Abstract

An algorithm for calculating the field distribution of a high numerical aperture Fresnel zone plate (FZP) in stratified media is presented, which is based on the vector angular spectrum method. The diffraction problem of FZP is solved for the case of a multilayer film with planar interfaces perpendicular to the optical axis. The solution is obtained in a rigorous mathematical manner and it satisfies the homogeneous wave equations. The electric strength vector of the transmitted and reflected field in the multilayer media is obtained for any polarized beam normally incident onto a binary phase circular FZP. For radially-, azimuthally- and linearly-polarized beam, the electric field in the focal region can be simplified as double or single integral, which can be readily used for numerical computation.

© 2015 Optical Society of America

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    [Crossref]
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2014 (3)

Y. Zhang, C. Zheng, Y. Zhuang, and X. Ruan, “Analysis of near-field subwavelength focusing of hybrid amplitude–phase Fresnel zone plates under radially polarized illumination,” J. Opt. 16(1), 015703 (2014).
[Crossref]

Y. Zhang, C. Zheng, and Y. Zhuang, “Effect of the shadowing in high-numerical-aperture binary phase Fresnel zone plates,” Opt. Commun. 317, 88–92 (2014).
[Crossref]

Y. Zhang, H. An, D. Zhang, G. Cui, and X. Ruan, “Diffraction theory of high numerical aperture subwavelength circular binary phase Fresnel zone plate,” Opt. Express 22(22), 27425–27436 (2014).
[Crossref] [PubMed]

2012 (1)

Y. Zhang and Y. Dai, “Multifocal optical trapping using counter-propagating radially-polarized beams,” Opt. Commun. 285(5), 725–730 (2012).
[Crossref]

2011 (4)

2010 (2)

2009 (4)

2008 (2)

2007 (1)

Y. Zhang, J. Chen, and X. Ye, “Multilevel phase Fresnel zone plate lens as a near-field optical element,” Opt. Commun. 269(2), 271–273 (2007).
[Crossref]

2006 (1)

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
[Crossref]

2005 (2)

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97(5), 053105 (2005).
[Crossref]

Y. Zhang and C. Zheng, “Axial intensity distribution behind a Fresnel zone plate,” Opt. Laser Technol. 37(1), 77–80 (2005).
[Crossref]

2004 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

1999 (3)

H. C. Gerritsen and C. J. De Grauw, “Imaging of optically thick specimen using two-photon excitation microscopy,” Microsc. Res. Tech. 47(3), 206–209 (1999).
[Crossref] [PubMed]

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

E. D. Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature 401(6756), 895–898 (1999).
[Crossref]

1997 (1)

1993 (1)

1991 (1)

O. Carnal, M. Sigel, T. Sleator, H. Takuma, and J. Mlynek, “Imaging and focusing of atoms by a Fresnel zone plate,” Phys. Rev. Lett. 67(23), 3231–3234 (1991).
[Crossref] [PubMed]

1973 (1)

1959 (1)

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

Acebal, P.

An, H.

André, J. M.

Barrett, H. H.

Barrett, R.

E. D. Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature 401(6756), 895–898 (1999).
[Crossref]

Billy, L.

Blaya, S.

Braat, J. J. M.

Bryngdahl, O.

Cabrini, S.

E. D. Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature 401(6756), 895–898 (1999).
[Crossref]

Cai, Z.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
[Crossref]

Carnal, O.

O. Carnal, M. Sigel, T. Sleator, H. Takuma, and J. Mlynek, “Imaging and focusing of atoms by a Fresnel zone plate,” Phys. Rev. Lett. 67(23), 3231–3234 (1991).
[Crossref] [PubMed]

Carretero, L.

Cases, Á. M.

Chen, J.

Y. Zhang, J. Chen, and X. Ye, “Multilevel phase Fresnel zone plate lens as a near-field optical element,” Opt. Commun. 269(2), 271–273 (2007).
[Crossref]

Chen, Z.

Cheng, M.

Crozier, K. B.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

Cui, G.

Dai, Y.

Y. Zhang and Y. Dai, “Multifocal optical trapping using counter-propagating radially-polarized beams,” Opt. Commun. 285(5), 725–730 (2012).
[Crossref]

David, C.

De Grauw, C. J.

H. C. Gerritsen and C. J. De Grauw, “Imaging of optically thick specimen using two-photon excitation microscopy,” Microsc. Res. Tech. 47(3), 206–209 (1999).
[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]

Elings, V. B.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

Ersoy, O. K.

Escarré, S. B.

Fabrizio, E. D.

E. D. Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature 401(6756), 895–898 (1999).
[Crossref]

Fimia, A.

Gentili, M.

E. D. Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature 401(6756), 895–898 (1999).
[Crossref]

Gerritsen, H. C.

H. C. Gerritsen and C. J. De Grauw, “Imaging of optically thick specimen using two-photon excitation microscopy,” Microsc. Res. Tech. 47(3), 206–209 (1999).
[Crossref] [PubMed]

Ghislain, L. P.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

Gil, A. F.

Goldberg, B. B.

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97(5), 053105 (2005).
[Crossref]

González, P. A.

Helseth, L. E.

L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281(8), 1981–1985 (2008).
[Crossref]

Horrigan, F. A.

Ippolito, S. B.

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97(5), 053105 (2005).
[Crossref]

Kaulich, B.

E. D. Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature 401(6756), 895–898 (1999).
[Crossref]

Kim, H. C.

Kino, G. S.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

Ko, H.

Lau, S. P.

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, X. F.

R. G. Mote, S. F. Yu, W. Zhou, and X. 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]

López, L. C.

Madrigal, R.

Madrigal, R. F.

Manalis, S. R.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

Minne, S. C.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

Mlynek, J.

O. Carnal, M. Sigel, T. Sleator, H. Takuma, and J. Mlynek, “Imaging and focusing of atoms by a Fresnel zone plate,” Phys. Rev. Lett. 67(23), 3231–3234 (1991).
[Crossref] [PubMed]

Molina, M. P.

Mote, R. G.

R. G. Mote, S. F. Yu, W. Zhou, and X. 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]

R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime--new insights,” Opt. Express 16(13), 9554–9564 (2008).
[Crossref] [PubMed]

Murciano, A.

Ng, B. K.

Nugent, K. A.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
[Crossref]

Paterson, D.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
[Crossref]

Peele, A. G.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
[Crossref]

Pereira, S. F.

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]

Quate, C. F.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

Quiney, H. M.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
[Crossref]

Richards, B.

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

Romanato, F.

E. D. Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature 401(6756), 895–898 (1999).
[Crossref]

Ruan, X.

Y. Zhang, C. Zheng, Y. Zhuang, and X. Ruan, “Analysis of near-field subwavelength focusing of hybrid amplitude–phase Fresnel zone plates under radially polarized illumination,” J. Opt. 16(1), 015703 (2014).
[Crossref]

Y. Zhang, H. An, D. Zhang, G. Cui, and X. Ruan, “Diffraction theory of high numerical aperture subwavelength circular binary phase Fresnel zone plate,” Opt. Express 22(22), 27425–27436 (2014).
[Crossref] [PubMed]

Saidani, M.

Sammar, A.

Sarkar, S. S.

Schmitz, M.

Sigel, M.

O. Carnal, M. Sigel, T. Sleator, H. Takuma, and J. Mlynek, “Imaging and focusing of atoms by a Fresnel zone plate,” Phys. Rev. Lett. 67(23), 3231–3234 (1991).
[Crossref] [PubMed]

Sleator, T.

O. Carnal, M. Sigel, T. Sleator, H. Takuma, and J. Mlynek, “Imaging and focusing of atoms by a Fresnel zone plate,” Phys. Rev. Lett. 67(23), 3231–3234 (1991).
[Crossref] [PubMed]

Solak, H. H.

Srisungsitthisunti, P.

Susini, J.

E. D. Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature 401(6756), 895–898 (1999).
[Crossref]

Takuma, H.

O. Carnal, M. Sigel, T. Sleator, H. Takuma, and J. Mlynek, “Imaging and focusing of atoms by a Fresnel zone plate,” Phys. Rev. Lett. 67(23), 3231–3234 (1991).
[Crossref] [PubMed]

Ünlü, M. S.

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97(5), 053105 (2005).
[Crossref]

van de Nes, A. S.

van der Veen, J. F.

Wilder, K.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999).
[Crossref]

Wolf, E.

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

Xu, X.

Ye, X.

Y. Zhang, J. Chen, and X. Ye, “Multilevel phase Fresnel zone plate lens as a near-field optical element,” Opt. Commun. 269(2), 271–273 (2007).
[Crossref]

Yu, S. F.

R. G. Mote, S. F. Yu, W. Zhou, and X. 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]

R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime--new insights,” Opt. Express 16(13), 9554–9564 (2008).
[Crossref] [PubMed]

Zhang, B.

Zhang, D.

Zhang, Y.

Y. Zhang, H. An, D. Zhang, G. Cui, and X. Ruan, “Diffraction theory of high numerical aperture subwavelength circular binary phase Fresnel zone plate,” Opt. Express 22(22), 27425–27436 (2014).
[Crossref] [PubMed]

Y. Zhang, C. Zheng, and Y. Zhuang, “Effect of the shadowing in high-numerical-aperture binary phase Fresnel zone plates,” Opt. Commun. 317, 88–92 (2014).
[Crossref]

Y. Zhang, C. Zheng, Y. Zhuang, and X. Ruan, “Analysis of near-field subwavelength focusing of hybrid amplitude–phase Fresnel zone plates under radially polarized illumination,” J. Opt. 16(1), 015703 (2014).
[Crossref]

Y. Zhang and Y. Dai, “Multifocal optical trapping using counter-propagating radially-polarized beams,” Opt. Commun. 285(5), 725–730 (2012).
[Crossref]

Y. Zhang, J. Chen, and X. Ye, “Multilevel phase Fresnel zone plate lens as a near-field optical element,” Opt. Commun. 269(2), 271–273 (2007).
[Crossref]

Y. Zhang and C. Zheng, “Axial intensity distribution behind a Fresnel zone plate,” Opt. Laser Technol. 37(1), 77–80 (2005).
[Crossref]

Zhao, D.

Zheng, C.

Y. Zhang, C. Zheng, Y. Zhuang, and X. Ruan, “Analysis of near-field subwavelength focusing of hybrid amplitude–phase Fresnel zone plates under radially polarized illumination,” J. Opt. 16(1), 015703 (2014).
[Crossref]

Y. Zhang, C. Zheng, and Y. Zhuang, “Effect of the shadowing in high-numerical-aperture binary phase Fresnel zone plates,” Opt. Commun. 317, 88–92 (2014).
[Crossref]

Y. Zhang and C. Zheng, “Axial intensity distribution behind a Fresnel zone plate,” Opt. Laser Technol. 37(1), 77–80 (2005).
[Crossref]

Zhou, W.

R. G. Mote, S. F. Yu, W. Zhou, and X. 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]

R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime--new insights,” Opt. Express 16(13), 9554–9564 (2008).
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Figures (3)

Fig. 1
Fig. 1 Cross-section diagram of the studied focusing system with a binary phase circular FZP. The FZP's pattern is etched in a glass film and the etching depth is w. The exit pupil is located in the plane of z = 0. The image space is a multilayer structure of plane parallel film where several medium transitions can be encountered at z = di. The first medium at the exit pupil has electric permittivity ε1 and the final medium has electric permittivity εL. All of materials are nonmagnetic. The origin of coordinates is positioned at the center of the exit pupil.
Fig. 2
Fig. 2 The normalized intensity distributions of light focused by a binary π-phase-shifted FZP through a solid film of d = 10 μm. (a) is the case along the optical z axis, (b) and (c) are the cases along the x and y axes, respectively, in the focal plane of f = 14.98 μm. The red and blue curves are obtained by the analytical model and by the FDTD method, respectively.
Fig. 3
Fig. 3 Reflection and transmission of wave propagation through a slab j.

Equations (30)

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r j = jλ f d + (jλ/2) 2 ,j=1,2,,2N+1
t(r)={ t rid = t 12 t 23 exp(iknw) 1+ r 12 r 23 exp(2iknw) , r 2m r< r 2m+1 t gro = 2 1+ n 1 exp(ikw), r 2m1 r< r 2m
r 12 = 1 n 1 1+ n 1 , r 23 = n n 1 n+ n 1 , t 12 = 2 1+n , t 23 = 2n n+ n 1
E x (ρ,θ,z)= 0 0 A x ( k x , k y ) exp[i( k x x+ k y y+ k z z)]d k x d k y , E y (ρ,θ,z)= 0 0 A y ( k x , k y ) exp[i( k x x+ k y y+ k z z)]d k x d k y , E z (ρ,θ,z)= 0 0 [ k x k z A x ( k x , k y )+ k y k z A y ( k x , k y ) ] exp[i( k x x+ k y y+ k z z)]d k x d k y .
A x = 1 (2π) 2 0 0 E x (r,φ)t(r) exp[i( k x x+ k y y)]dxdy, A y = 1 (2π) 2 0 0 E y (r,φ)t(r) exp[i( k x x+ k y y)]dxdy,
E ρ (ρ,θ,z)= 0 0 2π [ A ξ cos( k η θ) A η sin( k η θ)]exp[iρ k ξ cos( k η θ)]exp(i k ξ z) k ξ d k ξ d k η , E θ (ρ,θ,z)= 0 0 2π [ A ξ sin( k η θ)+ A η cos( k η θ)]exp[iρ k ξ cos( k η θ)]exp(i k z z) k ξ d k ξ d k η , E z (ρ,θ,z)= 0 0 2π k ξ A ξ k z exp[iρ k ξ cos( k η θ)]exp(i k z z) k ξ d k ξ d k η ,
A ξ ( k ξ , k η )= 1 (2π) 2 0 0 2π [ E r (r,φ) cos(φ k η ) E φ (r,φ)sin(φ k η )] ×t(r)exp[ir k ξ cos(φ k η )]rdrdφ, A η ( k ξ , k η )= 1 (2π) 2 0 0 2π [ E r (r,φ) sin(φ k η )+ E φ (r,φ)cos(φ k η )] ×t(r)exp[ir k ξ cos(φ k η )]rdrdφ.
E ρ,j = 0 0 2π { ( [ B j p+ exp[i k z,j (z d j1 )]+ B j p exp[i k z,j (z d j1 )] ) cos( k η,j θ) ( [ B j s+ exp[i k z,j (z d j1 )]+ B j s exp[i k z,j (z d j1 )] )sin( k η,j θ) } ×exp[iρ k ξ,j cos( k η,j θ)] k ξ,j d k ξ,j d k η,j , E θ,j = 0 0 2π { ( [ B j p+ exp[i k z,j (z d j1 )]+ B j p exp[i k z,j (z d j1 )] ) sin( k η,j θ) ( [ B j s+ exp[i k z,j (z d j1 )]+ B j s exp[i k z,j (z d j1 )] )cos( k η,j θ) } ×exp[iρ k ξ,j cos( k η,j θ)] k ξ,j d k ξ,j d k η,j , E z,j = 0 0 2π ( [ B j p+ exp[i k z,j (z d j1 )]+ B j p exp[i k z,j (z d j1 )] ) × k ξ,j k z,j exp[iρ k ξ,j cos( k η,j θ)] k ξ,j d k ξ,j d k η,j ,
k ξ,j+1 = k ξ,j k ξ , k η,j+1 = k η,j k η .
E ρ,j = 0 0 2π { ( [ B j p+ exp[i k z,j (z d j1 )]+ B j p exp[i k z,j (z d j1 )] ) cos( k η θ) ( [ B j s+ exp[i k z,j (z d j1 )]+ B j s exp[i k z,j (z d j1 )] )sin( k η θ) } ×exp[iρ k ξ cos( k η θ)] k ξ d k ξ d k η , E θ,j = 0 0 2π { ( [ B j p+ exp[i k z,j (z d j1 )]+ B j p exp[i k z,j (z d j1 )] ) sin( k η θ) +( [ B j s+ exp[i k z,j (z d j1 )]+ B j s exp[i k z,j (z d j1 )] )cos( k η θ) } ×exp[iρ k ξ cos( k η θ)] k ξ d k ξ d k η , E z,j = 0 0 2π k ξ k z,j ( [ B j p+ exp[i k z,j (z d j1 )]+ B j p exp[i k z,j (z d j1 )] ) ×exp[iρ k ξ cos( k η θ)] k ξ d k ξ d k η ,
A ξ = i 2π C( k ξ ), A η =0,
C( k ξ )= t rid m=0 N r 2m r 2m+1 J 1 (r k ξ )rdr + t gro m=1 N r 2m1 r 2m J 1 (r k ξ )rdr .
0 2π cos(nτ) exp[ir k ξ cosτ]dτ=2 πi n J n (r k ξ ),
E ρ,j (ρ,z)= 0 C( k ξ ){ T j p+ ( k ξ )exp[i k z,j (z d j1 )]+ T j p ( k ξ )exp[i k z,j (z d j1 )] } J 1 (ρ k ξ ) k ξ d k ξ , E z,j (ρ,z)=i 0 k ξ C( k ξ ) k z,j { T j p+ ( k ξ )exp[i k z,j (z d j1 )]+ T j p ( k ξ )exp[i k z,j (z d j1 )] } J 0 (ρ k ξ ) k ξ d k ξ .
A ξ =0, A η = i 2π C( k ξ ).
E θ,j (ρ,z)= 0 C( k ξ ){ T j s+ ( k ξ )exp[i k z,j (z d j1 )]+ T j s ( k ξ )exp[i k z,j (z d j1 )] } J 1 (ρ k ξ ) k ξ d k ξ .
A ξ = cos k η 2π D( k ξ ), A ξ = sin k η 2π D( k ξ ),
D( k ξ )= t rid m=0 N r 2m r 2m+1 J 0 (r k ξ )rdr + t gro m=1 N r 2m1 r 2m J 0 (r k ξ )rdr .
D( k ξ )= t rid m=0 N [ r 2m+1 2 J 1 ( r 2m+1 k ξ ) r 2m+1 k ξ r 2m 2 J 1 ( r 2m k ξ ) r 2m k ξ ] + t gro m=1 N [ r 2m 2 J 1 ( r 2m k ξ ) r 2m k ξ r 2m1 2 J 1 ( r 2m1 k ξ ) r 2m1 k ξ ] .
E ρ,j (ρ,θ,z)= 1 2 cosθ 0 { ( T j p+ ( k ξ )exp[i k z,j (z d j1 )]+ T j p ( k ξ )exp[i k z,j (z d j1 )] ) ×[ J 0 (ρ k ξ ) J 2 (ρ k ξ )]+( T j s+ ( k ξ )exp[i k z,j (z d j1 )]+ T j s ( k ξ )exp[i k z,j (z d j1 )] ) ×[ J 0 (ρ k ξ )+ J 2 (ρ k ξ )] }D( k ξ ) k ξ d k ξ , E θ,j (ρ,θ,z)= 1 2 sinθ 0 { ( T j p+ ( k ξ )exp[i k z,j (z d j1 )]+ T j p ( k ξ )exp[i k z,j (z d j1 )] ) ×[ J 0 (ρ k ξ )+ J 2 (ρ k ξ )]+( T j s+ ( k ξ )exp[i k z,j (z d j1 )]+ T j s ( k ξ )exp[i k z,j (z d j1 )] ) ×[ J 0 (ρ k ξ ) J 2 (ρ k ξ )] }D( k ξ ) k ξ d k ξ , E z,j (ρ,θ,z)=icosθ 0 { T j p+ ( k ξ )exp[i k z,j (z d j1 )]+ T j p ( k ξ )exp[i k z,j (z d j1 )] } × k ξ D( k ξ ) k z,j J 1 (ρ k ξ ) k ξ d k ξ .
E x,j = I 0 + I 2 cos2θ, E y,j = I 2 sin2θ, E z,j =2i I 1 cosθ
I 0 = 1 2 0 { [ T j s+ + T j p+ ]exp[i k z,j (z d j1 )]+[ T j s + T j p )]exp[i k z,j (z d j1 )] }D( k ξ ) J 0 (ρ k ξ ) k ξ d k ξ I 2 = 1 2 0 { [ T j s+ T j p+ ]exp[i k z,j (z d j1 )]+[ T j s T j p ]exp[i k z,j (z d j1 )] }D( k ξ ) J 2 (ρ k ξ ) k ξ d k ξ I 1 = 1 2 0 { T j p+ exp[i k z,j (z d j1 )]+ T j p exp[i k z,j (z d j1 )] } k ξ D( k ξ ) k z,j J 1 (ρ k ξ ) k ξ d k ξ .
I(0,0,z)= | 1 2 0 exp(i k z k 2 k ξ 2 z)D( k ξ ) k ξ d k ξ | 2 .
( T j s/p+ T j s/p )= M j,j+1 s/p ( T j+1 s/p+ T j+1 s/p ),
M j,j+1 s/p = Q j 1 P j,j+1 s/p , P j,j+1 s/p = 1 t j,j+1 s/p [ 1 r j,j+1 s/p r j,j+1 s/p 1 ], Q j =[ exp(i k z,j b j ) 0 0 exp(i k z,j b j ) ].
r j,j+1 s = k z,j k z,j+1 k z,j + k z,j+1 , t j,j+1 s = 2 k z,j k z,j + k z,j+1 , fortheTEpolarization, r j,j+1 p = ε j+1 k z,j ε j k z,j+1 ε j+1 k z,j + ε j k z,j+1 , t j,j+1 p = k j+1 k j 2 ε j+1 k z,j ε j+1 k z,j + ε j k z,j+1 ,fortheTMpolarization.
( 1 T 1 s/p )=( j=1 L M j,j+1 s/p )( T L s/p+ 0 ) = M s/p ( T L s/p+ 0 ),
T L s/p+ =1/ m 11 s/p , T 1 s/p = m 21 s/p / m 11 s/p .
( T j s/p+ T j s/p )= n=j L M n,n+1 s/p ( T L s/p+ 0 ) .
B j p± ( k ξ , k η )= A ξ ( k ξ , k η ) T j p± ( k r ), B j s± ( k ξ , k η )= A η ( k ξ , k η ) T j s± ( k r ).

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