Abstract

Since their invention in 1986, optical tweezers have become a popular manipulation and force measurement tool in cellular and molecular biology. However, until recently there has not been a sophisticated model for optical tweezers on trapping cells in the ray-optics regime. We present a model for optical tweezers to calculate the optical force upon a spherically symmetric multilayer sphere representing a common biological cell. A numerical simulation of this model shows that not only is the magnitude of the optical force upon a Chinese hamster ovary cell significantly three times smaller than that upon a polystyrene bead of the same size, but the distribution of the optical force upon a cell is also much different from that upon a uniform particle, and there is a 30% difference in the optical trapping stiffness of these two cases. Furthermore, under a small variant condition for the refractive indices of any adjacent layers of the sphere, this model provides a simple approximation to calculate the optical force and the stiffness of an optical tweezers system.

© 2006 Optical Society of America

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References

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  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986).
    [CrossRef] [PubMed]
  2. S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
    [CrossRef] [PubMed]
  3. M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
    [CrossRef]
  4. M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
    [CrossRef]
  5. Y. Wakamoto, I. Inoue, H. Moriguchi, and K. Yasuda, "Analysis of single-cell differences by use of an on-chip microculture system and optical trapping," Fresenius J. Anal. Chem. 371, 276-281 (2001).
    [CrossRef] [PubMed]
  6. G. Hummer and A. Szabo, "Free-energy reconstruction from nonequilibrium single-molecule pulling experiments," Proc. Natl. Acad. Sci. U.S.A. 98, 3658-3661 (2001).
    [CrossRef] [PubMed]
  7. A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
    [CrossRef] [PubMed]
  8. Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
    [CrossRef]
  9. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
    [CrossRef]
  10. F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
    [CrossRef]
  11. A. Rohrbach and E. H. K. Stelzer, "Optical trapping of dielectric particles in arbitrary fields," J. Opt. Soc. Am. A 18, 839-153 (2001).
    [CrossRef]
  12. R. C. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13, 3707-3718 (2005).
    [CrossRef] [PubMed]
  13. Y.-R. Chang, L. Hsu, and S. Chi, "Optical trapping of a spherically symmetric Rayleigh sphere: a model for optical tweezers upon cells," Opt. Commun. 246, 97-105 (2005).
    [CrossRef]
  14. A. Brunsting and P. F. Mullaney, "Differential light scattering from spherical mammalian cells," Biophys. J. 14, 439-453 (1974).
    [CrossRef] [PubMed]
  15. M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
    [CrossRef] [PubMed]

2005 (2)

R. C. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13, 3707-3718 (2005).
[CrossRef] [PubMed]

Y.-R. Chang, L. Hsu, and S. Chi, "Optical trapping of a spherically symmetric Rayleigh sphere: a model for optical tweezers upon cells," Opt. Commun. 246, 97-105 (2005).
[CrossRef]

2003 (2)

M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
[CrossRef]

M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
[CrossRef]

2001 (4)

Y. Wakamoto, I. Inoue, H. Moriguchi, and K. Yasuda, "Analysis of single-cell differences by use of an on-chip microculture system and optical trapping," Fresenius J. Anal. Chem. 371, 276-281 (2001).
[CrossRef] [PubMed]

G. Hummer and A. Szabo, "Free-energy reconstruction from nonequilibrium single-molecule pulling experiments," Proc. Natl. Acad. Sci. U.S.A. 98, 3658-3661 (2001).
[CrossRef] [PubMed]

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, "Optical trapping of dielectric particles in arbitrary fields," J. Opt. Soc. Am. A 18, 839-153 (2001).
[CrossRef]

2000 (1)

M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
[CrossRef] [PubMed]

1996 (1)

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

1994 (1)

F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

1992 (1)

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

1986 (1)

1974 (1)

A. Brunsting and P. F. Mullaney, "Differential light scattering from spherical mammalian cells," Biophys. J. 14, 439-453 (1974).
[CrossRef] [PubMed]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Asakura, T.

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Ashkin, A.

Barlow, C.

M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Bhatia, S. N.

M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
[CrossRef]

Birkbeck, A.

M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
[CrossRef]

Bjorkholm, J. E.

Brunsting, A.

A. Brunsting and P. F. Mullaney, "Differential light scattering from spherical mammalian cells," Biophys. J. 14, 439-453 (1974).
[CrossRef] [PubMed]

Chang, Y.-R.

Y.-R. Chang, L. Hsu, and S. Chi, "Optical trapping of a spherically symmetric Rayleigh sphere: a model for optical tweezers upon cells," Opt. Commun. 246, 97-105 (2005).
[CrossRef]

Chi, S.

Y.-R. Chang, L. Hsu, and S. Chi, "Optical trapping of a spherically symmetric Rayleigh sphere: a model for optical tweezers upon cells," Opt. Commun. 246, 97-105 (2005).
[CrossRef]

Choi, I. S.

M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
[CrossRef] [PubMed]

Chu, S.

Dziedzic, J. M.

Esener, S.

M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
[CrossRef]

M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
[CrossRef]

Flynn, R.

M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
[CrossRef]

Gauthier, R. C.

R. C. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13, 3707-3718 (2005).
[CrossRef] [PubMed]

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

Gouesbet, G.

F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

Grehan, G.

F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

Grover, C. P.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

Grover, S. C.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

Harada, Y.

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Hsu, L.

Y.-R. Chang, L. Hsu, and S. Chi, "Optical trapping of a spherically symmetric Rayleigh sphere: a model for optical tweezers upon cells," Opt. Commun. 246, 97-105 (2005).
[CrossRef]

Hummer, G.

G. Hummer and A. Szabo, "Free-energy reconstruction from nonequilibrium single-molecule pulling experiments," Proc. Natl. Acad. Sci. U.S.A. 98, 3658-3661 (2001).
[CrossRef] [PubMed]

Inoue, I.

Y. Wakamoto, I. Inoue, H. Moriguchi, and K. Yasuda, "Analysis of single-cell differences by use of an on-chip microculture system and optical trapping," Fresenius J. Anal. Chem. 371, 276-281 (2001).
[CrossRef] [PubMed]

Liang, M. N.

M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
[CrossRef] [PubMed]

Metallo, S. J.

M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
[CrossRef] [PubMed]

Moriguchi, H.

Y. Wakamoto, I. Inoue, H. Moriguchi, and K. Yasuda, "Analysis of single-cell differences by use of an on-chip microculture system and optical trapping," Fresenius J. Anal. Chem. 371, 276-281 (2001).
[CrossRef] [PubMed]

Mullaney, P. F.

A. Brunsting and P. F. Mullaney, "Differential light scattering from spherical mammalian cells," Biophys. J. 14, 439-453 (1974).
[CrossRef] [PubMed]

Ozkan, C.

M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
[CrossRef]

Ozkan, M.

M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
[CrossRef]

M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
[CrossRef]

Pisanic, T.

M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
[CrossRef]

Prentiss, M.

M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
[CrossRef] [PubMed]

Ren, F.

F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

Rohrbach, A.

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Scheel, J.

M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
[CrossRef]

Skirtach, A. G.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

Smith, S. P.

M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
[CrossRef] [PubMed]

Stelzer, E. H. K.

Szabo, A.

G. Hummer and A. Szabo, "Free-energy reconstruction from nonequilibrium single-molecule pulling experiments," Proc. Natl. Acad. Sci. U.S.A. 98, 3658-3661 (2001).
[CrossRef] [PubMed]

Wakamoto, Y.

Y. Wakamoto, I. Inoue, H. Moriguchi, and K. Yasuda, "Analysis of single-cell differences by use of an on-chip microculture system and optical trapping," Fresenius J. Anal. Chem. 371, 276-281 (2001).
[CrossRef] [PubMed]

Wang, M.

M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
[CrossRef]

Whitesides, G. M.

M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
[CrossRef] [PubMed]

Yasuda, K.

Y. Wakamoto, I. Inoue, H. Moriguchi, and K. Yasuda, "Analysis of single-cell differences by use of an on-chip microculture system and optical trapping," Fresenius J. Anal. Chem. 371, 276-281 (2001).
[CrossRef] [PubMed]

Biomed. Microdevices (1)

M. Ozkan, M. Wang, C. Ozkan, R. Flynn, A. Birkbeck, and S. Esener, "Optical manipulation of objects and biological cells in microfluidic devices," Biomed. Microdevices 5, 61-67 (2003).
[CrossRef]

Biophys. J. (2)

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

A. Brunsting and P. F. Mullaney, "Differential light scattering from spherical mammalian cells," Biophys. J. 14, 439-453 (1974).
[CrossRef] [PubMed]

Fresenius J. Anal. Chem. (1)

Y. Wakamoto, I. Inoue, H. Moriguchi, and K. Yasuda, "Analysis of single-cell differences by use of an on-chip microculture system and optical trapping," Fresenius J. Anal. Chem. 371, 276-281 (2001).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

J. Biomed. Opt. (1)

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

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

Langmuir (1)

M. Ozkan, T. Pisanic, J. Scheel, C. Barlow, S. Esener, and S. N. Bhatia, "Electro-optical platform for the manipulation of live cells," Langmuir 19, 1532-1538 (2003).
[CrossRef]

Opt. Commun. (3)

F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz-Mie theory, and associated resonance effects," Opt. Commun. 108, 343-354 (1994).
[CrossRef]

Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Y.-R. Chang, L. Hsu, and S. Chi, "Optical trapping of a spherically symmetric Rayleigh sphere: a model for optical tweezers upon cells," Opt. Commun. 246, 97-105 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

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

M. N. Liang, S. P. Smith, S. J. Metallo, I. S. Choi, M. Prentiss, and G. M. Whitesides, "Measuring the forces involved in polyvalent adhesion of uropathogenic Escherichia coli to mannose-presenting surfaces," Proc. Natl. Acad. Sci. U.S.A. 97, 13092-13096 (2000).
[CrossRef] [PubMed]

G. Hummer and A. Szabo, "Free-energy reconstruction from nonequilibrium single-molecule pulling experiments," Proc. Natl. Acad. Sci. U.S.A. 98, 3658-3661 (2001).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Scattering schemes of (a) a single incident ray scattered by the proposed multilayer spherically symmetric sphere and (b) the photon fluxes between the ( k - 1 ) th , kth, and ( k + 1 ) th layers.

Fig. 2
Fig. 2

Schematic of a bundle of rays focused at the origin point and their relative locations with a multilayer sphere.

Fig. 3
Fig. 3

Magnitude and direction of the optical force upon a CHO cell in the XY and XZ planes.

Fig. 4
Fig. 4

Optical force versus the displacements of trapped uniform and nonuniform cells showing (a) the relationships between the x component of optical force and the displacements on the x axis of the trapped CHO cells, and (b) the ones between the z component of optical force and the displacements on the z axis. The gray curve and dashed curve represent the numerical results of nonuniform cells with the exact matrix method and with approximation under a small variant condition, respectively. The dashed-point curve represents the numerical result of a uniform virtual cell.

Tables (1)

Tables Icon

Table 1 Numerical Results of Stiffness a

Equations (23)

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

F opt = h λ med ( p i - p scat ) ,
p refl = p 0 R exp ( i Δθ refl )
p refr = p 0 ( 1 - R ) exp ( i Δθ refr ) ,
p s , k , k = p i , k , k R k exp [ i ( - π + 2 ϕ k ) ] + p i , k , k + 1 ( 1 - R k ) exp [ - i ( θ k - ϕ k ) ] ,
p s , k , k + 1 = p i , k , k ( 1 - R k ) exp [ - i ( θ k - ϕ k ) ] + p i , k , k + 1 R k exp [ i ( π - 2 θ k ) ] ,
sin θ k = r k + 1 r k sin ϕ k + 1 = r k + 1 r k n k + 2 n k + 1 sin θ k + 1 = r N r k n N + 1 n k + 1 sin θ N ,
sin ϕ k = n k + 1 n k sin θ k = r N r k n N + 1 n k sin θ N ,
[ p i , k + 1 p s , k + 1 ] = [ exp [ i ( θ k - ϕ k ) ] 1 - R k R k exp [ i ( θ k + ϕ k ) ] 1 - R k - R k exp [ - i ( θ k + ϕ k ) ] 1 - R k ( 1 - 2 R k ) exp [ - i ( θ k - ϕ k ) ] 1 - R k ] [ p i , k p s , k ] .
[ p i , N + 1 p s , N + 1 ] = [ exp [ i ( θ N - ϕ N ) ] 1 - R N R N exp [ i ( θ N + ϕ N ) ] 1 - R N - R N exp [ - i ( θ N + ϕ N ) ] 1 - R N ( 1 - 2 R N ) exp [ - i ( θ N + ϕ N ) ] 1 - R N ] [ exp [ i ( θ N - 1 - ϕ N - 1 ) ] 1 - R N - 1 R N - 1 exp [ i ( θ N - 1 + ϕ N - 1 ) ] 1 - R N - 1 - R N - 1 exp [ - i ( θ N - 1 + ϕ N - 1 ) ] 1 - R N - 1 ( 1 - 2 R N - 1 ) exp [ - i ( θ N - 1 - ϕ N - 1 ) ] 1 - R N - 1 ] [ exp [ i ( θ j - ϕ j ) ] 1 - R j R j exp [ i ( θ j + ϕ j ) ] 1 - R j - R N - 1 exp [ - i ( θ N - 1 + ϕ N - 1 ) ] 1 - R N - 1 ( 1 - 2 R j ) exp [ - i ( θ j - ϕ j ) ] 1 - R j ] [ p i , j p s , j ] = [ M 11 ( j ) M 12 ( j ) M 21 ( j ) M 22 ( j ) ] [ p i , j p s , j ] ,
r N r m - 1 n N + 1 n m sin θ N 1.
p s , N + 1 = M 21 ( m ) + M 22 ( m ) M 11 ( m ) + M 12 ( m ) p i , N + 1 .
F ray = p i , N + 1 h λ med [ Re ( 1 - M 21 ( m ) + M 22 ( m ) M 11 ( m ) + M 12 ( m ) ) z ^ - Im ( M 21 ( m ) + M 22 ( m ) M 11 ( m ) + M 12 ( m ) ) y ^ ] ,
p scat = p i { 1 + R exp ( - i 2 θ ) - ( 1 - R ) 2 exp [ - 2 i ( θ - ϕ ) ] 1 + R exp ( 2 i ϕ ) } ,
F ray , 1 = p i h λ med ( 1 + R cos 2 θ - ( 1 - R ) 2 [ cos ( 2 θ - 2 ϕ ) + R cos 2 θ ] 1 + R 2 + 2 R cos 2 ϕ ) z ^ + p i h λ med ( R sin 2 θ - ( 1 - R ) 2 [ sin ( 2 θ - 2 ϕ ) + R sin 2 θ ] 1 + R 2 + 2 r cos 2 ϕ ) y ^ .
[ p i , N + 1 p s , N + 1 ] [ exp [ i ( θ N - ϕ N ) ] 1 - R N R N exp [ i ( θ N + ϕ N ) ] 1 - R N - R N exp [ - i ( θ N + ϕ N ) ] 1 - R N ( 1 - 2 R N ) exp [ - i ( θ N - ϕ N ) ] 1 - R N ] [ exp ( i Δ ) 0 0 exp ( - i Δ ) ] [ p i , m p s , m ] ,
Δ = k = m N - 1 ( θ k - ϕ k ) - r m r N { d n med / [ r n 2 ( r ) ] } n ( r ) 1 - { d n med / [ r n ( r ) ] } 2 d r .
p s , N + 1 p i , N + 1 ( 1 - R N exp ( - i 2 θ N ) + ( 1 - R N ) 2 exp { - 2 i [ θ N - ( ϕ N - Δ ) ] } 1 + R N exp [ 2 i ( ϕ N - Δ ) ] ) ,
F ray p i , N + 1 h λ med ( 1 + R N cos 2 θ N - ( 1 - R N ) 2 { cos [ 2 θ N - 2 ( ϕ N - Δ ) ] + R N cos 2 θ N } 1 + R N 2 + 2 R N cos [ 2 ( ϕ N - Δ ) ] ) z ^ + p i , N + 1 h λ med ( R N  sin  2 θ N - ( 1 - R N ) 2 { sin [ 2 θ N - 2 ( ϕ N - Δ ) ] + R N  sin  2 θ N } 1 + R N 2 + 2 R N cos [ 2 ( ϕ N - Δ ) ] ) y ^ .
d = [ - x 0 ( cos 2 Θ + sin 2 Θ sin 2 Φ ) + y 0 ( sin 2 Θ cos Φ sin Φ ) + z 0 ( cos Θ sin Θ cos Φ ) ] x ^ + [ x 0 ( sin 2 Θ cos Φ sin Φ ) - y 0 ( cos 2 Θ + sin 2 Θ cos 2 Φ ) + z 0 ( cos Θ sin Θ sin Φ ) ] y ^ + [ x 0 ( cos Θ sin Θ cos Φ ) + y 0 ( cos Θ sin Θ sin Φ ) - z 0 sin 2 Θ ] z ^ .
F tot = 0 2 π 0 Θ m F ray [ p i ( Θ , Φ ) , z ^ ( Θ , Φ ) , d ( Θ , Φ ) , n ( r ) ] dΘdΦ ,
k x = k y = F tot , x x 0 | ( 0 , 0 , 0 ) = F tot , y y 0 | ( 0 , 0 , 0 ) ,
k z = F tot , z z 0 | ( 0 , 0 , 0 ) ,
r cell = ( 1.38 ± 0.02 ) r nuc + ( 0.03 ± 0.05 ) .

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