Transport models for wave propagation in scattering media with nonlinear absorption Joseph KraislerWei LiKui RenJohn C. Schotland

2025-05-06 0 0 428.01KB 22 页 10玖币

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Transport models for wave propagation in scattering

media with nonlinear absorption

Joseph Kraisler∗Wei Li†Kui Ren‡John C. Schotland§

Yimin Zhong¶

Abstract

This work considers the propagation of high-frequency waves in highly-scattering

media where physical absorption of a nonlinear nature occurs. Using the classical

tools of the Wigner transform and multiscale analysis, we derive semilinear radiative

transport models for the phase-space intensity and the diﬀusive limits of such transport

models. As an application, we consider an inverse problem for the semilinear transport

equation, where we reconstruct the absorption coeﬃcients of the equation from a

functional of its solution. We obtain a uniqueness result on the inverse problem.

Key words. wave propagation, nonlinear media, nonlinear absorption, semilinear radiative trans-

port equation, semilinear diﬀusion equation, inverse problem

1 Introduction

The derivation of kinetic models for wave propagation in highly-scattering media is a classical

subject [32,15] that has received signiﬁcant attention in the past two decades due to its

importance in many emerging applications [11,13,25,39,41,35]. A signiﬁcant amount

of progress has been made on both the mathematical justiﬁcation of the derivation (such

as those based on multiscale analysis of the Wigner transform) [6,21,24,30,37,49] and

computational validation of the derived kinetic models [7,9,33,47,53]; see [1,2,4,5,12,23,

25,28] and references therein for additional investigations in this ﬁeld. The obtained models

for imaging in complex media have also been utilized in many diﬀerent settings [8,9,13,16].

∗Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027;

jek2199@columbia.edu

†Department of Mathematical Sciences, Depaul University, Chicago, IL 60604; wei.li@depaul.edu

‡Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY

10027;kr2002@columbia.edu

§Department of Mathematics and Department of Physics, Yale University, New Haven, CT 06511;

john.schotland@yale.edu

¶Department of Mathematics and Statistics, Auburn University, Auburn, AL 36830;

yimin.zhong@auburn.edu

arXiv:2210.17024v1 [math.AP] 31 Oct 2022

In this work, we are interested in developing kinetic models when nonlinear absorption

occurs during wave propagation [10,38,48,57]. We are mainly motivated by applications

where reconstructing the absorption of the underlying medium from internal or boundary

observations is of practical interest. Such applications include, for instance, the case of

reconstructing the two-photon absorption coeﬃcient of biological tissues with optical or

photoacoustic measurements [20,36,42,44,45,51,54,55,56].

Our derivation will be carried out in the frequency domain, where wave propagation is

described by the standard Helmholtz equation. The nonlinear absorption mechanism we are

interested we study is modeled as a zeroth-order perturbation to the second-order Helmholtz

diﬀerential operator. This is the essential factor that makes it possible for us to perform

the standard calculations in this ﬁeld for our nonlinear problem. Even though this is a

mathematically less attractive nonlinearity to study, the derivation does provide us a formal

justiﬁcation of the semilinear radiative transport model, see, for instance (20), used in many

applications. We refer interested readers to [22] for the derivation of transport models for a

diﬀerent type of nonlinearity that makes use a mean-ﬁeld approximation.

The rest of this paper is organized as follows. In Section 2, we derive the radiative

transport model for media with quadratic and higher-order absorption. We then discuss the

diﬀusive limit of the derived transport models in Section 3. As an application, we study

in Section 4an inverse medium problem for our semilinear radiative transport equation.

Concluding remarks are oﬀered in Section 5.

2 Derivation of the transport equation

For simplicity, we ﬁrst consider the case of quadratic nonlinear absorption. This could serve

as a model of light propagation in media with two-photon absorption [51,54]. We will then

generalize the result to the case of a general polynomial nonlinearity. Let the wave ﬁeld

p(z, x) be the solution to the scalar wave equation in the time-harmonic form, that is,

∆xp+∂2p

∂z2+k2n2p= 0,(1)

where ∆xis the transverse Laplacian in x∈Rd(d≥1), kis the wave number, and

n=n(z, x, p) is the refractive index. We assume that the refractive index takes the form

n2= 1 −2σV (z

) + ik−1µe

K(z, x)|p|2,(2)

where Vis a real bounded stationary random ﬁeld with zero mean, with `xand `zare

the transverse and longitudinal correlation lengths of the random ﬁeld, respectively. The

deterministic function e

Kis non-negative and measures the strength of the second-order

absorption. The parameters σand µare the scaling factors quantifying the amplitudes

of the ﬂuctuation and the second-order absorption, respectively. Assuming that the ﬁeld p

possesses a beam-like structure propagating in zdirection and neglecting the back scattering,

we may write p(z, x) = eikzψ(z, x) with complex amplitude ψ(z, x) satisﬁng the following

equation

∂2ψ

∂z2+ 2ik ∂ψ

∂z + ∆xψ+k2(n2−1)ψ= 0 .(3)

Let Lxand Lzbe the characteristic lengths of propagation in xand zdirections, respectively.

We rescale the variables x7→ Lxxand z7→ Lzzand x,zare now O(1). Then with the

newly-deﬁned variables, we may write (n2−1) as

n2−1 = −2σV (Lzz

,Lxx

) + ik−1µe

K(Lzz, Lxx)|ψ|2.(4)

The equation (3) now becomes

∂2ψ

∂z2+ 2 ik

∂ψ

∂z +1

∆xψ−2k2σV (Lzz

,Lxx

)ψ+ikµ e

K(Lzz, Lxx)|ψ|2ψ= 0 .(5)

With the small aperture assumption that LxLz, we can formally approximate the above

equation by the paraxial wave equation

i∂ψ

∂z +Lz

2kL2

∆xψ−kLzσV (Lzz

,Lxx

)ψ+iLz

2µe

K(Lzz, Lxx)|ψ|2ψ= 0 .(6)

Our derivation works in the regime where the longitudinal propagation distance Lzis much

larger than the correlation length `zand the correlation length is much larger than the

wavelength, that is Lz`zand `zλ:= 2π

k. We, therefore, introduce the small parameter

ε, and assume the scaling relations in the weak-coupling regime:

=`z

=ε1, k`2

x=`z, σ =1

k`z

√ε, µ ∝1

.(7)

Let us denote the rescaled wave ﬁeld by ψε(z, x) and take K(z, x) = Lzµe

K(Lzz, Lxx). Then

the paraxial wave equation turns into

i∂ψε

∂z +ε

2∆xψε−1

√εV(z

ε,x

ε)ψε+i

2K(z, x)|ψε|2ψε= 0.(8)

We then take the Wigner transform of ψε:

Wε(z, x,k) = ZRd

eik·yψε(x−εy

2, z)ψε(x+εy

2, z)dy

(2π)d,(9)

It is then standard to check Wε(z, x,k) satisﬁes the Liouville equation:

∂Wε

∂z +k· ∇xWε+LV,εWε+LK,εWε= 0,(10)

where

LV,εWε=i

√εZRd

e−ip·x/ε Wε(z, x,k+p

2)−Wε(z, x,k−p

2)b

V(z

ε,p)dp

(2π)d,

LK,εWε=1

2ZRd

e−ip·xWε(z, x,k+εp

2) + Wε(z, x,k−εp

2)b

Sε(z, p)dp

(2π)d.

(11)

Here Sε(z, x) is deﬁned as

Sε(z, x) := K(z, x)|ψε(z, x)|2=K(z, x)ZRd

Wε(z, x,k)dk,(12)

while b

Vand b

Sεdenote the Fourier transform (x→p) of Vand Sεrespectively. We use the

standard Fourier transform deﬁnition

f(p) = ZRd

eip·xf(x)dx

(2π)d.

2.1 Multiscale expansion

In order to ﬁnd the asymptotic limit as ε→0, we introduce y=x/ε as the fast variable

and denote Wε(z, x,k) = Wε(z, x,y,k). Formally we write Wεin its asymptotic expansion

in ε:

Wε(z, x,y,k) = W0(z, x,y,k) + √εW1(z, x,y,k) + εW2(z, x,y,k) + ··· .

Using (12), we may also expand Sε(z, x) = Sε(z, x,y) accordingly as

Sε(z, x,y) = S0(z, x,y) + √εS1(z, x,y) + εS2(z, x,y) + ··· .

where

S0(z, x,y) = K(z, x)ZRd

W0(z, x,y,k)dk.

We can now plug in the transform ∇x→ ∇x+1

ε∇yinto (10) to conclude that the leading

order equation at O(ε−1) implies k· ∇yW0= 0. This is equivalent to

W0(z, x,y,k) = W0(z, x,k),and S0(z, x,y) = S0(z, x).

For the order of O(ε−1/2), we have

k· ∇yW1+αW1+iZRd

e−ip·y(W0(z, x,k+p

2)−W0(z, x,k−p

2))b

V(z

ε,p)dp

(2π)d= 0 ,(13)

where α→0+. This gives that the Fourier transform of W1,c

W1, is

W1(z, x,p,k) = (W0(z, x,k+p

2)−W0(z, x,k−p

2))b

V(z

ε,p)

p·k+iα .(14)

Finally, we derive the equation for O(1) terms. To handle the nonlinearity, we let s>d+ 2

and assume exist positive constants C1,C2and C3such that

kW0(z, x,·)kC1(Rd)+kW0(z, x,·)kL1(Rd)< C1,∀(z, x)∈Rd+1,



ZRd

W0(z, ·,k)dk



Hs(Rd)

< C2,∀z∈R,

kK(z, ·)kHs(Rd)< C3,∀z∈R.

Then we have that

W0(z, x,k−ε

2p) + W0(z, x,k+ε

2p) = 2W0(z, x,k) + O(ε|p|),(15)

where O(ε|p|) has a uniform constant. Moreover, we have that

ZRd

|p||b

S0(z, p)|dp

(2π)d

≤1

(2π)2dZRd

|p|2

(1 + |p|2)sdpZRd

(1 + |p|2)s

2|b

S0(z, p)|2dp

(2π)2dZRd

|p|2

(1 + |p|2)sdpkS0(z, ·)k2

Hs(Rd)

≤CskK(z, ·)k2

Hs(Rd)



ZRd

W0(z, ·,k)dk



Hs(Rd)

(16)

is also uniformly bounded, where Cs>0 is a constant depending only on s. The last

inequality holds since s>d+ 2 implies s > d/2. Therefore the O(1) term is

∂W0

∂z +k· ∇xW0+k· ∇yW2

+iZRd

e−ip·y(W1(z, x,y,k+p

2)−W1(z, x,y,k−p

2))b

V(z

ε,p)dp

(2π)d

+W0(z, x,k)S0(z, x) = 0 .

(17)

In order to close the equation, we still need to add the orthogonal relation between W0and

W2, that is, E[k· ∇yW2] = 0. Hence, we have

∂W0

∂z +k· ∇xW0

+EiZRd

e−ip·y(W1(z, x,y,k+p

2)−W1(z, x,y,k−p

2))b

V(z

ε,p)dp

(2π)d

+W0(z, x,k)S0(z, x) = 0 .

(18)

Let Rbe the correlation function of V, and assume that the power spectrum satisﬁes

E[b

V(z, p)b

V(z, q)] = (2π)db

R(p)δ(p+q).(19)

Then the expectation term in (18) converges weakly to

EiZRd

e−ip·x/ε(W1(z, x,x

ε,k+p

2)−W1(z, x,x

ε,k−p

2))b

V(z

ε,p)dp

(2π)d

→4πZRdb

R(p−k)[W0(z, x,k)−W0(z, x,p)]δ(|k|2− |p|2)dp.

Therefore the ﬁnal radiative transport equation of W0is

∂W0

∂z +k· ∇xW0+ 4πZRdb

R(p−k)[W0(z, x,k)−W0(z, x,p)]δ(|k|2− |p|2)dp

+K(z, x)ZRd

W0(z, x,k0)dk0W0(z, x,k)=0.

(20)

The last term on the left side of (20) stands for the quadratic absorption, which shows that

the absorption coeﬃcient linearly depends on the angular average of W0(x, z, k).

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TransportmodelsforwavepropagationinscatteringmediawithnonlinearabsorptionJosephKraisler*WeiLiKuiRenJohnC.Schotland§YiminZhong¶AbstractThisworkconsidersthepropagationofhigh-frequencywavesinhighly-scatteringmediawherephysicalabsorptionofanonlinearnatureoccurs.UsingtheclassicaltoolsoftheWignertransfo...

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Transport models for wave propagation in scattering media with nonlinear absorption Joseph KraislerWei LiKui RenJohn C. Schotland

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