Forward-peaked scattering of polarized light: erratum

Julia P. Clark; Arnold D. Kim

doi:10.1364/OL.40.002020

Abstract

We intend to correct the typographical errors that occurred in our recent Letter [Opt. Lett. 39, 6422 (2014) [CrossRef] ].

In what follows, we correct the errors that appeared in [1].

The integrals in Eqs. (6) and (20) are each missing a normalization factor of $1 / 2$ . Equation (6) should read

\frac{1}{2} \int_{0}^{π} a_{1} (Θ) \sin Θ d Θ = 1,

and Eq. (20) should read

α_{j}^{(2 k)} = \frac{1}{2} \int_{0}^{π} {(1 - \cos Θ)}^{k} a_{j} (Θ) \sin Θ d Θ .

The second and third diagonal entries of the $4 \times 4$ matrix $L_{0}$ given in Eq. (16) are incorrect. The correct result for $L_{0}$ is

L_{0} = - [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 - \frac{{\bar{a}}_{2}^{(0)} + {\bar{a}}_{3}^{(0)}}{2} & 0 & 0 \\ 0 & 0 & 1 - \frac{{\bar{a}}_{2}^{(0)} + {\bar{a}}_{3}^{(0)}}{2} & 0 \\ 0 & 0 & 0 & 1 - {\bar{a}}_{4}^{(0)} \end{matrix}] .

The second term in the definition of $L_{2}$ given in Eq. (18) contains several errors: there is a missing scalar factor of 2, and the (2, 2), (2, 3), and (3, 2) entries are incorrect. The correct definition of $L_{2}$ is

L_{2} = \frac{1}{2} Δ_{\hat{Ω}} [\begin{matrix} 1 - g & 0 & 0 & 0 \\ 0 & {\bar{c}}^{(2)} & 0 & 0 \\ 0 & 0 & {\bar{c}}^{(2)} & 0 \\ 0 & 0 & 0 & {\bar{a}}_{4}^{(2)} \end{matrix}] + 2 {\bar{c}}^{(2)} [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 - \cot^{2} θ & \frac{\cos θ}{\sin^{2} θ} \partial_{φ} & 0 \\ - \frac{\cos θ}{\sin^{2} θ} \partial_{φ} & 1 - \cot^{2} θ & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] .

As a consequence of the errata in Eq. (18), which are addressed above, Eqs. (23) and (24) have errors. The corrected equations are

\hat{Ω} \cdot \nabla Q + μ_{a} Q + μ_{s} (1 - γ^{(0)}) Q - \frac{1}{2} μ_{s} γ^{(2)} Δ_{\hat{Ω}} Q - 2 γ^{(2)} (1 - \cot^{2} θ) Q - 2 γ^{(2)} \frac{\cos θ}{\sin^{2} θ} \partial_{φ} U = 0,

and

\hat{Ω} \cdot \nabla U + μ_{a} U + μ_{s} (1 - α_{3}^{(0)}) U - \frac{1}{2} μ_{s} {\bar{c}}^{(2)} Δ_{\hat{Ω}} U + 2 γ^{(2)} (1 - \cot^{2} θ) U - 2 γ^{(2)} \frac{\cos θ}{\sin^{2} θ} θ \partial_{φ} Q = 0 .

In these corrected equations, we have introduced $γ^{(k)} = (α_{2}^{(k)} + α_{3}^{(k)}) / 2$ for $k = 0$ , 2.

Reference

1. J. Clark and A. D. Kim, Opt. Lett. 39, 6422 (2014). [CrossRef]

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Equations (6)

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\frac{1}{2} \int_{0}^{π} a_{1} (Θ) \sin Θ d Θ = 1,

α_{j}^{(2 k)} = \frac{1}{2} \int_{0}^{π} {(1 - \cos Θ)}^{k} a_{j} (Θ) \sin Θ d Θ .

L_{0} = - [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 - \frac{{\bar{a}}_{2}^{(0)} + {\bar{a}}_{3}^{(0)}}{2} & 0 & 0 \\ 0 & 0 & 1 - \frac{{\bar{a}}_{2}^{(0)} + {\bar{a}}_{3}^{(0)}}{2} & 0 \\ 0 & 0 & 0 & 1 - {\bar{a}}_{4}^{(0)} \end{matrix}] .

L_{2} = \frac{1}{2} Δ_{\hat{Ω}} [\begin{matrix} 1 - g & 0 & 0 & 0 \\ 0 & {\bar{c}}^{(2)} & 0 & 0 \\ 0 & 0 & {\bar{c}}^{(2)} & 0 \\ 0 & 0 & 0 & {\bar{a}}_{4}^{(2)} \end{matrix}] + 2 {\bar{c}}^{(2)} [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 - \cot^{2} θ & \frac{\cos θ}{\sin^{2} θ} \partial_{φ} & 0 \\ - \frac{\cos θ}{\sin^{2} θ} \partial_{φ} & 1 - \cot^{2} θ & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] .

\hat{Ω} \cdot \nabla Q + μ_{a} Q + μ_{s} (1 - γ^{(0)}) Q - \frac{1}{2} μ_{s} γ^{(2)} Δ_{\hat{Ω}} Q - 2 γ^{(2)} (1 - \cot^{2} θ) Q - 2 γ^{(2)} \frac{\cos θ}{\sin^{2} θ} \partial_{φ} U = 0,

\hat{Ω} \cdot \nabla U + μ_{a} U + μ_{s} (1 - α_{3}^{(0)}) U - \frac{1}{2} μ_{s} {\bar{c}}^{(2)} Δ_{\hat{Ω}} U + 2 γ^{(2)} (1 - \cot^{2} θ) U - 2 γ^{(2)} \frac{\cos θ}{\sin^{2} θ} θ \partial_{φ} Q = 0 .

Abstract

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Equations (6)

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