Penetration depth of low-coherence enhanced backscattered light in. sub-diffusion regime

Transcription

1 Penetration depth of low-coherence enhanced backcattered light in ub-diffuion regime Hariharan Subramanian, Prabhakar Pradhan, Young L. Kim and Vadim Backman Biomedical Engineering Department, Northwetern Univerity, Evanton, IL Abtract: The mechanim of photon propagation in random media in the diffuive multiple cattering regime have been previouly tudied uing diffuion approximation. However, imilar undertanding in the low-order (ub-diffuion) cattering regime i not colete due to difficultie in tracking photon that undergo very few cattering event. Recent development in low-coherence enhanced backcattering (LEBS) overcome thee difficultie and enable probing photon that travel very hort ditance and undergo only a few cattering event. In LEBS, enhanced backcattering i oberved under illumination with patial coherence length L c le than the cattering mean free path l. In order to undertand the mechanim of photon propagation in LEBS in the ubdiffuion regime, it i ierative to develop analytical and numerical model that decribe the tatitical propertie of photon trajectorie. Here we derive the probability ditribution of penetration depth of LEBS photon and report Monte Carlo numerical imulation to upport our analytical reult. Our reult demontrate that, urpriingly, the tranport of photon that undergo low-order cattering event ha only weak dependence on the optical propertie of the medium ( l and aniotropy factor g) and trong dependence on the patial coherence length of illumination, L c relative to thoe in the diffuion regime. More iortantly, thee low order cattering photon typically penetrate le than l into the medium due to low patial coherence length of illumination and their penetration depth i proportional to the one-third power of the coherence 3 volume (i.e. l π L ] 1/ ). [ PACS number: 4.6.Be, e, Cc 1

2 I. INTRODUCTION Mot biological tiue are multilayered ytem that require depth-elective meaurement to obtain clinically ueful information [1-6]. Currently, a number of optical technique baed on backcattered light are under development for uch depthelective tiue characteriation or imaging. In order to exploit an optical technique in a biomedical etting, a proper knowledge of the photon trajectorie within the ale before being backcattered i eential. Thi information can be characteried by the ditribution of photon at different depth, herein called penetration depth ditribution, which provide information about the probability that a photon penetrate a certain depth before being detected. The penetration depth ditribution, in turn, can be conveniently characteried by the effective penetration depth (depth correponding to the peak of the probability ditribution curve). Several group have ued numerical and analytical model to tudy the penetration depth of backcattering photon in tiue in a multiple cattering medium [7-10]. In particular, Wei et al. ued lattice random walk model to obtain the tatitical propertie of the penetration depth of photon emitted from a bulk tiue [7, 8]. The depth ditribution of photon in a random cattering medium with the thickne of approximately 10 tranport mean free path ( l ) wa calculated by Durian [9]. Recently, Zaccanti et al. [10] derived analytical expreion for the time reolved probability of photon penetrating a certain depth in a diffuive medium, before being reemitted. Although the penetration depth of photon ha been well tudied in a diffuive multiple cattering regime, imilar undertanding in the low-order cattering regime i not colete. Thi i in part due to the difficultie in collecting photon that undergo only a few cattering event. Recently, we have developed low-coherence enhanced

3 backcattering (LEBS), which patially filter longer traveling photon and collect only photon that travel very hort ditance and undergo a few order of cattering. In thi paper, we ue Monte Carlo numerical imulation to tudy the propagation of photon that contribute to LEBS; and alo report development of a correponding analytical model to decribe the penetration depth ditribution and effective penetration depth of thee photon. Enhanced backcattering (EBS, alo known a coherent backcattering) i a phenomenon in which coherent photon traveling along exact time-revered path interfere contructively to produce an enhanced intenity peak in the direction cloe to backcattering. Therefore, theoretically, the intenity of the EBS peak in the backward direction can be a high a twice the diffued background. Typically, the angular width of the EBS peak i proportional to λ l /, where λ i the wavelength of light and l i the tranport mean free path length [11, 1]. Although the EBS phenomenon ha been extenively tudied in a variety of non-biological media [13-0], the invetigation of EBS in biological tiue ha been extremely limited [1-3]. A biological tiue i a weakly cattering medium ( l >> λ ) with l ranging between 0.5 to mm. The invetigation of EBS in uch weakly cattering media ha been exceedingly difficult due to very mall width of EBS peak (e.g., w hm ~ o for l ~1 mm). On the contrary, low coherence EBS (LEBS) overcome all of the major limitation that have prevented the widepread application of EBS in tiue optic [4, 5]. The LEBS peak i obtained by combining the EBS meaurement with low patial coherence, broadband illumination. In our previou tudie we howed that low patial coherence illumination (patial coherence length L < l << l ) behave a a patial filter that dephae the conjugated time- c 3

4 revered path outide the patial coherence area and thu reject longer path length [4, 5]. Thi give rie to EBS peak that are broadened by more than two order of magnitude coared to the width of conventional EBS peak, facilitating their experimental meaurement [4, 5]. We have alo previouly hown that LEBS open up the feaibility of tudying tiue optical propertie at a elected depth: LEBS can electively probe hort traveling photon from the top tiue layer ( μm, e.g. mucoa and epithelium,) by rejecting long traveling photon from the underlying (tromal/connective) tiue. Finally, we have hown that LEBS pectrocopy can reliably identify the earliet precancerou alteration in the colon and pancrea [4, 5] uing LEBS propertie uch a it pectral and angular ditribution. A dicued above, in LEBS, low-patial coherence illumination act a a patial filter that reject longer traveling photon. Therefore, the penetration depth of LEBS photon can be controlled externally by changing the patial coherence length of illumination, L c. In order to increae the enitivity of LEBS meaurement to pecific tiue depth it i iortant to know the relationhip between Lc and the depth of penetration of LEBS photon. The penetration depth in turn i characteried by tudying the penetration depth ditribution of the hort traveling LEBS photon p() (where, p() i the probability of photon to penetrate a certain depth before being detected), and their effective penetration depth ( ). Unlike the long traveling photon, the hort traveling LEBS photon ( r < l << l, where r i the radial ditance at which photon emerge) typically undergo very few cattering event; hence, the numerical model and analytical expreion addreed within the diffuion approximation for r >> l > l cannot be ued to tudy the mechanim of photon propagation in LEBS. Therefore, we ued numerical 4

5 Monte Carlo (MC) imulation to model low-order cattered photon in order to tudy the mechanim of photon propagation a a function of coherence length L c, aniotropy coefficient g and cattering mean free path length l. We alo developed an analytical model for p() and of LEBS photon, i.e. photon exiting at radial ditance r l << l < with L < l << l. c In order to tudy the penetration depth of LEBS photon, we perform the following: Firt, we ue the numerical Monte Carlo imulation to calculate the penetration depth ditribution p() and the effective penetration depth ( ) of the photon that form the LEBS peak. We tudy thi for different coherence length (L c ) of illumination, and for media with different cattering mean free path (l ) and aniotropie (g). Second, we ue a double cattering analytical model to develop analytical expreion for p() and, and how that the analytical expreion of p() and coare well with the correponding numerical reult for the parameter regime when L < l << l. Finally, we demontrate that both p() and of the exiting photon in c thi regime ( L < l << l ) exhibit a priori urpriing behavior, that i, only weak c dependence on optical propertie (l and g) and trong dependence on Lc relative to diffuive regime. Thi reult i in contrary to the general undertanding of the propertie of p() and oberved in the diffuive regime. II. METHODS A. Numerical model uing Monte Carlo imulation Monte Carlo (MC) imulation have been commonly ued to model photon tranport in biological media [6, 7], and alo to model EBS phenomenon indirectly [19, 5

6 8-30]. A it i challenging to imulate the time-reveral of photon and it interference effect explicitly uing MC method, EBS angular profile I (θ ) are generally calculated by uing the fundamental relationhip between I (θ ) and the radial ditribution p(r) with p(r) being the probability of photon to emerge from the urface at a radial ditance r. That i, I (θ ) i an integral tranform of the radial intenity EBS ditribution of p(r), where, in turn, p(r) i obtained from the MC imulation [1]: EBS EBS I EBS r ( q ) rp ( r) exp( iπr in θ / λ dr ) where r i the radial ditance at which the photon emerge and q r (1) i the projection of the wave vector onto the plane orthogonal to the backward direction. In order to explore the depth of penetration of LEBS photon and it dependence on the optical propertie of a medium, we ue a MC imulation method developed by Wang et al. [6]. Although, the propagation of photon and it dependence on optical propertie have been well tudied in the diffuion regime uing MC imulation, here we ue MC to tudy the low order cattering, particularly when the photon undergo minimum of double cattering event and then exit within a narrow radial ditance ( r < l << l ). The double cattering i of ignificance becaue in EBS the minimum number of cattering event i double cattering. The ingle cattering event contribute only to the incoherent baeline and not to the EBS peak formation. We have recently demontrated a direct experimental evidence that double cattering i the minimal cattering event neceary to generate an EBS peak in a dicrete random medium [31]. In thi tudy, we alo howed that LEBS iolate double cattering from higher order 6

7 cattering when L c i on the order of the cattering mean free path l of light in the medium ( l = l (1 g) ). Decription of the MC imulation i given in detail elewhere [6, 7]. In brief, we launch an infinitely narrow photon beam coniting of photon packet into a homogeneou diordered ingle layered medium with thickne much greater than the patial extent of the photon ditribution (thickne of the medium = 50mm). We vary the cattering mean free path l between μm and the aniotropy factor g between We aume aborption to be negligible (aborption length, l a = 1000 cm). We record the trajectorie of all photon that undergo two and higher order cattering event, and exit the ale at an angle < 3 from the direction of backcattering. We obtain the penetration depth ditribution of photon in the axial direction (p()) uing a twodimenional grid ytem whoe grid eparation in the r and direction were δ r = μm and δ = 5 μm, repectively, with the total number of grid N r = N = Furthermore, to account for the number of cattering event ( n ), we etup a eparate two-dimenional grid ytem with δ r = μm, interval between cattering eventδ = 1, total number of grid N r = 1000 and the total number of cattering event N = 500, repectively. Therefore, we can obtain the cattering ditribution p(n ), penetration depth ditribution p() and the effective penetration depth a a function of radial ditance r. We alo calculate p(n ), p(), and a a function of patial coherence length L c, by incorporating the effect of low patial coherence illumination on EBS in the numerical model. In thi cae, the angular profile of LEBS I (θ ) can be expreed a [5], LEBS n n I LEBS ( θ ) = C( r) rp( r)exp( i π rθ λ) dr, () 0 7

8 where C r) = J ( r / L ) /( r / L ) i the degree of patial coherence of illumination with ( 1 c c the firt order Beel function J 1 [3]. A C (r) i a decay function of r, it act a a patial filter allowing only photon emerging within it effective coherence area (~ L c ) to contribute to p(r). Therefore, we can obtain p() and a a function of r or L c. The following ection dicue in detail the derivation of the analytical expreion of p() and from a double cattering analytical model, and it coarion with the reult of our numerical imulation. B. Analytical derivation of p(), and We derive the expreion for p() and of photon that contribute to LEBS peak on the bai of a double-cattering analytical model of backcattering photon. Previou experimental tudie [31] and the numerical reult of the Monte Carlo imulation, which will be dicued in detail below (Section III.A), demontrate that LEBS peak from a low patial coherence illumination are mainly generated by the photon that predominantly undergo double cattering event. Hence, we ue the double cattering analytical model to derive the expreion for p() and and to verify our reult from the numerical imulation. 1. p() and a a function of radial ditance r: The probability of radial ditribution of photon exiting a medium p(r) due to double cattering event can be expreed a [33], p( r) = r 0 0 d' d' ' + ( ' ' ' ) exp[ μ ( r + ( ' ' ' ) + ' + ' ')] μ F( θ) μ F( π θ), (3) 8

9 where r i the radial ditance at which photon emerge, ' and '' are the vertical ditance from the urface to the catterer, F (θ ) i the phae function of ingle cattering with θ = tan 1 ( r /( ' ' ' )), and μ 1/ l ) i the cattering coefficient. A ( chematic picture of the cattering geometry i hown in Fig. 1. In our tudy, we ue the Henyey-Greentein cattering phae function, 1 1 g F( θ ) =. (4) 4π ( 1+ g g coθ ) 3 / To obtain the expreion of p() and of a double cattering photon from Eq. (), we perform the following: We define a new variable = ' ' ' and ''' = '' + '. The coordinate ytem, in the above double cattering model, can be tranformed to a coordinate ytem, uing a Jacobian tranformation. We then approximate the penetration depth of the double cattering event a ~, a one of the cattering event occur much cloer to the urface of the medium than the other cattering event when the exit ditance r of the majority of photon are retricted due to the finite value of L c ( r L < l << l ). Indeed, Fig. 1 illutrate that in order for the photon to undergo, c double cattering event within a mall r, one of the cattering event mut occur very cloe to the urface of the medium ( ' 0) (approximation validated in Section III.B). Therefore, the double cattering expreion can be rewritten a, dd' p( r) = r + 0 Integrating over 0 '' ''' exp[ μ ( r + + ' '')] μ F( θ ) μ F( π θ ). (5) ''' in Eq. (5) we obtain, d p( r) = p( r, ) d = exp[ μ ( r + )] μ F( θ ) μ F( π θ ). (6) μ r

10 10 From Eq. (6) it follow that for a given r, the penetration depth ditribution ) ( p can be written a, ) ( ) ( )] ( exp[ 1 ) ( θ π μ θ μ μ μ + + = F F r r r p. (7) Subtituting ) / ( tan 1 r = θ, the phae function Eq. (4) can be rewritten a, 3 / ) ( + + = r g g g F π θ. (8) Becaue the phae function F i motly uniform around the backward direction, i.e. π θ ~, we approximate, 1 ), ( λ θ π F. Then the penetration depth ditribution at a given r, p( r) become, 3 / 1 1 )] ( exp[ 1 ) ( = r g g g r r r p μ π μ. (9) Eq. (9) i the depth ditribution of photon that undergo double cattering event and exit the medium in the backward direction at radial ditance l l r << <. The effective penetration depth = i the olution of the following equation. 0 ) ( = = d dp. (10) From the Eq. (9) and (10) we obtain, 0 ) (1 1 3 = g g g r r g g g r r μ μ μ (11) Solving the above cubic equation (Eq. (11)) for, we obtain the exact olution for the effective penetration depth of double cattering photon:

11 1 ( r g, μ ) = 3μ (1 g) where 3 [ B( r, g, μ ) + B ( r, g, μ ) + 4A ( r, g, μ )] / 3 4 3gr μ A( r, g, μ ) = (1 g) 4 +, and (1 g) 6 45gr μ B( r, g, μ ) = (1 g) (1 g) 4 / 3 A( r, g, μ ) 1/ 3 3 [ B( r, g, μ ) + B ( r, g, μ ) + 4A ( r, g, μ )] + 1/ 3, (1) 3μ The dependence of on r (from Eq. (1) ) for μ r 0 i approximately linear. Here we are intereted in in the regime relevant for LEBS: r / l < 1, (1 g) ( μ r) 0, and ( g) ( μ r) ~ 1. To ee the leading behavior of in thi regime, we expand the right ide of Eq. (1) in term of A and B, when A / B << 1 and obtain ( )[ ] 1/ 3 (3μ (1 g) ) B( r, g, μ ) [ l π ] 1/ 3. Thi can be re-written a: 1/ 3 g ( r g, l ) r / 3. (13) (1 g) The above equation ilie that the effective penetration depth of a LEBS photon i proportional to the (1/3) power of the volume of a virtual cylinder whoe area i formed by a circle of radiu r and height l. In the cae for biological tiue (i.e., g ~ 1), Eq. (13) / 3 r can be rewritten a,, l l where l = l (1 g). Thi provide a critical value of r in the unit of l for the double cattering regime; i.e., for cattering event dominate coared to higher order cattering event. r < l, double 11

12 . p() and a a function of patial coherence length L c : To calculate the dependence of p () and on coherence length L c, we firt weight the Eq. (9) and (1) by the coherence function C r, L ), and integrate over r : ( c r= 0 p( L ) = p( r) C ( r) dr, (14) c c L ( L c r= 0 g, μ ) = ( r g, μ ) C ( r) dr. (15) Lc Eq. (14) and (15) repreent the analytical expreion for the penetration depth ditribution, and effective penetration depth of photon that predominantly undergo double cattering event in LEBS. Under low-coherence illumination with patial coherence length L < l << l : μ r < 1 and C ( r, L ) c dr ~ dr Lc. In thi low-coherence regime, integration in Eq. (15) can be performed analytically. [ l π L ] 1/ 3 lc 1/ 3 g ( Lc g, l ) ~ ( r g, L ) C( r, Lc ) dr / 3 c 0 (1 g) (16) Eq. (16) ilie that the effective penetration depth of a LEBS photon i proportional to the 1/3 power of an effective coherence volume in a large parameter pace. For exale, at g = 0.7 and l = 100 µm, the plot of log( ) veru log( π l L c ) ha a lope of 0.8 (~1/3). c III. RESULTS AND DISCUSSIONS A. Scattering and penetration depth ditribution - numerical tudie 1

13 The probability with which a photon catter, p(n ), and the depth to which it penetrate, p(), before it exit the medium at radial ditance r l << l < i dicued in thi ection. A tated throughout thi paper, we conider a low-coherence regime: L < l << l. We performed numerical imulation uing MC (Section II.A) for media c with different optical propertie (l = μm and g = ) in order to obtain p(n ) and p(). A an illutration, here we dicu the reult obtained for a medium with l = 100 μm and g = 0.9 and 0.7. Figure (a) and (b) how p(n ) for photon that exit at r l << l < for g = 0.9 and 0.7. For r = 5 μm (r/l = 0.05), it can be clearly een that the photon predominantly undergo double cattering event (Fig. (a)). Thi can be een from a harp peak in p(n ) for n =. However, a r increae (r>5 μm) the probability of collected photon to undergo higher order cattering (n > ) increae. Typically in a medium coniting of mall particle (g << 1), photon undergo iotropic cattering and hence penetrate hallower ditance than in the medium with large aniotropy factor (g ~ 0.9). A a reult, the photon propagating in a ale with mall g undergo relatively few cattering event before exiting the ale. Thi effect can be een in Fig. (b), where the cattering ditribution p(n ) i obtained for a medium with g = 0.7. In thi cae, the hape of p(n ) a a function of n i coniderably harper than p(n ) for g = 0.9 (r = 5 μm), illutrating that the photon have higher probability of exiting the medium after undergoing double cattering event. It i alo intereting to note that for mall particle, the probability of two and three cattering event of photon are coarable at r = 50 μm. 13

14 In the cae of LEBS, the coherence area within which a photon exit a medium i controlled by the L c of the light ource. The plot of p(n ) for three different value of L c for ale with g = 0.9 and 0.7 are hown in Fig. (c) and (d). Within a narrow coherence area defined by L < l << l (e.g., L c = 5 μm), it can be een that the c majority of the photon experience double cattering while the probability of collecting photon undergoing higher order of cattering i exponentially low. However, for L c = 50 μm the probabilitie of 3 and 4 cattering event are coarable to that of double cattering. Thee reult are critical to the following dicuion a they validate our ue of the double cattering model to derive the analytical expreion for p() and in the low-coherence regime ( L < l << l ). c Figure 3 how numerical imulation of p() for two et of optical propertie (l =100 μm, g=0.9 and l =100 μm, g=0.7) at different radial ditance r ( r = 5 μm, 5 μm, 50 μm ) and patial coherence length L c ( L c = 5 μm, 5 μm, 50 μm). For r = 5 μm << l, the photon typically penetrate a hallow ditance into the medium, which i, iortantly, le than the cattering mean free path of the medium l. Alo, for a contant g and l the penetration depth of the photon increae with increae in the radial ditance r at which photon exit the medium (Fig. 3). However, when the reult of p() are coared to the medium with different optical property (g = 0.7, l = 100 μm), the change in p() i coniderably le ignificant ( < 5%). Thi indicate that p() i only weakly dependent on the optical propertie of the medium for mall radial ditance r l << l <. On the other hand, p() how a trong dependence on r, and the hape of p() vary ignificantly (> 50 %) for different r. Similarly, the penetration depth ditribution at 14

15 different L c alo how a relatively weak dependence on optical propertie, and trong dependence on L c (Fig. 3(c) and 3(d)). Thi weak dependence of penetration depth on optical propertie for photon exiting at r < << wa further verified by our, Lc l l numerical imulation for other value of l and g of the medium (data not hown). It i intereting to note that within a narrow coherence area defined by L < l << l (e.g., L c c = 5 µm), the majority of photon penetrate only to a hallow depth ( ~ 5 µm < l ). However, a L c increae, the photon have a higher probability of penetrating deeper into the medium. From thee reult, we conclude that the tiue depth that are predominantly aled by the LEBS photon can be controlled by varying the patial coherence length of illumination L c, and the reulting penetration depth of the photon i eentially inenitive to the pecific of the tiue optical propertie. Alo, the LEBS photon typically penetrate a hallow ditance which i le than the cattering mean free path l of the medium. B. Coarion of the reult of numerical imulation and analytical model Here we coare the analytical expreion of p() and a a function of r (Eq. (9) and Eq. (1)) and L c (Eq. (14) and Eq. (15)) with the correponding numerical imulation in the low-coherence regime: L < l << l. A a repreentative illutration, c the analytical and numerical reult are hown for a medium with l = 100 μm and g = 0.9. Firt we validate our hypothei tated in Section II.B1 that in the double cattering regime when r < <<, the ditance from the urface of the medium to, Lc l l 15

16 one of the catterer i negligibly mall relative to that of the other catterer, i.e. the vertical ditance to the deeper catterer i everal order greater than the other catterer (either ' << ' ' or '' << ' ). To validate thi hypothei, we ued MC imulation analogou to the one dicued in Section II.A. Thi time, however, we followed photon that undergo only ingle cattering. Fig. 4 how the plot of p() of photon that undergo ingle cattering and thoe undergoing double cattering event for r = 5 µm. It i een that the hape of p() of a ingle cattering photon i everal time harper than that of double cattering photon. Thi confirm that for r l << l <, the ditance to the firt catterer i negligibly maller than the ditance of the econd catterer which i located much deeper within the medium. That i, in order for the photon to undergo double cattering and exit within a narrow radial ditance r l << l < (e.g., retricted by L < l << l ), one of the cattering event mut occur cloe to the urface of the c medium. Hence, for a photon undergoing double cattering, the difference in vertical ditance between the two catter can be taken a the penetration depth of thi photon. The validation of thi aution will be iortant for the validation of the p() obtained uing the analytical model with the prediction of numerical imulation. Fig. 5(a) coare p() given by Eq. (9) and the one obtained by the numerical model for r = 5 μm, 5 μm and 50 μm. A een in the double cattering regime (i.e., r < << ), the prediction of the analytical model are in good agreement with, Lc l l thoe of the numerical imulation with root mean quare error (RMSE) of le than 0.5 %. Similarly, p() obtained for L c = 5 μm, 5 μm and 50 μm indicate that the analytical expreion derived from the double cattering model (Eq. (14)) can aptly decribe the ditribution obtained from the numerical model (RMSE < 0.4 %) (Fig. 5(b)). Even 16

17 though the numerical model take into account higher order cattering event, Fig. 5(a) and 5(b) clearly how that for r < <<, the analytical and numerical reult are in, Lc l l good agreement. Thi reult further confirm that for r < <<, the photon, Lc l l predominantly undergo double cattering event, and Eq. (9) and Eq. (14)can accurately model the penetration depth ditribution of the photon. Figure 6 coare of LEBS photon predicted by the numerical model and analytical expreion (Eq. (15)). Here, i obtained a a function of L c for two different media with g = 0.7, g = 0.9, and l = 100 μm. It can be een from thi plot that in the low-coherence regime ( L < l << l ), the prediction of by the analytical c double-cattering model are in good agreement with thoe of the numerical imulation for all L < l (RMSE < 5 μm). Thi good agreement i due to the fact that, a dicued c above, double cattering dominate in thi regime. Furthermore, even if a photon undergoe a higher order cattering, the condition L < l << l and backcattering light c collection retrict the majority of the backcattered photon to go through only one backcattering event. Therefore, higher order cattering event only broaden the probability depth ditribution p() coared to the purely double cattering event wherea the value of. remain approximately unchanged. However, for larger coherence length (L c ~ l ; e.g. L c = 90 μm), obtained by the analytical model deviate from the one obtained by the numerical imulation due to emergent effect of higher order cattering event (n > 5) and fail for L >> l > l. We conclude that in c the low-coherence regime, which i the ubject of our invetigation, the analytical model 17

18 enable accurate prediction of both p() and, and, thu, can be ued to model p() and of LEBS photon. Figure 7 how the dependence of on the optical propertie of a medium (l and g) and the patial coherence length of illumination L c uing the analytical model (Eq. (15)). The figure are plotted for different value of g ( ) and l (80 μm 500 μm) for a contant L c (L c = 5 μm). A een, how a relatively weak dependence on the optical propertie of the medium when L < l << l (Fig. 7(a) and 7(b)). However, c a hown in Fig. 6, depend primarily on L c. Thi property of i critical for LEBS meaurement in biomedical application a it enable probing a given phyical depth of a biological tiue. That i, by adjuting the L c of a light ource, it hould be poible to collect photon propagating into a tiue up to the depth of interet regardle of pecific optical propertie of a given tiue ale. It i alo noticed that for a given L c, g [ l ] 1/ 3 which i a much lower varying function of g and l than (1 g) 1/ 3 ( g, l ) / 3 another length cale frequently ued to decribe light tranport in tiue, l l = ( 1 g). Experimental realiation to obtain the information about the depth of penetration of LEBS photon can be ilemented in different way a follow. a) The depth to which a photon penetrate can be experimentally etimated by varying the thickne of the ale. Varying the thickne provide a ile method for quantifying the contribution of different depth to the LEBS ignal [34]. b) Time reolved meaurement can alo be ued to ae the penetration depth by meauring hort light pule backcattered from the ale without any ale preparation. The depth of a photon inide the ale can then be experimentally gated baed on the time of flight of uch 18

19 hort light pule [35]. We are currently conidering a number of uch experimental methodologie to ilement our analytical derivation in potential experiment. IV. CONCLUSIONS We have derived an analytical model of the penetration depth ditribution p() and effective penetration depth of photon that generate an LEBS peak (patial coherence length L < l << l ), in the ub-diffuive cattering regime. We have c performed numerical Monte Carlo imulation to upport our analytical reult. The reult from the analytical model are in good agreement with thoe obtained from the Monte Carlo imulation. Our reult demontrate that [ ] 1/ 3 / 3 1/ 3 ( g (1 g) ) l π L, i.e. of the LEBS photon i approximately proportional to the 1/3 power of an effective coherence volume, [ ] 1/ 3 l π in an experimentally relevant parameter regime. More L c iortantly, LEBS photon typically penetrate le than the cattering mean free path of c the medium l (that i, < l ) when L < l << l. Furthermore, the analytical c calculation and numerical imulation how trong dependence of on L c (that can be controlled externally) and relatively weak dependence to tiue optical propertie ( l, g), which ugget the poibility of uing LEBS for depth-elective analyi of weakly cattering media uch a biological tiue. 19

20 Acknowledgement Thi tudy wa upported by National Health Intitute grant R01CA11315, R01EB00368, and Coulter Foundation Award. Y.L. Kim wa upported by a National Cancer Intitute training grant R5 CA A1. P. Pradhan wa upported by the Cancer Reearch Prevention Foundation (CRPF) and the Center for Cancer Nanotechnology Excellence (CCNE), Northwetern Univerity, Evanton, IL. 0

21 Reference: [1] S. G. Demo and R. R. Alfano, "Optical polariation imaging," Applied Optic, vol. 36, pp , [] Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, "Simultaneou meaurement of angular and pectral propertie of light cattering for characteriation of tiue microarchitecture and it alteration in early precancer," IEEE Journal of Selected Topic in Quantum Electronic, vol. 9, pp , 003. [3] A. Wax, C. H. Yang, M. G. Muller, R. Nine, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Daari, and M. S. Feld, "In itu detection of neoplatic tranformation and chemopreventive effect in rat eophagu epithelium uing angle-reolved low-coherence interferometry," Cancer Reearch, vol. 63, pp , 003. [4] S. L. Jacque, J. C. Ramella-Roman, and K. Lee, "Imaging kin pathology with polaried light," Journal of Biomedical Optic, vol. 7, pp , 00. [5] A. Wax, C. H. Yang, V. Backman, K. Badiadegan, C. W. Boone, R. R. Daari, and M. S. Feld, "Cellular organiation and ubtructure meaured uing anglereolved low-coherence interferometry," Biophyical Journal, vol. 8, pp , 00. [6] K. Sokolov, R. Dreek, K. Goage, and R. Richard-Kortum, "Reflectance pectrocopy with polaried light: i it enitive to cellular and nuclear morphology," Optic Expre, vol. 5, pp , [7] D. J. Bicout and G. H. Wei, "A meaure of photon penetration into tiue in diffuion model," Optic Communication, vol. 158, pp. 13-0, [8] G. H. Wei, "Statitical propertie of the penetration of photon into a emiinfinite turbid medium: a random-walk analyi," Applied Optic, vol. 37, pp , [9] D. J. Durian, "Penetration Depth for Diffuing-Wave Spectrocopy," Applied Optic, vol. 34, pp , [10] S. Del Bianco, F. Martelli, and G. Zaccanti, "Penetration depth of light re-emitted by a diffuive medium: theoretical and experimental invetigation," Phyic in Medicine and Biology, vol. 47, pp , 00. [11] M. B. Vandermark, M. P. Vanalbada, and A. Lagendijk, "Light-Scattering in Strongly Scattering Media - Multiple-Scattering and Weak Localiation," Phyical Review B, vol. 37, pp , [1] P. E. Wolf, G. Maret, E. Akkerman, and R. Maynard, "Optical Coherent Backcattering by Random-Media - an Experimental-Study," Journal De Phyique, vol. 49, pp , [13] Y. Kuga and A. Ihimaru, "Retroreflectance from a Dene Ditribution of Spherical-Particle," Journal of the Optical Society of America a-optic Image Science and Viion, vol. 1, pp , [14] M. P. Vanalbada and A. Lagendijk, "Obervation of Weak Localiation of Light in a Random Medium," Phyical Review Letter, vol. 55, pp ,

22 [15] P. E. Wolf and G. Maret, "Weak Localiation and Coherent Backcattering of Photon in Diordered Media," Phyical Review Letter, vol. 55, pp , [16] D. S. Wierma, M. P. Vanalbada, and A. Lagendijk, "Coherent Backcattering of Light from Alifying Random-Media," Phyical Review Letter, vol. 75, pp , [17] D. S. Wierma, M. P. Vanalbada, B. A. Vantiggelen, and A. Lagendijk, "Experimental-Evidence for Recurrent Multiple-Scattering Event of Light in Diordered Media," Phyical Review Letter, vol. 74, pp , [18] R. Sapiena, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wierma, "Aniotropic weak localiation of light," Phyical Review Letter, vol. 9, pp. -, 004. [19] G. Labeyrie, D. Delande, C. A. Muller, C. Miniatura, and R. Kaier, "Coherent backcattering of light by an inhomogeneou cloud of cold atom," Phyical Review A, vol. 67, pp. -, 003. [0] J. Huang, N. Eradat, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, "Anomalou coherent backcattering of light from opal photonic crytal," Phyical Review Letter, vol. 86, pp , 001. [1] G. Yoon, D. N. G. Roy, and R. C. Straight, "Coherent Backcattering in Biological Media - Meaurement and Etimation of Optical-Propertie," Applied Optic, vol. 3, pp , [] K. M. Yoo, F. Liu, and R. R. Alfano, "Biological-Material Probed by the Teoral and Angular Profile of the Backcattered Ultrafat Laer-Pule," Journal of the Optical Society of America B-Optical Phyic, vol. 7, pp , [3] K. M. Yoo, G. C. Tang, and R. R. Alfano, "Coherent Backcattering of Light from Biological Tiue," Applied Optic, vol. 9, pp , [4] Y. L. Kim, Y. Liu, V. M. Turhitky, R. K. Wali, H. K. Roy, and V. Backman, "Depth-reolved low-coherence enhanced backcattering," Optic Letter, vol. 30, pp , 005. [5] Y. L. Kim, Y. Liu, V. M. Turhitky, H. K. Roy, R. K. Wali, and V. Backman, "Coherent backcattering pectrocopy," Optic Letter, vol. 9, pp , 004. [6] L. H. Wang, S. L. Jacque, and L. Q. Zheng, "Mcml - Monte-Carlo Modeling of Light Tranport in Multilayered Tiue," Couter Method and Program in Biomedicine, vol. 47, pp , [7] S. P. Lin, L. H. Wang, S. L. Jacque, and F. K. Tittel, "Meaurement of tiue optical propertie by the ue of oblique-incidence optical fiber reflectometry," Applied Optic, vol. 36, pp , [8] M. H. Eddowe, T. N. Mill, and D. T. Delpy, "Monte-Carlo Simulation of Coherent Backcatter for Identification of the Optical Coefficient of Biological Tiue in-vivo," Applied Optic, vol. 34, pp , [9] R. Lenke, R. Tweer, and G. Maret, "Coherent backcattering of turbid ale containing large Mie phere," Journal of Optic a-pure and Applied Optic, vol. 4, pp , 00.

23 [30] H. Subramanian, P. Pradhan, Y. L. Kim, Y. Liu, X. Li and V. Backman, "Modeling low-coherence enhanced backcattering (LEBS) uing Monte Carlo imulation," Applied Optic, vol. 45, pp , 006. [31] Y. L. Kim, P. Pradhan, H. Subramanian, Y. Liu, M. H. Kim and Vadim Backman, "Origin of low-coherence enhanced backcattering," Optic Letter, vol. 31, pp , 006. [3] M. Born and E. Wolf, Principle of optic : electromagnetic theory of propagation, interference and diffraction of light, 7th (expanded) ed. Cambridge ; New York: Cambridge Univerity Pre, [33] M. J. Rakovic and G. W. Kattawar, "Theoretical analyi of polariation pattern from incoherent backcattering of light," Applied Optic, vol. 37, pp , [34] D. S. Wierma, P. Bartolini, A. Lagendijk, and R. Righini, "Localiation of light in a diordered medium," Nature, vol. 390, pp , [35] M. Storer, P. Gro, C. M. Aegerter, and G. Maret, "Obervation of the critical regime near Anderon localiation of light," Phyical Review Letter, vol. 96, art.no ,

24 Figure Caption: FIG 1: (Color online) A chematic picture of a photon that undergo double cattering and exit within a very mall radial ditance r. ' i the vertical ditance of the firt catterer from the urface of the medium, ' ' i the vertical ditance of the econd catter and θ i the cattering angle. In order for the photon to undergo double cattering and exit within the narrow radial ditance ( r < l << l, l - cattering mean free path of the l medium, l =, g aniotropy factor), one of the cattering event occur cloer to 1 g the urface of the medium. That i ' << ' ' or ' ' << '. FIG : (Color) The cattering ditribution of the photon p(n ) v number of cattering n from the numerical imulation for r < l << l and L c < l << l ( Lc - patial coherence length of illumination) for two different media with aniotropy factor g = 0.9 and 0.7 at contant l = 100 µm. The photon predominantly undergo double cattering at mall r and L c. However, the contribution from the double cattering photon decreae with increae in r and L c. FIG 3: (Color) The penetration depth ditribution p() of the photon v depth for r < l << l and L c < l << l for two different media with aniotropy factor g = 0.9 and 0.7 at contant l = 100 µm. The p() of the photon predicted by the numerical imulation ugget a trong dependence on r and L c and relatively weak dependence on the optical propertie l and g. FIG 4: (Color online) The penetration depth ditribution p() of ingle cattering photon plotted againt thoe from double cattering photon at different depth. At r < l << l, the firt catterer i located cloer to the urface of the medium while the econd catterer i located everal order deeper than the firt catterer. Hence, the difference in vertical ditance of the two catter can be taken a the penetration depth of the photon. FIG 5: (Color) Coarion of penetration depth ditribution p() of photon obtained from numerical imulation and analytical expreion. (a) The ditribution are obtained for a medium with g = 0.9 and l = 100 µm for a radial ditance r = 5 µm and 5 µm. (b) The ditribution are obtained for a medium with g = 0.9 and l = 100 µm for a patial coherence length, L c = 5 µm and 5 µm. FIG 6: (Color) Coarion between the effective penetration depth of the photon that form a LEBS peak predicted by the numerical model and analytical expreion (Eq. (15)). i obtained a a function of L c for two different media with aniotropy factor, g = 0.7 and g = 0.9 and a contant l (l = 100 μm). The agreement between the numerical model and analytical expreion decreae with the increae in L c a the photon undergo higher order cattering. 4

25 FIG 7: (Color online) The dependence of on individual optical propertie plotted againt l, and g. (a) increae lowly between the aniotropy value of 0.7 and 0.9 after which increae harply. (b) depend on l only lightly over the range 100 µm and 500 µm that i relevant to the biological ytem. 5

26 Figure: r d' ' θ '' θ d'' FIG 1 6

27 (a) (b) (c) (d) FIG 7

28 (a) (b) (c) (d) FIG 3 8

29 FIG 4 9

30 (a) (b) FIG 5 30

31 1/ 3 / 3 ( g (1 g) ) [ l ] 1/ 3 π L c FIG 6 31

32 (a) (b) FIG 7 3