Coherent wave field ( $m = E [E]$ )

In wave propagation in a random medium, the coherent field is the expectation of the complex wave amplitude,

m (r) = E [E (r)]

. The remaining field

E - m

has zero expectation. This is the coherent and diffuse wave fields decomposition. Statistical homogeneity of the medium does not make

m

spatially constant when the incident field is localized: translation symmetry applies to the response together with its incident data. The coherent component can attenuate while the total ensemble-averaged wave intensity remains substantial.

Table of contents
- Gaussian coherent-field propagation in the Markov approximation Coherent wave field
- Weak-fluctuation memory equation for the coherent field Coherent wave field
  - Quadratic refractive-index shift in the coherent field Weak-fluctuation memory equation for the coherent field

Gaussian coherent-field propagation in the Markov approximation ( $m = e^{- γ (x - x_{0})} E_{free}$ )

 0  0

For the linear weak-index parabolic wave equation, a Markov approximation with integrated covariance

B (η) = \int_{R} C (s, η) d s

gives

m_{x} = \frac{i}{2 k} m_{zz} - γm, γ = \frac{k ^{2} μ ^{2} B ( 0 )}{2} .

(1)

The drift follows by converting the multiplicative Stratonovich integral

ik μ E \circ d B

to an Itô integral for a Brownian field with transverse covariance

B

. For a Gaussian beam with one transverse coordinate entering the random region at

x_{0}

m (x, z) = e^{- γ (x - x_{0})} E_{free} (x, z)

. The coherent field retains transverse beam structure. The white-noise closure needs short longitudinal correlations and the corresponding separation of propagation scales; Gaussian one-point statistics alone are insufficient.

Weak-fluctuation memory equation for the coherent field

 0  0

For a parabolic wave equation with

n = 1 + μ W

, a zero-mean stationary real field

W

of unit variance, and

L_{0} = i \partial_{z}^{2} / (2 k)

, finite-correlation perturbation theory gives

m_{x} = L_{0} m + \frac{ik μ ^{2}}{2} m - k^{2} μ^{2} \int_{x_{0}}^{x} \int_{R} C (x - s, z - ζ) G_{x - s} (z - ζ) m (s, ζ) d ζ d s + O (μ^{3}) .

(1)

Here

m

is the coherent field,

C

is the autocorrelation function of a random field and

G

is the one-coordinate Fresnel propagator. Expand the random solution once using the Duhamel principle, multiply by

W

and average to obtain the memory term. The local phase term comes from the

μ^{2} W^{2}

term in

n^{2}

. The expansion is for fixed propagation distances with suitable covariance regularity and moment bounds. A Markov approximation is an additional scale assumption that can turn this integral equation into a local attenuation equation.