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Derivation of an equivalent Mueller matrix associated to an absorbing, emitting and multiply scattering plane medium
Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
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Derivation of an equivalent Mueller matrix associated to an absorbing, emitting and multiply scattering plane medium R. Vaillon Centre de Thermique de Lyon (CETHIL), UMR CNRS 5008, INSA de Lyon, 69621 Villeurbanne Cedex, France
Abstract Measurements of scattering matrix of particles in a medium are likely to provide su/cient information on size, structure and optical properties of particles through the confrontation with appropriate and accurate theoretical models. To allow direct comparison with measurements, when the semi-transparent medium is not optically thin, nonhomogeneous, absorbing, emitting and multiply scattering, the whole system may be modelled as a single optical device and an equivalent Mueller matrix may be derived. This is performed in the case of a plane parallel semi-transparent medium containing randomly oriented symmetric particles. The transfer of polarized light is solved by using the vector discrete ordinates method. The Mueller matrix elements are obtained as functions of directions of transmitted or re3ected radiation from the simulation of four virtual experiments. The e4ects of temperature and radiative properties on the Mueller matrix are investigated through the comparison of its elements with those of the scattering matrix of particles. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Mueller matrix; Multiple scattering; Particles characterization; Polarization; VDOM
1. Introduction Analysis of polarization state of radiation by theoretical and experimental methods has been carried out for characterization of scattering media in many diverse ;elds such as astrophysics, atmospheric radiation, remote sensing, combustion and heat transfer. Recent studies account for the nonspherical shape of many particles and thus leading to new particle diagnostic methods [1– 4]. A state-of-the-art description of the subject is given in the recent treatise by Mishchenko et al. [1]. The interaction of a particle or an assembly of particles with a beam of polarized light can be described as a transformation of the incident Stokes parameters into transformed ones thanks to E-mail address: [email protected] (R. Vaillon). 0022-4073/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 1 ) 0 0 2 2 6 - 6
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R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
Nomenclature I Iˆ I; Q; U; V Kabs Kext L M Ib J M; N P S SR w Z
Stokes parameters vector (W m−2 m−1 ) Stokes intensities vector (W m−2 m−1 sr −1 ) Stokes parameters (W m−2 m−1 ) (1 × 4) absorption (emission) vector (m−1 ) (4 × 4) extinction matrix (m−1 ) (4 × 4) rotation matrix (4 × 4) Mueller matrix Planck function (W m−2 m−1 sr −1 ) number of sub-layers number of azimuths and latitudes degree of polarization (4 × 4) scattering matrix (4 × 4) rotated scattering matrix quadrature weight (4 × 4) phase matrix
Greek letters = cos ’ ’
direction cosine polar angle azimuthal angle direction of propagation scattering angle
Subscript abs ext inc part sca surr t 0
absorption extinction incident radiation relative to the particles scattering (scattered radiation) relative to the purely absorbing surroundings transformed (transmitted or re3ected) radiation relative to the incident beam
the scattering matrix which contains information related to the optical properties, size, shape and composition of particles. Consequently, a detailed measurement and interpretation of scattering matrix elements may be used to characterize the physical properties of particles [2– 4]. If the sample is optically thin, the comparison of the scattering matrix deduced from theory with measurements is valid. However, if the medium is optically thick and nonhomogeneous, the model must take into account the transfer of polarized light inside the sample as it is expected that
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
149
multiple scattering, internal thermal emission for hot media, absorption by the surroundings may alter polarization information. These e4ects must be quanti;ed and may be inserted into an equivalent Mueller matrix accounting for the whole system to allow a direct comparison with measurements. A method of derivation of such a matrix is presented in the case of a plane-parallel semi-transparent sample containing randomly oriented symmetric particles. The theoretical formulation related to the experimental situation is presented as well as the basic concepts of the change of polarization induced by scattering and its e4ect on the polarized-vector-radiative transfer equation (VRTE). The third part deals with the numerical derivation of the Mueller matrix elements obtained from the simulation of four virtual experimental situations and of the implementation of the vector discrete ordinates method (VDOM) as a VRTE solution method. The numerical model is validated against reference results and other solution methods. Mueller matrix elements are analyzed as functions of directions of transmitted or re3ected radiation for various medium temperatures and radiative properties. For the sake of simplicity, only the small spherical particles have been considered, i.e., the Rayleigh scattering matrices are used.
2. Theoretical formulation 2.1. Modelling of the experimental setup Experimental systems used to measure the scattering matrix elements of an assembly of particles generally consist of a source, a set of optics and a detector (Fig. 1). Some optics serve to direct the beam to the medium and to collect the transformed beam and others are used to modify the polarization states of the beam before and after it crosses the sample [4,5]. If the medium is optically thin, measurements can be directly compared with the theoretical scattering matrix of the particles assembly without any restriction. However, if the medium is optically thick, single scattering cannot be further considered: multiply scattering, emission for hot media and absorption may alter polarization information, so that the transfer of polarized light must be included in the modelling. As a consequence, without changing the experimental procedure, the
Fig. 1. Schematic of a scattering or Mueller matrix measurement setup.
150
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
semi-transparent medium may be considered as a single optical device and its Mueller matrix may be derived and then directly compared with measurements. For the sake of simplicity, we consider a plane parallel semi-in;nite sample of thickness zmax which emits, absorbs and scatters radiation (Fig. 1). In order to separate the e4ects of boundary emission and re3ection on the change in polarization of radiation inside the medium, both boundaries are assumed transparent. Additional absorption, re3ection, transmission, and refraction phenomena at the boundaries will be included in future studies. The (z=0) boundary is subjected to a normal collimated irradiation representing the incident beam. The objective is to calculate the elements of the Mueller matrix, which represents the change of polarization state of radiation through the relation: It (’; ) = M (’; ; particles and slab radiative properties;: : : ;)Iinc (’0 ;
0 );
(1)
where It (’; ) is the transformed (transmitted at z = zmax or re3ected at z = 0) Stokes vector, Iinc (’0 ; 0 ) is the incident Stokes vector. M (’; ) is the Mueller matrix associated to the sample and depends on the direction of observation (’; ) and contains the information related to the optical properties, morphology, structure and orientations of particles and may also depend on other radiative properties related to the slab. Thus, the transformed Stokes vector will be obtained from a solution of the polarized radiative transfer which includes local polarization changes due to particles via the rotated scattering matrix concept. 2.2. Rotated scattering matrix A monochromatic transverse electromagnetic wave is fully described by the Stokes parameters, which constitute the Stokes vector I = {I; Q; U; V } (W m−2 m−1 ), where for simplicity, wavelength dependence is omitted. Complete de;nitions and basic properties of Stokes parameters can be found in Bohren and Hu4man [6]. After a scattering process by a single particle or a small volume-element of particles, the state of polarization of the incident wave is modi;ed, and the relation between incident and scattered Stokes parameters is given by [6] Isca =
1 SI inc ; kw R 2
(2)
where Iinc and Isca are the incident and scattered Stokes vectors, respectively, de;ned with respect to the scattering plane (containing the incident and scattered directions). kw is the wave number and R is the distance from the observation point to the small volume. S is the scattering matrix, which depends on the incident and scattered wave directions, on the electromagnetic properties and on the size and the shape of the particle. Fundamental properties of the scattering matrix are given by van de Hulst [7] and Hovenier and van der Mee [8]. To describe the transfer of polarized light, Stokes vectors are de;ned in the (x; y; z) ;xed reference frame. We consider a right-handed Cartesian coordinates system, so that Stokes vectors are de;ned with respect to the meridian plane containing the direction of propagation (Fig. 2). The relation between the incident and scattered Stokes vectors becomes [8] Isca =
1 1 SR Iinc = [L( − i2 )SL(−i1 )]Iinc ; 2 kw R kw R2
(3)
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
151
Fig. 2. Reference frames for polarization.
where SR is called the rotated scattering matrix, i1 is the angle between the meridian plane (O; inc ; z) and the scattering plane (O; inc ; sca ); i2 is the angle between the meridian plane (O; sca ; z) and the scattering plane. L is a rotation matrix which depends on the rotation angles cosines expressed as function of the polar angle ’ relative to the z coordinate axis, of the azimuth of the waves propagation directions and of the scattering angle [1,8] (Fig. 2). 2.3. Vector radiative transfer equation (VRTE) Scattering is assumed to be independent and the Stokes parameters can be incoherently added without restrictions [7]. This allows the use of the radiative transfer equation to solve problems of multiple light scattering by an assembly of particles. In this frame, the VRTE, which expresses the variations of Stokes parameters along path ds in the direction of propagation , in an emitting– absorbing and scattering media, is given by [9] d Iˆ(s; ) ˆ Z (s; ; )Iˆ(s; ) d ; (4) = −Kext (s; )I (s; ) + Kabs (s; )Ib (s) + ds 4 where Iˆ(s; ) is the Stokes intensities vector (W m−2 m−1 sr −1 ) at point s for the direction , Kext (s; ) is the average extinction matrix at point s for the direction , Kabs (s; ) is the average absorption (emission) vector at point s for the direction , Ib (s) is the Planck function at point s, Z (s; ; ) is the normalized average phase matrix [1] at point s for radiation coming from direction and scattered in the direction . The averaging is necessary when the medium is comprised of particles with di4erent sizes, shapes and refractive indices, and should be performed considering a proper normalization scheme [1]. In the case of randomly oriented symmetric particles, the radiation extinction process is simpli;ed while it is independent of the direction and polarization of the incident beam. As a consequence, the extinction matrix has a diagonal structure [1,9] Kext = diag{; ; ; }:
(5)
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=part +surr (m−1 ) is the extinction coe/cient related to particles and possibly to purely absorbing surrounding medium (gas, glass). The absorption vector, simply written as [1,9] Kabs = (part + surr ){1; 0; 0; 0}t ;
(6)
takes into account the absorption by the medium as well as by the particles. The normalized average phase matrix is given by [1] s s SR = L( − i2 )SL(−i1 ); (7) Z= 4 4 where s (m−1 ) is the scattering coe/cient. In the case of randomly oriented symmetric particles, the scattering matrix has a block-diagonal structure with six independent elements [1,7] a1 () b1 () 0 0 b () a () 0 0 2 1 S() = (8) ; 0 0 a3 () b2 () 0 0 −b2 () a4 () where a1 () is the phase function which satis;es the normalization condition [9]. To solve the problem, the collimated beam may be separated from all other contributions to the Stokes vector in the slab [10]. Boundary conditions associated to the experimental con;guration and in accordance with our assumptions are simply given by Iˆ(z = 0 (resp: zmax ); = cos ’ ¿ 0 (resp: ¡ 0); ) = {0; 0; 0; 0}t :
(9)
3. Calculation of the equivalent Mueller matrix 3.1. Matrix elements The Mueller matrix elements are obtained through the simulation of four successive virtual experiments corresponding to four polarization states of the incident beam : unpolarized (UP; ◦ ◦ Iinc = (1; 0; 0; 0)Iinc ), 0 linearly polarized (LP1; Iinc = (1; 1; 0; 0)Iinc ); 45 linearly polarized (LP2; Iinc = (1; 0; 1; 0)Iinc ), and right circularly polarized (RCP; Iinc = (1; 0; 0; 1)Iinc ). Taking into account the relation existing between the incident and transformed Stokes vectors via the equivalent Mueller matrix (Eq. (1)), the transformed Stokes parameters are given by 1 m11 0 m 21 It; UP = M Iinc = (10) Iinc ; 0 m31 0 m41 1 m11 + m12 m12 1 m +m m 22 21 22 I = I + It; LP1 = M Iinc = inc Iinc ; t; UP 0 m31 + m32 m32 0 m41 + m42 m42
(11)
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
1 m11 + m13 m13 0 m +m m 23 21 23 It; LP2 = M Iinc = Iinc = It; UP + Iinc ; 1 m31 + m33 m33 0 m41 + m43 m43 1 m11 + m14 m14 0 m +m m 24 21 24 It; RCP = M Iinc = Iinc = It; UP + Iinc ; 0 m31 + m34 m34 m41 + m44 m44 1
153
(12)
(13)
where the dependence on incident and transformed directions have been omitted for simplicity. Mueller matrix elements are simply deduced by: m11 m12 m13 m14 m 21 m22 m23 m24 M = m31 m32 m33 m34 m41 m42 m43 m44 =
1 ((It; UP )(It; LP1 − It; UP )(It; LP2 − It; UP )(It; RCP − It; UP )): Iinc
(14)
The transformed (transmitted or re3ected) Stokes-3ux densities-vectors (It (W m−2 m−1 )) comes from the Stokes-intensities-vectors (Iˆt (W m−2 sr −1 m−1 )) which are evaluated by solving the VRTE for the four selected polarization states of the incident beam. This is performed by using the VDOM. 3.2. VRTE solution using VDOM Recently, we have developed a VDOM code [11] to solve the problem using our previous experience concerning DOM [12]. The mains steps involved in the VDOM formulation are as following. Spatial domain is discretized into J -sub-layers. The discretization of the angular domain is performed using a PCA angular quadrature [13], which divides the domains of the polar and azimuthal angles into N latitudes each having M azimuths. The resulting form of the VRTE (Eq. (4)) is given by P zj "j Iˆjp − p (15) wp Zjpp Iˆjp + Zjp0 Iinc e 0 ; = −Kext; jp Iˆjp + Kabs; jp Ib; j + "z p =1 where j and p are indices, respectively, corresponding to the sub-layer and to the direction p under consideration, the notation "j Iˆjp is the di4erence of intensity vectors between the top and the bottom of the sub-layer j, the integral over directions in the in-scattering source term has been replaced by a numerical quadrature, (wp )p∈[1; NM ] are the weights of the quadrature, and the last term represents the scattering in direction p of the attenuated Stokes (3ux) vector of the beam (Iinc ) at sub-layer j.
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Because of the replacement of angular integrals by numerical quadratures, the in-scattering term of Eq. (15) has to be adjusted to ensure the normalization condition for the phase function by using a corrective factor for each direction of the quadrature set [9]. Other numerical features related to the quadrature set and spatial interpolation are described in Ref. [11]. The set of four coupled equations (15) is solved iteratively for each direction of the quadrature to give re3ected It (z = 0; =2 ¡ ’ ¡ ) and transmitted It (z = zmax ; 0 ¡ ’ ¡ =2) Stokes intensity vectors. Stokes-3ux densities-vectors are deduced through the relation It (m ) = wm | m |Iˆt (m ). The procedure is repeated for the four polarization states of the incident beam used for the determination of the equivalent Mueller matrix elements.
4. Results and discussion 4.1. Validation Tests of our VDOM code have been performed recently against analytic solutions for the four Stokes intensities in the case of a purely Raleigh scattering atmosphere [14]. Predictions of our VRTE solver showed a very good agreement with the reference results [11]. Cases where internal emission is included con;rmed these good performances by comparing our results with those of the adding–doubling method by Evans and Stephens [15]. 4.2. E@ects of the radiative properties on the Mueller matrix For the sake of simplicity, we consider Rayleigh scattering and no absorption in the medium (surr = 0). Temperature (Tm ) and radiative properties are assumed to be uniform in the layer. To evaluate the in3uence of radiative properties on the calculated Mueller matrix, variations of its elements with the transformed direction (’; ) are compared with those of the phase matrix Z (Eq. (7)). Since our interest is in the di4use part of the transformed radiation, each element of a matrix A is normalized by a11 evaluated for the direction of the angular discretization which is the closest to the incident direction. It is straightforward that the resulting Mueller matrices conserve the block diagonal structure of the original scattering matrix (Eq. (8)). Since the results provide a great amount of information, only the case of normal incidence (’0 = ◦ 0 ) of the beam is treated here. Fig. 3(a) represents the angular variations of element m11 of the equivalent-normalized-Mueller matrix when the optical thickness of the slab (&max =zmax ) is equal to unity. This element accounts for the angular distribution of the transmitted and re3ected radiations. The e4ects of multiple scattering and absorption, which depend on the radiation paths through the layer, are signi;cant. Di4erences between transmitted and re3ected-backscattered-radiations patterns are observed, also in comparison with the phase function of Rayleigh single scattering. For identical ◦ reasons, m12 , m21 and m22 are a4ected, more particularly for polar angles near 90 for which radiation paths are longer. The equalities m21 = m12 and m11 = m22 observed for the original scattering matrix are broken (Fig. 3(a), (b) and (d)). This trend is not encountered when the optical thickness of the slab is much lower (Fig. 4(a)): variations of m11 for &max = 0:1 are identical to those of the Rayleigh ◦ scattering phase function, except for angles near 90 . It is shown that Mueller matrix elements (m11
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
155
Fig. 3. Normalized Mueller matrix elements [(a) m11 ; (b) m12 — solid symbols — and m21 — open symbols —; (d) m22 ] and degree of polarization (c) of transmitted and re3ected radiations in the case of Rayleigh scattering with &max = 1 and 0 = 1.
and m22 in Fig. 4(a)) deviate from those of the Rayleigh scattering matrix when the optical thickness of the slab increases. The in3uence of the optical thickness on the degree of polarization is shown in Fig. 4(b). The results are quite meaningful since a deviation of m22 from m11 is generally considered as an indication of the presence of nonspherical particles [1,6]: it is obvious here that such a conclusion would lead to a great mistake. The same tendencies, not shown here, are observed for the m33 and m44 diagonal elements. A case where thermal emission is present (Tm = 400 K; ! = s = = 0:5), reported in Fig. 3, exhibits comparative behaviours. The global e4ect of multiple scattering, absorption and internal thermal emission on the degree of polarization is depicted in Fig. 3(c).
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R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
Fig. 4. Normalized Mueller matrix elements [(a) m11 — solid symbols — and m22 — open symbols —] and degree of polarization (b) of transmitted and re3ected radiations in the case of Rayleigh scattering with 0 = 1 for various sample optical thicknesses.
5. Conclusion We have presented a method of determination of the Mueller matrix associated to a semitransparent plane parallel sample containing randomly oriented symmetric particles. The procedure is based on the simulation of four virtual experiments of transfer of polarized light through the sample by using a VDOM code. The e4ects of multiple scattering, absorption and emission in connection with the geometry of the sample have been proved to be very signi;cant on the Mueller matrix elements. In the future, the e4ects of boundary re3ection, absorption and transmission will be included and the analysis will be extended to the larger Lorenz–Mie spheres and the nonspherical particles. References [1] Mishchenko MI, Hovenier JW, Travis LD. Light scattering by nonspherical particles. San Diego: Academic Press, 2000. [2] Volten H, Mu˜noz O, Rol E, de Haan J, Vassen W, Hovenier JW. Laboratory measurements of scattering matrices of irregular mineral particles. Proceedings of the Fifth International Conference on Light Scattering by Nonspherical Particles, Halifax, 2000. p. 41– 44. [3] Manickavasagam S, MengRucS MP. Scattering matrix elements of fractal-like soot agglomerates. Appl Opt 1997;36:1337–51. [4] Manickavasagam S, Govindan R, MengRucS MP. Estimating the morphology of soot agglomerates by measuring their scattering matrix elements. In: Proceedings of the ASME Heat Transfer Division, HTD 352, Vol. 2, 1997. [5] Kuik F, Stammes P, Hovenier JW. Experimental determination of scattering matrices of water droplets and quartz particles. Appl Opt 1991;30:4872–81. [6] Bohren F, Hu4man DR. Absorption and scattering of light by small particles. New York: Wiley, 1983. [7] van de Hulst HC. Light scattering of small particles. New York: Wiley, 1957.
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[8] Hovenier JW, van der Mee CVM. Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere. Astron Astrophys 1983;128:1–16. [9] Haferman JL. A polarized multi-dimensional discrete-ordinates radiative transfer model for remote sensing applications. Ph.D. thesis, University of Iowa, Iowa City, Iowa, 1995. [10] Modest MF. Oblique collimated irradiation of an absorbing, scattering, plane-parallel layer. JQSRT 1991;45:309–12. [11] Vaillon R, Sacadura JF, MengRucS MP. Analysis of polarization state of radiation intensity in an absorbing, emitting and scattering medium. Proceedings of the Third European Thermal Sciences Conference, Heidelberg, Germany, 2000. p. 593–598. [12] Vaillon R, Lallemand M, Lemonnier D. Radiative heat transfer in orthogonal curvilinear coordinates using the discrete ordinates method. JQSRT 1996;55:7–17. [13] Fiveland WA. Discrete ordinates methods for radiative heat transfer in isotropically and anisotropically scattering media. ASME J Heat Transfer 1987;109:809–12. [14] Coulson KL, Dave JV, Sekera J. Tables related to radiation emerging from a planetary atmosphere with Rayleigh scattering. Berkeley: University of California Press, 1960. [15] Evans KF, Stephens GL. A new polarized atmospheric radiative transfer model. JQSRT 1991;46:413–23.
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Derivation of an equivalent Mueller matrix associated to an absorbing, emitting and multiply scattering plane medium R. Vaillon Centre de Thermique de Lyon (CETHIL), UMR CNRS 5008, INSA de Lyon, 69621 Villeurbanne Cedex, France
Abstract Measurements of scattering matrix of particles in a medium are likely to provide su/cient information on size, structure and optical properties of particles through the confrontation with appropriate and accurate theoretical models. To allow direct comparison with measurements, when the semi-transparent medium is not optically thin, nonhomogeneous, absorbing, emitting and multiply scattering, the whole system may be modelled as a single optical device and an equivalent Mueller matrix may be derived. This is performed in the case of a plane parallel semi-transparent medium containing randomly oriented symmetric particles. The transfer of polarized light is solved by using the vector discrete ordinates method. The Mueller matrix elements are obtained as functions of directions of transmitted or re3ected radiation from the simulation of four virtual experiments. The e4ects of temperature and radiative properties on the Mueller matrix are investigated through the comparison of its elements with those of the scattering matrix of particles. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Mueller matrix; Multiple scattering; Particles characterization; Polarization; VDOM
1. Introduction Analysis of polarization state of radiation by theoretical and experimental methods has been carried out for characterization of scattering media in many diverse ;elds such as astrophysics, atmospheric radiation, remote sensing, combustion and heat transfer. Recent studies account for the nonspherical shape of many particles and thus leading to new particle diagnostic methods [1– 4]. A state-of-the-art description of the subject is given in the recent treatise by Mishchenko et al. [1]. The interaction of a particle or an assembly of particles with a beam of polarized light can be described as a transformation of the incident Stokes parameters into transformed ones thanks to E-mail address: [email protected] (R. Vaillon). 0022-4073/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 1 ) 0 0 2 2 6 - 6
148
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
Nomenclature I Iˆ I; Q; U; V Kabs Kext L M Ib J M; N P S SR w Z
Stokes parameters vector (W m−2 m−1 ) Stokes intensities vector (W m−2 m−1 sr −1 ) Stokes parameters (W m−2 m−1 ) (1 × 4) absorption (emission) vector (m−1 ) (4 × 4) extinction matrix (m−1 ) (4 × 4) rotation matrix (4 × 4) Mueller matrix Planck function (W m−2 m−1 sr −1 ) number of sub-layers number of azimuths and latitudes degree of polarization (4 × 4) scattering matrix (4 × 4) rotated scattering matrix quadrature weight (4 × 4) phase matrix
Greek letters = cos ’ ’
direction cosine polar angle azimuthal angle direction of propagation scattering angle
Subscript abs ext inc part sca surr t 0
absorption extinction incident radiation relative to the particles scattering (scattered radiation) relative to the purely absorbing surroundings transformed (transmitted or re3ected) radiation relative to the incident beam
the scattering matrix which contains information related to the optical properties, size, shape and composition of particles. Consequently, a detailed measurement and interpretation of scattering matrix elements may be used to characterize the physical properties of particles [2– 4]. If the sample is optically thin, the comparison of the scattering matrix deduced from theory with measurements is valid. However, if the medium is optically thick and nonhomogeneous, the model must take into account the transfer of polarized light inside the sample as it is expected that
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
149
multiple scattering, internal thermal emission for hot media, absorption by the surroundings may alter polarization information. These e4ects must be quanti;ed and may be inserted into an equivalent Mueller matrix accounting for the whole system to allow a direct comparison with measurements. A method of derivation of such a matrix is presented in the case of a plane-parallel semi-transparent sample containing randomly oriented symmetric particles. The theoretical formulation related to the experimental situation is presented as well as the basic concepts of the change of polarization induced by scattering and its e4ect on the polarized-vector-radiative transfer equation (VRTE). The third part deals with the numerical derivation of the Mueller matrix elements obtained from the simulation of four virtual experimental situations and of the implementation of the vector discrete ordinates method (VDOM) as a VRTE solution method. The numerical model is validated against reference results and other solution methods. Mueller matrix elements are analyzed as functions of directions of transmitted or re3ected radiation for various medium temperatures and radiative properties. For the sake of simplicity, only the small spherical particles have been considered, i.e., the Rayleigh scattering matrices are used.
2. Theoretical formulation 2.1. Modelling of the experimental setup Experimental systems used to measure the scattering matrix elements of an assembly of particles generally consist of a source, a set of optics and a detector (Fig. 1). Some optics serve to direct the beam to the medium and to collect the transformed beam and others are used to modify the polarization states of the beam before and after it crosses the sample [4,5]. If the medium is optically thin, measurements can be directly compared with the theoretical scattering matrix of the particles assembly without any restriction. However, if the medium is optically thick, single scattering cannot be further considered: multiply scattering, emission for hot media and absorption may alter polarization information, so that the transfer of polarized light must be included in the modelling. As a consequence, without changing the experimental procedure, the
Fig. 1. Schematic of a scattering or Mueller matrix measurement setup.
150
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
semi-transparent medium may be considered as a single optical device and its Mueller matrix may be derived and then directly compared with measurements. For the sake of simplicity, we consider a plane parallel semi-in;nite sample of thickness zmax which emits, absorbs and scatters radiation (Fig. 1). In order to separate the e4ects of boundary emission and re3ection on the change in polarization of radiation inside the medium, both boundaries are assumed transparent. Additional absorption, re3ection, transmission, and refraction phenomena at the boundaries will be included in future studies. The (z=0) boundary is subjected to a normal collimated irradiation representing the incident beam. The objective is to calculate the elements of the Mueller matrix, which represents the change of polarization state of radiation through the relation: It (’; ) = M (’; ; particles and slab radiative properties;: : : ;)Iinc (’0 ;
0 );
(1)
where It (’; ) is the transformed (transmitted at z = zmax or re3ected at z = 0) Stokes vector, Iinc (’0 ; 0 ) is the incident Stokes vector. M (’; ) is the Mueller matrix associated to the sample and depends on the direction of observation (’; ) and contains the information related to the optical properties, morphology, structure and orientations of particles and may also depend on other radiative properties related to the slab. Thus, the transformed Stokes vector will be obtained from a solution of the polarized radiative transfer which includes local polarization changes due to particles via the rotated scattering matrix concept. 2.2. Rotated scattering matrix A monochromatic transverse electromagnetic wave is fully described by the Stokes parameters, which constitute the Stokes vector I = {I; Q; U; V } (W m−2 m−1 ), where for simplicity, wavelength dependence is omitted. Complete de;nitions and basic properties of Stokes parameters can be found in Bohren and Hu4man [6]. After a scattering process by a single particle or a small volume-element of particles, the state of polarization of the incident wave is modi;ed, and the relation between incident and scattered Stokes parameters is given by [6] Isca =
1 SI inc ; kw R 2
(2)
where Iinc and Isca are the incident and scattered Stokes vectors, respectively, de;ned with respect to the scattering plane (containing the incident and scattered directions). kw is the wave number and R is the distance from the observation point to the small volume. S is the scattering matrix, which depends on the incident and scattered wave directions, on the electromagnetic properties and on the size and the shape of the particle. Fundamental properties of the scattering matrix are given by van de Hulst [7] and Hovenier and van der Mee [8]. To describe the transfer of polarized light, Stokes vectors are de;ned in the (x; y; z) ;xed reference frame. We consider a right-handed Cartesian coordinates system, so that Stokes vectors are de;ned with respect to the meridian plane containing the direction of propagation (Fig. 2). The relation between the incident and scattered Stokes vectors becomes [8] Isca =
1 1 SR Iinc = [L( − i2 )SL(−i1 )]Iinc ; 2 kw R kw R2
(3)
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
151
Fig. 2. Reference frames for polarization.
where SR is called the rotated scattering matrix, i1 is the angle between the meridian plane (O; inc ; z) and the scattering plane (O; inc ; sca ); i2 is the angle between the meridian plane (O; sca ; z) and the scattering plane. L is a rotation matrix which depends on the rotation angles cosines expressed as function of the polar angle ’ relative to the z coordinate axis, of the azimuth of the waves propagation directions and of the scattering angle [1,8] (Fig. 2). 2.3. Vector radiative transfer equation (VRTE) Scattering is assumed to be independent and the Stokes parameters can be incoherently added without restrictions [7]. This allows the use of the radiative transfer equation to solve problems of multiple light scattering by an assembly of particles. In this frame, the VRTE, which expresses the variations of Stokes parameters along path ds in the direction of propagation , in an emitting– absorbing and scattering media, is given by [9] d Iˆ(s; ) ˆ Z (s; ; )Iˆ(s; ) d ; (4) = −Kext (s; )I (s; ) + Kabs (s; )Ib (s) + ds 4 where Iˆ(s; ) is the Stokes intensities vector (W m−2 m−1 sr −1 ) at point s for the direction , Kext (s; ) is the average extinction matrix at point s for the direction , Kabs (s; ) is the average absorption (emission) vector at point s for the direction , Ib (s) is the Planck function at point s, Z (s; ; ) is the normalized average phase matrix [1] at point s for radiation coming from direction and scattered in the direction . The averaging is necessary when the medium is comprised of particles with di4erent sizes, shapes and refractive indices, and should be performed considering a proper normalization scheme [1]. In the case of randomly oriented symmetric particles, the radiation extinction process is simpli;ed while it is independent of the direction and polarization of the incident beam. As a consequence, the extinction matrix has a diagonal structure [1,9] Kext = diag{; ; ; }:
(5)
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=part +surr (m−1 ) is the extinction coe/cient related to particles and possibly to purely absorbing surrounding medium (gas, glass). The absorption vector, simply written as [1,9] Kabs = (part + surr ){1; 0; 0; 0}t ;
(6)
takes into account the absorption by the medium as well as by the particles. The normalized average phase matrix is given by [1] s s SR = L( − i2 )SL(−i1 ); (7) Z= 4 4 where s (m−1 ) is the scattering coe/cient. In the case of randomly oriented symmetric particles, the scattering matrix has a block-diagonal structure with six independent elements [1,7] a1 () b1 () 0 0 b () a () 0 0 2 1 S() = (8) ; 0 0 a3 () b2 () 0 0 −b2 () a4 () where a1 () is the phase function which satis;es the normalization condition [9]. To solve the problem, the collimated beam may be separated from all other contributions to the Stokes vector in the slab [10]. Boundary conditions associated to the experimental con;guration and in accordance with our assumptions are simply given by Iˆ(z = 0 (resp: zmax ); = cos ’ ¿ 0 (resp: ¡ 0); ) = {0; 0; 0; 0}t :
(9)
3. Calculation of the equivalent Mueller matrix 3.1. Matrix elements The Mueller matrix elements are obtained through the simulation of four successive virtual experiments corresponding to four polarization states of the incident beam : unpolarized (UP; ◦ ◦ Iinc = (1; 0; 0; 0)Iinc ), 0 linearly polarized (LP1; Iinc = (1; 1; 0; 0)Iinc ); 45 linearly polarized (LP2; Iinc = (1; 0; 1; 0)Iinc ), and right circularly polarized (RCP; Iinc = (1; 0; 0; 1)Iinc ). Taking into account the relation existing between the incident and transformed Stokes vectors via the equivalent Mueller matrix (Eq. (1)), the transformed Stokes parameters are given by 1 m11 0 m 21 It; UP = M Iinc = (10) Iinc ; 0 m31 0 m41 1 m11 + m12 m12 1 m +m m 22 21 22 I = I + It; LP1 = M Iinc = inc Iinc ; t; UP 0 m31 + m32 m32 0 m41 + m42 m42
(11)
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
1 m11 + m13 m13 0 m +m m 23 21 23 It; LP2 = M Iinc = Iinc = It; UP + Iinc ; 1 m31 + m33 m33 0 m41 + m43 m43 1 m11 + m14 m14 0 m +m m 24 21 24 It; RCP = M Iinc = Iinc = It; UP + Iinc ; 0 m31 + m34 m34 m41 + m44 m44 1
153
(12)
(13)
where the dependence on incident and transformed directions have been omitted for simplicity. Mueller matrix elements are simply deduced by: m11 m12 m13 m14 m 21 m22 m23 m24 M = m31 m32 m33 m34 m41 m42 m43 m44 =
1 ((It; UP )(It; LP1 − It; UP )(It; LP2 − It; UP )(It; RCP − It; UP )): Iinc
(14)
The transformed (transmitted or re3ected) Stokes-3ux densities-vectors (It (W m−2 m−1 )) comes from the Stokes-intensities-vectors (Iˆt (W m−2 sr −1 m−1 )) which are evaluated by solving the VRTE for the four selected polarization states of the incident beam. This is performed by using the VDOM. 3.2. VRTE solution using VDOM Recently, we have developed a VDOM code [11] to solve the problem using our previous experience concerning DOM [12]. The mains steps involved in the VDOM formulation are as following. Spatial domain is discretized into J -sub-layers. The discretization of the angular domain is performed using a PCA angular quadrature [13], which divides the domains of the polar and azimuthal angles into N latitudes each having M azimuths. The resulting form of the VRTE (Eq. (4)) is given by P zj "j Iˆjp − p (15) wp Zjpp Iˆjp + Zjp0 Iinc e 0 ; = −Kext; jp Iˆjp + Kabs; jp Ib; j + "z p =1 where j and p are indices, respectively, corresponding to the sub-layer and to the direction p under consideration, the notation "j Iˆjp is the di4erence of intensity vectors between the top and the bottom of the sub-layer j, the integral over directions in the in-scattering source term has been replaced by a numerical quadrature, (wp )p∈[1; NM ] are the weights of the quadrature, and the last term represents the scattering in direction p of the attenuated Stokes (3ux) vector of the beam (Iinc ) at sub-layer j.
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Because of the replacement of angular integrals by numerical quadratures, the in-scattering term of Eq. (15) has to be adjusted to ensure the normalization condition for the phase function by using a corrective factor for each direction of the quadrature set [9]. Other numerical features related to the quadrature set and spatial interpolation are described in Ref. [11]. The set of four coupled equations (15) is solved iteratively for each direction of the quadrature to give re3ected It (z = 0; =2 ¡ ’ ¡ ) and transmitted It (z = zmax ; 0 ¡ ’ ¡ =2) Stokes intensity vectors. Stokes-3ux densities-vectors are deduced through the relation It (m ) = wm | m |Iˆt (m ). The procedure is repeated for the four polarization states of the incident beam used for the determination of the equivalent Mueller matrix elements.
4. Results and discussion 4.1. Validation Tests of our VDOM code have been performed recently against analytic solutions for the four Stokes intensities in the case of a purely Raleigh scattering atmosphere [14]. Predictions of our VRTE solver showed a very good agreement with the reference results [11]. Cases where internal emission is included con;rmed these good performances by comparing our results with those of the adding–doubling method by Evans and Stephens [15]. 4.2. E@ects of the radiative properties on the Mueller matrix For the sake of simplicity, we consider Rayleigh scattering and no absorption in the medium (surr = 0). Temperature (Tm ) and radiative properties are assumed to be uniform in the layer. To evaluate the in3uence of radiative properties on the calculated Mueller matrix, variations of its elements with the transformed direction (’; ) are compared with those of the phase matrix Z (Eq. (7)). Since our interest is in the di4use part of the transformed radiation, each element of a matrix A is normalized by a11 evaluated for the direction of the angular discretization which is the closest to the incident direction. It is straightforward that the resulting Mueller matrices conserve the block diagonal structure of the original scattering matrix (Eq. (8)). Since the results provide a great amount of information, only the case of normal incidence (’0 = ◦ 0 ) of the beam is treated here. Fig. 3(a) represents the angular variations of element m11 of the equivalent-normalized-Mueller matrix when the optical thickness of the slab (&max =zmax ) is equal to unity. This element accounts for the angular distribution of the transmitted and re3ected radiations. The e4ects of multiple scattering and absorption, which depend on the radiation paths through the layer, are signi;cant. Di4erences between transmitted and re3ected-backscattered-radiations patterns are observed, also in comparison with the phase function of Rayleigh single scattering. For identical ◦ reasons, m12 , m21 and m22 are a4ected, more particularly for polar angles near 90 for which radiation paths are longer. The equalities m21 = m12 and m11 = m22 observed for the original scattering matrix are broken (Fig. 3(a), (b) and (d)). This trend is not encountered when the optical thickness of the slab is much lower (Fig. 4(a)): variations of m11 for &max = 0:1 are identical to those of the Rayleigh ◦ scattering phase function, except for angles near 90 . It is shown that Mueller matrix elements (m11
R. Vaillon / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 147 – 157
155
Fig. 3. Normalized Mueller matrix elements [(a) m11 ; (b) m12 — solid symbols — and m21 — open symbols —; (d) m22 ] and degree of polarization (c) of transmitted and re3ected radiations in the case of Rayleigh scattering with &max = 1 and 0 = 1.
and m22 in Fig. 4(a)) deviate from those of the Rayleigh scattering matrix when the optical thickness of the slab increases. The in3uence of the optical thickness on the degree of polarization is shown in Fig. 4(b). The results are quite meaningful since a deviation of m22 from m11 is generally considered as an indication of the presence of nonspherical particles [1,6]: it is obvious here that such a conclusion would lead to a great mistake. The same tendencies, not shown here, are observed for the m33 and m44 diagonal elements. A case where thermal emission is present (Tm = 400 K; ! = s = = 0:5), reported in Fig. 3, exhibits comparative behaviours. The global e4ect of multiple scattering, absorption and internal thermal emission on the degree of polarization is depicted in Fig. 3(c).
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Fig. 4. Normalized Mueller matrix elements [(a) m11 — solid symbols — and m22 — open symbols —] and degree of polarization (b) of transmitted and re3ected radiations in the case of Rayleigh scattering with 0 = 1 for various sample optical thicknesses.
5. Conclusion We have presented a method of determination of the Mueller matrix associated to a semitransparent plane parallel sample containing randomly oriented symmetric particles. The procedure is based on the simulation of four virtual experiments of transfer of polarized light through the sample by using a VDOM code. The e4ects of multiple scattering, absorption and emission in connection with the geometry of the sample have been proved to be very signi;cant on the Mueller matrix elements. In the future, the e4ects of boundary re3ection, absorption and transmission will be included and the analysis will be extended to the larger Lorenz–Mie spheres and the nonspherical particles. References [1] Mishchenko MI, Hovenier JW, Travis LD. Light scattering by nonspherical particles. San Diego: Academic Press, 2000. [2] Volten H, Mu˜noz O, Rol E, de Haan J, Vassen W, Hovenier JW. Laboratory measurements of scattering matrices of irregular mineral particles. Proceedings of the Fifth International Conference on Light Scattering by Nonspherical Particles, Halifax, 2000. p. 41– 44. [3] Manickavasagam S, MengRucS MP. Scattering matrix elements of fractal-like soot agglomerates. Appl Opt 1997;36:1337–51. [4] Manickavasagam S, Govindan R, MengRucS MP. Estimating the morphology of soot agglomerates by measuring their scattering matrix elements. In: Proceedings of the ASME Heat Transfer Division, HTD 352, Vol. 2, 1997. [5] Kuik F, Stammes P, Hovenier JW. Experimental determination of scattering matrices of water droplets and quartz particles. Appl Opt 1991;30:4872–81. [6] Bohren F, Hu4man DR. Absorption and scattering of light by small particles. New York: Wiley, 1983. [7] van de Hulst HC. Light scattering of small particles. New York: Wiley, 1957.
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