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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
1. Inroduction
The transformation from an incident polarization state into an exiting polarization state which occurs during a linear interaction between light and matter (e.g. transmission through a retarder or polarizer, or reflection from a thin film) is commonly described using one of two polarization calculi, the Jones calculus or the Mueller calculus. The Jones calculus, which contains absolute amplitude and phase information, is more useful for describing fully polarized light and non-depolarizing optical devices [1–3]. All Jones matrices are also expressible as Mueller matrices and these non-depolarizing Mueller matrices are well understood [4, 5]. The Mueller calculus is a more general representation for polarization interactions, which applies to incoherent states and describes polarized, partially polarized, or unpolarized light and can quantify depolarization. Among its other properties, a Mueller matrix describes how incident polarized light is depolarized for any incident state and describes the variations of depolarization with polarization state which occur. A single Jones matrix cannot describe depolarization. Sets of Jones matrices can describe depolarization (e.g., Jones matrices as a function of wavelength or position) but such Jones matrices cannot be measured in a straightforward and convenient manner to characterize the depolarization of an optical element, liquid crystal cell, scattering surface, or other depolarizing sample [6, 7]. Thus Mueller matrices are almost always used to characterize depolarization
Depolarization is the reduction of the degree of polarization of light. In the Mueller calculus depolarization can be pictured as a coupling of polarized into unpolarized light, where polarized light is incident and the exiting Stokes vector can be mathematically separated into a fully polarized and an unpolarized Stokes vector. Depolarization of some optical devices has been described (e.g. liquid crystals [8]), and commercial depolarizers are available. However, reports of depolarization measurements in the literature (see, e.g., [9–20]) have been relatively limited. This limited number of published depolarization measurements may in part be due to the only recent commercialization of Mueller matrix polarimeters, which offer the most straightforward way to measure depolarization. Lenses, mirrors, filters, and other typical optical elements exhibit very small amounts of depolarization, typically less than a few tenths of a percent. In contrast, the depolarization of most diffusely reflecting objects such as paints, metal and wood surfaces, natural materials, etc, is significant, varying from a few percent to 100% (i.e. complete depolarization).
A single-valued depolarization metric, the depolarization index, has been introduced to describe the degree to which a Mueller matrix depolarizes incident states [7, 21]. However, such a single number metric cannot describe the full complexity of depolarization associated with a Mueller matrix.
This capability of Mueller matrices to define depolarization for all incident states is useful, particularly when strong depolarization occurs [22]. Depolarization is associated with a reduction in the degree of polarization (DoP) of incident states. Here, DoP surfaces and maps are introduced to quantify and visualize the complete depolarizing properties of a Mueller matrix for all incident states.
Practically, DoP is a measure of the randomness of polarization in a light beam, a property characterized by how much of this beam may be blocked by a polarizer. Mathematically, on the Poincaré sphere, the DoP represents the distance of a normalized Stokes vector’s last three components from the origin. The surface of the unit Poincaré sphere has DoP =1 and represents all fully polarized states [23]. A depolarizing interaction causes fully polarized Stokes states on the surface of the Poincaré sphere to emerge with DoP≤1. The DoP surface plots these exiting DoP values along the same radial vectors from the origin that define the corresponding input states. A similar plot was discussed in a different context by Williams [10]. A DoP map is a two-dimensional contour plot of the DoP surface versus two coordinates defining input polarization states (i.e. polarization ellipse orientation and ellipticity).
This paper investigates the way that a general depolarizing Mueller matrix affects DoP surface and map shapes, thus providing new methods for classifying Mueller matrices and their behaviors. The DoP surfaces are explored by examining separately the changes imparted to the surface shape by the diattenuation, retardance, polarizance, and depolarization properties encoded in a Mueller matrix. DoP maps and surfaces are introduced in Section 2. The denominator of the DoP function, which contains diattenuation properties of the Mueller matrix, is treated in Section 3. The retardance and part of the depolarizing properties of the Mueller matrix are covered in Section 4, via a singular valued decomposition, while Section 5 addresses the additional depolarization effect of polarizance on the DoP surface. Section 6 presents an analysis of complete DoP surfaces and maps. Section 7 addresses the physical realizability of a Mueller matrix using a criterion related to the DoP numerator and denominator. Section 8 explores a family of Mueller matrices and the evolution of their DoP surfaces and maps to better understand how features of each plot relate to specific portions of the Mueller matrix.
Describing depolarization using the DoP of polarized states offers significant advantages over single-valued methods. DoP surfaces and maps provide an insightful visualization of how a Mueller matrix depolarizes all incident polarized states. The information content of DoP surfaces and maps is greater than other depolarization metrics. For example, maxima and minima in the output DoP are readily observed with DoP surfaces, and the values of these extremes easily quantified with DoP maps. DoP surfaces and maps are thorough in describing the behaviors of depolarizing Mueller matrices and are examined in detail here.
2. The degree of polarization
The degree of polarization (DoP) characterizes the randomness of a polarization state. The DoP of Stokes vector S=(S0,S1,S2,S3) is
When DoP =0, the light is unpolarized and all ideal polarizers block half the beam. When DoP =1, the beam is completely polarized and some ideal polarizer, either linear, circular, or elliptical, will completely block the beam. Thus, (1+ DoP)/2 is the fraction of a beam that can be blocked by an ideal polarizer.
Normalized Stokes vectors with DoP = 1 are fully polarized states which lie on the surface of the unit Poincaré sphere. The Stokes vectors on the surface of the Poincaré sphere can be parameterized as
where θ is the polarization orientation (one half the longitude on a globe), and φ is the latitude on the Poincaré sphere. For example, (θ, φ)=(0°, 0°) is horizontal linearly polarized light, (45°, 0°) is 45° polarized light, and (θ, 90°) represents right circularly polarized light for all θ. The Degree of Circular Polarization (DoCP) of a Stokes vector is defined as DoCP = S3/S0 = Sin(φ). A circular polarizer will block (1+|DoCP|)/2 of a beam; the polarizer’s helicity is right or left depending on whether the state is in the upper or lower hemisphere of the Poincaré sphere, respectively.
3. Degree of polarization surfaces and maps
The DoP surface for a Mueller matrix, M, is formed by moving normalized Stokes vectors, S, on the surface of the Poincaré sphere radially inward to a distance DoP(S′=M-S) from the origin, plotted for all incident S on the surface of the Poincaré sphere, given in Eq. (2). The DoP surface results from the product of a scalar, the DoP, and a vector, (S1, S2, S3), formed from the last three elements of the normalized Stokes vector,
for all (++)1/2=1.
The DoP map for a Mueller matrix is a contour plot of the DoP of exiting light as a function of the incident polarized state and represents a “flattened” DoP surface. In this paper, the DoP map is plotted with axes θ (polarization ellipse major axis orientation) and DoCP, but there is some flexibility in the choice of parameterization of the polarized Stokes vectors. In general the DoP map provides easier visualization of maxima, minima, saddles, and other features of the depolarization variation than the DoP surface.
Figure 1 shows a DoP surface and its corresponding DoP map for an example Mueller matrix with depolarization,
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.
The distance from the origin to the DoP surface is the output DoP for the corresponding incident state on the Poincaré sphere. Remember that the output polarization state is in general different from the input polarization state, and this output state information is not contained in the DoP Surface. Where the surface is pinched toward the origin, those incident Stokes states are more depolarized by the Mueller matrix. In this example, (160°, 0.1) is depolarized the most and (175°, -0.7) is depolarized the least.
The DoP surface is analyzed here by decomposing the surface into terms relating to specific polarization properties of the Mueller matrix. The output Stokes vectors for M as a function of the input Stokes vector, S, are
Substituting Eq. (5) into Eq. (3), the scalar part of the DoP surface takes the rather cumbersome form
4. The denominator of the DoP surface
Comparing Eqs. (3) and (6) shows that the denominator of the DoP surface function is
the output flux, S 0′, for input S. Note that S 0′ depends only on the first row of M. The m00 element is the intensity throughput of the Mueller matrix for unpolarized light and describes losses associated with absorption, reflection or scattering. The elements m01, m02, and m03 characterize the diattenuation, the tendency of M to act as a partial polarizer [23].
The shape of S 0′(M,S), the denominator surface, is a limaÇon of revolution [24],
The simplest polar form of a limaÇon is
plotted in Fig. 2 as ξ varies. LimaÇons belong to a more general family of curves, the botanic curves, with polar form
where “c” describes the number of “petals” of the botanic curve. Limaçons are botanic curves of order c=1 with only a single petal.
Fig. 2. Family of limaçons. DoP denominator surfaces are limaçons rotated about their axis of symmetry.
Consider the depolarizing Mueller matrix M 0 constructed from the sum of horizontal linear polarizer and 45° linear polarizer Mueller matrices,
which physically corresponds to covering an aperture with equal fractions of the two polarizers. The denominator of the DoP surface for M0 is
which represents the intensity transmittance for any input Stokes vector, and is plotted in Fig. 3. Note the dimple in the limaçon shape corresponding to minimum transmittance, which occurs along the axis of rotational symmetry of the denominator surface. This axis of symmetry defines the diattenuation axis through the Poincaré sphere.
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.
The Mueller matrix elements m01, m02, and m03 define three degrees of freedom, two of which determine the diattenuation axis orientation and latitude as
The diattenuation axis passes through the maximum and the minimum (dimple) of the DoP denominator surface, as well as the Poincaré sphere points yielding these transmittance extema, Tmax=((2θ)axis, ϕaxis) and Tmin= (π+(2θ)axis, -ϕaxis), respectively. The third degree of freedom defined by Mueller elements m01, m02, and m03 describes the diattenuation (also “diattenuation magnitude”), d, as
which describes the degree to which M is a partial polarizer, given by the maximum and minimum transmittance [23]. For ideal polarizers (d =1), the dimple in the DoP denominator surface becomes a cusp with its point at the origin.
5. The degree of polarization numerator
The DoP numerator surface, described in Eq. (6), is more complex than the denominator surface because it depends on S1′,S2′, and S3′ and includes a square root of the sum of three terms. The DoP numerator depends on the twelve elements in the bottom three rows of the Mueller matrix, which encode the polarizance, retardance, and depolarization properties of the matrix [7, 23].
Consider the last three Mueller matrix rows separated into the sum of two matrices:
The first column of M is the polarizance vector, P, the output Stokes vector when unpolarized light is incident,
The lower right 3×3 submatrix of M,
is an orthogonal matrix (real, unitary [25]) for retarders. Diattenuators, as well as depolarizers (uniform and non-uniform), generate Hermitian [25] Q.
Submatrix Q can be decomposed into the product of a unitary matrix and a Hermitian matrix in two ways [26, 27],
Unitary matrices represent rotations and preserve magnitude (i.e. impart no contraction or expansion to a surface). Retarders rotate the Poincaré sphere about the fast and slow axis of the retarder, so Mueller matrices and Q matrices for retarders are unitary.
The Hermitian component, H or H’, of submatrix Q (see Eq. (18)) is diagonalizable into the form of a similarity transformation [25]
where U1 and U2 are unitary and D is a diagonal matrix of H’s eigenvalues. The three diagonal elements of D operate on a sphere to contract the sphere along three orthogonal coordinate axes. The amount of contraction, 1-DoP(S′), is related to the amount of depolarization. Consider the three cases D1 , with all diagonal elements equal, D2 , with two diagonal elements equal, and D3 , with all three diagonal elements different,
D1 shrinks the Poincaré sphere uniformly into a smaller sphere. The corresponding Mueller matrix, UD, given by
is termed a uniform depolarizer Mueller matrix since UD depolarizes all incident states equally. D2 and D3 indicate nonuniform contractions along orthogonal axes associated with nonuniform depolarization. Figures 4–6 show the depolarization-contracted surfaces and cross-sections associated with D1 , D2 , and D3 , respectively. The cross sections all belong to the family of botanic curves of order c=2 (see Eq. (10)), having two petals, where the circle in Fig. 4 is the special case of ξ=0.
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}.
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}.
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}.
The singular value decomposition (SVD) of Q is obtained by substituting Eq. (19) into Eq. (18),
The product of two unitary matrices ( U3 or U0 ) is unitary just as successive rotations can be replaced by a single rotation. Note that the retardance components U1 and U2 of the decomposition are the same regardless of the decomposition order chosen in Eq. (18). The form U1DU2 relates the retardance and depolarization to the shape of the DoP numerator surface. Matrix U2 rotates the Poincaré sphere prior to D contracting it along three axes. This is followed by a final rotation U1 that rotates polarization states into their final output states. Since U1 (being unitary) preserves vector magnitude, and the DoP surface is plotted in terms of incident Stokes states, U1 bears no effect (rotation or otherwise) on the DoP numerator surface. U1 does however affect a related DoP surface representation plotted in terms of output Stokes states.
In the SVD of M1 of Eq. (4),
neither U1 nor U2 affects the DoP numerator shape, and only U2 affects the orientation. The contracting effect of D1 (along axes first rotated by U2 ) is shown in Fig. 7. The Poincaré sphere is contracted along three orthogonal axes to radii of 0.7477, 0.5769 and 0.3981. The first contraction occurs along an axis specified by the top row of U2 , and so forth. The resulting surface is not always convex.
The SVD handles singular matrices, such as ideal polarizers, without complication (e.g. Q0 of matrix M0 in Eq. (11)).
6. Effect of polarizance on DoP numerator
The DoP numerator surface depends on the sum of Q and the polarizance as shown in Eqs. (6) and (15). This combination drags and distorts the Q-surface in the direction of the polarizance vector (Eq. (16)), moving the origin a distance equal to the magnitude of the polarizance vector, P. Before the effect of polarizance, the DoP numerator surface coincides with the Q-surface, and the square root in the DoP numerator represents the magnitudes of the radial vectors to this surface. Because the summation of P with Q occurs inside the DoP numerator square root, the induced effect on the DoP numerator surface is not merely a pure translation, but rather a translation and distortion. This transformation is animated for M0 (Eq. (11)) in Fig. 8, and for M1 (Eq. (4)) in Fig. 9, where the final shapes in Figs. 8 and 9 are the DoP numerator surfaces for M0 and M1 , respectively.
Fig. 8. (1.33 MB) Animation showing the effect of polarizance on the numerator surface for Mueller matrix M0.
Fig. 9. (1.33MB) Animation showing the effect of polarizance on the numerator surface for Mueller matrix M1.
Formation of the DoP numerator surface is visualized in three steps, a rotation by U2 , the subsequent contraction of the Poincaré sphere in three orthogonal Stokes dimensions by D, and the shift and distortion (associated with the DoP numerator square root) in the direction of the three-component polarizance vector.
7. DoP surface and DoP map
The DoP surface is the quotient of the DoP numerator and DoP denominator surfaces. For example, Fig. 1(a) is the DoP surface for M1 (Eq. (4)), and Fig. 10 is the DoP surface for M0 (Eq. (11)).
Figure 11 contains the DoP map for Mueller matrix M0 (the DoP map for M1 is Fig. 1(b)).
DoP maps and surfaces may exhibit one or two maxima and one or two minima. These maxima or minima may be degenerate for entire circles of incident states around the Poincaré sphere, as in the minimum of Fig. 11 for M0 . Two local maxima also appear in the DoP map for M0 . M1 has two minima and two maxima. After an extensive search, cases where the DoP maps contain three or more maxima or minima have not been found.
8. Physical realizability of a Mueller matrix
When the output DoP for any incident state is less than zero or greater than one, that matrix is not a physically realizable Mueller matrix. Many relationships among Mueller matrix elements to ensure physical realizability have been published [28–30]. Such relationships can be understood geometrically in terms of the numerator and denominator surfaces. The numerator surface must not protrude from the denominator surface at any point or the DoP at that point is greater than one. The two surfaces may be tangent and the DoP is one at points of tangency. Similarly, the origin of the Poincaré sphere must lie within or on both surfaces, or the DoP will be negative. When the numerator surface passes through the origin, the corresponding state is completely depolarized.
Plotting the DoP numerator and denominator surfaces together for a given Mueller matrix aids the visualization of these geometrical relationships and is useful for establishing physical realizability. For example, the numerator and denominator surfaces for M1 (Eq. (4)) are plotted in Fig. 12. The denominator surface surrounds the numerator surface, so M1 is physical.
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.
Consider a family of matrices connecting the ideal depolarizer with M0 parameterized by coefficient i,
As i increases the numerator surface uniformly grows starting as a single point at the origin. Figure 13 animates the growth of the numerator surface within the denominator surface for i ranging from 0.0 to 0.7. When i = 0.5 the numerator surface is tangent to the denominator surface at the two maxima of Fig. 11. When i exceeds 0.5, the numerator surface protrudes from the denominator surface and the Mueller matrices are non-physical.
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.
9. A family of DoP maps
A family of Mueller matrices will be analyzed to demonstrate how certain matrix properties relate to properties of DoP surfaces and maps. Observing the metamorphosis of DoP surfaces and maps clarifies the depolarizing effects of the matrix family. Consider the family of Mueller matrices
whose members are formed by summation of a horizontal linear polarizer and another linear polarizer, LP(α), with its polarization axis at an angle α to horizontal. Figure 14 shows the DoP surfaces for each member of this family plotted within the Poincaré sphere, and Fig. 15 the corresponding DoP maps. Figure 16 animates the evolution of DoP surfaces.
This family always has two maxima with DoP = 1 which occur for Stokes vectors orthogonal to the two polarizer axes. An incident state orthogonal to one of the polarizers is completely blocked by that polarizer and light only transmits through the other polarizer, thus emerging with DoP = 1. States aligned with either polarizer also partially transmit through the second polarizer (unless the two polarizers are orthogonal) and the contribution of these two beams reduces the DoP. Since one of the polarizers forming every member of Eq. (25) is a horizontal linear polarizer, every DoP surface and map in this family shows a maximum DoP = 1 for vertically polarized light. The other DoP maximum rotates around the Poincaré sphere as the second polarizer rotates. Half way between these maxima is a valley of increasing width and depth (see Fig. 14). When the second polarizer axis is at 90° to the first, the output DoP maxima occur for input vertically and horizontally polarized light and the valley reaches a DoP = 0. The DoP surface in this case separates into two tangent spheres. The DoP = 0 output states occur for input states on the great circle of the Poincaré sphere through 45°, right circular, 135°, and left circular states.
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.
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.
10. Conclusion
DoP maps and surfaces represent the variation of depolarization of a Mueller matrix for all fully polarized incident states. For non-depolarizing Mueller matrices, the DoP surface is the unit sphere and the DoP map is unity everywhere. For depolarizing Mueller matrices the DoP surface contracts toward the origin by an amount equal to the depolarization for that incident state. Thus, the maps and surfaces indicate the variation of depolarization with incident state.
Twelve of sixteen Mueller matrix degrees of freedom affect the shape of DoP surfaces, with three related to the DoP denominator arising from the diattenuation properties encoded in the first row of the Mueller matrix. Two of three denominator degrees of freedom define a diattenuation axis and the third the diattenuation magnitude which relates to the size of the dimple in the limaçon cross-section of the denominator surface.
The DoP numerator surface function has nine degrees of freedom relating to depolarization, retardance, and polarizance. The unitary matrix U1 in the singular value decomposition of Mueller submatrix Q is a rotation of the output polarization state due to retardance which does not change the DoP surface. Three depolarization degrees of freedom in matrix D contract the Poincaré sphere along three axes rotated from the Stokes basis states by the retardance effects of unitary matrix U2 . The polarizance vector acts via an additive term that drags and distorts the surface in the direction of the polarizance vector. DoP surfaces may also be plotted in terms of the output Stokes states rather than input Stokes states (making visible the rotation effects of U1 ) but this representation is not explored here.
For a physically realizable Mueller matrix, the DoP numerator surface must lie entirely within the denominator surface and must incorporate the Poincaré sphere origin (either on or within the surface). Physically realizable DoP maps have only been observed with one or two maxima and one or two minima. It is postulated that the DoP maps cannot have more than two maxima or minima, but such a conjecture is not proven here.
DoP surfaces and DoP maps yield a detailed picture of depolarization and are of useful for understanding and classifying depolarizing Mueller matrices.
Acknowledgments
The authors wish to thank Sandia National Laboratories for its support of this work. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. This paper is a section of a dissertation submitted in partial fulfillment of the requirements for a PhD degree in Optical Sciences at the University of Arizona. The authors acknowledge the help of Bridget Ford and Eric Shields. Special thanks go to Justin Wolfe and Neil Beaudry of the Optical Sciences Center, University of Arizona, for their assistance.
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