Abstract

In this paper, we present a procedure to separate aggregates of overlapped particles in digital holograms, based on a focus plane analysis applied to each particle. The method can be applied either on phase or on amplitude objects, according that each object has a border in one focus plane. Numerical simulations are performed to quantify the robustness of the process by increasing the overlapping areas between the particles. The separation algorithm is successfully demonstrated experimentally on different types of aggregates.

© 2013 OSA

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2013 (2)

2012 (2)

I. Moon, B. Javidi, F. Yi, D. Boss, and P. Marquet, “Automated statistical quantification of three-dimensional morphology and mean corpuscular hemoglobin of multiple red blood cells,” Opt. Express20(9), 10295–10309 (2012).
[CrossRef] [PubMed]

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging multiple colloidal particles by fitting electromagnetic scattering solutions to digital holograms,” J. Quant. Spectrosc. Radiat. Transf.113(18), 2482–2489 (2012).
[CrossRef]

2011 (3)

2010 (1)

C. Jung and C. Kim, “Segmenting clustered nuclei using H-minima transform-based marker extraction and contour parameterization,” IEEE Trans. Biomed. Eng.57(10), 2600–2604 (2010).
[CrossRef] [PubMed]

2009 (1)

J. Cheng and J. C. Rajapakse, “Segmentation of clustered nuclei with shape markers and marking function,” IEEE Trans. Biomed. Eng.56(3), 741–748 (2009).
[CrossRef] [PubMed]

2008 (3)

N. Callens, C. Minetti, G. Coupier, M.-A. Mader, F. Dubois, C. Misbah, and T. Podgorski, “Hydrodynamic lift of vesicles under shear flow in microgravity,” Europhys. Lett.83(2), 24002 (2008).
[CrossRef]

M. Smereka and I. Duleba, “Circular object detection using a modified Hough transform,” Int. J. Appl. Math. Comput. Sci.18(1), 85–91 (2008).
[CrossRef]

B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt.47(4), A52–A61 (2008).
[CrossRef] [PubMed]

2007 (2)

B. Parvin, Q. Yang, J. Han, H. Chang, B. Rydberg, and M. H. Barcellos-Hoff, “Iterative voting for inference of structural saliency and characterization of subcellular events,” IEEE Trans. Image Process.16(3), 615–623 (2007).
[CrossRef] [PubMed]

S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express15(4), 1505–1512 (2007).
[CrossRef] [PubMed]

2006 (2)

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express14(13), 5895–5908 (2006).
[CrossRef] [PubMed]

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J. Biomed. Opt.11(5), 054032 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging13, 145–165 (2004).

2003 (1)

2002 (1)

U. Schnars and W. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13(9), R85–R101 (2002).
[CrossRef]

1997 (1)

N. Malpica, C. O. de Solórzano, J. J. Vaquero, A. Santos, I. Vallcorba, J. M. García-Sagredo, and F. del Pozo, “Applying watershed algorithms to the segmentation of clustered nuclei,” Cytometry28(4), 289–297 (1997).
[CrossRef] [PubMed]

1992 (1)

S. Beucher, “The watershed transformation applied to image segmentation,” Scanning Microsc. Suppl.6, 299–314 (1992).

1981 (1)

D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recognit.13(2), 111–122 (1981).
[CrossRef]

1980 (1)

M. Nazarathy and J. Shamir, “Fourier optics described by operator,” J. Opt. Soc. Am. A70(2), 150–159 (1980).
[CrossRef]

Allano, D.

Ballard, D. H.

D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recognit.13(2), 111–122 (1981).
[CrossRef]

Barcellos-Hoff, M. H.

B. Parvin, Q. Yang, J. Han, H. Chang, B. Rydberg, and M. H. Barcellos-Hoff, “Iterative voting for inference of structural saliency and characterization of subcellular events,” IEEE Trans. Image Process.16(3), 615–623 (2007).
[CrossRef] [PubMed]

Beucher, S.

S. Beucher, “The watershed transformation applied to image segmentation,” Scanning Microsc. Suppl.6, 299–314 (1992).

Boss, D.

Callens, N.

N. Callens, C. Minetti, G. Coupier, M.-A. Mader, F. Dubois, C. Misbah, and T. Podgorski, “Hydrodynamic lift of vesicles under shear flow in microgravity,” Europhys. Lett.83(2), 24002 (2008).
[CrossRef]

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express14(13), 5895–5908 (2006).
[CrossRef] [PubMed]

Chang, H.

B. Parvin, Q. Yang, J. Han, H. Chang, B. Rydberg, and M. H. Barcellos-Hoff, “Iterative voting for inference of structural saliency and characterization of subcellular events,” IEEE Trans. Image Process.16(3), 615–623 (2007).
[CrossRef] [PubMed]

Q. Wen, H. Chang, and B. Parvin, “A Delaunay triangulation approach for segmenting clumps of nuclei,” in Proceedings of IEEE Conference on Biomedical Imaging: From Nano to Macro (IEEE, 2009), pp 9–12.

Cheng, J.

J. Cheng and J. C. Rajapakse, “Segmentation of clustered nuclei with shape markers and marking function,” IEEE Trans. Biomed. Eng.56(3), 741–748 (2009).
[CrossRef] [PubMed]

Cheong, F. C.

Coëtmellec, S.

Colomb, T.

Coppola, G.

Corbin, F.

Coupier, G.

N. Callens, C. Minetti, G. Coupier, M.-A. Mader, F. Dubois, C. Misbah, and T. Podgorski, “Hydrodynamic lift of vesicles under shear flow in microgravity,” Europhys. Lett.83(2), 24002 (2008).
[CrossRef]

X. Grandchamp, G. Coupier, A. Srivastav, C. Minetti, and T. Podgorski, “Lift and down-gradient shear-induced diffusion in red blood cell suspensions,” Phys. Rev. Lett. (to be published).

Cuche, E.

De Nicola, S.

de Solórzano, C. O.

N. Malpica, C. O. de Solórzano, J. J. Vaquero, A. Santos, I. Vallcorba, J. M. García-Sagredo, and F. del Pozo, “Applying watershed algorithms to the segmentation of clustered nuclei,” Cytometry28(4), 289–297 (1997).
[CrossRef] [PubMed]

Debeir, O.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J. Biomed. Opt.11(5), 054032 (2006).
[CrossRef] [PubMed]

Decaestecker, C.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J. Biomed. Opt.11(5), 054032 (2006).
[CrossRef] [PubMed]

del Pozo, F.

N. Malpica, C. O. de Solórzano, J. J. Vaquero, A. Santos, I. Vallcorba, J. M. García-Sagredo, and F. del Pozo, “Applying watershed algorithms to the segmentation of clustered nuclei,” Cytometry28(4), 289–297 (1997).
[CrossRef] [PubMed]

Depeursinge, C.

Dimiduk, T. G.

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging multiple colloidal particles by fitting electromagnetic scattering solutions to digital holograms,” J. Quant. Spectrosc. Radiat. Transf.113(18), 2482–2489 (2012).
[CrossRef]

Dixon, L.

Dubois, F.

Duleba, I.

M. Smereka and I. Duleba, “Circular object detection using a modified Hough transform,” Int. J. Appl. Math. Comput. Sci.18(1), 85–91 (2008).
[CrossRef]

El Mallahi, A.

Emery, Y.

Ferraro, P.

Finizio, A.

Foucaut, J.-M.

Fung, J.

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging multiple colloidal particles by fitting electromagnetic scattering solutions to digital holograms,” J. Quant. Spectrosc. Radiat. Transf.113(18), 2482–2489 (2012).
[CrossRef]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express19(9), 8051–8065 (2011).
[CrossRef] [PubMed]

García-Sagredo, J. M.

N. Malpica, C. O. de Solórzano, J. J. Vaquero, A. Santos, I. Vallcorba, J. M. García-Sagredo, and F. del Pozo, “Applying watershed algorithms to the segmentation of clustered nuclei,” Cytometry28(4), 289–297 (1997).
[CrossRef] [PubMed]

Godard, G.

Grandchamp, X.

X. Grandchamp, G. Coupier, A. Srivastav, C. Minetti, and T. Podgorski, “Lift and down-gradient shear-induced diffusion in red blood cell suspensions,” Phys. Rev. Lett. (to be published).

Grier, D. G.

Grilli, S.

Han, J.

B. Parvin, Q. Yang, J. Han, H. Chang, B. Rydberg, and M. H. Barcellos-Hoff, “Iterative voting for inference of structural saliency and characterization of subcellular events,” IEEE Trans. Image Process.16(3), 615–623 (2007).
[CrossRef] [PubMed]

Javidi, B.

Jung, C.

C. Jung and C. Kim, “Segmenting clustered nuclei using H-minima transform-based marker extraction and contour parameterization,” IEEE Trans. Biomed. Eng.57(10), 2600–2604 (2010).
[CrossRef] [PubMed]

Jüptner, W.

U. Schnars and W. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13(9), R85–R101 (2002).
[CrossRef]

Kaz, D. M.

Kemper, B.

Kim, C.

C. Jung and C. Kim, “Segmenting clustered nuclei using H-minima transform-based marker extraction and contour parameterization,” IEEE Trans. Biomed. Eng.57(10), 2600–2604 (2010).
[CrossRef] [PubMed]

Kiss, R.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J. Biomed. Opt.11(5), 054032 (2006).
[CrossRef] [PubMed]

Lebrun, D.

Lecordier, B.

Lee, S.-H.

Legros, J.-C.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J. Biomed. Opt.11(5), 054032 (2006).
[CrossRef] [PubMed]

Long, F.

F. Long, H. Peng, and E. Myers, “Automatic segmentation of nuclei in 3D microscopy images of C. Elegans,” in Proceedings of IEEE Conference on Biomedical Imaging: From Nano to Macro (IEEE, 2007), pp 536–539.
[CrossRef]

Mader, M.-A.

N. Callens, C. Minetti, G. Coupier, M.-A. Mader, F. Dubois, C. Misbah, and T. Podgorski, “Hydrodynamic lift of vesicles under shear flow in microgravity,” Europhys. Lett.83(2), 24002 (2008).
[CrossRef]

Magistretti, P. J.

Magro, C.

Malek, M.

Malpica, N.

N. Malpica, C. O. de Solórzano, J. J. Vaquero, A. Santos, I. Vallcorba, J. M. García-Sagredo, and F. del Pozo, “Applying watershed algorithms to the segmentation of clustered nuclei,” Cytometry28(4), 289–297 (1997).
[CrossRef] [PubMed]

Manoharan, V. N.

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging multiple colloidal particles by fitting electromagnetic scattering solutions to digital holograms,” J. Quant. Spectrosc. Radiat. Transf.113(18), 2482–2489 (2012).
[CrossRef]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express19(9), 8051–8065 (2011).
[CrossRef] [PubMed]

Marquet, P.

Martin, K. E.

McGorty, R.

Minetti, C.

A. El Mallahi, C. Minetti, and F. Dubois, “Automated three-dimensional detection and classification of living organisms using digital holographic microscopy with partial spatial coherent source: Application to the monitoring of drinking water resources,” Appl. Opt.52(1), A68–A80 (2013).
[CrossRef] [PubMed]

N. Callens, C. Minetti, G. Coupier, M.-A. Mader, F. Dubois, C. Misbah, and T. Podgorski, “Hydrodynamic lift of vesicles under shear flow in microgravity,” Europhys. Lett.83(2), 24002 (2008).
[CrossRef]

X. Grandchamp, G. Coupier, A. Srivastav, C. Minetti, and T. Podgorski, “Lift and down-gradient shear-induced diffusion in red blood cell suspensions,” Phys. Rev. Lett. (to be published).

Misbah, C.

N. Callens, C. Minetti, G. Coupier, M.-A. Mader, F. Dubois, C. Misbah, and T. Podgorski, “Hydrodynamic lift of vesicles under shear flow in microgravity,” Europhys. Lett.83(2), 24002 (2008).
[CrossRef]

Monnom, O.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J. Biomed. Opt.11(5), 054032 (2006).
[CrossRef] [PubMed]

Moon, I.

Myers, E.

F. Long, H. Peng, and E. Myers, “Automatic segmentation of nuclei in 3D microscopy images of C. Elegans,” in Proceedings of IEEE Conference on Biomedical Imaging: From Nano to Macro (IEEE, 2007), pp 536–539.
[CrossRef]

Nazarathy, M.

M. Nazarathy and J. Shamir, “Fourier optics described by operator,” J. Opt. Soc. Am. A70(2), 150–159 (1980).
[CrossRef]

Parvin, B.

B. Parvin, Q. Yang, J. Han, H. Chang, B. Rydberg, and M. H. Barcellos-Hoff, “Iterative voting for inference of structural saliency and characterization of subcellular events,” IEEE Trans. Image Process.16(3), 615–623 (2007).
[CrossRef] [PubMed]

Q. Wen, H. Chang, and B. Parvin, “A Delaunay triangulation approach for segmenting clumps of nuclei,” in Proceedings of IEEE Conference on Biomedical Imaging: From Nano to Macro (IEEE, 2009), pp 9–12.

Peng, H.

F. Long, H. Peng, and E. Myers, “Automatic segmentation of nuclei in 3D microscopy images of C. Elegans,” in Proceedings of IEEE Conference on Biomedical Imaging: From Nano to Macro (IEEE, 2007), pp 536–539.
[CrossRef]

Perry, R. W.

J. Fung, R. W. Perry, T. G. Dimiduk, and V. N. Manoharan, “Imaging multiple colloidal particles by fitting electromagnetic scattering solutions to digital holograms,” J. Quant. Spectrosc. Radiat. Transf.113(18), 2482–2489 (2012).
[CrossRef]

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express19(9), 8051–8065 (2011).
[CrossRef] [PubMed]

Pierattini, G.

Podgorski, T.

N. Callens, C. Minetti, G. Coupier, M.-A. Mader, F. Dubois, C. Misbah, and T. Podgorski, “Hydrodynamic lift of vesicles under shear flow in microgravity,” Europhys. Lett.83(2), 24002 (2008).
[CrossRef]

X. Grandchamp, G. Coupier, A. Srivastav, C. Minetti, and T. Podgorski, “Lift and down-gradient shear-induced diffusion in red blood cell suspensions,” Phys. Rev. Lett. (to be published).

Rajapakse, J. C.

J. Cheng and J. C. Rajapakse, “Segmentation of clustered nuclei with shape markers and marking function,” IEEE Trans. Biomed. Eng.56(3), 741–748 (2009).
[CrossRef] [PubMed]

Rappaz, B.

Rydberg, B.

B. Parvin, Q. Yang, J. Han, H. Chang, B. Rydberg, and M. H. Barcellos-Hoff, “Iterative voting for inference of structural saliency and characterization of subcellular events,” IEEE Trans. Image Process.16(3), 615–623 (2007).
[CrossRef] [PubMed]

Sankur, B.

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging13, 145–165 (2004).

Santos, A.

N. Malpica, C. O. de Solórzano, J. J. Vaquero, A. Santos, I. Vallcorba, J. M. García-Sagredo, and F. del Pozo, “Applying watershed algorithms to the segmentation of clustered nuclei,” Cytometry28(4), 289–297 (1997).
[CrossRef] [PubMed]

Schnars, U.

U. Schnars and W. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13(9), R85–R101 (2002).
[CrossRef]

Schockaert, C.

Sezgin, M.

M. Sezgin and B. Sankur, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron. Imaging13, 145–165 (2004).

Shamir, J.

M. Nazarathy and J. Shamir, “Fourier optics described by operator,” J. Opt. Soc. Am. A70(2), 150–159 (1980).
[CrossRef]

Smereka, M.

M. Smereka and I. Duleba, “Circular object detection using a modified Hough transform,” Int. J. Appl. Math. Comput. Sci.18(1), 85–91 (2008).
[CrossRef]

Srivastav, A.

X. Grandchamp, G. Coupier, A. Srivastav, C. Minetti, and T. Podgorski, “Lift and down-gradient shear-induced diffusion in red blood cell suspensions,” Phys. Rev. Lett. (to be published).

Vallcorba, I.

N. Malpica, C. O. de Solórzano, J. J. Vaquero, A. Santos, I. Vallcorba, J. M. García-Sagredo, and F. del Pozo, “Applying watershed algorithms to the segmentation of clustered nuclei,” Cytometry28(4), 289–297 (1997).
[CrossRef] [PubMed]

Van Ham, P.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J. Biomed. Opt.11(5), 054032 (2006).
[CrossRef] [PubMed]

Vaquero, J. J.

N. Malpica, C. O. de Solórzano, J. J. Vaquero, A. Santos, I. Vallcorba, J. M. García-Sagredo, and F. del Pozo, “Applying watershed algorithms to the segmentation of clustered nuclei,” Cytometry28(4), 289–297 (1997).
[CrossRef] [PubMed]

von Bally, G.

Walle, F.

Wen, Q.

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Figures (9)

Fig. 1
Fig. 1

(a) Intensity image of a simulated aggregate of 4 particles, overlapping each other with an area of 64%. (b) 3D view of the position of the particles inside the simulated volume. The z axis is parallel to the main viewing direction (optical axis).

Fig. 2
Fig. 2

(a) Simulated aggregate of 4 spherical particles. The border in red is extracted from the smoothed border, with an opening operation, of the segmented aggregate. Perpendicularly to this border, successive rectangular ROIs are generated, e.g. in blue, for the first ROI (b) Based on this ROI, a mask is created in which the refocusing criterion is computed.

Fig. 3
Fig. 3

Estimated focus distance as a function of the border segment. The four highlighted horizontal steps correspond to the focus planes of the four simulated spherical particles constituting the aggregate.

Fig. 4
Fig. 4

Number of perimeter pixels corresponding to each computed focus plane.

Fig. 5
Fig. 5

Intensity images of different types of aggregates generated by the cluster generator, composed of (a) 4 spherical particles, (b) 4 prolate ellipsoidal particles (2 rotated by a 45° and 2 by a −45° from the x-axis), (c) 4 cubic particles, (d) 1 spherical, 1 ellipsoidal, and 2 cubic particles (whose one is rotated by a 45° from the x-axis), (e) 2 spherical and 2 cubic particles and (f) 2 spherical (of two different sizes) and 2 ellipsoidal particles (whose one is rotated by a 45° from the x-axis).

Fig. 6
Fig. 6

Separation method applied on aggregates constituted by opaque polyethylene spherical particles of different sizes. (a) Intensity image of an aggregate of two particles in the recorded plane (b) Automated separation and refocusing of the bottom particle at −370µm (error: 5µm). (c) Automated separation and refocusing of the top particle at 320µm (error: 6µm). (d) Intensity image of an aggregate of three particles in the recorded plane (e) Automated separation and refocusing of the top particle at −135µm (error: 3µm). (f) Automated separation and refocusing of the bottom particle at 95µm (error: 2µm). (g) Intensity image of an aggregate of four particles in the recorded plane (h) Automated separation and refocusing of the top particle at −125µm (error: 2µm). (i) Automated separation and refocusing of the bottom particle at −265µm (error: 4µm).

Fig. 7
Fig. 7

Cluster of two RBCs in the recorded plane (a) Intensity image (b) Phase image (c) Segmentation of the phase image of the aggregate showing that one big object is detected.

Fig. 8
Fig. 8

Separation procedure applied to RBCs cluster (a) Automated separation and refocusing of the left particle at −24µm (error: 0.8µm) - Intensity image (b) Automated separation and refocusing of the left particle at −24µm (error: 0.8µm) - Phase image (c) Automated separation and refocusing of the left particle at −76µm (error: 1.5µm) - Intensity image (d) Automated separation and refocusing of the left particle at −76µm (error: 1.5µm) - Phase image.

Fig. 9
Fig. 9

Cluster of two RBCs in two different orientations (a) Intensity image of the cluster in the recorded plane (b) Phase image of the cluster in the recorded plane (c) RBC on the right separated and refocused at −90µm (error: 2.4µm) – Intensity image (d) RBC on the right separated and refocused at −90µm (error: 2.4µm) – Phase image (e) RBC on the left separated and refocused at −42µm (error: 1.2µm) – Intensity image (f) RBC on the left separated and refocused at −42µm (error: 1.2µm) – Phase image.

Tables (1)

Tables Icon

Table 1 Reconstructed focus planes as a function of the area between overlapped particlesa

Equations (7)

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u(x',y',d)=R[d]u(x,y,0) =exp(jkd) F x',y' 1 Q[ λ 2 d] F ν x , ν y +1 u(x,y,0)
u(x',y',ε) F x',y' 1 ( 1 jk λ 2 ε 2 ( ν x 2 + ν y 2 ) ) F ν x , ν y +1 u(x,y,0) u(x',y',0) jk λ 2 ε 2 F x',y' 1 ( ν x 2 + ν y 2 ) F ν x , ν y +1 u(x,y,0)
u 0 z u z | z=0 u(x',y',ε)u(x,y,0) ε jk λ 2 2 F x',y' 1 ( ν x 2 + ν y 2 ) F ν x , ν y +1 u 0 (x,y)
Δ u 0 = 2 u 0 = (2πj) 2 F x',y' 1 ( ν x 2 + ν y 2 ) F ν x , ν y +1 u 0
j u 0 z = k λ 2 8 π 2 Δ u 0
u 0 z +.( u 0 v)=0
J= jk λ 2 8 π 2 u 0 u 0

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