Abstract

A solution is given for the problem of scattering of an arbitrary shaped beam by a multilayered sphere. Starting from Bromwich potentials and using the appropriate boundary conditions, we give expressions for the external and the internal fields. It is shown that the scattering coefficients can be generated from those established for a plane-wave illumination. Some numerical results that describe the scattering patterns and the radiation-pressure behavior when an incident Gaussian beam or a plane wave impinges on a multilayered sphere are presented.

© 1995 Optical Society of America

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References

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  1. L. Lorenz, “Lysbevaegelsen i og uden for en hal plane lysbôlger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1898).
  2. G. Mie, “Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
    [CrossRef]
  3. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
    [CrossRef]
  4. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  5. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  6. M. Kerker, The Scattering of Light, and Other Electromagnetic Radiations (Academic, New York, 1969).
  7. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  8. G. Gouesbet, B. Maheu, 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]
  9. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10.
  10. K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
    [CrossRef]
  11. K. F. Ren, G. Gréhan, 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]
  12. M. I. Angelova, B. Pouligny, G. M. Martinot-Lagarde, G. Gréhan, G. Gouesbet, “Stressing phospholipid membranes using mechanical effects of light,” Prog. Colloid Polym. Sci. (to be published).
  13. G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
    [CrossRef]
  14. F. Onofri, G. Gréhan, G. Gouesbet, T.-H. Xu, G. Brenn, C. Tropea, “Phase-Doppler anemometry with dual burst technique for particle refractive index measurements,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 11–14 July 1994.
  15. M. Schneider, E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379–2389 (1994).
    [CrossRef] [PubMed]
  16. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  17. L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
    [CrossRef] [PubMed]
  18. E. M. Khaled, S. C. Hill, P. W. Barber, “Light scattering by a coated sphere illuminated by a Gaussian beam,” Appl. Opt. 33, 3308–3314 (1994).
    [CrossRef] [PubMed]
  19. J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
    [CrossRef] [PubMed]
  20. G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
    [CrossRef]
  21. G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  22. G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. Off-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  23. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz-Mie theory. II. On-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  24. Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
    [CrossRef]
  25. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef] [PubMed]
  26. L. Kai, P. Massoli, A. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Partical Sizing, Yokohama, Japan, 1993.
  27. K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the GLMT, and associated resonances effects,” Opt. Commun. 108, 343–353 (1994).
    [CrossRef]
  28. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [CrossRef]

1995 (1)

1994 (7)

1993 (2)

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

1991 (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1990 (1)

1988 (2)

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, 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]

1985 (1)

1970 (1)

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

1951 (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

1898 (1)

L. Lorenz, “Lysbevaegelsen i og uden for en hal plane lysbôlger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1898).

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Angelova, M. I.

M. I. Angelova, B. Pouligny, G. M. Martinot-Lagarde, G. Gréhan, G. Gouesbet, “Stressing phospholipid membranes using mechanical effects of light,” Prog. Colloid Polym. Sci. (to be published).

Ashkin, A.

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

Barber, P. W.

Bhandari, R.

Bohren, C. F.

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

Brenn, G.

F. Onofri, G. Gréhan, G. Gouesbet, T.-H. Xu, G. Brenn, C. Tropea, “Phase-Doppler anemometry with dual burst technique for particle refractive index measurements,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 11–14 July 1994.

D’Alessio, A.

L. Kai, P. Massoli, A. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Partical Sizing, Yokohama, Japan, 1993.

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

Durst, F.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

Gouesbet, G.

K. F. Ren, G. Gréhan, 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, J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. Off-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

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

K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the GLMT, and associated resonances effects,” Opt. Commun. 108, 343–353 (1994).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10.

F. Onofri, G. Gréhan, G. Gouesbet, T.-H. Xu, G. Brenn, C. Tropea, “Phase-Doppler anemometry with dual burst technique for particle refractive index measurements,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 11–14 July 1994.

M. I. Angelova, B. Pouligny, G. M. Martinot-Lagarde, G. Gréhan, G. Gouesbet, “Stressing phospholipid membranes using mechanical effects of light,” Prog. Colloid Polym. Sci. (to be published).

Gréhan, G.

K. F. Ren, G. Gréhan, 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]

K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the GLMT, and associated resonances effects,” Opt. Commun. 108, 343–353 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10.

M. I. Angelova, B. Pouligny, G. M. Martinot-Lagarde, G. Gréhan, G. Gouesbet, “Stressing phospholipid membranes using mechanical effects of light,” Prog. Colloid Polym. Sci. (to be published).

F. Onofri, G. Gréhan, G. Gouesbet, T.-H. Xu, G. Brenn, C. Tropea, “Phase-Doppler anemometry with dual burst technique for particle refractive index measurements,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 11–14 July 1994.

Hill, S. C.

Hirleman, E. D.

Huffman, D. R.

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

Kai, L.

L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

L. Kai, P. Massoli, A. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Partical Sizing, Yokohama, Japan, 1993.

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

M. Kerker, The Scattering of Light, and Other Electromagnetic Radiations (Academic, New York, 1969).

Khaled, E. M.

Lock, J. A.

Lorenz, L.

L. Lorenz, “Lysbevaegelsen i og uden for en hal plane lysbôlger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1898).

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10.

Martinot-Lagarde, G. M.

M. I. Angelova, B. Pouligny, G. M. Martinot-Lagarde, G. Gréhan, G. Gouesbet, “Stressing phospholipid membranes using mechanical effects of light,” Prog. Colloid Polym. Sci. (to be published).

Massoli, P.

L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

L. Kai, P. Massoli, A. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Partical Sizing, Yokohama, Japan, 1993.

Mie, G.

G. Mie, “Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Naqwi, A.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

Onofri, F.

F. Onofri, G. Gréhan, G. Gouesbet, T.-H. Xu, G. Brenn, C. Tropea, “Phase-Doppler anemometry with dual burst technique for particle refractive index measurements,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 11–14 July 1994.

Pouligny, B.

M. I. Angelova, B. Pouligny, G. M. Martinot-Lagarde, G. Gréhan, G. Gouesbet, “Stressing phospholipid membranes using mechanical effects of light,” Prog. Colloid Polym. Sci. (to be published).

Ren, K. F.

K. F. Ren, G. Gréhan, 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]

K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the GLMT, and associated resonances effects,” Opt. Commun. 108, 343–353 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

Schneider, M.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tropea, C.

F. Onofri, G. Gréhan, G. Gouesbet, T.-H. Xu, G. Brenn, C. Tropea, “Phase-Doppler anemometry with dual burst technique for particle refractive index measurements,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 11–14 July 1994.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

Wang, Y. P.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wu, Z. S.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Xu, T.-H.

F. Onofri, G. Gréhan, G. Gouesbet, T.-H. Xu, G. Brenn, C. Tropea, “Phase-Doppler anemometry with dual burst technique for particle refractive index measurements,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 11–14 July 1994.

Ann. Phys. (2)

G. Mie, “Beitrâge zur optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

Appl. Opt. (5)

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Opt. (Paris) (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

K. F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the GLMT, and associated resonances effects,” Opt. Commun. 108, 343–353 (1994).
[CrossRef]

Part. Part. Syst. Charact. (2)

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

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

Radio Sci. (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Vidensk. Selk. Skr. (1)

L. Lorenz, “Lysbevaegelsen i og uden for en hal plane lysbôlger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1898).

Other (8)

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

M. Kerker, The Scattering of Light, and Other Electromagnetic Radiations (Academic, New York, 1969).

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

F. Onofri, G. Gréhan, G. Gouesbet, T.-H. Xu, G. Brenn, C. Tropea, “Phase-Doppler anemometry with dual burst technique for particle refractive index measurements,” presented at the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 11–14 July 1994.

M. I. Angelova, B. Pouligny, G. M. Martinot-Lagarde, G. Gréhan, G. Gouesbet, “Stressing phospholipid membranes using mechanical effects of light,” Prog. Colloid Polym. Sci. (to be published).

L. Kai, P. Massoli, A. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Partical Sizing, Yokohama, Japan, 1993.

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Figures (7)

Fig. 1
Fig. 1

Geometry of the stratified sphere.

Fig. 2
Fig. 2

Scattering diagrams for parallel polarization by a water droplet, a carbon sphere, a water-coated sphere (q = 1/2) by plane-wave and Gaussian-beam illuminations: λ = 0.6328 μm, 2ω0 = 20 μm, d = 10 μm, X = 0, Y = 0, Z = 0, m 1 = 1.33 + 0.0i, m 2 = 1.6 − 0.59i.

Fig. 3
Fig. 3

Scattering diagrams for parallel polarization by a water-coated carbon sphere (q = 1/2) for a Gaussian-beam illumination versus its position along the (OX) axis: λ = 0.6328 μm, 2ω0 = 20 μm, d = 10 μm, Y = 0, Z = 0, m 1 = 1.33 + 0.0i, m 2 = 1.6 − 0.59i.

Fig. 4
Fig. 4

Scattering diagrams for parallel polarization by a multilayered sphere: m j = m 1 + 0.5(m L m 1)[1 − cos(πt)], t = (j − 1)/(L − 1), x 1 = 0.0001, x L and x j = x 1 + (x L x 1)(j − 1)/(L − 1), j = 1, …, L, where x 1 = 0.001 x L , x L = 59.4, m 1 = 1.43 + i0.0, m L = 1.33 + i0.0, L = 100. Plane-wave and Gaussian-beam illuminations λ = 0.5145 μm. The beam-waist diameter is equal to the particle outer diameter, with X = 0, Y = 0, Z = 0.

Fig. 5
Fig. 5

Radiation-pressure cross sections C pr, Z versus particle locations along (OX) for several ratios of the water-coating radius to the carbon-core radius, with Gaussian-beam illumination. Parameters are the same as those of Fig. 3.

Fig. 6
Fig. 6

Radiation-pressure cross sections C pr, X versus particle locations along (OX) for several ratios of the water-coating radius to the carbon-core radius with Gaussian beam illumination. Parameters are the same as those of Fig. 3.

Fig. 7
Fig. 7

Radiation-pressure cross sections C pr, X for a two-layered particle (q = 0.95) versus its location along X with Gaussian-beam illumination. Parameters are the same as those of Fig. 3, except m 1 = 1.0 + 0.0i, m 2 = 1.2 + 0.0i.

Equations (101)

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

2 U r 2 + k 2 U + 1 r 2 sin θ θ sin θ U θ + 1 r 2 sin 2 θ 2 U φ 2 = 0 ,
k = ω ( μ ) 1 / 2 ,
U ( r , θ , φ ) = n = 1 m = - n + n c n m k [ Ψ n ( k r ) ξ n ( k r ) ] P n m ( cos θ ) exp ( i m φ ) ,
Ψ n ( k r ) = k r Ψ n ( 1 ) ( k r ) ,
ξ n ( k r ) = k r Ψ n ( 4 ) ( k r ) ,
Ψ n ( 1 ) ( k r ) = ( π 2 k r ) 1 / 2 J n + 1 / 2 ( k r ) ,
Ψ n ( 4 ) ( k r ) = ( π 2 k r ) 1 / 2 H n + 1 / 2 ( 2 ) ( k r ) ,
P n m ( cos θ ) = ( - 1 ) m ( sin θ ) m d m P n ( cos θ ) ( d cos θ ) m ,
H r , TM = E r , TE = 0.
E r , TM = 2 U TM r 2 + k 2 U TM ,
E θ , TM = 1 r 2 U TM r θ ,
E φ , TM = 1 r sin θ 2 U TM r φ ,
H θ , TM = - i ω r sin θ U TM φ ,
H φ , TM = i ω r U TM θ ,
E θ , TE = - i ω μ r sin θ U TE φ ,
E φ , TE = i ω μ r U TE θ ,
H r , TE = 2 U TE r 2 + k 2 U TE ,
H θ , TE = 1 r 2 U TE r θ ,
H φ , TE = 1 r sin θ 2 U TE r φ .
U TM i = E 0 k 0 n = 1 m = - n + n c n pw g n , TM m Ψ n ( k 0 r ) × P n m ( cos θ ) exp ( i m φ ) , U TE i = H 0 k 0 n = 1 m = - n + n c n pw g n , TE m Ψ n ( k 0 r ) × P n m ( cos θ ) exp ( i m φ ) ,
c n pw = 1 k 0 i n - 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) .
g n , TM m = ( 2 n + 1 ) 2 2 π 2 n ( n + 1 ) c n pw ( n - m ) ! ( n + m ) ! 0 π 0 2 π 0 E r i ( r , θ , φ ) E 0 × r Ψ n ( 1 ) ( k r ) P n m ( cos θ ) exp ( - i m φ ) × sin θ d θ d φ d ( k r ) ,
g n , TE m = ( 2 n + 1 ) 2 2 π 2 n ( n + 1 ) c n pw ( n - m ) ! ( n + m ) ! 0 π 0 2 π 0 H r i ( r , θ , φ ) H 0 × r Ψ n ( 1 ) ( k r ) P n m ( cos θ ) exp ( - i m φ ) × sin θ d θ d φ d ( k r ) .
E r i = k 0 E 0 n = 1 m = - n + n c n pw g n , TM m [ Ψ n ( k 0 r ) + Ψ n ( k 0 r ) ] × P n m ( cos θ ) exp ( i m φ ) ,
E θ i = E 0 r n = 1 m = - n + n c n pw [ g n , TM m Ψ n ( k 0 r ) τ n m ( cos θ ) + m g n , TE m Ψ n ( k 0 r ) Π n m ( cos θ ) ] exp ( i m φ ) ,
E φ i = i E 0 r n = 1 m = - n + n c n pw [ m g n , TM m Ψ n ( k 0 r ) Π n m ( cos θ ) + g n , TE m Ψ n ( k 0 r ) τ n m ( cos θ ) ] exp ( i m φ ) ,
H r i = k 0 H 0 n = 1 m = - n + n c n pw g n , TE m [ Ψ n ( k 0 r ) + Ψ n ( k 0 r ) ] × P n m ( cos θ ) exp ( i m φ ) ,
H θ i = - H 0 r n = 1 m = - n + n c n pw [ m g n , TM m Ψ n ( k 0 r ) Π n m ( cos θ ) - g n , TE m Ψ n ( k 0 r ) τ n m ( cos θ ) ] exp ( i m φ ) ,
H φ i = - i H 0 r n = 1 m = - n + n c n pw [ g n , TM m Ψ n ( k 0 r ) τ n m ( cos θ ) - m g n , TE m Ψ n ( k 0 r ) Π n m ( cos θ ) ] exp ( i m φ ) ,
τ n k ( cos θ ) = d d θ P n k ( cos θ )
Π n k ( cos θ ) = P n k ( cos θ ) sin θ .
U TM s = - E 0 k 0 n = 1 m = - n + n c n pw A n m ξ n ( k r ) × P n m ( cos θ ) exp ( i m φ ) ,
U TE s = - H 0 k 0 n = 1 m = - n + n c n pw B n m ξ n ( k r ) × P n m ( cos θ ) exp ( i m φ ) .
E r s = - k 0 E 0 n = 1 m = - n + n c n pw A n m [ ξ n ( k 0 r ) + ξ n ( k 0 r ) ] × P n m ( cos θ ) exp ( i m φ ) ,
E θ s = - E 0 r n = 1 m = - n + n c n pw [ A n m ξ n ( k 0 r ) τ n m ( cos θ ) + m B n m ξ n ( k 0 r ) Π n m ( cos θ ) ] exp ( i m φ ) ,
E φ s = - i E 0 r n = 1 m = - n + n c n pw [ m A n m ξ n ( k 0 r ) Π n m ( cos θ ) + B n m ξ n ( k 0 r ) τ n m ( cos θ ) ] exp ( i m φ ) ,
H r s = - k 0 H 0 n = 1 m = - n + n c n pw B n m [ ξ n ( k 0 r ) + ξ n ( k 0 r ) ] × P n m ( cos θ ) exp ( i m φ ) ,
H θ s = H 0 r n = 1 m = - n + n c n pw [ m A n m ξ n ( k 0 r ) Π n m ( cos θ ) - B n m ξ n ( k 0 r ) τ n m ( cos θ ) ] exp ( i m φ ) ,
H φ s = i H 0 r n = 1 m = - n + n c n pw [ A n m ξ n ( k 0 r ) τ n m ( cos θ ) - m B n m ξ n ( k 0 r ) Π n m ( cos θ ) ] exp ( i m φ ) .
U j TM = k j + 1 E 0 k j 2 n = 1 m = - n + n c n pw [ c j n m Ψ n ( k j r ) + e j n m χ n ( k j r ) ] × P n m ( cos θ ) exp ( i m φ ) , U j TE = k j + 1 H 0 k j 2 n = 1 m = - n + n c n pw [ d j n m Ψ n ( k j r ) + f j n m χ n ( k j r ) ] × P n m ( cos θ ) exp ( i m φ ) ,
Ψ n ( k r ) = ξ n ( k r ) - i χ n ( k r ) .
e 1 n m = f 1 n m = 0.
E j r = E j + 1 E 0 n = 1 m = - 1 + n c n pw { c j n m [ Ψ n ( k j r ) + Ψ n ( k j r ) ] + e j n m [ χ n ( k j r ) + χ n ( k j r ) ] P n m ( cos θ ) } exp ( i m φ ) ,
E j θ = E 0 r k j + 1 k j n = 1 m = - n + n c n pw { [ c j n m Ψ n ( k j r ) + e j n m χ n ( k j r ) ] × τ n m ( cos θ ) + m k j + 1 μ j k j μ j + 1 [ d j n m Ψ n ( k j r ) + f j n m χ n ( k j r ) ] × Π n m ( cos θ ) } exp ( i m φ ) ,
E j φ = i E 0 r k j + 1 k j n = 1 m = - n + n c n pw { m [ c j n m Ψ n ( k j r ) + c j n m χ n ( k j r ) ] × Π n m ( cos θ ) + k j + 1 μ j k j μ j + 1 [ d j n m Ψ n ( k j r ) + f j n m χ n ( k j r ) ] × τ n m ( cos θ ) } exp ( i m φ ) ,
H j r = k j + 1 H 0 n = 1 m = - n + n c n pw { d j n m [ Ψ n ( k j r ) + Ψ n ( k j r ) ] + f j n m [ χ n ( k j r ) + χ n ( k j r ) ] P n m ( cos θ ) } exp ( i m φ ) ,
H j θ = - H 0 r n = 1 m = - n + n c n pw { m μ j + 1 μ j [ c j n m Ψ n ( k j r ) + e j n m χ n ( k j r ) ] × Π n m ( cos θ ) - k j + 1 k j [ d j n m Ψ n ( k j r ) + f j n m χ n ( k j r ) × τ n m ( cos θ ) } exp ( i m φ ) ,
H j φ = - i H 0 r n = 1 m = - n + n c n pw { μ j + 1 μ j [ c j n m Ψ n ( k j r ) + e j n m χ n ( k j r ) ] × τ n m ( cos θ ) - m k j + 1 μ j k j μ j + 1 [ d j n m Ψ n ( k j r ) + f j n m χ n ( k j r ) ] × Π n m ( cos θ ) } exp ( i m φ ) .
V L θ , X ( k L r L ) = V θ , X i ( k 0 r L ) + V θ , X s ( k 0 r L ) ,
k 0 k L [ c L n m Ψ n ( k L r L ) + e L n m χ n ( k L r L ) ] = [ g n , TM m Ψ n ( k 0 r L ) - A n m ξ n ( k 0 r L ) ,
μ 0 μ L [ c L n m Ψ n ( k L r L ) + e L n m χ n ( k L r L ) ] = [ g n , TM m Ψ n ( k 0 r L ) - A n m ξ n ( k 0 r L ) ] ,
μ L μ 0 k 0 2 k L 2 [ ( d L n m Ψ n ( k L r L ) + f L n m χ n ( k L r L ) ] = [ g n , TE m Ψ n ( k 0 r L ) - B n m ξ n ( k 0 r L ) ] ,
k 0 k L [ d L n m Ψ n ( k L r L ) + f L n m χ n ( k L r L ) ] = [ g n , TE m Ψ n ( k 0 r L ) - B n m ξ n ( k 0 r L ) .
V 1 θ , X ( k 1 r 1 ) = V 2 θ , X ( k 2 r 1 ) ,
k 2 k 1 c 1 n m Ψ n ( k 1 r 1 ) = k 3 k 2 [ c 2 n m Ψ n ( k 2 r 1 ) + e 2 n m χ n ( k 2 r 1 ) ] ,
μ 2 μ 1 c 1 n m Ψ n ( k 1 r 1 ) = μ 3 μ 2 [ c 2 n m Ψ n ( k 2 r 1 ) + e 2 n m χ n ( k 2 r 1 ) ] ,
k 2 2 μ 3 k 1 2 μ 2 d 1 n m Ψ n ( k 1 r 1 ) = k 2 3 μ 2 k 2 2 μ 1 [ d 2 n m Ψ n ( k 2 r 1 ) + f 2 n m χ n ( k 2 r 1 ) ] ,
k 2 k 1 d 1 n m Ψ n ( k 1 r 1 ) = k 3 k 2 [ d 2 n m Ψ n ( k 2 r 1 ) + f 2 n m χ n ( k 2 r 1 ) ] .
V ( j - 1 ) θ , X ( k j - 1 r j - 1 ) = V j θ , X ( k j r j - 1 ) ,
k j k j - 1 [ c ( j - 1 ) n m Ψ n ( k j - 1 r j - 1 ) + e ( j - 1 ) n m χ n ( k j - 1 r j - 1 ) ] = k j + 1 k j [ c j n m Ψ n ( k j r j - 1 ) + e j n m χ n ( k j r j - 1 ) ] ,
μ j μ j - 1 [ c ( j - 1 ) n m Ψ n ( k j - 1 r j - 1 ) + e ( j - 1 ) n m χ n ( k j - 1 r j - 1 ) ] = μ j + 1 μ j [ c j n m Ψ n ( k j r j - 1 ) + e j n m χ n ( k j r j - 1 ) ] ,
μ j + 1 k j 2 μ j k j - 1 2 [ d ( j - 1 ) n m Ψ n ( k j - 1 r j - 1 ) + f ( j - 1 ) n m χ n ( k j - 1 r j - 1 ) ] = μ j k j + 1 2 μ j - 1 k j 2 [ d j n m Ψ n ( k j r j - 1 ) + f j n m χ n ( k j r j - 1 ) ] ,
k j k j - 1 [ d ( j - 1 ) n m Ψ n ( k j - 1 r j - 1 ) + f ( j - 1 ) n m χ n ( k j - 1 r j - 1 ) ] = k j + 1 k j [ d j n m Ψ n ( k j r j - 1 ) + f j n m χ n ( k j r j - 1 ) ] .
M j = k j + 1 k j ,             U j = μ j + 1 μ j ,             x j = k j r j ,
M L = k 0 k L ,             M 1 = k 2 k 1 ,             U L = μ 0 μ L ,             U 1 = μ 2 μ 1 .
R j n = - e j n m c j n m ,             Q j n = - f j n m d j n m ,
R ( j - 1 ) n c ( j - 1 ) n m c j n m = - e ( j - 1 ) n m c j n m ,             Q ( j - 1 ) n d ( j - 1 ) n m d j n m = - f ( j - 1 ) n m d j n m .
M 1 c 1 n m c 2 n m Ψ n ( x 1 ) = M 2 [ Ψ n ( M 1 x 1 ) - R 2 n χ n ( M 1 x 1 ) ] ,
U 1 c 1 n m c 2 n m Ψ n ( x 1 ) = U 2 [ Ψ n ( M 1 x 1 ) - R 2 n χ n ( M 1 x 1 ) ] ,
M 1 2 U 2 d 1 n m d 2 n m Ψ n ( x 1 ) = M 2 2 U 1 [ Ψ n ( M 1 x 1 ) - Q 2 n χ n ( M 1 x 1 ) ] ,
M 1 d 1 n m d 2 n m [ Ψ n ( x 1 ) = M 2 [ Ψ n ( M 1 x 1 ) - Q 2 n X n ( M 1 x 1 ) ] ,
M j - 1 c ( j - 1 ) n m c j n m [ Ψ n ( x j - 1 ) - R ( j - 1 ) n χ n ( x j - 1 ) ] = M j [ Ψ n ( M j x j - 1 ) - R j n χ n ( M j x j - 1 ) ] ,
U j - 1 c ( j - 1 ) n m c j n m [ Ψ n ( x j - 1 ) - R ( j - 1 ) n χ n ( x j - 1 ) ] = U j [ Ψ n ( M j x j - 1 ) - R j n χ n ( M j x j - 1 ) ] ,
M j - 1 2 U j d ( j - 1 ) n m d j n m [ Ψ n ( x j - 1 ) - Q ( j - 1 ) n χ n ( x j - 1 ) ] = M j 2 U j - 1 [ Ψ n ( M j x j - 1 ) - Q j n χ n ( M j x j - 1 ) ] ,
M j - 1 d ( j - 1 ) n m d j n m [ Ψ n ( x j - 1 ) - Q ( j - 1 ) n χ n ( x j - 1 ) ] = M j [ Ψ n ( M j x j - 1 ) - Q j n χ n ( M j x j - 1 ) ] ,
M L [ Ψ n ( x L ) - R L n χ n ( x L ) ] = 1 c L n m [ g n , TM m Ψ n ( M L x L ) - A n m ξ n ( M L x L ) ] ,
U L [ Ψ n ( x L ) - R L n χ n ( x L ) ] = 1 c L n m [ g n , TM m Ψ n ( M L x L ) - A n m ξ n ( M L x L ) ] ,
M L 2 [ Ψ n ( x L ) - Q L n χ n ( x L ) ] = U L d L n m [ g n , TE m Ψ n ( M L x L ) - B n m ξ n ( M L x L ) ] ,
M L [ Ψ n ( x L ) - Q L n χ n ( x L ) ] = 1 d L n m [ g n , TE m Ψ n ( M L x L ) - B n m ξ n ( M L x L ) ] .
H n ( x j ) = Ψ n ( x j ) - R j n χ n ( x j ) Ψ n ( x j ) - R j n χ n ( x j ) , K n ( x j ) = Ψ n ( x j ) - Q j n χ n ( x j ) Ψ n ( x j ) - Q j n χ n ( x j ) ,
e 1 n m = 0 ,             R 1 n = - e 1 n m c 1 n m = 0 ,             H n ( x 1 ) = Ψ n ( x 1 ) Ψ n ( x 1 ) , R 2 n = M 1 U 2 Ψ n ( x 1 ) Ψ n ( M 1 x 1 ) - M 2 U 1 Ψ n ( x 1 ) Ψ n ( M 1 x 1 ) M 1 U 2 Ψ n ( x 1 ) χ n ( M 1 x 1 ) - M 2 U 1 Ψ n ( x 1 ) χ n ( M 1 x 1 ) , R j n = M j - 1 U j Ψ n ( M j x j - 1 ) H n ( x j - 1 ) - M j U j - 1 Ψ n ( M j x j - 1 ) M j - 1 U j χ n ( M j x j - 1 ) H n ( x j - 1 ) - M j U j - 1 χ n ( M j x j - 1 ) ,
H n ( x j ) = Ψ n ( x j ) - R j n χ n ( x j ) Ψ n ( x j ) - R j n χ n ( x j ) ,             j = 2 , , L , A n m = g n , TM m M L Ψ n ( M L x L ) H n ( x L ) - U L Ψ n ( M L x L ) M L ξ n ( M L x L ) H n ( x L ) - U L ξ n ( M L x L ) ,
f 1 n m = 0 ,             Q 1 n = - f 1 n m d 1 n m = 0 ,             K n ( x 1 ) = Ψ n ( x 1 ) Ψ n ( x 1 ) , Q 2 n = M 2 U 1 Ψ n ( x 1 ) Ψ n ( M 1 x 1 ) - M 1 U 2 Ψ n ( x 1 ) Ψ n ( M 1 x 1 ) M 2 U 1 Ψ n ( x 1 ) χ n ( M 1 x 1 ) - M 1 U 2 Ψ n ( x 1 ) χ n ( M 1 x 1 ) , Q j n = M j U j - 1 Ψ n ( M j x j - 1 ) K n ( x j - 1 ) - M j - 1 U j Ψ n ( M j x j - 1 ) M j U j - 1 χ n ( M j x j - 1 ) K n ( x j - 1 ) - M j - 1 U j χ n ( M j x j - 1 ) ,
K n ( x j ) = Ψ n ( x j ) - Q j n χ n ( x j ) Ψ n ( x j ) - Q j n χ n ( x j ) ,             j = 2 , , L , B n m = g n , TE m U L Ψ n ( M L x L ) K n ( x L ) - M L Ψ n ( M L x L ) U L ξ n ( M L x L ) K n ( x L ) - M L ξ n ( M L x L ) .
A n m = g n , TM m a n ,
B n m = g n , TE m b n ,
E θ = i E 0 k 0 r exp ( - i k r ) S 2 ,
E φ = - E 0 k 0 r exp ( - i k r ) S 1 ,
S 1 = n = 1 m = - n + n 2 n + 1 n ( n + 1 ) × [ m a n g n , TM m Π n m ( cos θ ) + i b n g n , TE m τ n m ( cos θ ) ] × exp ( i m φ ) ,
S 2 = n = 1 m = - n + n 2 n + 1 n ( n + 1 ) × [ a n g n , TM m τ n m ( cos θ ) + i m b n g n , TE m Π n m ( cos θ ) ] × exp ( i m φ ) .
c L n m = g n , TM m U L [ Ψ n ( M L x L ) - a n ξ n ( M L x L ) Ψ n ( x L ) - R L n χ n ( x L ) ] , e L n m = - R L n c L n m , c ( j - 1 ) n m = c j n m U j U j - 1 [ Ψ n ( M j x j - 1 ) - R j n χ n ( M j x j - 1 ) Ψ n ( x j - 1 ) - R ( j - 1 ) n χ n ( x j - 1 ) ] , e ( j - 1 ) n m = - R ( j - 1 ) n c ( j - 1 ) n m ,             j = L , , 2.
d L n m = g n , TM m U L M L 2 [ Ψ n ( M L x L ) - b n ξ n ( M L x L ) Ψ n ( x L ) - Q L n χ n ( x L ) ] , f L n m = - Q L n d L n m , d ( j - 1 ) n m = d j n m M j 2 U j - 1 M j - 1 2 U j [ Ψ n ( M j x j - 1 ) - Q j n χ n ( M j x j - 1 ) Ψ n ( x j - 1 ) - Q ( j - 1 n ) χ n ( x j - 1 ) ] , f ( j - 1 ) n m = - Q ( j - 1 ) n d ( j - 1 ) n m ,             j = L , , 2.
c L n m = g n , TM m c L n , e L n m = - R L n c L n m , c ( j - 1 ) n m = c j n m c j n , e ( j - 1 ) n m = - R ( j - 1 ) n c ( j - 1 ) n m ,             j = L , , 2.
c L n = 1 U L [ Ψ n ( M L x L ) - a n ξ n ( M L x L ) Ψ n ( x L ) - R L n χ n ( x L ) ] ,
c ( j - 1 ) n = U j U j - 1 [ Ψ n ( M j x j - 1 ) - R j n χ n ( M j x j - 1 ) Ψ n ( x j - 1 ) - R ( j - 1 ) n χ n ( x j - 1 ) ] .
c ( j - 1 ) n m = g n , TM m c L n c ( L - 1 ) n c ( j + 1 ) n c j n ,
D n 1 ( x ) = Ψ n ( x ) Ψ n ( x ) ,             D n 2 ( x ) = χ n ( x ) χ n ( x ) ,             D n 3 ( x ) = ξ n ( x ) ξ n ( n ) .
e 1 n m = 0 ,             R 1 n = - e 1 n m c 1 n m = 0 ,             H n ( x 1 ) = D n 1 ( x 1 ) , R j n = Ψ n ( M j x j - 1 ) χ n ( M j x j - 1 ) M j - 1 U j H n ( x j - 1 ) - M j U j - 1 D n 1 ( M j x j - 1 ) M j - 1 U j H j ( x j - 1 ) - M j U j - 1 D n 2 ( M j x j - 1 ) , H n ( x j ) = Ψ n ( x j ) χ n ( x j ) D n 1 ( x j ) Ψ n ( x j ) χ n ( x j ) - R j n - R j n D n 2 ( x j ) Ψ n ( x j ) χ n ( x j ) - R j n ,             j = 2 , , L ,
A n m = g n , TM m Ψ n ( M L x L ) ξ n ( M L x L ) M L H n ( x L ) - U L D n 1 ( M j x L ) M L H n ( x L ) - U L D n 3 ( M L x L ) ,
f 1 n m = 0 ,             Q 1 n = - f 1 n m d 1 n m = 0 ,             K n ( x 1 ) = D n 1 ( x 1 ) , Q j n = Ψ n ( M j x j - 1 ) χ n ( M j x j - 1 ) M j U j - 1 K n ( x j - 1 ) - M j - 1 U j D n 1 ( M j x j - 1 ) M j U j - 1 K j ( x j - 1 ) - M j - 1 U j D n 2 ( M j x j - 1 ) , K n ( x j ) = Ψ n ( x j ) χ n ( x j ) D n 1 ( x j ) Ψ n ( x j ) χ n ( x j ) - Q j n - Q j n D n 2 ( x j ) Ψ n ( x j ) χ n ( x j ) - Q j n ,             j = 2 , , L ,
B n m = g n , TM m Ψ n ( M L x L ) ξ n ( M L x L ) U L K n ( x L ) - M L D n 1 ( M j x L ) U L K n ( x L ) - M L D n 3 ( M L x L ) .

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