Abstract

This article depicts a general case of recording a grating hologram, with the distance between the object and the hologram being arbitrarily chosen. The universality of this approach does not exclude the possibility of restoring the imaginary and real images of the grating and their self-image sequences. In the case under consideration, restoring the hologram of the grating results in the reconstruction of both imaginary and real grating images and the sequences of their self-images in ± 1st diffraction orders. An important feature of reconstructing a grating hologram is that the self-image sequences for both imaginary and real images extend to the real image area. The ratios derived in the article also show that the restoration of the grating self-image sequences occurs in the 0-th diffraction order.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Ser. 9(56), 401–407 (1836).
  2. Lord Rayleigh, “One coping diffraction grating and one some phenomena connected therewith,” Phil. Mag. Ser. 11(67), 196–205 (1881).
  3. M. Wolfke, “Periodic objects of infinite aperture,” Ann. Phys. 345(1), 194–200 (1913).
    [Crossref]
  4. F. Zernike, “Microscope imaging,” Physik. Z. 36, 848–854 (1935).
  5. H. H. Hopkins, “Talbot effect with infinite aperture objects,” Proc. Roy. Soc. (London) ser. A 217, 408–413 (1953).
  6. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel Images. I. Plane Periodic Objects in Monochromatic Light,” J. Opt. Soc. Am. 55(4), 373–381 (1965).
    [Crossref]
  7. W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. 57(6), 772–778 (1967).
    [Crossref]
  8. R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30(1), 73–84 (1983).
    [Crossref]
  9. K. Hane and C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. 4(4), 706–711 (1987).
    [Crossref]
  10. P. Latimer, “Talbot plane patterns: grating images or interference effects?” Appl. Opt. 32(7), 1078–1083 (1993).
    [Crossref]
  11. P. Latimer, “Use of the Talbot effect to couple the phases of lasers,” Appl. Phys. Lett. 62(3), 217–218 (1993).
    [Crossref]
  12. L. Liu, “Talbot and Lau effects on incident beam of arbitrary wavefront, and their use,” Appl. Opt. 28(21), 4668–4677 (1989).
    [Crossref]
  13. J. Ebbeni, “Nouveaux aspects du phenomene de moire. I,” Nouv. Rev. Opt. Appl. 1(5), 333–342 (1970).
    [Crossref]
  14. Y. Nakano and K. Murata, “Measurements of phase objects using the Talbot effect and moiré techniques,” Appl. Opt. 23(14), 2296–2299 (1984).
    [Crossref]
  15. J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photonics 5(1), 83–130 (2013).
    [Crossref]
  16. S. Teng, Z. Chen, T. Zhou, and C. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. 24(6), 1656–1665 (2007).
    [Crossref]
  17. J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98(8), 081108 (2011).
    [Crossref]
  18. Z. Benkö, “New considerations on Talbot’s bands,” Am. J. Phys. 68(6), 513–520 (2000).
    [Crossref]
  19. H. Konitz, “Zur Frage der selbastalbildung (Talbot – effekt) Periodischer Objecte unter holographischen Bedingungen,” Optik 66(3), 197–204 (1984).
  20. H. Konitz, S. Boseck, and R. Lasch, “Holographischen Experimente zum Talbot – Effekt von Streifengittern,” Optik. 69(3), 91–93 (1985).
  21. A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Appl. Phys. 74(12), 7039–7043 (1993).
    [Crossref]
  22. A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Opt. 25(1), 3–8 (1994).
    [Crossref]
  23. Y. K. Ismanov, T. D. Tynyshova, and Z. K. Aidaraliev, “Wide-range holographic interferometer,” Opt. Eng. 57(12), 1 (2019).
    [Crossref]
  24. A. Maripov, Y. Ismanov, and K. Omyrzakov, “Four-channel wide-range holographic interferometer,” Proc. SPIE 5144, 606–610 (2003).
    [Crossref]
  25. A. Maripov and Y. Ismanov, “Interferometer based on the Talbot effect in holography,” J. Opt. 26(1), 25–28 (1995).
    [Crossref]
  26. Y. Kh. Ismanov and A. Maripov, “Holographic Talbot Interferometer,” Proc. SPIE 4149, 213–220 (2000).
    [Crossref]
  27. D. Gabor, “Microscopy by reconstructed wave fronts,” Proc. Phys. Soc., London, Sect. B 64(6), 449–469 (1951).
    [Crossref]

2019 (1)

Y. K. Ismanov, T. D. Tynyshova, and Z. K. Aidaraliev, “Wide-range holographic interferometer,” Opt. Eng. 57(12), 1 (2019).
[Crossref]

2013 (1)

J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photonics 5(1), 83–130 (2013).
[Crossref]

2011 (1)

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98(8), 081108 (2011).
[Crossref]

2007 (1)

S. Teng, Z. Chen, T. Zhou, and C. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. 24(6), 1656–1665 (2007).
[Crossref]

2003 (1)

A. Maripov, Y. Ismanov, and K. Omyrzakov, “Four-channel wide-range holographic interferometer,” Proc. SPIE 5144, 606–610 (2003).
[Crossref]

2000 (2)

Y. Kh. Ismanov and A. Maripov, “Holographic Talbot Interferometer,” Proc. SPIE 4149, 213–220 (2000).
[Crossref]

Z. Benkö, “New considerations on Talbot’s bands,” Am. J. Phys. 68(6), 513–520 (2000).
[Crossref]

1995 (1)

A. Maripov and Y. Ismanov, “Interferometer based on the Talbot effect in holography,” J. Opt. 26(1), 25–28 (1995).
[Crossref]

1994 (1)

A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Opt. 25(1), 3–8 (1994).
[Crossref]

1993 (3)

A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Appl. Phys. 74(12), 7039–7043 (1993).
[Crossref]

P. Latimer, “Talbot plane patterns: grating images or interference effects?” Appl. Opt. 32(7), 1078–1083 (1993).
[Crossref]

P. Latimer, “Use of the Talbot effect to couple the phases of lasers,” Appl. Phys. Lett. 62(3), 217–218 (1993).
[Crossref]

1989 (1)

1987 (1)

K. Hane and C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. 4(4), 706–711 (1987).
[Crossref]

1985 (1)

H. Konitz, S. Boseck, and R. Lasch, “Holographischen Experimente zum Talbot – Effekt von Streifengittern,” Optik. 69(3), 91–93 (1985).

1984 (2)

H. Konitz, “Zur Frage der selbastalbildung (Talbot – effekt) Periodischer Objecte unter holographischen Bedingungen,” Optik 66(3), 197–204 (1984).

Y. Nakano and K. Murata, “Measurements of phase objects using the Talbot effect and moiré techniques,” Appl. Opt. 23(14), 2296–2299 (1984).
[Crossref]

1983 (1)

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30(1), 73–84 (1983).
[Crossref]

1970 (1)

J. Ebbeni, “Nouveaux aspects du phenomene de moire. I,” Nouv. Rev. Opt. Appl. 1(5), 333–342 (1970).
[Crossref]

1967 (1)

1965 (1)

1953 (1)

H. H. Hopkins, “Talbot effect with infinite aperture objects,” Proc. Roy. Soc. (London) ser. A 217, 408–413 (1953).

1951 (1)

D. Gabor, “Microscopy by reconstructed wave fronts,” Proc. Phys. Soc., London, Sect. B 64(6), 449–469 (1951).
[Crossref]

1935 (1)

F. Zernike, “Microscope imaging,” Physik. Z. 36, 848–854 (1935).

1913 (1)

M. Wolfke, “Periodic objects of infinite aperture,” Ann. Phys. 345(1), 194–200 (1913).
[Crossref]

1881 (1)

Lord Rayleigh, “One coping diffraction grating and one some phenomena connected therewith,” Phil. Mag. Ser. 11(67), 196–205 (1881).

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Ser. 9(56), 401–407 (1836).

Aidaraliev, Z. K.

Y. K. Ismanov, T. D. Tynyshova, and Z. K. Aidaraliev, “Wide-range holographic interferometer,” Opt. Eng. 57(12), 1 (2019).
[Crossref]

Benkö, Z.

Z. Benkö, “New considerations on Talbot’s bands,” Am. J. Phys. 68(6), 513–520 (2000).
[Crossref]

Boseck, S.

H. Konitz, S. Boseck, and R. Lasch, “Holographischen Experimente zum Talbot – Effekt von Streifengittern,” Optik. 69(3), 91–93 (1985).

Chen, H.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98(8), 081108 (2011).
[Crossref]

Chen, Z.

S. Teng, Z. Chen, T. Zhou, and C. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. 24(6), 1656–1665 (2007).
[Crossref]

Cheng, C.

S. Teng, Z. Chen, T. Zhou, and C. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. 24(6), 1656–1665 (2007).
[Crossref]

Du, S.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98(8), 081108 (2011).
[Crossref]

Ebbeni, J.

J. Ebbeni, “Nouveaux aspects du phenomene de moire. I,” Nouv. Rev. Opt. Appl. 1(5), 333–342 (1970).
[Crossref]

Gabor, D.

D. Gabor, “Microscopy by reconstructed wave fronts,” Proc. Phys. Soc., London, Sect. B 64(6), 449–469 (1951).
[Crossref]

Grover, C. P.

K. Hane and C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. 4(4), 706–711 (1987).
[Crossref]

Hane, K.

K. Hane and C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. 4(4), 706–711 (1987).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, “Talbot effect with infinite aperture objects,” Proc. Roy. Soc. (London) ser. A 217, 408–413 (1953).

Ismanov, Y.

A. Maripov, Y. Ismanov, and K. Omyrzakov, “Four-channel wide-range holographic interferometer,” Proc. SPIE 5144, 606–610 (2003).
[Crossref]

A. Maripov and Y. Ismanov, “Interferometer based on the Talbot effect in holography,” J. Opt. 26(1), 25–28 (1995).
[Crossref]

A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Opt. 25(1), 3–8 (1994).
[Crossref]

A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Appl. Phys. 74(12), 7039–7043 (1993).
[Crossref]

Ismanov, Y. K.

Y. K. Ismanov, T. D. Tynyshova, and Z. K. Aidaraliev, “Wide-range holographic interferometer,” Opt. Eng. 57(12), 1 (2019).
[Crossref]

Ismanov, Y. Kh.

Y. Kh. Ismanov and A. Maripov, “Holographic Talbot Interferometer,” Proc. SPIE 4149, 213–220 (2000).
[Crossref]

Jozwicki, R.

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30(1), 73–84 (1983).
[Crossref]

Konitz, H.

H. Konitz, S. Boseck, and R. Lasch, “Holographischen Experimente zum Talbot – Effekt von Streifengittern,” Optik. 69(3), 91–93 (1985).

H. Konitz, “Zur Frage der selbastalbildung (Talbot – effekt) Periodischer Objecte unter holographischen Bedingungen,” Optik 66(3), 197–204 (1984).

Lasch, R.

H. Konitz, S. Boseck, and R. Lasch, “Holographischen Experimente zum Talbot – Effekt von Streifengittern,” Optik. 69(3), 91–93 (1985).

Latimer, P.

P. Latimer, “Talbot plane patterns: grating images or interference effects?” Appl. Opt. 32(7), 1078–1083 (1993).
[Crossref]

P. Latimer, “Use of the Talbot effect to couple the phases of lasers,” Appl. Phys. Lett. 62(3), 217–218 (1993).
[Crossref]

Liu, L.

Maripov, A.

A. Maripov, Y. Ismanov, and K. Omyrzakov, “Four-channel wide-range holographic interferometer,” Proc. SPIE 5144, 606–610 (2003).
[Crossref]

Y. Kh. Ismanov and A. Maripov, “Holographic Talbot Interferometer,” Proc. SPIE 4149, 213–220 (2000).
[Crossref]

A. Maripov and Y. Ismanov, “Interferometer based on the Talbot effect in holography,” J. Opt. 26(1), 25–28 (1995).
[Crossref]

A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Opt. 25(1), 3–8 (1994).
[Crossref]

A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Appl. Phys. 74(12), 7039–7043 (1993).
[Crossref]

Montgomery, W. D.

Murata, K.

Nakano, Y.

Omyrzakov, K.

A. Maripov, Y. Ismanov, and K. Omyrzakov, “Four-channel wide-range holographic interferometer,” Proc. SPIE 5144, 606–610 (2003).
[Crossref]

Rayleigh, Lord

Lord Rayleigh, “One coping diffraction grating and one some phenomena connected therewith,” Phil. Mag. Ser. 11(67), 196–205 (1881).

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Ser. 9(56), 401–407 (1836).

Teng, S.

S. Teng, Z. Chen, T. Zhou, and C. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. 24(6), 1656–1665 (2007).
[Crossref]

Tynyshova, T. D.

Y. K. Ismanov, T. D. Tynyshova, and Z. K. Aidaraliev, “Wide-range holographic interferometer,” Opt. Eng. 57(12), 1 (2019).
[Crossref]

Wen, J.

J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photonics 5(1), 83–130 (2013).
[Crossref]

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98(8), 081108 (2011).
[Crossref]

Winthrop, J. T.

Wolfke, M.

M. Wolfke, “Periodic objects of infinite aperture,” Ann. Phys. 345(1), 194–200 (1913).
[Crossref]

Worthington, C. R.

Xiao, M.

J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photonics 5(1), 83–130 (2013).
[Crossref]

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98(8), 081108 (2011).
[Crossref]

Zernike, F.

F. Zernike, “Microscope imaging,” Physik. Z. 36, 848–854 (1935).

Zhang, Y.

J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photonics 5(1), 83–130 (2013).
[Crossref]

Zhou, T.

S. Teng, Z. Chen, T. Zhou, and C. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. 24(6), 1656–1665 (2007).
[Crossref]

Adv. Opt. Photonics (1)

J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photonics 5(1), 83–130 (2013).
[Crossref]

Am. J. Phys. (1)

Z. Benkö, “New considerations on Talbot’s bands,” Am. J. Phys. 68(6), 513–520 (2000).
[Crossref]

Ann. Phys. (1)

M. Wolfke, “Periodic objects of infinite aperture,” Ann. Phys. 345(1), 194–200 (1913).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

P. Latimer, “Use of the Talbot effect to couple the phases of lasers,” Appl. Phys. Lett. 62(3), 217–218 (1993).
[Crossref]

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98(8), 081108 (2011).
[Crossref]

J. Appl. Phys. (1)

A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Appl. Phys. 74(12), 7039–7043 (1993).
[Crossref]

J. Opt. (2)

A. Maripov and Y. Ismanov, “The Talbot effect (a self – imaging phenomenon) in holography,” J. Opt. 25(1), 3–8 (1994).
[Crossref]

A. Maripov and Y. Ismanov, “Interferometer based on the Talbot effect in holography,” J. Opt. 26(1), 25–28 (1995).
[Crossref]

J. Opt. Soc. Am. (4)

J. T. Winthrop and C. R. Worthington, “Theory of Fresnel Images. I. Plane Periodic Objects in Monochromatic Light,” J. Opt. Soc. Am. 55(4), 373–381 (1965).
[Crossref]

W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. 57(6), 772–778 (1967).
[Crossref]

S. Teng, Z. Chen, T. Zhou, and C. Cheng, “Quasi-Talbot effect of a grating in the deep Fresnel diffraction region,” J. Opt. Soc. Am. 24(6), 1656–1665 (2007).
[Crossref]

K. Hane and C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. 4(4), 706–711 (1987).
[Crossref]

Nouv. Rev. Opt. Appl. (1)

J. Ebbeni, “Nouveaux aspects du phenomene de moire. I,” Nouv. Rev. Opt. Appl. 1(5), 333–342 (1970).
[Crossref]

Opt. Acta (1)

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30(1), 73–84 (1983).
[Crossref]

Opt. Eng. (1)

Y. K. Ismanov, T. D. Tynyshova, and Z. K. Aidaraliev, “Wide-range holographic interferometer,” Opt. Eng. 57(12), 1 (2019).
[Crossref]

Optik (1)

H. Konitz, “Zur Frage der selbastalbildung (Talbot – effekt) Periodischer Objecte unter holographischen Bedingungen,” Optik 66(3), 197–204 (1984).

Optik. (1)

H. Konitz, S. Boseck, and R. Lasch, “Holographischen Experimente zum Talbot – Effekt von Streifengittern,” Optik. 69(3), 91–93 (1985).

Phil. Mag. Ser. (2)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Ser. 9(56), 401–407 (1836).

Lord Rayleigh, “One coping diffraction grating and one some phenomena connected therewith,” Phil. Mag. Ser. 11(67), 196–205 (1881).

Physik. Z. (1)

F. Zernike, “Microscope imaging,” Physik. Z. 36, 848–854 (1935).

Proc. Phys. Soc., London, Sect. B (1)

D. Gabor, “Microscopy by reconstructed wave fronts,” Proc. Phys. Soc., London, Sect. B 64(6), 449–469 (1951).
[Crossref]

Proc. Roy. Soc. (London) ser. A (1)

H. H. Hopkins, “Talbot effect with infinite aperture objects,” Proc. Roy. Soc. (London) ser. A 217, 408–413 (1953).

Proc. SPIE (2)

A. Maripov, Y. Ismanov, and K. Omyrzakov, “Four-channel wide-range holographic interferometer,” Proc. SPIE 5144, 606–610 (2003).
[Crossref]

Y. Kh. Ismanov and A. Maripov, “Holographic Talbot Interferometer,” Proc. SPIE 4149, 213–220 (2000).
[Crossref]

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Figures (2)

Fig. 1.
Fig. 1. Scheme of hologram recording of the grating. G is the grating; R is the reference wave; ${V_0}$ is the object wave.
Fig. 2.
Fig. 2. Scheme of hologram recovery of the grating. G is the imaginary image of the grating; SI are positions of the self-image planes in the imaginary image area; SR are positions of self-image planes in the real image area; H is the hologram; $t = \frac{{2{d^2}}}{\lambda }$ is the self-imaging constant; ${z_H}$ is the distance between the reconstructed image and the hologram plane.

Equations (19)

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V 0 ( x 1 , y 1 , z 1 , ) = a exp ( i k z 1 ) .
V 0 ( x 0 , y 0 , z 0 ) = a exp [ i k ( z 0 cos φ + x 0 sin φ ) ] .
V 0 ( x 0 , y 0 , z 0 ) = a exp ( i k x 0 sin φ ) .
V 0 ( x 0 , y 0 , z 0 + ) = σ ( x 0 ) a exp ( i k x 0 sin φ ) .
V 0 ( x 0 , y 0 , z 0 + ) = a exp ( i k x 0 sin φ ) n = b n exp ( 2 π i n x 0 / d ) .
V z ( x , y , z ) = exp ( i k z ) i k z V ( x 0 , y 0 , z 0 + ) exp { i π λ z [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } d x 0 d y 0 .
V z ( x , y , z ) = λ 2 exp ( i k z ) 2 π i exp ( i π / 2 ) exp ( 0 , 5 i k z sin 2 φ ) exp ( i k x sin φ ) × × n = b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] = L exp ( i k x sin φ ) n = b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] .
R = A exp ( i k z ) .
σ = σ 0 + σ n .
σ 0 = b 0 , σ n = n = , n 0 b n exp [ i 2 x π ( n x d n 2 λ z 2 d 2 ) ]
V z ( x , y , z ) = V 0 ( x , y , z ) + V n ( x , y , z ) = L exp ( i k x sin φ ) b 0 + L exp ( i k x sin φ ) × × n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ]
I ( x , y ) = | R + V z | 2 = | R + V 0 + V n | 2 = ( R + V 0 + V n ) ( R + V 0 + V n ) = A 2 + + A exp ( i k z ) { L b 0 exp ( i k x sin φ ) + L b 0 exp ( i k x sin φ ) × × n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] } + A exp ( i k z ) { L b 0 exp ( i k x sin φ ) + + L exp ( i k x sin φ ) n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] } + + L b 0 exp ( i k x sin φ ) L b 0 exp ( i k x sin φ ) + L b 0 exp ( i k x sin φ ) L exp ( i k x sin φ ) × × n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + L exp ( i k x sin φ ) L b 0 exp ( i k x sin φ ) × × n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + L exp ( i k x sin φ ) n = , n 0 exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + + L exp ( i k x sin φ ) n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] = A 2 + A exp ( i k z ) { L b 0 + L × × n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] } exp ( i k x sin φ ) + A exp ( i k z ) { L b 0 + L × × n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] } exp ( i k x sin φ ) + λ 4 4 π 2 b 0 2 + λ 4 4 π 2 b 0 × × n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + λ 4 4 π 2 b 0 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + + λ 4 4 π 2 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] = = A 2 + A exp ( i k z ) L n = b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] exp ( i k x sin φ ) + + A exp ( i k z ) L n = b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] exp ( i k x sin φ ) + λ 4 4 π 2 b 0 2 + λ 4 4 π 2 b 0 × × n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + λ 4 4 π 2 b 0 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + + λ 4 4 π 2 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ]
λ 4 4 π 2 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ]
ρ ( x , y ) I ( x , y ) = A exp ( i k z ) { L b 0 + L n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] } × × exp ( i k x sin φ ) + A exp ( i k z ) { L b 0 + L n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] } × × exp ( i k x sin φ ) + λ 4 4 π 2 b 0 2 + λ 4 4 π 2 b 0 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + + λ 4 4 π 2 b 0 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] = A exp ( i k z ) L n = b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] × × exp ( i k x sin φ ) + A exp ( i k z ) L n = b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] exp ( i k x sin φ ) + λ 4 4 π 2 b 0 2 + + λ 4 4 π 2 b 0 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] + λ 4 4 π 2 b 0 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ]
L n = b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] exp ( i k x sin φ ) .
λ 4 4 π 2 b 0 2 , λ 2 4 π 2 b 0 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] and λ 2 4 π 2 b 0 n = , n 0 b n exp [ i 2 π ( n x d n 2 λ z 2 d 2 ) ] .
z 1 = z H + 2 d 2 λ k .
z 1 = 2 d 2 λ n + [ z H int ( z H t ) t ] .
z 1 = 2 d 2 λ n + [ z H int ( z H t ) t ] .