From (1.2.70) to (1.3.7), the second order coherent reflected wave is in Java

Make QR Code ISO/IEC18004 in Java From (1.2.70) to (1.3.7), the second order coherent reflected wave is
From (1.2.70) to (1.3.7), the second order coherent reflected wave is
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(E~2\r))
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= ikLL:r-L+ikiZZ
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[e(kiZ)ji~)(kd) + h(kiz)h~(ki-l)]
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(1.3.9a)
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if incident wave is e(-kiz) polarized and
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(E~2)(r)) = ik'-L:f-L+ikiZZ [e(kiZ)]~~(kd) + h(kiz)j~2~(ki-l)]
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if incident wave is
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(1.3.9b)
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h( -kiz )
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polarized. Thus in general, there can be cross-
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1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
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polarizations in the coherent reflection depending on the statistical symmetry of the rough surface.
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Coherent Reflection for Isotropic Surfaces
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For the special case of isotropic surfaces, the spectral density has no preferred direction.
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= W(lk-i1) = W(kp)
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(1.3.10)
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Then in (1.3.8)-(1.3.10) we have
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2 W(k-i - ki-i) = W (2 + k pi - 2kpk pi COS(<Pk - <Pi) ) kp
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(1.3.11)
The integrands in (1.3.8) and (1.3.10) depend on <Pk only through COS(<Pk-<Pi) and sin(<pk - <Pi). We use the property that
21r d<pksin(<pk - <pi)f(cos(<Pk - <Pi)) = 0
(1.3.12)
for any function f of COS(<pk - <Pi). We have
J~t~(ku) = J~7:(ku)
(1.3.13)
Thus the coherent reflection has no cross-polarization for isotropic surfaces. For the special case of Gaussian correlation function, we have
(1.3.14) (1.3.15)
where
(1.3.16)
We next use the following integrals
21f 21f
eXcoso cosnBdB = In(x)
coso
(1.3.17) (1.3.18) (1.3.19)
r21r eX
sin2 BdB =
h (x)
excosocos2BdB = Io(x) - h (x) -X
302 Emissivities of FOllr Stokes Parameters
where In are the n-th order modified Bessel functions. Then
f-(2) (k t eh j~~(ki)
= 0 = 0
(1.3.20) (1.3.21 )
2 o 2 f-(2)(ko) = _ k tZ (k 1 -k ) h2z2 ee t (kiz+klzi)2
dk k ppexp
2 2 _ (k p +kpt )Z2)
. {Io(x) [-k 1Zi
+ (kr - k )] -(kr _ k 1z + k z
k 2) h(x)
k1zk
+ k1 k z
(1.3.22)
3.2 Emissivities of Four Stokes Parameters
We will calculate the emissivities of the four Stokes parameters by using SPM to the second-order. This includes the first-order scattering amplitude squared and the second-order coherent reflectivity. For formulas of emissivity of four stokes parameters in terms of bistatic scattering coefficients refer to 3, Section 5.4 of Volume 1. First consider an incident wave that is e( -kiz ) polarized-that is, horizontally polarized. Then, to the required order, we obtain
EsCf) = E~O) (r)
+ E~l) cn + E~2) (r) Rhoe(kiZ)eiki.lo1'.l+ik,.z + dkjeik.lor.l+ik,ziF(k.1
(1.3.24)
- ki.l)
[e(kz)J~;)(k.1' ki.l) + h(kz)J,~~) (k.1' ki.l)]
o dk.1 eik .l 1'.l+ik,z
[e(kz)f~;)(k.1'
ki.l)
+ 1;(kz)J,~~)(k.1' kiJJ] (1.3.25)
1 2-D RANDOM ROUGH SURFACE SCATTERING BASED ON SPM
HsCr ) = - Rho h(kiz)eiku or .dikuz + ~
ry ry
dk.1 eik -L or +ik,ZiF(k.1 - ki.1)
(1) --:: ' [ -h(kz)Jee (k.1, ki.1)
(1) + e(kz)Jhe (k.1' -ki.1) ]
dk.1 eik-L or +ik,z [-h(kz)f~;) (k.1' k i.1) + e(kz)Jh~) (k.1' k i.1)]
(1.3.26)
The incident and scattered power per unit area as received by the surface, respectively, are
Si . z = _ cos ()i
2ry 1 -* (58 z) ="2 Re(E 8 x H 8 ) z
_ 1 - - Re 2
E(O) 8
(1.3.27)
H(O)* 8
1 z+- Re 2
E(O) 8
(H(2)*) 8
, z
1 +- Re (E(2)) 2 8
H(O)* 8
1 z+- R e 2
(E(l) 8
H(l)*) 8
_ cos ()i IRhol - 2;} {2 + 2 Re (-(2)* (ki.1) )} Rho fee -
ry Jk
dk.1 W (k.1 ':5:ck
ki.1)cOS()8
We have used and k z = kCOS()k. Thus the emissivity of horizontally polarized wave is, in the direction of (()i, 1f + rPi),
[If~;) (k.1' ki.1)1 2 + Ifh~) (k.1' ki.1W] (1.3.28) the substitutions k x = kp cos rPk, k y = kp sin rPk, k p = k sin ()k,
eh(()i,1f + rPd
IRhol2 - 2 Re (RhoJi;) * (ki.1))
1 - --() cos i
1r 2 /
d()ksin()k
21r drPkk 2 cos2 ()kW(k.1 - -ki.1)
(1) 2 [ Ifee (k.1' ki.1)
(1) I + Ifhe (k.1, -ki.1) I2]
(1.3.29)
For the case of vertically polarized incident wave we have
E 8 (f) = Rvoh(kiz ) +
dk.1 eik -L or +ik,ZiF(k.1 - k i.1)
[e(kz)f~~) (k.1' k i.1) + h(kz)Jh~ (k.1' k u )] +
, (2) , (2) + h(kz)Jhh (k.1, -ki.1) ]
dk.1 eik-L or +ik,z
. [ e(kz)Jeh (k.1' ki.1)
(1.3.30)
3.2 Emissivities of Four Stokes Parameters
The magnetic field can be calculated similarly. Thus, the emissivity of a vertically polarized wave is
(1.3.31)
We clarify the notations for polarization and observation directions. For rough surface scattering using dyadic Green's function representation, we have been using e(k z ) and h(k z ). However, for volume scattering and also for emissivity calculations we have used h({), q;) and v({), q;). The relations between the two sets of notations are as follows
+ sin () sin q;y + cos () Z v({), q;) = v = cos () cos q;x + cos () sin q;y - sin ()z h({), q;) = 11, = - sinq;x + cosq;y e(k z ) = xsinq; - ycosq; = -h({),q;) 11,( k z ) = - cos () cos q;x - cos () sin q;y + sin ()z = -v((), q;) e( -k z ) = -h(1f - (), q;) 11,( -k z ) = cos () cos q;x + cos () sin q;y + sin ()z = -v( 1f - (), q;)
k((), q;)
sin () cos q;x
(1.3.32) (1.3.33) (1.3.34) (1.3.35) (1.3.36) (1.3.37) (1.3.38) (1.3.39)
v( 1f -
= - cos () cos q;x
- cos () sin q;y - sin {)z
cos {)iZ = Sob =
For the present case, since the incident wave direction is
ki =
sin {)i cos q;ix
+ sin {)i sin q;iY So =
k( 1f -
()i,
q;i)
(1.3.40)
the observed emissivity is in the direction
k({)i' 1f + q;i)
(1.3.41)