Abstract

A finite-difference time-domain code is used to model near-zone electromagnetic probe fields of subwavelength dimensions and the interactions of these fields with a dielectric sample. The magnitude and the phase of the electric and the magnetic fields are determined in the region in which the energy leaving the probe interacts with the sample. An angular-spectrum code is then used to propagate the electric field into the far zone, in which signal detection takes place. TE and TM polarizations in a two-dimensional waveguide are modeled. We examine the effects of scanning the probe over a surface asperity in a dielectric sample. Two different far-zone detection schemes (total-energy detection and differential detection with a split-cell detector) are studied. When the probe scans a well, total-energy detection by TE polarization yields the closest estimate of the well’s actual width, whereas differential detection by TM polarization yields the sharpest profile of the well’s edges. Differential detection is shown to be less sensitive to variations in the probe-to-sample separations during a scan and has minimal distortions with both TE and TM polarizations.

© 1995 Optical Society of America

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References

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  1. D. W. Pohl, D. Courjon, eds., Near Field Optics, NATO Advanced Scientific Institutes Series E242 (Kluwer, Dordrecht, The Netherlands, 1993).
    [CrossRef]
  2. E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
    [CrossRef] [PubMed]
  3. F. Froehlich, T. D. Milster, R. Uber, “High-resolution lithography with a near-field scanning sub-wavelength aperture,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 312–317 (1992).
    [CrossRef]
  4. E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
    [CrossRef]
  5. E. L. Buckland, P. J. Moyer, M. A. Paesler, “Resolution in collection-mode scanning optical microscopy,” J. Appl. Phys. 73, 1018–1028 (1993).
    [CrossRef]
  6. E. Betzig, A. Harootunian, A. Lewis, M. Isaacson, “Near-field diffraction by a slit: implications for superresolution microscopy,” Appl. Opt. 25, 1890–1900 (1986).
    [CrossRef] [PubMed]
  7. Y. Leviatan, “Study of near-zone fields of a small aperture,” J. Appl. Phys. 60, 1577–1583 (1986).
    [CrossRef]
  8. A. Roberts, “Near-zone fields behind circular apertures in thick, perfectly conducting screens,” J. Appl. Phys. 65, 2896–2899 (1989).
    [CrossRef]
  9. A. Roberts, “Small hole coupling of radiation into a near-field probe,” J. Appl. Phys. 70, 4045–4049 (1991).
    [CrossRef]
  10. D. A. Christensen, “Analysis of near field tip patterns including object interactions using finite-difference-time-domain calculations,” J. Ultramicrosc. (to be published).
  11. C. Girard, A. Dereux, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
    [CrossRef]
  12. C. Girard, X. Bouju, “Self-consistent study of dynamical and polarization effects in near-field optical microscopy,” J. Opt. Soc. Am. B 9, 298–305 (1992).
    [CrossRef]
  13. D. Van Labeke, D. Barchiesi, “Scanning-tunneling optical microscopy: a theoretical macroscopic approach,” J. Opt. Soc. Am. A 9, 732–739 (1992).
    [CrossRef]
  14. C. Girard, A. Dereux, “Optical spectroscopy of a surface at the nanometer scale: a theoretical study in real space,” Phys. Rev. B 49, 11344–11351 (1994).
    [CrossRef]
  15. L. Novotny, D. W. Pohl, P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
    [CrossRef]
  16. A. Dereux, D. W. Pohl, “The 90° prism edge as a model SNOM probe: near-field, photon tunneling, and far-field properties,” Ref. 1, pp. 189–198.
  17. J. B. Judkins, R. W. Ziolkowski, “FDTD modeling of nonperfect metallic thin film gratings,” submitted to J. Opt. Soc. Am. A.
  18. J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, J. Judkins, “Numerical analysis of a two dimensional near-field probe,” J. Ultramicrosc. (to be published).
  19. K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, Boca Raton, Fla., 1993), p. 32.
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48–51.
  21. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), pp. 40–80.
  22. W. D. Stanley, Digital Signal Processing (Reston, Reston, Va., 1975), p. 237.
  23. J. E. Harvey, “Light scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).
  24. C. C. Johnson, Field and Wave Electrodynamics (McGraw-Hill, New York, 1965), pp. 331–335.
  25. A. Sommerfeld, Optics (Academic, New York, 1954), pp. 273–289.
  26. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 148–152.
  27. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984), p. 80.
  28. J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 43.

1994 (2)

C. Girard, A. Dereux, “Optical spectroscopy of a surface at the nanometer scale: a theoretical study in real space,” Phys. Rev. B 49, 11344–11351 (1994).
[CrossRef]

L. Novotny, D. W. Pohl, P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
[CrossRef]

1993 (1)

E. L. Buckland, P. J. Moyer, M. A. Paesler, “Resolution in collection-mode scanning optical microscopy,” J. Appl. Phys. 73, 1018–1028 (1993).
[CrossRef]

1992 (4)

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

D. Van Labeke, D. Barchiesi, “Scanning-tunneling optical microscopy: a theoretical macroscopic approach,” J. Opt. Soc. Am. A 9, 732–739 (1992).
[CrossRef]

C. Girard, X. Bouju, “Self-consistent study of dynamical and polarization effects in near-field optical microscopy,” J. Opt. Soc. Am. B 9, 298–305 (1992).
[CrossRef]

1991 (1)

A. Roberts, “Small hole coupling of radiation into a near-field probe,” J. Appl. Phys. 70, 4045–4049 (1991).
[CrossRef]

1990 (1)

C. Girard, A. Dereux, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

1989 (1)

A. Roberts, “Near-zone fields behind circular apertures in thick, perfectly conducting screens,” J. Appl. Phys. 65, 2896–2899 (1989).
[CrossRef]

1986 (2)

Barchiesi, D.

Betzig, E.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

E. Betzig, A. Harootunian, A. Lewis, M. Isaacson, “Near-field diffraction by a slit: implications for superresolution microscopy,” Appl. Opt. 25, 1890–1900 (1986).
[CrossRef] [PubMed]

Bouju, X.

Buckland, E. L.

E. L. Buckland, P. J. Moyer, M. A. Paesler, “Resolution in collection-mode scanning optical microscopy,” J. Appl. Phys. 73, 1018–1028 (1993).
[CrossRef]

Christensen, D. A.

D. A. Christensen, “Analysis of near field tip patterns including object interactions using finite-difference-time-domain calculations,” J. Ultramicrosc. (to be published).

Dereux, A.

C. Girard, A. Dereux, “Optical spectroscopy of a surface at the nanometer scale: a theoretical study in real space,” Phys. Rev. B 49, 11344–11351 (1994).
[CrossRef]

C. Girard, A. Dereux, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

A. Dereux, D. W. Pohl, “The 90° prism edge as a model SNOM probe: near-field, photon tunneling, and far-field properties,” Ref. 1, pp. 189–198.

Finn, P. L.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Froehlich, F.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, J. Judkins, “Numerical analysis of a two dimensional near-field probe,” J. Ultramicrosc. (to be published).

F. Froehlich, T. D. Milster, R. Uber, “High-resolution lithography with a near-field scanning sub-wavelength aperture,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 312–317 (1992).
[CrossRef]

Gaskill, J.

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 43.

Girard, C.

C. Girard, A. Dereux, “Optical spectroscopy of a surface at the nanometer scale: a theoretical study in real space,” Phys. Rev. B 49, 11344–11351 (1994).
[CrossRef]

C. Girard, X. Bouju, “Self-consistent study of dynamical and polarization effects in near-field optical microscopy,” J. Opt. Soc. Am. B 9, 298–305 (1992).
[CrossRef]

C. Girard, A. Dereux, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48–51.

Gyorgy, E. M.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Harootunian, A.

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 148–152.

Harvey, J. E.

J. E. Harvey, “Light scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).

Isaacson, M.

Johnson, C. C.

C. C. Johnson, Field and Wave Electrodynamics (McGraw-Hill, New York, 1965), pp. 331–335.

Judkins, J.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, J. Judkins, “Numerical analysis of a two dimensional near-field probe,” J. Ultramicrosc. (to be published).

Judkins, J. B.

J. B. Judkins, R. W. Ziolkowski, “FDTD modeling of nonperfect metallic thin film gratings,” submitted to J. Opt. Soc. Am. A.

Kann, J. L.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, J. Judkins, “Numerical analysis of a two dimensional near-field probe,” J. Ultramicrosc. (to be published).

Kunz, K. S.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, Boca Raton, Fla., 1993), p. 32.

Leviatan, Y.

Y. Leviatan, “Study of near-zone fields of a small aperture,” J. Appl. Phys. 60, 1577–1583 (1986).
[CrossRef]

Lewis, A.

Luebbers, R. J.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, Boca Raton, Fla., 1993), p. 32.

Milster, T. D.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, J. Judkins, “Numerical analysis of a two dimensional near-field probe,” J. Ultramicrosc. (to be published).

F. Froehlich, T. D. Milster, R. Uber, “High-resolution lithography with a near-field scanning sub-wavelength aperture,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 312–317 (1992).
[CrossRef]

Moyer, P. J.

E. L. Buckland, P. J. Moyer, M. A. Paesler, “Resolution in collection-mode scanning optical microscopy,” J. Appl. Phys. 73, 1018–1028 (1993).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), pp. 40–80.

Novotny, L.

Paesler, M. A.

E. L. Buckland, P. J. Moyer, M. A. Paesler, “Resolution in collection-mode scanning optical microscopy,” J. Appl. Phys. 73, 1018–1028 (1993).
[CrossRef]

Pohl, D. W.

L. Novotny, D. W. Pohl, P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
[CrossRef]

A. Dereux, D. W. Pohl, “The 90° prism edge as a model SNOM probe: near-field, photon tunneling, and far-field properties,” Ref. 1, pp. 189–198.

Regli, P.

Roberts, A.

A. Roberts, “Small hole coupling of radiation into a near-field probe,” J. Appl. Phys. 70, 4045–4049 (1991).
[CrossRef]

A. Roberts, “Near-zone fields behind circular apertures in thick, perfectly conducting screens,” J. Appl. Phys. 65, 2896–2899 (1989).
[CrossRef]

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984), p. 80.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), pp. 273–289.

Stanley, W. D.

W. D. Stanley, Digital Signal Processing (Reston, Reston, Va., 1975), p. 237.

Trautman, J. K.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Uber, R.

F. Froehlich, T. D. Milster, R. Uber, “High-resolution lithography with a near-field scanning sub-wavelength aperture,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 312–317 (1992).
[CrossRef]

Van Labeke, D.

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984), p. 80.

Wolfe, R.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Ziolkowski, R. W.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, J. Judkins, “Numerical analysis of a two dimensional near-field probe,” J. Ultramicrosc. (to be published).

J. B. Judkins, R. W. Ziolkowski, “FDTD modeling of nonperfect metallic thin film gratings,” submitted to J. Opt. Soc. Am. A.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. L. Finn, “Near-field magneto-optics and high density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

J. Appl. Phys. (4)

E. L. Buckland, P. J. Moyer, M. A. Paesler, “Resolution in collection-mode scanning optical microscopy,” J. Appl. Phys. 73, 1018–1028 (1993).
[CrossRef]

Y. Leviatan, “Study of near-zone fields of a small aperture,” J. Appl. Phys. 60, 1577–1583 (1986).
[CrossRef]

A. Roberts, “Near-zone fields behind circular apertures in thick, perfectly conducting screens,” J. Appl. Phys. 65, 2896–2899 (1989).
[CrossRef]

A. Roberts, “Small hole coupling of radiation into a near-field probe,” J. Appl. Phys. 70, 4045–4049 (1991).
[CrossRef]

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

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

Phys. Rev. B (2)

C. Girard, A. Dereux, “Optical spectroscopy of a surface at the nanometer scale: a theoretical study in real space,” Phys. Rev. B 49, 11344–11351 (1994).
[CrossRef]

C. Girard, A. Dereux, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

Science (1)

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Other (16)

F. Froehlich, T. D. Milster, R. Uber, “High-resolution lithography with a near-field scanning sub-wavelength aperture,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 312–317 (1992).
[CrossRef]

D. W. Pohl, D. Courjon, eds., Near Field Optics, NATO Advanced Scientific Institutes Series E242 (Kluwer, Dordrecht, The Netherlands, 1993).
[CrossRef]

D. A. Christensen, “Analysis of near field tip patterns including object interactions using finite-difference-time-domain calculations,” J. Ultramicrosc. (to be published).

A. Dereux, D. W. Pohl, “The 90° prism edge as a model SNOM probe: near-field, photon tunneling, and far-field properties,” Ref. 1, pp. 189–198.

J. B. Judkins, R. W. Ziolkowski, “FDTD modeling of nonperfect metallic thin film gratings,” submitted to J. Opt. Soc. Am. A.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, J. Judkins, “Numerical analysis of a two dimensional near-field probe,” J. Ultramicrosc. (to be published).

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, Boca Raton, Fla., 1993), p. 32.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48–51.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), pp. 40–80.

W. D. Stanley, Digital Signal Processing (Reston, Reston, Va., 1975), p. 237.

J. E. Harvey, “Light scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).

C. C. Johnson, Field and Wave Electrodynamics (McGraw-Hill, New York, 1965), pp. 331–335.

A. Sommerfeld, Optics (Academic, New York, 1954), pp. 273–289.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 148–152.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984), p. 80.

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 43.

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Figures (10)

Fig. 1
Fig. 1

Geometry of our hybrid FDTD/angular-spectrum model. A Pyrex sample is placed a nominal distance h behind the subwavelength aperture. A well is present on the sample’s surface. Scanning is implemented by lateral shifting of the center of the well with respect to the aperture.

Fig. 2
Fig. 2

Magnitude plots of all nonzero components E radiating through a 100-nm-wide aperture into air: (a) magnitude of Ex (TM polarization), (b) magnitude of Ez (TM polarization), (c) magnitude of Ey (TE polarization). Magnitude behind the aperture: solid curves, 1 nm; dotted curves, 35 nm; dashed curves, 50 nm. Normalization is done with respect to the maximum value for the case of z = 1 nm.

Fig. 3
Fig. 3

Phase plots of all nonzero components of E radiating through a 100-nm-wide aperture into air: (a) phase of Ex (TM polarization), (b) phase of Ez (TM polarization), (c) phase of Ey (TE polarization). Phase behind the aperture: solid curves, 1 nm; dashed curves, 50 nm. All the values are in radians and are normalized by 2π.

Fig. 4
Fig. 4

Integrated values of |E|2 on a plane 85 nm behind the 100-nm-wide aperture with (a) TE and (b) TM polarization. A flat dielectric sample of index n is placed in proximity to the aperture. The curves indicate the total amount of electric-field energy coupled into the dielectric. The aperture-to-dielectric spacing is shown on the horizontal axis. Normalization is done with respect to the maximum integrated value.

Fig. 5
Fig. 5

Plots of the total detected power τ at a plane 10 μm past the aperture as the aperture is scanned across a well. The polarization is (a) TE and (b) TM. Aperture width: solid curves, 100 nm; dotted curves, 40 nm. The signal τ is plotted for (a) xs = ±200 nm and (b) xs = ±400 nm.

Fig. 6
Fig. 6

Plots of the normalized differential signal δ at a plane 10 μm past the aperture as the aperture is scanned across a well. The polarization is (a) TE and (b) TM. Aperture width: solid curves, 100 nm; dashed curves, 40 nm.

Fig. 7
Fig. 7

Plots of the integrated normalized differential signal Δ at a plane 10 μm past the aperture as the aperture is scanned across a well. The polarization is (a) TE and (b) TM. Aperture width: solid curves, 100 nm; dashed curves, 40 nm.

Fig. 8
Fig. 8

Plots of the total detected power τ at a plane 10 μm past the aperture as the aperture is scanned across a well with TM polarization. Aperture width: solid curve, 100 nm; dashed curve, 40 nm. For the case of the 100-nm aperture h is set to 35 nm; for that of the 40-nm aperture h is set to 25 nm.

Fig. 9
Fig. 9

Comparison of total and differential detection schemes as h is randomly varied during the scan. Aperture width, 40 nm. The polarization is (a) TE and (b) TM. Solid curves, total detected signal τ; dashed curves, integrated normalized differential signal Δ.

Fig. 10
Fig. 10

Computed images of the well when approximations are applied to the near-zone electric field. All three plots show calculated values of the integrated normalized differential signal Δ by (a) method 1, (b) method 2 with TE polarization, (c) method 2 with TM polarization. Aperture width, 100 nm.

Tables (3)

Tables Icon

Table 1 Constants Relating Phase Fronts of the Tangential Components of E to an Aspheric Surface

Tables Icon

Table 2 Comparison of Total and Differential Detectiona

Tables Icon

Table 3 Predicted Well Parameters with Approximate Methods for Calculating Δ (d = 100 nm)a

Equations (4)

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

ϕ ( x ) = c x 2 1 + [ 1 - ( 1 + k ) c x 2 ] 1 / 2 + A x 4 ,
τ ( x s ; z 0 ) = - E ( x , x s ; z 0 ) 2 d x
δ ( x s ; z 0 ) = - E ( x , x s ; z 0 ) 2 d x - - 0 E ( x , x s ; z 0 ) 2 d x - E ( x , x s ; z 0 ) 2 d x ,
Δ ( x s ; z 0 ) = - x s δ ( x s ; z 0 ) d x s .

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