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= kixx + kiyY. The spatial
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(3.4.86)
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G(x, y, z) =
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n,=-Ns n2=-Ns
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R n,n2 =
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J(x - n1 ax)2 + (y - n2ay)2 + z2
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and we also truncate at N s . The reciprocal lattice domain, or the spectral domain solution is
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G(x, y, z) = 20
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ei[(kiX+2:~1 )X+(kiy+2:~2)YJeikzI112Izl k zl,l2
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and truncation is at N r . Also Im(kzl11J :::0:
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3 SCATTERING AND EMISSION BY A PERIODIC ROUGH SURFACE
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On the other hand, in Ewald's method,
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G(x, y, z) = G 1 (x, y, z)
where
+ G2 (x, y, z)
(3.4.90)
G 1 (x,y,z)
= 40
ei[(k,x+2:~' )X+(kiy+2:~2)y] ---k---zlt l 2
11=-N, 12 =-N l
{ eikzll12zerfc ( -i k;~2 _
EZ) + e-ikzlll2zerfc ( -i k;~2 + EZ) }
(3.4.91)
(3.4.92) where truncations of G 1 and G 2 are done respectively at N 1 and N 2 . Asymptotically, e- W2 erfc(w) ---+ r;;:;. (3.4.93)
y7fW
Thus both series of G 1 (x, y) and G2(X, y) have exponential decay: G 1 (x, y) exp -
---+
e:~'
f e:~2 f]
(3.4.94)
G 2 (x,y)
---+
exp [- ((n 1 ax )2
+ (n 2ay )2) E 2]
,,[ ) 2
(3.4.95)
For the 1-D case, the two series are respectively e- ( ~ and e-[(n,a x )2]E2. The two series have the same exponential decay rate if E = Vii/ax. For the 2-D case, we choose splitting parameter E such that
Vaxay
k iy
(3.4.96)
In the numerical simulations, we use the following parameters: ,\ = 1, ax =
0.95A,
= 0.95'\, k ix = Re(ko)sinBicosq)i, N 2 = 2, E = 1T = 1.866,\-1. axa y
= Re(ko)sinBisinq)i, N 1 =
4.4 Numerical Results
Case (a) (1) (c) (d)
250 10 10 300
300 1000 1000 1000
Spatial 0.202 + iO.317 -0.164 + iO.069 0.0566 + iO.473 0.0609 + iO.465
Spectral
0.202 + iO.317 -.0.166 + iO.071 0.0163 + iO.457 0.0216 + iO.450
0.356 + iO.334 -0.623 + iO.071 0.162 + iO.468 0.175 + iO.460
-0.153 - iO.0170 0.457 -0.146 -iO.010 -0.154 - iO.0106
Ewald
0.202 + iO.317 -0.165 + iO.071 0.0162 + iO.457 0.0206 + iO.449
Table 3.4.1 Computation of the periodic Green's function using Ewald's method.
We use very few terms in the Ewald summation. The slight difference in the results could be due to the accuracy in computing the complementary error function of complex arguments. The results for the four cases considered below are tabulated in Table 3.4.1. Case (a) Lossy medium: ko = 2;(1 + iO.01), Bi = 45 , rPi = 25 , Z = 0, x = 0.48..\, Y = -0.91..\. In Fig. 3.4.2, we plot the convergence tests of the spatial solution (dotted line) and the spectral solution (solid line) for the real part of the Green's function as a function of N s and N r respectively. There are good convergence for both spatial and spectral solutions. Case (b) Normal incidence: k o = 2;, B = 0 , , rPi = 25 , Z = 0.1..\, x = i 0.48..\, Y = -0.91..\. In Fig. 3.4.3, we plot the convergence tests of the spatial solution (dotted line) and the spectral solution (solid line) for the real part of the Green's function as a function of N s and N r respectively. There is good convergence for the spectral solution. Because z is not equal to zero, there is good exponential decay for the evanescent Floquet modes in the spectral solution. On the other hand, the spatial solution needs many more terms. Case (c) Oblique incidence: z i= 0, ko = 2;, Bi = 45 , rPi = 25 , z = 0.1..\, x = 0.48..\, Y = -0.91..\. In Fig. 3.4.4, we plot the convergence tests. There is good convergence for the spectral solution. On the other hand, the spatial solution does not converge even for N s = 1000. Case (d) Oblique incidence: z = 0, k o = 2;, Bi = 45 , rPi = 25 , z = 0, x = 0.48..\, Y = -0.91..\. In Fig. 3.4.5, we plot the convergence tests. Neither spatial solution nor spectral solution converge well. However, the spectral solution shows a better convergence.
It is important to emphasize that Ewald's method requires very few terms for all the four cases considered. As shown in Table 3.4.1, the results of Ewald's method are also accurate.
3 SCATTERING AND EMISSION BY A PERIODIC ROUGH SURFACE
0.4 1
0.35 0.3 CD 0.25 '0
-----,-----,----.-------,------,------,----;====.======]1
spatial solution spectral solution
IV VVVV
0:0.15 0.1 0.05
Ns and Nr
Figure 3.4.2 Convergence of the spatial solution (dotted line) and spectral solution (solid (1 +iO.01), (}i = 45 , 1Ji = 25 , = 0, x = 0.48.\, line) for case (a) ~ Lossy medium: k o = y = -0.91.\.