Abstract

We optimize a general class of complete multispectral polarimeters with respect to signal-to-noise ratio, stability against alignment errors, and the minimization of errors regarding a given set of polarization states. The class of polarimeters that are dealt with consists of at least four polarization optics each with a multispectral detector. A polarization optic is made of an azimuthal oriented wave plate and a polarizing filter. A general, but not unique, analytic solution that minimizes signal-to-noise ratio is introduced for a polarimeter that incorporates four simultaneous measurements with four independent optics. The optics consist of four sufficient wave plates, where at least one is a quarter-wave plate. The solution is stable with respect to the retardance of the quarter-wave plate; therefore, it can be applied to real-world cases where the retardance deviates from λ/4. The solution is a set of seven rotational parameters that depends on the given retardances of the wave plates. It can be applied to a broad range of real world cases. A numerical method for the optimization of arbitrary polarimeters of the type discussed is also presented and applied for two cases. First, the class of polarimeters that were analytically dealt with are further optimized with respect to stability and error performance with respect to linear polarized states. Then a multispectral case for a polarimeter that consists of four optics with real achromatic wave plates is presented. This case was used as the theoretical background for the development of the Airborne Multi-Spectral Sunphoto- and Polarimeter (AMSSP), which is an instrument for the German research aircraft HALO.

© 2009 Optical Society of America

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References

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  1. A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus” (2006), arXiv:math-ph/0412061v3.
  2. R. C. Jones, “A new calculus for the treatment of optical systems V. A more general formulation, and description of another calculus,” J. Opt. Soc. Am. 37, 107-110 (1947).
    [CrossRef]
  3. E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  5. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802-804 (2000).
    [CrossRef]
  6. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34, 1651-1655 (1995).
    [CrossRef]
  7. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41, 619-630 (2002).
    [CrossRef] [PubMed]
  8. J. S. Tyo, “Optimum linear combination strategy for an n-channel polarization-sensitive imaging or vision system,” J. Opt. Soc. Am. A 15, 359-366 (1998).
    [CrossRef]
  9. R. Preusker and T. Ruhtz, “Airborne Multi-Spectral Sunphoto- & Polarimeter (AMSSP),” Deutschen Forschungsgemeinschaft proposal.
  10. T. Ruhtz, “Development Guidelines URMS/AMSSP Version 1.01, 7.” (Free University Berlin, Institute for Space Sciences, 2008), thomas.ruhtz@fu-berlin.de
  11. ITTVIS, “IDL,” http://www.ittvis.com.

2002 (1)

2000 (1)

1998 (1)

1995 (1)

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

1947 (1)

Aiello, A.

A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus” (2006), arXiv:math-ph/0412061v3.

Ambirajan, A.

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Dereniak, E. L.

Descour, M. R.

Jones, R. C.

Kemme, S. A.

Look, D. C.

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

Mandel, L.

E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Phipps, G. S.

Preusker, R.

R. Preusker and T. Ruhtz, “Airborne Multi-Spectral Sunphoto- & Polarimeter (AMSSP),” Deutschen Forschungsgemeinschaft proposal.

Ruhtz, T.

R. Preusker and T. Ruhtz, “Airborne Multi-Spectral Sunphoto- & Polarimeter (AMSSP),” Deutschen Forschungsgemeinschaft proposal.

T. Ruhtz, “Development Guidelines URMS/AMSSP Version 1.01, 7.” (Free University Berlin, Institute for Space Sciences, 2008), thomas.ruhtz@fu-berlin.de

Sabatke,

Sweatt, W. C.

Tyo, J. S.

Woerdman, J. P.

A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus” (2006), arXiv:math-ph/0412061v3.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

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

Opt. Lett. (1)

Other (6)

A. Aiello and J. P. Woerdman, “Linear algebra for Mueller calculus” (2006), arXiv:math-ph/0412061v3.

E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

R. Preusker and T. Ruhtz, “Airborne Multi-Spectral Sunphoto- & Polarimeter (AMSSP),” Deutschen Forschungsgemeinschaft proposal.

T. Ruhtz, “Development Guidelines URMS/AMSSP Version 1.01, 7.” (Free University Berlin, Institute for Space Sciences, 2008), thomas.ruhtz@fu-berlin.de

ITTVIS, “IDL,” http://www.ittvis.com.

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Figures (8)

Fig. 1
Fig. 1

Sketch of the instrument design with four entrance optics.

Fig. 2
Fig. 2

Four points on the surface of the Poincaré sphere that represent a polarimeter with minimal condition number. The polarimeter consists of four equal quarter-wave plates and four polarizing filters.

Fig. 3
Fig. 3

Optimal solution for a polarimeter with one 1 / 4 and three 1 / 6 wave plates; δ = π / 3 and φ = 1 2 sin 1 ( 1 3 sin ( π / 3 ) ) . The surface of the Poincaré sphere is drawn using the Mollweide projection.

Fig. 4
Fig. 4

Offset Δ θ ( δ ) from the original values θ i = 0 ° , 60 ° , 120 ° for arbitrary sufficient wave plates.

Fig. 5
Fig. 5

Behavior of the condition number with respect to the two angular parameters of one optical setup. The marked point represents a solution with minimal condition number for all parameters. Their values can be found in Table 1.

Fig. 6
Fig. 6

Numerically obtained result for an optimal polarimeter with additional minimization of measurement errors for fully linear polarized states. The Arabic numbers indicate the number of the parameter set from Table 1.

Fig. 7
Fig. 7

(a) Norm of the error vector ϵ s for the numerically obtained result and (b) the analytic solution Eq. (10) from Section 3.

Fig. 8
Fig. 8

Multispectral measurement of the spectral dependency of the four achromatic wave plates used.

Tables (2)

Tables Icon

Table 1 Numerically Obtained Values for the Polarimeter Shown in Fig. 6 a

Tables Icon

Table 2 Optimized Parameters for the Multispectral Case

Equations (20)

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S = J S J .
P ( θ ) R ( θ ) F R ( θ ) = ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) ( 1 0 0 0 ) ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) = ( cos 2 ( θ ) cos ( θ ) sin ( θ ) cos ( θ ) sin ( θ ) sin 2 ( θ ) ) .
W ( θ , δ ) R ( θ ) W ( δ ) R ( θ ) = R ( θ ) ( 1 0 0 e i δ ) R ( θ ) .
O ( δ , φ p , φ w p ) P ( φ p ) W ( φ p + φ w p , δ ) .
I = T r ( O S O ) O S .
O = 1 2 ( 1 sin ( 2 φ p ) cos 2 ( δ 2 ) + sin 2 ( δ 2 ) sin ( 2 ( φ p + 2 φ w ) ) 2 cos ( φ w ) sin ( δ ) sin ( φ w ) cos ( 2 φ p ) cos 2 ( δ 2 ) + cos ( 2 ( φ p + 2 φ w ) ) sin 2 ( δ 2 ) ) .
I = A S S = A 1 I .
c ( x ) = x x 1 .
0 = φ V sin ( δ ) cos ( 2 φ ) [ sin ( δ ) sin ( 2 φ ) + 1 ] [ 3 sin ( δ ) sin ( 2 φ ) 1 ] .
φ = 1 2 sin 1 ( 1 3 sin ( δ ) 1 ) .
A = ( 1 2 0 1 2 0 1 2 0 0 1 2 1 2 3 4 0 1 4 1 2 3 4 0 1 4 ) .
O ( θ = 0 , δ = π 2 , φ = 1 2 sin 1 ( 1 3 ) ) = O ( θ , δ i , φ = 1 2 sin 1 ( 1 3 sin ( δ i ) ) ) .
Δ = 1 + | ϵ ( b c ( A ) ) T | c ( b ) .
f ( δ λ , p ) = 1 2 λ i 1 g λ i | ( c ( A λ ) 3 , Δ ( A λ ) ) T | 2 .
ϵ S ( S ) = { A 1 ( i ( ϵ b i b i A ) ) } S .
s i ( x + ϵ ) s i ( x ) + ( x s i ( x ) ) T ( x + ϵ x ) = s i ( x ) + ( x s i ( x ) ) T ϵ .
ϵ S = S ( x + ϵ ) S ( x ) G b ϵ b systematic + G I ϵ I random .
ϵ R 7 + 4 + 4 = 15 ,     G b R 4 × 11 ,     ϵ b R 11 ,     G I R 4 x 4 ,     ϵ I R 4 ,
[ G b ] i j = b j S i ( b , I ) = b j [ A 1 ( b ) I ] i ,
[ G I ] i j = I j S i ( b , I ) = I j [ A 1 ( b ) I ] i .

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