## Stokes parameters in the unfolding of an optical vortex through a birefringent crystal

Optics Express, Vol. 14, Issue 23, pp. 11402-11411 (2006)

http://dx.doi.org/10.1364/OE.14.011402

Acrobat PDF (2277 KB)

### Abstract

Following our earlier work (F. Flossmann *et al.*, Phys. Rev. Lett. **95** 253901 (2005)), we describe the fine polarization structure of a beam containing optical vortices propagating through a birefringent crystal, both experimentally and theoretically.We emphasize here the zero surfaces of the Stokes parameters in three-dimensional space, two transverse dimensions and the third corresponding to optical path length in the crystal. We find that the complicated network of polarization singularities reported earlier – lines of circular polarization (C lines) and surfaces of linear polarization (L surfaces) – can be understood naturally in terms of the zeros of the Stokes parameters.

© 2006 Optical Society of America

## 1. Introduction

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

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

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

6. K. O’Holleran, M.J. Padgett, and M.R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express **14**3039–3044 (2006). [CrossRef] [PubMed]

7. M.V. Berry and M.R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A **456**2059–2079 (2000). [CrossRef]

8. L. Allen, M. Beijersbergen, R.J.C. Spreeuw, and J.P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes”, Phys. Rev. A **45**8185–8189 (1992). [CrossRef] [PubMed]

9. F. Flossmann, U.T. Schwarz, and M. Maier “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. **250**218–230 (2005). [CrossRef]

*πn*, where

*n*is the integer vortex strength (charge).

10. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**233901 (2003). [CrossRef] [PubMed]

11. J.V. Hajnal, “Observation on singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. Lond. A **430**413–421 (1990). [CrossRef]

12. 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]

13. M.V. Berry and M.R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. Lond. A **459**1261–1292 (2003). [CrossRef]

15. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “Fine structure of singular beams in crystals: colours and polarization,” *J. Opt. A: Pure Appl. Opt.* **6**S217–S228 (2004). [CrossRef]

16. 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, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. **95**253901 (2005). [CrossRef] [PubMed]

## 2. Interpretation using Stokes Parameters

**E**=(

*E*

_{x},

*E*

_{y}) describes the polarization ellipse, parameterized by its overall size (intensity), angle of major axis

*α*, and a shape parameter

*ω*, defined such that |tan

*ω*| is the ratio of minor and major ellipse axes, and sign

*ω*is the ellipse handedness (see Figure 1(a)). Up to a phase factor, the polarization vector is therefore

*I*is the overall intensity |

**E**|

^{2},

*α*and

*ω*are the ellipse parameters, and

*β*and

*δ*are a pair of equivalent parameters [19], which prove to be more useful in our analysis. The unnormalized Stokes parameters are defined [18, 19]

*α*,

*ω*, and the third, the complementary parameters

*β*,

*δ*. Since

*S*

_{i}/

*S*

_{0}(

*i*=1,2,3)- this is the Poincaré sphere (Figure 1). From Eq. (2), angles 2

*α*and 2

*ω*are the azimuth and colatitude angles on the Poincaré sphere with

*S*

_{3}as the axis.

*δ*and

*β*may be understood as the corresponding half-angles for azimuth and latitude with respect to

*S*

_{1}as the axis. Physically,

*S*

_{0}is simply the field intensity

*I*, and

*S*

_{3}/

*S*

_{0}=sin 2

*ω*is the ellipticity, signed by handedness: positive for right-handed elliptical polarization, negative for left-handed. When

*S*

_{3}=0, the polarization is linear, so the zero contour surface of

*S*

_{3}is the L surface [1, 4

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

20. I. Freund, A.I. Mokhun, M.S. Soskin, O.V. Angelsky, and I.I. Mokhun, “Stokes singularity relations,” Opt. Lett. **27**545–547 (2002). [CrossRef]

*α*=arg(

*S*

_{1}+

*iS*

_{2})/2 is the angle of the polarization ellipse major axis, and is undefined when S1=S2=0; this is the condition for circular polarization (where

*ω*=±

*π*/2, |

*S*

_{3}|=

*S*

_{0}). The polarization singularity positions are therefore determined by the zero surfaces of the three Stokes parameters: L surfaces by

*S*

_{3}=0, and C lines by the intersection of the surfaces

*S*

_{1}=0 and

*S*

_{2}=0.

16. 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]

*w*

_{0}) is incident in the plane containing the surface normal and the optic axis, implying that the spatial shift between the two beam components is in the direction of one of the crystal eigenpolarizations, and the difference in geometrical path length is negligible compared to the difference in optical path length. The phase shift therefore arises from the difference in refractive index, i.e. Λ=2

*π*(

*n*

_{2}-

*n*

_{1})

*l*/

*λ*

_{vac}, where

*l*is the average path length of the two rays. We will comment in the final section on what happens for crystals with optical activity and absorption, but in our experiment, these effects are negligible.

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

**E**

_{in}(

*x*,

*y*)=

*ψ*(

*x*,

*y*)

**d**

_{in}, where

*ψ*(

*x*,

*y*) is the complex scalar amplitude of the incident field, and

**d**

_{in}is a unit vector representing the initial polarization, written in terms of

*β*

_{in}and

*δ*

_{in}by appropriate modification of Eq. (1). In the crystal, the two polarizations which propagate in the crystal are

**d**

_{1},

**d**

_{2}(defined to be in the

*x*,

*y*-directions), the path length-dependent phase shift is Λ and transverse shifts (in the

*x*-direction) ±

*s*result in a total outgoing field

*ψ*

_{±}=

*ψ*(

*x*±

*s*,

*y*) :

*S*_{0}and*S*_{1}are independent of Λ (since the only Λ-dependence in Eq. (3) is in the phase of the two Cartesian components);- At
*x*=±*s*,*y*=0,*S*_{1}≠0, but*S*_{2}=*S*_{3}=0; - The only effect of shifting
*δ*_{in}is effectively to shift Λ; *S*_{2},*S*_{3}are identical, except for a quarter-period Λ-shift.

*S*

_{1}, but transforms

*S*

_{2}to

*S*

_{3}, and

*S*

_{3}to -

*S*

_{2}. In Stokes space, this is illustrated by a rotation of the Poincaré sphere by

*π*/2 about the

*S*

_{1}-axis, corresponding to a change of

*π*/4 in

*δ*. This is represented in Fig. 3, in which experimental plots are compared for

**d**

_{in}oriented at 45° (Fig. 3(a)) and right handed circular polarized (Fig. 3(b)), where

*δ*

_{in}differs

*π*/4 and Λ by

*π*/2. On changing Λ by

*π*, the pattern changes in a way analogous to a half-wave plate (as observed in Ref. [16

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

*S*

_{1}is unchanged, but

*S*

_{2}and

*S*

_{3}change sign.

*ψ*(

*x*,

*y*). In Ref. [16

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

*ψ*

_{1}=(

*x*+

*iy*)exp(-(

*x*

^{2}+

*y*

^{2})/

*S*

_{1},

*S*

_{2},

*S*

_{3}. Using Eq. (4), the zeros occur at the following

*x*,

*y*,Λ- coordinates (where, for simplicity here and hereafter,

*δ*

_{in}=0) :

*S*

_{2},

*S*

_{3}=0 assume that sin 2

*β*≠0, i.e.

**d**

_{in}≠

**d**

_{1}or

**d**

_{2}.) As expected, the

*S*

_{1}=0 surface is independent of Λ. The solution for

*x*and

*y*is transcendental, depending only on the input

*β*

_{in}and the splitting

*s*. On the other hand, the surfaces

*S*

_{2}=0 and

*S*

_{3}=0 are independent of

*β*

_{in}, and are mathematically easily determined: each constant-Λ section is a circle, with radius |(co) secΛ| and center

*x*=0,

*y*=(co) tan Λ. Such a circle for

*S*

_{3}=0 is plotted as the L line in Fig. 3(c); the experimental results, although topologically the same, do not give circles, probably due to lack of perfect rotational symmetry of the input beam. Various representations of the experimentally-determined ellipse parameters and zero contours of Stokes parameters, for a fixed choice of Λ, are plotted in Fig. 4.

*x*,

*y*,Λ-space, are plotted in Fig. 5. The surface for

*S*

_{1}=0 consists of a tube and a nearby sheet. The periodic surfaces

*S*

_{2},

*S*

_{3}=0 are identical, and topologically the same as Riemann’s periodic minimal surface [17]. The two surfaces intersect at (

*x*,

*y*)=(±

*s*,0), i.e. at the position of the shifted vortex in each of the two waves. Although topologically the same as Riemann’s example, using the expressions in Eq. (5), it can be shown that the surfaces are not themselves minimal (i.e. the mean curvature does not vanish everywhere [21]). The intersection of the translation-invariant

*S*

_{1}=0 surface with the surface

*S*

_{2}=0 gives rise to the C line geometry described in Ref. [16

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

*S*

_{1}=0 tube) and a transversally infinite C line (lying on the

*S*

_{1}=0 sheet). If

*β*

_{in}changes (this is the angle of the polarization if

**d**

_{in}is linearly polarized), the

*S*

_{2},

*S*

_{3}=0 surfaces do not change, but the

*S*

_{1}surface does; in particular, when

*β*

_{in}=45°, and the C lines reconnect, the

*S*

_{1}surface and tube touch along two self-intersection lines.

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

*ψ*

_{0}=exp(-(

*x*

^{2}+

*y*

^{2})/

*δ*

_{in}is assumed to be zero, and sin 2

*β*≠0.) The Λ-invariant

*S*

_{1}surface is just a single plane with constant

*x*, the

*S*

_{2}and

*S*

_{3}surfaces are planes occurring at constant Λ. Thus the output polarization field has Λ-layers alternating in handedness, with a single C line between two adjacent L surfaces. Since the C line lies in the transverse

*x*,

*y*-plane, its index is not defined [3

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

*S*

_{1},

*S*

_{2},

*S*

_{3}for an input Gaussian beam are plotted in Fig. 6.

*n*(assumed positive here for simplicity), i.e.

*ψ*

_{n}(

*x*,

*y*)=(

*x*+

*iy*)

^{n}exp(-(

*x*

^{2}+

*y*

^{2})/

*S*

_{1}…

*S*

_{3}are given by:

*n*=1), and, with appropriate limits, (6) (when

*n*=0). The zero contour for

*S*

_{1}is topologically the same as the

*n*=1 case described above, and the factor 1/

*n*just scales the beam waist

*w*

_{0}. However, there are differences in the zero contours of

*S*

_{2}and

*S*

_{3}. Although the sections of constant Λ are again circles, for each constant L there are now

*n*distinct solutions, resulting in exactly

*n*circles with different radii and centers. The corresponding surfaces are the same as those in the

*n*=1 case, but occur with an

*n*-fold multiplicity whilst being shifted in the Λ-direction and intersecting each other in a highly nongeneric and unstable way.

*n*=2, the Stokes parameter zero surfaces are plotted in Fig. 7, both experimentally (7(a)) and theoretically (7(b)). There are now six C lines, with four coiling around each other, with an overall pitch of 4

*π*instead of 2

*π*.

**E**, equivalent to stereographic projection of the Poincaré sphere in the positive

*S*

_{1}-direction, rather than the

*S*

_{3}direction, as is more usual [18, 13

13. M.V. Berry and M.R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. Lond. A **459**1261–1292 (2003). [CrossRef]

*ρ;*in particular, the Stokes zero contour surfaces are specified by Re

*ρ*=0 (

*S*

_{2}=0), Im

*ρ*=0 (

*S*

_{3}=0), and |

*ρ*|=1 (

*S*

_{1}=0), and

*ρ*has a zero or pole at the positions of the refracted vortices of the input field.

*n*axial vortices embedded), the Gaussian factor gives rise to a factor of exp(-4

*xs*/

*w*

^{2}); for the field

*ψ*

_{01}, this is multiplied by (

*x*+

*s*+

*iy*)

^{n}/(

*x*-

*s*+

*iy*)

^{n}.

*β*,2

*δ*).

*S*

_{1}has this privilege as the crystal eigenpolarizations have

*S*

_{1}=±

*S*

_{0},

*S*

_{2},

*S*

_{3}=0. The behavior of all incoming polarizations with the same

*β*(=arctan|

*d*

_{in,y}/

*d*

_{in,x}|) on the Poincaré sphere is the same. Also, the surface locus of polarizations lying on any great circle including the

*S*

_{1}-direction is the same as the surfaces

*S*

_{2}=0,

*S*

_{3}=0, up to a phase shift (corresponding to a surface Re

*ρ*exp(-

*iχ*) for some phase angle

*χ*). Finally, the loci of any pair of orthogonal polarizations corresponding to opposite points on the great circle perpendicular to the

*S*

_{1}-direction will be the same as the C lines, again up to a phase shift.

## 3. Discussion

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

13. M.V. Berry and M.R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. Lond. A **459**1261–1292 (2003). [CrossRef]

22. A. Volyar, V. Shvedov, T. Fadeyeva, A.S. Desyatnikov, D.N. Neshev, W. Krolikowski, and Y.S. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal” Opt. Express **14**3724–3729 (2006). [CrossRef]

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

23. M.R. Dennis, “Braided nodal lines in wave superpositions,” New J. Phys. **5**134 (2003). [CrossRef]

23. M.R. Dennis, “Braided nodal lines in wave superpositions,” New J. Phys. **5**134 (2003). [CrossRef]

*S*

_{2}=0 and

*S*

_{1}=0. Since the latter must be independent of Λ, a (pigtail) braid requires a transverse section of this surface to be a figure-8 [23

23. M.R. Dennis, “Braided nodal lines in wave superpositions,” New J. Phys. **5**134 (2003). [CrossRef]

*S*

_{2}=0 surface to cross the Λ-invariant

*S*

_{1}=0 surface three times for each Λ. This is clearly impossible, as contour surfaces intersect generically an even number of times, proving that braided C lines are impossible using Eq. (3).

6. K. O’Holleran, M.J. Padgett, and M.R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express **14**3039–3044 (2006). [CrossRef] [PubMed]

*S*

_{1}=0), and a pair of orthogonal helicoids along the same axis (

*S*

_{2},

*S*

_{3}=0): this results in a double helix of C lines with opposite sign and handedness. Although illustrative, this example cannot be physically realized in an experiment involving crystal birefringence, as both fields must be the same up to a spatial and phase shift.

**459**1261–1292 (2003). [CrossRef]

## Acknowledgements

## References and links

1. | J. F. Nye, |

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

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

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

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

6. | K. O’Holleran, M.J. Padgett, and M.R. Dennis, “Topology of optical vortex lines formed by the interference of three, four and five plane waves,” Opt. Express |

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

8. | L. Allen, M. Beijersbergen, R.J.C. Spreeuw, and J.P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes”, Phys. Rev. A |

9. | F. Flossmann, U.T. Schwarz, and M. Maier “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. |

10. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

11. | J.V. Hajnal, “Observation on singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. Lond. A |

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

13. | M.V. Berry and M.R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. Lond. A |

14. | M.V. Berry and M.R. Jeffrey, “Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics,” Prog. Opt. in press. |

15. | Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “Fine structure of singular beams in crystals: colours and polarization,” |

16. | 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. |

17. | B. Riemann, in |

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

19. | D.S. Kliger, J.W. Lewis, and C.E. Randall, |

20. | I. Freund, A.I. Mokhun, M.S. Soskin, O.V. Angelsky, and I.I. Mokhun, “Stokes singularity relations,” Opt. Lett. |

21. | D.J. Struik, |

22. | A. Volyar, V. Shvedov, T. Fadeyeva, A.S. Desyatnikov, D.N. Neshev, W. Krolikowski, and Y.S. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal” Opt. Express |

23. | M.R. Dennis, “Braided nodal lines in wave superpositions,” New J. Phys. |

24. | F. Flossmann, |

**OCIS Codes**

(260.1440) Physical optics : Birefringence

(260.2110) Physical optics : Electromagnetic optics

(260.5430) Physical optics : Polarization

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 29, 2006

Revised Manuscript: October 31, 2006

Manuscript Accepted: October 31, 2006

Published: November 13, 2006

**Citation**

Florian Flossmann, Ulrich T. Schwarz, Max Maier, and Mark R. Dennis, "Stokes parameters in the unfolding of an optical vortex through a birefringent crystal," Opt. Express **14**, 11402-11411 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11402

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### References

- J. F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations, (IoP Publishing, 1999).
- M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001). [CrossRef]
- J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983). [CrossRef]
- M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002). [CrossRef]
- M. V. Berry, "Singularities in waves and rays," in Les Houches, Session XXV - Physics of Defects, R. Balian, M. Kléman, and J.-P. Poirier, eds., (North-Holland, 1981).
- K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 143039-3044 (2006). [CrossRef] [PubMed]
- M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000). [CrossRef]
- L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992). [CrossRef] [PubMed]
- F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005). [CrossRef]
- R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003). [CrossRef] [PubMed]
- J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990). [CrossRef]
- A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 144208-4220 (2006). [CrossRef] [PubMed]
- M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003). [CrossRef]
- M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.
- Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004). [CrossRef]
- 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. 95253901 (2005). [CrossRef] [PubMed]
- B. Riemann, in Bernhard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlass, edited by H. Weber (Teubner, Leipzig, 1892) p. 301.
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).
- D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized light in optics and spectroscopy (Academic Press, San Diego, 1990).
- I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, "Stokes singularity relations," Opt. Lett. 27545-547 (2002). [CrossRef]
- D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988).
- A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, "Generation of single-charge optical vortices with an uniaxial crystal" Opt. Express 143724-3729 (2006). [CrossRef]
- M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003). [CrossRef]
- F. Flossmann, Singularit¨aten von Phase und Polarisation des Lichts (PhD Thesis, University of Regensburg, 2006).

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