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1896 J. Opt. Soc. Am. A/Vol. 26, No. 8 /August 2009 Alexander A. Kokhanovsky

Radiative properties of optically thickfluorescent turbid media

Alexander A. Kokhanovsky

Institute of Environmental Physics University of Bremen, D-28334 Bremen, Germany(alexk@iup.physik.uni-bremen.de)

Received January 21, 2009; revised April 17, 2009; accepted June 15, 2009;posted July 6, 2009 (Doc. ID 106520); published July 30, 2009

In this paper simple analytical equations for the reflection and transmission coefficients of fluorescent turbidmedia are given. The case of weakly absorbing optically thick media is considered (e.g., paper, textiles, tissues).The calculations are performed in the framework of the two-flux approximation for finite layers under mono-chromatic illumination conditions. The relationships of Kubelka–Munk parameters to the true absorption andtransport extinction coefficients of fluorescent turbid media are derived. The results can be used for the devel-opment of various optimization procedures in the paper and textile industries and also in the area of fluores-cence spectroscopy of turbid media. © 2009 Optical Society of America

OCIS codes: 170.1610, 260.2510, 000.3860, 260.1560.

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. INTRODUCTIONluorescent dyes are used in many applications includinghe enhancement of the brightness of turbid media suchs, e.g., paper and textiles in the visible [1–4]. The basicdea of the technique is quite simple. Indeed, light ab-orbed by a fluorescent dye in the ultraviolet part of thelectromagnetic spectrum can be re-emitted in the visibleeading to a brighter material as compared to the case ofhe same material but without application of fluorescentgents. Fluorescent dyes are also intensively used inedical diagnosis of human tissues [5,6]. Therefore, it is

f great interest to understand radiative transport in tur-id media taking account of the effects of fluorescence.everal models based on the study of the radiative trans-

er equation are available [6–8]. Monte Carlo methodsave also become popular in recent years [9–11].However, for technological applications analytical solu-

ions are of great importance. The analytical results ob-ained are only approximately valid, but they can be useds a guide for the improvement of technological processesn a fast and reliable way. The classical example is, for in-tance, Kubelka–Munk theory [12] (KMT). Although thisheory is approximate, it nevertheless correctly captureshe dependence of light reflectance on the geometricalhickness of a medium (e.g., paper and fabrics). KMT re-ies on the use of effective scattering s and absorption koefficients. These coefficients can also be related to localptical characteristics of turbid media such as the turbidedia absorption coefficient �abs and the transport extinc-

ion coefficient �exttr =�ext�1−g�, where �ext is the extinction

oefficient and g is the asymmetry parameter [13]. It fol-ows (see Appendix A) that k=2�abs and s=0.75�ext

tr . Gen-ralization of KMT in the case of fluorescent semi-infiniteedia was performed about 50 years ago [1]. Generaliza-

ions in the case of finite layers exist as well. In particu-ar, Fukshansky and Kazarinova [3] calculated the fluo-escence signal for the case of polychromatic illumination

1084-7529/09/081896-5/$15.00 © 2

onditions. The resulting equations appear quite complexnd require numerical calculations.The task of this paper is the generalization of KMT for

he case of monochromatic illumination of a sample by theoublet of lines �1 and �2 with the detection of the scat-ered signal at the wavelength �2��1. It is assumed thathe wavelength �2 is positioned outside the absorptionand of the fluorescent dye (e.g., in the visible). Clearly,he dependence of reflectance on �2 is governed by the fol-owing processes: (1) multiple light scattering at theavelength �1, (2) fluorescence (transition �1→�2), and

3) multiple light scattering at the wavelength �2 of lightrom both lines of the doublet.

. THEORYsing the general Kubelka–Munk approach, we assume

hat the incident diffused monochromatic light with in-ensity I0 illuminates the upper boundary of a scatteringayer �x=0� and that there is no light incident on the layert the bottom �x=L�. Here L is the geometrical thicknessf the scattering layer, x is the vertical coordinate, andhe coordinate axis OX is directed downward. These state-ents can be written mathematically as follows:

I�x = 0� = I0, J�x = L� = 0, �1�

here I is the downward flux and J is the upward flux.e can also introduce the diffuse reflection r and trans-ission t coefficients [14,15]:

r = J�0�/I0, t = I�L�/I0. �2�

he downward flux at the detection wavelength �2 is at-enuated by the extinction coefficient �, and this flux isnhanced due fluorescence effects, which can be describedy the function F�x�. The enhancement of the downwardux due to the backscattering of the upward flux shouldlso be taken into account. We assume that the corre-

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Alexander A. Kokhanovsky Vol. 26, No. 8 /August 2009 /J. Opt. Soc. Am. A 1897

ponding effective scattering coefficient at the detectionavelength �2 is equal to s. Then the difference k=�−sives the absorption coefficient at the same wavelength.hese statements can be written mathematically as fol-

ows:

I� = − εI + sJ + F�x�, �3�

here I� is the derivative of I with respect to x. Similarly,e obtain for the derivative of the upward light flux J

− J� = − εJ + sI + F�x�. �4�

he differentiation of the last equation gives

− J� = − εJ� + sI� + F��x�, �5�

here we assumed that the local optical characteristics ofhe medium do not depend on x. Therefore, one derives

J� − �2J = ��x�, �6�

here

� = �ε2 − s2, �7�

��x� = − �ε + s�F�x� − F��x�. �8�

he general solution of this equation can be presented assum of the solution of the homogeneous equation Jh

with ��x�=0] and the particular solution Jp. That is, weave

J = Jh + Jp. �9�

he solution of the homogeneous equation with boundaryonditions specified above can be written as follows:

Jh = c1 exp�− �x� + c2 exp��x�, �10�

here arbitrary coefficients c1 and c2 will be obtained at aater stage of the derivations. The particular solution Jpan be found if the analytical approximation for the func-ion F�x� is derived. We will assume [3,16,17] that

F�x� =1

2k1i1�x��12. �11�

ere k1 is the absorption coefficient at the wavelength �1,here the quantum transition (fluorescence) from theavelength �1 to the wavelength �2 with the dimension-

ess fluorescence strength (FS) �12 takes place. The func-ion i1�x� in Eq. (11) describes the average intensity ofultiply scattered light at the wavelength �1. The multi-

lier 1/2 appears as a result of the isotropy of the fluores-ence process. The average intensity i1�x� can be calcu-ated in the framework of the KMT:

i1�x� = ae−�x + be�x, �12�

here [2]

a =�1 + r1�I0

1 − r12 e−2�L

, b = −�1 + r1�r1e−2�LI0

1 − r12 e−2�L

, �13�

=��12−s1

2, and we assumed that the intensity of the inci-ent light is the same at wavelengths �1 and �2. The dif-erence in I0 at both wavelengths can be easily accountedor by multiplying FS by the ratio of incident intensities

n final equations. Here �1 and s1=�1−k1 are effective ex-inction and scattering coefficients, respectively, at theavelength �1, and k1 is the corresponding absorption co-fficient. The value of r1 is the reflectance of a semi-nfinite layer at the wavelength �1, which is given by theollowing expression in the framework of KMT [14,15,17]see Appendix A):

r1 =s1

ε1 + �. �14�

One derives, using Eqs. (8), (11), and (12),

��x� = Ae−�x + Be�x, �15�

here

A = −a

2k1�12�ε + s − ��, �16�

B = −b

2k1�12�ε + s + ��. �17�

herefore, the differential equation for the particular so-ution can be written in the following form:

Jp� − �2Jp = Ae−�x + Be�x. �18�

quation (18) can be solved using, e.g., the operatorethod [18]. That is, it follows that

Jp =1

�D2 − �2��Ae−�x + Be�x�, �19�

here the operator 1/�D2−�2� has the property [18]

1

�D2 − �2��Ae−�x� =

Ae−�x

�2 − �2 . �20�

herefore, one derives

Jp =1

�2 − �2 �Ae−�x + Be�x� �21�

nd we have, finally,

J�x� = c1 exp�− �x� + c2 exp��x� +Ae−�x + Be�x

�2 − �2 �22�

or the upward flux at any level in a fluorescent turbidedium.The downward flux is given by [see Eq. (4)]

I = s−1�εJ − J� − F�, �23�

r, after substitution of relevant analytical expressionsiven above [see Eqs. (11), (12), and (22)] and also J� de-ived from Eq. (22),

I�x� = D1e−�x + D2e�x + D3e−�x + D4e�x, �24�

here

D1 =�ε + ��c1

s, D2 =

�ε − ��c2

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1898 J. Opt. Soc. Am. A/Vol. 26, No. 8 /August 2009 Alexander A. Kokhanovsky

D3 =�ε + ��A

��2 − �2�s−

k1�12a

2s, D4 =

�ε − ��B

��2 − �2�s−

k1�12b

2s.

�25�

t follows that both upward �J� and downward �I� lightuxes are represented by the linear combinations of expo-ential functions with positive and negative argumentsexp�−Mx� and exp�Mx�, where M is given either by ���1�r ���2�]. All involved constants (except c1 and c2) are de-ned in expressions given above. Let us find integrationonstants c1 ,c2 now. For this we use boundary conditionsiven by Eq. (1). The result of straightforward algebraicalculations is

c1 = −q1 + p2c2

p1, c2 =

q1p1−1e−2�L − q2e−2�L

1 − p2p1−1e−2�L

, �26�

here

q1 =A�ε + �� + B�ε − ��

�2 − �2 −1

2k1�12�a + b� − sI0,

q2 =Ae−�L + Be�L

�2 − �2 , �27�

nd

p1 = ε + �, p2 = ε − �. �28�

his completes the solution. Therefore, we have obtainedhe downward and upward light fluxes inside the fluores-ence turbid medium and also at the boundaries. In par-icular, using Eqs. (2), (22), and (24) one can derive theollowing analytical results for the reflection and trans-ission coefficients for a finite layer of a fluorescent me-

ium:

r = c1 + c2 +A + B

�2 − �2 , �29�

t = D1e−�L + D2e�L + D3e−�L + D4e�L, �30�

here we assumed that I0=1. These analytical equationsan be used to study reflection and transmission of lighty fluorescent optically thick turbid media, depending onheir local optical characteristics (see Appendix A) andeometrical thickness. Also, a number of inverse problemsn the spirit of KMT can be formulated and solved.

The correctness of derived equations can be checkedtudying the well-known case of semi-infinite media1,2,17]. First of all, it follows from the equations givenbove that D2→0,D4→0, t→0 as �L→ ,�L→, as ithould be in this case. Also one derives, for semi-infiniteedia,

b = 0, B = 0, c2 = 0, a = 1 + r1, �31�

nd

A = −1

2�1 + r1��ε + s − ��k1�2, �32�

c1 = r2 +1 + r1

2�ε + ��k1�2 −

�ε + ��A

�ε + ����2 − �2�, �33�

here we used Eq. (14) (at the wavelength �2). Also it fol-ows that

r = c1 +A

�2 − �2 �34�

nd, therefore,

r = r2 +1 + r1

2�ε + ��k1�12 −

A

�ε + ���� − ��. �35�

sing Eq. (32), one derives, finally,

r = r2 +k1�12�1 + r1��1 + r2�

2�� + ��, �36�

here r2=s��+��−1 is the reflectance of a semi-infiniteayer at the observation wavelength �2 in the absence ofuorescence. Equation (36) for a semi-infinite mediumas been given by a number of authors [1,2,17], whichonfirms our calculations.

. CONCLUSIONShe two-flux theory of the fluorescence effects in turbidedia as presented by Morton [1] has been generalized to

ccount for the finite thickness of turbid media. The de-ived analytical equations are simple and can be used in aariety technological applications. It is assumed that theedium is illuminated by a doublet of lines with the

horter wavelength positioned in the absorption band ofhe fluorescent dye. The case of polychromatic illumina-ion condition is not considered to keep the equations asimple as possible. However, the generalization istraightforward.

PPENDIX A: KUBELKA–MUNKARAMETERS S AND K AND THEIRELATIONSHIP TO THE TRANSPORTXTINCTION AND ABSORPTIONOEFFICIENTS

t was shown [19,20] that asymptotic equations of the ra-iative transfer theory derived under the condition thathe optical thickness �=�extL→ and the single-cattering albedo �0=1−�abs /�ext→1 can be presented inform similar to that of the Kubelka–Munk equations.

hat is, it follows (see also Table 1) [20] that

ras =ras�1 − e−2p�

1 − l2e−2p , tas =�e−p

1 − l2e−2p , �A1�

rKM =rKM�1 − e−2p̄�

1 − rKM2 e−2p̄

, tKM =�1 − rKM

2 �e−p̄

1 − rKM2 e−2p̄

. �A2�

quations (A1) are derived using the asymptotic theory,nd Eqs. (A2) follow from KMT. Here subscripts “as” andKM” mean asymptotic theory and KMT, respectively. Thearameters in Eq. (A1) depend on the local optical char-

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Alexander A. Kokhanovsky Vol. 26, No. 8 /August 2009 /J. Opt. Soc. Am. A 1899

cteristics of the medium, namely, p=��, and one derives20–22] as �0→1,

� = �3�1 − �0��1 − g�, ras = 1 − 4�/�3�1 − g��,

l = ras, � = 1 − ras2 , = 1. �A3�

t follows for the Kubelka–Munk parameters that

rKM = 1 +k

s−�2

k

s+

k2

s2 , p̄ = �k2 + 2ksL. �A4�

n important point is that the asymptotic equations givenbove are derived from the exact radiative transfer equa-ion in the limit �0→1,�→. Therefore, all parameters inhe corresponding equations are defined via the opticalhickness �=�extL, �0, and the phase function. Theubelka–Munk parameters are empirical ones and can beetermined experimentally. However, they can not be cal-ulated from first principles (e.g., from Mie theory [13]).

The Kubelka–Munk equations become identical tohose of asymptotic theory in the limit �→ ,�0→1, ifKM=ras, p= p̄ or

1 +k

s−�2

k

s+

k2

s2 = 1 −4�

3�1 − g�, �k2 + 2ks = �kext.

�A5�

herefore, it follows that

�2k

s=

4�

3�1 − g�, �2ks = �kext, �A6�

here we accounted for the fact that k /s�1. So we have

Table 1. Radiative Characteristics of ThFramew

Radiative characteristic

Spherical albedo of a semi-infinite layer

Spherical albedo of a finite layer

Transmission coefficient

Downward light intensity

Upward light intensity

Average intensity

a�p̄=�k2+2ksL, v=�k2+2ksx, L is the thickness of the layer and x is the vertica

k

s=

8�1 − �0�

3�1 − g�, ks =

3�1 − �0��1 − g�

2kext

2 �A7�

r

k = 2�abs, s = 0.75�1 − g��ext. �A8�

ote that it follows in our approximation that ��0→1�ext��sca, where �sca is the scattering coefficient (e.g., de-ned in Mie theory). Then Eqs. (A8) transform to the fol-

owing formulas:

k = 2�abs, s = 0.75�1 − g��sca, �A9�

hich were suggested earlier [23] on the basis of numeri-al calculations. The relationships (A8) enable the calcu-ation of light scattering characteristics derived in thisaper (e.g., reflection and transmission coefficients) fromhe first principles via the Mie theory.

CKNOWLEDGMENTShe author thanks I. Hopkinson and R. Treloar for impor-

ant discussions with respect to radiative transport inuorescent turbid media.

EFERENCES1. T. H. Morton, “Fluorescent brightening agents on textiles:

elementary optical theory and its practical applications,” J.Soc. Dyers Colour. 79, 238–242 (1959).

2. E. Allen, “Fluorescent white dyes: calculation offluorescence from reflectivity values,” J. Opt. Soc. Am. 54,506–515 (1964).

3. L. Fukshansky and N. Kazarinova, “Extension of theKubelka-Munk theory of light propagation in intenselyscattering materials to fluorescent media,” J. Opt. Soc. Am.70, 1101–1111 (1980).

4. T. Shakespeare, “Colorant modeling for on-line paper

trongly Light-Scattering Layers in thef KMTa

Equation

r=s

ε+��1+m−�m2+2m,

m=ks

, ε=k+s, �=�ε2−s2

r=r�1−e−2p̄�

1−r2e−2p̄

t=�1−r

2�e−p̄

1−r2e−2p̄

I=�e−v−r

2ev−2p̄�I0

1−r2e−2p̄

J=�e−v−ev−2p̄�rI0

1−r2e−2p̄

i=I+J=�e−v−rev−2p̄��1+r�I0

1−r2e−2p̄

ate�.

ick Sork o

l coordin

1

1

1

1

1

1

1

1

1

12

2

2

2

1900 J. Opt. Soc. Am. A/Vol. 26, No. 8 /August 2009 Alexander A. Kokhanovsky

coloring: evaluations of models and an extension toKubelka-Munk model,” Ph.D. thesis (Tampere Universityof Technology, 2000).

5. J. Wu, M. S. Feld, and R. P. Rava, “Analytical model forextracting intrinsic fluorescence in turbid media,” Appl.Opt. 32, 3585–3595 (1993).

6. A. D. Klose and A. H. Hielscher, “Fluorescence tomographywith simulated data based on the equation of radiativetransfer,” Opt. Lett. 28, 1019–1021 (2003).

7. H. R. Gordon, “Diffuse reflectance of the ocean: the theoryof its augmentation by chlorophyll a fluorescence at685 nm,” Appl. Opt. 18, 5356–5367 (1979).

8. A. D. Klose, “Radiative transfer of luminescence light inbiological tissue,” in Light Scattering Reviews, A. A.Kokhanovsky, ed. (Springer-Praxis, 2009), Vol. 4, 347–405.

9. M. G. I. Muller, I. Georgakoudi, Q. Zhang, J. Wu, and M. S.Feld, “Intrinsic fluorescence spectroscopy in turbid media:disentangling effects of scattering and absorption,” Appl.Opt. 40, 4633–4646 (2001).

0. Q. Zhang, M. G. Muller, J. Wu, and M. S. Feld, “Turbidity-free fluorescence spectroscopy of biological tissue,” Opt.Lett. 25, 1451–1453 (2000).

1. J. Swartling, A. Pifferi, A. M. K. Enejder, and S. Anderson-Engels, “Accelerated Monte Carlo models to simulatefluorescence spectra from layered tissues,” J. Opt. Soc. Am.20, 714–727 (2003).

2. P. Kubelka and F. Munk, “Ein Beitrag Zur Optik derFarbanstriche,” Z. fur Techn. Physik 12, 593–601 (1931).

3. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

4. P. Kubelka, “New contributions to the optics of intenselylight-scattering materials. Part I,” J. Opt. Soc. Am. 38,448–457 (1948).

5. P. Kubelka, “New contributions to the optics of intenselylight-scattering materials. Part II: Nonhomogeneouslayers,” J. Opt. Soc. Am. 44, 330–335 (1954).

6. T. Shakespeare and J. Shakespeare, “A fluorescentextension to the Kubelka-Munk model,” Color Res. Appl.28, 4–14 (2002).

7. S. D. Howison and R. J. Lawrence, “Fluorescent transfer oflight in dyed materials,” SIAM J. Appl. Math. 53, 447–458(1993).

8. L. E. Elsgolts, Differential Equations (Gordon & Breach,1961).

9. A. A. Kokhanovsky, Cloud Optics (Springer, 2006).0. A. A. Kokhanovsky, “Physical interpretation and accuracy

of the Kubelka-Munk theory,” J. Phys. D 40, 2210–2216(2007).

1. A. A. Kokhanovsky, Light Scattering Media Optics(Springer-Praxis, 2004).

2. A. A. Kokhanovsky, “Asymptotic radiative transfer,” inLight Scattering Reviews, A. A. Kokhanovsky, ed.(Springer-Praxis, 2006), Vol. 1, pp. 253–290.

3. P. S. Mudgett and L. W. Richards, “Multiple scatteringcalculations for technology,” Appl. Opt. 10, 1485–1502(1971).