Generation of arbitrary vector fields based on a pair of orthogonal elliptically polarized base vectors

Danfeng Xu; Bing Gu; Guanghao Rui; Qiwen Zhan; Yiping Cui

doi:10.1364/OE.24.004177

1. Introduction

As an intrinsic and fundamental vectorial nature of light, the polarization plays an important role in focusing properties, propagation behaviors, and the light-matter interactions. In the past decades, manipulating polarization of light field has attracted growing interests owing to the fact that vectorial optical fields with spatially variant states of polarization (SoPs) exhibit fascinating properties and intriguing applications in optical tweezers [1, 2], optical microscopy [3, 4], optical micro-fabrication [5, 6], nonlinear optics [7, 8] etc.

Until now, many methods have been proposed to generate a variety of types of vector fields with the desired polarization distribution. For examples, Pohl [9] generated the radially polarized beams by using an intracavity axial birefrigent component. Bomzon et al. [10] obtained radially and azimuthally polarized beams by space-variant dielectric subwavelength gratings. Beresna et al. [11] demonstrated the generation of radially polarized optical vortex by femtosecond laser nanostructuring of glass. Marrucci et al. [12] generated vector fields based on q-plate devices. Actually, the q-plate can be considered as system introducing opposite topological phases on the two orthogonal circular polarizations [13]. Wang et al. [14] determined a radially polarized beam with controllable state of spatial coherence by using a rotating ground-glass plate and a radial polarization converter. Lerman et al. [15] created hybrid polarized beams by transmitting radially polarized light through a wave plate. Liu et al. [16] reported the generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer. Rong et al. [17] yielded arbitrary vector beams by means of two cascaded liquid crystal spatial light modulators (SLMs). Saucedo-Orozco et al. [18] presented the generation of unconventional polarization from light scattered by metallic cylinders under conical incidence. Chen et al. [19] generated an arbitrary space-variant vector field with structured polarization and phase distributions. Moreno et al. [20] demonstrated the arbitrary vector fields using a polarization optical Fourier processor. Chen et al. [21] demonstrated an interfermometric approach to generate arbitrary cylindrical vector beams on the higher order Poincaré sphere. Adopting a common-path interferometer implemented with a SLM, Wang’s group generated a variety of vector fields with localized linear polarization, such as cylindrical vector fields [22], elliptic-symmetry vector fields [23], parabolic-symmetry vector fields [24], and hyperbolic-symmetry vector fields [25].

As is well known, much attentions focus on the generation of vector fields based on the combination of a pair of orthogonally polarized base vectors. A localized linearly polarized vector field can be created by the linear combination of left-handed (LH) and right-handed (RH) circularly polarizations with opposite topological charges [13, 22, 26]. Full Poincaré beams are constructed from a coaxial superposition of a fundamental Gaussian mode and a spiral-phase Laguerre-Gauss mode having orthogonal polarizations [27]. Hybridly polarized vector fields are generated by the superposition of a pair of orthogonal linear polarizations with spiral phase [28].

The goal of the presented work is to extend the existing vector fields. For this purpose, we develop a generalization of arbitrary vector fields with hybrid polarization, named hybrid elliptically polarized vector fields. The existing vector fields, including the localized linearly polarized vector field [19–25 ] and the hybrid polarized vector field [28], are the interesting special cases of the presented work. Theoretically, we present an arbitrary vector field with hybrid polarization based on the combination of a pair of orthogonal elliptically polarized base vectors with spiral phase on the Poincaré sphere. Experimentally, we demonstrate the generation of such a kind of vector fields in a common path interferometer. Numerically, we also examine the tight focusing properties of the hybrid elliptically polarized vector fields.

2. Basic principle

To illustrate and characterize the polarization of light, a three-dimensional Poincaré sphere is a simple and visual geometric representation [29]. As shown in Fig. 1, a given point (2ψ, 2χ) on the Poincaré sphere Σ represents one kind of polarization, namely, circular polarization, linear polarization, or elliptical polarization. The angle 2ψ defines the longitude specifies the orientation of polarization while the angle 2χ is the latitude specifies the ellipticity. The SoP of the point (2ψ, 2χ) on Σ can be described by the unit vector as follows [28]

\begin{array}{l} \vec{S} (2 ψ, 2 χ) = & \frac{1}{\sqrt{2}} [sin (χ + π / 4) e^{- i ψ} + cos (χ + π / 4) e^{i ψ}] {\vec{e}}_{x} \\ + i \frac{1}{\sqrt{2}} [sin (χ + π / 4) e^{- i ψ} - cos (χ + π / 4) e^{i ψ}] {\vec{e}}_{y}, \end{array}

where e⃗_x and e⃗_y could be regarded as a pair of orthogonal linearly polarized base vectors along the x and y directions, respectively. It is noted that two SoPs on the same sphere Σ with (2ψ, 2χ) and (2ψ + π, −2χ) can also be served as a pair of orthogonal base vectors because of <S⃗(2ψ, 2χ)|S⃗(2ψ + π, −2χ)> = 0 from Eq. (1). Typically, three pairs of orthogonal base vectors on Σ are illustrated in Fig. 1.

Fig. 1 Poincaré sphere Σ and three pairs of orthogonal base vectors on Σ.

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It is well known that a localized linearly polarized vector field can be created by the linear combination of LH and RH circularly polarized spiral phase beams with opposite topological charges m = ±1 [22], as displayed in Fig. 2(a). Besides, a hybrid polarized vector field can be generated by the superposition of a pair of orthogonal linear polarizations [28], as illustrated in Fig. 2(b). Analogously, we propose the generation of arbitrary vector fields based on a pair of orthogonal elliptically polarized base vectors with spiral phase, as shown in Fig. 2(c). In a cylindrical coordinate system, an arbitrary vector field with hybrid polarization can be written as

\vec{E} (r, ϕ, 0) = E_{0} (r) [e^{- i δ (r, ϕ)} \vec{S} (2 ψ, 2 χ) + e^{i δ (r, ϕ)} \vec{S} (2 ψ + π, - 2 χ)],

Here the polar coordinate system (r, ϕ) describes the plane of the generated vector field, E ₀(r) stands for the axially symmetric amplitude distribution of the vector field, the space-variant phase δ(r, ϕ) is a function of r and ϕ. In principle, δ(r, ϕ) can have arbitrary spatial distributions. Generally, the phase factors of e ^{−iδ(r, ϕ)} and e ^{iδ(r, ϕ)} could be interpreted as having opposite topological charges.

Fig. 2 Generation of arbitrary vector fields based on the superposition of a pair of orthogonal base vectors with the space-variant phase of δ(ϕ) = mϕ (m = ±1, as represented by the arrow tips in the large circle in the center of each graph). (a) A local linearly polarized vector field superposed with the orthogonal LH and RH circularly polarizations. (b) A hybrid polarized vector field superposed with the orthogonal linear polarizations. (c) A hybrid elliptically polarized vector field superposed with the orthogonal elliptically polarizations.

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Inserting Eq. (1) into Eq. (2), one gets

\vec{E} (r, ϕ, 0) = E_{0} (r) [e^{i χ} cos δ (r, ϕ) {\vec{e}}_{x}^{'} + i e^{- i χ} sin δ (r, ϕ) {\vec{e}}_{y}^{'}],

where e⃗′_x = (cosψ − sinψ)e⃗_x + (cosψ + sinψ)e⃗_y and e⃗′_y = −(cosψ + sinψ)e⃗_x + (cosψ − sinψ)e⃗_y. Here unit vectors e⃗′_x and e⃗′_y with <e⃗′_x|e⃗′_y> = 0 are a pair of orthogonal linearly polarized base vectors. Interestingly, the created vector field described by Eq. (3) is only dependent on both the latitude angle 2χ and the space-variant phase δ(r, ϕ) but is independent on the longitude angle 2ψ on the Poincaré sphere. In principle, by modifying the parameters of both 2 χ and δ(r, ϕ), any desired vector field based on the combination of a pair of orthogonal elliptically polarized base vectors could be achieved.

Some interesting special cases of Eq. (3) are as follows. For the case of χ = π/4, as an example, two orthogonal base vectors of (−π/2, π/2) and (π/2, −π/2) are at the north and south poles on Σ as shown in Fig. 1, representing RH and LH circular polarizations, respectively. As a result, Eq. (3) reduces to the localized linearly polarized vector field as

\vec{E} (r, ϕ, 0) = E_{0}^{'} (r) [cos δ (r, ϕ) {\vec{e}}_{x}^{'} + sin δ (r, ϕ) {\vec{e}}_{y}^{'}],

where E′ ₀(r) = E ₀(r)e ^iπ/4 is the amplitude of the vector field. Equation (4) is in agreement with the reported result [22]. Accordingly, the generated principle of the localized linearly polarized vector field is shown in Fig. 2(a).

In addition, when χ = 0, for instance, a pair of orthogonal linear polarizations is located at the equator with (−π/2, 0) and (π/2, 0) on Σ as displayed in Fig. 1. In this case, we deduce the hybrid polarized vector field from Eq. (3) as

\vec{E} (r, ϕ, 0) = E_{0} (r) [cos δ (r, ϕ) {\vec{e}}_{x}^{'} + i sin δ (r, ϕ) {\vec{e}}_{y}^{'}] .

The obtained result is consistent with the one reported previously [28]. As illustrated in Fig. 2(b), this hybrid polarized vector field can be superposed with a pair of orthogonal linear polarizations with the space-variant phase of δ(ϕ) = ϕ.

In fact, the linear and circular polarizations are two special extreme cases. The more general polarized field is elliptical polarization. As shown in Fig. 1, any point (2ψ, 2χ) on Σ corresponds to the elliptical polarization, except for the points at the two poles (2ψ, ±π/2) and at the equator (2ψ, 0). As an example, we take a pair of orthogonal elliptical polarizations located at two points of (−π/2, π/4) and (π/2, −π/4) on Σ as displayed in Fig. 1, which can be obtained by two identical λ/3 waveplates as we will show below. Accordingly, Fig. 2(c) demonstrates the generation of this kind of vector fields by using a pair of orthogonal elliptically polarized base vectors with the space-variant phase of δ(ϕ) = ϕ.

3. Experimental demonstration

To experimentally generate the above-mentioned arbitrary vector fields, a flexible approach in a common path interferometer with the aid of a 4 f system [22] is adopted. The experimental arrangement is shown in Fig. 3. A linearly polarized laser beam at a wavelength of 532 nm is expanded and collimated to obtain a nearly uniform-intensity distribution. Then the beam is diffracted by a computer-controlled holographic grating on a transmissive phase-only SLM with 1024×768 pixels (each pixel has a 14 μm × 14 μm size), in which the space-variant phase δ(r, ϕ) is set. For the sake of simplify, we take δ(r, ϕ) = mϕ + φ ₀, where m and φ ₀ are the azimuthal topological charge and the initial phase of the vector field, respectively. Subsequently, the diffracted light with ±1th orders comes into a 4 f system composed of two identical lenses (L1 and L2). Then the two parts have orthogonally polarization states using a pair of identical waveplates (W) in the vicinity of the Fourier plane of the 4 f system. For instance, we can use two λ/4, λ/3, and λ/2 waveplates to obtain orthogonal circular, elliptical, and linear polarizations, respectively. Finally, two orthogonally polarized parts are combined by both the lens (L2) and a Ronchi phase grating (G), and give rise to the generation of arbitrary vector fields as the output. The period of holographic grating is adjusted to match with that of G as much as possible. The Ronchi phase grating with the period of 40 lines/mm has a ∼ 50% diffraction efficiency for the ±1 orders. The generated vector field is characterized and analyzed by a detector (D) (Beamview, Coherent Inc.). The experimental details can be found elsewhere [6–8 , 22–25 ]. Note that the experimental system has great flexibility but the conversion efficiency is low. Nevertheless, the purpose of the experiment is to verify our theory.

Fig. 3 Schematic of experimental arrangement for generating the arbitrary vector field based on the combination of a pair of orthogonal polarized base vectors with spiral phase on the Poincaré sphere.

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Experimentally, in order to obtain a pair of orthogonal elliptically polarized base vectors using two identical waveplates, some special attentions should be considered. Theoretically, we define ϑ as the incident polarization direction of the ±1th order diffracted light with relative to the fast axis of the waveplate. Based on the Jones Matrix of arbitrary waveplates [29, 30], one can easily get the arbitrary vector fields generated by the experimental arrangement shown in Fig. 3. We find the relationship between the angle ϑ and the phase difference of the waveplate Δφ by

sin (2 ϑ) = \sqrt{\frac{{tan}^{2} \frac{Δ φ}{2} + 1}{2 {tan}^{2} \frac{Δ φ}{2}}}

for 2nπ + π/2 ≤ Δφ ≤ 2nπ + 3π/2 and Δφ ≠ 2nπ + π, here n is an integer. This means, to get a pair of orthogonal polarized base vectors using two identical waveplates, one should set the prescribed angle +ϑ with respect to the +1th order diffracted light and a waveplate while −ϑ for the −1th order diffracted light and another waveplate, and vice versa. Furthermore, compared with the expressions of the generated vector field by Poincaré sphere (see Eq. (3)) and by Jones Matrix approaches, we find the conversion formula between the latitude angle 2χ in Poincaré spherical coordinate system and the phase difference of the waveplate Δφ by

2 χ = arcsin (\sqrt{2} cos \frac{Δ φ}{2}) .

In principle, arbitrary two identical waveplates could be adopted in the experimental arrangement shown in Fig. 3. Accordingly, the latitude angle 0 ≤ 2χ ≤ π/2 in Poincaré spherical coordinate system could be obtained. Hence, an arbitrary vector field described by Eq. (3) could be generated.

To demonstrate the arbitrary vector fields based on the combination of a pair of orthogonal elliptically polarized base vectors on the Poincaré sphere, as an example, we use two λ/3 waveplates in the experimental arrangement shown in Fig. 3. In experiments, we set $ϑ = arcsin (\sqrt{2 / 3}) / 2$ by Eq. (6) with the known phase difference Δφ = 2π/3 for the λ/3 waveplate. Besides, we obtain χ = π/8 by Eq. (7). Accordingly, the locations of the orthogonal elliptically polarized base vectors on Σ are, as example, (−π/2, π/4) and (π/2, −π/4) shown in Fig. 1. Thus, we obtain the expression of the hybrid elliptically polarized vector field from Eq. (3) as

\vec{E} (r, ϕ, 0) = A_{0} circ (r / r_{0}) [cos (m ϕ + φ_{0}) {\vec{e}}_{x}^{'} + e^{i π / 4} sin (m ϕ + φ_{0}) {\vec{e}}_{y}^{'}],

where A ₀ is a constant, circ(·) is the circular function, and r ₀ is the radius of the vector field.

Theoretically, the normalized Stokes parameters of the vector field described by Eq. (8) are calculated as s ₁ = cos(2mϕ + 2φ ₀), $s_{2} = sin (2 m ϕ + 2 φ_{0}) / \sqrt{2}$ , and $s_{3} = sin (2 m ϕ + 2 φ_{0}) / \sqrt{2}$ . Experimentally, we use the quarter-wave retarder polarizer method [29] to measure the Stokes parameters of the generated vector fields. To validate the theoretical predictions, the hybrid elliptically polarized vector fields with different parameters of m and φ ₀ are generated experimentally. As an example, Fig. 4 shows the experimentally measured intensity pattern and normalized Stokes parameters of the generated vector field with m = 1 and φ ₀ = 0. For comparison, the corresponding theoretically results are displayed in the first row of Fig. 4. The distribution of SoP is also superimposed with the theoretical intensity pattern. Obviously, the experimentally measured results are in good agreement with the theoretically predictions. Besides, we also experimentally generate the localized linearly polarized vector field and the hybrid polarized vector field using λ/4 and λ/2 waveplates in our experimental system, respectively. The results are consistent with the ones reported previously [19–25 , 28]. Briefly speaking, we experimentally demonstrate the generation of the hybrid elliptically polarized vector field based on the combination of a pair of orthogonal elliptically polarized base vectors with spiral phase.

Fig. 4 Theoretically predicted and experimentally measured intensity patterns and normalized Stokes parameters of hybrid elliptically polarized vector fields with m = 1 and φ ₀ = 0. The dimension of all of these images is 5mm × 5mm.

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4. Tight focusing properties

Due to the presence of the degree of freedom χ on Poincaré sphere, the arbitrary vector field with the expression given in Eq. (3) may have richer focusing properties compared with the linear and hybrid polarized vector fields [31, 32]. For the sake of simplify, we take the space-variant phase δ(r, ϕ) = mϕ + φ ₀ and the amplitude distribution E ₀(r) = A ₀circ(r/r ₀) of the vector field as described in Eq. (3). According to the Richards-Wolf vectorial diffraction method [33] and using Bessel identities, we obtain the three-dimensional electric field in the focal region of an aplanatic lens as

\begin{array}{l} E_{x} (ρ, φ, z) = & i^{m - 1} c_{0} \int_{0}^{α} sin θ \sqrt{cos θ} P (θ) e^{i k z cos θ} \\ \times {2 (1 + cos θ) e^{i χ} J_{m} (ξ) cos (m φ + φ_{0}) \\ + (1 - cos θ) [(e^{i χ} - i e^{- i χ}) J_{m + 2} (ξ) cos (m φ + 2 φ + φ_{0}) \\ + (e^{i χ} + i e^{- i χ}) J_{m - 2} (ξ) cos (m φ - 2 φ + φ_{0})]} d θ, \end{array}

\begin{array}{l} E_{y} (ρ, φ, z) = & i^{m - 1} c_{0} \int_{0}^{α} sin θ \sqrt{cos θ} P (θ) e^{i k z cos θ} \\ \times {2 i (1 + cos θ) e^{- i χ} J_{m} (ξ) sin (m φ + φ_{0}) \\ + (1 - cos θ) [(e^{i χ} - i e^{- i χ}) J_{m + 2} (ξ) sin (m φ + 2 φ + φ_{0}) \\ - (e^{i χ} + i e^{- i χ}) J_{m - 2} (ξ) sin (m φ - 2 φ + φ_{0})]} d θ, \end{array}

\begin{array}{l} E_{z} (ρ, φ, z) = & 2 i^{m} c_{0} \int_{0}^{α} {sin}^{2} θ \sqrt{cos θ} P (θ) \\ \times [(e^{i χ} - i e^{- i χ}) J_{m + 1} (ξ) cos (m φ + φ + φ_{0}) \\ - (e^{i χ} + i e^{- i χ}) J_{m - 1} (ξ) cos (m φ - φ + φ_{0})] d θ, \end{array}

Here ξ = kρ sinθ and c ₀ = πA ₀ f/λ. ρ, φ, and z are the polar radius, azimuthal angle, and longitudinal position of the observation point in the cylindrical coordinate, respectively. J_m(·) is the Bessel function of the first kind of order m. k = 2π/λ is the wavenumber, λ is the wavelength, f is the focal length of the objective lens, α = arcsin(NA) is the maximum angle obtained by the numerical aperture (NA) of the objective lens. P(θ) is the pupil apodization function. For a uniform-intensity illumination, P(θ) = circ(β sinθ/sinα) is chosen in the entire analysis in this work, where β is the pupil filling factor defined as the ratio of the pupil radius to the beam waist. Setting χ = π/4 in Eq. (9) , one obtains the focal field of a localized linearly polarized vector field, which is coincident with the one reported previously [31, 34].

To explore the tight focusing properties of the hybrid elliptically polarized vector fields, we perform the numerical simulation using Eq. (9) with m = 1, φ ₀ = 0, NA = 0.8, and β = 1. Figure 5 shows the transverse I_T = |E_x|² + |E_y|² (top row), longitudinal I_Z = |E_z|² (middle row), and total intensity I = |E_x|² + |E_y|² + |E_z|² (lower row) patterns of vector fields with different values of χ at the focus in the x − y plane (z = 0). The intensity patterns are normalized by the maximum of the total intensity I ^max(x, y, 0). For the sake of comparison, the tightly focused intensities of both the hybrid polarized vector field (χ = 0) and the radially polarized beam (χ = π/4) are also listed in Fig. 5. In fact, the related results were reported previously [31, 32] and are included here for completeness. By tuning the value of χ from 0 to π/4, the ellipticity of orthogonal elliptically polarized base vectors increases from 0 to 1. Accordingly, the orthogonal polarized base vector experiences linear, elliptical, and circular polarizations. As a result, the intensity patterns of the focused hybrid elliptically polarized vector field change from a square-like profile with a fourfold rotational symmetry to a focal spot with a circular symmetry. More interestingly, the hybrid elliptically polarized vector field with χ = π/16 forms a flattop focusing with square-like pattern, similar to the circularly flattop focusing of the focused generalized cylindrical vector beam [35]. Beside the degree of freedom χ of the hybrid elliptically polarized vector field, the parameters m, φ ₀, NA, and P(θ) can be used to engineer the tight focusing field.

Fig. 5 Normalized intensity patterns of tightly focused vector fields with m = 1, φ ₀ = 0, NA = 0.8, β = 1, and different values of χ in the x − y plane (z = 0). The intensity patterns are normalized by I ^max(x, y, 0).

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It is shown that the longitudinal intensity of the tight focusing vector fields become stronger compared with the transverse intensity when the value of χ increases from 0 to π/4. To illustrate the contribution of the longitudinal intensity in the tight focused vector fields, the ratio of the maximum intensities of the longitudinal and transverse components $η = I_{Z}^{max} (x, y, 0) / I_{T}^{max} (x, y, 0)$ of the tightly focused vector fields with m = 1, φ ₀ = 0, β = 1, and different values of NA versus the latitude angle 2χ is plotted in Fig. 6. As expected, the contribution of the longitudinal intensity increases as the values of both NA and 2χ increase. Due to the non-propagating properties of the longitudinal component of the focal field under high-NA focusing, direct detection of the tightly focused focal field is very challenging and requires near-field optical imaging capabilities. Limited by the available resources, we only perform the numerical simulations of the tightly focused vector fields. Nevertheless, numerical results demonstrate that the additional degree of freedom 2χ provided by arbitrary vector field with hybrid polarization allows one to engineer the focusing field.

Fig. 6 The ratio $η = I_{Z}^{max} (x, y, 0) / I_{T}^{max} (x, y, 0)$ of the tightly focused vector fields with m = 1, φ ₀ = 0, β = 1, and different values of NA versus the latitude angle 2χ.

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5. Conclusion

In summary, we proposed and demonstrated a new type of vector fields, named hybrid elliptically polarized vector fields. Theoretically, we proposed an arbitrary vector field with hybrid polarization based on the combination of a pair of orthogonal elliptically polarized base vectors with spiral phase on the Poincaré sphere. The proposed vector field is only dependent on the latitude angle 2χ but is independent on the longitude angle 2ψ on the Poincaré sphere. Adopting the common path interferometric arrangement for generating this kind of vector fields, we presented the conversion relationship between the latitude angle 2χ in Poincaré spherical co-ordinate system and the phase difference of the waveplate Δφ. Experimentally, using two λ/3 waveplates in a common path interferometer, we generated the hybrid elliptically polarized vector field based on the combination of a pair of orthogonal elliptically polarized base vectors. Besides, we numerically investigated the tight focusing properties of the hybrid elliptically polarized vector fields. We demonstrated that the additional degree of freedom 2χ provided by arbitrary vector field with hybrid polarization allows one to control the spatial structure of polarization and to engineer the focusing field.

Acknowledgments

This work was supported by the National Science Foundation of China (Grant Nos: 11474052, 11504049) and the National Key Basic Research Program of China (Grant No: 2015CB352002).

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Generation of arbitrary vector fields based on a pair of orthogonal elliptically polarized base vectors

Abstract

1. Introduction

2. Basic principle

3. Experimental demonstration

4. Tight focusing properties

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (11)

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