Abstract

We present a novel technique for three-dimensional (3D) image processing of complex fields. It consists in inverting the coherent image formation by filtering the complex spectrum with a realistic 3D coherent transfer function (CTF) of a high-NA digital holographic microscope. By combining scattering theory and signal processing, the method is demonstrated to yield the reconstruction of a scattering object field. Experimental reconstructions in phase and amplitude are presented under non-design imaging conditions. The suggested technique is best suited for an implementation in high-resolution diffraction tomography based on sample or illumination rotation.

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2010 (3)

2009 (3)

2008 (1)

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

2007 (2)

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

S. S. Kou and C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15(21), 13,640–13,648 (2007).
[CrossRef]

2006 (4)

2004 (1)

R. Arimoto and J. M. Murray, “A common aberration with water-immersion objective lenses,” J. Microsc. 216, 49–51 (2004).
[CrossRef] [PubMed]

2001 (1)

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98(7), 3790–3795 (2001).
[CrossRef] [PubMed]

2000 (1)

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

1999 (3)

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
[CrossRef]

M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

1994 (1)

1993 (1)

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

1992 (1)

1991 (1)

1982 (1)

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[CrossRef] [PubMed]

1979 (1)

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. 9(1), 62–66 (1979).
[CrossRef]

1968 (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Agard, D. A.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98(7), 3790–3795 (2001).
[CrossRef] [PubMed]

Arimoto, R.

R. Arimoto and J. M. Murray, “A common aberration with water-immersion objective lenses,” J. Microsc. 216, 49–51 (2004).
[CrossRef] [PubMed]

Aspert, N.

Badizadegan, K.

Barber, P. W.

Born, M.

M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

Braat, J.

Brady, D. J.

Charrière, F.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14(10), 4300–4306 (2006).
[CrossRef] [PubMed]

Charrire, F.

Choi, K.

Choi, W.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[CrossRef] [PubMed]

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

Chowdhury, D. Q.

Colomb, T.

Conchello, J. A.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Conchello, J.-A.

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Cotte, Y.

Cuche, E.

Cui, X. Q.

Dasari, R. R.

Depeursinge, C.

Devaney, A.

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[CrossRef] [PubMed]

Diez-Silva, M.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

Fang-Yen, C.

Feld, M. S.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[CrossRef] [PubMed]

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

Fink, H.-W.

Frisken Gibson, S.

Gehri, F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

Gustafsson, M. G. L.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98(7), 3790–3795 (2001).
[CrossRef] [PubMed]

Hanser, B.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98(7), 3790–3795 (2001).
[CrossRef] [PubMed]

Heng, X.

Hill, S. C.

Horisaki, R.

Kam, Z.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98(7), 3790–3795 (2001).
[CrossRef] [PubMed]

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Knapp, D. W.

Kou, S. S.

S. S. Kou and C. J. R. Sheppard, “Image formation in holographic tomography: high-aperture imaging conditions,” Appl. Opt. 48(34), H168–H175 (2009).
[CrossRef] [PubMed]

S. S. Kou and C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15(21), 13,640–13,648 (2007).
[CrossRef]

Kühn, J.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14(10), 4300–4306 (2006).
[CrossRef] [PubMed]

Lanni, F.

Latychevskaia, T.

Lim, S.

Lock, J. A.

Lykotrafitis, G.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

Marian, A.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

Marks, D. L.

Marquet, P.

McDowell, E. J.

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11(3), 1056–1067 (1994).
[CrossRef]

Montfort, F.

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

F. Montfort, F. Charrire, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23(11), 2944–2953 (2006).
[CrossRef]

Murray, J. M.

R. Arimoto and J. M. Murray, “A common aberration with water-immersion objective lenses,” J. Microsc. 216, 49–51 (2004).
[CrossRef] [PubMed]

Nehorai, A.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

Otsu, N.

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. 9(1), 62–66 (1979).
[CrossRef]

Park, Y.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

Pavillon, N.

Popescu, G.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

Preza, C.

Psaltis, D.

Sarder, P.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

Sedat, J. W.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98(7), 3790–3795 (2001).
[CrossRef] [PubMed]

Shaffer, E.

Sheppard, C. J.

S. S. Kou and C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15(21), 13,640–13,648 (2007).
[CrossRef]

Sheppard, C. J. R.

Sung, Y.

Suresh, S.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

Thomas, L. J.

Toy, M. F.

Wolf, E.

M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

Wu, J. G.

Yang, C. H.

Yaqoob, Z.

Appl. Opt. (5)

IEEE Signal Process. Mag. (1)

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. 9(1), 62–66 (1979).
[CrossRef]

J. Microsc. (2)

A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. 225, 156–169 (2007).
[CrossRef] [PubMed]

R. Arimoto and J. M. Murray, “A common aberration with water-immersion objective lenses,” J. Microsc. 216, 49–51 (2004).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Methods (1)

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
[CrossRef] [PubMed]

Opt. Express (7)

Opt. Lett. (1)

Proc. Natl. Acad. Sci. U.S.A. (2)

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. 105(37), 13730–13735 (2008).
[CrossRef] [PubMed]

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98(7), 3790–3795 (2001).
[CrossRef] [PubMed]

Ultrason. Imaging (1)

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[CrossRef] [PubMed]

Other (5)

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

M. Born and E. Wolf, Principles of Optics , 7th ed. (Cambridge University Press, 1999).

I. Bergoend, C. Arfire, N. Pavillon, and C. Depeursinge, “Diffraction tomography for biological cells imaging using digital holographic microscopy,” in Laser Applications in Life Sciences , SPIE vol. 7376 (2010).

Y. Cotte and C. Depeursinge, “Measurement of the complex amplitude point spread function by a diffracting circular aperture,” in Focus on Microscopy , Advanced linear and non-linear imaging, pp. TU–AF2–PAR–D (2009).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (9)

Fig. 1
Fig. 1

Optical transfer of a point source in real and reciprocal space. In scheme (a), a practical Abbe imaging system with holographic reconstruction. In scheme (b), full Ewald’s sphere under Born approximation in the reciprocal object plane according to Eq. (11).

Fig. 2
Fig. 2

Experimental configuration. (a) Optical setup: LD, laser diode; BS, beam splitter; M, mirror; DL, delay line; SF, spatial filter; ND, neutral density filter; L, lens; TL, tube lens; CL, condenser lens; MO, microscope objective; RS, rotatable specimen. (b) RS with complex point source in MO design and non-design conditions. Insert: SEM image of nano-metric aperture (⌀≈75nm) in aluminum film at 150000× magnification. (c) RS with objects (see section 3) in experimental conditions with incident light k 0 along optical axis.

Fig. 3
Fig. 3

Measurement of complex point source in MO design and MO non-design imaging conditions (cf. Fig. 2). The experimental APSF sections in (a) yield design MO imaging conditions (θ = 0°), whereas sections in (b) yield non-design conditions (θ = 15°). The left side images show |h| central sections and the right images arg[h], respectively. Colorbar, Scalebars: 2μm.

Fig. 4
Fig. 4

Experimental 3D CTF in different imaging conditions. The experimental CTF in (a) obtained for MO design imaging conditions (θ = 0°), whereas the CTF depicted in (b) is obtained for non-design conditions (θ = 15°), according to Fig. 2. The upper row shows the top view on the CTF and bottom row shows the side view through the CTF for kx = 0, respectively. Colorbars.

Fig. 5
Fig. 5

Scheme of reciprocal space. In image (a), a 1D coherent transfer function as given by the complex point source [see Eq. (12)]. In image (b), image’s spectrum with background illumination k⃗ 0 ≠ 0.

Fig. 6
Fig. 6

Complex fields of polystyrene microspheres in water at a tilt angle of θ = 15°, according to Fig. 2. The main images show the raw amplitude (a) and the deconvolved amplitude (b) with background ROI-1 and object ROI-2 with circle ⌀ ≈ 5.8μm. The inserts in ROI-3 show the phase parts, respectively. Colorbar, Scalebar: 4μm.

Fig. 7
Fig. 7

Human RBCs in phase. Images (a) and (c) show the phase images of two RBCs at θ = 0. Unprocessed images are labeled ’RAW’ and the label ’PROCESSED’ indicates the deconvolved phase for τ = 1. The profiles in (b) and (d) compare the according height differences of images above, at central sections (indicated by flashes). The error bars indicate the level of phase noise (≈ 0.1rad). Images (d)–(f) show the top view on RBC (c), processed with different τ (expressed in units of A ^ δ ( s ) ). Colorbars, Scalebars: 4μ m.

Fig. 8
Fig. 8

Inclined human RBCs in phase at various tilt angles θ. Images (a)–(f) are unprocessed and images (g)–(l) are pure phase deconvolved τ = 1. According to Fig. 2, the axis of rotation ’x’ is indicated. Colorbar, Scalebars: 4μm.

Fig. 9
Fig. 9

3D rendered images of two human RBC at θ = 0. Images (a), shows |U| in 3D-space in the middle row. Bottom and top rows show the sections through central RBC positions (indicated by flashes). Accordingly, the field |o (s)| is represented in (b), and |n| in (c) (uncalibrated levels), respectively. The RBCs’ positions are compared to an oval area of 6μm × 2.5μm, based on measurements from section 3.2. Colorbars.

Equations (24)

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U ( r 2 ) = o ( r 1 ) h ( r 2 r 1 ) d x 1 d y 1 d z 1 ,
{ U ( r 2 ) } = U ( r 2 ) exp [ i 2 π ( k r 2 ) ] d x 2 d y 2 d z 2 .
k = | k | = 2 π ν = 2 π λ .
{ U ( r 2 ) } G ( k ) = { o } O ( k ) { h } c ( k ) .
k x y , c = k sin α ,
k z , c = k ( 1 cos α ) ,
o ( r 1 ) = O ( k ) exp [ i 2 π ( k r 1 ) ] d k x d k y d k z = 1 { G ( k ) c ( k ) } .
c ˜ ( k ) = { c if | c | > τ 1 exp [ i arg [ c ] ] if | c | τ .
U ( r 2 ) = A ( r 2 ) exp [ i Φ ( r 2 ) ] = exp ( i k d z 2 ) i d λ Ψ ( x 2 , y 2 ) exp [ i π d z 2 λ [ ( x 2 x 2 ) 2 + ( y 2 y 2 ) ] 2 ] d x 2 d y 2 ,
h k 0 ( r 2 ) = A δ , k 0 ( s ) ( r 2 ) . exp [ i Φ δ , k 0 ( s ) ( r 2 ) ] ,
K = k k 0 ,
c ( k k 0 ) = A ^ δ ( s ) ( k k 0 ) exp [ i Φ ^ δ ( s ) ( k k 0 ) ] ,
o k 0 ( r 1 ) = o k 0 ( i ) ( r 1 ) + o k 0 ( s ) ( r 1 ) ,
o k 0 ( s ) ( r 1 ) = A k 0 ( s ) ( r 1 ) exp [ i Φ k 0 ( s ) ( r 1 ) ] ,
{ o k 0 ( s ) ( r 1 ) } = G ( k k 0 ) { o k 0 ( i ) ( r 1 ) h k 0 ( r 2 r 1 ) d x 1 d y 1 d z 1 } c ( k k 0 ) .
o k 0 ( i ) ( r 1 ) = A ( i ) exp [ i k 0 r 1 ] ,
{ o k 0 ( s ) ( r 1 ) } = G ( k k 0 ) A ^ ( i ) δ ( k k 0 ) c ( k k 0 ) .
G ( k k 0 ) A ^ ( k k 0 ) exp [ i Φ ^ ( k k 0 ) ] = { A ^ ( s ) ( k = k 0 ) + A ^ ( i ) if k = k 0 A ^ ( s ) ( k k 0 ) exp [ i Φ ^ ( k k 0 ) ] if k k 0 ,
{ o k 0 ( s ) ( r 1 ) } = A ^ ( s ) ( k k 0 ) exp [ i Φ ^ ( k k 0 ) ] A ^ δ ( s ) ( k k 0 ) exp [ i Φ ^ δ ( k k 0 ) ] .
{ o k 0 ( s ) ( r 1 ) } = G ( k k 0 ) 1 { h k 0 ( r 1 ) + o k 0 ( i ) ( r 1 ) } = A ^ ( k k 0 ) exp [ i Φ ^ ( k k 0 ) ] A ^ δ ( s ) ( k k 0 ) exp [ i Φ ^ δ ( k k 0 ) ] + A ^ ( i ) δ ( k k 0 ) .
{ F k 0 ( r 1 ) } = i π k z { U k 0 ( s ) ( x 1 , y 1 , z 1 = z ± ) } exp [ i k z z ± ] .
{ F k 0 ( r 1 ) } = i π { o k 0 ( s ) ( r 1 ) } .
n ( r 1 ) = [ n m 2 F L B ( r 1 ) / k 2 ] 1 / 2 ,
F L B ( r 1 ) = 1 ( 2 π ) 3 K 2 k F ^ ( K ) exp [ i K r 1 ] d K x d K y d K z ,

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