Polarization lidar: Corrections of instrumental effects

Jens Biele; Georg Beyerle; Gerd Baumgarten

doi:10.1364/OE.7.000427

Optics Express
Vol. 7,
Issue 12,
pp. 427-435
(2000)
•https://doi.org/10.1364/OE.7.000427

Polarization lidar: Corrections of instrumental effects

Jens Biele, Georg Beyerle, and Gerd Baumgarten

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Jens Biele, Georg Beyerle, and Gerd Baumgarten, "Polarization lidar: Corrections of instrumental effects," Opt. Express 7, 427-435 (2000)

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Abstract

An algorithm for correcting instrumental effects in polarization lidar studies is discussed. Cross-talk between the perpendicular and parallel polarization channels and imperfect polarization of the transmitted laser beam are taken into account. On the basis of the Mueller formalism it is shown that - with certain assumptions - the combined effects of imperfect polarization of the transmitted laser pulse, non-ideal properties of transmitter and receiver optics and cross-talk between parallel and perpendicular polarization channels can be described by a single parameter, which is essentially the overall system depolarization.

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Figures (1)

Fig. 1.

Fig. 1. Quasi-linear correlation between

S_{‖}^{m}

and S_m _⊥ on 20 February 1997 in an almost purely liquid polar stratospheric cloud (PSC), illustrating the correction of instrumental cross-talk between the parallel and perpendicular channels. All circles: uncorrected raw data; red circles: selected raw data with

δ_{m}^{A}

<0.015; filled red circles: raw data with a corrected S _⊥ consistent with 1. Dashed line: linear relation corresponding to δ ^C=0.0217. Note that the uncertainty of a single data point ranges from 0.1 to 0.5 (a few errorbars are shown for reference)

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Equations (44)

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I = 〈 E_{∥} \cdot E_{∥}^{*} + E_{⊥} \cdot E_{⊥}^{*} 〉

Q = 〈 E_{∥} \cdot E_{∥}^{*} - E_{⊥} \cdot E_{⊥}^{*} 〉

U = 〈 E_{∥} \cdot E_{⊥}^{*} + E_{⊥} \cdot E_{∥}^{*} 〉

V = \sqrt{- 1} 〈 E_{∥} \cdot E_{⊥}^{*} - E_{⊥} \cdot E_{∥}^{*} 〉 .

δ = \frac{i_{⊥}}{i_{∥}},

δ = \frac{I - Q}{I + Q} .

i = c β (z) T^{2} (z) / z^{2}

s = \frac{β^{R} + β^{A}}{β^{R}} = 1 + \frac{β^{A}}{β^{R}} .

S_{∥, ⊥, T} = \frac{β_{∥, ⊥, T}^{R} + β_{∥, ⊥, T}^{A}}{β_{∥, ⊥, T}^{R}} = 1 + \frac{β_{∥, ⊥, T}^{A}}{β_{∥, ⊥, T}^{R}}

δ^{V} = \frac{i_{⊥}}{i_{∥}} = \frac{β_{⊥}}{β_{∥}} = \frac{S_{⊥}}{S_{∥}} \frac{β_{⊥}^{R}}{β_{∥}^{R}} \equiv \frac{S_{⊥}}{S_{∥}} δ^{R}

δ^{A} = \frac{β_{⊥}^{A}}{β_{∥}^{A}} = \frac{S_{⊥} - 1}{S_{∥} - 1} \frac{β_{⊥}^{R}}{β_{∥}^{R}} = \frac{S_{⊥} - 1}{S_{∥} - 1} δ^{R}

= \frac{(1 + δ^{R}) δ^{V} S_{T} - (1 + δ^{V}) δ^{R}}{(1 + δ^{R}) S_{T} - (1 + δ^{V})} .

F_{s} (180^{\circ}) = β_{T} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{(1 - δ^{V})}{(1 + δ^{V})} & 0 & 0 \\ 0 & 0 & F_{33} & 0 \\ 0 & 0 & 0 & F_{44} \end{matrix}) .

F_{p, ∥} = \frac{1}{2} (\begin{matrix} 1 & 1 - B^{∥} & 0 & 0 \\ 1 - B^{∥} & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})

F_{p, ⊥} = \frac{1}{2} (\begin{matrix} 1 & - (1 - B^{⊥}) & 0 & 0 \\ - (1 - B^{⊥}) & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) .

i_{∥} \propto {[\frac{1}{2} (\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) F_{s} (\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix})]}_{1 . component}

= \frac{β_{T}}{2} (1 + \frac{1 - δ^{V}}{1 + δ^{V}}) = \frac{1}{1 + δ^{V}} β_{T} = β_{∥}

i_{⊥} \propto {[\frac{1}{2} (\begin{matrix} 1 & - 1 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) F_{s} (\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix})]}_{1 . component}

= \frac{β_{T}}{2} (1 - \frac{1 - δ^{V}}{1 + δ^{V}}) = \frac{δ^{V}}{1 + δ^{V}} β_{T} = β_{⊥} .

i_{∥}^{m} \propto {[\frac{1}{2} (\begin{matrix} 1 & (1 - B^{∥}) & 0 & 0 \\ (1 - B^{∥}) & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) F_{s} (\begin{matrix} 1 \\ 1 - α \\ 0 \\ 0 \end{matrix})]}_{1 . component}

= \frac{β_{T}}{2} (1 + \frac{1 - δ^{V}}{1 + δ^{V}} (1 - α) (1 - B^{∥})) = β_{∥}^{m}

i_{⊥}^{m} \propto {[\frac{1}{2} (\begin{matrix} 1 & - (1 - B^{⊥}) & 0 & 0 \\ - (1 - B^{⊥}) & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) F_{s} (\begin{matrix} 1 \\ 1 - α \\ 0 \\ 0 \end{matrix})]}_{1 . component}

\frac{β_{T}}{2} (1 - \frac{1 - δ^{V}}{1 + δ^{V}} (1 - α) (1 - B^{⊥})) = β_{⊥}^{m} .

(1 - 2 {\tilde{δ}}^{C}) \equiv (1 - α) (1 - B^{∥})

(1 - 2 δ^{C}) \equiv (1 - α) (1 - B^{⊥}) .

i_{∥}^{m} = (1 - {\tilde{δ}}^{C}) i_{∥} + {\tilde{δ}}^{C} i_{⊥}

i_{⊥}^{m} = δ^{C} i_{∥} + (1 - δ^{C}) i_{⊥}

β_{∥}^{m} = (1 - {\tilde{δ}}^{C}) β_{∥} + {\tilde{δ}}^{C} β_{⊥}

β_{⊥}^{m} = δ^{C} β_{∥} + (1 - δ^{C}) β_{⊥} .

β_{∥}^{m} \approx β_{∥}

β_{⊥}^{m} = δ^{C} β_{∥} + (1 - δ^{C}) β_{⊥} .

S_{⊥}^{m} = \frac{β_{⊥}^{A, m} + β_{⊥}^{R, m}}{β_{⊥}^{R, m}}

= \frac{S_{∥} δ^{C} + S_{⊥} (1 - δ^{C}) δ^{R}}{δ^{C} + (1 - δ^{C}) δ^{R}}

S_{∥}^{m} = \frac{β_{∥}^{A, m} + β_{∥}^{R, m}}{β_{∥}^{R, m}}

\approx S_{∥} .

S_{⊥} \approx (1 + \frac{δ^{C}}{δ^{R}}) S_{⊥}^{m} - \frac{δ^{C}}{δ^{R}} S_{∥}^{m}

S_{∥} \approx S_{∥}^{m} .

S_{⊥}^{m} \approx \frac{S_{∥} δ^{C} + δ^{R}}{δ_{C} + δ^{R}}

\approx \frac{δ^{C}}{δ^{C} + δ^{R}} S_{∥}^{m} + \frac{δ^{R}}{δ^{C} + δ^{R}}

δ^{C} = \frac{δ^{R} (S_{⊥}^{m} - 1)}{S_{∥}^{m} - S_{⊥}^{m}} .

δ^{V} = \frac{i_{⊥}}{i_{∥}} .

δ_{m}^{V} = k \frac{i_{⊥}^{m}}{i_{∥}^{m}} = k (δ^{C} + (1 - δ^{C}) δ^{V})

δ^{V} = δ_{m}^{V} \frac{δ^{C} / δ^{R} + (1 - δ^{C})}{1 - δ^{C}} - \frac{δ^{C}}{1 - δ^{C}}

\approx δ_{m}^{V} (δ^{C} / δ^{R} + 1) - δ^{C} .

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Equations (44)

Optics Express