## Longitudinal polarized focusing of radially polarized sinh-Gaussian beam |

Optics Express, Vol. 21, Issue 11, pp. 13193-13198 (2013)

http://dx.doi.org/10.1364/OE.21.013193

Acrobat PDF (1314 KB)

### Abstract

We consider the focusing performance of a radially polarized sinh-Gaussian beam. The sinh-Gaussian beam can be considered as superposition of a series of eccentric Gaussian beam. Based on the Richards-Wolf formulas, high beam quality and subwavelength focusing are achieved for the radially polarized incident sinh-Gaussian beam. Therefore, sinh-Gaussian beam can be applied in the focusing system with high numerical aperture to achieve focusing with superresolution.

© 2013 OSA

## 1. Introduction

1. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. **43**, 4322–4327 (2004) [CrossRef] [PubMed] .

4. X. P. Li, Y. Y. Cao, and M. Gu, “Superresolution-focal-volume induced 3.0 Tbytes/disk capacity by focusing a radially polarized beam,” Opt. Lett. **36**, 2510–2512 (2011) [CrossRef] [PubMed] .

5. M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. **22**, 1905–1907 (1997) [CrossRef] .

*λ*/NA, where

*λ*is the wavelength of incident light. However, in order to obtain small spot size, the wavelength

*λ*can not be shortened unlimitedly and the effective NA is limited by lens structure even for immersion lens system. Therefore, in the case of special wavelength

*λ*and NA of optical imaging system, achieving a higher lateral resolution of focusing spot of an objective or focusing optical system is still challenging in the field of optical direct writing, optical data storage and so on [6

6. H. Wang, L. Shi, B. Luḱyanchuk, 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] .

8. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. **36**, 4335–4337 (2011) [CrossRef] [PubMed] .

5. M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. **22**, 1905–1907 (1997) [CrossRef] .

6. H. Wang, L. Shi, B. Luḱyanchuk, 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. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express **7**, 77–87 (2000) [CrossRef] [PubMed] .

10. L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. **191**, 161–172 (2001) [CrossRef] .

5. M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. **22**, 1905–1907 (1997) [CrossRef] .

6. H. Wang, L. Shi, B. Luḱyanchuk, 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. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express **7**, 77–87 (2000) [CrossRef] [PubMed] .

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

1. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. **43**, 4322–4327 (2004) [CrossRef] [PubMed] .

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

14. Q.F. Tan, K. Cheng, Z.H. Zhou, and G.F. Jin, “Diffractive superresolution elements for radially polarized light,” J. Opt. Soc. Am. A **27**, 1355–1360 (2010) [CrossRef] .

*et al*firstly experimentally demonstrated that a radially polarized incident beam can be focused to a spot size significantly smaller than for linear polarization [11

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

**2**, 501–505 (2008) [CrossRef] .

**2**, 501–505 (2008) [CrossRef] .

15. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A **24**, 1793–1798 (2007) [CrossRef] .

18. K. Huang, P. Shi, G. W. Cao, X. Ke Li, B. Zhang, and Y. P. Li, “Vector-vortex Bessel-Gauss beams and their tightly focusing properties,” Opt. Lett. **36**, 888–890 (2011) [CrossRef] [PubMed] .

19. Q.G. Sun, K.Y. Zhou, G.Y. Fang, G.Q. Zhang, Z.J. Liu, and S.T. Liu, “Hollow sinh-Gaussian beams and their paraxial properties,” Opt. Express **20**, 9682–9691 (2012) [CrossRef] [PubMed] .

20. 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] .

## 2. Hollow sinh-Gaussian beams and Richards-Wolf formulas

19. Q.G. Sun, K.Y. Zhou, G.Y. Fang, G.Q. Zhang, Z.J. Liu, and S.T. Liu, “Hollow sinh-Gaussian beams and their paraxial properties,” Opt. Express **20**, 9682–9691 (2012) [CrossRef] [PubMed] .

*m*(

*m*= 0, 1, 2, ···) is the order of the hollow sinh-Gaussian beam. Obviously, for

*m*= 0, the beam governed by Eq. (1) is conventional Gaussian beam. However, a new kind of sinh-Gaussian beam is obtained when

*m*is greater than 1.0.

*θ*is determined by NA of objective lens and 0 ≤

*θ*≤ arcsin(NA/

*n*). Where,

*n*= 1.0 is the index of refraction of free space. Based on series, Eq. (1) can be considered as the sum of hollow Gaussian beam. Number of hollow Gaussian beam and its characteristic are determined by parameter

*m*and

*ω*

_{0}, simultaneously.

*ω*

_{0}and

*m*. In order to describe the relation of the amplitude distribution and the parameters of sinh-Gaussian intuitively, the normalized amplitude distribution of sinh-Gaussian with different value of parameter

*ω*

_{0}and

*m*are shown in Fig. 1 for focusing lens with NA = 0.95. It is easy to see that the position of maximal amplitude of hollow sinh-Gaussian beam is shifted to right while the value of

*ω*

_{0}or

*m*are increasing. Therefore, one can control the amplitude distribution of hollow sinh-Gaussian beam by choosing

*ω*

_{0}or

*m*reasonably.

*z*= 0 of high NA objective lens can be calculated by using the Richards-Wolf formulas [20

20. 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] .

**2**, 501–505 (2008) [CrossRef] .

*E*(

_{r}*r*,

*z*) and

*E*(

_{z}*r*,

*z*) are radial and longitudinal electric field component near the focus

*z*= 0, respectively. Considering that

*E*(

_{r}*r*,

*z*) and

*E*(

_{z}*r*,

*z*) are orthogonal, the total electric energy density is given as

*η*is defined as

*η*= Φ

*/(Φ*

_{z}*+ Φ*

_{z}*), where*

_{r}*i*=

*r*and

*z*) and

*r*

_{0}is the first zero point in the distribution of focusing electric density at focal plane, respectively. However, in order to include total energy of incident beam, the upper range of integration is taken as 4

*λ*, which is larger than

*r*

_{0}, for the calculation of

*η*in this paper.

## 3. Focusing property of a radially polarized sinh-Gaussian beam

*n*= 1 is the index of refraction of free space. Therefore,

*α*is approximately equal to 71.8° for Eqs. (2) and (3). Based on Eqs. (2) and (3), the focusing performance of the radially polarized sinh-Gaussian beam is shown in Fig. 2. Figures 2(a1), 2(b1) and 2(c1) display the lateral normalized intensity distribution on focal plane

*z*= 0. From Figs. 2(a1), 2(b1) and 2(c1), one can see that focusing is achieved in Figs. 2(b1) and 2(c1). In the case of (0.125, 4), lens with NA = 0.95 fail to focus the incident radially polarized sinh-Gaussian beam at the focal spot. As shown in Figs. 2(a2)–2(a4), 2(b2)–2(b4), and 2(c2)–2(c4), one can easily know that the total focused field is determined by radial and longitudinal components. The lateral intensity distribution of radial polarized component has a intensity null on the axial or on focal plane. Moreover, the position of maximum of radial component is most near the axis. However, the longitudinal component has a intensity maximum on the axis. Therefore, the maximal intensity of radial and longitudinal component directly determine the distribution and spot size of focused field. If the maximum of intensity occurs in radial component, one can not obtain focusing at focal spot, which shown as black curve in Fig. 2(a1) or contour plot of Fig. 2(a4). Otherwise, one can achieve beam focusing at focal plane. The amplitude distribution of cross section of incident sinh-Gaussian beams used in Fig. 2 are shown in Fig. 1. The black, red curve in Fig. 1(b), and the pink curve in Fig. 1(a) are governed by (0.125, 4), (0.25, 4) and (0.25, 8), respectively. Obviously, the radius of incident sinh-Gaussian beam is increasing gradually from Figs. 2(a1), to 2(b1), and to 2(c1). In Figs. 2(a1)–(a3), the incident beam is described as black curve in Fig.1(b). Obviously, the amplitude of sinh-Gaussian beam is approximately zero, when

*θ*is larger than 0.6 rad. Therefore, it can be considered as focusing by lens with low NA. In the view of transfer function, the incident beam described by black curve in Fig. 1(b) include much of components with low frequency. Therefore, the focusing field exhibits large lateral width at focal plane. The high frequency components of incident beam is increasing for large

*ω*

_{0}and

*m*in Eq. (1). Therefore, the circular focusing spot appears and the lateral size of the focusing spot is decreasing for large

*ω*

_{0}and

*m*.

*J*

_{1}(0) is zero. It is shown by red solid curves in Figs. 2(a1), 2(b1) and 2(c1). The maximal intensity don’t locate at focal spot will enlarge the size of focused spot. Meanwhile, in Eq. (3),

*E*(0, 0) is nozero. Therefore, the focusing is determined by the weight of longitudinal component. Obviously, numerical results indicate that the weight of longitudinal electric field component at focused spot will increase for large value of parameters (

_{z}*m*,

*ω*

_{0}). Usually, longitudinal electric field component dominates the focusing field at focal plane, while

*m*> 4 and

*ω*

_{0}> 0.25 are satisfied, simultaneously. Actually, for fixed value of

*m*(or

*ω*

_{0}), the focused field and subwavelength focused spot can be achieved if large value of

*ω*

_{0}(or

*m*) is selected. However, extreme large value of

*m*and

*ω*

_{0}is not recommended based on the numerical results. It is worth noting that incident energy should be confined in the pupil of lens when one determined the value of parameter of sinh-Gaussian beam. Therefore, considering the previous designed annual filter, most incident energy of sinh-Gaussian beam lie in the region near the edge of the lens.

*ω*

_{0}and

*m*of incident sinh-Gaussian beam. In order to further elaborate the focusing performance of sinh-Gaussian beam focused by the lens with large NA, the focusing characteristics, such as FWHM and beam quality

*η*, are tabulated in Table 1 for different value of (

*ω*

_{0},

*m*). In the case of (0.125, 4) and (0.250, 2), the maximal amplitude of incident sinh-Gaussian beam is near the origin of coordinate and the focusing spot is vanished at focal plane

*z*= 0. Therefore, in this case, the FWHM of focusing field can’t be defined. It is obvious that the focusing field with null central intensity appears for small value of (

*ω*

_{0},

*m*). On the other hand, for fixed value of

*ω*

_{0}(or

*m*), the beam quality

*η*is increasing while

*m*(or

*ω*

_{0}) is increasing. Therefore, in order to obtain longitudinal polarized focusing, larger

*ω*

_{0}or

*m*is essential in experiment. The focusing characteristics of incident beam with large

*ω*

_{0}and

*m*is similar with that of beam with annulus apodizer [21

21. K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express **18**, 4518–4525 (2010) [CrossRef] [PubMed] .

*λ*and beam quality

*η*of 83.4%. Moreover, superresolution is achieved in the case. Obviously, the FWHM and beam quality are superior to that in the case of (0.25, 6). It is more important that radial polarized sinh-Gaussian can be adopted to realize focusing with superresolution.

**2**, 501–505 (2008) [CrossRef] .

17. Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A **29**, 2439–2443 (2012) [CrossRef] .

*λ*, 0.68

*λ*and 0.78

*λ*for Bessel, Bessel-Gaussian [6

**2**, 501–505 (2008) [CrossRef] .

*λ*in Fig. 2(c1). Although the FWHM is 1.13

*λ*for sin-Gaussian beam with (0.25, 4), small FWHM can be achieved as shown in Tab. 1. Therefore, the radial sinh-Gaussian beam show the superiority in focusing with superresolution.

## 4. Conclusion

*m*and

*ω*

_{0}, their focusing characteristics are similar with that in focusing system with small NA. However, for large

*m*and

*ω*

_{0}, the focusing performance is similar with that of beam with annulus apodizer. One can obtain sub-wavelength focusing spot and overcome diffraction limit. The profile of sinh-Gaussian beam is similar with hollow Gaussian beam for large

*m*and

*ω*

_{0}. Therefore, focusing performance of sinh-Gaussian beam can be considered as that of annular pattern of incident beam and the sinh-Gaussian beam can be introduced to achieve super resolution.

## Acknowledgments

## References and links

1. | C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. |

2. | T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. |

3. | V. V. Kotlyar, S. S. Stafeev, L. O’Faolain, and V. A. Soifer, “Tight focusing with a binary microaxicon,” Opt. Lett. |

4. | X. P. Li, Y. Y. Cao, and M. Gu, “Superresolution-focal-volume induced 3.0 Tbytes/disk capacity by focusing a radially polarized beam,” Opt. Lett. |

5. | M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. |

6. | H. Wang, L. Shi, B. Luḱyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics |

7. | Y. J. Zhang and J. P. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express |

8. | K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. |

9. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express |

10. | L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. |

11. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light
beam,” Phys. Rev. Lett. |

12. | G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express |

13. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

14. | Q.F. Tan, K. Cheng, Z.H. Zhou, and G.F. Jin, “Diffractive superresolution elements for radially polarized light,” J. Opt. Soc. Am. A |

15. | Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A |

16. | S. Vyas, M. Niwa, Y. Kozawa, and S. Sato, “Diffractive properties of obstructed vector Laguerre-Gaussian beam under tight focusing condition,” J. Opt. Soc. Am. A |

17. | Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A |

18. | K. Huang, P. Shi, G. W. Cao, X. Ke Li, B. Zhang, and Y. P. Li, “Vector-vortex Bessel-Gauss beams and their tightly focusing properties,” Opt. Lett. |

19. | Q.G. Sun, K.Y. Zhou, G.Y. Fang, G.Q. Zhang, Z.J. Liu, and S.T. Liu, “Hollow sinh-Gaussian beams and their paraxial properties,” Opt. Express |

20. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. A |

21. | K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(140.3300) Lasers and laser optics : Laser beam shaping

(260.5430) Physical optics : Polarization

(350.5030) Other areas of optics : Phase

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 11, 2013

Revised Manuscript: May 14, 2013

Manuscript Accepted: May 14, 2013

Published: May 23, 2013

**Virtual Issues**

Vol. 8, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Jie Lin, Yuan Ma, Peng Jin, Graham Davies, and Jiubin Tan, "Longitudinal polarized focusing of radially polarized sinh-Gaussian beam," Opt. Express **21**, 13193-13198 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-11-13193

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### References

- C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt.43, 4322–4327 (2004). [CrossRef] [PubMed]
- T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun.272, 314–319 (2007). [CrossRef]
- V. V. Kotlyar, S. S. Stafeev, L. O’Faolain, and V. A. Soifer, “Tight focusing with a binary microaxicon,” Opt. Lett.36, 3100–3102 (2011). [CrossRef] [PubMed]
- X. P. Li, Y. Y. Cao, and M. Gu, “Superresolution-focal-volume induced 3.0 Tbytes/disk capacity by focusing a radially polarized beam,” Opt. Lett.36, 2510–2512 (2011). [CrossRef] [PubMed]
- M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett.22, 1905–1907 (1997). [CrossRef]
- H. Wang, L. Shi, B. Luḱyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2, 501–505 (2008). [CrossRef]
- Y. J. Zhang and J. P. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express17, 3698–3706 (2009). [CrossRef] [PubMed]
- K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett.36, 4335–4337 (2011). [CrossRef] [PubMed]
- K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express7, 77–87 (2000). [CrossRef] [PubMed]
- L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun.191, 161–172 (2001). [CrossRef]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003). [CrossRef] [PubMed]
- G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express16, 4567–4581 (2008). [CrossRef] [PubMed]
- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1, 1–57 (2009). [CrossRef]
- Q.F. Tan, K. Cheng, Z.H. Zhou, and G.F. Jin, “Diffractive superresolution elements for radially polarized light,” J. Opt. Soc. Am. A27, 1355–1360 (2010). [CrossRef]
- Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A24, 1793–1798 (2007). [CrossRef]
- S. Vyas, M. Niwa, Y. Kozawa, and S. Sato, “Diffractive properties of obstructed vector Laguerre-Gaussian beam under tight focusing condition,” J. Opt. Soc. Am. A28, 1387–1394 (2011). [CrossRef]
- Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A29, 2439–2443 (2012). [CrossRef]
- K. Huang, P. Shi, G. W. Cao, X. Ke Li, B. Zhang, and Y. P. Li, “Vector-vortex Bessel-Gauss beams and their tightly focusing properties,” Opt. Lett.36, 888–890 (2011). [CrossRef] [PubMed]
- Q.G. Sun, K.Y. Zhou, G.Y. Fang, G.Q. Zhang, Z.J. Liu, and S.T. Liu, “Hollow sinh-Gaussian beams and their paraxial properties,” Opt. Express20, 9682–9691 (2012). [CrossRef] [PubMed]
- B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. A253, 358–379 (1959). [CrossRef]
- K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express18, 4518–4525 (2010). [CrossRef] [PubMed]

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