Abstract

A nonlinear signal-processing model is derived for the optical recording channel based on scalar diffraction theory. In this model, the signal waveform is written in closed form as an explicit function of the channel bits that are stored on an optical disk, thereby comprising both linear and nonlinear terms. Its explicit dependence on the channel bits makes this model well suited for signal-processing purposes. With the model it is also convenient to assess the importance of nonlinear contributions to the signal waveform. The model is applied for one-dimensional optical storage as well as for two-dimensional (2D) optical storage in which bits are arranged on a 2D hexagonal lattice. Signal folding is addressed as a typical nonlinear issue in 2D optical storage and can be eliminated by recording of pit marks of sizes considerably smaller than the size of the hexagonal bit cell. Further simplifications of the model with only a limited number of channel parameters are also derived.

© 2003 Optical Society of America

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References

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  1. J. Pasman, “Vector theory of diffraction,” in Principles of Optical Disc Systems, E. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), Chap. 3, pp. 88–124.
  2. J. M. Brok, H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49, 1811–1829 (2002).
    [CrossRef]
  3. H. H. Hopkins, “Diffraction theory of laser read-out systems for optical video discs,” J. Opt. Soc. Am. 69, 4–24 (1979).
    [CrossRef]
  4. J. Braat, “Read-out of optical discs,” in Principles of Optical Disc Systems, E. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), Chap. 2, pp. 7–87.
  5. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (Mc-Graw Hill, New York, 1996).
  6. “120 mm DVD-Read-Only Disk,” 3rd ed., ECAM Standard 267 (April2001), www.ecma.ch .
  7. S. Kobayashi, “Nonlinear model for an optical read-only-memory disk readout channel based on an edge-spread function,” Appl. Opt. 41, 2679–2685 (2002).
    [CrossRef] [PubMed]
  8. H. Pozidis, J. W. M. Bergmans, W. M. J. Coene, “Modeling and compensation of asymmetry in optical recording,” IEEE Trans. Commun. 50, 2052–2063 (2002).
    [CrossRef]
  9. T. D. Milster, “New way to describe diffraction from optical disks,” Appl. Opt. 37, 6878–6883 (1998).
    [CrossRef]
  10. P. W. Nutter, C. D. Wright, “A new technique for the prediction and correction of nonlinearities in simulated optical readout waveforms,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 82–84 (2002).
    [CrossRef]
  11. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, Oxford, 1981).
  12. W. M. J. Coene, “Two-dimensional optical storage,” in Optical Data Storage 2003, M. O’Neill, N. Miyagawa, eds., Proc. SPIE5069, 90–92 (2003).
  13. A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).
  14. T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
    [CrossRef]
  15. B. Stek, R. Otte, T. Jansen, D. Modrie, “Advanced signalprocessing for the Bluray disc system,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 263–265.
  16. W. Weeks, R. E. Blahut, “The capacity and coding gain of certain checkerboard codes,” IEEE Trans. Inf. Theory 44, 1193–1203 (1998).
    [CrossRef]
  17. T. Kato, S. Taira, T. Maeda, Y. Katayama, T. Nishiya, “Two-dimensional run-length-limited code and partial response maximum likelihood system with multi-track recording,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 51–53.
  18. J. W. M. Bergmans, Digital Baseband Transmission and Recording (Kluwer Academic, Dordrecht, The Netherlands, 1996).
    [CrossRef]

2002

J. M. Brok, H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49, 1811–1829 (2002).
[CrossRef]

S. Kobayashi, “Nonlinear model for an optical read-only-memory disk readout channel based on an edge-spread function,” Appl. Opt. 41, 2679–2685 (2002).
[CrossRef] [PubMed]

H. Pozidis, J. W. M. Bergmans, W. M. J. Coene, “Modeling and compensation of asymmetry in optical recording,” IEEE Trans. Commun. 50, 2052–2063 (2002).
[CrossRef]

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

1998

W. Weeks, R. E. Blahut, “The capacity and coding gain of certain checkerboard codes,” IEEE Trans. Inf. Theory 44, 1193–1203 (1998).
[CrossRef]

T. D. Milster, “New way to describe diffraction from optical disks,” Appl. Opt. 37, 6878–6883 (1998).
[CrossRef]

1979

Bergmans, J. W. M.

H. Pozidis, J. W. M. Bergmans, W. M. J. Coene, “Modeling and compensation of asymmetry in optical recording,” IEEE Trans. Commun. 50, 2052–2063 (2002).
[CrossRef]

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

J. W. M. Bergmans, Digital Baseband Transmission and Recording (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

Blahut, R. E.

W. Weeks, R. E. Blahut, “The capacity and coding gain of certain checkerboard codes,” IEEE Trans. Inf. Theory 44, 1193–1203 (1998).
[CrossRef]

Braat, J.

J. Braat, “Read-out of optical discs,” in Principles of Optical Disc Systems, E. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), Chap. 2, pp. 7–87.

Brok, J. M.

J. M. Brok, H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49, 1811–1829 (2002).
[CrossRef]

Busch, C.

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

Coene, W. M. J.

H. Pozidis, J. W. M. Bergmans, W. M. J. Coene, “Modeling and compensation of asymmetry in optical recording,” IEEE Trans. Commun. 50, 2052–2063 (2002).
[CrossRef]

W. M. J. Coene, “Two-dimensional optical storage,” in Optical Data Storage 2003, M. O’Neill, N. Miyagawa, eds., Proc. SPIE5069, 90–92 (2003).

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

Conway, T.

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, Oxford, 1981).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (Mc-Graw Hill, New York, 1996).

Hattori, M.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

Hekstra, A. P.

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

Hopkins, H. H.

Immink, A. H. J.

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

Jansen, T.

B. Stek, R. Otte, T. Jansen, D. Modrie, “Advanced signalprocessing for the Bluray disc system,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 263–265.

Kahlman, J.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

Katayama, Y.

T. Kato, S. Taira, T. Maeda, Y. Katayama, T. Nishiya, “Two-dimensional run-length-limited code and partial response maximum likelihood system with multi-track recording,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 51–53.

Kato, T.

T. Kato, S. Taira, T. Maeda, Y. Katayama, T. Nishiya, “Two-dimensional run-length-limited code and partial response maximum likelihood system with multi-track recording,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 51–53.

Kobayashi, S.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

S. Kobayashi, “Nonlinear model for an optical read-only-memory disk readout channel based on an edge-spread function,” Appl. Opt. 41, 2679–2685 (2002).
[CrossRef] [PubMed]

Maeda, T.

T. Kato, S. Taira, T. Maeda, Y. Katayama, T. Nishiya, “Two-dimensional run-length-limited code and partial response maximum likelihood system with multi-track recording,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 51–53.

Milster, T. D.

Modrie, D.

B. Stek, R. Otte, T. Jansen, D. Modrie, “Advanced signalprocessing for the Bluray disc system,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 263–265.

Narahara, T.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

Nishiya, T.

T. Kato, S. Taira, T. Maeda, Y. Katayama, T. Nishiya, “Two-dimensional run-length-limited code and partial response maximum likelihood system with multi-track recording,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 51–53.

Nutter, P. W.

P. W. Nutter, C. D. Wright, “A new technique for the prediction and correction of nonlinearities in simulated optical readout waveforms,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 82–84 (2002).
[CrossRef]

Otte, R.

B. Stek, R. Otte, T. Jansen, D. Modrie, “Advanced signalprocessing for the Bluray disc system,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 263–265.

Pasman, J.

J. Pasman, “Vector theory of diffraction,” in Principles of Optical Disc Systems, E. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), Chap. 3, pp. 88–124.

Pozidis, H.

H. Pozidis, J. W. M. Bergmans, W. M. J. Coene, “Modeling and compensation of asymmetry in optical recording,” IEEE Trans. Commun. 50, 2052–2063 (2002).
[CrossRef]

Riani, J.

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

Shimpuku, Y.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

Stek, B.

B. Stek, R. Otte, T. Jansen, D. Modrie, “Advanced signalprocessing for the Bluray disc system,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 263–265.

Taira, S.

T. Kato, S. Taira, T. Maeda, Y. Katayama, T. Nishiya, “Two-dimensional run-length-limited code and partial response maximum likelihood system with multi-track recording,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 51–53.

Urbach, H. P.

J. M. Brok, H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49, 1811–1829 (2002).
[CrossRef]

van Beneden, S. J. L.

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

van den Enden, G.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

van der Lee, A. M.

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

van Dijk, M.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

van Woudenberg, R.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

Weeks, W.

W. Weeks, R. E. Blahut, “The capacity and coding gain of certain checkerboard codes,” IEEE Trans. Inf. Theory 44, 1193–1203 (1998).
[CrossRef]

Wright, C. D.

P. W. Nutter, C. D. Wright, “A new technique for the prediction and correction of nonlinearities in simulated optical readout waveforms,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 82–84 (2002).
[CrossRef]

Appl. Opt.

IEEE Trans. Commun.

H. Pozidis, J. W. M. Bergmans, W. M. J. Coene, “Modeling and compensation of asymmetry in optical recording,” IEEE Trans. Commun. 50, 2052–2063 (2002).
[CrossRef]

IEEE Trans. Inf. Theory

W. Weeks, R. E. Blahut, “The capacity and coding gain of certain checkerboard codes,” IEEE Trans. Inf. Theory 44, 1193–1203 (1998).
[CrossRef]

J. Mod. Opt.

J. M. Brok, H. P. Urbach, “Simulation of polarization effects in diffraction problems of optical recording,” J. Mod. Opt. 49, 1811–1829 (2002).
[CrossRef]

J. Opt. Soc. Am.

Jpn. J. Appl. Phys.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. van den Enden, J. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. Part 1 39, 912–919 (2002).
[CrossRef]

Other

B. Stek, R. Otte, T. Jansen, D. Modrie, “Advanced signalprocessing for the Bluray disc system,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 263–265.

T. Kato, S. Taira, T. Maeda, Y. Katayama, T. Nishiya, “Two-dimensional run-length-limited code and partial response maximum likelihood system with multi-track recording,” in Joint International Symposium on Optical Memory and Optical Data Storage ISOM/ODS 2002 (IEEE/LEOS, Piscataway, New Jersey, 2002), pp. 51–53.

J. W. M. Bergmans, Digital Baseband Transmission and Recording (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

J. Pasman, “Vector theory of diffraction,” in Principles of Optical Disc Systems, E. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), Chap. 3, pp. 88–124.

J. Braat, “Read-out of optical discs,” in Principles of Optical Disc Systems, E. R. Pike, ed. (Adam Hilger, Bristol, UK, 1985), Chap. 2, pp. 7–87.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (Mc-Graw Hill, New York, 1996).

“120 mm DVD-Read-Only Disk,” 3rd ed., ECAM Standard 267 (April2001), www.ecma.ch .

P. W. Nutter, C. D. Wright, “A new technique for the prediction and correction of nonlinearities in simulated optical readout waveforms,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 82–84 (2002).
[CrossRef]

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, Oxford, 1981).

W. M. J. Coene, “Two-dimensional optical storage,” in Optical Data Storage 2003, M. O’Neill, N. Miyagawa, eds., Proc. SPIE5069, 90–92 (2003).

A. H. J. Immink, W. M. J. Coene, A. M. van der Lee, C. Busch, A. P. Hekstra, J. W. M. Bergmans, J. Riani, S. J. L. van Beneden, T. Conway, “Signal processing and coding for two-dimensional optical storage,” in GLOBECOM 2003—2003 IEEE Global Telecommunications Conference (IEEE, Piscataway, New Jersey, to be published).

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Figures (10)

Fig. 1
Fig. 1

Coefficients for linear interference kernels 〈ψ L j 〉 for a DVD pick-up unit (λ = 650 nm; NA = 0.60). Channel bit length T = 133 nm; radial pit width, 260 nm. The laser spot is centered at bit position j = 0.

Fig. 2
Fig. 2

Coefficients for nonlinear interference kernels 〈ψ k j 〉 for a DVD pick-up unit (λ = 650 nm; NA = 0.60). Channel bit length, T = 133 nm; radial pit width, 260 nm. The laser spot is centered at bit position j = 0.

Fig. 3
Fig. 3

Simulated signal waveform according to the model of Section 2 for a DVD pick-up unit (λ = 650 nm; NA = 0.60). Channel bit length, T = 133 nm; radial pit width, 260 nm. The channel bit sequence is -11T-5T-5T-3T-3T-11T-. The complete signal waveform, the linear approximation (with constant term 1 and the linear contributions with 〈ψ L j 〉), and the separate nonlinear (NL) contributions related to 〈ψ k j 〉, where |k - j| = 0, 1, 2 and 3, are shown.

Fig. 4
Fig. 4

Simulated eye pattern according to the model of Section 2 for a DVD pick-up unit (λ = 650 nm; NA = 0.60). Channel bit length, T = 133 nm; radial pit width, 260 nm.

Fig. 5
Fig. 5

Schematic format for 2D optical storage (for simplicity, a seven-row broad spiral is shown). Each hexagon corresponds to a bit cell (white for bit 0, gray for bit 1).

Fig. 6
Fig. 6

Signal patterns for a 2D hexagonal bit lattice (drawn as a function of the cluster type) for λ = 405 nm and NA of 0.85 with lattice parameter a H = 165 nm and diameters of pit holes b ph = 122.5 nm (left) and b ph = 165 nm (right).

Fig. 7
Fig. 7

Signal patterns for a 2D hexagonal bit lattice (drawn as a function of the cluster type) for λ = 405 nm and NA of 0.85 with lattice parameter a H = 138 nm and diameters of the pit holes b ph = 102.5 nm (left) and b ph = 138 nm (right).

Fig. 8
Fig. 8

Two types of bilinear interference of wave fronts on the seven-bit hexagonal cluster: self-interference s 0,0 and s 1,1; cross interference x 0,1 and x 1,1(1).

Fig. 9
Fig. 9

Classification of neighboring bits (relative to the central bit at shell 0) in successive bit shells at distances equal to 1, 3, and 4 for shells 1, 2, and 3, respectively (the shell numbers are indicated).

Fig. 10
Fig. 10

Signal patterns (as a function of the cluster type) with lattice parameter a H = 165 nm and diameter of the pit hole b ph = 122.5 nm. The solid curves represent the average signal value for a four-parameter (single-shell) model, averaged for all possible clusters that have a fixed number of nearest neighbors. The stars represent different signal values, which depend on the number of two bit pairs of neighbor bits [denoted by the parameter p 1,1(1)] for the four-parameter model. The dashed curves (with open circles) represent the averaged signal levels for the full model (with five shells).

Tables (1)

Tables Icon

Table 1 Values of Nearest-Neighbor Bit Pairs [p 1,1(1) at Distance d = 1]

Equations (24)

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

IRp=CA |FTRΩpR-RprR|2dΩ,
Ik=n hk-nbn.
HΩ=2πarccosΩΩc-ΩΩc1-ΩΩc21/2 ΩΩc=0 Ω>Ωc.
Ωc=2NA/λ,
rR=1+j ajWR-Rj.
uj=1+bj2,
aj=ujexpiϕ-1.
|ψ=FTRΩpR-RprR.
ϕ|φ=CA ϕ*ΩφΩdΩ.
I=ψ|ψ.
|ψ=|ψL+j aj|ψj.
|ψL=FTRΩpR-Rp.
|ψj=FTRΩpR-RpWR-Rj.
I=ψL|ψL+j ajψL|ψj+j aj*ψj|ψL+j,k ak*ajψk|ψj.
ψL|ψj=ψj|ψL= |pR-Rp|2WR-RjdR,
ψj|ψk=ψk|ψj*.
I=1-j cjuj+jk ej,kujuk,
cj=21-cosϕψL|ψj-ψj|ψj
ej,k=21-cosϕReψj|ψk.
rR=1+j ujW˜ϕR-Rj,
W˜ϕR-Rj=expiϕR-Rj-1WR-Rj.
bph 50%=3/π aH.
I=1-21-cosϕu0l0-s0,0-21-cosϕj=1Nshell njlj-sj,j+41-cosϕu0j=1Nshell njx0,j+41-cosϕj=1NshellijNshelldDi,j pi,jdxi,jd.
I1-21-cosϕu0l0-s0,0-21-cosϕn1l1-s1,1+41-cosϕu0n1x0,1+41-cosϕp1,11x1,11.

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