- Email: [email protected]
Symmetry relations for the phase matrix of horizontally oriented particles
ARTICLE IN PRESS Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Symmetry relations for the phase matrix of horizontally oriented particles ˜ oz b, J.W. Hovenier a, O. Mun a b
Astronomical Institute ‘‘Anton Pannekoek’’, University of Amsterdam, P.O. Box 94249, 1090GE Amsterdam, The Netherlands Instituto de Astrofı´sica de Andalucı´a, CSIC, Glorieta de la Astronomı´a s/n, 18080 Granada, Spain
a r t i c l e in fo
abstract
Article history: Received 14 January 2010 Accepted 2 March 2010
Seven symmetry relations, with a wide range of validity, are presented for the phase matrix of horizontally oriented particles. Three of these relations constitute a fundamental basis from which four others can be derived by making combinations. All seven relations can be used for many types of particles, including hexagonal plates and columns as well as spheroids, cylinders and cubes. Some applications are also pointed out. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Light scattering Particles Horizontal orientation Symmetry relations Phase matrix
1. Introduction and basic concepts Scattering of light by a small volume element inside a medium filled with particles can be described by means of the four by four phase matrix of the volume element. This real matrix transforms the Stokes parameters of an incident beam of light into the Stokes parameters of a beam of scattered light, where the meridian planes of the incident and scattered beams act as planes of reference for the Stokes parameters. The phase matrix plays a fundamental role not only in calculations for optically thin media, but also in studies of multiple scattering in optically thick media. In theoretical studies the phase matrix occurs, for instance, as the kernel of the equation of radiative transfer [1–7]. Non-spherical particles can be randomly oriented in three dimensional space, but they can also be horizontally oriented. Well-known examples are provided by several types of hydrometeors, like raindrops and snow flakes, and by ice crystals in planetary atmospheres. Various shapes occur, for example, hexagonal plates and columns
Corresponding author. Tel.: + 34 958121311; fax: + 34 958814530.
˜ oz). E-mail address: [email protected] (O. Mun 0022-4073/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2010.03.001
[8–10]. Symmetry relations for the phase matrix of collections of particles that are randomly oriented in three dimensional space have been presented in an earlier paper [1]. These relations were derived from properties of the scattering matrix which transforms Stokes parameters having the scattering plane as a plane of reference. For randomly oriented particles this scattering matrix is a relative simple function of directions in three dimensional space, since it is rotationally symmetric with respect to the direction of incident radiation. For collections of horizontally oriented particles this is generally not the case and, as a result, the phase matrix is more complicated. The main purpose of this paper is to show by elementary symmetry considerations that various symmetry relations still hold for the phase matrices of volume elements filled with collections of horizontally oriented particles. These relations are based on symmetries in time and space. We consider scattering of (quasi) monochromatic light, or other electromagnetic radiation, without change of wavelength by particles that have a plane of symmetry. Such particles are identical to their mirror images. We first consider collections of hexagonal plates with two broad flat sides in horizontal planes, but with random orientation in such planes, so that there is rotational symmetry about the vertical. To consider the transfer of polarized light in a
ARTICLE IN PRESS 1450
˜oz / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453 J.W. Hovenier, O. Mun
Z N ϕ' -ϕ θ
θ'
P2 P1 O
Y
ϕ ϕ'
X Fig. 1. Scattering by a local volume-element at O. Points N, P1 and P2 are located on a unit sphere. The direction of the incident light is OP1 and that of the scattered light is OP2.
scattering medium like a planetary atmosphere we use a right handed Cartesian coordinate system with axes x, y, z, fixed in space. We call the x, y plane the horizontal plane and the positive z-axis the local vertical. Let us now imagine that a local small volume element containing a collection of independently scattering particles is located in the origin, O, of the coordinate system (see Fig. 1). The collection consists of identical particles or a homogeneous mixture of such collections due to e.g. a distribution in size, shape or refractive index of the particles. The direction of propagation of a beam is specified by an angle, y, ð0 r y r pÞ which it makes with the upward normal and an azimuth angle j, ð0 r j r 2pÞ. The latter is measured clockwise from the positive x-axis when looking in the direction of the upward normal. To describe the state of polarization of a beam we use Stokes parameters, I, Q, U and V, with the meridian plane as the plane of reference [11,12,2,5–7]). The four Stokes parameters can be made elements of a real column vector, called the Stokes vector. The directions of the incident and scattered beams are represented in Fig. 1 by the points P1 and P2, respectively, on the surface of a unit sphere, having O at its center. Suppose light is incident on the volume element in a direction (u0 ,j0 ) and is scattered once into a direction 0 (u,j), where u ¼ cosy and u0 ¼ cosy . Both lie in the range from 1 to +1, where the latter value corresponds to the perpendicularly downward direction. This scattering process at a certain moment can now be described as a transformation of the Stokes vector, Ii ðu0 ,j0 Þ, of the incident light into the Stokes vector, Is ðu,jÞ, of the scattered light. Because of the rotational symmetry of the collection of particles about the z-axis only differences of azimuthal angles need to be used. So we can write
where c is a non-directionally dependent scalar, which can be used for normalization purposes, and the real 4 4 matrix Zðu,u0 ,jj0 Þ is the phase matrix of the volume element. Its elements are denoted as i-j, where i is the row number and j the column number.
Is ðu,jÞ ¼ cZðu,u0 ,jj0 ÞIi ðu0 ,j0 Þ
0 ~ ,jj0 ÞP, Zðu0 ,u,j0 jÞ ¼ PZðu,u
ð1Þ
2. Symmetry relations The problem we wish to address now is whether symmetry considerations will allow us to establish relations between two phase matrices that only differ in values of the three directional variables, u, u0 and jj0 . For this purpose we will consider three fundamental symmetry operations and combinations thereof. 2.1. Reciprocity The reciprocity principle corresponds to invariance of the ratio cause/effect under inversion of time [3]. In the context of light scattering by particles this means that a relation may exist between the phase matrices corresponding to an original situation [see Fig. 1] and a situation in which (i) the incident light travels in the inverse direction of the originally scattered light and (ii) the scattered light travels in the inverse direction of the originally incident light. When polarization is ignored this simply means that the phase functions, i.e. the 1–1 elements of the two phase matrices, are the same. For polarized light, however, we must take into account that, according to the definitions of Stokes parameters, the sign of the third Stokes parameter switches when the light travels in the opposite direction [1,2]. The resulting reciprocity relation is [Cf. Fig. 1] ð2Þ
ARTICLE IN PRESS ˜oz / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453 J.W. Hovenier, O. Mun
where the diagonal matrix P=diag (1,1, 1,1). In the righthand side of Eq. (2) pre- and postmultiplication by P amounts to changing the signs of the elements 1–3, 2–3, 4–3, 3–1, 3–2, and 3–4 of the matrix in the middle, and a tilde above a matrix symbol stands for the transposed matrix. A formal proof of Eq. (2) can be obtained by first applying the reciprocity principle for a single particle and then adding the phase matrices of the individual particles in a volume element [13,14]. Sufficient conditions for the validity of this proof are that there is no magnetic field and the dielectric, permeability and conductivity tensors of all particles are symmetric, which includes the common cases of diagonal matrices and scalars. We shall henceforth assume that Eq. (2) holds. It should be noted that this is a very fundamental relation and its validity has nothing to do with the question whether the scattering matrix depends only on the scattering angle or on more directional parameters [14,3]. In this paper we consider reciprocity relations to be symmetry relations, just like relations based on symmetries in space. 2.2. Mirror symmetry
experiment the incident beam i2 which is the mirror image of i1 will give rise to a beam r2 of scattered light, which is the mirror image of r1. Now i1 and i2 differ only in (i) the sign of the position angle of the major axis of the polarization ellipse and (ii) the sense in which the polarization ellipse is traced, since mirroring causes right-handedness to become left-handedness and vice versa. According to the definitions of Stokes parameters these two differences correspond to sign switches in the third and fourth Stokes parameters of the two incident beams. Because of symmetry this must also hold for the two scattered beams (see Fig. 2). Hence, we have in the first experiment for the Stokes vector of the scattered beam [cf. Eq. (1)] Is ðu,j1 Þ ¼ cZðu,u0 ,j1 j0 ÞIi ðu0 ,j0 Þ,
ð3Þ
and in the second experiment Is ðu,j2 Þ ¼ PQIs ðu,j1 Þ ¼ cZðu,u0 ,j2 j0 ÞPQIi ðu0 ,j0 Þ,
ð4Þ
where the diagonal matrix Q=diag (1,1,1, 1), so that PQ=QP=diag (1,1,1,1). By premultiplying Eq. (4) by PQ and using j2 j0 ¼ j0 j1 we obtain Is ðu,j1 Þ ¼ cPQZðu,u0 ,j0 j1 ÞPQIi ðu0 ,j0 Þ:
It is clear from our assumptions that there is mirror symmetry with respect to the meridian plane of incidence. So when in Fig. 2 in a first experiment the incident beam i1 gives rise to a beam r1 of scattered light, then in a second
1451
ð5Þ
On comparing this equation with Eq. (3) we find the following mirror symmetry relation for the phase matrix Zðu,u0 ,j0 jÞ ¼ PQZðu,u0 ,jj0 ÞQP:
ð6Þ
local vertical
i2 i1 r1
r2
ϕ1
ϕ2
ϕ0
Fig. 2. Illustration of the mirror symmetry relation for the phase matrix. If the incident beam i1 gives rise (among others) to the beam of scattered light r1, then the incident beam i2, which is the mirror image of i1 with respect to the plane of incidence, gives rise (among others) to the beam of scattered light r2, which is the mirror image of r1 with respect to the plane of incidence. The position angles of the polarization ellipses of the incident light (dots) and scattered light (small arcs) are also indicated, as well as the sense in which the four polarization ellipses are traced. [After Hovenier [16]].
ARTICLE IN PRESS 1452
˜oz / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453 J.W. Hovenier, O. Mun
Note that pre-and postmultiplication of a 4 4 matrix by PQ causes a sign switch of the elements 1–3, 1–4, 2–3, 2–4, 3–1, 3–2, 4–1, and 4–2 of the matrix in the middle. So it follows from Eq. (6) that these elements of Zðu,u0 ,jj0 Þ are odd functions of ðjj0 Þ and the other 8 elements are even functions of ðjj0 Þ. In a Fourier series expansion of Zðu,u0 ,jj0 Þ in ðjj0 Þ the first eight of the elements above will, therefore, contain only sine terms or vanish and the remaining eight elements only cosine terms and in general an azimuth independent term.
2.3. Rotational symmetry Simultaneous rotation of the meridian planes of incidence and scattering about a vertical axis through an arbitrary angle gives no new relations since we have already assumed that we have azimuthal symmetry. But rotation about a horizontal axis over an angle p, i.e. turning the collection of particles as well as the directions of the incident and scattered light upside down gives a new symmetry relation, though the scattering situation remains physically the same. The reason for this new relation is that the sense in which the azimuth is measured is fixed in space so that jj0 must be transformed into j0 j [1]. Thus, we have the symmetry relation: Zðu,u0 ,j0 jÞ ¼ Zðu,u0 ,jj0 Þ:
ð7Þ
2.4. Combinations of symmetries By combining the three fundamental symmetry relations (2), (6) and (7) four other symmetry relations can be derived. For example, if Eqs. (2) and (6) are both valid it is clear that switching the sign of jj0 in Eq. (2) will give 0 ~ Zðu0 ,u,jj0 Þ ¼ Q Zðu,u ,jj0 ÞQ :
ð8Þ
Similarly, combining Eqs. (2) and (7) gives 0 ~ Zðu0 ,u,jj0 Þ ¼ PZðu,u ,jj0 ÞP:
ð9Þ
Eqs. (6) and (7) can be combined into Zðu,u0 ,jj0 Þ ¼ PQZðu,u0 ,jj0 ÞQP:
ð10Þ
Finally, Eqs. (2), (6) and (7) give 0 ~ Zðu0 ,u,j0 jÞ ¼ Q Zðu,u ,jj0 ÞQ ,
ð11Þ
which can of course also be viewed as a combination of Eqs. (6) and (9). We have thus found seven symmetry relations that hold for arbitrary directions of the incident and scattered light, namely Eqs. (2) and (6)–(11). Each of the three fundamental equations (2), (6), and (7) has a simple explanation in terms of symmetry in time and space. Evidently, it is also possible to choose another set of three equations, from which the other four can be derived by making combinations, as shown by Eqs. (6), (9) and (10). Each of these relations shows the result of a simple algebraic operation, namely (i) interchanging j and j0 , (ii) interchanging u and u0 and (iii) changing the signs of u and u0 simultaneously.
3. Concluding remarks So far we have only considered hexagonal plates with their broad flat surfaces lying in horizontal planes, but with random orientation in such planes. For this case we found three fundamental symmetry relations from which four others could easily be derived by making combinations. It is, however, clear from the symmetry considerations in the preceding section that the same seven relations, i.e. Eqs. (2) and (6)–(11), will hold for many other kinds of particles. For example, the boundaries of the broad flat sides of the plates do not need to be hexagonal, but may be circular, rectangular, pentagonal, etc. Furthermore, we may have e.g. oblate spheroids with their long axes in the horizontal plane. Also allowed are hexagonal columns with their long axes randomly oriented in the horizontal plane and either randomly rotated about their long axes or with two rectangular facets parallel to the horizontal plane. Finally, roughness of the surfaces and inhomogeneities will often not destroy the fundamental symmetries. So the seven symmetry relations are valid for many kinds of horizontally oriented particles, especially various model particles [10]. For natural particles they will often hold in good approximation, as long as the conditions are approximately satisfied. The particles need not be perfectly horizontally oriented. For example, the flat surfaces of the hexagonal plates that we discussed earlier may be tilted from the horizontal plane in such a way that the three fundamental symmetry relations still hold, so that all seven symmetry relations are valid. This is important for falling hydrometeors [9] and in particular for fluttering ice crystals. All seven symmetry relations for the phase matrix are exactly the same as for particles that are randomly oriented in three dimensional space and have a plane of symmetry (see Hovenier [1]). But the proofs in this paper are more directly based on symmetry considerations without making use of the scattering matrix. All seven symmetry relations are independent of the normalization of the phase matrix by means of a nondirectionally dependent constant. So, for example, the average of the one-one element of the phase matrix over all directions may be unity or equal to the albedo of single scattering. Takano and Liou [15] reported only three symmetry relations for the phase matrix of horizontally oriented particles, which correspond to our Eqs. (6), (7) and (10). Moreover, their derivations of these equations were more complicated than ours. No attempt has been made to present a complete survey of all possible kinds of particles and orientations including very artificial ones. The simple fundamental symmetry operations presented in this paper, together with the illustrative examples, will probably enable most readers to quickly determine which symmetry relations for the phase matrix are valid in a case which is not explicitly mentioned in this paper. The symmetry relations for the phase matrix can be used, for example, for theoretical purposes and to reduce or check numerical work. For instance, if the phase matrix
ARTICLE IN PRESS ˜oz / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453 J.W. Hovenier, O. Mun
is sought for a fixed value of jj0 (or for a specific component of the Fourier series expansion of the azimuth dependence) and a number of values of u and u0 spread in the ranges from 1 to + 1, the computations can be restricted to non-negative values of u and absolute values of u0 smaller than or equal to u. If we also restrict the range of jj0 to run from zero to p the computational labor can be reduced by a significant factor, up to about 8, when the symmetry relations (6), (9) and (10) are used.
Acknowledgment It is a pleasure to thank Dr. M.I. Mishchenko for fruitful discussions on the subject of this paper.
References [1] Hovenier JW. Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles. J Atmos Sci 1969;26: 488–99. [2] Hovenier JW, van der Mee CVM, Domke H. Transfer of polarized light in planetary atmospheres: basic concepts and practical methods. Dordrecht, Berlin: Kluwer, Springer; 2004. [3] van de Hulst HC. Multiple light scattering. Tables, formulas and applications. San Diego: Academic Press; 1980. [4] Mishchenko MI. Multiple scattering of polarized light in anisotropic plane- parallel media. Transp Theory and Stat Phys 1990;19:293–316.
1453
[5] Mishchenko MI, Hovenier JW, Travis LD. Light scattering by nonspherical particles: theory measurements and applications. San Diego: Academic Press; 2000. [6] Mishchenko MI, Travis LD, Lacis AA. Scattering, absorption, and emission of light by small particles. Cambridge: Cambridge University Press; 2002. [7] Mishchenko MI, Travis LD, Lacis AA. Multiple scattering of light by particles: radiative transfer and coherent backscattering. Cambridge: Cambridge University Press; 2006. [8] Liou KN, Takano Y, Yang P. Light scattering and radiative transfer in ice crystal clouds: applications to climate research. In: Mishchenko MI, Hovenier JW, Travis LD, editors. Light Scattering by nonspherical particles. Theory, measurements and applications. San Diego: Academic Press; 2000. p. 417–49. [9] Aydin K. Centimeter and millimeter wave scattering from nonspherical hydrometeors. In: Mishchenko MI, Hovenier JW, Travis LD, editors. Light scattering by nonspherical particles. Theory, measurements and applications. San Diego: Academic Press; 2000. p. 451–79. [10] Baran AJ. A review of the light scattering properties of cirrus. JQSRT 2009;110:1239–60. [11] Chandrasekhar S. Radiative transfer. London: Oxford University Press; 1950. [12] 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. [13] Saxon DS. Tensor scattering matrix for the electromagnetic field. Phys Rev 1995;100:1771–5. [14] Mishchenko MI. Transfer of polarized infrared radiation in optically anisotropic media: application to horizontally oriented ice crystals: comment. J Opt Soc Am A 1994;11:1376–7. [15] Takano Y, Liou KN. Transfer of polarized infrared radiation in optically anisotropic media: application to horizontally oriented ice crystals. J Opt Soc Am A 1993;10:1243–56. [16] Hovenier JW. Principles of symmetry for polarization studies of planets. Astron Atrophys 1970;7:86–90.
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Symmetry relations for the phase matrix of horizontally oriented particles ˜ oz b, J.W. Hovenier a, O. Mun a b
Astronomical Institute ‘‘Anton Pannekoek’’, University of Amsterdam, P.O. Box 94249, 1090GE Amsterdam, The Netherlands Instituto de Astrofı´sica de Andalucı´a, CSIC, Glorieta de la Astronomı´a s/n, 18080 Granada, Spain
a r t i c l e in fo
abstract
Article history: Received 14 January 2010 Accepted 2 March 2010
Seven symmetry relations, with a wide range of validity, are presented for the phase matrix of horizontally oriented particles. Three of these relations constitute a fundamental basis from which four others can be derived by making combinations. All seven relations can be used for many types of particles, including hexagonal plates and columns as well as spheroids, cylinders and cubes. Some applications are also pointed out. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Light scattering Particles Horizontal orientation Symmetry relations Phase matrix
1. Introduction and basic concepts Scattering of light by a small volume element inside a medium filled with particles can be described by means of the four by four phase matrix of the volume element. This real matrix transforms the Stokes parameters of an incident beam of light into the Stokes parameters of a beam of scattered light, where the meridian planes of the incident and scattered beams act as planes of reference for the Stokes parameters. The phase matrix plays a fundamental role not only in calculations for optically thin media, but also in studies of multiple scattering in optically thick media. In theoretical studies the phase matrix occurs, for instance, as the kernel of the equation of radiative transfer [1–7]. Non-spherical particles can be randomly oriented in three dimensional space, but they can also be horizontally oriented. Well-known examples are provided by several types of hydrometeors, like raindrops and snow flakes, and by ice crystals in planetary atmospheres. Various shapes occur, for example, hexagonal plates and columns
Corresponding author. Tel.: + 34 958121311; fax: + 34 958814530.
˜ oz). E-mail address: [email protected] (O. Mun 0022-4073/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2010.03.001
[8–10]. Symmetry relations for the phase matrix of collections of particles that are randomly oriented in three dimensional space have been presented in an earlier paper [1]. These relations were derived from properties of the scattering matrix which transforms Stokes parameters having the scattering plane as a plane of reference. For randomly oriented particles this scattering matrix is a relative simple function of directions in three dimensional space, since it is rotationally symmetric with respect to the direction of incident radiation. For collections of horizontally oriented particles this is generally not the case and, as a result, the phase matrix is more complicated. The main purpose of this paper is to show by elementary symmetry considerations that various symmetry relations still hold for the phase matrices of volume elements filled with collections of horizontally oriented particles. These relations are based on symmetries in time and space. We consider scattering of (quasi) monochromatic light, or other electromagnetic radiation, without change of wavelength by particles that have a plane of symmetry. Such particles are identical to their mirror images. We first consider collections of hexagonal plates with two broad flat sides in horizontal planes, but with random orientation in such planes, so that there is rotational symmetry about the vertical. To consider the transfer of polarized light in a
ARTICLE IN PRESS 1450
˜oz / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453 J.W. Hovenier, O. Mun
Z N ϕ' -ϕ θ
θ'
P2 P1 O
Y
ϕ ϕ'
X Fig. 1. Scattering by a local volume-element at O. Points N, P1 and P2 are located on a unit sphere. The direction of the incident light is OP1 and that of the scattered light is OP2.
scattering medium like a planetary atmosphere we use a right handed Cartesian coordinate system with axes x, y, z, fixed in space. We call the x, y plane the horizontal plane and the positive z-axis the local vertical. Let us now imagine that a local small volume element containing a collection of independently scattering particles is located in the origin, O, of the coordinate system (see Fig. 1). The collection consists of identical particles or a homogeneous mixture of such collections due to e.g. a distribution in size, shape or refractive index of the particles. The direction of propagation of a beam is specified by an angle, y, ð0 r y r pÞ which it makes with the upward normal and an azimuth angle j, ð0 r j r 2pÞ. The latter is measured clockwise from the positive x-axis when looking in the direction of the upward normal. To describe the state of polarization of a beam we use Stokes parameters, I, Q, U and V, with the meridian plane as the plane of reference [11,12,2,5–7]). The four Stokes parameters can be made elements of a real column vector, called the Stokes vector. The directions of the incident and scattered beams are represented in Fig. 1 by the points P1 and P2, respectively, on the surface of a unit sphere, having O at its center. Suppose light is incident on the volume element in a direction (u0 ,j0 ) and is scattered once into a direction 0 (u,j), where u ¼ cosy and u0 ¼ cosy . Both lie in the range from 1 to +1, where the latter value corresponds to the perpendicularly downward direction. This scattering process at a certain moment can now be described as a transformation of the Stokes vector, Ii ðu0 ,j0 Þ, of the incident light into the Stokes vector, Is ðu,jÞ, of the scattered light. Because of the rotational symmetry of the collection of particles about the z-axis only differences of azimuthal angles need to be used. So we can write
where c is a non-directionally dependent scalar, which can be used for normalization purposes, and the real 4 4 matrix Zðu,u0 ,jj0 Þ is the phase matrix of the volume element. Its elements are denoted as i-j, where i is the row number and j the column number.
Is ðu,jÞ ¼ cZðu,u0 ,jj0 ÞIi ðu0 ,j0 Þ
0 ~ ,jj0 ÞP, Zðu0 ,u,j0 jÞ ¼ PZðu,u
ð1Þ
2. Symmetry relations The problem we wish to address now is whether symmetry considerations will allow us to establish relations between two phase matrices that only differ in values of the three directional variables, u, u0 and jj0 . For this purpose we will consider three fundamental symmetry operations and combinations thereof. 2.1. Reciprocity The reciprocity principle corresponds to invariance of the ratio cause/effect under inversion of time [3]. In the context of light scattering by particles this means that a relation may exist between the phase matrices corresponding to an original situation [see Fig. 1] and a situation in which (i) the incident light travels in the inverse direction of the originally scattered light and (ii) the scattered light travels in the inverse direction of the originally incident light. When polarization is ignored this simply means that the phase functions, i.e. the 1–1 elements of the two phase matrices, are the same. For polarized light, however, we must take into account that, according to the definitions of Stokes parameters, the sign of the third Stokes parameter switches when the light travels in the opposite direction [1,2]. The resulting reciprocity relation is [Cf. Fig. 1] ð2Þ
ARTICLE IN PRESS ˜oz / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453 J.W. Hovenier, O. Mun
where the diagonal matrix P=diag (1,1, 1,1). In the righthand side of Eq. (2) pre- and postmultiplication by P amounts to changing the signs of the elements 1–3, 2–3, 4–3, 3–1, 3–2, and 3–4 of the matrix in the middle, and a tilde above a matrix symbol stands for the transposed matrix. A formal proof of Eq. (2) can be obtained by first applying the reciprocity principle for a single particle and then adding the phase matrices of the individual particles in a volume element [13,14]. Sufficient conditions for the validity of this proof are that there is no magnetic field and the dielectric, permeability and conductivity tensors of all particles are symmetric, which includes the common cases of diagonal matrices and scalars. We shall henceforth assume that Eq. (2) holds. It should be noted that this is a very fundamental relation and its validity has nothing to do with the question whether the scattering matrix depends only on the scattering angle or on more directional parameters [14,3]. In this paper we consider reciprocity relations to be symmetry relations, just like relations based on symmetries in space. 2.2. Mirror symmetry
experiment the incident beam i2 which is the mirror image of i1 will give rise to a beam r2 of scattered light, which is the mirror image of r1. Now i1 and i2 differ only in (i) the sign of the position angle of the major axis of the polarization ellipse and (ii) the sense in which the polarization ellipse is traced, since mirroring causes right-handedness to become left-handedness and vice versa. According to the definitions of Stokes parameters these two differences correspond to sign switches in the third and fourth Stokes parameters of the two incident beams. Because of symmetry this must also hold for the two scattered beams (see Fig. 2). Hence, we have in the first experiment for the Stokes vector of the scattered beam [cf. Eq. (1)] Is ðu,j1 Þ ¼ cZðu,u0 ,j1 j0 ÞIi ðu0 ,j0 Þ,
ð3Þ
and in the second experiment Is ðu,j2 Þ ¼ PQIs ðu,j1 Þ ¼ cZðu,u0 ,j2 j0 ÞPQIi ðu0 ,j0 Þ,
ð4Þ
where the diagonal matrix Q=diag (1,1,1, 1), so that PQ=QP=diag (1,1,1,1). By premultiplying Eq. (4) by PQ and using j2 j0 ¼ j0 j1 we obtain Is ðu,j1 Þ ¼ cPQZðu,u0 ,j0 j1 ÞPQIi ðu0 ,j0 Þ:
It is clear from our assumptions that there is mirror symmetry with respect to the meridian plane of incidence. So when in Fig. 2 in a first experiment the incident beam i1 gives rise to a beam r1 of scattered light, then in a second
1451
ð5Þ
On comparing this equation with Eq. (3) we find the following mirror symmetry relation for the phase matrix Zðu,u0 ,j0 jÞ ¼ PQZðu,u0 ,jj0 ÞQP:
ð6Þ
local vertical
i2 i1 r1
r2
ϕ1
ϕ2
ϕ0
Fig. 2. Illustration of the mirror symmetry relation for the phase matrix. If the incident beam i1 gives rise (among others) to the beam of scattered light r1, then the incident beam i2, which is the mirror image of i1 with respect to the plane of incidence, gives rise (among others) to the beam of scattered light r2, which is the mirror image of r1 with respect to the plane of incidence. The position angles of the polarization ellipses of the incident light (dots) and scattered light (small arcs) are also indicated, as well as the sense in which the four polarization ellipses are traced. [After Hovenier [16]].
ARTICLE IN PRESS 1452
˜oz / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453 J.W. Hovenier, O. Mun
Note that pre-and postmultiplication of a 4 4 matrix by PQ causes a sign switch of the elements 1–3, 1–4, 2–3, 2–4, 3–1, 3–2, 4–1, and 4–2 of the matrix in the middle. So it follows from Eq. (6) that these elements of Zðu,u0 ,jj0 Þ are odd functions of ðjj0 Þ and the other 8 elements are even functions of ðjj0 Þ. In a Fourier series expansion of Zðu,u0 ,jj0 Þ in ðjj0 Þ the first eight of the elements above will, therefore, contain only sine terms or vanish and the remaining eight elements only cosine terms and in general an azimuth independent term.
2.3. Rotational symmetry Simultaneous rotation of the meridian planes of incidence and scattering about a vertical axis through an arbitrary angle gives no new relations since we have already assumed that we have azimuthal symmetry. But rotation about a horizontal axis over an angle p, i.e. turning the collection of particles as well as the directions of the incident and scattered light upside down gives a new symmetry relation, though the scattering situation remains physically the same. The reason for this new relation is that the sense in which the azimuth is measured is fixed in space so that jj0 must be transformed into j0 j [1]. Thus, we have the symmetry relation: Zðu,u0 ,j0 jÞ ¼ Zðu,u0 ,jj0 Þ:
ð7Þ
2.4. Combinations of symmetries By combining the three fundamental symmetry relations (2), (6) and (7) four other symmetry relations can be derived. For example, if Eqs. (2) and (6) are both valid it is clear that switching the sign of jj0 in Eq. (2) will give 0 ~ Zðu0 ,u,jj0 Þ ¼ Q Zðu,u ,jj0 ÞQ :
ð8Þ
Similarly, combining Eqs. (2) and (7) gives 0 ~ Zðu0 ,u,jj0 Þ ¼ PZðu,u ,jj0 ÞP:
ð9Þ
Eqs. (6) and (7) can be combined into Zðu,u0 ,jj0 Þ ¼ PQZðu,u0 ,jj0 ÞQP:
ð10Þ
Finally, Eqs. (2), (6) and (7) give 0 ~ Zðu0 ,u,j0 jÞ ¼ Q Zðu,u ,jj0 ÞQ ,
ð11Þ
which can of course also be viewed as a combination of Eqs. (6) and (9). We have thus found seven symmetry relations that hold for arbitrary directions of the incident and scattered light, namely Eqs. (2) and (6)–(11). Each of the three fundamental equations (2), (6), and (7) has a simple explanation in terms of symmetry in time and space. Evidently, it is also possible to choose another set of three equations, from which the other four can be derived by making combinations, as shown by Eqs. (6), (9) and (10). Each of these relations shows the result of a simple algebraic operation, namely (i) interchanging j and j0 , (ii) interchanging u and u0 and (iii) changing the signs of u and u0 simultaneously.
3. Concluding remarks So far we have only considered hexagonal plates with their broad flat surfaces lying in horizontal planes, but with random orientation in such planes. For this case we found three fundamental symmetry relations from which four others could easily be derived by making combinations. It is, however, clear from the symmetry considerations in the preceding section that the same seven relations, i.e. Eqs. (2) and (6)–(11), will hold for many other kinds of particles. For example, the boundaries of the broad flat sides of the plates do not need to be hexagonal, but may be circular, rectangular, pentagonal, etc. Furthermore, we may have e.g. oblate spheroids with their long axes in the horizontal plane. Also allowed are hexagonal columns with their long axes randomly oriented in the horizontal plane and either randomly rotated about their long axes or with two rectangular facets parallel to the horizontal plane. Finally, roughness of the surfaces and inhomogeneities will often not destroy the fundamental symmetries. So the seven symmetry relations are valid for many kinds of horizontally oriented particles, especially various model particles [10]. For natural particles they will often hold in good approximation, as long as the conditions are approximately satisfied. The particles need not be perfectly horizontally oriented. For example, the flat surfaces of the hexagonal plates that we discussed earlier may be tilted from the horizontal plane in such a way that the three fundamental symmetry relations still hold, so that all seven symmetry relations are valid. This is important for falling hydrometeors [9] and in particular for fluttering ice crystals. All seven symmetry relations for the phase matrix are exactly the same as for particles that are randomly oriented in three dimensional space and have a plane of symmetry (see Hovenier [1]). But the proofs in this paper are more directly based on symmetry considerations without making use of the scattering matrix. All seven symmetry relations are independent of the normalization of the phase matrix by means of a nondirectionally dependent constant. So, for example, the average of the one-one element of the phase matrix over all directions may be unity or equal to the albedo of single scattering. Takano and Liou [15] reported only three symmetry relations for the phase matrix of horizontally oriented particles, which correspond to our Eqs. (6), (7) and (10). Moreover, their derivations of these equations were more complicated than ours. No attempt has been made to present a complete survey of all possible kinds of particles and orientations including very artificial ones. The simple fundamental symmetry operations presented in this paper, together with the illustrative examples, will probably enable most readers to quickly determine which symmetry relations for the phase matrix are valid in a case which is not explicitly mentioned in this paper. The symmetry relations for the phase matrix can be used, for example, for theoretical purposes and to reduce or check numerical work. For instance, if the phase matrix
ARTICLE IN PRESS ˜oz / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 1449–1453 J.W. Hovenier, O. Mun
is sought for a fixed value of jj0 (or for a specific component of the Fourier series expansion of the azimuth dependence) and a number of values of u and u0 spread in the ranges from 1 to + 1, the computations can be restricted to non-negative values of u and absolute values of u0 smaller than or equal to u. If we also restrict the range of jj0 to run from zero to p the computational labor can be reduced by a significant factor, up to about 8, when the symmetry relations (6), (9) and (10) are used.
Acknowledgment It is a pleasure to thank Dr. M.I. Mishchenko for fruitful discussions on the subject of this paper.
References [1] Hovenier JW. Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles. J Atmos Sci 1969;26: 488–99. [2] Hovenier JW, van der Mee CVM, Domke H. Transfer of polarized light in planetary atmospheres: basic concepts and practical methods. Dordrecht, Berlin: Kluwer, Springer; 2004. [3] van de Hulst HC. Multiple light scattering. Tables, formulas and applications. San Diego: Academic Press; 1980. [4] Mishchenko MI. Multiple scattering of polarized light in anisotropic plane- parallel media. Transp Theory and Stat Phys 1990;19:293–316.
1453
[5] Mishchenko MI, Hovenier JW, Travis LD. Light scattering by nonspherical particles: theory measurements and applications. San Diego: Academic Press; 2000. [6] Mishchenko MI, Travis LD, Lacis AA. Scattering, absorption, and emission of light by small particles. Cambridge: Cambridge University Press; 2002. [7] Mishchenko MI, Travis LD, Lacis AA. Multiple scattering of light by particles: radiative transfer and coherent backscattering. Cambridge: Cambridge University Press; 2006. [8] Liou KN, Takano Y, Yang P. Light scattering and radiative transfer in ice crystal clouds: applications to climate research. In: Mishchenko MI, Hovenier JW, Travis LD, editors. Light Scattering by nonspherical particles. Theory, measurements and applications. San Diego: Academic Press; 2000. p. 417–49. [9] Aydin K. Centimeter and millimeter wave scattering from nonspherical hydrometeors. In: Mishchenko MI, Hovenier JW, Travis LD, editors. Light scattering by nonspherical particles. Theory, measurements and applications. San Diego: Academic Press; 2000. p. 451–79. [10] Baran AJ. A review of the light scattering properties of cirrus. JQSRT 2009;110:1239–60. [11] Chandrasekhar S. Radiative transfer. London: Oxford University Press; 1950. [12] 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. [13] Saxon DS. Tensor scattering matrix for the electromagnetic field. Phys Rev 1995;100:1771–5. [14] Mishchenko MI. Transfer of polarized infrared radiation in optically anisotropic media: application to horizontally oriented ice crystals: comment. J Opt Soc Am A 1994;11:1376–7. [15] Takano Y, Liou KN. Transfer of polarized infrared radiation in optically anisotropic media: application to horizontally oriented ice crystals. J Opt Soc Am A 1993;10:1243–56. [16] Hovenier JW. Principles of symmetry for polarization studies of planets. Astron Atrophys 1970;7:86–90.