one section, one defines quadrature sample points and weighting values over each section, separately, since the Gaussian quadrature will not select the end points of integration We conclude our analysis by presenting some numerical results computed with the null-field method with vector Mie potentials distributed on auxiliary curves This system of vector functions is used to construct the solution to the scattering problem of a conducting axisymmetric particle The scatterer is a spherical particle with axisymmetric roughness The stochas- 294 CHAPTER X NFM IN ELECTROMAGNETICS DSCS lE+02 lE+01 lE+00 lE-01 lE-02 lE-03 • MMP lE-04 y — NFM lE-05 1 1 1 1 , 1 r—1 • • 30 60 90 120 150 180 (a) Scattering angle (deg) DSCS (b) 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.32 Differential scattering cross-section for an cylinder with M = 1.5, kaD = 10 and L/D = 1/10 The orientation of the particle is given by: (a) a = 0*=* and /3 = 0° and (b) a = 0° and /3 = 45° 295 CONVENTIONAL NULL-FIELD METHOD lE+Ol DSCS • localized sources lE+00 —distributed j sources lE-01 phi == 90 ° lE-02 lE-03 lE-04 lE-05 j phi = ° \ 1 F 1—' • — 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.33 Differential scattering cross-section for dielectric ellipsoids with M = 1.5 and size parameters: kga = 1.2, kab = 1.4 and ksc = lE+Ol DSCS lE+00 s -polarized lE-01 lE-02 lE-03 polari zed lE-04 lE-05 j • DDA —NFM lE-06 -T— ' r—T 1 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.34 Normalized differential scattering cross-section for a dielectric cube of size parameter 41/X = and refractive index M = 1.5 296 CHAPTER X NFM IN ELECTROMAGNETICS DSCS ^^'^^ p-polarized lE-01 lE-02 s-polarized ^ • A —B lE-03 r—1 r—1 r—1 r— 1 » • • 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.35 Differential scattering cross-section computed in case A and B for a rough sphere with a mean size parameter ka = and a standard deviation of kS = 0.2 tic shape of a rough sphere with Gaussian statistics was parametrized by the mean radius a, the standard deviation and the angular roughness scale 7c In order to compute the mean scattered intensity and its statistical properties one must solve the scattering problem for each sample Gaussian random surface In this case the convergence of the algorithm for a single realization of the surface profile is crucial for the whole analysis For this reason we consider a given sample random surface The method which use the system of vector Mie potentials will be referred to as case A, while the algorithm based on the discrete sources system (10.63) is referred to as case B We consider a distorted sphere with a mean size parameter of fca = and an angular roughness scale of 7^ = 0.5 The differential scattering cross-section will be evaluated for a plane wave incidence in the azimuthal plane (^ = 0° The direction of propagation of the incident wave is along the symmetry axis of the particle Two values are chosen for the standard deviation of the surface heights k6 = 0.2 and 0.4 Figures 10.35 and 10.36 show the field patterns for cases A and B In case A the discrete sources are distributed on an auxiliary surface ~ , homothetic to 5, with an homothetic ratio of 0.3 and 0.5, respectively The number of source points is taken to be 76 and 88, respectively In case B the discrete sources are distributed on an auxiliary curve L~, homothetic to L, with an homothetic ratio of 0.7 and 0.8, respectively In contrast to case A, a total of 22 and 28 sources, respectively, is sufficient to obtain the desired, accuracy 297 NOTES AND COMMENTS DSCS lE+OO s-polarized 1E~01 lE-02 : • A p-pol ari zed lE-03 J— I — I — r _ ^• ' — — ' f" ' r ' — > — — • — • —B _—,—,—j — 30 60 90 120 150 180 Scattering angle (deg) FIGURE 10.36 Differential scattering cross-section computed in case A and B for a rough sphere with a mean size parameter ka = and a standard deviation of kS = 0.4 As a result, the storage required to solve the electromagnetic scattering problem is substantially reduced Comparing case A with case B it is also observed that a significant saving of computational time is obtained when using the discrete sources system (10.63) NOTES AND COMMENTS The single spherical-coordinate null-field method was initially introduced by Waterman [155] as a technique for computing electromagnetic scattering by conducting objects Later, Bates [9] studied the same problem, followed by Waterman's applications of his method to electromagnetic scattering from dielectrics [156] and to acoustic scattering [157] An improved formulation of the electromagnetic scattering problem was also published by Waterman [158] Almost all existing computer codes assume rotational symmetric shapes both smooth, e.g spheroids and so-called Chebyshev particles, and sharp edged, e.g finite circular cylinders (see Mugnai and Wiscombe, [112], [113] and Barber and Hill [8]) Special techniques have been developed for improving the numerical stability of the method for particle with large size parameters or extreme geometries(see, e.g Lakhtakia et al, [88], Mishchenko and Travis [107], Wielaard et al [161] and Schulz et al [134]) Lakhtakia et al [89] used the null-field method to compute the scattering by chiral nonspherical particles, while Schneider and Peden [133], and 298 CHAPTER X NFM IN ELECTROMAGNETICS Wriedt and Doicu [165] investigated the scattering by particles without rotational symmetry Extension of the null-field method for arbitrary number of scatterers was first addressed by Peterson and Strom [122], [123] They applied the boundary conditions on all of the scatterers simultaneously resulting in a system of coupled linear matrix equations Later, Chew [21] and Chew et [26] reformulated the problem in a simple way to obtain a recursive procedure This method [153] and its variation [24], [154] have been applied to scattering problems and their potentials as faster than 0{N^) algorithms have been demonstrated Other contributions concerning light-scattering calculations by a cluster of particles are due to Borghese et al [14] and Mackowski [98] Various other extensions of the null-field method have been developed over time Among these, it is worthwhile to mention the application of the method to multilayered and composite scatterers by Peterson and Strom [124], [125], Hizal [72], Aydin and Hizal [4], Str5m and Zheng [140], Zheng [172] and Zheng and Str5m [173] Chew [22] has given a more general formulation for the multilayered scatterers An excellent review of the nullfield method was given by Mishchenko et [110] The null-field method with discrete sources has been discussed by Doicu and Wriedt [44]-[46] ... (y) = n(y) X J n(y) x V x ^ n (x) x g (x) ^* (x, y,fe)d5 (x) (10.26) Let M^g = Then n x (^^^g)* = and £{y) = V X y* n (x) x g* (x) ^ (x, y,fc)d5 (x) , W(y) = (l/jfc)V x 5(y) s- (10.27) define an electromagneticfieldwith... "max — 15 "max = 20 "max — 25 "max — 30 4.0e - 2.0e - 19 5.4e - 1.8e - 18 6.8e - 2.3e - 18 8.2e - 2.9e - 1.6e + 5.3e - 1.6e + 1.6e - 1.6e + 4.7e - 1.6e + 1.2e - 4.4e0 9.0e - 10 4.4e0 1.4e - 17... Y,a^[nx^l + Xnx{nxii>l)] f^=i (10.7) + b^ [nx^l + Xnx (nx^l)], X> 0, solves the null-field equations (h^w.nx {nx*^))2,s = i < e , n x (n x $ ^ ) ) g , (10.8) (h^jv.nx ( n x * ^ ) > , s = ; ( e , n x