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

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2013 (1)

2012 (1)

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]

2009 (3)

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

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]

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]

2008 (3)

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater.7(6), 435–441 (2008).
[CrossRef] [PubMed]

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. Photonics2(8), 501–505 (2008).
[CrossRef]

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]

2007 (4)

2006 (2)

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. Math. Gen.39(22), 6965–6977 (2006).
[CrossRef]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science311(5758), 189–193 (2006).
[CrossRef] [PubMed]

2005 (1)

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5(9), 1726–1729 (2005).
[CrossRef] [PubMed]

2003 (3)

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

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A311(2-3), 77–81 (2003).
[CrossRef]

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

2000 (3)

1999 (1)

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun.165(4-6), 267–278 (1999).

1991 (1)

1986 (1)

1982 (1)

1977 (1)

1972 (1)

1959 (1)

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]

1952 (1)

G. T. D. Francia, “Super-gain antennas and optical resolving power,” Nuovo Cim.9(Suppl.), 426–438 (1952).

Andres, P.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun.165(4-6), 267–278 (1999).

Bainier, C.

Berry, M. V.

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. Math. Gen.39(22), 6965–6977 (2006).
[CrossRef]

Brown, T. G.

Carter, W. H.

Chad, J. E.

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]

Chen, Y.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanoscale array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

Chong, C. T.

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. Photonics2(8), 501–505 (2008).
[CrossRef]

Courjon, D.

Cox, I. J.

Deng, D.

Dennis, M. R.

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]

Dorn, R.

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

Francia, G. T. D.

G. T. D. Francia, “Super-gain antennas and optical resolving power,” Nuovo Cim.9(Suppl.), 426–438 (1952).

Golub, I.

Grosjean, T.

Guo, Q.

Hegedus, Z. S.

Hewlett, S. J.

Huang, F. M.

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

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanoscale array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

Javier Garcia de Abajo, F.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanoscale array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

Kalosha, V. P.

Kasdin, N. J.

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

Kotlyar, V. V.

Kovalev, A. A.

Kowalczyk, M.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun.165(4-6), 267–278 (1999).

Leuchs, G.

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

Li, J.

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]

Li, Z. F.

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]

Lindberg, J.

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]

Liu, J.

Liu, Y.

Liu, Z.

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater.7(6), 435–441 (2008).
[CrossRef] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5(9), 1726–1729 (2005).
[CrossRef] [PubMed]

Lu, B.

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]

Lukyanchuk, B.

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. Photonics2(8), 501–505 (2008).
[CrossRef]

Martinez-Corral, M.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun.165(4-6), 267–278 (1999).

Mote, R. G.

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]

Mugnai, D.

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A311(2-3), 77–81 (2003).
[CrossRef]

O’Faolain, L.

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science311(5758), 189–193 (2006).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Pikus, Y.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5(9), 1726–1729 (2005).
[CrossRef] [PubMed]

Popescu, S.

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. Math. Gen.39(22), 6965–6977 (2006).
[CrossRef]

Quabis, S.

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

Ranfagni, A.

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A311(2-3), 77–81 (2003).
[CrossRef]

Richards, B.

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]

Rogers, E. T. F.

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]

Roy, T.

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]

Ruggeri, R.

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A311(2-3), 77–81 (2003).
[CrossRef]

Sarafis, V.

Savo, S.

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]

Sheppard, C.

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. Photonics2(8), 501–505 (2008).
[CrossRef]

Sheppard, C. J. R.

Shi, L.

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. Photonics2(8), 501–505 (2008).
[CrossRef]

Siegman, A. E.

Spergel, D. N.

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

Srituravanich, W.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5(9), 1726–1729 (2005).
[CrossRef] [PubMed]

Stafeev, S. S.

Steele, J. M.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5(9), 1726–1729 (2005).
[CrossRef] [PubMed]

Sun, C.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5(9), 1726–1729 (2005).
[CrossRef] [PubMed]

Tan, J.

Török, P.

Vanderbei, R. J.

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

Varga, P.

Wang, H.

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. Photonics2(8), 501–505 (2008).
[CrossRef]

Wang, Y.

Wilson, T.

Wolf, E.

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]

Youngworth, K. S.

Yu, S. F.

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]

Zapata-Rodriguez, C. J.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun.165(4-6), 267–278 (1999).

Zhang, X.

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater.7(6), 435–441 (2008).
[CrossRef] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5(9), 1726–1729 (2005).
[CrossRef] [PubMed]

Zheludev, N.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanoscale array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

Zheludev, N. I.

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]

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

Zhou, W.

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]

Zhu, S.

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]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanoscale array,” Appl. Phys. Lett.90(9), 091119 (2007).
[CrossRef]

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]

Astrophys. J. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Phys. Math. Gen. (1)

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. Math. Gen.39(22), 6965–6977 (2006).
[CrossRef]

Nano Lett. (2)

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

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5(9), 1726–1729 (2005).
[CrossRef] [PubMed]

Nat. Mater. (2)

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater.7(6), 435–441 (2008).
[CrossRef] [PubMed]

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]

Nat. Photonics (1)

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. Photonics2(8), 501–505 (2008).
[CrossRef]

Nuovo Cim. (1)

G. T. D. Francia, “Super-gain antennas and optical resolving power,” Nuovo Cim.9(Suppl.), 426–438 (1952).

Opt. Commun. (2)

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun.165(4-6), 267–278 (1999).

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]

Opt. Express (1)

Opt. Lett. (5)

Phys. Lett. A (1)

D. Mugnai, A. Ranfagni, and R. Ruggeri, “Pupils with super-resolution,” Phys. Lett. A311(2-3), 77–81 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of subwavelength focusing by a binary super-oscillatory lens with the radially polarized vector beam.

Fig. 2
Fig. 2

Subwavelength focusing by super-oscillatory lens with radially polarized beam: (a), (b) the total electric energy density distribution in optimal focal plane at z = zo; (c), (d) the electric energy density distribution along the radial direction; (e), (f) the transmission function of super-oscillatory lens; (a), (c), (e) for SOL1, and (b), (d), (f) for SOL4, respectively.

Fig. 3
Fig. 3

Comparison of total electric energy density distributions for SOL2 calculated using Eq. (12) and Eq. (16), respectively.

Fig. 4
Fig. 4

Comparison of electric energy density distributions calculated by using Eq. (12) and Eq. (17), respectively.

Tables (1)

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Table 1 Parameter and Performance of Optimized Binary SOLs

Equations (22)

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[ E x (x,y,z) E y (x,y,z) E z (x,y,z) ]= [ A x (m,n) A y (m,n) m A x (m,n)+n A y (m,n) q(m,n) ]exp{ j2π[ mx+ny+q(m,n)z ] }dmdn,
[ A x (m,n) A y (m,n) ]= [ E i,x (x,y,0) E i,y (x,y,0) ]exp[ j2π(mx+ny) ]dxdy ,
[ E i,x (r,φ,0) E i,y (r,φ,0) ]=t(r)g(r)[ cosφ sinφ ],
g(r)=exp( β 0 2 r 2 a 2 ) J 1 ( 2 β 0 r a ),
[ A x (l,ϕ) A y (l,ϕ) ]= 0 0 2π [ E i,x (r,φ,0) E i,y (r,φ,0) ]exp{ j2π[ lrcos(ϕφ) ] }rdrdφ ,
[ E x (r,φ,z) E y (r,φ,z) E z (r,φ,z) ]= 0 0 2π [ A x (l,ϕ) A y (l,ϕ) l A x (l,ϕ)cosϕ+ A y (l,ϕ)sinϕ q(l) ] ×exp{ j2π[ lrcos(ϕφ)+q(l)z ] }ldldϕ.
0 2π cos(nϑ)exp[jρcos(ϑγ)]dϑ =2π j n J n (ρ)cos(nγ) 0 2π sin(nϑ)exp[jρcos(ϑγ)]dϑ =2π j n J n (ρ)sin(nγ)
[ A x (l,ϕ) A y (l,ϕ) ]=j[ cosϕ sinϕ ] A r,1 (l),
A r,1 (l)= 0 t(r)g(r) J 1 (2πlr)2πrdr .
[ E x (r,φ,z) E y (r,φ,z) E z (r,φ,z) ]= 0 [ cosφ J 1 (2πlr) sinφ J 1 (2πlr) j l q(l) J 0 (2πlr) ] A r,1 (l)exp[ j2πq(l)z ]2πldl .
{ E r = E x cosφ+ E y sinφ E φ = E y cosφ E x sinφ
{ E r (r,z)= 0 A r,1 (l)exp[ j2πq(l)z ] J 1 (2πlr)2πldl E φ (r,z)=0 E z (r,z)=j 0 l q(l) A r,1 (l)exp[ j2πq(l)z ] J 0 (2πlr)2πldl
{ E x (r,z)= 0 A x,0 (l)exp[ j2πq(l)z ] J 0 (2πlr)2πldl E y (r,z)=0 E z (r,φ,z)=jcosφ 0 l q(l) A x,0 (l)exp[ j2πq(l)z ] J 1 (2πlr)2πldl
A x,0 (l)= 0 t(r)g(r) J 0 (2πlr)2πrdr ,
min | E( d 0 /2,z;T) E(0,z;T) | 2 s.t. | E(r,z;T) E(0,z;T) | 2 0.2, d 0 rκ d 0 T=[ t 1 , t 2 ,..., t N ]{0,1} 0.60sin[ tan 1 ( a z ) ]0.95
[ E r (r,φ,z) E φ (r,φ,z) E z (r,φ,z) ]= 1 2π 0 0 2π t(ρ)g(ρ)[ z 0 (cosψΔx+sinψΔy) ] × jku1 u 3 exp(jku)ρdρdψ,
{ E r (r,z)= 0 α l 0 (θ) t FZP (θ) ϕ FZP (θ) cos 3/2 θsin(2θ) J 1 (krsinθ)exp(jkzcosθ)dθ E φ (r,z)=0 E z (r,z)=2j 0 α l 0 (θ) t FZP (θ) ϕ FZP (θ) cos 3/2 θ sin 2 θ J 0 (krsinθ)exp(jkzcosθ)dθ
t FZP (θ)={ 1, θ 2m <θ θ 2m+1 1, θ 2m+1 <θ θ 2m+2
ϕ FZP (θ)=exp[ jkf( 1 1 cosθ ) ],
l 0 ( θ )=exp[ β 0 2 ( tanθ tanα ) 2 ] J 1 ( 2 β 0 tanθ tanα ).
{ E r (r,z)= 0 α l 0 (θ) cos 1/2 θsin(2θ) J 1 (krsinθ)exp(jkzcosθ)dθ E φ (r,z)=0 E z (r,z)=2j 0 α l 0 (θ) cos 1/2 θ sin 2 θ J 0 (krsinθ)exp(jkzcosθ)dθ
l 0 ( θ )=exp[ β 0 2 ( sinθ sinα ) 2 ] J 1 ( 2 β 0 sinθ sinα ),

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