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Time-independent polarized radiation in radiators with scattering media
Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 167 – 174
www.elsevier.com/locate/jqsrt
Time-independent polarized radiation in radiators with scattering media I.A. Vasilieva∗ Institute for High Energy Densities, Russian Academy of Sciences, Izhorskaya 13/19, Moscow 125412, Russia Received 7 July 2003; accepted 17 December 2003
Abstract The necessary condition of stationarity in radiators with scattering media is deduced. It is obtained from the fact that all primary emitted radiation disappears with probability unity when radiation is time-independent. The disappearance due to the absorption by matter and to escape from radiator is taken into account. The condition of stationarity includes the characteristics of radiation polarization, the coe4cients of medium absorption, and the elements of the Green’s function matrix. The relations between intensity components of radiation are derived from the condition. The results may be used in investigations of various radiators with scattering media. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Radiation; Polarization; Stationarity; Intensity; The Green’s function matrix
Usually emission and disappearance of radiation occur simultaneously and continuously in the radiators. Radiation may primarily appear as a result of conversion of matter energy into energy of radiation
Fax: +7-95-485-7990. E-mail address: [email protected] (I.A. Vasilieva).
0022-4073/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2003.12.017
168
I.A. Vasilieva / Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 167 – 174
The obtained relations between the radiation components gave us new useful possibilities of radiation description in the very various cases [1]. As the relations are exact, they were used for determination of computation errors when radiation was described by solution of the radiation transfer equations. As the relations may be used for Gnding individual components of the observed radiation, they are useful in the various spectral experiments. The stationary relations have been successfully used for experimental studies of the spectral-line proGles in dusty radiating gas and for measurements of the temperature of strongly scattering solid material. It was shown that the relations may help to investigate the near-electrode region in heavy-current discharge and to determine the relative atomic-level populations in dusty plasma. Polarization of radiation was not taken into account. Below, the condition of stationarity will be derived when the polarization is important. Possible polarization of the primary radiation and the in
(1)
In the case of the natural radiation one has Q = U = V = 0. In the case of the linearly polarized radiation at = 0 one has U = V = 0. Let us introduce the following designations: q ≡ cos 2 cos 2; u ≡ cos 2 sin 2; v ≡ sin 2:
(2)
The Stokes parameters may be presented as four components of the intensity vector I = I × [1; q; u; v]T : T
The sign describes the transposition of the vector.
(3)
I.A. Vasilieva / Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 167 – 174
169
Fig. 1. Diagram of a radiator. r; r —radius–vectors of the points within the volume v; r0 ; r0 —radius–vectors of the points at the surface S; u, (−u), u , (−u ), u0 —unit vectors.
Suppose an emitting, absorbing, and scattering medium is within the volume v enclosed by the surface S (see Fig. 1). Let pv be the
(4)
Let IS be the
GVQ
GVU
GVV
This matrix describes the change of the Stokes parameters when radiation is passing from the point (r ; u ) to the point (r; u) by arbitrary paths inside the volume v, see Fig. 1. Every element of the matrix G describes the transformation of the parameters. The second index of the element designs the transformed parameter, related to the initial point (r ; u ), and the Grst index designs the obtained parameter in the point (r; u). The matrices G and their elements depend on all optical and geometric characteristics of the radiator. At the same time, the matrices G do not depend on the characteristics of the radiation and can be used for description of arbitrary radiation. Let us describe the possibilities of disappearance of some arbitrary chosen
170
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Suppose the coe4cient of absorption kabs does not depend on direction and on polarization of the radiation. It may only depend on the place (r) in the radiator. Secondly, the chosen radiation disappears when it arrives at the surface S from the volume v. This fact complies with the requirement, mentioned above, that the intensity IS of the radiation coming into the radiator from the outside does not depend on the analyzed radiation. Let the chosen radiation has in the point r the direction u and its intensity is equal to I. The disappearance in
(6)
Similarly, the arrival of the same radiation at the arbitrary point (r0 ; u0 ) of the surface S is determined by the expression iS [(r; u) → (r0 ; u0 )] = G[(r; u) → (r0 ; u0 )]I(r; u):
(7)
The values of the partial scalar intensities iv [(r; u) → (r ; u )] and iS [(r; u) → (r0 ; u0 )] are determined by the product of the Grst line of the G-matrix [Eq. (5)] by the column of the I-vector [Eq. (3)]: iv [(r; u) → (r ; u )] = I (r; u){GII [(r; u) → (r ; u )] + GIQ [(r; u) → (r ; u )]q(r; u) +GIU [(r; u) → (r ; u )]u(r; u) + GIV [(r; u) → (r ; u )]v(r; u)};
(8)
iS [(r; u) → (r0 ; u0 )] = I (r; u){GII [(r; u) → (r0 ; u0 )] + GIQ [(r; u) → (r0 ; u0 )]q(r; u) +GIU [(r; u) → (r0 ; u0 )]u(r; u) + GIV [(r; u) → (r0 ; u0 )]v(r; u)}:
(9)
Eqs. (8) and (9) described the intensities completely when the Grst element GII includes the radiation that interacts and does not interact with medium between initial and Gnal points of radiation transfer. These
(10)
Similarly, the part of the same
(11)
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171
To obtain the part of analyzed radiation that disappears everywhere inside the radiator and on the surface, it is necessary to integrate the part DV [(r; u) → (r ; u )] over the volume v, the part DS [(r; u) → (r0 ; u0 )] over the surface S, and the both parts over the corresponding solid angles. At last, one has to add together the obtained values. If the analyzed radiation disappears completely, the result must be equal to unity. As was said previously, the complete disappearance of the arbitrary observed radiation is the necessary condition of the time-independent radiation in arbitrary radiator. Then, the necessary condition of stationarity may be presented as follows: dv du Dv [(r; u) → (r ; u )] + dS du0 DS [(r; u) → (r0 ; u0 )] = 1: (12) v
4
S
2
The parts Dv [(r; u) → (r ; u )] and DS [(r; u) → (r0 ; u0 )] have the probabilistic sense. Namely, every part is equal to the density of the probability that the analyzed radiation will arrive via arbitrary paths inside the volume v at the vicinity of (r ; u ) or (r0 ; u0 ) and will disappear there in the units of volume or surface. Then, Eq. (12) means that the probability of the complete disappearance of the radiator from arbitrary part of the radiation is a certain event. It follows from Eqs. (10) and (11) that the condition of stationarity Eq. (12) includes the characteristics of the polarization of the chosen observed radiation (q; u; v), the absorption coe4cient (kabs ), and the elements of the Grst line of the Green’s function matrix Eq. (5). If the condition of stationarity Eq. (12) is not fulGlled the radiation cannot be time-independent. This condition is necessary, but it is not su4cient. Really, for instance, if the sources of the radiation attenuate, the resulting radiation attenuates also when the condition is fulGlled. The condition of stationarity Eq. (12) is exact when the theory of linear radiation transfer is applicable, scattering is elastic and coe4cients of absorption do not depend on polarization and direction of radiation. The condition does not depend on geometry and dimensions of a radiator, on frequency, on the concrete characteristics of scattering and absorption, on their change in volume. It relates to arbitrary place and direction of the radiation inside the radiator. At the same time, the Green’s functions depend on all these circumstances. Let the polarization of the radiation in some place and the absorption coe4cients inside volume of the radiator be known. Then, Eqs. (10)–(12) represent the necessary relations between the elements of the Grst lines of the Green’s function matrices in the stationary case. The Green’s function matrixes are analyzed and used in various cases of stationary radiation [7–11]. The necessary condition Eq. (12) has to be fulGlled in diIerent conditions when the above mentioned suppositions were satisGed. It may be used in practice diIerently, for instance, for the veriGcation of the calculations of the Green’s function matrixes. The obtained condition of stationarity permits to determine the relations between the components of scalar radiation intensities. The observed intensity in the arbitrary chosen point (r; u) may be presented as the sum of the partial intensity components. Each partial component is determined by the primary radiation in the points (r ; u ) or (r0 ; u0 ) and by passing this radiation to the point of observation. Let us designate these partial components as iv [(r ; u ) → (r; u)] and iS [(r0 ; u0 ) → (r; u)]. The partial components are determined by the equations similar to Eqs. (6)–(9) when using the corresponding primary radiation instead of intensity I(r; u). To deduce the relations between the components, it is necessary to connect the partial scalar intensities with the parts of disappearance Dv , DS that are included in Eq. (12). Let us use the condition (Eq. (12)) for the description of disappearance of the radiation that has in the point r
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the direction (−u). The direction of this disappearing radiation is opposite to the direction of the observed in (r; u) radiation. In this case the disappearance of radiation is described by the parts Dv [(r; −u) → (r ; −u )] and DS [(r; −u) → (r0 ; −u0 )] in Eq. (12). The necessary connections between iv [(r ; u ) → (r; u)], iS [(r0 ; u0 ) → (r; u)], on the one hand, and Dv [(r; −u) → (r ; −u )], DS [(r; −u) → (r0 ; −u0 )], on the other hand, may be obtained by relations similar to Eqs. (8)–(11). In these equations the observed intensities iv , iS in the points (r; u) are determined by the primary radiation pv in the points (r ; u ). It gives us the possibility to introduce the values iv , iS into the condition of stationarity Eq. (12). As a result, the following relation between the components of the radiation intensities is obtained in the general case: kabs (r )iv [(r ; u ) → (r; u)] gv [(r ; u ) → (r; u)] dv d(−u ) ; u ) p (r v v 4 iS [(r0 ; u0 ) → (r; u)] gS [(r0 ; u0 ) → (r; u)] = 1: + dS d(−u0 ) (13) IS (r0 ; u0 ) S 2 In
G↑↓II + G↑↓IQ qp (r; −u) + G↑↓IU up (r; −u) + G↑↓IV vp (r; −u) ; GII + GIQ qp (r ; u ) + GIU up (r ; u ) + GIV vp (r ; u ) G↑↓II + G↑↓IQ qp (r; −u) + G↑↓IU up (r; −u) + G↑↓IV vp (r; −u) : GII + GIQ q(r0 ; u0 ) + GIU u(r0 ; u0 ) + GIV v(r0 ; u0 )
(14)
(15)
Here, the Green’s function matrix G↑↓ describes the transfer of disappearing radiation from (r; −u). The arguments of the functions Gmk in gV are equal to [(r ; u ) → (r; u)] and in gS −[(r0 ; u0 ) → (r; u)]. The arguments of the functions G↑↓mk in gV are equal to [(r; −u) → (r ; −u )] and in gS −[(r; −u) → (r0 ; −u0 )]. When the relation of the reciprocity for the phase matrices is true [12] it is correct for the Green’s function matrices also [7,8]. Then, in the case of the natural primary radiation one has gV = gS = 1 because qp = up = vp = 0 and G↑↓II = GII . In practice, the general relation is seldom required but it permits to obtain the relations in the various useful particular cases. Let us discuss one of them. Let the surface S consist of the radiating Srad and of non-radiating parts. The observed scalar radiation I in the (r; u) may be presented in this case as follows: I (r; u) = Iv (r; u) + ISrad; scat (r; u) + IS (r0 ; u)exp[ − t(r0 → r)]:
(16)
Here, Iv is the radiation in (r; u) that is determined by the primary radiation of the whole of volume v; ISrad; scat is the radiation that is determined by the whole of radiating surface Srad after scattering inside the volume v; IS exp[ − t(r0 → r)] is the radiation that is directly coming at (r; u) from the point r0 of the surface S (see Fig. 1); t(r0 → r) is the optical density between the points r0 and r. Often it is supposed that characteristics of media and the primary sources of radiation inside the volume v and at the radiating surface Srad are constant. Besides, the functions gv and gS do not depend on (r; r0 ; u; u0 ) in some cases. Then, it follows from the general relation between the
I.A. Vasilieva / Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 167 – 174
173
components of scalar intensities Eq. (13): (r; u) IS IV (r; u) × gV (r; u) + rad; scat × gS (r; u) + DS; ex (r; −u) + exp[ − t(r → r0 )] = 1: pV =kabs IS
(17)
Here, DS; ex (r; −u) is the part of the radiation from (r; −u) that is coming at non-radiating part of the surface S after scattering inside the volume v. The values of t and DS; ex may be measured in experiments [1]. The value of the optical density t(r → r0 ) may be determined by measuring the attenuation of radiation when it coming from r to r0 . The value of DS; ex (r; −u) may be determined if one illuminates the object at the point r in the direction (−u) and measures the scattered-radiation
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Conclusion (1) The necessary condition of stationarity of polarized radiation in scattering media is obtained in the integral form Eq. (12). The condition describes the connection between the elements of the Grst lines of the Green’s function matrices when the polarization of analyzed radiation and the coe4cients of absorption of media are known. (2) The relation between the intensity components of polarized radiation is deduced in the integral form Eq. (13). The useful simple relation for some particular cases is expressed by Eq. (17). (3) The condition of stationarity and the relations between intensity components are general. It means that the corresponding expressions do not depend on concrete characteristics of radiators. That is why they may be used in the various circumstances. The condition of stationarity and the relations between intensity components are exact when the theory of linear radiation transfer is applicable, scattering is elastic, and the coe4cients of absorption do not depend on polarization and direction of radiation. (4) It has been shown that when the primary radiation is not polarized, the condition of stationarity and the relations between intensity component do not diIer from the case when polarization is not taken into account. It is true though the natural radiation is polarized in scattering media usually. References [1] Vasil’eva IA. Stationary radiation of objects with scattering media. Physics-Uspekhi 2001;44:1255–82. [2] Vasil’eva IA. Relations between thermal radiation components of stationary scattering radiator. JQSRT 2000;66: 223–42. [3] Vasil’eva IA. Correlations between the components of stationary radiation in scattering media. High Temperature 2000;38:254–63. [4] Vasilieva IA. Radiation components of steady-state scattering dusty plasma at low pressure. Radiat EIects Defects Solids 2001;154:61–87. [5] Apresyan LA, Kravtsov YA. Radiation transfer theory: statistical and wave aspects. Amsterdam: Harwood Publishers; 1996. [6] Chandrasekar S. Radiative transfer. Oxford: Clarendon Press; 1953. [7] Domke H, Yanovitskij EG. A simple computational method for internal polarized radiation Gelds of Gnite slab atmospheres. JQSRT 1981;26:389–96. [8] Zege EP, Chaikovskaya LI. New approach to the polarized radiative transfer problem. JQSRT 1996;55:19–31. [9] Zege EP, Chaikovskaya LI. Approximate theory of linearly polarized light propagation through a scattering medium. JQSRT 2000;64:413–35. [10] Siewert CE. A discrete-ordinates solution for radiative-transfer models that include polarization eIects. JQSRT 2000;64:227–54. [11] Domke H. Biorthogonality and radiative transfer in Gnite slab atmospheres. JQSRT 1983;30:119–29. [12] Hovenier JW. Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles. J Atmos Sci 1969;26:488–99. [13] Bohren CF, HuIman DR. Absorption and scattering of light by small particles. New York: Wiley; 1986.
www.elsevier.com/locate/jqsrt
Time-independent polarized radiation in radiators with scattering media I.A. Vasilieva∗ Institute for High Energy Densities, Russian Academy of Sciences, Izhorskaya 13/19, Moscow 125412, Russia Received 7 July 2003; accepted 17 December 2003
Abstract The necessary condition of stationarity in radiators with scattering media is deduced. It is obtained from the fact that all primary emitted radiation disappears with probability unity when radiation is time-independent. The disappearance due to the absorption by matter and to escape from radiator is taken into account. The condition of stationarity includes the characteristics of radiation polarization, the coe4cients of medium absorption, and the elements of the Green’s function matrix. The relations between intensity components of radiation are derived from the condition. The results may be used in investigations of various radiators with scattering media. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Radiation; Polarization; Stationarity; Intensity; The Green’s function matrix
Usually emission and disappearance of radiation occur simultaneously and continuously in the radiators. Radiation may primarily appear as a result of conversion of matter energy into energy of radiation
Fax: +7-95-485-7990. E-mail address: [email protected] (I.A. Vasilieva).
0022-4073/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2003.12.017
168
I.A. Vasilieva / Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 167 – 174
The obtained relations between the radiation components gave us new useful possibilities of radiation description in the very various cases [1]. As the relations are exact, they were used for determination of computation errors when radiation was described by solution of the radiation transfer equations. As the relations may be used for Gnding individual components of the observed radiation, they are useful in the various spectral experiments. The stationary relations have been successfully used for experimental studies of the spectral-line proGles in dusty radiating gas and for measurements of the temperature of strongly scattering solid material. It was shown that the relations may help to investigate the near-electrode region in heavy-current discharge and to determine the relative atomic-level populations in dusty plasma. Polarization of radiation was not taken into account. Below, the condition of stationarity will be derived when the polarization is important. Possible polarization of the primary radiation and the in
(1)
In the case of the natural radiation one has Q = U = V = 0. In the case of the linearly polarized radiation at = 0 one has U = V = 0. Let us introduce the following designations: q ≡ cos 2 cos 2; u ≡ cos 2 sin 2; v ≡ sin 2:
(2)
The Stokes parameters may be presented as four components of the intensity vector I = I × [1; q; u; v]T : T
The sign describes the transposition of the vector.
(3)
I.A. Vasilieva / Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 167 – 174
169
Fig. 1. Diagram of a radiator. r; r —radius–vectors of the points within the volume v; r0 ; r0 —radius–vectors of the points at the surface S; u, (−u), u , (−u ), u0 —unit vectors.
Suppose an emitting, absorbing, and scattering medium is within the volume v enclosed by the surface S (see Fig. 1). Let pv be the
(4)
Let IS be the
GVQ
GVU
GVV
This matrix describes the change of the Stokes parameters when radiation is passing from the point (r ; u ) to the point (r; u) by arbitrary paths inside the volume v, see Fig. 1. Every element of the matrix G describes the transformation of the parameters. The second index of the element designs the transformed parameter, related to the initial point (r ; u ), and the Grst index designs the obtained parameter in the point (r; u). The matrices G and their elements depend on all optical and geometric characteristics of the radiator. At the same time, the matrices G do not depend on the characteristics of the radiation and can be used for description of arbitrary radiation. Let us describe the possibilities of disappearance of some arbitrary chosen
170
I.A. Vasilieva / Journal of Quantitative Spectroscopy & Radiative Transfer 87 (2004) 167 – 174
Suppose the coe4cient of absorption kabs does not depend on direction and on polarization of the radiation. It may only depend on the place (r) in the radiator. Secondly, the chosen radiation disappears when it arrives at the surface S from the volume v. This fact complies with the requirement, mentioned above, that the intensity IS of the radiation coming into the radiator from the outside does not depend on the analyzed radiation. Let the chosen radiation has in the point r the direction u and its intensity is equal to I. The disappearance in
(6)
Similarly, the arrival of the same radiation at the arbitrary point (r0 ; u0 ) of the surface S is determined by the expression iS [(r; u) → (r0 ; u0 )] = G[(r; u) → (r0 ; u0 )]I(r; u):
(7)
The values of the partial scalar intensities iv [(r; u) → (r ; u )] and iS [(r; u) → (r0 ; u0 )] are determined by the product of the Grst line of the G-matrix [Eq. (5)] by the column of the I-vector [Eq. (3)]: iv [(r; u) → (r ; u )] = I (r; u){GII [(r; u) → (r ; u )] + GIQ [(r; u) → (r ; u )]q(r; u) +GIU [(r; u) → (r ; u )]u(r; u) + GIV [(r; u) → (r ; u )]v(r; u)};
(8)
iS [(r; u) → (r0 ; u0 )] = I (r; u){GII [(r; u) → (r0 ; u0 )] + GIQ [(r; u) → (r0 ; u0 )]q(r; u) +GIU [(r; u) → (r0 ; u0 )]u(r; u) + GIV [(r; u) → (r0 ; u0 )]v(r; u)}:
(9)
Eqs. (8) and (9) described the intensities completely when the Grst element GII includes the radiation that interacts and does not interact with medium between initial and Gnal points of radiation transfer. These
(10)
Similarly, the part of the same
(11)
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171
To obtain the part of analyzed radiation that disappears everywhere inside the radiator and on the surface, it is necessary to integrate the part DV [(r; u) → (r ; u )] over the volume v, the part DS [(r; u) → (r0 ; u0 )] over the surface S, and the both parts over the corresponding solid angles. At last, one has to add together the obtained values. If the analyzed radiation disappears completely, the result must be equal to unity. As was said previously, the complete disappearance of the arbitrary observed radiation is the necessary condition of the time-independent radiation in arbitrary radiator. Then, the necessary condition of stationarity may be presented as follows: dv du Dv [(r; u) → (r ; u )] + dS du0 DS [(r; u) → (r0 ; u0 )] = 1: (12) v
4
S
2
The parts Dv [(r; u) → (r ; u )] and DS [(r; u) → (r0 ; u0 )] have the probabilistic sense. Namely, every part is equal to the density of the probability that the analyzed radiation will arrive via arbitrary paths inside the volume v at the vicinity of (r ; u ) or (r0 ; u0 ) and will disappear there in the units of volume or surface. Then, Eq. (12) means that the probability of the complete disappearance of the radiator from arbitrary part of the radiation is a certain event. It follows from Eqs. (10) and (11) that the condition of stationarity Eq. (12) includes the characteristics of the polarization of the chosen observed radiation (q; u; v), the absorption coe4cient (kabs ), and the elements of the Grst line of the Green’s function matrix Eq. (5). If the condition of stationarity Eq. (12) is not fulGlled the radiation cannot be time-independent. This condition is necessary, but it is not su4cient. Really, for instance, if the sources of the radiation attenuate, the resulting radiation attenuates also when the condition is fulGlled. The condition of stationarity Eq. (12) is exact when the theory of linear radiation transfer is applicable, scattering is elastic and coe4cients of absorption do not depend on polarization and direction of radiation. The condition does not depend on geometry and dimensions of a radiator, on frequency, on the concrete characteristics of scattering and absorption, on their change in volume. It relates to arbitrary place and direction of the radiation inside the radiator. At the same time, the Green’s functions depend on all these circumstances. Let the polarization of the radiation in some place and the absorption coe4cients inside volume of the radiator be known. Then, Eqs. (10)–(12) represent the necessary relations between the elements of the Grst lines of the Green’s function matrices in the stationary case. The Green’s function matrixes are analyzed and used in various cases of stationary radiation [7–11]. The necessary condition Eq. (12) has to be fulGlled in diIerent conditions when the above mentioned suppositions were satisGed. It may be used in practice diIerently, for instance, for the veriGcation of the calculations of the Green’s function matrixes. The obtained condition of stationarity permits to determine the relations between the components of scalar radiation intensities. The observed intensity in the arbitrary chosen point (r; u) may be presented as the sum of the partial intensity components. Each partial component is determined by the primary radiation in the points (r ; u ) or (r0 ; u0 ) and by passing this radiation to the point of observation. Let us designate these partial components as iv [(r ; u ) → (r; u)] and iS [(r0 ; u0 ) → (r; u)]. The partial components are determined by the equations similar to Eqs. (6)–(9) when using the corresponding primary radiation instead of intensity I(r; u). To deduce the relations between the components, it is necessary to connect the partial scalar intensities with the parts of disappearance Dv , DS that are included in Eq. (12). Let us use the condition (Eq. (12)) for the description of disappearance of the radiation that has in the point r
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the direction (−u). The direction of this disappearing radiation is opposite to the direction of the observed in (r; u) radiation. In this case the disappearance of radiation is described by the parts Dv [(r; −u) → (r ; −u )] and DS [(r; −u) → (r0 ; −u0 )] in Eq. (12). The necessary connections between iv [(r ; u ) → (r; u)], iS [(r0 ; u0 ) → (r; u)], on the one hand, and Dv [(r; −u) → (r ; −u )], DS [(r; −u) → (r0 ; −u0 )], on the other hand, may be obtained by relations similar to Eqs. (8)–(11). In these equations the observed intensities iv , iS in the points (r; u) are determined by the primary radiation pv in the points (r ; u ). It gives us the possibility to introduce the values iv , iS into the condition of stationarity Eq. (12). As a result, the following relation between the components of the radiation intensities is obtained in the general case: kabs (r )iv [(r ; u ) → (r; u)] gv [(r ; u ) → (r; u)] dv d(−u ) ; u ) p (r v v 4 iS [(r0 ; u0 ) → (r; u)] gS [(r0 ; u0 ) → (r; u)] = 1: + dS d(−u0 ) (13) IS (r0 ; u0 ) S 2 In
G↑↓II + G↑↓IQ qp (r; −u) + G↑↓IU up (r; −u) + G↑↓IV vp (r; −u) ; GII + GIQ qp (r ; u ) + GIU up (r ; u ) + GIV vp (r ; u ) G↑↓II + G↑↓IQ qp (r; −u) + G↑↓IU up (r; −u) + G↑↓IV vp (r; −u) : GII + GIQ q(r0 ; u0 ) + GIU u(r0 ; u0 ) + GIV v(r0 ; u0 )
(14)
(15)
Here, the Green’s function matrix G↑↓ describes the transfer of disappearing radiation from (r; −u). The arguments of the functions Gmk in gV are equal to [(r ; u ) → (r; u)] and in gS −[(r0 ; u0 ) → (r; u)]. The arguments of the functions G↑↓mk in gV are equal to [(r; −u) → (r ; −u )] and in gS −[(r; −u) → (r0 ; −u0 )]. When the relation of the reciprocity for the phase matrices is true [12] it is correct for the Green’s function matrices also [7,8]. Then, in the case of the natural primary radiation one has gV = gS = 1 because qp = up = vp = 0 and G↑↓II = GII . In practice, the general relation is seldom required but it permits to obtain the relations in the various useful particular cases. Let us discuss one of them. Let the surface S consist of the radiating Srad and of non-radiating parts. The observed scalar radiation I in the (r; u) may be presented in this case as follows: I (r; u) = Iv (r; u) + ISrad; scat (r; u) + IS (r0 ; u)exp[ − t(r0 → r)]:
(16)
Here, Iv is the radiation in (r; u) that is determined by the primary radiation of the whole of volume v; ISrad; scat is the radiation that is determined by the whole of radiating surface Srad after scattering inside the volume v; IS exp[ − t(r0 → r)] is the radiation that is directly coming at (r; u) from the point r0 of the surface S (see Fig. 1); t(r0 → r) is the optical density between the points r0 and r. Often it is supposed that characteristics of media and the primary sources of radiation inside the volume v and at the radiating surface Srad are constant. Besides, the functions gv and gS do not depend on (r; r0 ; u; u0 ) in some cases. Then, it follows from the general relation between the
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components of scalar intensities Eq. (13): (r; u) IS IV (r; u) × gV (r; u) + rad; scat × gS (r; u) + DS; ex (r; −u) + exp[ − t(r → r0 )] = 1: pV =kabs IS
(17)
Here, DS; ex (r; −u) is the part of the radiation from (r; −u) that is coming at non-radiating part of the surface S after scattering inside the volume v. The values of t and DS; ex may be measured in experiments [1]. The value of the optical density t(r → r0 ) may be determined by measuring the attenuation of radiation when it coming from r to r0 . The value of DS; ex (r; −u) may be determined if one illuminates the object at the point r in the direction (−u) and measures the scattered-radiation
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Conclusion (1) The necessary condition of stationarity of polarized radiation in scattering media is obtained in the integral form Eq. (12). The condition describes the connection between the elements of the Grst lines of the Green’s function matrices when the polarization of analyzed radiation and the coe4cients of absorption of media are known. (2) The relation between the intensity components of polarized radiation is deduced in the integral form Eq. (13). The useful simple relation for some particular cases is expressed by Eq. (17). (3) The condition of stationarity and the relations between intensity components are general. It means that the corresponding expressions do not depend on concrete characteristics of radiators. That is why they may be used in the various circumstances. The condition of stationarity and the relations between intensity components are exact when the theory of linear radiation transfer is applicable, scattering is elastic, and the coe4cients of absorption do not depend on polarization and direction of radiation. (4) It has been shown that when the primary radiation is not polarized, the condition of stationarity and the relations between intensity component do not diIer from the case when polarization is not taken into account. It is true though the natural radiation is polarized in scattering media usually. References [1] Vasil’eva IA. Stationary radiation of objects with scattering media. Physics-Uspekhi 2001;44:1255–82. [2] Vasil’eva IA. Relations between thermal radiation components of stationary scattering radiator. JQSRT 2000;66: 223–42. [3] Vasil’eva IA. Correlations between the components of stationary radiation in scattering media. High Temperature 2000;38:254–63. [4] Vasilieva IA. Radiation components of steady-state scattering dusty plasma at low pressure. Radiat EIects Defects Solids 2001;154:61–87. [5] Apresyan LA, Kravtsov YA. Radiation transfer theory: statistical and wave aspects. Amsterdam: Harwood Publishers; 1996. [6] Chandrasekar S. Radiative transfer. Oxford: Clarendon Press; 1953. [7] Domke H, Yanovitskij EG. A simple computational method for internal polarized radiation Gelds of Gnite slab atmospheres. JQSRT 1981;26:389–96. [8] Zege EP, Chaikovskaya LI. New approach to the polarized radiative transfer problem. JQSRT 1996;55:19–31. [9] Zege EP, Chaikovskaya LI. Approximate theory of linearly polarized light propagation through a scattering medium. JQSRT 2000;64:413–35. [10] Siewert CE. A discrete-ordinates solution for radiative-transfer models that include polarization eIects. JQSRT 2000;64:227–54. [11] Domke H. Biorthogonality and radiative transfer in Gnite slab atmospheres. JQSRT 1983;30:119–29. [12] Hovenier JW. Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles. J Atmos Sci 1969;26:488–99. [13] Bohren CF, HuIman DR. Absorption and scattering of light by small particles. New York: Wiley; 1986.