Scalar Wave Equation Presentation

Here, to make things simple, we shall only consider the scalar wave equation

together with the initial conditions

In Section 2.3.2, it was shown that this problem is equivalent to the random variable problem described by formula (2.57). We shall assume that the refractive index n(r) is a stationary random function of position and is time independent. The assumption of strict stationarity (i.e., not only for the two first moments) is essential. We separate now the constant mean value of n2(r) and its random part.

Here e is a dimensionless small positive parameter characterizing the relative strength of index fluctuations. Equation 2.59 can now be rewritten as

where (n2(r)) has been incorporated into 1/c2.

The stationary random function is written in terms of its FT and fi(r), which is a random valued measure

The Laplace transformation (LT) of Equation 2.62, taking into account the initial conditions (2.60), is

The FT of this equation is

Method of Diagrams for Green Function

Equations 2.62 and 2.64 are both of the type

where L0 is a nonrandom operator whose inverse G(0) = L— called the unperturbed propagator (or unperturbed Green’s function), is known, and L1 is a random operator. In the space (r) domain

acting as an integral convolution operator. In wave number (k) domain

acting as an integral convolution operator.

In r-domain L1 is diagonal operator and L0 is not; it is the converse in k-domain. The solution of Equation 2.66 is now formally expanded in powers of e yielding

(L0 + eL1)-1 = G is called the perturbed propagator (or perturbed Green’s function).

Let us represent the perturbation series for G with the aid of diagrams, which will be called bare diagrams to discriminate between them and other drossed diagrams to be introduced afterward. We make the following conventions:

  • 1. The unperturbed propagator G(0)(r,r/) is represented by a solid line r r'.
  • 2. The random operator — eL1 is represented by a dot •.
  • 3. Operators act to the right.

If so, we may write

Let us write down explicity a few terms of the perturbation series in r-domain and in k-domain

where 6(k k') is Dirac’s measure.

In order to help the interpretation of bare diagrams, it is sometimes useful to introduce subscripts under certain elements:

If so, the dashed curve will connect the concrete points for which jj,(rj) and /j(r2) (or /j,(k kj) and ji(kj k')) are inside the integrals, that is,

or

We now give the physical interpretation of the perturbation expansion. The r-space diagrams correspond to multiple scattering of the wave at points rj, r2, ... ,rN. The k-space diagrams correspond to multiple interactions between Fourier components of the wave and of the random inhomogeneities; at each vortex of a diagram, a Fourier component kp of the wave function interacts with a Fourier component (kp+j kp) of the random inhomogeneities, giving, as a result, a Fourier component kp+j = kp+jkp + kp of the wave function. Both viewpoints are useful—the first one, particularly for single or double scattering, and the second one for multiple scattering, because of the wave vector conservation conditions.

In future description, we also need the expansion of the perturbed double propagator G 0 G*, that is, the tensor product of the perturbed propagator and its complex conjugate. In r-space

In k-space

This expansion can also be written in terms of diagrams:

If we make the convention that operators of the lower line are the complex conjugate of the usual ones, for example

we can present the mean perturbed propagator following Reference 23.

 
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