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where Imn,gr is the intensity operator as given by (4.2.15). Note that in (4.2.18), we have integration over intermediate variables and summation over intermediate dyadic vector components. Quite often the Bethe-Salpeter equation is written in terms of the field covariance rather than the field correlation. The field covariance is defined by the following diagram

(4.2.19)

(4.2.20)

GotFocus, KeyDown, KeyUp, LayoutUpdated, Loaded, LostFocus, MouseEnter, MouseLeave, MouseLeftButtonDown, MouseLeftButtonUp, MouseMove, SizeChanged

The Bethe-Salpeter equation is an exact equation for the second moment of the field. However, the intensity operator as given by (4.2.15) is in the form of an infinite series. It must be approximated to give tractable solutions. A common approximation known as the ladder approximation is to retain only the first term of the series expansion in (4.2.15):

5(1'1 -

1'2)5(1'~ - ~)C(1'1 -~)

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(4.2.21)

The Bethe-Salpeter equation then becomes (4.2.22) To see which terms of the Neumann series for the field covariance are included in the ladder approximation, we iterate (4.2.22) to obtain (4.2.23) Thus under the ladder approximation, the field covariance contains all the "ladder-like" terms. On the other hand, the distorted Born approximation consists of only the first term in (4.2.23).

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Dyson's equation for the mean field and the Bethe-Salpeter equation for the covariance have been derived and are exact equations for the respective statistical quantities. The exact field equations satisfy energy conservation. However, when approximations are imposed upon the Dyson and Bethe-Salpeter equations to the mass and intensity operators respectively, the results may be inconsistent with the requirement of the conservation of energy. Therefore, the question of having approximations on Dyson and Bethe-Salpeter equations that are consistent with energy conservation is an important one. In this section it is shown that the ladder approximation and the nonlinear approximation satisfy energy conservation exactly. The general relation between mass and intensity operators for energy conservation is also established. The consistency in conservation law is derived for the case of real permittivity; that is, both (E) and E are real. f Consider scalar wave in a continuous random medium. The wave equation is

(V 2 + k 2 )4{r) = _w 2JLE fCr)4J(r)

V ['ljJ*(r)V'ljJ(r) - 'ljJ(r)V'~J*(r)]

(4.2.24)

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+ (k 2 -

k*2)'ljJ(r)'ljJ*(r)

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(4.2.25)

These methods for the solution of the differential equation pY = F ( Y , X ) (IV.3.1) with Y(0)= Yo and X ( 0 ) = X, have all been developed from the general k-step finite difference equation

Equation (4.2.25) is an energy conservation law where the quantity in square brackets defines an energy flux vector, S1/;:

(4.2.26)

ApplyTemplate, Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, Focus, GetAnimationBaseValue, GetHashCode, GetTemplateChild, GetType, GetValue, HitTest, InitializeFromXaml, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, OnApplyTemplate, ReleaseMouseCapture, Select, SetBinding, SetValue, ToString, TransformToVisual, UpdateLayout

-(V 2 + ,,~2)('ljJ(0)) = Q(O, 1) ('ljJ(1)) _(V 2 + k2)(G(0, 1))

(4.2.27) (4.2.28)

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