## Abstract

In this paper, the general formula for tightly focusing radially polarized beams (RPB) superposed with off-axis vortex arrays is derived based on Richard-Wolf vector diffraction theory. The off-axis vortex breaks the rotational symmetry of the energy flow along the axial direction and leads to the spatial redistribution of intensity within the focal plane. The dependence of the consequent focal intensity redistribution on the off-axis distance of vortices as well as the numerical aperture of the lens is theoretically studied. Based on this intriguing feature, generation of equilateral-polygon-like flat-top focus (EPFF) with a flat-top area on the level of sub-*λ*^{2} is realized. The demonstrated method provides new opportunities for focus shaping and holds great potentials in optical manipulation and laser fabrication.

© 2017 Optical Society of America

## 1. Introduction

Owing to their peculiar properties, vector beams with cylindrical symmetry in polarization such as radially polarized beams (RPB) and azimuthally polarized beams (APB) emerge as a research hot topic [1]. As one of their most pronounced properties, enriched focal field components with different strengths and characteristic spatial distributions can be generated under tightly focused condition by a high numerical aperture (NA) objective. For example, it has been shown that the focal field of a tightly focused RPB is dominant by a strong longitudinal field component with a focal area approaching 0.16*λ*^{2}, significantly smaller than 0.26*λ*^{2} of a linearly polarized beam [2]. In addition to a sub-diffraction focal area, configuring the relative weighting factor between the longitudinal and transverse field components, a three-dimensional orientated focal polarization can even be obtained [3]. These exceptional properties by focusing vector beams have mediated a variety of advanced applications including high resolution microscopy [4], optical trapping [5, 6], electron acceleration [7] and optical data storage [8].

The fusion with beam engineering techniques including pupil plane phase or amplitude masks offers additional flexibility to further advance focal manipulation and focus shaping. Flat-top focus [9], optical needle [10], optical chain [11] and multiple focal spots [12,13] have been successfully demonstrated using a diffractive optical element with several concentric phase or amplitude modulated zones at the pupil plane. In addition, optical vortex carrying spiral phase distributions can break the rotational symmetry of polarization distribution of cylindrical polarized vector beams allowing versatile means for focus shaping. Complicated focus shaping such as optical hole [14], subwavelength hollow channel focus and multiple circularly polarized focal spots along the optical axis [15], subwavelength transversely polarized focal needle [16] can be generated by tightly focusing cylindrical vector beams concentrically superposed with an optical vortex. However, this kind of beam shaping method is restricted to only produce circular-symmetric focus patterns. Generation of non-circular-symmetric fields or even arbitrary shaped fields to meet diverse demands from different aspects such as material processing, particle manipulation and optical imaging remains elusive.

In this paper, we demonstrate the generation of non-circular-symmetric equilateral-polygon-like flat-top focus (EPFF) by tightly focusing a RPB superposed with off-axis vortex arrays. The general formula for tightly focusing RPBs superposed with off-axis vortex arrays is derived based on Richard-Wolf vector diffraction theory. The dependence of focusing characteristics on the position of off-axis vortices nested in the incident beam and NA of objective lens is theoretically studied. By judiciously adjusting the arrangement and positions of off-axis vortices in the incident beam, multiple EPFF with variant non-circular-symmetric shapes can be realized.

## 2. Principle

Without loss of generality, a triangle-shaped EPFF is taken as an example to illuminate the principle. The focus shaping setup is schemed in Figs. 1(a)-1(c). Figure 1(a) shows the arrangement of the three off-axis vortices. A linearly polarized Gaussian beam is first incident to the designed phase plate arranged with three off-axis vortices without changing the polarization state of the beam, and is converted to a RPB through a polarization converter, which is then tightly focused by an objective lens, as schemed in Fig. 1(c). Asymmetry energy flow in the axial direction induced by off-axis vortices leads to the intensity redistribution towards the designed EPFF at the focal plane [Fig. 1(b)].

The incident RPB can be expressed as

*w*is the waist radius of the incident Gaussian beam, (

*ρ, ϕ*) is the polar coordinate in the object space and ${\widehat{e}}_{\rho}$ is the unit vector along the radial direction.

*ρ*can be obtained by $\rho \text{=}f\mathrm{sin}\theta $ where

*f*is the focal length of the objective lens.

Assume an off-axis vortex is centered at the point P_{1} in the object space and it can be expressed as *ρ*_{1}e^{j}^{φ}^{1}. In this case the incident beam carrying the vortex can be expressed as [17]:

*m*

_{1}is the topological charge (TC) of the vortex.

Similarly, the expression of the incident beam carrying multiple off-axis vortices, whose locations are centered at P_{1}, P_{2}, …, P_{N} with TC are *m*_{1}, *m*_{2}, …, *m*_{N}, respectively, can be given as

*A*

_{n}denotes the coefficient determined by the product of all terms in Eq. (3).

The focal field components can be obtained according to the Richards–Wolf vector diffraction theory [18]:

_{0}= 1 is the refraction index in the image space.

Substituting Eq. (1) and Eq. (4) into Eq. (6), we obtained the expression of tightly focused RPB superposed with multiple vortices with different off-axis locations and variant TCs as

*l = M-n*, and the

*J*(

_{l}*kr*sin

*θ*) is the

*l*-th order Bessel function of the first kind.

## 3. Focal characteristics

From Eq. (8) to Eq. (10), each field components in the focal region can be obtained. Three off-axis vortices locate equally at a distance with respect to the pupil center with an identical TC of *m* = 1. To unveil the influence of the location of off-axis vortices with respect to the pupil center and NA of the objective lens on the focal fields, we study the intensity distribution along the *y* axis in the focal plane. The Poynting vector field is also discussed to reveal the focusing energy flow under the influence of off-axis vortices.

#### 3.1 Influence of the off-axis distance r_{0} of vortices

Figure 2(a) shows the relationship between the off-axis distance *r*_{0} of three vortices in the incident beam and the focal intensity distribution along the *y* axis under the condition of NA = 0.2, 0.4, 0.6 and 0.8, respectively. When NA = 0.2, the intensity distribution along *y*<0 experiences an attenuation and then is followed by a restoration as *r*_{0} increases. There is a critical position at about *r*_{0} = 0.6 where the intensity attenuation ends and the restoration begins. Around this critical position, the transverse field slightly decreases in size as *r*_{0} increases. The focal field is ‘expelled’ outside away from the pupil center when *r*_{0} is close to the optical axis and vice versa. It is evident in the transverse intensity distribution along the *y* axis at different *r*_{0} as shown in Fig. 2(a). It is noted that the critical position decreases as the NA of the objective lens increases.

Figure 2(b) shows the transverse field at NA = 0.2 from top to bottom with *r*_{0} = 0.2*w*, 0.4*w*, 0.6*w* and 0.8*w*, respectively. The second and third column corresponds to their radial and azimuthal components, respectively. When the three off-axis vortices are located near the center of the incident RPB, the intensity of the radial component is coupled to the azimuthal component. As vortices move away from the pupil center, the intensity of the radial component increases and that of the azimuthal component reduces. It means the coupling from the radial component to the azimuthal component gradually reduces with the increase of off-axis distance *r*_{0}. In the meantime, the intensity profile of the overall transverse field varies from a ring to three separated lobes gradually [Fig. 2(b)]. When *r*_{0} reaches the critical position of *r*_{0} = 0.6, the off-axis vortices induced focal intensity redistribution is pronounced. These intensity lobes in the transverse component are distinctively isolated and the intensity profile presents triangle shape, which is rotated by an angle compared with the arrangement of the off-axis vortices shown in Fig. 1(a).

#### 3.2 Influence of numerical aperture of the objective lens

The focal intensity distribution along the *y* axis focused by variant NA of the objective lens is studied for three vortices with off-axis distance *r*_{0} = 0.2*w*, 0.4*w*, 0.6*w* and 0.8*w*, respectively, as shown in Fig. 3(a). With the NA increasing, the transverse size of the focus decreases quickly at beginning and gradually saturates due to the nonlinear relationship between the aperture angle and the NA. When *r*_{0} = 0.2*w*, there is also a critical position which intensifies the asymmetry of the intensity distribution for *y* > 0 and *y* < 0. It is worth noting that the critical position also moves to smaller NA as the off-axis distance *r*_{0} increases.

The transverse intensity at *r*_{0} = 0.8*w* focused by an objective lens with NA = 0.2, 0.4, 0.6 and 0.8, respectively, is shown in Fig. 3(b) from top to bottom with different scale. At this case, the critical point almost vanishes. When the NA is small and approaching the critical value, the off-axis vortices induced intensity redistribution is pronounced and the intensity profile of the transverse field follows the triangle shape. When the NA is large and away from the critical value, the influence of the off-axis vortices is reduced and the dominant radial component follows a ring shape. Figure 3(b) also shows the azimuthal component is weakened with increasing of NA for the reason that much more incident RPB is coupled into the longitudinal polarized beam leading to enhanced focusing longitudinal component.

#### 3.3 Energy flow

Besides the analysis of the intensity distribution in the focal plane, the time-averaged Poynting vector field and the energy flow is studied here to unveil insights into the focus shaping by off-axis vortex arrays. According to Maxwell’s equations, the radial, azimuthal and longitudinal components of magnetic fields should follow

*c*is the speed of light in vacuum, the asterisk and Re ($\overrightarrow{E}\times {\overrightarrow{H}}^{\ast}$) denote the operation of complex conjugation and real part of ($\overrightarrow{E}\times {\overrightarrow{H}}^{\ast}$), respectively.

The normalized Poynting vector field and energy flow lines are obtained under the condition of NA = 0.6 and *r*_{0} = 0.1*w*, 0.5*w*, 0.9*w*, respectively as shown in Fig. 4. The critical position under the condition of NA = 0.6 is approximately *r*_{0} = 0.5*w* as indicated in Fig. 2(a). For a small *r*_{0} such as 0.1*w*, the influence of the off-axis vortices on the axial Poynting vector field is negligible, as seen in Fig. 4(a). As *r*_{0} approaches the critical position of *r*_{0} = 0.5*w*, the Poynting vector field in *y*>0 almost disappeared as a consequence of deconstructive interference. In the meantime, energy flow lines change directions which is evidenced in the red frame of Fig. 4(b). When *r*_{0} increases from 0.1*w* to 0.9*w*, the energy flow lines within the yellow frame of Fig. 4(a) change from the concave shape to the convex shape within the red frame of Fig. 4(b), and recovers to the concave shape as shown in the yellow frame of Fig. 4(c). As a consequence of the changed energy flow in the axial direction, it raises the spatial redistribution of intensity in the focal plane.

## 4. Generation of equilateral-polygon-like flat-top focus

From the above analysis, it is clear that off-axis vortices can break the circular symmetry and shape the intensity profile of transverse field components following the arrangement of the off-axis vortices. It should be noted that the proposed approach for the flat-top focus is mainly applied to the radially polarized beam with balanced transverse and longitudinal components in the tight focus. To generate an EPFF, judicious choice of *r*_{0} and NA is of crucial importance to tailor the focus shape and balance the transverse and longitudinal field components.

Figure 5 shows the flat-top regular-triangle-like focus generated by using the configuration of three off-axis vortices arranged in Fig. 1(a) at the condition of NA = 0.7, *r*_{0} = 0.8*w* and *m* = 1. Figures 5(a)-5(c) depict the azimuthal, radial and longitudinal field distribution. Since the longitudinal component exhibits complementary distribution to the radial component with a comparable strength, the total intensity pattern with a flap-top area of 0.31*λ*^{2} is demonstrated as shown in Fig. 5(d). Figures 5(e) and 5(f) show the intensity distribution in the *x*-*z* plane and the *y*-*z* plane, respectively. The flat-top length in Fig. 5(d) along the *x* axis and the *y* axis are 0.63*λ* and 0.60*λ*, respectively.

By using the same method, we can obtain other EPFF as shown in Fig. 6. From left to right, they are, in turn, the arrangement of off-axis vortices in the incident beam, radial component, azimuthal component, longitudinal component and total intensity distribution in the *x*-*y* plane. From top to bottom, the number N of vortices used in tailoring the EPFF are in turn 2, 4, 5 and 6, respectively. It’s apparent that flat-top focus including bar-type-like, square-like, pentagon-like and hexagon-like can be generated.

## 5. Conclusion

In this paper, the generation of non-circular-symmetric equilateral-polygon-like flat-top focus is demonstrated. A general formula for focal field components of a tightly focused RPB superposed with off-axis vortex arrays possessing multi-location and multi-TC has been derived. The location of off-axis vortices with respect to the pupil center is critical to introduce significant focal intensity redistributions for the subsequent focus shaping. In addition, the characteristic field distributions in the focal region are found to be dependent on the NA of the objective lens, which can influence the optimized location of the off-axis vortices. By judiciously choosing the arrangement of the off-axis vortices, off-axis distance and NA of the objective lens, flat-top foci including bar-, triangle-, square-, pentagon- and hexagon-shape have been successfully demonstrated. With non-circular symmetric shapes and sub-wavelength focal area, the proposed method may find potential applications in laser fabrication with controllable-shape and shape-dependent materials [19].

## Funding

National Natural Science Foundation of China (NSFC) (No. 61275133, 61522504).

## References and links

**1. **Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics **1**(1), 1–57 (2009).

**2. **R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**(23), 233901 (2003). [PubMed]

**3. **X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. **3**(8), 998 (2012). [PubMed]

**4. **S. Segawa, Y. Kozawa, and S. Sato, “Resolution enhancement of confocal microscopy by subtraction method with vector beams,” Opt. Lett. **39**(11), 3118–3121 (2014). [PubMed]

**5. **L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. **37**(10), 1694–1696 (2012). [PubMed]

**6. **Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express **18**(10), 10828–10833 (2010). [PubMed]

**7. **J. Xu, Z. J. Yang, J. X. Li, and W. P. Zang, “Electron acceleration by a tightly focused cylindrical vector Gaussian beam,” Laser Phys. Lett. **14**(2), 025301 (2017).

**8. **V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. **6**, 7706 (2015). [PubMed]

**9. **T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express **16**(10), 7203–7213 (2008). [PubMed]

**10. **H. Wang, L. Shi, and B. Lukyanchuk, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics **2**(6), 501–505 (2008).

**11. **Y. Zhao, Q. Zhan, Y. Zhang, and Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. **30**(8), 848–850 (2005). [PubMed]

**12. **X. Wang, L. Gong, Z. Zhu, B. Gu, and Q. Zhan, “Creation of identical multiple focal spots with three-dimensional arbitrary shifting,” Opt. Express **25**(15), 17737–17745 (2017). [PubMed]

**13. **Y. Yu and Q. Zhan, “Creation of identical multiple focal spots with prescribed axial distribution,” Sci. Rep. **5**, 14673 (2015). [PubMed]

**14. **L. Z. Rao, J. X. Pu, Z. Y. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. **41**(3), 241–246 (2009).

**15. **L. Wei and H. P. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. **18**(6), 25–28 (2016).

**16. **F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. **5**, 9977 (2015). [PubMed]

**17. **G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt. **40**(1), 73–87 (1993).

**18. **B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959).

**19. **J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science **325**(5947), 1513–1515 (2009). [PubMed]