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

Optics Express, Vol. 16, Issue 2, pp. 695-709 (2008)

http://dx.doi.org/10.1364/OE.16.000695

Acrobat PDF (1104 KB)

### Abstract

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.

© 2008 Optical Society of America

## 1. Introduction

3. M.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. **42**, 219–276 (2001). [CrossRef]

4. L. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. **39**, 291–372 (1999). [CrossRef]

*polarization singularities*[5–9

5. J.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A **389**, 279–290 (1983). [CrossRef]

10. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. **198**, 21–27 (2001). [CrossRef]

*statistical*approach [7

7. M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A **457**, 141–155 (2001). [CrossRef]

9. M.R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. **213**201–221 (2002). [CrossRef]

11. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. **101**, 247–264 (1993). [CrossRef]

*deterministic*way. There is a number of laboratory methods for generating and manipulating phase [3

3. M.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. **42**, 219–276 (2001). [CrossRef]

4. L. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. **39**, 291–372 (1999). [CrossRef]

12. E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. **47**, 215–289 (2005). [CrossRef]

13. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express **14**, 4208–4220 (2006). [CrossRef] [PubMed]

14. K. Yu. Bliokh, “Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect,” Phys. Rev. Lett. **97**, 043901 (2006). [CrossRef] [PubMed]

14. K. Yu. Bliokh, “Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect,” Phys. Rev. Lett. **97**, 043901 (2006). [CrossRef] [PubMed]

15. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. **95**253901 (2005). [CrossRef] [PubMed]

16. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express **14**, 11402–11411 (2006). [CrossRef] [PubMed]

15. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. **95**253901 (2005). [CrossRef] [PubMed]

16. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express **14**, 11402–11411 (2006). [CrossRef] [PubMed]

15. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. **95**253901 (2005). [CrossRef] [PubMed]

16. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express **14**, 11402–11411 (2006). [CrossRef] [PubMed]

12. E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. **47**, 215–289 (2005). [CrossRef]

13. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express **14**, 4208–4220 (2006). [CrossRef] [PubMed]

**95**253901 (2005). [CrossRef] [PubMed]

**14**, 11402–11411 (2006). [CrossRef] [PubMed]

*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.

## 2. General theory

### 2.1 Statement of the problem

*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

*z*=0 and some dynamical equation describing the evolution of the field along the

*z*axis.

**r**be described through the three-component normalized Stokes vector,

**s**=

**s**(

**r**),

**s**

^{2}=1, representing the polarization state on the Poincaré sphere. Then, the Cauchy problem is given by the initial Stokes vector distribution,

**s**,

*z*) is a matrix operator which relates the Stokes-vectors values in two neighbor points. Although equations similar to Eq. (2) are well-established in classical polarimetry [18–26], they usually assume the uniform polarization distribution in the transverse plane,

**s**

_{0}(

*x*,

*y*)=const, making the polarization evolution effectively

*one-dimensional*,

**s**=

**s**(

*z*). In contrast, a non-uniform initial distribution (1) in our problem,

**s**

_{0}(

*x*,

*y*)≠const, makes the polarization distribution essentially

*three-dimensional*,

**s**=

**s**(

*x*,

*y*,

*z*). Despite this, in order to find the complete distribution of the polarization in 3D space,

**s**(

**r**), one need to integrate an effectively

*ordinary*differential equation (2) with initial conditions (1) for each point (

*x*,

*y*).

8. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. **201**, 251–270 (2002). [CrossRef]

**14**, 11402–11411 (2006). [CrossRef] [PubMed]

*x*,

*y*) plane (2D space) and their evolution along the

*z*axis. Note also, that polarization singularities are essentially determined by the third component of the Stokes vector - conditions (3) are equivalent to |

*s*

_{3}|=1, or

*s*

_{3}=

*χ*, where

*χ*=±1 is the wave helicity which indicates the sign of polarization in the C-point. Having a solution of the problem Eqs. (1) and (2),

**s**=

**s**(

**r**), one immediately gets the space distribution of all polarization singularities from Eqs. (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:

*I*

_{0}(

*x*,

*y*)=

*I*(

*x*,

*y*,0)≠0 (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

*z*≠0 as well:

*I*(

*x*,

*y*,

*z*)≠0.

*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 (

*x*,

*y*) plane,

*w*≫

*λ*. At the same time, the medium anisotropy is characterized by a typical difference between the dielectric constants corresponding to the normal modes,

*ν*≪1. Then, diffraction and refraction effects are negligible if: (i) the propagation distance is much smaller than the typical diffraction distance (the Rayleigh range),

*Z*≪

*Z*=

_{R}*w*

^{2}/

*λ*and (ii) the propagation distance is much smaller than the distance at which the double refraction of the anisotropic medium causes transverse shifts comparable with

*w*, i.e.

*Z*≪

*Z*≡

_{D}*w*/

*ν*. Note that the characteristic distance of the polarization evolution due to Eq. (2),

*Z*=

_{P}*λ*/

*ν*, is much smaller than

*Z*and can be small as compared to

_{D}*Z*. Thus, our approach is effective within the range of distances

_{R}*λ*⋍0.6µm and width

*w*⋍1mm propagating through Quartz (where anisotropy is

*ν*⋍0.03), we have

*Z*~1.5m,

_{R}*Z*~30mm, and

_{D}*Z*⋍0.02mm. This gives the propagation range 0.02mm≤

_{P}*z*≪30mm.

### 2.2 Equation for the Stokes vector evolution

**s**, let us start with the 4-component Stokes vector,

*S⃗*. Hereafter, 4-component vectors are indicated by arrows, and the last three components of a 4-vector form usual 3-component vector, so that

*S⃗*=(

*S*

_{0},

*S*

_{1},

*S*

_{2},

*S*

_{3})≡(

*S*

_{0},

**S**). In the most general case of a linear anisotropic medium the Stokes vector S⃗ obeys the following evolution equation [18–26]:

*ε*

_{0}

*Î*

_{2}is the main, isotropic part proportional to the unit matrix

*Î*

_{2}=diag (1,1),

*is a small anisotropic part (which effectively represents the differential Jones matrix), and we assume Im*ν ^

*ε*

_{0}=0 (small dissipation is ascribed to the anisotropic term). The differential Mueller matrix can be represented as [19

19. R.M.A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus,” J. Opt. Soc. Am. **68**, 1756–1767 (1978). [CrossRef]

24. R. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. **116**, 285–294 (2006). [CrossRef]

26. Y.A. Kravtsov, B. Bieg, and K.Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A **24**, 3388–3396 (2007). [CrossRef]

*G⃗*=(

*G*

_{0},

**G**) is expressed via components of the dielectric tensor (7) as

*k*

_{0}is the wave number in vacuum, and components of quantities (7)–(9) can be

*z*- dependent. Vector

*G⃗*gives decomposition of the anisotropy tensor

*, Eq. (6), with respect to the basis of Pauli matrices, and establishes close relations between polarization optics (Mueller and Jones calculus) and relativistic problems with the Lorentz-group symmetry [20*ν ^

20. C.S. Brown and A.E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. **34**, 1625–1635 (1995). [CrossRef]

24. R. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. **116**, 285–294 (2006). [CrossRef]

26. Y.A. Kravtsov, B. Bieg, and K.Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A **24**, 3388–3396 (2007). [CrossRef]

26. Y.A. Kravtsov, B. Bieg, and K.Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A **24**, 3388–3396 (2007). [CrossRef]

**s**=

**S**/

*S*

_{0}. As a result, we find that the 3-component Stokes vector obeys the following equation [26

**24**, 3388–3396 (2007). [CrossRef]

**G**≡

**Ω**and Im

**G**≡

**Σ**. This equation represents the basic evolution equation (2) in the general case. [In terms of Eq. (2), the matrix m̂ is given by m

_{ij}=-

*e*Ω

_{ijk}_{k}-

*e*

_{ijk}*e*

_{klm}*s*Σ

_{l}_{m}, where indices take values 1, 2, 3 and

*e*is the unit antisymmetric tensor.] Thus, all the evolution on the Poincaré sphere can be described by the precession equation (10) which includes two real vectors

_{ijk}**Ω**and

**Σ**responsible for the birefringent and dichroic effects, respectively. Equation (10) conserves the absolute value of the normalized Stokes vector under the evolution: ∂

**s**

^{2}/∂

*z*=0. The common attenuation, Im

*G*

_{0}, naturally, does not affect the normalized Stokes vector and is absent in Eq. (10). Note also that Eq. (10) resembles the Landau-Lifshitz equation describing the nonlinear spin precession in ferromagnets [33], but, in contrast to the latter, Eq. (10) contains two

*different*effective fields

**Ω**and

**Σ**. In inhomogeneous medium

**Ω**=

**Ω**(

*z*) and

**Σ**=

**Σ**(

*z*).

### 2.3 Solutions in a homogeneous birefringent medium

22. H. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. **80**, 1888–1891 (1998). [CrossRef]

23. S.E. Segre, “New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma,” J. Opt. Soc. Am. A **18**, 2601–2606 (2001). [CrossRef]

25. K.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A **75**, 053821 (2007). [CrossRef]

*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,

**ω**=(0, 0,±1) and

**ω**=(ω

_{1},ω

_{2},0) correspond, respectively, to the cases of circularly- and linearly-birefringent medium. Equation (11) with initial condition (1) can be easily integrated:

**s**(

*x*,

*y*,

*z*)=

*R̂*

_{ω}(Ω

*z*)

**s**

_{0}(

*x*,

*y*), where

*R̂*

_{ω}(Ω

*z*) is the operator of rotation about

**ω**on the angle Ω

*z*. Using the Rodrigues rotation formula [34], we arrive at

*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),

*s*

_{3}=const, when

**ω**=(0, 0,±1). Thus, C-lines (

*s*

_{3}=±1) and L-surfaces (

*s*

_{3}=0) are parallel to the

*z*axis in this case and are trivially determined by the initial distribution (1) with Eqs. (3) and (4):

**95**253901 (2005). [CrossRef] [PubMed]

**14**, 11402–11411 (2006). [CrossRef] [PubMed]

*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

*π*/2Ω 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

**ω**=(ω

_{1}, ω

_{2}, 0) can be reduced to

**ω**=(1,0,0) by a fixed rotation of coordinate axes in the (

*x*,

*y*) plane, which brings the anisotropy tensor

*, Eq. (7), to the principal axes. Then, substituting solution (12) with*ν ^

**ω**=(1,0,0) into Eqs. (3) and (4), we obtain equations determining C-lines and L-surfaces:

### 2.4 Solutions in a homogeneous dichroic medium

**Ω**=0), Eq. (10) takes the form

**s**

^{±}=±

**σ**(where

**σ**=

**Σ**/Σ), which determine eigenmodes of the medium. However, in contrast to the birefringent-medium case, solution on

**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

**Σ**=const, which yields the solution (see Appendix):

*A*

_{0}=

**s**

_{0}

**σ**. As seen from Eq. (18), all solutions with

**s**

_{0}≠

**s**

^{-}move in the plane given by vectors

**s**

_{0}and

**σ**, and tend exponentially to

**s**

^{+}point. Supplied with the initial distribution (1), Eq. (18) gives the Stokes-vector distribution in 3D space.

*z*both in circularly and linearly dichroic medium. More precisely, in a circularly-dichroic medium,

**σ**=(0,0,±1), C-points in the (

*x*,

*y*) plane do not evolve with

*z*since circular polarizations correspond to “stationary” solutions

**s**

^{±}in this case. Thus C-lines in 3D space are parallel to the

*z*axis and determined analogously to Eq. (13):

*x*,

*y*) plane evolve with

*z*, and L-surfaces in 3D space are determined by Eq. (4) with (18):

**σ**=(1, 0, 0) in the principal-axes coordinate frame, L-surfaces are parallel to the

*z*axis and are given by Eq. (14):

*x*,

*y*) plane evolve with

*z*, and 3D C-lines are determined by Eq. (3) with (18), i.e.

*A*

_{0}=

*s*

_{01}. Resolving of this equation yields:

30. T. Opartny and J. Perina, “Non-image-forming polarization optical devices and Lorentz transformation — an analogy,” Phys. Lett. A **181**, 199–202 (1993). [CrossRef]

31. D. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A **219**, 26–32 (1996). [CrossRef]

## 3. Application: Evolution of a vectorial vortex

### 3.1 Initial polarization distribution

*vectorial vortex*, which possesses a singularity in the polarization distribution [12

12. E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. **47**, 215–289 (2005). [CrossRef]

13. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express **14**, 4208–4220 (2006). [CrossRef] [PubMed]

*ρ*,

*φ*) are the polar coordinates in the (

*x*,

*y*) plane,

*f*(

*ρ*) is a radial distribution function, |

*f*(

*ρ*)|≤1,

*m*=±1,±2,… 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 |

*m*-1| -fold rotational symmetry (one turn is effectively compensated by a 2

*π*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 |

*m*-2| -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 |

*mπ*-2

*π*|mod

*π*. Note that the cases

*m*=1 and

*m*=2 are peculiar: the vectorial vortex represents an azimuthally-symmetric Stokes-vector and polarization distributions, respectively.

*ρ*=

*ρ*and

_{C}*ρ*=

*ρ*, such that

_{L}*f*(

*ρ*)=

_{C}*χ*=±1 and

*f*(

*ρ*)=0 correspond to C-and L-type singularities in the initial distribution (25). We will concentrate on a simple case of a single C-point with the right-hand circular polarization at the origin, so that

_{L}*f*(0)=

*χ*and 0<|

*f*(

*ρ*)|<1 at

*ρ*>0. This case can be modeled using the function

*ρ** characterizes the radial scale of the distribution. The vectorial vortex (25) and (26) with the right-hand polarization in the center,

*χ*=1, is characterized by the optical vortex (phase singularity) in the left-hand polarized component of the field, and vice versa. Below, we will examine the behavior of polarization singularities in homogeneous anisotropic media considered in Sections 2.3 and 2.4, assuming the initial polarization distribution of Eqs. (25) and (26).

### 3.2 Homogeneous linearly-birefringent medium

*φ*=const and given by equations

*n*=0,1,…, 2

*m*-1, signs “±” in the second equation correspond to even and odd

*n*, respectively. Note that tan(Ω

*z*) 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

*π*/2Ω, 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

*s*

_{3}) 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,

*m*=±1, are generic [5–9

5. J.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A **389**, 279–290 (1983). [CrossRef]

**95**253901 (2005). [CrossRef] [PubMed]

*topological charge*by tracking the polarization singularities evolution. Topological charge,

*γ*, characterizes each C-point in plane

*z*=const and is equal to the product of its local azimuthal index,

*m*, and sign of the polarization,

*χ*=

*s*

_{3}:

*γ*=

*mχ*. The total topological charge,

*z*- independent [7

7. M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A **457**, 141–155 (2001). [CrossRef]

9. M.R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. **213**201–221 (2002). [CrossRef]

**95**253901 (2005). [CrossRef] [PubMed]

*a*indicates different C-points and summation is taken over whole plane

*z*=const. Figure 3 demonstrates polarization and Stokes-vector distributions, Eq. (12), for a vectorial vortex evolving in a linearly birefringent medium corresponding to Fig. 2. Conservation of Γ=3 is seen — initial C-point with

*m*=3 and

*χ*=1 is split into three C-points with

*m*=1 and

*χ*=1; then, crossing of L-surface causes simultaneous flip of helicity and azimuthal index: three C-points with

*m*=-1 and

*χ*=-1 occur; finally, they merge into single C-point with

*m*=-3 and

*χ*=-1.

### 3.3 Homogeneous dichroic media

*z*, Eq. (19), whereas the behavior of L-surfaces is given by Eq. (21) with Eq. (25):

*z*<0 or

*z*>0 half-space when

*σ*

_{3}>0 or

*σ*

_{3}<0, respectively. Structure of the polarization singularities with initial distribution (26) in a circularly-dichroic medium is shown in Fig. 4a.

*φ*=const:

*n*=0,1,…, 2

*m*-1, and signs “∓” in the second equation correspond to even and odd

*n*, respectively. It is easily seen that solutions with even and odd

*n*are realized, respectively, at

*z*<0 and

*z*>0, 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.

## 4. Conclusion

*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 parallel-transported along the ray [26

**24**, 3388–3396 (2007). [CrossRef]

## Appendix: Solution of equation (17)

**s**can be represented as

*A*=

**sσ**and vector

**B**=(

**s**×

**σ**). From Eq. (17), it can be easily seen that they obey equations

## Acknowledgements

## References and links

1. | M.V. Berry, “Singularities in waves and rays,” in R. Balian, M. Kléman, and J.-P. Poirier, editors, |

2. | J.F. Nye, |

3. | M.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. |

4. | L. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. |

5. | J.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A |

6. | J.F. Nye and J.V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London A |

7. | M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A |

8. | I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. |

9. | M.R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. |

10. | J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. |

11. | I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. |

12. | E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. |

13. | A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express |

14. | K. Yu. Bliokh, “Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect,” Phys. Rev. Lett. |

15. | F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. |

16. | F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express |

17. | R.M.A. Azzam and N.M. Bashara, |

18. | C. Brosseau, |

19. | R.M.A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus,” J. Opt. Soc. Am. |

20. | C.S. Brown and A.E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. |

21. | C. Brosseau, “Evolution of the Stokes parameters in optically anisotropic media,” Opt. Lett. |

22. | H. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. |

23. | S.E. Segre, “New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma,” J. Opt. Soc. Am. A |

24. | R. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. |

25. | K.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A |

26. | Y.A. Kravtsov, B. Bieg, and K.Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A |

27. | R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. |

28. | R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. |

29. | S.R. Cloude, “Group theory and polarization algebra,” Optik |

30. | T. Opartny and J. Perina, “Non-image-forming polarization optical devices and Lorentz transformation — an analogy,” Phys. Lett. A |

31. | D. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A |

32. | V.B. Berestetskii, E.M. Lifshits, and L.P. Pitaevskii, |

33. | C.P. Slichter, |

34. | H. Goldstein, C. Poole, and J. Safko, |

35. | A.D. Kiselev, “Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells,” J. Phys.: Condens. Matter |

**OCIS Codes**

(260.1180) Physical optics : Crystal optics

(260.2130) Physical optics : Ellipsometry and polarimetry

(260.5430) Physical optics : Polarization

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 29, 2007

Revised Manuscript: December 13, 2007

Manuscript Accepted: December 15, 2007

Published: January 8, 2008

**Citation**

Konstantin Yu. Bliokh, Avi Niv, Vladimir Kleiner, and Erez Hasman, "Singular polarimetry: Evolution of polarization
singularities in electromagnetic waves
propagating in a weakly anisotropic medium," Opt. Express **16**, 695-709 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-695

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### References

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- A. D. Kiselev, "Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells," J. Phys.: Condens. Matter 19, 246102 (2007). [CrossRef]

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