Abstract

Channeled polarimeters measure polarization by modulating the measured intensity in order to create polarization-dependent channels that can be demodulated to reveal the desired polarization information. A number of channeled systems have been described in the past, but their proposed designs often unintentionally sacrifice optimality for ease of algebraic reconstruction. To obtain more optimal systems, a generalized treatment of channeled polarimeters is required. This paper describes methods that enable handling of multi-domain modulations and reconstruction of polarization information using linear algebra. We make practical choices regarding use of either Fourier or direct channels to make these methods more immediately useful. Employing the introduced concepts to optimize existing systems often results in superficial system changes, like changing the order, orientation, thickness, or spacing of polarization elements. For the two examples we consider, we were able to reduce noise in the reconstruction to 34.1% and 57.9% of the original design values.

© 2014 Optical Society of America

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References

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2012

M. W. Kudenov, M. J. Escuti, N. Hagen, E. L. Dereniak, and K. Oka, “Snapshot imaging Mueller matrix polarimeter using polarization gratings,” Opt. Lett. 37, 1367–1369 (2012).
[CrossRef]

A. S. Alenin and J. S. Tyo, “Task-specific snapshot Mueller matrix channeled spectropolarimeter optimization,” Proc. SPIE 8364, 836402 (2012).
[CrossRef]

2011

C. F. LaCasse, R. A. Chipman, and J. S. Tyo, “Band limited data reconstruction in modulated polarimeters,” Opt. Express 19, 14976–14989 (2011).
[CrossRef]

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” Proc. SPIE 8160, 81600K (2011).
[CrossRef]

2008

2007

2006

2003

2002

2000

1999

1996

Alenin, A. S.

A. S. Alenin and J. S. Tyo, “Task-specific snapshot Mueller matrix channeled spectropolarimeter optimization,” Proc. SPIE 8364, 836402 (2012).
[CrossRef]

Alfano, R.

Cairns, B.

Cariou, J.

Chenault, D. B.

Chipman, R. A.

Davis, A.

Demos, S.

Dereniak, E. L.

Descour, M. R.

Diner, D. J.

Dubreuil, M.

Engheta, N.

Escuti, M. J.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Vol. 1, Chap. 7, pp. 179–217.

Gerhart, G. R.

Goldstein, D. L.

Gutt, G.

Hagen, N.

Hagen, N. A.

Hancock, B.

Hayakawa, M.

Jeune, B. L.

Kaneko, T.

Kato, T.

Kemme, S. A.

Kudenov, M. W.

LaCasse, C. F.

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” Proc. SPIE 8160, 81600K (2011).
[CrossRef]

C. F. LaCasse, R. A. Chipman, and J. S. Tyo, “Band limited data reconstruction in modulated polarimeters,” Opt. Express 19, 14976–14989 (2011).
[CrossRef]

Lemaillet, P.

Naito, H.

Oka, K.

Okabe, H.

Phipps, G. S.

Pugh, J. E. N.

Radousky, H.

Ririe, T.

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” Proc. SPIE 8160, 81600K (2011).
[CrossRef]

Rivet, S.

Rowe, M. P.

Sabatke, D. S.

Shaw, J. A.

Sweatt, W. C.

Taniguchi, A.

Twietmeyer, K. M.

Tyo, J. S.

Appl. Opt.

Opt. Express

Opt. Lett.

Proc. SPIE

A. S. Alenin and J. S. Tyo, “Task-specific snapshot Mueller matrix channeled spectropolarimeter optimization,” Proc. SPIE 8364, 836402 (2012).
[CrossRef]

C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” Proc. SPIE 8160, 81600K (2011).
[CrossRef]

Other

R. A. Chipman, Handbook of Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 1, Chap. 15, pp. 15.1–15.41.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Vol. 1, Chap. 7, pp. 179–217.

R. A. Chipman, Handbook of Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 1, Chap. 14, pp. 14.1–14.44.

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Figures (9)

Fig. 1.
Fig. 1.

First four FPMs. The circles represent the polar form of the coefficients—the direction of the radius line contains the phase information as: right =+1, up =+j, left =1, down =j. Each FPM has an omitted weight of 2M. If ξi=ξj, then δ(ξ±+ξiξj±) and δ(ξ±ξi+ξj±) will combine and change the magnitude of the impulse at that frequency. The side brackets denoted with 2/4/8/16 can be used as crop guidelines for obtaining FPMs for M=1/2/3/4.

Fig. 2.
Fig. 2.

Snapshot systems. Case (a) provides no modulation. Case (e) is not straightforward to implement physically. Case pairs (b)/(d), (c)/(g), and (f)/(h) are essentially equivalent.

Fig. 3.
Fig. 3.

Hagen’s [12] polarimeter, d̲=(12510). The matrix containing 21 cropped channels can be seen between the two horizontal lines and has an EWV of 355, including the other 16 channels lowers the EWV to 187. These extra channels must be measured to prevent aliasing. The distinction is whether the data contained within these channels is used in reconstruction, after the Fourier transform of the measured intensity was found.

Fig. 4.
Fig. 4.

First two elements swapped. d̲=(21510), with all 37 channels used. EWV is lowered to 130(4/7).

Fig. 5.
Fig. 5.

Further thickness adjustments can produce an optimal with d̲=(21411) and EWV=121, while keeping the same number of channels.

Fig. 6.
Fig. 6.

+ for different configurations.

Fig. 7.
Fig. 7.

Top row: ξ/η plane of channels (the number inside each channel corresponds to the number of Mueller elements contained within). Bottom row: +.

Fig. 8.
Fig. 8.

Frequency grid of the Mueller modulation. ξ and η are the x and y axes, respectively.

Fig. 9.
Fig. 9.

Optimal spatial–spatial polarimeter.

Tables (5)

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Table 1. Channeled System Designs with the Lowest EWV for a Given Number of Channelsa

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Table 2. Hagen [12] Polarimeter’s Channels

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Table 3. Hagen’s [12] Proposed Reconstruction

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Table 4. Minimum Achievable EWV for Ny×Nx Channel Arrangement

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Table 5. Results for Different Number of Temporal Snapshotsa

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

S̲=[s0s1s2s3]=[IH+IVIHIVI+45I45IRIL].
s0s12+s22+s32,
S̲out=S̲in,
=[m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33].
A̲n=[a0a1a2a3]nT.
=[A̲0A̲1A̲N1]T.
G̲n=[g0g1g2g3]nT
D̲n=A̲nG̲n=[a0g0a0g3a3g0a3g3]nT,
M̲=vec()=[m00m03m30m33]T.
=[D̲0D̲1D̲N1]T.
M̲^=(I̲+n̲).
k=4i+j
i=k/4,
j=k4i.
1(x)δ(ξ),
cos(2πξix)12[δ(ξ+ξi)+δ(ξξi)],
sin(2πξix)j2[δ(ξ+ξi)δ(ξξi)].
fM(x)=m=1Mcossin(2πξmx),
M[f̲1f̲2f̲M],
M[o̲1o̲2o̲M],
M12Mexp[jπ2(MMT)].
total=M0+M1++ML1,
q̲{τ}q̲{ω}q̲{ξ}q̲{η}.
vec(q̲{ξe1}*nq̲{ηe2}*nq̲{ξe3}*nq̲{ηe4}),
q̲mij=vec(q̲gi*q̲aj).
=[q̲{τ,ω,ξ,η};m00Tq̲{τ,ω,ξ,η};m01Tq̲{τ,ω,ξ,η};m02Tq̲{τ,ω,ξ,η};m03Tq̲{τ,ω,ξ,η};m10Tq̲{τ,ω,ξ,η};m11Tq̲{τ,ω,ξ,η};m12Tq̲{τ,ω,ξ,η};m13Tq̲{τ,ω,ξ,η};m20Tq̲{τ,ω,ξ,η};m21Tq̲{τ,ω,ξ,η};m22Tq̲{τ,ω,ξ,η};m23Tq̲{τ,ω,ξ,η};m30Tq̲{τ,ω,ξ,η};m31Tq̲{τ,ω,ξ,η};m32Tq̲{τ,ω,ξ,η};m33T]T
F{C̲}=F{M̲},
M̲=F1{+F{C̲}}.
total=[t0Tt1TtN1T]T.
=Σ̳,
+=Σ̳+,
I(ϑ⃗)=i=03j=03fai(ϑ⃗)mij(ϑ⃗)fgj(ϑ⃗),
EWV=tr[+]=k=0151/σ+,k2,
f̲G̲=[1cos(2πϑ1ϑ˜1)sin(2πϑ1ϑ˜1)cos(2πϑ2ϑ˜2)sin(2πϑ1ϑ˜1)sin(2πϑ2ϑ˜2)],
f̲A̲=[1cos(2πϑ3ϑ˜3)sin(2πϑ3ϑ˜3)cos(2πϑ4ϑ˜4)sin(2πϑ3ϑ˜3)sin(2πϑ4ϑ˜4)],
EWVmin=n̲Gn̲A,
d̲=(ϑ˜1ϑ˜2ϑ˜3ϑ˜4),
NC=1+2i=14di.
f̲G̲=[1cos(c1σ)sin(c1σ)cos(c2σ)sin(c1σ)cos(c2σ)],
f̲A̲=[1cos(c4σ)sin(c3σ)sin(c4σ)cos(c3σ)sin(c4σ)].
ciσ=2πτiσ=2πdodiλoBσ,τi=dodiλoB
Nx=1+2i=14ϑ˜i(ϑi=?x),
Ny=1+2i=14ϑ˜i(ϑi=?y).
f̲G̲=[1cos(2πy)sin(2πy)cos(2πx)sin(2πy)sin(2πx)],
f̲A̲=[1cos(4πx)sin(4πx)cos(4πy)sin(4πx)sin(4πy)].
θ=[AF{A̲}]*[F{G̲}TG].

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