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  • Vol. 8, Iss. 7 — Aug. 1, 2013
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Longitudinal polarized focusing of radially polarized sinh-Gaussian beam

Jie Lin, Yuan Ma, Peng Jin, Graham Davies, and Jiubin Tan  »View Author Affiliations


Optics Express, Vol. 21, Issue 11, pp. 13193-13198 (2013)
http://dx.doi.org/10.1364/OE.21.013193


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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

With the development of modern microscopical and micro-fabrication technique, optical imaging system and optical tomography generally call for high lateral resolution [1

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

4

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] .

]. However, when propagation of light wave is perturbed, such as encountering an obstacle or focusing by phase element, diffraction occurs. Based on Abbe’s principle, airy disk is formed at the focal plane of lens [5

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] .

]. For an aplanatic imaging lens with numerical aperture NA, the finite focusing spot size of plane incident light beam is governed by 0.51λ/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

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

].

In order to obtain focusing spot with higher resolution, several methods have been proposed based on far field apodization technique and handling of evanescent waves with near field diffraction structure [5

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

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

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

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

]. For example, binary phase filter has been widely studied and the structured illumination has been applied to optical lithography and confocal microscopy [5

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] .

]. However, as the key property of optical beam, the polarization state of incident light should be considered [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] .

,9

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

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

]. Therefore, based on polarization of light, a variety of filter were proposed to improve focusing characteristics [1

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

, 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] .

, 14

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] .

]. For example, Dorn 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] .

]. For a high NA objective, the needle of longitudinally polarized light was achieved by using binary optics for the first time [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] .

]. For radially polarized beam, the focus of research is the design of the binary phase or complex amplitude apodizers [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] .

]. However, the mode or intensity profile of incident beam only be considered to achieve focusing with high resolution recently. For example, the sharper focal spot was achieved by higher order radially polarized Laguerre-Gaussian beam and hollow Bessel-Gaussian beam [15

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

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] .

].

Recently, one kind of Hermite-sinusoidal-Gaussian beams, which named sinh-Gaussian beam, has been received intensive attentions [19

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] .

]. The similarity of hollow sinh-Gaussian (HsG) beam and higher-order Laguerre-Gaussian beam is the intensity null at center. Therefore, a radially polarized HsG beam will show similar focusing characteristics comparing with higher-order Laguerre-Gaussian beam. In this paper, we demonstrate the focusing performance of radially polarized hollow sinh-Gaussian beam by high NA objective lens based on Richards and Wolf’s theory [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] .

]. Our calculated results indicate that subwavelength focusing can be obtained and one can achieve longitudinal polarized focusing simultaneously.

In Sec. 2, the hollow sinh-Gaussian beam and Richards-Wolf formulas are introduced. The numerical results of the focusing radially polarized sinh-Gaussian beam are analyzed in Sec. 3. Some discussions and conclusion are given in Sec. 4.

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

For a high NA objective lens, the electric field of hollow sinh-Gaussian incident beam at a pupil can be defined as follows [19

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] .

]:
Emω0(θ)=sinhm(sinθω0)exp(sin2θω02),
(1)
where 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 ω0, simultaneously.

Obviously, as defined in Eq. (1), the amplitude distribution of hollow sinh-Gaussian beam is determined by ω0 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 ω0 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 ω0 or m are increasing. Therefore, one can control the amplitude distribution of hollow sinh-Gaussian beam by choosing ω0 or m reasonably.

Fig. 1 Normalized radial electric field distribution at z = 0 for different value of parameter ω0 and m. (a) for ω = 0.25 and (b) for m = 4.

For radially polarized sinh-Gaussian incident beam, the electric field near the focus 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] .

]. For radial polarized sinh-Gaussian beam governed by Eq. (1), the focusing field is simplified as [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] .

]:
Er(r,z)=A0αcos1/2θsin(2θ)Em(θ)J1(krsinθ)exp(ikzcosθ)dθ,
(2)
Ez(r,z)=2iA0αcos1/2θsin2θEm(θ)J0(krsinθ)exp(ikzcosθ)dθ.
(3)
where, Er(r, z) and Ez(r, z) are radial and longitudinal electric field component near the focus z = 0, respectively. Considering that Er(r, z) and Ez(r, z) are orthogonal, the total electric energy density is given as Et2(r,z)=Er2(r,z)+Ez2(r,z). In order to describe the focusing property, the full-width at half-maximum (FWHM) at focal plane will be investigated. Meanwhile, in order to describe longitudinal component of the focused field, the beam quality η is defined as η = Φz/(Φz + Φr), where Φi=2π0r0|Ei(r,0)|2rdr (i = r and z) and r0 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 4λ, which is larger than r0, for the calculation of η in this paper.

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

In order to obtain the focusing property of a radially polarized sinh-Gaussian beam, the numerical aperture of the focusing lens is considered as NA = 0.95 and 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 ω0 and m in Eq. (1). Therefore, the circular focusing spot appears and the lateral size of the focusing spot is decreasing for large ω0 and m.

Fig. 2 Focusing performance of radially polarized sinh-Gaussian beam Emω0 with (ω0, m) focused by lens with NA = 0.95. (a1)–(a4) for (0.125, 4), (b1)–(b4) for (0.25, 4) and (c1)–(c4) for (0.25, 8), respectively. (a1), (b1) and (c1) are the density distribution on the focal plane z = 0. The red, green and black curve represents the radial component, longitudinal component and the total electric energy density, respectively. (a2)–(a4), (b2)–(b4), (c2)–(c4), contour of the electrical energy density distribution on the rz plane. Where, (a2), (b2) and (c2) represent the distribution of the radial field component. (a3), (b3) and (c3) represent the distribution of the longitudinal field component. (a4), (b4) and (c4) represent the distribution of the total field component.

Based on results shown in Fig. 2, one can find that the resolution of focusing spot is increasing with the increasing values of parameters ω0 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 (ω0, 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 (ω0, m). On the other hand, for fixed value of ω0 (or m), the beam quality η is increasing while m (or ω0) is increasing. Therefore, in order to obtain longitudinal polarized focusing, larger ω0 or m is essential in experiment. The focusing characteristics of incident beam with large ω0 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] .

]. For example, in the case of (0.25, 8), focusing spot appears at focal spot with FWHM of 0.47λ 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.

Table 1. Focusing performance of sinh-Gaussian beam with different parameters focused by lens with NA=0.95

table-icon
View This Table

Fig. 3 The density distribution on the focal plane z = 0 for three different radial polarized incident beam. (a) for first order Bessel beam J1, (b) for Bessel-Gaussian beam and (c) for Gaussian beam, respectively.

4. Conclusion

In summary, we have clarified the focusing performance of radial polarized sinh-Gaussian beam by lens with large NA based on the Richards-Wolf’s formulas. The FWHM and beam quality are analyzed for different parameters of sinh-Gaussian beam. High beam quality and subwavelength focusing are achieved, simultaneously. For small m and ω0, their focusing characteristics are similar with that in focusing system with small NA. However, for large m and ω0, 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 ω0. 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

This work was funded by the Program for New Century Excellent Talents in University (Grant No. NECT100059), the National Natural Science Foundation of China (Grants No. 61008039 and 51275111), the Research Fund for the Doctoral Program of Higher Education of China (Grants No. 20102304110006 and 20102302120008) and the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.2012017).

References and links

1.

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

2.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007) [CrossRef] .

3.

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] .

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] .

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] .

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 17, 3698–3706 (2009) [CrossRef] [PubMed] .

8.

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

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] .

11.

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

12.

G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008) [CrossRef] [PubMed] .

13.

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

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] .

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] .

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 28, 1387–1394 (2011) [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] .

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] .

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] .

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

  1. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt.43, 4322–4327 (2004). [CrossRef] [PubMed]
  2. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun.272, 314–319 (2007). [CrossRef]
  3. 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]
  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]
  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. Photonics2, 501–505 (2008). [CrossRef]
  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. Express17, 3698–3706 (2009). [CrossRef] [PubMed]
  8. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett.36, 4335–4337 (2011). [CrossRef] [PubMed]
  9. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express7, 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]
  11. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003). [CrossRef] [PubMed]
  12. 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]
  13. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1, 1–57 (2009). [CrossRef]
  14. 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]
  15. 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]
  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. A28, 1387–1394 (2011). [CrossRef]
  17. Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A29, 2439–2443 (2012). [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. Express20, 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. A253, 358–379 (1959). [CrossRef]
  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. Express18, 4518–4525 (2010). [CrossRef] [PubMed]

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