Abstract

On the basis of generalized Lorenz–Mie theory, the Debye series expansion (DSE) for radiation pressure forces (RPF) exerted on a multilayered sphere induced by focused beams is introduced. The DSE can isolate the contribution of each scattering process to RPF, and give a physical explanation of RPF. Typically, the RPF induced by a Gaussian beam is studied. The DSE is employed to the simulation of RPF corresponding to different scattering processes (diffraction, reflection, refraction, etc.) in detail, and gives the physical mechanism of RPF. The effects of various parameters, such as scattering mode p, beam position, and radius of core for coated spheres, to RPF is researched.

© 2010 Optical Society of America

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  2. A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283-285 (1971).
    [CrossRef]
  3. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517-1520 (1987).
    [CrossRef] [PubMed]
  4. 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]
  5. A. Ashkin and J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803-1814 (1981).
    [CrossRef] [PubMed]
  6. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
    [CrossRef]
  7. P. Domachuk, M. Cronin-Golomb, B. Eggleton, S. Mutzenich, G. Rosengarten, and A. Mitchell, “Application of optical trapping to beam manipulation in optofluidics,” Opt. Express 13, 7265-7275 (2005).
    [CrossRef] [PubMed]
  8. V. Bormuth, A. Jannasch, M. Ander, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, “Optical trapping of coated microspheres,” Opt. Express 16, 13831-13844 (2008).
    [CrossRef] [PubMed]
  9. J. Harris and G. McConnell, “Optical trapping and manipulation of live T cells with a low numerical aperture lens,” Opt. Express 16, 14036-14043 (2008).
    [CrossRef] [PubMed]
  10. D. C. Appleyard and M. J. Lang, “Optical trapping of cells and control of multiple particles through silicon,” Biophys. J. 93, 651a (2007).
  11. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769-771 (1987).
    [CrossRef] [PubMed]
  12. K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35, 2702-2710 (1996).
    [CrossRef] [PubMed]
  13. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108, 343-354 (1994).
    [CrossRef]
  14. G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. 59, 6-8 (1976).
    [CrossRef]
  15. G. Roosen, “A theoretical and experimental study of the stable equilibrium position of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189-194 (1977).
    [CrossRef]
  16. 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]
  17. R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922-1930 (1992).
    [CrossRef]
  18. W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735-1748 (1994).
    [CrossRef] [PubMed]
  19. W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715-717 (1993)
    [CrossRef]
  20. P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273-285 (1998).
    [CrossRef]
  21. K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930-932.
    [CrossRef] [PubMed]
  22. P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
    [CrossRef] [PubMed]
  23. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427-1443(1988).
    [CrossRef]
  24. G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998-1003 (1990).
    [CrossRef]
  25. G. Gouesbet and J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516-2525 (1994).
    [CrossRef]
  26. G. Gréhan, B. Maheu, and G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539-3548 (1986).
    [CrossRef] [PubMed]
  27. J. A. Lock and G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503-2515 (1994).
    [CrossRef]
  28. B. Maheu, G. Gréhan, and G. Gouesbet, “Generalized Lorenz-Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23-25 (1987).
    [CrossRef] [PubMed]
  29. K. F. Ren, G. Gouesbet, and G. Gréhan, “The integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218-4225 (1998).
    [CrossRef]
  30. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532-2544(2004).
    [CrossRef] [PubMed]
  31. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force,” Appl. Opt. 43, 2545-2554 (2004).
    [CrossRef] [PubMed]
  32. 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]
  33. X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).
  34. Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).
  35. F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113-7124 (1995).
    [CrossRef] [PubMed]
  36. H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169-179 (1998).
    [CrossRef]
  37. P. Debije, “Das elektromagnetische feld um einen zylinder und die theorie des regenbogens,” Phys. Z. 9, 775-778 (1908).
  38. E. A. Hovenac and J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781-795 (1992).
    [CrossRef]
  39. J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677-4690 (1994).
    [CrossRef] [PubMed]
  40. R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series for light scattering by a multilayered sphere,” Appl. Opt. 45, 1260-1270 (2006).
    [CrossRef] [PubMed]
  41. R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series of normally incident plane wave scattering by an infinite multi-layered cylinder,” Appl. Opt. 45, 6255-6262 (2006).
    [CrossRef] [PubMed]
  42. J. A. Lock and C. L. Adler, “Debye-series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1316-1328 (1997).
    [CrossRef]
  43. R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602 (2009).
    [CrossRef]
  44. J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594-5603 (2005).
    [CrossRef] [PubMed]
  45. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  46. R. Li, X. Han, L. Shi, K. F. Ren, and H. Jiang, “Debye series for Gaussian beam scattering by a multilayered sphere,” Appl. Opt. 46, 4804-4812 (2007).
    [CrossRef] [PubMed]
  47. K. F. Ren, G. Gréhan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072-2079 (1994).
    [CrossRef]
  48. F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
    [CrossRef]

2009

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602 (2009).
[CrossRef]

2008

2007

R. Li, X. Han, L. Shi, K. F. Ren, and H. Jiang, “Debye series for Gaussian beam scattering by a multilayered sphere,” Appl. Opt. 46, 4804-4812 (2007).
[CrossRef] [PubMed]

F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

D. C. Appleyard and M. J. Lang, “Optical trapping of cells and control of multiple particles through silicon,” Biophys. J. 93, 651a (2007).

2006

2005

2004

2002

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

2000

X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).

1998

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169-179 (1998).
[CrossRef]

P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273-285 (1998).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “The integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

1997

1996

1995

1994

1993

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715-717 (1993)
[CrossRef]

1992

1990

1989

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]

1988

1987

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769-771 (1987).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, and G. Gouesbet, “Generalized Lorenz-Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23-25 (1987).
[CrossRef] [PubMed]

1986

1981

1979

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

1977

G. Roosen, “A theoretical and experimental study of the stable equilibrium position of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189-194 (1977).
[CrossRef]

1976

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. 59, 6-8 (1976).
[CrossRef]

1971

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283-285 (1971).
[CrossRef]

1970

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

1908

P. Debije, “Das elektromagnetische feld um einen zylinder und die theorie des regenbogens,” Phys. Z. 9, 775-778 (1908).

Adler, C. L.

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]

Ander, M.

Appleyard, D. C.

D. C. Appleyard and M. J. Lang, “Optical trapping of cells and control of multiple particles through silicon,” Biophys. J. 93, 651a (2007).

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 235, 1517-1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769-771 (1987).
[CrossRef] [PubMed]

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]

A. Ashkin and J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803-1814 (1981).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283-285 (1971).
[CrossRef]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156-159 (1970).
[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]

Berns, M. W.

W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735-1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715-717 (1993)
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930-932.
[CrossRef] [PubMed]

Bormuth, V.

Brevik, I.

Cai, X.

F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

Chaumet, P. C.

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

Chen, J.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Cheng, B.

X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).

Chu, S.

Cronin-Golomb, M.

Cui, G.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Davis, L. W.

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

Debije, P.

P. Debije, “Das elektromagnetische feld um einen zylinder und die theorie des regenbogens,” Phys. Z. 9, 775-778 (1908).

Domachuk, P.

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769-771 (1987).
[CrossRef] [PubMed]

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]

A. Ashkin and J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803-1814 (1981).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283-285 (1971).
[CrossRef]

Eggleton, B.

Gouesbet, G.

F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “The integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169-179 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35, 2702-2710 (1996).
[CrossRef] [PubMed]

F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113-7124 (1995).
[CrossRef] [PubMed]

G. Gouesbet and J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516-2525 (1994).
[CrossRef]

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

K. F. Ren, G. Gréhan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072-2079 (1994).
[CrossRef]

J. A. Lock and G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503-2515 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998-1003 (1990).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427-1443(1988).
[CrossRef]

B. Maheu, G. Gréhan, and G. Gouesbet, “Generalized Lorenz-Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23-25 (1987).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, and G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539-3548 (1986).
[CrossRef] [PubMed]

Gréhan, G.

F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “The integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169-179 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35, 2702-2710 (1996).
[CrossRef] [PubMed]

F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113-7124 (1995).
[CrossRef] [PubMed]

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

K. F. Ren, G. Gréhan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072-2079 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998-1003 (1990).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427-1443(1988).
[CrossRef]

B. Maheu, G. Gréhan, and G. Gouesbet, “Generalized Lorenz-Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23-25 (1987).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, and G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539-3548 (1986).
[CrossRef] [PubMed]

Guo, H.

X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).

Gussgard, R.

Han, X.

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602 (2009).
[CrossRef]

R. Li, X. Han, L. Shi, K. F. Ren, and H. Jiang, “Debye series for Gaussian beam scattering by a multilayered sphere,” Appl. Opt. 46, 4804-4812 (2007).
[CrossRef] [PubMed]

R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series for light scattering by a multilayered sphere,” Appl. Opt. 45, 1260-1270 (2006).
[CrossRef] [PubMed]

R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series of normally incident plane wave scattering by an infinite multi-layered cylinder,” Appl. Opt. 45, 6255-6262 (2006).
[CrossRef] [PubMed]

X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).

Harris, J.

Hovenac, E. A.

Howard, J.

Imbert, C.

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. 59, 6-8 (1976).
[CrossRef]

Jamison, J. M.

Jannasch, A.

Jiang, H.

Jonáš, A.

P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273-285 (1998).
[CrossRef]

Lang, M. J.

D. C. Appleyard and M. J. Lang, “Optical trapping of cells and control of multiple particles through silicon,” Biophys. J. 93, 651a (2007).

Li, R.

Li, Y.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Li, Z.

X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).

Lin, C. Y.

Lindmo, T.

Liška, M.

P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273-285 (1998).
[CrossRef]

Liu, H.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Lock, J. A.

J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594-5603 (2005).
[CrossRef] [PubMed]

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532-2544(2004).
[CrossRef] [PubMed]

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force,” Appl. Opt. 43, 2545-2554 (2004).
[CrossRef] [PubMed]

J. A. Lock and C. L. Adler, “Debye-series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1316-1328 (1997).
[CrossRef]

G. Gouesbet and J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516-2525 (1994).
[CrossRef]

J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677-4690 (1994).
[CrossRef] [PubMed]

J. A. Lock and G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503-2515 (1994).
[CrossRef]

E. A. Hovenac and J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781-795 (1992).
[CrossRef]

Maheu, B.

McConnell, G.

Mitchell, A.

Mutzenich, S.

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Nieto-Vesperinas, M.

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

Onofri, F.

Polaert, H.

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169-179 (1998).
[CrossRef]

Rahmani, A.

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

Ren, K. F.

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602 (2009).
[CrossRef]

R. Li, X. Han, L. Shi, K. F. Ren, and H. Jiang, “Debye series for Gaussian beam scattering by a multilayered sphere,” Appl. Opt. 46, 4804-4812 (2007).
[CrossRef] [PubMed]

F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series for light scattering by a multilayered sphere,” Appl. Opt. 45, 1260-1270 (2006).
[CrossRef] [PubMed]

R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series of normally incident plane wave scattering by an infinite multi-layered cylinder,” Appl. Opt. 45, 6255-6262 (2006).
[CrossRef] [PubMed]

K. F. Ren, G. Gouesbet, and G. Gréhan, “The integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35, 2702-2710 (1996).
[CrossRef] [PubMed]

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

K. F. Ren, G. Gréhan, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072-2079 (1994).
[CrossRef]

Roosen, G.

G. Roosen, “A theoretical and experimental study of the stable equilibrium position of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189-194 (1977).
[CrossRef]

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. 59, 6-8 (1976).
[CrossRef]

Rosengarten, G.

Schäffer, E.

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]

Shi, L.

Sonek, G. J.

W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735-1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715-717 (1993)
[CrossRef]

Šrámek, L.

P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273-285 (1998).
[CrossRef]

Sun, Q.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Svoboda, K.

van Blaaderen, A.

van Kats, C. M.

Wright, W. H.

W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735-1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715-717 (1993)
[CrossRef]

Xu, F.

F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

Xu, J.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Yamane, T.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769-771 (1987).
[CrossRef] [PubMed]

Yao, X.

X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).

Zemánek, P.

P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273-285 (1998).
[CrossRef]

Zhang, D.

X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).

Zhang, Y.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Zhao, L.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Acta Opt. Sin.

X. Yao, Z. Li, H. Guo, B. Cheng, X. Han, and D. Zhang, “Analysis and calculation of the optical force on a double-layer dielectric sphere,” Acta Opt. Sin. 20, 1305-1310 (2000).

Acta Phys. Sin.

Y. Zhang, Y. Li, L. Zhao, H. Liu, J. Chen, G. Cui, J. Xu, and Q. Sun, “Transverse optical trapping force of absorbing double-layer spherical particles in a Gaussian beam,” Acta Phys. Sin. 58, 258-263 (2009).

Appl. Opt.

A. Ashkin and J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803-1814 (1981).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, and M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735-1748 (1994).
[CrossRef] [PubMed]

J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677-4690 (1994).
[CrossRef] [PubMed]

F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113-7124 (1995).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35, 2702-2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gouesbet, and G. Gréhan, “The integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

G. Gréhan, B. Maheu, and G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539-3548 (1986).
[CrossRef] [PubMed]

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532-2544(2004).
[CrossRef] [PubMed]

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force,” Appl. Opt. 43, 2545-2554 (2004).
[CrossRef] [PubMed]

J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594-5603 (2005).
[CrossRef] [PubMed]

R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series for light scattering by a multilayered sphere,” Appl. Opt. 45, 1260-1270 (2006).
[CrossRef] [PubMed]

R. Li, X. Han, H. Jiang, and K. F. Ren, “Debye series of normally incident plane wave scattering by an infinite multi-layered cylinder,” Appl. Opt. 45, 6255-6262 (2006).
[CrossRef] [PubMed]

R. Li, X. Han, L. Shi, K. F. Ren, and H. Jiang, “Debye series for Gaussian beam scattering by a multilayered sphere,” Appl. Opt. 46, 4804-4812 (2007).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, and G. Gouesbet, “Generalized Lorenz-Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23-25 (1987).
[CrossRef] [PubMed]

Appl. Phys. Lett.

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715-717 (1993)
[CrossRef]

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283-285 (1971).
[CrossRef]

Biophys. J.

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]

D. C. Appleyard and M. J. Lang, “Optical trapping of cells and control of multiple particles through silicon,” Biophys. J. 93, 651a (2007).

J. Appl. Phys.

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. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nature

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared-laser beams,” Nature 330, 769-771 (1987).
[CrossRef] [PubMed]

Opt. Commun.

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

G. Roosen, “A theoretical and experimental study of the stable equilibrium position of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189-194 (1977).
[CrossRef]

P. Zemánek, A. Jonáš, L. Šrámek, and M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273-285 (1998).
[CrossRef]

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169-179 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett.

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. 59, 6-8 (1976).
[CrossRef]

Phys. Rev. A

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

Phys. Rev. E

F. Xu, K. F. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007).
[CrossRef]

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602 (2009).
[CrossRef]

Phys. Rev. Lett.

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

Phys. Z.

P. Debije, “Das elektromagnetische feld um einen zylinder und die theorie des regenbogens,” Phys. Z. 9, 775-778 (1908).

Rev. Sci. Instrum.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Science

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Longitudinal RPCS versus beam-waist center with parameters p max and x 0 = y 0 = 0 .

Fig. 2
Fig. 2

Longitudinal RPCS C p r , z versus z 0 corresponding to single mode p, x 0 = y 0 = 0 .

Fig. 3
Fig. 3

Scattered intensity for homogeneous sphere with same parameters as Fig. 2.

Fig. 4
Fig. 4

Transverse RPCS versus beam-waist center with parameters p max , z 0 = 0 , (a)  y 0 = 0 , and (b)  x 0 = 0 .

Fig. 5
Fig. 5

Transverse RPCS C p r , x versus x 0 corresponding to single mode p, y 0 = z 0 = 0 .

Fig. 6
Fig. 6

Transverse RPCS C p r , y versus y 0 corresponding to single mode p, x 0 = z 0 = 0 .

Fig. 7
Fig. 7

Scattered intensity for homogeneous sphere at off-axis incidence.

Fig. 8
Fig. 8

Transverse RPCS C p r , x versus x 0 for water-covered carbon with different ratio q of inner and outer radius, y 0 = z 0 = 0 .

Fig. 9
Fig. 9

Transverse RPCS C p r , x versus x 0 for water-covered carbons of various q with parameters mode p, y 0 = z 0 = 0 .

Fig. 10
Fig. 10

Transverse RPCS C p r , y versus y 0 for a three-layered sphere.

Equations (23)

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

C pr , z = λ 2 π n = 1 Re { 1 n + 1 ( A n g n 0 , TM g n + 1 , TM 0 * + B n g n 0 , TE g n + 1 , TE 0 * ) + m = 1 n [ 1 ( n + 1 ) 2 ( n + m + 1 ) ! ( n m ) ! ( A n g n , TM m g n + 1 , TM m * + A n g n , TM m g n + 1 , TM m * + B n g n , TE m g n + 1 , TE m * + B n g n , TE m g n + 1 , TE m * ) + m 2 n + 1 n 2 ( n + 1 ) 2 ( n + m ) ! ( n m ) ! C n ( g n , TM m g n , TE m * g n , TM m g n , TE m * ) ] } ,
C p r , x = Re ( C ) , C p r , y = Im ( C ) ,
C = λ 2 2 π n = 1 { ( 2 n + 2 ) ! ( n + 1 ) 2 F n n + 1 + m = 1 n ( n + m ) ! ( n m ) ! 1 ( n + 1 ) 2 [ F n m + 1 n + m + 1 n m + 1 F n m + 2 n + 1 n 2 ( C n g n , TM m 1 g n , TE m * C n g n , TM m g n + 1 , TE m + 1 * + C n * g n , TE m 1 g n , TM m * C n * g n , TE m g n , TM m + 1 * ) ] } ,
F n m = A n g n , TM m 1 g n + 1 , TM m * + B n g n , TE m 1 g n + 1 , TE m * + A n + 1 , TM m * g n , TM m + 1 * + B n m * g n + 1 , TE m g n , TE m + 1 * ,
A n = a n l + a n + 1 l * 2 a n l a n + 1 l * , B n = b n l + b n + 1 l * 2 b n l b n + 1 l * , C n = i ( a n l + b n + 1 l * 2 a n l b n + 1 l * ) .
( g n , TM m i g n , TE m ) = i Q Z n m e i Q γ 2 + i k z 0 1 2 [ e i ( m 1 ) ϕ 0 J m 1 ( 2 Q ρ n ρ 0 ) ± e i ( m + 1 ) ϕ 0 J m + 1 ( 2 Q ρ n ρ 0 ) ] ,
k = 2 π λ , l = k w 0 2 , s = 1 k w 0 ,
z n 0 = 2 n ( n + 1 ) i 2 n + 1 , z n m = ( 2 i 2 n + 1 ) | m | 1 ,
Q = 1 i 2 z 0 / l , ρ n = n + 1 / 2 k , γ = ρ n 2 + ρ 0 2 ,
ξ 0 = x 0 w 0 , η 0 = y 0 w 0 , ρ 0 = ξ 0 2 + η 0 2 , ϕ 0 = ξ 0 η 0 ,
g n = i Q exp [ i Q ρ n 2 ω n 2 ] exp ( i k z 0 ) .
a n l b n l } = 1 2 ( 1 Q n l ) ,
Q n j = R n j + 1 , j , j + 1 + T n j + 1 , j Q n j 1 T n j , j + 1 p = 1 ( R n j , j + 1 , j Q n j 1 ) p 1 ,
T n j + 1 , j = m j m j + 1 2 i D n j ,
R n j + 1 , j , j + 1 = α ξ n ( 1 ) ( m j + 1 k a ) ξ n ( 1 ) ( m j k a ) β ξ n ( 1 ) ( m j + 1 k a ) ξ n ( 1 ) ( m j k a ) D n j ,
T n j , j + 1 = 2 i D n j ,
R n j , j + 1 , j = α ξ n ( 2 ) ( m j + 1 k a ) ξ n ( 2 ) ( m j k a ) β ξ n ( 2 ) ( m j + 1 k a ) ξ n ( 2 ) ( m j k a ) D n j
D n j = α ξ n ( 2 ) ( m j + 1 k a ) ξ n ( 1 ) ( m j k a ) + β ξ n ( 2 ) ( m j + 1 k a ) ξ n ( 1 ) ( m j k a ) ,
α = { 1 TE   wave m j m j + 1 TM  wave , β = { m j m j + 1 TE  wave 1 TM  wave ,
ξ n ( 1 ) ( m k r ) = m k r h n ( 1 ) ( m k r ) , ξ n ( 2 ) ( m k r ) = m k r h n ( 2 ) ( m k r ) .
a n 2 b n 2 } = 1 2 T n 32 Q n 1 T n 23 ,
Q n 1 = R n 212 + T n 21 T n 12 1 R n 121 .
p max

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