Abstract

We derive and apply formulas that employ the vector-wave addition theorem and rotation matrices for quantitative calculations of both radial and axial optical forces exerted on particles trapped in arbitrarily shaped tweezer beams. For the tightly focused beams encountered in optical tweezers, we shall highlight the importance of formulating the optical forces and beam symmetries in terms of the irradiance and total beam power. A major interest of the addition theorem treatment of optical forces is that it opens up the possibility of modeling a wide variety of beam shapes while automatically ensuring that the beams satisfy the Maxwell equations. In some of the first numerical applications of our method, we shall illustrate that resonance effects play an important role in the axial trapping position of particles comparable in size with the wavelength of the trapping beam.

© 2005 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. J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, "Entropic attraction and repulsion in binary colloids probed with a line optical tweezer," Phys. Rev. Lett. 82, 4352-4355 (1999).
    [CrossRef]
  3. L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
    [CrossRef]
  4. H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, "Optical trapping of three-dimensional structures using dynamic holograms," Opt. Express 1, 3562-3567 (2003).
    [CrossRef]
  5. G. Sinclair, J. Leach, P. Jordan, G. Gibson, E. Yao, Z. J. Laczik, M. J. Padgett, and J. Courtial, "Interactive application in holographic optical tweezers of a multiplane Gerchberg-Saxton algorithm for three-dimensional light shaping," Opt. Express 12, 1665-1670 (2004).
    [CrossRef] [PubMed]
  6. J. Leach, G. Sinclair, P. Jordan, J. Courtil, and M. J. Padgett, "3D manipulation of particles into crystal structures using holographic optical tweezers," Opt. Express 12, 220-226 (2004).
    [CrossRef] [PubMed]
  7. A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 2335, 1517-1520 (1987).
    [CrossRef]
  8. 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]
  9. T. Tlusty, A. Meller, and R. Bar-Ziv, "Optical gradient forces of strongly localized fields," Phys. Rev. Lett. 81, 1738-1741 (1998).
    [CrossRef]
  10. G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, "Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton," Biophys. J. 81, 43-56 (2001).
    [CrossRef] [PubMed]
  11. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  12. W. Chew, Waves and Fields in Inhomogeneous Media, Series on Electromagnetic Waves (IEEE, 1990).
  13. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  14. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  15. G. Gouesebet, J. A. Lock, and G. Gréhan, "Partial-wave representations of laser beams for use in light-scattering calculations," Appl. Opt. 34, 2133-2143 (1995).
    [CrossRef]
  16. B. Stout, J.-C. Auger, and J. Lafait, "Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity," J. Mod. Opt. 48, 2105-2128 (2001).
  17. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1957).
  18. P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material," Ann. Phys. (Leipzig) 30, 57-136 (1909).
    [CrossRef]
  19. M. I. Mishchenko, "Radiation force caused by scattering, absorption, and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 811-816 (2001).
    [CrossRef]
  20. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).
  21. S. Stein, "Addition theorems for spherical wave function," Q. Appl. Math. 19, 15-24 (1961).
  22. O. R. Cruzan, "Translation addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).
  23. B. Stout, J. C. Auger, and J. Lafait, "A transfer matrix approach to local field calculations in multiple-scattering problems," J. Mod. Opt. 49, 2129-2152 (2002).
    [CrossRef]

2004 (2)

2003 (2)

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, "Optical trapping of three-dimensional structures using dynamic holograms," Opt. Express 1, 3562-3567 (2003).
[CrossRef]

2002 (1)

B. Stout, J. C. Auger, and J. Lafait, "A transfer matrix approach to local field calculations in multiple-scattering problems," J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

2001 (3)

M. I. Mishchenko, "Radiation force caused by scattering, absorption, and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 811-816 (2001).
[CrossRef]

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, "Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton," Biophys. J. 81, 43-56 (2001).
[CrossRef] [PubMed]

B. Stout, J.-C. Auger, and J. Lafait, "Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity," J. Mod. Opt. 48, 2105-2128 (2001).

1999 (1)

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, "Entropic attraction and repulsion in binary colloids probed with a line optical tweezer," Phys. Rev. Lett. 82, 4352-4355 (1999).
[CrossRef]

1998 (1)

T. Tlusty, A. Meller, and R. Bar-Ziv, "Optical gradient forces of strongly localized fields," Phys. Rev. Lett. 81, 1738-1741 (1998).
[CrossRef]

1995 (1)

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]

1987 (1)

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 2335, 1517-1520 (1987).
[CrossRef]

1986 (1)

1979 (1)

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

1962 (1)

O. R. Cruzan, "Translation addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

1961 (1)

S. Stein, "Addition theorems for spherical wave function," Q. Appl. Math. 19, 15-24 (1961).

1909 (1)

P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material," Ann. Phys. (Leipzig) 30, 57-136 (1909).
[CrossRef]

Arit, J.

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

Ashkin, A.

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. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 2335, 1517-1520 (1987).
[CrossRef]

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]

Auger, J. C.

B. Stout, J. C. Auger, and J. Lafait, "A transfer matrix approach to local field calculations in multiple-scattering problems," J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

Auger, J.-C.

B. Stout, J.-C. Auger, and J. Lafait, "Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity," J. Mod. Opt. 48, 2105-2128 (2001).

Bar-Ziv, R.

T. Tlusty, A. Meller, and R. Bar-Ziv, "Optical gradient forces of strongly localized fields," Phys. Rev. Lett. 81, 1738-1741 (1998).
[CrossRef]

Bjorkholm, J. E.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Chew, W.

W. Chew, Waves and Fields in Inhomogeneous Media, Series on Electromagnetic Waves (IEEE, 1990).

Chu, S.

Courtial, J.

Courtil, J.

Crocker, J. C.

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, "Entropic attraction and repulsion in binary colloids probed with a line optical tweezer," Phys. Rev. Lett. 82, 4352-4355 (1999).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, "Translation addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Debye, P.

P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material," Ann. Phys. (Leipzig) 30, 57-136 (1909).
[CrossRef]

Dholakia, K.

H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, "Optical trapping of three-dimensional structures using dynamic holograms," Opt. Express 1, 3562-3567 (2003).
[CrossRef]

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

Dinsmore, A. D.

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, "Entropic attraction and repulsion in binary colloids probed with a line optical tweezer," Phys. Rev. Lett. 82, 4352-4355 (1999).
[CrossRef]

Dultz, W.

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

Dziedzic, J. M.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).

Gallet, F.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, "Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton," Biophys. J. 81, 43-56 (2001).
[CrossRef] [PubMed]

Gibson, G.

Gouesebet, G.

Gréhan, G.

Hénon, S.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, "Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton," Biophys. J. 81, 43-56 (2001).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Jordan, P.

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Laczik, Z. J.

Lafait, J.

B. Stout, J. C. Auger, and J. Lafait, "A transfer matrix approach to local field calculations in multiple-scattering problems," J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, "Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity," J. Mod. Opt. 48, 2105-2128 (2001).

Leach, J.

Lenormand, G.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, "Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton," Biophys. J. 81, 43-56 (2001).
[CrossRef] [PubMed]

Lock, J. A.

MacDonald, M. P

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

Matteo, J. A.

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, "Entropic attraction and repulsion in binary colloids probed with a line optical tweezer," Phys. Rev. Lett. 82, 4352-4355 (1999).
[CrossRef]

McGloin, D.

H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, "Optical trapping of three-dimensional structures using dynamic holograms," Opt. Express 1, 3562-3567 (2003).
[CrossRef]

Meller, A.

T. Tlusty, A. Meller, and R. Bar-Ziv, "Optical gradient forces of strongly localized fields," Phys. Rev. Lett. 81, 1738-1741 (1998).
[CrossRef]

Melville, H.

H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, "Optical trapping of three-dimensional structures using dynamic holograms," Opt. Express 1, 3562-3567 (2003).
[CrossRef]

Milne, G. F.

H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, "Optical trapping of three-dimensional structures using dynamic holograms," Opt. Express 1, 3562-3567 (2003).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, "Radiation force caused by scattering, absorption, and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 811-816 (2001).
[CrossRef]

Padgett, M. J.

Paterson, L.

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

Richert, A.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, "Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton," Biophys. J. 81, 43-56 (2001).
[CrossRef] [PubMed]

Schmitzer, H.

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Sibbett, W.

H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, "Optical trapping of three-dimensional structures using dynamic holograms," Opt. Express 1, 3562-3567 (2003).
[CrossRef]

Sibett, W.

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

Siméon, J.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, "Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton," Biophys. J. 81, 43-56 (2001).
[CrossRef] [PubMed]

Sinclair, G.

Spalding, G. C.

H. Melville, G. F. Milne, G. C. Spalding, W. Sibbett, K. Dholakia, and D. McGloin, "Optical trapping of three-dimensional structures using dynamic holograms," Opt. Express 1, 3562-3567 (2003).
[CrossRef]

Stein, S.

S. Stein, "Addition theorems for spherical wave function," Q. Appl. Math. 19, 15-24 (1961).

Stout, B.

B. Stout, J. C. Auger, and J. Lafait, "A transfer matrix approach to local field calculations in multiple-scattering problems," J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J.-C. Auger, and J. Lafait, "Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity," J. Mod. Opt. 48, 2105-2128 (2001).

Tlusty, T.

T. Tlusty, A. Meller, and R. Bar-Ziv, "Optical gradient forces of strongly localized fields," Phys. Rev. Lett. 81, 1738-1741 (1998).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1957).

Yao, E.

Yodh, A. G.

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, "Entropic attraction and repulsion in binary colloids probed with a line optical tweezer," Phys. Rev. Lett. 82, 4352-4355 (1999).
[CrossRef]

Ann. Phys. (1)

P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material," Ann. Phys. (Leipzig) 30, 57-136 (1909).
[CrossRef]

Appl. Opt. (1)

Biophys. J. (2)

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, "Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton," Biophys. J. 81, 43-56 (2001).
[CrossRef] [PubMed]

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]

J. Mod. Opt. (3)

B. Stout, J.-C. Auger, and J. Lafait, "Individual and aggregate scattering matrices and cross-sections: conservation laws and reciprocity," J. Mod. Opt. 48, 2105-2128 (2001).

B. Stout, J. C. Auger, and J. Lafait, "A transfer matrix approach to local field calculations in multiple-scattering problems," J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

L. Paterson, M. P MacDonald, J. Arit, W. Dultz, H. Schmitzer, W. Sibett, and K. Dholakia, "Controlled simultaneous rotation of multiple optically trapped particles," J. Mod. Opt. 50, 1591-1601 (2003).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

M. I. Mishchenko, "Radiation force caused by scattering, absorption, and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 811-816 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Phys. Rev. Lett. (2)

T. Tlusty, A. Meller, and R. Bar-Ziv, "Optical gradient forces of strongly localized fields," Phys. Rev. Lett. 81, 1738-1741 (1998).
[CrossRef]

J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, "Entropic attraction and repulsion in binary colloids probed with a line optical tweezer," Phys. Rev. Lett. 82, 4352-4355 (1999).
[CrossRef]

Q. Appl. Math. (2)

S. Stein, "Addition theorems for spherical wave function," Q. Appl. Math. 19, 15-24 (1961).

O. R. Cruzan, "Translation addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

Science (1)

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 2335, 1517-1520 (1987).
[CrossRef]

Other (5)

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

W. Chew, Waves and Fields in Inhomogeneous Media, Series on Electromagnetic Waves (IEEE, 1990).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1960).

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1957).

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Figures (4)

Fig. 1
Fig. 1

Focal-plane irradiance and intensity of a y ̂ -polarized fifth-order Davis beam with s = 1 π .

Fig. 2
Fig. 2

Axial trapping efficiency, Q p ( z ) , for spheres of radii R = 2.1 ( μ m ) composed of silica ( n s = 1.46 ) and latex ( n s = 1.59 ) immersed in water ( n b = 1.32 ) and exposed to a y ̂ -polarized fifth-order Davis beam.

Fig. 3
Fig. 3

Trapping positions for silica and latex spheres in water in a y ̂ -polarized, s = 1 π , fifth-order Davis beam, as a function of the sphere radius.

Fig. 4
Fig. 4

(a) Radial trapping efficiency, Q ρ , for a R = 2.1 ( μ m ) silica as a function of the radial displacements along the x ̂ and y ̂ axes at the trapping position, z tr 0.75 ( μ m ) . (b) Radial trapping efficiency, Q ρ , for x ̂ and y ̂ axis displacements of a R = 0.2 ( μ m ) silica sphere evaluated at its trapping position z tr 0.43 ( μ m ) .

Equations (84)

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

× × E ( r ) k 2 E ( r ) = 0 ,
M n m ( k r ) h n ( k r ) X n m ( θ , ϕ ) ,
N n m ( k r ) 1 k r { n ( n + 1 ) h n ( k r ) Y n m ( θ , ϕ ) + [ k r h n ( k r ) ] Z n m ( θ , ϕ ) } ,
X n m ( θ , ϕ ) Z n m ( θ , ϕ ) × r ̂ = γ n m C n m ( θ , ϕ ) ,
Y n m ( θ , ϕ ) r ̂ Y n m ( θ , ϕ ) γ n m n ( n + 1 ) P n m ( θ , ϕ ) ,
Z n m ( θ , ϕ ) r Y n m ( θ , ϕ ) n ( n + 1 ) = r ̂ × X n m ( θ , ϕ ) = γ n m B n m ( θ , ϕ ) ,
E i ( r ) = A n , m R g { [ M n , m ( k r ) ] } a n , m M + R g { [ N n , m ( k r ) ] } a n , m N A R g { [ M ( k r ) , N ( k r ) ] } a A R g { Ψ t ( k r ) } a ,
Ψ ( k r ) = [ M 1 , 1 ( k r ) , M 1 , 0 ( k r ) , M 1 , 1 ( k r ) , , N 1 , 1 ( k r ) , N 1 , 0 ( k r ) , N 1 , 1 ( k r ) , ] .
H i ( r ) = 1 i ω μ b μ 0 × E i ( r ) = A i ω μ b μ 0 R g { × Ψ t ( k r ) } a = A ( ε b ϵ 0 μ b μ 0 ) 1 2 R g { [ N ( k r ) , M ( k r ) ] } a ,
[ p ] n m M 4 π X n m ( k ̂ i ) e ̂ i 4 π i n X n m * ( k ̂ i ) e ̂ i ,
[ p ] n m N = 4 π Z n m ( k ̂ i ) e ̂ i 4 π i n 1 Z n m * ( k ̂ i ) e ̂ i ,
A 2 2 S i ( 0 ) ( μ b μ 0 ε b ϵ 0 ) 1 2 ,
Re { [ a ] 1 , 1 N , * [ a ] 1 , 1 M [ a ] 1 , 1 N , * [ a ] 1 , 1 M } = 6 π .
u ̂ i S i ( 0 ) S i ( 0 ) ,
I ( r ) S i ( r ) u ̂ i ,
P i = I ( x , y , z ) d x d y z = const . = 0 ρ d ρ 0 2 π d ϕ I ( ρ , ϕ , z ) z = const . ,
[ a ] n , m A = [ g ] n [ p ] n , m A , A = M , N ,
s 1 k w 0 w 0 2 z R tan θ d 2 ,
[ p ] n , 1 M 4 π X n 1 ( 0 , 0 ) e ̂ i = i n π ( 2 n + 1 ) ( i θ ̂ + ϕ ̂ ) e ̂ i ,
[ p ] n , 1 N 4 π Z n 1 ( 0 , 0 ) e ̂ i = i n π ( 2 n + 1 ) ( i θ ̂ + ϕ ̂ ) e ̂ i ,
[ p ] n , 1 M = i n π ( 2 n + 1 ) ( i θ ̂ ϕ ̂ ) e ̂ i ,
[ p ] n , 1 N = i n π ( 2 n + 1 ) ( i θ ̂ + ϕ ̂ ) e ̂ i ,
I ( 0 ) φ k 2 P i π ,
P i = | 2 π k 2 0 I ( r ) k ρ ( k d ρ ) z = const . .
φ = [ 2 0 I ( r ) I ( 0 ) k ρ ( k d ρ ) z = const . ] 1 .
I g ( r ) = I g ( 0 ) [ w 0 w ( z ) ] 2 exp [ 2 ρ 2 w 2 ( z ) ] ;
w ( z ) = w 0 [ 1 + ( z z R ) 2 ] 1 2 ,
P i g = π 2 w 0 2 I g ( 0 ) = π 2 1 k 2 s 2 I g ( 0 ) ,
φ = { p , q = 0 ( N max 1 ) 2 [ g ] 2 p + 1 [ g ] 2 q + 1 ( 4 p + 3 ) ( 4 q + 3 ) ( 2 p 1 ) ! ! ( 2 p + 2 ) ! ! ( 2 q 1 ) ! ! ( 2 q + 2 ) ! ! ( 1 ) p q + 2 p = 1 N max 2 q = 0 ( N max 1 ) 2 [ g ] 2 p [ g ] 2 q + 1 ( 2 q + 1 ) ! ! ( 2 q ) ! ! ( 2 p 1 ) ! ! ( 2 p ) ! ! ( 4 p + 1 ) ( 4 q + 3 ) 2 p ( 2 p + 1 ) ( 2 q + 1 ) ( 2 q + 2 ) ( 1 ) p q + 1 } 1 .
[ g 1 ] n = exp [ s 2 ( n 1 ) ( n + 2 ) ] ,
[ g 3 ] n = [ g 1 ] n + exp [ s 2 ( n 1 ) ( n + 2 ) ] ( n 1 ) ( n + 2 ) s 4 [ 3 ( n 1 ) ( 2 + n ) s 2 ] ,
[ g 5 ] n = [ g 3 ] n + exp [ s 2 ( n 1 ) ( n + 2 ) ] { ( n 1 ) 2 ( n + 2 ) 2 s 8 [ 10 5 ( n 1 ) ( 2 + n ) s 2 + 0.5 ( n 1 ) 2 ( n + 2 ) 2 s 4 ] } ,
σ s ( θ i , ϕ i ) = 0 π sin θ d θ 0 2 π d ϕ d σ s ( θ , ϕ , θ i , ϕ i ) d Ω ,
d σ s ( θ , ϕ , θ i , ϕ i ) d Ω lim r r 2 r ̂ S s ( r ) I ,
d σ s ( x ) d Ω lim r r 2 r ̂ S s ( r ) I ( 0 ) = lim r r 2 2 ( ε b ϵ 0 μ b μ 0 ) 1 2 E s * ( r ) E s ( r ) I ( 0 ) .
E s ( r ) = A R g { Ψ t [ k ( r x ) ] } f ( x ) ,
σ s ( x ) = 1 k 2 f ( x ) f ( x ) .
f ( x ) = T a ( x ) ,
a ( x ) = J ( k x ) a ,
σ s ( x ) = 1 k 2 a J ( k x ) T T J ( k x ) a .
P s ( x ) = P i k 2 π φ σ s ( x ) P i Q s ( x ) ,
Q s ( x ) k 2 φ π σ s ( x ) = φ π a J ( k x ) T T J ( k x ) a ,
F = Γ T n ̂ d s ,
T i , j ( r ) = 1 2 Re { ε b ϵ 0 E i * ( r ) E j ( r ) + 1 μ b μ 0 B i * ( r ) B j ( r ) 1 2 [ ε b ϵ 0 E ( r ) 2 + B ( r ) 2 μ b μ 0 ] δ i , j } .
F I ( 0 ) v b σ f ( r ) = P i v b k 2 φ π σ f ( r ) P i v b Q f ( r ) ,
σ f = σ r σ a ,
σ a = v b 4 I ( 0 ) Ω r ̂ { ε b ϵ 0 E s * E s + 1 μ b μ 0 B s * B s } d Ω = r 1 2 I ( 0 ) r 2 ( ε b ϵ 0 μ b μ 0 ) 1 2 Ω r ̂ E s * E s d Ω .
B s ( r ) = r ( ε b ϵ 0 μ b μ 0 ) 1 2 r ̂ × E s .
σ r = r v b 4 I ( 0 ) r 2 { ε b ϵ 0 Ω r ̂ Re { E s * E i + E i * E s } d Ω + 1 μ b μ 0 Ω r ̂ Re { B s * B i + B i * B s } d Ω } = 1 2 I ( 0 ) lim r r 2 ( ε b ϵ 0 μ b μ 0 ) 1 2 Ω r ̂ Re { E s * E i + E i * E s } d Ω ,
σ p z ̂ σ r z ̂ σ a σ r g σ s ,
g 1 σ s z ̂ σ a = 1 2 I ( 0 ) σ s ( ε b ϵ 0 μ b μ 0 ) 1 2 lim r r 2 Ω z ̂ r ̂ E s * E s d Ω = 1 k 2 σ s f ( x ) { d Ω cos θ [ X ( θ , ϕ ) Z ( θ , ϕ ) ] [ X * ( θ , ϕ ) , Z * ( θ , ϕ ) ] } f ( x ) = 1 σ s k 2 f ( x ) Υ f ( x ) ,
Υ = [ Ξ ϴ ϴ Ξ ] ,
ϴ n m , ν μ = m n ( n + 1 ) δ m , μ δ n , ν ,
Ξ n m , ν μ = δ m , μ i ( 2 n + 1 ) ( 2 ν + 1 ) [ δ ν , n 1 n ( n 2 1 ) ( n 2 m 2 ) δ ν , n + 1 ν ( ν 2 1 ) ( ν 2 m 2 ) ] .
σ r ( x ) z ̂ σ r = 1 2 I ( 0 ) ( ε b ϵ 0 μ b μ 0 ) 1 2 lim r r 2 Ω z ̂ r ̂ Re { E s * E i + E i * E s } d Ω = 1 k 2 Re { a ( x ) Υ f ( x ) } .
σ p ( x ) = 1 k 2 Re { a J ( k x ) Υ T J ( k x ) a } 1 k 2 a J ( k x ) T Υ T J ( k x ) a ,
Q p ( x ) = φ π Re { a J ( k x ) Υ T J ( k x ) a } φ π a J ( k x ) T Υ T J ( k x ) a ,
F p = P i v b Q p ( x ) .
1 v b n b 3 10 8 N W = n b 10 3 pN mW 4.4 pN mW ,
σ r σ e = 1 k 2 Re { p T p } .
F ρ ρ ̂ F = P i v b ρ ̂ Q f = P i v b ρ ̂ ( Q r Q a ) .
ρ ̂ Q a = k 2 φ 2 π I ( 0 ) ( ε b ϵ 0 μ b μ 0 ) 1 2 lim r r 2 Ω ( ρ ̂ r ̂ ) E s * E s d Ω = φ π f ( x ) D ( ϕ , π 2 , 0 ) { d Ω cos θ [ X ( θ , ϕ ) Z ( θ , ϕ ) ] [ X * ( θ , ϕ ) , Z * ( θ , ϕ ) ] } D ( ϕ , π 2 , 0 ) f ( x ) ,
D ( α , β , γ ) [ D ( α , β , γ ) 0 0 D ( α , β , γ ) ] ,
[ D ( α , β , γ ) ] ν μ , n m = δ n , ν exp ( i μ α ) d μ m ( n ) ( β ) exp ( i m γ ) .
Λ ( θ , ϕ ) D ( ϕ , θ , 0 ) Υ D ( ϕ , θ , 0 ) ,
Q ρ ρ ̂ Q f = φ π Re [ a J ( k x ) T Λ ( π 2 , ϕ ) J ( k x ) a + a J ( k x ) Λ ( π 2 , ϕ ) T J ( k x ) a ] φ π a J ( k x ) T Λ ( π 2 , ϕ ) T J ( k x ) a ,
Y n m ( θ , ϕ ) = γ n m n ( n + 1 ) P n m ( cos θ ) exp ( i m ϕ ) r ̂ Y n m ( θ , ϕ ) r ̂ ,
X n m ( θ , ϕ ) = γ n m [ i m sin θ P n m ( cos θ ) θ ̂ d d θ P n m ( cos θ ) ϕ ̂ ] exp ( i m ϕ ) [ i u ¯ n m ( cos θ ) θ ̂ s ¯ n m ( cos θ ) ϕ ̂ ] exp ( i m ϕ ) ,
Z n m ( θ , ϕ ) = γ n m [ d d θ P n m ( cos θ ) θ ̂ + i m sin θ P n m ( cos θ ) ϕ ̂ ] exp ( i m ϕ ) [ s ¯ n m ( cos θ ) θ ̂ + i u ¯ n m ( cos θ ) ϕ ̂ ] exp ( i m ϕ ) ,
γ n m [ ( 2 n + 1 ) ( n m ) ! 4 π n ( n + 1 ) ( n + m ) ! ] 1 2 .
R g { Ψ ( k r ) } = R g { Ψ t ( k r ) } J ( k x ) , r j ,
J ( k x ) [ A ¯ ( k x ) B ¯ ( k x ) B ¯ ( k x ) A ¯ ( k x ) ] .
d μ m ( n ) ( β ) = [ ( n + μ ) ! ( n μ ) ! ( n + m ) ! ( n m ) ] 1 2 ( cos β 2 ) m + μ ( sin β 2 ) m μ P n μ ( μ m , m + μ ) ( cos β ) .
I ( r ) = 1 2 Re { E i * × H i } z ̂ = 1 2 ( ε b ϵ 0 μ b μ 0 ) 1 2 A 2 n , m ; ν μ Re { i [ a n m M , * R g { M n m * ( k r ) } + a n m N , * R g { N n m * ( k r ) } ] × [ a ν μ M R g { N ν μ ( k r ) } + a ν μ N R g { M ν μ ( k r ) } ] z ̂ } .
δ P NM = 0 r d r 0 2 π d ϕ Re { i [ a n m N , * a ν μ M R g { N n m * ( k r ) } × R g { N ν μ ( k r ) } ] } z ̂ θ = π 2 = A 2 π k 2 ( ε b ϵ 0 μ b μ 0 ) 1 2 n , ν , m = ± 1 Re { [ a ] n m N * [ a ] ν m M } u ¯ n m ( 0 ) u ¯ ν m ( 0 ) m × { n ( n + 1 ) 0 j n ( x ) x [ x j ν ( x ) ] d x + ν ( ν + 1 ) 0 [ x j n ( x ) ] j ν ( x ) x d x } ,
u ¯ n m ( 0 ) u ¯ ν m ( 0 ) = ( 1 ) ( n ν ) 2 4 π ( 2 n + 1 ) ( 2 ν + 1 ) ( n 2 ) ! ! ( n + 1 ) ! ! ( ν 2 ) ! ! ( ν + 1 ) ! ! .
n ( n + 1 ) 0 j n ( x ) x [ x j ν ( x ) ] d x + ν ( ν + 1 ) 0 [ x j n ( x ) ] j ν ( x ) x d x
= ( 1 ) ( n ν ) 2 ,
δ P NM = I ( 0 ) 2 k 2 p , q = 0 ( N max 1 ) 2 m = ± 1 Re [ { [ a ] 2 p + 1 , m N , * [ a ] 2 q + 1 , m M } ] m × ( 4 p + 3 ) ( 2 q + 3 ) ( 2 p 1 ) ! ! ( 2 p + 2 ) ! ! ( 2 q 1 ) ! ! ( 2 q + 2 ) ! ! ,
δ P NM = I ( 0 ) π k 2 p , q = 0 ( N max 1 ) 2 g 2 p + 1 g 2 q + 1 ( 4 p + 3 ) ( 4 q + 3 ) ( 2 p 1 ) ! ! ( 2 p + 2 ) ! ! ( 2 q 1 ) ! ! ( 2 q + 2 ) ! ! ( 1 ) p q .
R g { N 1 m * ( k r ) } × R g { N 1 μ ( k r ) } z ̂ = 2 j 1 ( k r ) [ k r j 1 ( k r ) ] Y 1 m * ( r ̂ ) X 1 μ ( r ̂ ) k 2 r 2 2 [ k r j 1 ( k r ) ] j 1 ( k r ) Y 1 μ ( r ̂ ) X 1 m * ( r ̂ ) k 2 r 2 [ ( k r j ν ( k r ) ] ) 2 Z n m * ( θ , ϕ ) X ν μ ( θ , ϕ ) r ̂ k 2 r 2 .
lim x 0 j 1 ( x ) x 3 , lim x 0 [ x j ν ( x ) ] 2 x 3 ,
1 2 π 0 2 π d ϕ m , μ = 1 1 a 1 m N , * a 1 μ M R g { N 1 m * ( 0 ) } × R g { N 1 μ ( 0 ) } z ̂ θ = π 2 = i 8 9 m = ± 1 m a 1 , m N , * a 1 , m M u ¯ 1 m ( 0 ) u ¯ 1 m ( 0 ) = i 6 π { a 1 , 1 N , * a 1 , 1 M a 1 , 1 N , * a 1 , 1 M } ,
I ( 0 ) = 1 2 ( ε b ϵ 0 μ b μ 0 ) 1 2 A 2 1 6 π Re { a 1 , 1 N , * a 1 , 1 M a 1 , 1 N , * a 1 , 1 M } = S ( 0 ) 6 π Re { a 1 , 1 N , * a 1 , 1 M a 1 , 1 N , * a 1 , 1 M } ,

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