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Degree of polarization surfaces and maps for analysis of depolarization

B. DeBoo, J. Sasian, and R. Chipman

Open Access Open Access

Abstract

The concept of degree of polarization surfaces is introduced as an aid to classifying the depolarization properties of Mueller matrices. Degree of polarization surfaces provide a visualization of the dependence of depolarization on incident polarization state. The surfaces result from a non-uniform contraction of the Poincaré sphere corresponding to the depolarization properties encoded in a Mueller matrix. For a given Mueller matrix, the degree of polarization surface is defined by moving each point on the unit Poincaré sphere radially inward until its distance from the origin equals the output state degree of polarization for the corresponding input state. Of the sixteen elements in a Mueller matrix, twelve contribute to the shape of the degree of polarization surface, yielding a complex family of surfaces. The surface shapes associated with the numerator and denominator of the degree of polarization function are analyzed separately. Protrusion of the numerator surface through the denominator surface at any point indicates non-physical Mueller matrices. Degree of polarization maps are plots of the degree of polarization on flat projections of the sphere. These maps reveal depolarization patterns in a manner well suited for quantifying the degree of polarization variations, making degree of polarization surfaces and maps valuable tools for categorizing and classifying the depolarization properties of Mueller matrices.

©2004 Optical Society of America

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Supplementary Material (4)

Media 1: MPG (1330 KB)     
Media 2: MPG (1328 KB)     
Media 3: MPG (967 KB)     
Media 4: MPG (1457 KB)     

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Figures (16)
Fig. 1.
Fig. 1. Degree of polarization surface (a) and degree of polarization map (b) of a typical depolarizing Mueller matrix showing two local maxima, two local minima, and two saddles.

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Fig. 2.
Fig. 2. Family of limaçons. DoP denominator surfaces are limaçons rotated about their axis of symmetry.

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Fig. 3.
Fig. 3. DoP denominator surface for Mueller matrix M0 of Eq. (11). The red line denotes the diattenuation axis, the axis of rotational symmetry of the surface.

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Fig. 4.
Fig. 4. Surface shape and associated cross section after contraction of unit Poincaré sphere by uniform depolarization matrix of type D1 with {a,a,a}={0.7,0.7,0.7}.

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Fig. 5.
Fig. 5. Surface shape (a) and associated cross sections (b), (c), (d), after contraction of unit Poincaré sphere by nonuniform depolarization matrix of type D2 with {a,b,a}={0.6,0.2,0.6}.

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Fig. 6.
Fig. 6. Surface shape (a) and associated cross sections (b), (c), (d), after contraction of unit Poincaré sphere by nonuniform depolarization matrix of type D3 with {a,b,c}={0.4,0.2,0.8}.

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Fig. 7.
Fig. 7. Final shape of surface from interaction of Q1 with incident fully polarized Stokes vectors.

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Fig. 8.
Fig. 8. (1.33 MB) Animation showing the effect of polarizance on the numerator surface for Mueller matrix M0.

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Fig. 9.
Fig. 9. (1.33MB) Animation showing the effect of polarizance on the numerator surface for Mueller matrix M1.

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Fig. 10.
Fig. 10. DoP surface for Mueller matrix M0.

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Fig. 11.
Fig. 11. DoP map for Mueller matrix M0.

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Fig. 12.
Fig. 12. DoP numerator surface (colored) plotted within DoP denominator surface (wireframe). The numerator surface must lie entirely within the denominator surface for physically realizable Mueller matrices, to maintain a DoP less than or equal to one.

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Fig. 13.
Fig. 13. (967 KB) DoP numerator surface plotted within DoP denominator surface, as Mueller matrix parameter “i” of Eq. (24) is increased in increments of 0.1 from zero to 0.7.

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Fig. 14.
Fig. 14. Family of DoP surfaces for LP(0)+LP(jπ/12). The DoP surfaces are plotted within the normalized Poincaré sphere (unit radius) to ensure that the DoP remains physical (radius less than one) throughout the family.

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Fig. 15.
Fig. 15. DoP maps for Mueller matrix family of Eq. (25). Note that the map always has two maxima and one mimimum (except for the degenerate case of a sphere). Also note that the scale changes between plots to better convey information using the same number of contours.

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Fig. 16.
Fig. 16. (1.46 MB) Evolution of DoP surfaces through members of Mueller matrix family in Eq. (25).

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Equations (26)

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DoP ( S ) = S 1 2 + S 2 2 + S 3 2 S 0 .
S ( θ , φ ) = ( 1 Cos ⁡ ( 2 θ ) Cos ⁡ ( φ ) Sin ⁡ ( 2 θ ) Cos ⁡ ( φ ) Sin ⁡ ( φ ) ) ,
DoPSurface ( M , S ) = S 1 ′ ( M , S ) 2 + S 2 ′ ( M , S ) 2 + S 3 ′ ( M , S ) 2 S 0 ′ ( M , S ) ( S 1 , S 2 , S 3 ) ,
M 1 = ( 1.0 − 0.226 0.069 0.196 − 0.030 0.052 0.357 − 0.336 0.069 − 0.454 − 0.266 − 0.194 0.196 − 0.336 0.194 0.584 ) .
( S 0 ′ S 1 ′ S 2 ′ S 3 ′ ) = ( m 00 m 01 m 02 m 03 m 10 m 11 m 12 m 23 m 20 m 21 m 22 m 23 m 30 m 31 m 32 m 33 ) · ( S 0 S 1 S 2 S 3 ) = ( m 00 S 0 + m 01 S 1 + m 02 S 2 + m 03 S 3 m 10 S 0 + m 11 S 1 + m 12 S 2 + m 13 S 3 m 20 S 0 + m 21 S 1 + m 22 S 2 + m 23 S 3 m 30 S 0 + m 31 S 1 + m 32 S 2 + m 33 S 3 ) .
DoP Surface ( M , S ) = DoP Numerator DoP Deno min ator = ( ∑ k = 1 3 ( m k 0 S 0 + m k 1 S 1 + m k 2 S 2 + m k 3 S 3 ) 2 ) 1 2 m 00 S 0 + m 01 S 1 + m 02 S 2 + m 03 S 3 .
S 0 ′ ( M , S ) = m 00 S 0 + m 01 S 1 + m 02 S 2 + m 03 S 3 ,
DoPDenominator ( M , S ) = m 00 + m 01 Cos ⁡ ( 2 θ ) Cos ⁡ ( φ ) + m 02 Sin ⁡ ( 2 θ ) Cos ⁡ ( φ ) + m 03 Sin ⁡ ( φ ) .
r = 1 + ξ Cos ⁡ ( α ) , ( 0 ≤ α < 2 π ) ,
r = 1 + ξ Cos ⁡ ( cα ) , ( 0 ≤ α < 2 π ) ,
M 0 = 1 2 ( 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ) + 1 2 ( 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ) = ( 1 0.5 0.5 0 0 . 5 0.5 0 0 0 . 5 0 0.5 0 0 0 0 0 ) ,
DoP Denominator ( M 0 , S ) = 1 + 0.5 Cos ⁡ ( 2 θ ) Cos ⁡ ( φ ) + 0.5 Sin ⁡ ( 2 θ ) Cos ⁡ ( φ ) ,
( ( 2 θ ) axis , ϕ axis ) = ( ArcCos [ m 01 m 01 2 + m 02 2 ] , ArcSin [ m 03 m 01 2 + m 02 2 + m 03 2 ] ) .
d = m 01 2 + m 02 2 + m 03 2 m 00 = T max − T min T max + T min ,
( … … … … m 10 m 11 m 12 m 13 m 20 m 21 m 22 m 23 m 30 m 31 m 32 m 33 ) = ( … … … … m 10 0 0 0 m 20 0 0 0 m 30 0 0 0 ) + ( … … … … 0 m 11 m 12 m 13 0 m 21 m 22 m 23 0 m 31 m 32 m 33 ) .
P ( M ) = M · ( 1 0 0 0 ) = ( m 00 m 10 m 20 m 30 ) .
Q = ( m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ) ,
Q = HU = U ′ H ′ .
H = U 1 D U 1 − 1 or H ′ = U 2 − 1 D U 2 ,
D 1 = ( a 0 0 0 a 0 0 0 a ) ; D 2 = ( a 0 0 0 a 0 0 0 b ) ; D 3 = ( a 0 0 0 b 0 0 0 c ) .
UD = ( 1 0 0 0 0 a 0 0 0 0 a 0 0 0 0 a )
Q = ( U 1 D U 1 − 1 ) U 3 = U 0 ( U 2 − 1 D U 2 ) = U 1 D U 2 .
Q 1 = ( 0.4224 0.2741 0.8640 − 0.0474 − 0.9452 0.3230 − 0.9052 0.1774 0.3863 ) • ( 0.7477 0 0 0 0.5769 0 0 0 0.3981 ) • ( 0.4652 − 0.0166 − 0 . 8850 0.6665 0.6645 0.3379 − 0.5825 0.7471 − 0.3202 )
= U 1 D U 2 ,
M ′ = ( 1 i i 0 i i 0 0 i 0 i 0 0 0 0 0 ) .
M j = LP ( 0 ) + LP ( j π 12 ) , j ∊ { 0,1 , … 11 }
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