Abstract

A simple method for generating 2D binary amplitude structure with additive superimposition of mutually orthogonal 1D amplitude gratings is proposed. Its implementation requires software generated three binary amplitude gratings, i.e., the crossed Ronchi, checker board and 1D Ronchi gratings with aspect ratio equal to 0.5. Their computer processing involves only two steps. First the checker grating is multiplied by a high frequency 1D grating. Next the product is added to the crossed grating. In result 3-level transmittance (0, 0.5, 1) hybrid diffraction structure is obtained. The intermediate level results from the use of a dense 1D grating. The zero diffraction order, well separated from the rest of the spectrum, consists of crossed spectra of additively superimposed 1D Ronchi gratings. Detailed heuristic explanation of the process aided by spectrum domain analyses is presented. Additionally, simulations and experiments conducted in the Fresnel diffraction field exemplify the invented structure properties in comparison with the multiplicative superimposition crossed Ronchi grating. Up to authors’ best knowledge the Fresnel field (self-imaging phenomenon or Talbot effect) properties of 2D periodic structure with additive superimposition of component 1D gratings have not been published in the literature.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  7. K. Patorski, Ł. Służewski, and M. Trusiak, “Single-shot 3 × 3 beam grating interferometry for self-imaging free extended range wave front sensing,” Opt. Lett. 41(18), 4417–4420 (2016).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  23. K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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2017 (3)

K. Patorski, L. Sluzewski, M. Trusiak, and K. Pokorski, “Generation of phase edge singularities by coplanar three-beam interference and their detection,” Opt. Express 25(3), 2432–2445 (2017).
[Crossref] [PubMed]

S. Rasoulli, D. Hebri, and A. M. Khazaei, “Investigation of various behaviors of near- and far-filed diffractions from multiplicatively separable structures in the x and y directions, and a detailed study of the near-field diffraction patterns of 2D multiplicatively separable periodic structures using the contrast variation method,” J. Opt. 19(9), 095601 (2017).
[Crossref]

S. Rasouli and D. Hebri, “Contrast enhanced quarter-Talbot images,” J. Opt. Soc. Am. A 34(12), 2145–2156 (2017).
[Crossref] [PubMed]

2016 (3)

2015 (3)

2014 (2)

K. Patorski, M. Trusiak, and K. Pokorski, “„Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

2013 (2)

2012 (1)

2009 (1)

J. Salinas-Luna, F. Granados-Augustin, A. Cornejo-Rodriguez, E. Luna, J. J. Sanchez-Escobar, and J. M. Hernandez-Cid, “Ronchi test with variable-frequency rulings,” Opt. Eng. 48(1), 013604 (2009).
[Crossref]

2000 (1)

1999 (2)

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

H. Canabal and E. Bernabeu, “Phase extraction methods for analysis of crossed fringe patterns,” Proc. SPIE 3744, 231–240 (1999).
[Crossref]

1997 (1)

1982 (1)

K. Patorski, L. Wronkowski, and M. Dobosz, “„Some properties of Fresnel image of a square wave amplitude grating,” Opt. Acta (Lond.) 29(5), 565–567 (1982).
[Crossref]

1971 (2)

S. Yokozeki and T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10(7), 1575–1580 (1971).
[Crossref] [PubMed]

A. W. Lohmann and D. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2(9), 413–415 (1971).
[Crossref]

1964 (2)

1881 (1)

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected herewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

1836 (1)

F. Talbot, “„Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Bai, J.

Bernabeu, E.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

H. Canabal and E. Bernabeu, “Phase extraction methods for analysis of crossed fringe patterns,” Proc. SPIE 3744, 231–240 (1999).
[Crossref]

Bhattacharya, S.

Bravo-Medina, B.

Bryngdahl, O.

Canabal, H.

H. Canabal and E. Bernabeu, “Phase extraction methods for analysis of crossed fringe patterns,” Proc. SPIE 3744, 231–240 (1999).
[Crossref]

Cornejo-Rodriguez, A.

J. Salinas-Luna, F. Granados-Augustin, A. Cornejo-Rodriguez, E. Luna, J. J. Sanchez-Escobar, and J. M. Hernandez-Cid, “Ronchi test with variable-frequency rulings,” Opt. Eng. 48(1), 013604 (2009).
[Crossref]

Crespo, D.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

Dobosz, M.

K. Patorski, L. Wronkowski, and M. Dobosz, “„Some properties of Fresnel image of a square wave amplitude grating,” Opt. Acta (Lond.) 29(5), 565–567 (1982).
[Crossref]

Ferrari, J. A.

Flores, J. L.

Garcia-Lechuga, L.

N.-I. Toto-Arellano, J. M. Miranda Gomez, L. Garcia-Lechuga, A. Montes-Perez, G. Rodriguez Zurita, A. Martinez Garcia, and J. A. Martinez Dominguez, “Diffraction theory of binary amplitude and phase gratings with application for Ronchi test,” Optik (Stuttg.) 126(23), 3717–3727 (2015).
[Crossref]

Granados-Augustin, F.

J. Salinas-Luna, F. Granados-Augustin, A. Cornejo-Rodriguez, E. Luna, J. J. Sanchez-Escobar, and J. M. Hernandez-Cid, “Ronchi test with variable-frequency rulings,” Opt. Eng. 48(1), 013604 (2009).
[Crossref]

Guérineau, N.

Han, P.

P. Han and J. Wang, “Talbot images and Talbot spectra of a 2D orthogonal periodicity structure,” J. Opt. 18(5), 055606 (2016).
[Crossref]

He, A.-Z.

Hebri, D.

S. Rasoulli, D. Hebri, and A. M. Khazaei, “Investigation of various behaviors of near- and far-filed diffractions from multiplicatively separable structures in the x and y directions, and a detailed study of the near-field diffraction patterns of 2D multiplicatively separable periodic structures using the contrast variation method,” J. Opt. 19(9), 095601 (2017).
[Crossref]

S. Rasouli and D. Hebri, “Contrast enhanced quarter-Talbot images,” J. Opt. Soc. Am. A 34(12), 2145–2156 (2017).
[Crossref] [PubMed]

Hernandez-Cid, J. M.

J. Salinas-Luna, F. Granados-Augustin, A. Cornejo-Rodriguez, E. Luna, J. J. Sanchez-Escobar, and J. M. Hernandez-Cid, “Ronchi test with variable-frequency rulings,” Opt. Eng. 48(1), 013604 (2009).
[Crossref]

Jiang, J.

Khazaei, A. M.

S. Rasoulli, D. Hebri, and A. M. Khazaei, “Investigation of various behaviors of near- and far-filed diffractions from multiplicatively separable structures in the x and y directions, and a detailed study of the near-field diffraction patterns of 2D multiplicatively separable periodic structures using the contrast variation method,” J. Opt. 19(9), 095601 (2017).
[Crossref]

Kim, H.-W.

Kim, S.-K.

Lee, B.

Lee, H.

Li, Z.-H.

Ling, T.

Liu, D.

Lohmann, A. W.

A. W. Lohmann and D. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2(9), 413–415 (1971).
[Crossref]

Luna, E.

J. Salinas-Luna, F. Granados-Augustin, A. Cornejo-Rodriguez, E. Luna, J. J. Sanchez-Escobar, and J. M. Hernandez-Cid, “Ronchi test with variable-frequency rulings,” Opt. Eng. 48(1), 013604 (2009).
[Crossref]

Martinez Dominguez, J. A.

N.-I. Toto-Arellano, J. M. Miranda Gomez, L. Garcia-Lechuga, A. Montes-Perez, G. Rodriguez Zurita, A. Martinez Garcia, and J. A. Martinez Dominguez, “Diffraction theory of binary amplitude and phase gratings with application for Ronchi test,” Optik (Stuttg.) 126(23), 3717–3727 (2015).
[Crossref]

Martinez Garcia, A.

N.-I. Toto-Arellano, J. M. Miranda Gomez, L. Garcia-Lechuga, A. Montes-Perez, G. Rodriguez Zurita, A. Martinez Garcia, and J. A. Martinez Dominguez, “Diffraction theory of binary amplitude and phase gratings with application for Ronchi test,” Optik (Stuttg.) 126(23), 3717–3727 (2015).
[Crossref]

Miranda Gomez, J. M.

N.-I. Toto-Arellano, J. M. Miranda Gomez, L. Garcia-Lechuga, A. Montes-Perez, G. Rodriguez Zurita, A. Martinez Garcia, and J. A. Martinez Dominguez, “Diffraction theory of binary amplitude and phase gratings with application for Ronchi test,” Optik (Stuttg.) 126(23), 3717–3727 (2015).
[Crossref]

Montes-Perez, A.

N.-I. Toto-Arellano, J. M. Miranda Gomez, L. Garcia-Lechuga, A. Montes-Perez, G. Rodriguez Zurita, A. Martinez Garcia, and J. A. Martinez Dominguez, “Diffraction theory of binary amplitude and phase gratings with application for Ronchi test,” Optik (Stuttg.) 126(23), 3717–3727 (2015).
[Crossref]

Patorski, K.

K. Patorski, L. Sluzewski, M. Trusiak, and K. Pokorski, “Generation of phase edge singularities by coplanar three-beam interference and their detection,” Opt. Express 25(3), 2432–2445 (2017).
[Crossref] [PubMed]

K. Patorski, Ł. Służewski, and M. Trusiak, “Single-shot 3 × 3 beam grating interferometry for self-imaging free extended range wave front sensing,” Opt. Lett. 41(18), 4417–4420 (2016).
[Crossref] [PubMed]

K. Patorski, M. Trusiak, and K. Pokorski, “„Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
[Crossref] [PubMed]

K. Patorski, L. Wronkowski, and M. Dobosz, “„Some properties of Fresnel image of a square wave amplitude grating,” Opt. Acta (Lond.) 29(5), 565–567 (1982).
[Crossref]

Pokorski, K.

K. Patorski, L. Sluzewski, M. Trusiak, and K. Pokorski, “Generation of phase edge singularities by coplanar three-beam interference and their detection,” Opt. Express 25(3), 2432–2445 (2017).
[Crossref] [PubMed]

K. Patorski, M. Trusiak, and K. Pokorski, “„Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
[Crossref] [PubMed]

Primot, J.

Quiroga, J. A.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

Rasouli, S.

Rasoulli, S.

S. Rasoulli, D. Hebri, and A. M. Khazaei, “Investigation of various behaviors of near- and far-filed diffractions from multiplicatively separable structures in the x and y directions, and a detailed study of the near-field diffraction patterns of 2D multiplicatively separable periodic structures using the contrast variation method,” J. Opt. 19(9), 095601 (2017).
[Crossref]

Rayleigh, L.

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected herewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

Rodriguez Zurita, G.

N.-I. Toto-Arellano, J. M. Miranda Gomez, L. Garcia-Lechuga, A. Montes-Perez, G. Rodriguez Zurita, A. Martinez Garcia, and J. A. Martinez Dominguez, “Diffraction theory of binary amplitude and phase gratings with application for Ronchi test,” Optik (Stuttg.) 126(23), 3717–3727 (2015).
[Crossref]

Ronchi, V.

Salinas-Luna, J.

J. Salinas-Luna, F. Granados-Augustin, A. Cornejo-Rodriguez, E. Luna, J. J. Sanchez-Escobar, and J. M. Hernandez-Cid, “Ronchi test with variable-frequency rulings,” Opt. Eng. 48(1), 013604 (2009).
[Crossref]

Sanchez-Escobar, J. J.

J. Salinas-Luna, F. Granados-Augustin, A. Cornejo-Rodriguez, E. Luna, J. J. Sanchez-Escobar, and J. M. Hernandez-Cid, “Ronchi test with variable-frequency rulings,” Opt. Eng. 48(1), 013604 (2009).
[Crossref]

Saveljev, V.

Shen, Y.

Silva, D.

A. W. Lohmann and D. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2(9), 413–415 (1971).
[Crossref]

Sirohi, R. S.

Sluzewski, L.

Song, Y.

Sun, N.

Suzuki, T.

Talbot, F.

F. Talbot, “„Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Toto-Arellano, N.-I.

N.-I. Toto-Arellano, J. M. Miranda Gomez, L. Garcia-Lechuga, A. Montes-Perez, G. Rodriguez Zurita, A. Martinez Garcia, and J. A. Martinez Dominguez, “Diffraction theory of binary amplitude and phase gratings with application for Ronchi test,” Optik (Stuttg.) 126(23), 3717–3727 (2015).
[Crossref]

Trusiak, M.

Wang, J.

Wronkowski, L.

K. Patorski, L. Wronkowski, and M. Dobosz, “„Some properties of Fresnel image of a square wave amplitude grating,” Opt. Acta (Lond.) 29(5), 565–567 (1982).
[Crossref]

Yang, Y.

Yokozeki, S.

Yue, X.

Appl. Opt. (7)

J. Opt. (2)

P. Han and J. Wang, “Talbot images and Talbot spectra of a 2D orthogonal periodicity structure,” J. Opt. 18(5), 055606 (2016).
[Crossref]

S. Rasoulli, D. Hebri, and A. M. Khazaei, “Investigation of various behaviors of near- and far-filed diffractions from multiplicatively separable structures in the x and y directions, and a detailed study of the near-field diffraction patterns of 2D multiplicatively separable periodic structures using the contrast variation method,” J. Opt. 19(9), 095601 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Acta (Lond.) (1)

K. Patorski, L. Wronkowski, and M. Dobosz, “„Some properties of Fresnel image of a square wave amplitude grating,” Opt. Acta (Lond.) 29(5), 565–567 (1982).
[Crossref]

Opt. Commun. (1)

A. W. Lohmann and D. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2(9), 413–415 (1971).
[Crossref]

Opt. Eng. (2)

J. Salinas-Luna, F. Granados-Augustin, A. Cornejo-Rodriguez, E. Luna, J. J. Sanchez-Escobar, and J. M. Hernandez-Cid, “Ronchi test with variable-frequency rulings,” Opt. Eng. 48(1), 013604 (2009).
[Crossref]

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38(6), 974–982 (1999).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Optik (Stuttg.) (1)

N.-I. Toto-Arellano, J. M. Miranda Gomez, L. Garcia-Lechuga, A. Montes-Perez, G. Rodriguez Zurita, A. Martinez Garcia, and J. A. Martinez Dominguez, “Diffraction theory of binary amplitude and phase gratings with application for Ronchi test,” Optik (Stuttg.) 126(23), 3717–3727 (2015).
[Crossref]

Philos. Mag. (2)

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected herewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

F. Talbot, “„Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Proc. SPIE (3)

K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

H. Canabal and E. Bernabeu, “Phase extraction methods for analysis of crossed fringe patterns,” Proc. SPIE 3744, 231–240 (1999).
[Crossref]

K. Patorski, M. Trusiak, and K. Pokorski, “„Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).
[Crossref]

Other (4)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf ed. (North-Holland, 1989), vol. 27, pp. 1–108.

K. Patorski and M. Kujawinska, Handbook of the Moiré Fringe Technique, (Elsevier, 1993).

I. Amidror, Theory of the Moiré Phenomenon (Springer, 2007), vol. I and II.

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007).

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Figures (14)

Fig. 1
Fig. 1 Magnified central part of: (a) the crossed Ronchi grating and (b) the modulus of its Fourier transform. Four lowest, spatial beating formed side orders are encircled in (b), see text.
Fig. 2
Fig. 2 Magnified central part of: (a) the checker grating and (b) the modulus of its Fourier transform. Four lowest fundamental harmonics are encircled in (b).
Fig. 3
Fig. 3 Magnified central part of: (a) the product of the checker grating, Fig. 2(a), and 1D Ronchi grating (dense vertical lines); (b) the modulus of the Fourier transform of (a).
Fig. 4
Fig. 4 Magnified central part of: (a) the result of adding the structures shown in Figs. 1(a) and 3(a); (b) the modulus of the Fourier transform of (a).
Fig. 5
Fig. 5 Magnified central part of (a) the sum of the product of the crossed grating and the 1D (line) grating added to the checker grating; (b) the modulus of its Fourier transform.
Fig. 6
Fig. 6 Magnified central part of (a) direct addition of the crossed and checker gratings; (b) the modulus of its spectrum.
Fig. 7
Fig. 7 Central regions of the Fresnel field intensity patterns at the distances z = d2/λ (a) and z = (1.5)d2/λ (b) from multiplicatively superimposed two orthogonal cosinusoidal amplitude gratings.
Fig. 8
Fig. 8 Central regions of the Fresnel field intensity patterns at the distances z = d2/λ (a) and z = (1.5)d2/λ (b) from additively superimposed two orthogonal cosinusoidal amplitude gratings.
Fig. 9
Fig. 9 Central regions of the Fresnel field intensity patterns calculated for the distances z = d2/λ (a) and z = (1.5)d2/λ (b) from multiplicatively superimposed two orthogonal binary amplitude Ronchi gratings. Eleven positive and eleven negative side diffraction orders together with the zero order for both orthogonal directions x and y were taken into calculations.
Fig. 10
Fig. 10 Central regions of the Fresnel field intensity patterns at the distances z = d2/λ (a) and z = (1.5)d2/λ (b) generated by additively superimposed two orthogonal 1D binary amplitude gratings.
Fig. 11
Fig. 11 Central region of the intensity distribution at the plane of orthogonally, additively superimposed binary grating (nonlinear display for enhanced visualization of diversified intensity levels).
Fig. 12
Fig. 12 Recorded experimental Fresnel diffraction images formed by the diffraction orders from the spectrum central region (composed of two mutually orthogonal directions of diffraction orders) of the developed 3-level binary amplitude grating (numerically calculated spectrum is shown in Fig. 4(b)). Figure 12(a) shows the intensity pattern of one of the self-images whereas image Fig. 12(b) was recorded in one of the self-image intermediate planes.
Fig. 13
Fig. 13 Binary amplitude crossed Ronchi (a) and checker grating (c) together with their Fourier transform modulus (b), and (d), respectively. Gratings can be considered as built of transparent and opaque square elements of the same dimensions. Figures 13(b) and 13(d) show enlarged central parts of spatial frequency spectra (displayed with the same nonlinear scale for their enhanced visualization).
Fig. 14
Fig. 14 Cross-sections through the calculated modulus of Fourier transforms presented in Fig. 13(b) and 13(d) shown in linear scale. Horizontal cross-sections through the zero order of the crossed Ronchi and checker gratings are shown in Figs. 14(a) and 14(b), respectively, and 45 deg diagonal cross sections through the zero order of the crossed Ronchi and checker gratings are shown in Figs. 14(c) and 14(d), respectively.

Equations (7)

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E cm ( x,y,z )=[ a 0 + a 1 exp{i( 2π d x2π λ d 2 z)}+ a 1 exp{i( 2π d x+2π λ d 2 z)}] *[ a 0 + a 1 exp{i( 2π d y2π λ d 2 z)}+ a 1 exp{i( 2π d y+2π λ d 2 z)}],
I(x,y,z)= E cm ( x,y,z ) E cm *( x,y,z ).
E ca ( x,y,z )=[ a 0 + a 1 exp{i( 2π d x2π λ d 2 z)}+ a 1 exp{i( 2π d x+2π λ d 2 z)}] +[ a 0 + a 1 exp{i( 2π d y2π λ d 2 z)}+ a 1 exp{i( 2π d y+2π λ d 2 z)}].
E bm ( x,y,z )= m n exp{i2π[( m x d m 2 λz 2 d 2 )+(n y d n 2 λz 2 d 2 )]}.
E ba ( x,y,z )= m exp{ i2π(m x d m 2 λz 2 d 2 )}+ n exp{ i2π(n y d n 2 λz 2 d 2 )}.
a m a n =( 1 mn π 2 )sin( mπh d )sin( nπh d );
( a m a n ) checker =[1+ (1) m+n ] a m a n ;

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