Single-scattering solutions to radiative transfer in infinite turbid media

Transcription

1 M. L. Shendeleva Vol. 3, No. / November 23 / J. Opt. Soc. Am. A 269 Single-cattering olution to radiative tranfer in infinite turbid media Margarita L. Shendeleva Photonic Procee Department, Intitute of Phyic, 46 Propect Nauki, Kiev 328, Ukraine (hend@iop.kiev.ua) Received July 7, 23; revied September 9, 23; accepted September, 23; poted September, 23 (Doc. ID 945); publihed October 2, 23 An intantaneou point ource generating a light field in an infinite turbid medium with aniotropic individual catterer i conidered. Single-cattering olution are ought a the firt term of a erie expanion of the olution of the radiative tranfer equation in ucceive cattering order. A imple formula for a ingle-cattering olution in media with an arbitrary axially ymmetric phae function i derived. Application of thi formula are hown for the Henyey Greentein phae function, an ellipoidal phae function, and a linear phae function. In addition, the ingle-cattering term of the RTE olution derived by Kholin [h. Vych. Mat. i Mat. Fy. 4, 26 (964)] for media with a phae function repreented by a finite erie in Legendre polynomial i conidered in more detail. 23 Optical Society of America OCIS code: (7.366) Light propagation in tiue; (29.42) Multiple cattering; (6.553) Pule propagation and temporal oliton; (3.562) Radiative tranfer. INTRODUCTION The radiative tranfer equation (RTE) i widely ued in variou field of phyic, including cattering-media optic, geophyic, and atrophyic. One of the method of olving the RTE involve the expanion of the radiance in a erie of ucceive cattering order called a Neumann erie. The firt term of the erie, which i proportional to the cattering coefficient, repreent a ingle-cattering olution that decribe the field undergoing one cattering event on the way from the ource to the obervation point. Thi paper conider uch ingle-cattering olution. Single-cattering olution are ued in many application, uch a ingle-cattering optical tomography [], nonline-of-ight (NLOS) communication [2], and real-time rendering [3]. A number of olution to the RTE, repreented a an expanion in the number of cattering event, have been derived that conider intantaneou point ource. For media with iotropic cattering, a time-domain olution wa obtained by Monin [4]. An approximate yet accurate olution for iotropically cattering media wa derived by Paachen [5]. For media with aniotropic cattering, a olution wa obtained by Kholin [6] and Kholin and co-worker [7]. Recurive formula for calculating ucceive cattering order were derived by Fomenko and Shvart for media with iotropic cattering [8] and by Fomenko et al. [9] for media with aniotropic cattering. A olution expreed a a Legendre polynomial expanion wa obtained by Ganapol []. Note that the ingle-cattering olution can be derived in a more traightforward manner. To do o, one can ue the property of an ellipoid that tate that the um of ditance between the foci and any point on the ellipoidal urface i contant. The olution i obtained after ome geometrical conideration and the ue of prolate pheroidal coordinate. Thi method wa applied by Sato [], who firt obtained a inglecattering olution for iotropically cattering media. The method of pheroidal coordinate i alo commonly ued to model repone in NLOS communication. Thi paper explore the approach propoed by Paachen [5] for iotropically cattering media. In the next ection, the approach i modified to include media with aniotropic cattering. Although the equation become more complicated in comparion to the iotropic cae, the firt two term correponding to the ballitic peak and to ingle cattering are found to be tractable. For ingle cattering, a imple integral formula i obtained. In Section 3, 4, and 6, thi formula i applied to obtain the olution for a few common phae function. Section 5 decribe a ingle-cattering olution, which i the firt term of the expanion of the olution derived by Kholin [6] for media with a polynomial phae function. It i hown that thi ingle-cattering olution can be ued for a polynomial approximation of the olution for media with the Henyey Greentein (HG) phae function. Section 7 ummarize the reult. 2. DERIVATION OF A SINGLE-SCATTERING SOLUTION FOR MEDIA WITH ANISOTROPIC SCATTERING The RTE for media with aniotropic cattering i written a I r; t;ŝ ŝ I r;t; ŝμ v t μ a I r; t;ŝ μ I r; ~ t;ŝ S r;t; () v where I r;t; ŝ (Wm 2 r ) i the radiance at time t and pace point r in the direction pecified by a unit vector, ŝ; μ m i the cattering coefficient, and μ a m i the aborption coefficient, both expreed through the cattering mean free path l and the aborption mean free path l a a μ l and μ a l a, repectively; S r; t Wm 2 r i the rate of /3/269-6$5./ 23 Optical Society of America

2 27 J. Opt. Soc. Am. A / Vol. 3, No. / November 23 M. L. Shendeleva a ource power per unit urface area per unit olid angle; and v i the peed of light in the medium, v c n, where c i the peed of light in the vacuum and n i the refractive index. In thi paper, an iotropic intantaneou point ource repreented by delta function in time and pace i conidered uch that where S r;t S δ rδt; (2) S N ħωv (3) with N the number of photon emitted at t and ħω a photon R energy. Thu, in a nonaborbing medium, we impoe IdVdΩ N ħωv, where the integral i over all pace and a full olid angle. Note alo that the RTE olution with an intantaneou ource ha the aborption a an exponential factor, exp μ a vt; thu, without lo of generality, the derivation can be conidered for a nonaborbing cae. On the right-hand ide of Eq. (), I ~ denote the integral over a olid angle: ~I pŝ ŝ I r;t; ŝ dŝ : (4) Here pŝ ŝ i a phae function normalized a pŝ ŝ dŝ (5) with the element of a olid angle dŝ in ΘdΘdφ, where Θ and φ are the polar and azimuthal angle in a pherical coordinate ytem. The angle-averaged intenity i [5] Īr;t I r; t;ŝ dŝ : (6) Note that the literature alo conider the fluence, Φr;t, which i related to the angle-averaged intenity a Φr; t Īr;t. In the cae of iotropic cattering, when pŝ ŝ, one obtain ~ I Ī. For aniotropic cattering Ī R ~ Idŝ. Repreenting the radiance and angle-averaged intenitie a a erie in order of cattering, that i, and and I r; t;ŝ X N ~I r;t; ŝ X N Ī r;t X N I N r; t;ŝ (7) ~I N r;t; ŝ; Ī N r;t; (8) the recurive equation for partial intenitie are derived, v t I l N r; t;ŝ IN ~ r;t; ŝ (9) l for N ; 2;, and v t I l r;t; ŝ S r; t: () v Thee equation can be integrated to obtain I N r;t; ŝ e r l IN ~ r r l ŝ; t r v; ŝdr ; () I r;t; ŝ v e r l S r r ŝ; t r vdr : (2) In what follow, the ource i conidered in the form of Eq. (2), with S (Wm r) omitted; thu the angle-averaged intenity will have unit of m 3. Subtituting Eq. (2) into Eq. (2), one obtain the ballitic peak I r;t; ŝ a found in [5] and e r l δ r r ŝδvt r dr e vt l δ r vtŝ; (3) Ī r; t e vt l δvt r: (4) 2 r For ingle-cattering intenity, Eq. () yield, for N, I r;t; ŝ e r l I ~ r r l ŝ; t r v; ŝdr : (5) Subtituting Eq. (3) into Eq. (5), I i derived a I r;t;ŝ e r l I r r l ŝ;t r v;ŝ pŝ ŝ dŝ dr e r r l δ r r ŝ r l ŝδvt r r pŝ ŝ dr dr dŝ : (6) Integrating both ide of Eq. (6) over the olid angle decribed by ŝ, one obtain the angle-averaged ingle-cattering intenity Ī r;t e r r l l r 2 δ r r r δvt r r r2 pˆr ˆr d r d r ; (7) where ˆr and ˆr denote the unit vector in the direction r and r. A imilar expreion for noniotropic ingle cattering wa derived by Sato [2]. For iotropic cattering, thi expreion reduce to Ī r;t l e r r l r r 2 δ r r r δvt r r d r d r ; (8)

3 M. L. Shendeleva Vol. 3, No. / November 23 / J. Opt. Soc. Am. A 27 which wa obtained by Paachen [5]. With the aid of δ-function, Eq. (7) and (8) can be given a imple geometrical interpretation. Equalitie r r vt and r r r mean that all ingle-cattering event lie on the urface of a prolate ellipoid whoe foci are eparated by r and whoe longer axi i equal to vt, a hown in Fig.. For iotropic cattering, the integral in Eq. (8) can be evaluated. Performing the firt integration over r and then integration over r, one tranform the ingle-cattering intenity to Ī io r;t dξ e vt l l vt 2 r 2 2vtrξ (9) for vt > r, where ξ co α, with α the angle between r and r, a hown in Fig.. After evaluation, the ingle iotropic cattering intenity i obtained a Ī io r;t e vt l vt r l vtr ln Hvt r; (2) vt r where Hx denote the Heaviide tep function. The ame reult wa obtained by Paachen [5] with the ue of the Fourier tranform. An identical expreion for ingle iotropic cattering, apart from the exponential factor, wa derived much earlier by Sato [] in eimology. Note that Sato ued the firt cattering approximation from the very beginning and thu left only term of the firt order of the cattering coefficient. Conider now Eq. (7) for aniotropic cattering. Denoting the angle between r and r by θ, a hown in Fig., one tranform Eq. (7) to Ī r; t e vt l l pco θdξ vt 2 r 2 2vtrξ : (2) Here it i important to notice the relation between the angle θ and the angle α, which can be found from geometrical conideration a follow. With the introduction of the new variable a; c, and e a r 2c, vt 2a, and c a e and uing the relation for an ellipe, S r a ex; (22) r a ex; (23) r r r vt Fig.. Single-cattering geometry. The ource S and the obervation point R are located at the foci of an ellipoid, and the cattering event take place at point P on the ellipoid urface. x P θ α R where x i the ditance from the center of the ellipe along the big axi, a hown in Fig., the expreion for angle are obtained a ξ co α c x a ex ; (24) μ co θ 2c2 a 2 e 2 x 2 a 2 e 2 x 2 : (25) Section 3 and 4 conider the application of Eq. (2) for two commonly ued phae function. 3. HG PHASE FUNCTION The HG [3] phae function i defined a p HG co θ g 2 g 2 ; (26) 3 2 2g co θ where g i the parameter, jgj <, which i called the aniotropy factor, ince g μ 2π R p HG μμdμ. The phae function i normalized to unity, 2π R pμdμ. Subtituting Eq. (26) into Eq. (2), one can derive the inglecattering intenity in the form Ī HG r; t g2 e vt l 2πl v 2 t 2 r 2 χ 2 2 dχ v 2 t 2 r 2 χ 2 g 2 4gv 2 t 2 r ; (27) for vt > r. Reilly and Warde [2] obtained the ame expreion by a different method by in the field of NLOS communication. One can check numerically that thi olution atifie the normalization condition vt Ī r 2 dr vt l e vt l : (28) Figure 2 how the plot of the ingle-cattering intenity, ĪHG, for variou aniotropy factor. 4. ELLIPSOIDAL PHASE FUNCTION It would be ueful to find a phae function for which the integral in Eq. (2) can be evaluated. One poible choice i to ue the power of one intead of 3 2 in the denominator of Eq. (26). Impoing the normalization condition, Eq. (5), one obtain the phae function p el co θ 2π ln ς ς ς ; (29) ς 2 2ς co θ where jςj <. Since the equation of thi function contain the equation of an ellipe in polar coordinate, thi phae function i called ellipoidal [4]. Such a function ha already been ued in biomedical application [5]. Note that the aniotropy factor μ i related to ζ in a more complicated way in comparion to the HG phae function:

4 272 J. Opt. Soc. Am. A / Vol. 3, No. / November 23 M. L. Shendeleva I,cm 3 8 g.9 g.7 6 g.5 g.3 4 g t, n Fig. 2. Single-cattering olution for media with the HG phae function for variou aniotropy factor, hown in the figure. Here l.5 cm, r. cm, and v 3 cm n. μ ς2 2ς ln ς ς : (3) Subtituting the phae function in Eq. (29) into Eq. (2) and performing the integration, one obtain for the inglecattering intenity r; t ςe vt l 2πl r ς ln ς p ς v 2 t 2 ς 2 4r 2 ς p! v 2 t 2 ς 2 4r 2 ς ςr ln p ; (3) v 2 t 2 ς 2 4r 2 ς ςr Ī el for vt > r. For iotropic cattering, ς, thi reduce to Eq. (2). A comparion of the olution for media with the HG phae function and media with the ellipoidal phae function, given by Eq. (27) and (3), repectively, i hown in Fig. 3. For a given g μ, the correponding ζ i found numerically from Eq. (3). Thu g.9 correpond to ζ Figure 4 how the ratio of angle-averaged intenitie Īel ĪHG for variou aniotropy factor. Note that thi ratio i independent of l and depend only on r vt and g. 5. POLYNOMIAL PHASE FUNCTION Empirically found phae function are frequently approximated with polynomial. The Legendre polynomial form a convenient bai for uch an approximation; thu the phae function i expreed in the form I,cm g.9 HG el t, n Fig. 3. Comparion of ingle-cattering olution for media with the HG phae function and the ellipoidal phae function for aniotropy factor μ.9, where l.5 cm, r. cm, and v 3 cm n. Fig. 4. Ratio of angle-averaged intenitie Īel ĪHG for variou aniotropy factor. Here l S.5 cm, r. cm, and v 3 cm n. X N p pl co θ b k P k co θ; (32) k where P k x are Legendre polynomial, and b k are the expanion coefficient, where b. The upercript indicate thi i a polynomial phae function. For finite N, Kholin et al. [7] found the following olution for the ingle-cattering intenity: Ī Kh ie vt l X N 2 rvtl k b k Q 2 k z Q 2 k z ; (33) where Q k z are the Legendre function of the econd kind, z η iε are point in the complex plane on two bank of the cut ;, ε i a poitive number, ε, and η r vt: (34) Note that the notation ha changed from the original. Taking into account the propertie of Q k function [6], one obtain Q k z iπ 2 P kη η 2 ln P η k η W k η; (35) where W k denote the polynomial W k η Xk m P m ηp k m η k m for k>, W. Thu Eq. (33) i converted to the form Ī Kh e vt l η X N 2πrvtl 2 ln η k (36) b k P 2 XN kη b k P k ηw k η ; k (37) for vt > r. Conider, for example, the ue of polynomial to approximate the HG phae function. It i known that the HG phae function ha the following expanion in Legendre polynomial: p HG μ X k 2k g k P k μ: (38) Suppoe the HG phae function i approximated with a finite number of term, a

5 M. L. Shendeleva Vol. 3, No. / November 23 / J. Opt. Soc. Am. A 273 Phae function g.7 HG pl, N 6 pl, N p HG pl μ Fig. 5. HG phae function and it approximation with polynomial phae function, Eq. (39), with N 6 and N and aniotropy factor g.7. X N k 2k g k P k μ; (39) where the upercript GH-pl indicate thi equation i a polynomial approximation to the HG phae function. The minimal number of term ufficient for approximation depend on the aniotropy factor, a wa pointed out in [7]. When the number of term i too mall, the approximate phae function can have negative value. Thi effect i hown in Fig. 5, which how the approximation of the HG phae function with Eq. (39) for N 6 and N. It can be een that for g.7 the approximation with N 6 exhibit nonphyical negative value, while N look ufficient. Figure 6 how the ingle-cattering olution correponding to the polynomial phae function, Eq. (37), and to the HG phae function, Eq. (27), for two ditance from the ource. It can be een that at a very mall ditance (r l.5, the left peak), the Kholin olution how ocillation related to the ocillation of the polynomial approximation hown in Fig. 5. The ocillation of ingle-cattering olution are wahed out for larger ditance (r l.5, the right peak), and the olution approach the olution for media with the HG phae function. 6. LINEAR PHASE FUNCTION A linear phae function correpond to N in Eq. (32): I,cm 3 µ p lin co θ 3g co θ : (4) 7 HG 8 6 pl, N 6 pl, N t, n Fig. 6. Single-cattering olution correponding to the HG phae function and the polynomial phae function with N 6 and N for two ditance from the ource, where l.5 cm, v 3 cm n, and g.7. The left peak correpond to r.25 cm, and the right peak correpond to r.25 cm. To enure that thi phae function i poitive, the aniotropy factor g hould be retricted to the value jgj 3. The ingle-cattering olution can then be found from Eq. (37) for N : Ī lin e vt l η 3gη 2 ln rvtl η 6gη : (4) The ame expreion i obtained from Eq. (2) after evaluation of the integral. An identical expreion wa alo derived by Fomenko et al. [9]. 7. CONCLUSION Following the approach of Paachen [5], the RTE in media with aniotropic cattering i decompoed into a number of recurive equation correponding to ucceive order of cattering. It i found that the equation for the firt cattering order can be olved, and thu the analytic formula for inglecattering intenity i derived for media with an arbitrary, axially ymmetric phae function, Eq. (2). With thi formula, a number of olution for media with common phae function are derived. Some of thee olution are validated by comparion with thoe in the literature. Specifically, the ingle-cattering intenity correponding to the HG phae function, Eq. (27), i found to be identical to the olution derived by Reilly and Warde [2] by a different method. The ingle-cattering angle-averaged intenity correponding to a linear phae function i found to be the ame a that derived from Kholin olution for a polynomial phae function, Eq. (4). In addition, it i found that the ingle-cattering olution correponding to an ellipoidal phae function, Eq. (3), ha a imple analytical form. Thi olution i compared with the olution for media with the HG phae function. Kholin ingle-cattering olution, Eq. (33), i the firt term of the RTE olution derived by Kholin and co-worker for all ucceive order of cattering [6,7]. In practice, the phae function given by imple formula, uch a the HG or ellipoidal formula, frequently do not provide a cloe approximation to oberved data. In thi context, the polynomial phae function, conidered by Kholin, provide more flexibility. It i noteworthy that for a phae function repreented by a finite Legendre polynomial erie, a ingle-cattering olution i found in a imple form that i expreed via a finite erie containing Legendre polynomial, Eq. (37). In turbid media, a ingle-cattering repone dominate at hort time interval after the wave-front arrival and at hort ditance (up to a few cattering mean free path). It i hown that the hape of the ingle-cattering intenity varie according to the hape of the phae function and not jut it aniotropy factor. The relatively imple and exact analytical olution derived in thi paper can be ued to model repone in application baed on meaurement of ingle- and lowcattering order. REFERENCES. L. Florecu, J. C. Schotland, and V. A. Markel, Single-cattering optical tomography, Phy. Rev. E 79, 3667 (29). 2. D. M. Reilly and C. Warde, Temporal characteritic of inglecatter radiation, J. Opt. Soc. Am. 69, (979). 3. V. Pegoraro and S. G. Parker, An analytical olution to ingle cattering in homogeneou participating media, Comput. Graph. Forum 28, (29).

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