Abstract

Grating lenses for optical branching were designed by introducing a phase-modulation function. The lenses have the dual functions of focusing and branching. The phase-modulation function was optimized to balance the optical intensity values for the required branchings and to suppress those for undesirable branchings. Phase-modulated grating lenses, with three and five branchings, were fabricated by using a computer and an electron-beam exposure system. The focusing and branching characteristics obtained by these lenses show excellent agreement with the theoretically predicted profiles.

© 1993 Optical Society of America

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References

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  1. H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a novel coupler for optical circuits,” IEEE J. Quantum Electron. QE-14, 794–796 (1975).
    [Crossref]
  2. W. B. Veldkamp, “Laser beam profile shaping with interlaced binary diffraction gratings,” Appl. Opt. 21, 3209–3212 (1982).
    [Crossref] [PubMed]
  3. G. Hatakoshi, K. Goto, “Grating lenses for the semiconductor laser wavelength,” Appl. Opt. 24, 4307–4311 (1985).
    [Crossref] [PubMed]
  4. G. Hatakoshi, M. Kawachi, K. Terashima, Y. Uematsu, A. Amano, K. Ueda, “Grating axicon for collimating Cerenkov radiation waves,” Opt. Lett. 15, 1336–1338 (1990).
    [Crossref] [PubMed]
  5. C. Kojima, K. Miyahara, K. Hasegawa, T. Otobe, H. Ooki, “In-line holographic lenses of high numerical aperture,” Jpn. J. Appl. Phys. 20, Suppl. 20-1, 199–204 (1980).
  6. G. Hatakoshi, M. Yoshimi, K. Goto, “Off-axis grating lenses fabricated by EB lithography,” in Technical Digest, Fourth International Conference on Integrated Optics and Optical Fiber Communication, (Institute of Electronics and Communication Engineers, Tokyo, 1983), paper 29A2-2.
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  9. K. Goto, K. Mori, G. Hatakoshi, S. Takahashi, “Spherical grating objective lenses for optical disk pickups,” Jpn. J. Appl. Phys. 26, Suppl. 26-4, 135–140 (1987).
  10. T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
    [Crossref] [PubMed]
  11. J. Jahns, S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29, 931–936 (1990).
    [Crossref] [PubMed]
  12. K. Tastani, A. Marrakchi, S. F. Habiby, W. M. Hubbard, H. Gilchrist, R. E. Nahory, “Binary phase Fresnel lenses for generation of two-dimensional beam arrays,” Appl. Opt. 30, 1347–1354 (1991).
    [Crossref]
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  14. T. Fujita, H. Nishihara, J. Koyama, “Fabrication of micro lenses using electron-beam lithography,” Opt. Lett. 6, 613–615 (1981).
    [Crossref] [PubMed]
  15. H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [Crossref]
  16. H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [Crossref]
  17. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
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    [Crossref]
  19. J. N. Mait, “Design of Dammann grating for two-dimensional, nonseparable, noncentrosymmetric responses,” Opt. Lett. 14, 196–198 (1989).
    [Crossref] [PubMed]
  20. W. B. Veldkamp, J. R. Leger, G. J. Swanson, “Coherent summation of laser beams using binary phase gratings,” Opt. Lett. 11, 303–305 (1986).
    [Crossref] [PubMed]
  21. J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).
  22. J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
    [Crossref]
  23. G. Hatakoshi, “Investigation of grating lenses for optical branching and mixing,” in Extended Abstracts of the 32nd Spring Meeting, (Japan Society of Applied Physics and Related Societies, Tokyo, 1985), paper 31p-P-10.
  24. G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for optical branching,” in Technical Digest, Third Microoptics Conference, (Japan Society of Applied Physics, Tokyo, 1991), paper F2.
  25. G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for two-dimensional optical branching,” Jpn. J. Appl. Phys. 31, 1606–1610 (1992).
    [Crossref]
  26. G. Hatakoshi, H. Fujima, K. Goto, “Waveguide grating lenses for optical couplers,” Appl. Opt. 23, 1749–1753 (1984).
    [Crossref] [PubMed]
  27. H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).
  28. T. Fujita, H. Nishihara, J. Koyama, “Blazed gratings and Fresnel lenses fabricated by electron-beam lithography,” Opt. Lett. 7, 578–580 (1982).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

1992 (1)

G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for two-dimensional optical branching,” Jpn. J. Appl. Phys. 31, 1606–1610 (1992).
[Crossref]

1991 (1)

1990 (4)

1989 (3)

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

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

J. N. Mait, “Design of Dammann grating for two-dimensional, nonseparable, noncentrosymmetric responses,” Opt. Lett. 14, 196–198 (1989).
[Crossref] [PubMed]

1987 (2)

K. Goto, K. Mori, G. Hatakoshi, S. Takahashi, “Spherical grating objective lenses for optical disk pickups,” Jpn. J. Appl. Phys. 26, Suppl. 26-4, 135–140 (1987).

T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
[Crossref] [PubMed]

1986 (1)

1985 (1)

1984 (3)

1982 (2)

1981 (1)

1980 (1)

C. Kojima, K. Miyahara, K. Hasegawa, T. Otobe, H. Ooki, “In-line holographic lenses of high numerical aperture,” Jpn. J. Appl. Phys. 20, Suppl. 20-1, 199–204 (1980).

1977 (1)

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

1975 (1)

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a novel coupler for optical circuits,” IEEE J. Quantum Electron. QE-14, 794–796 (1975).
[Crossref]

1974 (1)

1971 (1)

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[Crossref]

1970 (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

Amano, A.

Dammann, H.

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

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[Crossref]

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

Downs, M. M.

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

Engel, A.

Fujima, H.

Fujita, T.

Gilchrist, H.

Gortler, K.

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[Crossref]

Goto, K.

K. Goto, K. Mori, G. Hatakoshi, S. Takahashi, “Spherical grating objective lenses for optical disk pickups,” Jpn. J. Appl. Phys. 26, Suppl. 26-4, 135–140 (1987).

G. Hatakoshi, K. Goto, “Grating lenses for the semiconductor laser wavelength,” Appl. Opt. 24, 4307–4311 (1985).
[Crossref] [PubMed]

G. Hatakoshi, H. Fujima, K. Goto, “Waveguide grating lenses for optical couplers,” Appl. Opt. 23, 1749–1753 (1984).
[Crossref] [PubMed]

G. Hatakoshi, M. Yoshimi, K. Goto, “Off-axis grating lenses fabricated by EB lithography,” in Technical Digest, Fourth International Conference on Integrated Optics and Optical Fiber Communication, (Institute of Electronics and Communication Engineers, Tokyo, 1983), paper 29A2-2.

Habiby, S. F.

Hasegawa, K.

C. Kojima, K. Miyahara, K. Hasegawa, T. Otobe, H. Ooki, “In-line holographic lenses of high numerical aperture,” Jpn. J. Appl. Phys. 20, Suppl. 20-1, 199–204 (1980).

Hatakoshi, G.

G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for two-dimensional optical branching,” Jpn. J. Appl. Phys. 31, 1606–1610 (1992).
[Crossref]

G. Hatakoshi, M. Kawachi, K. Terashima, Y. Uematsu, A. Amano, K. Ueda, “Grating axicon for collimating Cerenkov radiation waves,” Opt. Lett. 15, 1336–1338 (1990).
[Crossref] [PubMed]

K. Goto, K. Mori, G. Hatakoshi, S. Takahashi, “Spherical grating objective lenses for optical disk pickups,” Jpn. J. Appl. Phys. 26, Suppl. 26-4, 135–140 (1987).

G. Hatakoshi, K. Goto, “Grating lenses for the semiconductor laser wavelength,” Appl. Opt. 24, 4307–4311 (1985).
[Crossref] [PubMed]

G. Hatakoshi, H. Fujima, K. Goto, “Waveguide grating lenses for optical couplers,” Appl. Opt. 23, 1749–1753 (1984).
[Crossref] [PubMed]

G. Hatakoshi, M. Yoshimi, K. Goto, “Off-axis grating lenses fabricated by EB lithography,” in Technical Digest, Fourth International Conference on Integrated Optics and Optical Fiber Communication, (Institute of Electronics and Communication Engineers, Tokyo, 1983), paper 29A2-2.

G. Hatakoshi, “Investigation of grating lenses for optical branching and mixing,” in Extended Abstracts of the 32nd Spring Meeting, (Japan Society of Applied Physics and Related Societies, Tokyo, 1985), paper 31p-P-10.

G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for optical branching,” in Technical Digest, Third Microoptics Conference, (Japan Society of Applied Physics, Tokyo, 1991), paper F2.

Herziger, G.

Hubbard, W. M.

Inohara, S.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a novel coupler for optical circuits,” IEEE J. Quantum Electron. QE-14, 794–796 (1975).
[Crossref]

Jahns, J.

J. Jahns, S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29, 931–936 (1990).
[Crossref] [PubMed]

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

Kamiya, T.

Kawachi, M.

Klotz, E.

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

Kodate, K.

Kojima, C.

C. Kojima, K. Miyahara, K. Hasegawa, T. Otobe, H. Ooki, “In-line holographic lenses of high numerical aperture,” Jpn. J. Appl. Phys. 20, Suppl. 20-1, 199–204 (1980).

Koyama, J.

Krackhardt, U.

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

Leger, J. R.

Mait, J. N.

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

J. N. Mait, “Design of Dammann grating for two-dimensional, nonseparable, noncentrosymmetric responses,” Opt. Lett. 14, 196–198 (1989).
[Crossref] [PubMed]

J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).

Marrakchi, A.

Miyahara, K.

C. Kojima, K. Miyahara, K. Hasegawa, T. Otobe, H. Ooki, “In-line holographic lenses of high numerical aperture,” Jpn. J. Appl. Phys. 20, Suppl. 20-1, 199–204 (1980).

Mori, K.

K. Goto, K. Mori, G. Hatakoshi, S. Takahashi, “Spherical grating objective lenses for optical disk pickups,” Jpn. J. Appl. Phys. 26, Suppl. 26-4, 135–140 (1987).

Nahory, R. E.

Nakamura, M.

G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for two-dimensional optical branching,” Jpn. J. Appl. Phys. 31, 1606–1610 (1992).
[Crossref]

G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for optical branching,” in Technical Digest, Third Microoptics Conference, (Japan Society of Applied Physics, Tokyo, 1991), paper F2.

Nishihara, H.

Nukui, K.

Ooki, H.

C. Kojima, K. Miyahara, K. Hasegawa, T. Otobe, H. Ooki, “In-line holographic lenses of high numerical aperture,” Jpn. J. Appl. Phys. 20, Suppl. 20-1, 199–204 (1980).

Otobe, T.

C. Kojima, K. Miyahara, K. Hasegawa, T. Otobe, H. Ooki, “In-line holographic lenses of high numerical aperture,” Jpn. J. Appl. Phys. 20, Suppl. 20-1, 199–204 (1980).

Prise, M. E.

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

Saheki, T.

Setsune, K.

Shiono, T.

Steffen, J.

Streibl, N.

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

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

Suhara, T.

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a novel coupler for optical circuits,” IEEE J. Quantum Electron. QE-14, 794–796 (1975).
[Crossref]

Swanson, G. J.

Takahashi, S.

K. Goto, K. Mori, G. Hatakoshi, S. Takahashi, “Spherical grating objective lenses for optical disk pickups,” Jpn. J. Appl. Phys. 26, Suppl. 26-4, 135–140 (1987).

Takei, T.

Takenaka, H.

Tastani, K.

Tatsumi, K.

Terashima, K.

Ueda, K.

Uematsu, Y.

Veldkamp, W. B.

Walker, S. J.

J. Jahns, S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29, 931–936 (1990).
[Crossref] [PubMed]

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

Wasa, K.

Yamazaki, O.

Yoshimi, M.

G. Hatakoshi, M. Yoshimi, K. Goto, “Off-axis grating lenses fabricated by EB lithography,” in Technical Digest, Fourth International Conference on Integrated Optics and Optical Fiber Communication, (Institute of Electronics and Communication Engineers, Tokyo, 1983), paper 29A2-2.

Appl. Opt. (9)

IEEE J. Quantum Electron. (1)

H. Nishihara, S. Inohara, T. Suhara, J. Koyama, “Holocoupler: a novel coupler for optical circuits,” IEEE J. Quantum Electron. QE-14, 794–796 (1975).
[Crossref]

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

Jpn. J. Appl. Phys. (3)

G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for two-dimensional optical branching,” Jpn. J. Appl. Phys. 31, 1606–1610 (1992).
[Crossref]

K. Goto, K. Mori, G. Hatakoshi, S. Takahashi, “Spherical grating objective lenses for optical disk pickups,” Jpn. J. Appl. Phys. 26, Suppl. 26-4, 135–140 (1987).

C. Kojima, K. Miyahara, K. Hasegawa, T. Otobe, H. Ooki, “In-line holographic lenses of high numerical aperture,” Jpn. J. Appl. Phys. 20, Suppl. 20-1, 199–204 (1980).

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. (2)

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[Crossref]

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

Opt. Eng. (1)

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

Opt. Lett. (6)

Optik (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

Other (4)

G. Hatakoshi, “Investigation of grating lenses for optical branching and mixing,” in Extended Abstracts of the 32nd Spring Meeting, (Japan Society of Applied Physics and Related Societies, Tokyo, 1985), paper 31p-P-10.

G. Hatakoshi, M. Nakamura, “Phase-modulated grating lenses for optical branching,” in Technical Digest, Third Microoptics Conference, (Japan Society of Applied Physics, Tokyo, 1991), paper F2.

G. Hatakoshi, M. Yoshimi, K. Goto, “Off-axis grating lenses fabricated by EB lithography,” in Technical Digest, Fourth International Conference on Integrated Optics and Optical Fiber Communication, (Institute of Electronics and Communication Engineers, Tokyo, 1983), paper 29A2-2.

J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).

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Figures (8)

Fig. 1
Fig. 1

Configuration of a grating lens for optical branching.

Fig. 2
Fig. 2

Schematic intensity distributions in the ξ direction at the focal plane. Optical intensity I(ξ, 0) is given by the product of these two functions.

Fig. 3
Fig. 3

Examples of phase-modulation function Ω1(x) for five branchings: (a), (b) βn = 0; (c)–(f) βn = 3 − n.

Fig. 4
Fig. 4

Phase-modulation functions Ω1(x) (upper) and calculated intensity distributions (lower) for diffracted waves: (a) three branchings, (b) five branchings.

Fig. 5
Fig. 5

Grating patterns for a phase-modulated grating lens. The right-hand side of each figure represents a cross section of the grating: (a) binary grating; (b) blazed grating.

Fig. 6
Fig. 6

Microscopic photographs of phase-modulated grating lenses with (a) three and (b) five branchings.

Fig. 7
Fig. 7

Measured intensity distributions in the ξ direction at the focal plane for the grating lenses shown in Fig. 6.

Fig. 8
Fig. 8

Comparison between two types of branching grating lenses. The photographs show the grating-lens patterns, and the graphs show the intensity distributions for diffracted light waves. The condition given by Eq. (2) is not satisfied in (a), while the grating pattern shown in (b) is designed to satisfy Eq. (2).

Tables (1)

Tables Icon

Table 1 Calculated Examples for Optimized Phase Constants and Resulting Peak Intensities

Equations (29)

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Ω ( x , y ) = Ω 0 ( x , y ) + Ω 1 ( x ) ,
Ω 0 ( x , y ) = Ω 0 i ( x , y ) + ( 2 π / λ ) l 0 ( x , y ) ,
l 0 ( x , y ) = ( x 2 + y 2 + f 2 ) 1 / 2 ,
Λ = λ f / d ,
e ( ξ , η ) = ( i / λ ) exp [ i Ω 0 i ( x , y ) + i Ω ( x , y ) ] × { exp [ i k 0 l ( x , y , ξ , η ) ] / l ( x , y , ξ , η ) } d x d y ,
l ( x , y , ξ , η ) = [ ( x ξ ) 2 + ( y η ) 2 + f 2 ] 1 / 2 .
I ( ξ , η ) = | [ i / ( λ f ) ] exp [ i Ω 1 ( x ) ] × exp [ i k 0 ( ξ x + η y ) / f ] d x d y | 2 .
Ω 1 ( x + m Λ ) = Ω 1 ( x ) ,
I ( ξ , η ) = | ( c / A ) sinc [ B η / ( λ f ) ] × [ Λ a ( ξ ) m = 0 M 1 exp ( i k 0 ξ m Λ / f ) + b ( ξ ) ] | 2 ,
c = A B / ( λ f ) ,
a ( ξ ) = ( 1 / Λ ) 0 Λ exp [ i ( Ω 1 + k 0 ξ x / f ) ] d x ,
b ( ξ ) = M Λ A exp [ i ( Ω 1 + k 0 ξ x / f ) ] d x ,
M = Int ( A / Λ ) ,
sinc ( x ) = sin ( π x ) / ( π x ) ,
I ( ξ , η ) = c 2 | a ( ξ ) | 2 [ sinc ( M ξ / d ) / sinc ( ξ / d ) ] 2 × sinc 2 [ B η / ( λ f ) ] .
exp [ i Ω 1 ( x ) ] = m a ( m d ) exp ( i 2 π m x / Λ ) .
Ω 1 n ( x ) = 2 π [ α n + β n ( x Λ / 2 ) / Λ ] , n = 1 , , N ,
a ( ξ ) = ( 1 / N ) n = 1 N exp { i π [ 2 α n β n + ( 2 n 1 ) × ( ξ + β n d ) / ( N d ) ] } sinc [ ( ξ + β n d ) / ( N d ) ] .
a ( ξ ) = ( 1 / 5 ) sinc [ ξ / ( 5 d ) ] × n = 1 N exp { i π [ 2 α n + ( 2 n 1 ) ξ / ( 5 d ) ] } .
a ( 2 d ) = a ( d ) = a ( 0 ) = a ( d ) = a ( 2 d ) .
α 3 = 0 , α 2 = α 4 , α 1 = α 5 .
| 5 a ( 0 ) | 2 = 9 + 4 ( a 1 + a 2 ) + 8 a 12 ,
| 5 a ( d ) | 2 = sinc 2 ( 1 / 5 ) [ 5 + 2 ( c 1 + c 2 ) + 4 ( c 2 a 1 + c 1 a 2 ) + 8 c 1 c 2 a 12 ] ,
| 5 a ( 2 d ) | 2 = sinc 2 ( 2 / 5 ) [ 5 + 2 ( c 1 + c 2 ) + 4 ( c 1 a 1 + c 2 a 2 ) + 8 c 1 c 2 a 12 ] ,
c 1 = cos ( 2 π / 5 ) , c 2 = cos ( 4 π / 5 ) ,
a 1 = cos ( 2 πα 1 ) , a 2 = cos ( 2 πα 2 ) , a 12 = cos [ 2 π ( α 1 α 2 ) ] .
α 1 = ± 0 . 171 , α 2 = 0 . 254 ,
α 1 = ± 0 . 318 , α 2 = 0 . 432 .
β n = 3 n , n = 1 , , 5 .

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