Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media

13 Pages·1996·0.68 MB·English

Fluorescent diffuse photon density waves
in homogeneous and heterogeneous
turbid media: analytic solutions and applications
X.D.Li,M.A.O’Leary,D.A.Boas,B.Chance,andA.G.Yodh
Wepresentanalyticsolutionsforfluorescentdiffusephotondensitywavesoriginatingfromfluorophores
distributedinthickturbidmedia. Solutionsarederivedforahomogeneousturbidmediumcontaining
auniformdistributionoffluorophoresandforasystemthatishomogeneousexceptforthepresenceofa
single spherical inhomogeneity. Generally the inhomogeneity has fluorophore concentration, and
lifetime and optical properties that differ from those of the background. The analytic solutions are
verifiedbynumericalcalculationsandareusedtodeterminethefluorophorelifetimeandconcentration
changesrequiredfortheaccuratedetectionofinhomogeneitiesinbiologicallyrelevantsystems. The
relativesensitivitiesofabsorptionandfluorescencemethodsarecompared. r1996OpticalSocietyof
America
1. Introduction in a rapidly growing tumor142. Thus functional im-
Duringthepastfewyearsmanyschemeshavebeen aging of the variations of fluorophore concentration
developed to detect and image heterogeneities in andlifetimemayprovideusefulclinicalinformation.
biological tissues with diffusing near-IR light.1 Much effort has been devoted to formulating the
Mostofthesestudieshavecenteredonthedetection forwardproblemoffluorescentlightinturbidmedia.
ofvariationsintheabsorptionandscatteringcoeffi- Anintegralforminthetimedomainhasbeengiven
cients in tissue and tissue phantoms. Recently, bySevick-Muracaandco-workers15,16forthemigra-
however,fluorescentcontrastagentshavealsobeen tionoffluorescentlightinturbidmedia;theseinves-
consideredasameanstoenhancethespecificityand tigators have also numerically solved this integral
sensitivity in tumor detection.2–9 Fluorescent life- equation in specific two-dimensional homogeneous
time-basedquantitationofdifferentbiologicalparam- andheterogeneoussystems. Amathematicalmodel
eters such as tissue oxygenation pO ,10 pH value,11 for frequency-domain fluorescent-light propagation
2
and intracellular calcium concentration 3Ca214 1Ref. in semi-infinite homogeneous turbid media was ob-
122 have been proposed by several investigators. tainedbyPattersonandPogue3usingzeroboundary
Localizedfluorophoreconcentrationorlifetimevaria- conditionsandassumingthatthereducedscattering
tion may signal an increased number of small blood coefficients are the same for the excitation and the
vesselsassociatedwithrapidtumorgrowth,13orthe emission light. Finally a useful algorithm was de-
altered concentration of a fluorescence quencher veloped by Wu et al.17 to deconvolve fluorescent
thataffectsthefluorophorelifetime1e.g.,thelowered emission spectra from tissue reflectance measure-
concentrationofoxygenfromthehighoxygenuptake
ments.
In this paper we derive more general analytic
modelstounderstandtheresponseoffluorophoresin
AllauthorsarewiththeUniversityofPennsylvania,Philadel- homogeneous and heterogeneous turbid media.
phia, Pennsylvania 19104. X. D. Li, M.A. O’Leary, and D.A. Our discussion is divided into three sections. In
BoasarewiththeDepartmentofPhysicsandAstronomyandthe Subsection 2.A. we assume that the turbid medium
DepartmentofBiochemistryandBiophysics. B.Chanceiswith isinfiniteandhomogeneousandthatthefluorophore
theDepartmentofBiochemistryandBiophysics. A.G.Yodhis
concentration1numberdensity2andlifetimearecon-
withtheDepartmentofPhysicsandAstronomy.
stant. In Subsection 2.B. we assume that the me-
Received31July1995;revisedmanuscriptreceived27Febru-
dium is infinite and heterogeneous, consisting of a
ary1996.
0003-6935@96@193746-13$10.00@0 spherical heterogeneity in an otherwise homoge-
r1996OpticalSocietyofAmerica neous turbid medium. The fluorophore concentra-
3746 APPLIEDOPTICS @ Vol.35,No.19 @ 1July1996
tion and lifetime are piecewise continuous in these Here f1r,t2 is the photon fluence rate 3photons@
models, simulating situations wherein fluorophores 1cm2s24,visthespeedoflightintheturbidmedium,
preferentially accumulate in tumors and may have µ is the absorption coefficient, D 5 v@3µ8 is the
a s
environmentally sensitive lifetimes. In Subsection photon diffusion coefficient where µ8 is the reduced
s
2.C. we focus on analytic solutions for semi-infinite scattering coefficient, and S1r,t2 is the source term
homogeneous systems. Numerical calculations that is the number of photons emitted at positon r
where finite difference methods are used are per- andtimetperunitvolumeperunittime.
formedinthiscasetoverifytheanalyticsolutions. Forapointsourceatpositionr withanintensity
s
The analytic solutions reveal the explicit depen- modulation frequency f, we have S1r,t2 5 3M 1M
dc 0
dence of fluorescent diffusive waves on the fluoro- exp12ivt24d1r2r 2,whereM andM arethedcand
s dc 0
phore concentration, the lifetime, as well as the acpartsofthesourcestrength,respectively;visthe
optical properties of the medium. In Section 3 angular modulation frequency, and v 5 2pf. The
thesesolutionsareappliedtoexposesignal-to-noise resultantDPDWcanbewrittenasthesumofdcand
requirementsfordetectingheterogeneitiesthatmay acparts,i.e.,f1r,t25f 1r21f 1r2exp12ivt2. Inthis
dc 0
ariseinpracticalsituations. paperwefocusontheaccomponent. Inaninfinite
Although the analytic solutions derived in this homogeneousmediumtheacsolutiontoEq.112is20
paper are limited to some specific medium geom-
etries 1e.g., infinite and semi-infinite homogeneous f 1r,r25vM0exp1ik0r2rs02. 122
media or heterogeneous media containing a single 0 s D 4p0r2r 0
s
spherical object2, the results presented here are
useful because they are exact and enable rapid Here the wave number k is complex, k2 5
computations for signal-to-noise assessment over a 12vµ 1iv2@D. We see that the solution of the
a
broad range of experimental parameters. For ex- excitationDPDWinaninfinitehomogeneoussystem
ample,thecomputationwhenanalyticsolutionsare isadampedsphericalwave.
used for 100 arbitrary source–detector pairs in an In this paper we treat fluorophore and chromo-
infiniteheterogeneoussystemcontainingaspherical phore absorptions separately, but we assume that
object takes ,4 s of CPU time on a Sun Sparc10 the change of scattering coefficient caused by the
workstation. Finite difference calculations, on the fluorophore is negligible for notational simplicity.
other hand, take ,20 min of CPU time for each 1However, the fluorophore scattering effect is easily
sourceposition.18 Least-squaredfittingprocedures incorporatedintotheanalyticsolutions.2 Absorption
for determination of fluorophore concentration and fromtissuechromophoresischaracterizedbyabsorp-
lifetime based on these analytic solutions are thus tioncoefficientµ c. Absorptionfromexogenousfluo-
a
possiblewithinreasonableCPUtimes. rophores is characterized by an additional absorp-
tioncoefficientsN,whereN istheconcentrationof
t t
fluorophoreandsisthefluorophoreabsorptioncross
2. Theory
section 1which is essentially the same as the fluoro-
A. PhotonDiffusioninInfiniteHomogeneousMedia phoreextinctioncoefficient,seeSection32. Thusin
First we consider an infinite homogeneous turbid the presence of fluorophores the total absorption
medium with a spatially uniform distribution of coefficient is the sum of these two coefficients, i.e.,
fluorophores. An intensity-modulated point light µa5µac1sNt.
source with an optical wavelength that falls within
the absorption band of the fluorophores creates an
2. SolutionsforFluorescentDiffusiveWaves
excitation diffuse photon density wave 1DPDW2 that
Supposethatanintensity-modulatedpointsourceis
excites the fluorophores. The fluorescent photons
atpositionr inahomogeneoussystem. Thefluoro-
then add to create a fluorescent DPDW originating s
phores in the medium are excited by the incident
fromthesourcefluorophoredistribution. Thefluo-
DPDW, f , given in Eq. 122. Each excited fluoro-
rescentradiationisassumedtobewellseparatedin 0
phore then acts as a secondary point source of
energy from that of incident photons so that we can
fluorescentdiffusivewaves3Fig.11a24. Treatingfluo-
safely ignore the possibility of the excitation of
rophoresastwo-levelquantumsystemsandignoring
fluorphores by the fluorescent reemission. We also
saturation effects, we find that the steady-state
assume that the excited fluorphores have a single
numberdensityofexcitedfluorophoresatpositionr
lifetime.
is21
1. SolutionforaPointSource
sNh
t
Inhighlyscatteringmediasuchasbiologicaltissue, Ne1r,rs25G2ivf01r,rs2, 132
lightpropagationiswellapproximatedbythephoton
diffusionequationthatweadoptasacentralassump- wheresisthefluorophoreabsorptioncrosssectionat
tionofourdiscussion,19i.e., source wavelength l , N is the fluorophore concen-
ex t
tration, G 5 1@t is the decay rate of an excited
vµ 1 ›f1r,t2 v
a fluorophore,whichissimplythereciprocaloffluoro-
=2f1r,t22 f1r,t22 52 S1r,t2. 112
D D ›t D phorelifetimet,andf istheincidentphotonfluence
0
1July1996 @ Vol.35,No.19 @ APPLIEDOPTICS 3747
Fig. 1. Diagrammatic representation of fluorescent DPDW generated by fluorophores 1a2 in a homogeneous system and 1b2–1f2 in a
heterogeneous system. 1a2 The fluorophore is excited by the excitation DPDW, and the reemission reaches the detector. 1b2 The
fluorophoreinsideisexcited,andthereemissionreachesthedetector. 1c2ThefluorophoreoutsideisexcitedbytheincidentDPDW,and
thereemissionreachesthedetectorwithoutbeingscatteredbythesphere. 1d2ThefluorophoreoutsideisexcitedbytheincidentDPDW,
butthereemissionisscatteredbythespherebeforereachingthedetector. 1e2ThefluorophoreoutsideisexcitedbythescatteredDPDW,
andthereemissionreachesthedetectorwithoutbeingscatteredbythesphere. 1f2Thefluorophoreoutsideisexcitedbythescattered
DPDW,butthereemissionisscatteredbythespherebeforereachingthedetector.
rate 3Eq. 1224; h is the fluorescence quantum yield. NotethatthefluorescentDPDWisasuperimposi-
For notational simplicity hereafter we assume ex- tion of two spherical waves with different DPDW
cited fluorophores decay only through the radiation wavenumbers: OnehastheDPDWwavenumberk
channel,i.e.,h5100%. Theseexcitedfluorophores at the source wavelength l , and the other has the
ex
decay with a rate G, and thus the source term for DPDW wave number k at the fluorescence wave-
f
fluorescentDPDWatpositionris length l . The fluorescent DPDW is basically pro-
fl
portional to the fluorophore concentration 1DPDW
sNt wavenumberkalsodependsonfluorophoreconcen-
S 1r,r25GN1r,r25 f 1r,r2. 142
f s e s 12ivt 0 s tration2 and is explicitly related to the lifetime by a
factorof1@112ivt2. Notethatarelatedresultfora
Each fluorescence source generates a fluorescent planar homogeneous medium has been derived in
DPDWthatpropagatestothedetectoratpositionr. onedimensionbyTrombergetal.21
ThedetectedfluorescentDPDWisfoundbyintegrat-
ingoverallfluorescencesources,i.e.,
B. PhotonDiffusioninInfiniteHeterogeneousMedia
ContainingaSphericalHeterogeneity
e v exp1ik 0r2r 02
f 1r,r25 S 1r ,r2 f 1 dr . 152 Asphericalobjectembeddedinanotherwisehomoge-
0f s f 1 s Df 4p0r2r10 1 neousturbidmediummaybeofpracticalinterestas
a model for a tumor embedded in normal biological
Whenthefluorophoredistributionandlifetimeare
tissues. The analytic solution for fluorescent
constant, one can solve the integral exactly by ex-
DPDW’s in this case is useful in determining the
panding the spherical waves in terms of spherical
fluorophore concentration contrast and lifetime
Bessel functions and spherical harmonics and then
changesnecessaryfordetectionoftheobject. Boas
byusingtheappropriateorthogonalityrelations1see
et al. have shown that the solution of DPDW’s for a
AppendixA2. Theanalyticexpressionforthefluores-
point source outside the sphere in a heterogeneous
centDPDWinahomogeneousmediumis
medium can be obtained by applying appropriate
boundary conditions on the surface of the spherical
v2M sN 1
0 t object.22 We treat the analogous problem for the
f 1r,r25
0f s DD 12ivtk22k2 fluorescentDPDW.
f f
The problem is more complicated, however, be-
3exp1ik0r2r 02 exp1ik 0r2r 024
3 s 2 f s . 162 cause the fluorescent point sources are now distrib-
4p0r2r 0 4p0r2r 0 utedinspace. Ingeneral,theincidentlightsource
s s
3748 APPLIEDOPTICS @ Vol.35,No.19 @ 1July1996
at l is outside the sphere whereas the fluorescent Physically we demand that the DPDW be finite at
ex
sourcesemittingatl canbeinsideandoutsidethe theorigin1r502andvanishatinfinity1r=‘2.
fl
sphere. WecalculatethefluorescentDPDWgener- Forapointsourceoutsidethespheretheresultant
atedbyapointsourceinsideandoutsidethesphere DPDWsolutionsoutsideandinsidethesphereare
and then integrate the fluorescent DPDW over the
correspondingsourcedistributiontoobtainthetotal f 1r,r25f 1r,r21f 1r,r25f 1r,r2
1 s 10 s 1sc s 10 s
fluorescentDPDW.
o
1 A 1r2h1121k r2Y 1V2, 172
Before proceeding, we specify the following nota- lm s l 1 lm
lm
tionconvention: Allbackgroundquantitiesoutside
thespherearedenotedbysubscript1,e.g.,f1,k1,D1, f21r,rs25oBlm1rs2jl1k2r2Ylm1V2, 182
and all quantities inside the sphere are denoted by lm
subscript2,e.g.,f ,k ,D . Furthermoreallfluores-
2 2 2
where f is the incident wave with background
cence-related quantities are denoted by an extra 10
subscriptf,e.g.,f ,f ,k ,k ,D ,D . Forsimplic- parameters3Eq.1224,i.e.,
1f 2f 1f 2f 1f 2f
ity we take the index of refraction n 5 n . This
1 2 vM exp1ik 0r2r 02
enablesustoignoretheopticalreflectionandrefrac- f 1r,r25 0 1 s ,
tioneffectsonthesurfaceofthesphere. Thespeed 10 s D 4p0r2r 0
1 s
of light is then the same everywhere, v1 5 v2 5 v. andf1scisthescatteredwave. Thefunctionshl1121k1r2
Differences in the indices of refraction can be ac-
and j1k r2aresphericalHankelfunctionsofthefirst
l 2
counted for by following the procedures set forth by
kind and spherical Bessel functions, respectively.
Haskelletal.23andAronson.24
TheY 1V2termsaresphericalharmonics,andk and
Forreaderswhoarenotinterestedinthedetailsof lm 1
k are DPDW wave numbers. We found the coeffi-
the derivation, the main results are in Eq. 1202 1the 2
cients A and B by requiring that the diffusive
solution for the fluorescent DPDW generated by lm lm
photonfluencerateandnormalfluxbecontinuousat
fluorophores inside a spherical object2, Eq. 1242 1the
thesurfaceofthesphere,i.e.,
solution for the fluorescent DPDW generated by
fluorophores outside the spherical object2, and Eq. f 1a,r25f 1a,r2,
1 s 2 s
12523thesumofEqs.1202and1242,whichisthesolution
for the fluorescent DPDW in an infinite hetero- ›f11r,rs2 ›f21r,rs2
D 5D . 192
geneousturbidmediumcontainingasphericalobject4. 1 ›r 2 ›r
Equations 1262 and 1272 are solutions for the fluores- 0r5a 0r5a
cent DPDW in a semi-infinite homogeneous system Inthiscasewefindthat
in which zero fluence rate boundary conditions and
extrapolated zero fluence rate boundary conditions 2ik
1
areused,respectively. A 1r25v Qh1121k r2Y *1V2M , 1102
lm s D l l 1 s lm s 0
1
1. SolutionforaPointSource
B 1r25vRh1121k r2Y *1V2M , 1112
Consideraninfinitesystemwithasphericalhetero- lm s l l 1 s lm s 0
geneityofradiusaattheoriginandapointsourceat where Q and R are constants that depend on the
l l
rs. Ingeneralwehavetwopossiblesourceconfigu- optical properties at lex, modulation frequency, and
rations: a point source outside the sphere 1rs.a2 the radius of the embedded spherical object 1a2.
and inside the sphere 1r ,a2 as shown in Fig. 2.
s TheirdefinitionsareinAppendixB.
Forthecaseofapointsourceinsidethesphereat
r1r ,a2, the solutions for the DPDW outside and
s s
insidethesphereareoftheforms
o
F 1r,r25 C 1r2h1121k r2Y 1V2, 1122
1 s lm s l 1 lm
lm
F 1r,r25F 1r,r21F 1r,r2
2 s 20 s 2sc s
o
5F 1r,r21 D 1r2j1k r2Y 1V2. 1132
20 s lm s l 2 lm
lm
1a2 1b2
Fig. 2. Two source configurations in a heterogeneous medium Using the same boundary conditions for the fluence
containing a spherical object. 1a2 The source is outside the rateandnormalfluxonthesurfaceofthesphere3Eq.
sphere,andthesolutionofDPDWoutsidethesphere3Eq.1724isthe 1924,weobtain
sumofthehomogeneousincidentpart,f ,andthescatteredpart,
10
f1sc. Thesolutioninsidethesphereisf2asgivenbyEq.182. 1b2 C 1r25vR j1k r2Y *1V2M , 1142
Thesourceisinsidethesphere,andthesolutionofDPDWoutside lm s l l 2 s lm s 0
thesphereisF13Eq.11224. Thesolutioninsidethesphereisthe 2ik
sum of the homogeneous incident part, F20, and the scattered D 1r25v 2S j1k r2Y *1V2M , 1152
part,F2sc,asgivenbyEq.1132. lm s D2 l l 2 s lm s 0
1July1996 @ Vol.35,No.19 @ APPLIEDOPTICS 3749
where S is a constant that depends on the optical tion r outside the sphere and generated by a fluoro-
l
properties at l , modulation frequency, and the phore outside the sphere at r with a source term
ex 1
radiusoftheobject. ItsdefinitionisgiveninAppen- S 1r ,r2isoftheformofEq.172,i.e.,
1f 1 s
dixB. R isthesameasinEq.1112.
l
df 1r,r ,r25df 1r,r ,r21df 1r,r ,r2
1f 1 s 10f 1 s 1scf 1 s
2. SolutionsforFluorescentDiffusiveWaves
o
Weassumethatthefluorophoredistributionispiece- 5df10f1r,r1,rs21 dAlmf1r12
wiseuniform,i.e.,theoutsideandinsidefluorophore lm
concentration N and N may be different but they 3h1121k r2Y 1V2, 1212
1 2 l 1f lm
are constant. We also assume that the excitation
pointsource1atl 2isoutsidethesphereatr1r .a2. where
ex s s
The source term for fluorophores outside and inside
thespherewillbeoftheform df10f1r,r1,rs25f10f1r,r12S1f1r1,rs2dr1
v exp1ik 0r2r 02
sN 1f 1
1 5
S1f1r1,rs2512ivt f11r1,rs2, 1162 D1f 4p0r2r10
1
3S 1r ,r2dr , 1222
1f 1 s 1
sN
2
S 1r ,r25 f 1r ,r2, 1172
2f 2 s 12ivt2 2 2 s dAlmf1r125f1scf1r,r12S1f1r1,rs2dr1
3 2ik 4
respectively, where f11r1,rs2 is given by Eq. 172 and 5 v 1fQ fh1121k r 2Y *1V 2
f 1r ,r2isgivenbyEq.182. D l l 1f 1 lm 1
2 2 s 1f
The fluorescent DPDW measured at position r
3S 1r ,r2dr , 1232
outside the sphere and generated by a fluorophore 1f 1 s 1
inside the sphere at position r with a source term
2 anddr isthevolumeelementaroundpositionr .
S 1r ,r2isoftheformofEq.1122,i.e., 1 1
2f 2 s The physical processes producing the fluorescent
o DPDW from fluorophores outside the sphere are
dF 1r,r ,r25 dC f1r 2h1121k r2Y 1V2, 1182
1f 2 s lm 2 l 1f lm showninFigs.11c2–11f2. IntegratingEq.1212overall
lm
spaceoutsidethesphere1i.e.,overallpossibler 2,we
1
where obtain the contribution to the fluorescent DPDW
fromfluorophoresoutsidethesphere,i.e.,
dC f1r 25vR fj1k r 2Y *1V 2S 1r ,r2dr , 1192
lm 2 l l 2f 2 lm 2 2f 2 s 2
e sN
1
anddr isthevolumeelementaroundpositionr . f 1r,r25 3f 1r,r 21f 1r,r 24
2 2 1f s 10f 1 1scf 1 12ivt
Figure 11b2 shows that the fluorophore inside the outside 1
sphere at r is excited by incident DPDW f 1r ,r2
2 2 2 s 33f 1r ,r21f 1r ,r24dr
andthereemissionpropagatesfromr toapointatr 10 1 s 1sc 1 s 1
2
outside the sphere. Integrating Eq. 1182 over the v2M sN 1
0 1
entire space inside the sphere 1i.e., over all possible 5
D D 12ivt k 22k 2
r 2,wederivethetotalcontributiontothefluorescent 1 1f 1 1 1f
2
DPDWfromfluorophoresinsidethesphere,i.e., 53exp1ik10r2rs02 exp1ik1f0r2rs024
3 2
e 4p0r2rs0 4p0r2rs0
F 1r,r25 dF 1r,r ,r2
1f s 1f 2 s 3 o
Inside 1 1ik1f2 Qlfhl1121k1fr2hl1121k1frs2
lm
sN a2
5M0v212iv2t k 22k 2 112ik12oQlhl1121k1r2hl1121k1rs2
2 2 2f lm
o
o 11ik 2 QEh1121k r2h1121k r2
3 53k j1k a2j81k a22k j81k a2 1 l l l 1f l 1 s
2f l 2 l 2f 2 l 2
lm
lm
4
o
3jl1k2fa24RlRlfhl1121k1fr2hl1121k1rs2 112ik12 Flhl1121k1fr2hl1121k1rs2
lm
3Y 1V2Y *1V26, 1202 6
lm lm s 3Y *1V2Y 1V2 , 1242
lm s lm
where constant R f has the same functional form as
l
R exceptthatitdependsontheopticalpropertiesat wheretheconstantQ fhasthesamefunctionalform
l l
l whereas R depends on the optical properties at asQ exceptthatitdependsontheopticalproperties
fl l l
l . ItisdefinedinAppendixB. atl whereastheconstantQ dependsontheoptical
ex fl l
Next we consider the fluorophores outside the propertiesatl . ConstantsE andF dependonthe
ex l l
sphere. The fluorescent DPDW measured at posi- opticalpropertiesatl andtheopticalpropertiesat
ex
3750 APPLIEDOPTICS @ Vol.35,No.19 @ 1July1996
l . The definitions of these constants are given in phores at the z 5 0 plane and the corresponding
fl
AppendixB. imageboundaryoftheimagefluorophoresatthez5
Notethatthesphericalregionisdevoidofoutside 214@3µ82plane. Therearenofluorophoresorimage
s
fluorophores. The contribution from the outside fluorophoresinsidethisgap. Thereforetheintegral
fluorophores may be constructed by subtracting the equation of the form of Eq. 152 cannot be evaluated
solution for a sphere 1containing the outside fluoro- analytically. However, using a less accurate zero
phore2fromthehomogeneoussolution—thefirsttwo boundarycondition1wherethephotonfluencerateis
terms in the braces in Eq. 1242 3see also Eq. 162 and zero at the physical boundary2, we can calculate the
Fig.11a24. Allothertermsareduetothepresenceof integralanalytically1seeAppendixC2,i.e.,
thesphericalobjectandcorrespondtovariouscombi-
nations of the other three paths in Figs. 11d2–11f2. f si2z.b.c.1r,r25f 1r,r22f 1r,rzbc2 1262
0f s 0f s 0f s
Equation 1242 is exact, but unfortunately after the
algebraicsimplificationssomeofthetermsinEq.1242 wherer isthepositionoftherealsourceandrzbcis
s s
arise from combinations of different paths, and the
the position of the image source with respect to the
physicaloriginsofthesetermsarenotalwayseasily
zero boundary; f 1r,r2 and f 1r,rzbc2 are of the
visualizedasinFigs.11c2–11f2. 0f s 0f s
Finally the total fluorescent DPDW outside the formofEq.162.
sphere, f fl1r,r2, generated by all the fluoro- We have found that the extrapolated zero bound-
hetero s
phoresinsideandoutsidethesphere,isgivenbythe ary condition can be implemented if we neglect the
sumofEqs.1202and1242: contribution of the gap. Neglecting the gap effect,
wecantakethesolutioninaninfinitehomogeneous
f fl1r,r25f 1r,r21F 1r,r2. 1252 system for a source at r and the solution for an
hetero s 1f s 1f s s
image source at rebc, then superpose these two
s
Note that in Eq. 1252, F 1r,r2 3see Eq. 12024 is the
1f s solutions. TheresultantfluorescentDPDW1f si2e.b.c2
fluorescentDPDWgeneratedbyafluorescingsphere 0f
forasemi-infinitehomogeneoussystemwithextrapo-
with no background fluorophores whereas f 1r,r2
1f s latedzeroboundaryconditionshasthesameformas
3see Eq. 12424 is the sum of the background term and
Eq.1262:
the scattering terms. The total fluorescent DPDW
isthenthesumofF 1r,r2andf 1r,r23Eq.12524.
1f s 1f s f si2e.b.c.1r,r25f 1r,r22f 1r,rebc2, 1272
0f s 0f s 0f s
In the forward calculation we take advantage of
thesymmetryofthesystem,e.g.,ifthelightsourceis
where r is the position of the real source and the
onthezaxis,wehaveazimuthalasymmetryandwe s
rebc is the position of the image source with respect
can remove the summation over m in Eqs. 1202 and s
to the extrapolated zero boundary; f 1r,r2 and
1242. 0f s
f 1r,rebc2areoftheformofEq.162.
0f s
C. Semi-infiniteMediaandNumericalVerificationof Thisisanextensionoftheresultthatweobtained
AnalyticSolutions in Appendix C using zero boundary conditions to
Intheabovediscussionwerestrictedourattentionto approximatetheexactzeropartialcurrentboundary
infinitehomogeneousandheterogeneousturbidme- conditions. Equation 1272 turns out to be a very
dia. The standard procedure for solving the diffu- goodapproximation. Numericalcalculationsoffluo-
sionequationforasemi-infinitehomogeneousturbid rescentDPDW’satdifferentmodulationfrequencies
medium is to approximate the exact zero partial 1f2 where a finite difference algorithhm is used25
current boundary condition with the extrapolated have been performed to verify Eq. 1272 for two
zero boundary condition.23 For the excitation fluorophore lifetimes 11 and 2 ns2. The basic proce-
DPDW, we satisfy the extrapolated zero boundary dureistosolvethediffusionequation3Eq.1124forthe
condition by using the method of images, i.e., the fluorescenceDPDWinthetimedomainbyusingthe
DPDWinthemediumiscalculatedfromthesuperpo- exact zero partial current boundary condition23 and
sition of the DPDW’s generated by the real and the then Fourier transform the data into the frequency
imagesources. Thissituationbecomesmorecompli- domain. Inthecalculationsthefluorophoreconcen-
cated for the fluorescent DPDW. The fluorescent tration is assumed to be N 5 0.1 µM. The source
t
DPDW is generated by a uniform distribution of and the detector are placed on the surface of the
fluorophores 1in half-space2, and therefore we must medium with a 2-cm separation, and the optical
consider a uniform distribution of image fluoro- properties1µ ,µ 8,µ ,µ 8,s2aregiveninTable1.
a1 s1 a1f s1f
phorestocalculatethefluorescentDPDW. Suppose Results are shown in Fig. 3 in which we plot the
fluorophores distribute in the upper space, z . 0. theoreticalamplitudeandphasecalculatedbyusing
Thephysicalboundaryofthefluorophoresisthex–y Eq. 1272 and the amplitude and phase calculated by
plane at z 5 0. For extrapolated zero boundary usingthefinitedifferencemethod. Theratioofthe
conditions,wherethephotonfluencerateisapproxi- normalizedamplitudeandtheresidualofthephase
matedtobezerointhex–yplaneatz512@3µ82,the are also shown in Fig. 3. We find that the analytic
s
resultant image fluorophores then distribute in the solutionofthefluorescentDPDWforasemi-infinite
lower space, z , 214@3µ82. In this case a gap homogeneous system using an extrapolated zero
s
appearsbetweenthephysicalboundaryofthefluoro- boundary condition 3Eq. 12724 agrees with the finite
1July1996 @ Vol.35,No.19 @ APPLIEDOPTICS 3751
Table1. ChromophoreOpticalPropertiesatl andl andOtherParametersa
ex fl
µa1c µs18 µa2c µs28 µa1fc µs1f8 µa2fc µs2f8 s1cm21M212 f 1MHz2 rsd1cm2
0.02 8.0 0.04 10.0 0.025 8.0 0.05 10.0 63104 200 6
aTheopticalpropertiesareinunitsofcm21;r isthesource–detectorseparation.
sd
differenceresultswithanaccuracyofbetterthan3% latedzeroboundarycondition3Eq.12724. Thetrade-
in amplitude and 1.2° in phase up to an 800-MHz off of using a smaller grid size is a much longer
modulation frequency. Using a zero boundary con- computation time. As we mentioned in Section 1,
dition 3Eq. 12624 for the same system, we find that a the analytic calculation is much faster than the
littlelargerdiscrepancy,inparticular,,4%inampli- numerical calculation. For t 5 1 ns in the above
tudeand1.5°inphase,wasobserved. Thesediscrep- calculations, the analytic method takes ,0.1 s of
ancies are mainly due to a numerical computation CPUtime1modulationfrequencyf variesfrom0to1
error. If we reduce the grid size from 1@µ8 to GHzwithastepof50MHz2,butthefinitedifference
s
1@12µ82 in the finite difference calculations, we find method takes ,25 min. For a longer lifetime 1e.g.,
s
that these discrepancies decrease, e.g., less than t52 ns2 the numerical method requires a longer
1.2% in amplitude and less than 1° in phase for a CPU time 1e.g., ,1 h2 whereas the analytic method
semi-infinitehomogeneoussystemusinganextrapo- stilltakes,0.1sofCPUtime.
Asforthesemi-infiniteheterogeneoussystem,the
situationwillbemuchmorecomplicatedbecausewe
mustconsidernotonlytheimagesourcebutalsothe
image object as well as the image fluorophores.
Nevertheless in analogy with semi-infinite homoge-
neousmediaobservations,onecanbeencouragedto
employ the same basic procedure as used in the
aboveanalysis. Finitedifferencecalculationsofthe
fluorescent DPDW in a semi-infinite heterogeneous
turbidmediacontainingasphericalobjecthavebeen
performed for different fluorophore concentration
and lifetime contrasts. We found good agreement
1a2 1b2
betweenthefinitedifferenceresultsandtheresults
calculated by the analytic solutions, following the
samebasicprocedureasforthesemi-infinitehomoge-
neousmedia. Considerasemi-infinitesystemwith
a 0.6-cm radius object 3 cm deep in the medium.
Thesourceandthedetectorareplacedonthesurface
ofthemediumandseparatedby2cm. Anaccuracy
ofbetterthan1.8%inamplitudeand1.2°inphaseto
an800-MHzmodulationfrequencyisobtainedwhen
theopticalpropertiesgiveninTable1areused.
1c2 1d2
Fig. 3. Comparison of the analytic solution 3Eq. 12724 and the 3. Applications
finitedifferencemethodforasemi-infinitehomogeneoussystem. Analytic solutions reveal the manner in which fluo-
Thesourceanddetectorareplacedonthesurfaceofthemedium2 rescent DPDW’s are related to the spatial distribu-
cmapart. 1a2Amplitudenormalizedbythedcamplitude1f50
tion of fluorophore lifetimes and concentrations.
MHz2. Thesolidcurveandtheopencirclesrepresentnormalized
Wenowapplythesesolutionstoseveralproblemsof
amplitudesfort51nsascalculatedbytheanalyticsolutionsand
practical importance. First, we discuss the fitting
thefinitedifferencemethod,respectively. Thedashedcurveand
procedure for determination of the fluorophore life-
filled circles represent normalized amplitudes for t 5 2 ns as
calculated by the analytic solutions and the finite difference time and concentration in homogeneous media.
method, respectively. 1b2 Ratio of the normalized amplitude Next we calculate the fluorophore concentration
calculated by the finite difference methods and the normalized contrast and the lifetime variation necessary to
amplitudecalculatedbyanalyticsolutions1solidcurve,t51ns; detect an object embedded in an otherwise homoge-
dashed curve, t52 ns2. 1c2 Phase with respect to the dc phase neous medium. Finally we compare the sensitivi-
1which is zero2. The solid curve and the open circles represent ties of fluorophores as absorption contrast and fluo-
phasesfort51nscalculatedbyanalyticsolutionsandthefinite
rescentcontrastforthedetectionofsphericalobjects
difference method, respectively. The dashed curve and filled
inturbidmedia.
circles represent phases for t 5 2 ns calculated by analytic
In general we take the fluorophore absorption
solutionsandthefinitedifferencemethod,respectively. 1d2Phase
crosssection 1s2tobe ,10216cm2fornear-IRexcita-
residuesthatarethedifferencebetweenthephasecalculatedby
thefinitedifferencemethodandthephasecalculatedbyanalytic tionlight.26 Theabsorptioncrosssectionisrelated
solutions1solidcurve,t51ns;dashedcurve,t52ns2. to the extinction coefficient by Avogadro’s constant,
3752 APPLIEDOPTICS @ Vol.35,No.19 @ 1July1996
i.e.,E1cm21M2125s1cm223NAvog31023. Intermsof measured before adding fluorophores into the sys-
the extinction coefficient the above absorption cross tem to reduce the number of free parameters in the
sectionis6.03104cm21M21. Unlessstatedother- fit.
wise, we assume this value for the fluorophore Using the same optical parameters and source–
absorption cross section. In the following simula- detector separation 16 cm2 as in Fig. 4, we calculate
tionsthefluorophoreconcentrationinhomogeneous the amplitude and phase of the fluorescent DPDW
media is assumed to be N 50.1 µM. The back- for N 5 0.1 µM and t 5 1.0 ns at multiple modula-
t t
ground concentration 1outside the sphere2 in the tionfrequencies10–500MHzin50-MHzsteps2. We
heterogeneousmediaisalsoassumedtobeN 50.1 then add 2% random noise to the amplitude and 1°
1
µM. Differentconcentrationsinsidethesphereare random noise to the phase. The noisy data are
considered. We choose the concentration to be in fitted with the analytic solution 3Eq. 1624 where we
the micromolar regime for three reasons: 112 Such assume that all the parameters are known except
concentrations are realistically expected for experi- thefluorophoreconcentrationandlifetime. Forsix
ments in vivo.7 122 Fluorescence saturation effects independent least-square fittings, we obtain the
can be ignored. 132 The fluorophore absorption is fluorophore concentration, Nfit 5 10.100260.00072
t
comparable with the chromophore absorption when µM and lifetime tfit 5 11.00360.0072 ns. For the
thefluorophoreisusedasanabsorptioncontrast. data with 5% amplitude noise and 5° phase noise,
the fitted results are still very close to real values,
A. FittingProcedureforDeterminationofFluorophore e.g., Nfit 5 10.1003 6 0.00172 µM and tfit 5
t
ConcentrationandLifetimeinHomogeneousSystems 11.00560.0162ns.
Fluorophorelifetimeisagoodindicatorofthefluoro- AnothermethodsuggestedbySevick-Muracaand
phoretissueenvironment1e.g.,thepHorpO value2. co-workers for lifetime measurements in homoge-
2
One way to determine the fluorophore lifetime in neousturbidmediaistointroduceareferencefluoro-
homogeneousmediaistomeasurethephaseand@or phorewithaknownlifetime. Theypointedoutthat
the amplitude of the fluorescent DPDW at different the unknown fluorophore lifetime can then be ob-
modulationfrequencies. Thepredicteddependence tained from the relative phase shift between the
oftheamplitudeandphaseoffluorescentDPDWon fluorescentDPDWgeneratedfromfluorophoreswith
the modulation frequency 1f2 for three different unknownlifetimeandthefluorescentDPDWgener-
lifetimes is illustrated in Fig. 4. The differences ated from reference fluorophores with a known life-
in the curves indicate the sensitivity to fluoro- time.16 Thisrelationshipiseasilyverifiedforhomo-
phore lifetime t. The optical properties 1µ ,µ 8, geneous media where the analytic solution is used.
a1 s1
µ ,µ 8,s2,thesource–detectorseparation1r 2,and In Eq. 162 we see that the phase of the fluorescent
a1f s1f sd
the modulation frequence 1f2 used in the simulation DPDWcausedbythefluorophorelifetimeistan211vt2.
are given in Table 1. With experimental data, we Ifweassumethattheopticalpropertiesarethesame
can obtain accurate lifetimes and concentration by for the fluorophore with lifetime t 1unknown2 and
2
fittingtheamplitudeandphasecurvesofthefluores- the fluorophore with lifetime t 1known2, we obtain
1
cent DPDW with the analytic solution 3Eq. 1624 at therelativephaseshiftu :
21
multiplemodulationfrequencies. Insuchcasesthe
tissue optical properties at lex and lfl should be u 5tan213v1t22t1245tan211vt 22tan211vt 2,
21 11v2t t 2 1
1 2
1282
which is in agreement with the result in Ref. 16.
For a homogeneous system, Eq. 1282 provides a
simplerelationshipbetweenthephasesofthefluores-
cent DPDW’s generated by two kinds of fluorophore
withdifferentlifetimes,anditisvalidaslongasthe
theopticalpropertiesofthemediumarethesamefor
thetwofluorophores.
1a2 1b2 B. RequiredContrastofFluorophoreConcentrationand
Fig. 4. Fluorescent DPDW 3Eq. 1624 versus the modulation fre- LifetimeforDetectionandCharacterizationofaSpherical
quency 1f2 in an infinite homogeneous turbid medium for three Heterogeneity
differentlifetimes1t51,2,and5ns2: 1a2normalizedamplitude, The variations in fluorophore distribution and life-
1b2phaseshift. Theamplitudeisnormalizedbytheamplitudeat time in normal and malignant tissues may give
zeromodulationfrequency1f502foreachcurveatthecorrespond-
enhancedsensitivityandspecificityfortumordetec-
ing fluorophore lifetime, and the phase is phase shifted with
tion. UsingEq.1252,wecalculatetheamplitudeand
respect to the phase at the zero modulation frequency. Solid
the phase of the fluorescent DPDW for a heteroge-
curves,t51ns;dashedcurves,t52ns;dash–dottedcurve,t55
neous system containing a spherical object. The
ns. Source–detector separation r , 6 cm. The optical proper-
sd
ties1µa1c,µs18forincidentDPDWatlexandµa1fc,µs1f8forfluores- required phase precision to detect such an object
centDPDWatlfl2aregiveninTable1. with fluorescent DPDW’s is determined from the
1July1996 @ Vol.35,No.19 @ APPLIEDOPTICS 3753
position-dependent phase difference between the
fluorescent DPDW in the heterogeneous medium
and the fluorescent DPDW in the homogeneous
mediumwiththesamebackgroundopticalproperties.
The required amplitude precision is found from a
similar position-dependent normalized amplitude
0f fl@f 0 3see Eqs. 162 and 12524. Figure 5 shows
hetero 0f
contour plots of the spatially dependent normalized
amplitudeandthephaseshiftforanobjectof0.6-cm
radius at the origin in a medium with the optical
propertiesgiveninTable1. Thesourceislocated3 Fig. 6. Fluorescent DPDW versus fluorophore concentration
cm from the object. A detector would scan in the contrastinaheterogeneoussystem. Thesourceandthedetector
planethatcontainsthesourceandthecenterofthe are separated by 6 cm, and the object is halfway between
object. Inthecalculationthelifetimesofthefluoro- them. In the simulation the lifetimes inside and outside the
phore outside and inside the sphere are assumed to sphere are assumed to be equal 1t15t251 ns2. The system
parametersaregiveninTable1. Theamplitudeisnormalizedby
be equal, i.e., t 5 t 5 1 ns, and the concentration
1 2 theamplitudeinahomogeneoussystem,andthephaseshiftis
contrast N @N is assumed to be 5. We see that
2 1 withrespecttothephaseinahomogeneoussystem.
differential amplitude and phase signals from an
objectof0.6-cmradius1within,4cmoftheobject2is
detectable with the 0.1° phase and 0.1% amplitude gives enhanced sensitivity to the detection of the
measurement precisions achievable with current heterogeneity. In Fig. 7 we show the lifetime ratio
technologies. 1t @t 2 dependence of fluorescent DPDW’s for three
2 1
Some fluorophores may be tumor specific and the different size objects, again centered between the
fluorophore concentration inside the tumor may be sourceandthedetector,usingtheparametersgiven
higher than in the surrounding normal tissues. in Table 1. We assume the fluorophore concentra-
Normalizedamplitudesandphaseshiftsofthefluo- tioncontrast,N @N 55,inthecalculationandfind
2 1
rescent DPDW versus fluorophore contrast N @N thatwhen0.2#t @t #1.8theamplitudeandphase
2 1 2 1
are calculated for objects of three different sizes by provide signals above the defined noise level for the
usingtheparametersgiveninTable1. Theobjects detectionofanobjectof0.5-cmradius.
are always centered between the source and the If we assume that all the parameters are known
detector. Results are shown in Fig. 6. We find except the fluorophore concentration and lifetime
that,ifoursystemislimitedby0.1%amplitudenoise insidetheobject,e.g.,N andt ,wecanusethedata
2 2
and0.1°phasenoise,acontrast,N2@N1,5,givesus at multiple modulation frequencies 10–500 MHz in
differentiable fluorescence signals for an object of 50-MHz steps, added with 2% amplitude noise and
0.5-cmradius. 1° phase noise2 to fit for the concentration and the
The fluorophore lifetime can be altered by the lifetime. In this case we find that N and t ob-
2 2
environment.27 This is another possible indicator tained from the fitting procedure are close to their
for tumor detection.28 If the fluorophore lifetime realvalues,i.e.,N fit510.50860.0152µMandt fit5
2 2
insidetheobject1t22increases,thefluorescentDPDW 11.53060.0602 ns where the real concentration and
amplitude decreases and the phase shift resulting lifetimeareN 50.5µMandt 51.5ns. Theerrors
2 2
fromt saturatesforvt :1. If,ontheotherhand, in N fit and tfit are the standard errors for six
2 2 2 f
t decreases and vt 9 1, the amplitude and the independentfittings. Inthecalculationaspherical
2 2
phase are no longer sensitive to t . We see, how-
2
ever, that a regime exists wherein the change of t
2
Fig.7. FluorescentDPDWversusafluorophorelifetimechange
1a2 1b2 in a heterogeneous system. An object is centered between the
Fig.5. Contourplotsinaplanethatcontainsthesourceandthe sourceandthedetectorforasource–detectorseparation,r 56
sd
objectillustratingtherequiredphaseandamplitudeprecisionto cm. Weassumethatthelifetimeoutsidethesphereist 51ns
1
detectandlocalizeanobjectof0.6-cmradius. 1a2Contoursofthe and that the fluorophore concentration contrast, N2@N1, is 5.
normalized amplitude. 1b2 Contours of the phase shift. The TheparametersaregiveninTable1. Theamplitudeisnormal-
optical properties are given in Table 1. Fluorophore lifetimes izedbytheamplitudeinahomogeneoussystem,andthephase
insideandoutsidethesphereareassumedtobeequal1t15t251 shiftiswithrespecttothephaseinahomogeneoussystem. The
ns2,andthefluorophoreconcentrationcontrastisassumedtobe5. nosielevelisindicatedbythedashed–dottedcurves.
3754 APPLIEDOPTICS @ Vol.35,No.19 @ 1July1996
object of 0.6-cm radius is centered between the tude for the excitation DPDW is 0f @f 0, and the
1 10
source and the detector, and the source–detector phaseshiftisthephasedifferencebetweenthesetwo
separationis6cm. Backgroundfluorophoreconcen- DPDW’s3seeEqs.172and1624. Thenormalizedampli-
tration and lifetime are assumed to be N 5 0.1 µM tude and phase shift for the fluorescent DPDW are
1
andt 51.0ns,respectively. Theopticalproperties the same as in Subsection 3.B. We find that for
1
are given in Table 1. Note that in practice the smaller fluorophore absorption cross sections, e.g.,
fittingproceduretoobtainthefluorophoreconcentra- s#63104cm21M21,thefluorescentDPDWgivesa
tion and lifetime inside the object is more difficult, better sensitivity, whereas the excitation DPDW
because there are more unknown parameters such gives a better sensitivity for bigger fluorophore ab-
as the optical properties inside the object and the sorptioncrosssections,e.g.,s$63104cm21M21.
sizeandthepositionoftheobject. Thusthefitting Inthediscussionabove,weassumedthatthenoise
procedure above for heterogeneous media provides levels for the excitation DPDW and fluorescent
best-caseestimates. DPDWarecomparable1e.g.,1%amplitudenoiseand
0.5°phasenoise2andthatthenoiseismainlydueto
such factors as the source and detector positional
C. ComparisonofAbsorptionandFluorescence errors rather than the shot noise. For a 0.1-mW
SensitivitiesofFluorophoresinHeterogeneousMedia light source the excitation DPDW fluence rate is
Fluorophores increase the local absorption coeffi- ,1011 photons@1cm2 s2 at the detector for a 6-cm
source–detector separation in a homogeneous me-
cient by an amount sN as discussed in Section 2.
t
diumorinaheterogeneousmediumwithanobjectof
Inaturbidsystem,andinthepresenceofaspherical
,0.5-cmradiuscenteredbetweenthesourceandthe
object, the excitation DPDW amplitude 3Eqs. 172 and
detector. The fluorescent DPDW fluence rate in
1824 decreases exponentially as the absorption coeffi-
thiscaseislowerthantheexcitationDPDWfluence
cientoftheobjectincreases. Anincreaseinfluoro-
rate by ,2–3 orders of magnitude when reasonable
phore absorption cross section 1s2 will thus enhance
concentrationsandfluorophoreabsorptioncrosssec-
the total absorption coefficient. As for the fluores-
tionsareused,e..g,sN ,µ c. Evenwiththelossof
centDPDW,anincreaseinsintroducestwocompet- t a
2 orders of magnitude of photon-fluence rate result-
ingeffects: TheamplitudeoftheexcitationDPDW
ingfromthedetectionareaandtheopticalcoupling,
that is related to the fluorescence source terms 3Eq.
we find that our detection of heterogeneities is still
1162 and 11724 decreases exponentially as s increases,
notshotnoiselimitedineithercase.
but the fluorescent DPDW amplitude increases lin-
early as s increases. The phase of the excitation
DPDW and fluorescent DPDW always increases
4. Conclusions
when s increases. Our calculations of the normal-
Wehavepresentedanalyticsolutionsforfluorescent
izedamplitudeandphaseshiftoftheexcitationand
diffuse photon density waves in homogeneous and
fluorescent DPDW’s versus fluorophore absorption
heterogeneous turbid media. Using these results
cross section are shown in Fig. 8. Here we use the
and considering a simplified heterogeneous system
parameters 1except the absorption cross section s2
containingasinglesphericalobject,weconcludethat
given in Table 1. The fluorophore concentration
contrastisN @N 55,andtheobjectradiusis0.5cm. approximatelyfivefoldfluorophoreconcentrationcon-
1 2
trast and lifetime variation of a factor of 0.2–1.8
Fluorophorelifetimesinsideandoutsideoftheobject
provideanobservabledifferentialsignaltodetectan
aretakentobeequal11ns2. Thenormalizedampli-
object of 0.5-cm radius centered between the source
and the detector with r 5 6 cm. This frequency-
sd
domain analysis can easily be applied to time-
domain experiments by Fourier transformation of
the time-domain data. Tomographic image recon-
struction techniques can also be used for fluores-
cencelifetimeimaginginturbidmedia.29
AppendixA. BasicRelationsUsedintheDerivationof
AnalyticSolutions
The Green’s function can be expanded in terms of
sphericalBesselfunctionsandsphericalharmonics:
Fig. 8. Comparison of absorption and fluorescence of fluoro-
phores in a heterogeneous medium. An object is centered be-
exp1ik0r 2r 02
tweenthesourceandthedetectorforrsd56cm. Weassumethat G1r ,r ,k25 1 2 5iko j1kr 2
thelifetimesoutsideandinsidethesphereareequal1t15t251 1 2 4p0r 2r 0 l ,
1 2 lm
ns2 and that the fluorophore concentration contrast, N2@N1, is
5. The parameters are given in Table 1. The amplitude is
3h1121kr 2Y *1V 2Y 1V 2, 1A12
normalized by the corresponding amplitude in a homogeneous l . lm 1 lm 2
system,andthephaseshiftiswithrespecttothecorresponding
phaseinahomogeneoussystem. wherer 5min1r ,r 2,r 5max1r ,r 2. Theorthogo-
, 1 2 . 1 2
1July1996 @ Vol.35,No.19 @ APPLIEDOPTICS 3755