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Dropping the quadratic contrast is an additional weak-fluctuation assumption, not a consequence of small propagation angles. Although the linearized random potential generates attenuation of order

μ^{2}

, it does not retain every effect of that order in the literal finite-correlation index: the discarded quadratic contrast can also produce a mean wave phase shift. This linearization must precede a Gaussian white noise limit: the square of ideal white noise has no ordinary pointwise meaning.

The split-step Fourier method alternates free-space diffraction,

U_{h} = e^{h D}

, with a random phase screen,

E (x + h, z) ≃ U_{h} [e^{i ϕ_{h} (z)} E (x, z)], ϕ_{h} (z) = k_{0} μ \int_{x}^{x + h} W (s, z) d s .

(3)

For jointly Gaussian random fields, Gaussian phase averaging gives

E e^{i ϕ_{h}} = e^{- Var (ϕ_{h}) /2}

. More generally, a product of

p

fields and

q

conjugate fields picks up

exp (i \sum_{a} ϵ_{a} ϕ_{h} (z_{a}))

, with signs

ϵ_{a} = + 1

- 1

. Its screen average is determined entirely by the wave phase covariance matrix. The deterministic diffraction step acts on each coordinate, with opposite signs on factors formed by complex conjugation. This is the basis of the field-moment equations.

There is an important closure qualification. A stationary Gaussian random field need not have independent longitudinal increments. For finite-correlation fluctuations, the unlinearized parabolic wave equation instead gives

\partial_{x} m = D m + i k_{0} μ E [W E] + \frac{i k _{0} μ ^{2}}{2} E [W^{2} E] .

(4)

In the weak-fluctuation model, the exact first-moment equation is

\partial_{x} m = D m + i k_{0} μ E [W E], m = E E .

(5)

The last term cannot in general be replaced by a constant times

m

. An exact generic solution of the linearized model is

m (x) = E [T exp (\int_{0}^{x} (D + i k_{0} μ W (s, \cdot)) d s)] E_{0},

(6)

where

T

denotes time ordering of the propagation operators. For the unlinearized model, add

i k_{0} μ^{2} W^{2} /2

inside the propagation generator. The covariance function of the medium is needed to evaluate this expression; One-point Gaussian distributions alone would not even determine the joint wave phase statistics.

For example, omit diffraction and take a longitudinal autocovariance function

R (s) = E [W (s, z) W (0, z)]

. Direct Gaussian phase averaging gives

m (x, z) = E_{0} (z) exp [- k_{0}^{2} μ^{2} \int_{0}^{x} (x - s) R (s) d s], \partial_{x} m = - k_{0}^{2} μ^{2} (\int_{0}^{x} R (s) d s) m .

(7)

Even here, unit-variance stationary Gaussian random fields with different

R

give different answers. In the general problem the diffraction and multiplication operators do not commute, so this scalar attenuation cannot simply be multiplied by

U_{x}

without an additional approximation.

The standard closed answer uses the Markov approximation for a random medium, made explicit in part ii. Let the longitudinally integrated covariance kernel be

B (r) = \int_{R} E [W (s, z + r) W (0, z)] d s .

(8)

Replace the medium by longitudinal Gaussian white noise with this strength. A screen of thickness

h

then has

E [ϕ_{h} (z_{a}) ϕ_{h} (z_{b})] = k_{0}^{2} μ^{2} h B (z_{a} - z_{b})

and is independent of the incoming field. For the moment

M = E \prod_{a} E_{a}^{(ϵ_{a})}

, expanding both steps to order

h

gives

\partial_{x} M = \frac{i}{2 k _{0}} \sum_{a} ϵ_{a} Δ_{z_{a}} M - \frac{k _{0}^{2} μ ^{2}}{2} \sum_{a, b} ϵ_{a} ϵ_{b} B (z_{a} - z_{b}) M .

(9)

In particular, the coherent attenuation in a white-noise random medium and its Fresnel propagator solution are

\partial_{x} m = \frac{i}{2 k _{0}} Δ_{⊥} m - γm, γ = \frac{k _{0}^{2} μ ^{2} B ( 0 )}{2} .

(10)

m (x) = e^{- γ x} U_{x} E_{0} .

(11)

Equivalently, write

W d x = d β_{x}

, where

β_{x} (z)

is Brownian motion in

x

with transverse covariance kernel

E [d β_{x} (z) d β_{x} (z^{'})] = B (z - z^{'}) d x

. The Stratonovich integral formulation is

d E = D E d x + i k_{0} μ E \circ d β_{x}

. Its Itô integral form is

d E = (D E - γ E) d x + i k_{0} μ E d β_{x} .

(12)

The mean of the Itô integral vanishes, independently confirming the attenuation drift. For

x > 0

in two transverse dimensions,

(U_{x} E_{0}) (z) = \frac{k _{0}}{2 πi x} \int_{R^{2}} exp (\frac{i k _{0} ∣ z - z ^{'} ∣ ^{2}}{2 x}) E_{0} (z^{'}) d z^{'} .

(13)

Thus a unit plane wave has

m = e^{- γ x}

. For a general incident wave envelope, the Fresnel propagator supplies its spreading. The attenuation is redistribution between the coherent and diffuse wave fields, rather than wave absorption: for a field and its conjugate at the same point, the screen contribution in the second-moment equation cancels.

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