Abstract

The video stream captured by an in-line holographic microscope can be analyzed on a frame-by-frame basis to track individual colloidal particles’ three-dimensional motions with nanometer resolution. In this work, we compare the performance of two complementary analysis techniques, one based on fitting to the exact Lorenz-Mie theory and the other based on phenomenological interpretation of the scattered light field reconstructed with Rayleigh-Sommerfeld back-propagation. Although Lorenz-Mie tracking provides more information and is inherently more precise, Rayleigh-Sommerfeld reconstruction is faster and more general. The two techniques agree quantitatively on colloidal spheres’ in-plane positions. Their systematic differences in axial tracking can be explained in terms of the illuminated objects’ light scattering properties.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]

2010

2009

F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009).
[CrossRef]

F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

2008

P. Messmer, P. J. Mullowney, and B. E. Granger, “GPULib: GPU computing in high-level languages,” Comput. Sci. Eng. 10, 70–73 (2008).
[CrossRef]

2007

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

S.-H. Lee, and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007).
[CrossRef] [PubMed]

2006

2005

2003

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

2002

C. Gosse, and V. Croquette, “Magnetic tweezers: Micromanipulation and force measurement at the molecular level,” Biophys. J. 82, 3314–3329 (2002).
[CrossRef] [PubMed]

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

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[CrossRef] [PubMed]

1999

1998

E. R. Dufresne, and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

1997

T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

1996

J. C. Crocker, and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

1995

T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, “Stretching of a single tethered polymer in a uniform flow,” Science 268, 83–87 (1995).
[CrossRef] [PubMed]

1994

J. C. Crocker, and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
[CrossRef] [PubMed]

1980

1976

1967

Castaneda, R.

J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62–67 (2006).
[CrossRef]

Cheong, F. C.

F. C. Cheong, and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express 18, 6555–6562 (2010).
[CrossRef] [PubMed]

F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009).
[CrossRef]

F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

Chu, S.

T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, “Stretching of a single tethered polymer in a uniform flow,” Science 268, 83–87 (1995).
[CrossRef] [PubMed]

Crocker, J. C.

J. C. Crocker, and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

J. C. Crocker, and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
[CrossRef] [PubMed]

Croquette, V.

C. Gosse, and V. Croquette, “Magnetic tweezers: Micromanipulation and force measurement at the molecular level,” Biophys. J. 82, 3314–3329 (2002).
[CrossRef] [PubMed]

Cuche, E.

Depeursinge, C.

Dixon, L.

F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009).
[CrossRef]

Dreyfus, R.

F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009).
[CrossRef]

Duarte, S.

F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

Dubois, F.

Dufresne, E. R.

E. R. Dufresne, and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

Emery, Y.

Evans, D. J.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[CrossRef] [PubMed]

Ganesan, K.

T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

Garcia-Sucerquia, J.

J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62–67 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef] [PubMed]

Gosse, C.

C. Gosse, and V. Croquette, “Magnetic tweezers: Micromanipulation and force measurement at the molecular level,” Biophys. J. 82, 3314–3329 (2002).
[CrossRef] [PubMed]

Granger, B. E.

P. Messmer, P. J. Mullowney, and B. E. Granger, “GPULib: GPU computing in high-level languages,” Comput. Sci. Eng. 10, 70–73 (2008).
[CrossRef]

Grier, D. G.

F. C. Cheong, and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express 18, 6555–6562 (2010).
[CrossRef] [PubMed]

K. Xiao, and D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010).
[CrossRef] [PubMed]

F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009).
[CrossRef]

F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

S.-H. Lee, and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007).
[CrossRef] [PubMed]

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005).
[CrossRef] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

E. R. Dufresne, and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

J. C. Crocker, and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

J. C. Crocker, and D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
[CrossRef] [PubMed]

Grill, A.

F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009).
[CrossRef]

Jericho, M. H.

Jericho, S. K.

Joannes, L.

Jüptner, W. P. O.

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

Katz, J.

Kim, S.-H.

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

Klages, P.

Kreuzer, H. J.

Kuo, S. C.

T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

Ladavac, K.

Larson, R. G.

T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, “Stretching of a single tethered polymer in a uniform flow,” Science 268, 83–87 (1995).
[CrossRef] [PubMed]

Lee, S.-H.

F. C. Cheong, S. Duarte, S.-H. Lee, and D. G. Grier, “Holographic microrheology of polysaccharides from Streptococcus mutans biofilms,” Rheol. Acta 48, 109–115 (2009).
[CrossRef]

S.-H. Lee, and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007).
[CrossRef] [PubMed]

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005).
[CrossRef] [PubMed]

Legros, J. C.

Lentz, W. J.

Magistretti, P. J.

Malkiel, E.

Marquet, P.

Mason, T. G.

T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

Messmer, P.

P. Messmer, P. J. Mullowney, and B. E. Granger, “GPULib: GPU computing in high-level languages,” Comput. Sci. Eng. 10, 70–73 (2008).
[CrossRef]

Mittag, E.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[CrossRef] [PubMed]

Mullowney, P. J.

P. Messmer, P. J. Mullowney, and B. E. Granger, “GPULib: GPU computing in high-level languages,” Comput. Sci. Eng. 10, 70–73 (2008).
[CrossRef]

Perkins, T. T.

T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, “Stretching of a single tethered polymer in a uniform flow,” Science 268, 83–87 (1995).
[CrossRef] [PubMed]

Polin, M.

Ramírez, J. H.

J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62–67 (2006).
[CrossRef]

Rappaz, B.

Roichman, Y.

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13(15), 5831–5845 (2005).
[CrossRef] [PubMed]

Schnars, U.

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

Searles, D. J.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[CrossRef] [PubMed]

Sevick, E. M.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[CrossRef] [PubMed]

Sheng, J.

Sherman, G. C.

Smith, D. E.

T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, “Stretching of a single tethered polymer in a uniform flow,” Science 268, 83–87 (1995).
[CrossRef] [PubMed]

Sun, B.

F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009).
[CrossRef]

van Blaaderen, A.

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

van Oostrum, P.

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

van Zanten, J. H.

T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

Wang, G. M.

G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, “Experimental demonstration of violations of the second law of thermodynmaics for small systems and short time scales,” Phys. Rev. Lett. 89, 050601 (2002).
[CrossRef] [PubMed]

Wirtz, D.

T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

Wiscombe, W. J.

Xiao, K.

K. Xiao, and D. G. Grier, “Multidimensional optical fractionation with holographic verification,” Phys. Rev. Lett. 104, 028302 (2010).
[CrossRef] [PubMed]

F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

F. C. Cheong, B. Sun, R. Dreyfus, A. Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13,071–13,079 (2009).
[CrossRef]

Xu, W.

Yang, S.-M.

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

Yi, G.-R.

S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18,275–18,282 (2007).

Appl. Opt.

Biophys. J.

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[CrossRef] [PubMed]

Comput. Sci. Eng.

P. Messmer, P. J. Mullowney, and B. E. Granger, “GPULib: GPU computing in high-level languages,” Comput. Sci. Eng. 10, 70–73 (2008).
[CrossRef]

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[CrossRef]

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F. C. Cheong, K. Xiao, and D. G. Grier, “Characterization of individual milk fat globules with holographic video microscopy,” J. Dairy Sci. 92, 95–99 (2009).
[CrossRef]

J. Opt. Soc. Am.

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[CrossRef]

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[CrossRef] [PubMed]

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J. Garcia-Sucerquia, J. H. Ramírez, and R. Castaneda, “Incoherent recovering of the spatial resolution in digital holography,” Opt. Commun. 260, 62–67 (2006).
[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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Supplementary Material (1)

» Media 1: MOV (663 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) In-line holographic video microscope. A collimated laser beam illuminates the sample. Light scattered by the sample interferes with the unscattered portion of the beam in the focal plane of the objective lens. The interference pattern is magnified, recorded and then analyzed to obtain measurements of the particle’s position. (b) Unprocessed hologram recorded by the video camera. (c) The corresponding normalized hologram.

Fig. 2.
Fig. 2.

Lorenz-Mie particle tracking and characterization. The upper image is the normalized hologram b(r) of a 1.51 µm diameter polystyrene sphere in water. The lower image is a fit to the Lorenz-Mie theory. The solid curve is the azimuthal average b(r) of the measured intensity around the center identified by the fit. Dashed curves indicate the azimuthal standard deviation of the hologram’s values, and indicates the measurement error. Plot points show the corresponding radial profiles of the fit. Error bars on the fit values are smaller than the plot symbols.

Fig. 3.
Fig. 3.

Rayleigh-Sommerfeld back-propagation. (a) Volumetric reconstruction of the scattered intensity due to a single colloidal sphere, colored by intensity. The diverging rays arise from the spurious mirror image of the sphere in the focal plane. (b) Volumetric reconstructions of 22 individual 1.58 µm diameter silica spheres organized in body center crystalline lattice with holographic optical tweezers in distilled water. Colored regions indicate the isosurface of the brightest 1 percent of reconstructed voxels. (Media 1)

Fig. 4.
Fig. 4.

Rayleigh-Sommerfeld back-propagation of aspherical objects. (a) Hologram of a colloidal silica sphere in water. (b) Detail of the brightest region of the volumetric reconstruction showing a symmetric structure. (c) Hologram of a silica sphere that is partially coated with 40 nm of permalloy. (d) Volumetric reconstruction showing asymmetric structure. Scale bars in (b) and (d) indicate 1 µm.

Fig. 5.
Fig. 5.

Comparison of Lorenz-Mie and Rayleigh-Sommerfeld particle-tracking algorithms. (a) Trajectory of a colloidal silica sphere at 1/30 s intervals obtained with the two strategies. Each point indicates the position of the sphere in one holographic snapshot as estimated by the Lorenz-Mie (LM) and Rayleigh-Sommerfeld (RS) approaches. (b) Distribution of differences Δx and Δy in the in-plane position estimated by the two strategies. (c) Mean difference Δz(z LM) in the axial position as a function of the Lorenz-Mie estimate z LM, obtained from 10,000 measurements. (d) Mean difference 〈Δz(ap )〉 in axial position as a function of sphere radius ap for polystyrene (PS) spheres (circles) and silica (SiO2) spheres (squares). Solid curves are predictions of Lorenz-Mie theory.

Equations (19)

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E 0 ( r , z ) = u 0 ( r ) e i k z ε ̂ 0 ,
E s ( r , z ) = E s ( r , z ) ε ̂ ( r , z )
I ( r ) = E 0 ( r , 0 ) + E s ( r , 0 ) 2
= u 0 2 ( r ) + 2 { u 0 ( r ) E s ( r , 0 ) ε ̂ 0 * · ε ̂ ( r , 0 ) } + E s ( r , 0 ) 2 .
b ( r ) 1 + 2 { E R ( r , 0 ) } + E R ( r , 0 ) 2 ,
E s ( r , 0 ) = u 0 ( r p ) f s ( k ( r r p ) ) ,
b ( r ) = 1 + 2 { f s ( k ( r r p ) ) · ε ̂ 0 } + f s ( k ( r r p ) ) 2 .
f s ( k r ) = Σ n = 1 n c i n ( 2 n + 1 ) n ( n + 1 ) ( i α n N e 1 n ( 3 ) ( k r ) β n M o 1 n ( 3 ) ( k r ) )
α n = m 2 j n ( m k a p ) [ k a p j n ( k a p ) ] j n ( k a p ) [ m k a p j n ( m k a p ) ] m 2 j n ( m k a p ) [ k a p h n ( 1 ) ( k a p ) ] h n ( 1 ) ( k a p ) [ m k a p j n ( m k a p ) ] and
β n = j n ( m k a p ) [ k a p j n ( k a p ) ] j n ( k a p ) [ m k a p j n ( m k a p ) ] j n ( m k a p ) [ k a p h n ( 1 ) ( k a p ) ] h n ( 1 ) ( k a p ) [ m k a p j n ( m k a p ) ] ,
b ( r ) 1 2 { E R ( r , 0 ) } .
E s ( r , z ) = E s ( r , 0 ) h z ( r )
h z ( r ) = 1 2 π z e ikR R ,
U ( q , z ) = U ( q , 0 ) H ( q , z ) ,
U ( q , z ) = E s ( r , z ) e i q · r d 2 r
H ( q , z ) = e iz ( k 2 q 2 ) 1 2 .
B ( q ) U R ( q , 0 ) + U R * ( q , 0 ) ,
B ( q ) H ( q , z ) = U R ( q , z ) + U R * ( q , z )
E s ( r , z ) e ikz 4 π 2 B ( q ) H ( q , z ) e i q · r d 2 q .

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