Abstract

The transmission of light through an infinite slit in a thick perfectly conducting screen is investigated. The spatial distribution of the near-field energy flux is determined through the formulation of four coupled integral equations, which are solved numerically. Transmission coefficients calculated by this method are in agreement with those determined by an alternative formulation. The results theoretically demonstrate the feasibility of near-field superresolution microscopy, in which the collimated radiation passed by an aperture is used to circumvent the diffraction limit of conventional optics, and further suggest the feasibility of near-field superresolution acoustic imaging.

© 1986 Optical Society of America

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References

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  1. H. A. Bethe, “Theory of Diffraction by Small Holes,” Phys. Rev. 66, 163 (1944).
    [CrossRef]
  2. C. M. Butler, Y. Rahmat-Samii, R. Mittra, “Electromagnetic Penetration Through Apertures in Conducting Surfaces,” IEEE Trans. Antennas Propag. AP-26, 82 (1978).
    [CrossRef]
  3. E. A. Ash, G. Nicholls, “Super-Resolution Aperture Scanning Microscope,” Nature London 237, 510 (1972).
    [CrossRef] [PubMed]
  4. R. F. Harrington, D. T. Auckland, “Electromagnetic Transmission Through Narrow Slots in Thick Conducting Screens,” IEEE Trans. Antennas Propag. AP-28, 616 (1980).
    [CrossRef]
  5. F. L. Neerhoff, G. Mur, “Diffraction by a Slit in a Thick Screen,” Appl. Sci. Res. 28, 73 (1973).
  6. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 812–818.
  7. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 480, 496.
  8. G. Binnig, H. Rohrer, “Scanning Tunneling Microscopy,” Helv. Phys. Acta 55, 726 (1982).
  9. J. R. Matey, J. Blanc, “Scanning Capacitance Microscopy,” J. Appl. Phys. 57, 1437 (1985).
    [CrossRef]
  10. A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å Spatial Resolution Light Microscope,” Ultramicroscopy 13, 227 (1984).
    [CrossRef]
  11. E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, E. Kratschmer, “Near-Field Scanning Optical Microscopy (NSOM): Development and Biophysical Applications,” Biophys. J. 49, 269 (1986).
    [CrossRef] [PubMed]
  12. U. Ch. Fischer, “Optical Characteristics of 0.1 μm Circular Apertures in a Metal Film as Light Sources for Scanning Ultramicroscopy,” J. Vac. Sci. Technol. B3, 386 (1985).
  13. D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording With Resolution λ/20,” Appl. Phys. Lett. 44, 652 (1984).
    [CrossRef]
  14. G. A. Massey, J. A. Davis, S. M. Katnik, E. Omon, “Subwavelength Resolution Far-Infrared Microscopy,” Appl. Opt. 24, 1498 (1985).
    [CrossRef] [PubMed]
  15. A. Barraud, “Polymerization in Langmuir-Blodgett Films and Resist Applications,” Thin Solid Films 99, 317 (1983).
    [CrossRef]
  16. U. Ch. Fischer, H. P. Zingsheim, “Submicroscopic Pattern Replication With Visible Light,” J. Vac. Sci. Technol. 19, 881 (1981).
    [CrossRef]
  17. K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
    [CrossRef]
  18. B. Hadimioglu, C. F. Quate, “Water Acoustic Microscopy at Suboptical Wavelengths,” Appl. Phys. Lett. 43, 1006 (1983).
    [CrossRef]
  19. J. W. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, “Optical Properties of Metals,” Phys. Daten 18-2, 74 (1981).
  20. G. A. Massey, “Microscopy and Pattern Generation With Scanned Evanescent Waves,” Appl. Opt. 23, 658 (1984).
    [CrossRef] [PubMed]
  21. Y. Leviatan, Israel Institute of Technology; private communication.
  22. N. A. McDonald, “Electric and Magnetic Coupling through Small Apertures in Shield Walls of Any Thickness,” IEEE Trans. Microwave Theory Tech. MTT-20, 689 (1972).
    [CrossRef]

1986 (1)

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, E. Kratschmer, “Near-Field Scanning Optical Microscopy (NSOM): Development and Biophysical Applications,” Biophys. J. 49, 269 (1986).
[CrossRef] [PubMed]

1985 (3)

U. Ch. Fischer, “Optical Characteristics of 0.1 μm Circular Apertures in a Metal Film as Light Sources for Scanning Ultramicroscopy,” J. Vac. Sci. Technol. B3, 386 (1985).

G. A. Massey, J. A. Davis, S. M. Katnik, E. Omon, “Subwavelength Resolution Far-Infrared Microscopy,” Appl. Opt. 24, 1498 (1985).
[CrossRef] [PubMed]

J. R. Matey, J. Blanc, “Scanning Capacitance Microscopy,” J. Appl. Phys. 57, 1437 (1985).
[CrossRef]

1984 (3)

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å Spatial Resolution Light Microscope,” Ultramicroscopy 13, 227 (1984).
[CrossRef]

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording With Resolution λ/20,” Appl. Phys. Lett. 44, 652 (1984).
[CrossRef]

G. A. Massey, “Microscopy and Pattern Generation With Scanned Evanescent Waves,” Appl. Opt. 23, 658 (1984).
[CrossRef] [PubMed]

1983 (2)

B. Hadimioglu, C. F. Quate, “Water Acoustic Microscopy at Suboptical Wavelengths,” Appl. Phys. Lett. 43, 1006 (1983).
[CrossRef]

A. Barraud, “Polymerization in Langmuir-Blodgett Films and Resist Applications,” Thin Solid Films 99, 317 (1983).
[CrossRef]

1982 (1)

G. Binnig, H. Rohrer, “Scanning Tunneling Microscopy,” Helv. Phys. Acta 55, 726 (1982).

1981 (2)

U. Ch. Fischer, H. P. Zingsheim, “Submicroscopic Pattern Replication With Visible Light,” J. Vac. Sci. Technol. 19, 881 (1981).
[CrossRef]

J. W. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, “Optical Properties of Metals,” Phys. Daten 18-2, 74 (1981).

1980 (2)

K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
[CrossRef]

R. F. Harrington, D. T. Auckland, “Electromagnetic Transmission Through Narrow Slots in Thick Conducting Screens,” IEEE Trans. Antennas Propag. AP-28, 616 (1980).
[CrossRef]

1978 (1)

C. M. Butler, Y. Rahmat-Samii, R. Mittra, “Electromagnetic Penetration Through Apertures in Conducting Surfaces,” IEEE Trans. Antennas Propag. AP-26, 82 (1978).
[CrossRef]

1973 (1)

F. L. Neerhoff, G. Mur, “Diffraction by a Slit in a Thick Screen,” Appl. Sci. Res. 28, 73 (1973).

1972 (2)

E. A. Ash, G. Nicholls, “Super-Resolution Aperture Scanning Microscope,” Nature London 237, 510 (1972).
[CrossRef] [PubMed]

N. A. McDonald, “Electric and Magnetic Coupling through Small Apertures in Shield Walls of Any Thickness,” IEEE Trans. Microwave Theory Tech. MTT-20, 689 (1972).
[CrossRef]

1944 (1)

H. A. Bethe, “Theory of Diffraction by Small Holes,” Phys. Rev. 66, 163 (1944).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 480, 496.

Ash, E. A.

E. A. Ash, G. Nicholls, “Super-Resolution Aperture Scanning Microscope,” Nature London 237, 510 (1972).
[CrossRef] [PubMed]

Auckland, D. T.

R. F. Harrington, D. T. Auckland, “Electromagnetic Transmission Through Narrow Slots in Thick Conducting Screens,” IEEE Trans. Antennas Propag. AP-28, 616 (1980).
[CrossRef]

Barraud, A.

A. Barraud, “Polymerization in Langmuir-Blodgett Films and Resist Applications,” Thin Solid Films 99, 317 (1983).
[CrossRef]

Bethe, H. A.

H. A. Bethe, “Theory of Diffraction by Small Holes,” Phys. Rev. 66, 163 (1944).
[CrossRef]

Betzig, E.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, E. Kratschmer, “Near-Field Scanning Optical Microscopy (NSOM): Development and Biophysical Applications,” Biophys. J. 49, 269 (1986).
[CrossRef] [PubMed]

Binnig, G.

G. Binnig, H. Rohrer, “Scanning Tunneling Microscopy,” Helv. Phys. Acta 55, 726 (1982).

Blanc, J.

J. R. Matey, J. Blanc, “Scanning Capacitance Microscopy,” J. Appl. Phys. 57, 1437 (1985).
[CrossRef]

Butler, C. M.

C. M. Butler, Y. Rahmat-Samii, R. Mittra, “Electromagnetic Penetration Through Apertures in Conducting Surfaces,” IEEE Trans. Antennas Propag. AP-26, 82 (1978).
[CrossRef]

Davis, J. A.

Denk, W.

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording With Resolution λ/20,” Appl. Phys. Lett. 44, 652 (1984).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 812–818.

Fischer, U. Ch.

U. Ch. Fischer, “Optical Characteristics of 0.1 μm Circular Apertures in a Metal Film as Light Sources for Scanning Ultramicroscopy,” J. Vac. Sci. Technol. B3, 386 (1985).

U. Ch. Fischer, H. P. Zingsheim, “Submicroscopic Pattern Replication With Visible Light,” J. Vac. Sci. Technol. 19, 881 (1981).
[CrossRef]

Hadimioglu, B.

B. Hadimioglu, C. F. Quate, “Water Acoustic Microscopy at Suboptical Wavelengths,” Appl. Phys. Lett. 43, 1006 (1983).
[CrossRef]

Harootunian, A.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, E. Kratschmer, “Near-Field Scanning Optical Microscopy (NSOM): Development and Biophysical Applications,” Biophys. J. 49, 269 (1986).
[CrossRef] [PubMed]

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å Spatial Resolution Light Microscope,” Ultramicroscopy 13, 227 (1984).
[CrossRef]

Harrington, R. F.

R. F. Harrington, D. T. Auckland, “Electromagnetic Transmission Through Narrow Slots in Thick Conducting Screens,” IEEE Trans. Antennas Propag. AP-28, 616 (1980).
[CrossRef]

Isaacson, M.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, E. Kratschmer, “Near-Field Scanning Optical Microscopy (NSOM): Development and Biophysical Applications,” Biophys. J. 49, 269 (1986).
[CrossRef] [PubMed]

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å Spatial Resolution Light Microscope,” Ultramicroscopy 13, 227 (1984).
[CrossRef]

Katnik, S. M.

Kemmerer, C. T.

K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
[CrossRef]

Koch, E. E.

J. W. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, “Optical Properties of Metals,” Phys. Daten 18-2, 74 (1981).

Krafka, C.

J. W. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, “Optical Properties of Metals,” Phys. Daten 18-2, 74 (1981).

Kratschmer, E.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, E. Kratschmer, “Near-Field Scanning Optical Microscopy (NSOM): Development and Biophysical Applications,” Biophys. J. 49, 269 (1986).
[CrossRef] [PubMed]

Lamberte, V. E.

K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
[CrossRef]

Lanz, M.

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording With Resolution λ/20,” Appl. Phys. Lett. 44, 652 (1984).
[CrossRef]

Leviatan, Y.

Y. Leviatan, Israel Institute of Technology; private communication.

Lewis, A.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, E. Kratschmer, “Near-Field Scanning Optical Microscopy (NSOM): Development and Biophysical Applications,” Biophys. J. 49, 269 (1986).
[CrossRef] [PubMed]

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å Spatial Resolution Light Microscope,” Ultramicroscopy 13, 227 (1984).
[CrossRef]

Lynch, D. W.

J. W. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, “Optical Properties of Metals,” Phys. Daten 18-2, 74 (1981).

Massey, G. A.

Matey, J. R.

J. R. Matey, J. Blanc, “Scanning Capacitance Microscopy,” J. Appl. Phys. 57, 1437 (1985).
[CrossRef]

McDonald, N. A.

N. A. McDonald, “Electric and Magnetic Coupling through Small Apertures in Shield Walls of Any Thickness,” IEEE Trans. Microwave Theory Tech. MTT-20, 689 (1972).
[CrossRef]

Mittra, R.

C. M. Butler, Y. Rahmat-Samii, R. Mittra, “Electromagnetic Penetration Through Apertures in Conducting Surfaces,” IEEE Trans. Antennas Propag. AP-26, 82 (1978).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 812–818.

Mur, G.

F. L. Neerhoff, G. Mur, “Diffraction by a Slit in a Thick Screen,” Appl. Sci. Res. 28, 73 (1973).

Muray, A.

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å Spatial Resolution Light Microscope,” Ultramicroscopy 13, 227 (1984).
[CrossRef]

Neerhoff, F. L.

F. L. Neerhoff, G. Mur, “Diffraction by a Slit in a Thick Screen,” Appl. Sci. Res. 28, 73 (1973).

Nicholls, G.

E. A. Ash, G. Nicholls, “Super-Resolution Aperture Scanning Microscope,” Nature London 237, 510 (1972).
[CrossRef] [PubMed]

Omon, E.

Pohl, D. W.

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording With Resolution λ/20,” Appl. Phys. Lett. 44, 652 (1984).
[CrossRef]

Quate, C. F.

B. Hadimioglu, C. F. Quate, “Water Acoustic Microscopy at Suboptical Wavelengths,” Appl. Phys. Lett. 43, 1006 (1983).
[CrossRef]

Rahmat-Samii, Y.

C. M. Butler, Y. Rahmat-Samii, R. Mittra, “Electromagnetic Penetration Through Apertures in Conducting Surfaces,” IEEE Trans. Antennas Propag. AP-26, 82 (1978).
[CrossRef]

Rohrer, H.

G. Binnig, H. Rohrer, “Scanning Tunneling Microscopy,” Helv. Phys. Acta 55, 726 (1982).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 480, 496.

Tai, K. L.

K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
[CrossRef]

Timko, A. G.

K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
[CrossRef]

Vadimsky, R. G.

K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
[CrossRef]

Wagner, J. S.

K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
[CrossRef]

Weaver, J. W.

J. W. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, “Optical Properties of Metals,” Phys. Daten 18-2, 74 (1981).

Zingsheim, H. P.

U. Ch. Fischer, H. P. Zingsheim, “Submicroscopic Pattern Replication With Visible Light,” J. Vac. Sci. Technol. 19, 881 (1981).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

D. W. Pohl, W. Denk, M. Lanz, “Optical Stethoscopy: Image Recording With Resolution λ/20,” Appl. Phys. Lett. 44, 652 (1984).
[CrossRef]

B. Hadimioglu, C. F. Quate, “Water Acoustic Microscopy at Suboptical Wavelengths,” Appl. Phys. Lett. 43, 1006 (1983).
[CrossRef]

Appl. Sci. Res. (1)

F. L. Neerhoff, G. Mur, “Diffraction by a Slit in a Thick Screen,” Appl. Sci. Res. 28, 73 (1973).

Biophys. J. (1)

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, E. Kratschmer, “Near-Field Scanning Optical Microscopy (NSOM): Development and Biophysical Applications,” Biophys. J. 49, 269 (1986).
[CrossRef] [PubMed]

Helv. Phys. Acta (1)

G. Binnig, H. Rohrer, “Scanning Tunneling Microscopy,” Helv. Phys. Acta 55, 726 (1982).

IEEE Trans. Antennas Propag. (2)

C. M. Butler, Y. Rahmat-Samii, R. Mittra, “Electromagnetic Penetration Through Apertures in Conducting Surfaces,” IEEE Trans. Antennas Propag. AP-26, 82 (1978).
[CrossRef]

R. F. Harrington, D. T. Auckland, “Electromagnetic Transmission Through Narrow Slots in Thick Conducting Screens,” IEEE Trans. Antennas Propag. AP-28, 616 (1980).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

N. A. McDonald, “Electric and Magnetic Coupling through Small Apertures in Shield Walls of Any Thickness,” IEEE Trans. Microwave Theory Tech. MTT-20, 689 (1972).
[CrossRef]

J. Appl. Phys. (1)

J. R. Matey, J. Blanc, “Scanning Capacitance Microscopy,” J. Appl. Phys. 57, 1437 (1985).
[CrossRef]

J. Vac. Sci. Technol. (3)

U. Ch. Fischer, “Optical Characteristics of 0.1 μm Circular Apertures in a Metal Film as Light Sources for Scanning Ultramicroscopy,” J. Vac. Sci. Technol. B3, 386 (1985).

U. Ch. Fischer, H. P. Zingsheim, “Submicroscopic Pattern Replication With Visible Light,” J. Vac. Sci. Technol. 19, 881 (1981).
[CrossRef]

K. L. Tai, R. G. Vadimsky, C. T. Kemmerer, J. S. Wagner, V. E. Lamberte, A. G. Timko, “Submicron Optical Lithography Using an Inorganic Resist/Polymer Bilevel Scheme,” J. Vac. Sci. Technol. 17(5), 1169 (1980).
[CrossRef]

Nature London (1)

E. A. Ash, G. Nicholls, “Super-Resolution Aperture Scanning Microscope,” Nature London 237, 510 (1972).
[CrossRef] [PubMed]

Phys. Daten (1)

J. W. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, “Optical Properties of Metals,” Phys. Daten 18-2, 74 (1981).

Phys. Rev. (1)

H. A. Bethe, “Theory of Diffraction by Small Holes,” Phys. Rev. 66, 163 (1944).
[CrossRef]

Thin Solid Films (1)

A. Barraud, “Polymerization in Langmuir-Blodgett Films and Resist Applications,” Thin Solid Films 99, 317 (1983).
[CrossRef]

Ultramicroscopy (1)

A. Lewis, M. Isaacson, A. Harootunian, A. Muray, “Development of a 500 Å Spatial Resolution Light Microscope,” Ultramicroscopy 13, 227 (1984).
[CrossRef]

Other (3)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 812–818.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 480, 496.

Y. Leviatan, Israel Institute of Technology; private communication.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the near-field scanning optical microscope.

Fig. 2
Fig. 2

Model for diffraction by a slit in a thick screen.

Fig. 3
Fig. 3

Regions of integration used for the boundary value problem.

Fig. 4
Fig. 4

Transmission coefficients for λ = 5000-Å radiation as a function of screen thickness for slits of λ/2.5, λ/5, λ/10, and λ/20 widths.

Fig. 5
Fig. 5

Distributions of Sz, the component of the energy flux normal to the screen, transmitted by slits of width (A) λ/20; (B) λ/10; (C) λ/5; (D) λ/2.5 for λ = 5000-Å radiation.

Fig. 6
Fig. 6

Rate of spreading of λ = 5000-Å radiation past the screen as determined by the FWHM of Sz for slits of λ/2.5, λ/5, λ/10, and λ/20 widths.

Equations (91)

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H ( x , y , z , t ) = U ( x , z ) exp ( - i ω t ) e ^ y .
E x ( x , z ) = - i c ω z U ( x , z ) ;
E y ( x , z ) = 0 ;
E x ( x , z ) = - i c ω x U ( x , z ) .
( 2 + k j 2 ) U j = 0             ( j = 1 , 2 , 3 ) .
U 1 ( x , z ) = U i ( x , z ) + U r ( x , z ) + U d ( x , z ) ,
( 2 + k 1 2 ) U i , r , d = 0.
U i ( x , z ) = exp [ i k 1 ( x sin θ - z cos θ ) ]             θ < π / 2.
U r ( x , z ) = U i ( x , 2 b - z ) .
n U = 0 at the conductor surface ,
U d , U 3 o ( 1 / R ) as R ;
n U d , n U 3 o ( 1 / R ) as R .
U 1 ( x , z ) z b + = U 2 ( x , z ) z b -             for x < a ; continuity of tangential H
U 2 ( x , z ) z 0 + = U 3 ( x , z ) z 0 -             for x < a ;
2 z U 1 ( x , z ) z b + = 1 z U 2 ( x , z ) z b -             for x < a ; continuity of tangential E
3 z U 2 ( x , z ) z 0 + = 2 z U 3 ( x , z ) z 0 -             for x < a .
( 2 + k j 2 ) G j = - δ ( x - x , z - z )             ( j = 1 , 2 , 3 ) ,
U ( x , z ) = Boundary ( G n U - n G ) d S ,
z G 1 ( x , z ) z b + = 0             for x < ;
z G 3 ( x , z ) z 0 - = 0             for x < ;
x G 2 ( x , z ) x a - = 0             for 0 < z < b ;
x G 2 ( x , z ) x - a + = 0             for 0 < z < b .
G 1 , G 3 ( r , t ) exp [ - i ( ω t - k · r ) ] as r .
G 1 ( x , z ; x , z ) = i 4 [ H 0 ( 1 ) ( k 1 R ) + H 0 ( 1 ) ( k 1 R ) ] ,
G 3 ( x , z ; x , z ) = i 4 [ H 0 ( 1 ) ( k 3 R ) + H 0 ( 1 ) ( k 3 R ) ] ,
R = ( x - x ) 2 + ( z - z ) 2 ,
R = ( x - x ) 2 + ( z + z - 2 b ) 2 ,
R = ( x - x ) 2 + ( z + z ) 2 .
G 2 ( x , z ; x z ) = i 4 a γ 0 exp ( i γ 0 z - z ) + i 2 a m = 1 γ m - 1 × cos [ m π ( x + a ) 2 a ] cos [ m π ( x + a ) 2 a ] × exp ( i γ m z - z ) ,
γ m = k 2 2 - ( m π 2 a ) 2             Re ( γ m ) 0             Im ( γ m ) 0.
U d ( x , z ) = - - a a ( 1 2 ) G 1 ( x , z ; x , b ) D U b ( x ) d x             for b < z < ,
U 3 ( x , z ) = - a a ( 3 2 ) G 3 ( x , z ; x , 0 ) D U 0 ( x ) d x             for - < z < 0 ,
U 2 ( x , z ) = - - a a [ G 2 ( x , z ; x , 0 ) D U 0 ( x ) - U 0 ( x ) z G 2 ( x , z ; x z ) z 0 + ] d x + - a a [ G 2 ( x , z ; x , b ) D U b ( x ) - U b ( x ) z G 2 ( x , z ; x z ) z b - ] d x             for x < a and 0 < z < b ,
U 0 ( x ) U 2 ( x , z ) z 0 + ,
D U 0 ( x ) z U 2 ( x , z ) z 0 + ,
U b ( x ) U 2 ( x , z ) z b - ,
D U b ( x ) z U 2 ( x , z ) z b - .
2 U b i ( x ) - U b ( x ) = - a a ( 1 2 ) G 1 ( x , b ; x , b ) D U b ( x ) d x             for x < a ,
U 0 ( x ) = - a a ( 3 2 ) G 3 ( x , 0 ; x , 0 ) D U 0 ( x ) d x             for x < a ,
1 2 U b ( x ) = - - a a [ G 2 ( x , b ; x , 0 ) D U 0 ( x ) - U 0 ( x ) z G 2 ( x , b ; x , z ) z 0 + ] d x + - a a G 0 ( x , b ; x , b ) D U b ( x ) d x             for x < a ,
1 2 U 0 ( x ) = - a a [ G 2 ( x , 0 ; x , b ) D U b ( x ) - U b ( x ) z G 2 ( x , 0 ; x , z ) z b - ] d x - - a a G 0 ( x , 0 ; x , 0 ) D U 0 ( x ) d x             for x < a ,
U b i ( x ) = exp [ i k 1 ( x sin θ - b cos θ ) ]             θ < π 2 .
X j = 2 a ( j - 1 2 ) N - a             j = 1 , 2 , , N .
2 U b i ( x k ) - U b ( x k ) = j = 1 j k N ( 1 2 ) C j G 1 ( x k , b ; x b ) D U b ( x ) d x + ( 1 2 ) C k G 1 ( x k , b ; x , b ) D U b ( x ) d x .
2 U b i ( x k ) - U b ( x k ) = j = 1 j k N 2 a N 1 2 G 1 ( x k , b ; ξ j , b ) D U b ( ξ j ) + 1 2 [ C k G 1 ( x k , b ; x , b ) d x ] D U b ( ξ k ) .
ξ j x j
2 U b i ( x k ) - U b ( x k ) j = 1 N s k , j I D U b ( x j )             k = 1 , 2 , , N ,
s k , j I = 2 a N 1 2 G 1 ( x k , b ; x j , b )             for j k ,
s k , k I = 1 2 C k G 1 ( x k , b ; x , b ) d x
2 U b i - U b = S I D U b ,
( U b i ) j = U b i ( x j )             j = 1 , 2 , , N
s k , j I = i a N 1 2 H 0 ( 1 ) ( 2 k 1 a N k - j )             for j k ;
s k , k I = i a N 1 2 { H 0 ( 1 ) ( k 1 a N ) + π 2 [ H 0 ( k 1 a N ) H 1 ( 1 ) ( k 1 a N ) - H 1 ( k 1 a N ) H 0 ( 1 ) ( k 1 a N ) ] } .
U 0 = S III D U 0 ,
s k , j III = i a N 3 2 H 0 ( 1 ) ( 2 k 3 a N k - j )             for j k ,
s k , k III = i a N 3 2 { H 0 ( 1 ) ( k 3 a N ) + π 2 [ H 0 ( k 3 a N ) H 1 ( 1 ) ( k 3 a N ) - H 1 ( k 3 a N ) H 0 ( 1 ) ( k 3 a N ) ] } .
½ U b = - R II D U 0 + D II U 0 + S II D U b ,
½ U 0 = - S II D U 0 + D II U b + R II D U b ,
r k , j II = i 2 N γ 0 exp ( i γ 0 b ) + i N m = 1 1 γ m T m j , k exp ( i γ m b ) ,
d k , j II = 1 2 N exp ( i γ 0 b ) + 1 N m = 1 T m j , k exp ( i γ m b ) ,
s k , j II = i 2 N γ 0 + i N m = 1 1 γ m T m j , k ,
T m j , k = 2 N m π cos [ m π ( j - 1 / 2 ) N ] cos [ m π ( k - 1 / 2 ) N ] sin ( m π 2 N ) .
A · B = C ,
A m n = { ( D II S III - R II ) m , n for m , n { 1 , 2 , , N } , ( S II + ½ S I ) m , n - N for m { 1 , 2 , , N } n { N + 1 , N + 2 , , 2 N } , ( S II + ½ S III ) m - N , n for m { N + 1 , N + 2 , , 2 N } n { 1 , 2 , , N } , ( D II S I - R II ) m - N , n - N for m , n { N + 1 , N + 2 , , 2 N } ,
B n = { ( D U 0 ) n for n { 1 , 2 , , N } , ( D U b ) n - N for n { N + 1 , N + 2 , , 2 N } ,
C n = { ( U b i ) n for n { 1 , 2 , , N } , ( 2 D II U b i ) n - N for n { N + 1 , N + 2 , , 2 N } .
H ( x , z ) i a N 3 2 j = 1 N H 0 ( 1 ) [ k 3 ( x - x j ) 2 + z 2 ] ( D U 0 ) j e ^ y ,
E x ( x , z ) - a N 3 2 j = 1 N z ( x - x j ) 2 + z 2 H 1 ( 1 ) [ k 3 ( x - x j ) 2 + z 2 ] ( D U 0 ) j .
E y ( x , z ) = 0 ,
E z ( x , z ) a N 3 2 j = 1 N x - x j ( x - x j ) 2 + z 2 H 1 ( 1 ) [ k 3 ( x - x j ) 2 + z 2 ] ( D U 0 ) j .
S = c 16 π ( E × H * + E * × H ) = c 8 π Re ( E × H * ) .
S z S z i = - 1 cos θ Re ( E x H y * ) .
T = - 1 4 a cos θ - a a [ lim z 0 - ( E x H y * + E x * H y ) ] d x .
T = i 1 3 4 k 1 a 2 2 cos θ lim z 0 - [ I 1 - I 2 ] ,
I 1 = - a a d x { [ - a a z G 3 ( x , z ; x , 0 ) D U 0 ( x ) d x ] × [ - a a G 3 * ( x , z ; x , 0 ) D U 0 * ( x ) d x ] } ,
I 2 = - a a d x { [ - a a z G 3 * ( x , z ; x , 0 ) D U 0 * ( x ) d x ] × [ - a a G 3 ( x , z ; x , 0 ) D U 0 ( x ) d x ] } .
T = i 1 3 4 k 1 a 2 2 cos θ - a a d x D U 0 ( x ) - a a d x D U 0 * ( x ) [ I 4 - I 3 ] ,
I 3 ( x , x ) = lim z 0 - - a a G 3 ( x , z ; x , 0 ) z G 3 * ( x , z ; x , 0 ) d x ,
I 4 ( x , x ) = lim z 0 - - a a z G 3 ( x , z ; x , 0 ) G 3 * ( x , z ; x , 0 ) d x .
I 3 = i 2 H 0 ( 1 ) ( k 3 x - x ) ,
I 4 = - i 2 H 0 ( 2 ) ( k 3 x - x ) .
T = a 2 1 2 N 2 k 1 a 2 2 cos θ i = 1 N j = 1 N J 0 ( 2 k 3 a N i - j ) × { Re [ ( D U 0 ) i ] Re [ ( D U 0 ) j ] + Im [ ( D U 0 ) i ] Im [ ( D U 0 ) j ] }
z > 2 a N
T 1 k a at resonance
( 2 + k 2 ) P = 0 ;
k = ρ ( ω / c ) ;
u = i ω ρ P .
n P = 0 at the screen .
E = E 0 exp ( - 2 × 1.841 d / a ) ,
E = E 0 exp ( - d / δ ) ,
2 a 4 × 1.841 δ .

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