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Fresnel diffraction describes diffraction when the quadratic propagation wave phase across an aperture must be retained. Its Fresnel propagator becomes the Fraunhofer diffraction description when that curvature is negligible.

A Fresnel diffraction pattern is a map of how propagation changes the wave interference of light from an aperture. It can resemble the opening, develop bright and dark structure near its edges, and eventually approach a Fraunhofer diffraction pattern. The controlling comparison involves wavelength, opening size and propagation distance together; a distance large compared with the opening need not give the far field.

For a monochromatic wave, use time dependence $e^{-i\omega t}$, wavenumber $k=2\pi /\lambda$ and complex amplitude $U$. Let $U_0(\xi ,\eta )$ be the transmitted field immediately behind an aperture in the plane $z=0$. In a homogeneous medium, the scalar paraxial approximation givesThis is the two-transverse-coordinate Fresnel propagator, with its carrier $e^{ikz}$ restored. It adds complex amplitudes before taking their squared complex modulus to obtain light intensity. It is a scalar paraxial approximation, rather than a full vector solution of Maxwell equations.^[ref]

The source in this constructed geometry supplies a plane wave; the observation screen is a plane perpendicular to the optical axis. The right panel isolates the aperture-dependent wave phase that distinguishes Fresnel diffraction from Fraunhofer diffraction. Its vertical scale is in radians, and the drawing's lengths are schematic.

Keeping this wave phase is useful when transporting a field between nearby optical components: both the shape and the complex amplitude matter for subsequent wave interference. A particularly striking feature is that propagation can create a dark centre even from a uniformly illuminated symmetric aperture; the incident wave amplitudes need not be unequal for their outgoing contributions to cancel.

The quadratic wave phase follows from an ordinary distance calculation. Write ${\rho }^2=(x-\xi {)}^2+(y-\eta {)}^2$. The Taylor expansion of the distance to a screen point isIn the slowly varying propagation prefactor, $r\simeq z$ suffices. In the complex exponential $e^{ikr}$, however, a small distance error is multiplied by the large wavenumber $k$. Retaining the quadratic term while dropping the quartic term requires both $\rho /z\ll 1$ and $k{\rho }^4/(8z^3)\ll 1$ over the contributing aperture and the screen region being considered. The second condition is a wave phase accuracy requirement; the first is the geometric paraxial approximation.

Expanding the retained quadratic term separates three effects:The outside quadratic factor changes the wave phase at each screen point but has unit complex modulus, so it leaves that screen's light intensity unchanged. The quadratic factor inside the integral changes relative wave phases between contributions from different aperture points, so it changes their wave interference. The remaining linear factor supplies a Fourier transform. Fraunhofer diffraction drops the aperture quadratic factor when it is nearly constant; it does not require dropping the screen quadratic factor when predicting a later coherent field.

There is also a useful proof from the Helmholtz equation. Substituting $U=e^{ikz}u$ gives $u_{zz}+2ik{u}_z+u_{xx}+u_{yy}=0$. Neglecting $u_{zz}$ yields the parabolic wave equation. A transverse Fourier transform evolves each spatial frequency $\mathbf{\nu }$ by $\exp [-iz|\mathbf{\nu }{|}^2/(2k)]$, using the transform kernel $e^{-i\mathbf{\nu }\cdot \mathbf{x}}$. Its inverse Fourier transform is the quadratic Fresnel propagator kernel above. Thus quadratic wave phase in space and quadratic wave phase in spatial frequency are two views of the same propagation.

In two transverse dimensions, the Fresnel propagator has kernel $k_0(2\pi ix{)}^{-1}\exp (i{k}_0|\mathbf{z}-{\mathbf{z}}^{\prime }{|}^2/(2x))$ for $x>0$. It evolves the homogeneous parabolic wave equation.

Holography records the wave interference between a holographic object wave and a known holographic reference wave, then uses that record to reconstruct a wavefront. The record is a hologram. A photographic plate or photodetector measures light intensity; the relative wave phase is encoded in the positions and contrast of interference fringes, rather than measured directly as an optical oscillation. Reillumination makes the recorded pattern act as a diffraction grating that produces an image-bearing field.

The useful change of viewpoint is to preserve a field that can propagate again, instead of only a focused brightness picture. Recovered complex amplitude permits refocusing with the Fresnel propagator; a spatially extended hologram can also supply different viewing directions. The recoverable information is limited by the collected aperture, recorded spatial frequencies, optical coherence and separation of unwanted diffraction orders.

The Huygens-Fresnel principle describes propagation by coherent addition of secondary-wave contributions from a wavefront or aperture. Their complex amplitudes, including relative wave phases, must be summed before calculating light intensity. The Fresnel propagator follows by retaining quadratic propagation wave phase in the paraxial approximation.

For one transverse coordinate, the parabolic wave equation $2ik{E}_x+E_{zz}=0$ has the Fresnel propagator kernelThe square root uses $\sqrt{1/i}=e^{-i\pi /4}$. With $\hat{f}(\nu )=(2\pi {)}^{-1}\int f(z)e^{-i\nu z}dz$, its Fourier transform multiplier is $e^{-i{\nu }^{2}x/(2k)}$. This distinguishes the one-transverse-coordinate prefactor from the two-coordinate kernel. The operator is unitary on $L^2(\mathbb{R})$, although its spatial integral is interpreted as an oscillatory integral.

Free propagation is convolution with a chirp, $K_L(x)=C_L{e}^{ikL}{e}^{ik{x}^2/(2L)}$. The convolution theorem therefore turns it into multiplication by its Fourier transform. Complete the square:The Gaussian integral then givesThus the wavevector-dependent phase isup to an overall phase convention; after removing the carrier it is $-L{k}_{\perp }^2/(2k)$. This also follows from $k_z=\sqrt{k^2-k_{\perp }^2}\simeq k-k_{\perp }^2/(2k)$.

Normalization qualification: the source keeps a two-dimensional-style prefactor while integrating over only one transverse coordinate. With its literal $C_L=ik/(2\pi L)$, there is an additional $q$-independent factor, rather than solely a unit-modulus multiplier. A properly normalized one-dimensional Fresnel propagator has $C_L=\sqrt{k/(2\pi iL)}$, which cancels the square-root factor above and makes propagation a pure Fourier-domain phase. For example, the literal source kernel multiplies a plane-wave Fourier mode by a factor of magnitude $\sqrt{k/(2\pi L)}$, not one. The spatially dependent phase and the proportional Fourier-transform results remain unchanged. ^[ref]

Set $\psi =e^{ikx+\mu {\chi }_1+O({\mu }^2)}$. Substitution into the Helmholtz equation gives the first Rytov approximation equationFor the forward parabolic approximation, omit ${\partial }_x^2{\chi }_1$. With zero perturbation at $x=0$, the Fresnel propagator then givesA jointly Gaussian random field makes ${\chi }_1$ a centered complex Gaussian variable. The complex Gaussian exponential moment therefore gives the first-Rytov coherent meanThe second moment here has no complex conjugation. It cannot be replaced by the positive intensity variance $\mathbb{E}[|{\chi }_1{|}^2]$.

For example, let $\hat{C}(u,v;\mathbf{\kappa })$ be the transverse spectrum of the covariance between depths $u$ and $v$. The Fourier multiplier of the Fresnel propagator is $e^{-i(x-u)|\mathbf{\kappa }{|}^2/(2k)}$, soThis supplies the requested mean in terms of medium statistics. For the longitudinal white-noise model, $\hat{C}=\delta (u-v)\Phi$, which reduces the double longitudinal integral to one.

The exponential is the mean of the first-order logarithmic model. It is not a complete second-order perturbation expansion of the true mean: the omitted logarithmic term ${\mu }^2{\chi }_2$ can itself have a nonzero mean. Strictly at first order, $\mathbb{E}[\psi ]=e^{ikx}+O({\mu }^2)$. As an independent check, the closed white-noise parabolic model gives exactlyfor plane-wave input. A truncated first-Rytov covariance expression with transverse diffraction need not equal this exact mean; omitted logarithmic terms account for the difference.

Write $\mathbf{r}=(y,z)$ and prescribe a deterministic entrance field $E_0(\mathbf{r})$. Substitution of the carrier into the Helmholtz equation givesThe usual weak-index paraxial model drops $E_{xx}$ and linearizes $n^2-1=2\mu W+O({\mu }^2)$, givingAll the mean-field formulas below use this model. If the ${\mu }^2{W}^2$ term in $n^2$ is retained, its extra mean and fluctuations must also be included; Gaussian averaging of the linear random potential alone is then insufficient.

For a colored medium, taking expectations gives the exact propagation identityStationarity and Gaussianity alone do not close the last term. To make its dependence precise, let $C(s,\mathbf{r})=⟨W(x+s,{\mathbf{r}}_0+\mathbf{r})W(x,{\mathbf{r}}_0)⟩$ and let $U_W(x,s;\mathbf{r},{\mathbf{r}}^{\prime })$ be the random paraxial propagator. Assuming a jointly Gaussian random field, rather than only Gaussian one-point marginals, the functional derivative of the solution isThe Furutsu–Novikov formula therefore yieldsThis is a propagation equation for the mean, with the remaining correlation displayed rather than silently replacing it by a product of means.

The general solution can be written explicitly as a Gaussian-averaged phase-screen product. Put $\Delta =x/N$, $x_j=j\Delta$, ${\mathbf{r}}_N=\mathbf{r}$ and letbe the integral kernel of the Fresnel propagator. Successive free-propagation and random-phase steps, followed by Gaussian averaging, give the colored Gaussian paraxial mean propagatorFor smooth colored media this is the time-slicing representation, with oscillatory integrals understood by the usual Fresnel regularization. At finite $N$ the exponential follows directly from the characteristic function of the jointly Gaussian phase sum. It exhibits why a general transverse covariance cannot be replaced by its value at zero separation along every diffraction path.

For completeness, the unlinearized paraxial model has $E_x=DE+iaWE+i(k{\mu }^2/2)W^{2}E$ and the corresponding mean equation includes $i(k{\mu }^2/2)⟨W^{2}E⟩$. Its colored-medium solution uses the same time-slicing integral, replacing the Gaussian factor byThis follows by completing the square in the finite-dimensional Gaussian integral, with the determinant branch continuous from $\mu =0$. It supplies the generic colored-medium mean when the $W^2$ term is retained. A square of ideal white noise is not defined by the covariance in part (b); that limit uses the linear random-potential model and its stated drift.

If transverse diffraction of the fluctuations is neglected, the simpler straight-ray approximation givesIndeed $E=E_0\exp [ia{\int }_0^{x}W(s,\mathbf{r})ds]$ and the variance of its phase is $2a^2{\int }_0^x(x-s)C(s,0)ds$. If $W$ is constant transversely, this formula is exact with $E_0$ replaced by the free diffracted field $e^{xD}{E}_0$. For example $W=Z$, a single standard normal variable constant throughout space, satisfies stationarity and unit variance but gives $m=e^{-a^2{x}^2/2}{e}^{xD}{E}_0$. Thus a constant-rate exponential attenuation is not a consequence of part (a)'s assumptions alone. It becomes the closed paraxial result under the longitudinal white-noise assumption in the next part.

Interpret the longitudinal delta covariance as a white-noise limit. It is a generalized random field, not a pointwise unit-variance function. Let $B_x(\mathbf{r})$ have independent longitudinal increments withThe smooth-medium limit gives the Stratonovich equation $dE=DEdx+iaE\circ dB$. Conversion to Itô form contributes the local variance drift:The Itô stochastic integral has zero mean, so coherent attenuation in a white-noise random medium satisfiesHere $U_x=e^{ix{\Delta }_{\perp }/(2k)}$ is the Fresnel propagator. Equivalently,For a constant entrance envelope this is simply its constant value times $e^{-\gamma x}$.

Put $B_n(r)={\mu }^{2}A(r)$, the transverse covariance coefficient of the refractive-index fluctuations. The spectrum required here is the transverse spectrum of that longitudinal covariance coefficient, or equivalently the longitudinal-zero-frequency slice of a prelimit stationary medium. Radial symmetry refers to the transverse plane; exact white noise in one direction is not full three-dimensional isotropy. With the Cartesian Fourier convention,angular integration gives $S_F(\nu )=2\pi {\int }_0^{\infty }{B}_n(r)J_0(\nu r)rdr$. Its inverse givesThe printed transform hint mixes two different normalizations. Its radial transform and inverse constitute the self-reciprocal Hankel pair, whose spectrum is $S_H=S_F/(2\pi )$, rather than the unnormalized Cartesian transform also displayed there. Using that Hankel convention consistently yields insteadThese are the same physical solution after converting spectra. The distinction is Fourier-Hankel normalization for an isotropic spectrum; $J_0(0)=1$ fixes the zero-separation covariance in either convention.

Write $m=⟨E⟩$ and $L_0=i{\partial }_z^2/(2k)$. Substituting $n=1+\mu W$ into part (a), without prematurely replacing $n^2$ by its linear part, givesTaking the expectation yields the exact mean equation within the paraxial approximation:The correlation term cannot be discarded just because $⟨W⟩=0$: the field depends on the same random medium as $W$. Nor can $⟨W^{2}E⟩$ be factored exactly from the given variance alone. The printed statistics do not specify the autocorrelation function of a random field, so they do not determine a unique local closed equation for $m$.

A useful covariance-dependent equation follows to second order in $\mu$. Defineand letbe the Fresnel propagator kernel for one transverse coordinate. The Duhamel principle applied to the linear fluctuation term giveswhere $E^{(0)}=E_{\mathrm{free}}$. Multiply by $W(x,z)$ and average. Also $⟨W(x,z{)}^{2}E(x,z)⟩=E^{(0)}(x,z)+O(\mu )$. Since $m=E^{(0)}+O({\mu }^2)$, replacing $E^{(0)}$ by $m$ in terms already multiplied by ${\mu }^2$ preserves second-order accuracy. We obtain the weak-fluctuation memory equation for the coherent fieldThis perturbation statement is for fixed propagation intervals with sufficient covariance regularity and moment bounds. The kernel is understood as an oscillatory integral. If the field is jointly Gaussian, odd perturbative contributions vanish and the remainder can be improved under the corresponding expansion hypotheses. Joint Gaussianity itself still leaves the covariance unspecified. The term $ik{\mu }^{2}m/2$ is the quadratic refractive-index shift in the coherent field; it comes from the mean of $W^2$.

For the usual local answer, impose the additional Markov approximation for a random medium. Its longitudinal correlation length must be much shorter than the distance on which the envelope varies, and diffraction across that correlation distance must not resolve substantial transverse covariance variation. Away from the entry layer, replace the slowly varying factors in the memory integral by their local values and extend the longitudinal integral to infinity. Define the integrated covarianceFor an integrable real stationary covariance, $C(s,0)$ is even, and $B(0)\ge 0$ is its zero-longitudinal-frequency spectral density. The conventional leading weak-index model keeps only the linear random potential $ik\mu W$; in that model the local first-moment equation isThe factor $1/2$ can be checked independently in the white-noise model. If ${\mathcal{B}}_x(z)$ is a Brownian field with $⟨d{\mathcal{B}}_x(z)d{\mathcal{B}}_x(z^{\prime })⟩=B(z-z^{\prime })dx$, writeConverting the Stratonovich integral to an Itô integral adds $-k^2{\mu }^{2}B(0)Edx/2$. The remaining stochastic integral has zero mean, giving the boxed equation andIf the finite-correlation model retains $n^2$ consistently through order ${\mu }^2$, the local second-order expansion also has $+ik{\mu }^{2}m/2$, adding the phase factor $e^{ik{\mu }^2(x-x_0)/2}$. This phase is computed before the white-noise idealization: squaring ideal Gaussian white noise is not the finite-variance operation $W^2$.

The need for a correlation assumption can be seen without any closure argument. The stationary Gaussian random field $W(x,z)=Z$, with a single standard normal variable $Z$, has all the printed one-point statistics. In the linear weak-index model it givesIts attenuation is quadratic in propagation distance, whereas the Markov model gives linear-distance exponential attenuation. The local attenuation equation requires the additional correlation/Markov assumption; it does not follow from statistical homogeneity and unit variance alone.

Use the time convention $e^{-i\omega t}$. The free-space Helmholtz equation, with $\psi =e^{ikx}E$, gives the exact reduced equationFor a forward plane wave at angle $\alpha$, $E_x=O(k{\alpha }^{2}E)$ and $E_{zz}=O(k^2{\alpha }^{2}E)$, whereas $E_{xx}=O(k^2{\alpha }^{4}E)$. Thus the paraxial approximation neglects $E_{xx}$ relative to $2ik{E}_x$. The reduced field satisfies the parabolic wave equationWith the Fourier transform convention used below and its inverse, this becomes an ordinary differential equation for each transverse wavenumber:Consequently the initial-value solution isEquivalently, evaluating the oscillatory Gaussian integral gives the one-dimensional transverse Fresnel propagation formulaThe square-root branch has $\sqrt{1/i}=e^{-i\pi /4}$. The formula is an oscillatory integral for general data, or the Fresnel propagator acting on square-integrable functions. It approaches the initial field as $x\downarrow 0$. A plane wave has transverse wavenumber $\nu =k\sin \alpha$; the reduced axial phase $-{\nu }^{2}x/(2k)$ agrees with $k(\cos \alpha -1)x$ through order ${\alpha }^2$. This also checks the sign. The paraxial approximation applies to $|\nu |\ll k$.

At each point the Gaussian process has a standard normal marginal distribution. Its characteristic function gives the coherent screen fieldIt is constant in $z$, so its Fourier transform is $m\delta (\nu )$. Applying the Fresnel propagator to this zero-transverse-wavenumber component gives the ensemble-averaged reduced spectrumThus the reduced mean is independent of $x$. The physical mean $⟨\psi (x,z)⟩=e^{ikx}{e}^{-{\beta }^2/2}$ still has its carrier phase. The coherent attenuation by a Gaussian phase screen takes place at the screen, with no further coherent attenuation in homogeneous space.

The full spectral acoustic flux depends on the second moment, not the square of this mean. For a stationary process, the generalized cross-spectral correlation isIndeed its propagation factors cancel on $\nu ={\nu }^{\prime }$. This explains why the ensemble-averaged spectral acoustic flux in (b) is also independent of $x$, even for nonzero transverse wavenumbers whose individual complex amplitudes change phase. The coherent atom has weight $|m{|}^2=e^{-{\beta }^2}$; the full stationary phase-screen power spectrum also contains fluctuations. Under the decay assumptions in (b), their integrated weight is $1-e^{-{\beta }^2}$. One must use these weights in a spectral measure of a stationary random field, rather than square a Dirac delta distribution. A phase-only screen preserves the leading-order total acoustic energy flux, although it reduces its coherent part.

For the Fourier transform convention of equation 3, insert the given power spectrum as $S_F$. With $t=\nu L/2$,The Fourier transform of a Gaussian, or the order-zero Hankel transform, also givesThus the coherent attenuation in a white-noise random medium isFor a unit incident plane wave, the Fresnel propagator leaves the initial envelope unchanged, so the answer reduces to $\boxed{m=e^{-k_0^2{\mu }^{2}Lx/(2\pi )}}$. If instead the given numerical function is interpreted as the self-reciprocal Hankel transform spectrum suggested by the printed equations 4 and 5, then $Q(r)=2{\mu }^{2}L{e}^{-r^2/L^2}$ and the alternative isThe factor $2\pi$ ambiguity is present in the original PDF. No unique numerical attenuation coefficient follows until the power spectrum convention is fixed. The covariance strength $Q(0)$ has units of length in this longitudinal white noise model, making both attenuation exponents dimensionless.

Interpret the longitudinal Dirac delta function as the Markov approximation for a random medium. The two arguments on the left of the printed covariance must be $x_1$ and $x_2$. The transverse covariance kernel is $B(|{\mathbf{z}}_1-{\mathbf{z}}_2|)$. Here statistical isotropy means transverse isotropy: a medium with a distinguished longitudinal white noise direction and a smooth transverse covariance is not literally isotropic in all three directions. Also, ideal Gaussian white noise replaces the original finite-variance field; it cannot simultaneously satisfy a pointwise normalization $\mathbb{E}{W}^2=1$.

Write $Q(r)={\mu }^{2}B(r)$ for the covariance strength of the refractive index fluctuations. Define the transverse power spectrum using the unnormalized forward Fourier transform printed in equation 3:This is the spectrum of the fluctuations, excluding the deterministic mean index. Equivalently it is the zero-longitudinal-frequency slice of the three-dimensional fluctuation spectrum before the Markov approximation. The angular integral is $2\pi J_0(\nu r)$, where $J_0$ is the Bessel function of the first kind. Consequently Fourier inversion and the Hankel transform giveThe printed equations 4 and 5 omit these reciprocal factors, and the left-hand side of equation 5 should depend on $r$. They are instead a consistent order-zero Hankel transform pair if their $F$ denotes $S_H=S_F/(2\pi )$.

Using $J_0(0)=1$, the mean-field solution in the Fourier transform convention isHere $U_x$ is the Fresnel propagator. Finiteness of the integral ensures a finite screen wave phase variance. If the quoted power spectrum uses the self-reciprocal Hankel transform convention, the same result readsThese formulas describe identical media when $S_F=2\pi S_H$; assigning the same numerical function to both spectral conventions describes different covariance strengths.

Use the time-harmonic wave convention $\psi e^{-i\omega t}$ and write $\mathbf{z}=(y,z)$, $E(0,\mathbf{z})=E_0(\mathbf{z})$, with a deterministic incident wave envelope. The printed speed ratio is inconsistent with $k=k_{0}n$: the refractive index used below is $n=c_0/c$. Substituting $\psi =e^{i{k}_{0}x}E$ in the Helmholtz equation givesThe paraxial approximation discards ${\partial }_x^{2}E$. The usual weak-fluctuation model also linearizes $n^2-1=2\mu W+O({\mu }^2{W}^2)$, givingDropping 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 ${\mu }^2$, 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, $U_h=e^{hD}$, with a random phase screen,For jointly Gaussian random fields, Gaussian phase averaging gives $\mathbb{E}{e}^{i{\varphi }_h}=e^{-\mathrm{Var}({\varphi }_h)/2}$. More generally, a product of $p$ fields and $q$ conjugate fields picks up $\exp (i{\sum }_a{ϵ}_a{\varphi }_h({\mathbf{z}}_a))$, with signs ${ϵ}_a=+1$ or $-1$. 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 givesIn the weak-fluctuation model, the exact first-moment equation isThe last term cannot in general be replaced by a constant times $m$. An exact generic solution of the linearized model iswhere $\mathcal{T}$ denotes time ordering of the propagation operators. For the unlinearized model, add $i{k}_0{\mu }^2{W}^2/2$ 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 $R(s)=\mathbb{E}[W(s,\mathbf{z})W(0,\mathbf{z})]$. Direct Gaussian phase averaging givesEven 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 beReplace the medium by longitudinal Gaussian white noise with this strength. A screen of thickness $h$ then has $\mathbb{E}[{\varphi }_h({\mathbf{z}}_a){\varphi }_h({\mathbf{z}}_b)]=k_0^2{\mu }^{2}hB({\mathbf{z}}_a-{\mathbf{z}}_b)$ and is independent of the incoming field. For the moment $M=\mathbb{E}{\prod }_a{E}_a^{({ϵ}_a)}$, expanding both steps to order $h$ givesIn particular, the coherent attenuation in a white-noise random medium and its Fresnel propagator solution areEquivalently, write $Wdx=d{\beta }_x$, where ${\beta }_x(\mathbf{z})$ is Brownian motion in $x$ with transverse covariance kernel $\mathbb{E}[d{\beta }_x(\mathbf{z})d{\beta }_x({\mathbf{z}}^{\prime })]=B(\mathbf{z}-{\mathbf{z}}^{\prime })dx$. The Stratonovich integral formulation is $dE=DEdx+i{k}_0\mu E\circ d{\beta }_x$. Its Itô integral form isThe mean of the Itô integral vanishes, independently confirming the attenuation drift. For $x>0$ in two transverse dimensions,Thus a unit plane wave has $m=e^{-\gamma x}$. 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.

For a parabolic wave equation with $n=1+\mu W$, a zero-mean stationary real field $W$ of unit variance, and $L_0=i{\partial }_z^2/(2k)$, finite-correlation perturbation theory givesHere $m$ 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 ${\mu }^2{W}^2$ 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.