## Abstract

We present and demonstrate a novel method for engineering the radial-variant polarization on the incident field to achieve a needle of transversally polarized field without any pupil filters. We generate a new kind of localized linearly-polarized vector fields with distributions of states of polarization (SoPs) describing by the radius to the power *p* and explore its tight focusing, nonparaxial focusing, and paraxial focusing properties. By tuning the power *p*, we obtain the needle-like focal field with hybrid SoPs and give the formula for describing the length of the needle. Experimentally, we systematically investigate both the intensity distributions and the polarization evolution of the optical needle by paraxial focusing the generated vector field. Such an optical needle, which enhances the light-matter interaction, has intriguing applications in optical microma-chining and nonlinear optics.

© 2013 Optical Society of America

## 1. Introduction

In recent years, the needle-like field with a small transverse spot and a long depth of focus (DOF) has received extensively interest because of its fascinating properties and intriguing applications, such as super-resolution imaging, optical nanofabrication, and optical trapping and manipulation [1–7]. During the past few years, researchers have established various methods to create needle-like fields. For examples, Wang *et al*. [8] achieved a needle of longitudinally polarized light by focusing a radially polarized Bessel-Gaussian beam with a combination of a binary-phase optical element and a high-numerical aperture (NA) lens. Yuan *et al*. [9] presented a needle-like field by tightly focusing an azimuthally polarized beam with a high-NA lens and a multibelt spiral phase hologram. Hu *et al*. [10] generated a super-length optical needle by focusing hybridly polarized vector beams with an annular high-NA lens. Kitamura *et al*. [11] demonstrated the needle-like focus generation by radially polarized halo beams. Guo *et al*. [12] reported the optical needle by tightly focusing of a higher-order radially-polarized beam transmitting through multi-zone binary phase pupil filters. In a word, the above-mentioned needle-like fields are achieved by means of transmitting a vector beam through a multibelt pupil filters and then tight focusing by a high-NA lens. It is noteworthy that all these methods suffer from the energy loss of pupil filters and the sidelobe intensity. Hence, it is desirable to enhance the conversion efficiency and to suppress the sidelobe intensity of the needle-like field.

In this work, we present a new method for engineering the radial-variant polarization on the incident field to achieve a needle of transversally polarized field without any pupil filters. To the best of our knowledge, this is the first report of needle-like field with nearly 100% conversion efficiency and negligible sidelobe intensity. Here the conversion efficiency refers to the power ratio of the needle-like field to the generated vector field instead of the linearly polarized beam from the laser. We study the optical needle by tight focusing, nonparaxial focusing, and paraxial focusing of the engineered vector field. Furthermore, we demonstrate both the intensity distributions and the polarization evolution of the needle by paraxial focusing the generated vector field.

## 2. Generation of the radial-variant vector field

Following the same method presented by Wang *et al*. [13–15], we generate the radial-variant vector fields by a common path interferometer implemented with a spatial light modulator (SLM) as shown in Fig. 1. A linearly polarized CW laser beam at a wavelength of 532 nm is expanded and collimated to obtain a nearly uniform-intensity distribution. Then the beam is diffracted by a computer-controlled SLM, in which the additional phase is modeled as *δ*(*r*) = 2*π*(*r/r*_{0})* ^{p}*. Subsequently, the diffracted light with ±1th orders are converted into the left-handed (LH) and right-handed (RH) circularly polarized beams by a pair of orthogonal

*λ*/4 wave plates, respectively. Finally, the designed vector fields are generated in the output plane of the 4

*f*system by combining the ±1th orders with the aid of both the lens and a Ronchi phase grating. It is emphasized that the polarization distribution of the generated vector field depends on the power

*p*of

*r*.

The electric field distribution of the generated radial-variant vector field can be written as

*δ*= 2

*π*(

*r/r*

_{0})

*. Here*

^{p}*A*

_{0}is a constant, circ(·) is the circular function,

*r*is the polar radius,

*r*

_{0}is the radius of the vector field, and the nonnegative integer

*p*is the power.

*ê*and

_{x}*ê*are the unit vectors in the Cartesian coordinate system, respective describing the localized linearly polarized vector fields with their directions of SoPs along the

_{y}*x*and

*y*axes. In the case of

*p*= 0, Eq. (1) describes the linearly polarized scalar field. Note that the radial-variant vector field belongs a kind of localized linearly-polarized field and always exhibits the cylindrical symmetry for any value of

*p*.

It is found that the intensity patterns of all generated vector fields exhibit nearly uniform-intensity distribution and are indistinguishable for different values of *p*. After passing through a polarizer, as shown in Fig. 2, the intensity patterns exhibit the cylindrical symmetry with concentric extinction rings, indicating that the SoPs of the generated vector fields are indeed radial variant. As displayed in Fig. 2, the number of the extinction rings is equal to 2. The radii of the two extinction rings are determined by
${r}_{1}=\sqrt[p]{1/4}{r}_{0}$ and
${r}_{2}=\sqrt[p]{3/4}{r}_{0}$. Specially, the result for *p* = 1 is consistent with the one reported previously [14]. The radius of all the generated vector fields is measured to be *r*_{0} = 2.72 ± 0.03 mm.

## 3. Tight focusing of the radial-variant vector field

It is very interesting to exploit the tight focusing of the generated radial-variant vector field. According to the Richards-Wolf vectorial diffraction method [16, 17] and similar to the investigation in [18], we obtain the three-dimensional (3D) electric field in the focal region of an aplanatic lens as

*ρ*,

*φ*, and

*z*are the polar radius, azimuthal angle, and longitudinal position in the cylindrical coordinate system for the observation point, respectively.

*c*

_{0}=

*πfA*

_{0}

*/λ*,

*k*= 2

*π/λ*,

*α*= sin

^{−1}(

*NA*) is the maximal angle obtained by the NA of the objective lens,

*f*=

*r*

_{0}/

*NA*is the focal length of the high NA objective,

*λ*is the wavelength of the laser,

*β*is the pupil filling factor defining as the radio of the pupil radius to the beam radius, and

*J*(

_{m}*x*) is the Bessel function of the first kind of order

*m*. It is emphasized that the coordinate origin

*z*= 0 is at the lens’ geometrical focus.

To explore the tight focusing properties of the generated vector fields, we perform the numerical simulation using Eq. (2) by taking *β* = 1 and *NA* = 0.8. Figure 3 shows the intensity patterns of the tightly focused radial-variant vector fields with different values of *p* in the XZ plane (*y* = 0). As displayed in Figs. 3(a)–3(c), the intensity forms almost twinborn foci along the optical axis sandwiching a dark region in the lens’ geometrical focus. This is anticipated for the following reasons: (i) the destructive interference among the annular patterns with opposite polarization direction in the cross-section of the radial-variant vector field; (ii) the radial-variant vector field focused by a lens is equivalent to two lenses with different focal lengths [19]. Interestingly, it is found that for *p* = 8 the nearly uniform-intensity nondiffracting light occurs owing to the superposition of multi foci. Consequently, the focal volume exhibits the needle-like profile with negligible sidelobe intensity, as shown in Fig. 3(d). Without any additional optical element, the needle-like field is achieved by tightly focusing of radial-variant vector field, in contrast to the ones reported previously with the aid of multibelt pupil filters [8–12].

Now we study the size of the optical needle generated by the tightly focused vector field with *β* = 1, *p* = 8, and *NA* = 0.8, as shown in Fig. 4. The global intensity profile along the *z*-axis is displayed in Fig. 4(a). We define the DOF as the FWHM of its intensity profile in the longitudinal direction. We find an empirical expression for estimating the length of the optical needle as

*NA*-dependent DOF values by the numerical simulations and Eq. (3) are illustrated in the inset of Fig. 4(a) by circles and the solid line, respectively. Obviously, this empirical formula is applicable for estimating the value of DOF. Figure 4(b) shows the transverse, longitudinal, and global intensities of the tightly focused vector field at the lens’ geometrical focal plane, respectively. It is noteworthy that the longitudinal intensity is very small (∼ 5%) compared with the global intensity. Accordingly, the longitudinal intensity has little influence on the global intensity distribution. That is to say, the achieved needle is a nondiffracting transversally polarized field. The transverse FWHM of the optical needle is nearly uniform and is measured to be 0.93

*λ*for

*NA*= 0.8, which is larger than the diffraction limit for this focusing lens

*λ*/(2

*NA*) = 0.625

*λ*. We obtain the needle-like field by tightly focusing the unperturbed vector field directly, in contrast to the reported optical needles breaking the diffraction limit with the aid of multibelt pupil filters [8–12]. Accordingly, the reported needle-like field has intriguing applications in optical micromachining and nonlinear optics instead of super-resolution imaging.

## 4. Nonparaxial focusing of the radial-variant vector field

Based on the vectorial Rayleigh-Sommerfeld formulas under the weak nonparaxial approximation, similar to the investigations in [20], we obtain the 3D electric field for the nonparaxial focusing of the radial-variant vector field in free space along the +*z* direction as

*γ*=

*kρ/ξ*, $\xi =\sqrt{{(z+f)}^{2}+{\rho}^{2}}$,

*τ*=

*k*/(2

*ξ*) −

*k*/(2

*f*). Note that the coordinate origin

*z*= 0 is at the lens’ geometrical focus and

*r*is the polar radius in the plane of the lens.

To illustrate the nonparaxial focusing of the radial-variant vector field, numerical simulations have been performed using Eq. (4) by taking *λ* = 532 nm, *r*_{0} = 2.5 mm, *f* = 10 mm (or *NA* = 0.25), and *p* = 8. Figures 5(a) and 5(b) show the intensity distributions along the *z* axis (*x* = *y* = 0) and *x* axis (*z* = *y* = 0), respectively. For comparison, the intensity distributions obtained by Eq. (2) are also displayed by the circles in Fig. 5. Obviously, the results obtained by the Rayleigh-Sommerfeld formulas and the Richards-Wolf method are identical, confirming that Eq. (4) holds true under the weak nonparaxial condition (*NA* ≤ 0.3). Analysis of intensity distributions reveals the following: (i) the value of DOF is nonlinear increasing function of *f*, as shown in the inset of Fig. 5(a); (ii) the generated field is nondiffracting over 70*λ* with the transverse FWHM of ∼ 2.8*λ*. Using the conversion relation *f* = *r*_{0}/*NA*, we revise the DOF of the nondiffracting field from Eq. (3) as

## 5. Paraxial focusing of the radial-variant vector field

Under the paraxial approximation, one gets $\sqrt{{(z+f)}^{2}+{\rho}^{2}}\approx z+f$ and ignores the longitudinal component. Accordingly, we determine the paraxial propagation of a radial-variant vector field from Eq. (4) as follows

*η*=

*z*+

*f*and

*μ*=

*k*/(2

*η*) −

*k*/(2

*f*). Apparently, the field distribution is independent of the azimuthal angle

*φ*, indicating that the field preserves the cylindrical symmetry at any propagation position.

Figure 6 illustrates the paraxial intensity distributions of the vector field in the focal region when *λ* = 532 nm, *r*_{0} = 2.72 mm, *f* = 175 mm, and *p* = 8. Figure 6(a) displays the simulation intensity pattern through the lens’ geometrical focus in the XZ plane (*y* = 0). One can see that the paraxial focusing of the engineered radial-variant vector field is uniform and the generated field is nondiffracting over a long range about 8.8 mm. Figures 6(b) and 6(c) show the measured and simulated transverse intensity patterns at planes (1–5) with *d* = 1 mm marked in Fig. 6(a), respectively. Apparently, the experimental results are in good agreement with the numerical simulations, implying that we indeed observed the optical needle generated by the focused radial-variant vector field.

Next, we study the polarization evolution of this nondiffracting field. We rewrite the electric field of the focused vector field from Eq. (6) as

with*σ̂*

_{+}and

*σ̂*

_{−}are the unit vectors of the LH and RH circular polarizations, respectively. In general, the LH and RH circularly polarized components have different amounts (i.e., |

*E*

_{+}| ≠ |

*E*

_{−}|). Hence the focused radial-variant vector field exhibits the hybrid SoPs, as shown in Fig. 7(e). Specially, when

*z*= 0 one gets |

*E*

_{+}| ≡ |

*E*

_{−}| from Eq. (8), suggesting the focused radial-variant vector field with localized linear polarization at the lens’ geometrical focal plane.

To identify the polarization evolution of the optical needle shown in Fig. 6(a), a polarizer is used. As examples, Figs. 7(a)–7(d) show the measured and simulated transverse intensity patterns behind a polarizer at planes (3 and 5) marked in Fig. 6(a). Obviously, the measured results are consistent with the numerical simulations. When a polarizer in the same direction at different planes (1–5), we find that the intensity patterns are distinguishable, as illustrated in Figs. 7(a)–7(d). This suggests that not only the intensity but also the polarization of the optical needle keeps approximately invariant within the range of the depth of focus. By calculating the Stokes parameters, we map the distribution of SoPs at planes (1–5) marked in Fig. 6(a), as shown in Fig. 7(e). It can be seen that the nondiffracting field exhibits the hybrid SoPs mainly originating from the localized elliptical polarization at each observational plane, different from the localized linearly-polarized field at the lens’ geometrical focus. As shown in Fig. 7(e), the chiralities of the intensity patterns at both sides of the lens’ geometrical focus are opposite. For instance, the field at the center part of the observational plane exhibits the RH and LH polarization before and after the lens’ geometrical focus, respectively. Besides, the distribution of SoPs at each observational plane is radial variant and preserves the cylindrical symmetry. For an example, the field on the axis is RH polarization at the plane (1). As the radial distance increases, the ellipticity decreases from positive to negative. Accordingly, the chirality changes from RH polarization to LH polarization.

## 6. Conclusion

In summary, we have proposed a method for engineering the radial-variant polarization on the incident field to achieve a needle of transversally polarized field without any pupil filters, in which both the intensity and the polarization keeps approximately invariant within the range of the depth of focus. We have modeled and generated the radial-variant vector fields with distributions of SoPs describing by *r ^{p}*. We have systematically investigated the propagation behaviors of the focused radial-variant vector fields, including its tight focusing, nonparaxial focusing, and paraxial focusing properties. We have achieved the needle-like focal field by focusing the radial-variant vector field with

*p*= 8 and presented the empirical formula for describing the depth of focus. Under the paraxial focusing, we have demonstrated both theoretically and experimentally that not only the intensity but also the polarization of the optical needle keeps approximately invariant within the range of the depth of focus. Such a nondiffracting transversally polarized field, which enhances both the distance and the intensity of the light-matter interaction, has intriguing applications in optical micromachining and nonlinear optics.

## Acknowledgments

This work was supported by the National Science Foundation of China (Grants: 11174160 and 10947004), Jiangsu Key Laboratory for Optoelectronics (Grant: 1640703061-9), and the Program for New Century Excellent Talents in University (Grant: NCET-10-0503).

## References and links

**1. **J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express **18**, 21965–21972 (2010). [CrossRef] [PubMed]

**2. **J. Lin, K. Yin, Y. Li, and J. Tan, “Achievement of longitudinally polarized focusing with long focal depth by amplitude modulation,” Opt. Lett. **36**, 1185–1187 (2011). [CrossRef] [PubMed]

**3. **J. Wang, W. Chen, and Q. Zhan, “Three-dimensional focus engineering using dipole array radiation pattern,” Opt. Commun. **284**, 2668–2671 (2011). [CrossRef]

**4. **H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express **20**, 14891–14905 (2012). [CrossRef] [PubMed]

**5. **T. Grosjean and I. Gauthier, “Longitudinally polarized electric and magnetic optical nano-needles of ultra high lengths,” Opt. Commun. **294**, 333–337 (2013). [CrossRef]

**6. **L. Guo, C. Min, S. Wei, and X. Yuan, “Polarization and amplitude hybrid modulation of longitudinally polarized subwavelength-sized optical needle,” Chin. Opt. Lett. **11**, 052601 (2013). [CrossRef]

**7. **E. T. F. Rogers, S. Savo, J. Lindberg, T. Roy, M. R. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. **102**, 031108 (2013). [CrossRef]

**8. **H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in
vacuum using binary optics,” Nat. Photonics **2**, 501–505 (2008). [CrossRef]

**9. **G. H. Yuan, S. B. Wei, and X. C. Yuan, “Nondiffracting transversally polarized beam,” Opt. Lett. **36**, 3479–3481 (2011). [CrossRef] [PubMed]

**10. **K. Hu, Z. Chen, and J. Pu, “Generation of super-length optical needle by focusing hybridly polarized vector beams through a dielectric interface,” Opt. Lett. **37**, 3303–3305 (2012). [CrossRef]

**11. **K. Kitamura, M. Nishimoto, K. Sakai, and S. Noda, “Needle-like focus generation by radially polarized halo beams emitted by photonic-crystal ring-cavity laser,” Appl. Phys. Lett. **101**, 221103 (2012). [CrossRef]

**12. **H. Guo, X. Weng, M. Jiang, Y. Zhao, G. Sui, Q. Hu, Y. Wang, and S. Zhuang, “Tight focusing of a higher-order radially polarized beam transmitting through multi-zone binary phase pupil filters,” Opt. Express **21**, 5363–5372 (2013). [CrossRef] [PubMed]

**13. **X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. **32**, 3549–3551 (2007). [CrossRef] [PubMed]

**14. **X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. **105**, 253602 (2010). [CrossRef]

**15. **H. Chen, J. Hao, B. F. Zhang, J. Xu, J. P. Ding, and H. T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. **36**, 3179–3181 (2011). [CrossRef] [PubMed]

**16. **B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II.
Structure of the image field in an aplanatic system,” Proc. R. Soc.
A **253**, 358–379 (1959). [CrossRef]

**17. **K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**, 77–87 (2000). [CrossRef] [PubMed]

**18. **K. Hu, Z. Chen, and J. Pu, “Tight focusing properties of hybridly polarized vector beams,” J. Opt. Soc. Am. A **29**, 1099–1104 (2012). [CrossRef]

**19. **J. W. Goodman, *Introduction to Fourier Optics*, 3rd ed.
(Roberts and Company, 2005), Sec. 5.

**20. **V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A **27**, 372–380 (2010). [CrossRef]