Abstract

We apply the techniques of digital holography to obtain microscopic three-dimensional images of biological cells. The optical system is capable of microscopic holography with diffraction-limited resolution by projecting a magnified image of a microscopic hologram plane onto a CCD plane. Two-wavelength phase-imaging digital holography is applied to produce unwrapped phase images of biological cells. The method of three-wavelength phase imaging is proposed to extend the axial range and reduce the effect of phase noise. These results demonstrate the effectiveness of digital holography in high-resolution biological microscopy.

© 2006 Optical Society of America

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  1. S. Seebacher, W. Osten, and W. Jueptner, "Measuring shape and deformation of small objects using digital holography," in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 3479, 104-115 (1998).
    [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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. I. Yamaguchi, T. Matsumura, and J. Kato, "Phase-shifting color digital holography," Opt. Lett. 27, 1108-1110 (2002).
    [CrossRef]
  22. J. Kato, I. Yamaguchi, and T. Matsumura, "Multicolor digital holography with an achromatic phase shifter," Opt. Lett. 27, 1403-1405 (2002).
    [CrossRef]
  23. N. Demoli, D. Vukicevic, and M. Torzynski, "Dynamic digital holographic interferometry with three wavelengths," Opt. Express 11, 767-774 (2003).
    [CrossRef] [PubMed]
  24. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  25. I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]
  26. T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23, 1221-1223 (1998).
    [CrossRef]
  27. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
    [CrossRef] [PubMed]
  28. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, "Algorithm for reconstruction of digital holograms with adjustable magnification," Opt. Lett. 29, 1668-1670 (2004).
    [CrossRef] [PubMed]
  29. M. Jacquot, P. Sandoz, and G. Tribillon, "High resolution digital holography," Opt. Commun. 190, 87-94 (2001).
    [CrossRef]
  30. In principle, it is actually possible to extend the unambiguous axial range beyond beat wavelength Lambda12 by using two-wavelength phase imaging, though with a stricter requirement on the phase measurement accuracy; see P. de Groot, "Extending the unambiguous range of two-color interferometers," Appl. Opt. 33, 5948-5953 (1994).

2005 (1)

2004 (4)

2003 (4)

2002 (3)

2001 (5)

M. Jacquot, P. Sandoz, and G. Tribillon, "High resolution digital holography," Opt. Commun. 190, 87-94 (2001).
[CrossRef]

S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, "Whole optical wave fields reconstruction by digital holography," Opt. Express 9, 294-302 (2001).
[CrossRef] [PubMed]

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. USA 98, 11,301-11,305 (2001).
[CrossRef]

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image formation in phase-shifting digital holography and applications to microscopy," Appl. Opt. 40, 6177-6686 (2001).
[CrossRef]

1999 (2)

1998 (4)

1997 (1)

1994 (1)

1992 (1)

1987 (1)

1984 (1)

Akkin, T.

Alfieri, D.

Badizadegan, K.

Barty, A.

Bevilacqua, F.

Boyer, K.

Cheng, Y. Y.

Colomb, T.

Coppola, G.

Creath, K.

Cuche, E.

Cuevas, F. J.

Cullen, D.

Dakoff, A.

Dasari, R. R.

Dave, D.

De Nicola, S.

Deflores, L. P.

Deiwick, M.

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Delere, H.

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Demoli, N.

Depeursinge, C.

Diller, K. R.

Dirksen, D.

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Droste, H.

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Emery, Y.

Feld, M. S.

Ferraro, P.

Finizio, A.

Gass, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Grilli, S.

Haddad, W. S.

Iwai, H.

Jacquot, M.

M. Jacquot, P. Sandoz, and G. Tribillon, "High resolution digital holography," Opt. Commun. 190, 87-94 (2001).
[CrossRef]

Jericho, M. H.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. USA 98, 11,301-11,305 (2001).
[CrossRef]

Jueptner, W.

S. Seebacher, W. Osten, and W. Jueptner, "Measuring shape and deformation of small objects using digital holography," in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 3479, 104-115 (1998).
[CrossRef]

Jueptner, W. P. O.

U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Kato, J.

Kemper, B.

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Kim, M. K.

Kreuzer, H. J.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. USA 98, 11,301-11,305 (2001).
[CrossRef]

Longworth, J. W.

Magistretti, P. J.

Magro, C.

Malacara, D.

Marquet, P.

Marroquin, J. L.

Matsumura, T.

McPherson, A.

Meinertzhagen, I. A.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. USA 98, 11,301-11,305 (2001).
[CrossRef]

Meucci, R.

Milner, T. E.

Mizuno, J.

Nugent, K. A.

Ohta, S.

Osten, W.

S. Seebacher, W. Osten, and W. Jueptner, "Measuring shape and deformation of small objects using digital holography," in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 3479, 104-115 (1998).
[CrossRef]

Paganin, D.

Pierattini, G.

Popescu, G.

Rappaz, B.

Rhodes, C. K.

Roberts, A.

Rylander, C. G.

Sandoz, P.

M. Jacquot, P. Sandoz, and G. Tribillon, "High resolution digital holography," Opt. Commun. 190, 87-94 (2001).
[CrossRef]

Scheld, H. H.

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

U. Schnars, "Direct phase determination in hologram interferometry with use of digitally recorded holograms," J. Opt. Soc. Am. A 11, 2011-2015 (1994).
[CrossRef]

Schofield, M. A.

Seebacher, S.

S. Seebacher, W. Osten, and W. Jueptner, "Measuring shape and deformation of small objects using digital holography," in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 3479, 104-115 (1998).
[CrossRef]

Servin, M.

Solem, J. C.

Torzynski, M.

Tribillon, G.

M. Jacquot, P. Sandoz, and G. Tribillon, "High resolution digital holography," Opt. Commun. 190, 87-94 (2001).
[CrossRef]

Vaughan, J. C.

von Bally, G.

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Vukicevic, D.

Welch, A. J.

Wyant, J. C.

Xu, W.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. USA 98, 11,301-11,305 (2001).
[CrossRef]

Yamaguchi, I.

Yaroslavsky, L. P.

Zhang, F.

Zhang, T.

Zhu, Y.

Appl. Opt. (7)

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

Meas. Sci. Technol. (1)

U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Opt. Commun. (1)

M. Jacquot, P. Sandoz, and G. Tribillon, "High resolution digital holography," Opt. Commun. 190, 87-94 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (1)

D. Dirksen, H. Droste, B. Kemper, H. Delere, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Opt. Lett. (13)

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. 30, 468-470 (2005).
[CrossRef] [PubMed]

M. A. Schofield and Y. Zhu, "Fast phase unwrapping algorithm for interferometric applications," Opt. Lett. 28, 1194-1196 (2003).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

C. G. Rylander, D. Dave, T. Akkin, T. E. Milner, K. R. Diller, and A. J. Welch, "Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy," Opt. Lett. 29, 1509-1511 (2004).
[CrossRef] [PubMed]

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, "Fourier phase microscopy for investigation of biological structures and dynamics," Opt. Lett. 29, 2503-2505 (2004).
[CrossRef] [PubMed]

J. Gass, A. Dakoff, and M. K. Kim, "Phase imaging without 2pi-ambiguity by multiple-wavelength digital holography," Opt. Lett. 28, 1141-1143 (2003).
[CrossRef] [PubMed]

I. Yamaguchi, T. Matsumura, and J. Kato, "Phase-shifting color digital holography," Opt. Lett. 27, 1108-1110 (2002).
[CrossRef]

J. Kato, I. Yamaguchi, and T. Matsumura, "Multicolor digital holography with an achromatic phase shifter," Opt. Lett. 27, 1403-1405 (2002).
[CrossRef]

I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
[CrossRef] [PubMed]

T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23, 1221-1223 (1998).
[CrossRef]

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
[CrossRef] [PubMed]

F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, "Algorithm for reconstruction of digital holograms with adjustable magnification," Opt. Lett. 29, 1668-1670 (2004).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. USA 98, 11,301-11,305 (2001).
[CrossRef]

Proc. SPIE (1)

S. Seebacher, W. Osten, and W. Jueptner, "Measuring shape and deformation of small objects using digital holography," in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, and W. P. O. Jueptner, eds., Proc. SPIE 3479, 104-115 (1998).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

In principle, it is actually possible to extend the unambiguous axial range beyond beat wavelength Lambda12 by using two-wavelength phase imaging, though with a stricter requirement on the phase measurement accuracy; see P. de Groot, "Extending the unambiguous range of two-color interferometers," Appl. Opt. 33, 5948-5953 (1994).

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Figures (9)

Fig. 1
Fig. 1

Apparatus for digital holography: BS1–BS3, beam splitters and combiners; M1, mirror; OBJ, object; REF, reference; other abbreviations defined in text.

Fig. 2
Fig. 2

High-resolution holographic image reconstruction of the resolution target of area 88 µm × 88 µm with 360 × 360 pixels : (a) direct image when the object is on plane H, illuminated by a laser; (b) reconstructed amplitude image; (c) phase image; (d) phase image in perspective view.

Fig. 3
Fig. 3

Holographic imaging of onion cells of area 88 µm × 88 µm with 360 × 360 pixels : (a) conventional microscope image, whose scale is somewhat different from those of the holographic images; (b) direct image; (c) amplitude image; (d) phase image.

Fig. 4
Fig. 4

Simulation of two-wavelength phase-imaging digital holography: (a) height profile z 1 ( x ) of a 10 µm high incline, derived from phase φ 1 ( x ) of λ 1 = 0.532 µm ; (b) z 2 ( x ) derived from phase φ 2 ( x ) of λ 2 = 0.633 µm ; (c) difference phase map φ 12 = φ 1 - φ 2 ; (d) coarse map, z 12 ( x ) , with beat wavelength Λ 12 = 3.33 µm ; (e) z 12 ( x ) , where z 12 ( x ) is divided into integer multiples of λ 1 ; (f) z 12 ( x ) , where z 1 ( x ) is pasted onto z 12 ( x ) ; (g) fine map, z 12 ( x ) , where most of the spikes in z 12 ( x ) are removed by comparison with z 12 ( x ) . The vertical axis is 5.0 µm full scale in every figure, except for (c), where the vertical range is - 2 π to + 2 π .

Fig. 5
Fig. 5

Experimental profiles of two-wavelength phase-imaging digital holography. The descriptions of the individual figures are the same as for Fig. 6.

Fig. 6
Fig. 6

Two-dimensional profiles of a resolution target, 157 × 157 µm with 360 × 360 pixels , from two-wavelength phase-imaging digital holography: (a) phase map z 1 ( x , y ) derived from phase φ 1 ( x , y ) of λ 1 = 0.532 µm ; (b) phase map z 2 ( x , y ) derived from phase φ 2 ( x , y ) of λ 2 = 0.633 µm ; (c) coarse map z 12 ( x , y ) ; (d) fine map z 12 ( x , y ) .

Fig. 7
Fig. 7

Two-dimensional profiles of onion cells, 193 µm × 193 µm with 360 × 360 pixels , from two-wavelength phase imaging digital holography: (a) phase map z 1 ( x , y ) derived from phase φ 1 ( x , y ) of λ 1 = 0.532 µm ; (b) phase map z 2 ( x , y ) derived from phase φ 2 ( x , y ) of λ 2 = 0.633 µm ; (c) coarse map z 12 ( x , y ) ; (d) fine map z 12 ( x , y ) .

Fig. 8
Fig. 8

Simulation of three-wavelength phase-imaging digital holography: (a) height profile z 1 ( x ) of a 10 µm high incline, derived from phase φ 1 ( x ) of λ 1 = 0.62 µm ; (b) z 2 ( x ) derived from phase φ 2 ( x ) of λ 2 = 0.58 µm ; (c) coarse map z 12 ( x ) of beat wavelength Λ 12 = 8.99 µm ; (d) fine map z 12 ( x ) ; (e) z 3 ( x ) derived from phase φ 3 ( x ) of λ 3 = 0.50 µm ; (f) coarse map z 13 ( x ) of beat wavelength Λ 13 = 2.58 µm ; (g) coarse map z 23 ( x ) of beat wavelength Λ 23 = 3.63 µm ; (h) intermediate fine map z 13 23 , where z 13 ( x ) is pasted onto z 13 23 ( x ) = z 12 ( x ) ; (i) final fine map z 13 23 ( x ) , where z 1 ( x ) is pasted onto z 13 23 ( x ) ; (j) noise in (c), z 12 - z ( x ) , where z ( x ) is the actual height profile; (k) noise in (d), z 12 ( x ) - z ( x ) ; (l) noise in (h), z 13 23 ( x ) - z ( x ) ; (m) noise in (i), z 13 23 ( x ) - z ( x ) . The vertical axis is 10.0 µm full scale in (a)–(i) and 1.0 µm full scale in (j)–(m).

Fig. 9
Fig. 9

Object space for reflection and transmission holography.

Equations (19)

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

E ( x , y ) = E h S ( x , y ; z 1 ) ,
S ( x , y ; z ) = - i k 2 π z exp [ i k z + i k 2 z ( x 2 + y 2 ] ,
E ( x , y ) = E ( x , y ) exp [ - i k 2 f ( x 2 + y 2 ) ] .
E c ( x c , y c ) = E S ( x c , y c ; z 2 ) .
E c ( x c , y c ) = - k 2 4 π 2 z 1 z 2 exp [ i k ( z 1 + z 2 ) + i k 2 z 2 ( x c 2 + y c 2 ) ] d x h d y h E h ( x h , y h ) exp [ i k 2 z 1 ( x h 2 + y h 2 ] d x d y exp [ i k 2 ( 1 z 1 + 1 z 2 - 1 f ) ( x 2 + y 2 ) - i k ( x h z 1 + x c z 2 ) x - i k ( y h z 1 + y c z 2 ) y ] .
1 z 1 + 1 z 2 = 1 f , d x exp ( i k x ) = 2 πδ ( k ) ;
E c ( x c , y c ) = - z 1 z 2 exp [ i k ( z 1 + z 2 ) ] exp [ i k 2 ( z 2 - f ) ( x c 2 + y c 2 ) ] E h ( - z 1 z 2 x h , - z 1 z 2 y h ) .
exp [ i k 2 ( z 2 - f ) ( x c 2 + y c 2 ) ]
E 0 ( x 0 , y 0 ) = ε 0 δ ( x 0 - X 0 , y 0 - Y 0 ) .
E Ho ( x h , y h ) = ε 0 exp { i k 2 z 0 [ ( x h - X 0 ) 2 + ( y h - Y 0 ) 2 ] } .
E H r ( x h , y h ) = ε r exp [ i ( k x x h + k y y h ) ] ,
I h ( x h , y h ) = E h 2 = ε r 2 + ε 0 2 + ε r ε 0 * exp { - i k 2 z 0 [ ( x h - X 0 ) 2 + ( y h - Y 0 ) 2 ] + i ( k x x h + k y y h ) } + ε r * ε 0 exp { i k 2 z 0 [ ( x h - X 0 ) 2 + ( y h - Y 0 ) 2 ] - i ( k x x h + k y y h ) } .
I h ( α , β ) = ε r 2 + ε 0 2 + ε r ε 0 * exp { - i k 2 z 0 [ ( αΔ - X 0 ) 2 + ( βΔ - Y 0 ) 2 ] + i ( α k x + β k y ) Δ } + ε r * ε 0 exp { i k 2 z 0 [ ( αΔ - X 0 ) 2 + ( βΔ - Y 0 ) 2 ] - i ( α k x + β k y ) Δ } .
H ( α , β ) I h ( α , β ) E H r * ( α , β ) = ε r 2 ε 0 * exp { - i k 2 z 0 [ ( αΔ - X 0 ) 2 + ( βΔ - Y 0 ) 2 ] } .
E i ( γ , δ ) = H S ( γ , δ ; z i ) = - i k 2 π z i exp ( i k z i ) α , β = 0 N x - 1 Δ 2 H ( α , β ) exp { i k Δ 2 2 z i [ ( α - γ ) 2 + ( β - δ ) 2 ] } .
E i ( γ , δ ) = - i k Δ 2 2 π z i ε r 2 ε 0 * exp ( i k z i ) exp { i k 2 z 0 [ ( γ 2 Δ 2 - X 0 2 ) + ( δ 2 Δ 2 - Y 0 2 ) ] } α , β = 0 N x - 1 exp { i k Δ z 0 [ α ( γΔ - X 0 ) + β ( δΔ - Y 0 ) ] } = - i k Δ 2 2 π z i ε r 2 ε 0 * exp ( i k z i ) exp { i k 2 z 0 [ ( γ 2 Δ 2 - X 0 2 ) + ( δ 2 Δ 2 - Y 0 2 ) ] } exp { - i k a x 2 z 0 [ ( γΔ - X 0 ) + ( δΔ - Y 0 ) ] } sin ( k a x / 2 z 0 ) ( γΔ - X 0 ) sin ( k Δ / 2 z 0 ) ( γΔ - X 0 ) sin ( k a x / 2 z 0 ) ( δΔ - Y 0 ) sin ( k Δ / 2 z 0 ) ( δΔ - Y 0 )
- i k a x 2 2 π z i ε r 2 ε 0 * exp ( i k z i ) δ γ , X 0 / Δ δ δ , Y 0 / Δ .
z 0 > a x 2 N x λ .
r exp ( 2 i k a ) - r     exp [ 2 i k ( a + n b ) ] = - 2 i r sin ( k n b ) exp [ i k ( 2 a + n b ) ] .

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