Abstract

Mueller matrix polarimeters (MMPs) are designed to probe the polarization properties of optical scattering processes. When using a MMP for a detection, discrimination, classification, or identification task, a user considers certain elements of the Mueller matrix. The usual way of performing this task is to measure the full Mueller matrix and discard the unused elements. For polarimeter designs with speed, miniaturization, or other constraints, it may be desirable to have a system with reduced dimensionality that measures only elements of the Mueller matrix that are important in a particular application as efficiently as possible. In this paper, we develop a framework that allows partial MMPs to be analyzed. Quantitative metrics are developed by considering geometrical relationships between the space spanned by a particular MMP and the space occupied by the scene components. The method is generalized to allow the effects of noise to be considered. The results are general and can also be used to optimize complete and overspecified MMPs for performing specific tasks, as well.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  32. R. A. Chipman, “Degrees of freedom in depolarizing Mueller matrices,” Proc. SPIE  6682, 66820I (2007).
    [CrossRef]
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    [CrossRef]
  37. H. Hotelling, “Relations between two sets of variates,” Biometrika  28, 312–377 (1936).
  38. In some cases it might be advantageous to generate or analyze partially polarized states. However, the difficulty of doing this in an active system will preclude its use except in specialized situations. The obvious exception to this is the generation or analysis of an unpolarized state. The term m00 can be measured with a single measurement when both generator and analyzer are unpolarized. The elements of the first row m0j can be measured in two measurements by analyzing an unpolarized state. The elements of the first column mi0 can be obtained in two measurements by generating an unpolarized state.
  39. G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11–29.
  40. S. J. Johnson, “Use of partial polarimetry in material discrimination,” Master’s thesis (University of Arizona, Tucson, 2009).
  41. D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
    [CrossRef]
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    [CrossRef]

2009 (1)

J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Designing partial Mueller matrix polarimeters,” Proc. SPIE  7461, 74610V (2009).
[CrossRef]

2008 (4)

2007 (4)

2006 (3)

2005 (2)

2004 (1)

2003 (2)

2002 (3)

2000 (3)

J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable retardance polarimeters,” Opt. Lett.  25, 1198–2000 (2000).
[CrossRef]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A  17, 328–334 (2000).
[CrossRef]

1999 (1)

1998 (1)

1995 (2)

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng.  34, 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.  34, 1656–1659 (1995).
[CrossRef]

1994 (1)

1988 (1)

1987 (1)

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt.  34, 569–575 (1987).
[CrossRef]

1986 (1)

S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttgart)  75, 26–36 (1986).

1985 (1)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta  32, 259–261 (1985).
[CrossRef]

1981 (1)

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun.  38, 159–161 (1981).
[CrossRef]

1936 (1)

H. Hotelling, “Relations between two sets of variates,” Biometrika  28, 312–377 (1936).

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng.  34, 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.  34, 1656–1659 (1995).
[CrossRef]

Anderson, D. G.

Artal, P.

Atondo-Rubio, G.

Azzam, R. M. A.

Barakat, R.

D. G. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A  11, 2305–2319 (1994).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun.  38, 159–161 (1981).
[CrossRef]

Beaudry, N.

Beaudry, N. A.

Bernabeu, E.

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Bueno, J. M.

Chenault, D. B.

Chipman, R.

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttgart)  75, 26–36 (1986).

Dainty, C.

Daubecheis, I.

I. Daubecheis, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992), Chap. 3, pp. 53–106.
[CrossRef]

De Martino, A.

DeBoo, B. J.

Dereniak, E.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Descour, M. R.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Drèvillon, B.

Elminyawi, I. M.

El-Saba, A. M.

Espinosa-Luna, R.

Garcia, J. P.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Garcia-Caurel, E.

Gil, J. J.

J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A  17, 328–334 (2000).
[CrossRef]

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta  32, 259–261 (1985).
[CrossRef]

Goldstein, D. H.

J. S. Tyo, D. H. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt.  45, 5453–5469 (2006).
[CrossRef] [PubMed]

D. G. Jones, D. H. Goldstein, and J. C. Spaulding, “Reflective and polarimetric characteristics of urban materials,” Proc. SPIE  6240, 62400A (2006).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11–29.

Goudail, F.

F. Goudail, “Comparative study of the best achievable contrast in scalar, Stokes and Mueller images,” in First NanoCharm Workshop on Advanced Polarimetric Instrumentation (Multifunctional Nanomaterials Characterization Exploiting EllipsoMetry and Polarimetry (NanoCharM), 2009).

Hayat, M. M.

Hoover, B. G.

Hotelling, H.

H. Hotelling, “Relations between two sets of variates,” Biometrika  28, 312–377 (1936).

Johnson, S. J.

J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Designing partial Mueller matrix polarimeters,” Proc. SPIE  7461, 74610V (2009).
[CrossRef]

S. J. Johnson, “Use of partial polarimetry in material discrimination,” Master’s thesis (University of Arizona, Tucson, 2009).

Jones, D. G.

D. G. Jones, D. H. Goldstein, and J. C. Spaulding, “Reflective and polarimetric characteristics of urban materials,” Proc. SPIE  6240, 62400A (2006).
[CrossRef]

Kemme, S. A.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Kim, Y.-K.

Lara, D.

Laude, B.

Lee, J.-S.

J.-S. Lee and E. Pottier, Polarimetric Radar Imaging: From Basics to Applications (CRC, 2009).
[CrossRef]

Locke, A. M.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Look, D. C.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.  34, 1656–1659 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng.  34, 1651–1655 (1995).
[CrossRef]

Park, R.

Phipps, G. S.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Pottier, E.

J.-S. Lee and E. Pottier, Polarimetric Radar Imaging: From Basics to Applications (CRC, 2009).
[CrossRef]

Sabatke, D. S.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Salyer, D.

Sasian, J. M.

Sassen, K.

K. Sassen, “Polarization in lidar: a review,” Proc. SPIE  5158, 151–160 (2003).
[CrossRef]

Savenkov, S. N.

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng.  41, 965–972 (2002).
[CrossRef]

Shaw, J. A.

Simon, R.

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt.  34, 569–575 (1987).
[CrossRef]

Smith, M. H.

Spaulding, J. C.

D. G. Jones, D. H. Goldstein, and J. C. Spaulding, “Reflective and polarimetric characteristics of urban materials,” Proc. SPIE  6240, 62400A (2006).
[CrossRef]

Sweatt, W. C.

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Twietmeyer, K.

Tyo, J. S.

van Loan, C. F.

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11–29.

Vaughn, I. J.

Wang, Z.

Zhao, Y.

Appl. Opt. (7)

Biometrika (1)

H. Hotelling, “Relations between two sets of variates,” Biometrika  28, 312–377 (1936).

J. Mod. Opt. (1)

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt.  34, 569–575 (1987).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Acta (1)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta  32, 259–261 (1985).
[CrossRef]

Opt. Commun. (1)

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun.  38, 159–161 (1981).
[CrossRef]

Opt. Eng. (3)

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng.  41, 965–972 (2002).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng.  34, 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng.  34, 1656–1659 (1995).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Optik (Stuttgart) (1)

S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttgart)  75, 26–36 (1986).

Proc. SPIE (6)

R. A. Chipman, “Metrics for depolarization,” Proc. SPIE  5888, 58880L (2005).
[CrossRef]

R. A. Chipman, “Degrees of freedom in depolarizing Mueller matrices,” Proc. SPIE  6682, 66820I (2007).
[CrossRef]

D. G. Jones, D. H. Goldstein, and J. C. Spaulding, “Reflective and polarimetric characteristics of urban materials,” Proc. SPIE  6240, 62400A (2006).
[CrossRef]

K. Sassen, “Polarization in lidar: a review,” Proc. SPIE  5158, 151–160 (2003).
[CrossRef]

J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Designing partial Mueller matrix polarimeters,” Proc. SPIE  7461, 74610V (2009).
[CrossRef]

D. S. Sabatke, A. M. Locke, M. R. Descour, W. C. Sweatt, J. P. Garcia, E. Dereniak, S. A. Kemme, and G. S. Phipps, “Figures of merit for complete Stokes polarimeters,” Proc. SPIE  4133, 75–81 (2000).
[CrossRef]

Other (8)

I. Daubecheis, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, 1992), Chap. 3, pp. 53–106.
[CrossRef]

R. A. Chipman, “Polarimetry,” in “Handbook of Optics,” M.Bass, ed. (McGraw-Hill, 1995), Vol.  2, Chap. 22.

J.-S. Lee and E. Pottier, Polarimetric Radar Imaging: From Basics to Applications (CRC, 2009).
[CrossRef]

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

F. Goudail, “Comparative study of the best achievable contrast in scalar, Stokes and Mueller images,” in First NanoCharm Workshop on Advanced Polarimetric Instrumentation (Multifunctional Nanomaterials Characterization Exploiting EllipsoMetry and Polarimetry (NanoCharM), 2009).

In some cases it might be advantageous to generate or analyze partially polarized states. However, the difficulty of doing this in an active system will preclude its use except in specialized situations. The obvious exception to this is the generation or analysis of an unpolarized state. The term m00 can be measured with a single measurement when both generator and analyzer are unpolarized. The elements of the first row m0j can be measured in two measurements by analyzing an unpolarized state. The elements of the first column mi0 can be obtained in two measurements by generating an unpolarized state.

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983), Chap. 2, pp. 11–29.

S. J. Johnson, “Use of partial polarimetry in material discrimination,” Master’s thesis (University of Arizona, Tucson, 2009).

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Figures (3)

Fig. 1
Fig. 1

pMMP with only two receiver channels. This concept uses a single polarizing beam splitter in the analyzer. When coupled with the preceding retarders, the PSA can analyze an arbitrary pair of orthogonally polarized elliptical states.

Fig. 2
Fig. 2

General schematic of a MMP.

Fig. 3
Fig. 3

These two one-dimensional scene spaces have the same angle with respect to the sensor space. However, the space in (a) projects into a direction in sensor space with poor SNR and the scene space in (b) projects into a direction in sensor space with good SNR.

Equations (27)

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

S = [ s 0 s 1 s 2 s 3 ] = [ I 0 I x I y I 45 I 45 I L I R ] .
S o = M S i ,
M = [ 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 ] .
I = S A T M S G .
X = [ I 1 I N ] = [ S A ( 1 ) T M S G ( 1 ) S A ( N ) T M S G ( N ) ] .
S A = [ s 0 s 1 s 2 s 3 ] T ,
S G = [ σ 0 σ 1 σ 2 σ 3 ] T ,
I i j = i , j = 0 3 s i m i j σ j .
Δ i j = S G ( j ) ( S A ( i ) ) T .
I i j = M , Δ i j = Δ i j T M ,
Δ i j = vec ( Δ i j ) T ,
M = vec ( M ) ,
W T M = [ Δ 1 Δ N ] M = X ,
M = W X = [ Z Δ ( 1 ) Z Δ ( N ) ] X = i = 1 N x i Z Δ ( i ) .
( s 1 , out ) 2 + ( s 2 , out ) 2 + ( s 3 , out ) 2 ( s 0 , out ) S i n .
R span { Δ i } i = 1 N 16 .
{ Z Δ i } i = 1 N
cos θ 1 = | e i j Δ 1 | = 1 2 .
s A , i = ± 1 2 , s G , j = ± 1 ,
Σ n = E [ W T n n T W ] = σ 2 W T W ,
σ m ^ 2 = m ^ T Σ n m ^ .
W = U S V T .
SNR = 1 σ 2 u ^ i T Σ N u ^ i = 1 σ 2 u ^ i T U S 1 V T V S 1 U T u ^ = | s i | 2 σ 2 ,
S = [ α i s ^ 1 α M s ^ M ] ,
S = T s V ,
SNR i = s i 2 σ 2 | t i | 2 t i T | t i | Σ N t i | t i | = s i 2 σ 2 [ ( W T W T t ^ i ) T ( W T W T t ^ i ) ] 2 ( W T W T t ^ i ) T W T W ( W T W T t ^ i ) = s i 2 σ 2 [ t ^ i T I W T I W t ^ i ] 2 t ^ i T I W T Σ N I W t ^ i .
I W = W T ( ) W T .

Metrics