Develop qrcode on java
using java toinsert qr barcode on asp.net web,windows application
Derivation of Radiative Transfer Equation from Ladder Approximation
Barcode writer for java
generate, create barcode none on java projects
In this section, we illustrate the derivation of the radiative transfer equation from the ladder approximation. To simplify the derivation, we use isotropic point scatterers. We assume that the extinction coefficient of the coherent wave is much smaller than the wavenumber. Consider a wave 'lj)inc("r) incident on a single point scatterer located at l'a' Then the total field 'lj) is given by
Java bar code recognizer in java
Using Barcode decoder for Java Control to read, scan read, scan image in Java applications.
'1/)(1') = 'lj)inc(r)
Control qr codes data for c#.net
to get qr code iso/iec18004 and qr barcode data, size, image with visual c# barcode sdk
+ f 1_ r
QR barcode library for .net
using barcode printer for asp.net webform control to generate, create qr barcode image in asp.net webform applications.
Draw qr code for .net
using .net crystal todeploy qr code jis x 0510 in asp.net web,windows application
1'lj)inc(ra )
Control qr-code data for vb
qr-codes data with visual basic.net
Control qr image on java
using java topaint qr codes with asp.net web,windows application
where f is the scattering amplitude and k is the wavenumber of the background medium. Since C(O)(r) = exp(ih)j41IT is the Green's function for the unperturbed problem, (5.5.1) can be written as
Control ean / ucc - 14 data on java
ean 128 barcode data in java
Java code 128b integrating in java
using barcode encoding for java control to generate, create code 128b image in java applications.
Control ean-13 data on java
to build gtin - 13 and ean-13 data, size, image with java barcode sdk
= 'ljJinc(r) + ./df'
Control qr code 2d barcode data on java
to include denso qr bar code and quick response code data, size, image with java barcode sdk
Leitcode barcode library with java
use java leitcode printer topaint leitcode for java
C(O) (1', r')To,{r', 1''')'ljJinc ("r")
Control qr code 2d barcode image for .net
using an asp.net form tocreate qrcode with asp.net web,windows application
Control datamatrix data on .net
to receive data matrix and barcode data matrix data, size, image with .net barcode sdk
Tn (1", 1''') = 47f f 0(1" - 1'0') 0(1''' - 1'0')
Control pdf417 image on .net
using barcode creation for aspx.cs page control to generate, create pdf417 2d barcode image in aspx.cs page applications.
is the transition operator for the point scatterer at C (I)) (1' 1") = -(-,'----;-:, 47flr oiklr-r'l
read code 128 code set b with none
Using Barcode Control SDK for None Control to generate, create, read, scan barcode image in None applications.
Control qr code 2d barcode size with excel spreadsheets
to display qr code iso/iec18004 and qr code 2d barcode data, size, image with microsoft excel barcode sdk
rn .
Control code 128 image in office excel
using microsoft excel touse uss code 128 on asp.net web,windows application
We also use (5.5.4)
In diagrammatic notation, let (5.5.5) Then (5.5.2) becomes '1/) =
+ --0
where - - stands for C(O). For the case of isotropic point scatterers, we a,ssume that the particle positions are independent. For the first and second moment equations, retaining the first terms in the mass operator and the intensity operator leads, respectively, to
= --
+ ~'C'-----
for the Foldy-approximated Dyson's equation, and
(CC*) =
5 Derivation of Radiative Transfer Equation
for the ladder-approximated Bethe-Salpeter equation, where the diagrammatic nota.tions are used. In (5.5.7) and (5.5.8), stands for transition operator and a solid line joining two denotes that the two belong to the same scatterer. Equation (5.5.7) in analytic form is
(G(1',1' o)) = G(O) (1', 1'o)+no
dfi df 1
df2 G(O) (1', 1'I)Ti(1'1' 1'2)(G(1'2' 1'0)) (5.5.9)
and (5.5.8) in analytic form is
(G(1',1'o)G*(r', 1'~)) = (G(1', 1'0)) (G*(r', ~))
+ no
dfi df 1 df2 df3
(G(1', 1'1)) (G*(1", 1'2) )Ti (1'1' 1'3)Tt(1'2, 1'4)(G(1'3' 1'o)G*(1'4, 1'~))
Using (5.5.3) in (5.5.9) and (5.5.10), we have
(G(1',1' o)) = G(O) (1', 1'0)
+ 47rn of
dfi G(O) (1', 1'i)(G(1'i, 1'0))
(G(1', 1'o)G*(r', 1'~)) = (G(1', 1'0)) (G*(r', 1'~)) + n o(47r)2IfI 2 dfi(G(1', 1'i))(G*(r', 1'i)) (G(1'i, 1'o)G*(1'i, ~)) (5.5.12)
The Foldy-approximated Dyson's equation in (5.5.11) can be solved readily. Operating on (5.5.11) by (\72 + k 2 ), we have
+ k 2 + 47rn of) (G(1', 1'0))
= -<5(1' - 1'0)
Hence, the effective propagation constant K obeys the equation
K 2 = k 2 + 47rn of
K = k
+ 27r~of
and the at tenuation rate
"'e for the coherent wave is "'e = 2K" = 47rkn o1m f (5.5.16) Generally the extinction rate "'e is much smaller than the wavenumber K'
Next consider a plane wave incident upon a slab of point scatterers of thickness d in the direction ((}i, <Pi) (Fig. 5.5.1). The slab of scatterers lies between z = 0 and z = -d. Let k be the wavenumber of the upper halfspace (region 0) and the lower half-space (region 2) and also the wavenumber
Regioll 0
z =-d
Regioll 1
Rt'gioll 2
Figure 5.5.1 Scattering of waves by a slab of isotropic point scatterers.
of the background medium of the slab (region 1). It is assumed that the concentration of particles is low so that K' is approximately equal to k. The mean field in region 1 is (5.5.18) where
K i = k sin Oi cos 4>/i~
+ k sin Oi sin 4>i'l1 -
K i;):
K iz = (K 2 - k 2 sin 2 0;) 1/2 ::::
+ iK" ILi
and ILi = cos 0i. The approximate relation in (5.5.20) is a result of (5.5.17) that attenuation rate is much less than the wavenumber. Let h(r) denote intensity in region 0, which is a summation of all the ladder terms. Diagrammatically,
(5.5.21) where
= mean Green's function (C ll (r, r')) with
both observation and source points in region 1
= mean = mean
field 'ljJ m
Green's function (COl (r, r')) with observation point in region 0 and source in region 1
.5 Derivation of Radiative Transfer Equation
Note that (5.5.21) is consistent with (5.5.8). The mean Green's function (C II ) is
(C ll (r'1,'h))
= 4ei;lrl-r~11 = 8 i
7r1'1-72 7r
dkl.. ikl-.(ru-r;)+iIClzl-Z21
(5.5.22) with kl.. = kxx + kyY and K z = (K2 - k;, - k;)1/2. Because the observation point is in far field in region 0, the mean Green's function (COl (1', 1")) is e ik l' _ (C 011',1' = - e -iK ., .1" (- -')) (5.5.23)
where K s denotes scattered direction and is
K s = k sin Os cos <P.,x + k sin Os sin <PsY + Kzsz
K zs =
J K2 -
k 2 sin 2 Os ::::::' k cos Os
+ i K"
with lis = cos Os' Next we define [] as
h with ends stripped. Hence diagrammatically,
By summing the ladder series, the operator equation for [] is
We also let
(1'1, 1'2;r:l, 1'4) = [ ]
:1 I
To solve the ladder intensity, let
(1'1,1'2;1'3,1'4) = F(1'I,1'2)8(1'I-1':J)8(1'2 -1'tl) + Tl. o (47r)2IfI 2 8(1'1 - 1'2) 8(1'1 - 1':l) 8(1'2 - 1'4)
On substituting (5.5.28) into (5.5.27) and using the transition operator for point scatterers as given by (5.5.3a), the F function in (5.5.28) obeys the equation
F(1'I,1'2) =