## A new type of vector fields with hybrid states of polarization

Optics Express, Vol. 18, Issue 10, pp. 10786-10795 (2010)

http://dx.doi.org/10.1364/OE.18.010786

Acrobat PDF (4585 KB)

### Abstract

We present an idea based on Poincaré sphere and demonstrate the creation of a new type of vector fields, which have hybrid states of polarization. Such a type of hybridly polarized vector fields have completely different property from the reported scalar and vector fields. The novel vector fields are anticipated to result in new effects, phenomena, and applications.

© 2010 Optical Society of America

## 1. Introduction

1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. **1**, 1–57 (2009). [CrossRef]

4. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express **7**, 77–87 (2000). [CrossRef] [PubMed]

4. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express **7**, 77–87 (2000). [CrossRef] [PubMed]

5. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express **10**, 324–331 (2002). [PubMed]

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

10. P. Wróbel, J. Pniewski, T. J. Antosiewicz, and T. Szoplik, “Focusing Radially Polarized Light by a Concentrically Corrugated Silver Film without a Hole,” Phys. Rev. Lett. **102**, 103902 (2009). [CrossRef]

4. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express **7**, 77–87 (2000). [CrossRef] [PubMed]

5. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express **10**, 324–331 (2002). [PubMed]

21. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. **265**,
411–417 (2006). [CrossRef]

5. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express **10**, 324–331 (2002). [PubMed]

21. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. **265**,
411–417 (2006). [CrossRef]

22. N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. **279**, 229–234 (2007). [CrossRef]

23. X. L. Wang, J. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage
generated from cylindrical vector beams,” Opt. Commun. **282**, 3421–3425 (2009). [CrossRef]

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

3. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial
light modulator and a common path interferometric arrangement,” Opt. Lett. **32**, 3549–3551 (2007). [CrossRef] [PubMed]

24. M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. **11**, 065204 (2009). [CrossRef]

24. M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. **11**, 065204 (2009). [CrossRef]

25. Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. **30**, 3063–3065 (2005). [CrossRef] [PubMed]

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

3. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial
light modulator and a common path interferometric arrangement,” Opt. Lett. **32**, 3549–3551 (2007). [CrossRef] [PubMed]

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

30. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. **27**, 285–287 (2002). [CrossRef]

34. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. **184**, 67–71 (2000). [CrossRef]

35. C. F. Li, “Physical evidence for a new symmetry axis of electromagnetic beams,” Phys. Rev. A. **79**, 053819 (2009). [CrossRef]

*ħ*

*k*for the circular polarizations. Not only the scalar fields with homogeneous polarization but also the reported vector fields with inhomogeneous local linear polarization have zero spatial gradient in spin angular momentum. In the present article, we present an idea based on Poincaré sphere [36] and create a new type of vector fields with the hybrid SoPs, which might have spatial-variant spin-angular momentum. Such a new type of vector fields are more general vector fields than the vector fields with local linear polarization. This type of novel vector fields we presented could be expected to result in new effects and phenomena [37

37. M. Onoda, S. Murakami, and N. Nagaosa, “Geometrical aspects in optical wave-packet dynamics,” Phys. Rev. E. **74**, 066610 (2006). [CrossRef]

38. K. Yu. Bliokh and Yu. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E. **75**, 066609 (2007). [CrossRef]

## 2. Basic Principle

*s*

_{1},

*s*

_{2}and

*s*

_{3}denote the Stokes parameters of a point

**S**on ∑ in Cartesian coordinate system (satisfying

*s*

^{2}

_{1}+

*s*

^{2}

_{2}+

*s*

^{2}

_{3}= 1), and 2

*α*and 2

*ϕ*stand for the latitude and longitude angles of this point in spherical coordinate system, respectively. So the point on ∑ can be defined by (2

*ϕ*, 2

*α*). The factor 2 in 2

*α*and 2

*ϕ*is used to ensure that one point on ∑ corresponds to a unique SoP and vice versa.

*β*and the angle

*ϕ*. The ellipticity

*β*determines the shape of the polarization ellipse, which is defined by

*β*= tan

*α*, where the positive or negative sign of

*β*distinguishes the RH or LH rotation. The angle

*ϕ*specifies the orientation of the polarization ellipse. Based on Eq. (1), the SoPs at some special points on ∑ are shown in Fig. 1(b). The north and south poles on ∑ correspond to the RH and LH circular polarizations, respectively. The SoP at any point in the equator on ∑ is linearly polarized. For instance, a pair of points {(0,0), (

*π*,0)} correspond to two orthogonal linear polarizations in the

*x*and

*y*directions. A pair of points {(

*π*/2,0), (3

*π*/2,0)} represent two orthogonal linear polarizations with the angles of ±

*π*/4 with the +

*x*direction. Except for at the two poles and at the points in the equator, any point in the 2

*ϕ*meridian circle (composed of the 2

*ϕ*meridian and its opposite meridian) on ∑ corresponds to the elliptical polarization. The RH (LH) polarizations are represented by points on ∑ which lie above (below) the equator. In addition, at any pair of points on ∑ with the inverse symmetry with respect to the origin, two SoPs can also be served as a pair of orthogonal base vectors, since

**〈**Ŝ(2

*ϕ*, 2

*α*)

**∣**Ŝ(2

*ϕ*+

*π*,−2

*α*)〉 = 0 from Eq. (1).

3. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial
light modulator and a common path interferometric arrangement,” Opt. Lett. **32**, 3549–3551 (2007). [CrossRef] [PubMed]

*t*(

*x*,

*y*)=[1+

*γ*cos(2

*π*

*f*

_{0}

*x*+

*δ*)]/2, where the additional phase distribution

*δ*is the function of the azimuthal angle

*φ*only as

*δ*=

*mφ*+

*φ*

_{0}(

*m*and

*φ*

_{0}are the topological charge and the initial phase, respectively). (ii) Two

*λ*/4 waveplates behind the spatial filter

**F**in the Fourier spatial-frequency plane of the 4f system transfer the ±1st-order linearly polarized fields from SLM into the RH and LH circularly polarized fields. It should be pointed that

*x*and

*y*are the coordinates in the Cartesian coordinate system attached in the SLM plane (the input plane of the 4f system) and

*φ*is the azimuthal angle in the corresponding polar coordinate system. Since the input plane and the output plane are the object plane and image planes each other in the 4f system, the output plane and the input plane are allowed to use the same coordinates in both the Cartesian coordinate system and the polar coordinate system. Ultimately, the SoP of the created vector field in the output plane can be written by the unit vector

**P̂**(

*ρ*,

*φ*) as

*ρ*and

*φ*are the polar radius and the azimuthal angle in the polar coordinate system attached in the output (or input) plane of the 4f system, respectively. We can find from Eq. (2b) that the SoP at any position in the field cross section is linearly polarized because the

*x*and

*y*components have the same phase, in particular, depends on the azimuthal angle

*φ*only because

*δ*is the function of

*φ*only. Consequently, the vector fields created in our previous work [3

**32**, 3549–3551 (2007). [CrossRef] [PubMed]

**P̂**(

*ρ*,

*φ*) as

*δ*is still defined by

*δ*=

*mφ*+

*φ*

_{0}, as mentioned above. In the SoPs of the created vector field described by Eq. (3), the value of

*ϕ*determines which pair of orthogonal linear polarization as the base vectors. We can find from Eq. (3a) that the pair of base vectors {cos

*ϕ*

**ê**

_{x}+ sin

*ϕ*

**ê**

_{y},−sin

*ϕ*

**ê**

_{x}+ cos

*ϕ*

**ê**

_{y}} correspond to the pair of points {(2

*ϕ*,0), (2

*ϕ*+

*π*,0)} in the equator on ∑. The SoPs of the created vector field described by Eq. (3b) are all in the meridian circle of 2

*ϕ*+

*π*/2, which is orthogonal to the connecting line between the two points {(2

*ϕ*,0), (2

*ϕ*+

*π*,0)}, due to the presence of the factor

*ϕ*+

*π*/4 in Eq. (3b).

## 3. Experimental realization

**32**, 3549–3551 (2007). [CrossRef] [PubMed]

*λ*/4 waveplates behind

**F**are used to transfer the +1st and -1st order fields diffracted by SLM into the RH and LH circularly polarized fields as a pair of orthogonal base vectors [3

**32**, 3549–3551 (2007). [CrossRef] [PubMed]

**F**as a pair of base vectors, two

*λ*/2 waveplates should be used, as the experimental arrangement shown in Fig. 2. Since the linearly polarized field has the distinguishable direction, in principle, a finite pairs of orthogonal linearly polarized fields can be found in the equator on ∑ shown in Fig. 1. In experiment, different pair of orthogonal linearly polarized base vectors can be realized by changing the orientations of two

*λ*/2 waveplates behind

**F**. So in the creation of the vector fields, using the pair of two orthogonal linearly polarized base vectors should be more flexible than using the orthogonal RH and LH circularly polarized base vectors.

*ϕ*in Eq. (3) is the azimuthal angle of the polar coordinate system in the spatial-frequency plane of the 4f system and specifies the polarization directions of the orthogonal linearly polarized fields generated by the

*λ*/2 waveplates in the ±1st order paths, in particular, 2

*ϕ*can characterize the meridian circle on ∑. (ii)

*φ*in

*δ*of Eq. (3) can indicate the azimuthal angle of the polar coordinate systems attached in both the input plane and the output plane of the 4f system. In addition, the pairs of orthogonal arrows in the corner in each figure show the directions of two orthogonal linearly polarized base vectors for creating the hybridly polarized vector fields.

*ϕ*= −

*π*/4 in Eq. (3) and assuming

*δ*=

*φ*(

*m*=1 and

*φ*

_{0}= 0). In this situation, the pair of orthogonal linearly polarized base vectors is

*π*/2,0), (

*π*/2,0)} in the equator on ∑ shown in Fig. 1. The sketch drawing of SoPs in the cross section of the created vector field is shown in Fig. 3(a).

*π*. In detail, the local SoPs are radially polarized at any location in the radial directions of

*φ*= 0,

*π*/2,

*π*, and 3

*π*/2. The local SoPs are LH circularly polarized along the radial directions of

*φ*=

*π*/4 and 5

*π*/4, while are RH circularly polarized along the radial directions of

*φ*= 3

*π*/4 and 7

*π*/4. At any location in the first and third quadrants, the local SoP is LH elliptically polarized, while at any location in the second and fourth quadrants, the local SoP is RH elliptically polarized. For the elliptically polarized states, the major axis of any polarization ellipse in the ranges of

*φ*∈ (0,±

*π*/4) and

*φ*∈ (

*π*,

*π*±

*π*/4) is in the radial direction of

*φ*= 0 (or

*π*), while that in the ranges of

*φ*∈ (

*π*/2,

*π*/2±

*π*/4) and

*φ*∈ (3

*π*/2,3

*π*/2±

*π*/4) is in the radial direction of

*φ*=

*π*/2 (or 3

*π*/2). In fact, the SoPs of this vector field corresponds to the points in the meridian circle of 2

*ϕ*= 0 on ∑. For comparison, Fig. 3(b) shows the distribution of SoPs of the well-known radially polarized field.

*δ*=

*φ*are shown in the first row of Fig. 4. For comparison, the radially polarized vector field is also shown in the second row of Fig. 4. If no polarizer is used, both vector fields have no distinction in intensity pattern. When a polarizer is used, however, the situations are quite different. For the hybridly polarized vector field, two intensity patterns behind the horizontal and vertical polarizers are recognizable, the extinction directions orthogonal to the direction of the polarizer, and are the same as that for the radially polarized vector field. For the hybridly polarized vector field, the intensity patterns behind the

*π*/4 and 3

*π*/4 polarizers are unrecognizable and have no extinction direction because the field components in the two polarization directions are identical at any location in the cross section, in particular, both intensities are a half of that without a polarizer. In contrast, for the radially polarized vector field, the intensity patterns behind the polarizer are always recognizable with the extinction direction orthogonal to the direction of the polarizer.

*δ*=

*φ*, as shown in Fig. 5. The four pairs of base vectors are {

**ê**

_{x},

**ê**

_{y}},

**ê**

_{y}, −

**ê**

_{x}}, and

*π*,0)}, {(

*π*/2,0), (3

*π*/2,0)}, {(

*π*,0), (2

*π*,0)} and {(3

*π*/2,0), (5

*π*/2,0)}, in the equator on ∑ in Fig. 1, respectively. If no polarizer is used, all the four intensity patterns have no distinction direction (here we do not show). If a horizontal polarizer is used, the two intensity patterns for the first and third pairs of base vectors are unrecognizable without the extinction direction and both intensities are a half of that without a polarizer. In contrast, the intensity patterns behind a horizontal polarizer for the second and fourth pairs of base vectors become recognizable with the extinction directions parallel and orthogonal to the direction of the polarizer, respectively. All the phenomena with and without a polarizer can be easily understood by the schematic distributions of SoPs in the first row of Fig. 5.

*φ*= 0 is linearly polarized with the direction of

**ê**

_{y}], and corresponds to the point (

*π*/2,0) [(

*π*,0)] in the equator on ∑. Within the range of

*φ*∈ (0,

*π*/4), the SoP is the LH elliptical polarization with its major axis of polarization ellipse in the direction of

**ê**

_{y}]. In the radial direction of

*φ*=

*π*/4, the SoP is LH circularly polarized. Within the range of

*φ*∈ (

*π*/4,

*π*/2), the SoP is still the LH elliptical polarization with its major axis of the polarization ellipse in the direction of

**ê**

_{x}]. At the radial direction of

*φ*=

*π*/2, the SoP becomes linearly polarized again, while its polarization direction is in the direction of

**ê**

_{x}]. Within the range of

*φ*∈ (

*π*/2,3

*π*/4), the SoP becomes RH elliptically polarized, and the major axis of its polarization ellipse is the same as the linearly polarized direction in the radial direction of

*φ*=

*π*/2. In the radial direction of

*φ*= 3

*π*/4, the SoP is RH circularly polarized. Within the range of

*φ*∈ (3

*π*/4,

*π*), the SoP is RH elliptically polarized and the major axis of its polarization ellipse is in the direction of

**ê**

_{y}]. At the radial direction of

*φ*=

*π*, the SoP becomes linearly polarized again and its polarization direction is in the direction of

**ê**

_{y}]. The variation of SoPs with

*φ*ranging from

*π*to 2

*π*is very similar to the situation when

*φ*varies from 0 to

*π*. For the vector field created by the first [second] pair of base vectors, the evolution process of SoPs with

*φ*ranging from 0 to

*π*corresponds to the point move in the meridian circle of 2

*ϕ*=

*π*/2 [2

*ϕ*= 0] on ∑, which starts from the equator point of (

*π*/2, 0) [(

*π*, 0)], then pass through orderly the south pole, the equator point of (3

*π*/2, 0) [(0, 0)] and the north pole, and finally backs to the starting equator point of (

*π*/2, 0) [(

*π*, 0)]. For the variation of SoPs with

*φ*from

*π*to 2

*π*, the corresponding point move in the meridian circle of 2

*ϕ*=

*π*/2 on ∑ will experience the same evolution process as mentioned above. The residual two situations, we will not give the detailed descriptions.

*φ*

_{0}= 0 only. We now investigate the creation of hybridly polarized vector fields for different values of

*φ*

_{0}. As examples, the topological charge

*m*is still taken to be

*m*= 1 and the pair of orthogonal linearly polarized base vectors is still to be {

**ê**

_{x},

**ê**

_{y}}. As shown in Fig. 6, the intensity patterns for four different values of

*φ*

_{0}, when no polarizer is used, have no difference. The intensity patterns passing through the horizonal or vertical polarizer are also unrecognizable and have no extinction direction, and their intensity is a half of that without the polarizer. In contrast, when a

*π*/4 or 3

*π*/4 polarizer is used, the intensity patterns behind the polarizer become discriminable. For four vector fields, the extinction direction with the

*π*/4 polarizer is always orthogonal to that with the 3

*π*/4 polarizer. For the

*π*/4 and 3

*π*/4 polarizers, the extinction directions of the intensity patterns behind the polarizer for the vector field of

*φ*

_{0}= 0 are in the vertical and horizonal directions, respectively. For the other three vector fields of

*φ*

_{0}=

*π*/4,

*π*/2, and 3

*π*/4, the extinction directions are clockwise rotated by the angles of

*π*/4,

*π*/2, and 3

*π*/4, respectively. We carefully investigate with Eq. (3) to find that SoP in the radial direction of

*φ*for the vector field of

*φ*

_{0}= 0 is the same as that in the radial directions of

*φ*−

*π*/4,

*φ*−

*π*/2, and

*φ*− 3

*π*/4 for the vector fields of

*φ*

_{0}=

*π*/4,

*π*/2, and 3

*π*/4, respectively. The intensity patterns behind the polarizer are easily understood by the distributions of SoPs of four vector fields, as mentioned above. The SoPs of the vector field of

*φ*

_{0}= 0 have been shown in the upper row in Fig. 5.

*m*is larger than unity. As an example, we consider the situation of

*m*= 2 and

*φ*

_{0}= 0, and the pair of orthogonal linearly polarized base vectors {

**ê**

_{x},

**ê**

_{y}}. The experimental results are shown in the upper row in Fig. 7. For comparison, the bottom row in Fig. 7 gives also the local linearly polarized vector field with

*m*= 2 and

*φ*

_{0}= 0 based on the RH and LH circular polarized base vectors, by the method presented in Ref. 3

**32**, 3549–3551 (2007). [CrossRef] [PubMed]

*π*/4 or 3

*π*/4 polarizer is used, the intensity patterns behind the polarizer for both vector fields are distinguishable due to the different extinction directions, as shown in Fig. 7. The local linearly polarized vector field with

*m*= 2 and

*φ*

_{0}= 0 has always the extinction directions, provided that a polarizer is used. It should be pointed out that for both vector fields, the number of extinction directions are the same as the topological charge

*m*.

## 4. Summary

39. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef] [PubMed]

40. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. **100**, 013602 (2008). [CrossRef] [PubMed]

## Acknowledgements

## References and links

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39. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

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**OCIS Codes**

(260.5430) Physical optics : Polarization

(070.6120) Fourier optics and signal processing : Spatial light modulators

**History**

Original Manuscript: February 1, 2010

Revised Manuscript: March 26, 2010

Manuscript Accepted: April 12, 2010

Published: May 10, 2010

**Virtual Issues**

Unconventional Polarization States of Light (2010) *Optics Express*

**Citation**

Hui-Tian Wang, Xi-Lin Wang, Yongnan Li, Jing Chen, Cheng-Shan Guo, and Jianping Ding, "A new type of vector fields with hybrid states of polarization," Opt. Express **18**, 10786-10795 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-10-10786

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