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For a parabolic wave equation driven by longitudinal Gaussian white noise with transverse covariance kernel $B$, the mean envelope obeys ${\partial }_{x}m=i{\Delta }_{\perp }m/(2k_0)-\gamma m$. This closure requires independent increments in the longitudinal direction; statistical homogeneity and one-point Gaussian distributions alone are insufficient.

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.

A fundamental Gaussian beam is an ideal beam of light whose transverse wave amplitude has a Gaussian function profile. Here consider a circularly symmetric beam in vacuum, with propagation along $z$, two transverse coordinates $x,y$, and radius $r=\sqrt{x^2+y^2}$ from the optical axis. With the beam waist at $z=0$, total power $P>0$, and vacuum wavelength $\lambda$, its time-averaged light intensity, meaning irradiance in $\mathrm{W}{\mathrm{m}}^{-2}$, isHere $w_0$ is the minimum beam radius, and $z_R$ is the Rayleigh range. Far from the beam waist, $w(z)\sim \theta |z|$ with beam divergence half-angle $\theta =\lambda /(\pi w_0)$. These are the fundamental circular Gaussian beam laws within the scalar paraxial approximation, appropriate when $w_0\gg \lambda$ and the beam divergence is small. The underlying envelope solves the parabolic wave equation exactly; this does not make it an exact vector solution of Maxwell equations. ^[ref].

The radius convention matters. Writing $U$ for the complex wave amplitude envelope, the same $w(z)$ is the wave amplitude $1/e$ radius and the light intensity $1/e^2$ radius:At $r=w(z)$ these ratios are $e^{-1}\approx 0.367879$ and $e^{-2}\approx 0.135335$, respectively. The light intensity $1/e$ radius is instead $w(z)/\sqrt{2}$. None of these radii is a sharp edge: the ideal Gaussian function has nonzero tails at every finite radius. Kogelnik and Li explicitly use the wave amplitude $1/e$ radius in their fundamental-mode treatment. ^[ref].

For a numerical example, take vacuum wavelength $\lambda =632.8\mathrm{nm}$ and beam waist radius $w_0=0.500\mathrm{mm}$. Substitution gives Rayleigh range $z_R\approx 1.24114754\mathrm{m}$ and far-field beam divergence half-angle $\theta \approx 0.402852992\mathrm{mrad}$. At $z=z_R$, the beam radius is $w=\sqrt{2}{w}_0\approx 0.707107\mathrm{mm}$ and the on-axis irradiance is half its waist value. At the same wavelength, halving the beam waist radius to $0.250\mathrm{mm}$ quarters the Rayleigh range to $0.310286885\mathrm{m}$ and doubles the beam divergence half-angle to $0.805705984\mathrm{mrad}$. At fixed power, that smaller beam waist also quadruples the peak irradiance.

The contours in the left panel are $r=\pm w(z)$ in a plane through the optical axis; they mark an light intensity level, not an outer boundary. The right panel compares the exact model radii with the far-field asymptotes $w=\theta z$. The horizontal and vertical axes use different length units, so the drawn slopes are not physical angles.

The normalization expresses conserved power during free propagation. In transverse polar coordinates, an annulus has area $2\pi rdr$, soA proof outline is the substitution $s=2r^2/w(z{)}^2$ in this integral. Spreading reduces peak irradiance as $w^{-2}$ while increasing transverse area as $w^2$, leaving power unchanged. In particular, the disk of radius $w(z)$ contains $1-e^{-2}\approx 0.8646647168$, or $86.4665\%$, of the total power; the local light intensity at its rim is only $13.5335\%$ of the on-axis light intensity. These are different quantities.

The left panel compares normalized wave amplitude magnitude and irradiance; the right panel integrates the transverse area to show enclosed power. For an ideal absorbing circular aperture of radius $a$ centered on the optical axis, the same fraction gives the power passing the aperture at that plane. An aperture of radius $w$ clips about $13.5335\%$ of the incident power; radius $2w$ clips only $e^{-8}\approx 0.000335463$, or $0.0335463\%$. After clipping, extra diffraction means the transmitted field no longer obeys the unmodified ideal Gaussian beam law. Every finite aperture cuts some ideal tail. Kogelnik and Li distinguish their large-aperture beam analysis from their finite-aperture diffraction analysis. ^[ref].

The useful tradeoff is that a small bright spot cannot remain narrow indefinitely. An optical lens can create a smaller beam waist, but diffraction then makes the beam spread faster. The particularly simple relation $w_0\theta =\lambda /\pi$ ties localization directly to beam divergence. A proof outline for the propagation law starts with $U(x,y,0)=U_0{e}^{-(x^2+y^2)/w_0^2}$. The parabolic wave equation separates into two transverse factors, and a Gaussian integral in each coordinate givesHere $q$ is dimensionless and $k$ is the wavenumber; the carrier convention is $e^{i(kz-\omega t)}$, with angular frequency $\omega$. Taking the squared complex modulus gives the stated $w(z)$ and its $w^{-2}$ light intensity scaling. Each transverse factor contributes $q^{-1/2}$, explaining the $q^{-1}$ prefactor here and the different prefactor in the preserved Gaussian beam with one transverse coordinate.

Historically, Gaussian beam theory developed alongside early laser resonator theory and lens-guided transmission. In their 1966 review, Kogelnik and Li credit Boyd and Gordon, and Goubau and Schwering, with early beam treatments; they summarize earlier work rather than claim the whole construction as a new discovery. Their paper brings beam propagation, focusing and finite-aperture diffraction into a common practical account. ^[ref].

For the free parabolic wave equation $E_x=i{E}_{zz}/(2k)$, input $E(0,z)=e^{-A{z}^2}$ with $\mathrm{Re}A>0$ propagates asThe branch is continuous from $q(0{)}^{-1/2}=1$. One-dimensional transverse Fresnel propagation or two Gaussian integrals prove the formula. With $A=D^{-2}+ik/(2F)$, the squared envelope is $|q{|}^{-1}\exp [-2z^2/(D^2|q{|}^2)]$. The amplitude power is $-1/2$ because there is only one transverse coordinate. A two-transverse-coordinate Gaussian beam instead has power $-1$ when both coordinates share the same $q$.

For the linear weak-index parabolic wave equation, a Markov approximation with integrated covariance $B(\eta )={\int }_{\mathbb{R}}C(s,\eta )ds$ givesThe drift follows by converting the multiplicative Stratonovich integral $ik\mu E\circ d\mathcal{B}$ 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 $x_0$, $m(x,z)=e^{-\gamma (x-x_0)}{E}_{\mathrm{free}}(x,z)$. 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.

A parabolic wave equation driven by longitudinal Gaussian white noise with transverse covariance kernel $C$ has the displayed closed mutual coherence equation. The noise is the derivative of a longitudinal Brownian field with transverse covariance; the physical smooth-noise limit uses the Stratonovich integral. Conversion to the Itô integral supplies the attenuation drift and the cross variation supplies the $C({\mathbf{s}}_1-{\mathbf{s}}_2)$ term. For unit plane-wave incidence and transverse statistical homogeneity, $\Gamma =\exp [-k^2{\mu }^{2}x(C(0)-C({\mathbf{s}}_1-{\mathbf{s}}_2))]$. General Gaussian colored media do not have this local closure.

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.

The paraxial approximation describes waves whose propagation directions make small angles with a preferred axis. Factoring out a carrier $e^{ikx}$ and neglecting the envelope's second longitudinal derivative turns the Helmholtz equation into a parabolic wave equation and suppresses backward propagation.

Use positive carrier wavenumber $k$ and retain waves traveling mainly in the positive $x$ direction. Factor out their rapid wave phase by writing $\psi =e^{ikx}E(x,z)$. Substitution into the Helmholtz equation is exact and givesFor a transverse Fourier mode $e^{i\nu z}$ at a small angle, $|\nu |/k\ll 1$ and its forward longitudinal wavenumber isThus the envelope varies along $x$ on scale $k/{\nu }^2$, much more slowly than the carrier wavelength, and $|E_{xx}|/|k{E}_x|=O({\nu }^2/k^2)$. Dropping this smaller term derives the parabolic wave equation,With transform $\hat{E}(x,\nu )=\int E(x,z)e^{-i\nu z}dz$, its free propagation isThis is a one-way paraxial approximation, not a replacement for backward waves or large-angle Helmholtz components. Its Fourier multiplier has unit modulus and therefore does not attenuate any admitted component.

Inverse Fourier transform of the expected value givesindependent of both coordinates. Its squared modulus is the coherent scalar wave intensity $e^{-{\sigma }^2}$, which must not be confused with the mean wave intensity.

For the latter use the stationary two-point statistics. Let $F_0(d\nu )$ be the uncentered spectral measure of a stationary random field of $E(0,z)$, normalized bySince $|E(0,z)|=1$, its total mass is $\int F_0(d\nu )=1$. Under free paraxial propagation each spectral component is multiplied by $e^{-i{\nu }^{2}x/(2k)}$; in its wave-field correlation this wave phase multiplies its complex conjugate and cancels. The stationary intensity under free paraxial propagation is consequentlyStationarity is essential to this pointwise statement: it makes the spectral covariance diagonal. Conservation of integrated power alone would not show that a particular spatial point keeps the same scalar wave intensity.

The same result follows locally from $I_x+J_z=0$, where $J=\mathrm{Im}(E^{\ast }{E}_z)/k$. Taking expectations makes $⟨J⟩$ independent of $z$, so ${\partial }_x⟨I⟩=0$. Thus coherent scalar wave intensity and mean wave intensity are both constant in propagation, with different values. The incoherent contribution has mean $1-e^{-{\sigma }^2}$. Individual realizations can show focusing and changing scalar wave intensity even though the expected value remains constant. All these propagation statements concern the free parabolic wave equation from part (a).

Write $\psi =e^{ikx}E(x,\mathbf{s})$, $\mathbf{s}=(y,z)$. In the weak-contrast approximation $n^2\simeq 1+2\mu W$, neglecting $E_{xx}$ relative to $k{E}_x$ gives the parabolic wave equationFor the mutual coherence $\Gamma (x;{\mathbf{s}}_1,{\mathbf{s}}_2)=\mathbb{E}[E_1{E}_2^{\ast }]$, differentiating gives the exact moment equation within this wave approximation,This is an evolution equation but not a closed one. Even a jointly Gaussian random field does not make $W$ independent of $E$, which depends on its history. Transverse statistical homogeneity alone therefore does not close the second moment. A covariance or longitudinal Markov approximation is additional information.

Under the white-in-$x$ assumption introduced in the next part, interpret $Wdx=d{\mathcal{B}}_x$, a longitudinal Brownian field with transverse covariance $C$, and use the Stratonovich integral as the smooth-medium limit. Conversion to the Itô integral yieldsThe product rule with quadratic variation gives the closed mutual coherence in a white-noise parabolic medium equation,The two attenuation drifts contribute $-k^2{\mu }^{2}C(0)$ and the cross variation contributes $+k^2{\mu }^{2}C({\mathbf{s}}_1-{\mathbf{s}}_2)$. The coefficient $C$ is the white-noise covariance intensity; it replaces, rather than literally retains, the original finite pointwise variance of $W$.

Assume the relative permittivity is real and the field decays sufficiently at transverse infinity, or impose boundary conditions with zero transverse flux. Then $L(x)$ is self-adjoint on its transverse domain by integration by parts. The scalar parabolic wave equation givesThis is norm conservation for the scalar parabolic wave equation. It remains true when $L$ depends explicitly on $x$; its instantaneous self-adjointness suffices. Equivalently, the local balance isSince $|E|=|U|$, the invariant can also be writtenFor a unit transverse polarization in a fixed reference medium with impedance $Z_0=\sqrt{{\mu }_0/{ϵ}_0}$, the leading forward paraxial Poynting vector has $S_x\simeq |E{|}^2/(2Z_0)$. Thus the scalar model conserves its leading reference-normalized power $P_0=\mathcal{N}/(2Z_0)$. This is the intended consequence of the Hermitian-operator hint. Complex permittivity would add absorption or gain and invalidate this norm invariant.

Two qualifications are necessary for the printed physical interpretation. First, $(1,0,1)$ is neither a unit vector nor transverse to nearly axial propagation: its longitudinal electric component is as large as its transverse one. In a homogeneous source-free medium, the example $\mathbf{E}=(1,0,1)e^{ikx}$ satisfies the componentwise Helmholtz equation but violates $\nabla \cdot \mathbf{E}=0$. It therefore is not the stated near-axial electromagnetic plane wave. The same scalar calculation formally conserves $\int |\mathbf{U}{|}^2=2\mathcal{N}$ for that fixed vector, but does not make it a physical Maxwell field. A unit transverse polarization, for example $(0,0,1)$, is the consistent paraxial choice, with only small longitudinal corrections.

Second, for varying local refractive index $n(x)=\sqrt{{ϵ}_{\mathrm{rel}}(x)}$, the quoted local plane-wave impedance relation gives $Z(x)=Z_0/n(x)$ and hence a flux proportional to $n(x)|E{|}^2$, not merely $|E{|}^2$. The scalar norm proof alone cannot make this exactly constant for arbitrary $n(x)$. For example a transversely uniform envelope obeying the printed parabolic equation hasso its intensity is constant while the locally inferred plane-wave flux changes with $n$. This supplies a direct obstruction to the literal exact-flux claim within the supplied scalar assumptions.

The consistent physical statement follows from Poynting theorem: for a lossless Maxwell solution and no lateral energy leakage,For a slowly varying transverse plane wave, the corresponding forward WKB field has amplitude proportional to $n^{-1/2}$, making $n|E{|}^2$ constant. In one dimension this is also encoded in the exact conserved flux $(2{\mu }_0\omega {)}^{-1}\mathrm{Im}(E^{\ast }{E}_x)$. Thus the requested physical conservation holds for the corrected lossless Maxwell description, while the given scalar equation proves the reference-normalized paraxial invariant above. The supplied fixed longitudinal polarization and arbitrary local-impedance inference cannot be used to claim more.

Factor the forward carrier as $E=e^{ikx}U$. Substitution into the scalar Helmholtz equation gives the exact envelope equationIf $|U_{xx}|\ll 2k|U_x|$, drop the second longitudinal derivative to obtain the parabolic wave equationTypical sufficient conditions are a transverse width $w$ with $kw\gg 1$, longitudinal envelope and material variation on scales much longer than $k^{-1}$, and weak refractive contrast so that factoring out the free-space carrier leaves a slowly varying envelope. A paraxial ordering has transverse propagation angles $O(\eta )$ and ${ϵ}_{\mathrm{rel}}-1=O({\eta }^2)$, with $U_x=O(k{\eta }^{2}U)$ and $U_{xx}$ higher order. A large refractive contrast may require a different, medium-adapted carrier even when the ray direction is nearly axial.

The paraxial approximation retains transverse diffraction and slowly accumulated refraction, but neglects backward propagation, strong reflection from rapid material changes, wide-angle propagation and, in this scalar model, polarization coupling. The supplied vector Helmholtz equation is itself a model assumption: in a source-free inhomogeneous dielectric, Maxwell equations additionally require $\nabla \cdot (ϵ\mathbf{E})=0$, and generally supply a longitudinal-gradient term when the vector wave equation is expanded. An independent scalar component is justified only with a compatible polarization and the relevant small-gradient approximation.

Use the time-harmonic wave convention $\mathrm{Re}(\psi e^{-i\omega t})$, with a positive background wavenumber $k$. The scalar Helmholtz equation for a constant-density acoustic model isWriting $\psi =e^{ikx}E$ gives the exact envelope equationThe paraxial approximation neglects $E_{xx}$ relative to $2ik{E}_x$, giving the parabolic wave equationThe choice of carrier and the direction of propagation matter: this is a forward, slowly varying envelope approximation. Sufficient scale conditions are $|E_{xx}|\ll 2k|E_x|$, transverse spectral components $|p|\ll k$, and medium/envelope variation on longitudinal scales large compared with $1/k$. With the carrier fixed at the background $k$, small $|n-1|$ makes the refractive phase vary slowly too. Large-angle propagation, appreciable backscattering, or rapid longitudinal variation violates the approximation. Acoustic models with variable mass density can have additional gradient terms, so the scalar Helmholtz equation itself is a model assumption. For the Gaussian beam in part (b), useful initial conditions are $kD\gg 1$ and $D/|F|\ll 1$, with the second condition absent for an initially uncurved beam. A narrow angular spectrum, rather than merely the label “Gaussian”, justifies the paraxial approximation.

PutIn free space, the parabolic wave equation is $E_x=i{E}_{zz}/(2k)$. Under the Fourier transform convention $\hat{E}(p)=\int E(z)e^{-ipz}dz$, it becomesThe Gaussian integral is legitimate because $\mathrm{Re}A=1/D^2>0$. Invert the transform after multiplying by $e^{-i{p}^{2}x/(2k)}$. A second Gaussian integral, or equivalently one-dimensional transverse Fresnel propagation, givesChoose the square-root branch continuously from $q(0{)}^{-1/2}=1$; for real $x\ge 0$ there is no zero of $q$. This gives the correct incident field at $x=0$, and direct differentiation verifies the free parabolic wave equation.

For clarity, the squared envelope magnitude isThe Gaussian beam with one transverse coordinate remains Gaussian, with one-transverse-coordinate amplitude factor $q^{-1/2}$, not the $q^{-1}$ of a beam with two transverse coordinates. The negative initial quadratic phase produces focusing for $F>0$; diffraction prevents a singularity at $x=F$. These expressions describe the paraxial approximation to free propagation, rather than an exact unrestricted Helmholtz equation beam.

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 $|\nu |\ll k$.

Put $\beta =k\xi \mu$. The random phase screen producesInterpret the normality assumption as a jointly Gaussian process, not merely Gaussian one-point marginal distributions. Write its normalized autocorrelation as $\rho (\zeta )=⟨W(z+\zeta )W(z)⟩$. The difference $W(z+\zeta )-W(z)$ is a zero-mean Gaussian random variable with variance $2(1-\rho (\zeta ))$. Its characteristic function therefore gives the Gaussian phase-screen correlationWithout joint Gaussianity the autocorrelation alone does not determine this expectation.

For an acoustic pressure amplitude $\psi$, with background mass density ${\rho }_0$, the time-averaged acoustic energy flux isAn acoustic potential convention changes the dimensional prefactor. The physical field includes the carrier $e^{ikx}$: a transverse Fourier transform mode has axial wavenumber ${\kappa }_{\nu }=k-{\nu }^2/(2k)$ in the parabolic wave equation. Thus its spectral acoustic flux is proportional to ${\kappa }_{\nu }|\hat{E}(x,\nu ){|}^2$, or to $k|\hat{E}(x,\nu ){|}^2$ at leading paraxial order. Differentiating the reduced field alone would omit the carrier contribution.

The infinite stationary illumination has a spectral measure of a stationary random field, rather than a finite ordinary value of $⟨|\hat{E}{|}^2⟩$ at each $\nu$. To specify the normalization, truncate the screen to an aperture of length $L$, propagate this truncated initial field, and setThe propagation multiplier has modulus one. Using the Gaussian phase-screen correlation and the overlap of the two aperture intervals givesMultiplication by ${\kappa }_{\nu }/(2{\rho }_0\omega )$ gives the ensemble-averaged pressure-normalized modal acoustic energy flux for this finite-aperture convention.

The infinite-aperture spectral normalization consistent with the specified Fourier transform isThe integral and limit are understood as spectral measures of a stationary random field when necessary. By Parseval's identity, this is power per unit transverse length: ${\int }_{\mathbb{R}}{S}_E(d\nu )=C_E(0)=1$. The ensemble-averaged axial spectral acoustic flux is independent of $x$:Here the notation includes Dirac delta distributions in the spectral measure of a stationary random field; the leading-order total flux is $I_{\mathrm{inc}}$. The ${\kappa }_{\nu }$ correction is meaningful within $|\nu |\ll k$, the range of the paraxial approximation.

If $\rho (\zeta )\to 0$ and $e^{{\beta }^2\rho (\zeta )}-1$ is integrable, the stationary phase-screen power spectrum separates into coherent and diffuse wave fields:The coherent fraction is $e^{-{\beta }^2}$ and the diffuse fraction is $1-e^{-{\beta }^2}$. If additionally $|\beta |\ll 1$, the stationary phase-screen power spectrum is $(1-{\beta }^2)\delta +{\beta }^2\hat{\rho }+O({\beta }^4)$ with the same Fourier transform convention. Small $\mu$ alone does not imply small accumulated phase $k\xi \mu$.

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.