Generation of arbitrary vector vortex beams on hybrid-order Poincar&#x00E9; sphere based on liquid crystal device

Shaozhen Lou; Yaqin Zhou; Yide Yuan; Tiegang Lin; Fan Fan; Xiaoqian Wang; Huihui Huang; Shuangchun Wen

doi:10.1364/OE.27.008596

1. Introduction

Polarization and phase are playing important roles on properties of light. Conventional polarization states have homogeneous spatial distribution which includes linear, circular, and elliptical polarizations. The surfaces of identical phase present simple structures such as flat, cylindrical and spherical surfaces. In the recent years, the vector beam which has a spatially inhomogeneous polarization state and the vortex beam with spiral wavefronts have attracted much attention. With this special optical property, the vector beam exhibits excellent potential to be used in many applications such as nanoscale optical imaging, and optical manipulation [1,2]. The vortex beam can be used for dichroism studying and data transmission [3,4].

Polarization and phase of vector vortex beams have spatial distribution which cannot be described by the Poincaré sphere (PS) [5,6]. To represent the vector vortex beams in the framework of the PS, higher-order Poincaré sphere (HOPS) was proposed and has offered great utility such as describing higher-order Pancharatnam-Berry geometric phase [7–11]. Shortly after, hybrid-order Poincaré sphere (HyOPS) came up as a supplement for the situation of the two orthogonal states with different topological charges [12,13]. HOPS and HyOPS are complete models to describe the evolution of phase and polarization of wave propagating in inhomogeneous anisotropic media. They are of great importance in the study of vector vortex light.

A vector vortex beam provides more degrees of freedom in beam manipulation. That means we can create a beam with more special structures to satisfy the needs of different applications [14,15]. Various approaches to generate vector vortex beams have been proposed, but most of them have some defects. Generally speaking, it can be divided into two categories: one is capable of producing complete sets of beams with poor controllability and complicated operation; the other is highly controllable but cannot produce complete sets of beams.

The method based on modified interference of different modes includes complex structure and beam merging [16,17], which leads to low controllability. Another way by using the combination of a metasurface which is difficult to fabricate and untunable once fabricated and a spiral phase plate for generating a vector beam and transforming it into a vector vortex beam needs mechanical adjustment [18–20]. That way causes much inconvenience and defect in practical application. The spatial light modulator can be used to generate arbitrary beam [21–23], but the complexity and the high cost of the system limit its range of application. Due to the good electrical controllability of liquid crystal devices, it is a common mode to generate vector vortex beams by liquid crystal devices [24–26]. However, the types of beams produced in previous work on the evolution of HyOPS based on metasurfaces is costly because of the diffculty of metasurfaces’ manufacturing process and is based on manual operation which leads to incalculable inaccuracy in experiment [27]. The precious work based on liquid crystal device is also limited and cannot achieve a complete set of beam states by totally electrical controlled [28]. To meet the requirements of the application and study, a flexible generation method of high efficiency and integratable structure need to be put forward. The theory and experiments are focused on fully electronic-controlled generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere based on liquid crystal device.

2. Theoretical basis

In our work, a system based on electrically controlled liquid crystal cell to achieve the fully electrical-controlled evolution on the HyOPS has been proposed. The system can convert a homogeneous polarized light beam into a vector vortex beam in one step and generate the phase difference to further adjust the state of polarization. When changing the position of the point on the HyOPS, we merely need to alter the voltage across the cell which can depress the defects resulting from manual operation on the optical devices in the experiment by fully electrical control. It is equipped with many advantages such as integration, easy preparation, low cost, better flexibility and controllability. The system can generate a complete set of beam states on the HyOPS and provide fundamental optical system support for optical tweezers, vector vortex beam optical communications and other applications of structured beam.

We now use a HyOPS to describe the polarization and phase of a vector vortex beam. Any polarization state of monochromatic vector vortex beams can be mapped on surface of hybrid-order Poincaré sphere. Here, two orthogonal bases, $| N_{l} 〉$ and $| S_{m} 〉$ , serve as the north and south poles where l and m are the topological charges. The beam can be described as a superposition of the orthogonal bases with coefficients $ψ_{N}^{l}$ and $ψ_{S}^{m}$ , in the paraxial approximation.

| ψ_{l, m} 〉 = ψ_{N}^{l} | N_{l} 〉 + ψ_{s}^{m} | S_{m} 〉,

where

| N_{l} 〉 = \sqrt{2} / 2 * ({\hat{e}}_{x} + i {\hat{e}}_{y}) \exp (i l φ)

and

| S_{m} 〉 = \sqrt{2} / 2 * ({\hat{e}}_{x} - i {\hat{e}}_{y}) \exp (i m φ) .

Here,

{\hat{e}}_{x}

and

{\hat{e}}_{y}

are the unit vectors along the x and y axes in the Cartesian coordinate system. And

φ

is the angle in the polar coordinate system.

Then we map the polarization states on the hybrid-order Poincaré sphere by representing the Stokes parameters in the sphere’s Cartesian coordinates, and the Stokes parameters are defined as:

S_{0}^{(l, m)} = | ψ_{N}^{l} |^{2} + | ψ_{S}^{m} |^{2},

S_{1}^{l, m} = 2 | ψ_{N}^{l} | | ψ_{S}^{m} | \cos ϕ,

S_{2}^{l, m} = 2 | ψ_{N}^{l} | | ψ_{S}^{m} | \sin ϕ,

S_{3}^{l, m} = | ψ_{N}^{l} |^{2} - | ψ_{S}^{m} |^{2},

where

ϕ = \arg (| ψ_{N}^{l} |) - arg (| ψ_{S}^{m} |)

,

| ψ_{N}^{l} |^{2}

and

| ψ_{S}^{m} |^{2}

are the intensities of

| N_{l} 〉

and

| S_{m} 〉

, respectively.

A HyOPS with l = 0 and m = 2 is shown in Fig. 1. $(θ, ϕ)$ are the spherical coordinates. The north pole $| N_{l} 〉$ and south pole $| S_{m} 〉$ represent left-handed and right-handed circularly polarized eigenstates with topological charges of l and m respectively. The point $| H_{l, m} 〉$ , $| V_{l, m} 〉$ , $| D_{l, m} 〉$ and $| A_{l, m} 〉$ represent the horizontal, vertical, diagonal and antidiagonal polarization bases, respectively. The point coordinate on the HyOPS is electrically controlled by the liquid crystal cells. The movement direction of the point while voltage increasing is showed by navy blue arrowhead.

Fig. 1 Schematic illustration of the HyOPS.

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3. Experimental setup

Most methods of generating vector vortex beams are carried out step by step for vector part and vortex part which lead to more potential errors in terms of matching two parts. To generate a vortex phase, numerous methods have been developed so far, such as spiral phase plates [29,30], spatial light modulators [31–33], diffractive elements and so on [34–37]. However, a q-plate cell can convert Gauss beam to vector beam and generate a vortex phase in one step. Compared with other methods, it is equipped with higher efficiency and controllability with less complicated structures.

Therefore, we establish an experimental setup based on electrically controlled liquid crystal cell to generate arbitrary vector vortex beams on the HyOPS and the system is shown in Fig. 2(a).

Fig. 2 (a). The system to generate arbitrary vector vortex beams. (b). The system to measure the relationship between retardation and the voltage across the cell.

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A Glan laser polarizer (GLP) with optical axis = 90° and a quarter-wave plate (QWP1) with optical axis = 45° convert the laser beam from the He–Ne laser (beam waist size $w_{0}$ = 0.7 mm and operational wavelength λ = 632.8 nm) into a circular polarized beam. Next, the q-plate cell (QP) with $q = 1$ and $α_{0} = π / 2$ converts Gauss beam to vector vortex beam. Then two quarter-wave plates (QWP2 with optical axis = 45° and QWP3 with optical axis = 135°) together with a wave plate cell are used to generate an electrically controlled phase retardation. QWP4 and GLP2 are used to measure the Stokes parameters.

Next, we use the expression to illustrate the specific function of q-plate and wave plate. The Jones vector of the electric field associated with the input wave is given by

| ψ 〉 = \frac{\sqrt{2}}{2} ({\hat{e}}_{x} + i σ {\hat{e}}_{y}) \exp (i l φ) .

The beam in the inhomogeneous anisotropic media (q-plate) can be written as

\begin{array}{l} | ψ 〉 = \cos \frac{δ}{2} \frac{\sqrt{2}}{2} ({\hat{e}}_{x} + i σ {\hat{e}}_{y}) \exp (i l φ) + \\ s i n \frac{δ}{2} \frac{\sqrt{2}}{2} ({\hat{e}}_{x} - i σ {\hat{e}}_{y}) \exp (i m φ) \exp [i (2 σ α_{0} - \frac{π}{2})], \end{array}

where

σ = + 1

represents the left-handed circular (LHC) polarization,

σ = - 1

represents the right-handed circular (RHC) polarization, the phase retardation value

δ = 2 πdΔn / λ_{0}

,

α_{0}

is a constant angle specifying the initial orientation on the axis x, and q is a constant specifying the topological charge and

m = l + 2 σ q

. Obviously, two parts of the expression represent the north and south poles of HyOPS with coefficients

\cos δ / 2

and

\sin δ / 2

and the beam is a superposition of them. When l = 0 and

m \neq 0

, the phase term

\exp (il φ)

is a constant and the value of phase term

\exp (im φ)

depends on the azimuth.

To generate arbitrary vector vortex beams on HyOPS, we need adjust the phase difference between two terms in Eq. (7) by using a wave plate cell generating a phase retardation $Γ$ . A controllable phase term $exp (iΓ)$ is added to $| ψ 〉$ :

\begin{array}{l} | ψ 〉 = \cos \frac{δ}{2} \frac{\sqrt{2}}{2} ({\hat{e}}_{x} + i σ {\hat{e}}_{y}) \exp (i l φ) + \sin \frac{δ}{2} ({\hat{e}}_{x} - i σ {\hat{e}}_{y}) \\ \times \exp (i m φ) \exp [i (2 σ α_{0} - \frac{π}{2})] exp (i Γ) . \end{array}

Both of phase retardation $δ$ and $Γ$ are electrically controlled. What’s more, they correspond to the spherical coordinates one by one. A concise relationship between $(θ, ϕ)$ and $(δ, Γ)$ can be established as $θ = δ$ and $ϕ = - (π / 2 + Γ)$ .

So, we can easily and accurately control the parameter $θ$ of the point coordinate on the HyOPS by changing the voltage across the q-plate cell to modulate the intensities of two parts (LHC part and RHC part) of the beam and control the parameter $ϕ$ using wave plate cell through the same way.

To study the relationship between retardation and the voltage, we place the wave plate cell between two orthogonal polarizers (the axis intersection angles between the cell and polarizers are 45°) and use a light intensity meter recording the intensity of emergent light to calculate the transmissivity. Figure 2(b) shows the measure system.

Transmission of a ECB cell is given by $T = \sin^{2} (δ / 2)$ , where $δ = 2 πdΔn / λ$ . Here, T is the transmittance calculated by the recorded data of light intensity and $δ$ is the phase retardation value. Through $δ = 2 πdΔn / λ$ , we can get the relationship between retardation and the voltage. Using the relationship showed in $T = \sin^{2} (δ / 2)$ , $θ = δ$ and $ϕ = - (π / 2 + Γ)$ , we can get the relationship between the parameter and the voltage. The result is shown in Fig. 3. To acquire the relationship between retardation and the voltage of the q-plate, we used a polarization grating measuring the intensities of two parts (LHC part and RHC part) of the beam under different voltages. Because the manufacturing process of the q-plate cell is exactly the same as the wave plate cell except the orientation and the optical axis of the latter cell is vertical, the two curves are very similar but opposite.

Fig. 3 The relationship between the parameter and the voltage. Cell information: spacer: 4 μm, photoalignment material: 0.5% SD1, LC material: 5CB.

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The fabrication of liquid crystal cell is described in the following procedure [38,39]. An alignment layer was prepared by coating 0.5% wt/wt solution of SD1 in N.N. dimethyl formamide (DMF) on conductive glass substrate. Then the substrate was baked at 100 degrees Celsius for 5 minutes to evaporate excess solvent. After the substrate became cool, put the spacers sprayed uniformly over one the substrate. Afterwards two substrates were assembled to make the cell and a liquid crystal injection port was kept. Next, the cell was sealed up and pressed for some time. Lastly, the cell was exposured for photo-alignment and the LC material was injected. Note that the result of photo-alignment is a major factor affecting cell quality.

Here we introduced the photo-alignment method of q-plate cell in detail. First, the glass substrate was fixed on to a rotating stage as in the experimental setup shown in Fig. 4. A UV lamp (Omnicure S1000) was used to illuminate the alignment layer. Two lenses were introduced in the light path to broaden the beam size and then an arrow light beam has been projected onto the substrate by a line mask with a cylindrical lens. Then after a wire-grid polarizer with axis at $π / 2$ to the x-axis (shown in Fig. 4) was placed in the light path to provide linearly polarized light. The rotating stage with SD1 substrate was rotated at the speed of one degree per minute. Thus, the half circle of rotation (i.e. 180 degrees) of the SD1 substrate provides the alignment, which aligns the easy axis at the azimuthal angle with the axial symmetry of $q = 1$ , $α_{0} = π / 2$ .

Fig. 4 Experiment setup for the patterned photo-alignment.

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We verify the polarization of the generated vector beams. Stokes parameters S1, S2, and S3 are measured by QWP4 and GLP2 in Fig. 2(a) [8]. They are given by

s_{1} = \frac{I (0 °, 0 °) - I (90 °, 90 °)}{I (0 °, 0 °) + I (90 °, 90 °)},

s_{2} = \frac{I (45 °, 45 °) - I (135 °, 135 °)}{I (45 °, 45 °) + I (135 °, 135 °)},

s_{3} = \frac{I (- 45 °, 0 °) - I (45 °, 0 °)}{I (- 45 °, 0 °) + I (45 °, 0 °)} .

Here, $I (α, β)$ is the intensity of light measured by a CCD, where $α$ and $β$ are, respectively, the optical axis directions of QWP and GLP with respect to the vertical direction. The positive angle is chosen as clockwise. By extracting the parameters from the Stokes parameters and utilizing the relationship between Stokes parameters and polarization, the polarization states on the cross section of the generated vector beam are acquired. Figure 5. shows theoretical and experimental polarization states of eight points on the HyOPS. It is clear that the experimental results show good agreement with the theory. By depicting the polarization distribution, it is proved intuitively that the generated beams are the desired vector part.

Fig. 5 Polarization and intensity distribution of the theoretical and experimental results of vector vortex beams (The left pictures of each points are theoretical results).

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The left column shows the results of the points $(-1, 0, 0)$ , $(1, 0, 0)$ , $(0, 1, 0)$ and $(0, -1, 0)$ on the HyOPS in order from top to bottom. They are points on the equator of HyOPS which means coefficients $ψ_{N}^{l}$ and $ψ_{S}^{m}$ are equal and the polarization states are linear. In fact, the linear polarization always appears in the part where the intensities of two parts of the beam are equal. The pictures of the spots are very similar because the changing of voltage across the wave plate cell has no effect on light intensity. The right column shows the results of the points $(0, 0, 1)$ , $(0, 0, -1)$ , $(0, \sqrt{2} / 2, \sqrt{2} / 2)$ and $(- \sqrt{2} / 2, 0, \sqrt{2} / 2)$ on the HyOPS. Obviously, two poles are circularly polarized and they are pure gaussian beam and vortex beam respectively. The specific applied voltages and spherical coordinates are listed in the Table 1.

Table 1. Applied voltages and spherical coordinates in the HyOPS of each case shown in Fig. 5.

View Table

There are some deviations in Fig. 5. The main reason is the cells that we made are not perfect, such as the thickness of liquid crystal is not completely uniform and the result of photo-alignment is not flawless, which cause the phase distribution of the beam not completely desired. Another reason for the errors is the difficulty of making all the pictures locate at the same pixel, which leads to inaccuracy of calculation.

4. Conclusion

In summary, an arbitrary vector vortex beam on the HyOPS have been realized by using a q-plate cell which can change the topological charges and phase of initial beam, besides, the wave plate cell also can alter phase difference. By using LC cells as the core components of the system, we can reduce cost and complexity and improve adjust accuracy and realize quickly switching among different points on the surface of HyOPS. What's more, switching process is completely electrically controlled which make the system possess possible of integration. In addition, this method makes the foundation for the further development of the transformation of different HyOPSs and shows the potential of electrical controlled liquid crystal devices in structural beams generation.

Funding

National Natural Science Foundation of China (61605046, 61704053, 11574079), Natural Science Foundation of Hunan Province (2018JJ3069), and Fundamental Research Funds for the Central Universities of China.

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(S₁, S₂, S ₃)	$(θ, ϕ)$	Voltage (Vpp)	(S₁, S₂, S ₃)	$(θ, ϕ)$	Voltage (Vpp)
(−1, 0, 0)	(π/2, π)	4.04, 1.98	(0, 0, 1)	(0, null)	1.53, null
(1, 0, 0)	(π/2, 0)	4.04, 4.05	(0, 0, −1)	(π, null)	2.55, null
(0, 1, 0)	(π/2, π/2)	4.04, 1.53	$(0, \sqrt{2} / 2, \sqrt{2} / 2)$	(π/4, π/2)	6.83, 1.53
(0, −1, 0)	(π/2, 3π/2)	4.04, 2.55	$(- \sqrt{2} / 2, 0, \sqrt{2} / 2)$	(π/4, π)	6.83, 1.98

Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere based on liquid crystal device

Abstract

1. Introduction

2. Theoretical basis

3. Experimental setup

4. Conclusion

Funding

References

Cited By

Figures (5)

Tables (1)

Equations (12)

Optics Express