## Abstract

We report on the generation of a spiraling zero order Bessel beam by means of conventional axicon and the phase hologram. Obtained results are in a fairly good agreement with the theoretical predictions.

©2010 Optical Society of America

In 1987 Durnin and co-workers gave the first experimental demonstration of so called conical wave [1]. The simplest example of a conical wave is the monochromatic zeroth-order Bessel beam, which intensity cross-section comprises a central bright spot surrounded by rings spaced according to Bessel function *J*
_{0}. Later, using compensated spatial light modulator, the achromatic zero-order Bessel beam was demonstrated experimentally [2]. Such beam has intensity cross-section which is independent of wavelength. In general all common Bessel beams including ones of higher order [3] propagate straight in isotropic medium with the constant transversal intensity distribution. In the meantime, beams which intensity distributions are periodic with propagation distance are also known. This includes propagation-invariant beams constructed from properly selected and superposed Laguerre-Gauss modes [4], superposition of Bessel modes [5, 6], helicoidal beams [7], “snake” beams [8], and many others [9–14]. In this letter we report results on the formation of the spiraling zero order Bessel like beam in isotropic medium. The intensity pattern of this beam coincide with common zero order Bessel beam but in addition is laterally displaced with respect to optical axis and spirals as a whole around this axis while the beam propagates along it. However the beam itself does not rotate in time.

Recently we suggested that such light field can be experimentally created using an axicon and an appropriate hologram [15]. The latter introduces equal phase shift of the spiral form and is characterized by the transfer function exp[*i*Φ], where Φ is the function of a light phase modulation defined in polar coordinates (*ρ*, *ϕ*) as Φ = *β*Δcos(*ϕ* - 2*πρ*/*P*)]. Parameter Δ is responsible for the Bessel beam lateral displacement, *P* is the periodicity of the phase modulation along *ρ*, *β* = *kΓ* denotes a transverse component of a plane wave vector **k**, and *γ* is a deflection angle of this wave beyond an axicon. It was predicted that if such optical system is illuminated by the Gaussian beam with a waist *w*, the light field intensity in a transverse plane at the distance *z* beyond the axicon, under the approximation of stationary phase, is proportional to

where *s* = *β* [Δ^{2} + *r*
^{2} - 2*r*Δcos(*ψ* - 2*πγz*/*P*)]^{1/2}, (*r*, *ψ*) are polar coordinates of the considered plane, and *w* is the radius of the Gaussian beam incident upon the hologram. One can see that the beam peak spirals as *z* increases. The radius of the trajectory is Δ and one revolution is performed over distance *L* = *P*/*γ*.

In this work we demonstrate that spiraling zero order Bessel beam is not just another peculiar theoretical construction of the electromagnetic field but is experimentally available beam. To that end we have examined (i) how the lateral displacement of the considered beam depends on the modulation depth of the hologram’s phase, and (ii) how the spiraling period of this beam depends on the hologram’s period *P*.

The experiment has been performed using common helium-neon (He-Ne) laser with radiation wavelength of *λ* = 632.8 nm. The hologram was simulated by means of a spatial light modulator (SLM) Holoeye LC-R 2500 with pixel pitch 19 *μ*m, XGA resolution 1024 × 768 pixels, and a palette of 256 phases (8 bits per pixel). The careful analysis revealed that the maximum phase shift introduced by this device was Φ_{max} = *ϕ*
_{255} - *ϕ*
_{0} ≈ 3*π*/2 rad for the radiation of He-Ne laser. Bessel-like beam was formed by the axicon with the apex angle of 172.3° and a material refractive index *n _{a}* = 1.5. Such axicon deflects normaly incident plane waves at an angle

*γ*≈ 33.5 mrad. The sketch of the experiment is outlined in Fig. 1. The laser beam passes through a linear polarizer Pol, a 3 × magnifying telescope, and reflects out from computer driven SLM at approximately 6° angle with respect to the incident beam. The Gaussian beam diameter (FWHM) in front of the SLM was ≈ 6.7 mm. The 4

*f*imaging system composed of two lenses each with focal length of 250 mm was used to image the modulated wavefront onto the axicon entrance plane. The particular attention has been paid to superpose the center of a simulated hologram with the apex angle of the axicon. Finally a set of measurements were carried out to record the light intensity distribution beyond the axicon at the different distances from it.

The registration was performed by means of the lens with the focal length 44 mm and the charge-coupled-device (CCD). The magnification (7.26×) was chosen so that the diameter of the imaged Bessel beam peak was approximately more than 10× the pixel size of the detection camera. A lens and the CCD were mounted on the translation stage which moved along the beam propagation axis. In order to eliminate the artificial beam displacement detection related to the mechanical imperfections of the moving part of the optical setup the two different Bessel beam images were captured at each stage position. The first image was captured for unperturbed Bessel beam and served as reference (SLM worked as ordinary mirror). Then the hologram was simulated and the displaced Bessel beam image was captured. The true beam displacement was obtained from numerical analysis of these two images.

In order to found out how the lateral displacement of the Bessel beam depends on the beam phase modulation, four holograms Φ = Φ_{0}cos(*ϕ* - 2*πρ*/*P*) with different amplitudes Φ_{0} of the light phase modulation have been calculated and tested by the SLM. The theoretical lateral displacement of the Bessel beam was calculated as Δ_{theory} = Φ_{0}/*β* = Φ_{0}
*λ*/(2*πγ*). Corresponding measurements have been performed at the distance approximately 1 cm beyond the axicon, and experimental values of the lateral displacements Δ_{exp.} have been found from subsequent numerical analysis of the captured images. The results are presented in Fig. 2. On the left column the profiles of phase modulation and the phase holograms (insets) are shown. The depth of phase modulation in insets is visualized by means of the different picture contrast. On the right column the experimentally registered intensity distributions are shown. The optical axis is located at the center of each picture, the solid circle indicates the trajectory of the beam peak motion around the optical axis as the beam propagate along it. The calculated Δ_{theory} and measured Δ_{exp} lateral displacements are shown on each graph. One can see that experimental results are in fairly good agreement with predicted values. It is worth to note that as the beam displacement is directly proportional to the depth of phase modulation one cannot exceed the value Δ_{max} = *π*/*β* without significant distortion of the beam structure. The phase is defined unambiguously only for Φ_{0} < *π*, and as soon as we exceed this value the distribution of light intensity become irregular.

Next, three holograms with the same amplitude of phase modulation (Φ_{0} < *π*), but with different spiral period *P* have been calculated and examined. The period *P* determines the distance over which spiraling zero order Bessel beam makes one revolution. For each hologram a set of 208 images were captured and subsequently three dimensional structure of light field was reconstructed. The example of such structure is presented in Fig. 3(b). For clarity only center part of the beam is shown. The period of beam rotation was estimated by the following procedure. First the beam peak position as the function of distance beyond the axicon was extracted in the form *c*(*z*) = *x _{c}*(

*z*) +

*iy*(

_{c}*z*). Then the graph sin(arg(

*c*) +

*ϕ*

_{0}) versus

*z*was plotted [Fig. 3(a)]. The constant phase shift

*ϕ*

_{0}was chosen to approximate experimental data by the function sin(2

*πz*/

*L*

_{exp.}). This allowed us to estimate the quantity of interest

*L*

_{exp.}. The results are presented in Table 1. One can see that the experimental results are in good agreement with values predicted by the theory. In addition it has been verified that the length

*L*

_{exp}does not depend on the Φ

_{0}. However the clear Bessel beam structure was distorted quickly for some critical distances

*z*

_{crit}. beyond the axicon. We assume that the Bessel beam is fully distorted when two peaks with the same intensities are observed. It was confirmed experimentally that greater Φ

_{0}causes smaller

*z*

_{crit.}[15]. For instance for

*P*= 0.5 mm, and Φ

_{0}= 1.7–2.3 rad,

*z*

_{crit.}was found varying approximately from 90 mm to 50 mm. The specific range of Φ

_{0}values was chosen so that

*z*

_{crit.}was within Bessel beam zone (≈ 100 mm).

In summary, by using the conventional axicon and the hologram combination, the spiraling zero-order Bessel like beam was experimentally generated and analyzed for the first time. Obtained results are in a good agreement with the theoretical predictions. The spiral beam potentially can attract interest in the material processing, micro-fabrication, plasma physics, and particles manipulation applications. The higher order spiraling Bessel beams also could be a matter of interest.

## Acknowledgements

The authors acknowledge financial support from the Lithuanian State Science and Studies Foundation project “ConTeX” (contract No. B-30/2009), and are thankful to M. Vengris for providing the SLM, Š. Jablonskas for the manufacturing of the axicon, and students A. Šimkutė, D. Birenis, R. Kasparavičius, V. Šiburskis for their technical assistance.

## References and links

**1. **J. Durnin, J.J. Miceli, and J.H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

**2. **J. Leach, G.M. Gibson, M.J. Padgett, E. Esposito, G. McConnell, A.J. Wright, and J. Girkin, “Generattion of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express **14**, 5581–5585 (2006). [CrossRef] [PubMed]

**3. **J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. **177**, 297–301 (2000). [CrossRef]

**4. **Y.Y. Schechner, R. Piestun, and J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E **54**, R50–R53 (1996). [CrossRef]

**5. **V.V. Kotlyar, V.A. Soifer, and S.N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: Rotating images,” J. Mod. Opt. **44**, 1409–1416 (1997). [CrossRef]

**6. **P. Pääkkänen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, “Experimental demonstration with diffractive optics,” J. Mod. Opt. **45**, 2355–2369 (1998). [CrossRef]

**7. **C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. **124**, 131–140 (1996). [CrossRef]

**8. **J. Rosen, B. Salik, and A. Yariv, “Pseudo-nondiffracting beams generby radial harmonic functions,” J. Opt. Soc. Am. A **12**, 2446–2457 (1995). [CrossRef]

**9. **R. Piestun, B. Spector, and J. Shamir, “Wave fields in three dimensions: Analysis and Sybthesis,” J. Opt. Soc. Am. A **13**, 1837–1848 (1996). [CrossRef]

**10. **S. Ch́vez-Cerda, G.S. McDonald, and G.H.C. New, “Nondiffracting beams: traveling, standing, rotating and spiral waves,” Opt. Commun. **123**, 225–233 (1996). [CrossRef]

**11. **R. Piestun, B. Spector, and J. Shamir, “Unconventional light distributions in tree-dimensional domains,” J. Mod. Opt. **43**, 1495–1507 (1996). [CrossRef]

**12. **C.A. Alonzo, P.J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express **13**, 1749–1760 (2005). [CrossRef] [PubMed]

**13. **R. Piestun and J. Shamir, “Synthesis of three-dimesnional light fields and applications,” Proceedings of the IEEE **90**, 222–244 (2002). [CrossRef]

**14. **P. Polynkin, M. Kolesik, J.V. Moloney, G.A. Siviloglou, and D.N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science **324**, 229–232 (2009). [CrossRef] [PubMed]

**15. **V. Jarutis, A. Matijošius, P. Di Trapani, and A. Piskarskas, “Spiraling zero-order Bessel beam,” Opt. Lett. **34**, 2129–2131 (2009). [CrossRef] [PubMed]