Abstract

In this paper, we consider the forward problem in depolarization by optical systems. That is, we seek a compact parameterization that allows us to take an arbitrary “pure” optical system (namely one defined by a single Mueller–Jones matrix) and model all possible ways in which that system can depolarize light. We model this structure using compound unitary transformations and illustrate physical interpretation of the parameters involved by considering four examples, the family of depolarizers generated by scattering by random nonspherical particle clouds. We then turn attention to circular polarizers before considering all ways in which mirror reflection can cause depolarization. Finally, we consider a numerical example applied to a published Mueller matrix for backscatter from chiral turbid media.

© 2013 Optical Society of America

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References

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  1. J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  2. M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).
  3. J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.
  4. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
    [CrossRef]
  5. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
    [CrossRef]
  6. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
    [CrossRef]
  7. R. Ossikovski, M. Foldyna, C. Fallet, and A. De Martino, “Experimental evidence for naturally occurring nondiagonal depolarizers,” Opt. Lett. 34, 2426–2428 (2009).
    [CrossRef]
  8. R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010).
    [CrossRef]
  9. S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).
  10. M. Zerrad, J. Sorrentini, G. Soriano, and C. Amra, “Gradual loss of polarization in light scattered from rough surfaces: electromagnetic prediction,” Opt. Express 18, 15832–15843 (2010).
    [CrossRef]
  11. S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).
  12. F. D. Murnaghan, The Unitary and Rotation Groups (Spartan Books, 1962).
  13. S. R. Cloude, “Entropy of the Amsterdam light scattering database,” J. Quant. Spectrosc. Radiat. Transfer 110, 1665–1676 (2009).
    [CrossRef]
  14. I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
    [CrossRef]
  15. S. Manhas, M. K. Swami, P. Buddhiwant, P. K. Gupta, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190–202 (2006).
    [CrossRef]

2011

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

2010

2009

2007

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
[CrossRef]

2006

1996

1986

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

Amra, C.

Buddhiwant, P.

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “Entropy of the Amsterdam light scattering database,” J. Quant. Spectrosc. Radiat. Transfer 110, 1665–1676 (2009).
[CrossRef]

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

S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).

De Martino, A.

Domke, H.

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

Fallet, C.

Foldyna, M.

Gil, J. J.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

Gupta, P. K.

Guyot, S.

Hovenier, J. W.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

Lu, S.-Y.

Manhas, S.

Mischenko, M. I.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

Murnaghan, F. D.

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan Books, 1962).

Ossikovski, R.

San José, I.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

Singh, K.

Soriano, G.

Sorrentini, J.

Swami, M. K.

Travis, L. D.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

van der Mee, C.

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

Zerrad, M.

Eur. Phys. J. Appl. Phys.

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

S. R. Cloude, “Entropy of the Amsterdam light scattering database,” J. Quant. Spectrosc. Radiat. Transfer 110, 1665–1676 (2009).
[CrossRef]

Opt. Commun.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

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

Other

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres (Kluwer Academic, 2004), Vol. 318.

S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan Books, 1962).

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Equations (45)

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[M]=[m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33]m̲=[m00m10m33].
t̲=[Q]m̲m̲=[Q]1t̲.
[Q]=12[Q1Q2Q3Q4Q2Q1iQ4iQ3Q3iQ4Q1iQ2Q4iQ3iQ2Q1][Q]1=[Q]*TQ1=[1000010000100001]Q2=[01001000000i00i0]Q3=[0010000i10000i00]Q4=[000100i00i001000].
t̲=[t00t10t33][T]=[t00t01t02t03t01*t11t12t13t02*t12*t22t23t03*t13*t23*t33]=[U4][D][U4]*T.
[J]=[j11j12j21j22]=k0[1001]+k1[1001]+k2[0110]+k3[0ii0]kiC.
k̲=[k0k1k2k3]=a[cosαeiδ1cosβsinαeiδ2cosγsinβsinαeiδ3sinγsinβsinαeiδ4]=ae̲e̲*Te̲=1,
[J]=[j11j12j21j22]j11=a(cosαeiδ1+cosβsinαeiδ2),j12=asinβsinα(cosγeiδ3+isinγeiδ4),j21=asinβsinα(cosγeiδ3isinγeiδ4),j22=a(cosαeiδ1cosβsinαeiδ2).
[T1]=λ1e̲1e̲1*T.
[TD]=λ1e̲1e̲1*T+(λ2e̲2e̲2*T+λ3e̲3e̲3*T+λ4e̲4e̲4*T).
k̲=[k0k1k2]=λ1[cosαeiδ1cosβsinαeiδ2sinβsinαeiδ3]=λ1e̲1.
[U3R]e̲1=[100].
[U3R]=[cosαsinα0sinαcosα0001][1000cosβsinβ0sinβcosβ]·[eiδ1000eiδ2000eiδ3].
[U3R]=[e̲1*Te̲2r*Te̲3r*T][U3R]*T=[e̲1e̲2re̲3r]=[cosαeiδ1sinαeiδ10cosβsinαeiδ2cosβcosαeiδ2sinβeiδ2sinβsinαeiδ3sinβcosαeiδ3cosβeiδ3],
[U3R]*T=[e̲1e̲2re̲3r]=[cosαeiδ1sinαeiδ10sinαeiδ2cosαeiδ20001].
[U3R][U3]=[10̲T0̲U2].
U2(ϕ,σ)=[cosϕsinϕeiσsinϕeiσcosϕ]{0ϕπ2πσ<π.
[U3(ϕ,σ)]=[U3R]*T[10̲T0̲U2(ϕ,σ)].
[TD]=[U3(ϕ,σ)][λ1000λ2000λ3][U3(ϕ,σ)]*Tλ1λ2,30R[U3(ϕ,σ)]=[e̲1e̲2e̲3]e̲1=[cosαeiδ1cosβsinαeiδ2sinβsinαeiδ3]e̲2=[cosϕsinαeiδ1cosϕcosβcosαeiδ2sinϕsinβei(σ+δ2)cosϕsinβcosαeiδ3+sinϕcosβei(σ+δ3)]e̲3=[sinϕsinαei(δ1σ)cosϕsinβeiδ2sinϕcosβcosαei(δ2σ)cosϕcosβeiδ3sinϕsinβcosαei(δ3σ)].
[U4R]e̲1=[1000]T.
[U4R]=[cosαsinα00sinαcosα0000100001][10000cosβsinβ00sinβcosβ00001]·[1000010000cosγsinγ00sinγcosγ][eiδ10000eiδ20000eiδ30000eiδ4].
[U4R]=[e̲1*Te̲2*Te̲3*Te̲4*T][U4R]*T=[e̲1e̲2e̲3e̲4]e̲1=[cosαeiδ1cosβsinαeiδ2cosγsinβsinαeiδ3sinγsinβsinαeiδ4],e̲2=[sinαeiδ1cosβcosαeiδ2cosγsinβcosαeiδ3sinγsinβcosαeiδ4]e̲3=[0sinβeiδ2cosγcosβeiδ3sinγcosβeiδ4],e̲4=[00sinγeiδ3cosγeiδ4],
[U4R][U4]=[10̲T0̲U3][U4]=[U4R]*T[10̲T0̲U3],
U3(ϕi,σi)=U23(ϕ3,σ3)U212(ϕ2,σ2)U13(ϕ1,σ1)U13=[cosϕ10sinϕ1eiσ1010sinϕ1eiσ10cosϕ1]U12=[cosϕ2sinϕ2eiσ20sinϕ2eiσ2cosϕ20001]U23=[1000cosϕ3sinϕ3eiσ30sinϕ3eiσ3cosϕ3].
[TD]=[U4(ϕi,σi)][λ10000λ20000λ30000λ4][U4(ϕi,σi)]*Tλ1λ2,3,40R,[U4(ϕi,σi)]=[e̲1e̲2e̲3e̲4]=[U4R]*T[10̲T0̲U3(ϕi,σi)],U3(ϕi,σi)=[1000cosϕ3sinϕ3eiσ30sinϕ3eiσ3cosϕ3]·[cosϕ2sinϕ2eiσ20sinϕ2eiσ2cosϕ20001]·[cosϕ10sinϕ1eiσ1010sinϕ1eiσ10cosϕ1].
[TD]=[U4(α,δ)][λ10000λ20000λ30000λ4][U4(α,δ)]*T[U4(α,δ)]=[cosαsinαeiδ00sinαeiδcosα0000100001].
[TD]=[U4(α,δ)][λ1λ2000000000000000][U4(α,δ)]*T+[λ20000λ20000λ30000λ4]=[T1]+[T2],
[M]=[M1]+[MN]=m[1sin2αcosδ00sin2αcosδ10000cos2αsin2αsinδ00sin2αsinδcos2α]+[n000000n110000n220000n33],m=12(λ1λ2),n00=12(2λ2+λ3+λ4),n11=12(2λ2λ3λ4)n22=12(λ3λ4),n33=12(λ3λ4).
[J]=[U2][D][U2]*T=[cosψsinψeiδsinψeiδcosψ][1000][cosψsinψeiδsinψeiδcosψ]=[cos2ψcosψsinψeiδcosψsinψeiδsin2ψ].
k̲=12[1cos2ψsin2ψcosδsin2ψsinδ]{α=45°δ1=δ2=δ3=δ4=0{β=ψγ=δ.
[e̲1e̲2e̲3]=12[1cosϕsinϕeiσ02sinϕeiσ12cosϕ±1±cosϕsinϕeiσ].
[M1]=[100±100000000±1001],[M2]=[112sin2ϕcosσ12sin2ϕsinσcos2ϕ12sin2ϕcosσsin2ϕ012sin2ϕcosσ12sin2ϕsinσ0sin2ϕ12sin2ϕsinσcos2ϕ12sin2ϕcosσ12sin2ϕsinσcos2ϕ],[M3]=[112sin2ϕcosσ12sin2ϕsinσsin2ϕ12sin2ϕcosσcos2ϕ012sin2ϕcosσ12sin2ϕsinσ0cos2ϕ12sin2ϕsinσsin2ϕ12sin2ϕcosσ12sin2ϕsinσcos2ϕ].
[M]=i=13λi[Mi]=12[m00m01m02m03m01m11m12m13m02m12m22m23m03m13m23m33],m00=λ1+λ2+λ3,m01=12sin2ϕcosσ(λ2λ3),m02=12sin2ϕsinσ(λ2λ3),m03=λ1λ2cos2ϕλ3sin2ϕ,m11=λ2sin2ϕ+λ3cos2ϕ,m12=0,m13=12sin2ϕcosσ(λ3λ2),m22=λ2sin2ϕλ3cos2ϕ,m23=12sin2ϕsinσ(λ2λ3),m33=λ1+cos2ϕ(λ2λ3),
[M]=12[λ1+λ2+λ300λ1λ20λ30000λ30λ1λ200λ1+λ2λ3]=12[λ100λ100000000λ100λ1]+12[λ200λ200000000λ200λ2]+[λ30000λ30000λ30000λ3].
k̲=[cosαsinα0].
[e̲1e̲2e̲3]=[cosαcosϕsinαsinϕsinαeiσsinαcosϕcosαsinϕcosαeiσ0sinϕeiσcosϕ].
[M1]=[1sin2α00sin2α10000cos2α0000cos2α],[M2]=[1sin2αcos2ϕcosαsin2ϕsinσsinαsin2ϕcosσsin2αcos2ϕcos2ϕsinαsin2ϕsinσcosαsin2ϕcosσcosαsin2ϕsinσsinαsin2ϕsinσsin2ϕcos2ϕcos2α0sinαsin2ϕcosσcosαsin2ϕcosσ0sin2ϕcos2ϕcos2α],[M3]=[1sin2αsin2ϕcosαsin2ϕsinσsinαsin2ϕcosσsin2αsin2ϕcos2ϕsinαsin2ϕsinσcosαsin2ϕcosσcosαsin2ϕsinσsinαsin2ϕsinσcos2ϕsin2ϕcos2α0sinαsin2ϕcosσcosαsin2ϕcosσ0cos2ϕsin2ϕcos2α].
[M1]=[1000010000100001],[M2]=[100sin2ϕcosσ0cos2ϕsin2ϕsinσ00sin2ϕsinσcos2ϕ0sin2ϕcosσ001],[M3]=[100sin2ϕcosσ0cos2ϕsin2ϕsinσ00sin2ϕsinσcos2ϕ0sin2ϕcosσ001],
[M]=i=13λi[Mi]=12[m00m01m02m03m01m11m12m13m02m12m22m23m03m13m23m33],m00=λ1+λ2+λ3,m01=m02=m13=m23=0,m03=sin2ϕcosσ(λ3λ2),m11=λ1+cos2ϕ(λ2λ3),m12=sin2ϕsinσ(λ3+λ2),m22=λ1+cos2ϕ(λ2λ3),m33=λ2+λ3λ1,
[M]=12[λ1+λ2+λ30000λ1+λ2λ30000λ2λ3λ10000λ2+λ3λ1],
[T]=[λ2000λ1000λ3].
[M]=[1.0000.1150.0660.0230.1110.7590.0610.0010.0180.1510.4350.1390.0460.0060.1280.334].
abs([T])=[0.4950.1750.0420.1070.1751.2640.0570.0240.0420.0570.0700.0060.1070.0240.0060.171][0.4950.17500.1070.1751.26400.02400000.1070.02400.171].
[M]=[1.0000.1170.0250.0120.1170.8230.1100.0030.0250.1100.4870.1380.0120.0030.1380.310]λ=1,0.378,0.104.
e̲1=[0.2130.622i0.7530.020i0.008][M1]=[1.0000.2650.0190.0080.2650.9990.0040.0370.0190.0040.9090.3210.0080.0370.3210.908].
[U3]=[10000.9490.147i0.28000.147i0.2800.949]{ϕ=18.45°σ=62.30°.

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