Abstract

A simplified Fourier optics approach is applied to study how near-field optical disks retrieve evanescent signals through random nanostructure. The statistical properties of the random nanostructures are used to realize the general behavior of near-field optical disks. The mechanism of its super-resolution capability and an analytical expression of the readout contrast of near-field optical disks with random apertures are derived. The resolution of near-field optical disk is determined by the size of the random nanostructure.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. Y. H. Fu, F. H. Ho, W.-C. Hsu, S.-Y. Tsai and D. P. Tsai, "Nonlinear optical properties of the Au-SiO nanocomposite superresolution near-field thin film," Jpn. J. Appl. Phys. 43, 5020-5023 (2004).
    [CrossRef]
  31. H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
    [CrossRef]
  32. T. Shima, M. Kuwahara, T. Fukaya, T. Nakano and J. Tominaga, "Super-resolutional readout disk with metal-free phthalocyanine recording layer," Jpn. J. Appl. Phys. 43, L88-L90 (2004).
    [CrossRef]
  33. T. Kikukawa, T. Kato, H. Shingai and H. Utsunomiya, "High-density read-only memory disc with super resolution reflective layer," Jpn. J. Appl. Phys. 40, 1624-1628 (2001).
    [CrossRef]
  34. J. W. Fang, C. C. Wu, A. Liao. W. C. Lin and D. P. Tsai, "Implementation of practical super-resolution near-field structure system using commercial drive," Jap. J. Appl. Phys. 45, 1383-1384 (2006).
    [CrossRef]

2006

J. W. Fang, C. C. Wu, A. Liao. W. C. Lin and D. P. Tsai, "Implementation of practical super-resolution near-field structure system using commercial drive," Jap. J. Appl. Phys. 45, 1383-1384 (2006).
[CrossRef]

2005

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Enhanced resolution induced by random silver nanoparticles in near-field optical disks," Opt. Commun. 246, 561-567 (2005).
[CrossRef]

J. M. Ji, L. P. Shi, K. G. Lim, X. S. Miao, H. X. Yang and T. C. Chong, "Enhanced scattering of random-distribution nanoparticles and evanescent field in super-resolution near-field structure," Jpn. J. Appl. Phys. 44, 3620-3622 (2005).
[CrossRef]

M.-Y. Ng and W.-C. Liu, "Super-resolution and frequency-dependent efficiency of near-field optical disks with silver nanoparticles," Opt. Exp. 13, 9422-9430 (2005).
[CrossRef]

2004

F. Zhang, W. Xu, Y. Wang and F. Gan, "Static optical recording properties of super-resolution near-field structure with Bismuth mask layer," Jpn. J. Appl. Phys. 43, 7802-7806 (2004).
[CrossRef]

Y. H. Fu, F. H. Ho, W.-C. Hsu, S.-Y. Tsai and D. P. Tsai, "Nonlinear optical properties of the Au-SiO nanocomposite superresolution near-field thin film," Jpn. J. Appl. Phys. 43, 5020-5023 (2004).
[CrossRef]

W.-C. Liu, M.-Y. Ng and D. P. Tsai, "Surface plasmon effects on the far-field signals of AgOx-type super-resolution near-field structure," Jpn. J. Appl. Phys. 43, 4713-4717 (2004).
[CrossRef]

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Near- and far-field optical properties of embedded scatters in AgOx-type super-resolution near-field structures," Scanning,  26, 102-105 (2004).

T. Shima, M. Kuwahara, T. Fukaya, T. Nakano and J. Tominaga, "Super-resolutional readout disk with metal-free phthalocyanine recording layer," Jpn. J. Appl. Phys. 43, L88-L90 (2004).
[CrossRef]

2003

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

F. H. Ho, H. H. Chang, Y. H. Lin, B.-M. Chen, S.-Y. Wang, and D. P. Tsai, "Functional structures of AgOx thin film for near-field recording," Jpn. J. Appl. Phys., Part 1 42,1000-1004 (2003).
[CrossRef]

W.-C. Liu and D. P. Tsai, "Nonlinear near-field optical effects of the AgOx-Type super-resolution near-field structure," Jpn. J. Appl. Phys. 42, Part 1, 1031-1032 (2003).
[CrossRef]

2002

L. P. Shi, T. C. Chong, H. B. Yao, P. K. Tan and X. S. Miao, "Super-resolution near-field optical disk with an additional localized surface plasmon coupling layer," J. Appl. Phys. 91, 10209-10211 (2002).
[CrossRef]

T. Kikukawa, T. Nakano, T. Shima, and J. Tominaga, "Rigid bubble bit formation and huge signal enhancement in super-resolution near-field structure disk with platinum-oxide layer," Appl. Phys. Lett. 81, 4697-4699 (2002).
[CrossRef]

2001

W.-C. Liu, C.-Y. Wen, K.-H. Chen, W. C. Lin, and D. P. Tsai, "Near-field images of the AgOx-type super-resolution near-field structure," Appl. Phys. Lett. 78, 685-687 (2001).
[CrossRef]

F. H. Ho, W. Y. Lin, H. H. Chang, Y. H. Lin, W.-C. Liu and D. P. Tsai, "Nonlinear optical absorption in the AgOx-type super-resolution near-field structure," Jap. J. Appl. Phys. 40, 4101-4102 (2001).
[CrossRef]

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

T. Kikukawa, T. Kato, H. Shingai and H. Utsunomiya, "High-density read-only memory disc with super resolution reflective layer," Jpn. J. Appl. Phys. 40, 1624-1628 (2001).
[CrossRef]

1998

J. Tominaga, T. Nakano, and N. Atoda, "An approach for recording and readout beyond the diffraction limit with an Sb thin film," Appl. Phys. Lett. 73,2078-2080 (1998).
[CrossRef]

1996

H. Awano, S. Ohnuki, H. Shirai ans N. Ohta, "Magnetic domain expansion readout for amplification of an ultra high density magneto-optical recording signal," Appl. Phys. Lett. 69, 4257-4259 (1996).
[CrossRef]

1994

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, "Photon scanning tunneling microscopy images of optical excitations of fractal metal colloid clusters," Phys. Rev. Lett. 72, 4149-4152 (1994).
[CrossRef] [PubMed]

1993

K. Yasuda, M. Ono, K. Aratani, A. Fukumoto and M. Kaneko, "Premastered optical disk by superresolution," Jpn. J. Appl. Phys. 32, 5210-5213 (1993)
[CrossRef]

1992

1985

1984

D. W. Pohl, W. Denk, and M. Lanz, "Optical spectroscopy: Image recording with resolution lambda /20, " Appl. Phys. Lett. 44, 651-653 (1984).
[CrossRef]

1972

E.A. Ash and G. Nicholls, " Super-resolution aperture scanning microscope," Nature 237, 510-512 (1972)
[CrossRef] [PubMed]

1928

E. H. Synge, "An application of piezoelectricity to microscopy," Philos. Mag. 6, 356-362 (1928).

Aratani, K.

K. Yasuda, M. Ono, K. Aratani, A. Fukumoto and M. Kaneko, "Premastered optical disk by superresolution," Jpn. J. Appl. Phys. 32, 5210-5213 (1993)
[CrossRef]

Ash, E.A.

E.A. Ash and G. Nicholls, " Super-resolution aperture scanning microscope," Nature 237, 510-512 (1972)
[CrossRef] [PubMed]

Atoda, N.

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

J. Tominaga, T. Nakano, and N. Atoda, "An approach for recording and readout beyond the diffraction limit with an Sb thin film," Appl. Phys. Lett. 73,2078-2080 (1998).
[CrossRef]

Awano, H.

H. Awano, S. Ohnuki, H. Shirai ans N. Ohta, "Magnetic domain expansion readout for amplification of an ultra high density magneto-optical recording signal," Appl. Phys. Lett. 69, 4257-4259 (1996).
[CrossRef]

Büchel, D.

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

Chang, C.-J.

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

Chang, H. H.

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

F. H. Ho, H. H. Chang, Y. H. Lin, B.-M. Chen, S.-Y. Wang, and D. P. Tsai, "Functional structures of AgOx thin film for near-field recording," Jpn. J. Appl. Phys., Part 1 42,1000-1004 (2003).
[CrossRef]

F. H. Ho, W. Y. Lin, H. H. Chang, Y. H. Lin, W.-C. Liu and D. P. Tsai, "Nonlinear optical absorption in the AgOx-type super-resolution near-field structure," Jap. J. Appl. Phys. 40, 4101-4102 (2001).
[CrossRef]

Chen, B.-M.

F. H. Ho, H. H. Chang, Y. H. Lin, B.-M. Chen, S.-Y. Wang, and D. P. Tsai, "Functional structures of AgOx thin film for near-field recording," Jpn. J. Appl. Phys., Part 1 42,1000-1004 (2003).
[CrossRef]

Chen, K. H.

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

Chen, K.-H.

W.-C. Liu, C.-Y. Wen, K.-H. Chen, W. C. Lin, and D. P. Tsai, "Near-field images of the AgOx-type super-resolution near-field structure," Appl. Phys. Lett. 78, 685-687 (2001).
[CrossRef]

Chiang, H.-H.

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

Chong, T. C.

J. M. Ji, L. P. Shi, K. G. Lim, X. S. Miao, H. X. Yang and T. C. Chong, "Enhanced scattering of random-distribution nanoparticles and evanescent field in super-resolution near-field structure," Jpn. J. Appl. Phys. 44, 3620-3622 (2005).
[CrossRef]

L. P. Shi, T. C. Chong, H. B. Yao, P. K. Tan and X. S. Miao, "Super-resolution near-field optical disk with an additional localized surface plasmon coupling layer," J. Appl. Phys. 91, 10209-10211 (2002).
[CrossRef]

Chu, T. C.

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Enhanced resolution induced by random silver nanoparticles in near-field optical disks," Opt. Commun. 246, 561-567 (2005).
[CrossRef]

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Near- and far-field optical properties of embedded scatters in AgOx-type super-resolution near-field structures," Scanning,  26, 102-105 (2004).

Courjon, D.

Denk, W.

D. W. Pohl, W. Denk, and M. Lanz, "Optical spectroscopy: Image recording with resolution lambda /20, " Appl. Phys. Lett. 44, 651-653 (1984).
[CrossRef]

Depasse, F.

Fang, J. W.

J. W. Fang, C. C. Wu, A. Liao. W. C. Lin and D. P. Tsai, "Implementation of practical super-resolution near-field structure system using commercial drive," Jap. J. Appl. Phys. 45, 1383-1384 (2006).
[CrossRef]

Fu, Y. H.

Y. H. Fu, F. H. Ho, W.-C. Hsu, S.-Y. Tsai and D. P. Tsai, "Nonlinear optical properties of the Au-SiO nanocomposite superresolution near-field thin film," Jpn. J. Appl. Phys. 43, 5020-5023 (2004).
[CrossRef]

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

Fukaya, T.

T. Shima, M. Kuwahara, T. Fukaya, T. Nakano and J. Tominaga, "Super-resolutional readout disk with metal-free phthalocyanine recording layer," Jpn. J. Appl. Phys. 43, L88-L90 (2004).
[CrossRef]

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

Fukumoto, A.

K. Yasuda, M. Ono, K. Aratani, A. Fukumoto and M. Kaneko, "Premastered optical disk by superresolution," Jpn. J. Appl. Phys. 32, 5210-5213 (1993)
[CrossRef]

Gan, F.

F. Zhang, W. Xu, Y. Wang and F. Gan, "Static optical recording properties of super-resolution near-field structure with Bismuth mask layer," Jpn. J. Appl. Phys. 43, 7802-7806 (2004).
[CrossRef]

Ho, F. H.

Y. H. Fu, F. H. Ho, W.-C. Hsu, S.-Y. Tsai and D. P. Tsai, "Nonlinear optical properties of the Au-SiO nanocomposite superresolution near-field thin film," Jpn. J. Appl. Phys. 43, 5020-5023 (2004).
[CrossRef]

F. H. Ho, H. H. Chang, Y. H. Lin, B.-M. Chen, S.-Y. Wang, and D. P. Tsai, "Functional structures of AgOx thin film for near-field recording," Jpn. J. Appl. Phys., Part 1 42,1000-1004 (2003).
[CrossRef]

F. H. Ho, W. Y. Lin, H. H. Chang, Y. H. Lin, W.-C. Liu and D. P. Tsai, "Nonlinear optical absorption in the AgOx-type super-resolution near-field structure," Jap. J. Appl. Phys. 40, 4101-4102 (2001).
[CrossRef]

Hsu, S.-P.

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

Hsu, W.-C.

Y. H. Fu, F. H. Ho, W.-C. Hsu, S.-Y. Tsai and D. P. Tsai, "Nonlinear optical properties of the Au-SiO nanocomposite superresolution near-field thin film," Jpn. J. Appl. Phys. 43, 5020-5023 (2004).
[CrossRef]

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

Hung, T.-T.

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

Ji, J. M.

J. M. Ji, L. P. Shi, K. G. Lim, X. S. Miao, H. X. Yang and T. C. Chong, "Enhanced scattering of random-distribution nanoparticles and evanescent field in super-resolution near-field structure," Jpn. J. Appl. Phys. 44, 3620-3622 (2005).
[CrossRef]

Kaneko, M.

K. Yasuda, M. Ono, K. Aratani, A. Fukumoto and M. Kaneko, "Premastered optical disk by superresolution," Jpn. J. Appl. Phys. 32, 5210-5213 (1993)
[CrossRef]

Kao, T. S.

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

Kato, T.

T. Kikukawa, T. Kato, H. Shingai and H. Utsunomiya, "High-density read-only memory disc with super resolution reflective layer," Jpn. J. Appl. Phys. 40, 1624-1628 (2001).
[CrossRef]

Kikukawa, T.

T. Kikukawa, T. Nakano, T. Shima, and J. Tominaga, "Rigid bubble bit formation and huge signal enhancement in super-resolution near-field structure disk with platinum-oxide layer," Appl. Phys. Lett. 81, 4697-4699 (2002).
[CrossRef]

T. Kikukawa, T. Kato, H. Shingai and H. Utsunomiya, "High-density read-only memory disc with super resolution reflective layer," Jpn. J. Appl. Phys. 40, 1624-1628 (2001).
[CrossRef]

Kovacs, J.

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, "Photon scanning tunneling microscopy images of optical excitations of fractal metal colloid clusters," Phys. Rev. Lett. 72, 4149-4152 (1994).
[CrossRef] [PubMed]

Kuo, P. C.

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

Kuwahara, M.

T. Shima, M. Kuwahara, T. Fukaya, T. Nakano and J. Tominaga, "Super-resolutional readout disk with metal-free phthalocyanine recording layer," Jpn. J. Appl. Phys. 43, L88-L90 (2004).
[CrossRef]

Lanz, M.

D. W. Pohl, W. Denk, and M. Lanz, "Optical spectroscopy: Image recording with resolution lambda /20, " Appl. Phys. Lett. 44, 651-653 (1984).
[CrossRef]

Liao, A.

J. W. Fang, C. C. Wu, A. Liao. W. C. Lin and D. P. Tsai, "Implementation of practical super-resolution near-field structure system using commercial drive," Jap. J. Appl. Phys. 45, 1383-1384 (2006).
[CrossRef]

Lim, K. G.

J. M. Ji, L. P. Shi, K. G. Lim, X. S. Miao, H. X. Yang and T. C. Chong, "Enhanced scattering of random-distribution nanoparticles and evanescent field in super-resolution near-field structure," Jpn. J. Appl. Phys. 44, 3620-3622 (2005).
[CrossRef]

Lin, W. C.

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

W.-C. Liu, C.-Y. Wen, K.-H. Chen, W. C. Lin, and D. P. Tsai, "Near-field images of the AgOx-type super-resolution near-field structure," Appl. Phys. Lett. 78, 685-687 (2001).
[CrossRef]

Lin, W. Y.

F. H. Ho, W. Y. Lin, H. H. Chang, Y. H. Lin, W.-C. Liu and D. P. Tsai, "Nonlinear optical absorption in the AgOx-type super-resolution near-field structure," Jap. J. Appl. Phys. 40, 4101-4102 (2001).
[CrossRef]

Lin, Y. H.

F. H. Ho, H. H. Chang, Y. H. Lin, B.-M. Chen, S.-Y. Wang, and D. P. Tsai, "Functional structures of AgOx thin film for near-field recording," Jpn. J. Appl. Phys., Part 1 42,1000-1004 (2003).
[CrossRef]

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

F. H. Ho, W. Y. Lin, H. H. Chang, Y. H. Lin, W.-C. Liu and D. P. Tsai, "Nonlinear optical absorption in the AgOx-type super-resolution near-field structure," Jap. J. Appl. Phys. 40, 4101-4102 (2001).
[CrossRef]

Liu, W.-C.

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Enhanced resolution induced by random silver nanoparticles in near-field optical disks," Opt. Commun. 246, 561-567 (2005).
[CrossRef]

M.-Y. Ng and W.-C. Liu, "Super-resolution and frequency-dependent efficiency of near-field optical disks with silver nanoparticles," Opt. Exp. 13, 9422-9430 (2005).
[CrossRef]

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Near- and far-field optical properties of embedded scatters in AgOx-type super-resolution near-field structures," Scanning,  26, 102-105 (2004).

W.-C. Liu, M.-Y. Ng and D. P. Tsai, "Surface plasmon effects on the far-field signals of AgOx-type super-resolution near-field structure," Jpn. J. Appl. Phys. 43, 4713-4717 (2004).
[CrossRef]

W.-C. Liu and D. P. Tsai, "Nonlinear near-field optical effects of the AgOx-Type super-resolution near-field structure," Jpn. J. Appl. Phys. 42, Part 1, 1031-1032 (2003).
[CrossRef]

F. H. Ho, W. Y. Lin, H. H. Chang, Y. H. Lin, W.-C. Liu and D. P. Tsai, "Nonlinear optical absorption in the AgOx-type super-resolution near-field structure," Jap. J. Appl. Phys. 40, 4101-4102 (2001).
[CrossRef]

W.-C. Liu, C.-Y. Wen, K.-H. Chen, W. C. Lin, and D. P. Tsai, "Near-field images of the AgOx-type super-resolution near-field structure," Appl. Phys. Lett. 78, 685-687 (2001).
[CrossRef]

Miao, X. S.

J. M. Ji, L. P. Shi, K. G. Lim, X. S. Miao, H. X. Yang and T. C. Chong, "Enhanced scattering of random-distribution nanoparticles and evanescent field in super-resolution near-field structure," Jpn. J. Appl. Phys. 44, 3620-3622 (2005).
[CrossRef]

L. P. Shi, T. C. Chong, H. B. Yao, P. K. Tan and X. S. Miao, "Super-resolution near-field optical disk with an additional localized surface plasmon coupling layer," J. Appl. Phys. 91, 10209-10211 (2002).
[CrossRef]

Moskovits, M.

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, "Photon scanning tunneling microscopy images of optical excitations of fractal metal colloid clusters," Phys. Rev. Lett. 72, 4149-4152 (1994).
[CrossRef] [PubMed]

Nakano, T.

T. Shima, M. Kuwahara, T. Fukaya, T. Nakano and J. Tominaga, "Super-resolutional readout disk with metal-free phthalocyanine recording layer," Jpn. J. Appl. Phys. 43, L88-L90 (2004).
[CrossRef]

T. Kikukawa, T. Nakano, T. Shima, and J. Tominaga, "Rigid bubble bit formation and huge signal enhancement in super-resolution near-field structure disk with platinum-oxide layer," Appl. Phys. Lett. 81, 4697-4699 (2002).
[CrossRef]

J. Tominaga, T. Nakano, and N. Atoda, "An approach for recording and readout beyond the diffraction limit with an Sb thin film," Appl. Phys. Lett. 73,2078-2080 (1998).
[CrossRef]

Ng, M.-Y.

M.-Y. Ng and W.-C. Liu, "Super-resolution and frequency-dependent efficiency of near-field optical disks with silver nanoparticles," Opt. Exp. 13, 9422-9430 (2005).
[CrossRef]

W.-C. Liu, M.-Y. Ng and D. P. Tsai, "Surface plasmon effects on the far-field signals of AgOx-type super-resolution near-field structure," Jpn. J. Appl. Phys. 43, 4713-4717 (2004).
[CrossRef]

Nicholls, G.

E.A. Ash and G. Nicholls, " Super-resolution aperture scanning microscope," Nature 237, 510-512 (1972)
[CrossRef] [PubMed]

Nieto-Vesperinas, M.

Ohnuki, S.

H. Awano, S. Ohnuki, H. Shirai ans N. Ohta, "Magnetic domain expansion readout for amplification of an ultra high density magneto-optical recording signal," Appl. Phys. Lett. 69, 4257-4259 (1996).
[CrossRef]

Ono, M.

K. Yasuda, M. Ono, K. Aratani, A. Fukumoto and M. Kaneko, "Premastered optical disk by superresolution," Jpn. J. Appl. Phys. 32, 5210-5213 (1993)
[CrossRef]

Pohl, D. W.

D. W. Pohl, W. Denk, and M. Lanz, "Optical spectroscopy: Image recording with resolution lambda /20, " Appl. Phys. Lett. 44, 651-653 (1984).
[CrossRef]

Shalaev, V. M.

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, "Photon scanning tunneling microscopy images of optical excitations of fractal metal colloid clusters," Phys. Rev. Lett. 72, 4149-4152 (1994).
[CrossRef] [PubMed]

Shi, L. P.

J. M. Ji, L. P. Shi, K. G. Lim, X. S. Miao, H. X. Yang and T. C. Chong, "Enhanced scattering of random-distribution nanoparticles and evanescent field in super-resolution near-field structure," Jpn. J. Appl. Phys. 44, 3620-3622 (2005).
[CrossRef]

L. P. Shi, T. C. Chong, H. B. Yao, P. K. Tan and X. S. Miao, "Super-resolution near-field optical disk with an additional localized surface plasmon coupling layer," J. Appl. Phys. 91, 10209-10211 (2002).
[CrossRef]

Shima, T.

T. Shima, M. Kuwahara, T. Fukaya, T. Nakano and J. Tominaga, "Super-resolutional readout disk with metal-free phthalocyanine recording layer," Jpn. J. Appl. Phys. 43, L88-L90 (2004).
[CrossRef]

T. Kikukawa, T. Nakano, T. Shima, and J. Tominaga, "Rigid bubble bit formation and huge signal enhancement in super-resolution near-field structure disk with platinum-oxide layer," Appl. Phys. Lett. 81, 4697-4699 (2002).
[CrossRef]

Shinbori, S.

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

Shingai, H.

T. Kikukawa, T. Kato, H. Shingai and H. Utsunomiya, "High-density read-only memory disc with super resolution reflective layer," Jpn. J. Appl. Phys. 40, 1624-1628 (2001).
[CrossRef]

Synge, E. H.

E. H. Synge, "An application of piezoelectricity to microscopy," Philos. Mag. 6, 356-362 (1928).

Tan, P. K.

L. P. Shi, T. C. Chong, H. B. Yao, P. K. Tan and X. S. Miao, "Super-resolution near-field optical disk with an additional localized surface plasmon coupling layer," J. Appl. Phys. 91, 10209-10211 (2002).
[CrossRef]

Tominaga, J.

T. Shima, M. Kuwahara, T. Fukaya, T. Nakano and J. Tominaga, "Super-resolutional readout disk with metal-free phthalocyanine recording layer," Jpn. J. Appl. Phys. 43, L88-L90 (2004).
[CrossRef]

T. Kikukawa, T. Nakano, T. Shima, and J. Tominaga, "Rigid bubble bit formation and huge signal enhancement in super-resolution near-field structure disk with platinum-oxide layer," Appl. Phys. Lett. 81, 4697-4699 (2002).
[CrossRef]

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

J. Tominaga, T. Nakano, and N. Atoda, "An approach for recording and readout beyond the diffraction limit with an Sb thin film," Appl. Phys. Lett. 73,2078-2080 (1998).
[CrossRef]

Tsai, D. P.

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Enhanced resolution induced by random silver nanoparticles in near-field optical disks," Opt. Commun. 246, 561-567 (2005).
[CrossRef]

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Near- and far-field optical properties of embedded scatters in AgOx-type super-resolution near-field structures," Scanning,  26, 102-105 (2004).

Y. H. Fu, F. H. Ho, W.-C. Hsu, S.-Y. Tsai and D. P. Tsai, "Nonlinear optical properties of the Au-SiO nanocomposite superresolution near-field thin film," Jpn. J. Appl. Phys. 43, 5020-5023 (2004).
[CrossRef]

W.-C. Liu, M.-Y. Ng and D. P. Tsai, "Surface plasmon effects on the far-field signals of AgOx-type super-resolution near-field structure," Jpn. J. Appl. Phys. 43, 4713-4717 (2004).
[CrossRef]

W.-C. Liu and D. P. Tsai, "Nonlinear near-field optical effects of the AgOx-Type super-resolution near-field structure," Jpn. J. Appl. Phys. 42, Part 1, 1031-1032 (2003).
[CrossRef]

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

F. H. Ho, H. H. Chang, Y. H. Lin, B.-M. Chen, S.-Y. Wang, and D. P. Tsai, "Functional structures of AgOx thin film for near-field recording," Jpn. J. Appl. Phys., Part 1 42,1000-1004 (2003).
[CrossRef]

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

W.-C. Liu, C.-Y. Wen, K.-H. Chen, W. C. Lin, and D. P. Tsai, "Near-field images of the AgOx-type super-resolution near-field structure," Appl. Phys. Lett. 78, 685-687 (2001).
[CrossRef]

F. H. Ho, W. Y. Lin, H. H. Chang, Y. H. Lin, W.-C. Liu and D. P. Tsai, "Nonlinear optical absorption in the AgOx-type super-resolution near-field structure," Jap. J. Appl. Phys. 40, 4101-4102 (2001).
[CrossRef]

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, "Photon scanning tunneling microscopy images of optical excitations of fractal metal colloid clusters," Phys. Rev. Lett. 72, 4149-4152 (1994).
[CrossRef] [PubMed]

Tsai, S.-Y.

Y. H. Fu, F. H. Ho, W.-C. Hsu, S.-Y. Tsai and D. P. Tsai, "Nonlinear optical properties of the Au-SiO nanocomposite superresolution near-field thin film," Jpn. J. Appl. Phys. 43, 5020-5023 (2004).
[CrossRef]

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

Tseng, M.-R.

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

Utsunomiya, H.

T. Kikukawa, T. Kato, H. Shingai and H. Utsunomiya, "High-density read-only memory disc with super resolution reflective layer," Jpn. J. Appl. Phys. 40, 1624-1628 (2001).
[CrossRef]

Vigoureux, J. M.

Wang, S.-Y.

F. H. Ho, H. H. Chang, Y. H. Lin, B.-M. Chen, S.-Y. Wang, and D. P. Tsai, "Functional structures of AgOx thin film for near-field recording," Jpn. J. Appl. Phys., Part 1 42,1000-1004 (2003).
[CrossRef]

Wang, Y.

F. Zhang, W. Xu, Y. Wang and F. Gan, "Static optical recording properties of super-resolution near-field structure with Bismuth mask layer," Jpn. J. Appl. Phys. 43, 7802-7806 (2004).
[CrossRef]

Wang, Z.

D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, "Photon scanning tunneling microscopy images of optical excitations of fractal metal colloid clusters," Phys. Rev. Lett. 72, 4149-4152 (1994).
[CrossRef] [PubMed]

Wen, C.-Y.

W.-C. Liu, C.-Y. Wen, K.-H. Chen, W. C. Lin, and D. P. Tsai, "Near-field images of the AgOx-type super-resolution near-field structure," Appl. Phys. Lett. 78, 685-687 (2001).
[CrossRef]

Wolf, E.

Wu, C. C.

J. W. Fang, C. C. Wu, A. Liao. W. C. Lin and D. P. Tsai, "Implementation of practical super-resolution near-field structure system using commercial drive," Jap. J. Appl. Phys. 45, 1383-1384 (2006).
[CrossRef]

Wu, C. T.

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

Xu, W.

F. Zhang, W. Xu, Y. Wang and F. Gan, "Static optical recording properties of super-resolution near-field structure with Bismuth mask layer," Jpn. J. Appl. Phys. 43, 7802-7806 (2004).
[CrossRef]

Yang, H. X.

J. M. Ji, L. P. Shi, K. G. Lim, X. S. Miao, H. X. Yang and T. C. Chong, "Enhanced scattering of random-distribution nanoparticles and evanescent field in super-resolution near-field structure," Jpn. J. Appl. Phys. 44, 3620-3622 (2005).
[CrossRef]

Yao, H. B.

L. P. Shi, T. C. Chong, H. B. Yao, P. K. Tan and X. S. Miao, "Super-resolution near-field optical disk with an additional localized surface plasmon coupling layer," J. Appl. Phys. 91, 10209-10211 (2002).
[CrossRef]

Yasuda, K.

K. Yasuda, M. Ono, K. Aratani, A. Fukumoto and M. Kaneko, "Premastered optical disk by superresolution," Jpn. J. Appl. Phys. 32, 5210-5213 (1993)
[CrossRef]

Zhang, F.

F. Zhang, W. Xu, Y. Wang and F. Gan, "Static optical recording properties of super-resolution near-field structure with Bismuth mask layer," Jpn. J. Appl. Phys. 43, 7802-7806 (2004).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

H. Awano, S. Ohnuki, H. Shirai ans N. Ohta, "Magnetic domain expansion readout for amplification of an ultra high density magneto-optical recording signal," Appl. Phys. Lett. 69, 4257-4259 (1996).
[CrossRef]

J. Tominaga, T. Nakano, and N. Atoda, "An approach for recording and readout beyond the diffraction limit with an Sb thin film," Appl. Phys. Lett. 73,2078-2080 (1998).
[CrossRef]

D. W. Pohl, W. Denk, and M. Lanz, "Optical spectroscopy: Image recording with resolution lambda /20, " Appl. Phys. Lett. 44, 651-653 (1984).
[CrossRef]

T. Kikukawa, T. Nakano, T. Shima, and J. Tominaga, "Rigid bubble bit formation and huge signal enhancement in super-resolution near-field structure disk with platinum-oxide layer," Appl. Phys. Lett. 81, 4697-4699 (2002).
[CrossRef]

W.-C. Liu, C.-Y. Wen, K.-H. Chen, W. C. Lin, and D. P. Tsai, "Near-field images of the AgOx-type super-resolution near-field structure," Appl. Phys. Lett. 78, 685-687 (2001).
[CrossRef]

J. Appl. Phys.

L. P. Shi, T. C. Chong, H. B. Yao, P. K. Tan and X. S. Miao, "Super-resolution near-field optical disk with an additional localized surface plasmon coupling layer," J. Appl. Phys. 91, 10209-10211 (2002).
[CrossRef]

T. Fukaya, D. Büchel, S. Shinbori, J. Tominaga, N. Atoda, D. P. Tsai and W. C. Lin, "Micro-optical nonlinearity of a silver oxide layer," J. Appl. Phys. 89, 6139-6144 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Jap. J. Appl. Phys.

F. H. Ho, W. Y. Lin, H. H. Chang, Y. H. Lin, W.-C. Liu and D. P. Tsai, "Nonlinear optical absorption in the AgOx-type super-resolution near-field structure," Jap. J. Appl. Phys. 40, 4101-4102 (2001).
[CrossRef]

J. W. Fang, C. C. Wu, A. Liao. W. C. Lin and D. P. Tsai, "Implementation of practical super-resolution near-field structure system using commercial drive," Jap. J. Appl. Phys. 45, 1383-1384 (2006).
[CrossRef]

Jpn. J. Appl. Phys

F. H. Ho, H. H. Chang, Y. H. Lin, B.-M. Chen, S.-Y. Wang, and D. P. Tsai, "Functional structures of AgOx thin film for near-field recording," Jpn. J. Appl. Phys., Part 1 42,1000-1004 (2003).
[CrossRef]

Jpn. J. Appl. Phys.

W.-C. Liu and D. P. Tsai, "Nonlinear near-field optical effects of the AgOx-Type super-resolution near-field structure," Jpn. J. Appl. Phys. 42, Part 1, 1031-1032 (2003).
[CrossRef]

W.-C. Liu, M.-Y. Ng and D. P. Tsai, "Surface plasmon effects on the far-field signals of AgOx-type super-resolution near-field structure," Jpn. J. Appl. Phys. 43, 4713-4717 (2004).
[CrossRef]

J. M. Ji, L. P. Shi, K. G. Lim, X. S. Miao, H. X. Yang and T. C. Chong, "Enhanced scattering of random-distribution nanoparticles and evanescent field in super-resolution near-field structure," Jpn. J. Appl. Phys. 44, 3620-3622 (2005).
[CrossRef]

F. Zhang, W. Xu, Y. Wang and F. Gan, "Static optical recording properties of super-resolution near-field structure with Bismuth mask layer," Jpn. J. Appl. Phys. 43, 7802-7806 (2004).
[CrossRef]

Y. H. Fu, F. H. Ho, W.-C. Hsu, S.-Y. Tsai and D. P. Tsai, "Nonlinear optical properties of the Au-SiO nanocomposite superresolution near-field thin film," Jpn. J. Appl. Phys. 43, 5020-5023 (2004).
[CrossRef]

H.-H. Chiang, W.-C. Hsu, S.-Y. Tsai, M.-R. Tseng, S.-P. Hsu, T.-T. Hung, C.-J. Chang and P. C. Kuo, "Thermal and optical properties of organic dyes for duper-resolution recordable disks," Jpn. J. Appl. Phys. 42, 997-999 (2003).
[CrossRef]

T. Shima, M. Kuwahara, T. Fukaya, T. Nakano and J. Tominaga, "Super-resolutional readout disk with metal-free phthalocyanine recording layer," Jpn. J. Appl. Phys. 43, L88-L90 (2004).
[CrossRef]

T. Kikukawa, T. Kato, H. Shingai and H. Utsunomiya, "High-density read-only memory disc with super resolution reflective layer," Jpn. J. Appl. Phys. 40, 1624-1628 (2001).
[CrossRef]

W. C. Lin, T. S. Kao, H. H. Chang, Y. H. Lin, Y. H. Fu, C. T. Wu, K. H. Chen and D. P. Tsai, "Study of a Super-resolution optical structure: polycarbonate/ZnS-SiO2/ZnO/ZnS-SiO2/Ge2Sb2Te5/ZnS-SiO2," Jpn. J. Appl. Phys. 42, 1029-1030 (2003).
[CrossRef]

K. Yasuda, M. Ono, K. Aratani, A. Fukumoto and M. Kaneko, "Premastered optical disk by superresolution," Jpn. J. Appl. Phys. 32, 5210-5213 (1993)
[CrossRef]

Nature

E.A. Ash and G. Nicholls, " Super-resolution aperture scanning microscope," Nature 237, 510-512 (1972)
[CrossRef] [PubMed]

Opt. Commun.

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Enhanced resolution induced by random silver nanoparticles in near-field optical disks," Opt. Commun. 246, 561-567 (2005).
[CrossRef]

Opt. Exp.

M.-Y. Ng and W.-C. Liu, "Super-resolution and frequency-dependent efficiency of near-field optical disks with silver nanoparticles," Opt. Exp. 13, 9422-9430 (2005).
[CrossRef]

Philos. Mag.

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Phys. Rev. Lett.

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Scanning

T. C. Chu, W.-C. Liu, and D. P. Tsai, "Near- and far-field optical properties of embedded scatters in AgOx-type super-resolution near-field structures," Scanning,  26, 102-105 (2004).

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S. Kawata, M. Ohtsu and M. Irie, ed., Nano-Optics (Springer, Berlin Heidelberg, 2002).

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Figures (7)

Fig. 1.
Fig. 1.

Model of the near-field optical disk with a random nanostructure.

Fig. 2
Fig. 2

Illustration of readout contrast in spatial frequency domain: (a) Aperture size D is close Gaussian beam width a. (b) Aperture size D is much smaller than the beam width a.

Fig. 3
Fig. 3

Readout contrast signals of analytical results (solid line) and numerical results (mark) with various aperture size for (a) for normal transmitting (kx = 0) cases and (b) N.A. = 0.6 cases.

Fig. 4
Fig. 4

Numerical readout contrast signals of dilute case for (a) kx = 0, and (b) N.A. = 0.6 cases. The coverage rate is 65%.

Fig. 5
Fig. 5

(a) Waveform of single frequency case for kx = 0 and D = 0.1λ ; (b) Numerical waveform of two-frequency case for kx = 0 and D = 0.1λ .

Fig. 5
Fig. 5

A realistic model of near-field optical disk with random nanostructures.

Fig. 6
Fig. 6

Readout contrasts of numerical results for N.A. = 0.6 for a model of near-field optical disk.

Equations (23)

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A ( k x ) = ʃ U x 0 exp ( i k x x ) dx ,
U x z = ʃ A ( k x ) exp ( i ( k x x + k z z ) ) d k x ,
I ( k x ) = A ( k x ) 2 .
I ( k x ) = 1 4 a π 7 2 [ ʃ e x ' 2 a 2 G ( x ' ) F ( x ' ) e i k x x ' dx ' ] * [ ʃ e x 2 a 2 G ( x ) F ( x ) e i k x x dx ] ,
G ( x ) = 1 2 ( 1 + cos ( gx ) ) ,
G ( x ) = 1 2 ( 1 cos ( gx ) ) .
Δ I ( k x ) = 1 4 a π 7 2 ʃ ʃ dxdx ' e x 2 x ' 2 a 2 ( cos ( gx ) + cos ( gx ' ) ) F ( x ) F ( x ' ) e i k x ( x x ' ) .
Δ I ( k x ) = 1 4 a π 7 2 ʃ ʃ dxdx ' e x 2 x ' 2 a 2 ( cos ( gx ) + cos ( gx ' ) ) F ( x ) F ( x ' ) e i k x ( x x ' ) .
= 1 4 a π 7 2 e g 2 a 2 8 { h 2 [ exp ( a 2 ( k x g 2 ) 2 2 ) + exp ( a 2 ( k x + g 2 ) 2 2 ) ]
+ σ 2 l 2 a 2 + l 2 [ exp ( a 2 l 2 ( k x g 2 ) 2 2 ( 2 a 2 + l 2 ) ) + exp ( a 2 l 2 ( k x + g 2 ) 2 2 ( 2 a 2 + l 2 ) ) ] } .
Δ I 2 ( k x ) = ( 1 4 a π 7 2 ) 2 ʃ ʃ ʃ ʃ d x 1 d x 1 ' d x 2 d x 2 ' e x 1 2 x 1 ' 2 x 2 2 x 2 ' 2 a 2 ( cos ( g x 1 ) + cos ( g x 1 ' ) )
· ( cos ( g x 2 ) + cos ( g x 2 ' ) ) e i k x ( x 1 x 1 ' + x 2 x 2 ' ) F ( x 1 ) F ( x 1 ' ) F ( x 2 ) F ( x 2 ' ) .
F ( x 1 ) F ( x 1 ' ) F ( x 2 ) F ( x 2 ' ) = i , j , l , m F x 1 x i F ( x 1 ' , x j ) F x 2 x l F ( x 2 ' , x m ) ,
Δ I 2 ( k x ) = i , j , l , m Δ I 2 k x x i x j x l x m .
Δ I 2 ( k x ) = i , j , l , m ʃ ʃ ʃ ʃ d x i d x j d x l d x m Δ I 2 k x x i x j x l x m P ( x i ) P ( x j ) P ( x l ) P ( x m )
= C ʃ d x i Δ I 2 k x x i = x j = x l = x m P ( x i ) + O ( C 2 ) ,
F ( x i ) = i = 1 N e ( x x i ) 2 D 2 ,
Δ I 2 ( k x ) = aD 4 π 3 ( a 2 + D 2 ) ( e a 2 D 2 ( k x + g 2 ) 2 2 ( a 2 + D 2 ) + e a 2 D 2 ( k x g 2 ) 2 2 ( a 2 + D 2 ) )
· [ ( a 2 + D 2 ) 2 3 2 D [ e g 2 a 2 4 + e g 2 a 2 D 2 4 ( a 2 + D 2 ) ] ] 1 2 .
G ( x ) = 1 2 ( 1 + cos ( g ( x + l ) ) ) .
I l x i = 1 4 π 5 2 ( a D 2 a 2 + D 2 ) const · + cos ( g ( a 2 x i a 2 + D 2 + l ) ) · e 2 x i 2 a 2 D 2 ( k x + g ) 2 4 a 2 D 2 k x 2 4 ( a 2 + D 2 ) + cos ( g ( a 2 x i a 2 + D 2 + l ) ) · e 2 x i 2 a 2 D 2 ( k x g ) 2 4 a 2 D 2 k x 2 4 ( a 2 + D 2 ) + cos ( 2 g ( a 2 x i a 2 + D 2 + l ) ) · e 2 x i 2 a 2 D 2 ( k x + g ) 2 4 a 2 D 2 ( k x g ) 2 4 ( a 2 + D 2 )
I 2 ( l ) ~ e 11 a 2 D 2 g 2 16 ( a 2 + D 2 ) ( e a 2 D 2 ( k x + g 4 ) 2 ( a 2 + D 2 ) + e a 2 D 2 ( k x g 4 ) 2 ( a 2 + D 2 ) )
· ( cos ( 3 g · l ) e 9 a 4 g 2 16 ( a 2 + D 2 ) + cos ( g · l ) e a 4 g 2 16 ( a 2 + D 2 ) ) + O ( e a 2 g 2 4 ) .

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