## Abstract

We propose a method for controlling the local spin and orbital angular momentum (SAM and OAM) of a focused light beam in a uniaxial crystal by means of Pockels effect. For an input circularly polarized Bessel-Gaussian (BG) beam, both the local SAM and OAM of the output beam are circularly symmetric, their patterns and peak values vary with the applied electric field *E*_{0}. Let the output beam pass through a quarter-wave plate, the OAM keeps while the SAM varies. The local SAM density is nearly directly proportional to ± sin(2*φ*), where *φ* is the azimuthal angle and the signs are dependent on the radius and *E*_{0}.

© 2012 OSA

## 1. Introduction

It is well known that light carries spin and orbital angular momenta [1]. Orbital angular momentum (OAM) arises from the spatial profile of light intensity and the phase, while spin angular momentum (SAM) associates with circular polarization [1]. Recently, angular momentum has attracted considerable attention owing to its applications including optical manipulation [2–4], quantum information [5], and biosciences [6,7]. The global angular momentum (the average value over the section of beam) plays an important role in these applications when the whole beam is taken into account. The local angular momentum, on the contrary, is important for the experiments accessing only portions of beam [8]. As demonstrated by Garcés-Chávez and associates, SAM (or OAM) transferred to a small particle in a big beam is proportional to the local SAM (or OAM) instead of the global one [9]. They further found that a high-index particle trapped off-axis can rotate around the beam axis in the opposite direction to the beam helicity under appropriate conditions [10]. Although the management of the global SAM (or OAM) is relatively straightforward [11], the control of the local SAM (or OAM) density is much more complex. The interference of several vortex modes and the superposition of two orthogonal linearly polarized beams are two of the common methods of this control [8,12,13]. By changing topological charges and relative amplitudes of the light beams, various SAM and OAM density patterns can be obtained [14]. However, these methods are not suitable for rapid control, which may be useful in optical manipulation [15].

In this paper, we introduce a simple method to control the local SAM and OAM in a single uniaxial crystal with Pockels effect. This kind of crystal is well known for its wide use in the generation of optical singularity [16–18] and mode conversion of light beam [19]. Very recently, we utilized the crystal to generate and control high order multiring modes [20]. When a circularly polarized rotationally symmetric beam propagates along the optical axis of the crystal, SAM will transfer to OAM, but the total angular momentum conserves [21]. As illustrated by Chen, with a dc electric field applied on the crystal, the conversion between SAM and OAM is electrically controllable [22]. Here, both the local and global properties of SAM and OAM of the output light beam are studied in detail and their variations with the applied electric field are demonstrated.

## 2. Theory

Suppose that a focused vectorial beam of 632.8 nm propagates along the optical axis of a uniaxial crystal strontium barium niobate (SBN), which has a large electro-optical coefficient *γ*_{33} = 1340 pm/V [23], and a dc electric field is applied along the same direction. Following [20], the wave equation is

*ε*and

_{o}*ε*being the permittivity of ordinary and extraordinary waves in the uniaxial crystal, respectively. It has been previously shown that, a circularly polarized Gaussian beam propagating along the optical axis of a uniaxial crystal has low conversion efficiency from SAM to OAM (lower than 50%) [24]. In contrast, the experiment showed that a Bessel-Gaussian (BG)-like beam can obtain efficiency near 100% [18]. BG beams can also obtain high conversion efficiency [25], for example 99.61% as shown below. So, we choose BG beams for our investigation. For a

_{e}*m*-order right-handed circularly polarized paraxial BG beam, its field at the incident surface of SBN crystal (

*z*= 0) is ${E}_{in\perp}(r,\phi ,0)={J}_{m}(\beta r)\mathrm{exp}(-{r}^{2}/{w}_{0}^{2})$

_{$\mathrm{exp}(-im\phi )(X+iY)/\sqrt{2}$}, where

*β*and

*w*

_{0}are the transverse component of wave-number and beam waist, respectively

_{.}Following the method proposed in [25], the right- and left-handed circularly polarized (RHP and LHP) components of output light field can be expressed as

*β*= 0 and

*m*= 0, Eqs. (2a) and (2b) reduce to that describing the propagation of Gaussian beam through a uniaxial crystal. They also describe the propagation of non-diffracting Bessel beam when

*w*

_{0}→∞.

The angular momentum of a light beam is given by the sum of spin and orbital contributions in paraxial regime. If a particle is held off-axis, SAM results in the rotation of the particle around its own axis, while OAM causes the rotation around the beam axis. SAM and OAM are mechanically equivalent for the particle held on the beam axis, where the center of the particle cannot move [26]. The cross product of radius vector **r** and linear momentum density **P** gives the angular momentum density. And the angular momentum density in the propagation direction is *j _{z}* =

*l*+

_{z}*s*, with [8]

_{z}*S*

_{3}is the Stokes parameter,

*E*and

_{x}*E*are the light field components along two orthogonal directions. By the relations ${E}_{x}=\left({E}_{+}+{E}_{-}\right)/\sqrt{2}$ and

_{y}*E*, we get

_{y$=i\left({E}_{+}-{E}_{-}\right)/\sqrt{2}$}*l*= -

_{z}*ε*

_{0}/(2

*w*) [

*m*|

*A*

_{1}(

*r*,

*z*)|

^{2}+ (

*m*-2)|

*A*

_{2}(

*r*,

*z*)|

^{2}],

*s*=

_{z}*ε*

_{0}/(2

*w*) [|

*A*

_{1}(

*r*,

*z*)|

^{2}-|

*A*

_{2}(

*r*,

*z*)|

^{2}], where

*A*

_{1}(

*r*,

*z*) and

*A*

_{2}(

*r*,

*z*) are the azimuthal independent parts of

*E*

_{+}(

*r*,

*φ*,

*z*) and

*E*

_{-}(

*r*,

*φ*,

*z*) in Eqs. (2a) and (2b), respectively. The output local SAM and OAM densities rely on the local intensities of RHP and LHP components of the output beam. This can be easily understood by taking the output beam as a superposition of two circularly polarized waves, then performing an addition of their OAM but a subtraction between their SAM [8]. The average SAM (or OAM) is obtained by integrating the SAM (or OAM) density over the whole cross section of the beam. However, it is not an easy work. A much simple method for calculating the average SAM per photon in the uniaxial crystal is proposed in [25]. For a BG

*incident beam, the average SAM can be calculated and is with the form*

_{m}*I*(

_{m}*x*) stands for the modified Bessel function of first kind. By employing the conservation law of angular momentum, we get the average OAM per photonSAM and OAM are not independent of each other, thus we can control SAM and OAM simultaneously. In the next section, we will use numerical method to investigate the influence of Pockels effect on the local (density) and global (average) properties of SAM and OAM in the output beam.

## 3. The electric control of local and global angular momentum of the output beam

To study the electric dependence of the local and global angular momentum, and of the conversion between SAM and OAM, we first consider the lowest order BG beam with *β* = 1 μm^{−1} and *w*_{0} = 10 μm at the input plane (*z* = 0), propagating along the optical axis of SBN crystal. The parameters of the beam are in paraxial regime [27]. BG beam is a well known diffraction-free beam [28]. However, it is non-diffracting for a distance D = *kw*_{0}/*β* only. The intensity distribution of BG beam cannot keep its initial profile and becomes single-ringed after a propagation distance *z*>>D [28]. In our case, the length of the crystal *L* = 8 mm is much longer than D, thus the total intensity pattern of output light beam is a single ring when *E*_{0} = 0, which is evident in Fig. 1(b)
. The pattern hardly changes when the applied electric field varies from −1.6883 to 3.55 kV/mm. The local SAM and OAM densities have circular symmetry. Figure 1(c) shows the dependences of normalized SAM (blue line) and OAM (red line) densities on radius for different *E*_{0}, from which one sees that, the densities can be tuned by the applied electric field. This is because the Pockels effect modulates the diffraction lengths of both ordinary and extraordinary beam components [24]. And hence the RHP and LHP components change with *E*_{0} [shown by Eqs. (2a) and (2b)]. As a result, the local SAM and OAM densities become electrically tunable. At *E*_{0} = −1.6883 kV/mm the OAM vanishes.

Then a single ring emerges in OAM density pattern and grows gradually. The peak value of the ring increases with *E*_{0} and reaches its maximum at *E*_{0} = 1 kV/mm. At the same time, the SAM density changes from positive to negative one. The reverse process occurs when *E*_{0} continues increasing. The distributions of OAM and SAM densities are interesting for some special values of *E*_{0.} For example, at *E*_{0} = 3.55 kV/mm, the OAM density is not longer single-ringed but double-ringed. There are two rings with opposite signs in the SAM pattern when *E*_{0} = −0.35 and 2.26 kV/mm. That is to say, the SAMs located at these two rings have different orientations. Therefore, two absorbing or birefringent particles trapped in these two different rings (off axis) may have opposite rotation directions. The sign of the inner ring is positive while that of the outer ring is negative at *E*_{0} = −0.35 kV/mm. The signs of these two rings reverse at *E*_{0} = 2.26 kV/mm. Form Fig. 1(b) and 1(c) one concludes that, the patterns of SAM and OAM densities are quite different with that of total intensity.

In the following, we investigate the control of the global SAM and OAM of the output beam, and compare them with the local ones. For a RHP BG_{0} incident beam, the global SAM and OAM of the output beam can be calculated according to Eqs. (4) and (5). And their dependences on the applied electric field *E*_{0} are shown in Fig. 2
for *w*_{0} = 10 μm, where the blue and green lines represent the global SAM and OAM, respectively. As illustrated by Fig. 2, the conversion between SAM and OAM is electrically controllable. When *E*_{0} varies from −1.6883 to 0.87 kV/mm, SAM transfers to OAM gradually. The conversion efficiency of such a transformation defined as a ratio of the energy of LHP component to the total energy of output light beam, namely, *η* = (1-*S _{z}*)/2, has a maximum value more than 90% at

*E*

_{0}= 0.87 kV/mm. When further increasing

*E*

_{0}, OAM returns back to SAM. By comparing Fig. 1(c) and 2, one finds that, the applied electric field for obtaining the maximum value of global OAM (

*E*

_{0}= 0.87 kV/mm) is smaller than that corresponding to the maximum peak in OAM density pattern (

*E*

_{0}= 1 kV/mm). Although the ring in the OAM density pattern has a relatively smaller peak value at

*E*

_{0}= 0.87 kV/mm, it has a larger size, which leads to a maximum global OAM. One can further conclude that, light beam may have different local angular momentum density patterns even though its global angular momentum is identical. Besides, the local angular momentum changes in both magnitude and sign over the beam profile, while the global one describes the whole beam by a single value. Thus the signs of the local angular momentum density in some regions can be different with the global one.

We find that the conversion efficiency from SAM to OAM depends not only on applied electric field *E*_{0} but also on beam waist *w*_{0}. Figure 3
shows the efficiency as a function of *E*_{0} and *w*_{0} for *L* = 8 mm and *β* = 1 μm^{−1}. One sees from Fig. 3 that, the maximum conversion efficiency increases with *w*_{0}, reaches 99.61% when *w*_{0} = 50μm. And the applied electric field has only a small change for different *w*_{0} at the maximum conversion efficiency.

## 4. The electric control of local and global angular momentum of the light beam after SBN and QWP

When a quarter-wave plate (QWP) with its fast axis being 45° to *x*-axis is placed near the output surface of SBN crystal, it will transfer RHP and LHP components to horizontal and vertical ones, respectively, and causes a phase difference of π/2 between the horizontal and vertical components [23]. Therefore, the light field after QWP becomes

*r*,

*z*) = |

*A*

_{2}(

*r*,

*z*)|/

*|A*

_{1}(

*r*,

*z*)| and

*δ*(

*r*,

*φ*,

*z*) = arg(

*A*

_{2}(

*r*,

*z*))-arg(

*A*

_{1}(

*r*,

*z*)) + 2

*φ*-π/2. After some simple calculations, we find that the total intensity and the OAM keep the same as those before QWP. The SAM, in contrast, shows new features. The global SAM vanishes whatever the applied electric field is. Of particular interesting, the local SAM density is nonzero, except for

*E*

_{0}= −1.6883 kV/mm. As shown by the Jones representation in Eq. (6), the phase difference between the horizontal and vertical components of light field depends on the azimuthal angle, while the parameter Ψ is independent of it. Therefore the parameter

*S*

_{3}= sin(2Ψ)sin

*δ*has 2-fold rotational symmetry around the axis of light beam, which means that the SAM density has lost its circular symmetry. The parameters Ψ and

*δ*rely on both radius

*r*and the applied electric field

*E*

_{0}, and the sign of sin

*δ*would change with

*r*and

*E*

_{0}. So, the SAM density will change its magnitude and sign against

*r*or

*E*

_{0}. Figure 4 shows the electric dependence of normalized SAM density for a RHP BG

_{0}incident beam, where the beam waist of the incident beam is chosen to be 10 μm in order to display the SAM density patterns in good quality. For

*E*

_{0}in range of −1.6883 and −0.35 kV/mm, the SAM density is closely proportional to -sin(2

*φ*), and has a ring shape pattern. The peak value of the ring increases with

*E*

_{0}and reaches its maximum at

*E*

_{0}= −0.35 kV/mm. Then it decreases when

*E*

_{0}further increases and vanish at

*E*

_{0}= 2.26 kV/mm. When

*E*

_{0}varies from −0.35 to 2.26 kV/mm, an outer ring emerges in the density pattern and grows gradually. And the SAM located at this ring is nearly directly proportional to sin(2

*φ*). The peak value of the outer ring reaches its maximum at

*E*

_{0}= 2.26 kV/mm, when the inner ring disappears and a new ring with a radius larger than the old outer ring begins to emerge. It is worth noting that the peak values of inner and outer rings are equal at

*E*

_{0}= 1 and 3.55 kV/mm. However, the signs of the SAM densities for these two cases are opposite. For higher-order BG incident beams, the patterns of SAM density have similar behavior, changing with

*E*

_{0}, if

*L*>>D.

## 5. Conclusion

We have investigated the electro-optical control of the local SAM and OAM of light beam by focusing a RHP BG* _{m}* beam into a SBN crystal with an applied electric field

*E*

_{0}along its optical axis. For a BG

_{0}beam incident onto a 8 mm long crystal, SAM converts to OAM with a very high conversion efficiency, for example 99.61% when

*w*

_{0}= 50μm. The patterns of SAM and OAM densities of the output light beam are ring-shaped and electrically controllable. When the output beam passes through a QWP, the OAM keeps the same, while the SAM shows new features. The local SAM density loses its circular symmetry and is proportional to ± sin(2

*φ*) with signs dependent on the radius and

*E*

_{0}.

## Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC) (grant 90921009).

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