1. Introduction

Paraxial beams can carry three types of angular momentum (AM): spin AM (SAM), intrinsic orbital AM (IOAM) and extrinsic orbital AM (EOAM). The SAM is determined by circular polarization state. The IOAM is associated with azimuthal phase structure. It is well known that light beam possessing IOAM exhibits a helical phase front in the mathematical form of exp(ilφ), where φ is the azimuthal angle and l is the topological charge [1]. The EOAM is arisen from the transverse shift of the beam center from the coordinate origin, which can be analogous to mechanical AM of a classical particle and is given by the cross-product of the transverse position of the beam center and its momentum. Recently, considerable interest has been exhibited in spin-orbit interactions (SOI) of light [2], in which the spin (circular polarization) of light can affect and control its spatial degrees of freedom. In SOI effect, the SAM can both interact with the IOAM and EOAM. The interaction between the SAM and IOAM leads to spin controllable optical vortices generation and transformation. In turn, the interaction between the SAM and EOAM results in spin Hall effect (SHE). Examples of the SOI of light include SHE of laser beam reflected or refracted at a dielectric interface [3–5], spin dependent optical vortex generation [6] and SHE [7–9] at the tight focus, spin controlled unidirectional propagation of the surface or waveguide modes in coupling between transversely-propagating spin light and the evanescent tails of the surface or waveguide mode [10,11], and the SOI in artificial structures such as metamaterial and metasurface [12], etc.

Propagation of laser beam in a uniaxial crystal is also a representative process having SOI. Since A. Ciattoni et al. [13,14] found that a particular circularly polarized beam propagating along the optic axis in a uniaxial crystal can generate a vortex with a reversed circular polarization, numerous studies of SOI in this polarization conversion process have been carried out [15–22]. This spin to orbit conversion was widely used to realize the generation of vortex beams [14,20] and the change of the Bessel beam order [19]. However, in all these previous studies, the initial input light beams have cylindrical symmetry, such as Gaussian beam [14], circular Airy beam [16], Bessel-Gaussian beam [17], Bessel beam [19,20], circular Airy vortex beam [21], Laguerre-Gaussian beam [22], etc. In these cases, the total system preserves cylindrical symmetry so that the output light after passing through the uniaxial crystal doesn’t exhibit spin dependent symmetry breaking. Therefore, in all these previous studies, the SOI produces only spin-to-IOAM conversion, and no interaction between the SAM and EOAM, i.e., SHE, occurs. As a result, the SHE of laser beam propagating in uniaxial crystal is rarely studied. Physically, only when the cylindrical symmetry is broken, a spin-orbit coupled system can produce spin dependent symmetry breaking. This can be obtained via simple symmetry analysis. Therefore, to explore the SHE of beams propagating in a uniaxial crystal, in Refs. [23,24], the uniaxial crystal was placed tilted with respect to the beam axis to break the cylindrical symmetry of the system; and in Ref. [25], electro-optic effect is employed to break the cylindrical symmetry of the system. In their works, however, the input laser beam was chosen to be the simplest kind of beams, Gaussian beams. It is well known, as possessing more adjustable degrees of freedom, structured light beams can generate many novel kinds of light-matter interactions, which can provide unprecedented new technologies and opportunities for the development of numerous research fields, such as microimaging, topological photonics, optical manipulating, nonlinear optics, etc. So, investigating the SHE of structured light beams propagating in uniaxial crystal is of fundamental interests. Employing the structured light beams, the spatial structure distribution of the input beam can be constructed artificially, hence the way of cylindrical symmetry breaking can be very flexible, which can exhibit a variety of spatial distributions. This adjustable symmetry breaking can be used to realize the flexible control of the spatial distribution of photon spin, even customize the spin distribution according to the requirements of practical applications. This may open a new perspective on investigating the SHE of light beams propagating in uniaxial crystal.

Therefore, in this paper, as an example, we investigate the SOI of a kind of novel structured light beam, grafted vortex beam (GVB), under that the beam propagates along the optic axis of the uniaxial crystal. In this case, the cylindrical symmetry of the system is broken by the grafted helical phases. Hence a SHE determined by the spatial phase structure emerges during the propagation. The SHE and the evolution of the local AM of the GVB propagating in the uniaxial crystal with and without linear electro-optic (EO) effect is investigated. It is found that the SHE and the evolution of the local AM of the GVB are controllable both by changing the topological charge of the GVB and by modulating the DC electric field of the linear EO effect. It is also found that the spin separation in our SHE can reach the same size of the light intensity pattern, hence, is very large. It is well known that a large spin separation of the same size as the beam pattern can also be obtained when a light beam reflected near the Brewster angle on an air-prism interface. In this case, however, the reflection coefficient is near zero, hence the energy efficiency is very low. As a result, the figure of merit (the product of the energy efficiency and the ratio of the spin separation with respect to the intensity pattern size) is usually only ∼10⁻⁵ for Gaussian incident beams near the Brewster angle [24]. However, in our work, a figure of merit 1.172 is obtained, which is 5 orders of magnitude larger than the figure of merit in the case that a light beam reflected near the Brewster angle. Our results provide an opportunity to flexibly regulate the spin photons with a very high energy efficiency.

2. Grafted vortex beam

In general, a vortex has a complex exponential term, exp(iψ), where ψ is a phase function. For vortex beam with standard spiral phase, ψ=lφ, where φ is the azimuthal angle and l is the topological charge. While, the GVB [26,27] means grafting two or more different spiral phases in the angular direction. Its phase function reads:

 

Fig. 1. The schematic diagram of grafting two different spiral phases.

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Recently, Hao Zhang and his coauthors [26] found that the GVB has unique control ability of orbital angular momentum (OAM). Usually, the OAM distribution (local OAM) of an optical vortex strongly depends on the intensity, which results in difficulty in OAM independent modulation. However, by grafting different spiral phases, it is found that the OAM distribution and magnitude can be manipulated flexibly without changing the light intensity significantly [26]. It provides an ingenious method to control the local OAM and can promote potential applications in optical manipulation. Moreover, the GVB is a kind of superimposed OAM state. As it is found that the superimposed OAM state is of importance both in classical physics and quantum sciences [28], light beams possessing superimposed OAM state have recently received extensive attention from researchers [29–31]. By expanding the grafted complex exponential term $T({\psi _{\textrm{GVB}}}) = \exp (i{\psi _{\textrm{GVB}}})$ to be a superposition of a series of standard spiral phase vortices, i.e., $T({\psi _{\textrm{GVB}}}) = \sum\nolimits_{n ={-} \infty }^\infty {{F_n}\exp(in\varphi )}$ with ${F_n} = {1 / {2\pi }}\int_0^{2\pi } {T({\psi _{\textrm{GVB}}})} \exp({ - in\varphi } )d\varphi$ being the Fourier expansion coefficient, the relative power of the nth order OAM mode on the OAM spectrum of the GVB can be obtained to be ${P_n} = {|{{F_n}} |^2}$. Figure 2 shows the OAM spectrum of the GVB with l₁ = −3 and l₂ = 5. It can be seen that the GVB is a superimposed OAM state whose power is mainly concentrated in the −9^th – 10^th order OAM modes (the power of the OAM modes out of this range is very low).

 

Fig. 2. The OAM spectrum of the GVB with l₁ = −3 and l₂ = 5.

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In this paper, to show the SHE of the GVB propagating in uniaxial crystal and explore its electrically controlled local AM evolution by linear EO effect, grafted Gauss vortex beam grafting two different spiral phases is employed as the incident beam, whose expression has the form

3. Propagation theory of GVB in uniaxial crystal

We use the paraxial vector theory developed by A. Ciattoni et al. [32] to addresses the propagation of the GVB in uniaxial crystal. When laser beam propagates in an anisotropic medium, the electric field obeys the following equation:

Equations (6)–(9) show that during the SOI, the nth-order RHCP component of the input light is partly converted to LHCP light with the vortex order increasing by 2, and the nth-order LHCP component of the input light is partly converted to LHCP light with the vortex order decreasing by 2. By using these equations, we can calculate the electric field distribution of the beam at arbitrary observation plane z, hence can explore the evolution of the GVB in uniaxial crystal as well as the evolution of the local AM. The OAM and SAM densities in z direction is given by ${l_z} = ({{{{\varepsilon_0}} / {2\omega }}} ){\mathop{\rm Im}\nolimits} ({E_x^\ast {\partial_\varphi }{E_x} + E_y^\ast {\partial_\varphi }{E_y}} )$ and ${s_z} = ({{{{\varepsilon_0}} / \omega }} ){\mathop{\rm Im}\nolimits} ({E_x^\ast {E_y}} )$, respectively [33]. Here, ${\varepsilon _0}$ is the permittivity of vacuum, $\omega$ is the angular frequency of the incident light, and ${E_x}$ and ${E_y}$ are the x and y polarized components of ${\bf E}({r,\varphi ,z} )$, respectively. By using the relations ${E_x} = {{[{{E_ + }({r,\varphi ,z} )+ {E_ - }({r,\varphi ,z} )} ]} / {\sqrt 2 }}$ and ${E_y} = i{{[{{E_ + }({r,\varphi ,z} )- {E_ - }({r,\varphi ,z} )} ]} / {\sqrt 2 }}$, we have

According to the refraction index ellipsoid theory, the influence of linear EO effect is changing the refraction index of the crystal. In this paper, we consider a simple case of linear EO effect, in which the SBN crystal is chosen as the working one and the DC electric field E_DC is applied along the optic axis. In this case, the three components of the linear EO polarization [34] is $P_x^{EO} ={-} {\varepsilon _0}n_o^4{r_{13}}{E_{DC}}{E_x}({r,\varphi ,z} )$, $P_y^{EO} ={-} {\varepsilon _0}n_o^4{r_{13}}{E_{DC}}{E_y}({r,\varphi ,z} )$ and $P_z^{EO} ={-} {\varepsilon _0}n_e^4{r_{33}}{E_{DC}}{E_z}({r,\varphi ,z} )$, respectively. It implies that the effective ordinary and extraordinary refractive indices become $n_o^{eff} = {n_o} - {{n_o^3{r_{13}}{E_{DC}}} / 2}$ and $n_e^{eff} = {n_e} - {{n_e^3{r_{33}}{E_{DC}}} / 2}$, respectively. With this knowledge, we can further investigate the electrically controlled evolution of the GVB in uniaxial crystal under linear EO effect.

4. Numerical calculation and discussion

To numerically calculate the propagation of the GVB in uniaxial crystal and explore its SHE, we consider that the polarization of the incident GVB is along x axis, i.e., ${\bf E}({r,\varphi ,0} )= {E_0}{\hat{e}_x}\exp ({{{ - {r^2}} / {{s^2}}}} )T({{\psi_{\textrm{GVB}}}} )$, thus ${E_{0 + }} = {E_{0 - }} = {{{E_0}} / {\sqrt 2 }}$. For simplicity, we set ${E_0} = \sqrt 2$, i.e., ${E_{0 + }} = {E_{0 - }} = 1$. The wavelength of the incident light is chosen 632.8 nm, where the SBN crystal has large EO coefficient ${r_{33}} = 1340{{\textrm{pm}} / \textrm{V}},{r_{13}} = 66{{\textrm{pm}} / \textrm{V}}$ and refractive indices ${n_o} = 2.3117,{n_e} = 2.2987$. Moreover, the waist width is chosen to be s = 15um.

Figure 3 gives the intensity (first line), OAM density (second line) and SAM density (third line) distributions of GVB with l₁=-l₂ = 1 at different propagation distances. Although the distribution pattern evolution is complex, the physics behind is clear, and some meaningful insights can be obtained. Firstly, the center of mass of the intensity distribution moves along negative x direction during propagation. It is due to that the beam phase is tilted towards the negative x-axis because of the positive topological charge (l₁ = 1) of the “scion” and the negative topological charge (l₂=−1) of the “rootstock”. Secondly, the OAM density distributions are spatially nonuniform but odd symmetric about the x-axis, so that the total (global) OAM is always zero during the propagation; and though the evolution of the OAM density is rapid, the shape of the intensity distribution is almost unchanged, implying that the OAM distribution can be changed without changing the light intensity significantly. Thirdly, the SAM density distributions are also spatially nonuniform and odd symmetric about the x-axis. This is of interest and implies that the SHE emerges during the propagation. In the SAM density distributions, there are two irregular leaf-shaped patterns, and the signs of the SAM density value of the top and bottom patterns are positive and negative, respectively. It means that the LHCP and RHCP components are shifted to positive and negative y direction, respectively. Physically, if a SOI system has cylindrical symmetry, it cannot exhibit spin dependent symmetry breaking, hence the SOI can only produce spin-to-IOAM conversion and the SHE cannot occur. In our work, the cylindrical symmetry of the system is broken by the tilted beam phase towards the negative x-axis of the GVB. This is the physical origin of the SHE of our work.

 

Fig. 3. The intensity (first line), OAM density (second line) and SAM density (third line) distributions at different propagation distances under l₁=-l₂ = 1, where (a1)-(c1), (a2)-(c2) and (a3)-(c3) are the results of z = 5 mm, z = 10 mm and z = 15 mm, respectively. The OAM and SAM densities are the results relative to ${{{\varepsilon _0}} / {2\omega }}$.

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Based on the conservation of total AM, a simple and intuitive explanation of this SHE can be provided. Because of the tilted phase of the GVB, there exist a deflection angle $\theta ({{l_1},{l_2}} )$ of the beam axis with respect to the optic axis, i.e., z-axis, resulting in nonzero transverse projection ${s_ \bot }$ of single photon SAM s on the xy plane. In the case that a LHCP photon is converted to RHCP one via SOI, the variation of the SAM transverse projection is $2{s_ \bot }$. Therefore, to preserve the conservation of the total AM, an additional OAM ${l_ + } ={-} 2{s_ \bot }$ should be acquired to offset the change of the SAM (see Fig. 4(a)). Conversely, in the case that a RHCP photon is converted to LHCP one via SOI, the variation of the SAM transverse projection is $- 2{s_ \bot }$, hence an additional OAM ${l_ - } = 2{s_ \bot }$ should be acquired to preserve the conservation of the total AM (see Fig. 4(b)). Since ${l_ + }$ and ${l_{-} }$ are nonzero and their signs are opposite, the center of mass of the LHCP and RHCP photons have to move along opposite directions on the xy plane. Hence the SHE occurs. This is a general explanation suitable for any combination of ${l_1}$ and ${l_2}$. For the particular case of l₁=-l₂ = 1 discussed in Fig. 3, the transverse unit vector ${\hat{e}_ \bot }$ will coincide with the unit vector ${\hat{e}_x}$ of the x-axis. In this case, the negative additional OAM ${l_ + } ={-} 2{s_ \bot }$ causes the center of mass of the RHCP photon to move in negative y direction; and the positive additional OAM ${l_ - } = 2{s_ \bot }$ causes the center of mass of the LHCP photon to move in positive y direction. This is well agreed with the calculated results of Fig. 3. For more general cases of ${l_1} \ne - {l_2}$, the transverse unit vector ${\hat{e}_ \bot }$ will not coincide with the unit vector ${\hat{e}_{x}}$ of the x-axis. In these cases, the SHE can still occur, but instead of separating along the y-axis, the spin photons separate along the direction perpendicular to ${\hat{e}_ \bot }$. It should be noted here that, strictly, the system does not have rotational symmetry with respect to the beam axis because of the noncoincidence of the beam axis and the optic axis. However, due to that the deflection angle θ is very small, the breaking of rotational symmetry is so slight that can cause only negligible nonconservation of the total AM (i.e., the variation of the AM caused by symmetry breaking is significantly less than the variation of the AM caused by SOI), therefore we consider here that the total AM is conserved in the SOI process. The above explanation, though not very rigorous, is simple and intuitive.

 

Fig. 4. The schematic diagram of the change of AM during the SOI. (a)/(b) is the case that a LHCP/ RHCP photon is converted to RHCP/LHCP one via SOI, where s and ${s_ \bot }$ are the SAM and its transverse projection on the xy plane of LHCP (blue line) and RHCP (green line) photons, l₊ and l_- are the additional OAM acquired in order to preserve the conservation of the total AM during the SOI, and θ is the deflection angle of the beam axis with respect to the z-axis caused by the tilted phase of the GVB.

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Obviously, it can be found from Fig. 4 that when the deflection angle θ increases, the change of SAM's transverse projection during SOI process increases, thus to preserve the conservation of the total AM, l₊ and l_- should be increased. As the lager ${l_ + }$ and ${l_ - }$ imply lager spin separation, according to the above explanation, it's easy to draw a conclusion that when the grafted topological charge l₁=-l₂ > 1 (the deflection angle θ increases with the increase of topological charge), the SHE will become more pronounced and, the larger the topological charge l₁=-l₂ is, the more pronounced the SHE becomes. This is unequivocally supported by the results of Fig. 5, from which we can see that when l₁=-l₂ = 2 the major lobes of the LHCP and RHCP components are coincident and their side lobes exhibit slight different (it means that the SHE is slight); when l₁=-l₂ = 3 the major lobes of the LHCP and RHCP components are still coincident but their side lobes exhibit more significant different (it means that the SHE becomes more pronounced); and when l₁=-l₂ = 4 the major lobes of the LHCP and RHCP components are not coincident at all (the strongest lobe of the LHCP component is the side lobe in the positive y-axis direction, while the strongest lobe of the RHCP component is the side lobe in the negative y-axis direction, meaning that the SHE becomes more significant). Therefore, the SHE can be controlled effectively by changing the topological charge of the grafted spiral phases.

 

Fig. 5. The intensity distributions of the total field (first line), LHCP component (second line) and RHCP component (third line) at z = 10 mm for different GVBs, where (a1)-(c1), (a2)-(c2) and (a3)-(c3) are the results of l₁=-l₂ = 2, l₁=-l₂ = 3 and l₁=-l₂ = 4, respectively.

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To show more detailed information about the influence of the grafted topological charge on the AM evolution, the OAM and SAM density distributions at z = 10 mm for different GVBs are calculated, and the results are shown in Fig. 6. It can be seen clearly that both the OAM and SAM density distributions can be changed by changing the topological charge of the grafted spiral phases.

 

Fig. 6. The OAM density (first line) and SAM density (second line) distributions at z = 10 mm for different GVBs, where (a1)-(b1), (a2)-(b2) and (a3)-(b3) are the results of l₁=-l₂ = 2, l₁=-l₂ = 3 and l₁=-l₂ = 4, respectively. The OAM and SAM densities are the results relative to ${{{\varepsilon _0}} / {2\omega }}$.

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Obviously, from Fig. 4, we can further obtain that because there exists transverse shift of the photons undergoing polarization conversion (LHCP/RHCP converts to RHCP/LHCP), if the number of photons undergoing polarization conversion increases, the transverse shift of the center of mass of the overall LHCP component or RHCP component should increase, implying the SHE can be further influenced by changing the polarization conversion efficiency. As it is well known that the linear EO effect is an effective way to manipulate the polarization conversion efficiency of the light beam propagating in uniaxial crystal, we further investigate the influences of the linear EO effect on the SHE. For this, the intensity distributions of the LHCP and RHCP components, and the SAM density distributions at z = 5 mm under l₁=-l₂ = 4 and different dc electric field E_DC are calculated. The numerical results are shown in Fig. 7. It can be seen that when E_DC = −16.87 kV/cm, the SHE no longer occurs. It is due to that the birefringence of the SBN crystal is erased when E_DC = −16.87 kV/cm, resulting in no spin-orbit coupling during the propagation. When the dc electric field is positive, with the increase of the dc electric field, more and more energy of the LHCP component will be concentrated on the sidelobe above the x-axis, while more and more energy of the RHCP component will be concentrated on the sidelobe below the x-axis, implying the SHE become more and more pronounced. This is due to that a positive dc electric field increase the birefringence of the SBN crystal, resulting in the increasing of the polarization conversion efficiency. Therefore, the SHE can be enhanced or suppressed simply by controlling the dc electric field. However, here we should note that the polarization conversion efficiency cannot increase to 100% with the continuous increase of the dc electric field. It has been found that an incident LHCP (RHCP) OAM mode with Gaussian space envelope can only convert to RHCP (LHCP) OAM mode with a limit conversion efficiency of 50% [18]. Therefore, the conversion efficiency of the grafted Gauss vortex beam investigated here is limited by 50%, since the arbitrary nth order OAM mode ${E_{0 \pm }}{F_n}\exp ({{{ - {r^2}} / {{s^2}}}} ){e^{in\varphi }}$ on the GVB OAM spectrum is a Gaussian vortex. It has been demonstrated that light beams with different space envelope (Bessel-Gaussian, Laguerre–Gaussian and Hermite–Gaussian) have different conversion efficiencies of SAM to OAM when propagating in uniaxial crystals [17,35]. So, if the light beam is other kind of GVB with other kind of space envelope, the situation can be different so that the SHE can be further enhanced by applying larger dc electric field.

 

Fig. 7. The intensity distributions of the LHCP component (first line) and RHCP component (second line), and the SAM density distributions (third line) at z = 5 mm under l₁=-l₂ = 4 and different dc electric field, where (a1) -(c1), (a2)-(c2), (a3)-(c3) and (a4)-(c4) are the results of E_DC = −16.87 kV/cm, E_DC = 0 kV/cm, E_DC = 20 kV/cm and E_DC = 60 kV/cm, respectively. The SAM densities are the results relative to ${{{\varepsilon _0}} / {2\omega }}$.

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In the SHE, for a given z plane, the transverse shifts of the center of mass of the LHCP and RHCP components along y-axis under the case of l₁=-l₂ are ${\delta _ \pm } = \int\!\!\!\int {y|{E_ \pm }({r,\varphi ,z} ){|^2}} dxdy/\int\!\!\!\int |{E_ \pm }(r,$$\varphi ,z ){|^2} dxdy$. According to this expression, the transverse shifts at z = 5 mm under l₁=-l₂ = 4 and E_DC = 60 kV/cm (corresponding to the intensity distribution results of Fig. 7(a4) and Fig. 7(b4)) are calculated to be ${\delta _ + } = 17\,{\mathrm{\mu} \mathrm{m}}$ and ${\delta _ - } ={-} 17\,{\mathrm{\mu} \mathrm{m}}$. As optical diffraction can affect both the spin separation in a SHE and the size of the light pattern, therefore the ratio of the spin separation with respect to the size of the light pattern is a good parameter to evaluate the SHE, which can eliminate the effect of the optical diffraction. Because the intensity distribution pattern is complex, to compare the calculated spin separation and the size of the light pattern, we extract the light intensity distribution on the circle whose center is the origin point O of the coordinate system and radius (${r_0} = 72.1\,{\mathrm{\mu} \mathrm{m}}$) is the distance between point O and the point with the strongest light intensity. The position of the circle is shown by the white dotted lines in Fig. 8(a) and Fig. 8(b); and the extracted intensity distribution of the LHCP component, RHCP component and the total field on the circle are shown in Fig. 8(c) by the red line, orange line and blue line, respectively. In Fig. 8(c), $S(\varphi )= {r_0}\varphi$ represents the length of the arc on the circle corresponding to the angle $\varphi$. We find that the half-height width of the first sidelobe of the LHCP component above the x-axis is $\varpi = 29\,{\mathrm{\mu} \mathrm{m}}$, therefore, the ratio of the spin separation $\Delta = 34\,{\mathrm{\mu} \mathrm{m}}$ with respect to this half-height width is 1.172, implying the spin separation is about the same size of the light intensity spot, hence, very large.

 

Fig. 8. The intensity distributions of the (a) LHCP component and (b) RHCP component at z = 5 mm under l₁=-l₂ = 4 and E_DC = 60 kV/cm, in which the white dotted line is a circle whose center and radius are respective the origin point O of the coordinate system and the distance between point O and the point with the strongest light intensity. (c) The intensity distributions of the LHCP component (red line), RHCP component (orange line) and the total field (blue line) on the white dotted line circle.

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It is well known that a large spin separation can also be obtained when a horizontally polarized beam reflected near the Brewster angle on an air-prism interface. Around the Brewster angle, however, the reflection coefficient is near zero. Therefore, only a small part of the incident power can be reflected. Since the reflectivity (energy efficiency) appears in the denominator in the expression of the spin separation, the larger the spin separation, the lower the energy efficiency. Amplifying the spin separation via weak measurement techniques also suffer from the same problem of low energy efficiency. To make a comprehensive assessment of spin separation and energy efficiency, in Ref. [24] a figure of merit $F = ({{W_ + } + {W_ - }} )\tau$ is introduced, where ${W_{ +{/} - }}$ is the energy efficiency of the LHCP/RHCP component and $\tau$ is the ratio of the spin separation with respect to the intensity spot size (the half-height width of the Gauss spot). Usually, for an air-prism interface, F is only ∼10⁻⁵ for Gaussian incident beams near the Brewster angle. Here in our work, as the intensity distribution pattern is complex, we consider the half-height width of the first sidelobe $\varpi = 29\,{\mathrm{\mu} \mathrm{m}}$ as the parameter to describe the intensity spot size, hence we have $\tau = 1.172$. If anti-reflection film is plated on the interface of the uniaxial crystal, the energy efficiency may reach ${W_ + } + {W_ - } = 100\%$. Therefore, a very high figure of merit F = 1.172 can be obtained.

Further, from Fig. 8(c) we can see that there exist five lobes on the intensity pattern. For the middle lobe and the two second side lobes above and below the x-axis, the intensity distributions of the LHCP and RHCP components are almost the same. This fact reduces the spin separation. In actual measurement, blocking these intensity lobes with an angular filter will enlarge the spin separation, hence make the observation of the SHE more intuitive. Of course, this angular blocking is a way to improve the spin separation by reducing energy efficiency. From this point of view, it has the same effect as the weak measurement technique. The schematic diagram of filtering the output beam is shown in Fig. 9, in which Fig. 9(a) is the light intensity distribution before filtering, Fig. 9(b) is the transmission coefficient distribution of the filter, Fig. 9(c) is the light intensity distribution after filtering. In order to filter out the middle lobe and the two second side lobes completely and retain the other lobes unchanged, we select the transmittance coefficient of the angular filter T = 1 when $0.61\pi \le \varphi \le 0.88\pi$ and $1.12\pi \le \varphi \le 1.39\pi$; and T = 0 when the angle $\varphi$ is other values. Figure 9(c) gives the filtered intensity distributions of the LHCP component and RHCP component at z = 5 mm under l₁=-l₂ = 4 and E_DC = 60 kV/cm. Under these parameters, the transverse shifts of the filtered beam are calculated to be ${\delta _ + } = 28.4\,{\mathrm{\mu} \mathrm{m}}$ and ${\delta _ - } ={-} 28.4\,{\mathrm{\mu} \mathrm{m}}$ so that the corresponding spin separation is $\Delta = 56.8\,{\mathrm{\mu} \mathrm{m}}$ and the ratio of the spin separation with respect to the half-height width $\varpi = 29{\mathrm{\mu} \mathrm{m}}$ is $\tau = 1.959$. Through the filtering, the proportion of the light energy blocked is calculated to be 43.7%, hence the energy efficiency is ${W_ + } + {W_ - } = 0.563$. It is still a very large energy efficiency comparing to the case of a beam reflected near the Brewster angle. After filtering, the figure of merit is finally calculated to be F = 1.1, which is slightly low than the figure of merit without angular filtering.

 

Fig. 9. The schematic diagram of filtering the output beam with an angular filter, in which (a) is the light intensity distribution before filtering, (b) is the transmission coefficient distribution of the filter, (c) is the light intensity distribution after filtering.

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In above, the SHE under l₁ = - l₂ was investigated. In the following, we further discuss the cases of l₁ ≠- l₂. For this, we calculate the intensity distributions of the total field, LHCP component and RHCP component, and the SAM density distributions at z = 5 mm under E_DC = 60 kV/cm and l₁ ≠- l₂. Figures 10(a1) - (a4) are the results under l₁ = 3, l₂=- 4, in which the Fig. 10(a1) is the intensity distribution of the total field, Fig. 10(a2) is the intensity distribution of LHCP component, Fig. 10(a3) is the intensity distribution of RHCP component, and Fig. 10(a4) is the SAM density distribution. Figures 10(b1) - (b4) are the results under l₁ = 4, l₂=- 5, in which the Fig. 10(b1) is the intensity distribution of the total field, Fig. 10(b2) is the intensity distribution of LHCP component, Fig. 10(b3) is the intensity distribution of RHCP component, and Fig. 10(b4) is the SAM density distribution. It can be seen clearly that the SHE can still occur when l₁ ≠- l₂. However, different with the cases of l₁ = - l₂, under l₁ ≠- l₂, the intensity distributions of the total field as well as the SAM density distributions will not have any symmetry about the x-axis. This is due to that the initial symmetry between the “scion” and the “rootstock” is broken by l₁ ≠- l₂.

 

Fig. 10. The intensity distributions of (a1) the total field, (a2) LHCP component and (a3) RHCP component, and (a4) the SAM density distribution at z = 5 mm under l₁ = 3, l₂=- 4 and E_DC = 60 kV/cm. The intensity distributions of (b1) the total field, (b2) LHCP component and (b3) RHCP component, and (b4) the SAM density distribution at z = 5 mm under l₁ = 4, l₂=- 5 and E_DC = 60 kV/cm. The SAM densities are the results relative to ${{{\varepsilon _0}} / {2\omega }}$.

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Until now, we have investigated the propagation of grafted Gauss vortex beams in uniaxial crystals theoretically. Then, we further explain how to generate a grafted Gauss vortex beam experimentally. For this, we first calculate the two-dimension Fourier transform of the nth-order OAM component ${E_{0 \pm }}\exp ({{{ - {r^2}} / {{s^2}}}} ){F_n}\exp ({in\varphi } )$ of the incident GVB to be ${i^n}{e^{in\theta }}\tilde{E}_ \pm ^{(n )}(k )$, where $\theta$ is the azimuth angle of the transverse component of the wave vector. Therefore, the Fourier transform of the grafted Gauss vortex beam is given by ${\cal F}\left\{ {{\mathbf E}\left( {r,\varphi ,0} \right)} \right\} = \sum\limits_n {{i^n}{e^{in\theta }}[\tilde{E}_ + ^{\left( n \right)}\left( k \right)} {\hat{e}_ + } + \tilde{E}_ - ^{\left( n \right)}\left( k \right){\hat{e}_ - }]$. Having this Fourier transform expression, one can now generate the grafted Gauss vortex beam via Fourier space filtering method. The proposed experimental diagram is shown Fig. 11, in which the polarizer is utilized to control the polarization of the laser beam to match the polarization requirement of the spatial light modulator (SLM). The beam will be first expanded and collimated after the beam expander. Then, the expanded beam will be reflected by the SLM uploaded with predesigned holograms that contain the formation of the Fourier transform of the grafted Gauss vortex beam. The desired holograms loading to the SLM can be generated by the interference between the Fourier transform and a plane wave. Through these steps, the Fourier transform of the grafted Gauss vortex beam is obtained. Then, a lens L3 with focal length f can be used to perform the inverse Fourier transform. Therefore, finally, a grafted Gauss vortex beam can be obtained on the focal plane of L3, which can be chosen to be the input plane of the uniaxial crystal.

 

Fig. 11. The schematic diagram that can be used to generate the grafted Gauss vortex beam.

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5. Conclusion

In conclusion, we have theoretically investigated the propagation of GVB in uniaxial crystals. It is found that since the cylindrical symmetry of the system is broken by the spatial phase structure of the GVB, a SHE determined by the spatial phase structure emerges during the propagation. It is demonstrated that the SHE can be modulated by changing the topological charges of the grafted spiral phases. This is of interest and could inspire researchers that the SHE of light beams in uniaxial crystals can be modulated by manipulating the spatial structure of the input beams, hence open a new perspective on investigating the SHE of structured light beams in uniaxial crystals. Moreover, if linear EO effect is employed, it is found that the SHE can also be modulated (enhanced or suppressed) by controlling the dc electric field of the EO effect. These results offer novel regulation capabilities of photon spin via uniaxial crystals and may be useful in the fields of spin photonics and optical vortices.