Real-space Green's function calculation for the solution of the diffusion equation in stratified turbid media

2046 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 Tualle et al.

Real-space Green’s function calculationfor the solution of the diffusion

equation in stratified turbid media

Jean-Michel Tualle, Jerome Prat, Eric Tinet, and Sigrid Avrillier

Laboratoire de Physique des Lasers, Universite Paris XIII, avenue J.-B. Clement, 93430 Villetaneuse, France

Received January 28, 2000; revised manuscript received June 5, 2000; accepted June 29, 2000

We have derived the space–time Green’s function for the diffusion equation in layered turbid media, startingfrom the case of a planar interface between two random scattering media. This new approach for workingdirectly in real space permits highly efficient numerical processing, which is a decisive criterion for the feasi-bility of the inverse problem in biomedical optics. The results obtained by this method in the case of a two-layered medium are compared with Monte Carlo simulations. © 2000 Optical Society of America[S0740-3232(00)00711-0]

OCIS codes: 030.5620, 110.7050, 290.1990.

1. INTRODUCTIONIn this paper we study the diffusion equation in stratifiedturbid media. One comes across this type of problem inmany fields of physics, and especially in the biomedicalfield,1–3 where one typically finds sequences of stronglyscattering layers such as skin/fat, skin/fat/muscle, orskin/skull/brain. The availability of a fast and accuratemethod for solving the direct problem would be essentialfor solving the inverse problem, which is to obtain thecharacteristics of the various layers from space- and time-resolved measurements of the light backscattered by themedium.

The use of a transverse-space Fourier transform4–7

does not lead to efficient numerical processing. A basicproblem indeed arises from the strong dynamic of the so-lution of the problem. Recently Mochi et al.,8 using spec-tral theory, obtained an asymptotic expansion of this so-lution in the time domain, but only at the location of thesource. We present here a new approach to deal withplanar interfaces between two random scattering media.This approach has allowed us to establish efficient ana-lytical expressions for the solution of the diffusion equa-tion in two-layered scattering media, which are usable inan arbitrary space–time area and with absorbing media.

2. CASE OF A PLANAR INTERFACEBETWEEN TWO SEMI-INFINITESCATTERING MEDIALet us first consider a diffusion process in a semi-infinitehomogeneous scattering medium containing a point lightsource S and having a flat interface with a completely ab-sorbing outer medium. The method of images9–11 con-sists in replacing this problem with another problem inan infinite scattering medium containing, in addition toS, a negative image source Rf S symmetrical to the realone about the interface. In the case of a distribution of

0740-3232/2000/112046-10$15.00 ©

real sources, the operator Rf associates with this distribu-tion a distribution of fictitious image sources.

Let us now consider two semi-infinite scattering mediaseparated by a flat interface at z 5 0 and a point sourcelocated at z 5 Z . 0 in medium 1 (Fig. 1). Within thediffusion approximation, these two media can be deter-mined by the diffusion constant Di , the absorption coeffi-cient mai

, the group velocity ci , and the index ni (i5 1, 2).

The boundary conditions for the energy density fare4,9,12

]

]zf~r, 01, t ! 5 a

]

]zf~r, 02, t !,

f~r, 01, t ! 5 bf~r, 02, t !. a 5 D2 /D1 ,

b 5 n12/n2

2. (1)

Following an approach very similar to the previous one,we tried to express the solution to this problem in the fol-lowing form:

f 5 G1~r, z 2 Z, t ! 1 G1 ^ SR~z . 0 !,

f 5 G2 ^ ST ~z , 0 !, (2)

where G1 and G2 are the Green’s functions for infinitemedia having the properties of media 1 and 2, respec-tively. The convolution product ^ applies here to thespace and time variables, and r is the transverse spacevector. From the simple fact that these expressions areobtained from source distributions, they obviously satisfymost of the requirement: The diffusion equation is sat-isfied in each half-space, and the causality is satisfied, asis the behavior at infinity. The last point is to choose thesources SR and ST , localized, respectively, in the half-space z,0 and in the half-space z.0, in order to matchthe boundary conditions [Eq. (1)].

We finally obtained13 the following expressions forthese source distributions:

2000 Optical Society of America

Tualle et al. Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 2047

Fig. 1. Generalization of the method of images in the case of a flat interface between two scattering media. The virtual source consistsof an image point source associated with a more complex source distribution.

SR~r, z, Z, t ! 5 d ~r!d ~z 1 Z !d ~t ! 2 4aD2u~2z 2 Z !

3 G2~2 !(r, a~z 1 Z !, t),

ST~r, z, Z, t ! 5 24D2

a@G1

~1 !~r, Z, t !d ~z ! 1 a21u~z !

3 G1~2 !~r, Z 1 a21z, t !#, (3)

where a 5 a/b, G (m) 5 (]m/]zm)G(r, z, t), and u is theHeaviside function.

We show in Appendix A that the energy density f cal-culated from Eqs. (2) with these source distributionsmatches the required boundary conditions. Thus if weare interested in the solution in medium 2, we can replacethe problem involving interface with another problem inan infinite medium having the same properties as me-dium 2, but replace the real source with the source distri-bution ST , located in this new infinite medium in thehalf-space z . 0. In the same way, if we are more par-ticularly interested in the light backscattered in medium1, we can replace the problem involving interface with an-other problem in an infinite medium having the sameproperties as medium 1, but add to the real source thevery specific distribution of image sources SR , located inthe half-space z , 0 (Fig. 1). We thus have a generali-zation of the method of image that we recognize herewhen ma2

→ `.This result can be generalized to the case of an arbi-

trary source distribution s by introducing the reflectanceand transmittance operators, Rm and Tm , respectively,which associate with s the sources representing the pres-ence of the interface:

Rms 5 E dz8SR~r, z, z8, t ! ^ s~r, z8, t !,

Tms 5 E dz8ST~r, z, z8, t ! ^ s~r, z8, t !, (4)

where the convolution product ^ is restricted to thetransverse space and time variables r, t.

3. TWO-LAYERED MEDIAA. Formal Derivation of the Energy Density fWe have shown that an interface between two scatteringmedia can be replaced by a specific source distribution,described by operators such as Rm or Tm . The iterationof such operators allows the treatment of multilayeredmedia. We will now illustrate this point with the case ofa two-layered medium: A scattering medium 1 of finitethickness d having a flat interface with a nonscatteringouter medium covers a semi-infinite scattering medium 2(Fig. 2). The interface between the two media is at z5 0, and the source S(r, z, t) 5 d (r)d (z 2 Z)d (t) is lo-cated in medium 1 at z 5 Z , d. In the framework ofoptics in a scattering medium, this source, if placed at adistance of the order of the transport mean free path ltr15 @1/(3D1) 2 ma1#21 under the free surface, can be usedto simulate the behavior of a laser pulse incident on me-dium 1.14

To model the free surface, it has been shown10,14–16

that a useful approach is to set the diffuse fluence rate tozero at an extrapolated boundary surface located at z5 d 1 zb 5 d 1 (2/3)ltr1 in the case of index adaptation.We can then use the method of images with respect to thisextrapolated boundary, as well as the corresponding op-erator Rf @Rf f(z) 5 2f(2(d 1 z0) 2 z)#.

To model the interface between the two scattering me-dia, we will use the generalization of the method of im-ages described above and the operator Rm .

The energy density f of this problem can then be de-scribed as the field corresponding to a source distributionSf located in medium 1, considered infinite:

Sf~r, z, t ! [ H F1 1 ~1 1 Rf!(n51

`

~Rf Rm!nG3 ~1 1 Rf!SJ ~r, z, t !. (5)

In Eq. (5), n represents the number of round trips thediffuse light makes between the two interfaces (Fig. 2).The effect of the highest-order terms of this series on theevaluation of physical quantities, such as, for example,the light backscattered at the free surface, vanishes very


quickly; and we can generally make do with the first term(n 5 1) or the first two terms (n 5 1, 2). At the limitma2 → `, it is easy to verify that this distribution coin-cides with the solution for a slab of scattering medium.14

Figure 3 is a schematic of distribution Sf . One can seeon this figure how our method can highlight the contribu-tion of each boundary to the signal, which is the linear su-perposition of the signal emitted from sources linked di-rectly to the boundary positions. This highlight can bequite helpful for the inverse problem.

To get a full solution, all we have to do now is to ex-press the energy density f corresponding to the sourceSf :

f~r, t ! 5 G1 ^ Sf . (6)

In Eq. (6) the G1 ^ (1 1 Rf)S term corresponds to thesolution f0 for a semi-infinite medium 1 with a freeinterface.14 To evaluate the other terms, we let

jn~r, z, Z, t ! [ ~RfRm!nS~r, z, Z, t !,

fn~r, z, Z, t ! 5 G1 ^ jn .

With these two quantities, the light emitted by thesources defined in Eq. (5) can simply be put in the form

f~r, z, t ! 5 f0~r, z, t ! 1 (n51

`

fn~r, z, Z, t !

2 fn(r, 2~d 1 zb!2z, Z, t)

2 fn(r, z, 2~d 1 zb!2Z, t)

1 fn(r, 2~d 1 zb!

2 z, 2~d 1 zb!2Z, t). (7)

We proved that (see Appendix B)

Fig. 2. Case of a scattering medium 1 of finite thickness d hav-ing a flat interface with a nonscattering outer medium andplaced on a semi-infinite scattering medium 2. The diffuse lightinteracts alternately with the two interfaces, making n roundtrips.

jn~r, z, Z, t ! 5 ~21 !nd ~r!d ~z 2 zn!d ~t !

1 (k50

n21

pnk~z 2 zn!k

3 G2~k12 !(r, a~z 2 zn!, t)u~z 2 zn!, (8)

where zn 5 2n(d 1 zb) 1 Z 5 zn21 1 2(d 1 zb) andwhere the coefficients pn

k are defined by recurrence:

pn0 5 ~21 !n114anD2 ,

pn11k 5 2pn

k 22a

kpn

k21, if k , n,

pn11n 5 2

2a

npn

n21, (9)

and the fn terms can therefore be put in the form

fn~r, z, Z, t ! 5 ~21 !nG1~r, z 2 zn , t !

1 (k50

n21

pnkIn

k~r, z, Z, t !, (10)

with

Fig. 3. Infinite-medium problem equivalent to the problem il-lustrated in Fig. 2. The contribution of each boundary to thesignal can clearly be seen. This highlight can be really helpfulfor the inverse problem.


Ink~r, z, Z, t ! 5 E

0

`

dz8E0

t

dtc ~r, t, t!z8k

3 G1~z 2 z8 2 zn , t 2 t!G2~k12 !~az8,t!,

(11)where

c ~r, t, t! 5exp$2r2/@4D1c1~t 2 t! 1 4D2c2t#%

4p@D1c1~t 2 t! 1 D2c2t#

3 exp@2ma1c1~t 2 t! 2 ma2c2t#,and where Gi is the one-dimensional Green’s functionfound by canceling the absorption coefficient

Gi~z, t ! 5 ciu~t !exp@2z2/~4Dicit !#

~4pDicit !1/2 .

Considering the causality properties of the Green’s func-tion, we have t P @0, t# in expression (11), which hintsthat we may use a limited expansion of c in powers of twith an efficiency that increases as the contrast betweenthe diffusion parameters of the two media becomesmaller. By setting

c 5 (l50

`

h l~r, t !t l, (12)

we can write

Ink 5 (

l50

`

h l~r, t !f nkl~t !, (13)

with

f nkl 5 E

2`

`

dtE0

`

dz8t lz8kG1~z 2 z8 2 zn , t 2 t!

3 G2~k12 !~az8, t!.

This quantity can be evaluated analytically (see AppendixC):

f nkl 5 c1 (

i50

l

~21 !l~k 1 i !!

4D2

1

kQkl

i S AD1c1

kD k111i22l

3 S A4D1c1t

kD 2l21

i2l21 erfcS zn 2 z

A4D1c1tD , (14)

where k 5 1 1 aAD1 /D2. The quantity im erfc(z) is(21)m times the mth primitive of the error function,which is calculated very simply by recurrence.17 We alsohave

Qk00 5 2S 2

1

AD2c2D k11

,

Qk~l11 !i 5

k 1 1 1 i 2 2l

2Qkl

i 2a

2AD2c2

Qkli21,

Qk~l11 !0 5

k 1 1 2 2l

2Qkl

0 ,

Qk~l11 !~l11 ! 5 2

a

2AD2c2

Qkll , for i Þ 0, l 1 1. (15)

The f nkl(t) functions can be computed very quickly, all

the more so as they are only time dependent. Moreover,the h l(r, t) terms are very simple in form, and the In

k ex-pression (13) can be summed without difficulty. Beforeillustrating this approach with a numerical example, itshould be noted that the physically relevant quantity isnot necessarily the field f(r, t) calculated by Eqs. (6) butcan be a derivative quantity such as the reflectance at thefree surface4,15:

R~r, t ! 5 Af~r, d, t ! 1 BD1

]

]zf~r, d, t !, (16)

where A and B can be adjusted to take into account thenumerical aperture at which the surface is observed. Wecompute the quantity defined in Eq. (16) noting that in or-der to derive Eq. (14) with respect to z, we only have toreplace the i2l21 erfc@(zn 2 z)/A4D1c1t# term in f n

kl(t)with

1

A4D1c1ti2l22 erfcS zn 2 z

A4D1c1tD .

B. Practical Calculation of the Time- and Space-Resolved Reflectance at the Free SurfaceLet us summarize the algorithm that we have used forthe practical calculation of the reflectance: To calculatef from Eq. (7) we have to evaluate the fn given by equa-tion (10), that is, the pn

k and the Ink 5 ( l50

` h l(r, t)f nkl(t),

where the f nkl are given by Eq. (14).

There is finally a triple summation to perform over n, k,and l. The first step is to choose the number Nmax ofround trips (for odd n 5 1...Nmax) and the order Lmax ofthe development of c. The convenient values of Nmax andLmax will be discussed in the conclusion and are very oftenof the order of 1 or 2 for Nmax and of the order of 6 forLmax . Then the knowledge of the optical coefficients al-lows the computation of the pn

k and Qkli .

Concerning the computation of pnk , where n is running

from 1 to Nmax and k from 0 to n 2 1, recurrence relations(9) can be used. In coefficients Qkl

i , l is running from 0 toLmax , i from 0 to l, and k from 1 to Nmax . Equations (15)can be used for computing first Qk0

0 , then Qkli .

The only term depending on zn in f nkl is the im erfc(x)

term: For a given n and a given t, one has to computethis function at point x 5 (z 2 zn)/A4D1c1t for n runningfrom 22 to 2Lmax 2 1. Taking into account the fact thatwe are calculating the reflectance from Eq. (16), we cancalculate the quantity

Vm 5 Aim erfc~x ! 1BD1

2AD1c1tim21 erfc~x !,

where m 5 21...2Lmax 2 1.This quantity allows the evaluation of the terms

f nRkl (t):


f nRkl 5 c1(

i50

l

~21 !l~k 1 i !!

4D2

1

kQkl

i S AD1c1

kD k111i22l

3 S A4D1c1t

kD 2l21

V2l21 .

At this point, one can easily see that it is possible to per-form first the summations over n and k and then the sum-mation over l, which reduces the computation time andallows a straightforward determination of the conver-gence with l.

The terms h1(r, t) are given by

h l~r, t ! 5exp~2ma1c1t !

4pD1texpS 2

r2

4D1c1t D (k50

2l

bkl~r, t !,

with

b00 5 1,

bk~l11 ! 5

2ma1c1bkl 1 kD1 2 D2

D1tb~k21 !l 2

D1 2 D2

4D12c1t

b~k22 !l

l 1 1, ~bkl 5 0 if k , 0 or k . l !.

4. COMPARISON WITH MONTE CARLOSIMULATIONSWe will now give a numerical example to show that thismethod can be applied with a relatively high contrast.To do this, we assume14 that A 5 0. The speed of light isc 5 c0 /n with index n 5 1.33 for both media. We take atransport mean free path ltr1 5 1 mm and an absorptioncoefficient ma1 5 0.026 cm21 for medium 1 and ltr2

Fig. 4. Time-resolved reflectance obtained at r 5 2.5 mm at thefree surface of a scattering medium 1 (n1 5 1.33, ma1

5 0.026 cm21, ltr1 5 1 mm) of thickness d 5 6 mm, placed onsemi-infinite scattering medium 2 (n2 5 1.33, ma2

5 0.05 cm21,ltr2 5 1/3 mm). The figure shows the effect of the number ofround trips n of the light between the two interfaces. These re-sults obtained by the diffusion approximation are compared withMonte-Carlo simulations (noisy curves).

5 1/3 mm and ma2 5 0.05 cm21 for medium 2. Thethickness of the first layer is d 5 6 mm, and the bound-ary conditions that we set are those usually adopted withthe diffusion approximation4,12,14,18: b 5 1 for adaptingthe index between the two media, and a 5 D2 /D1 . Thecontrast chosen here between the parameters of the twolayers is therefore high but is still very far from the limitsof the method, since series (13) converges for Lmax 5 6.The whole time- and space-resolved reflectance map wascalculated for a 256 3 256 grid (r P @0, 3.5 cm#, tP @0, 2.5 ns#) in less than 1 s on a Pentium II 300 MHz,which is a time gain of three orders of magnitude over theusual methods.

Figure 4 shows the time variation of the reflectancefunction R(r, t) for r 5 2.5 cm obtained by our methodfor n running up to 0 (semi-infinite medium), 1, and 2,which gives an idea of the correction provided by the sec-ond term. Note the causality effect reflected in the

delayed addition of the correction due to the second roundtrip (n 5 2) to the signal due to the first term (n 5 1).These results are compared with a Monte Carlosimulation.19 The phase function used for the latter is aHenyey–Greenstein function10 with an anisotropy factorg 5 0.8. These comparisons are made in the case of thetwo-layered medium described above but also in the caseof a semi-infinite medium 1. Since the lower layer ismore diffusive, the energy tends to remain trapped, whichexplains the greater intensity for the semi-infinite prob-lem (n 5 0). The prefactor B is of course the same inboth cases, so that this very good comparison between ournumerical results and Monte Carlo simulations is rel-evant and demonstrates the accuracy of the method.

5. DISCUSSION AND CONCLUSIONThis generalization of the method of images for the case ofinterfaces between two scattering media gives an explicitformulation of the solution in real space. This method al-lows the derivation of a fast-converging series for efficientnumerical calculation. In comparison with the usualmethods, we are thus able to reduce the computation timeof the reflectance at the free surface by a factor of at least50. This time gain is essential for performing the inverseproblem on stratified media and to achieve new medicaldiagnostic methods that would be effectively usable.

The construction of a solution was presented here inthe case of a two-layered medium by the iteration of ourgeneralized method of images, and we could show how thecontribution of each boundary to the signal was clearly


isolated. This fact is of major interest for the inverseproblem, and this approach could reveal itself to be a re-ally powerful tool for studying propagation problems instratified media. The next step is to perform such inter-actions in the case of multilayered media, in a way verysimilar to the method described in this paper.

APPENDIX ATo prove that the energy density f from Eqs. (2), usingthe sources SR and ST given in Eq. (3), fits boundary con-ditions (1), we first derived a property of the Green’s func-tion G in a uniform infinite medium: G satisfies the dif-fusion equation

]

c]tG 2 DDG 1 maG 5 d ~r!d ~z !d ~t !. (A1)

The quantity

G 5 E2`

`

G~r, z, t !dz,

which depends only on r and t, obviously satisfies the two-dimensional diffusion equation

]

c]tG 2 DD tG 1 maG 5 d ~r!d ~t !,

where D t corresponds to the transverse coordinates.However, applying the operator *2`

02

dz 1 *01` dz to Eq.

(A1) leads to

]

]zG~r, 02, t ! 2

]

]zG~r, 01, t !

51D S ]

c]t2 DD t 1 maD G,

since G (1)(z) 5 ]/]zG(z) vanishes at large z, so that

G ~1 !~r, 02, t ! 2 G ~1 !~r, 01, t ! 51

Dd ~r!d ~t !;

and since the function G (1) is an odd one, we can write

2D]

]zG~r, 02, t ! 5 d ~r!d ~t !. (A2)

Let us now calculate the energy density f from Eqs. (2),using the sources SR and ST given in Eqs. (3):

For z . 0, one can directly obtain

f~r, z, t ! 5 G1~r, z 2 Z, t ! 1 G1~r, z 1 Z, t !

2 4aD2E0

`

dlG1~r, z 1 Z 1 bl, t !

^ G2~2 !~r, 2al, t !,

where ^ is the convolution product restricted to r and tand where the role of the Heaviside function has beentaken into account by using an integration over a newvariable l. With use of an integration by parts, Eq. (A2)leads to

f~r, z, t ! 5 G1~r, z 2 Z, t ! 2 2D2

3 E0

`

dl@bG1~1 !~r, z 1 Z 1 bl, t !

^ G2~1 !~r, 2al, t ! 1 aG1~r, z 1 Z 1 bl, t !

^ G2~2 !~r, 2al, t !#. (A3)

For z , 0, in the same way we can directly write

f~r, z, t ! 5 24D2

aG1

~1 !~r, Z, t ! ^ G2~r, z, t !

24a21D2E0

`

dlG1~2 !~r, Z 1 bl, t !

^ G2~r, z 2 al, t !.

An integration by parts leads to

f~r, z, t ! 5 24D2E0

`

dl@G1~1 !~r, Z 1 bl, t !

^ G2~1 !~r, z 2 al, t !]. (A4)

It is easy to show that Eqs. (A3) and (A4) satisfy bound-ary conditions (3) when the following relation, which is adirect consequence of Eq. (A2), is used:

22D2E0

`

dl]

]l@G1~r, Z 1 Ul, t ! ^ G2

~1 !~r,2Vl, t !#

5 G1~r, Z, t ! ~; U, V . 0 !.

From the uniqueness of the solution, we can conclude thatthe energy density here defined is the solution of ourproblem.

APPENDIX BOur purpose here is to calculate the quantity

jn~r, z, Z, t ! [ ~Rf Rm!nS~r, z, Z, t !,

that is, to demonstrate formula (8)

jn~r, z, Z, t ! 5 ~21 !nd ~r!d ~z 2 zn!d ~t !

1 (k50

n21

pnk~z 2 zn!k

3 G2~k12 !(r, a~z 2 zn!, t)u~z 2 zn!.

First, since

Rf f ~z ! 5 2f(2~d 1 z0! 2 z)

and

RmS 5 E dz8SR~r, z, z8, t ! ^ S~r, z8, t !,

we can write the recurrence relation:


jn~r, z, Z, t ! [ ~RfRm!jn21~r, z, Z, t !

5 2E dz8SR~r, 2~d 1 z0! 2 z, z8, t !

^ jn21~r, z8, Z, t !. (B1)

For n 5 1 and for S(r, z, Z, t) 5 d (r)d (z 2 Z)d (t),

j1~r, z, Z, t ! 5 2SR(r, 2~d 1 z0! 2 z, Z, t);

that is,

j1~r, z, Z, t ! 5 2d ~r!d(Z 1 2~d 1 z0! 2 z)d ~t !

1 4aD2u(z 2 Z 2 2~d 1 z0!)

3 G2~2 !(r, a[Z 1 2~d 1 z0! 2 z], t),

in which we already recognize, for n 5 1 and using obvi-ous symmetry properties, the formula (8) that we are try-ing to demonstrate here. To finish the demonstration ofEq. (8), we have only to estimate recurrence relation (B1).

Assuming from Eq. (8) that

jn21~r, z, Z, t ! 5 ~21 !n21d ~r!d ~z 2 zn21!d ~t !

1 (k50

n22

pn21k ~z 2 zn21!k

3 G2~k12 !(r, a~z 2 zn21!, t)

3 u~z 2 zn21!,

we first have to calculate

I 5 Rmjn21~r, z, Z, t !

5 SE dz8SR~r, z, z8, t ! ^ jn21~r, z8, Z, t !

5 I1 1 I2 1 I3 1 I4 ,

where

I1 5 E dz8@d ~r!d ~z 1 z8!d ~t !#

^ @~21 !n21d ~r!d ~z8 2 zn21!d ~t !#,

I2 5 (k50

n22 E dz8@d ~r!d ~z 1 z8!d ~t !#

^ @ pn21k ~z8 2 zn21!k

3 G2~k12 !(r, a~z8 2 zn21!, t)u~z8 2 zn21!#,

I3 5 E dz8@24aD2u~2z 2 z8!

3 G2~2 !(r, a~z 1 z8!, t)#

^ @~21 !n21d ~r!d ~z8 2 zn21!d ~t !#,

I4 5 (k50

n22 E dz8@24aD2u~2z 2 z8!

3 G2~2 !(r, a~z 1 z8!, t)#

^ @ pn21k ~z8 2 zn21!k

3 G2~k12 !(r, a~z8 2 zn21!, t)u~z8 2 zn21!#.

The first three integrals are straightforward:

I1 5 ~21 !n21d ~r!d ~z 1 zn21!d ~t !,

I2 5 (k50

n22

pn21k ~2z 2 zn21!k

3 G2~k12 !(r, a~2z 2 zn21!, t)u~2z 2 zn21!,

I3 5 ~21 !n4aD2u~2z 2 zn21!

3 G2~2 !(r, a~z 1 zn21!, t).

Concerning the last term I4 , we have, after integrationover r,

I4 5 24aD2

exp~2r2/4D2c2t !

4pD2c2t

3 exp~2ma2c2t !u~2z 2 zn21!

3 Ezn21

2z

dz8pn21k ~z8 2 zn21!k

3 E dtG2~2 !(a~z 1 z8!, t 2 t)

3 G2~k12 !(a~z8 2 zn21!, t).

The integration over t can be performed analytically,leading to the term

21

2D2G2

~k13 !(2a~z 1 zn21!, t).

As this term does not depend on z8, the last integrationreduces to

Ezn21

2z

~z8 2 zn21!kdz8 5~z 2 zn21!k11

k 1 1.

We finally have for I4

I4 5 2a~z 2 zn21!k11

k 1 1u~2z 2 zn21!

3 G2~k13 !(r, 2a~z 1 zn21!, t).

Applying Rf all these terms leads to

I8 5 RfRmjn21~r, z, Z, t ! 5 I18 1 I28 1 I38 1 I48 ,

with


I18 5 ~21 !nd ~r!d ~z 2 zn!d ~t !,

I28 5 (k50

n22

2pn21k ~z 2 zn!k

3 G2~k12 !(r, a~z 2 zn!, t)u~z 2 zn!,

I38 5 ~21 !n114aD2u~z 2 zn!

3 G2~2 !(r, a~z 2 zn!, t),

I48 5 22a~z 2 zn!k11

k 1 1u~z 2 zn!

3 G2~k13 !(r, a~z 2 zn!, t).

Linking all these terms together gives Eq. (8).

APPENDIX CWe want to calculate

f nkl~t ! 5 E

0

`

dz8E2`

`

dt t lz8kG1~z 2 z8 2 zn , t 2 t!

3 G2~k12 !~az8, t!.

The Fourier transform of

Gi~z, t ! 5 ciu~t !exp$2@z2/~4Dicit !#%

A4pDicit

is

Gi~z, v! 5ADici

2DiAivexpS 2

Aiv

ADici

uzu D .

In the same way,

Gi~k12 !~z, v! 5 2

1

2Di~sign z !k12S 2

Aiv

ADiciD k11

3 expS 2Aiv

ADici

uzu D .

We also have

f nkl~v! 5 E

0

`

dz8z8kG1~z 2 z8 2 zn , v!~21 !l

3dl

d~iv!l G2~k12 !~az8, v!.

We set

dl

d~iv!l G2~k12 !~az, v!

51

2D2~sign z !k12

3 (i50

l

Qkli Aivk111i22luzui expS 2a

Aiv

AD2c2

uzu D ,

where the Q coefficients are defined through the followingrecurrence relationship (a . 0):

dl11

d~iv!l11 G2~k12 !~az, v!

51

2D2~sign z !k12

3 (i50

l11

Qk~l11 !i Aivk211i22luzui expS 2a

Aiv

AD2c2

uzu D5

1

2D2~sign z !k12(

i50

l k 1 1 1 i 2 2l

2

3 Qkli Aivk211i22luzui expS 2a

Aiv

AD2c2

uzu D1

1

2D2~sign z !k12(

i50

l2a

2AD2c2

3 Qkli Aivk1i22luzui11 expS 2a

Aiv

AD2c2

uzu D ,

which is equivalent to

Qk00 5 2S 2

1

AD2c2D k11

,

Qk~l11 !i 5

k 1 1 1 i 2 2l

2Qkl

i 2a

2AD2c2

Qkli21,

Qk~l11 !0 5

k 1 1 2 2l

2Qkl

i ,

Qk~l11 !~l11 ! 5 2

a

2AD2c2

Qkll .

Finally, as the Heaviside functions set z8 . 0 and z2 z8 2 zn , 0, we can write

f nkl~v! 5

1

2D2(i50

l

~21 !lE0

`

dz8Qkli z8k1i

AD1c1

2D1Aiv

3 Aivk111i22l expF2Aiv

AD1c1

~kz8 1 zn 2 z !G ,

with k 5 1 1 a@(D1c1)/(D2c2)#1/2. We then write


f nkl~v! 5

1

2D2(i50

l

~21 !lE0

`

dz8Qkli z8k1i

3AD1c1

2D1AivS 2

AD1c1

kD k111i22l

3 S d

dz8D k111i22l

3 expF2Aiv

AD1c1

~kz8 1 zn 2 z !G ,

with a k 1 1 1 i 2 2l time integration by parts, we have

f nkl~v! 5

1

2D2(i50

l

~ 2 1 !l~k 1 i !!

~2l 2 1 !!

3 E0

`

dz8Qkli z82l21

AD1c1

2D1Aiv

3 S AD1c1

kD k111i12l

3 expF2Aiv

AD1c1

~kz8 1 zn 2 z !G .

We continue to integrate (2l 2 1) times to obtain

f nkl~v! 5 (

i50

l

~21 !l~k 1 i !!

2D2E

0

`

dz8Qkli S AD1c1

kD k111i22l

3 Ez8

`

dz1Ez1

`

dz2 ¯Ez2l

`

dz2l21

AD1c1

2D1Aiv

3 expF2Aiv

AD1c1

~kz2l21 1 zn 2 z !G .

Coming back to real time,

f nkl~t ! 5 (

i50

l

~21 !l~k 1 i !!

2D2E

0

`

dz8Qkli S AD1c1

kD k111i22l

3 Ez8

`

dz1Ez1

`

dz2 ¯Ez2l

`

dz2l21c1

3

expF2~kz2l21 1 zn 2 z !2

4D1c1t GA4pD1c1t

.

Setting xj 5 k/A4D1c1t zj , we have

f nkl~t ! 5 (

i50

l

~21 !lc1

~k 1 i !!

2D2

1

A4pD1c1t

3 Qkli S AD1c1

kD k111i22lS A4D1c1t

kD 2l

3 E0

`

dx8Ez8

`

dx1Ez1

`

dx2¯Ez2l

`

dx2l21

3 expF2S x2l21 1zn 2 z

A4D1c1tD 2G ,

in which we recognize

im erfc~a ! 52

ApE

a

`

dx0Ex0

`

dx1¯Exm21

`

dxm exp~2xm2 !

52

ApE

0

`

dx0Ex0

`

dx1¯Exm21

`

dxm

3 exp@2~xm 1 a !2#

so that

f nkl~t ! 5 c1(

i50

l

~21 !l~k 1 i !!

2D2

1

A4pD1c1t

3 Qkli S AD1c1

kD k111i22l

3 S A4D1c1t

kD 2l Ap

2i2l21 erfcS zn 2 z

A4D1c1tD ,

or, equivalently,

f nkl~t ! 5 c1(

i50

l

~21 !l~k 1 i !!

4D2

1

kQkl

i S AD1c1

kD k111i22l

3 S A4D1c1t

kD 2l21

i2l21 erfcS zn 2 z

A4D1c1tD .

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Real-space Green's function calculation for the solution of the diffusion equation in stratified turbid media