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Fresnel boundary and interface conditions for polarized radiative transfer in a multilayer medium
Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
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Fresnel boundary and interface conditions for polarized radiative transfer in a multilayer medium R.D.M. Garcia Instituto de Estudos Avanc- ados, Trevo Cel. Av. Jose´ Alberto Albano do Amarante no. 1, Sa~ o Jose´ dos Campos, SP 12228-001, Brazil
a r t i c l e in f o
abstract
Article history: Received 12 July 2011 Received in revised form 17 October 2011 Accepted 24 November 2011 Available online 3 December 2011
In many applications of the theory of radiative transfer, it is important to consider the changes in the index of refraction that occur when the physical domain being studied consists of material regions with distinct optical properties. When polarization effects are taken into account, the radiation field is described by a vector of four components known as the Stokes vector. At an interface between two different material regions, the reflected and transmitted Stokes vectors are related to the incident Stokes vector by means of reflection and transmission matrices, which are derived from the Fresnel formulas for the amplitude coefficients of reflection and transmission. Having seen that many works on polarized radiative transfer that allow for changes in the index of refraction exhibit discrepancies in their expressions for the transmission matrix, we present in this work a careful derivation of the relations between the reflected and transmitted Stokes vectors and the Stokes vector incident on an interface. We obtain a general form of a transmission factor that is required to ensure conservation of energy and we show that most of the discrepancies encountered in existing works are associated with the use of improper forms of this factor. In addition, we derive explicit and compact expressions for the Fresnel boundary and interface conditions appropriate to the study of polarized radiative transfer in a multilayer medium. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Radiative transfer Polarization Fresnel conditions Index of refraction
1. Introduction Along with its continued use in the fields of astrophysics [1,2], atmospheric sciences [3,4], remote sensing [5], mechanical engineering [6,7], hydrological optics [8], and electromagnetic-wave propagation [9], radiative transfer has been applied to new fields of study, such as biomedical optics [10], optical tomography [11], radiation protection [12], analysis of masterpiece paintings [13], modeling of lamp envelope and optical fiber fabrication processes [14,15], and modeling of photovoltaic modules [16]. In many cases, to obtain a good description of the radiation field, it is important to consider the changes in the index of refraction that occur when the physical
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domain being studied consists of material regions with distinct optical properties. For the scalar case, where only the radiation intensity is of interest and polarization effects can be neglected, some authors have been able to treat a change in the index of refraction when an interface between two dissimilar planar media is crossed [17–19]. Other authors have considered systems with two or more planar interfaces [20–30]. While working on an extension of previous papers on the scalar case [28–30] to the case with polarization, we were able to find various works where polarized radiative transfer with changes in the index of refraction is considered and discussed in some detail. When polarization effects are taken into account, one needs to work with a vector of four components—the Stokes vector—instead of a single quantity, as in the scalar case, and the Fresnel
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
formulas for the amplitude coefficients of reflection and transmission for the parallel and perpendicular components of the electric field. Surprisingly, we have found discrepancies in most of the works that we have examined [5,31–40], especially in the expressions for the transmission matrix. And so, starting with the Fresnel formulas, we present in this work a detailed derivation of the reflection and transmission matrices that relate the reflected and transmitted Stokes vectors to the Stokes vector incident on an interface. We also derive the Fresnel boundary and interface conditions appropriate to the study of polarized radiative transfer in a multilayer medium.
2. Fundamentals In this section, we summarize the theoretical background of radiative transfer with polarization. Details can be found in many sources, e.g. [1,2,41–44]. We also discuss briefly the Fresnel formulas for the reflection and transmission coefficients at an interface between two dissimilar media [45,46].
2.1. The radiative transfer equation (RTE) Following Siewert [47], we write the equation that describes polarized radiative transfer in a homogeneous plane-parallel medium as Z Z @ $ 2p 1 0 m Iðt, m, fÞ þIðt, m, fÞ ¼ Pðm, m0 , ff Þ @t 4p 0 1 0
0
Iðt, m0 , f Þ dm0 df þ Sðt, m, fÞ,
ð1Þ
where the optical variable t 2 ðtmin , tmax Þ is used to define the position in the medium and m 2 ½1; 1, with m ¼ cos y, and f 2 ½0; 2p are, respectively, the polar and azimuthal variables that specify the direction of propagation. Here, m is measured with respect to the positive t-axis. Furthermore, 0 1 I BQ C B C I¼B C ð2Þ @UA V is the Stokes vector that characterizes the radiation field, $ is the single-scattering albedo, and Pðm, m0 , ff0 Þ is the phase matrix [48] for scattering from an initial direction 0 ðm0 , f Þ to a final direction ðm, fÞ. The Stokes parameters I, Q, U, and V in Eq. (2) depend on t, m, and f, and can be expressed in terms of complex oscillatory functions El and Er. Making use of l and r to represent unit vectors along directions respectively parallel and perpendicular to a reference plane (called meridian plane [2]), defined by the direction of propagation and the t-axis, and choosing the orientation of l and r such that r l is in the direction of propagation, we can write the electric field at a given position as E ¼ El l þ Er r:
ð3Þ
307
The Stokes parameters are then defined in terms of the El and Er components of the electric field as [41,42] 1 0 1 0 1 0 Il þIr El Enl þ Er Enr I C B C BQ C B El Enl Er Enr C B Il Ir C B C B C B C ð4Þ B C¼B n C, n n C ) B B @ U A @ El Er þ Er El A @ 2RfEl Er g A V
iðEl Enr Er Enl Þ
2IfEr Enl g
where n denotes the operation of complex conjugation, i is the imaginary unit, and Rfzg and Ifzg are the real and imaginary parts of a complex variable z. Even though Eq. (4) is strictly true only for a monochromatic beam, it can still be applied to a quasi-monochromatic beam if the components on the right side are understood as time averages [2]. Finally, the internal source Sðt, m, fÞ in Eq. (1) can be either an emission source or the first-collision source resulting from a decomposition of the Stokes vector into uncollided and collided components that is frequently used for solving problems driven by monodirectional external sources. To close this subsection, we believe it important to mention that some authors use definitions of the Stokes vector that differ slightly from the one adopted in this work or, sometimes, definitions based on different sets of Stokes parameters. One frequently used alternative representation makes use of the intensities Il and Ir in the place of I and Q. Also, in some works, V is defined as the negative of our V. Sometimes, a scalar appears multiplying the right side of Eq. (4). When that scalar does not depend on the medium properties, no change is required in the expressions for the interface relations that are presented in Section 3 of our work. Should the scalar depend on the medium properties, the interface relation for transmission will have to be modified accordingly. 2.2. The Fresnel formulas In general, upon reaching an interface between two media with different properties, radiation is partially reflected and partially transmitted. Consider a beam of radiation traveling in medium a (characterized by an index of refraction na). When that beam reaches the interface with medium b (characterized by an index of refraction nb), part of it is reflected specularly (we assume a smooth interface) and part is refracted along a direction determined by Snell’s law, na sin Wa ¼ nb sin Wb ,
ð5Þ
where Wa , Wb are angles measured with respect to the normal at the interface. The angle Wa defines the initial direction of propagation in medium a and the angle Wb the direction of propagation in medium b, after refraction. The amplitude coefficients of reflection and transmission are given by the Fresnel formulas, which are usually written as [45,46] Rl ¼ ERl =EIl ¼
tanðWa Wb Þ , tanðWa þ Wb Þ
Rr ¼ ERr =EIr ¼
sinðWa Wb Þ , sinðWa þ Wb Þ
ð6aÞ
ð6bÞ
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T l ¼ ETl =EIl ¼
2 sin Wb cos Wa , sinðWa þ Wb ÞcosðWa Wb Þ
ð6cÞ
2 sin Wb cos Wa , sinðWa þ Wb Þ
ð6dÞ
and T r ¼ ETr =EIr ¼
where the superscripts I, R, and T affixed to El and Er mean ‘‘incident’’, ‘‘reflected’’, and ‘‘transmitted’’. At this point, a comment on the way Eqs. (6a) and (6b) have been written is important. As discussed many years ago by Friedmann and Sandhu [49], the theory is unable to determine univocally the signs that should be used on the right sides of these equations, out of the four possible choices ( þ þ , þ , þ , and ). Physically, a change of sign on the right side of Eq. (6a) [or Eq. (6b)] means a change of p in the phase of the parallel (or the perpendicular) component of the reflected electric field with respect to the phase of the parallel (or the perpendicular) component of the incident electric field. The choice of signs used in Eqs. (6a) and (6b), i.e. þ, is the choice used by most authors of books on this subject and is in agreement with the Nebraska ellipsometry conventions [50]. However, this choice leads to a physical inconsistency [50], as discussed next. The reflection and transmission coefficients defined by Eqs. (6) can also be written in the alternative (and sometimes more convenient) form Rl ¼
nb cos Wa na cos Wb , nb cos Wa þna cos Wb
ð7aÞ
Rr ¼
na cos Wa nb cos Wb , na cos Wa þ nb cos Wb
ð7bÞ
2na cos Wa , Tl ¼ nb cos Wa þ na cos Wb
ð7cÞ
and Tr ¼
2na cos Wa : na cos Wa þ nb cos Wb
3. Interface relations for the Stokes vector To obtain general formulas relating the reflected and transmitted Stokes vectors to the incident Stokes vector at an interface, we consider, as in Section 2.2, that the radiation beam is traveling initially in medium a and, upon reaching the interface between media a and b, is partially reflected back to medium a and partially transmitted into medium b. 3.1. The reflection and transmission matrices Working with Eq. (4) and the first equalities in Eqs. (6) we find that the reflected Stokes vector IR is related to the incident Stokes vector II by IR ¼ Rab II , where 01 n n 2ðRl Rl þ Rr Rr Þ B1 B 2ðRl Rnl Rr Rnr Þ Rab ¼ B B 0 @ 0
ð9Þ
n n 1 2ðRl Rl Rr Rr Þ n n 1 2ðRl Rl þ Rr Rr Þ
0
0
RfRl Rnr g
0
IfRl Rnr g
0
0
and
C C C IfRl Rnr g C A RfRl Rnr g
ð10Þ is the reflection matrix. The subscript ab attached to R indicates radiation traveling in medium a that is reflected by medium b. Similarly, we find that the transmitted Stokes vector IT is related to the incident Stokes vector II by IT ¼ Tab II ,
ð8bÞ
Clearly, this implies a phase difference of p between the parallel and perpendicular components of the electric field upon reflection. Such a phase difference is unjustified on physical grounds, since the parallel and perpendicular components become indistinguishable when the incidence is normal to the interface. The fact that a change of p in the phase of the reflected electric field with respect to the phase of the incident electric field is expected when nb 4na [51] suggests that using Eq. (6a) [and consequently Eq. (7a)] with a change in sign would be the most consistent choice from a physical point of view. In spite of its strong appeal, the ‘‘physically consistent’’ sign choice was rejected by the ellipsometry community
1
0
ð7dÞ
For the case of normal incidence, Wa ¼ Wb ¼ 0, and so Eqs. (7a) and (7b) reduce to n na Rl ¼ b ð8aÞ nb þ na na nb : Rr ¼ na þ nb
because it involves a change of coordinate system for describing the incident and reflected fields [50]. In fact, using an extension for the case with polarization of a previous work on the analytical discrete-ordinates (ADO) method for solving scalar radiative transfer problems subject to Fresnel boundary and interface conditions [30], we have found that the ‘‘physically consistent’’ sign choice gives rise to a sign flip on the computed U and V Stokes parameters at either side of a Fresnel interface, when the cosine of the polar angle m-0 from above and from below. On the other hand, the ‘‘conventional’’ sign choice used in this work yields continuous results for all of the Stokes parameters at both sides of a Fresnel interface. Details will be given in a future work.
ð11Þ
where 01 n n 2ðT l T l þ T r T r Þ B1 B 2ðT l T nl T r T nr Þ Tab ¼ f T B B 0 @ 0
n n 1 2ðT l T l T r T r Þ n n 1 ðT T þ T T r rÞ 2 l l
0 0
0 0 RfT l T nr g IfT l T nr g
0
1
C C 0 C n C IfT l T r g A n RfT l T r g
ð12Þ is the transmission matrix. The subscript ab attached to T indicates radiation traveling in medium a that is transmitted into medium b. In Eq. (12), fT is a factor that takes into account the change of properties when going from medium a to medium b, and can be determined from the requirement of conservation of energy, as will be shown shortly.
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The reflection and transmission matrices defined by Eqs. (10) and (12) can be written in a more explicit way when both indices of refraction (na and nb) are real. For such purpose, it is necessary to distinguish between two cases. The first case comprises two distinct situations: (i) na =nb r1 or (ii) na =nb 4 1 and Wa o Wc , where Wc ¼ arcsinðnb =na Þ is the critical angle [45,46]. In this case, all of the reflection and transmission coefficients are real, and so, with the help of Eqs. (7a) and (7b) and the definition nab ¼ na =nb ,
ð13Þ
we can readily write the basic quantities that appear in the reflection matrix as 2 cos Wa nab cos Wb , ð14aÞ Rl Rnl ¼ cos Wa þnab cos Wb Rr Rnr ¼
nab cos Wa cos Wb nab cos Wa þ cos Wb
RfRl Rnr g ¼
2
,
cos Wa nab cos Wb cos Wa þ nab cos Wb
ð14bÞ
nab cos Wa cos Wb , nab cos Wa þ cos Wb ð14cÞ
and IfRl Rnr g ¼ 0:
ð14dÞ
For the transmission matrix, we get, with the help of Eqs. (7c) and (7d), 2 2nab cos Wa T l T nl ¼ , ð15aÞ cos Wa þnab cos Wb T r T nr ¼
2nab cos Wa nab cos Wa þ cos Wb
2
,
ð15bÞ
4n2ab cos2 Wa , ðcos Wa þnab cos Wb Þðnab cos Wa þ cos Wb Þ
RfT l T nr g ¼
ð15cÞ
and IfT l T nr g ¼ 0:
ð15dÞ
To determine the transmission factor fT, we consider that the direction of propagation of the beam in medium a is in a cone defined by a differential solid angle dXa about Xa . When the beam strikes an elementary area dA on the interface with medium b, part of it is reflected specularly and part is refracted into a cone defined by a differential solid angle dXb about Xb . Since Il and Ir, the parallel and perpendicular components of the intensity, are independent of each other, conservation of energy implies that IIx dXa ðdA
cos
Wa Þ ¼ IRx dXa ðdA
cos
Wa Þ þf T ITx dXb ðdA
cos Wb Þ, ð16Þ
for both x¼l and x ¼r. Here, as before, the superscripts I, R, and T indicate ‘‘incident’’, ‘‘reflected’’, and ‘‘transmitted’’. In addition, dA cos Wa and dA cos Wb are the projections of the elementary area dA into planes perpendicular to the directions Xa and Xb . Using dXa ¼ sin Wa dWa df and dXb ¼ sin Wb dWb df and noting that IRx ¼ Rx Rnx IIx
ð17aÞ
309
and ITx ¼ T x T nx IIx ,
ð17bÞ
for x ¼l and x¼r, we find from Eq. (16) cos Wa sin Wa dWa 1Rx Rnx , fT ¼ cos Wb sin Wb dWb T x T nx
ð18Þ
for x¼l and x¼r. Now, multiplying Snell’s law by its differential, we can show that n2a cos Wa sin Wa dWa ¼ n2b cos Wb sinWb dWb :
ð19Þ
Using this result in Eq. (18), we find that fT can be expressed as 1Rx Rnx , ð20Þ f T ¼ n2ba n T xT x for x¼l and x ¼r. Using Eqs. (14a), (14b), (15a) and (15b), we find that both forms of Eq. (20) [i.e. Eq. (20) for x ¼l and Eq. (20) for x¼ r] reduce to cos Wb f T ¼ n3ba , ð21Þ cos Wa a result that was also obtained in a recent work by Sommersten et al. [38], using physical arguments. Simply stated, fT can be understood as the product of three factors: one that adjusts for the change in the power of the beam when an interface is crossed, another that is given by the ratio between the areas obtained by projecting a unit area lying on the interface into planes perpendicular to the directions of incidence and refraction, and one that takes into account the angular redistribution of radiation caused by refraction. The second case to consider when both indices of refraction na and nb are real is defined by na =nb 4 1 and Wa Z Wc and is known in the literature as total reflection [45,46], since there is no transmission of radiation from medium a to medium b in this case. It can be concluded from Snell’s law that the angle of refraction Wb is purely imaginary for this case, and so the algebra has to be carried out in complex mode. We get, for the basic quantities that appear in the reflection matrix, Rl Rnl ¼ 1,
ð22aÞ
Rr Rnr ¼ 1,
ð22bÞ
RfRl Rnr g ¼
2 sin4 Wa 1, 1ð1þ n2ba Þcos2 Wa
ð22cÞ
and 2
IfRl Rnr g ¼
2
2 cos Wa sin Wa ðsin Wa n2ba Þ1=2 : 1ð1 þn2ba Þcos2 Wa
ð22dÞ
Also, both forms of Eq. (20) yield fT ¼0 for this case, and so Eq. (12) reduces to Tab ¼ 0,
ð23Þ
which confirms that there is no transmission of radiation from medium a to medium b when na =nb 4 1 and Wa Z Wc . Finally, we note that to be more explicit in our notation we should have written the reflection and transmission matrices defined by Eqs. (10) and (12) as
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Rab ðnab ,cos Wa ,cos Wb Þ and Tab ðnab ,cos Wa ,cos Wb Þ, and keep in mind that nab, Wa , and Wb are related by Snell’s law. 3.2. Comparison with other works In this subsection, we compare our results for the elements of the reflection and transmission matrices that are reported in Section 3.1 with results of other authors. In general, except when otherwise noted, we have observed agreement with regard to the elements of the reflection matrix. However, in the case of the transmission matrix we have detected many discrepancies, as discussed next. We begin with the paper by Tsang and Kong [31] and the book by Tsang et al. [5]. In spite of some slight changes in notation, the final results reported in these references are the same, and so we focus on a comparison of our results with the results of the latter [5], which is more explicit than the former [31]. The transmission matrix of these authors is given by Eq. (13), Section 6, Chapter 3 of Tsang et al. [5]. Working out this equation using the explicit definitions of the reflection and transmission coefficients given by Eqs. (9a) and (9b), Section 6, Chapter 2, of that same book [5], we found that it is equivalent to our result. However, in their analysis of the interface relations, these authors [5] have defined the Stokes vector in a way that differs from ours. They have used, in our notation, 0 1 Il B 1 B Ir C C I¼ B ð24Þ C, Z@ U A V where Z is the wave impedance of the medium, which is given by Z ¼ ðmm =EÞ1=2 for a dielectric. Here, mm is the magnetic permeability and E the electric permittivity of the medium. After taking the presence of the factor 1=Z in Eq. (24) into account, we concluded that, in order to agree with our result, the transmission matrix of Tsang et al. [5] should be multiplied by ðE2 =E1 Þ1=2 , where E1 is used by the authors [5] to denote the electric permittivity of the medium that contains the incident beam and E2 the electric permittivity of the medium that contains the transmitted beam. Note that, for dielectric media, ðE2 =E1 Þ1=2 n2 =n1 . Continuing with our comparison, we note that there is no mention of the transmission factor fT that is needed to ensure conservation of energy in the work of Kattawar and Adams [32]. This is also the case of a more recent work by Zhai et al. [33]. With regard to the work by Lam and Ishimaru [34], the transmission matrix that is expressed by Eq. (16) of their work [34] should be multiplied by ðnq =np Þ2 to agree with our result. Note that, in their notation, np denotes the index of refraction of the medium that contains the incident beam and nq the index of refraction of the medium that contains the transmitted beam. It should also be pointed out in regard to that work [34] that the term on the right side of the second equality in their Eq. (4) is missing a minus sign, and so the Stokes parameter V
of these authors, although defined in the same way as we do in this work, ended up being the negative of what we have. Concerning the work of Chami et al. [35], which is specific for the case of an atmosphere–ocean interface, we first note that the V component of the Stokes vector has been neglected and so these authors have worked with a Stokes vector of only three components: I, Q, and U. With regard to the reflection matrix, there are some corrections to be made in their work [35]: the squares in their Eqs. (8) and (9) should be removed and a minus sign included on the right side of their Eq. (8). The transmission matrix, which is given by Eq. (11) of their work [35], will agree with our result only after multiplication by ðnw =na Þ2 , where nw denotes the index of refraction of water and na 1 the index of refraction of air. The work of He et al. [36] is also specific for an atmosphere–ocean interface. The reflection and transmission matrices given by their Eqs. (36) and (37) agree with ours for incidence from the atmosphere side. However, for incidence from the ocean side, nw (the index of refraction of water) needs to be replaced by 1=nw in their equations. In the work of Bordier et al. [37], a Stokes-vector representation based on the Stokes parameters Il, Ir, U, and V has been used. A simple correction is required in the work of these authors: the (1,1) and (2,2) elements of the reflection and transmission matrices [Eqs. (32) and (33) of their work] should be squared. Moreover, to agree with our result, their transmission matrix needs to be multiplied by ðnj =ni Þ2 , where transmission from medium i to medium j is assumed by these authors [37]. Sommersten et al. [38] have also used the Stokes parameters Il, Ir, U, and V to define their Stokes vector. These authors have selected the sign of Eq. (7a) in the ‘‘physically consistent’’ way mentioned at the end of Section 2.2. This makes the signs of the (3,3), (3,4), (4,3), and (4,4) elements of the reflection matrix that is given by Eq. (17) of their work [38] to be the opposite of what we have. Other than that difference, which is caused by a choice of sign that departs from the recommended choice [50], the formulation of these authors for the reflection and transmission matrices is in perfect agreement with our formulation. We believe, however, that a mistake was made by these authors in the derivation of the reflection coefficients for the case of total reflection. We have found that the correct forms of these coefficients can be obtained by interchanging and multiplying by 1 the right sides of their Eqs. (19) and (20). Next, we comment on the work of Ota et al. [39], which is also specific for the case of an atmosphere–ocean interface. We concluded that, in order to agree with our result, the transmission matrix given by their Eq. (A.4) ~ 2 sin2 bÞ1=2 =cos b, should be multiplied by the factor ðm ~ is the index of refraction of water relative to air where m and b is the angle of incidence from the atmosphere. In addition, we have found that the (3,3), (3,4), (4,3), and (4,4) elements of the matrix for total reflection given by their Eq. (A.8) are incorrect. To see this, it is sufficient to note that the right sides of their Eqs. (A.9), when squared and added, do not yield unity, as required. In the notation of these authors [39], the correct quantities that define
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
311
the (3,3), (3,4), (4,3), and (4,4) elements of their matrix for total reflection are given by cos d ¼
~ 2 sin4 b 2m ~ sin2 bcos2 b m 2
1
ð25aÞ
and sin d ¼
~ cos b sin2 bðm ~ 2 sin2 b1Þ1=2 2m : 2 2 ~ sin bcos2 b m
ð25bÞ
Completing the analysis of the series of works that we have examined, we have found that the transmission matrix used by Zhai et al. [40] needs to be multiplied by an ðnt =ni Þ2 factor to agree with our result. In the notation of these authors, ni denotes the refractive index of the medium of incidence and nt that of the medium to where radiation is transmitted. Finally, we illustrate in Figs. 1–6, for the case of light crossing an air–water interface, the effects of the two most frequent discrepancies in the works that we have examined. In these figures, we plot the basic elements of the transmission matrix Tab given by Eq. (12) against the angle of incidence Wa . The solid lines represent the correct elements obtained from Eqs. (12) and (15), the dashed
Fig. 3. The (3,3) element of the transmission matrix for light going from air to water.
Fig. 4. The (1,1) element of the transmission matrix for light going from water to air.
Fig. 1. The (1,1) element of the transmission matrix for light going from air to water.
Fig. 5. The (1,2) element of the transmission matrix for light going from water to air.
Fig. 2. The (1,2) element of the transmission matrix for light going from air to water.
lines represent the elements obtained when fT is replaced by f T =n2ba in Eq. (12), and the dot-dashed lines represent the elements obtained when the transmission factor fT is neglected in Eq. (12). Figs. 1–3 are for incidence from the air side (n ¼1.0), while Figs. 4–6 are for incidence from the
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To provide some numerical results that could be useful for checking future computational implementations, we report in Tables 1 and 2, for the considered air–water interface, the basic elements of the reflection and transmission matrices for selected values of the incidence angle (Wa ).
water side (n¼ 1.338). The curves plotted in Figs. 1–6 indicate that the transmission of radiation to the ocean in radiative transfer simulations of the atmosphere–ocean system is underestimated when either fT or n2ba is omitted in the expression for the Fresnel transmission matrix.
3.3. A preliminary investigation of the effect of the discrepancies in the transmission matrix on multiple scattering calculations Since the discrepancies in the transmission matrix that are discussed in detail in Section 3.2 affect even the scalar case for which polarization effects are not taken into account, we thought it interesting to carry out a computational simulation of a very simple scalar case defined by a single layer, in order to estimate the influence of the two most frequent discrepancies (fT missing or replaced by f T =n2ba ) on important integral quantities such as reflectance and transmittance. We thus used the scalar ADO method reported in a recent work [29] to compute the reflectance and transmittance of a water layer with index of refraction n¼ 1.338 that is illuminated on the top by a uniform beam of radiation traveling in an external
Fig. 6. The (3,3) element of the transmission matrix for light going from water to air.
Table 1 Basic elements of the reflection matrix for an air–water interface. Direction
Wa (deg.)
(1,1) element
(1,2) element
(3,3) element
Air-Water
0 15 30 45 60 75 90
2.0899909( 2) 2.0955742( 2) 2.1979938( 2) 2.8529127( 2) 6.0630179( 2) 2.1394058( 1) 1.0
0.0 2.2079861( 3) 9.7003547( 3) 2.5599246( 2) 5.6383185( 2) 1.0309504( 1) 0.0
2.0899909( 2) 2.0839096( 2) 1.9723610( 2) 1.2593240( 2) 2.2292487( 2) 1.8746195( 1) 1.0
Water-Air
0 15 30 45 60 75 90
2.0899909( 2) 2.1095570( 2) 2.6242311( 2) 1.4886311( 1) 1.0 1.0 1.0
0.0 4.1327359( 3) 2.1475043( 2) 9.1222168( 2) 0.0 0.0 0.0
2.0899909( 2) 2.0686797( 2) 1.5082486( 2) 1.1763818( 1) 8.4319194( 1) 9.4398799( 1) 1.0
(3,4) element 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.3761264( 1) 3.2997980( 1) 0.0
Table 2 Basic elements of the transmission matrix for an air–water interface. Direction
Wa (deg.)
(1,1) element
(1,2) element
(3,3) element
Air-Water
0 15 30 45 60 75 90
1.7528281 1.7527281 1.7508945 1.7391699 1.6817012 1.4072382 0.0
0.0 3.9528339( 3) 1.7366002( 2) 4.5828897( 2) 1.0093966( 1) 1.8456528( 1) 0.0
1.7528281 1.7527237 1.7508084 1.7385660 1.6786691 1.3950824 0.0
Water-Air
0 15 30 45 60 75 90
5.4690874( 1) 5.4679945( 1) 5.4392457( 1) 4.7543066( 1) 0.0 0.0 0.0
0.0 2.3084764( 3) 1.1995596( 2) 5.0955159( 2) 0.0 0.0 0.0
5.4690874( 1) 5.4679457( 1) 5.4379228( 1) 4.7269217( 1) 0.0 0.0 0.0
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
medium with index of refraction n ¼1.0. At the bottom of the layer, a black surface is simulated by assuming the presence of an external medium with an index of refraction identical to that of the layer, i.e. n ¼1.338. The external media are non-participating and scattering in the water layer is described by the Henyey–Greenstein phase function [52], which depends only on one parameter, the so-called asymmetry factor g.
313
We report in Tables 3–5 the reflectance and the transmittance of the water layer for three choices of g: a case dominated by forward scattering (g¼0.9), a case dominated by backward scattering (g ¼ 0:7), and the isotropic scattering case (g ¼0.0). For each choice of g, we have reported numerical results for three choices of the cosine of the polar angle of the incident beam (m0 ¼ 0:2, 0.5, and 1.0) and three choices of the optical
Table 3 The reflectance and the transmittance for the case $ ¼ 1:0 and g ¼ 0.9.
m0
D
Reflectance
Transmittance
Correct fT
fT missing
f T ( f T =n2ba
Correct fT
fT missing
f T ( f T =n2ba
0.2
0.1 1.0 10.0
3.005348( 1) 3.122315( 1) 4.927353( 1)
2.998677( 1) 3.013018( 1) 3.234328( 1)
3.001992( 1) 3.067327( 1) 4.075591( 1)
6.994652( 1) 6.877685( 1) 5.072647( 1)
8.575925( 2) 8.432516( 2) 6.219415( 2)
3.907094( 1) 3.841759( 1) 2.833495( 1)
0.5
0.1 1.0 10.0
6.146126( 2) 7.402524( 2) 3.019589( 1)
6.085776( 2) 6.429823( 2) 1.267146( 1)
6.109441( 2) 6.811243( 2) 1.954323( 1)
9.385387( 1) 9.259748( 1) 6.980411( 1)
2.570055( 1) 2.535650( 1) 1.911486( 1)
5.242519( 1) 5.172338( 1) 3.899140( 1)
1.0
0.1 1.0 10.0
2.142050( 2) 2.884992( 2) 2.180722( 1)
2.111724( 2) 2.421885( 2) 1.032146( 1)
2.119070( 2) 2.534065( 2) 1.310370( 1)
9.785795( 1) 9.711501( 1) 7.819278( 1)
4.085336( 1) 4.054320( 1) 3.264362( 1)
5.466179( 1) 5.424680( 1) 4.367716( 1)
Table 4 The reflectance and the transmittance for the case $ ¼ 1:0 and g ¼ 0:7.
m0
D
Reflectance
Transmittance
Correct fT
fT missing
f T ( f T =n2ba
Correct fT
fT missing
f T ( f T =n2ba
0.2
0.1 1.0 10.0
3.356045( 1) 5.601499( 1) 9.072452( 1)
3.041675( 1) 3.316983( 1) 3.742545( 1)
3.197885( 1) 4.452158( 1) 6.390973( 1)
6.643955( 1) 4.398501( 1) 9.275483( 2)
8.145947( 2) 5.392866( 2) 1.137238( 2)
3.711201( 1) 2.456928( 1) 5.181128( 2)
0.5
0.1 1.0 10.0
1.142017( 1) 4.188899( 1) 8.764800( 1)
7.529997( 2) 1.587345( 1) 2.840390( 1)
9.055431( 2) 2.607480( 1) 5.163501( 1)
8.857983( 1) 5.811101( 1) 1.235200( 1)
2.425633( 1) 1.591287( 1) 3.382419( 2)
4.947920( 1) 3.245983( 1) 6.899618( 2)
1.0
0.1 1.0 10.0
8.408298( 2) 4.055740( 1) 8.694572( 1)
4.727734( 2) 1.814922( 1) 3.751523( 1)
5.619291( 2) 2.357723( 1) 4.948896( 1)
9.159170( 1) 5.944260( 1) 1.305428( 1)
3.823735( 1) 2.481587( 1) 5.449852( 2)
5.116157( 1) 3.320363( 1) 7.291901( 2)
Table 5 The reflectance and the transmittance for the case $ ¼ 1:0 and g ¼ 0.0.
m0
D
Reflectance
Transmittance
Correct fT
fT missing
f T ( f T =n2ba
Correct fT
fT missing
f T ( f T =n2ba
0.2
0.1 1.0 10.0
3.182911( 1) 4.891585( 1) 8.635459( 1)
3.020447( 1) 3.229942( 1) 3.688967( 1)
3.101176( 1) 4.055612( 1) 6.146877( 1)
6.817089( 1) 5.108415( 1) 1.364541( 1)
8.358221( 2) 6.263270( 2) 1.673021( 2)
3.807910( 1) 2.853474( 1) 7.622092( 2)
0.5
0.1 1.0 10.0
8.298818( 2) 2.997128( 1) 8.112362( 1)
6.675260( 2) 1.260995( 1) 2.661730( 1)
7.311898( 2) 1.941777( 1) 4.799060( 1)
9.170118( 1) 7.002872( 1) 1.887638( 1)
2.511106( 1) 1.917637( 1) 5.169027( 2)
5.122273( 1) 3.911686( 1) 1.054402( 1)
1.0
0.1 1.0 10.0
3.893032( 2) 2.325834( 1) 7.855518( 1)
2.842718( 2) 1.092727( 1) 3.401238( 1)
3.097139( 2) 1.391427( 1) 4.480215( 1)
9.610697( 1) 7.674166( 1) 2.144482( 1)
4.012237( 1) 3.203781( 1) 8.952699( 2)
5.368373( 1) 4.286659( 1) 1.197871( 1)
314
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
depth of the layer (D ¼ 0:1, 1.0, and 10.0). The singlescattering albedo $ was taken to be equal to 1.0 in the water layer. Other values of $ were also studied but we give here results only for $ ¼ 1:0 to explicitly show that the reflectance and the transmittance add to one in the conservative case only when the correct transmission factor fT is used. It can be observed that the values obtained for the reflectance and transmittance when fT is missing or is replaced by f T =n2ba always underestimate the correct values. For a missing fT the deviations can be as large as 67% in the reflectance and 87% in the transmittance. For f T =n2ba replacing fT the maximum deviations are 43% in the reflectance and 44% in the transmittance.
As done for the scalar case [26,28–30], we can now make use of the Heaviside function ( 1, x Z 0, HðxÞ ¼ ð29Þ 0, x o 0, to write Eqs. (27) and (28) in a more explicit and compact way. We get, for m 2 ð0; 1 and f 2 ½0; 2p, Ia ðtab ,m, fÞ ¼ Xðnab , mÞIa ðtab , m, fÞ þ Yðnab , mÞIb ½tab ,f ðnab , mÞ, f
ð30aÞ and Ib ðtab , m, fÞ ¼ Xðnba , mÞIb ðtab ,m, fÞ þYðnba , mÞIa ½tab ,f ðnba , mÞ, f,
ð30bÞ where (
4. Boundary and interface conditions for the RTE Xðn, mÞ ¼
4.1. A single interface
Gðn, mÞ,
n r 1,
Gðn, mÞH½mmc ðnÞ þ Cðn, mÞf1H½mmc ðnÞg,
n Z 1,
ð31aÞ Let t 2 ðta , tab Þ define the location of medium a and t 2 ðtab , tb Þ the location of medium b. The position tab corresponds to the interface between these media. To completely define the interface conditions for the Stokes vector at tab , we need to distinguish what happens just on the left of tab from what happens just on the right of tab . We also need to relate the polar variable m 2 ½1; 1 used in Eq. (1) to cos Wa and cos Wb , the cosines of the incident and refracted angles. For this purpose, we separate the range of m into two half-ranges, 7 m 2 ð0; 1, and simply use either cos Wa ¼ m or cos Wb ¼ m in Eqs. (14), (15) and (22) as appropriate to each case. Note that when we take, for example, cos Wa ¼ m in these equations Snell’s law implies that cos Wb ¼ f ðnab , mÞ, where f ðn, mÞ ¼ ½1n2 ð1m2 Þ1=2 :
Ia ðtab ,m, fÞ ¼ Rab ½nab , m,f ðnab , mÞIa ðtab , m, fÞ
Dðn, mÞ,
n r1,
Dðn, mÞH½mmc ðnÞ,
n Z1:
ð27Þ
mc ðnÞ ¼ ð11=n2 Þ1=2
ð32Þ
is the cosine of the critical angle and the non-zero elements of the 4 4 matrices Gðn, mÞ, Cðn, mÞ, and Dðn, mÞ are given by ( ) 1 mnf ðn, mÞ 2 nmf ðn, mÞ 2 þ G11 ðn, mÞ ¼ G22 ðn, mÞ ¼ , 2 nm þf ðn, mÞ m þ nf ðn, mÞ ð33aÞ G12 ðn, mÞ ¼ G21 ðn, mÞ ¼
1 2
(
mnf ðn, mÞ 2 nmf ðn, mÞ nm þ f ðn, mÞ m þ nf ðn, mÞ
mnf ðn, mÞ nmf ðn, mÞ , m þ nf ðn, mÞ nm þ f ðn, mÞ
ð33cÞ
G11 ðn, mÞ ¼ G22 ðn, mÞ ¼ 1, G33 ðn, mÞ ¼ G44 ðn, mÞ ¼
ð34aÞ
2ð1m2 Þ2 1, 1ð1 þ 1=n2 Þm2
G43 ðn, mÞ ¼ G34 ðn, mÞ ¼
ð34bÞ
2mð1m2 Þ½m2c ðnÞm2 1=2 , 1ð1 þ 1=n2 Þm2 (
1
½m þ nf ðn, mÞ2
þ
ð34cÞ )
1 ½nm þ f ðn, mÞ2
,
ð35aÞ
ð28Þ
for m 2 ð0; 1 and f 2 ½0; 2p. The interpretation of this equation is similar to that of Eq. (27): radiation traveling away from the interface tab in medium b is made up of two parts, one that is due to reflection of the radiation traveling toward the interface tab in medium b and another that is due to transmission from medium a to medium b.
2 ) , ð33bÞ
D11 ðn, mÞ ¼ D22 ðn, mÞ ¼ 2n3 mf ðn, mÞ
Ib ðtab , m, fÞ ¼ Rba ½nba , m,f ðnba , mÞIb ðtab ,m, fÞ
ð31bÞ
Here,
G33 ðn, mÞ ¼ G44 ðn, mÞ ¼
for m 2 ð0; 1 and f 2 ½0; 2p. This equation simply states that radiation traveling away from the interface tab in medium a is made up of two parts: one that is due to reflection of the radiation traveling toward the interface tab in medium a and another that is due to transmission from medium b to medium a. Just on the right of tab , we get
þTab ½nab ,f ðnba , mÞ, mIa ½tab ,f ðnba , mÞ, f,
Yðn, mÞ ¼
(
ð26Þ
And so, just on the left of tab , we get, after using the full notation of the reflection and transmission matrices given at the end of Section 3.1,
þTba ½nba ,f ðnab , mÞ, mIb ½tab ,f ðnab , mÞ, f,
and
( D12 ðn, mÞ ¼ D21 ðn, mÞ ¼ 2n3 mf ðn, mÞ
) 1 , ½m þ nf ðn, mÞ2 ½nm þ f ðn, mÞ2 1
ð35bÞ and D33 ðn, mÞ ¼ D44 ðn, mÞ ¼
4n3 mf ðn, mÞ : ½m þ nf ðn, mÞ½nm þ f ðn, mÞ
ð35cÞ
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
It should be noted that the above ð1; 1Þ elements are in perfect agreement with the results reported for the scalar case [28,53], where polarization effects are neglected. Also, it can be shown that lim Gðn, mÞ ¼ lim Cðn, mÞ ¼ diagf1; 1,1; 1g,
m-mc ðnÞ
m-mc ðnÞ
ð36Þ
for n Z1. In addition, since lim f ðn, mÞ ¼ m,
ð37aÞ
lim Xðn, mÞ ¼ 0,
ð37bÞ
n-1
n-1
and lim Yðn, mÞ ¼ diagf1; 1,1; 1g,
n-1
ð37cÞ
it is easy to see that Eqs. (30) reduce to the usual continuity conditions Ia ðtab , 7 m, fÞ ¼ Ib ðtab , 7 m, fÞ,
ð38Þ
for m 2 ð0; 1 and f 2 ½0; 2p, when na ¼nb.
4.2. A multilayer system We now consider a medium composed of K dissimilar layers. In each layer, the radiation field is described by Eq. (1), which we repeat here with a subscript k added, to indicate that the equation applies to a specific layer k. We thus write, for k ¼ 1; 2, . . . ,K, Z Z @ $ 2p 1 0 m Ik ðt, m, fÞ þIk ðt, m, fÞ ¼ k Pk ðm, m0 , ff Þ @t 4p 0 1 0
0
Ik ðt, m0 , f Þ dm0 df þ Sk ðt, m, fÞ,
ð39Þ
where t 2 ðak1 ,ak Þ, m 2 ½1; 1, and f 2 ½0; 2p. In our notation, a0 defines the location of the surface of the first layer, a1 ,a2 , . . . ,aK1 define the locations of the interfaces between the layers, and aK is the location of the surface of the last layer. An index of refraction nk is associated to layer k. With regard to the boundary (surface) condition for the first layer, we assume that a known distribution of radiation given by a Stokes vector W0 ðm, fÞ strikes the surface located at t ¼ a0 , coming from an external medium (t oa0 ) characterized by an index of refraction n0. Adapting the notation used in Section 4.1 to our present situation and replacing the unknown Stokes vector in the transmission term of Eq. (30b) by the known incident Stokes vector, we can write I1 ða0 , m, fÞ ¼ Xðn1;0 , mÞI1 ða0 ,m, fÞ þYðn1;0 , mÞW0 ½f ðn1;0 , mÞ, f, ð40Þ for m 2 ð0; 1 and f 2 ½0; 2p. In general, we define nk,k0 ¼ nk =nk0 . At each of the interfaces, we rewrite Eqs. (30a) and (30b) as
Ik þ 1 ½ak ,f ðnk,k þ 1 , mÞ, f
and Ik þ 1 ðak , m, fÞ ¼ Xðnk þ 1,k , mÞIk þ 1 ðak ,m, fÞ þ Yðnk þ 1,k , mÞIk ½ak ,f ðnk þ 1,k , mÞ, f,
ð41aÞ
ð41bÞ
for m 2 ð0; 1, f 2 ½0; 2p, and k ¼ 1; 2, . . . ,K1. Finally, with regard to the boundary (surface) condition for the last layer, we assume that a known distribution of radiation given by a Stokes vector WK ðm, fÞ strikes the surface located at t ¼ aK , coming from an external medium (t 4 aK ) characterized by an index of refraction nK þ 1 . Similarly to what was done for the first layer, we can adapt the notation used in Section 4.1 and replace the unknown Stokes vector in the transmission term of Eq. (30a) by the known incident Stokes vector, to obtain IK ðaK ,m, fÞ ¼ XðnK,K þ 1 , mÞIK ðaK , m, fÞ þYðnK,K þ 1 , mÞWK ½f ðnK,K þ 1 , mÞ, f,
ð42Þ
for m 2 ð0; 1 and f 2 ½0; 2p. To close this section, we note that Kovtanyuk and Prokhorov [54] have recently reported a mathematical analysis of the azimuthally independent case of the polarized radiative-transfer problem for a layered medium with Fresnel interface conditions. As discussed elsewhere [55], the azimuthally independent case can be decoupled into two 2-vector problems (one for the I and Q components and another for the U and V components of the Stokes vector). The work by Kovtanyuk and Prokhorov [54] treats only the IQ problem (in the alternative Il Ir formulation). We have checked that the 2 2 reflection and transmission matrices used by these authors are consistent with our results, provided the quantities n and ci ðnÞ in the formulas for the amplitude coefficients of reflection and transmission given on p. 2008 of their work [54] are replaced by 9n9 and 9ci ðnÞ9. 5. Conclusions In this work, we have presented what we believe to be a careful derivation of the relations between the reflected and transmitted Stokes vectors and the Stokes vector incident on an interface between two dissimilar media. In particular, we have obtained a general form of a transmission factor that is required to ensure conservation of energy and we have shown that most of the discrepancies encountered in published works are due to the use of improper forms of this factor. The new result obtained for the transmission factor helps to explain, in a rigorous way, the phenomenon of total reflection that occurs for angles of incidence larger than a critical angle, when the index of refraction of the medium of incidence is larger than that of the medium of transmission. Finally, we have also derived in this work explicit and compact forms of the Fresnel boundary and interface conditions appropriate to the study of polarized radiation transport in a multilayer medium.
Acknowledgments
Ik ðak ,m, fÞ ¼ Xðnk,k þ 1 , mÞIk ðak , m, fÞ þYðnk,k þ 1 , mÞ
315
This work was supported by CNPq of Brazil.
316
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
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Fresnel boundary and interface conditions for polarized radiative transfer in a multilayer medium R.D.M. Garcia Instituto de Estudos Avanc- ados, Trevo Cel. Av. Jose´ Alberto Albano do Amarante no. 1, Sa~ o Jose´ dos Campos, SP 12228-001, Brazil
a r t i c l e in f o
abstract
Article history: Received 12 July 2011 Received in revised form 17 October 2011 Accepted 24 November 2011 Available online 3 December 2011
In many applications of the theory of radiative transfer, it is important to consider the changes in the index of refraction that occur when the physical domain being studied consists of material regions with distinct optical properties. When polarization effects are taken into account, the radiation field is described by a vector of four components known as the Stokes vector. At an interface between two different material regions, the reflected and transmitted Stokes vectors are related to the incident Stokes vector by means of reflection and transmission matrices, which are derived from the Fresnel formulas for the amplitude coefficients of reflection and transmission. Having seen that many works on polarized radiative transfer that allow for changes in the index of refraction exhibit discrepancies in their expressions for the transmission matrix, we present in this work a careful derivation of the relations between the reflected and transmitted Stokes vectors and the Stokes vector incident on an interface. We obtain a general form of a transmission factor that is required to ensure conservation of energy and we show that most of the discrepancies encountered in existing works are associated with the use of improper forms of this factor. In addition, we derive explicit and compact expressions for the Fresnel boundary and interface conditions appropriate to the study of polarized radiative transfer in a multilayer medium. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Radiative transfer Polarization Fresnel conditions Index of refraction
1. Introduction Along with its continued use in the fields of astrophysics [1,2], atmospheric sciences [3,4], remote sensing [5], mechanical engineering [6,7], hydrological optics [8], and electromagnetic-wave propagation [9], radiative transfer has been applied to new fields of study, such as biomedical optics [10], optical tomography [11], radiation protection [12], analysis of masterpiece paintings [13], modeling of lamp envelope and optical fiber fabrication processes [14,15], and modeling of photovoltaic modules [16]. In many cases, to obtain a good description of the radiation field, it is important to consider the changes in the index of refraction that occur when the physical
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domain being studied consists of material regions with distinct optical properties. For the scalar case, where only the radiation intensity is of interest and polarization effects can be neglected, some authors have been able to treat a change in the index of refraction when an interface between two dissimilar planar media is crossed [17–19]. Other authors have considered systems with two or more planar interfaces [20–30]. While working on an extension of previous papers on the scalar case [28–30] to the case with polarization, we were able to find various works where polarized radiative transfer with changes in the index of refraction is considered and discussed in some detail. When polarization effects are taken into account, one needs to work with a vector of four components—the Stokes vector—instead of a single quantity, as in the scalar case, and the Fresnel
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formulas for the amplitude coefficients of reflection and transmission for the parallel and perpendicular components of the electric field. Surprisingly, we have found discrepancies in most of the works that we have examined [5,31–40], especially in the expressions for the transmission matrix. And so, starting with the Fresnel formulas, we present in this work a detailed derivation of the reflection and transmission matrices that relate the reflected and transmitted Stokes vectors to the Stokes vector incident on an interface. We also derive the Fresnel boundary and interface conditions appropriate to the study of polarized radiative transfer in a multilayer medium.
2. Fundamentals In this section, we summarize the theoretical background of radiative transfer with polarization. Details can be found in many sources, e.g. [1,2,41–44]. We also discuss briefly the Fresnel formulas for the reflection and transmission coefficients at an interface between two dissimilar media [45,46].
2.1. The radiative transfer equation (RTE) Following Siewert [47], we write the equation that describes polarized radiative transfer in a homogeneous plane-parallel medium as Z Z @ $ 2p 1 0 m Iðt, m, fÞ þIðt, m, fÞ ¼ Pðm, m0 , ff Þ @t 4p 0 1 0
0
Iðt, m0 , f Þ dm0 df þ Sðt, m, fÞ,
ð1Þ
where the optical variable t 2 ðtmin , tmax Þ is used to define the position in the medium and m 2 ½1; 1, with m ¼ cos y, and f 2 ½0; 2p are, respectively, the polar and azimuthal variables that specify the direction of propagation. Here, m is measured with respect to the positive t-axis. Furthermore, 0 1 I BQ C B C I¼B C ð2Þ @UA V is the Stokes vector that characterizes the radiation field, $ is the single-scattering albedo, and Pðm, m0 , ff0 Þ is the phase matrix [48] for scattering from an initial direction 0 ðm0 , f Þ to a final direction ðm, fÞ. The Stokes parameters I, Q, U, and V in Eq. (2) depend on t, m, and f, and can be expressed in terms of complex oscillatory functions El and Er. Making use of l and r to represent unit vectors along directions respectively parallel and perpendicular to a reference plane (called meridian plane [2]), defined by the direction of propagation and the t-axis, and choosing the orientation of l and r such that r l is in the direction of propagation, we can write the electric field at a given position as E ¼ El l þ Er r:
ð3Þ
307
The Stokes parameters are then defined in terms of the El and Er components of the electric field as [41,42] 1 0 1 0 1 0 Il þIr El Enl þ Er Enr I C B C BQ C B El Enl Er Enr C B Il Ir C B C B C B C ð4Þ B C¼B n C, n n C ) B B @ U A @ El Er þ Er El A @ 2RfEl Er g A V
iðEl Enr Er Enl Þ
2IfEr Enl g
where n denotes the operation of complex conjugation, i is the imaginary unit, and Rfzg and Ifzg are the real and imaginary parts of a complex variable z. Even though Eq. (4) is strictly true only for a monochromatic beam, it can still be applied to a quasi-monochromatic beam if the components on the right side are understood as time averages [2]. Finally, the internal source Sðt, m, fÞ in Eq. (1) can be either an emission source or the first-collision source resulting from a decomposition of the Stokes vector into uncollided and collided components that is frequently used for solving problems driven by monodirectional external sources. To close this subsection, we believe it important to mention that some authors use definitions of the Stokes vector that differ slightly from the one adopted in this work or, sometimes, definitions based on different sets of Stokes parameters. One frequently used alternative representation makes use of the intensities Il and Ir in the place of I and Q. Also, in some works, V is defined as the negative of our V. Sometimes, a scalar appears multiplying the right side of Eq. (4). When that scalar does not depend on the medium properties, no change is required in the expressions for the interface relations that are presented in Section 3 of our work. Should the scalar depend on the medium properties, the interface relation for transmission will have to be modified accordingly. 2.2. The Fresnel formulas In general, upon reaching an interface between two media with different properties, radiation is partially reflected and partially transmitted. Consider a beam of radiation traveling in medium a (characterized by an index of refraction na). When that beam reaches the interface with medium b (characterized by an index of refraction nb), part of it is reflected specularly (we assume a smooth interface) and part is refracted along a direction determined by Snell’s law, na sin Wa ¼ nb sin Wb ,
ð5Þ
where Wa , Wb are angles measured with respect to the normal at the interface. The angle Wa defines the initial direction of propagation in medium a and the angle Wb the direction of propagation in medium b, after refraction. The amplitude coefficients of reflection and transmission are given by the Fresnel formulas, which are usually written as [45,46] Rl ¼ ERl =EIl ¼
tanðWa Wb Þ , tanðWa þ Wb Þ
Rr ¼ ERr =EIr ¼
sinðWa Wb Þ , sinðWa þ Wb Þ
ð6aÞ
ð6bÞ
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T l ¼ ETl =EIl ¼
2 sin Wb cos Wa , sinðWa þ Wb ÞcosðWa Wb Þ
ð6cÞ
2 sin Wb cos Wa , sinðWa þ Wb Þ
ð6dÞ
and T r ¼ ETr =EIr ¼
where the superscripts I, R, and T affixed to El and Er mean ‘‘incident’’, ‘‘reflected’’, and ‘‘transmitted’’. At this point, a comment on the way Eqs. (6a) and (6b) have been written is important. As discussed many years ago by Friedmann and Sandhu [49], the theory is unable to determine univocally the signs that should be used on the right sides of these equations, out of the four possible choices ( þ þ , þ , þ , and ). Physically, a change of sign on the right side of Eq. (6a) [or Eq. (6b)] means a change of p in the phase of the parallel (or the perpendicular) component of the reflected electric field with respect to the phase of the parallel (or the perpendicular) component of the incident electric field. The choice of signs used in Eqs. (6a) and (6b), i.e. þ, is the choice used by most authors of books on this subject and is in agreement with the Nebraska ellipsometry conventions [50]. However, this choice leads to a physical inconsistency [50], as discussed next. The reflection and transmission coefficients defined by Eqs. (6) can also be written in the alternative (and sometimes more convenient) form Rl ¼
nb cos Wa na cos Wb , nb cos Wa þna cos Wb
ð7aÞ
Rr ¼
na cos Wa nb cos Wb , na cos Wa þ nb cos Wb
ð7bÞ
2na cos Wa , Tl ¼ nb cos Wa þ na cos Wb
ð7cÞ
and Tr ¼
2na cos Wa : na cos Wa þ nb cos Wb
3. Interface relations for the Stokes vector To obtain general formulas relating the reflected and transmitted Stokes vectors to the incident Stokes vector at an interface, we consider, as in Section 2.2, that the radiation beam is traveling initially in medium a and, upon reaching the interface between media a and b, is partially reflected back to medium a and partially transmitted into medium b. 3.1. The reflection and transmission matrices Working with Eq. (4) and the first equalities in Eqs. (6) we find that the reflected Stokes vector IR is related to the incident Stokes vector II by IR ¼ Rab II , where 01 n n 2ðRl Rl þ Rr Rr Þ B1 B 2ðRl Rnl Rr Rnr Þ Rab ¼ B B 0 @ 0
ð9Þ
n n 1 2ðRl Rl Rr Rr Þ n n 1 2ðRl Rl þ Rr Rr Þ
0
0
RfRl Rnr g
0
IfRl Rnr g
0
0
and
C C C IfRl Rnr g C A RfRl Rnr g
ð10Þ is the reflection matrix. The subscript ab attached to R indicates radiation traveling in medium a that is reflected by medium b. Similarly, we find that the transmitted Stokes vector IT is related to the incident Stokes vector II by IT ¼ Tab II ,
ð8bÞ
Clearly, this implies a phase difference of p between the parallel and perpendicular components of the electric field upon reflection. Such a phase difference is unjustified on physical grounds, since the parallel and perpendicular components become indistinguishable when the incidence is normal to the interface. The fact that a change of p in the phase of the reflected electric field with respect to the phase of the incident electric field is expected when nb 4na [51] suggests that using Eq. (6a) [and consequently Eq. (7a)] with a change in sign would be the most consistent choice from a physical point of view. In spite of its strong appeal, the ‘‘physically consistent’’ sign choice was rejected by the ellipsometry community
1
0
ð7dÞ
For the case of normal incidence, Wa ¼ Wb ¼ 0, and so Eqs. (7a) and (7b) reduce to n na Rl ¼ b ð8aÞ nb þ na na nb : Rr ¼ na þ nb
because it involves a change of coordinate system for describing the incident and reflected fields [50]. In fact, using an extension for the case with polarization of a previous work on the analytical discrete-ordinates (ADO) method for solving scalar radiative transfer problems subject to Fresnel boundary and interface conditions [30], we have found that the ‘‘physically consistent’’ sign choice gives rise to a sign flip on the computed U and V Stokes parameters at either side of a Fresnel interface, when the cosine of the polar angle m-0 from above and from below. On the other hand, the ‘‘conventional’’ sign choice used in this work yields continuous results for all of the Stokes parameters at both sides of a Fresnel interface. Details will be given in a future work.
ð11Þ
where 01 n n 2ðT l T l þ T r T r Þ B1 B 2ðT l T nl T r T nr Þ Tab ¼ f T B B 0 @ 0
n n 1 2ðT l T l T r T r Þ n n 1 ðT T þ T T r rÞ 2 l l
0 0
0 0 RfT l T nr g IfT l T nr g
0
1
C C 0 C n C IfT l T r g A n RfT l T r g
ð12Þ is the transmission matrix. The subscript ab attached to T indicates radiation traveling in medium a that is transmitted into medium b. In Eq. (12), fT is a factor that takes into account the change of properties when going from medium a to medium b, and can be determined from the requirement of conservation of energy, as will be shown shortly.
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The reflection and transmission matrices defined by Eqs. (10) and (12) can be written in a more explicit way when both indices of refraction (na and nb) are real. For such purpose, it is necessary to distinguish between two cases. The first case comprises two distinct situations: (i) na =nb r1 or (ii) na =nb 4 1 and Wa o Wc , where Wc ¼ arcsinðnb =na Þ is the critical angle [45,46]. In this case, all of the reflection and transmission coefficients are real, and so, with the help of Eqs. (7a) and (7b) and the definition nab ¼ na =nb ,
ð13Þ
we can readily write the basic quantities that appear in the reflection matrix as 2 cos Wa nab cos Wb , ð14aÞ Rl Rnl ¼ cos Wa þnab cos Wb Rr Rnr ¼
nab cos Wa cos Wb nab cos Wa þ cos Wb
RfRl Rnr g ¼
2
,
cos Wa nab cos Wb cos Wa þ nab cos Wb
ð14bÞ
nab cos Wa cos Wb , nab cos Wa þ cos Wb ð14cÞ
and IfRl Rnr g ¼ 0:
ð14dÞ
For the transmission matrix, we get, with the help of Eqs. (7c) and (7d), 2 2nab cos Wa T l T nl ¼ , ð15aÞ cos Wa þnab cos Wb T r T nr ¼
2nab cos Wa nab cos Wa þ cos Wb
2
,
ð15bÞ
4n2ab cos2 Wa , ðcos Wa þnab cos Wb Þðnab cos Wa þ cos Wb Þ
RfT l T nr g ¼
ð15cÞ
and IfT l T nr g ¼ 0:
ð15dÞ
To determine the transmission factor fT, we consider that the direction of propagation of the beam in medium a is in a cone defined by a differential solid angle dXa about Xa . When the beam strikes an elementary area dA on the interface with medium b, part of it is reflected specularly and part is refracted into a cone defined by a differential solid angle dXb about Xb . Since Il and Ir, the parallel and perpendicular components of the intensity, are independent of each other, conservation of energy implies that IIx dXa ðdA
cos
Wa Þ ¼ IRx dXa ðdA
cos
Wa Þ þf T ITx dXb ðdA
cos Wb Þ, ð16Þ
for both x¼l and x ¼r. Here, as before, the superscripts I, R, and T indicate ‘‘incident’’, ‘‘reflected’’, and ‘‘transmitted’’. In addition, dA cos Wa and dA cos Wb are the projections of the elementary area dA into planes perpendicular to the directions Xa and Xb . Using dXa ¼ sin Wa dWa df and dXb ¼ sin Wb dWb df and noting that IRx ¼ Rx Rnx IIx
ð17aÞ
309
and ITx ¼ T x T nx IIx ,
ð17bÞ
for x ¼l and x¼r, we find from Eq. (16) cos Wa sin Wa dWa 1Rx Rnx , fT ¼ cos Wb sin Wb dWb T x T nx
ð18Þ
for x¼l and x¼r. Now, multiplying Snell’s law by its differential, we can show that n2a cos Wa sin Wa dWa ¼ n2b cos Wb sinWb dWb :
ð19Þ
Using this result in Eq. (18), we find that fT can be expressed as 1Rx Rnx , ð20Þ f T ¼ n2ba n T xT x for x¼l and x ¼r. Using Eqs. (14a), (14b), (15a) and (15b), we find that both forms of Eq. (20) [i.e. Eq. (20) for x ¼l and Eq. (20) for x¼ r] reduce to cos Wb f T ¼ n3ba , ð21Þ cos Wa a result that was also obtained in a recent work by Sommersten et al. [38], using physical arguments. Simply stated, fT can be understood as the product of three factors: one that adjusts for the change in the power of the beam when an interface is crossed, another that is given by the ratio between the areas obtained by projecting a unit area lying on the interface into planes perpendicular to the directions of incidence and refraction, and one that takes into account the angular redistribution of radiation caused by refraction. The second case to consider when both indices of refraction na and nb are real is defined by na =nb 4 1 and Wa Z Wc and is known in the literature as total reflection [45,46], since there is no transmission of radiation from medium a to medium b in this case. It can be concluded from Snell’s law that the angle of refraction Wb is purely imaginary for this case, and so the algebra has to be carried out in complex mode. We get, for the basic quantities that appear in the reflection matrix, Rl Rnl ¼ 1,
ð22aÞ
Rr Rnr ¼ 1,
ð22bÞ
RfRl Rnr g ¼
2 sin4 Wa 1, 1ð1þ n2ba Þcos2 Wa
ð22cÞ
and 2
IfRl Rnr g ¼
2
2 cos Wa sin Wa ðsin Wa n2ba Þ1=2 : 1ð1 þn2ba Þcos2 Wa
ð22dÞ
Also, both forms of Eq. (20) yield fT ¼0 for this case, and so Eq. (12) reduces to Tab ¼ 0,
ð23Þ
which confirms that there is no transmission of radiation from medium a to medium b when na =nb 4 1 and Wa Z Wc . Finally, we note that to be more explicit in our notation we should have written the reflection and transmission matrices defined by Eqs. (10) and (12) as
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Rab ðnab ,cos Wa ,cos Wb Þ and Tab ðnab ,cos Wa ,cos Wb Þ, and keep in mind that nab, Wa , and Wb are related by Snell’s law. 3.2. Comparison with other works In this subsection, we compare our results for the elements of the reflection and transmission matrices that are reported in Section 3.1 with results of other authors. In general, except when otherwise noted, we have observed agreement with regard to the elements of the reflection matrix. However, in the case of the transmission matrix we have detected many discrepancies, as discussed next. We begin with the paper by Tsang and Kong [31] and the book by Tsang et al. [5]. In spite of some slight changes in notation, the final results reported in these references are the same, and so we focus on a comparison of our results with the results of the latter [5], which is more explicit than the former [31]. The transmission matrix of these authors is given by Eq. (13), Section 6, Chapter 3 of Tsang et al. [5]. Working out this equation using the explicit definitions of the reflection and transmission coefficients given by Eqs. (9a) and (9b), Section 6, Chapter 2, of that same book [5], we found that it is equivalent to our result. However, in their analysis of the interface relations, these authors [5] have defined the Stokes vector in a way that differs from ours. They have used, in our notation, 0 1 Il B 1 B Ir C C I¼ B ð24Þ C, Z@ U A V where Z is the wave impedance of the medium, which is given by Z ¼ ðmm =EÞ1=2 for a dielectric. Here, mm is the magnetic permeability and E the electric permittivity of the medium. After taking the presence of the factor 1=Z in Eq. (24) into account, we concluded that, in order to agree with our result, the transmission matrix of Tsang et al. [5] should be multiplied by ðE2 =E1 Þ1=2 , where E1 is used by the authors [5] to denote the electric permittivity of the medium that contains the incident beam and E2 the electric permittivity of the medium that contains the transmitted beam. Note that, for dielectric media, ðE2 =E1 Þ1=2 n2 =n1 . Continuing with our comparison, we note that there is no mention of the transmission factor fT that is needed to ensure conservation of energy in the work of Kattawar and Adams [32]. This is also the case of a more recent work by Zhai et al. [33]. With regard to the work by Lam and Ishimaru [34], the transmission matrix that is expressed by Eq. (16) of their work [34] should be multiplied by ðnq =np Þ2 to agree with our result. Note that, in their notation, np denotes the index of refraction of the medium that contains the incident beam and nq the index of refraction of the medium that contains the transmitted beam. It should also be pointed out in regard to that work [34] that the term on the right side of the second equality in their Eq. (4) is missing a minus sign, and so the Stokes parameter V
of these authors, although defined in the same way as we do in this work, ended up being the negative of what we have. Concerning the work of Chami et al. [35], which is specific for the case of an atmosphere–ocean interface, we first note that the V component of the Stokes vector has been neglected and so these authors have worked with a Stokes vector of only three components: I, Q, and U. With regard to the reflection matrix, there are some corrections to be made in their work [35]: the squares in their Eqs. (8) and (9) should be removed and a minus sign included on the right side of their Eq. (8). The transmission matrix, which is given by Eq. (11) of their work [35], will agree with our result only after multiplication by ðnw =na Þ2 , where nw denotes the index of refraction of water and na 1 the index of refraction of air. The work of He et al. [36] is also specific for an atmosphere–ocean interface. The reflection and transmission matrices given by their Eqs. (36) and (37) agree with ours for incidence from the atmosphere side. However, for incidence from the ocean side, nw (the index of refraction of water) needs to be replaced by 1=nw in their equations. In the work of Bordier et al. [37], a Stokes-vector representation based on the Stokes parameters Il, Ir, U, and V has been used. A simple correction is required in the work of these authors: the (1,1) and (2,2) elements of the reflection and transmission matrices [Eqs. (32) and (33) of their work] should be squared. Moreover, to agree with our result, their transmission matrix needs to be multiplied by ðnj =ni Þ2 , where transmission from medium i to medium j is assumed by these authors [37]. Sommersten et al. [38] have also used the Stokes parameters Il, Ir, U, and V to define their Stokes vector. These authors have selected the sign of Eq. (7a) in the ‘‘physically consistent’’ way mentioned at the end of Section 2.2. This makes the signs of the (3,3), (3,4), (4,3), and (4,4) elements of the reflection matrix that is given by Eq. (17) of their work [38] to be the opposite of what we have. Other than that difference, which is caused by a choice of sign that departs from the recommended choice [50], the formulation of these authors for the reflection and transmission matrices is in perfect agreement with our formulation. We believe, however, that a mistake was made by these authors in the derivation of the reflection coefficients for the case of total reflection. We have found that the correct forms of these coefficients can be obtained by interchanging and multiplying by 1 the right sides of their Eqs. (19) and (20). Next, we comment on the work of Ota et al. [39], which is also specific for the case of an atmosphere–ocean interface. We concluded that, in order to agree with our result, the transmission matrix given by their Eq. (A.4) ~ 2 sin2 bÞ1=2 =cos b, should be multiplied by the factor ðm ~ is the index of refraction of water relative to air where m and b is the angle of incidence from the atmosphere. In addition, we have found that the (3,3), (3,4), (4,3), and (4,4) elements of the matrix for total reflection given by their Eq. (A.8) are incorrect. To see this, it is sufficient to note that the right sides of their Eqs. (A.9), when squared and added, do not yield unity, as required. In the notation of these authors [39], the correct quantities that define
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311
the (3,3), (3,4), (4,3), and (4,4) elements of their matrix for total reflection are given by cos d ¼
~ 2 sin4 b 2m ~ sin2 bcos2 b m 2
1
ð25aÞ
and sin d ¼
~ cos b sin2 bðm ~ 2 sin2 b1Þ1=2 2m : 2 2 ~ sin bcos2 b m
ð25bÞ
Completing the analysis of the series of works that we have examined, we have found that the transmission matrix used by Zhai et al. [40] needs to be multiplied by an ðnt =ni Þ2 factor to agree with our result. In the notation of these authors, ni denotes the refractive index of the medium of incidence and nt that of the medium to where radiation is transmitted. Finally, we illustrate in Figs. 1–6, for the case of light crossing an air–water interface, the effects of the two most frequent discrepancies in the works that we have examined. In these figures, we plot the basic elements of the transmission matrix Tab given by Eq. (12) against the angle of incidence Wa . The solid lines represent the correct elements obtained from Eqs. (12) and (15), the dashed
Fig. 3. The (3,3) element of the transmission matrix for light going from air to water.
Fig. 4. The (1,1) element of the transmission matrix for light going from water to air.
Fig. 1. The (1,1) element of the transmission matrix for light going from air to water.
Fig. 5. The (1,2) element of the transmission matrix for light going from water to air.
Fig. 2. The (1,2) element of the transmission matrix for light going from air to water.
lines represent the elements obtained when fT is replaced by f T =n2ba in Eq. (12), and the dot-dashed lines represent the elements obtained when the transmission factor fT is neglected in Eq. (12). Figs. 1–3 are for incidence from the air side (n ¼1.0), while Figs. 4–6 are for incidence from the
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To provide some numerical results that could be useful for checking future computational implementations, we report in Tables 1 and 2, for the considered air–water interface, the basic elements of the reflection and transmission matrices for selected values of the incidence angle (Wa ).
water side (n¼ 1.338). The curves plotted in Figs. 1–6 indicate that the transmission of radiation to the ocean in radiative transfer simulations of the atmosphere–ocean system is underestimated when either fT or n2ba is omitted in the expression for the Fresnel transmission matrix.
3.3. A preliminary investigation of the effect of the discrepancies in the transmission matrix on multiple scattering calculations Since the discrepancies in the transmission matrix that are discussed in detail in Section 3.2 affect even the scalar case for which polarization effects are not taken into account, we thought it interesting to carry out a computational simulation of a very simple scalar case defined by a single layer, in order to estimate the influence of the two most frequent discrepancies (fT missing or replaced by f T =n2ba ) on important integral quantities such as reflectance and transmittance. We thus used the scalar ADO method reported in a recent work [29] to compute the reflectance and transmittance of a water layer with index of refraction n¼ 1.338 that is illuminated on the top by a uniform beam of radiation traveling in an external
Fig. 6. The (3,3) element of the transmission matrix for light going from water to air.
Table 1 Basic elements of the reflection matrix for an air–water interface. Direction
Wa (deg.)
(1,1) element
(1,2) element
(3,3) element
Air-Water
0 15 30 45 60 75 90
2.0899909( 2) 2.0955742( 2) 2.1979938( 2) 2.8529127( 2) 6.0630179( 2) 2.1394058( 1) 1.0
0.0 2.2079861( 3) 9.7003547( 3) 2.5599246( 2) 5.6383185( 2) 1.0309504( 1) 0.0
2.0899909( 2) 2.0839096( 2) 1.9723610( 2) 1.2593240( 2) 2.2292487( 2) 1.8746195( 1) 1.0
Water-Air
0 15 30 45 60 75 90
2.0899909( 2) 2.1095570( 2) 2.6242311( 2) 1.4886311( 1) 1.0 1.0 1.0
0.0 4.1327359( 3) 2.1475043( 2) 9.1222168( 2) 0.0 0.0 0.0
2.0899909( 2) 2.0686797( 2) 1.5082486( 2) 1.1763818( 1) 8.4319194( 1) 9.4398799( 1) 1.0
(3,4) element 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.3761264( 1) 3.2997980( 1) 0.0
Table 2 Basic elements of the transmission matrix for an air–water interface. Direction
Wa (deg.)
(1,1) element
(1,2) element
(3,3) element
Air-Water
0 15 30 45 60 75 90
1.7528281 1.7527281 1.7508945 1.7391699 1.6817012 1.4072382 0.0
0.0 3.9528339( 3) 1.7366002( 2) 4.5828897( 2) 1.0093966( 1) 1.8456528( 1) 0.0
1.7528281 1.7527237 1.7508084 1.7385660 1.6786691 1.3950824 0.0
Water-Air
0 15 30 45 60 75 90
5.4690874( 1) 5.4679945( 1) 5.4392457( 1) 4.7543066( 1) 0.0 0.0 0.0
0.0 2.3084764( 3) 1.1995596( 2) 5.0955159( 2) 0.0 0.0 0.0
5.4690874( 1) 5.4679457( 1) 5.4379228( 1) 4.7269217( 1) 0.0 0.0 0.0
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
medium with index of refraction n ¼1.0. At the bottom of the layer, a black surface is simulated by assuming the presence of an external medium with an index of refraction identical to that of the layer, i.e. n ¼1.338. The external media are non-participating and scattering in the water layer is described by the Henyey–Greenstein phase function [52], which depends only on one parameter, the so-called asymmetry factor g.
313
We report in Tables 3–5 the reflectance and the transmittance of the water layer for three choices of g: a case dominated by forward scattering (g¼0.9), a case dominated by backward scattering (g ¼ 0:7), and the isotropic scattering case (g ¼0.0). For each choice of g, we have reported numerical results for three choices of the cosine of the polar angle of the incident beam (m0 ¼ 0:2, 0.5, and 1.0) and three choices of the optical
Table 3 The reflectance and the transmittance for the case $ ¼ 1:0 and g ¼ 0.9.
m0
D
Reflectance
Transmittance
Correct fT
fT missing
f T ( f T =n2ba
Correct fT
fT missing
f T ( f T =n2ba
0.2
0.1 1.0 10.0
3.005348( 1) 3.122315( 1) 4.927353( 1)
2.998677( 1) 3.013018( 1) 3.234328( 1)
3.001992( 1) 3.067327( 1) 4.075591( 1)
6.994652( 1) 6.877685( 1) 5.072647( 1)
8.575925( 2) 8.432516( 2) 6.219415( 2)
3.907094( 1) 3.841759( 1) 2.833495( 1)
0.5
0.1 1.0 10.0
6.146126( 2) 7.402524( 2) 3.019589( 1)
6.085776( 2) 6.429823( 2) 1.267146( 1)
6.109441( 2) 6.811243( 2) 1.954323( 1)
9.385387( 1) 9.259748( 1) 6.980411( 1)
2.570055( 1) 2.535650( 1) 1.911486( 1)
5.242519( 1) 5.172338( 1) 3.899140( 1)
1.0
0.1 1.0 10.0
2.142050( 2) 2.884992( 2) 2.180722( 1)
2.111724( 2) 2.421885( 2) 1.032146( 1)
2.119070( 2) 2.534065( 2) 1.310370( 1)
9.785795( 1) 9.711501( 1) 7.819278( 1)
4.085336( 1) 4.054320( 1) 3.264362( 1)
5.466179( 1) 5.424680( 1) 4.367716( 1)
Table 4 The reflectance and the transmittance for the case $ ¼ 1:0 and g ¼ 0:7.
m0
D
Reflectance
Transmittance
Correct fT
fT missing
f T ( f T =n2ba
Correct fT
fT missing
f T ( f T =n2ba
0.2
0.1 1.0 10.0
3.356045( 1) 5.601499( 1) 9.072452( 1)
3.041675( 1) 3.316983( 1) 3.742545( 1)
3.197885( 1) 4.452158( 1) 6.390973( 1)
6.643955( 1) 4.398501( 1) 9.275483( 2)
8.145947( 2) 5.392866( 2) 1.137238( 2)
3.711201( 1) 2.456928( 1) 5.181128( 2)
0.5
0.1 1.0 10.0
1.142017( 1) 4.188899( 1) 8.764800( 1)
7.529997( 2) 1.587345( 1) 2.840390( 1)
9.055431( 2) 2.607480( 1) 5.163501( 1)
8.857983( 1) 5.811101( 1) 1.235200( 1)
2.425633( 1) 1.591287( 1) 3.382419( 2)
4.947920( 1) 3.245983( 1) 6.899618( 2)
1.0
0.1 1.0 10.0
8.408298( 2) 4.055740( 1) 8.694572( 1)
4.727734( 2) 1.814922( 1) 3.751523( 1)
5.619291( 2) 2.357723( 1) 4.948896( 1)
9.159170( 1) 5.944260( 1) 1.305428( 1)
3.823735( 1) 2.481587( 1) 5.449852( 2)
5.116157( 1) 3.320363( 1) 7.291901( 2)
Table 5 The reflectance and the transmittance for the case $ ¼ 1:0 and g ¼ 0.0.
m0
D
Reflectance
Transmittance
Correct fT
fT missing
f T ( f T =n2ba
Correct fT
fT missing
f T ( f T =n2ba
0.2
0.1 1.0 10.0
3.182911( 1) 4.891585( 1) 8.635459( 1)
3.020447( 1) 3.229942( 1) 3.688967( 1)
3.101176( 1) 4.055612( 1) 6.146877( 1)
6.817089( 1) 5.108415( 1) 1.364541( 1)
8.358221( 2) 6.263270( 2) 1.673021( 2)
3.807910( 1) 2.853474( 1) 7.622092( 2)
0.5
0.1 1.0 10.0
8.298818( 2) 2.997128( 1) 8.112362( 1)
6.675260( 2) 1.260995( 1) 2.661730( 1)
7.311898( 2) 1.941777( 1) 4.799060( 1)
9.170118( 1) 7.002872( 1) 1.887638( 1)
2.511106( 1) 1.917637( 1) 5.169027( 2)
5.122273( 1) 3.911686( 1) 1.054402( 1)
1.0
0.1 1.0 10.0
3.893032( 2) 2.325834( 1) 7.855518( 1)
2.842718( 2) 1.092727( 1) 3.401238( 1)
3.097139( 2) 1.391427( 1) 4.480215( 1)
9.610697( 1) 7.674166( 1) 2.144482( 1)
4.012237( 1) 3.203781( 1) 8.952699( 2)
5.368373( 1) 4.286659( 1) 1.197871( 1)
314
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
depth of the layer (D ¼ 0:1, 1.0, and 10.0). The singlescattering albedo $ was taken to be equal to 1.0 in the water layer. Other values of $ were also studied but we give here results only for $ ¼ 1:0 to explicitly show that the reflectance and the transmittance add to one in the conservative case only when the correct transmission factor fT is used. It can be observed that the values obtained for the reflectance and transmittance when fT is missing or is replaced by f T =n2ba always underestimate the correct values. For a missing fT the deviations can be as large as 67% in the reflectance and 87% in the transmittance. For f T =n2ba replacing fT the maximum deviations are 43% in the reflectance and 44% in the transmittance.
As done for the scalar case [26,28–30], we can now make use of the Heaviside function ( 1, x Z 0, HðxÞ ¼ ð29Þ 0, x o 0, to write Eqs. (27) and (28) in a more explicit and compact way. We get, for m 2 ð0; 1 and f 2 ½0; 2p, Ia ðtab ,m, fÞ ¼ Xðnab , mÞIa ðtab , m, fÞ þ Yðnab , mÞIb ½tab ,f ðnab , mÞ, f
ð30aÞ and Ib ðtab , m, fÞ ¼ Xðnba , mÞIb ðtab ,m, fÞ þYðnba , mÞIa ½tab ,f ðnba , mÞ, f,
ð30bÞ where (
4. Boundary and interface conditions for the RTE Xðn, mÞ ¼
4.1. A single interface
Gðn, mÞ,
n r 1,
Gðn, mÞH½mmc ðnÞ þ Cðn, mÞf1H½mmc ðnÞg,
n Z 1,
ð31aÞ Let t 2 ðta , tab Þ define the location of medium a and t 2 ðtab , tb Þ the location of medium b. The position tab corresponds to the interface between these media. To completely define the interface conditions for the Stokes vector at tab , we need to distinguish what happens just on the left of tab from what happens just on the right of tab . We also need to relate the polar variable m 2 ½1; 1 used in Eq. (1) to cos Wa and cos Wb , the cosines of the incident and refracted angles. For this purpose, we separate the range of m into two half-ranges, 7 m 2 ð0; 1, and simply use either cos Wa ¼ m or cos Wb ¼ m in Eqs. (14), (15) and (22) as appropriate to each case. Note that when we take, for example, cos Wa ¼ m in these equations Snell’s law implies that cos Wb ¼ f ðnab , mÞ, where f ðn, mÞ ¼ ½1n2 ð1m2 Þ1=2 :
Ia ðtab ,m, fÞ ¼ Rab ½nab , m,f ðnab , mÞIa ðtab , m, fÞ
Dðn, mÞ,
n r1,
Dðn, mÞH½mmc ðnÞ,
n Z1:
ð27Þ
mc ðnÞ ¼ ð11=n2 Þ1=2
ð32Þ
is the cosine of the critical angle and the non-zero elements of the 4 4 matrices Gðn, mÞ, Cðn, mÞ, and Dðn, mÞ are given by ( ) 1 mnf ðn, mÞ 2 nmf ðn, mÞ 2 þ G11 ðn, mÞ ¼ G22 ðn, mÞ ¼ , 2 nm þf ðn, mÞ m þ nf ðn, mÞ ð33aÞ G12 ðn, mÞ ¼ G21 ðn, mÞ ¼
1 2
(
mnf ðn, mÞ 2 nmf ðn, mÞ nm þ f ðn, mÞ m þ nf ðn, mÞ
mnf ðn, mÞ nmf ðn, mÞ , m þ nf ðn, mÞ nm þ f ðn, mÞ
ð33cÞ
G11 ðn, mÞ ¼ G22 ðn, mÞ ¼ 1, G33 ðn, mÞ ¼ G44 ðn, mÞ ¼
ð34aÞ
2ð1m2 Þ2 1, 1ð1 þ 1=n2 Þm2
G43 ðn, mÞ ¼ G34 ðn, mÞ ¼
ð34bÞ
2mð1m2 Þ½m2c ðnÞm2 1=2 , 1ð1 þ 1=n2 Þm2 (
1
½m þ nf ðn, mÞ2
þ
ð34cÞ )
1 ½nm þ f ðn, mÞ2
,
ð35aÞ
ð28Þ
for m 2 ð0; 1 and f 2 ½0; 2p. The interpretation of this equation is similar to that of Eq. (27): radiation traveling away from the interface tab in medium b is made up of two parts, one that is due to reflection of the radiation traveling toward the interface tab in medium b and another that is due to transmission from medium a to medium b.
2 ) , ð33bÞ
D11 ðn, mÞ ¼ D22 ðn, mÞ ¼ 2n3 mf ðn, mÞ
Ib ðtab , m, fÞ ¼ Rba ½nba , m,f ðnba , mÞIb ðtab ,m, fÞ
ð31bÞ
Here,
G33 ðn, mÞ ¼ G44 ðn, mÞ ¼
for m 2 ð0; 1 and f 2 ½0; 2p. This equation simply states that radiation traveling away from the interface tab in medium a is made up of two parts: one that is due to reflection of the radiation traveling toward the interface tab in medium a and another that is due to transmission from medium b to medium a. Just on the right of tab , we get
þTab ½nab ,f ðnba , mÞ, mIa ½tab ,f ðnba , mÞ, f,
Yðn, mÞ ¼
(
ð26Þ
And so, just on the left of tab , we get, after using the full notation of the reflection and transmission matrices given at the end of Section 3.1,
þTba ½nba ,f ðnab , mÞ, mIb ½tab ,f ðnab , mÞ, f,
and
( D12 ðn, mÞ ¼ D21 ðn, mÞ ¼ 2n3 mf ðn, mÞ
) 1 , ½m þ nf ðn, mÞ2 ½nm þ f ðn, mÞ2 1
ð35bÞ and D33 ðn, mÞ ¼ D44 ðn, mÞ ¼
4n3 mf ðn, mÞ : ½m þ nf ðn, mÞ½nm þ f ðn, mÞ
ð35cÞ
R.D.M. Garcia / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 306–317
It should be noted that the above ð1; 1Þ elements are in perfect agreement with the results reported for the scalar case [28,53], where polarization effects are neglected. Also, it can be shown that lim Gðn, mÞ ¼ lim Cðn, mÞ ¼ diagf1; 1,1; 1g,
m-mc ðnÞ
m-mc ðnÞ
ð36Þ
for n Z1. In addition, since lim f ðn, mÞ ¼ m,
ð37aÞ
lim Xðn, mÞ ¼ 0,
ð37bÞ
n-1
n-1
and lim Yðn, mÞ ¼ diagf1; 1,1; 1g,
n-1
ð37cÞ
it is easy to see that Eqs. (30) reduce to the usual continuity conditions Ia ðtab , 7 m, fÞ ¼ Ib ðtab , 7 m, fÞ,
ð38Þ
for m 2 ð0; 1 and f 2 ½0; 2p, when na ¼nb.
4.2. A multilayer system We now consider a medium composed of K dissimilar layers. In each layer, the radiation field is described by Eq. (1), which we repeat here with a subscript k added, to indicate that the equation applies to a specific layer k. We thus write, for k ¼ 1; 2, . . . ,K, Z Z @ $ 2p 1 0 m Ik ðt, m, fÞ þIk ðt, m, fÞ ¼ k Pk ðm, m0 , ff Þ @t 4p 0 1 0
0
Ik ðt, m0 , f Þ dm0 df þ Sk ðt, m, fÞ,
ð39Þ
where t 2 ðak1 ,ak Þ, m 2 ½1; 1, and f 2 ½0; 2p. In our notation, a0 defines the location of the surface of the first layer, a1 ,a2 , . . . ,aK1 define the locations of the interfaces between the layers, and aK is the location of the surface of the last layer. An index of refraction nk is associated to layer k. With regard to the boundary (surface) condition for the first layer, we assume that a known distribution of radiation given by a Stokes vector W0 ðm, fÞ strikes the surface located at t ¼ a0 , coming from an external medium (t oa0 ) characterized by an index of refraction n0. Adapting the notation used in Section 4.1 to our present situation and replacing the unknown Stokes vector in the transmission term of Eq. (30b) by the known incident Stokes vector, we can write I1 ða0 , m, fÞ ¼ Xðn1;0 , mÞI1 ða0 ,m, fÞ þYðn1;0 , mÞW0 ½f ðn1;0 , mÞ, f, ð40Þ for m 2 ð0; 1 and f 2 ½0; 2p. In general, we define nk,k0 ¼ nk =nk0 . At each of the interfaces, we rewrite Eqs. (30a) and (30b) as
Ik þ 1 ½ak ,f ðnk,k þ 1 , mÞ, f
and Ik þ 1 ðak , m, fÞ ¼ Xðnk þ 1,k , mÞIk þ 1 ðak ,m, fÞ þ Yðnk þ 1,k , mÞIk ½ak ,f ðnk þ 1,k , mÞ, f,
ð41aÞ
ð41bÞ
for m 2 ð0; 1, f 2 ½0; 2p, and k ¼ 1; 2, . . . ,K1. Finally, with regard to the boundary (surface) condition for the last layer, we assume that a known distribution of radiation given by a Stokes vector WK ðm, fÞ strikes the surface located at t ¼ aK , coming from an external medium (t 4 aK ) characterized by an index of refraction nK þ 1 . Similarly to what was done for the first layer, we can adapt the notation used in Section 4.1 and replace the unknown Stokes vector in the transmission term of Eq. (30a) by the known incident Stokes vector, to obtain IK ðaK ,m, fÞ ¼ XðnK,K þ 1 , mÞIK ðaK , m, fÞ þYðnK,K þ 1 , mÞWK ½f ðnK,K þ 1 , mÞ, f,
ð42Þ
for m 2 ð0; 1 and f 2 ½0; 2p. To close this section, we note that Kovtanyuk and Prokhorov [54] have recently reported a mathematical analysis of the azimuthally independent case of the polarized radiative-transfer problem for a layered medium with Fresnel interface conditions. As discussed elsewhere [55], the azimuthally independent case can be decoupled into two 2-vector problems (one for the I and Q components and another for the U and V components of the Stokes vector). The work by Kovtanyuk and Prokhorov [54] treats only the IQ problem (in the alternative Il Ir formulation). We have checked that the 2 2 reflection and transmission matrices used by these authors are consistent with our results, provided the quantities n and ci ðnÞ in the formulas for the amplitude coefficients of reflection and transmission given on p. 2008 of their work [54] are replaced by 9n9 and 9ci ðnÞ9. 5. Conclusions In this work, we have presented what we believe to be a careful derivation of the relations between the reflected and transmitted Stokes vectors and the Stokes vector incident on an interface between two dissimilar media. In particular, we have obtained a general form of a transmission factor that is required to ensure conservation of energy and we have shown that most of the discrepancies encountered in published works are due to the use of improper forms of this factor. The new result obtained for the transmission factor helps to explain, in a rigorous way, the phenomenon of total reflection that occurs for angles of incidence larger than a critical angle, when the index of refraction of the medium of incidence is larger than that of the medium of transmission. Finally, we have also derived in this work explicit and compact forms of the Fresnel boundary and interface conditions appropriate to the study of polarized radiation transport in a multilayer medium.
Acknowledgments
Ik ðak ,m, fÞ ¼ Xðnk,k þ 1 , mÞIk ðak , m, fÞ þYðnk,k þ 1 , mÞ
315
This work was supported by CNPq of Brazil.
316
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