## Polarization singularities in superposition of vector beams |

Optics Express, Vol. 21, Issue 7, pp. 8972-8986 (2013)

http://dx.doi.org/10.1364/OE.21.008972

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### Abstract

We present a systematic study of the superposition of two vector Laguerre-Gaussian (LG) beams. Propagation depended field distribution obtained from the superposition of two vector LG beams has many interesting features of intensity and polarization. Characteristic inhomogeneous polarization distribution of the vector LG beam appears in the form of azimuthally modulated intensity and polarization distributions in the superposition of the beams. We found that the array of polarization singular points, whose number depends upon the azimuthal indices of the two beams, evolves during propagation of the field. The position and number of C-points generated in the field were analyzed using Stokes singularity relations. Novel intensity and polarization patterns obtained from the superposition of two vector LG beams may find applications in the field of molecular imaging, optical manipulation, atom optics, and optical lattices.

© 2013 OSA

## 1. Introduction

13. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. **31**(6), 820–822 (2006). [CrossRef] [PubMed]

15. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics **2**(8), 501–505 (2008). [CrossRef]

16. X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun **3**, 998 (2012). [CrossRef] [PubMed]

17. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. **9**(3), 78 (2007). [CrossRef]

18. G. M. Lerman and U. Levy, “Radial polarization interferometer,” Opt. Express **17**(25), 23234–23246 (2009). [CrossRef] [PubMed]

## 2. Vector Laguerre-Gaussian beam

2. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A **15**(10), 2705–2711 (1998). [CrossRef]

19. Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A **29**(11), 2439–2443 (2012). [CrossRef] [PubMed]

*r*is the radius,

*ϕ*is the azimuthal angle,

*z*is the distance from the beam waist,

*p*is the radial mode index,

*m*is the azimuthal mode index and,

**i**

*, and*

_{ϕ}**i**

*are the unit vectors of electric field for the azimuthal and radial directions, respectively. The term*

_{r}*Y*(

_{p,m}*r, z*) is given by

*w*is the minimum beam radius at

_{0}*z*= 0,

*R*(

*z*) = (

*z*

^{2}

*+ z*

_{R}

^{2})/

*z*is the radius of curvature of the wavefront,

*z*

_{R}=

*kw*

_{0}

^{2}/2 is the Rayleigh length,

*k*is the wave number, (2

*p +*|

*m*| + 1)

*ψ*(

*z*) is the Gouy phase shift with

*ψ*(

*z*) =

*tan*

^{−1}(

*z*/

*z*

_{R}), and

*m,*a vector LG beam can have four different polarization distributions, which are in this study, classified as type-I, II, III and IV as denoted in Eq. (1). These four polarization distributions have different angular dependence of the local polarization direction. At each point on the beam cross section, while the local state of polarization is linear there is an azimuthal variation in the polarization direction. The local angle

*θ*(

_{p}*ϕ*) of polarization vector for different polarization types of vector LG modes are shown in Table 1. In all of these cases, the direction of polarization vector is undefined at the beam centre, which can also be viewed as an isolated vector point singularity (V-point). In general, a V-point is characterized by its winding number or Poincaŕe-Hopf index, which is quantified by the net rotation angle (divided by 2π) of the polarization vectors surrounding the V-point [20

20. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. **201**(4-6), 251–270 (2002). [CrossRef]

*m*, the V-point possesses the winding number

*η*= ±

*m*. The sign of the winding number corresponds to the angular dependence of

*θ*for four different types of vector LG beams as shown in Table 1. It is important to note that type I and III have the same sign of

_{p}*η*(

*=*+

*m*). Similarly, type II and IV have the same sign of

*η*(

*=*-

*m*) opposite that of type I and III.

*m*is equal to zero, then the beam is reduced to a linearly polarized Gaussian beam with the angle of

*θ*(

_{p}*ϕ*) = π/2 (for type-I and II) or the angle of

*θ*(

_{p}*ϕ*)

*= 0*(for type-III and IV). For

*m*= 1, types I and III correspond to well-known azimuthally and radially polarized beams respectively, whereas types II and IV correspond to hybrid vector beams [see Fig. 1(a)]. For larger azimuthal index

*m,*vector beams with complicated polarization distribution can be obtained. Figure 1(b) illustrates the intensity and polarization distributions for

*m*= 2.

*m*have an identical intensity distribution but polarization patterns are different. This spatially varying polarization distribution together with the cylindrical symmetry provides unique properties to vector LG beams. The azimuthally dependent polarization distribution provides many interesting intensity and polarization features to vector LG beams which we will discuss in the following sections. In all the calculations, we assumed that the wavelength (

*λ*) is 632.8 nm and

*w*= 0.3 × 10

_{0}^{−3 }m.

## 3. Superposition of two vector Laguerre-Gaussian beams

*I*

_{01}and

*I*

_{02}are the intensities of two vector LG beams. The subscripts 1 and 2 in

**E**

_{p}_{1,}

_{m}_{1}and

**E**

_{p}_{2,}

_{m}_{2}represent the beam 1 and beam 2, respectively. Then, the general expression of the intensity distributions for the superposition of two vector LG beams is given bywhere

*S*(

*z*)

*=*cos[(

*2p*

_{1}

*- 2p*

_{2}

*+ m*

_{1}

*- m*

_{2})

*ψ*(

*z*)] is the Gouy phase dependent term.

*Q*(

*ϕ*) is the azimuthally varying term, which arises due to the difference of polarization distribution between two superposed beams. This term provides special features to the intensity and polarization distribution of the resultant field. For a specific azimuthal index

*m*, there are four different types of polarization distributions as mentioned previously and their superposition will give different effects on the resultant field. Superposing two vector LG beams with the azimuthal indices

*m*

_{1}and

*m*

_{2}can give sixteen combinations of polarization distributions, which can be summarized in the following six equations by avoiding duplication.

*η*of the two vector LG beams are same, then the term

*m*

_{1}−

*m*

_{2}always appears in

*Q*(

*ϕ*). On the contrary, if the sign of

*η*are opposite then term

*m*

_{1}+

*m*

_{2}appears. The additive and subtractive terms of azimuthal indices will produce a different number of intensity maxima and minima in the intensity distribution.

7. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express **17**(12), 9818–9827 (2009). [CrossRef] [PubMed]

*p*

_{1}=

*p*

_{2}= 0, we can obtain the condition for the formation of dark points around the beam axis. The condition to get dark points (

*I*= 0) requires

*Q*(

*ϕ*)

*S*(

*z*) = −1, which gives the radial position

*r*of dark points similar to the case for the superposition of scalar LG beams. Because both

_{d}*Q*(

*ϕ*) and

*S*(

*z*) are trigonometric functions, the radial position of dark points can be obtained only when either

*Q*(

*ϕ*) = 1 and

*S*(

*z*) = −1, or

*Q*(

*ϕ*) = −1 and

*S*(z) = 1 is satisfied aswhere

*w*(

*z*) which gives intensity modulation to the beam during propagation. For example, if

*m*

_{1}−

*m*

_{2}| dark points angularly distributes in the beam cross-section. The angular positions

*ϕ*

_{d}for these dark points are

*ϕ*

_{d}= 2

*n*π / |

*m*

_{1}–

*m*

_{2}| for

*Q*(

*ϕ*) = 1, or

*ϕ*

_{d}= (2

*n*+ 1)π / |

*m*

_{1}−

*m*

_{2}| for

*Q*(

*ϕ*) = −1, where

*n*= 0, 1, 2, …, |

*m*

_{1}−

*m*

_{2}|-1. Thus, the dark points surrounding the beam axis appear only when

*Q*(

*ϕ*) = ± 1 and

*Q*(

*ϕ*). The position of

*z*-planes which contain dark points (

*z*

_{d}) can be obtained from the condition

*S*(

*z*) = ± 1, bywhere

*q*= 0, ± 1, ± 2, …, and |

*q*| < |

*m*

_{1}−

*m*

_{2}| / 2. Therefore, the beam waist (

*q*= 0,

*z*

_{d}= 0) always contains dark points in any conditions. Additionally, multiple

*z*-planes on which the dark points appear within the finite

*z*range will be obtained when |

*m*

_{1}−

*m*

_{2}|> 2. Although the radius of dark points for the vector LG beam case presented in Eq. (5) is the same as the scalar LG beam case [7

7. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express **17**(12), 9818–9827 (2009). [CrossRef] [PubMed]

### 3.1. superposition of two vector Laguerre-Gaussian beams with the same azimuthal indices (m_{1} = m_{2})

#### (a) the same sign of *η*

*z*= 0 plane) for the superposition of two vector LG beams with

*m*

_{1}=

*m*

_{2}= 2 and polarization distribution (I

_{1}+ I

_{2}) for both beams. Here, we assume that the beams have equal intensity distribution, namely, α = 1. As expected in this case, the total intensity distribution is the same as that of the initial beams. Other combinations (II

_{1}+ II

_{2}, III

_{1}+ III

_{2}, IV

_{1}+ IV

_{2}, I

_{1}+ III

_{3}, and II

_{1}+ IV

_{2}) which have same sign of

*η*will also exhibit similar behaviors because the local polarizations of both beams are completely identical or orthogonal to each other.

#### (b) opposite sign of *η*

*z*= 0) for two beams with the same azimuthal indices (

*m*

_{1}=

*m*

_{2}= 2) but opposite sign of

*η*, namely, (I

_{1}+ II

_{2}). In this case, a petal shaped intensity pattern is obtained which is the same as that of the bright ring lattice obtained for a superposition of two scalar LG beams with equal but opposite topological charges [4

4. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express **15**(14), 8619–8625 (2007). [CrossRef] [PubMed]

*m*|. In other combinations (III

_{1}+ IV

_{2}, II

_{1}+ III

_{2}, IV

_{1}+ I

_{2}), the node positions for the superposed field differ in angle according to the combination of the beams as indicated in Eq. (4).

*η*. Apart from the value of the azimuthal index

*m*we can infer, from the above results that polarization distribution also plays a major role in deciding the intensity distribution of the superposed field. By choosing different polarization distributions and higher azimuthal indices, a variety of intensity patterns can be obtained.

### 3.2. Superposition of two vector Laguerre-Gaussian beams having different azimuthal indices (m_{1}≠m_{2})

#### (a) the same sign of *η*

*m*, the intensity distribution of the superposed field is modulated in the azimuthal direction according to the local polarization states of two beams.

*z*= 0 plane for the superposition of two beams with the same polarization orientations (type-I) and different azimuthal indices

*m*

_{1}= 1 and

*m*

_{2}= 2. In the present example, a single dark point appears at the azimuthal angle

*π*. This is consistent with the condition of dark point formation

*Q*(

*ϕ*)

*S*(z) = −1, where dark point is formed at the

*z*= 0. The radial position is equal to

*w*

_{0}, which can be obtained from Eq. (5). For higher values of azimuthal indices, the number of dark points will be larger. The number of dark points in the azimuthal direction depends on the difference of the azimuthal indices |

*m*

_{1}-

*m*

_{2}|. As shown in Eq. (4), other combinations are also possible which can give the similar situation of different azimuthal indices and the same sign of

*η.*

#### (b) opposite sign of *η*

*m*= 1 (type-I) and

_{1}*m*

_{2}= 2 (type-II). The resultant intensity distribution has three dark points around the beam axis. In this combination, according to Eq. (4), the number of dark points in the resultant intensity pattern is |

*m*

_{1}+

*m*

_{2}|, which are in contrast to the previous case. Although the time averaged intensity distribution of a scalar LG beams is identical to a vector LG beam with the same mode indices, the phase and polarization properties are different. This difference in the phase and the polarization properties appears in the intensity distribution when the two beams are superposed. In the case of superposition of two scalar LG beams, the spiral phase distribution is responsible for the azimuthal intensity variation. Whereas, in the superposition of vector LG beams, the angular variation of polarization direction is responsible for the azimuthal intensity variation.

## 4. Propagation of the superposed vector Laguerre-Gaussian beam

*xy*-plane and we assume that beams are propagating in the

*z*direction. Figure 4(a) shows the total intensity distributions for a superposition of two vector LG beams with

*m*

_{1}=

*m*

_{2}= 2 and polarization type-I and II. When the azimuthal indices of two beams are same, the resultant intensity distribution has a petal shape pattern. As previously discussed, this pattern is similar to the superposition of two scalar LG beams with equal and opposite topological charges, where bright and dark petals are obtained and their position remains fixed during propagation [4

4. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express **15**(14), 8619–8625 (2007). [CrossRef] [PubMed]

*Q(ϕ)*and

*S(z)*with different roles.

*Q(ϕ)*is responsible for the modulation of intensity in the azimuthal direction due to the interference between two beams with different polarization patterns. The tem

*S(z)*provides the Gouy phase shift to the superposed field. This term is unity in the case of the same azimuthal indices and hence there is no contribution to the superposed field. As a result, the intensity patters remain the same during propagation.

*m*

_{1}= 1 and

*m*

_{2}= 5) and the same sign of

*η*(type-I). Figure 4(c) shows the intensity distributions for two beams with

*m*

_{1}= 1 and

*m*

_{2}= 5 but opposite sign of

*η*. (type-I and II). In both cases, it can be seen that the intensity maxima and minima interchange their positions in the beam cross section during propagation. We can consider this variation as a flip in the intensity pattern. These intensity flips appear when |

*m*

_{1}-

*m*

_{2}| ˃ 2. This behavior is in contrast to the superposition of scalar beams, where dark points always appear in the superposed field.

*S*(

*z*) in Eq. (3). The peripheral dark point can be found on the

*z*-planes where the condition

*S*(

*z*) = ± 1 is satisfied. This condition for

*S*(

*z*), therefore, interchanges the azimuthal angles of the peripheral dark points, which are determined by

*Q*(

*ϕ*) =

*m*

_{1}= 1 with polarization type-I and

*m*

_{2}= 8 with polarization type-IV), Media 1 gives an animated view of the flip in the intensity distribution during propagation between the positions

*z = -*1.13

*z*to 1.13

_{R}*z*. We can clearly see that the superposed field accompanies an azimuthal intensity modulation as it propagates and the bright points switched to dark points and vice versa. The position of

_{R}*z*-plane containing dark points can be deduced from Eq. (6). For the case shown in Media 1 seven planes exist at the distance of

*z = -*4.38

*z*, −1.25

_{R}*z*−0.48

_{R},*z*, 0, 0.48

_{R}*z*1.25

_{R},*z*and 4.38

_{R},*z*respectively. From these results, the intrinsic feature of spatially varying polarization distribution in the vector beams is revealed.

_{R}## 5. Polarization singularities and complex Stokes field in superposition of two vector LG beams

20. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. **201**(4-6), 251–270 (2002). [CrossRef]

22. I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. **208**(4-6), 223–253 (2002). [CrossRef]

*E*and

_{x}*E*are the

_{y}*x*and

*y*components of the local electric field in the Cartesian coordinate basis and

*S*

_{1}

^{2}+

*S*

_{2}

^{2}+

*S*

_{3}

^{2}= 1.

20. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. **201**(4-6), 251–270 (2002). [CrossRef]

29. I. Freund, “Polarization flowers,” Opt. Commun. **199**(1-4), 47–63 (2001). [CrossRef]

*S*

_{1}=

*S*

_{2}= 0 and

*S*

_{3}= ± 1, where

*S*

_{3}= 1 and

*S*

_{3}= −1 correspond to right-handed and left-handed circular polarizations, respectively. On the other hand, L-lines are lines connecting points where polarization is linear (

*S*

_{3}= 0) and are embedded in an ellipse field where the handedness of elliptical polarization is undefined [26

26. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statics,” Opt. Commun. **213**(4-6), 201–221 (2002). [CrossRef]

28. A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. **27**(12), 995–997 (2002). [CrossRef] [PubMed]

30. T. H. Lu, Y. F. Chen, and K. F. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields,” Phys. Rev. A **76**(6), 063809 (2007). [CrossRef]

27. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. **27**(7), 545–547 (2002). [CrossRef] [PubMed]

*S*

_{12}representation, C-points and V-points appear as phase vortices [20

**201**(4-6), 251–270 (2002). [CrossRef]

22. I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. **208**(4-6), 223–253 (2002). [CrossRef]

*S*

_{2}= 0 and

*S*

_{3}= 0. The stokes index (winding number of Stokes vortices) for vector and elliptic field is given by

*σ*and

_{12}= 2η*σ*, where generally

_{12}= 2Ic*η*have integer value and

*Ic*have half integer value [20

**201**(4-6), 251–270 (2002). [CrossRef]

22. I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. **208**(4-6), 223–253 (2002). [CrossRef]

*φ*of complex Stokes field

_{12}*S*

_{12}for vector LG beams at

*z*= 0 plane, for two values of azimuthal indices (

*m*= 1, and

*m*= 2) as shown in Fig. 5. For easy identification, the polarization distribution is also shown. The center of the beam contains a Stokes vortex corresponding to a V-point, whose winding number

*σ*is equal to ± 2

_{12}*m*. From Figs. 5(a) and (b), it is clear that type-I and III have a Stokes vortex with

*σ*= + 2

_{12}*m*, and types II and IV, have a Stokes vortex with

*σ*= −2

_{12}*m*.

**i**

*and*

_{x}**i**

*are the unit vectors for the*

_{y}*x*and

*y*directions, respectively) as

*x*and

*y*components of a superposed field of two vector LG beams

*S*

_{3}given by Eq. (7), the condition for C-points (

*S*

_{3}= ± 1) yieldswhere the term

*C*

_{±}(

*ϕ*,

*z*) depends on the combination of the superposition as

*p*

_{1}=

*p*

_{2}= 0. In this case, the condition Eq. (9) is valid only when

*C*

_{±}(

*ϕ*,

*z*) = −1. Note that the plus and minus signs in the subscript of the notation

*C*

_{±}and in the right-hand side of Eq. (10) correspond to C-points with right-handed (

*S*

_{3}= 1) and left-handed (

*S*

_{3}= −1) circular polarizations, respectively. For any of the cases given by Eq. (10), the radial position of C-points

*r*

_{c}in the beam cross-section expressed byConsequently, the radial position of C-points represented by Eq. (11) is identical to radial position of the dark points given by Eq. (5). By contrast, the angular position

*ϕ*

_{C}_{±}of C-points depends on the handedness of polarization distribution of the superposed field. For example, for the cases of

*I*

_{1}+

*I*

_{2}and

*III*

_{1}+

*III*

_{2}in Eq. (10),

*n*= 1, 3, …, 2|

*m*

_{1}-

*m*

_{2}|-1 is an odd integer number. Due to the Gouy phase term (

*m*

_{1}-

*m*

_{2})

*ψ*(

*z*) in Eq. (12), the angular position of each C-point varies depending on the

*z*position, which shows a rotational behavior similar to the peripheral vortices formed by the superposition of two scalar LG beams [7

7. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express **17**(12), 9818–9827 (2009). [CrossRef] [PubMed]

*S*

_{3}= 1 and

*S*

_{3}= −1) are evenly distributed on a

*z*-plane. In addition, these C-points with opposite handedness will rotate in the opposite direction as the superposed beam propagates. In the following examples, we will see these polarization characteristics in the superposition of two vector LG beams.

### (a) the same sign of η

*η*but different azimuthal indices. Figure 6(a) shows the intensity distribution for the superposition of two vector LG beams (

*m*

_{1}= 1 with type-I, and

*m*

_{2}= 2 with type-I). The spatial polarization distribution corresponding to the superposed field is shown in Fig. 6(b), where the line and ellipse represent linear and elliptical polarizations respectively. At beam waist (

*z*= 0), the polarization is linear at every point. We can see that an elliptically polarized vector field is generated during propagation. A line of linear polarization (L-line), which intersects the beam axis horizontally, divides the elliptic field into two regions with opposite handedness [22

**208**(4-6), 223–253 (2002). [CrossRef]

26. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statics,” Opt. Commun. **213**(4-6), 201–221 (2002). [CrossRef]

*z*= 0 plane where each point on the beam cross-section is linearly polarized, an off-axis V-point (

*σ*= + 2) corresponding to a dark point is generated. In general, for the other azimuthal indices |

_{12}*m*

_{1}-

*m*

_{2}| off-axis V-points will be generated. During propagation, linearly polarized field is transformed into the elliptically polarized field. The off-axis V-point transforms into two C-points, lying each in the region of opposite handedness. For easy identification, the C-points are marked with the squares in Fig. 6(b). The number of C-points can be generally expressed as 2|

*m*

_{1}-

*m*

_{2}| and is two in this case. To determine the winding number of C-points in the superposed field, we plotted the phase of complex Stokes field of the combined beams in Fig. 6(c). As we can see from Fig. 6(c), the Stokes vortices generated in the two regions of opposite handedness have the same winding number (

*σ*= + 1). When the superposed field propagates through the beam waist (

_{12}*z*= 0), C-points in the two regions move in the opposite directions to each other as shown in Fig. 6(b). According to Eq. (12), the angular positions

*ϕ*

_{C}_{±}of these C-points are given by

*r*

_{c}=

*w*

_{0},

*ϕ*

_{C}_{±}= −π), where a pair of C-points is transformed into a dark point (V-point). Thus, at each

*z*-plane containing dark points indicated by Eq. (6), a pair of C-points with opposite handedness and same the winding number is always located at the same position, resulting in dark points. The phase map Fig. 6(c) clearly shows the presence and the motion of a

*S*vortex, which corresponds to a C-point with the same winding number in the superposed field. Media 2 shows the Stokes field corresponding to the field shown in Media 1. The change of the radial and angular position of the C-points is evident. C-points with opposite handedness and the same winding number move in the opposite directions and disappear at the point where a dark point is generated.

_{12}### (b) opposite sign of η

*η*are opposite and the azimuthal indices are different. Figure 7(a) shows the intensity distribution for two beams with (

*m*

_{1}= 1, and

*m*

_{2}= 2) and type-I and type-II polarization distributions. Figure 7(b) shows the spatial distribution of polarization at different

*z*-planes. In contrast to the previous case, the polarization distribution is entirely different and the number of C-points generated is equal to 2|

*m*

_{1}+

*m*

_{2}|. In this case, three L-lines are present which divides the total field into the regions of opposite handedness and each region has its own C-point. Figure 7(c) shows the phase map for complex Stokes field

*S*

_{12}. In contrast to the previous case at

*z*= 0 plane, three off-axis V-points (

*σ*= −2) are generated. In general |

_{12}*m*

_{1}+

*m*

_{2}| off-axis V-points will be generated. During propagation these three off-axis V-points are transformed into six C-points, which can be generally expressed as 2|

*m*

_{1}+

*m*

_{2}|. The numbers of C-points in the right and left handed polarization region are equal. Figure 7(c) shows that the Stokes vortices generated in the regions of opposite handedness have the same winding number (

*σ*= −1) which is opposite in sign to the previous case as shown in Fig. 6(c). The comparison between Figs. 6 and 7 reveals that the azimuthal indices of two beams are same but the polarization distribution of the resultant field and the associated singular points are different. The number of polarization singular points depends not only on the azimuthal indices but also on the sign of

_{12}*η*. By superposing two vector LG beams with different polarization distributions, different number of polarization singular points can be generated.

## 6. Discussion

## 7. Conclusions

*η*, |

*m*

_{1}-

*m*

_{2}| peripheral V-points are evolved. During propagation these peripheral V-points are transformed into 2|

*m*

_{1}-

*m*

_{2}| C-points. Whereas in the case of the superposition of two the vector LG beams with opposite sign of

*η*, |

*m*

_{1}+

*m*

_{2}| peripheral V-points are evolved. During propagation these peripheral V-points are transformed into 2|

*m*

_{1}+

*m*

_{2}| C-points. Radial and angular positions of C-points in the superposed field have been obtained from Stokes singularity relations. Novel structures of intensity and polarization can be generated from various combinations of azimuthal indices. Present results may have an important implication in understanding the geometry and topological properties of vector beams during propagation.

## Acknowledgments

## References and links

1. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

2. | A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A |

3. | J. Courtial, “Self-imaging beams and the Gouy effect,” Opt. Commun. |

4. | S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express |

5. | T. Ando, N. Matsumoto, Y. Ohtake, Yu. Takiguchi, and T. Inoue, “Structure of optical singularities in coaxial superposition of Laguerre-Gaussian mode,” J. Opt. Soc. Am. A |

6. | S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express |

7. | S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express |

8. | I. D. Maleev and G. A. Swartzlander Jr., “Composite optical vortices,” J. Opt. Soc. Am. B |

9. | M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science |

10. | A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Size selective trapping with optical “cogwheel” tweezers,” Opt. Express |

11. | D. Yang, J. Zhao, T. Zhao, and L. Kong, “Generation of rotating intensity blades by superposing optical vortex beams,” Opt. Commun. |

12. | L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science |

13. | Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. |

14. | Y. Q. Zhao, Q. Zhan, Y. L. Zhang, and Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. |

15. | H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics |

16. | X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun |

17. | C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. |

18. | G. M. Lerman and U. Levy, “Radial polarization interferometer,” Opt. Express |

19. | Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A |

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

21. | I. Freund, “Poincaré vortices,” Opt. Lett. |

22. | I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. |

23. | J. V. Hajnal, “Singularities in the transverse field of electromagnetic waves. I Theory,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

24. | J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observation on the electric field,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

25. | J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

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

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

28. | A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. |

29. | I. Freund, “Polarization flowers,” Opt. Commun. |

30. | T. H. Lu, Y. F. Chen, and K. F. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields,” Phys. Rev. A |

31. | Y. Luo and B. Lü, “Composite polarization singularities in superimposed Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A |

32. | I. Freund, “Möbius strip and twisted ribbon in intersecting Gauss-Laguerre beams,” Opt. Commun. |

33. | P. Kurzynowski, W. A. Woźniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express |

34. | P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt. |

35. | K. Yu. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express |

36. | E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A |

37. | 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 |

38. | E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. |

39. | A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express |

40. | M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. |

41. | O. V. Angelsky, A. G. Ushenko, Y. G. Ushenko, and Y. Y. Tomka, “Polarization singularities of biological tissues images,” J. Biomed. Opt. |

42. | Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. |

43. | R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express |

44. | G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(260.5430) Physical optics : Polarization

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 8, 2013

Revised Manuscript: March 15, 2013

Manuscript Accepted: March 19, 2013

Published: April 4, 2013

**Citation**

Sunil Vyas, Yuichi Kozawa, and Shunichi Sato, "Polarization singularities in superposition of vector beams," Opt. Express **21**, 8972-8986 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8972

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### References

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- P. Kurzynowski, W. A. Woźniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express20(24), 26755–26765 (2012). [CrossRef] [PubMed]
- P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt.2012, 517591 (2012).
- K. Yu. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express16(2), 695–709 (2008). [CrossRef] [PubMed]
- E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A82(6), 063832 (2010). [CrossRef]
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- A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18(10), 10777–10785 (2010). [CrossRef] [PubMed]
- M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett.102(3), 033902 (2009). [CrossRef] [PubMed]
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- R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express14(12), 5733–5745 (2006). [CrossRef] [PubMed]
- G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett.108(19), 190401 (2012). [CrossRef] [PubMed]

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