1. Introduction

Adherence photonic spin Hall effect (PSHE) refers to the separations of spin photons in both the real and momentum space owing to the geometric phases [1–3]. The real space PSHE occurs in paraxial light beams that reflected from or transmitted through a simple dielectric interface [4,5], a homogeneous single- or multi-layer optical slab [6,7], or a photonic crystal slab [8]. The spin separations of the real space PSHEs are generally smaller than half of the beam sizes, which are equal to the beam waists for Gaussian beams [9]. However, it was demonstrated in 2021 that a small incident angle can arise naturally a wave-vector-varying Pancharatnam-Berry (PB) phase gradient when a paraxial beam is reflected from or transmitted through a homogeneous media [10]. The PB phase gradient induces a spin separation up to 2.2 times beam waist with a highest energy efficiency of 86%. However, a larger spin separation for the real space PSHE is still challenge.

The momentum space PSHEs are generated mainly by arranging the orientations of anisotropic structure units of liquid crystals [11] and metasurfaces [12–15] to create spatially-varying PB phases. Since the size of the unit can be down to subwavelength scale, large momentum-space spin separations can be achieved [16,17]. The separation angle between light field components with opposite spin can be up to thousand times diverging angle of the fields [18]. The active manipulation of the momentum-space PSHE is appealing for its applications in light detection and ranging [19], free space optical communications [20], particle manipulations [21], laser processing [22], and beam steering [23–25]. However, the orientations of anisotropic structure units of both the liquid crystals [11] and metasurfaces can be merely altered after fabricated. Thus, the intrinsic deflection angles of spin photons cannot be tuned, although there are ways have been proposed to tune the average deflection angles via the control of spin-conversion efficiency [26–28]. Therefore, an effective tunable method for manipulating the propagation directions of spin photons is still lacked.

Here, a novel tunable momentum-space PSHE is proposed based on two cascaded liquid crystal (LC) PB lenses [29]. By moving one of the PB lens laterally, linear PB phase gradients can be generated, which are proportional linearly to the lateral relative displacements. The propagation directions of spin photons can thus be flexibly tuned and double-lines two-dimensional (2D) continue beam scanning is achieved with a scanning angle of 39°×39° and a diverging angle of 0.028°×0.028°.

2. Theory

To demonstrate the 2D beam scanning based on the PSHE, two identical PB lenses are employed as shown by Fig. 1(a,b). The orientations of the LC molecules of PB lenses vary along the radial direction: θ=k₀r/4f (see Fig. 1(c,d)), where k₀, r, and f are the wavenumber, radial coordinate, and focal length of PB lenses, respectively. The inhomogeneous orientations of PB lenses are fabricated by a photopatterning technique [30,31]. When transmitted through a PB lens, the polarization state and phase distribution of a light field will be modulated. The modulation can be described by a Jones matrix. In the circular polarization bases, the matrix is [32]

The PB lenses are supposed to be ideal half waveplates whose fast axes vary spatially. According to Eq. (1), the incident photons will flip their spin states and acquire additional geometric phases. The geometric phases are Φ_±=±2θ, resulting from the orientations variance of the LC molecules. These geometric phases will focus and defocus the left and right- circular polarized (RCP/LCP) incident light fields, respectively. Assume two PB lenses are overlapped without any gap, the transmitted field is ${\textbf E} = {\textbf M}(x - \Delta {x_1},y - \Delta {y_1}){\textbf M}(x - \Delta {x_2},y - \Delta {y_2})$ with Δx_1,2, Δy_1,2 being the displacements of the two PB lenses. Light fields suffer from cascaded geometric phase modulations and twice spin-flipped processes. Their polarization states are unchanged, and the focus and defocus effects of the two PB lenses are cancelled out. However, the lateral displacements induce linear phase gradients of ± k₀[(Δx₂-Δx₁)x + (Δy₁-Δy₁)y]/f for the RCP and LCP incident states. The phase gradients separate the incident linear polarized light field into RCP and LCP components, deflecting them toward opposite directions along both the x- and y-directions: θ_x= ±(Δx₂-Δx₁)/f and θ_y= ±(Δy₂-Δy₁)/f.

 

Fig. 1. The schematic of tunable photonic spin Hall effect based on a pair of PB lenses. (a,b) The beam deflections for RCP (a) and LCP (b) incident states. (c) The orientations of the LC molecules of PB lenses. (d) The intensity pattern of a PB lens under polarizing microscope.

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When the gap between two PB lenses is nonzero d≠0, the diffraction effect of light fields in the gap must be considered, which cannot be handled by the Jones matrix. Thus, we employ ray transfer matrix method. In the transfer matrix method, an optical ray is described by its position and slope. Considering the degeneration of the RCP and LCP states, the ray vector can be described by $[{\vec{r}_ + },{\vec{r^{\prime}}_ + },{\vec{r}_ - },{\vec{r^{\prime}}_ - }]$ with ${\vec{r}_ \pm } = {x_ \pm }{\hat{e}_x} + {y_ \pm }{\hat{e}_y}$, ${\vec{r}^{\prime}_ \pm } = {x^{\prime}_ \pm }{\hat{e}_x} + {y^{\prime}_ \pm }{\hat{e}_y}$. The subscript ± stands for the rays with RCP and LCP states. The ray transfer matrixes for the free space and PB lens respectively are [32]

Assume that the PB lenses have displacements with the respect to the reference line, which are $\Delta {\vec{r}_1} = \Delta {x_1}{\hat{e}_x} + \Delta {y_1}{\hat{e}_y}$ and $\Delta {\vec{r}_2} = \Delta {x_2}{\hat{e}_x} + \Delta {y_2}{\hat{e}_y}$ for the first and second PB lenses, respectively. For a collimated incident ray ${{\textbf R}_0} = [{\vec{r}_ + },0,{\vec{r}_ - },0]$ transmitted through the first PB lens, the ray vector becomes

Then, the ray propagates in the gap, in the front plane of the second PB lens, the ray is

After transmitted through the second PB lens, the ray becomes

One can find from Eq. (5) that, the relative displacement between two PB lenses will deflect optical rays with RCP and LCP states toward opposite directions. The displacement of the first PB lens causes an additional change of the deflection angle $\Delta {\vec{r}_1}d/{f^2}$, toward the same direction. According to Eq. (5), when the gap between two PB lenses is nonzero (d≠0), the ray slopes for the RCP and LCP states have a term of ${\vec{r}_ \pm }d/{f^2}$, which will focus the rays [33]. The focal lengths for the RCP and LCP states are identically ${f^2}/d$. However, the positions of the rays for the two states are different, resulting in difference of focal positions of the RCP and LCP states, which have distances of respectively 80 cm and 70 cm from the second PB lens.

3. Results

To demonstrate the PHSE, two LC-PB lenses (PBL25-633-F50, Lbtek) designed at 633 nm are used. The focal lengths of the PB lenses are ±50 mm, with a clear aperture of 20 mm. The modulation efficiency of a LC-PB lens is 98%, meaning that 2% incident light passes through the lens directly without modulation. Although the thicknesses of the LC layers are in several micrometers, the LC layers are sandwiched by two 1.6 mm thick glasses. When two PB lenses are closely placed, the gap between the LC layers is estimated as 5 mm. Thus, the focus and defocus effects of two PB lenses cannot be cancelled out.

To visualize the spin-dependent separation phenomena, we set the initial polarization as vertical polarization state and pass it through a quarter-wave plate (QWP) before entering the PB lenses (see Fig. 2(a)). The incident polarization state for the PB lenses is thus (1/2)^1/2[-isin2ϕ, 1 + icos2ϕ,]. The incident state varies from the RCP, linear polarization, LCP, linear polarization, and back to RCP states, when the QWP angle ϕ varies from -45° to 135° gradually. After transmitted through the PB lens pair, the RCP and LCP components will be separated spatially owing to the PSHE. As shown by Fig. 2(b,c), intensities of the RCP and LCP components of transmitted field change with the incident polarization state. The change relationships of I_RCP/LCP= 1/4|±cos2ϕ-isin2ϕ±i|², in good agreement with the experiment measurement results.

 

Fig. 2. (a) The schematic of the control of spin separation by incident polarization. (b,c) The energies (b) and intensity patterns (c) for LCP and RCP field components changing with the rotation of QWP. Insets in (b) show the polarization state for the QWP angle being -45°, -15°, 15°, 45°, 75°, 105°, 135°, respectively.

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The deflection of the RCP and LCP light fields along the x- and y-axes are controlled respectively by the relative displacements along the x- and y-axes. While the deflection angle varies the relative displacements, the intensity profiles are merely changed, as shown by Fig. 3(a,b), where the intensity profiles of the RCP and LCP field components for different relative displacements are given. Figure 3(c,d) shows the relationships between the deflection angles and the relative displacements along the x- and y-axes, respectively. The experimentally measured and theoretically predicted results are in good agreement. To tune the relative displacements, the first PB lens is fixed, only the second PB lens is moving. For the relative displacement |d_x,y|<8 mm, the first PB lens is fixed at the origin $\Delta {\vec{r}_1} = 0$, i.e., the centers for the first lens and incident optical light are overlapped. The deflection efficiencies are ±1.15 degree/mm for LCP and RCP states. For the relative displacement d_x> 8 mm and d_x< -8 mm, the first PB lens is fixed respectively at $\Delta {\vec{r}_1} ={\mp} 8\textrm{ mm }{\hat{e}_x}$, so that the light beam is illuminated at the edge of the first PB lens. The deflection efficiency is 1.32 degree/mm for LCP state, while it is only -1.05 degree/mm for the RCP state. The rule is also suitable for the displacement along the y-direction. The total deflection ranges are 39^o× 39° and 36^o× 36°, for the LCP and RCP states. The diverging angles for the LCP and RCP field components are measured, they are almost identically equal ∼0.028^o× 0.028°. Therefore, 1392 × 1392 numbers of resolvable deflection angles can be formed. A PB lens pair with a smaller focal length and a larger the aperture can effectively increase the deflection range.

 

Fig. 3. (a,b) The measured intensity profiles of the RCP and LCP field components for different relative displacements. (c,d) The relationships between the deflection angle and relative displacements along the x- and y-directions.

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Table 1 compares our 2D beam scanner with some conventional scanner. Compared to the MEMS mirror [34], cascaded metasurfaces [19], and optical phase array [35], our LC-based beam scanner takes the advantage of easy manufacture and simple structure. Compared to the other LC-based beam scanners [36,37], our beam scanner has larger scanning range.

Table 1. Summary of beam scanning

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Finally, we apply the tunable PSHE to laser processing. The experiment setup is shown in Fig. 4(a), where a 1064 nm picosecond laser with a pulse width of 7.5 ps, a repetition rate of 100kHz and an average power larger than 1 W is used. The light beam from the picosecond laser passes through a QWP to prepare RCP state, and then enters the PB lens pair. The second PB lens is mounted on a two-axis motorized linear stage. The deflected beam is focused on a 200 nm thick gold film by 20× objective. The gold film is coated on a glass substrate. An optical illuminated and imaging system is built to monitor the fabricated process. By programming the movement of the motorized stage, the beam scanning trajectory can be controlled and different patterns can be fabricated, as shown by Fig. 4(b-d). Since the spot size on the gold film is ∼5 µm (see the inset in Fig. 4(a)), the widths of fabricated lines are in the range of 4∼5 µm, with a ∼1 µm roughness. The roughness can be reduced by narrowing the spot size.

 

Fig. 4. (a) Experiment setup for laser processing based on the photonic spin Hall effect. (b,c) The fabricated patterns in the gold films. (d) The intensity profile along the red line in (b). The inset in (a) shows the focused spot pattern of a 20× objective.

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4. Conclusions

A novel PSHE has been demonstrated for flexible control of the spin separation in the momentum space. The PSHE is created by a pair of LC-PB lenses. Based on the ray transfer matrix method, the deflection angles for two spin components of the transmitted beam are derived, which vary linearly with the relative lateral displacements of the lens pair. A continuous double-lines 2D beam scanning is achieved with a two-axis motorized linear stage. The scanning angles are up to 39^o× 39°, with a diverging angle of 0.028^o× 0.028°. Using this beam scanning system, different patterns are fabricated on a 200 nm thickness gold film. Therefore, these findings not only pave a flexible way for the control of spin photons, but also provide an effective method for laser processing, Lidar, particle manipulation.