are of dimension 2L max and T is of dimension 2L max x in .NET

Writer datamatrix 2d barcode in .NET are of dimension 2L max and T is of dimension 2L max x
are of dimension 2L max and T is of dimension 2L max x
Visual .net data matrix encoder in .net
using barcode printing for visual .net control to generate, create data matrix barcode image in visual .net applications.
(2.7.11)
Data Matrix 2d Barcode recognizer on .net
Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications.
Using the T-matrix, the scattering amplitude dyad F(Os, cPs; Oi, cPi) can be calculated. For a plane wave incident in the direction ki = (Oi' cPi),
.net Vs 2010 Crystal bar code writer in .net
using barcode implementation for .net framework crystal control to generate, create barcode image in .net framework crystal applications.
-='LE'ne = ei E 0 e ik r ~ '
Include barcode on .net
generate, create barcode none with .net projects
the incident field coefficients are, from (1.4.67),
Control datamatrix size for c#.net
datamatrix 2d barcode size for visual c#
a mn
Data Matrix ECC200 barcode library on .net
generate, create ecc200 none with .net projects
(2.7.12)
Control data matrix barcode size in visual basic
data matrix barcodes size for vb
+ 1)) Z C -mn (0 i, 'f/i . ei E 0 'n "') ~ 'Yrnn n n + 1 1 aE(N) = (_1)m_ _(2n+1) inILmn(Oi,cPi) eiEo mn 'Ymn n(n + 1) i
Visual .net bar code integrated on .net
using barcode development for visual studio .net control to generate, create barcode image in visual studio .net applications.
E(M)
Code 128 Barcode barcode library for .net
using .net toproduce code128 in asp.net web,windows application
= (1)m -1 (2n - (
Visual Studio .NET 2d barcode drawer for .net
generate, create 2d matrix barcode none in .net projects
(2.7.13) (2.7.14)
Linear Barcode printing with .net
using visual .net crystal toproduce 1d on asp.net web,windows application
The asymptotic far-field solutions of M mn and N mn are given in (1.4.69a) and (1.4.69b). From (2.7.4), (2.7.5), (2.7.13), and (2.7.14), we have
Visual Studio .NET Crystal code 93 extended maker with .net
use visual .net crystal uss code 93 generation tointegrate uss code 93 on .net
kr-too
Code-128 integrating in office word
using microsoft word toinsert code-128c for asp.net web,windows application
lim a~r;:) M mn (kr, 0, cP)
Bar Code barcode library for java
use java barcode integration todraw bar code on java
"" [
Include data matrix on java
use java data matrix barcode printing toattach data matrix ecc200 in java
Tmnm'n l ( -1)
Control ean13 size in .net
ean-13 supplement 2 size on .net
(11)
Control code128b data in microsoft excel
code128b data in office excel
m'n'
Control pdf417 2d barcode image for .net
use .net winforms pdf-417 2d barcode writer todraw pdf 417 for .net
1 (2n' + 1) .n'~ - - '( '+ 1) . Z C-rn'nl(Oi, cPi) . eiEo '1rn'n' n n
Visual Studio .NET (WinForms) Crystal datamatrix writer for visual basic.net
using windows forms crystal toprint gs1 datamatrix barcode in asp.net web,windows application
7.1 T-Matrix and Relation to Scattering Amplitudes
Excel barcode 3/9 integrated for excel
use microsoft excel barcode code39 maker toproduce code-39 for microsoft excel
+T(12)
mnm'n'
(_1)m,_1_ (2n' + 1) . .n,ILm'n,(Oi,cPi) . . et "(m'n' n '( n , + 1) 't 't
A.E]
'-n-1
. "(mnCmn(O, cP)~et
kr---->oo
(2.7.15)
lim a~~) N mn(kr, 0, cP)
- L.J Tmnm'n' (-1)
_ " [ (21)
mInI
1 (2n' + 1) 'n'- - '(' 1)' 't C-m'n,(Oi, cPi) . eiEo "(m'n' n n +
T(22)
mnm'n'
(_1)m'
1 (2n'+1). 'n,B-m'n,(Oi,cPi) .A.E] . et 0 "(m'n' n '(' + 1) 't n 't
-1..) '-n 1 ikr . "(mn B mn (0 , 'I' 't kr e
(2.7.16)
Using (2.7.15) and (2.7.16) in (2.7.3) and the definition ~f the scattering amplitude dyad F gives the following relation expressing F in terms of the T-matrix elements
F 0, cP,; 0, cP ) =
,,47f
n,m,n',m'
(2.7.17)
Using the optical theorem, the extinction cross section for incident direction (Oi, cPi) can be calculated from (2.4.24). To find the scattering cross section for incident direction (Oi, cPi) and incident polarization (3, w,e note that (2.7.18) Substituting (2.7.17) in (2.7.18) and making use of orthogonality relations for vector spherical harmonics, we have
2 BASIC THEORY OF ELECTROMAGNETIC SCATTERING
167[2 ~{I L..J n' (7s/3(e i, i) = k2 L..J ~ z (-1) m ' "(-m'n'
n,m T(ll) C ,,(e. ~, [ mnm'n' -m n m'n'
.) . ~ + T(12)
mnm'n'
B-m'nl(ei, i)
.~] 1
+ I Lin' ( -1 )m' "(-m'n'
m'n'
[T,\;~~'n'C-m'n' (9i , <Pi) . h T,\;~~,}Lm'n;(9i , <Pi) . jJ]
(2.7.19)
with (3 = V o~h. Hence, given the T-matrix elements, the scattering matrix elements for F can be computed according to (2.7.17).
7.2 Unitarity and Symmetry
In this section, we shall derive the unitarity and symmetry properties of the T-matrix. The unitarity property is a result of energy conservation for a nonabsorptive scatterer, and symmetry is a result of reciprocity.
A. Unitarity
The total field is a summation of incident and scattered fields. We use (2.7.2) and (2.7.3) and also the fact that regular wave solution is a combination of outgoing waves (Hankel functions of first kind) and incoming waves (Hankel functions of second kind), that is, in = (h~l) + h~2 )/2. We use superscript (2) to denote that the spherical Hankel function of the seco~d kind is used in the vector wave function. Defining the scattering matrix S as
_ _ [T(ll) S = 1 + 2T = 1 + 2 T(21)
and using the expression for
E(r)
T(12)] T(22)
(2.7.20)
=i L{ (Sgl)a~(M) +SI\~2)a~(N))
E and E
in Section 7.1 gives the total field
Mz(kr,e, )
(Sz\:l)a~(M) +Sz\:2)a~(N ) Nz(kr,O, )}
(2.7.21)
+ ~ L {af'M) M~2\kr, 0, ) + af<N) N~2)(kr, 0, )}
7.2 Unitarity and Symmetry
In the far field, we make asymptotic approximations of the spherical vector
wave functions of (1.4.69a) and (1.4.69b). Thus,
_ eikr _ e-ikr _ lim E(r) = -k W 1 (O, ) + -k-W2(O, ) kr---+oo 2 r 2 r
where
(2.7.22)
Wl(O, ) =
mnm'n'
{(S~~~'n,a~~, + S~~~'n,a~;n') Cmn(O, )
(2.7.23) (2.7.24)
+ S(22) (N ) 'B (lJ A..)} -n-l + ( S(21) mnm'n,am'n' mnm'n,am'n' Z mn u, 'I' 'Ymn Z
W2(O, ) =
L {a~~Cmn(O, ) + a'::~ Bmn;O, )} 'Ymn i n+
and are perpendicular to f. The complex Poynting vector is then
2 S = E x H* = 417(:2 r 2) {IW1(t'1, )1 -IW2(O, )12
+e- 2ikr [W 2(O, ). W~(O, )]
_e 2ikr [Wl(O, ).w;(O, )]}
(2.7.25)
The sum of the latter two terms in the curly brackets is purely imaginary. Hence, the time-average Poynting's vector is
(8) = ~Re(E x H*) = 817(:r)2 {IW 1(O, )1 -IW2(O, )1 }
(2.7.26)
If the scattering object is lossless, i.e., the imaginary part of permittivity ts is zero, the scatterer is nonabsorptive and integration of (2.7.26) over a spherical surface at infinity should give zero. Hence,
dO {IW 1 (O, )1 -IW 2 (O, )1
(2.7.27)
Since
-* _ m(n+m)!Cmn(O, ) - (-1) (n _ m)! C-mn(O, ) -* m(n+m)!Bmn(O, ) = (-1) (n _ m)! B-mn(O, )
(2.7.28)
(2.7.29
Using (2.7.24), (2.7.28), and (2.7.29), and the orthogonality relation of vector spherical harmonics, we have