Abstract

The numerical solutions of binary-phase (0, π) gratings for one-dimensional array illuminators up to 32 are presented. Some fabrication errors, which are due to position-quantization errors, phase errors, dilation (or erosion) errors, and the side-slope error, are calculated and show that even-number array illuminators are superior to odd-number array illuminators when these fabrication errors are considered. One (0, π) binary-phase, 8 × 16 array illuminator made with the wet-chemical-etching method is given in this paper.

© 1995 Optical Society of America

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References

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  1. H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [Crossref]
  2. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [Crossref]
  3. U. Killat, G. Rabe, W. Rave, “Binary phase grating for star couples with a high splitting ratio,” Fiber Integrat. Opt. 4, 159–167 (1982).
    [Crossref]
  4. J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
    [Crossref]
  5. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illuminators in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
    [Crossref] [PubMed]
  6. R. L. Morrison, “Symmetries that simplify the design of spot-array phase gratings,” J. Opt. Soc. Am. A 9, 464–471 (1992).
    [Crossref]
  7. R. L. Morrison, S. L. Walker, T. J. Cloonan, “Beam array generation and holographic interconnections in a free-space optical switching network,” Appl. Opt. 32, 2512–2518 (1993).
    [Crossref] [PubMed]
  8. M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
    [Crossref] [PubMed]
  9. R. Brauer, F. Wyrowski, O. Bryngdahl, “Binarization of diffractive elements with nonperiodic structures,” Appl. Opt. 31, 2535–2540 (1992).
    [Crossref] [PubMed]
  10. J. N. Mait, “Design of binary- and multi-phase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
    [Crossref]
  11. M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
    [Crossref]
  12. P. Ehbets, H. P. Herzig, D. Prongué, M. T. Gale, “High-efficiency continuous surface-relief gratings for two-dimensional array generation,” Opt. Lett. 17, 908–910 (1992).
    [Crossref] [PubMed]
  13. M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).
  14. J. R. Leger, G. M. Morris, eds., “Diffractive optics: design, fabrication, and applications feature,” Appl. Opt. 32, 14 (1993).
  15. F. B. McCormick, T. J. Cloonan, A. L. Lentine, J. M. Sasian, R. L. Morrison, M. G. Beckman, S. L. Walker, M. J. Wojcik, S. J. Hinterlong, R. J. Crisci, R. A. Novotny, H. S. Hinton, “Five-stage free-space optical switching network with field-effect transistor self-electro-optic-effect-device smart-pixel arrays,” Appl. Opt. 33, 1601–1618 (1994).
    [Crossref] [PubMed]
  16. F. A. P. Tooley, S. Wakelin, M. R. Taghizadeh, “Interconnects for symmetric-self-electro-optic-effect-device cellularlogic image processors,” Appl. Opt. 33, 1398–1403 (1994).
    [Crossref] [PubMed]
  17. U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. 31, 27–37 (1992).
    [Crossref] [PubMed]
  18. X. Lu, Y. Wang, M. Wu, G. Jin, “The fabrication of a 25 × 25 multiple beam splitter,” Opt. Commun. 72, 157–162 (1989).
    [Crossref]
  19. J. Turunen, X. Bian, G. Jin, “Optimization of grating multi-beam splitters,” Acta Optica Sinica 8, 946–953 (1988).
  20. U. Krackhardt, N. Streibl, “Design of Dammann gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
    [Crossref]
  21. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann grating for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
  22. J. N. Mait, “Complex plane representation and design of array generators,” presented at the Optical Society of America Annual Meeting, San Jose, California, 3–8 November 1991.

1994 (2)

1993 (2)

1992 (7)

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illuminators in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[Crossref] [PubMed]

R. L. Morrison, “Symmetries that simplify the design of spot-array phase gratings,” J. Opt. Soc. Am. A 9, 464–471 (1992).
[Crossref]

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. 31, 27–37 (1992).
[Crossref] [PubMed]

R. Brauer, F. Wyrowski, O. Bryngdahl, “Binarization of diffractive elements with nonperiodic structures,” Appl. Opt. 31, 2535–2540 (1992).
[Crossref] [PubMed]

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

P. Ehbets, H. P. Herzig, D. Prongué, M. T. Gale, “High-efficiency continuous surface-relief gratings for two-dimensional array generation,” Opt. Lett. 17, 908–910 (1992).
[Crossref] [PubMed]

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

1990 (1)

J. N. Mait, “Design of binary- and multi-phase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
[Crossref]

1989 (6)

X. Lu, Y. Wang, M. Wu, G. Jin, “The fabrication of a 25 × 25 multiple beam splitter,” Opt. Commun. 72, 157–162 (1989).
[Crossref]

M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
[Crossref] [PubMed]

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[Crossref]

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[Crossref]

U. Krackhardt, N. Streibl, “Design of Dammann gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
[Crossref]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann grating for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

1988 (1)

J. Turunen, X. Bian, G. Jin, “Optimization of grating multi-beam splitters,” Acta Optica Sinica 8, 946–953 (1988).

1982 (1)

U. Killat, G. Rabe, W. Rave, “Binary phase grating for star couples with a high splitting ratio,” Fiber Integrat. Opt. 4, 159–167 (1982).
[Crossref]

1977 (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

Beckman, M. G.

Bian, X.

J. Turunen, X. Bian, G. Jin, “Optimization of grating multi-beam splitters,” Acta Optica Sinica 8, 946–953 (1988).

Brauer, R.

Bryngdahl, O.

Cloonan, T. J.

Crisci, R. J.

Dammann, H.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann grating for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Ehbets, P.

P. Ehbets, H. P. Herzig, D. Prongué, M. T. Gale, “High-efficiency continuous surface-relief gratings for two-dimensional array generation,” Opt. Lett. 17, 908–910 (1992).
[Crossref] [PubMed]

Ekberg, M.

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

Feldman, M. R.

Gale, M. T.

P. Ehbets, H. P. Herzig, D. Prongué, M. T. Gale, “High-efficiency continuous surface-relief gratings for two-dimensional array generation,” Opt. Lett. 17, 908–910 (1992).
[Crossref] [PubMed]

Guest, C. C.

Hård, S.

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

Herzig, H. P.

P. Ehbets, H. P. Herzig, D. Prongué, M. T. Gale, “High-efficiency continuous surface-relief gratings for two-dimensional array generation,” Opt. Lett. 17, 908–910 (1992).
[Crossref] [PubMed]

Hinterlong, S. J.

Hinton, H. S.

Ichikawa, H.

Jaakkola, T.

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann grating for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Jin, G.

X. Lu, Y. Wang, M. Wu, G. Jin, “The fabrication of a 25 × 25 multiple beam splitter,” Opt. Commun. 72, 157–162 (1989).
[Crossref]

J. Turunen, X. Bian, G. Jin, “Optimization of grating multi-beam splitters,” Acta Optica Sinica 8, 946–953 (1988).

Killat, U.

U. Killat, G. Rabe, W. Rave, “Binary phase grating for star couples with a high splitting ratio,” Fiber Integrat. Opt. 4, 159–167 (1982).
[Crossref]

Klotz, E.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

Krackhardt, U.

Kuisma, S.

Larson, M.

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

Leger, J. R.

J. R. Leger, G. M. Morris, eds., “Diffractive optics: design, fabrication, and applications feature,” Appl. Opt. 32, 14 (1993).

Lentine, A. L.

Lu, X.

X. Lu, Y. Wang, M. Wu, G. Jin, “The fabrication of a 25 × 25 multiple beam splitter,” Opt. Commun. 72, 157–162 (1989).
[Crossref]

Mait, J. N.

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. 31, 27–37 (1992).
[Crossref] [PubMed]

J. N. Mait, “Design of binary- and multi-phase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
[Crossref]

J. N. Mait, “Complex plane representation and design of array generators,” presented at the Optical Society of America Annual Meeting, San Jose, California, 3–8 November 1991.

McCormick, F. B.

Miller, J. M.

Morris, G. M.

J. R. Leger, G. M. Morris, eds., “Diffractive optics: design, fabrication, and applications feature,” Appl. Opt. 32, 14 (1993).

Morrison, R. L.

Noponen, E.

Novotny, R. A.

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann grating for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Prongué, D.

P. Ehbets, H. P. Herzig, D. Prongué, M. T. Gale, “High-efficiency continuous surface-relief gratings for two-dimensional array generation,” Opt. Lett. 17, 908–910 (1992).
[Crossref] [PubMed]

Rabe, G.

U. Killat, G. Rabe, W. Rave, “Binary phase grating for star couples with a high splitting ratio,” Fiber Integrat. Opt. 4, 159–167 (1982).
[Crossref]

Rave, W.

U. Killat, G. Rabe, W. Rave, “Binary phase grating for star couples with a high splitting ratio,” Fiber Integrat. Opt. 4, 159–167 (1982).
[Crossref]

Salin, A.

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[Crossref]

Sasian, J. M.

Streibl, N.

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. 31, 27–37 (1992).
[Crossref] [PubMed]

U. Krackhardt, N. Streibl, “Design of Dammann gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
[Crossref]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann grating for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[Crossref]

Taghizadeh, M. R.

F. A. P. Tooley, S. Wakelin, M. R. Taghizadeh, “Interconnects for symmetric-self-electro-optic-effect-device cellularlogic image processors,” Appl. Opt. 33, 1398–1403 (1994).
[Crossref] [PubMed]

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illuminators in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[Crossref] [PubMed]

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

Tooley, F. A. P.

Turunen, J.

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illuminators in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[Crossref] [PubMed]

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[Crossref]

J. Turunen, X. Bian, G. Jin, “Optimization of grating multi-beam splitters,” Acta Optica Sinica 8, 946–953 (1988).

Vasara, A.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illuminators in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[Crossref] [PubMed]

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[Crossref]

Wakelin, S.

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann grating for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Walker, S. L.

Wang, Y.

X. Lu, Y. Wang, M. Wu, G. Jin, “The fabrication of a 25 × 25 multiple beam splitter,” Opt. Commun. 72, 157–162 (1989).
[Crossref]

Westerholm, J.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illuminators in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[Crossref] [PubMed]

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[Crossref]

Wojcik, M. J.

Wu, M.

X. Lu, Y. Wang, M. Wu, G. Jin, “The fabrication of a 25 × 25 multiple beam splitter,” Opt. Commun. 72, 157–162 (1989).
[Crossref]

Wyrowski, F.

Acta Optica Sinica (1)

J. Turunen, X. Bian, G. Jin, “Optimization of grating multi-beam splitters,” Acta Optica Sinica 8, 946–953 (1988).

Appl. Opt. (7)

Fiber Integrat. Opt. (1)

U. Killat, G. Rabe, W. Rave, “Binary phase grating for star couples with a high splitting ratio,” Fiber Integrat. Opt. 4, 159–167 (1982).
[Crossref]

J. Mod. Opt. (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[Crossref]

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

R. L. Morrison, “Symmetries that simplify the design of spot-array phase gratings,” J. Opt. Soc. Am. A 9, 464–471 (1992).
[Crossref]

J. N. Mait, “Design of binary- and multi-phase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
[Crossref]

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

Opt. Commun. (1)

M. Ekberg, M. Larson, S. Hård, J. Turunen, M. R. Taghizadeh, J. Westerholm, A. Vasara, “Multilevel grating array illuminators manufactured by electron-beam lithography,” Opt. Commun. 88, 37–41 (1992).
[Crossref]

Opt. Lett. (1)

P. Ehbets, H. P. Herzig, D. Prongué, M. T. Gale, “High-efficiency continuous surface-relief gratings for two-dimensional array generation,” Opt. Lett. 17, 908–910 (1992).
[Crossref] [PubMed]

Opt. Commun. (3)

X. Lu, Y. Wang, M. Wu, G. Jin, “The fabrication of a 25 × 25 multiple beam splitter,” Opt. Commun. 72, 157–162 (1989).
[Crossref]

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[Crossref]

U. Krackhardt, N. Streibl, “Design of Dammann gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
[Crossref]

Opt. Comput. Process. (1)

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

Opt. Eng. (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann grating for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Opt. Lett. (1)

Other (1)

J. N. Mait, “Complex plane representation and design of array generators,” presented at the Optical Society of America Annual Meeting, San Jose, California, 3–8 November 1991.

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Figures (6)

Fig. 1
Fig. 1

Uniformity of an array generator that is attributable to the quantization levels in one period: The triangles represent quantization level 1000; the squares, quantization level 5000; and the circles, quantization level 10,000.

Fig. 2
Fig. 2

Uniformity of the array generator that is attributable to the phase error: θ = π + δ, for δ = 0.1 (triangles), 0.05 (squares), 0.01 (circles), respectively.

Fig. 3
Fig. 3

Uniformity of the array generator that is attributable to the dilation (or erosion) error. Three dilation sizes for one side wall are shown: W = 0.001 (triangles), 0.0001 (squares), 0.00001 (circles), respectively.

Fig. 4
Fig. 4

Uniformity of the array generator that is attributable to the side-wall error: Values for one side-wall size were W = 0.005 (triangles), 0.001 (squares), 0.0001 (circles), respectively.

Fig. 5
Fig. 5

Surface profile of one (8 × 16) binary-phase grating as measured with alpha step 200.

Fig. 6
Fig. 6

Experimental photograph for an 8 × 16 array illumination taken in the focal plane of the lens.

Tables (3)

Tables Icon

Table 1. Numerical Solutions for Some Binary-Phase (0, π) Gratings for 1D Array Illumination

Tables Icon

Table 2 Theoretical and Practical Results Showing the Relations between the Number of Transition Points K in One Period and Odd-Number (2N + 1), 1D Array Illuminators a

Tables Icon

Table 3 Theoretical and Practical Results Showing the Relations between the Number of Transition Points K in a Half-Period and Even-Number (2N) Array Illuminators a

Equations (16)

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

p ¯ ( u ) = p ( u ) repl ( u ) ,
P ¯ ( x ) = P ( x ) samp ( x ) .
ϕ = ( I 0 I av ) 2 + 2 i = 1 N ( I i I av ) 2 I 0 + 2 i = 1 N I i ,
I av = 1 2 N + 1 ( I 0 + 2 i = 1 N I i ) ,
η = I 0 + 2 i = 1 N I i .
ϕ = i = 1 N ( I 2 i 1 I av ) 2 i = 1 N I 2 i 1 ,
I a v = 1 2 N i = 1 N I 2 i 1 ,
η = k = 1 N I 2 i 1 .
Δ = min | x k + 1 x k | ,
uni = max ( I n ) min ( I n ) max ( I n ) + min ( I n ) ,
I 0 = [ 1 + 2 sin 2 θ 2 k = 1 K + 1 ( 1 ) k + 1 X k ] 2 + sin 2 θ [ k = 1 K + 1 ( 1 ) k + 1 X k ] 2 , I n = [ sin ( θ / 2 ) n π ] 2 { [ k = 1 K + 1 ( 1 ) k cos α k ] 2 + [ K = 1 K + 1 ( 1 ) k sin α k ] 2 } ,
I n X k = 16 ( 1 ) k n π { k = 1 K [ ( 1 ) k sin α k ] cos α k + [ k = 1 K + 1 ( 1 ) k cos α k ] ( sin α k ) } ,
I n X k = I n X k ,
I 0 = [ k = 1 K + 1 ( X k X K 1 ) cos θ k ] 2 + [ k = 1 K + 1 ( X k X K + 1 ) sin θ k ] 2 , I n = 1 4 n 2 π 2 [ ( k = 1 K + 1 p k ) 2 + ( K = 1 K + 1 q k ) 2 ] ,
p k = ( a k c k ) ( b k d k ) , q k = ( b k c k ) ( a k d k ) ,
a k = sin α k sin α k 1 , b k = cos α k cos α k 1 , c k = cos θ k , d k = sin θ k ,

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