## Vectorial design of super-oscillatory lens |

Optics Express, Vol. 21, Issue 13, pp. 15090-15101 (2013)

http://dx.doi.org/10.1364/OE.21.015090

Acrobat PDF (1736 KB)

### Abstract

A design and optimization method based on vectorial angular spectrum theory is proposed in this paper for the vectorial design of a super-oscillatory lens (SOL), so that the radially polarized vector beam can be tightly focused. The structure of a SOL is optimized using genetic algorithm and the computational process is accelerated using fast Hankel transform algorithm. The optimized results agree well with what is obtained using the vectorial Rayleigh-Sommerfeld diffraction integral. For an oil immersed SOL, a subwavelength focal spot of about 0.25 illumination wavelength without any significant side lobe can be created at a distance of 184.86μm away from a large SOL with a diameter of 1mm. The proposed vectorial design method can be used to efficiently design a SOL of arbitrary size illuminated by various vector beams, with the subwavelength hotspot located in a post-evanescent observation plane.

© 2013 OSA

## 1. Introduction

3. H. Wang, L. 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**(8), 501–505 (2008). [CrossRef]

4. I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. **72**(9), 1287–1291 (1982). [CrossRef]

5. Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A **3**(11), 704–716 (1986). [CrossRef]

6. D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A **311**(2-3), 77–81 (2003). [CrossRef]

7. R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Circularly symmetric apodization via starshaped masks,” Astrophys. J. **599**(1), 686–694 (2003). [CrossRef]

6. D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A **311**(2-3), 77–81 (2003). [CrossRef]

8. J. Liu, J. Tan, and Y. Wang, “Synthetic complex superresolving pupil filter based on double-beam phase modulation,” Appl. Opt. **47**(21), 3803–3807 (2008). [CrossRef] [PubMed]

9. T. Wilson and S. J. Hewlett, “Superresolution in confocal scanning microscopy,” Opt. Lett. **16**(14), 1062–1064 (1991). [CrossRef] [PubMed]

10. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. **11**(5), 432–435 (2012). [CrossRef] [PubMed]

11. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. **9**(3), 1249–1254 (2009). [CrossRef] [PubMed]

10. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. **11**(5), 432–435 (2012). [CrossRef] [PubMed]

18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic systems,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

19. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A **17**(11), 2090–2095 (2000). [CrossRef] [PubMed]

20. R. G. Mote, S. F. Yu, W. Zhou, and Z. F. Li, “Subwavelength focusing behavior of high numerical-aperture phase Fresnel zone plates under various polarization states,” Appl. Phys. Lett. **95**(19), 191113 (2009). [CrossRef]

21. V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. **52**(3), 330–339 (2013). [CrossRef] [PubMed]

3. H. Wang, L. 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**(8), 501–505 (2008). [CrossRef]

22. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

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

## 2. Integral representations based on vectorial angular spectrum theory

21. V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. **52**(3), 330–339 (2013). [CrossRef] [PubMed]

24. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. **32**(18), 2711–2713 (2007). [CrossRef] [PubMed]

25. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. **62**(10), 1195–1201 (1972). [CrossRef]

10. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. **11**(5), 432–435 (2012). [CrossRef] [PubMed]

11. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. **9**(3), 1249–1254 (2009). [CrossRef] [PubMed]

21. V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. **52**(3), 330–339 (2013). [CrossRef] [PubMed]

*x*,

*y*,

*z*) in the observation plane (

*z*>0), can be described, using the vectorial angular spectrum theory [24

24. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. **32**(18), 2711–2713 (2007). [CrossRef] [PubMed]

25. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. **62**(10), 1195–1201 (1972). [CrossRef]

*m*,

*n*,

*q*are the frequency components along x, y, and z directions, and

*q*(

*m*,

*n*) = (1/

*λ*

^{2}-

*m*

^{2}-

*n*

^{2})

^{1/2};

*λ*=

*λ*

_{0}/

*η*with

*λ*

_{0}being the vacuum wavelength, and

*η*the refractive index of the immersion medium;

*q*(

*m*,

*n*) = j(

*m*

^{2}+

*n*

^{2-}1/

*λ*

^{2})

^{1/2}if

*m*

^{2}+

*n*

^{2}>1/

*λ*

^{2};

*A*

_{x}_{,}

*(*

_{y}*m*,

*n*) are the angular spectrums of the electric field components in the mask plane (

*z*= 0), determined by [24

24. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. **32**(18), 2711–2713 (2007). [CrossRef] [PubMed]

25. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. **62**(10), 1195–1201 (1972). [CrossRef]

*E*

_{i}_{,}

*(*

_{x}*x*,

*y*) and

*E*

_{i}_{,}

*(*

_{y}*x*,

*y*) are the multiplications of the illumination electric components and the transmission function of SOL,

*t*(

*x*,

*y*); for a radially polarized beam, in a cylindrical coordinate system, one has [24

**32**(18), 2711–2713 (2007). [CrossRef] [PubMed]

*r*= (

*x*

^{2}+

*y*

^{2})

^{1/2}, and tan

*φ*=

*y*/

*x*;

*g*(

*r*) represents the amplitude distribution of the radially polarized beam, with the waist plane located in the aperture plane of SOL; for the widely used radially polarized Bessel-Gaussian beam,

*g*(

*r*) can be expressed as [3

3. H. Wang, L. 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**(8), 501–505 (2008). [CrossRef]

22. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

*β*

_{0}is the ratio of the mask radius,

*a*, to the beam waist,

*w*

_{0};

*J*

_{1}(·) denotes the first-order Bessel function of the first kind. The equivalent numerical aperture of SOL can also be defined as NA =

*η*sin

*α*, with

*α*being the maximum focusing semi-angle with respect to the z direction. Using Eq. (3), Eq. (2) is rewritten aswhere,

*l*= (

*m*

^{2}+

*n*

^{2})

^{1/2}and

18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic systems,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

*J*(·) is the

_{n}*n*th-order Bessel function of the first kind; Eq. (5) reduces towithInserting Eq. (8) into Eq. (6), and again using Eq. (7),

*E*

_{x}_{,}

*into*

_{y}*E*

_{r}_{,}

*in the transverse plane, via [22*

_{φ}22. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

*E*and

_{r}*E*have been expressed as the first-order and zeroth-order Hankel transforms, respectively; the radially, longitudinally polarized electric energy densities are calculated using |

_{z}*E*(

_{r}*r*,

*z*)|

^{2}and |

*E*(

_{z}*r*,

*z*)|

^{2}, while the total electric energy density (or light intensity) is calculated using |

**(**

*E**r*,

*z*)|

^{2}= |

*E*(

_{r}*r*,z)|

^{2}+ |

*E*(

_{z}*r*,

*z*)|

^{2}.

*E*(

_{r}*r*= 0,

*z*) = 0, and the on-axis electric field becomes purely longitudinally polarized.

*g*(

*r*) = exp(-

*r*

^{2}/

*w*

_{0}

^{2}), with

*w*

_{0}being the waist radius of the Gaussian beam in the aperture plane of SOL. It is obvious that when

*φ*=

*π*/2 or 3

*π*/2,

*E*(

_{z}*r*,

*φ*,

*z*) = 0, i.e., the longitudinally polarized component vanishes along the y direction. When Eq. (13) is compared with the result obtained using the scalar angular spectrum theory [10

**11**(5), 432–435 (2012). [CrossRef] [PubMed]

*E*, disappears in the regime of a scalar theory, and thus the circular symmetry of the transverse light field is maintained; however, when

_{z}*E*in Eq. (13) becomes pronounced, the vectorial nature of the incident beam must be considered. The derived integral formula, Eq. (13), is consistent with that in [21

_{z}**52**(3), 330–339 (2013). [CrossRef] [PubMed]

**7**(2), 77–87 (2000). [CrossRef] [PubMed]

18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic systems,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

## 3. Optimization of super-oscillatory lens for super-resolution focusing

*E*, is remarkably suppressed for the former with a high-NA SOL. The tighter focusing effect is similar to that in the lens-based optical system with a radially polarized beam [3

_{r}**2**(8), 501–505 (2008). [CrossRef]

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

**11**(5), 432–435 (2012). [CrossRef] [PubMed]

11. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. **9**(3), 1249–1254 (2009). [CrossRef] [PubMed]

*d*

_{0}restrains the full-width-at-half-maximum (FWHM) of the subwavelength hotspot (central main lobe); the radial width of the transition dark region between the central main lobe and surrounding large side lobes is set to be (

*κ*-1)

*d*

_{0}, among which the normalized maximum intensity is constrained to be no larger than 20% of the peak intensity of the central lobe;

*N*is the total number of rings contained in the SOL; the optimal distance is constrained to make the SOL have a high NA, e.g., between 0.60 and 0.95 in air, which is useful for rapidly reaching a subwavelength SOL. Following the basic configuration in [10

**11**(5), 432–435 (2012). [CrossRef] [PubMed]

*z*, it is coded using ten binary digits; the total length of the binary digits contained in one individual is hereby (

*N*+ 10). For a stable and fast convergence, GA is configured using two-point crossover, two-point mutation, and the elite selection strategy. Further, GA is set to hold a population of 40~60, with a crossover probability of 0.8, and a mutation probability of 0.12. It is found through numerical calculations that, using the above configurations, various microscale or macroscale SOLs can be steadily reached within several hundred iterations.

27. A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. **1**(1), 13–15 (1977). [CrossRef] [PubMed]

27. A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. **1**(1), 13–15 (1977). [CrossRef] [PubMed]

*η*= 1) or in oil immersion medium (

*η*= 1.515). The maximum iteration number is set to be 300. In Table 1, four SOLs from microscale to macroscale are optimized to create subwavelength hotspots far beyond the near-field region and all surpassing the Abbe’s diffraction resolution limit of 0.5

*λ*

_{0}/NA. SOL

_{1}is designed in air and the other three are designed in oil immersion medium. In order to describe the SOL (might contain several hundred rings) more concisely, the transmission

*t*is coded from the first ring (innermost) to the

_{i}*N*th ring (outermost) by continuously converting every four successive binary digits into one hexadecimal digit. For SOL

_{1}in Table 1, the transmissions of the first four rings, ‘1110’, has been coded to be ‘E’; while for SOL

_{2}, the first hexadecimal digit ‘9’ denotes the real transmissions of ‘1001’.

*D*and Δ

*r*represent the diameter of the SOL and the annular width (as also the minimum feature size), respectively.

*z*denotes the distance at which the optimal focal pattern is found. The diffraction patterns corresponding to SOL

_{o}_{1}and SOL

_{4}have been plotted in Figs. 2(a) and 2(b), respectively. The electric energy density distributions along the radial direction are shown in Figs. 2(c) and 2(d), corresponding to SOL

_{1}and SOL

_{4}, respectively. The real transmissions for SOL

_{1}and SOL

_{4}are plotted in Figs. 2(e) and 2(f), respectively; there are 22 and 55 transparent rings contained in SOL

_{1}and SOL

_{4}, respectively. When compared with the example in [10

**11**(5), 432–435 (2012). [CrossRef] [PubMed]

_{2~4}, which is sharper than the result in [10

**11**(5), 432–435 (2012). [CrossRef] [PubMed]

_{2~4}(less than 20% of the central peak intensity), in contrast to a pronounced side lode in [10

**11**(5), 432–435 (2012). [CrossRef] [PubMed]

_{2~4}are much larger (at least 1μm) than 200nm in [10

**11**(5), 432–435 (2012). [CrossRef] [PubMed]

**11**(5), 432–435 (2012). [CrossRef] [PubMed]

*E*. While in the x direction, both

_{x}*E*and

_{x}*E*contribute to the light field; thus, when |

_{z}*E*|/|

_{z}*E*| becomes pronounced, the circular symmetry of the hotspot is broken. The transverse focal pattern is generally in a ‘bone’ shape [21

_{x}**52**(3), 330–339 (2013). [CrossRef] [PubMed]

## 4. Theoretical validations and discussions

*x*=

*r*cos

*φ*-

*ρ*cos

*ψ*, Δ

*y*=

*r*sin

*φ*-

*ρ*sin

*ψ*,

*u*= (Δ

*x*

^{2}+ Δ

*y*

^{2}+

*z*

^{2})

^{1/2}, and the wave number

*k*= 2

*π*/

*λ*.

*g*(

*ρ*) denotes the amplitude distribution of the radially polarized beam, with

*ρ*being the normalized radial coordinate; from Eq. (4), let

*β*

_{0}= 1,

*g*(

*ρ*) = exp(-

*ρ*

^{2})

*J*

_{1}(2

*ρ*). Numerical calculation using Eq. (16) is rather inefficient, entailing extensive data storage as well as long computing time, and so a microscale SOL, i.e., SOL

_{2}, is chosen for comparison. The subwavelength focal spots calculated using Eq. (12) and Eq. (16) are shown in Fig. 3. The results agree well with each other, and the slight divergence among the higher-order side lobes might be caused by the insufficient discretization of the microstructure plate; however, the problem is much alleviated by using the fast Hankel transform algorithm in Section 3.

29. J. Li, S. Zhu, and B. Lu, “The rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture,” Opt. Commun. **282**(23), 4475–4480 (2009). [CrossRef]

29. J. Li, S. Zhu, and B. Lu, “The rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture,” Opt. Commun. **282**(23), 4475–4480 (2009). [CrossRef]

20. R. G. Mote, S. F. Yu, W. Zhou, and Z. F. Li, “Subwavelength focusing behavior of high numerical-aperture phase Fresnel zone plates under various polarization states,” Appl. Phys. Lett. **95**(19), 191113 (2009). [CrossRef]

**253**(1274), 358–379 (1959). [CrossRef]

20. R. G. Mote, S. F. Yu, W. Zhou, and Z. F. Li, “Subwavelength focusing behavior of high numerical-aperture phase Fresnel zone plates under various polarization states,” Appl. Phys. Lett. **95**(19), 191113 (2009). [CrossRef]

30. V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. **32**(24), 3540–3542 (2007). [CrossRef] [PubMed]

*α*is the maximum focusing semi-angle of FZP, with tan

*α*=

*a*/

*f*, and NA =

*η*sin

*α*;

*f*is the focal length of FZP. The transmission function of a binary phase FZP is described aswith

*m*= 0, 1, …,

*N*/2-1; tan

*θ*= (

_{n}*nλf*+

*n*

^{2}

*λ*

^{2}/4)

^{1/2}/

*f*,

*n*= 0, 1, …,

*N*; the phase function takes the formwhich is extracted from the aplanatic assumption [30

30. V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. **32**(24), 3540–3542 (2007). [CrossRef] [PubMed]

*l*

_{0}(

*θ*) denotes the amplitude distribution of a radially polarized beam; as for FZP,

*r*=

*f*tan

*θ*, Eq. (4) thus becomesThe result obtained using Eq. (17) agrees well with that calculated using Eq. (12), as shown in Fig. 4, and the parameters are

*λ*

_{0}= 633nm,

*N*= 16,

*f*= 0.5μm, and

*β*

_{0}= 1. Both are consistent with the rigorous electromagnetic simulation using FDTD [20

**95**(19), 191113 (2009). [CrossRef]

*λ*

_{0}/(2NA). Under a conventional high-NA aplanatic lens system with a radially polarized vector beam, the electric field in the focal region can be expressed as [3

**2**(8), 501–505 (2008). [CrossRef]

**7**(2), 77–87 (2000). [CrossRef] [PubMed]

*l*

_{0}(

*θ*) should be written asas the radial distance

*r*=

*f*sin

*θ*in the pupil plane, to obey the sine condition [3

**2**(8), 501–505 (2008). [CrossRef]

*η*= 1.515), FWHM of the radial focal spot is 0.68

*λ*

_{0}and 0.50

*λ*

_{0}, respectively. The subwavelength spots in Table 1 are about half of the above results. It should also be indicated that, SOL provides a practical way to beat the theoretical limit of 0.36

*λ*

_{0}/NA, which can be obtained under the known far-field lens- or mirror-based focusing systems [20

**95**(19), 191113 (2009). [CrossRef]

30. V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. **32**(24), 3540–3542 (2007). [CrossRef] [PubMed]

31. T. Grosjean, D. Courjon, and C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. **32**(8), 976–978 (2007). [CrossRef] [PubMed]

## 5. Conclusion

**2**(8), 501–505 (2008). [CrossRef]

## Acknowledgments

## References and links

1. | G. T. D. Francia, “Super-gain antennas and optical resolving power,” Nuovo Cim. |

2. | M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. |

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

4. | I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. |

5. | Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A |

6. | D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A |

7. | R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Circularly symmetric apodization via starshaped masks,” Astrophys. J. |

8. | J. Liu, J. Tan, and Y. Wang, “Synthetic complex superresolving pupil filter based on double-beam phase modulation,” Appl. Opt. |

9. | T. Wilson and S. J. Hewlett, “Superresolution in confocal scanning microscopy,” Opt. Lett. |

10. | E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. |

11. | F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. |

12. | F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanoscale array,” Appl. Phys. Lett. |

13. | M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. Math. Gen. |

14. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

15. | X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. |

16. | Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. |

17. | E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science |

18. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic systems,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

19. | P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A |

20. | R. G. Mote, S. F. Yu, W. Zhou, and Z. F. Li, “Subwavelength focusing behavior of high numerical-aperture phase Fresnel zone plates under various polarization states,” Appl. Phys. Lett. |

21. | V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. |

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

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

24. | D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. |

25. | W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. |

26. | D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning, first ed., Addison-Wesley Professional, Boston, 1989. |

27. | A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. |

28. | R. K. Luneburg, Mathematical theory of optics, University of California Press, Berkeley, 1966. |

29. | J. Li, S. Zhu, and B. Lu, “The rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture,” Opt. Commun. |

30. | V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. |

31. | T. Grosjean, D. Courjon, and C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. |

**OCIS Codes**

(050.1380) Diffraction and gratings : Binary optics

(050.1940) Diffraction and gratings : Diffraction

(100.6640) Image processing : Superresolution

(260.5430) Physical optics : Polarization

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 19, 2013

Revised Manuscript: May 21, 2013

Manuscript Accepted: May 28, 2013

Published: June 17, 2013

**Virtual Issues**

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

**Citation**

Tao Liu, Jiubin Tan, Jian Liu, and Hongting Wang, "Vectorial design of super-oscillatory lens," Opt. Express **21**, 15090-15101 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-13-15090

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### References

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