Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating through a weakly anisotropic medium

We describe the evolution of a paraxial electromagnetic wave characterizing by a non-uniform polarization distribution with singularities and propagating in a weakly anisotropic medium. Our approach is based on the Stokes vector evolution equation applied to a non-uniform initial polarization field. In the case of a homogeneous medium, this equation is integrated analytically. This yields a 3-dimensional distribution of the polarization parameters containing singularities, i.e. C-lines of circular polarization and L-surfaces of linear polarization. The general theory is applied to specific examples of the unfolding of a vectorial vortex in birefringent and dichroic media.


Introduction
Singular optics is an essential part of modern optics, which contributes to practically all fundamental wave phenomena [1−4]. Scalar wave fields are characterized by phase singularities (i.e., zeros of the intensity where the phase is undeterminate), such as optical vortices which have found a number of applications in classical and quantum optics [3,4]. Vector fields, e.g. electromagnetic or elastic waves, have more degrees of freedom and are also characterized by polarization singularities [5−9]. The generic types of the polarization singularities of transverse electromagnetic waves in 3D space are C-lines and L-surfaces, where the polarizations are, respectively, circular and linear. In the two cases, the polarization ellipse degenerates either to a circle (the eccentricity vanishes and the orientation is undeterminate) or to a line segment (the helicity vanishes and sign of polarization is undeterminate).
The wave field singularities may form a rich variety of structures in rather simple systems. Even interference of only three plane scalar waves results in a lattice of optical vortices [10]. Clearly, propagation in inhomogeneous or/and anisotropic media significantly modifies the wave interference patterns and, hence, gives rise to a quite tangled singularities structures. Therefore, the wave field singularities in complex media are frequently studied within the statistical approach [7,9,11]. At the same time, for various applications it is very important to describe behavior of the specific singularities explicitly, i.e. in a deterministic way. There is a number of laboratory methods for generating and manipulating phase [3,4] and polarization [12,13] singularities in electromagnetic fields. Therefore, one of the currently important problems is to know how singularities evolve as the wave propagates through a medium.
Propagation of uniformly-polarized paraxial beams with phase vortices in inhomogeneous or anisotropic media have been studied recently [14−16]. While such beams represent independent localized modes of a smoothly inhomogeneous isotropic medium [14], phase vortices become drastically unstable in an anisotropic medium [15,16]. Even in the simplest uniaxial homogeneous medium, the phase vortex disappears, giving way to an essentially space-variant polarization pattern with a variety of polarization singularities [15,16]. The following features are characteristic for this system: (i) an initial phase singularity, (ii) a uniform initial polarization, and (iii) a double refraction in the medium, which destroys the phase singularity and transforms in to a set of polarization singularities.
In the present paper we aim to investigate the dynamical behavior of the polarization singularities in a paraxial wave field propagating in a weakly anisotropic and, possibly, inhomogeneous medium. However, in contrast to [15,16], the problem is considered under opposite conditions: (i) a space-variant polarization pattern with polarization singularities in the incident field and (ii) absence of the phase singularities therein; (iii) we argue that double refraction is negligible in the system, while the variations of the normal modes parameters (phases and amplitudes) along the propagation direction lead to an effective dynamics of the polarization distribution and singularities. Our choice of the initial conditions is justified by two reasons. First, it is the presence of an effective technique using subwavelength gratings for generating arbitrary space-variant polarization patterns of the field [12,13]. Second, as it follows from [15,16], the phase singularities become unstable in anisotropic media, while the polarization ones experience a continuous evolution.
By applying a dynamical approach, well-established in standard polarimetry [17,18], to a space-variant polarization pattern, we develop a powerful method for studying 3D complex polarization distributions. Assuming paraxial approximation and weak anisotropy, our approach reduces a challenging wave problem to the solution of effectively ordinary differential equation for the Stokes vector evolution along the wave propagation direction [18−26]. The equation is integrated in a homogeneous medium analytically, and, despite its simple form, it reveals an intricate evolution of the polarization singularities when the wave propagates through the medium. Thus, our method brings together polarimetry and singular optics, thereby giving rise to singular polarimetry. It may have promising applications − space-variant polarization patterns with singularities can be more informative and sensitive with respect to the medium properties.

Statement of the problem
We will examine propagation of a paraxial monochromatic electromagnetic wave through a weakly anisotropic and, possibly, inhomogeneous (stratified) medium. We assume that the wave propagates along the z axis, whereas the polarization ellipse lies nearly in the ( ) , x y plane, so that one can apply Mueller or Jones calculus to the z -dependent evolution of polarization [17−19]. Under this assumption, the incident field is treated as a collection of parallel rays that have essentially independent phase and amplitude evolution. Mathematically, this means that we deal with a Cauchy problem with an initial distribution of the field in the ( ) , x y plane at 0 z = and some dynamical equation describing the evolution of the field along the z axis.
Let the polarization of the wave field at a point r be described through the threecomponent normalized Stokes vector, ( ) = s s r , 2 1 = s , representing the polarization state on the Poincaré sphere. Then, the Cauchy problem is given by the initial Stokes vector distribution, and a dynamical equation  [8,16]. Indeed, the north and south poles of the Poincaré sphere correspond to the right-and left-hand circularly polarized waves, while the equator represents linear polarizations with different orientations. Then, C-and L-type polarization singularities are determined, respectively, by the conditions From Eqs. (3) and (4) it is clear that in the generic case C-and L-singularities are, respectively, lines and surfaces in 3D space: the dimension of the singularity is the dimension of the space minus the number of constraints. Alternatively, one may refer to C-points and Llines in the ( )  3) and (4). Note also that Eq. (1) implies that there are no phase singularities in the initial field, i.e. the intensity of the wave does not vanish: , , 0 0 I x y I x y = ≠ (otherwise, the Stokes vector s would be undefined in nodal points).
As we will see, the dynamical equation (2) ensures that the nodal points cannot appear at 0 z ≠ as well: ( ) , , 0 I x y z ≠ . Our approach of z -dependent evolution, Eqs. (1) and (2), is justified assuming that the refraction and diffraction processes are negligible. Let the wave field be characterized by two scales: the wavelength λ and a typical scale of its transverse distribution in the ( )

Equation for the Stokes vector evolution
To derive the evolution equation (2) where M is the differential Mueller matrix (a 4 4 × real matrix) which summarizes optical properties of the medium. These are given by 2 2 × complex dielectric tensor: Here 0 2 I ε is the main, isotropic part proportional to the unit matrix ν is a small anisotropic part (which effectively represents the differential Jones matrix), and we assume 0 Im 0 ε = (small dissipation is ascribed to the anisotropic term).
Here the complex 4-component vector where 0 k is the wave number in vacuum, and components of quantities (7) is the unit matrix. The first, diagonal part, proportional to 0 Im G , describes the common attenuation of the field intensity. The second, symmetric part, related to components of Im G , describes the phenomenon of dichroism, i.e. the selective attenuation of different field components. Finally, the third, antisymmetric part of M , related to components of Re G , is responsible for the medium birefringence. The component 0 Re G does not contribute to the matrix (8), since it causes merely an additional total phase of the wave field, which does not affect the polarization state and is lost in the Stokes vector representation.
Evolution of the normalized 3-component Stokes vector can be derived immediately from Eqs. (6) and (8) where we denoted Re ≡ G Ω and Im ≡ G Σ . This equation represents the basic evolution equation (2)

Solutions in a homogeneous birefringent medium
In a homogeneous non-dissipative birefringent medium, Eq. (10) According to Eq. (11), as the wave propagates along the z axis, the Stokes vector s precesses with a constant spatial frequency Ω about the fixed direction / = Ω ω Ω . In terms of the medium properties, direction ω and absolute value Ω characterize, respectively, the type and the strength of the medium birefringence. In so doing, two "stationary" solutions ± = ± s ω on the Poincaré sphere correspond to mutually-orthogonal eigenmodes of the medium. In particular, , , 0 = ω ω ω correspond, respectively, to the cases of circularly-and linearly-birefringent medium. Equation (11)  In a circularly birefringent medium, distribution of polarization singularities in ( ) , x y plane does not vary when the wave propagates along the z axis. Indeed, the helicity of the wave, given by the third component of the Stokes vector, is invariant of Eqs. (11) and (12) and 03 0 s = .
On the contrary, polarization singularities evolve in a linearly-birefringent medium, cf. [15,16]. As it is clear from Eq. (12), this evolution is periodic in z with the period 2 / π Ω . In fact, L-lines and C-points in ( ) , x y plane come back to the initial locations after / π Ω period, corresponding to the half-wavelength plate. In so doing, C-points only change their signs after / π Ω period. One can also note that under propagation at / 2 π Ω distance (corresponding to the quarter-wavelength plate) C-points give their place to points of L-lines, while some points of L-lines give place to C-points. To determine the whole 3D structure of polarization singularities note that vector ( ) In contrast to Eqs. (13) and (14), these rather simple equations reveal non-trivial z -dependent dynamics of polarization singularities (see examples in Section 3.2).

Solutions in a homogeneous dichroic medium
In a homogeneous dichroic medium, with selective attenuation of modes but without a phase difference between them (i.e., 0 = Ω ), Eq. (10) takes the form Similarly to Eq. (11), this equation has two "stationary" solutions ± = ± s σ (where / = Σ σ Σ ), which determine eigenmodes of the medium. However, in contrast to the birefringent-medium case, solution − s is "unstable". As we will see, solutions of Eq. (17) move on the Poincaré sphere away from − s towards + s . Thus, the dichroic medium is a polarizer, in which only one mode (given by + s ) survives at long enough propagation distances. Equation (17) can be integrated analytically at const = Σ , which yields the solution (see Appendix): Thus, unlike birefringent medium, evolution of the Stokes vector and polarization singularities in dichroic medium is monotonic rather than periodic, see examples in Section 3.3. It is described by hyperbolic functions, which appear naturally in the Lorentzgroup representation of polarization optics [30,31].

Initial polarization distribution
As a characteristic example of initial space-variant polarization pattern, Eq. (1), we consider a vectorial vortex, which possesses a singularity in the polarization distribution [12,13]. The Stokes-vector distribution (1) of a vectorial vortex at the origin can be given as ( ) 03 s f ρ = .
is the integer number (the azimuthal index of the polarization distribution), and δ indicates a fixed angle between the distribution and x axis.
In the above distribution, the Stokes vector experiences m complete rotations along a loop path enclosing the vortex center. Therefore, the distribution possesses the 1 m − -fold rotational symmetry (one turn is effectively compensated by a 2π rotation of local radial vector), Fig. 1. At the same time, the corresponding polarization pattern (i.e. the distribution of polarization ellipses in the ( ) , x y plane) reveals 2 m − -fold symmetry, Fig. 1. This is because a complete turn of the Stokes vector corresponds to a half-turn of the polarization ellipse; as a result, the symmetry of the polarization distribution is characterized by the order of 2 mod mπ π π − . Note that the cases 1 m = and 2 m = are peculiar: the vectorial vortex represents an azimuthally-symmetric Stokes-vector and polarization distributions, respectively. Points

Homogeneous linearly-birefringent medium
Since the behavior of polarization singularities in a circularly-birefringent medium is trivial, Eqs. (13) and (14), let us consider the case of linearly-birefringent medium. Substituting the initial Stokes-vector distribution, Eqs. (25), into Eqs. (15) and (16) we obtain the equations describing C-lines and L-surfaces in space. For C-lines, this yields Here 0,1,..., 2 1 n m = − , signs " ± " in the second equation correspond to even and odd n , respectively. Note that ( ) tan z Ω is either positive or negative at each value of z , and, hence, only solutions with either even or odd n are valid each time. They alternate after a period of / 2 π Ω , whereas all the structure of polarization singularities (up to sign of the polarization) has a period of / π Ω . L-surfaces (28) separate C-lines with different helicities χ (and space areas with positive and negative 3 s ) and represent azimuthally-corrugated surfaces. Figure 2 shows an example of the polarization singularities described by Eqs. (29) and (28). The initial C-point of m th order splits into m branches under the evolution along z . This reveals an instability of higher-order C-points during evolution in an anisotropic medium. Only C-points with of a minimal order, 1 m = ± , are generic [5−9]. Here a indicates different C-points and summation is taken over whole plane const z = . Figure Thus, the surface lies in the 0 z < or 0 z > half-space when 3 0 σ > or 3 0 σ < , respectively.
Structure of the polarization singularities with initial distribution (26) in a circularly-dichroic medium is shown in Fig. 4a.  Fig. 4b. Thus, similarly to the case of linearly-birefringent medium, the initial C-point of the m th order is split into m C-points with unit azimuthal indices, Fig. 5. The total topological charge, Eq. (30), is also conserved in this process.

Conclusion
To summarize, we have developed an efficient formalism describing the evolution of nonuniformly polarized waves in anisotropic media. Provided that refraction and diffraction effects are negligible, our method reduces the initial wave problem to the integration of a simple differential equation for the Stokes-vector evolution. Polarization singularities are readily found in the resulting space distribution of the Stokes vector. The evolution equation has been integrated analytically for the characteristic cases of homogeneous birefringent and dichroic media.
We have applied the general formalism to the evolution of a polarization vortex in birefringent and dichroic media. The resulting space polarization patterns describe remarkable behavior of polarization singularities as the wave propagates in the medium. In particular, we showed the splitting of a higher-order vectorial vortex into a number of the minimal-order generic vortices and verified conservation of the topological charge under that process.
The fine behavior of the wave-field singularities in anisotropic media can be used for needs of polarimetry, and, perhaps, will enable one to increase the sensitivity of classical polarimetric methods.
Finally, though we considered a rectilinear propagation of the wave along the z axis, our approach is also valid for the geometrical-optics wave propagation along smooth curvilinear rays in large-scale inhomogeneous media. In this case, the problem is reduced to the same form if one involves a local ray coordinate system with basis vectors being paralleltransported along the ray [26].
Note added.-Recent paper [35] with related arguments came to our attention after submission of this work.
Appendix: Solution of equation (17) To integrate Eq. (17), note that the unit vector s can be represented as Here we introduced two auxiliary quantities: scalar A = sσ and vector ( ) = × B s σ . From Eq. (17), it can be easily seen that they obey equations ( ) By integrating Eq. (A2) we obtain