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Photonics
, Volume 8 (8) – Aug 4, 2021

/lp/multidisciplinary-digital-publishing-institute/geometric-interpretation-and-general-classification-of-three-GMJooFpX0c

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Communication Geometric Interpretation and General Classification of Three-Dimensional Polarization States through the Intrinsic Stokes Parameters José J. Gil Department of Applied Physics, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain; ppgil@unizar.es Abstract: In contrast with what happens for two-dimensional polarization states, defined as those whose electric field fluctuates in a fixed plane, which can readily be represented by means of the Poincaré sphere, the complete description of general three-dimensional polarization states involves nine measurable parameters, called the generalized Stokes parameters, so that the generalized Poincaré object takes the complicated form of an eight-dimensional quadric hypersurface. In this work, the geometric representation of general polarization states, described by means of a simple polarization object consti- tuted by the combination of an ellipsoid and a vector, is interpreted in terms of the in- trinsic Stokes parameters, which allows for a complete and systematic classification of polarization states in terms of meaningful rotationally invariant descriptors. Keywords: polarization optics; light scattering; polarimetry; depolarization; polarization object Citation: Gil, J.J. Geometric Interpretation and General Classification of Three-Dimensional 1. Introduction Polarization States through the The Poincaré sphere [1] provides a simple and meaningful representation of those Intrinsic Stokes Parameters. polarization states whose electric field fluctuates in a fixed plane (2D states). Despite the Photonics 2021, 8, 315. https://doi.org/ interest of such a particular type of polarization states, which is commonly applied in 10.3390/photonics8080315 many problems involving paraxial fields and is characterized through the conventional Received: 8 July 2021 four Stokes parameters [2], or (equivalently) by means of the 2 × 2 polarization matrix (or Accepted: 29 July 2021 coherency matrix) [3–8], the description of a general polarization state involves nine Published: 4 August 2021 generalized Stokes parameters [9–22] instead of the conventional four ones, and therefore their geometric representation through a generalized Poincaré sphere is determined by Publisher’s Note: MDPI stays an eight-dimensional object, which does not admit a simple geometric and physical in- neutral with regard to jurisdictional terpretation. claims in published maps and The relevance of the pioneering work of Soleilllet [4], who for the first time intro- institutional affiliations. duced a matrix structure that is equivalent to polarization matrices of 3D states has been discussed and emphasized by Arteaga and Nichols [23]. In the case of Mueller matrices, which represent linear transformations of 2D po- larization states, appropriate geometric representations have been introduced by means Copyright: © 2021 by the author. characteristic ellipsoids [24–32], thus avoiding the problem of interpreting the extremely Licensee MDPI, Basel, Switzerland. complicated 16-dimensional quadric object defined from the 16 elements of Mueller This article is an open access article matrices. distributed under the terms and In this work, the geometric representation introduced by Dennis [33] for general conditions of the Creative Commons Attribution (CC BY) license three-dimensional (3D) polarization states is studied and interpreted in terms of the (http://creativecommons.org/licenses intrinsic Stokes parameters [21] and other meaningful descriptors that are invariant un- /by/4.0/). der rotations of the reference frame [17–22,33–45]. The classification introduced in Refs. Photonics 2021, 8, 315. https://doi.org/10.3390/photonics8080315 www.mdpi.com/journal/photonics Photonics 2021, 8, 315 2 of 15 [20,46] is improved and completed in the light of the recent approaches on nonregularity [39,42,45,47,48], polarimetric dimension [41], spin of a polarization state [43,47,49] and interpretation of sets of orthogonal 3D polarization states [44]. The intrinsic geometric representation of a polarization state is determined by the polarization density object, which is constituted by the combination of an ellipsoid (the polarization density ellipsoid) and a vector (the spin density vector). Apart from the scale parameter given by the intensity of the state, the shape of the polarization density ellip- soid, the magnitude of the spin density vector and its relative orientation with respect to the symmetry axes of the polarization density ellipsoid describe completely the intrinsic properties of the polarization state, while the spatial orientation of the polarization den- sity object involves the three additional angular parameters required for its representa- tion with respect to the reference frame considered. That is, the complete information on a polarization state can be parameterized through the following set of nine parameters (whose definitions are summarized in Sec- tion 2): (1) the intensity, I, which plays the geometric role of a scale factor that determines the size of the polarization object; (2) the degree of linear polarization, P and the degree of directionality, P , which determine the shape of the polarization density ellipsoid; (3) ˆ ˆ ˆ ˆ the spin density vector [n (n ,n ,n ) ](three real parameters) associated with the O1 O 2 O 3 polarization state, whose magnitude and relative orientation with respect to the polari- zation density ellipsoid are fixed for each polarization state, and (4) the three orientation angles ( , , ) of the polarization density ellipsoid with respect to the Cartesian refer- ence frame XYZ considered [20]. It should be noted that the parameters describing a polarization state are defined for a given point r in space, and therefore they do not carry direct information on the direc- tion of propagation of the wave at that point. Thus, the name degree of directionality used for P refers to the stability of the plane containing the polarization ellipse, and not to the direction of propagation. Similarly to what happens with other intrinsic quantities in physics (such as, for in- stance, the tangential and normal components of the acceleration vector in kinematics) which give a natural a meaningful view of the physical quantities involved, the polari- zation object provides direct geometric representation of the intrinsic Stokes parameters (I ,P ,P ,n ˆ ,n ˆ ,n ˆ ) [20,21] (note that, even though the definition of the intrinsic Stokes l d O1 O 2 O 3 parameters involves P 3 instead of P , we omit the coefficient 1 3 for brevity and d d simplicity). The contents of this paper are organized as follows: Section 2 contains a summary of the concepts and notations that are necessary to make the required developments in further sections; the definition and discussion of the concept of polarization object is in- troduced in Section 3; Section 4 includes a classification of polarization states based on the peculiar geometric features of the polarization object, and Section 5 is devoted to the conclusions. 2. Materials and Methods All the second-order polarization properties of an electromagnetic wave, at a given point r in space, are embodied in the three-dimensional polarization matrix (or coherency matrix) R, whose mathematical structure is that of a 3 × 3 positive semidefinite Hermitian matrix determined by the second-order moments of the analytic signals (t ) (i 1, 2, 3) (complex random variables, assumed stationary, at least in wide sense) associated with the three fluctuating Cartesian components (referenced with respect to a laboratory ref- erence frame XYZ) of the electric field vector at point r: * * * t t t t t t 1 1 1 2 1 3 * * * R t t t t t t , (1) 2 1 2 2 2 3 * * * t t t t t t 3 1 3 2 3 3 Photonics 2021, 8, 315 3 of 15 where the superscript * indicates complex conjugate, while the brackets stand for time averaging over the measurement time. The analytic signal vector is defined as T † ε (t ) [ (t ), (t ), (t )] , so that R can be expressed as R ε (t )ε (t ) , where repre- 1 2 3 sents Kronecker product and the superscript † stands for conjugate transpose. When the fluctuations of (t ) (i 1, 2, 3) have Gaussian probability density func- tions, their second-order moments (the elements of R) characterize completely the statis- tical properties, so that the higher-order moments do not add complementary infor- mation, and R fully characterizes the polarization state. In the most general case, R only characterizes the second order polarization properties. The above indicated features are intimately linked to the fact, from its very defini- tion, R is Hermitian and positive semidefinite, and therefore R has the mathematical structure of a covariance matrix of the three random complex variables . [ (t ), (t ), (t )] 1 2 3 In fact, the diagonal elements of R are the respective (real and nonnegative) variances (t ) (t ) of (t ) , while the off-diagonal elements of R are the (complex-valued) co- i i i variances determined by the respective quantities (t ) (t ) . i j Since subsequent analyses involve a large number of peculiar quantities, vectors and matrices, these, together with the corresponding references, are summarized in Table A1. It has been proven that R can always be expressed as [20,21,33]: R QR Q , ˆ ˆ ˆ ˆ ˆ a in 2 in 2 a ,n i 1, 2, 3 1 O 3 O 2 i Oi T 1 (2) Q Q ˆ ˆ ˆ ˆ ˆ ˆ R I in 2 a in 2 , a a a 0 , , O O 3 2 O1 1 2 3 detQ 1 ˆ ˆ ˆ ˆ ˆ ˆ in 2 in 2 a a a a 1 O 2 O1 2 1 2 3 where I trR is the intensity and Q is the orthogonal matrix that diagonalizes the real part of R, so that the polarization matrix of the state under consideration takes the form R when the field variables are represented with respect to the given reference frame XYZ, and takes the form R when represented with respect to the intrinsic reference frame XOYOZO, which is characterized by the fact that the real part of R is a diagonal matrix, ReR I diag (a ˆ ,a ˆ ,a ˆ ) (with a ˆ a ˆ a ˆ 1 ) and where the convention a ˆ a ˆ a ˆ is O 1 2 3 1 2 3 1 2 3 taken without loss of generality [note that since R is positive semidefinite, then neces- sarily a ˆ 0 (i 1, 2, 3) ]. ˆ ˆ ˆ From a statistical point of view, the diagonal elements (a ,a ,a ) of the intrinsic 1 2 3 polarization density matrix R R I are dimensionless and represent the intensi- O O ty-normalized variances of the field variables (referenced with respect to XOYOZO), while the off-diagonal components represent the respective intensity-normalized covariances. The principal variances a can be expressed as follows in terms of the degree of linear polarization, P a ˆ a ˆ , and the degree of directionality, P 1 3a ˆ , [20,21]: l 2 1 d 3 2P 3P 2P 3P 21P d l d l a ˆ , a ˆ , a ˆ , (3) 1 2 3 6 6 6 so that the intrinsic polarization matrix R takes the form: 2P 3P d l in ˆ in ˆ O 3 O 2 0 P P 1 I 2P 3P l d d l ˆ ˆ R in in , , (4) O O 3 O1 2 2 2 2 3 ˆ ˆ ˆ P n n n 1 c O1 O 2 O 3 2 1P ˆ ˆ in in O 2 O1 ˆ ˆ ˆ where the set of six rotationally invariant parameters (I ,P ,P ,n ,n ,n ) constitute the l d O1 O 2 O 3 intrinsic Stokes parameters of the state R. The intrinsic representation of the spin density vector of R is given by n ˆ (n ˆ ,n ˆ ,n ˆ ) , which can also be parameterized through its O O1 O 2 O 3 magnitude ˆ P and its orientation angles ( , ) with respect to the axes XOYOZO O c n n Photonics 2021, 8, 315 4 of 15 ˆ ˆ ˆ along which lie the principal intensities a Ia , a Ia and a Ia , respectively 1 1 2 2 3 3 (Figure 1), i.e., n ˆ n ˆ sin cos , n ˆ n ˆ sin sin and n ˆ n ˆ cos . O1 n n O 2 n n O 3 n (a) (b) Figure 1. (a) Leaving aside the intensity, the polarization state is fully characterized by its polarization density object, composed of the polarization density ellipsoid E (with semiaxes a ˆ a ˆ a ˆ ), together with the spin density vector n . 1 2 3 O (b) n ˆ is determined by its magnitude n P and its orientation angles , with respect to the symmetry axes O O c n n XOYOZO of E. The polarization object can be expressed in terms of the intrinsic Stokes parameters (P ,P ,n ˆ ,n ˆ ,n ˆ ) l d O1 O 2 O 3 and therefore it has tensor character in the sense that it is invariant with respect to changes of the reference frame. When the polarization density object is referenced with respect to an arbitrary coordinate system, the orientation angles ( , , ) of the polarization density ellipsoid should be considered as additional parameters. Note that even though the spin density vector is dimensionless (and therefore it has no dimensions of angular momentum) it is a proper descriptor of the spin anisotropy of the polarization state to which it corresponds [47]. Furthermore, as with R and R , n ˆ O O is relative to a specific point r in space and the term density in this context indicates that it describes the intensity-normalized version n n I of the spin vector n associated O O O with the given state [43,47]. The degree of polarimetric purity (or degree of polarization) P [40] of R is related 3D to intrinsic Stokes parameters through the following weighted square average of the components of purity (P ,P ,P ) of R [38]: l c d 3 1 2 2 2 P P P P , (5) 3D l c d 4 4 2 2 where the quantity P P P that summarizes the combined contributions of P e c l and P to the overall purity P is called the degree of elliptical purity [39]. It is also c 3D worth recalling that P can also be expressed as the following equivalent weighted 3D quadratic average [41,43]: 3 1 2 2 2 2 P d P , d 3P P , (6) 3D c l d 4 2 where d represents the degree of intensity anisotropy [41,43] of the polarization state, while the degree of circular polarization P n ˆ gives a measure of the degree of spin c O anisotropy [43]. Moreover, the orthogonal matrix Q of the rotation transformation R QR Q depends on three angular parameters, and can be parameterized as follows in terms of the overall azimuth and elevation angles, φ and θ, respectively, of the axis Z of the ref- erence frame XYZ and the azimuth of the XY axes about the direction Z, all referenced with respect to the intrinsic reference frame XOYOZO: Photonics 2021, 8, 315 5 of 15 c s 0 c s 0 c c c s s c c s s c c s c 0 s Q s c 0 0 1 0 s c 0 c s c c s c s s c c s s O s 0 c (7) 0 0 1 0 0 1 s c s s c s sinx, c cosx x x Another relevant concept that will be used in the classification of polarization states in Section 4 is the degree of nonregularity [39,42] whose definition relies on the charac- teristic decomposition of R [34,35]: ˆ ˆ ˆ R PIR P P IR 1P IR , 1 p 2 1 m 2 u3D (8) 1 1 † † ˆ ˆ ˆ R U diag 1, 0, 0 U , R U diag 1,1, 0 U , R diag 1,1,1 , p m u3D 2 3 where U is the unitary matrix that diagonalizes R, while P and P are the indices of 1 2 polarimetric purity (IPP) defined as [36,37]: ˆ ˆ ˆ ˆ ˆ ˆ P , P 1 3 , 1 , (9) 1 2 1 2 3 1 2 3 ˆ ˆ ˆ ˆ ( , , ) being the eigenvalues of the polarization density matrix R R I , taken in 1 2 3 ˆ ˆ ˆ decreasing order ( ) , so that the IPP satisfy the peculiar nested inequalities 1 2 3 0 P P 1 and fully determine the quantitative structure of polarimetric randomness 1 1 of R [39]. Regarding the intensity-normalized components of the characteristic decom- position, R is a pure (totally polarized) state, the discriminating state R is given by p m an equiprobable incoherent mixture of the two eigenstates of R with largest eigenvalues ˆ ˆ ( , ) and the unpolarized component R lacks both intensity and spin aniso- u3D 1 2 tropies [43]. The pure component R has a well-defined polarization ellipse that evolves in a fixed plane. Except for the particular case that R is regular (see below), the electric field of the discriminating component does not fluctuate in a fixed plane. Both polarization plane and shape of the polarization ellipse of the unpolarized component ˆ ˆ R evolve fully randomly, so that R completely lacks anisotropy and is propor- u3D u3D tional to the identity matrix. Regular states are defined as those for which either P P (in which case R does 1 2 m not take place in the characteristic decomposition) or R is a real-valued matrix, in which case R lacks spin and takes the form of a 2D-unpolarized state ˆ ˆ ˆ R R (1 2) diag (1,1, 0) . Thus, R is a particular limiting situation of the general m u 2D u 2D ˆ ˆ case where ImR 0 and 0 P (R ) 1 2 [42,48]. In general, P P and P P , m c m 1 e 2 d where the equality P P (which implies P P and vice versa) is a characteristic and 1 e 2 d peculiar property of regular states. That is to say, R represents a nonregular state if and only if P P . The degree of nonregularity P of R is defined as follows by any of the 1 e N following expressions in terms of the components of purity of R [42]: P 4P PP R P P 1 4P R 4 3P P1P R , (10) N 2 1 l m 2 1 c m 2 1 d m P 0 if and only if R corresponds to a regular state. States with maximal nonreg- ularity, P 1 , are called perfect nonregular states, and necessarily satisfy P (R ) 1 2 N c m while they are equivalent of an equiprobable incoherent mixture of a circularly polarized state and a linearly polarized state whose electric field vibrates in a direction orthogonal to the plane containing the polarization circle of the circular component [42]. Photonics 2021, 8, 315 6 of 15 3. Results 3.1. The Polarization Object ˆ ˆ ˆ The nondimensional quantities (a ,a ,a ) can be considered as the semiaxes of the 1 2 3 polarization density ellipsoid, denoted by E, or inertia ellipsoid [33], associated with R , and leaving aside the intensity (I), which plays the role of a scale factor, E is determined by the intrinsic Stokes parameters P and P related to the intensity anisotropies [43]. l d The geometric parameterization is completed with the intrinsic representation n ˆ (n ˆ ,n ˆ ,n ˆ ) of the spin density vector of the state. O O1 O 2 O 3 Therefore, from Equations (2) and (4) it follows that a general three-dimensional polarization state admits a simple geometric representation through its associated po- larization density object, constituted by the combination of E and n , in such a manner that the center of E and the origin of n ˆ coincide with the point r to which the polariza- ˆ ˆ tion state corresponds. Thus, both the magnitude n P of n and its relative orien- O c O ˆ ˆ ˆ tation with respect to E (determined by the intrinsic components n , n and n ) are O1 O 2 O 3 fixed, in such a manner that the polarization density object has a fixed shape, while its orientation with respect to a given reference frame XYZ involves the three angles associ- ated with the rotation from the intrinsic axes XOYOZO to XYZ. As with the Poincaré sphere, the polarization density object has been defined through intensity-normalized quantities and therefore, it is the intensity-normalized version of the polarization object composed of the intensity ellipsoid E , with semiaxes (a ,a ,a ) , and the spin vector n In ˆ . 1 2 3 O O The polarization density object is represented in Figure 1 for a single point r and referenced with respect to XOYOZO. The expressions of the semiaxes of the polarization density ellipsoid in terms of P and P , allows for a direct analysis of certain important features of the polarization density object. When P 1 , the electric field of the state fluctuates in a fixed plane (2D n ˆ state) and the polarization density ellipsoid becomes an ellipse, while is orthogonal to . Moreover, P 1 implies P 1 , which means that all 2D states are regular states. d N A subclass of 2D states is that constituted by pure states (Figure 2), which are character- ized by P 1 , or equivalently 1 P P P P ), and include the limiting particular 3D d 2 1 e cases of linearly polarized states (P 1 , E degenerates into a simple straight segment, and therefore P 0 ), and circularly polarized states (P 1 and E takes the form of a c c circle). It should be stressed that pure states as well as any kind of 2D states (character- ized by P 1 ), constitute a particular subclass of 3D states. Thus, despite the obvious fact that polarization states are realized in the three-dimensional physical space, states with P 1 are called genuine 3D states (which may be regular or not) and are characterized by the fact that the three semiaxes of the associated polarization density ellipsoid are nonzero. Photonics 2021, 8, 315 7 of 15 (a) (b) (c) Figure 2. Polarization density object of a pure state (P 1) : linearly polarized state (P 1) (a); elliptically polarized e l state (0 P 1) (b), and circularly polarized state (P 1) (c). The spin density vector of pure states is orthogonal to c c the variance ellipse and is zero in the case of linearly polarized states. 3.2. Classification of Three-Dimensional Polarization States based on the Polarization Object The concept of polarization object, together with other polarization descriptors such as the CP, the IPP, P and P allows for a meaningful classification of both 2D states 3D N and genuine 3D states, where each category can also be interpreted in terms of the cor- responding characteristic decomposition. For the sake of clarity, the classification is performed through a pair of tables that correspond to 2D states (Table 1), and genuine 3D states (Table 2). 2D mixed states (or 2D partially polarized states) are characterized by 1 P P d 2 and P P 1 . The characteristic decomposition of a 2D mixed state consists of a com- 1 e bination of a pure state and a 2D partially polarized state Ru 2D , both components sharing a common polarization plane. The case of a regular discriminating state (P P 1,P 0) corresponds precisely to R (Figure 3b). u 2D 2 d 1 The types of genuine 3D states are summarized in Table 2, whose columns, from left to right, are devoted to the cases of (a) regular genuine 3D states (P 1,P 0) , whose d N discriminating component is a 2D unpolarized state, and (b) nonregular states (P 0) . The polarization planes of the eigenstates u ˆ and u ˆ associated with the respective ei- 1 2 ˆ ˆ ˆ ˆ genvalues and of R (which coincide with those of R , and are taken so as to 1 2 O ˆ ˆ ˆ satisfy ) are denoted by and , and they only coincide for regular 1 2 1 2 3 states. Due to the critical role played by the discriminating states for the interpretation of polarization states [47,48], Table 3 summarizes the main features of (from left to right) (a) regular discriminating states; (b) partially nonregular discriminating states, and (c) per- fect nonregular states. Regular discriminating states have the simple form R and are built by equi- u 2D probable incoherent compositions of arbitrary pairs of mutually orthogonal states with a common polarization plane, as for instance two linearly polarized states whose electric fields vibrate along two orthogonal directions embedded in the polarization plane. Nonregular discriminating states and are built by equiprobable incoherent compo- sitions of certain pairs of mutually orthogonal states with different polarization planes, including the combination of a linearly polarized state and an elliptically polarized state whose polarization planes are mutually orthogonal. The case of perfect nonregular states corresponds to the limiting situation equivalent to an equiprobable incoherent mixture of a linearly polarized state and a circularly po- larized state whose polarization planes are orthogonal. Photonics 2021, 8, 315 8 of 15 (a) (b) Figure 3. (a) Polarization object of a 2D mixed state (P 1,P P 1) . (b) When P 0 , the state is itself a regular d e 1 e discriminating state R IR (b). u 2D u 2D Table 1. Classification of 2D states (P 1) . P 1 (2D states) P 1,P P ,P 0,P 1 2 d 2 1 e N 3D P 1 (pure states) P 1 (2D mixed states) e e R R R PR 1P R p 1 p 1 u 2D P 1 0 P 1 P 0 l l l P 0 0 P 1 R R u 2D P 0 0 P 1 P 1 c c c Linear Elliptical Circular Partially polarized Unpolarized Indep. parameters: Indep. parameters: Indep. parameters: Indep. parameters: Indep. parameters: I , , , I , , I , , I ,P , , , I ,P ,P , , , c c l Principal variances: Principal variances: Principal variances: Principal variances: Principal variances: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 a a , a 1 0 a , 4aa P 0 a ,a a 1 2 0 a ˆ a ˆ a ˆ 0 a , a a 1 2 3 2 1 3 1 2 c 3 2 1 3 2 1 3 2 1 Spin density vector: Spin density vector: Spin density vector: Spin density vector: Spin density vector: ˆ 0n ˆ , n ˆ ˆ ˆ ˆ ˆ ˆ ˆ n 0 n 1, n n 1, n n 0, n O O O O O O O O O Polarization object: Polarization object: Polarization object: Polarization object: Polarization object: Figure 2a Figure 2b Figure 2c Figure 3a Figure 3b Characteristic decomp.: Characteristic decomp.: Characteristic decomposition: R PR 1P R R R 1 p 1 u 2D u 2D R R (P 0) Figure 4b (P 0) Figure 4a 1 (a) (b) Figure 4. Intrinsic representation of the characteristic decomposition of a mixed 2D state (P 1,P P 1) . (a) In gen- d e 1 eral, R it is given by the incoherent superposition of a pure state R I R and a 2D unpolarized state O pO pO R IR whose polarization planes coincide [Gil 2014a]. (b) When P 0 , then the pure component vanishes u 2D u 2D e and the state is itself 2D unpolarized state (the electric field fluctuates fully randomly in a fixed plane. XOYO). Photonics 2021, 8, 315 9 of 15 Table 2. Classification of genuine 3D states P 1 . P 1 (genuine 3D states) Regular 3D mixed states (P 0) N Nonregular 3D mixed states P 0 P P P P 0 P 1 P P P P N 1 e 2 d N 1 e 2 d Independent parameters: I ,P ,P ,P , , , Independent parameters: I ,P ,P ,P , , , , , l c d l d c n n ˆ ˆ ˆ ˆ ˆ ˆ Principal variances: 0 a a a Principal variances: 0 a a a 3 2 1 3 2 1 ˆ ˆ ˆ ˆ Spin density vector: 0 n n Z Spin density vector: 0 n , 0 PR 1 2, n P O O O O c m O 3 c Polarization object: Figure 5a Polarization object: Figure 5b Characteristic decomposition: Figure 6 Characteristic decomposition: Figure 7 R PR P P R 1P R R PR P P R 1P R R R 1 p 2 1 u 2D 2 u3D 1 p 2 1 m 2 u3D m u 2D (a) (b) Figure 5. Polarization density object of genuine 3D states . (a) The 3D state is regular when the spin density (P 1) ˆ ˆ vector n is orthogonal to the plane XOYO. (b) the state it is nonregular if and only if n is not orthogonal to XOYO. O O Figure 6. Characteristic decomposition of a regular genuine 3D state. The discriminating component is a 2D unpolarized ˆ ˆ state R whose polarization plane coincides with that of the pure component R and the 3D unpolarized state has u 2D pO nonzero contribution [20]. (P P 1) d 2 Figure 7. Characteristic decomposition of a nonregular 3D state. The discriminating component R is itself nonregular. Table 3. Classification of discriminating states (P 1,P 0) . 2 1 P 1,P 0 (discriminating states, R R ) 2 1 P 0 0 P 1 P 1 N N P P 1 P P P P P 0,P 1 d 2 1 e 2 d 1 2 P P 0,P 1 0 P 1 2 , 0 P 1 4 ,P 1 4 P P 1 4 ,P 1 2 c l d c l d l d c 2D unpolarized state Nonregular discriminating state Perfect nonregular state Independent parameters: I , , Independent parameters: I , , , ,P Independent parameters: I , , Principal variances: Principal variances: Principal variances: Photonics 2021, 8, 315 10 of 15 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ a 0, a a 1 2 0 a a a 1 2 , a a 1 2 a a 1 4 , a 1 2 3 2 1 3 2 1 2 3 3 2 1 Spin density vector: Spin density vector: Spin density vector: ˆ ˆ ˆ ˆ 0 n 1 2 n 1 2 n X n 0 O O O O Polarization object: Figure 8a Polarization object: Figure 8b Polarization object: Figure 8c (b) (c) (a) Figure 8. Polarization density object of a discriminating state R (P 0, P 1) : (a) zero spin corresponds uniquely to m 1 2 the regular case P (R ) 0 R R ; (b) R is nonregular if and only if P (R ) 0 , and (c) R is perfect c m m u 2D c m m m nonregular if and only if P (R ) 1 2 . c m 4. Discussion The mere qualitative properties of the polarization density object are sufficient to identify certain properties of the polarization state, while other features are linked to quantitative aspects. For instance, R corresponds to a 2D state if and only if at least one of the semiaxes of E is zero (P 1) ; R corresponds to a nonregular state if and only if n d O is not parallel to ZO. Moreover, R corresponds to a pure state if and only if the (quantita- 2 2 2 2 tive) condition P 1 is satisfied (recall that and P P P and P n ). From the e e c c O sole inspection of the polarization density object, the classification presented in Tables 1–3 can be synthesized as follows: P 1 (a ˆ 0) R is a 2D state the polarization density ellipsoid is an ellipse. d ´3 o P 1 R is a 2D mixed state. P 0 R is a 2D unpolarized state (i.e., R is a 2D discriminating state), R R (a ˆ a ˆ and n 0 ). u 2D 2 1 O ˆ ˆ o P 1 R is a pure state, a a P 4 . e 1 2 c P 1 R is a linearly polarized pure state, a ˆ 1 ( P 0) . l 1 c ˆ ˆ 0 P 1 R is an elliptically polarized pure state, 0 a a , with l 2 1 2 2 0 P 1P 1 . c l P 0 R is a circularly polarized pure state, a ˆ a ˆ 1 2 with P 1 . l 2 1 c P 1 (a 0) R is a genuine 3D state. d ´3 o n ˆ 0 or n ˆ Z R is a regular genuine 3D state. O O O ˆ ˆ o n 0 and n not parallel to Z R is a nonregular state. O O O o P P 0 The polarization density ellipsoid E of R is a sphere (a ˆ a ˆ a ˆ , l d 3 2 1 full intensity isotropy, d 0 ) P 0 n 0 (full intensity isotropy, d 0 , with nonzero spin). c O P 0 R is a 3D unpolarized state, R R (full intensity and spin isot- u3D ropy). o P 3P 1 R is a 3D discriminating state, R R ( a ˆ a ˆ a ˆ 1 2 , d l 2 3 1 0 P 1 2 and n not parallel to Z ). c O O Photonics 2021, 8, 315 11 of 15 P 1 2 R is a perfect nonregular state (n X , P P 1 4) c O O l d 5. Conclusions Given a polarization matrix R, its associated intrinsic form R is obtained through the rotation transformation (in the real space) that diagonalizes the real part of R. The three (real) diagonal elements of R , together with the three (pure imaginary) off-diagonal elements determine biunivocally the six intrinsic Stokes parameters ˆ ˆ ˆ (I ,P ,P ,n ,n ,n ) of the state, which have a direct physical interpretation, namely the l d O1 O 2 O 3 intensity, I, the degree of linear polarization, P , the degree of directionality, P (which l d is an objective measure of the degree of stability of the fluctuating polarization plane containing the polarization ellipse) and the three intrinsic components (n ˆ ,n ˆ ,n ˆ ) of O1 O 2 O 3 the spin density vector n of the state. Consequently, any polarization state is fully characterized through its associated intrinsic Stokes parameters, which are invariant under rotation transformations of the laboratory reference frame XYZ, together with a three-dimensional rotation (determined by three angular parameters, φ, θ, , that depend on the specific spatial orientation of XYZ). As with the conventional four Stokes parameters characterizing 2D states, these quantities have phenomenological nature, and therefore, are always measurable. ˆ ˆ ˆ The diagonal elements (a ,a ,a ) of the intrinsic polarization density matrix 1 2 3 R R I , (with the convention a ˆ a ˆ a ˆ , taken without loss of generality) are called O O 1 2 3 the principal variances (because of their statistical nature) and constitute the semiaxes of an ellipsoid (polarization density ellipsoid, denoted by E), which in turn determines the directions of the respective axes XOYOZO of the intrinsic reference frame of the state. Furthermore, the principal variances can be readily expressed in terms of the pair of intrinsic Stokes parameters P and P [see Equation (3)]. Leaving aside the intensity, I, l d which plays the geometric role a scale factor, the polarization density object is defined as the composition of the polarization density ellipsoid (which depends on P and P ) and l d n ˆ , so that the relative orientation of n ˆ with respect to XOYOZO is fixed and therefore O O the shape and features of the polarization density object are rotationally invariant. The approach presented solves the problem of representing geometrically, in a sim- ple and meaningful manner, all polarization states (three-dimensional, in general). The geometric features of the polarization density object are determined by the intrinsic Stokes parameters, which allows for a complete and systematic classification of polariza- tion states. Funding: This research received no external funding. Informed Consent Statement: Not applicable. Conflicts of Interest: The author declares no conflict of interest. Photonics 2021, 8, 315 12 of 15 Appendix A Table A1. Quantities and structures relative to three-dimensional polarization states. Structure or Definition Properties Physical Meaning Quantity Invariant under rota- tion and under the I trR (t ) (t ) Averaged power of the electromagnetic i i Intensity action of birefringent i1 wave at point r devices Polarization Hermitian positive Provides complete information on sec- matrix semidefinite ond-order polarization properties Hermitian positive Polarization Intensity-normalized polarization matrix. R R I semidefinite, with density matrix ˆ Formally equivalent to a density matrix trR 1 Relative weights of the spectral incoherent ˆ ˆ ˆ 1 components of R [20] 1 2 3 Eigenvalues of ˆ ˆ ˆ , , 1 2 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ R R R R R ˆ ˆ ˆ 1 p1 2 p 2 3 p 3 3 2 2 ˆ ˆ R u ˆ u ˆ (u ˆ : eigenvectors of R) pi i i i The IPP provide a complete quantitative Indices of po- characterization of the structure of polari- ˆ ˆ ˆ 0 P P 1 larimetric pu- P , P 1 3 1 2 1 2 1 2 3 metric purity [36,37] rity (IPP) Intrinsic representation of the polarization Represents the same state as R, but refer- state. enced with respect to the corresponding ˆ ˆ ˆ a in 2 in 2 1 O 3 O 2 Principal variances: R I in ˆ 2 a ˆ in ˆ 2 intrinsic reference frame. Intrinsic polar- O O 3 2 O1 in ˆ 2 in ˆ 2 a ˆ The off-diagonal elements are pure imagi- ization matrix O 2 O1 3 nary because O is defined through the Spin density vector: diagonalization of the real part of R [20,33]. ˆ ˆ ˆ n n I n ˆ ,n ,n O O O1 O 2 O 3 Semiaxes of the polarization density ellip- ˆ ˆ ˆ a a a 1 Principal vari- ˆ 1 2 3 a ˆ ,a ˆ ,a 1 2 3 soid [20,33]. a ˆ a ˆ a ˆ 1 ances of R O 1 2 3 Spin vector of the state, with dimensions of T 2 2 2 n n n n n ,n ,n O1 O 2 O 3 O O1 O 2 O 3 1 Spin vector intensity [33,47] 4a a 4a a 4a a 2 3 1 3 1 2 Spin density vector of the state (nondimen- T 2 2 2 ˆ ˆ ˆ Spin density n n n n n I n ˆ ,n ˆ ,n ˆ O1 O 2 O 3 O O O1 O 2 O 3 1 sional) [20,33] ˆ ˆ ˆ ˆ ˆ ˆ 4a a 4a a 4aa vector 2 3 1 3 1 2 Absolute value of the spin density vector. Is n ˆ n I n ˆ P a measure of the degree of circular polari- O O O c Spin density zation P of the state Rigid composition of E and n Intensity ellipsoid E , with I O Determines geometrically all intrinsic Polarization The orientation angles semiaxes a ,a ,a 1 2 3 properties of the state. object ( , ) of n with n n O and spin vector n respect to the sym- metry axes XOYOZO of Photonics 2021, 8, 315 13 of 15 E are fixed and in- variant Rigid composition of E polarization density ellipsoid E, and n ˆ ˆ ˆ ˆ with semiaxes a ,a ,a 1 2 3 The orientation angles Determines geometrically all intrinsic 2P 3P 2P 3P d l d l a ˆ , a ˆ 1 2 ( , ) of n ˆ with Polarization n n O 6 6 properties of the state, but I, as with the density object respect to the sym- 21P Poincaré sphere of 2D polarization states a ˆ metry axes XOYOZO of and spin density vector n ˆ E are fixed and in- variant , , determine the Orientation The angles that allow for representing the rotation from the in- angles of the , , polarization object with respect to a given trinsic reference frame polarization reference frame axes XOYOZO to an ar- object bitrary one. An objective measure of how close to a Degree of linear P a ˆ a ˆ l 2 1 0 P P 1 l d linearly polarized state the state is [20,21] polarization Degree of cir- An objective of how close to a circularly P n c O 2 2 P P 1 cular polariza- c l polarized state the state is [20,21] tion An objective measure of the degree of sta- bility of the plane containing the fluctuating Degree of di- ˆ P 1 3a polarization ellipse. Equivalently, a meas- d 3 0 P P 1 l d rectionality ure of the closeness of the 3D state to a 2D one [20,21] Intrinsic measurable quantities. 3 1 2 2 2 Have phenomenological nature: They are P P P 1 Intrinsic Stokes l c d ˆ ˆ ˆ I ,P ,P 3 ,n ,n ,n l d O1 O 2 O 3 4 4 always well defined, regardless of the un- parameters 2 2 2 2 ˆ ˆ ˆ P n n n c O1 O 2 O 3 derlying microscopic model considered [20,21] Dimensionality Determine the effective dimensions taking 2 2 3P P l d index, d and place in the state. D 1 : linearly polarized; d , D 3 2d I 0d 1, 1 D 3 polarimetric D 2 , 2D state; 2 D 3 , 3D state [41] I I dimension, D . An objective measure of how close to a pure 3 1 2 2 2 P P P P Degree of po- 3D l c d state R is. It is determined by: (1) IPP con- 4 4 0 P 1 larimetric pu- 3D tributions; (2) CP contributions and (3) 3 1 3 2 2 2 2 rity P P P d P Intensity and spin anisotropies [38,43] 3D 1 2 c 4 4 4 Determine completely Complete information carried by R in terms the polarization object Complete pa- of nine meaningful quantities: the six in- (I ,P ,P ,n ˆ ,n ˆ ,n ˆ , , , ) and its spatial orienta- l d O1 O 2 O 3 rameterization trinsic Stokes parameters and the three tion with respect to the of R orientation angles of the polarization den- laboratory reference sity object [20,21] frame R is polarimetrically In the case of 2D states becomes the well R PIR 1 p equivalent to an inco- known decomposition into an incoherent Characteristic ˆ ˆ P P IR 1P IR 2 1 m 2 u 3D herent composition of decomposition combination of a pure state and a 2D un- pure state R , a dis- polarized state R [39] p u2D Photonics 2021, 8, 315 14 of 15 criminating state R and a unpolarized state u 3D In general, the discriminating component is This is the general form different from R . 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Photonics – Multidisciplinary Digital Publishing Institute

**Published: ** Aug 4, 2021

**Keywords: **polarization optics; light scattering; polarimetry; depolarization; polarization object

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