Abstract

Digital in-line holography is used to visualize particle motion within a cylindrical micropipe. Analytical expression of the intensity distribution recorded in the CCD sensor plane is derived using the generalized Huygens-Fresnel integral associated with the ABCD matrices formalism. Holograms obtained in a 100µm in diameter micropipe are then reconstructed using fractional Fourier transformation. Astigmatism brought by the cylindrical micropipe is finally used to select a three dimensional region of interest in the microflow and thus to improve axial localization of objects located within a micropipe. Experimental results are presented and a short movie showing particle motion within a micropipe is given.

© 2010 OSA

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2009

N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Determination of 3D-region of interest using digital in-line holography with astigmatic Gaussian beams,” J. Europ. Opt. Soc. Rap. Public 4, 09038 (2009).
[CrossRef]

2008

2007

2006

2005

S. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro-channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12(6), 442–444 (2005).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. E. M. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22(11), 2569–2577 (2005).
[CrossRef]

2004

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15(4), 699–705 (2004).
[CrossRef]

D. Sinton, “Microscale flow visualization,” Microfluid. Nanofluid. 1(1), 2–21 (2004).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital holographic particle tracking velocimetry for 3-D transient flow around an obstacle in a narrow channel,” Opt. Rev. 11, 162–164 (2004).

2003

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206(20), 3657–3666 (2003).
[CrossRef] [PubMed]

W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. 28(3), 164–166 (2003).
[CrossRef] [PubMed]

2001

J. Darabi, M. M. Ohabi, and D. Devoe, “An electrohydrodynamic polarization micropump for electronic cooling,” J. Microelectromech. Syst. 10(1), 98–106 (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. U.S.A. 98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

2000

C. D. Meinhart and H. Zhang, “The flow structure inside a microfabricated inkjet printhead,” J. Microelectromech. Syst. 9(1), 67–75 (2000).
[CrossRef]

1997

1994

1993

1988

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[CrossRef]

1987

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4(10), 1931–1948 (1987).
[CrossRef]

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987).
[CrossRef]

1980

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
[CrossRef]

1970

Allano, D.

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. E. M. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22(11), 2569–2577 (2005).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15(4), 699–705 (2004).
[CrossRef]

Anraku, T.

S. Satake, T. Anraku, H. Kanamori, T. Kunugi, K. Sato, and T. Ito, “Measurement of three-dimensional flow in microchannel with complex shape by micro-digital-holographic particle-tracking velocimetry,” J. Heat Transfer 130(4), 042413 (2008).
[CrossRef]

Bagini, V.

Breazeale, M.

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[CrossRef]

Brunel, M.

Coëtmellec, S.

Collins, S. A.

Darabi, J.

J. Darabi, M. M. Ohabi, and D. Devoe, “An electrohydrodynamic polarization micropump for electronic cooling,” J. Microelectromech. Syst. 10(1), 98–106 (2001).
[CrossRef]

Devoe, D.

J. Darabi, M. M. Ohabi, and D. Devoe, “An electrohydrodynamic polarization micropump for electronic cooling,” J. Microelectromech. Syst. 10(1), 98–106 (2001).
[CrossRef]

Du, X.

Garcia-Sucerquia, J.

Hanson, S. G.

Ito, T.

S. Satake, T. Anraku, H. Kanamori, T. Kunugi, K. Sato, and T. Ito, “Measurement of three-dimensional flow in microchannel with complex shape by micro-digital-holographic particle-tracking velocimetry,” J. Heat Transfer 130(4), 042413 (2008).
[CrossRef]

S. Satake, H. Kanamori, T. Kunugi, K. Sato, T. Ito, and K. Yamamoto, “Parallel computing of a digital hologram and particle searching for microdigital-holographic particle-tracking velocimetry,” Appl. Opt. 46(4), 538–543 (2007).
[CrossRef] [PubMed]

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro-channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12(6), 442–444 (2005).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital holographic particle tracking velocimetry for 3-D transient flow around an obstacle in a narrow channel,” Opt. Rev. 11, 162–164 (2004).

Janssen, A. J. E. M.

Jericho, M. H.

Jericho, S. K.

Jüptner, W.

Kanamori, H.

S. Satake, T. Anraku, H. Kanamori, T. Kunugi, K. Sato, and T. Ito, “Measurement of three-dimensional flow in microchannel with complex shape by micro-digital-holographic particle-tracking velocimetry,” J. Heat Transfer 130(4), 042413 (2008).
[CrossRef]

S. Satake, H. Kanamori, T. Kunugi, K. Sato, T. Ito, and K. Yamamoto, “Parallel computing of a digital hologram and particle searching for microdigital-holographic particle-tracking velocimetry,” Appl. Opt. 46(4), 538–543 (2007).
[CrossRef] [PubMed]

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[CrossRef]

Katz, J.

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206(20), 3657–3666 (2003).
[CrossRef] [PubMed]

Kerr, F. H.

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987).
[CrossRef]

Klages, P.

Kreuzer, H. J.

Kunugi, T.

S. Satake, T. Anraku, H. Kanamori, T. Kunugi, K. Sato, and T. Ito, “Measurement of three-dimensional flow in microchannel with complex shape by micro-digital-holographic particle-tracking velocimetry,” J. Heat Transfer 130(4), 042413 (2008).
[CrossRef]

S. Satake, H. Kanamori, T. Kunugi, K. Sato, T. Ito, and K. Yamamoto, “Parallel computing of a digital hologram and particle searching for microdigital-holographic particle-tracking velocimetry,” Appl. Opt. 46(4), 538–543 (2007).
[CrossRef] [PubMed]

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro-channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12(6), 442–444 (2005).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital holographic particle tracking velocimetry for 3-D transient flow around an obstacle in a narrow channel,” Opt. Rev. 11, 162–164 (2004).

Lebrun, D.

Lohmann, A. W.

Malek, M.

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15(4), 699–705 (2004).
[CrossRef]

Malkiel, E.

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206(20), 3657–3666 (2003).
[CrossRef] [PubMed]

McBride, A. C.

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987).
[CrossRef]

Meinertzhagen, I. A.

W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. 28(3), 164–166 (2003).
[CrossRef] [PubMed]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Meinhart, C. D.

C. D. Meinhart and H. Zhang, “The flow structure inside a microfabricated inkjet printhead,” J. Microelectromech. Syst. 9(1), 67–75 (2000).
[CrossRef]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
[CrossRef]

Nicolas, F.

Ohabi, M. M.

J. Darabi, M. M. Ohabi, and D. Devoe, “An electrohydrodynamic polarization micropump for electronic cooling,” J. Microelectromech. Syst. 10(1), 98–106 (2001).
[CrossRef]

Özkul, C.

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15(4), 699–705 (2004).
[CrossRef]

Palma, C.

Pellat-Finet, P.

Satake, S.

S. Satake, T. Anraku, H. Kanamori, T. Kunugi, K. Sato, and T. Ito, “Measurement of three-dimensional flow in microchannel with complex shape by micro-digital-holographic particle-tracking velocimetry,” J. Heat Transfer 130(4), 042413 (2008).
[CrossRef]

S. Satake, H. Kanamori, T. Kunugi, K. Sato, T. Ito, and K. Yamamoto, “Parallel computing of a digital hologram and particle searching for microdigital-holographic particle-tracking velocimetry,” Appl. Opt. 46(4), 538–543 (2007).
[CrossRef] [PubMed]

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro-channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12(6), 442–444 (2005).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital holographic particle tracking velocimetry for 3-D transient flow around an obstacle in a narrow channel,” Opt. Rev. 11, 162–164 (2004).

Sato, K.

S. Satake, T. Anraku, H. Kanamori, T. Kunugi, K. Sato, and T. Ito, “Measurement of three-dimensional flow in microchannel with complex shape by micro-digital-holographic particle-tracking velocimetry,” J. Heat Transfer 130(4), 042413 (2008).
[CrossRef]

S. Satake, H. Kanamori, T. Kunugi, K. Sato, T. Ito, and K. Yamamoto, “Parallel computing of a digital hologram and particle searching for microdigital-holographic particle-tracking velocimetry,” Appl. Opt. 46(4), 538–543 (2007).
[CrossRef] [PubMed]

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro-channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12(6), 442–444 (2005).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital holographic particle tracking velocimetry for 3-D transient flow around an obstacle in a narrow channel,” Opt. Rev. 11, 162–164 (2004).

Schnars, U.

Sheng, J.

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206(20), 3657–3666 (2003).
[CrossRef] [PubMed]

Sinton, D.

D. Sinton, “Microscale flow visualization,” Microfluid. Nanofluid. 1(1), 2–21 (2004).
[CrossRef]

Strickler, J. R.

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206(20), 3657–3666 (2003).
[CrossRef] [PubMed]

Taniguchi, J.

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro-channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12(6), 442–444 (2005).
[CrossRef]

Verrier, N.

Wen, J. J.

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[CrossRef]

Xu, W.

Yamamoto, K.

Yura, H. T.

Zhang, H.

C. D. Meinhart and H. Zhang, “The flow structure inside a microfabricated inkjet printhead,” J. Microelectromech. Syst. 9(1), 67–75 (2000).
[CrossRef]

Zhao, D.

Appl. Opt.

IMA J. Appl. Math.

A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39(2), 159–175 (1987).
[CrossRef]

J. Acoust. Soc. Am.

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[CrossRef]

J. Europ. Opt. Soc. Rap. Public

N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Determination of 3D-region of interest using digital in-line holography with astigmatic Gaussian beams,” J. Europ. Opt. Soc. Rap. Public 4, 09038 (2009).
[CrossRef]

J. Exp. Biol.

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206(20), 3657–3666 (2003).
[CrossRef] [PubMed]

J. Heat Transfer

S. Satake, T. Anraku, H. Kanamori, T. Kunugi, K. Sato, and T. Ito, “Measurement of three-dimensional flow in microchannel with complex shape by micro-digital-holographic particle-tracking velocimetry,” J. Heat Transfer 130(4), 042413 (2008).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
[CrossRef]

J. Microelectromech. Syst.

J. Darabi, M. M. Ohabi, and D. Devoe, “An electrohydrodynamic polarization micropump for electronic cooling,” J. Microelectromech. Syst. 10(1), 98–106 (2001).
[CrossRef]

C. D. Meinhart and H. Zhang, “The flow structure inside a microfabricated inkjet printhead,” J. Microelectromech. Syst. 9(1), 67–75 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15(4), 699–705 (2004).
[CrossRef]

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[CrossRef]

Microfluid. Nanofluid.

D. Sinton, “Microscale flow visualization,” Microfluid. Nanofluid. 1(1), 2–21 (2004).
[CrossRef]

Opt. Lett.

Opt. Rev.

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital holographic particle tracking velocimetry for 3-D transient flow around an obstacle in a narrow channel,” Opt. Rev. 11, 162–164 (2004).

S. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro-channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12(6), 442–444 (2005).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A. 98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Other

C. S. Vikram, “Particle Field Holography” in Cambridge Studies in Modern optics (Cambridge U. Press, 1992).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Application in Optics and Signal Processing (Wiley, 2001).

Supplementary Material (1)

» Media 1: MPG (2242 KB)     

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Figures (11)

Fig. 1
Fig. 1

Experimental and recording set-up of digital in-line holograms

Fig. 2
Fig. 2

Close-up (not to scale) of the micropipe used

Fig. 3
Fig. 3

Holograms obtained with z p = 13 m m and z l 1 = 1 m m . (a) Experimental hologram. (b) Simulated hologram.

Fig. 4
Fig. 4

Comparison of the simulated and the experimental normalized intensity profiles

Fig. 5
Fig. 5

(a) Hologram obtained with z p = 11 m m and z l 1 = 3 m m . (b) Evolution of the fractional orders against observation position within the pipe.

Fig. 6
Fig. 6

(Media 1) FRFT reconstruction of the sequence of holograms proposed Fig. 5 with a x = 0.967 ,     a y = 0.972

Fig. 7
Fig. 7

3D-ROI experimental set-up.

Fig. 8
Fig. 8

Evolution of the fractional reconstruction orders in the microchannel.

Fig. 9
Fig. 9

(a) Hologram with a 3D-ROI. (b) FRFT reconstruction of one particle located in the ROI.

Fig. 10
Fig. 10

Evolution of the fractional reconstruction orders in the microchannel for z = 80 m m .

Fig. 11
Fig. 11

Estimation of the reconstruction distance error over the microchannel for different bias in fractional order values.

Equations (18)

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G ( μ , ν ) = exp ( μ 2 + ν 2 ω 2 ) ,
I ( x , y ) = 1 λ 2 B 2 x B 2 y ( | | 2 2 { O ¯ } + | O | 2 ) ,
( x , y ) = 2 G 1 ( ξ , η ) exp [ i π λ B 2 x ( A 2 x ξ 2 2 x ξ + D 2 x x 2 ) ] × exp [ i π λ B 2 y ( A 2 y η 2 2 y η + D 2 y y 2 ) ] d ξ d η .
( x , y ) exp [ π λ ( N x B 2 x x 2 + N y B 2 y y 2 ) ] exp [ i π λ ( M x B 2 x x 2 + M y B 2 y y 2 ) ] .
O ( x , y ) = 2 G 1 ( ξ , η ) T ( ξ , η ) exp [ i π λ B 2 x ( A 2 x ξ 2 2 x ξ + D 2 x x 2 ) ] × exp [ i π λ B 2 y ( A 2 y η 2 2 y η + D 2 y y 2 ) ] d ξ d η ,
T ( ξ , η ) = k = 1 N A k exp [ B k b 2 ( ξ 2 + R e l l 2 η 2 ) ]
O ( x , y ) exp [ i π λ ( D 2 x B 2 x x 2 + D 2 y B 2 y y 2 ) ] k = 1 N A k K 2 x e q K 2 y e q × exp [ π λ ( N x e q B 2 x x 2 + N y e q B 2 y y 2 ) ] exp [ i π λ ( M x e q B 2 x x 2 + M y e q B 2 y y 2 ) ] ,
α x , α y [ I ( x , y ) ] ( x a , y a ) = 2 N α x ( x , x a ) N α y ( y , y a ) I ( x , y ) d x d y ,
N α p ( x , x a ) = C ( α p ) exp ( i π x 2 + x a 2 s p 2 tan α p ) exp ( i 2 π x a x s p 2 sin α p ) .
φ = π λ [ ( M x D 2 x B 2 x ) x 2 + ( M y D 2 y B 2 y ) y 2 ] ,
φ a = π ( cot α x s x 2 x 2 + cot α y s y 2 y 2 ) .
φ a ± φ = 0.
a x = 2 π arctan [ λ B 2 x s x 2 ( M x D 2 x ) ] ,     a y = 2 π arctan [ λ B 2 y s y 2 ( M y D 2 y ) ] .
( x , y ) exp [ π λ ( N x B 2 x x 2 + N y B 2 y y 2 ) ] exp [ i π λ ( M x B 2 x x 2 + M y B 2 y y 2 ) ] ,
M x , y = D 2 x , y + ( π ω 1 x , y 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y A 2 x , y ) 1 + ( π ω 1 x , y 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y A 2 x , y ) 2 , N x , y = π ω 1 x , y 2 λ B 2 x , y 1 + ( π ω 1 x , y 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y A 2 x , y ) 2 .
1 ω 1 x e q 2 = 1 ω 1 x 2 + { B k } b 2 ;         1 ω 1 y e q 2 = 1 ω 1 y 2 + R e l l 2 { B k } b 2 1 R 1 x e q = 1 R 1 x + { B k } λ π b 2 ;         1 R 1 x e q = 1 R 1 x + R e l l 2 { B k } λ π b 2 ,
O ( x , y ) exp [ i π λ ( D 2 x B 2 x x 2 + D 2 y B 2 y y 2 ) ] k = 1 N A k K 2 x e q K 2 y e q × exp [ π λ ( N x e q B 2 x x 2 + N y e q B 2 y y 2 ) ] exp [ i π λ ( M x e q B 2 x x 2 + M y e q B 2 y y 2 ) ] ,
M x , y e q = ( π ω 1 x , y e q 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y e q A 2 x , y ) 1 + ( π ω 1 x , y e q 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y e q A 2 x , y ) 2 , N x , y e q = π ω 1 x , y e q 2 λ B 2 x , y 1 + ( π ω 1 x , y e q 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y e q A 2 x , y ) 2 .

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