Abstract

A new boundary integral equation method for solving the near field in three-dimensional vector form in scanning near-field optical microscopy (SNOM) using Borgnis potentials as auxiliary functions is presented. A boundary integral equation of the electromagnetic fields, expressed by Borgnis potentials, is derived based on Green’s theorem. The harmonic expansion in rotationally symmetric SNOM probe–sample systems is studied, and the three-dimensional electromagnetic problem is partly simplified into a two-dimensional one. The boundary conditions of Borgnis potentials both on dielectric boundaries and on perfectly conducting boundaries are derived. Relevant algorithms were studied, and a computer program was written. As an example, a SNOM probe–sample system composed of a round metal-covered probe and a sample with a flat surface has been numerically studied, and the computational results are given. This new method can be used efficiently for other electromagnetic field problems with round subwavelength structures.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. W. Pohl, W. Denk, M. Lanz, “Optical stethosocopy: image recording with resolution λ∕20,” Appl. Phys. Lett. 44, 651–655 (1984).
    [CrossRef]
  2. R. C. Reddick, R. J. Warmack, T. L. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767–770 (1989).
    [CrossRef]
  3. D. Courjon, K. Sarageddine, M. Spazer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
    [CrossRef]
  4. E. Betzig, P. L. Finn, J. S. Weiner, “Combined shearforce and near-field scanning optical microscopy,” Appl. Phys. Lett. 60, 2484–2486 (1992).
    [CrossRef]
  5. S. Bozhevolnyi, S. Berntsen, E. Bozherolnaya, “Extension of the macroscopic model for reflection near-field microscopy: regularization and image formation,” J. Opt. Soc. Am. A 11, 609–617 (1994).
    [CrossRef]
  6. B. Labani, C. Girard, D. Courjon, D. Van Labeke, “Optical interaction between a dielectric tip and a nanometric lattice: implications for near-field microscopy,” J. Opt. Soc. Am. B 7, 936–943 (1990).
    [CrossRef]
  7. M. Xiao, X. Chen, “Shear-force, constant-height, and constant-intensity imaging in scanning near field optical microscopy with s- and p-polarized incident light,” Opt. Eng. (Bellingham) 39, 2495–2500 (2000).
    [CrossRef]
  8. D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–195 (1995).
    [CrossRef]
  9. F. I. Baida, D. Van Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
    [CrossRef]
  10. R. Carminati, J. J. Greffet, “Influence of dielectric contrast and topography on the near field scattered by an inhomogeneous surface,” J. Opt. Soc. Am. A 12, 2716–2725 (1995).
    [CrossRef]
  11. P. J. Valle, J. J. Greffet, R. Carminati, “Optical contrast, topographic contrast and artifacts in illumination-mode scanning near-field optical microscopy,” J. Appl. Phys. 86, 648–656 (1999).
    [CrossRef]
  12. L. Novotny, D. W. Pohl, B. Hecht, “Scanning near-field optical probe with ultrasmall spot size,” Opt. Lett. 20, 970–972 (1995).
    [CrossRef] [PubMed]
  13. S. Bozhevolnyi, “Topographical artifacts and optical resolution in near-field optical microscopy,” J. Opt. Soc. Am. A 14, 2254–2259 (1997).
    [CrossRef]
  14. M. Tanaka, K. Tanaka, “Boundary integral equations for computer-aided design and simulations of near-field optics: two-dimensional optical manipulator,” J. Opt. Soc. Am. A 15, 101–108 (1998).
    [CrossRef]
  15. X. E. Wang, Z. Z. Fan, T. T. Tang, “Study on the power transmission and light spot size of optical probes in scanning near-field optical microscopes,” Opt. Commun. 235, 31–40 (2004).
    [CrossRef]
  16. A. Vainstein, Electromagnetic Waves (Soviet Radio Press, Moscow, 1957) (in Russian).
  17. Keqian Zhang, Dejie Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer-Verlag, Berlin, 1998).
    [CrossRef]
  18. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]

2004 (1)

X. E. Wang, Z. Z. Fan, T. T. Tang, “Study on the power transmission and light spot size of optical probes in scanning near-field optical microscopes,” Opt. Commun. 235, 31–40 (2004).
[CrossRef]

2003 (1)

F. I. Baida, D. Van Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

2000 (1)

M. Xiao, X. Chen, “Shear-force, constant-height, and constant-intensity imaging in scanning near field optical microscopy with s- and p-polarized incident light,” Opt. Eng. (Bellingham) 39, 2495–2500 (2000).
[CrossRef]

1999 (1)

P. J. Valle, J. J. Greffet, R. Carminati, “Optical contrast, topographic contrast and artifacts in illumination-mode scanning near-field optical microscopy,” J. Appl. Phys. 86, 648–656 (1999).
[CrossRef]

1998 (1)

1997 (1)

S. Bozhevolnyi, “Topographical artifacts and optical resolution in near-field optical microscopy,” J. Opt. Soc. Am. A 14, 2254–2259 (1997).
[CrossRef]

1995 (3)

1994 (1)

1992 (1)

E. Betzig, P. L. Finn, J. S. Weiner, “Combined shearforce and near-field scanning optical microscopy,” Appl. Phys. Lett. 60, 2484–2486 (1992).
[CrossRef]

1990 (1)

1989 (2)

R. C. Reddick, R. J. Warmack, T. L. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767–770 (1989).
[CrossRef]

D. Courjon, K. Sarageddine, M. Spazer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

1984 (1)

D. W. Pohl, W. Denk, M. Lanz, “Optical stethosocopy: image recording with resolution λ∕20,” Appl. Phys. Lett. 44, 651–655 (1984).
[CrossRef]

Baida, F. I.

F. I. Baida, D. Van Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

Berntsen, S.

Betzig, E.

E. Betzig, P. L. Finn, J. S. Weiner, “Combined shearforce and near-field scanning optical microscopy,” Appl. Phys. Lett. 60, 2484–2486 (1992).
[CrossRef]

Bozherolnaya, E.

Bozhevolnyi, S.

Carminati, R.

P. J. Valle, J. J. Greffet, R. Carminati, “Optical contrast, topographic contrast and artifacts in illumination-mode scanning near-field optical microscopy,” J. Appl. Phys. 86, 648–656 (1999).
[CrossRef]

R. Carminati, J. J. Greffet, “Influence of dielectric contrast and topography on the near field scattered by an inhomogeneous surface,” J. Opt. Soc. Am. A 12, 2716–2725 (1995).
[CrossRef]

Chen, X.

M. Xiao, X. Chen, “Shear-force, constant-height, and constant-intensity imaging in scanning near field optical microscopy with s- and p-polarized incident light,” Opt. Eng. (Bellingham) 39, 2495–2500 (2000).
[CrossRef]

Christensen, D. A.

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–195 (1995).
[CrossRef]

Courjon, D.

Denk, W.

D. W. Pohl, W. Denk, M. Lanz, “Optical stethosocopy: image recording with resolution λ∕20,” Appl. Phys. Lett. 44, 651–655 (1984).
[CrossRef]

Fan, Z. Z.

X. E. Wang, Z. Z. Fan, T. T. Tang, “Study on the power transmission and light spot size of optical probes in scanning near-field optical microscopes,” Opt. Commun. 235, 31–40 (2004).
[CrossRef]

Ferrell, T. L.

R. C. Reddick, R. J. Warmack, T. L. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767–770 (1989).
[CrossRef]

Finn, P. L.

E. Betzig, P. L. Finn, J. S. Weiner, “Combined shearforce and near-field scanning optical microscopy,” Appl. Phys. Lett. 60, 2484–2486 (1992).
[CrossRef]

Girard, C.

Greffet, J. J.

P. J. Valle, J. J. Greffet, R. Carminati, “Optical contrast, topographic contrast and artifacts in illumination-mode scanning near-field optical microscopy,” J. Appl. Phys. 86, 648–656 (1999).
[CrossRef]

R. Carminati, J. J. Greffet, “Influence of dielectric contrast and topography on the near field scattered by an inhomogeneous surface,” J. Opt. Soc. Am. A 12, 2716–2725 (1995).
[CrossRef]

Hecht, B.

Labani, B.

Lanz, M.

D. W. Pohl, W. Denk, M. Lanz, “Optical stethosocopy: image recording with resolution λ∕20,” Appl. Phys. Lett. 44, 651–655 (1984).
[CrossRef]

Li, Dejie

Keqian Zhang, Dejie Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer-Verlag, Berlin, 1998).
[CrossRef]

Novotny, L.

Pagani, Y.

F. I. Baida, D. Van Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

Pohl, D. W.

L. Novotny, D. W. Pohl, B. Hecht, “Scanning near-field optical probe with ultrasmall spot size,” Opt. Lett. 20, 970–972 (1995).
[CrossRef] [PubMed]

D. W. Pohl, W. Denk, M. Lanz, “Optical stethosocopy: image recording with resolution λ∕20,” Appl. Phys. Lett. 44, 651–655 (1984).
[CrossRef]

Reddick, R. C.

R. C. Reddick, R. J. Warmack, T. L. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767–770 (1989).
[CrossRef]

Sarageddine, K.

D. Courjon, K. Sarageddine, M. Spazer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Spazer, M.

D. Courjon, K. Sarageddine, M. Spazer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Tanaka, K.

Tanaka, M.

Tang, T. T.

X. E. Wang, Z. Z. Fan, T. T. Tang, “Study on the power transmission and light spot size of optical probes in scanning near-field optical microscopes,” Opt. Commun. 235, 31–40 (2004).
[CrossRef]

Vainstein, A.

A. Vainstein, Electromagnetic Waves (Soviet Radio Press, Moscow, 1957) (in Russian).

Valle, P. J.

P. J. Valle, J. J. Greffet, R. Carminati, “Optical contrast, topographic contrast and artifacts in illumination-mode scanning near-field optical microscopy,” J. Appl. Phys. 86, 648–656 (1999).
[CrossRef]

Van Labeke, D.

F. I. Baida, D. Van Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

B. Labani, C. Girard, D. Courjon, D. Van Labeke, “Optical interaction between a dielectric tip and a nanometric lattice: implications for near-field microscopy,” J. Opt. Soc. Am. B 7, 936–943 (1990).
[CrossRef]

Wang, X. E.

X. E. Wang, Z. Z. Fan, T. T. Tang, “Study on the power transmission and light spot size of optical probes in scanning near-field optical microscopes,” Opt. Commun. 235, 31–40 (2004).
[CrossRef]

Warmack, R. J.

R. C. Reddick, R. J. Warmack, T. L. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767–770 (1989).
[CrossRef]

Weiner, J. S.

E. Betzig, P. L. Finn, J. S. Weiner, “Combined shearforce and near-field scanning optical microscopy,” Appl. Phys. Lett. 60, 2484–2486 (1992).
[CrossRef]

Xiao, M.

M. Xiao, X. Chen, “Shear-force, constant-height, and constant-intensity imaging in scanning near field optical microscopy with s- and p-polarized incident light,” Opt. Eng. (Bellingham) 39, 2495–2500 (2000).
[CrossRef]

Zhang, Keqian

Keqian Zhang, Dejie Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer-Verlag, Berlin, 1998).
[CrossRef]

Appl. Phys. Lett. (2)

D. W. Pohl, W. Denk, M. Lanz, “Optical stethosocopy: image recording with resolution λ∕20,” Appl. Phys. Lett. 44, 651–655 (1984).
[CrossRef]

E. Betzig, P. L. Finn, J. S. Weiner, “Combined shearforce and near-field scanning optical microscopy,” Appl. Phys. Lett. 60, 2484–2486 (1992).
[CrossRef]

J. Appl. Phys. (1)

P. J. Valle, J. J. Greffet, R. Carminati, “Optical contrast, topographic contrast and artifacts in illumination-mode scanning near-field optical microscopy,” J. Appl. Phys. 86, 648–656 (1999).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

Opt. Commun. (3)

D. Courjon, K. Sarageddine, M. Spazer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

F. I. Baida, D. Van Labeke, Y. Pagani, “Body-of-revolution FDTD simulations of improved tip performance for scanning near-field optical microscopes,” Opt. Commun. 225, 241–252 (2003).
[CrossRef]

X. E. Wang, Z. Z. Fan, T. T. Tang, “Study on the power transmission and light spot size of optical probes in scanning near-field optical microscopes,” Opt. Commun. 235, 31–40 (2004).
[CrossRef]

Opt. Eng. (Bellingham) (1)

M. Xiao, X. Chen, “Shear-force, constant-height, and constant-intensity imaging in scanning near field optical microscopy with s- and p-polarized incident light,” Opt. Eng. (Bellingham) 39, 2495–2500 (2000).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

R. C. Reddick, R. J. Warmack, T. L. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767–770 (1989).
[CrossRef]

Ultramicroscopy (1)

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–195 (1995).
[CrossRef]

Other (3)

A. Vainstein, Electromagnetic Waves (Soviet Radio Press, Moscow, 1957) (in Russian).

Keqian Zhang, Dejie Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer-Verlag, Berlin, 1998).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Model of 3D SNOM system.

Fig. 2
Fig. 2

Local orthogonal coordinates.

Fig. 3
Fig. 3

Distributions of the solved Borgnis potentials and their derivatives on (a) the interface S w inside the probe and (b) the interface S s inside the sample.

Fig. 4
Fig. 4

Amplitude of the Borgnis potentials in the near-field region of the probe on meridian planes: (a) U on the plane ϕ = 0 ° , (b) V on the plane ϕ = 90 ° .

Fig. 5
Fig. 5

Electric and magnetic field intensity on the plane z = λ 0 200 in front of the probe: (a) electric field intensity, (b) magnetic field intensity.

Fig. 6
Fig. 6

Near-field electric and magnetic field intensity on meridian planes: (a) electric field intensity on the plane ϕ = 0 ° , (b) magnetic field intensity on the plane ϕ = 90 ° .

Tables (1)

Tables Icon

Table 1 Reflectivity and Transmittivity of the Probe for Different Positions of the Additional Matching Point

Equations (80)

Equations on this page are rendered with MathJax. Learn more.

E r = 2 U ( r , ϕ , z ) r z j ϖ μ r V ( r , ϕ , z ) ϕ ,
E ϕ = 2 U ( r , ϕ , z ) r ϕ z + j ϖ μ V ( r , ϕ , z ) r ,
E z = ( 2 z 2 + k 2 ) U ( r , ϕ , z ) ,
H r = 2 V ( r , ϕ , z ) r z + j ϖ ϵ r U ( r , ϕ , z ) ϕ ,
H ϕ = 2 V ( r , ϕ , z ) r ϕ z j ϖ ϵ U ( r , ϕ , z ) r ,
H z = ( 2 z 2 + k 2 ) V ( r , ϕ , z ) .
2 F i + k i 2 F i = 0 ( i = 1 , 2 , 3 ) ,
c F i ( r ) = S i [ G i ( r , r ) F i b ( r ) n F i b ( r ) G i ( r , r ) n ] d s ,
c = { 1 P in Q i 1 2 P on S i } ;
2 G i + k i 2 G i = δ ( r r ) ( i = 1 , 2 , 3 ) ,
G i ( r , r ) = 1 4 π exp ( j k i r r ) r r = 1 4 π exp { j k i [ r 2 + r 2 2 r r cos ( ϕ ϕ ) + ( z z ) 2 ] 1 2 } [ r 2 + r 2 2 r r cos ( ϕ ϕ ) + ( z z ) 2 ] 1 2 ,
1 2 F 1 b ( r ) = S w G 1 ( r , r ) F 1 b w ( r ) n F 1 b w ( r ) G 1 ( r , r ) n d s + S t G 1 ( r , r ) F 1 b t ( r ) n F 1 b t ( r ) G 1 ( r , r ) n d s ,
1 2 F 2 b ( r ) = S w G 2 ( r , r ) F 2 b w ( r ) n F 2 b w ( r ) G 2 ( r , r ) n d s + S t G 2 ( r , r ) F 2 b t ( r ) n F 2 b t ( r ) G 2 ( r , r ) n d s + S s G 2 ( r , r ) F 2 b s ( r ) n F 2 b s ( r ) G 2 ( r , r ) n d s ,
1 2 F 3 b s ( r ) = S s G 3 ( r , r ) F 3 b s ( r ) n F 3 b s ( r ) G 3 ( r , r ) n d s ,
F i b w = n = 1 N A n F i b + ( n ) ( r ) + n = 1 M R n F i b ( n ) ( r ) + F i b l ( i = 1 , 2 ) ,
1 2 F 1 b ± ( n ) ( r ) = S w [ G 1 ( r , r ) F 1 b ± ( n ) ( r ) n F 1 b ± ( n ) ( r ) G 1 ( r , r ) n ] d s + W in ± ( n ) ( r ) ;
S w [ G 1 ( r , r ) F 1 b ± ( n ) ( r ) n F 1 b ± ( n ) ( r ) G 1 ( r , r ) n ] d s
+ W in ± ( n ) ( r ) = 0 ,
W in ± ( n ) ( r ) = S 0 i [ G 1 ( r , r ) F 1 b ± ( n ) ( r ) z F 1 b ± ( n ) ( r ) G 1 ( r , r ) z ] d s .
1 2 F 2 b ± ( n ) ( r ) = S w [ G 2 ( r , r ) F 2 b ± ( n ) ( r ) n F 2 b ± ( n ) ( r ) G 2 ( r , r ) n ] d s + W out ± ( n ) ( r ) ,
S w [ G 2 ( r , r ) F 2 b ± ( n ) ( r ) n F 2 b ± ( n ) ( r ) G 2 ( r , r ) n ] d s
+ W out ± ( n ) ( r ) = 0 ,
W out ± ( n ) ( r ) = S 0 o [ G 2 ( r , r ) F 2 b ± ( n ) ( r ) z F 2 b ± ( n ) ( r ) G 2 ( r , r ) z ] d s .
n = 1 N A n W in + ( n ) ( r ) = 1 2 F 1 b p ( r ) n = 1 M R n W in ( n ) ( r ) + S w + S t [ G 1 ( r , r ) F 1 b p ( r ) n F 1 b p ( r ) G 1 ( r , r ) n ] d s ,
n = 1 N A n W out + ( n ) ( r ) = 1 2 F 2 b V ( r ) n = 1 M R n W out ( n ) ( r ) + S w + S t [ G 2 ( r , r ) F 2 b p ( r ) n F 2 b p ( r ) G 2 ( r , r ) n ] d s + S s [ G 2 ( r , r ) F 2 b s ( r ) n F 2 b s ( r ) G 2 ( r , r ) n ] d s ,
0 = 1 2 F 3 b s ( r ) + S s [ G 3 ( r , r ) F 3 b s ( r ) n F 3 b s ( r ) G 3 ( r , r ) n ] d s ,
F i b + ( n ) ( r ) = F i b + ( n ) ( r , z ) exp ( j ν 0 ϕ ) ( i = 1 , 2 ) ,
F i b ( n ) ( r ) = F i b ( n ) ( r , z ) exp ( j ν n ϕ ) ( i = 1 , 2 ) ,
F b ( r ) = ν F b ν ( r , z ) exp ( j ν ϕ ) ,
F 1 b p ( r ) = ν F 1 b ν p ( r , z ) exp ( j ν ϕ ) ,
n F 1 b p ( r ) = ν exp ( j ν ϕ ) n F 1 b ν p ( r , z ) .
exp ( j ν 0 ϕ ) n = 1 N A n W in + ( n ) ( r , z ) = n = 1 M [ R n W in ( n ) ( r , z ) exp ( j ν n ϕ ) ] + ν [ Q 1 b ν p ( r , z ) Q 1 b ν p ( r , z ) 1 2 F 1 b ν p ( r , z ) ] exp ( j ν ϕ ) ,
0 2 π exp ( j ν n ϕ ) exp ( j ν m ϕ ) d ϕ = { 2 π , n = m 0 , n m } .
F i b ( n ) ( r ) = F i b ( n ) ( r , z ) exp ( j ν 0 ϕ ) ( i = 1 , 2 ) ,
F b ( r ) = F b ν 0 ( r , z ) exp ( j ν 0 ϕ ) ,
F 1 b p ( r ) = F 1 b ν 0 p ( r , z ) exp ( j ν 0 ϕ ) ,
n F 1 b p ( r ) = exp ( j ν 0 ϕ ) n F 1 b ν 0 p ( r , z ) .
n = 1 N A n W in + ( n ) ( r , z ) = n = 1 M R n W in ( n ) ( r , z ) + Q 1 b ν 0 p ( r , z ) Q 1 b ν 0 p ( r , z ) 1 2 F 1 b p ( r , z ) .
n = 1 N A n W out + ( n ) ( r , z ) = n = 1 M R n W out ( n ) ( r , z ) + Q 2 b ν 0 p ( r , z ) Q 2 b ν 0 p ( r , z ) + Q 2 b ν 0 s ( r , z ) Q 2 b ν 0 s ( r , z ) 1 2 F 2 b p ( r , z ) ,
0 = Q 3 b ν 0 s ( r , z ) Q 3 b ν 0 s ( r , z ) 1 2 F 3 b p ( r , z ) ,
F = F l e l + F n e n .
F r = F e r = sin θ l F l + sin θ n F n ,
F z = F e z = cos θ l F l + cos θ n F n ,
2 F r z = F z e r = sin θ l cos θ l 2 F l 2 + sin ( θ l + θ n ) 2 F l n + sin θ n cos θ n 2 F n 2 ,
2 F z 2 = F z e z = cos 2 θ l 2 F l 2 + 2 cos θ l cos θ n 2 F l n + cos 2 θ n 2 F n 2 .
E r = cos θ l sin θ l 2 U l 2 + sin ( θ l + θ n ) 2 U l n + cos θ n sin θ n 2 U n 2 + ϖ μ ν 0 r V ,
E ϕ = j ν 0 r ( U l cos θ l + U n cos θ n ) + j ϖ μ ( V l sin θ l + V n sin θ n ) ,
E z = cos 2 θ l 2 U l 2 + 2 cos θ l cos θ n 2 U l n + cos 2 θ n 2 U n 2 + k 2 U ,
H r = cos θ l sin θ l 2 V l 2 + sin ( θ l + θ n ) 2 V l n + cos θ n sin θ n 2 V n 2 ϖ ϵ ν 0 r U ,
H ϕ = j ν 0 r ( V l cos θ l + V n cos θ n ) j ϖ ϵ ( U l sin θ l + U n sin θ n ) ,
H z = cos 2 θ l 2 V l 2 + 2 cos θ l cos θ n 2 V l n + cos 2 θ n 2 V n 2 + k 2 V .
E t = E r sin θ l + E z cos θ l ,
E t = cos θ l 2 U l 2 + cos θ n 2 U l n + cos θ l k 2 U + ϖ μ ν 0 r sin θ l V .
H t = cos θ l 2 V l 2 + cos θ n 2 V l n + cos θ l k 2 V ϖ ϵ ν 0 r sin θ l U .
E 1 t = E 2 t , E 1 ϕ = E 2 ϕ ,
H 1 t = H 2 t , H 1 ϕ = H 2 ϕ .
cos θ l 2 U 1 l 2 + cos θ n 2 U 1 l n + k 1 2 cos θ l U 1 + ϖ μ 1 ν 0 r sin θ l V 1
= cos θ l 2 U 2 l 2 + cos θ n 2 U 2 l n + k 2 2 cos θ l U 2 + ϖ μ 2 ν 0 r sin θ l V 2 ,
ν 0 r ( U 1 l cos θ l + U 1 n cos θ n ) + ϖ μ 1 ( V 1 l sin θ l + V 1 n sin θ n ) = ν 0 r ( U 2 l cos θ l + U 2 n cos θ n ) + ϖ μ 2 ( V 2 l sin θ l + V 2 n sin θ n ) ,
cos θ l 2 V 1 l 2 + cos θ n 2 V 1 l n + k 1 2 cos θ l V 1 ϖ ϵ 1 ν 0 r sin θ l U 1 = cos θ l 2 V 2 l 2 + cos θ n 2 V 2 l n + k 2 2 cos θ l V 2 ϖ ϵ 2 ν 0 r sin θ l U 2 ,
ν 0 r ( V 1 l cos θ l + V 1 n cos θ n ) ϖ ϵ 1 ( U 1 l sin θ l + U 1 n sin θ n )
= ν 0 r ( V 2 l cos θ l + V 2 n cos θ n ) ϖ ϵ 2 ( U 2 l sin θ l + U 2 n sin θ n ) ,
E t = 0 , E ϕ = 0 ,
cos θ l 2 U l 2 + cos θ n 2 U l n + k 2 cos θ l U + ϖ μ ν 0 r sin θ l V = 0 ,
ν 0 r ( U l cos θ l + U n cos θ n ) + ϖ μ ( V l sin θ l + V n sin θ n )
= 0 .
Radius of the uniform part of the probe : R = 0.25 λ 0 , Length of the tapered part : L = 0.5 λ 0 , Aperture size : Λ = 0.1 λ 0 , Tapered angle : α = 21.8 ° , Probe sample distance : Δ = 0.01 λ 0 , Refractive indices of the probe sample : n 1 = 1.5 , n 2 = 1.0 , n 3 = 2.0 .
n = 1 N A n W in + ( n ) ( r ) = 1 2 F 1 b p ( r ) n = 1 M R n W in ( n ) ( r ) + ν S w + S t [ exp ( j ν ϕ ) G 1 ( r , r , cos ( ϕ ϕ ) , z , z ) n F 1 b ν p ( r , z ) ] d s ν S w + S t [ exp ( j ν ϕ ) F 1 b ν p ( r , z ) n G 1 ( r , r , cos ( ϕ ϕ ) , z , z ) ] d s ,
W in + ( n ) ( r ) = S 0 i exp ( j ν 0 ϕ ) G 1 ( r , r , cos ( ϕ ϕ ) , z , z ) z F 1 b + ( n ) ( r , z ) d s S 0 i exp ( j ν 0 ϕ ) F 1 b + ( n ) ( r , z ) z G 1 ( r , r , cos ( ϕ ϕ ) , z , z ) d s ,
W in ( n ) ( r ) = S 0 i exp ( j ν n ϕ ) G 1 ( r , r , cos ( ϕ ϕ ) , z , z ) z F 1 b ( n ) ( r , z ) d s S 0 i exp ( j ν n ϕ ) F 1 b ( n ) ( r , z ) z G 1 ( r , r , cos ( ϕ ϕ ) , z , z ) d s .
d s = r d l d ϕ ,
ν S w + S t [ exp ( j ν ϕ ) G 1 ( r , r , cos ( ϕ ϕ ) , z , z ) n F 1 b ν p ( r , z ) ] r d l d ϕ = ν exp ( j ν ϕ ) S w + S t [ exp ( j ν ϕ ) G 1 ( r , r , cos ϕ , z , z ) n F 1 b ν p ( r , z ) ] r d l d ϕ = ν exp ( j ν ϕ ) Q 1 b ν p ( r , z ) ,
Q 1 b ν p ( r , z ) = S w + S t [ exp ( j ν ϕ ) G 1 ( r , r , cos ϕ , z , z ) n F 1 b ν p ( r , z ) ] r d l d ϕ .
ν S w + S t [ exp ( j ν ϕ ) F 1 b ν p ( r , z ) n G 1 ( r , r , cos ( ϕ ϕ ) , z , z ) ] d s = ν exp ( j ν ϕ ) Q 1 b ν p ( r , z ) ,
Q 1 b ν p ( r , z ) = S w + S t [ exp ( j ν ϕ ) F 1 b ν p ( r , z ) n G 1 ( r , r , cos ϕ , z , z ) ] r d l d ϕ .
W in + ( n ) ( r ) = W in + ( n ) ( r , z ) exp ( j ν 0 ϕ ) ,
W in ( n ) ( r ) = W in ( n ) ( r , z ) exp ( j ν n ϕ ) ,
W in + ( n ) ( r , z ) = S 0 i [ exp ( j ν 0 ϕ ) G 1 ( r , r , cos ϕ , z , z ) z F 1 b + ( n ) ( r , z ) ] r d l d ϕ S 0 i [ exp ( j ν 0 ϕ ) F 1 b + ( n ) ( r , z ) z G 1 ( r , r , cos ϕ , z , z ) ] r d l d ϕ ,
W in ( n ) ( r , z ) = S 0 i [ exp ( j ν n ϕ ) G 1 ( r , r , cos ϕ , z , z ) z F 1 b ( n ) ( r , z ) ] r d l d ϕ S 0 i [ exp ( j ν n ϕ ) F 1 b ( n ) ( r , z ) z G 1 ( r , r , cos ϕ , z , z ) ] r d l d ϕ .
exp ( j ν 0 ϕ ) n = 1 N A n W in + ( n ) ( r , z ) = n = 1 M [ R n W in ( n ) ( r , z ) exp ( j ν n ϕ ) ] + ν [ Q 1 b ν p ( r , z ) Q 1 b ν p ( r , z ) 1 2 F 1 b ν p ( r , z ) ] exp ( j ν ϕ ) .

Metrics