## Abstract

An imaging Stokes-vector polarimeter using liquid crystal variable retarders (LCVRs) has
been built and calibrated. Operating in five bands from 450 to
$700\text{\hspace{0.17em} nm}$, the polarimeter can
be changed quickly between narrow (12°) and wide
$\left(\sim \text{160}\xb0\right)$ fields of view. The instrument
is designed for studying the effects of differing sky polarization upon the measured
polarization of ground-based objects. LCVRs exhibit variations in retardance with ray
incidence angle and ray position in the aperture. Therefore LCVR-based Stokes
polarimeters exhibit unique calibration challenges not found in other systems. Careful
design and calibration of the instrument has achieved errors within
$\pm 1.5\%$. Clear-sky
measurements agree well with previously published data and cloudy data provide
opportunities to explore spatial and spectral variations in sky polarization.

© 2006 Optical Society of America

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### Equations (8)

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(1)
$${\left[\begin{array}{c}{S}_{0}\\ {S}_{1}\\ {S}_{2}\\ {S}_{3}\end{array}\right]}_{\text{output}}=\left[\begin{array}{cccc}{m}_{00}& {m}_{01}& {m}_{02}& {m}_{03}\\ {m}_{10}& {m}_{11}& {m}_{12}& {m}_{13}\\ {m}_{20}& {m}_{21}& {m}_{22}& {m}_{23}\\ {m}_{30}& {m}_{31}& {m}_{32}& {m}_{33}\end{array}\right]{\left[\begin{array}{c}{S}_{0}\\ {S}_{1}\\ {S}_{2}\\ {S}_{3}\end{array}\right]}_{\text{input}}.$$
(2)
$$\left[\begin{array}{c}{S}_{01}\\ {S}_{02}\\ {S}_{03}\\ {S}_{04}\end{array}\right]=\left[\begin{array}{cccc}{a}_{00}& {a}_{01}& {a}_{02}& {a}_{03}\\ {a}_{10}& {a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{20}& {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{30}& {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right]\left[\begin{array}{c}{S}_{0}\\ {S}_{1}\\ {S}_{2}\\ {S}_{3}\end{array}\right]=AS.$$
(3)
$$\left[\begin{array}{c}{S}_{0}\\ {S}_{1}\\ {S}_{2}\\ {S}_{3}\end{array}\right]=\left[\begin{array}{cccc}{m}_{00}& {m}_{01}& {m}_{02}& {m}_{03}\\ {m}_{10}& {m}_{11}& {m}_{12}& {m}_{13}\\ {m}_{20}& {m}_{21}& {m}_{22}& {m}_{23}\\ {m}_{30}& {m}_{31}& {m}_{32}& {m}_{33}\end{array}\right]\left[\begin{array}{c}{S}_{0\text{ray}}\\ {S}_{1\text{ray}}\\ {S}_{2\text{ray}}\\ {S}_{3\text{ray}}\end{array}\right]={M}_{1}{S}_{1}.$$
(4)
$$\left[\begin{array}{c}{S}_{0\text{total}}\\ {S}_{1\text{total}}\\ {S}_{2\text{total}}\\ {S}_{3\text{total}}\end{array}\right]={M}_{1}{S}_{1}+{M}_{2}{S}_{2}+{M}_{3}{S}_{3}+\cdots +{M}_{N}{S}_{N},$$
(5)
$$\left[\begin{array}{c}{S}_{0\text{total}}\\ {S}_{1\text{total}}\\ {S}_{2\text{total}}\\ {S}_{3\text{total}}\end{array}\right]=\left({M}_{1}+{M}_{2}+{M}_{3}+\cdots +{M}_{N}\right){S}^{\left(1\right)}=M\text{\hspace{0.17em}}{S}_{\text{input}},$$
(6)
$$\begin{array}{l}{\text{Image}}_{-90}\hfill \\ {\text{Image}}_{\text{0}}\hfill \\ {\text{Image}}_{+45}\hfill \\ {\text{Image}}_{-45}\hfill \end{array}\begin{array}{l}=1{a}_{00}+1{a}_{01}+0{a}_{02}+0{a}_{03},\hfill \\ =1{a}_{00}-1{a}_{01}+0{a}_{02}+0{a}_{03},\hfill \\ =1{a}_{00}+0{a}_{01}+1{a}_{02}+0{a}_{03},\hfill \\ =1{a}_{00}+0{a}_{01}-1{a}_{02}+0{a}_{03}.\hfill \end{array}$$
(7)
$$\begin{array}{l}{S}_{1\left(0\right)}\hfill \\ {S}_{1\left(-90\right)}\hfill \\ {S}_{1\left(+45\right)}\hfill \\ {S}_{1\left(-45\right)}\hfill \end{array}\begin{array}{l}=1{m}_{10}+1{m}_{11}+0{m}_{12}+0{m}_{13},\hfill \\ =1{m}_{10}-1{m}_{11}+0{m}_{12}+0{m}_{13},\hfill \\ =1{m}_{10}+0{m}_{11}+1{m}_{12}+0{m}_{13},\hfill \\ =1{m}_{10}+0{m}_{11}-1{m}_{12}+0{m}_{13}.\hfill \end{array}$$
(8)
$$\begin{array}{rrrr}1.0000& -0.0006& -0.0018& -0.0081\\ -0.0008& 1.0083& -0.0047& 0.0320\\ 0.0009& -0.0019& 1.0012& 0.0089\\ -0.0081& -0.0320& -0.0089& 1.0047\end{array}.$$