## Abstract

We demonstrate that the spatially diffractive properties of cylindrical vector beams could be controlled via linear interactions with anisotropic crystals. It is the first time to show experimentally that the diffraction of the vector beams can be either suppressed or enhanced significantly during propagation, depending on the sign of anisotropy. Importantly, it is also possible to create a linear non-spreading and shape-preserving vector beam, by vanishing its diffraction during propagation via strong anisotropy in a crystal. The manageable diffractive effect enables manipulating propagation dynamics of the circular Airy vector beams, i.e., their propagation trajectories can be dynamically controlled by weakening or enhancing self-acceleration of the Airy beam. We further demonstrate that the cylindrical vector beams with initially zero orbital angular momentum can be rotated either clockwise or anticlockwise, relying on the sign of the anisotropy.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The physical quantities of a light beam including amplitude, frequency, phase, and polarization have been widely explored and could be independently controlled with an optical component such as optical lens, diffractive element, polarizer, and phase plate. However, dynamic control of the optical quantities is still limited to scalar beams, whose polarizations are spatially homogeneous. In recent years, there has been an increasing interest in vector beams with spatially inhomogeneous polarization [1–3]. Among others, an important subclass is cylindrical vector (CV) beams, e.g., the typical Laguerre-Gaussian (LG) vector beams [4] and circular Airy vector beams [5,6] with spatially radial, azimuthal as well as spiral polarizations, as shown in Fig. 1. These CV modes are axially symmetric solutions to the vector Maxwell wave equation [4,7] and have been manifested as an important field owing to their unique characteristics. For instance, the radially polarized LG vector beams could be tightly focused [8,9], creating stronger longitudinal field and tighter optical spot as compared with that of a scalar beam; while the circular Airy vector beams could self-accelerate along a prescribed trajectory in free space, hence exhibiting unique feature of abruptly autofocusing [6,10,11]. Owing to their salient features, the CV vector beams have been employed in many fields such as particle trapping [12,13], high-resolution microscopy [14] and optical communications [15,16].

Over the past decade, many techniques were proposed to generate the CV beams. To summarize, they could be achieved either from laser intracavity, e.g., with few-mode fibers that enable emitting vector beams [17–19], or extracavity where scalar beams are converted into vector beams via wave plates [20,21], spatial light modulator (SLM) [3,22], metasurface [1,23], etc. It is worth noting that vector beams with discretionarily chosen polarizations could be manipulated by combining nanostructured elements with wave plates [24] or SLM [3]. Notwithstanding the progress of vector beam generations, it is obvious that the next challenges are in polarization propagation dynamics and interactions with materials, aiming for controlling the beam propagations [25]. However, to date, there is a lack of studies, particularly the experiments, addressing this issue. Therefore it is desirable and meaningful to investigate the spatially structured vector beam propagation as well as its interaction with structured materials, which is expected to yield new physics [26].

In this article, we demonstrate both theoretically and experimentally the linear interactions of CV beams with anisotropic crystals, and reveal an intriguing phenomenon that the spatial diffraction of CV beams could be significantly suppressed or enhanced during propagation, depending on the sign of anisotropy. Of particular interest, a linear non-spreading CV beam with radial polarization could be realized by vanishing its diffraction via strong interactions with anisotropic crystals, which is analogous to the non-dispersive waves propagating in a waveguide where the wave dispersion vanishes by engineering the waveguide dispersion [27]. Therefore, diffraction manipulations for the CV beam is possible. Here we utilize such phenomena to control the self-accelerating characteristics of the Airy waves, which, to our knowledge, has not been reported so far. In addition, we present the first experimental demonstration that the CV beams with initially zero orbital angular momentum could be rotated either clockwise or anticlockwise, which might find technological applications, e.g., in polarization-sensitive sensing.

## 2. Theory

We start by considering linear interactions of a CV beam with an anisotropic crystal whose dielectric tensor can be expressed as

*n*(

_{j}*j*=

*x*,

*y*,

*z*) is refractive index along the respective coordinates

*x*,

*y*and

*z*. In the following, we investigate the vector beam propagation along the

*z*axis, which is in accordance with the optical axis. In this scenario, we note that G. Cincotti et al derived paraxial wave equation governing the vectorial field behaviors in a uniaxial medium [28]. Here we generalize the theoretical model by considering the vector wave equation:

**E**is electric field, and

*µ*denotes the permeability of medium.

*t*is elapse time. With the assumption of constant permeability, we derive wave equations for the slowly varying vectorial fields inside the anisotropic crystal, given by

*E*(

_{j}*j*=

*x*,

*y*) represents the Cartesian polarization components of the vector light field.

*β*= 2

_{j}*k*

_{0}

*n*, where

_{j}*k*

_{0}= 2

*π/λ*is wavenumber with

*λ*being the carrier wavelength, and Δ

*β*=

*β*−

_{y}*β*. Here we define ${\gamma}_{j}=1-{n}_{j}^{2}/{n}_{z}^{2}$ as anisotropy parameter of the crystals. Clearly,

_{x}*γ*< 1, and the value can be either positive or negative, depending on the anisotropic crystals. The situation of

_{j}*γ*=

_{x}*γ*would lead to vanishing the exponential factor exp(±

_{y}*i*Δ

*βz*/2), in which case Eq. (3) is in accordance with the model shown in [28]. It is evident that these two polarization components are highly structural symmetry, and coupled as a result of the anisotropy

*γ*which plays the role of coupling strength. We note that the uncoupled paraxial wave equations for the slowly varying envelopes associated with the radial and azimuthal components of the vector field in an uniaxial crystal were derived [29]. It is worth mentioning that the theoretical model [Eq. (3)] considers a general situation where the refractive indexes of the anisotropic crystals have the general form of

_{j}*n*≠

_{x}*n*≠

_{y}*n*, not only limited to the previously demonstrated case of uniaxial crystal, i.e.,

_{z}*n*=

_{x}*n*.

_{y}More interestingly, Eq. (3) reveals that the diffraction of the induced extraordinary vector beams could be either enhanced or suppressed by anisotropy, depending on the sign of *γ _{j}*; while the ordinary vector beam only experiences normal refractive index and would not be affected by the anisotropy. It indicates that we are able to manipulate the diffraction dynamics of cylindrical vector beams with radial polarization, since it was found that the extraordinary vector beams are radially polarized while the ordinary vector beams are azimuthally polarized [29]. We further deduce the Fresnel diffraction factor for the CV beams with radial polarization, written as

*f*= (1 −

*γ*)/

*β*(here we have assumed

*γ*≃

_{x}*γ*=

_{y}*γ*). Clearly, increasing the anisotropy

*γ*would lead to a considerable decrease of the diffraction factor. As a consequence, it would significantly weaken the diffraction of the radially polarized CV beams in a crystal. Of particular interest, in an extreme case of

*γ*→ 1, it is expected that a linearly non-spreading vector beam could be created. These intriguing phenomena are manifested by simulations with incident CV beams which take the LG form [30]

*ϕ*= arctan(

*y*/

*x*); $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ are the unit vectors along the corresponding axis, respectively.

*E*

_{0}(

*r*) has a form of lowest-order LG function, and can be expressed as ${E}_{0}\left(r\right)=r/{\sigma}_{0}\mathrm{exp}\left(-{r}^{2}/{\sigma}_{0}^{2}\right)$ with

*σ*

_{0}denoting the beam width. The parameter

*θ*is polarization angle between the field vector and the radial direction [30], as illustrated in Fig. 1(a). For instance, the values of

*θ*=0,

*π*/2,

*π*/4 represent the radial, azimuthal, and spiral polarizations, as shown in Figs. 1(b)–1(d) respectively.

To see the non-diffractive characteristic of the CV beams, we perform simulations based on Eq. (3), setting the values: *γ* = 0.94 (with *n _{x}* =

*n*= 1.9929),

_{y}*σ*=20

_{o}*µ*m, and

*λ*= 632.8 nm. Here we neglect the slight biaxial effect (

*γ*≃

_{x}*γ*) on beam propagation, since it would affect the beam shape during propagation. With these conditions, it is shown from the outcome, see Fig. 2(a), that the radially polarized LG beam propagates over a long distance without any expanding, in contrary to the azimuthally polarized LG beam that starts diffracting seriously after a diffraction length of ${L}_{d}={k}_{0}{\sigma}_{0}^{2}=4$ mm, see Fig. 2(c). We emphasize that the created linearly shape-preserving vector beam is very different from the nonlinearly solitary beam, which is characterized by a balance between diffraction and nonlinearity [31]. However, this fascinating non-diffractive beam is analogous to the non-spreading wavepacket propagating in a waveguide where the wave dispersion vanishes by engineering the waveguide dispersion [27]. Here such a non-diffractive beam is realized in a crystal with strong anisotropy

_{y}*γ*, which, thereby, vanishes the beam diffraction over a large propagation distance. We further note that the crystals with strong anisotropy can also act as a polarization filter, since for a LG beam with spiral polarization, only the radial component is allowed for propagation without diffraction, while the azimuthal component would be filtered out owing to its diffraction loss, as illustrated in Fig. 2(b). We point out that this argument can be numerically confirmed by illustrating their states of polarization (SoP) of the output beams, which show the azimuthal component being completely filtered out when propagating through the crystal.

## 3. Experiment

In order to verify our theoretical predictions, we carried out experiments with a uniaxial crystal (YVO_{4}), whose refractive index are *n _{x}* =

*n*=1.9929 and

_{y}*n*=2.2154. In this case the resulting positive anisotropy strength is

_{z}*γ*= 0.191. The dimension of the crystal is

*x*×

*y*×

*z*= 4 mm × 4 mm ×30 mm. Firstly, we generated the radially polarized LG beams with setup presented in Fig. 3(a). A linearly polarized He-Ne laser working at the wavelength of 632.8 nm was expanded and then collimated. After propagating through a neutral density (ND) filter, the collimated beam illuminated a radial polarization converter (PC) (ARC optix, Switzerland), hence converting the spatially invariant beam into vector beam. The nearly lowest-order LG vector beam was obtained by focusing the radially polarized beam from the PC element, as seen from Figs. 3(b)–3(c). Figure 3(b) shows the measured intensity of the obtained LG vector beam; while Fig. 3(c) and 3(d) depict its horizontal and perpendicular polarization components respectively. These measurements suggest that a highly quality LG beam with radial polarization was generated in the system, with a beam width measured as

*σ*

_{0}= 9

*µ*m at the focal plane. Secondly, we investigated linear interactions of these LG beams with the crystal. After passing through a pinhole (PH) that is used to block the undesired light spot, the LG beam enters the crystal. Note that the pinhole is located at the focal plane (

*z*= 0), and the crystal is just behind the pinhole. The output facet of the crystal was imaged onto a CCD camera. The experimental result showing the intensity distribution after passing through the crystal (

*z*= 30 mm) is presented in Fig. 3(f). In comparison, the free-space propagation of the LG vector beam having the same initial size was considered, with the recorded intensity shown in Fig. 3(e). Clearly, it is observed that the LG beam is shape-preserving, and its diffraction is significantly suppressed inside the crystal. It is seen from Fig. 3(g) that for the case of free-space propagation, owing to serious wave diffraction, the beam width is expanded to 1.73 mm (full width at half maximum, FWHM). However, this effect of diffraction can be considerably weakened in a crystal, since the FWHM is measured only as 0.72 mm. We also considered a larger Rayleigh rang, e.g., increasing the incident beam width to

*σ*

_{0}= 14

*µ*m, the output beam width is expanded to 1.04 mm in free space, while in crystal, the FWHM of the output beam is measured as 0.39 mm (results not shown here).

We further point out that the cylindrical vector beam with radial polarization propagating in the uniaxial crystal gives the Fresnel diffraction factor as ${f}_{c}=1/\left(2{k}_{0}{n}_{z}^{2}/{n}_{j}\right)$ while for the case of free space propagation, the Fresnel diffraction factor is written as *f _{v}* = 1/2

*k*

_{0}). Therefore, the suppression factor can be given analytically by ${f}_{v}/{f}_{c}={n}_{z}^{2}/{n}_{j}$. For the YVO

_{4}crystal, we have

*f*/

_{v}*f*=2.46. In experiment, the suppression factor can be measured by the ratio of FWHM. For instance, in case of

_{c}*σ*

_{0}= 9

*µ*m, we have 1.73/0.72 =2.4, see Fig. 3(g); while in case of

*σ*

_{0}=14

*µ*m, we get 1.04/0.39 =2.6, both being very colse to the analytical results. It is expected that further diffraction suppression, i.e., the creation of a nearly non-diffracting beam with radial polarization, could be observed with stronger anisotropy of the crystals. The experimental results agree well with the simulations.

Next, we present the first application of the manageable diffractive effect of the CV beams to control propagation dynamics of the Airy beams, which is difficult to be achieved since the propagation trajectory of a specific Airy beam is prescribed and would not be changed during acceleration in free space [32–36]. To this end, we consider the circular Airy vector beam that could self-accelerate toward the axis center, thereby exhibiting unique feature of abruptly autofocusing [6,10,11] (We note that the propagation dynamics of the Airy-Hermite-Gaussian beams in uniaxial crystals was numerically investigated [37]). The circular Airy beam with radial polarization (*θ* = 0) takes the form of

*Ai*is the Airy function,

*r*

_{0}the initial radius of the Airy rings, and

*ω*determines the main lobe of the Airy rings.

*α*is the decay parameter. We utilize a setup similar to that used in [5] for generating the circular Airy vector beam. A linearly polarized He-Ne laser working at the wavelength of 632.8 nm was expanded and collimated, and was sent toward a phase-only reflecting spatial light modulator (SLM). The phase pattern, which includes both the amplitude and phase information of Fourier transform (FT) function of the circular Airy beam, was loaded to the SLM, using the method proposed by Davis, et al [38]. Here the FT function of the circular Airy beams is given analytically by [11]

*k*is the spatial frequency, and

*J*

_{0}(·) the zero-order Bessel function. A circular Airy beam with invariant polarization was then obtained in a focal plane (

*z*=0) of a lens, and was converted into radially polarized beam with polarization converter (PC) element. A CCD camera was used to measure the transverse intensity distribution at the plane

*z*> 0. Note that at the focal plane, an annular aperture was used to blocked the undesired intensity at the center of the patterns. Figure 4(a) shows experimentally the intensity distribution of the generated circular Airy pattern without polarizer; while Fig. 4(b) and 4(c) illustrate its horizontal and perpendicular polarization components respectively, indicating that a highly quality circular Airy beam with radial polarization (

*θ*= 0) was generated.

To observe the autofocusing property, we propagated the generated Airy beams in free space, and measured their intensity patterns at different planes, with results illustrated in Fig. 5(a). It shows that the Airy main ring self-accelerates toward the axis during propagation, exhibiting abruptly autofocusing at the position of *z _{f}* =105 mm. The experimental results match well with our simulations based on the parameters:

*r*

_{0}=0.37 mm,

*ω*= 40

*μ*m, and

*α*= 0.1. We then control the self-accelerating as well as autofocusing property of the Airy beams via crystals. To illustrate this point, we placed a crystal (YVO

_{4}) just behind the aperture, and measured the intensity distributions of the Airy beam after passing through the crystal, with results shown in Fig. 5(b) and 5(c), for two cases of crystal length. Interestingly, owing to the diffraction suppression, the acceleration of the beams is significantly weakened. As a result, the autofocusing position of the circular Airy beams is increased from

*z*= 105 mm to 114 mm, with a 10 mm-length crystal, see Fig. 5(b). We found that a longer crystal length would further increase the focusing position, as seen from Fig. 5(c) where the focusing position is increased to

_{f}*z*= 125 mm, with a 30 mm-length crystal, also illustrated in Fig. 6(a), showing clearly the increasing relation between the autofocusing position and the crystal length.

_{f}Moreover, it is interesting to find linear modulation of the circular Airy vector beams via controlling the anisotropy of the crystals. As shown in Fig. 6(b), the autofocusing position *z _{f}* is linearly increased with the value of

*γ*. This dynamical phenomenon could be achieved with a crystal whose refractive indexes can be electrically or optically controlled. It is worth mentioning that after autofocusing, the vector beam exhibits nearly non-diffracting when propagating in free space, as seen from Fig. 5. This is because the circular Airy vector beam undergoes abrupt autofocusing and, due to polarization singularity in the center, eventually evolves into the first-order Bessel-like pattern that could preserve its profile while propagating in free space.

Additionally, it is expected that the crystal with *γ* < 0 would enhance the diffraction of the vector beam with radial polarization, and hence strengthening self-acceleration of the Airy beams. To see this phenomenon, we repeated experiments investigating propagation of the Airy beams with radial polarization in a BBO crystal, with results shown in Fig. 7. However, self-acceleration enhancement of the Airy vector beam in the crystal is not found, since it is shown that the Airy beam self-accelerates in free space and autofocuses at *z* = 75 mm, as illustrated in Fig. 7(a); while the autofocusing position is increased to *z* = 85 mm when the Airy beam travels through a 30-mm BBO crystal at its early stage of propagation, see Fig. 7(b). This is because the Rayleigh range of the Airy beam in the crystal is larger than the one associated with the vacuum, and the anisotropy strength of BBO is insufficient to enhance the self-acceleration of the Airy beam. However, the phenomenon of self-acceleration enhancement of the Airy beams can be observed in the crystal with identical Rayleigh range, e.g., see the simulated results in Fig. 8. Obviously, in the case of *γ* = −0.38, see Fig. 8(a), the autofocusing position is shifted forward, while it is shifted backward with the value of *γ* = 0.24, see Fig. 8(c), as compared with the autofocusing position shown in Fig. 8(b) (*γ* = 0).

Finally, we reveal that the polarization rotations of the CV beams could be also controlled via anisotropy, as indicated from the couplings between two polarization components, see Eq. (3). It is worth mentioning that the radially (*θ* = 0) and azimuthally (*θ* = *π*/2) polarized beams propagate independently in crystals. Therefore, in order to explore the nontrivial coupling effects between the x- and y-component, the polarization angle of the CV beam should be limited to 0 < *θ < π*/2. It is emphasized that although the polarization behaviors in a uniaxial crystal were widely explored with scalar beams, the phenomenon has not been addressed by the use of a CV beam. We note that the coupled terms in Eq. (3) would rotate the energy as well as the polarization of the beams during propagation. To observe this effect, a CV beam with typically spiral polarization angle *θ* = *π*/4 was generated by elaborately interfering with the radially and the azimuthally polarized LG beams that were achieved from two separated PC elements, for details see the experimental setup shown in Fig. 9. Figure 10(i) show experimentally the intensity distributions of the horizontal component of the generated vector beams, suggesting that the expected lowest-order LG beam with *θ* = *π*/4 was generated. Figures 10(a)–10(c) illustrate the intensity distributions of the horizontal component after passing through the YVO_{4} crystal with crystal length of *L _{c}* =10 mm,

*L*=20 mm, and

_{c}*L*= 30 mm, respectively; while Figs. 10(d)–10(f) depict their corresponding simulations. Clearly, it is observed that the energy of the LG beam is clockwise rotated when propagating through the crystal, as compared with the initial one shown in Fig. 10(i). However, the energy is anticlockwise rotated when the beam traveling through the BBO crystal, as can be seen from Fig. 10(g) and 10(h), showing experimental and numerical results, respectively. The intriguing phenomena can be understood as follows. For a spirally polarized LG beam, which is a balanced superposition of the radial and the azimuthal vector beams, the induced ordinary and the extraordinary beams have different wave vectors, owing to different refractive indexes. These beams are coupled when propagating in the crystal, and they experience different diffractions. As a consequence, it would modify the spatial state of polarization (SoP) at the output end of the crystal. Here we plotted the SoP of the spirally polarized LG beams after passing through the YVO

_{c}_{4}and BBO crystals, see Fig. 11(a) and 11(c) respectively. Evidently, the polarization of the output beam is redistributed with respect to the initial SoP shown in Fig. 1(d). Interestingly, it clearly shows the clockwise energy flux in the case of YVO

_{4}crystal, see Fig. 11(a) and 11(b), and anticlockwise energy flux in the case of BBO crystal, see Fig. 11(c) and 11(d), which explains the experimental results shown in Fig. 10.

## 4. Conclusion

In conclusion, we have investigated the linear interactions of cylindrical vector beams with anisotropic medium, and demonstrated dynamic control for the diffraction properties, polarizations of the LG vector beams, as well as the self-accelerations of the circular Airy vector beams, by the use of anisotropy *γ*. Particularly, we have shown that a linear non-spreading and shape-preserving vector beam with radial polarization could be generated by eliminating its diffraction via strong interactions with the anisotropic crystals. It is worth mentioning that our demonstrations enable to dynamically manipulate the CV beams with a crystal whose refractive indexes can be electrically or optically controlled. This is of great interest and could find potential applications, e.g., in material processing [10] and polarization-sensitive sensing.

It is worth noting that our investigations concentrate on vector beam propagation along optical axis of the anisotropic crystals. Actually, previously investigations (e.g., see [39]) have also shown significantly anisotropic behaviors of optical beam when propagating along the direction perpendicular to the optical axis of the crystal. A nature extension of this work is to study anisotropic properties of vector beam (with or without cylindrical symmetry) that propagates along the direction perpendicular to the optical axis, which was not addressed before.

## Funding

National Key Research and Development Program of China (2017YFB1104500); National Natural Science Foundation of China (11704155, 61475067, 61735005); Natural Science Foundation of Guangdong Province (2017B030306009).

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