Abstract

We rigorously analyze the optical singularities and power flux in the near-field region of the novel superlenses reported in [Science 317, 927 (2007) ] For this purpose, we derive near-field expressions and a general criterion to classify the optical singularities in the vacuum, which are valid when the (s- or p-polarized) electromagnetic fields are generated by any planar field distribution with Cartesian or azimuthal symmetry. Such general results are particularized to the superlenses [Science 317, 927 (2007) ], for which we identify a sequence of optical vortices and saddles that arise from evanescent-field interference. While the saddles are always located around the focal region, the vortex locations depend on the source field. The features of the topological connection between vortices and saddles are also discussed.

© 2008 Optical Society of America

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References

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  1. E. Abbe, “Beitrage zur Theorie des Mikroskops und der Mikroskop ischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413-468 (1873).
    [CrossRef]
  2. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  3. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
    [CrossRef] [PubMed]
  4. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
    [CrossRef] [PubMed]
  5. A. Wong, E. Sarris, and G. Eleftheriades, “Metallic transmission screen for sub-wavelength focusing,” Electron. Lett. 43, 1402-1404 (2007).
    [CrossRef]
  6. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  7. R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290-1292 (2004).
    [CrossRef]
  8. R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927-929 (2007).
    [CrossRef] [PubMed]
  9. L. E. Helseth, “The almost perfect lens and focusing of evanescent waves,” Opt. Commun. 281, 1981-1985 (2008).
    [CrossRef]
  10. A. Y. Bekshaev and M. S. Soskin, “Tranverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332-348 (2007).
    [CrossRef]
  11. H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, and H. Blok, “Creation and annihilation of phase singularities near a sub-wavelength slit,” Opt. Express 11, 371-380 (2003).
    [CrossRef] [PubMed]
  12. G. D'Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 44, 043825 (2008).
    [CrossRef]
  13. M. C. Yang and K. J. Webb, “Poynting vector analysis of a superlens,” Opt. Lett. 30, 2382-2384 (2005).
    [CrossRef] [PubMed]
  14. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).
  15. In general, for two nonzero complex numbers a,b, it holds that Re[ab*]=∣ab∣cos(arg[a]−arg[b]) and Im[ab*]=∣ab∣sin(arg[a]−arg[b]), being ∣ab∣>0. In particular, Re[ab*]=0 if and only if arg[a]−arg[b]=+/-π/2, whereas Im[ab*]=0 if and only if arg[a]−arg[b]=0 or arg[a]−arg[b]=+/-π.
  16. I. S. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products (Academic, 1994).
  17. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. B 24, 2844-2849 (2007).
    [CrossRef]
  18. J. D. Weston, “Some remarks about the curl of a vector field,” The American Mathematical Monthly, April 1961, pp. 359-361.

2008 (2)

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

G. D'Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 44, 043825 (2008).
[CrossRef]

2007 (6)

A. Y. Bekshaev and M. S. Soskin, “Tranverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332-348 (2007).
[CrossRef]

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. B 24, 2844-2849 (2007).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

A. Wong, E. Sarris, and G. Eleftheriades, “Metallic transmission screen for sub-wavelength focusing,” Electron. Lett. 43, 1402-1404 (2007).
[CrossRef]

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927-929 (2007).
[CrossRef] [PubMed]

2005 (2)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

M. C. Yang and K. J. Webb, “Poynting vector analysis of a superlens,” Opt. Lett. 30, 2382-2384 (2005).
[CrossRef] [PubMed]

2004 (1)

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290-1292 (2004).
[CrossRef]

2003 (1)

2000 (1)

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

1873 (1)

E. Abbe, “Beitrage zur Theorie des Mikroskops und der Mikroskop ischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413-468 (1873).
[CrossRef]

Abbe, E.

E. Abbe, “Beitrage zur Theorie des Mikroskops und der Mikroskop ischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413-468 (1873).
[CrossRef]

Bekshaev, A. Y.

A. Y. Bekshaev and M. S. Soskin, “Tranverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332-348 (2007).
[CrossRef]

Bloemer, M.

G. D'Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 44, 043825 (2008).
[CrossRef]

Blok, H.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

D'Aguanno, G.

G. D'Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 44, 043825 (2008).
[CrossRef]

de Rosny, J.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Desyatnikov, A.

G. D'Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 44, 043825 (2008).
[CrossRef]

Eleftheriades, G.

A. Wong, E. Sarris, and G. Eleftheriades, “Metallic transmission screen for sub-wavelength focusing,” Electron. Lett. 43, 1402-1404 (2007).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Fink, M.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Gbur, G.

Gradshteyn, I. S.

I. S. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products (Academic, 1994).

Helseth, L. E.

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

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Lenstra, D.

Lerosey, G.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Mattiucci, N.

G. D'Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 44, 043825 (2008).
[CrossRef]

Merlin, R.

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927-929 (2007).
[CrossRef] [PubMed]

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290-1292 (2004).
[CrossRef]

Novitsky, A. V.

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. B 24, 2844-2849 (2007).
[CrossRef]

Novitsky, D. V.

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. B 24, 2844-2849 (2007).
[CrossRef]

Pendry, J. B.

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

Ryzhik, I.

I. S. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products (Academic, 1994).

Sarris, E.

A. Wong, E. Sarris, and G. Eleftheriades, “Metallic transmission screen for sub-wavelength focusing,” Electron. Lett. 43, 1402-1404 (2007).
[CrossRef]

Schouten, H. F.

Soskin, M. S.

A. Y. Bekshaev and M. S. Soskin, “Tranverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332-348 (2007).
[CrossRef]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Tourin, A.

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Visser, T. D.

Webb, K. J.

Weston, J. D.

J. D. Weston, “Some remarks about the curl of a vector field,” The American Mathematical Monthly, April 1961, pp. 359-361.

Wong, A.

A. Wong, E. Sarris, and G. Eleftheriades, “Metallic transmission screen for sub-wavelength focusing,” Electron. Lett. 43, 1402-1404 (2007).
[CrossRef]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Yang, M. C.

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290-1292 (2004).
[CrossRef]

Arch. Mikrosc. Anat. Entwicklungsmech. (1)

E. Abbe, “Beitrage zur Theorie des Mikroskops und der Mikroskop ischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413-468 (1873).
[CrossRef]

Electron. Lett. (1)

A. Wong, E. Sarris, and G. Eleftheriades, “Metallic transmission screen for sub-wavelength focusing,” Electron. Lett. 43, 1402-1404 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. B 24, 2844-2849 (2007).
[CrossRef]

Opt. Commun. (2)

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

A. Y. Bekshaev and M. S. Soskin, “Tranverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332-348 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

G. D'Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 44, 043825 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

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

Science (4)

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927-929 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315, 1120-1122 (2007).
[CrossRef] [PubMed]

Other (4)

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

In general, for two nonzero complex numbers a,b, it holds that Re[ab*]=∣ab∣cos(arg[a]−arg[b]) and Im[ab*]=∣ab∣sin(arg[a]−arg[b]), being ∣ab∣>0. In particular, Re[ab*]=0 if and only if arg[a]−arg[b]=+/-π/2, whereas Im[ab*]=0 if and only if arg[a]−arg[b]=0 or arg[a]−arg[b]=+/-π.

I. S. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products (Academic, 1994).

J. D. Weston, “Some remarks about the curl of a vector field,” The American Mathematical Monthly, April 1961, pp. 359-361.

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Figures (15)

Fig. 1
Fig. 1

log 10 S ( y , z ) assuming the source field given in Eq. (27). We can appreciate the superfocused image centered at ( 0 , L ) as well as optical phase singularities in the near-field region.

Fig. 2
Fig. 2

Electromagnetic fields generated by the source field given in Eq. (27). (a) log 10 E ( y , z ) , (b) log 10 H y ( y , z ) , (c) log 10 H z ( y , z ) , (d) log 10 H ( y , z ) .

Fig. 3
Fig. 3

Phase difference arg [ H y ( y , L ) ] arg [ H z ( y , L ) ] between the magnetic field components at the focal plane z = L [the source field given in Eq. (27) is assumed].

Fig. 4
Fig. 4

Vectorial map for the time-averaged Poynting vector around the singularities at z = L [the source field given in Eq. (27) is assumed]. The field lines for S are circles with a counter-clockwise rotation direction, which corresponds to right-handed vortices.

Fig. 5
Fig. 5

Vectorial map for the time-averaged Poynting vector around the singularities at z > L [the source field given in Eq. (27) is assumed]. The field lines for S are hyperbolas, which corresponds to saddle points.

Fig. 6
Fig. 6

Normalized time-averaged Poynting vector S S around the focal region for the source field given in Eq. (27). Each saddle is connected to the two nearest vortices, which have the same rotation direction (right-handed).

Fig. 7
Fig. 7

Map of S S showing the direction of the power flux in the near-field region [the source field given in Eq. (27) is assumed].

Fig. 8
Fig. 8

Normalized time-averaged Poynting vector S S around the focal region assuming the source field E x ( y , 0 ) = L 2 ( y 2 + L 2 ) e i q 0 y . Each saddle is connected to the two nearest vortices, which have the same rotation direction (left-handed). Note the change of the vortex rotation directions with respect to Fig. 6.

Fig. 9
Fig. 9

Electromagnetic fields generated by the source field E x ( y , 0 ) = L 2 ( y 2 + L 2 ) cos ( q 0 y ) . (a) log 10 S ( y , z ) . (b) Zoomed image of (a) showing the optical singularities at z = 0 . (c) log 10 H ( y , z ) . (d) Im [ H y ( y , 0 ) ( H z ( y , 0 ) ) * ] is alternating positive and negative when S ( y , 0 ) = 0 , which demonstrates the presence of vortices with alternating rotations at z = 0 .

Fig. 10
Fig. 10

Normalized time-averaged Poynting vector S S assuming the source field E x ( y , 0 ) = L 2 ( y 2 + L 2 ) cos ( q 0 y ) . Each saddle is connected to two consecutive vortices, which have alternating rotation directions.

Fig. 11
Fig. 11

Mapping of S S assuming the source field given in Eq. (32). Besides the optical singularities, it can be appreciated that the field lines for S are parallel to the straight line r = 0 around it.

Fig. 12
Fig. 12

Electromagnetic fields generated by the source field given in Eq. (32). (a) log 10 S ( r , z ) , for which we can appreciate the superfocused image centered at ( 0 , L ) . (b) Zoomed map for log 10 S ( r , z ) around z = 0 . (c) log 10 E ( r , z ) . (d) log 10 H ( r , z ) .

Fig. 13
Fig. 13

When r > 0 and S ( r , 0 ) = 0 , the value for Im [ H r ( r , 0 ) H z * ( r , 0 ) ] is positive or negative but never zero [which excludes the presence of sources or sinks at z = 0 for the source field considered in Eq. (32)].

Fig. 14
Fig. 14

Vectorial map for the time-averaged Poynting vector around the singularities at z > L assuming the source field given in Eq. (32). The field lines for S are hyperbolas, which corresponds to saddle points.

Fig. 15
Fig. 15

Poynting vector component S z ( r , 0 ) along r for the source field given in Eq. (32).

Equations (52)

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D = E = 0 ,
B = H = 0 ,
× E = i k η H ,
× H = i k η E ,
S = 1 2 Re [ E × H * ]
E ( y , z ) = E x ( y , z ) x ̂ ,
H ( y , z ) = H y ( y , z ) y ̂ + H z ( y , z ) z ̂ .
E x ( y , z ) = 1 2 π + + E x ( y , 0 ) e i [ q ( y y ) + κ ( q ) z ] d y d q ,
κ ( q ) = { k 2 q 2 , q < k i q 2 k 2 , q k } .
F ( q ) = + E x ( y , 0 ) e i q y d y .
E x ( y , z ) = 1 2 π + F ( q ) e i [ q y + κ ( q ) z ] d q .
H y ( y , z ) = 1 2 π k η + κ ( q ) F ( q ) e i [ q y + κ ( q ) z ] d q ,
H z ( y , z ) = 1 2 π k η + q F ( q ) e i [ q y + κ ( q ) z ] d q .
S ( y , z ) = S y ( y , z ) y ̂ + S z ( y , z ) z ̂ ,
S y ( y , z ) = 1 2 Re [ E x ( y , z ) H z * ( y , z ) ] ,
S z ( y , z ) = 1 2 Re [ E x ( y , z ) H y * ( y , z ) ] .
× S ( y , z ) = k η Im [ H y ( y , z ) H z * ( y , z ) ] x ̂ ,
E ( r , z ) = E ϕ ( r , z ) ϕ ̂ ,
H ( r , z ) = H r ( r , z ) r ̂ + H z ( r , z ) z ̂ .
E ϕ ( r , z ) = 0 + 0 + E ϕ ( r , 0 ) J 1 ( q r ) J 1 ( q r ) q r e i κ ( q ) z d r d q ,
C ( q ) = 0 + E ϕ ( r , 0 ) J 1 ( q r ) r d r .
E ϕ ( r , z ) = 0 + q C ( q ) J 1 ( q r ) e i κ ( q ) z d q .
H r ( r , z ) = 1 k η 0 + κ ( q ) q C ( q ) J 1 ( q r ) e i κ ( q ) z d q ,
H z ( r , z ) = i k η 0 + q 2 C ( q ) J 0 ( q r ) e i κ ( q ) z d q .
S ( r , z ) = S r ( r , z ) r ̂ + S z ( r , z ) z ̂ ,
S r ( r , z ) = 1 2 Re [ E ϕ ( r , z ) H z * ( r , z ) ] ,
S z ( r , z ) = 1 2 Re [ E ϕ ( r , z ) H r * ( r , z ) ] .
× S ( r , z ) = ( k η Im [ H r * ( r , z ) H z ( r , z ) ] + 1 r S z ( r , z ) ) ϕ ̂ .
E x ( y , 0 ) = L 2 y 2 + L 2 e i q 0 y ,
F ( q ) = π L e q q 0 L .
E x ( y , z ) = 1 2 [ E x ( y , z ) + E x ( y , z ) ] ,
H y ( y , z ) = 1 2 [ H y ( y , z ) + H y ( y , z ) ] ,
H z ( y , z ) = 1 2 [ H z ( y , z ) + H z ( y , z ) ] .
+ F ( q ) e i [ q y + κ ( q ) z ] d q = k k F ( q ) e i [ q y + κ ( q ) z ] d q + q k F ( q ) e i [ q y + κ ( q ) z ] d q
E x ( y , z ) = E x * ( y , z ) + e x ( y , z ) ,
H y ( y , z ) = H y * ( y , z ) + h y ( y , z ) ,
H z ( y , z ) = H z * ( y , z ) + h z ( y , z ) ,
e x ( y , z ) = i L k k e q q 0 L e i q y sin [ κ ( q ) z ] d q ,
h y ( y , z ) = L k η k k κ ( q ) e q q 0 L e i q y cos [ κ ( q ) z ] d q ,
h z ( y , z ) = i L k η k k q e q q 0 L e i q y sin [ κ ( q ) z ] d q .
E ϕ ( r , 0 ) = L 2 J 1 ( q 0 r ) L 2 + r 2 .
C ( q ) = { L 2 I 1 ( q L ) K 1 ( q 0 L ) , q < q 0 L 2 I 1 ( q 0 L ) K 1 ( q L ) , q q 0 } ,
S ( r , 0 ) = S z ( r , 0 ) z ̂ .
A ( y , z ) = M [ y y 0 z z 0 ] + [ f 1 ( y , z ) f 2 ( y , z ) ] ,
f k ( y , z ) = i , j 1 c i , j , k ( y y 0 ) i ( z z 0 ) j ,
M = P 1 J P .
[ y z ] = P [ y y 0 z z 0 ] ,
A ( y , z ) = P A ( y , z ) .
A ( y , z ) = J [ y z ] + [ g 1 ( y , z ) g 2 ( y , z ) ]
g k ( y , z ) = i , j 1 c i , j , k ( y ) i ( z ) j ,
× A ( y , z ) = × A ( y , z ) .
× A ( 0 , 0 ) = × ( J [ y z ] ) ( y , z ) = ( 0 , 0 ) .

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