## Abstract

We propose a generalized model for the creation of vector Bessel-Gauss (BG) beams having state of polarization (SoP) varying along the propagation direction. By engineering longitudinally varying Pancharatnam-Berry (PB) phases of two constituent components with orthogonal polarizations, we create zeroth- and higher-order vector BG beams having (i) uniform polarizations in the transverse plane that change along *z* following either the equator or meridian of the Poincaré sphere and (ii) inhomogeneous polarizations in the transverse plane that rotate during propagation along *z*. Moreover, we evaluate the self-healing capability of these vector BG beams after two disparate obstacles. The self-healing capability of spatial SoP information may enrich the application of BG beams in light-matter interaction, polarization metrology and microscopy.

© 2017 Optical Society of America

## 1. Introduction

The Bessel beams have attracted extensive research interests in the past decades, since it was originally proposed by Durnin as a solution of the Helmholtz equation [1,2]. Typically, such beams possess two remarkable features, the first one is that they remain their transverse intensity profiles whilst these beams propagate in free space, namely, the non-diffractive propagation [1,2]. Although ideal Bessel beams with infinite transverse extent and energy cannot be generated experimentally, some quasi-Bessel beams with finite size and energy have been experimentally demonstrated with a propagation ability through a large distance without diffraction [3–7]. In addition to non-diffraction, Bessel beams are also characterized by the other fascinating property of self-healing [8]. It has been shown that their intensity can be reconstructed after propagation through various kinds of obstacles [8–12]. Such intriguing features enable Bessel beams be applied in optical micromanipulation [13,14], microfabrication [15], microscopy [16,17] and optical communication [18] *et al*.

Recently, the vector beams with transversely inhomogeneous state of polarization (SoP) have been intensively studied because of their two salient merits: sharper focus and significant enhancement of longitudinal components [19–22]. Such beams have found increasing utilization in applications such as optical micromanipulation [23,24], microscopy [25,26], lithography [27], plasmonic focusing [28], quantum information and optical communication [29–31]. At the same time, the vector Bessel beams with cylindrically symmetric SoP distribution have been proposed with the help of spatial light modulator (SLM) [32,33] and combination of axicon and inhomogeneous waveplate (*q*-plate) [34–36]. The propagation and focusing properties of vector Bessel beams have been extensively studied [32,34–38]. The self-healing capacity of their intensity and transversely inhomogeneous SoP have also been demonstrated [33,39]. Only quite recently the research attention has somewhat focused to the vector Bessel beams with variant SoPs along optical axis [40–42]. The new capability could expand the application of Bessel beams in material processing and so on. However, the theoretical frame depicting the construction of such beams is still scarce. Meanwhile, the self-healing capacity of longitudinally variant feature of SoP has not been investigated.

In this paper, we present a frame for the construction of vector Bessel-Gauss (BG) beams, which show a remarkable property: their transverse SoPs change along the propagation direction. In our implementation, we create two constituent beams with longitudinally varying Pancharatnam-Berry (PB) phases [43,44], based on the transverse-to-longitudinal structuring strategy [45] produced by an axicon in concert with Sagnac interferometer including SLM, to synthesize vector BG beams with *z*-dependent SoPs. Furthermore, we explore the self-healing capability of these vector BG beams, by experimentally observing transverse intensity and Stokes parameters distributions at different propagating planes.

## 2. Theory analysis

To derive a generalized model, let us consider a paraxial light beam with frequency *ω* propagating along the *z* axis, of which the transverse electric field in cylindrical coordinates (*r*,*ϕ*,*z*) can be expressed as the superposition of a pair of orthogonally polarized bases

*E*

_{1,2}(

*r*,

*ϕ*,

*z*) are complex amplitudes of two polarized components, |Ψ

_{1,2}〉 refer to arbitrary polarization bases with 〈Ψ

_{1}|Ψ

_{2}〉 = 0. We assume that two constituent components have cylindrical eigenmodes with the first kind of Bessel function and a Gauss background expressed as

*E*

_{1,2}(

*r*,

*ϕ*,

*z*) =

*J*

_{|v}_{1,2}

*(k*

_{|}

_{r}_{1,2}

*r*)exp(-

*r*

^{2}/${w}_{a}^{2}$)exp(i

*v*

_{1,2}

*ϕ*)exp(i

*k*

_{z}_{1,2}

*z*). Where,

*J*

_{|v}_{1,2}

*(∙) are*

_{|}*v*

_{1,2}orders Bessel functions of the first kind,

*w*is the waist of the Gauss background, and

_{a}*k*and

_{r}*k*are the transverse and longitudinal components of wave vector, respectively.

_{z}Clearly, when |*v*_{1}|≠|*v*_{2}|, the superposition of Bessel beams with different orders can give rise to the variation of SoP along *z* aixs, which has been reported in [37]. Here, we consider the case of *v*_{1} = -*v*_{2}. For such a case, the electric field in Eq. (1) can be expressed as (omitting the common time factor e^{-i}* ^{ωt}*)

*E*

_{0}=

*J*

_{|v}_{1,2}

*(k*

_{|}

_{r}_{1,2}

*r*)exp(-

*r*

^{2}/${w}_{a}^{2}$)exp(i

*k*

_{z}_{1,2}

*z*) is the complex amplitude, |Ψ(

*r*,

*ϕ*,

*z*)〉 denotes the spatial SoP,

*u*

_{1,2}(

*r*,

*ϕ*,

*z*) are normalized amplitude profiles, and Φ

_{1,2}(

*r*,

*ϕ*,

*z*) are the phases independent of dynamic phase, namely, PB phases [46–49], respectively.

Equation (2) indicates that the spatial SoP closely depends on the normalized amplitudes *u*_{1,2}(*r*,*ϕ*,*z*) and PB phases Φ_{1,2}(*r*,*ϕ*,*z*). This means that we can construct vector BG beams with variant SoPs along propagation direction, by engineering three-dimensionally changed amplitude profiles or PB phases, especially introducing *z*-dependence. Comparing with the modulation on amplitude profile, it is flexible and stable to engineer the PB phases. Therefore, here we suppose two constituent components have the same amplitude profiles, i.e., *u*_{1} = *u*_{2}≡1/$\sqrt{2}$. In so doing, we focus on modulating three-dimensionally changed PB phases.

It is somewhat convenient to divide the three-dimensional modulation of PB phases into the transverse and longitudinal parts. First, for the transverse modulation part, considering the plane *z* = 0, one of the most typical selection is the spiral phase, i.e., Φ_{1,2}(*r*,*ϕ*,*z* = 0) = ± *mϕ*, where *m* denote the topological charge. With this modulation, if we choose the two circular polarizations (their SoPs denoted as |R〉 and |L〉) as polarization bases, i.e., |Ψ_{1}〉 = |R〉 and |Ψ_{2}〉 = |L〉, the SoP could be expressed as |Ψ(*r*,*ϕ*,*z* = 0)〉 = [exp(-i*mϕ*)|R〉 + exp(i*mϕ*)|L〉]/$\sqrt{2}$ , this means that the field has azimuthal-variant SoP of order *m* [47–49]. Moreover, we can also steer the SoP by varying polarization bases. For example, if we adopt two linear polarizations corresponding to points (0,2*ψ*) and (0,2*ψ* + π/2) on the Poincaré sphere, where *ψ* is the azimuthal angle of polarization ellipse, the SoP can be characterized as

Second, for the longitudinal modulation part, we utilize the transverse-to-longitudinal structuring strategy provided by an axicon [45], of which the principle is schematically shown in the inset (a) of Fig. 1. The two constituent components incident onto the axicon, having transversely varying PB phase distributions expressed as Φ_{1,2}(*r*,*ϕ*,*z* = 0) = ± (*mϕ* + π*r*/*d*) with a radial period of *d*, via the linear focusing of the axicon, are transformed into Bessel beams having longitudinally varying PB phases with a period of *Λ* = *d*/tan*α*, i.e., Φ_{1,2}(*r*,*ϕ*,*z*) = ± (*mϕ* + π*z*/*Λ*). Here *α* = (*n*-1)*θ*, *n* is the refractive index and *θ* is the opening angle of the axicon. It is clearly seen that, as *z* increases, the PB phases changes periodically, so, the synthesized beam varies SoP along propagation direction, namely, the constructed vector BG beam has *z*-dependent SoP.

## 3. Generation of vector BG beams with variant SoPs upon propagation

Figure 1 shows a schematic representation of the experimental setup, which includes two segments: the preparation of transverse PB phases and the transverse-to-longitudinal structuring. We utilize a modified Sagnac interferometer to prepare the transverse PB phases [51–53]. The input expanded Gauss beam (from Ar^{+} laser with *λ* = 514.5nm and *w _{a}* = 4mm) is firstly split into two orthogonally polarized components via a half-wave plate and polarizing beam splitter (PBS). Then these two components pass through the Sagnac interferometer that consists of two mirrors (M

_{1}and M

_{2}) and a reflective phase SLM (Holoeye, Pluto, placed along 45° with respect to the horizontal). On the SLM, we imprint carefully modified phase functions for two components. In fact, we record and reconstruct the spatial phase profiles by digital holography. The two linearly polarized components passing through the Sagnac interferometer are coaxially combined by the PBS, and then orderly pass through a quarter- or half-wave plate (fast axes oriented at 45° with respect to the horizontal), the 4

*f*system consisted by lenses L

_{3}, L

_{4}(

*f*= 40cm), and filter A. Note that, in this configuration, the SLM is posited at the front focal plane of L

_{3}. Via the 4

*f*system, the output constituent beams with transverse PB phases illuminate onto an axicon (Thorlabs, AX251-A) with opening angle 1° placed at the focal plane of L

_{4}. A 20 × microscope objective is used to project the constructed BG beam onto a CCD set on a linear motor. In experiment, we set

*d*= 1.8mm.

#### 3.1 Zeroth-order vector BG beams

We first construct two kinds of zeroth-order vector BG beams with *z*-dependent SoP, so that *m* = 0 and Φ_{1,2}(*r*,*ϕ*,*z*) = ± π*z*/*Λ*. For the first one, we choose two circular polarizations as bases. In experiment, correspondingly, we adopt a quarter-wave plate inserting after the PBS. It is expected that, for such a specific case, the constructed BG beam has linear SoP in cross section but variant polarization orientation along propagation direction, as schematically shown in Fig. 2(a). Figure 2(b) shows the intensity distribution in *y*-*z* plane. Figures 2(c)-2(g) display the transverse intensity distributions after a linear polarizer at planes *z*_{1} to *z*_{5}: 21cm, 23.8cm, 26.6cm, 29.5cm, 32.2cm, respectively. The arrows denote the polarization orientation of the polarizer. It is clear that the measured field presents a zeroth-order BG intensity profile, and keeps linear SoP in such a propagating distance. Importantly, in the longitudinal region from *z*_{1} to *z*_{5}, the polarization orientation gradually turns from horizontal to vertical, and returns horizontal again after propagating a distance about 11.2cm. This transformation trajectory of *z*-dependent SoP corresponds to the equator on the Poincaré sphere. It should be noted that, the SoP also slightly changes along radial direction, because the PB phases Φ_{1,2} have slightly shifted longitudinal wave numbers. However, when selecting the appropriate parameters, such radial change is very small. In addition, in the following content, we focus on the SoP properties of central spot that has maximal intensity. Therefore, we disregard the radial variation of SoP in the following content.

Figure 3 illustrates the second kind of zeroth-order vector BG beam when choosing the |H〉 and |V〉 states as bases. In this case, we insert a half-wave plate after the PBS. The PB phase difference between two spin components is ΔΦ(*r*,*ϕ*,*z*) = 2π*z*/*Λ*. As a result, the SoP of constructed beam transforms between two poles along a longitude line on the Poincaré sphere, that is, the polarization ellipticity changes with *z*. In experiment, to characterize the polarization ellipse, we measure the Stokes parameters (S_{0}, S_{1}, S_{2}, S_{3}). For a light field with intensity *I*, the Stokes parameters can be expressed as:

*I*(

_{ϕ}*ϕ*= 0°, 45°, 90°, 135°) indicates the intensity of linearly polarized component with azimuthal angle

*ϕ*, and

*I*and

_{R}*I*indicate the intensity of |R〉 and |L〉 components. Under this definitions, the Stokes parameters account for the ellipticity and azimuthal angles can be shown as 2

_{L}*χ*= arctan[S

_{3}/$\sqrt{{S}_{1}^{2}+{S}_{2}^{2}}$] and 2

*ψ*= arctan(S

_{2}/S

_{1}).

Figures 3(a)-3(e) show the transverse intensity and normalized Stokes parameters distributions at planes *z*_{1} to *z*_{5}. It can be directly seen that the Stokes parameters are almost homogeneous in the central spot. At plane *z* = *z*_{1}, S_{1} and S_{3} are nearly zeroth, but S_{2}≈1. This indicates that the SoP is linearly polarized along 45° with respect to the horizontal, which defined as the diagonal state |A〉 = (|H〉 + |V〉)/$\sqrt{2}$, so this SoP corresponds to the point A on the Poincaré sphere shown in Fig. 3(f). After propagating about 2.8cm, the Stokes parameters are changed to S_{1}≈0, S_{2}≈0, S_{3}≈1, namely, the SoP is transformed into the |R〉 state [denoted as point B on the Poincaré sphere in Fig. 3(f)]. As the beam propagates further, the SoP is orderly transformed into the diagonal [|D〉 = (-|H〉 + |V〉)/$\sqrt{2}$, linearly polarized along −45° with respective to horizontal] and |L〉 states, and then retrieval the |A〉 state at plane *z* = *z*_{5}. This transformation trajectory corresponds to the longitude line with 2*ψ* = 90° on the Poincaré sphere, as the red curve in Fig. 3(f). Importantly, these two kinds of zeroth-order vector BG beams demonstrate that, we can manipulate the transformation trajectory along any geodesic line of the Poincaré sphere by adopting appropriate bases.

#### 3.2 Higher-order vector BG beams

Next, we construct higher-order vector BG beams with transversely inhomogeneous SoP. The most common selections are the radially and azimuthally polarized beams. For generating these special SoPs, we set the PB phases as Φ_{1,2}(*r*,*ϕ*,*z*) = ± (*ϕ* + π*z*/*Λ*). It is obvious that, at the plane of *z* = 0, the SoP is radially polarized. As *z* increases to *Λ*/2, the SoP transforms into an azimuthal one. Figure 4(a) schematically shows the SoP transformation with *z*. To demonstrate this spatial property, we record the intensity profiles after a vertical polarizer, and show the results in Figs. 4(b)-4(f). Clearly, after the polarizer, the Bessel-Gauss modes change into Hermite-Gauss ones, with the dark line marking locally horizonal polarization, indicating the successfully constructing of first-order vector BG beam with *z*-dependent SoP.

We also construct higher-order vector BG beam with hybrid SoPs according to Eq. (3). Here we set the |H〉 and |V〉 states as the bases, i.e., *ψ* = 0, and Φ_{1,2}(*r*,*ϕ*,*z*) = ± (*ϕ* + π*z*/*Λ*). For such a case, the measured transverse intensity and local SoP distributions of the constructed BG beam at planes *z*_{1} to *z*_{5} are shown in Fig. 5. The red and green ellipses, denoting the RH and LH elliptical polarizations, are mapped from the local values of ellipticity and azimuthal angles calculated according to Stokes parameters. It can be clearly seen that, the ellipse changes its ellipticity and the direction of long axis two times in a circle. This means that the local SoP changes the ellipticity (tan*χ*) from |R〉 to |L〉 states, at the same time, the azimuthal angle of local SoP varies two times along azimuthal direction, indicating the local SoP of the constructed beam is azimuthal-variant. The transformation of local SoP in cross section can be represented as the longitude line 2*ψ* = 90° on the Poincaré sphere. Importantly, comparing the cross-sectional SoPs at different propagation distances presented in Fig. 5, one can clearly find that, the local SoP also transforms along the longitude line 2*ψ* = 90° with beam propagating. As a whole, in the three-dimensional coordinate system, the hybrid SoP periodically rotates along azimuthal direction with beam propagating. The rotation speed depends on the parameter *d*.

## 4. Self-healing of vector BG beams with variant SoPs upon propagation

#### 4.1 Zeroth-order vector BG beams

The self-healing capability of intensity and spatial SoP information of these vector BG beams passing through two disparate obstacles are experimentally evaluated. We first use a circle obstacle with diameter about *D* = 335μm, of which the position was adjusted, until it is coaxial with *z* axis. The optical setup is schematically shown as the inset (b) in Fig. 1, where the obstacle is set at *z*_{0} = 16.6cm. The obstructed field is shown in Fig. 6(a). The incident beam has intensity profile and spatial SoP information as shown in Fig. 3.

Figure 6(b) depicts the intensity profile in *y*-*z* plane from *z* = 17.5cm to 33cm. The transverse intensity and local SoP distributions at specific planes *z*_{1} to *z*_{5} are shown in Figs. 6(c)-6(g), where the red and green ellipses denot the RH and LH elliptical polarizations. It can be seen that, after propagating about 7.2cm, the intensity profile of the zeroth-order vector BG beam can be reconstructed, as shown in Fig. 6(c_{2}). Importantly, compared with the local SoP distributions, it is worthy note that the *z*-dependent SoP property can self-heal.

To analyze the influence of the size and shape of obstacle on the reconstruction of vector BG beam with *z*-dependent SoP, a disparate linear obstacle with diameter about *D* = 70μm that fabricated by a taper fiber is insert into the plane *z*_{0} = 16.6cm. The obstructed and reconstructed field intensity, as well as local SoP distributions are shown in Figs. 6(d)-6(f). As can be seen that, with the decrease of obstacle size, the reconstruction distance reduces [10–12]. Note that, because of the complete obstruction in vertical direction, the intensity distribution of the reconstructed beam actually appear perturbation along the orthogonal direction, as a result, the reconstructed field is no longer cylindrically symmetric. But, for such a case, after propagating a distance about 7.2cm, the low-order side lobes of the vector BG beam can reconstruct well, as shown in Figs. 6(f). However, for another one with a larger width, the fewer lobes can be reconstructed at the same propagation distance.

Furthermore, it is worth noting that, the *z*-dependent SoP property can also self-heal for such a kind of obstacle, because the constructed beam consist of two linear polarizations, of which the reconstruction can be understood as the interference of conical rays based on the geometric optical theory [10–12]. Meanwhile, the obstructing and interference processes both are irrelevant to SoP transformation, namely, no other PB phases are introduced onto two components in the obstructing and reconstructing process. Consequently, the transverse SoP and its longitudinal-variant feature retain. This self-healing capability is important in practical applications of light-matter interaction in micro scalar size.

#### 4.2 Second-order vector BG beams with hybrid SoP

We also investigate the reconstruction of higher-order vector BG beam. Here we take the second-order vector BG beam having hybrid SoP as example, the obstacle and its position are the same as that one shown in Fig. 6(d). Figure 7 shows the transverse intensity and local SoP distributions of the reconstructed vector BG beam at planes *z*_{1} to *z*_{5}. Comparing with Fig. 5, we can find that, similar to the intensity profile, the inhomogeneous SoP and the SoP *z*-dependence can also be robustly reconstructed. The self-healing of such beam can also be understood by the separate reconstruction of two constituent components. It should be pointed out that, for such a case, although the obstacle is infinite in vertical direction, the spiral phases carried by two constituent components dramatically enhance the transverse power flows, results in a superior reconstruction. As a result, the higher-order vector BG beam can robustly self-heal in such a propagation distance.

This degree of longitudinal variation of SoP and its self-healing can address many challenges in applications. For instance, combining it with the polarization response feature of material, it can be used for material process at special depth [54,55], and to improve the axial resolution in 3D optical microscopy [17]. In addition, it is noteworthy that the higher-order Bessel beams generally support quantitative OAM and SAM simultaneously. To some extent, with the SoP longitudinally varying, the SAM-OAM conversion maybe induced, and intriguing phenomena such as spin transport could be aroused [56].

## 5. Conclusions

In conclusion, we delineated a model for constructing vector BG beams having *z*-dependent SoP, by steering longitudinally varying PB phases, which were created from the transverse-to-longitudinal structuring of axicon. We experimentally demonstrated and created several vector BG beams with *z*-dependent SoP. Furthermore, we explored the self-healing of these vector BG beams by observing their transverse intensity and Stokes parameters distributions after propagating through two disparate obstacles. The results show that, similar to the self-healing of their intensity profiles, the spatial property of SoP, especially the SoP *z*-dependence can self-heal. The self-healing capability of SoP makes these vector BG beams particularly suited to the applications referring to light-matter interaction, optical microscopy and polarization metrology.

## 6. Funding and Acknowledgments

National Natural Science Foundations of China (NSFC) (11634010, 11404262, 61675168, 61377035, and U1630125); Natural Science Basic Research Plan in Shaanxi Province of China (2015JQ1026).

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