## Abstract

We generate nondiffracting Bessel beams whose polarization state varies with propagation distance. We use a reflective geometry where a single parallel-aligned spatial light modulator device is used to spatially modulate two orthogonal linear polarizations with two axicon phase profiles. Then, by adding an extra phase retardation radial profile between these linear states, we are able to modulate the state of polarization along the line focus of the axicon. We provide experimental results that demonstrate the polarization axial control with zero-order and higher order Bessel beams.

© 2015 Optical Society of America

Bessel beams are optical beams with a Bessel function electric-field profile that can propagate through large distances without diffraction. They were originally introduced by Durnin [1] and experimentally demonstrated [2] with a simple optical system consisting of a narrow annular slit and a lens placed one focal length away. Later, McLeod [3] proposed a refractive axicon (the conical lens). This refractive element [4] produces a line focus from an incident collimated beam that can be considered to be an approximately zero-order Bessel beam that has a finite propagation distance. A similar diffractive version of the axicon creates Bessel beams, using static computer generated holograms [5] or by means of controlled devices such as spatial light modulators (SLMs) [6–8]. A great number of works have dealt with Bessel beams because of their unique properties and interesting applications in microfabrication and materials processing [9], optical trapping and tweezing [10], and as information carriers for quantum communications [11]. A good review can be found in [12].

However, in all the techniques mentioned above for generating Bessel beams, the polarization state of the nondiffracting beam is the same as the polarization of the incident beam and remains constant with propagation distance. There have been other approaches that allow the generation of nondiffractive beams with a spatial distribution of polarizations, named vectorial Bessel beams. The typical cases have radial or azimuthal polarization states [13–16]. Again, these special polarization patterns remain constant along the propagation distance.

Here, on the contrary, we present a novel approach where we produce polarization diffractive axicons where the polarization state can be controlled over the propagation distance. In order to encode polarization dependence onto a beam using a single SLM, we use a reflective geometry.

The classical linear axicon is described with the phase function:

where $r$ is the radial coordinate. The grating period $d$ controls the diameter of the quasi-Bessel function beam. This phase function creates a beam with an approximate Bessel function profile that stays constant over a certain distance.The idea can be best explained using a ray approach, as shown in Fig. 1(a). Here, collimated light enters a diffractive axicon with a radial diffraction grating with period $d$. All of the rays are diffracted by the same angle $\theta $, and those at a common given radial distance $r$ from the optical axis intersect at a distance $z$ from the axicon that depends on the diffraction grating equation, where $\lambda =d\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta \approx dr/z$. In the figure, we show three sets of rays that intersect at three different distances. From this relation, the focus at plane $z$ is mainly created by the rays at radius $r$, related by $z=rd/\lambda $, and the maximum range of the Bessel beam is ${z}_{\mathrm{max}}={R}_{\mathrm{max}}d/\lambda $, where ${R}_{\mathrm{max}}$ is the maximum radius of the phase mask [6].

When this pattern is encoded onto a diffractive pixelated SLM, the maximum radius and the axicon period can be written as ${R}_{\mathrm{max}}=(N/2)\mathrm{\Delta}$, and $d=n\mathrm{\Delta}$, where $N\times N$ are the number of available pixels in the SLM to display the axicon, and $\mathrm{\Delta}$ is the pixel spacing. Then, the maximum range of the Bessel beam can be expressed simply as

However, the beams will all have the same polarization state. Moreover, if the axicon is created with a liquid crystal SLM, only the state of polarization parallel to the liquid crystal (LC) director will be focused.We can introduce polarization dependence if the axicon is illuminated with an arbitrary state of polarization, as shown in Fig. 1(b). Now, we place a birefringent material in front of one set of rays having one radius. The polarization state will then be altered depending on the initial polarization state and the thickness of the birefringent material. In the example in Fig. 1(b), the input state of polarization is linearly polarized light at 45°. In this case, we assume a half-wave retarder that covers the central portion of the axicon, leaving the outer part unaffected. In this situation, the first part of the line focus, closer to the axicon, will be linearly polarized at $-45\xb0$, while the final part of the line focus remains polarized at 45°, thus creating two line segments with orthogonal polarizations.

The same idea can be generalized, as indicated in Fig. 1(c). By tailoring the phase shift between two orthogonal linear polarizations as a function of radius, $\varphi (r)$, the polarization state of the propagated beam can be controlled. In the example in Fig. 1(c), we are adding a linear radial retardance variation. Therefore, the state of polarization will change continuously along the line focus. Note that arbitrary radial retardance functions $\varphi (r)$ can be implemented to produce an arbitrary modification of the state of polarization along the axial line focus.

However, this approach requires the ability to modulate the two components of the electric field. This cannot be achieved with a single liquid crystal SLM because only one linear polarization component (parallel to the LC director) is modulated.

Here, we adapt an experimental setup already demonstrated in Refs. [17,18], which is designed to modulate two orthogonal polarization components. The optical architecture is sketched in Fig. 2(a). A linearly polarized light beam is launched onto a transmissive parallel-aligned liquid crystal display (LCD), with the polarization direction at 45° with respect to the LC director axis. This way, the incoming beam is divided into two linear components with equal power, one parallel and another perpendicular to the LC director. While the first one is sensitive to the voltage applied to the display, the second one remains unaffected.

The system is based on dividing the LCD screen into two halves, where two different phase patterns are addressed, as shown in Fig. 2(b). The initial beam illuminates only the left half part of the screen, where an axicon is encoded as shown in the left pattern of Fig. 2(b). The axicon phase pattern addressed on the left side of the LCD is encoded onto the vertical linear polarization component, the one that is parallel to the LC director. Then, by means of a lens and a mirror, the beam is reflected back to the right part of the LCD.

The insertion of the quarter-wave plate (QWP), oriented at 45° with respect to the LC director, reverses the orientation of the initially vertical and horizontal linear polarization components. As a consequence, the initial horizontal linear polarization component, which is not affected by the LCD in the first passage, now becomes vertical linear polarization for the beam impinging back on the right side of the LCD. Therefore, it will be modulated by the phase pattern encoded on this side of the LCD. Here, we encode the same axicon phase pattern, but we add an additional phase that increases linearly with radius, as shown in the right pattern of Fig. 2(b).

As a result, the initial vertical linear polarization component, which carries the phase pattern encoded on the first passage through the LCD, now becomes horizontal linear polarization, which is not affected by the LCD on the second passage. In this way, we are able to encode different phase content onto the two orthogonal linear polarization components. Note that Fig. 2(b) shows the two different axicon patterns where the phase offset is different on one side compared with the other.

Finally, a nonpolarizing beam splitter is placed to separate the reflected beam from the incident beam.

We employ a transmissive parallel-aligned nematic LCD manufactured by Seiko Epson with 640 pixels x 480 pixels having dimensions of $\mathrm{\Delta}=42\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ [19]. Each pixel acts as an electrically controllable phase plate where the total phase shift exceeds $2\pi $ radians as a function of gray level at the argon laser wavelength of 514.5 nm. Because the LCD has to be divided into two halves, the effective array number to encode the axicons is $N=120$. We have encoded axicon gratings with periods $d=5\mathrm{\Delta}$. Therefore, from Eq. (2), the maximum range of the Bessel beam is over 1 m.

Experimental results are presented in Fig. 3. In order to produce the linear retardance variation, we added an additional radial linear phase, given by $\varphi (r)=2\pi r/D$, where $D=m\mathrm{\Delta}$. Hence, a linear variation of the state of polarization is obtained along the line focus created by the axicon. The distance $Z$ between two axial points with the same state of polarization corresponds to two radial coordinates with a $2\pi $ retardance variation. This results in the relation:

In our case, where the axicon patterns have a period of $d=5\mathrm{\Delta}$ and $D=60\mathrm{\Delta}$, the periodic variation in the state of polarization occurs over a distance of $Z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{m}$.The first row in Fig. 3 shows the experimental intensity pattern captured at a close distance from the beam-splitter output, which we denote as $z={z}_{0}$. The image indicated as “NoA” corresponds to the case without an analyzer. It shows the characteristic Bessel beam pattern. We adjusted a bias phase added to one of the axicon patterns in order to make the center of the beam in this plane linearly polarized, with orientation at $+45\xb0$. This is verified by analyzing the Bessel beam in this plane with a linear polarizer oriented at 0°, 45°, 90° and 135°, as well as with RCP and LCP circular polarizers. This is shown in the first row of Fig. 3. It is observed that the Bessel beam appears bright when the linear analyzer is oriented at $+45\xb0$, while it appears dark when the linear analyzer is oriented at 135°. As expected, the intensity is approximately one half when the linear analyzer is horizontally or vertically oriented, as well as for right (RCP) and left (LCP) circular analyzers.

Then, we experimentally verified the changes in the state of polarization along the axis. However, in order to capture the corresponding intensity patterns at precise distances and avoid any possible misalignment when moving the camera, we applied a recently developed technique that allows the virtual propagation of the beam using a fast Fresnel propagation algorithm [20,21]. In this way, we can evaluate the beam propagation without moving either the LCD or the camera [22]. We apply the algorithm to propagation distances of $Z/4$, $Z/2$, and $3Z/4$. The corresponding results are presented in the second, third, and fourth rows of Fig. 3.

The results without an analyzer (left column) show the nondiffracting characteristic propagation of the Bessel beam. However, there is a continuous linear change in the state of polarization, which becomes clear when the polarization analyzers are placed before the camera. For a propagation of $Z/4$, the $\pi /2$ retardance variation makes the center of the beam LCP polarized. This is verified as the intensity at the center of the beam remains constant when rotating the linear polarizer, and it appears bright and dark when the circular analyzers are employed. The results in the third and fourth rows, corresponding to propagation distances of $Z/2$ and $3Z/4$, demonstrate that the beam is linearly polarized at $-45\xb0$ and RCP polarized at these two planes, respectively. Therefore, these results verify the linear change in the state of polarization of the Bessel beam generated by this conical retarder axicon system.

In order to quantitatively verify this axial change in the state of polarization, we measured the relative intensity of the central peak of the Bessel beam for different axial distances when the final analyzer is oriented at $+45\xb0$. The images in the third column of Fig. 3 correspond to this situation. The state of polarization is changing linearly—from being linear at 45°, to LCP, to linear at $-45\xb0$, to RCP, and back to the original linear state at $+45\xb0$—as the propagation distance $z$ increases over a distance of $Z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{m}$ from the original plane at ${z}_{0}$. Therefore, the expected relative intensity when the analyzer is oriented at 45° should follow a ${\mathrm{cos}}^{2}(\pi z/Z)$ relationship as a function of the propagation distance $z$. Figure 4 shows this plot, together with the measured peak intensity. We can observe the expected intensity oscillation with propagation distance, and the experimental measured data match very well to the expected curve.

Higher order Bessel beams can be obtained by adding a spiral phase to the axicon phase [7] to create a vortex axicon phase plate. The phase function in Eq. (1) is modified to:

where $\varphi $ is the azimuth angle, and $\ell $ denotes the topological charge. The resulting vortex-creating axicon produces a nondiffracting beam that corresponds to the ${J}_{\ell}$ Bessel function, which shows a dark center and a surrounding ring of light whose diameter depends on the topological charge when $\ell \ne 0$, and that has the same focusing range as the original nondiffracting beam.Figure 5 shows experimental results equivalent to those presented in Fig. 3, but corresponding to the case of $\ell =3$. These results present the higher order Bessel beam, which is characterized by the presence of a singularity on the axis, thus becoming a vortex beam. However, this vortex line focus displays a variation in the state of polarization along the axis, in a similar way as the axicon presented in Fig. 3. The results in Fig. 4 verify again the continuous transition from linear polarization at 45°, to LCP, to linear polarization at $-45\xb0$, to RCP, as the beam is propagated from the reference initial plane at ${z}_{0}$ to planes at distances $Z/4$, $Z/2$, and $3Z/4$. As in the previous case, we verified in the laboratory this polarization variation, but the virtual propagation algorithm was applied to obtain these results at exact propagation distances and to avoid misalignments.

In conclusion, we introduce a new idea where the polarization state of a beam can vary along the propagation distance. Although we demonstrated a linear change in polarization with distance, any desired propagation map can be applied. We expect this new capability to be useful in a number of applications where axicons and polarization control are of interest, including material processing, polarimetry, microscopy, and optical communications.

## Funding

Ministerio de Ciencia e Innovación (MICINN) (FIS2012-39158-C02-02).

## Acknowledgment

I. M. and M. M. S.-L. acknowledge support from MICINN from Spain.

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