## Abstract

We present a detailed theoretical analysis of the formation of standing waves using cylindrically polarized vector Laguerre-Gaussian (LG) beams. It is shown that complex interplay between the radial and azimuthal polarization state can be used to realize different kinds of polarization gradients with cylindrically symmetric polarization distribution. Expressions for four different cases are presented and local dynamics of spatial polarization distribution is studied. We show cylindrically symmetric Sisyphus and corkscrew type polarization gradients can be obtained from vector LG beams. The optical landscape presented here with spatially periodic polarization patterns may find important applications in the field of atom optics, atom interferometry, atom lithography, and optical trapping.

© 2015 Optical Society of America

## 1. Introduction

Geometry and topology of polarization patterns in cylindrical vector beams are intricate in nature. These beams have unconventional polarization distribution in the sense that the polarization orientation at each point in the beam cross section has angular dependence. Cylindrical vector beams have wide application areas, ranging from microscopy to optical communications [1–4]. Among these beams, radially and azimuthally polarized beams are of particular interest due to their special tight focusing properties. A Laguerre-Gaussian (LG) beam can be a scalar LG beam with homogenous polarization distribution or a vector LG beam with inhomogenous polarization distribution. The polarization distribution of vector LG beams varies with the azimuthal mode index of the beam, and can show a variety of patterns. The most distinguished feature of vector LG beams is the cylindrically symmetric polarization distribution with a singularity at center of the beam [1, 2]. In the two dimensional spatial structure of vector beams in the transverse plane, this vector point singularity is known as a V-point. V-point singularities can also appear in superposition of vector LG beams. In such cases the V-point singularity can transform into polarization singularity triplet (two C-points with opposite handedness separated by L-line) during propagation, which results in a dynamic polarization pattern [5]. Superposition of vector beams can give us an idea about the connection and transformation between different kinds of polarization singularity in vector fields.

Spatially periodic fields are prerequisite for many of the important technological applications of optical fields, for example, optical lattices [6], atom lithography [7], and photonic crystals [8]. Intensity and phase gradients have been studied by many authors in various contexts [9–11]. Continuous change in the polarization state with change in the axial positon of the beam is considered as the polarization gradient [6, 12, 13]. Polarization gradients have regularly been used in optical lattices for sub-Doppler cooling of atoms. Most of the previous works in this direction has been done using plane waves and homogenous polarization distribution [6, 12], In this regard, the use of cylindrical vector beams is expected to offer new features to the polarization gradients due to the cylindrically symmetric and inhomogeneous polarization distributions of vector beams.

In this paper we present, to the best of our knowledge, the first report on the creation of polarization gradients from the superposition of cylindrical vector beams. Here, using numerical simulation we demonstrate that radially and azimuthally polarized beams can form standing waves with axially symmetric polarization properties, which is a promising candidate for realization of different kinds of polarization gradients. Basis of the present study is the superposition of counter propagating vector LG beams. We derive simple expressions of the superposed field and discuss all cases of standing waves in terms of spatial polarization properties. Superposition of vector LG beams exhibit new aspects that are absent in the superposition of plane waves with homogenous polarization. We obtained a cylindrically symmetric Sisyphus polarization gradient from the superposition of radially and azimuthally polarized LG beams propagating in counter-propagating geometry [14, 15]. The polarization state of the standing wave in this case changes from linear to circular polarization while the intensity and polarization distributions remain axially symmetric. We discuss another interesting case where the resultant field shows a corkscrew polarization gradient [13]. It is obtained from a particular combination of the radially and azimuthally polarized LG beams. In corkscrew polarization gradient, linear polarization states change their orientation as the axial position is changed. At some planes either radial or azimuthal polarization distributions appear. When both radial and azimuthal components are non-zero then a spirally polarized beam is obtained. Due to this effect, radial, azimuthal, and spiral type polarization distributions can be obtained at different axial position. The present results are quite attractive for a variety of applications such as optical trapping [16], atom lithography [17] and atomic superconducting quantum interference devices [18], in which the spatially periodic variation of polarization plays an important role.

## 2. Vector Laguerre-Gaussian beams

Consider a vector Laguerre-Gaussian beam propagating along the positive *z*-direction. A simple expression for a vector LG beams can be written as [5]

*exp(−iωt)*is assumed. The radial and axial dependence

*U*(

_{p,m}*r, z*) is given by

*r*is the radial direction,

*ϕ*is the azimuthal angle,

*z*is the distance from the beam waist,

*p*is the radial mode index, and

*m*is the azimuthal mode index.

*w*is the minimum beam radius at

_{0}*z*= 0,

*R*(

*z*) = (

*z*

^{2}

*+ z*

_{R}

^{2})/

*z*is the radius of curvature of the wavefront,

*z*

_{R}=

*kw*

_{0}

^{2}/2 is the Rayleigh length,

*k*is the wave number, (2

*p +*|

*m*| + 1)

*η*(

*z*) is the Gouy phase shift with

*η*(

*z*) = tan

^{−1}(

*z*/

*z*

_{R}), and$w\left(z\right)={w}_{0}\sqrt{1+{\left(z/{z}_{R}\right)}^{2}}$. ${L}_{p}^{\left|m\right|}$ is the generalized Laguerre polynomial. The second factor in Eq. (1) is responsible for the inhomogenous polarization distribution in the beam cross-section. The polarization components of the beam are defined by

**P**(

_{m}*ϕ*) as

*m,*there are four vector LG beams with different polarization distributions are possible as shown in Fig. 1. In all our calculations, we assumed wavelength (

*λ*) is 632.8 nm and

*w*= 0.8 mm. The dark core at the center of vector LG beam is due to the presence of a polarization singularity that can be characterized by the Poincaŕe-Hopf index, which is the winding number of rotation angle of the polarization vector surrounding the singularity [19].

_{0}For *m* = 1, type I and III correspond to azimuthally and radially polarized beams, respectively. It is well-known that radially and azimuthally polarized beams have identical amplitude distribution and the direction of local polarization at each point in the beam cross-section is orthogonal between the two beams. For simplicity, we assumed radial index *p* = 0 for all cases. The expression for an azimuthally polarized beam can be written as

*U*(

_{0,1}*r, z*) is given by

The expressions for an azimuthally and a radially polarized beams propagating in negative *z*-direction can be written as

***) represents the complex conjugate.

## 3. Superposition of counter propagating vector Laguerre-Gaussian beams

The superposition of two beams propagating in opposite directions forms a standing wave. The fundamental feature of the standing waves is the redistribution of energy with intensity minimum and maximum at the nodes and antinodes, respectively. In this section, we show four different cases of superposition of counter-propagating vector LG beams to obtain polarization gradients.

#### 3.1 Cylindrically polarized standing waves

A cylindrically polarized standing wave is the simplest case of superposition of two identical, counter-propagating vector LG beams. For example, the electric field of the superposed counter-propagating, radially polarized LG_{01} beam is written as

*z*\<<

*z*, both Gouy and curvature phases can be considered to be constant, thus the only dominant term in

_{R}*T*is

*kz*[15].

Obviously, the polarization state of the standing wave is identical to those of the two superposed beams at all transverse planes, while the intensity modulation along the *z* direction with a period of half wavelength appears as shown in Fig. 2. The polarization property of the standing wave is solely determined by the identical polarization distribution of the two beams. A standing wave with cylindrically symmetric polarization distribution can also be formed by the other combinations of vector beams such as two azimuthally polarized or higher order vector modes that can lead to standing waves with cylindrically symmetric polarization distributions [5].

#### 3.2 Cylindrically symmetric Sisyphus polarization gradient

A Sisyphus polarization gradient has a particular importance in the sub-Doppler cooling of atoms [6]. It is obtained by superposition of two plane waves propagating in opposite directions with orthogonal linear polarization states. The light intensity in the resultant field is constant everywhere and only the polarization state changes from circular to linear and back to circular as one moves a distance of *λ*/4. The change in ellipticity of polarization is the essence of the polarization gradient. A Sisyphus polarization gradient can be realized in many field configurations. As suggested by Eq. (3), since a vector LG beam with type-I (azimuthal) polarization has local polarization orthogonal to a vector LG beam with the same mode index and type-III (radial) polarization at all points, the counter-propagating superposition of type-I and type-III beams having the same mode index can produce a Sisyphus polarization gradient with cylindrically symmetry. It should be noted that the combination of type-II and type-IV polarization in Eq. (3) for vector LG beams is also the case producing cylindrically symmetric Sisyphus polarization gradient.

We consider here a superposition of radially and azimuthally polarized beams propagating in opposite directions. The total electric field obtained from the superposition can be written as

The resultant superposed electric field of the two beams is given by*kz*= π/4, 3π/4, 5π/4, and 7π/4 as shown in Fig. 3. Note that the light intensity is constant everywhere and only polarization state is modulated for different axial position. The ellipticity of the polarization changes with longitudinal distance and a Sisyphus type polarization gradient is obtained. Due to the orthogonal polarization states of the two superposed beams, a spirally polarized beam is obtained at the planes

*z*= 0,

*λ*/4,

*λ*/2, and 3

*λ*/4. In general the direction of the spiral polarization flips at

*z*= (2

*n*+ 1) λ/8 where circular polarization appears, and the handedness of elliptical/circular polarization flips at

*z*=

*nλ*/4. At these points, the field components acquire a spiral phase distribution, which will cause an orbital angular momentum with a unit topological charge. However, in terms of angular momentum of light [2], the handedness of circular polarization corresponding to spin angular momentum at these points is always balanced with orbital angular momentum, resulting in the total angular momentum, which is the sum of spin and orbital angular momentum [2], is therefore zero. This is in contrast to the case of the superposition of orthogonal linearly polarized plane waves where no such spiral phase distribution appears. We would like to stress that the field in present case is inhomogenously polarized and the

*local*polarization states follow exactly the same behavior as the polarization gradient obtained in the case of superposition of plane waves. Since we are discussing standing waves we define the handedness of elliptical/circular polarization with respect to the +

*z*direction, as opposed to the common practice of defining handedness with respect to the direction of beam propagation.

#### 3.3 Corkscrew polarization gradient

In a corkscrew polarization gradient, polarization state of the beam is linear everywhere but rotates in space. It is obtained from the superposition of counter-propagating circularly polarized plane waves with opposite handedness [13]. The resultant field has a dynamic polarization distribution with change in the axial position. The orientation of the polarization vector traces a corkscrew type of polarization pattern within each beam cross section. Due to change in polarization state, a polarization gradient is obtained in the axial direction.

To obtain a corkscrew polarization gradient from vector LG beams, a special combination of a radially and an azimuthally polarized beams is required. This combination is equivalent to the superposition of two circularly polarized scalar LG beams with opposite handedness, which can be obtained from the superposition of the radially and azimuthally polarized LG beams with a phase difference of π/2 [20]. Based on this equivalence, a corkscrew polarization gradient can be produced by the combination of the two vector LG beams. We consider two superposition of radially and azimuthally polarized beams propagating in the opposite direction with the requisite phase difference. The electric field of the circularly polarized LG beam obtained can be written as

where*σ*

^{−}represents the right handed circularly polarized LG beam propagating in the +

*z*direction and

*σ*

^{−*}represents the left handed circularly polarized LG beam propagating in the −

*z*direction. Note that

*σ*

^{−*}is the complex conjugate and therefore the time reversal of

*σ*

^{−}, so that the local rotation of the electric field as well as the propagation direction of the beam is reversed. The notation

*σ*

^{−*}was chosen to emphasize this fact and to distinguish it from ${\sigma}^{+}={E}^{RP}+i\text{\hspace{0.17em}}{E}^{AP}$, a left handed circularly polarized LG beam propagating in the + z direction. As discussed previously handedness is defined with respect to the + z direction and not with respect to the propagation direction of each beam (if handedness is defined with respect to the propagation direction,

*σ*

^{−}is a beam with right handed circular polarization propagating in the + z direction, and

*σ*

^{−*}is a beam with right handed circular polarization propagating in the –z direction). The superposition of these two beams results in a cylindrically symmetric field distribution. The total electric field obtained from superposition of the orthogonally circularly polarized LG beams propagating in opposite directions can be written asUsing above equations, resultant field can be written asThe polarization gradient has the cylindrical symmetry of intensity and polarization that gives quite a different effect as compared to plane waves. The local polarization direction in the transverse plane changes with the change in the axial position and this results in the corkscrew polarization gradient at each point in the beam cross section. The overall effect is that of a spirally polarized beam. In this configuration, the periodic change of the polarization distribution is obtained with change in the axial position as shown in Fig. 4.

A characteristic feature of this configuration is that the polarization state of the beam changes from radial to azimuthal via spiral polarization distribution at different *z*-planes. In contrast to the previous cases, the local polarization state is always linear and only the orientation of the local polarization changes with change in the axial position and a polarization gradient is obtained in the axial as well as the azimuthal direction. Due to the mutually orthogonal polarization for the counter-propagating beams, the doughnut shaped intensity distribution remains constant at all axial positions and only polarization is modulated. The doughnut shaped intensity distribution may provide some advantage in trapping of cold atoms [21, 22]. This optical arrangement is equivalent to the two counter-propagating scalar LG beams with the topological charge of same magnitude but opposite sign and opposite circular polarization [14, 15]. It is important to discuss here the singular state of the polarization at center of the beam. In addition to the formation of cylindrically symmetric corkscrew type polarization gradient, another prominent feature in this case is the dynamic nature of vector point singularity (V-point) at center of the beam, which changes its morphology from radial to azimuthal via spiral type of polarization distribution. This result also suggests that, due to the formation of the standing waves, topological features of the beams are also periodic. Namely, in a period of half wavelength, azimuthal component appears where the radial component is zero and vice versa.

#### 3.4 Circularly polarized standing waves

Using the same concept as discussed above, a circularly polarized standing wave with a cylindrically symmetric intensity distribution can be obtained from a particular combination of two radially and azimuthally polarized LG beams. The electric field of two circularly polarized LG beams with the same handedness propagating in opposite directions can be written as

Total electric field of the superposition of two LG beams propagating with identical circular polarization is given byUsing Eqs. (4) to (8), the above equation can be written asEquation (19) indicates that the resultant field is circular polarization produced by a phase difference of π/2 between radial and azimuthal component. In contrast to the cylindrically polarized standing waves (shown in 3.1), the intensity of the resultant field for both radial and azimuthal component is modulated by the*z*-dependent oscillatory term. Therefore, in this case, only the intensity is modulated in the axial direction whereas the polarization state of the beam remains fixed. The formation of the standing wave for this case is shown in Fig. 5.

The standing wave has rotational symmetry of intensity, and polarization. The light forces acting on the atoms in such a field will be highly symmetric which may offer some added advantage in the atom cooling experiments [6].

## 4. Discussion

The superposition of vector LG beams considered in this study can be classified into two types. The first is where the counter propagating beams result in an intensity modulation with fixed polarization states. The other situation is where the intensity distribution remains fixed but the polarization state is modulated as the beam propagates. Since the polarization state is mutually orthogonal for the two beams in the cylindrically symmetric Sisyphus and corkscrew polarization gradients, there is no variation in the intensity along the *z*-direction. Polarization gradient induced by the superposition of linear and circularly polarized Laguerre-Gaussian beams have been studied before [14–16]. We want to emphasize that the polarization gradients studied here have some generic features and that they can be realized by combinations of other vector beams as well. A common feature of all these standing waves is cylindrical symmetric intensity distribution with dark core at the center of the beam which is maintained at all axial positions. The annular region of the doughnut shaped intensity distribution of the vector LG beams has an additional advantage in creating dark beam traps [23], and quantum memory [24]. The dependence of the polarization and intensity distributions on the propagation distance makes these field distributions appropriate to use for axial trapping and manipulation of particles. Cylindrical symmetry of polarization distribution of vector LG beams allows us to produce new polarization distributions in the standing waves. For example, by extending the concept mentioned in section 3.3, the superposition of counter-propagating, two spirally polarized beams expressed as$[{E}^{RP}+{E}^{AP}]+[{E}^{R{P}^{*}}-{E}^{A{P}^{*}}]$ can give the resultant field$2\Psi [cos(kz){\widehat{e}}_{r}+isin(kz){\widehat{e}}_{\phi}]$, which shows radial and azimuthal polarization distributions in between the circular polarization states at different *z*-planes. An azimuthal Sisyphus polarization gradient can also be obtained from the combination of linearly polarized LG beams [14, 15]. All these structures suggest the versatility of the vector beams. These optical field distributions may find important applications in various disciplines of atom optics. We believe with the availability of various methods for the generation of high quality vector beams and techniques to precisely measure amplitude, phase, and polarization under different conditions experimental realization of these polarization gradients is possible [25–30].

## 5. Conclusions

In conclusion, we have investigated the formation of different kinds of standing waves formed by two counter propagating vector LG beams, and expressions are derived for the superposed field. Two important cases of intensity and polarization modulations are discussed in detail. Continuous modulation of polarization around the central singularity shows interesting features. Mutual conversion of scalar and vector beams provides many useful properties to the resulting field. It is shown that polarization gradient in the axial direction can be obtained by different combinations of radially and azimuthally polarized LG beams. The method present here is simple and straightforward and can be extended to the higher order vector beams where complexity may lead to new effects. The cylindrically symmetric intensity and polarization distributions presented here may be useful to explore new phenomena in atom optics, optical trapping, and guiding for atoms and other particles.

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