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OPTICS LETTERS / Vol. 39, No. 22 / November 15, 2014
Forward-peaked scattering of polarized light Julia P. Clark1,* and Arnold D. Kim1,2 1
Applied Mathematics Unit, School of Natural Sciences, University of California, Merced 95343, California, USA 2 e-mail: [email protected] *Corresponding author: [email protected]
Received July 29, 2014; revised September 24, 2014; accepted October 6, 2014; posted October 10, 2014 (Doc. ID 219992); published November 6, 2014 Polarized light propagation in a multiple scattering medium is governed by the vector radiative transfer equation. We analyze the vector radiative transfer equation in asymptotic limit of forward-peaked scattering and derive an approximate system of equations for the Stokes parameters, which we call the vector Fokker–Planck approximation. The vector Fokker–Planck approximation provides valuable insight into several outstanding issues regarding the forward-peaked scattering of polarized light such as the polarization memory phenomenon. © 2014 Optical Society of America OCIS codes: (030.5620) Radiative transfer; (290.4210) Multiple scattering; (290.5855) Scattering, polarization. http://dx.doi.org/10.1364/OL.39.006422
Radiative transfer theory governs the propagation of light in a multiple scattering medium [1]. For many multiple scattering media such as clouds, oceans, and biological tissues, scattering is forward-peaked. Forward-peaked scattering has been studied extensively for the case in which polarization is neglected leading to the Fokker– Planck approximation [2,3] and its generalizations [4–7]. In contrast, forward-peaked scattering for polarized light is less understood theoretically. Nonetheless, there are several studies that indicate the need for a theoretical framework to study forward-peaked scattering of polarized light. For example, it has been shown for forwardpeaked scattering media that the characteristic length scale for circular polarized light to depolarize is much larger than that for linear polarized light [8]. Another example is polarization memory [9], the phenomena in which backscattering of circular polarized light incident on a forward-peaked scattering medium largely retains its original circular polarization state. There have been theoretical explanations of these issues (cf. [10–12]). However, a systematic analysis of forward-peaked scattering for the vector radiative transfer equation remains outstanding. To study forward-peaked scattering of polarized light, we analyze the vector radiative transfer equation in the asymptotic limit of forward-peaked scattering and derive a reduced system of equations. We call this reduced system the vector Fokker–Planck approximation. Let I I; Q; U; V T denote the vector of Stokes parameters. The Stokes parameters describes the ellipticity, orientation, and degree of polarization of light as well as the total intensity of a light beam [1]. The vector radiative transfer equation ˆ · ∇I μa I − μs LI S Ω
(1)
governs I in a medium that absorbs, scatters, and emits ˆ is the direction (a vector on the unit light. Here, Ω sphere S2 ), r is the position, μa is the absorption coefficient, μs is the scattering coefficient, and S is a source. The scattering operator L is defined as Z LI −I
S2
ˆ Ω ˆ 0 IΩ ˆ 0 ; rdΩ ˆ 0; ZΩ;
(2)
0146-9592/14/226422-04$15.00/0
where Z is the scattering matrix that relates the Stokes ˆ 0 into the direction vector scattered from the direction Ω ˆ Ω. We seek an asymptotic approximation of L when ˆ Ω ˆ 0. scattering is forward-peaked about Ω Let Rσ; n denote the matrix that rotates I by angle σ about the vector n. The Stokes vector satisfies the ˆ Ω ˆ HσIΩ ˆ with rotation property Rσ; ΩI 2
1 60 Hσ 6 40 0
0 cos 2σ sin 2σ 0
0 − sin 2σ cos 2σ 0
3 0 07 7: 05 1
(3)
Note that H −1 σ H T σ. In terms of H, the scattering matrix is given by ˆ Ω ˆ 0 H T ΦFΘHχ: ZΩ;
(4)
ˆ Ω ˆ 0 g denote the scattering plane. The Let P spanfΩ; scattering matrix F gives Stokes vector scattered from ˆ 0 into the direction Ω ˆ for a reference frame the direction Ω that is defined with respect to P. The angles χ and Φ are ˆ 0 and RΦ; Ω ˆ are the rotation the angles such that Rχ; Ω matrices from the experimental frame to the scattering frame. When scatterers are randomly oriented F has the following block-diagonal form [13]: 2a
1
6b FΘ 4 1 0 0
b1 a2 0 0
0 0 a3 −b2
03 07 5: b2 a4
(5)
The component 0 ≤ a1 ≤ 1 is the scattering phase function and is normalized according to Z 0
π
a1 Θ sin ΘdΘ 1:
(6)
In addition, a1 ≥ jbi j for i 1; 2 and a1 ≥ jaj j for j 2; 3; 4. ˆ 0 and HΦIΩ ˆ are related The intensity HχIΩ by a rotation of Θ in P. Let J denote the total angular © 2014 Optical Society of America
November 15, 2014 / Vol. 39, No. 22 / OPTICS LETTERS
momentum operator, and define J s to be the component of J in P. The operator J s generates infinitesimal rotaˆ 0 can be extions in the scattering plane. Hence, HχIΩ 0 ˆ panded with respect to HΦIΩ about Θ 0 as ˆ 0 e−iΘJ s HΦIΩ ˆ HχIΩ
∞ X −iΘk k0
k!
ˆ J ks HΦIΩ: (7)
ˆ sin θ cos φ; sin θ sin φ; cos θ. We use the Let Ω Euler angles φ, θ, and Φ to calculate J s . In terms of the contravariant spherical components of the angular momentum operator J 0 , which are given by [14], i ∂ ∂ 1 ∂ J 0 p e∓iΦ cot θ i ∓ ; (8) ∂Φ ∂θ sin θ ∂φ 2 p and we have J s J 01 − J 0−1 ∕ 2. Thus, for integers m, n, and any smooth function ψ, we have 1 2π
Z
2π 0
einΦ J s2k1 e−imΦ ψθ; φdΦ 0;
(9)
except when m n −k; −k 2; …; k − 2; k. We substitute Eq. (4) and Eq. (7) into Eq. (2) and comˆ 0 sin ΘdΘdΦ with Θ ∈ 0; π pute the integral using dΩ and Φ ∈ 0; 2π. Using the property given in Eq. (9), we find that L simplifies to LI −I ×
1 2π
Z
2π
0
∞ X −1k
2k!
k0
H T Φ
ˆ F 2k J2k s HΦIΩdΦ:
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for j 1; 2; 3; 4. Moreover, we assume for forward¯ k peaked scattering that a¯ k , and that b¯ k for j j 2 ∼a 3 k 1; 2 are negligible compared to a¯ j for j 1; 2; 3; 4. These assumptions are consistent with Mie scattering for uniform dielectric spheres. For that case, we have a2 a1 and a3 a4 , so only a1 , a4 , b1 , and b2 are needed to specify F defined in Eq. (5). We show plots of these scattering matrix entries in Fig. 1 for the case in which the relative refractive index is m 1.4 and the size parameter is ka 4, where k is the wavenumber and a is the sphere radius. For this case, scattering is not completely forward-peaked in the sense of Eq. (14). Nonetheless, these results show the features of forward-peaked scattering. Namely, we observe that a1 and a4 are sharply peaked about cos Θ 1 and b1 and b2 are significantly smaller, quantified by the moments defined in Eqs. (12) and (13). In the limit of very large spheres [13], we obtain that b1 b2 0 identically, a1 a2 a3 a4 , and Eq. (14) is satisfied. In light of these assumptions on the entries of F 2k , we truncate the series in Eq. (10) after k 1, neglect terms involving b¯ 0 and b¯ 2 for j 1; 2, integrate each term, j j and obtain the approximation LI ≈ L0 I L2 I:
(15)
Here, the 4 × 4 matrix L0 is given by 2
0 60 L0 −6 40 0
0 1 − a¯ 0 2 0 0
0 0 1 − a¯ 0 3 0
3 0 0 7 7; 0 5 1 − a¯ 0 4
(16)
(10)
Here, F k is defined as 2
F k
a¯ k 1 6 b¯ k 6 1 6 4 0 0
b¯ k 1 a¯ k 2 0 0
0 0 ¯ak 3 b¯ k 2
3 0 0 7 7 ; ¯bk 7 5 2 a¯ k 4
(11)
with a¯ k j
1 2
Z
π 0
Θk aj Θ sin ΘdΘ;
(12)
Θk bj Θ sin ΘdΘ;
(13)
for j 1; 2; 3; 4, and 1 b¯ k j 2
Z
π 0
for j 1; 2. When scattering is forward-peaked, aj for j 1; 2; 3; 4 is peaked about Θ 0 and vanishes rapidly away from there. Consequently, we quantify forward-peakedness by assuming that k ¯ 2 ¯ 2 a¯ 0 ≫ ; j ≫ a j ≫ ≫ a j
(14)
Fig. 1. Entries of the scattering matrix F given in Eq. (5) for Mie scattering with relative refractive index m 1.4 and size parameter ka 4.0, with k denoting the wavenumber and a denoting the sphere radius. For Mie scattering, a2 a1 and a3 a4 , so we plot only a1 (upper left), a4 (upper right), b1 (lower left), and b2 (lower right). Using these scattering matrix entries, we numerically compute the moments defined 2 2 in Eq. (12) and obtain a0 4 0.9559, a1 0.24, and a4 0.1703. We also numerically compute the moments defined 0 2 in Eq. (13) and obtain b0 1 0.0187, b2 0.0504, b1 2 −0.0051, and b2 −0.0416.
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OPTICS LETTERS / Vol. 39, No. 22 / November 15, 2014
since a¯ 0 1 1. Since 0 ≤ a1 ≤ 1 and a1 ≥ jaj j for j 2; 3; 4, we determine that 1 − a¯ 0 j ≥ 0 for j 2; 3; 4. For Mie scattering, a2 a1 and a4 a3 , so 1 − a¯ 0 2 0. k k Moreover, the assumption that a¯ 2 ∼ a¯ 3 for forwardpeaked scattering implies that L0 is zero identically for forward-peaked Mie scattering. Let θ denote the polar angle and φ denote the azimuthal angle in the experimental reference frame. Using J s icos Φ cot θ∂Φ sin Φ∂θ − cos Φ csc θ∂φ ; (17) we find that 21
0
c¯ 2 1−g
0 0
0 3 0 7 7 0 7 5
60 1 6 L2 1 − gΔΩˆ 6 0 0 c¯ 2 4 1−g 2 a¯ 2 4 0 0 0 1−g 2 0 0 0 2 θ 6 − cot θ∂φ 0 −1 − cot c¯ 2 6 4 1 − cot2 θ cot θ∂φ 0 0 0
1 1 ∂ sin θ∂θ 2 ∂2φ ; sin θ θ sin θ
3 0 07 7; 05 0
(18)
(19)
denoting the orbital angular momentum operator, or spherical Laplacian. Since the major contribution to the integral defining the a¯ j2k moments defined in Eq. (12) is about Θ 0, we replace Θ2 ∕2 ≈ 1 − cos Θ and consider instead the moments Zπ αj2k 1 − cos Θk aj Θ sin ΘdΘ: (20) 0
α0 1
Note that 1 by Eq. (6) and α2 1 1 − g, where g is the anisotropy factor or mean cosine of the scattering angle. For the scattering phase function, a1 , the moments defined in Eq. (20) are the same used by Leakeas and Larsen [6] to quantify the assumption of forward-peakedness. Replacing L in Eq. (1) with Eq. (15), we obtain ˆ · ∇I μa I − μs L0 I − μs L2 I 0; Ω
(21)
which we call the vector Fokker–Planck approximation. We write the four equations contained in Eq. (21) for each of the Stokes parameters below. The total intensity I satisfies ˆ · ∇I μa I − 1 μs 1 − gΔ ˆ I 0; Ω Ω 2
(23)
ˆ · ∇U μa U μs 1 − α0 U − 1 μs c¯ 2 Δ ˆ U Ω Ω 3 2 2 2 2 − γ 1 − cot θU − γ cot θ∂φ Q 0;
(24)
and
2 respectively, with γ 2 α2 2 α3 ∕2. Equations (23) and (24) are coupled due to the rotations of references frames manifested by forward-peaked scattering that, in turn, modify Q and U. The Stokes parameter V satisfies
ˆ · ∇V μa V μs 1 − α0 V − 1 μs α2 Δ ˆ V 0: Ω Ω 4 4 2
¯ 2 with c¯ 2 a¯ 2 2 a 3 ∕2, and ΔΩˆ
ˆ · ∇Q μa Q μs 1 − α0 Q − 1 μs γ 2 Δ ˆ Q Ω Ω 2 2 2 2 2 γ 1 − cot θQ γ cot θ∂φ U 0;
(22)
which is the Fokker–Planck approximation established for the scalar radiative transfer equation. The new inverse length scale introduced here is the reduced scattering coefficient μ0s μs 1 − g. The Stokes parameters Q and U satisfy the system
(25)
The vector Fokker–Planck equation, defined as the system comprised of Eqs. (21)–(24), governs the Stokes parameters in a forward-peaked multiple scattering medium. For arbitrarily polarized light incident on a forward-peaked scattering medium, the vector Fokker– Planck approximation yields decoupled equations for the total intensity, I, circular polarization, V , and linear polarization, Q and U. These results are physically intuitive since forward-peaked scattering is mostly restricted to a narrow cone of directions about the forward direction. Consider the following special case as an example. Suppose we seek to solve the vector Fokker–Planck approximation in a plane-parallel slab 0 < z < z0 due to circular polarized incident normally on the boundary z 0, and no other source of light. Because Q and U are decoupled from I and V , we would only need to solve Eq. (22) and Eq. (25). Assuming that solutions of Eq. (22) exist, it is sufficient that μa ≥ 0 and μs 1 − g > 0 to establish positivity of solutions for positive source and boundary data since −ΔΩˆ is symmetric positive-definite [15]. Equation (25) takes the same form as Eq. (22), but with different coefficients. We have established already Mie scattering, that 1 − α0 4 ≥ 0. For forward-peaked 2 μs α2 4 > 0. Assuming that μs α4 > 0 is true in general, we find that the helicity of the light incident on the slab will be preserved since Eq. (25) is sign-definite. This result is consistent with the observation of polarization memory in which backscattered circular polarized light incident on a forward-peaked scattering medium largely retains its original helicity. We have derived a vector Fokker–Planck approximation for forward-peaked scattering of polarized light. This approximation yields a simplified system of equations for the Stokes parameters describing the polarization state of light. In particular, the vector Fokker–Planck approximation yields decoupled equations for Stokes parameters I and V , and a coupled system of equations for Stokes parameters Q and U. As a special case, we find that this approximation is consistent with observations and theoretical predictions of polarization memory for circular polarized light. Studying polarized light propagation in a forward-peaked scattering medium is analytically and computationally challenging. Consequently,
November 15, 2014 / Vol. 39, No. 22 / OPTICS LETTERS
the simplifications in the vector Fokker–Planck approximation should be very useful for studying light propagation in forward-peaked scattering media. The authors are grateful to Drs. Harish Bhat and Boaz Ilan for their helpful discussions during the preparation of this article. J. Clark acknowledges support from the UC Merced Applied Math graduate program. References 1. S. Chandrasekhar, Radiative Transfer (Dover, 1960). 2. G. C. Pomraning, Math. Models Methods Appl. Sci. 2, 21 (1992). 3. E. W. Larsen, Prog. Nucl. Energy 34, 413 (1999). 4. G. C. Pomraning, Nucl. Sci. Eng. 124, 390 (1996). 5. A. K. Prinja and G. C. Pomraning, Nucl. Sci. Eng. 137, 227 (2001). 6. C. L. Leakeas and E. W. Larsen, Nucl. Sci. Eng. 137, 236 (2001).
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7. P. González-Rodríguez and A. D. Kim, Appl. Opt. 47, 2599 (2008). 8. D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994). 9. F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, Phys. Rev. B 40, 9342 (1989). 10. E. E. Gorodnichev, A. I. Kuzovlev, and D. B. Rogozkin, JETP Lett. 68, 22 (1998). 11. A. D. Kim and M. Moscoso, Opt. Lett. 27, 1589 (2002). 12. M. Xu and R. R. Alfano, Phys. Rev. E 72, 065601(R) (2005). 13. H. C. van de Hulst, Scattering of Light by Small Particles, (Dover, 1957). 14. D. A. Varshalovich, A. N. Mokalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988). 15. K. M. Case and P. F. Zweifel, J. Math. Phys. 4, 1376 (1963).
OPTICS LETTERS / Vol. 39, No. 22 / November 15, 2014
Forward-peaked scattering of polarized light Julia P. Clark1,* and Arnold D. Kim1,2 1
Applied Mathematics Unit, School of Natural Sciences, University of California, Merced 95343, California, USA 2 e-mail: [email protected] *Corresponding author: [email protected]
Received July 29, 2014; revised September 24, 2014; accepted October 6, 2014; posted October 10, 2014 (Doc. ID 219992); published November 6, 2014 Polarized light propagation in a multiple scattering medium is governed by the vector radiative transfer equation. We analyze the vector radiative transfer equation in asymptotic limit of forward-peaked scattering and derive an approximate system of equations for the Stokes parameters, which we call the vector Fokker–Planck approximation. The vector Fokker–Planck approximation provides valuable insight into several outstanding issues regarding the forward-peaked scattering of polarized light such as the polarization memory phenomenon. © 2014 Optical Society of America OCIS codes: (030.5620) Radiative transfer; (290.4210) Multiple scattering; (290.5855) Scattering, polarization. http://dx.doi.org/10.1364/OL.39.006422
Radiative transfer theory governs the propagation of light in a multiple scattering medium [1]. For many multiple scattering media such as clouds, oceans, and biological tissues, scattering is forward-peaked. Forward-peaked scattering has been studied extensively for the case in which polarization is neglected leading to the Fokker– Planck approximation [2,3] and its generalizations [4–7]. In contrast, forward-peaked scattering for polarized light is less understood theoretically. Nonetheless, there are several studies that indicate the need for a theoretical framework to study forward-peaked scattering of polarized light. For example, it has been shown for forwardpeaked scattering media that the characteristic length scale for circular polarized light to depolarize is much larger than that for linear polarized light [8]. Another example is polarization memory [9], the phenomena in which backscattering of circular polarized light incident on a forward-peaked scattering medium largely retains its original circular polarization state. There have been theoretical explanations of these issues (cf. [10–12]). However, a systematic analysis of forward-peaked scattering for the vector radiative transfer equation remains outstanding. To study forward-peaked scattering of polarized light, we analyze the vector radiative transfer equation in the asymptotic limit of forward-peaked scattering and derive a reduced system of equations. We call this reduced system the vector Fokker–Planck approximation. Let I I; Q; U; V T denote the vector of Stokes parameters. The Stokes parameters describes the ellipticity, orientation, and degree of polarization of light as well as the total intensity of a light beam [1]. The vector radiative transfer equation ˆ · ∇I μa I − μs LI S Ω
(1)
governs I in a medium that absorbs, scatters, and emits ˆ is the direction (a vector on the unit light. Here, Ω sphere S2 ), r is the position, μa is the absorption coefficient, μs is the scattering coefficient, and S is a source. The scattering operator L is defined as Z LI −I
S2
ˆ Ω ˆ 0 IΩ ˆ 0 ; rdΩ ˆ 0; ZΩ;
(2)
0146-9592/14/226422-04$15.00/0
where Z is the scattering matrix that relates the Stokes ˆ 0 into the direction vector scattered from the direction Ω ˆ Ω. We seek an asymptotic approximation of L when ˆ Ω ˆ 0. scattering is forward-peaked about Ω Let Rσ; n denote the matrix that rotates I by angle σ about the vector n. The Stokes vector satisfies the ˆ Ω ˆ HσIΩ ˆ with rotation property Rσ; ΩI 2
1 60 Hσ 6 40 0
0 cos 2σ sin 2σ 0
0 − sin 2σ cos 2σ 0
3 0 07 7: 05 1
(3)
Note that H −1 σ H T σ. In terms of H, the scattering matrix is given by ˆ Ω ˆ 0 H T ΦFΘHχ: ZΩ;
(4)
ˆ Ω ˆ 0 g denote the scattering plane. The Let P spanfΩ; scattering matrix F gives Stokes vector scattered from ˆ 0 into the direction Ω ˆ for a reference frame the direction Ω that is defined with respect to P. The angles χ and Φ are ˆ 0 and RΦ; Ω ˆ are the rotation the angles such that Rχ; Ω matrices from the experimental frame to the scattering frame. When scatterers are randomly oriented F has the following block-diagonal form [13]: 2a
1
6b FΘ 4 1 0 0
b1 a2 0 0
0 0 a3 −b2
03 07 5: b2 a4
(5)
The component 0 ≤ a1 ≤ 1 is the scattering phase function and is normalized according to Z 0
π
a1 Θ sin ΘdΘ 1:
(6)
In addition, a1 ≥ jbi j for i 1; 2 and a1 ≥ jaj j for j 2; 3; 4. ˆ 0 and HΦIΩ ˆ are related The intensity HχIΩ by a rotation of Θ in P. Let J denote the total angular © 2014 Optical Society of America
November 15, 2014 / Vol. 39, No. 22 / OPTICS LETTERS
momentum operator, and define J s to be the component of J in P. The operator J s generates infinitesimal rotaˆ 0 can be extions in the scattering plane. Hence, HχIΩ 0 ˆ panded with respect to HΦIΩ about Θ 0 as ˆ 0 e−iΘJ s HΦIΩ ˆ HχIΩ
∞ X −iΘk k0
k!
ˆ J ks HΦIΩ: (7)
ˆ sin θ cos φ; sin θ sin φ; cos θ. We use the Let Ω Euler angles φ, θ, and Φ to calculate J s . In terms of the contravariant spherical components of the angular momentum operator J 0 , which are given by [14], i ∂ ∂ 1 ∂ J 0 p e∓iΦ cot θ i ∓ ; (8) ∂Φ ∂θ sin θ ∂φ 2 p and we have J s J 01 − J 0−1 ∕ 2. Thus, for integers m, n, and any smooth function ψ, we have 1 2π
Z
2π 0
einΦ J s2k1 e−imΦ ψθ; φdΦ 0;
(9)
except when m n −k; −k 2; …; k − 2; k. We substitute Eq. (4) and Eq. (7) into Eq. (2) and comˆ 0 sin ΘdΘdΦ with Θ ∈ 0; π pute the integral using dΩ and Φ ∈ 0; 2π. Using the property given in Eq. (9), we find that L simplifies to LI −I ×
1 2π
Z
2π
0
∞ X −1k
2k!
k0
H T Φ
ˆ F 2k J2k s HΦIΩdΦ:
6423
for j 1; 2; 3; 4. Moreover, we assume for forward¯ k peaked scattering that a¯ k , and that b¯ k for j j 2 ∼a 3 k 1; 2 are negligible compared to a¯ j for j 1; 2; 3; 4. These assumptions are consistent with Mie scattering for uniform dielectric spheres. For that case, we have a2 a1 and a3 a4 , so only a1 , a4 , b1 , and b2 are needed to specify F defined in Eq. (5). We show plots of these scattering matrix entries in Fig. 1 for the case in which the relative refractive index is m 1.4 and the size parameter is ka 4, where k is the wavenumber and a is the sphere radius. For this case, scattering is not completely forward-peaked in the sense of Eq. (14). Nonetheless, these results show the features of forward-peaked scattering. Namely, we observe that a1 and a4 are sharply peaked about cos Θ 1 and b1 and b2 are significantly smaller, quantified by the moments defined in Eqs. (12) and (13). In the limit of very large spheres [13], we obtain that b1 b2 0 identically, a1 a2 a3 a4 , and Eq. (14) is satisfied. In light of these assumptions on the entries of F 2k , we truncate the series in Eq. (10) after k 1, neglect terms involving b¯ 0 and b¯ 2 for j 1; 2, integrate each term, j j and obtain the approximation LI ≈ L0 I L2 I:
(15)
Here, the 4 × 4 matrix L0 is given by 2
0 60 L0 −6 40 0
0 1 − a¯ 0 2 0 0
0 0 1 − a¯ 0 3 0
3 0 0 7 7; 0 5 1 − a¯ 0 4
(16)
(10)
Here, F k is defined as 2
F k
a¯ k 1 6 b¯ k 6 1 6 4 0 0
b¯ k 1 a¯ k 2 0 0
0 0 ¯ak 3 b¯ k 2
3 0 0 7 7 ; ¯bk 7 5 2 a¯ k 4
(11)
with a¯ k j
1 2
Z
π 0
Θk aj Θ sin ΘdΘ;
(12)
Θk bj Θ sin ΘdΘ;
(13)
for j 1; 2; 3; 4, and 1 b¯ k j 2
Z
π 0
for j 1; 2. When scattering is forward-peaked, aj for j 1; 2; 3; 4 is peaked about Θ 0 and vanishes rapidly away from there. Consequently, we quantify forward-peakedness by assuming that k ¯ 2 ¯ 2 a¯ 0 ≫ ; j ≫ a j ≫ ≫ a j
(14)
Fig. 1. Entries of the scattering matrix F given in Eq. (5) for Mie scattering with relative refractive index m 1.4 and size parameter ka 4.0, with k denoting the wavenumber and a denoting the sphere radius. For Mie scattering, a2 a1 and a3 a4 , so we plot only a1 (upper left), a4 (upper right), b1 (lower left), and b2 (lower right). Using these scattering matrix entries, we numerically compute the moments defined 2 2 in Eq. (12) and obtain a0 4 0.9559, a1 0.24, and a4 0.1703. We also numerically compute the moments defined 0 2 in Eq. (13) and obtain b0 1 0.0187, b2 0.0504, b1 2 −0.0051, and b2 −0.0416.
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OPTICS LETTERS / Vol. 39, No. 22 / November 15, 2014
since a¯ 0 1 1. Since 0 ≤ a1 ≤ 1 and a1 ≥ jaj j for j 2; 3; 4, we determine that 1 − a¯ 0 j ≥ 0 for j 2; 3; 4. For Mie scattering, a2 a1 and a4 a3 , so 1 − a¯ 0 2 0. k k Moreover, the assumption that a¯ 2 ∼ a¯ 3 for forwardpeaked scattering implies that L0 is zero identically for forward-peaked Mie scattering. Let θ denote the polar angle and φ denote the azimuthal angle in the experimental reference frame. Using J s icos Φ cot θ∂Φ sin Φ∂θ − cos Φ csc θ∂φ ; (17) we find that 21
0
c¯ 2 1−g
0 0
0 3 0 7 7 0 7 5
60 1 6 L2 1 − gΔΩˆ 6 0 0 c¯ 2 4 1−g 2 a¯ 2 4 0 0 0 1−g 2 0 0 0 2 θ 6 − cot θ∂φ 0 −1 − cot c¯ 2 6 4 1 − cot2 θ cot θ∂φ 0 0 0
1 1 ∂ sin θ∂θ 2 ∂2φ ; sin θ θ sin θ
3 0 07 7; 05 0
(18)
(19)
denoting the orbital angular momentum operator, or spherical Laplacian. Since the major contribution to the integral defining the a¯ j2k moments defined in Eq. (12) is about Θ 0, we replace Θ2 ∕2 ≈ 1 − cos Θ and consider instead the moments Zπ αj2k 1 − cos Θk aj Θ sin ΘdΘ: (20) 0
α0 1
Note that 1 by Eq. (6) and α2 1 1 − g, where g is the anisotropy factor or mean cosine of the scattering angle. For the scattering phase function, a1 , the moments defined in Eq. (20) are the same used by Leakeas and Larsen [6] to quantify the assumption of forward-peakedness. Replacing L in Eq. (1) with Eq. (15), we obtain ˆ · ∇I μa I − μs L0 I − μs L2 I 0; Ω
(21)
which we call the vector Fokker–Planck approximation. We write the four equations contained in Eq. (21) for each of the Stokes parameters below. The total intensity I satisfies ˆ · ∇I μa I − 1 μs 1 − gΔ ˆ I 0; Ω Ω 2
(23)
ˆ · ∇U μa U μs 1 − α0 U − 1 μs c¯ 2 Δ ˆ U Ω Ω 3 2 2 2 2 − γ 1 − cot θU − γ cot θ∂φ Q 0;
(24)
and
2 respectively, with γ 2 α2 2 α3 ∕2. Equations (23) and (24) are coupled due to the rotations of references frames manifested by forward-peaked scattering that, in turn, modify Q and U. The Stokes parameter V satisfies
ˆ · ∇V μa V μs 1 − α0 V − 1 μs α2 Δ ˆ V 0: Ω Ω 4 4 2
¯ 2 with c¯ 2 a¯ 2 2 a 3 ∕2, and ΔΩˆ
ˆ · ∇Q μa Q μs 1 − α0 Q − 1 μs γ 2 Δ ˆ Q Ω Ω 2 2 2 2 2 γ 1 − cot θQ γ cot θ∂φ U 0;
(22)
which is the Fokker–Planck approximation established for the scalar radiative transfer equation. The new inverse length scale introduced here is the reduced scattering coefficient μ0s μs 1 − g. The Stokes parameters Q and U satisfy the system
(25)
The vector Fokker–Planck equation, defined as the system comprised of Eqs. (21)–(24), governs the Stokes parameters in a forward-peaked multiple scattering medium. For arbitrarily polarized light incident on a forward-peaked scattering medium, the vector Fokker– Planck approximation yields decoupled equations for the total intensity, I, circular polarization, V , and linear polarization, Q and U. These results are physically intuitive since forward-peaked scattering is mostly restricted to a narrow cone of directions about the forward direction. Consider the following special case as an example. Suppose we seek to solve the vector Fokker–Planck approximation in a plane-parallel slab 0 < z < z0 due to circular polarized incident normally on the boundary z 0, and no other source of light. Because Q and U are decoupled from I and V , we would only need to solve Eq. (22) and Eq. (25). Assuming that solutions of Eq. (22) exist, it is sufficient that μa ≥ 0 and μs 1 − g > 0 to establish positivity of solutions for positive source and boundary data since −ΔΩˆ is symmetric positive-definite [15]. Equation (25) takes the same form as Eq. (22), but with different coefficients. We have established already Mie scattering, that 1 − α0 4 ≥ 0. For forward-peaked 2 μs α2 4 > 0. Assuming that μs α4 > 0 is true in general, we find that the helicity of the light incident on the slab will be preserved since Eq. (25) is sign-definite. This result is consistent with the observation of polarization memory in which backscattered circular polarized light incident on a forward-peaked scattering medium largely retains its original helicity. We have derived a vector Fokker–Planck approximation for forward-peaked scattering of polarized light. This approximation yields a simplified system of equations for the Stokes parameters describing the polarization state of light. In particular, the vector Fokker–Planck approximation yields decoupled equations for Stokes parameters I and V , and a coupled system of equations for Stokes parameters Q and U. As a special case, we find that this approximation is consistent with observations and theoretical predictions of polarization memory for circular polarized light. Studying polarized light propagation in a forward-peaked scattering medium is analytically and computationally challenging. Consequently,
November 15, 2014 / Vol. 39, No. 22 / OPTICS LETTERS
the simplifications in the vector Fokker–Planck approximation should be very useful for studying light propagation in forward-peaked scattering media. The authors are grateful to Drs. Harish Bhat and Boaz Ilan for their helpful discussions during the preparation of this article. J. Clark acknowledges support from the UC Merced Applied Math graduate program. References 1. S. Chandrasekhar, Radiative Transfer (Dover, 1960). 2. G. C. Pomraning, Math. Models Methods Appl. Sci. 2, 21 (1992). 3. E. W. Larsen, Prog. Nucl. Energy 34, 413 (1999). 4. G. C. Pomraning, Nucl. Sci. Eng. 124, 390 (1996). 5. A. K. Prinja and G. C. Pomraning, Nucl. Sci. Eng. 137, 227 (2001). 6. C. L. Leakeas and E. W. Larsen, Nucl. Sci. Eng. 137, 236 (2001).
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