Depolarization of multiply scattered light in transmission through a turbid medium with large particles

Optics Communications 260 (2006) 30–45 www.elsevier.com/locate/optcom

Depolarization of multiply scattered light in transmission through a turbid medium with large particles E.E. Gorodnichev *, A.I. Kuzovlev, D.B. Rogozkin Moscow Engineering Physics Institute, Theoretical Physics, Kashirskoe Shosse, 31, 115409 Moscow, Russia Received 5 January 2005; received in revised form 30 August 2005; accepted 17 October 2005

Abstract An approximate analytical method for solving the vector radiative transfer equation is proposed. The method is based on the assumption that single scattering of light by large-scale inhomogeneities occurs predominantly through small angles. The method is applied to calculate the polarization state of multiply scattered light. The results obtained are discussed for various turbid media.  2005 Elsevier B.V. All rights reserved. PACS: 42.25.Dd; 42.25.Ja; 42.68.Ay Keywords: Multiple light scattering; Polarization

1. Introduction In the resent years, explaining the polarization effects in multiply scattering media with large-scale (size a is larger than wavelength k) inhomogeneities have been of special interest in connection with many applications. A large number of experimental and theoretical studies have been devoted to this problem [1–23]. New effects were revealed, in particular, the difference in the depolarization rates between linearly and circularly polarized beams of light [6–9,21–23]. In most theoretical studies dealing with multiple scattering of polarized light in turbid media, methods of numerical calculations are generally discussed [13–18,21]. Simple analytical results that could explain basic experimentally observed effects were not available until recently. Within the framework of simplifying assumptions, the first results along this line were obtained in [12,16,24–28]. In this study, we consider the depolarization of multiply scattered light in optically isotropic turbid media with large inhomogeneities. Two different mechanisms of depolarization, viz, the ‘‘geometrical’’ mechanism and the ‘‘dynamical’’ one, can be distinguished [26]. The ‘‘geometrical’’ mechanism is due to the Rytov rotation [29,30]. The plane of polarization turns simultaneously with the ray of light. The wave remains linearly polarized along the overall path of propagation. The depolarization observed in multiple scattering of linearly polarized light results from superposition of randomly oriented polarizations of the different rays. The ‘‘dynamical’’ mechanism [31] is due to the difference in amplitudes between two cross-polarized components of the single-scattered wave. By this mechanism, multiply scattered light depolarizes as the spread in amplitudes increases. Single scattering of light by large-scale inhomogeneities occurs predominantly through small angles (1  hcosci  1, where hcosci is the mean cosine of single-scattering angle c) [32,33]. In this case, the ‘‘geometrical’’ mechanism of depolarization is dominant as compared with the ‘‘dynamical’’ one [6–8,25]. This permits us to

*

Corresponding author. Tel.: +7 095 3239377; fax: +7 095 3243184. E-mail address: [email protected] (E.E. Gorodnichev).

0030-4018/$ - see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.10.042

E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

31

develop a procedure for decoupling the vector radiative transfer equation. This procedure is based on the selection of basic and additional modes. In the case of single scattering through small angles the interaction between the basic polarization modes appears to be weak. To a first approximation we can neglect this interaction. In the succeeding approximation, the interaction between the basic modes results in the excitation of the additional modes (overtones). Allowance for the overtones makes it possible to describe in detail the polarization state of multiply scattered light deep in the medium. With the method proposed, the polarization state of multiply scattered polarized light is calculated. The influence of the scatterer parameters on the depolarization process is discussed. 2. General relations Let a wide polarized beam of light be incident on a medium normally to its surface. The medium is assumed to be a statistically isotropic disordered ensemble of large-scale scatterers. The polarization state of scattered light is generally described by the four Stokes parameters [1,32–34] 0 1 I BQ C B C b ð1Þ S ¼ B C; @U A V where I ¼ hEk Ek þ E? E? i; Q ¼ hEk Ek  E? E? i;

ð2Þ

U ¼ hEk E? þ Ek E? i; V ¼ ihEk E?  Ek E? i.

The Stokes parameters and the components Ek and E? of the electric field appearing in Eq. (2) are defined in the system of unit vectors {ek = on/oh, e? = [ek,n], n} [34]. The unit vector n = (sinh cosu, sinh sinu, cosh) is the direction of propagation of the transverse electromagnetic wave, the vector ek lies in the plane formed by the vectors n0 and n (n0 is the internal normal to the surface), the vector e? is perpendicular to this plane (Fig. 1). The brackets h. . .i denote statistical averaging. The Stokes parameters are always defined with respect to a reference frame, in our case the system of unit vectors {ek, e?, n} or the plane {n, n0}. If the reference frame is rotated through angle a around the direction n, the transformation 0 of the old Stokes vector b S into the new Stokes vector b S is given by rotation matrix b L 0 b S ¼b LðaÞ b S;

ð3Þ

where 0

1

B0 B b LðaÞ ¼ B @0 0

0

0

cos a  sin a

sin a cos a

0

0

0

1

0C C C. 0A 1

Fig. 1. Coordinate system used to describe the direction of propagation and polarization state of the incident and scattered light.

ð4Þ

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E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

The Stokes parameters obey the vector radiative transfer equation [33,34],   Z o b ðn; n0 Þ b Sðz; n0 Þ; l þ rtot b S ðz; nÞ ¼ n0 dn0 Z oz

ð5Þ

where n0 is the number of scattering particles per unit volume, rtot = r + ra is the coefficient of total attenuation, r is the b ðn; n0 Þ entering into Eq. (5) is expressed in scattering coefficient, and ra is the absorption coefficient. The phase matrix Z 0 b terms of the scattering matrix F ðcos cÞ (cosc = nn ) as follows (see [1] and Appendix A). b ðn; n0 Þ ¼ b Lðp  bÞ Fb ðcos cÞ b Lðb0 Þ. Z

ð6Þ

The scattering matrix Fb ðcos cÞ describes the intrinsic properties of the medium. The matrix b Lðb Þ describes the transformation of the Stokes parameters of the incident light in going from the system of unit vectors fe0k ; e0? ; n0 g to the scattering plane, i.e., the plane formed by the vectors n and n 0 (see Fig. 1). The matrix b Lðp  bÞ corresponds to the inverse transformation from the scattering plane to the system of unit vectors {ek, e?, n} related to the direction of propagation of the scattered light. The angles entering into Eq. (6) are defined by the formulas 0

cos 2b ¼ 1  sin 2b ¼

2ð1  l02 Þð1  cos2 wÞ

; 2 1  ðnn0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1  l02 ðl 1  l2  l 1  l02 cos wÞ sin w 2

1  ðnn0 Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos c ¼ nn0 ¼ ll0 þ ð1  l2 Þð1  l02 Þ cos w; l0 ¼ n0 n0 ¼ cos h0 ;

l ¼ nn0 ¼ cos h;

;

ð7Þ

w ¼ u  u0 .

Functions cos 2b 0 and sin 2b 0 differ from functions cos 2b and sin 2b by the substitution of l for l 0 . For a macroscopically isotropic and symmetric medium, the scattering matrix Fb ðcos cÞ appearing in Eq. (6) has the block-diagonal structure (see, for example, [1,33]): 0 1 a1 ðcos cÞ b1 ðcos cÞ 0 0 B b ðcos cÞ a ðcos cÞ C 0 0 2 B 1 C ð8Þ Fb ðcos cÞ ¼ B C. @ 0 0 a3 ðcos cÞ b2 ðcos cÞ A 0

0

b2 ðcos cÞ

a4 ðcos cÞ

For the forward scattering (n = n 0 , cos c = 1), the matrix Fb is diagonal: Fb ð1Þ ¼ diagða1 ; a2 ; a2 ; a4 Þ and a2(1) = a3(1) [1]. For spherical scatterers of a given size, the matrix elements ai and bi are expressed in terms of the scattering amplitudes of single scattered cross-polarized waves [32] (see Appendix A). 1 a1 ðcos cÞ ¼ a2 ðcos cÞ ¼ ðjAk ðcos cÞj2 þ jA? ðcos cÞj2 Þ; ð9Þ 2 ð10Þ a3 ðcos cÞ ¼ a4 ðcos cÞ ¼ ReAk ðcos cÞA? ðcos cÞ; 1 ð11Þ b1 ðcos cÞ ¼ ðjAk ðcos cÞj2  jA? ðcos cÞj2 Þ; 2 ð12Þ b2 ðcos cÞ ¼ ImAk ðcos cÞA? ðcos cÞ; where Ak and A? are the scattering amplitudes of the wave components polarized, respectively, parallel and perpendicularly to the scattering plane, Ak(1) = A?(1). Note that the scattering coefficient r and the matrix element a1 appearing in matrix (8) are related by the equality Z r ¼ n0 dn0 a1 ðnn0 Þ. ð13Þ In many practical cases, the off-diagonal matrix element b2, that is responsible for interaction between the linear and circular polarizations, can be neglected [1]. Under this assumption, the equation for the fourth Stokes parameter V is separated from the equations for the other Stokes parameters, and, circularly and linearly polarized beams can be considered as independently propagating through the medium. In what follows, we assume that b2 = 0 and of the fourth Stokes parameter obeys the separate equation   Z o ð14Þ l þ rtot V ðz; lÞ ¼ n0 dn0 a4 ðnn0 ÞV ðz; l0 Þ. oz The vector radiative transfer equation for the other Stokes parameters (I, Q, U) can be transformed as follows.

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The intensity I is a scalar [35]. Under rotations, the intensity remains unchanged. In contrast, the second and third Stokes parameters (Q and U, respectively) are expressed in terms of each other under spatial rotations. From Eq. (4) it follows that Q0 ¼ Q cos 2a þ U sin 2a; U 0 ¼ Q sin 2a þ U cos 2a.

ð15Þ

In order to avoid coupling between the Stokes parameters Q and U we introduce new quantities [36,37]: 1 I 2 ¼ pffiffiffi ðQ  iU Þ. 2

ð16Þ

Unlike the Stokes parameters Q and U, either quantity defined by equality (16) is transformed via itself under rotations: I 02 ¼ expð2iaÞI 2 ;

I 02 ¼ expð2iaÞI 2 .

With regard to the preceding, in order to describe the polarization state of light, we will use the vector 0 1 1 0 1 pffiffi ðQ  iU Þ I2 2 C B C bI ¼ B I @ A ¼ @ I 0 A. p1ffiffi ðQ 2

þ iU Þ

ð17Þ

ð18Þ

I 2

b The transformation from the Stokes parameters I, Q, U to quantities I2, I0, I2 is performed with unitary matrix M 0 1 0 1 I I2 B C C bB ð19Þ @ I0 A ¼ M @ Q A; U

I 2 where 0

0 B b ¼ @1 M 0

1  piffiffi2 0 0 C A. p1ffiffi piffiffi p1ffiffi 2

2

ð20Þ

2

Using Eq. (19), we derive the following radiative transfer equation for bI   Z o ^ n0 ÞbI ðz; n0 Þ. l þ rtot bI ðz; nÞ ¼ n0 dn0 dðn; oz

ð21Þ

^ n0 Þ appearing in Eq. (21) is related to the matrix Z b ðn; n0 Þ (6) (more precisely, to matrix Z b with the The phase matrix dðn; last column and last row excluded) as follows: bZ bM b 1 . d^ ¼ M

ð22Þ

Instead of formula (6), we obtain 0 ^ n0 Þ ¼ c ^ dðn; Lðp  bÞf^ ðcos cÞLðb Þ;

ð23Þ

where 0

expð2iaÞ B c LðaÞ ¼ @ 0 0

0 1

0 0

1 C A

ð24Þ

0 expð2iaÞ

and the scattering matrix f^ takes the form 0 1 pffiffiffi a2 þ a3 2b1 a2  a3 pffiffiffi C 1 B pffiffiffi f^ ðcos cÞ ¼ @ 2b1 2b1 A . 2a 2 pffiffiffi 1 a2  a3 2b1 a2 þ a3 Explicitly, the phase matrix (23) is given by

ð25Þ

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E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

0

ða2 þ a3 Þ expð2ivþ Þ 1 B pffiffiffi 0 ^ dðn; n Þ ¼ @ 2b1 expð2ib0 Þ 2 ða2  a3 Þ expð2iv Þ

pffiffiffi 2b1 expð2ibÞ 2a1 pffiffiffi 2b1 expð2ibÞ

1 ða2  a3 Þ expð2iv Þ pffiffiffi C 2b1 expð2ib0 Þ A. ða2 þ a3 Þ expð2ivþ Þ

ð26Þ

Angles v± appearing in Eq. (26) are defined by formula v ¼ p  ðb  b0 Þ.

ð27Þ

Expression (26) shows clearly the advantage of representation (18). Single scattering from large-scale inhomogeneities occurs mainly through small angles [1,32,33]. In this case, the off-diagonal elements of matrix (26) are much smaller than the diagonal elements of this matrix (see, for example, [12,26]). In the first approximation, we can neglect the off-diagonal elements. In this approximation, vector equation (21) decomposes into the independent equations for each component of the vector bI (18). Coupling between the equations for the components of the vector bI arises only in the succeeding approximation, where the off-diagonal elements of matrix (26) are taken into account. 3. Basic modes The purpose of this section is to describe the polarization state of multiply scattered light under the assumption that the ^ n0 Þ can be neglected. Neglect of the off-diagonal elements of matrix elements off-diagonal elements of the matrix dðn; a1, a2, a3 and b1 (26) is based on the properties of quantities a1, a2, a3 and b1 of large-scale for inhomogeneities. The features of the scattering matrix of large inhomogeneities (spherical and nonspherical) have been discussed in many publications in the last few years (see, for example, [1,16,38]). The interest in this problem is caused by wide applications of optical methods for studying natural scattering media (aerosols, seawater, biological tissues, etc.). On the basis of measurements and numerical calculations, the main properties of the elements ai and bi were determined [1,38]. As single scattering by large inhomogeneities occurs predominantly through small angles (1  hcosci  1) [1,32,33,38], the off-diagonal elements in Eq. (26) (or in Eq. (25)) appears to be small as compared to the diagonal elements of the corresponding scattering matrix. The order of ratio between the off-diagonal elements and the diagonal ones can be estimated as follows. In the case of weak spherical scatterers (it is the case of the Born or Rayleigh–Gans approximation [32] kajn  1j  1, where k = 2p/k, a and n are the radius and the relative refractive index of the scatterers), the amplitudes Ak and Ak are related by the following equation [32] Ak ðcos cÞ ¼ A? ðcos cÞ cos c.

ð28Þ

In accordance with Eq. (28), the off-diagonal elements of matrix (25) (or (26)) for small c are of the following order of magnitude [1,32] 2

2

b1 jAk j  jA? j ¼  c2 ; a1 jAk j2 þ jA? j2 2

jAk  A? j a2  a3 ¼  c4 . a1 jAk j2 þ jA? j2 These estimations hold also true in the general case of a statistically isotropic ensemble of strong (‘‘non-Born’’) inhomogeneities. This can be easily verified by expanding the scattering-matrix elements in the generalized spherical functions [1,36]. The relationship between the elements of matrix (26) allows us to develop an iterative procedure for solving the vector radiative transfer equation (see Eq. (21)). As a first approximation, we neglect the off-diagonal elements of scattering matrix (26), and Eq. (21) falls into three independent equations 0 a2 þa3 1 expð2ivþ Þ 0 0   Z 2 o B Cb 0 a1 0 l þ rtot bI ðz; nÞ ¼ n0 dn0 @ ð29Þ A I ðz; n0 Þ. oz a2 þa3 0 0 expð2ivþ Þ 2 The first and third equations in Eq. (29) are different from each other only by the sign of their complex conjugation. The separate equations derived from Eq. (29) (along with the separate equation for V (see Eq. (14))) describe the propagation of the basic polarization modes in the medium. The scalar mode, the intensity Iscal, obeys the conventional radiative transfer equation   Z o l þ rtot I scal ðz; lÞ ¼ n0 dn0 a1 ðnn0 ÞI scal ðz; l0 Þ. ð30Þ oz

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35

The second equation following from Eq. (29) describes the basic mode of linear polarization. Separating out the phase factors in the angular dependence of I±2 (these factor are responsible for the transformation of I±2 under rotations), 1 I 2 ¼ pffiffiffi W ðz; lÞ expð2iuÞ; 2 we have the following equation for W [26]   Z o a2 ðnn0 Þ þ a3 ðnn0 Þ expð2iðvþ  wÞÞW ðz; l0 Þ. l þ rtot W ðz; lÞ ¼ n0 dn0 oz 2

ð31Þ

The equations for V and W (see Eqs. (14) and (31)) differ from the scalar transfer equation (Eq. (30)) by the form of the phase functions. The phase functions appearing in Eqs. (14) and (31) are a4 and (a2 + a3)exp(2i(v+  w))/2, respectively. The difference between these phase functions and phase function a1 entering into Eq. (30) gives rise to nonzero effective ‘‘absorption’’ in Eqs. (14) and (31) (even in the absence of true absorption). The effective ‘‘absorption’’ in Eqs. (14) and (31) is responsible for the additional attenuation of V and W as compared to the intensity Iscal and describes the effect of depolarization of circularly and linearly polarized light. There are two different mechanisms of wave depolarization in scattering by a random medium. These mechanisms were first pointed out by Kravtsov [29] within the framework of the study of wave propagation through a turbulent atmosphere. The ‘‘geometrical’’ mechanism of depolarization is due to RytovÕs rotation of the polarization plane [30]. According to [30], the plane of polarization turns, as the ray of light propagates along a nonplanar curve. The depolarization observed in multiple scattering of linearly polarized light in thick layers results from superposition of randomly oriented polarizations of the waves propagating along different random paths. Therefore, depolarization by the ‘‘geometrical’’ mechanism occurs simultaneously with isotropization of the beam of light over the directions of propagation, at depths of the order of several transport lengths ltr (ltr = l(1  hcos ci), where l = r1 is the mean free path) [7,25]. The situation is different in the case of circularly polarized light. Circularly polarized light can be presented as a superposition of two linearly cross-polarized waves shifted in phase by p/2. In multiple scattering, the Rytov effect results in the turn of the polarization plane of each linearly polarized component, but has no effect on the phase shift between them (the ray remains circularly polarized). Therefore, a circularly polarized wave propagating along any random trajectory is unaffected by the Rytov rotation (or, what is the same, by the ‘‘geometrical’’ mechanism). The pure geometrical depolarization can be obtained in the limit a1 = a2 = a3 and b1 = b2 = 0 (or, for spherical particles, Ak = A?). In this case, the phase matrix entering into Eq. (21) (or Eq. (29)) can be written as 0 1 expð2ivþ Þ 0 0 C ^ n0 Þ ¼ n0 a1 ðcos cÞLðp ^  b  b0 Þ ¼ n0 a1 ðcos cÞB 0 1 0 dðn; ð32Þ @ A; 0

0

expð2ivþ Þ

where v+ is defined by Eq. (27). Matrix (32) gives rise to depolarization due to multiple turns of the polarization plane as the direction of wave propagation changes randomly. The difference between diagonal elements ai, i = 1–4 and nonzero element b1 (or, for spherical particles, the difference between the amplitudes Ak and A?) are responsible for the ‘‘dynamical’’ mechanism of depolarization. Physically, the ‘‘dynamical’’ mechanism is due to the difference in amplitudes between two components of the single-scattered wave that are polarized, respectively, parallel and perpendicularly to the scattering plane. By this mechanism, multiply scattered light depolarizes as the spread in amplitudes increases. The ‘‘dynamical’’ depolarization occurs independently of the initial polarization of light. In particular, circularly polarized light depolarizes only due to the ‘‘dynamical’’ mechanism (the difference between a1 and a4 is responsible for depolarization of circularly polarized waves) [25,26]. In the case of the linearly polarized incident beam, the role of one or the other mechanism depends on the optical properties of the scattering particles, their size and shape. As shown below the geometrical mechanism can be either dominant [25,26] or as important as the dynamical mechanism of depolarization. In the range of relatively small angles the matrix elements a1, a2 and a3 differ little from each other (see, for example, [32]), and it is instructive to rewrite the phase function appearing in Eq. (31) in the form a þ a  a þ a  a2 þ a3 2 3 2 3 expð2iðvþ  wÞÞ ¼ a1 þ a1 ½expð2iðvþ  wÞÞ  1 þ  a1 þ  a1 ½expð2iðvþ  wÞÞ  1. 2 2 2 ð33Þ Each term appearing in the right hand side of identity (33) has its own physical meaning. If we neglect the difference between the diagonal elements a1, a2 and a3 of the scattering matrix and disregard the deviation of the spherical triangle ABC shown in Fig. 1 from a planar one (i.e., we assume v+ = w), equality (33) contains only the first term. In this approximation, the equation for W does not differ from the scalar transfer equation (Eq. (30)) and W coincides with Iscal. There is no depolarization of light in this approximation.

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E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

Depolarization is described by the second, third, and fourth terms in Eq. (33). The second term in Eq. (33) is responsible for the geometrical depolarization. This term is due to the deviation of the spherical triangle ABC (Fig. 1) from a planar triangle. The difference between the angles vþ  w ¼ p  b  b0  ðu  u0 Þ

ð34Þ

is the difference between the sum of angles in the spherical triangle and the sum of angles in a planar triangle [39]. The third term in Eq. (33) is due to the difference between the diagonal elements a1, a2 and a3. This term describes the dynamical depolarization. The combined effect of the geometrical and dynamical mechanisms is described by the fourth term in Eq. (33). In scattering by large inhomogeneities, the second, third, and fourth terms in Eq. (33) are small as compared with the first term. Therefore, the depolarization process occurs slowly, as a result of many acts of scattering. For large inhomogeneities, the fourth term in Eq. (33) is of minor importance. The relationship between the contributions from the geometrical and depolarization mechanisms depends on the specific angular dependence of the matrix elements ai ðcos cÞ (i = 1–3). In the case of diffusive propagation of light (z > ltr), the solutions of the transfer equations for the basic modes (see Eqs. (14), (30), (31)) can be sought in the form of the expansions in the corresponding spherical functions. In the scalar transfer theory this approach is well-known as the Pn approximation [41]. In the case of normal incidence of a wide beam, the intensity of light and the basic mode of circular polarization can be presented as expansions in the Legendre polynomials: X 2l þ 1 X 2l þ 1 I scal ðz; lÞP l ðlÞ; V ðz; lÞ ¼ V ðz; lÞP l ðlÞ. I scal ðz; lÞ ¼ ð35Þ 4p 4p l¼0 l¼0 The coefficients Iscal(z, l) and V(z, l) obey the following equations l oI scal ðz; l  1Þ ðl þ 1Þ oI scal ðz; l þ 1Þ þ þ ½r þ ra  n0 a1 ðlÞI scal ðz; lÞ ¼ 0; ð2l þ 1Þ oz ð2l þ 1Þ oz l oV ðz; l  1Þ ðl þ 1Þ oV ðz; l þ 1Þ þ þ ½r þ ra  n0 a4 ðlÞV ðz; lÞ ¼ 0; ð2l þ 1Þ oz ð2l þ 1Þ oz

ð36Þ ð37Þ

where a1;4 ðlÞ ¼ 2p

Z

1

dla1;4 ðlÞP l ðlÞ.

ð38Þ

1

The basic mode of linear polarization is expanded in terms of the generalized spherical functions [40] X 2l þ 1 W ðz; lÞP l22 ðlÞ. W ðz; lÞ ¼ 4p l¼2

ð39Þ

Substitution of Eq. (39) into Eq. (31) gives 2

l2  4 oW ðz; l  1Þ 4 oW ðz; lÞ ðl þ 1Þ  4 oW ðz; l þ 1Þ þ þ þ ½r þ ra  n0 aþ ðlÞW ðz; lÞ ¼ 0; lð2l þ 1Þ oz lðl þ 1Þ oz ðl þ 1Þð2l þ 1Þ oz where the coefficient a+(l) appearing in Eq. (40) is defined by equality  Z 1  a2 ðlÞ þ a3 ðlÞ l aþ ðlÞ ¼ 2p dl P 22 ðlÞ. 2 1

ð40Þ

ð41Þ

The explicit expressions for several first generalized spherical functions have the form 1 2 P 222 ðlÞ ¼ ð1 þ lÞ ; 4

1 2 P 322 ðlÞ ¼ ð1 þ lÞ ð3l  2Þ; 4

1 2 P 422 ðlÞ ¼ ð1 þ lÞ ð1  7l þ 7l2 Þ. 4

ð42Þ

The coefficients Iscal(z, l), V(z, l) and W(z, l) can be easily calculated for the asymptotic state of propagation at large depths. This limiting case is of practical importance for many applications [7–9,14,22]. Substituting Iscal(z, l), V(z, l) and W(z, l) in the exponential form (e.g., Iscal(z, l) = Iscal(l)exp(Iz)) into Eqs. (36), (37), and (40), we arrive at an eigenvalue problem. A minimal eigenvalue gives the attenuation coefficient of the corresponding mode at large depths. For the scalar transfer equation this point has been discussed in detail [41]. In the case of diffusive propagation of light the contributions from the higher terms to expansions (35), (39) appear to be small. Several first terms in these expansions play a dominant role.

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The asymptotic state of the basic polarization modes are determined by the expressions: CI ðP 0 ðlÞ þ aI P 1 ðlÞ þ bI P 2 ðlÞ þ Þ expðI zÞ; ð43Þ I scal ðz; lÞ ¼ C I UI ðlÞ expðI zÞ ¼ 4p CV ðP 0 ðlÞ þ aV P 1 ðlÞ þ bV P 2 ðlÞ þ Þ expðV zÞ; V ðz; lÞ ¼ C V UV ðlÞ expðV zÞ ¼ ð44Þ 4p 5C W 2 ðP 22 ðlÞ þ aW P 322 ðlÞ þ bW P 422 ðlÞ þ Þ expðW zÞ; ð45Þ W ðz; lÞ ¼ C W UW ðlÞ expðW zÞ ¼ 4p where I,V,W and UI,V,W are the corresponding eigenvalues and angular eigenfunctions. In the leading approximation the eigenvalues I,V and the coefficients aI,V, bI,V entering into Eqs. (43) and (44) have the following form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ra 2ra ð46Þ I 3ðrtr þ ra Þra ; aI

; bI

rtr þ ra r þ ra  n0 a1 ð2Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3ðra þ rð4Þ dep Þ ð4Þ ð4Þ ð4Þ ; V 3ðrtr þ rdep þ ra Þðra þ rdep Þ; aV t ð4Þ ð4Þ rtr þ rdep þ ra   ð4Þ 2 ra þ rdep bV ¼ ; ð47Þ r þ ra  n0 a4 ð2Þ where r ¼ n0 a1 ð0Þ; rtr ¼ rð1  hcos ciÞ ¼ r  n0 a1 ð1Þ; ð48Þ ð4Þ ð4Þ rtr ¼ n0 ða4 ð0Þ  a4 ð1ÞÞ; rdep ¼ r  n0 a4 ð0Þ. The corresponding parameters entering into Eq. (45) are determined by more complicated expressions: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 7 36 3ðr þ ra Þ  2n0 aþ ð3Þ  n0 aþ ð2Þ  ð3ðr þ ra Þ  2n0 aþ ð3Þ  n0 aþ ð2ÞÞ2  ðr þ ra  n0 aþ ð3ÞÞðr þ ra  n0 aþ ð2ÞÞ ; W

6 7

21 r þ ra  n0 aþ ð2Þ 2  ; aW

5 W 3



63 r þ ra  n0 aþ ð3Þ 1 r þ ra  n0 aþ ð2Þ 2  1;   bW

5 W 3 W 3 ð49Þ where the coefficients a+(l) are calculated with Eq. (41). Note that Eq. (49) permits us to calculate the values W, aW, bW within the framework of the ‘‘geometrical’’ approximation in the phase matrix. This approximation is in substitution of the approximate phase matrix (32) for the exact matrix (26). The values geom , ageom , bgeom can be obtained from Eq. (49), if we replace (a2(l) + a3(l))/2 by a1(l) in definition (41). W W W The difference between W and geom is a measure of the contribution from the ‘‘dynamical’’ mechanism to depolarization of W the linearly polarized beam. As follows from Eqs. (47), (49), depolarization of light manifests itself as additional effective ‘‘absorption’’ of the basic ð4Þ modes V and W. For V the coefficient of the additional ‘‘absorption’’ equals rdep . For W the corresponding coefficient 2 2 should be estimated as ½W =ðn0 ðaþ ð2Þ  aþ ð3ÞÞÞ  ra (or, roughly, as ½ðW =3rtr Þ  ra   rtr ). The coefficients CI,V,W can approximately be calculated by the following formula (see Appendix B), UI;V ;W ðl ¼ 1Þ C I;V ;W

. ð50Þ R1 2p 0 ldlU2I;V ;W ðlÞ Eq. (50) gives a good agreement with numerical calculations of the scalar intensity (see e.g. [33]) in the case of an absorbing medium. For CI the difference between Eq. (50) and the numerical results of [33] is less than 10% for ra > 0.1rtr and falls off with increasing ratio ra/rtr. The depolarization of the incident circularly (or linearly) polarized beam in the medium can be described by the degree of polarization [32,33]. Within the considered approximation, where only the basic modes Iscal, V and W are taken into account, the degree of polarization for circularly (PC) and linearly (PL) polarized beams is determined by the following equations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V C V ð1 þ laV Þ Q2 þ V 2 expðzðV  I ÞÞ;

PC ¼

ð51Þ I scal C I ð1 þ laI Þ I pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q2 þ U 2 W 5C W ð1 þ lÞ2 ð1 þ aW ð3l  2ÞÞ expðzðW  I ÞÞ.



ð52Þ PL ¼ I scal 4C I ð1 þ laI Þ I These equations are derived from Eqs. (43)–(45) with allowance for the first two terms in each expansion.

38

E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

Table 1 Integral parameters of the diagonal matrix elements for large scattering inhomogeneities (x = 2pa/k)

Latex spheres (x = 5) Latex spheres (x = 10) Water droplets (x = 5) Water droplets (x = 10) Cloud 1 [42] Sea water [43]

ð4Þ

ð4Þ

rtr =r

ðr  n0 aþ ð2ÞÞ=r

ðr  n0 aþ ð3ÞÞ=r

rdep =r

rtr =r

1.1 · 101 6.7 · 102 1.5 · 101 2.9 · 101 1.5 · 101 7.0 · 102

9.4 · 102 5.8 · 102 1.3 · 102 2.3 · 101 1.1 · 101 8.1 · 102

2.9 · 101 1.7 · 101 3.5 · 101 5.1 · 101 2.7 · 101 1.7 · 101

1.7 · 102 9.8 · 103 3.4 · 102 8.4 · 102 4.9 · 102 5.8 · 102

8.4 · 102 5.6 · 102 1.1 · 101 1.9 · 101 8.5 · 102 2.6 · 102

The numerical values of the scattering matrix parameters entering into Eqs. (43)–(52) are presented in Table 1. For latex particles in water (n = 1.19) and water droplets in air (n = 1.33) the elements of the scattering matrix were calculated with the Mie formulas [32]. The matrix elements for clouds and seawater were taken from [42,43]. According to Eqs. (51) and (52) the z dependence of the degree of polarization is primarily governed by the values of the attenuation coefficients I, V and W of the basic modes. These coefficients contain information on the depolarization properties of the medium. The numerical values of the attenuation coefficients and the other parameters appearing in the expressions for the basic modes V and W can be found in Tables 2 and 3. For comparison, the values geom that are determined within the ‘‘geometW rical’’ approximation (see Eq. (32)) are also presented in Table 2. In our calculations the scattering medium is assumed to be virtually nonabsorbing (ra  rdep,rtr). Our result obtained with the seawater scattering matrix should be considered as a modeling example, because natural seawater possesses rather great absorption (ra > rtr). 4. Additional modes Independent propagation of the basic polarization modes I, W, and V in the medium represents a first approximation. In the succeeding approximation the off-diagonal elements of the phase matrix should be taken account. The inclusion of the off-diagonal elements gives rise to additional contributions to the Stokes parameters I, Q, U. The calculations can be presented in the most compact form by introducing the polarization Green matrix, 0 1 G22 G20 G22 C b ¼B G ð53Þ G00 G02 A; @ G02 G22

G20

G22

ð0Þ This matrix allows one to express the vector bI of scattered light through parameters bI of the incident beam,

b bI ð0Þ ; bI ¼ G Table 2 Attenuation coefficients of the basic polarization modes V/r Latex spheres (x = 5) Latex spheres (x = 10) Water droplets (x = 5) Water droplets (x = 10) Cloud 1 Sea water

geom =r W

W/r 2

7.1 · 10 4.3 · 102 1.2 · 101 2.1 · 101 1.3 · 101 1.0 · 101

1

1.32 · 10 8.13 · 102 1.81 · 101 3.09 · 101 1.59 · 101 1.10 · 101

V/W 1

1.29 · 10 7.95 · 102 1.75 · 101 2.93 · 101 1.52 · 101 0.80 · 101

5.4 · 101 5.3 · 101 6.6 · 101 6.8 · 101 8.2 · 101 9.1 · 101

Table 3 Parameters of the basic polarization modes in the asymptotic state

Latex spheres (x = 5) Latex spheres (x = 10) Water droplets (x = 5) Water droplets (x = 10) Cloud 1 Sea water

CV

CW

aV

aW

3.08 3.13 3.66 2.51 2.70 2.30

1.55 1.55 1.56 1.56 1.56 1.56

0.74 0.69 0.25 1.38 1.14 1.68

0.18 0.19 0.22 0.27 0.25 0.30

E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

39

b obeys the vector radiative transfer equation (21). The matrix G b has the same symSimilarly to the vector bI , the matrix G ^ metry as the scattering matrix d~ (see Eq. (26)). The complex conjugation of the elements Gmn is by changing the sign of the index, G02 ¼ G02 ;

G20 ¼ G20 ;

G22 ¼ G22 .

ð54Þ

In the explicit form the equation for the elements Gkm can be written as  l

 Z o þ rtot Gkm ðz; nÞ ¼ n0 dn0 d~kk ðn; n0 ÞGkm ðz; n0 Þ þ Qkm ðz; nÞ; oz

where the sources Qkm have the form XZ dn0 d~kl ðn; n0 ÞGlm ðz; n0 Þ. Qkm ðz; nÞ ¼ n0

ð55Þ

ð56Þ

l6¼k

^ n0 Þ is assumed to be diagonal and the matrix G b describes independent In the first approximation, the phase matrix dðn; propagation of the basic modes and has the diagonal form, 0 1 W expð2iuÞ 0 0 C b B G 0 I scal 0 ð57Þ @ A. 0

0

W  expð2iuÞ

In the succeeding approximation, allowance for nonzero off-diagonal elements of the phase matrix (see Eq. (26)) genb matrix. These quantities can be termed additional modes (or overtones). erates the off-diagonal elements of the G The terms involving the basic modes make the main contribution to the sources (56), Z Qkm ðz; nÞ n0 dn0 d~km ðn; n0 ÞGmm ðz; n0 Þ. ð58Þ In the case of normal incidence, the off-diagonal elements Gkm (k 5 m) have the following azimuth dependence: 1 G02 ðz; nÞ ¼ pffiffiffi QW ðz; lÞ expð2iuÞ; 2 G22 ðz; nÞ ¼ wðz; lÞ expð2iuÞ;

1 G20 ðz; nÞ ¼ pffiffiffi Qun ðz; lÞ; 2

ð59Þ ð60Þ

where the functions QW, Qun and w obey the following equations 

 Z Z o 0 0 0 l þ rtot QW ðz; lÞ ¼ n0 dn a1 ðnn Þ expð2iwÞQW ðz; l Þ þ n0 dn0 b1 ðnn0 Þ expð2iðb0 þ wÞÞW ðz; l0 Þ; oz     Z Z 0 0 o 0 a2 ðnn Þ þ a3 ðnn Þ 0 l þ rtot Qun ðz; lÞ ¼ n0 dn expð2ivþ ÞQun ðz; l Þ þ n0 dn0 b1 ðnn0 Þ expð2ibÞI scal ðz; l0 Þ; oz 2     Z 0 0 o 0 a2 ðnn Þ þ a3 ðnn Þ l þ rtot wðz; lÞ ¼ n0 dn expð2iðvþ þ wÞÞwðz; l0 Þ oz 2   Z 0 0 0 a2 ðnn Þ  a3 ðnn Þ þ n0 dn expð2iðv þ wÞÞW ðz; l0 Þ. 2

ð61Þ ð62Þ

ð63Þ

As long as the scattered waves do not forget the initial polarization of the incident beam, the contribution of the overtones to the polarization state of light is a small effect. With increasing depths the depolarization of the incident beam continues to escalate and the contribution of the overtones becomes significant.1 In the asymptotic state the additional modes QW, Qun and w can be easily expressed in terms of the coefficients of the basic mode expansions (see Eqs. (43)–(45)). In this case QW, Qun and w are given by (see Appendix C),

1 If the incident beam is unpolarized, quantities W and V equal zero and the polarization state of light is governed only by the contribution of the overtone G20 [24].

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E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

5C W n0 b1 ð2Þ P 2 ðlÞ expðW zÞ; 4p r þ ra  n0 a1 ð2Þ 20 5C I bI n0 b1 ð2Þ P 2 ðlÞ expðI zÞ; Qun ðz; lÞ

4p r þ ra  n0 aþ ð2Þ 20 5C W n0 a ð2Þ wðz; lÞ

P 2 ðlÞ expðW zÞ; 4p r þ ra  n0 aþ ð2Þ þ 23 W 22

QW ðz; lÞ

ð64Þ ð65Þ ð66Þ

where b1 ð2Þ ¼ 2p

Z

1 1

P 220 ðlÞ

dlb1 ðlÞP 220 ðlÞ;

pffiffiffi 6 ð1  l2 Þ; ¼ 4

a ð2Þ ¼ 2p

Z

1

dl 1

a2 ðlÞ  a3 ðlÞ 2 P 22 ðlÞ; 2

ð67Þ

1 P 222 ðlÞ ¼ ð1  lÞ2 . 4

Eqs. (64)–(66) allow us to obtain a more accurate result for the Stokes vector takes the form 1 0 1 0 I I scal þ QW cos 2u C B C B @ Q A ¼ @ ðW þ wÞ cos 2u þ Qun A. U

ð68Þ

ðW þ wÞ sin 2u

As follows from Eqs. (64)–(66), the contributions of the overtones to the polarization state incorporate the small factors, n0 b1 ð2Þ=ðr  n0 a1 ð2Þ þ ra Þ, n0 b1 ð2Þ=ðr  n0 aþ ð2Þ þ ra Þ and n0 a ð2Þ=ðr  n0 aþ ð2Þ þ ra þ 2W =3Þ (see Table 4 and Fig. 2). Hence, the overtone contributions to the transmitted intensity I and the Stokes parameter U are always small quantities. The contribution to the Stokes parameter Q becomes evident only at very large depths, where the incident beam completely depolarizes. With allowance for the overtones the degree of polarization for linearly polarized light can be written as (compare with Eq. (52))

Table 4 Integral parameters of the off-diagonal elements of the scattering matrix (25) for large scattering inhomogeneities

Latex spheres (x = 5) Latex spheres (x = 10) Water droplets (x = 5) Water droplets (x = 10) Cloud 1 [42] Sea water [43]

n0b1(2)/r

n0a-(2)/r

1.0 · 102 2.0 · 103 2.0 · 103 2.3 · 102 1.4 · 102 1.6 · 102

4.0 · 103 2.0 · 103 8.0 · 103 1.8 · 102 4.8 · 102 4.0 · 102

0.50 0.45

2 0.40 0.35 0.30 0.25

1 3

0.20 0.15 0.10 0.05 0.00

ka 0

2

4

6

8

10

12

14

Fig. 2. Ratios n0 b1 ð2Þ=ðr  n0 a1 ð2ÞÞ (curve 1), n0 b1 ð2Þ=ðr  n0 aþ ð2ÞÞ (curve 2) and n0 a ð2Þ=ðr  n0 aþ ð2Þ þ 2W =3Þ (curve 3) as functions of radius a of the scattering particles. Latex spheres in water (n = 1.19), no absorption.

E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

PL

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W 2 þ 2Ww cos 4u þ 2WQun cos 2u þ Q2un I scal

1

QW cos 2u . I scal

41

ð69Þ

As depth z increases, the incident light depolarizes (Qun  W) and PL tends to the degree of polarization of unpolarized light, PL Pun = Qun/Iscal. 5. Discussion The results obtained above allow us to understand how the parameters of scattering particles influence on the depolarization of light. The integral parameters of the scattering matrix for various media with large inhomogeneities are presented in Tables 1 and 4 (x = ka = 2pa/k). For spherical scatterers the numerical calculations were performed with the Mie formulas [1,32]. For clouds and sea water we use the numerical [42] and experimental [43] data. As follows from Tables 1 and 4 the integral parameters of the off-diagonal elements of the scattering matrix appear to be small as compared with the corresponding parameters of the diagonal elements. Therefore, the contribution of the additional modes to the Stokes parameters appears to be relatively small in all considered cases. The relationship between the ‘‘geometrical’’ and ‘‘dynamical’’ mechanisms of depolarization depends on the radius of scatterers, their refractive index and the spread of inhomogeneities in sizes and shapes (see Table 2). As the size of particles increases and the relative refractive index approaches the unity, the role of the ‘‘geometrical’’ mechanism increases. The spread of particles in sizes (Table 2, Cloud 1) and deviation of scattering inhomogeneities from the spherical shape (Table 2, sea water) enhance the role of the ‘‘dynamical’’ mechanism. This law can best be appreciated from comparison of the depolarization coefficients W and geom . As follows from Table 2, the effect of slow decay of W circular polarization is typical only for large spherical scatterers. For media with inhomogeneities distributed in sizes and shapes the attenuation coefficients for circularly (V) and linearly (W) polarized modes are of the same order of magnitude. It is of interest to compare the results obtained above with the experimental data [7,8] and the results of numerical simulations [7]. In the case rdep  rtr, the linearly polarized light is depolarized more rapidly as compared with the circularly polarized light. This effect follows at once from our calculations and is well distinguished in a medium with no absorption, in particular, in a suspension of latex particles in water [7,8]. As shown in Fig. 3, the ratio between the attenuation coefficients calculated from V/W with Eqs. (47), (49) (solid curve) is in good agreement with data of experiments and Monte Carlo calculations [7]. Our results for the z dependence of the degree of polarization also correlate well with experimental data [8]. In experiments [8], the degree of polarization for circularly and linearly polarized beams transmitted through a thick slab was measured. The degree of polarization as a function of the slab thickness L is shown in Fig. 4. Our results presented in Fig. 4 are based on Eqs. (44) and (45). Instead of Eq. (43) we used the well-known formula for a non-absorbing medium (ra = 0) [33,34]

ε V /ε

1.4

W

1.2

- Monte Carlo calculations - experiment

1.0

0.8

0.6

0.4

ka 0

2

4

6

8

10

12

Fig. 3. Ratio V/W for a water suspension of latex particles as a function of their radius (n = 1.19).

42

E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45 P 1. 0

- PC - PL

0. 8

0. 6

0. 4

0. 2

0. 0

0

1

2

3

4

5

L / l tr

Fig. 4. Degree of polarization as a function of the thickness of the scattering slab (a suspension of latex particles in water, ka = 7). Symbols s, correspond to experimental data [8].

I scal ðl ¼ 1Þ ¼

·

ltr þ z0 ; pðL þ 2z0 Þ

where z0 0.71ltr is the extrapolated length [33]. Under the conditions of experiment [8], the ‘‘geometrical’’ mechanism of depolarization prevails and therefore linear polarization decays rapidly as compared with circular polarization. 6. Conclusion We present the analytical method for solving the vector radiation transfer equation. The representation for the Stokes vector that was first proposed in [36] is used. Our approach is based on the assumption that single scattering of light by large-scale inhomogeneities occurs predominantly through relatively small angles and the off-diagonal elements of the phase matrix (see Eq. (26)) are small as compared with the diagonal ones. This approximation allows us to decouple the vector radiative transfer equation. In the first approximation, we derive three independent equations for the basic modes, namely, for the intensity and for the basic modes of linear and circular polarizations. In the succeeding approximation, allowance for the off-diagonal elements of the phase matrix results in the excitation of the additional modes (overtones). Within the framework of this approach, the asymptotic behavior of the Stokes parameters of multiply scattered light is studied in the limiting case of large depths. The validity of our results is illustrated by comparison with data of experiments [7,8] and Monte Carlo simulation [7]. Acknowledgements We thank A. Borovoi and A. Kokhanovsky for interest in this work and helpful discussions. This work was supported by the Ministry of Education of the Russian Federation, Project NE02-3.2-203 and by President program Support of leading scientific schools 5898.2003.02. Appendix A Consider the scattering of electromagnetic wave by a single particle. Let unit vectors n and n 0 be the directions of propagation of the scattered and incident waves, respectively. For spherical particles the relationship between the components of the scattered and incident electric fields is given by [1,32,33] ! !

ðscÞ ðiÞ Ek Ek expðikrÞ Ak 0 ¼ ; ð70Þ ðscÞ ðiÞ r 0 A? E E ?

?

where r is the distance between the observation point and the scattering particle, k is the wave number in the medium surrounding the particle, Ak and A? are the scattering amplitudes of waves polarized, respectively, parallel and perpendicularly to the scattering plane. The components of the scattered and incident electric fields are also defined with respect to the scattering plane. The amplitudes Ak and A? depend on the scattering angle c (cosc = nn 0 ) as well as on the properties (size, refractive index) of the scattering particle. The amplitudes are the primary quantities that determines the single-scattering law.

E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

43

Using Eq. (70), we calculate the Stokes parameters (see Eqs. (1) and (2)) of the scattered and incident waves and obtain the following relation between them: ðscÞ ðiÞ 1 b S ¼ 2 Fb ðcos cÞ b S ; ð71Þ r ðscÞ ðiÞ S are the Stokes parameters of where Fb ðcos cÞ is the scattering matrix with the elements defined by Eqs. (9)–(12), b S and b the scattered and incident waves defined with respect to the scattering plane {n,n 0 }. ðscÞ ðiÞ S defined with respect to the scattering plane into the ‘‘laboratory’’ refTransforming the Stokes parameters b S and b erence frames we derive the following equation ðscÞ ðiÞ 1b b S ðnÞ ¼ 2 Z ðn; n0 Þ b S ðn0 Þ; r

ð72Þ

where b ðn; n0 Þ ¼ b Lðp  bÞ Fb ðcos cÞ b Lðb0 Þ. Z

ð73Þ

ðscÞ

The Stokes vector Sb ðnÞ is defined with respect to the system of unit vectors {ek, e?, n} (or, with respect to the reference ðiÞ S ðn0 Þ is defined with respect to the system of unit vectors fe0k ; e0? ; n0 g (or, with respect to plane {n,n0}). The Stokes vector b the reference plane {n 0 , n0}). Angles b and b 0 are defined by Eq. (7). For nonspherical particles, the off-diagonal elements of the amplitude matrix (see Eq. (70)) appears to be nonzero. However, the block-diagonal structure of the scattering matrix (see Eq. (8)) remains unchanged. The block-diagonal matrix results from averaging over orientations of particles [1,33]. In the case of nonspherical particles the diagonal elements a1, a2, a3 and a4 may differ from each other. Appendix B A solution to the radiative transfer equation can be written as 1 X ðnÞ ðnÞ ðnÞ I scal ðz; lÞ ¼ C I UI ðlÞ expðI zÞ;

ð74Þ

n¼0 ðnÞ

ðnÞ

where I and UI ðlÞ are the nth eigenvalue and the corresponding eigenfunction. In the asymptotic state the contribution from the term with n = 0 to Eq. (74) survives only. With allowance for the boundary condition 1 dð1  lÞ; l > 0; ð75Þ I scal ðz ¼ 0; lÞ ¼ 2p Rðl; 1Þ; l < 0; where R(l,1) is the reflectivity of the medium, and the orthogonality of the eigenfunctions we can obtain the following equality Z 1 Z 0 Z 0  2  2 ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ UI ðl ¼ 1Þ þ 2p ldlRðl; 1ÞUI ðlÞ ¼ 2pC I ldl U0 ðlÞ þ 2pC I ldl U0 ðlÞ . ð76Þ 1

1

0

Assuming that the ‘‘reflective’’ contributions in Eq. (76) cancel each other we find ð0Þ

ð0Þ

CI ¼

U0 ðl ¼ 1Þ  2 . R1 ð0Þ 2p 0 ldl U0 ðlÞ

ð77Þ

Comparison with the numerical results [33] shows that the maximum error of Eq. (77) is less than 20%. For ra > 0.1rtr the error is less than 10% (see Table 5). The data presented in Table 5 are obtained for the Henyey–Greenstein phase function [33] with the mean cosine hcos ci ¼ 0:875. Table 5 Numerical values of the coefficient CI for an absorbing medium r/rtot ra/rtr ð0Þ C I Eq. (77) ð0Þ C I [33]

0.6 3.2 2.50 2.52

0.8 1.6 2.64 2.66

0.9 0.9 2.82 2.86

0.95 0.4 3.02 3.12

0.99 0.08 3.46 3.82

44

E.E. Gorodnichev et al. / Optics Communications 260 (2006) 30–45

Appendix C The overtones QW, Qun and w can be presented as expansions in the generalized spherical functions X 2l þ 1 X 2l þ 1 QW ðz; lÞP l20 ðlÞ; Qun ðz; lÞ ¼ Qun ðz; lÞP l20 ðlÞ; QW ðz; lÞ ¼ 4p 4p l¼2 l¼2 X 2l þ 1 wðz; lÞP l22 ðlÞ. wðz; lÞ ¼ 4p l¼2

ð78Þ

Substitution of Eqs. (43), (45) and (78) into Eqs. (61)–(63) gives pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl þ 2Þðl  2Þ oQW ðz; l  1Þ ðl þ 3Þðl  1Þ oQW ðz; l þ 1Þ ð79Þ þ þ ½ðr  n0 a1 ðlÞÞ þ ra QW ðz; lÞ ¼ n0 b1 ðlÞW ðz; lÞ; oz oz ð2l þ 1Þ ð2l þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl þ 2Þðl  2Þ oQun ðz; l  1Þ ðl þ 3Þðl  1Þ oQun ðz; l þ 1Þ ð80Þ þ þ ½ðr  n0 aþ ðlÞÞ þ ra Qun ðz; lÞ ¼ n0 b1 ðlÞI scal ðz; lÞ; oz oz ð2l þ 1Þ ð2l þ 1Þ ðl þ 2Þðl  2Þ owðz; l  1Þ 4 owðz; lÞ ðl þ 3Þðl  1Þ owðz; l þ 1Þ  þ þ ½ðr  n0 aþ ðlÞÞ þ ra wðz; lÞ ¼ n0 a ðlÞW ðz; lÞ. lð2l þ 1Þ oz lðl þ 1Þ oz ðl þ 1Þð2l þ 1Þ oz ð81Þ In the asymptotic state the contribution from the first terms (l = 2) to expansions (78) plays a dominant role. Substituting the asymptotic expansions of Iscal and W into Eqs. (79)–(81) we derive at Eqs. (64)–(66). References [1] M.I. Mishchenko, J.W. Hovenier, L.D. Travis (Eds.), Light Scattering by Nonspherical Particles. Theory, Measurements and Applications, Academic Press, New York, 2000. [2] C.E. Mandt, L. Tsang, A. Ishimaru, JOSA A7 (1990) 585. [3] S.G. Demos, R.R. Alfano, Appl. Opt. 36 (1997) 150. [4] G.W. Kattawar, M.J. Rakovic, B.D. Cameron, Adv. Opt. Imag. Photon Migrat. 21 (1998) 105. [5] V.V. Tuchin, Laser Phys. 8 (1998) 807. [6] F.C. MacKintosh, J.X. Zhu, D.J. Pine, D.A. Weitz, Phys. Rev. B 40 (1989) 9342. [7] D. Bicout, C. Brosseau, A.S. Martinez, J.M. Schmitt, Phys. Rev. E 49 (1994) 1767. [8] V. Sankaran, M.J. Everett, D.J. Maitland, J.T. Walsh, Opt. Lett. 24 (1999) 1044. [9] N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, B.N. Singh, Opt. Commun. 222 (2003) 93. [10] D.A. Zimnyakov, Waves Random Media 10 (2000) 417. [11] A.D. Kim, M. Moscoso, Phys. Rev. E 64 (2001) 026612-1. [12] E.P. Zege, L.I. Chaikovskaya, Zh. Prikl. Spektrosk. 44 (1986) 996. [13] M.I. Mishchenko, Transp. Theory Stat. Phys. 19 (1990) 293. [14] J.M. Schmitt, A.H. Gandjbakhche, R.F. Bonner, Appl. Opt. 31 (1992) 6535. [15] P. Bruscaglioni, G. Zaccanti, A. Wei, Appl. Opt. 32 (1993) 6142. [16] E.P. Zege, L.I. Chaikovskaya, JQSRT 66 (2000) 413. [17] A.A. Kokhanovsky, JOSA 18 (2001) 883. [18] W. Cai, M. Lax, R.R. Alfano, Phys. Rev. E 63 (2000) 016606-1. [19] B.D. Cameron, M.J. Rakovic, M. Mehrubeoglu, et al., Opt. Lett. 23 (1998) 485. [20] S. Mujumdar, H. Ramachandran, Opt. Commun. 241 (2004) 1. [21] G. Yao, Opt. Commun. 241 (2004) 255. [22] N. Glosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, R.P. Singh, Phys. Rev. E 70 (2004) 066607. [23] Xiaohui Ni, R.R. Alfano, Opt. Lett. 29 (2004) 2773. [24] E.E. Gorodnichev, D.B. Rogozkin, JETP 80 (1995) 112. [25] E.E. Gorodnichev, A.I. Kuzovlev, D.B. Rogozkin, JETP Lett. 68 (1998) 22. [26] E.E. Gorodnichev, A.I. Kuzovlev, D.B. Rogozkin, Laser Phys. 9 (1999) 1210. [27] E.E. Gorodnichev, A.I. Kuzovlev, D.B. Rogozkin, JETP 88 (1999) 421. [28] E.E. Gorodnichev, A.I. Kuzovlev, D.B. Rogozkin, Opt. Spectrosc. 94 (2003) 273. [29] Yu.A. Kravtsov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 13 (1970) 281. [30] L.D. Landau, E.M. Lifshits, Electrodynamics of Continuous Media, Pergamon Press, New York, 1984 (Chapter 10, Section 85). [31] V.I. Tatarskii, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10 (1967) 1762. [32] R.G. Newton, Scattering Theory of Waves and Particles, McGraw-Hill, New York, 1966. [33] H.C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications, Academic Press, New York, 1980. [34] A. IshimaruWave Propagation and Scattering in Random Media, vols. 1,2, Academic Press, New York, 1978. [35] L.D. Landau, E.M. Lifshits, The Classical Theory of Fields, Pergamon Press, Oxford, 1975. [36] I. Kuscer, M. Ribaric, Opt. Acta 6 (1959) 42. [37] H. Domke, Astrofizika 15 (1975) 205. [38] A.A. Kokhanovsky, Polarization Optics of Random Media, Praxis Publishing, 2003.

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