A new type of vector fields with hybrid states of polarization

Xi-Lin Wang; Yongnan Li; Jing Chen; Cheng-Shan Guo; Jianping Ding; Hui-Tian Wang

doi:10.1364/OE.18.010786

1. Introduction

Besides intensity and phase, polarization is one important nature of light. Manipulating states of polarization (SoPs) of light has become a subject of rapidly growing interest, due to its unique features and novel applications in various realms, such as nonlinear optics, near-field optics, optical trapping and particle manipulation, imaging, spectroscopy, micromechanics, microfluidics, and biology. Most of past research dealt with scalar fields with spatially homogeneous SoPs, such as linear, elliptical, and circular polarizations. For the scalar fields, the SoPs at different locations of the field cross section are identical. The polarization as an additional freedom can be used to control optical field, such as the creation of the vector fields [1–3], which have spatially inhomogeneous SoPs. Most of the vector fields reported in literature are the radially polarized fields as a kind of typical vector fields [4–20], which have all the local SoPs to be linearly polarized and arranged along the radial direction in the field cross section. Focusing by an objective with high numerical-aperture (NA), the radially polarized field could create a non-propagating strong longitudinal electric field in the central region of the focal plane [4,5], and further this longitudinal component could result in a sharper focal spot than that from a scalar field [6–9]. This distinct feature makes the radially polarized field better for many applications [10–20]. Besides the radially polarized fields, another kind of typical vector fields is the azimuthally polarized fields, which can be highly focused into a hollow dark spot [4–6]. Due to the unique focusing property, the vector fields are widely used on focus engineering [5,21–23]. For instance, focusing an optical field with the combination of radial and azimuthal polarizations could be highly focused into an optical plat [5,21] or a three-dimensional optical cage [22,23]. Besides the radially and azimuthally polarized vector fields, there are other vector fields [2,3,24], which could be focused into the flower-like patterns [24].

Until now, several methods have been proposed to create vector fields. These methods can be divided into two kinds of active and passive schemes. The mostly active way is from the output of novel lasers with specially designed laser resonators [25–29]. The passive one is based on the wavefront reconstruction of the output field from the traditional lasers, with the aid of specially designed optical elements [2,3,6,30–33].

The creation of novel vector fields remains a great challenge and a paramount issue, due to the expectation of high flexibility in manipulating the SoPs of the optical field and in developing novel photonic devices and optical systems. All the vector fields as mentioned above have a common feature that the SoPs at all the locations in the field cross section are linearly polarized. These vector fields have powerful applications in many fields. It is widely accepted that spin angular momentum is associated with the SoPs [34,35], and is zero for the linear polarization and ±ħk 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 [37,38] that can expand the functionality and enhance the capability of photonic devices and optical systems.

2. Basic Principle

All the conditions considered in this paper are in the paraxial approximation. To describe all possible SoPs of a polarized field, Poincaré sphere is a simple and convenient geometric representation [36]. As shown in Fig. 1(a), the Poincaré sphere ∑ has a unit radius since we are not interested in light intensity here: s₁, s₂ and s₃ denote the Stokes parameters of a point S on ∑ in Cartesian coordinate system (satisfying s²₁ + s²₂ + s²₃ = 1), and 2α and 2ϕ stand for the latitude and longitude angles of this point in spherical coordinate system, respectively. So the point on ∑ can be defined by (2ϕ, 2α). The factor 2 in 2α and 2ϕ is used to ensure that one point on ∑ corresponds to a unique SoP and vice versa.

For any given polarized field, its SoP could be described by the combination of a pair of orthogonal base vectors. For instance, a pair of orthogonal base vectors are {ê_x, ê_y} (with 〈ê_x∣ê_y〉 = 0) in the Cartesian coordinate system. ê_x and ê_y describe the linearly polarized fields with their directions of vibration along the x and y axes. Another pair of orthogonal base vectors are right-handed (RH) and left-handed (LH) unit vectors {ê_r, ê_l} (with 〈ê_r∣ê_l〉 = 0), for characterizing the RH and LH circularly polarized fields. The two pairs of orthogonal base vectors, as mentioned above, are related by ${\hat{e}}_{x} = \frac{1}{\sqrt{2}} ({\hat{e}}_{r} + {\hat{e}}_{l})$ and ${\hat{e}}_{y} = - j \frac{1}{\sqrt{2}} ({\hat{e}}_{r} - {\hat{e}}_{l})$ as well as ${\hat{e}}_{r} = \frac{1}{\sqrt{2}} ({\hat{e}}_{x} + j {\hat{e}}_{y})$ and ${\hat{e}}_{l} = \frac{1}{\sqrt{2}} ({\hat{e}}_{x} - j {\hat{e}}_{y})$ . Thus the SoP at a given point (2ϕ, 2α) on ∑ can be described by the unit vector Ŝ(2ϕ, 2α), based on the pair of {ê_r, ê_l} or {ê_x, ê_y}, as follows

\hat{S} (2 ϕ, 2 α) = \sin (α + π / 4) \exp (- j ϕ) {\hat{e}}_{r} + \cos (α + π / 4) \exp (j ϕ) {\hat{e}}_{l}

Fig. 1. (a) Poincaré sphere ∑ and (b) different SoPs on Poincaré sphere ∑.

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In fact, the linearly and circularly polarized fields are two special yet extreme cases of light fields. A more general polarized field is elliptically polarized. Its SoP can be characterized by the polarization ellipse, which can be specified by two parameters: the ellipticity 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), (3π/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 2ϕ meridian circle (composed of the 2ϕ 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(2ϕ, 2α)∣Ŝ(2ϕ + π,−2α)〉 = 0 from Eq. (1).

Now let us look back on the creation of vector fields we reported [3], in which the experimental configuration has two key points: (i) The transmission function of the spatial light modulator (SLM) is designed as t(x,y)=[1+γ cos(2π f₀x+δ)]/2, where the additional phase distribution δ is the function of the azimuthal angle φ only as δ = mφ + φ₀ (m and φ₀ 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

\hat{P} (ρ, φ) = \frac{1}{\sqrt{2}} [\exp (- j δ) {\hat{e}}_{r} + \exp (j δ) {\hat{e}}_{l}]

= \cos δ {\hat{e}}_{x} + \sin δ {\hat{e}}_{y}

where 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], based on the orthogonal RH and LH circularly polarized fields (located at the north and south poles on ∑) as two base vectors, belong to a kind of local linearly polarized vector fields (in the equator on ∑). Therefore, we can imagine that if taking the orthogonal linear polarizations in the equator on ∑ as a pair of base vectors for creating the vector field, its SoPs should correspond to the point in the certain meridian circle on ∑. As a result, the SoP of the created vector field can be characterized by the unit vector P̂(ρ,φ) as

\hat{P} (ρ, φ) = \frac{1}{\sqrt{2}} (\cos ϕ {\hat{e}}_{x} + \sin ϕ {\hat{e}}_{y}) \exp (j δ) + \frac{1}{\sqrt{2}} (- \sin ϕ {\hat{e}}_{x} + \cos ϕ {\hat{e}}_{y}) \exp (- j δ)

= \cos (δ + π / 4) \exp [(- j (ϕ + π / 4)] {\hat{e}}_{r} + \sin (δ + π / 4) \exp [j (ϕ + π / 4)] {\hat{e}}_{l}

where δ is still defined by δ = mφ + φ₀, 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 ϕê_x + sin ϕê_y,−sin ϕê_x + cosϕê_y} correspond to the pair of points {(2ϕ ,0), (2ϕ +π,0)} in the equator on ∑. The SoPs of the created vector field described by Eq. (3b) are all in the meridian circle of 2ϕ +π/2, which is orthogonal to the connecting line between the two points {(2ϕ ,0), (2ϕ +π,0)}, due to the presence of the factor ϕ +π/4 in Eq. (3b).

3. Experimental realization

To realize a type of hybridly polarized vector fields (maybe consist of linear, circular and elliptical polarizations), the experimental arrangement is very similar to Fig. 1 in Ref. 3. Two 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]. Inasmuch as the RH and LH circularly polarized fields correspond to the north and south poles only on ∑ and both polarizations have no distinguishable direction, only one pair of base vectors can be chosen. In contrast, in the present work, to obtain two orthogonal linear polarized fields behind 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.

Fig. 2. Schematic diagram of experimental setup.

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We should emphasize that: (i) 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, 2ϕ 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.

First, we examine the case when setting ϕ = −π/4 in Eq. (3) and assuming δ = φ (m=1 and φ₀ = 0). In this situation, the pair of orthogonal linearly polarized base vectors is ${\frac{1}{\sqrt{2}} ({\hat{e}}_{x} - {\hat{e}}_{y}), \frac{1}{\sqrt{2}} ({\hat{e}}_{x} + {\hat{e}}_{y})}$ . The two linear polarizations correspond to the pair of points {(3π/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).

Fig. 3. Distribution of SoPs of the created vector field for ϕ = −π/4 and δ = φ (a) and the well-known radially polarized field (b).

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Evidently, the local SoP has no change in any radial direction while depend on the azimuthal angle π. In detail, the local SoPs are radially polarized at any location in the radial directions of φ = 0, π/2, π, and 3π/2. The local SoPs are LH circularly polarized along the radial directions of φ = π/4 and 5π/4, while are RH circularly polarized along the radial directions of φ = 3π/4 and 7π/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 ∈ (3π/2,3π/2±π/4) is in the radial direction of φ = π/2 (or 3π/2). In fact, the SoPs of this vector field corresponds to the points in the meridian circle of 2ϕ = 0 on ∑. For comparison, Fig. 3(b) shows the distribution of SoPs of the well-known radially polarized field.

Fig. 4. Intensity patterns for the vector fields without and with a polarizer. First and second rows are the hybridly polarized vector field with ϕ = −π/4 and δ = φ and the radially polarized field, respectively. Third row shows the directions of the polarizer.

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The intensity patterns of the created hybridly polarized vector field based on the pair of base vectors ${\frac{1}{\sqrt{2}} ({\hat{e}}_{x} - {\hat{e}}_{y}), \frac{1}{\sqrt{2}} ({\hat{e}}_{x} + {\hat{e}}_{y})}$ when 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 3π/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.

Second, we explore the creation of the hybridly polarized vector field for four pairs of orthogonal linearly polarized base vectors and still take δ = φ, as shown in Fig. 5. The four pairs of base vectors are {ê_x, ê_y}, ${\frac{1}{\sqrt{2}} ({\hat{e}}_{x} + {\hat{e}}_{y}), \frac{1}{\sqrt{2}} (- {\hat{e}}_{x} + {\hat{e}}_{y})}$ , {ê_y, −ê_x}, and ${\frac{1}{\sqrt{2}} (- {\hat{e}}_{x} + {\hat{e}}_{y}), - \frac{1}{\sqrt{2}} ({\hat{e}}_{x} + {\hat{e}}_{y})}$ , which correspond to the four pairs of points, {(0,0), (π,0)}, {(π/2,0), (3π/2,0)}, {(π,0), (2π,0)} and {(3π/2,0), (5π/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.

Fig. 5. Created four different vector fields with the directions of the λ/2 waveplate in the +1 order path along 0, π/4, π/2 and 3π/4, the SoPs (the first row) and the intensity patterns behind the horizontal polarizer (the second row).

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For the vector field created by the first [second] pair of base vectors, its SoP at any location along the radial direction of φ = 0 is linearly polarized with the direction of $\frac{1}{\sqrt{2}} ({\hat{e}}_{x} + {\hat{e}}_{y})$ [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 $\frac{1}{\sqrt{2}} ({\hat{e}}_{x} + {\hat{e}}_{y})$ [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 $\frac{1}{\sqrt{2}} ({- \hat{e}}_{x} + {\hat{e}}_{y})$ [−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 $\frac{1}{\sqrt{2}} ({- \hat{e}}_{x} + {\hat{e}}_{y})$ [−ê_x]. Within the range of φ ∈ (π/2,3π/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 = 3π/4, the SoP is RH circularly polarized. Within the range of φ ∈ (3π/4,π), the SoP is RH elliptically polarized and the major axis of its polarization ellipse is in the direction of $\frac{1}{\sqrt{2}} ({- \hat{e}}_{x} - {\hat{e}}_{y})$ [−ê_y]. At the radial direction of φ = π, the SoP becomes linearly polarized again and its polarization direction is in the direction of $\frac{1}{\sqrt{2}} ({- \hat{e}}_{x} - {\hat{e}}_{y})$ [−ê_y]. The variation of SoPs with φ ranging from π to 2π 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 2ϕ = π/2 [2ϕ = 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 (3π/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 2π, the corresponding point move in the meridian circle of 2ϕ = π/2 on ∑ will experience the same evolution process as mentioned above. The residual two situations, we will not give the detailed descriptions.

In the above, we explore the situation of φ₀ = 0 only. We now investigate the creation of hybridly polarized vector fields for different values of φ₀. 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 φ₀, 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 3π/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 3π/4 polarizer. For the π/4 and 3π/4 polarizers, the extinction directions of the intensity patterns behind the polarizer for the vector field of φ₀ = 0 are in the vertical and horizonal directions, respectively. For the other three vector fields of φ₀ = π/4, π/2, and 3π/4, the extinction directions are clockwise rotated by the angles of π/4, π/2, and 3π/4, respectively. We carefully investigate with Eq. (3) to find that SoP in the radial direction of φ for the vector field of φ₀ = 0 is the same as that in the radial directions of φ − π/4, φ − π/2, and φ − 3π/4 for the vector fields of φ₀ = π/4, π/2, and 3π/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 φ₀ = 0 have been shown in the upper row in Fig. 5.

Finally, we will explore the creation of hybridly polarized vector fields when the topological charge m is larger than unity. As an example, we consider the situation of m = 2 and φ₀ = 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 φ₀ = 0 based on the RH and LH circular polarized base vectors, by the method presented in Ref. 3. If no polarizer is used, both the vector fields have no difference. However, when a polarizer is inserted, the situations are quite different. When a horizontal or vertical polarizer is inserted, for the former the intensity patterns behind the polarizer have no extinction direction, while for the latter there are two extinction directions. When a π/4 or 3π/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 φ₀ = 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.

Fig. 6. Hybridly polarized vector fields created by the pair of base vectors {ê_x, ê_y} when m = 1, for φ₀ = 0, π/4, π/2, and 3p/4.

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Fig. 7. Hybridly polarized vector fields by the pair of base vectors {ê_x, ê_y} with φ₀ = 0 and m=2 (the upper row). For comparison, the local linearly polarized vector field with φ₀ = 0 and m = 2, by the method in Ref. 3 (the bottom row).

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4. Summary

In summary, we have proposed and demonstrated new type of vector fields, so-called hybridly polarized vector fields. The different SoPs including linear, elliptical, and circular polarizations have inhomogeneous distribution, resulting in the hybridly polarized vector fields that might have the feature of spatial-variant SoPs, i.e., non-zero gradient of SoPs. Such a type of hybridly polarized vector fields are completely different from the concepts of non-zero gradients of amplitude [39] and phase [40]. This unique feature can be expected to lead to new effect and phenomena in the interaction of light with matter.

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants 10934003 and 10874078, and the National Basic Research Program of China under Grant 2006CB921805.

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A new type of vector fields with hybrid states of polarization

Abstract

1. Introduction

2. Basic Principle

3. Experimental realization

4. Summary

Acknowledgements

References and links

Cited By

Figures (7)

Equations (7)

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