Abstract

We study the scattering of Gaussian beams by infinite cylinders in the framework of the so-called generalized Lorenz–Mie theory for cylinders. The general theory is expressed by using the theory of distributions. Several descriptions of the illuminating Gaussian beams are considered—i.e., Maxwellian beams at limited order, quasi-Gaussian beams defined by a plane-wave spectrum, and the cylindrical localized approximation—leading to different specific formulations. In the last two cases, the theory in terms of distributions reduces to theories expressed in terms of usual functions.

© 1997 Optical Society of America

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  1. 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]
  2. B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory forarbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
    [CrossRef]
  3. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” Chap. 10 of Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), pp. 339–384.
  4. G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
    [CrossRef]
  5. F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in anarbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
    [CrossRef] [PubMed]
  6. G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussianbeams: II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
    [CrossRef]
  7. F. Onofri, D. Blondel, G. Gréhan, G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayeredparticles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104–111 (1996).
    [CrossRef]
  8. H. Mignon, F. Onofri, G. Gréhan, G. Gouesbet, C. Tropea, “Mesure de particules non sphériques par phase-Doppler: cas des cylindres et des particules irrégulières,” in Recueil des actes du 4eme congrès francophone de vélocimètrie laser (Poitiers-Futuroscope, A. V. L., Poitiers, France, 1994), pp. 1–4.
  9. F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “Phase-Doppler measurements of non-spherical particles: cylindrical and multilayer particles,” in Proceedings of the 4th International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nuremberg, Germany, 1995), pp. 275–284.
  10. H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
    [CrossRef]
  11. G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, includinga discussion of Gaussian beams,” Part. Part. System Charact. 11, 299–308 (1994).
    [CrossRef]
  12. G. Gouesbet, G. Gréhan, “Interaction between a Gaussian beam and an infinite cylinder usingnon Σ-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
    [CrossRef]
  13. G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by usingthe theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
    [CrossRef]
  14. G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder witharbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
    [CrossRef]
  15. G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
    [CrossRef]
  16. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
    [CrossRef] [PubMed]
  17. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  18. G. Gouesbet, “The separability theorem revisited with application to light scatteringtheory,” J. Opt. (Paris) 26, 123–135 (1995).
    [CrossRef]
  19. E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussionof electromagnetic scattering theory,” J. Math. Phys. 9, 4705–4710 (1996).
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    [CrossRef]
  24. S. Kozaki, “Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder,” J. Opt. Soc. Am. 72, 1470–1474 (1982).
    [CrossRef]
  25. S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
    [CrossRef]
  26. S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conductingcylinder,” IEEE Trans. Antennas Propag. AP-30, 881–887 (1982).
    [CrossRef]
  27. T. C. K. Rao, “Scattering by a radially inhomogeneous cylindrical dielectric shelldue to an incident Gaussian beam,” Can. J. Phys. 67, 471–475 (1989).
    [CrossRef]
  28. E. Zimmermann, R. Dändliker, N. Souli, B. Krattiger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder comparedwith a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398–403 (1995).
    [CrossRef]
  29. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitelong homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
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  30. J. A. Lock, “Morphology-dependent resonances of an infinitely long circular cylinderilluminated by a diagonally incident plane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A 14, 653–651 (1997).
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  31. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  32. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamentalGaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  33. J. A. Lock, G. Gouesbet, “A rigorous justification of the localized approximation to the beamshape coefficients in the generalized Lorenz–Mie theory. I. On-axisbeams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  34. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scatteringcalculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef] [PubMed]
  35. G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
    [CrossRef]
  36. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1976).
  37. G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficientsin generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
    [CrossRef] [PubMed]
  38. G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finiteseries,” J. Opt. (Paris) 19, 35–48 (1988).
    [CrossRef]
  39. K. F. Ren, G. Gréhan, G. Gouesbet, “On prediction of reverse radiation pressure by generalized Lorenz–Mietheory,” Appl. Opt. 35, 2702–2710 (1996).
    [CrossRef] [PubMed]
  40. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  41. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical resultsusing a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef]
  42. 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]
  43. K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mietheory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
    [CrossRef]
  44. G. Gouesbet, J. A. Lock, “A rigorous justification of the localized approximation to the beamshape coefficients in the generalized Lorenz–Mie theory. II. Off-axisbeams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  45. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  46. K. F. Ren, G. Gouesbet, G. Gréhan, “Scattering of a Gaussian beam by an infinite cylinder: numerical result in GLMT-framework,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Técnico, Lisbon, Portugal, 1996), pp. 6.5.1–6.5.8.
  47. N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).
  48. F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
    [CrossRef]

1997 (5)

1996 (6)

G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficientsin generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, G. Gouesbet, “On prediction of reverse radiation pressure by generalized Lorenz–Mietheory,” Appl. Opt. 35, 2702–2710 (1996).
[CrossRef] [PubMed]

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
[CrossRef]

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussionof electromagnetic scattering theory,” J. Math. Phys. 9, 4705–4710 (1996).
[CrossRef]

F. Onofri, D. Blondel, G. Gréhan, G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayeredparticles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104–111 (1996).
[CrossRef]

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

1995 (7)

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by usingthe theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[CrossRef]

G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder witharbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

G. Gouesbet, “The separability theorem revisited with application to light scatteringtheory,” J. Opt. (Paris) 26, 123–135 (1995).
[CrossRef]

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussianbeams: II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

E. Zimmermann, R. Dändliker, N. Souli, B. Krattiger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder comparedwith a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398–403 (1995).
[CrossRef]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scatteringcalculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef] [PubMed]

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

1994 (6)

1990 (1)

1989 (2)

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamentalGaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

T. C. K. Rao, “Scattering by a radially inhomogeneous cylindrical dielectric shelldue to an incident Gaussian beam,” Can. J. Phys. 67, 471–475 (1989).
[CrossRef]

1988 (3)

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory forarbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (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, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finiteseries,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

1986 (1)

1984 (1)

1982 (4)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

S. Kozaki, “Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder,” J. Opt. Soc. Am. 72, 1470–1474 (1982).
[CrossRef]

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[CrossRef]

S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conductingcylinder,” IEEE Trans. Antennas Propag. AP-30, 881–887 (1982).
[CrossRef]

1979 (1)

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

1972 (1)

N. G. Alexopoulos, P. K. Park, “Scattering of waves with normal amplitude distribution from cylinders,” IEEE Trans. Antennas Propag. AP-20, 216–217 (1972).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamentalGaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

Alexopoulos, N. G.

N. G. Alexopoulos, P. K. Park, “Scattering of waves with normal amplitude distribution from cylinders,” IEEE Trans. Antennas Propag. AP-20, 216–217 (1972).
[CrossRef]

Allano, D.

Angelova, M. I.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussianbeams: II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1976).

Barton, J. P.

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamentalGaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

Blondel, D.

F. Onofri, D. Blondel, G. Gréhan, G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayeredparticles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104–111 (1996).
[CrossRef]

Butkov, E.

E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass.) 1968).

Dändliker, R.

Davis, L. W.

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

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Gauchet, N.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

Girasole, T.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

Gouesbet, G.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
[CrossRef]

G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
[CrossRef] [PubMed]

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
[CrossRef]

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussionof electromagnetic scattering theory,” J. Math. Phys. 9, 4705–4710 (1996).
[CrossRef]

G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficientsin generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, G. Gouesbet, “On prediction of reverse radiation pressure by generalized Lorenz–Mietheory,” Appl. Opt. 35, 2702–2710 (1996).
[CrossRef] [PubMed]

F. Onofri, D. Blondel, G. Gréhan, G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayeredparticles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104–111 (1996).
[CrossRef]

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

G. Gouesbet, “The separability theorem revisited with application to light scatteringtheory,” J. Opt. (Paris) 26, 123–135 (1995).
[CrossRef]

G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder witharbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussianbeams: II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scatteringcalculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef] [PubMed]

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by usingthe theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[CrossRef]

G. Gouesbet, G. Gréhan, “Interaction between a Gaussian beam and an infinite cylinder usingnon Σ-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
[CrossRef]

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

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

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, includinga discussion of Gaussian beams,” Part. Part. System Charact. 11, 299–308 (1994).
[CrossRef]

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

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory forarbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finiteseries,” 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. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical resultsusing a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

H. Mignon, F. Onofri, G. Gréhan, G. Gouesbet, C. Tropea, “Mesure de particules non sphériques par phase-Doppler: cas des cylindres et des particules irrégulières,” in Recueil des actes du 4eme congrès francophone de vélocimètrie laser (Poitiers-Futuroscope, A. V. L., Poitiers, France, 1994), pp. 1–4.

K. F. Ren, G. Gouesbet, G. Gréhan, “Scattering of a Gaussian beam by an infinite cylinder: numerical result in GLMT-framework,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Técnico, Lisbon, Portugal, 1996), pp. 6.5.1–6.5.8.

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “Phase-Doppler measurements of non-spherical particles: cylindrical and multilayer particles,” in Proceedings of the 4th International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nuremberg, Germany, 1995), pp. 275–284.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” Chap. 10 of Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), pp. 339–384.

Gréhan, G.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficientsin generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, G. Gouesbet, “On prediction of reverse radiation pressure by generalized Lorenz–Mietheory,” Appl. Opt. 35, 2702–2710 (1996).
[CrossRef] [PubMed]

F. Onofri, D. Blondel, G. Gréhan, G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayeredparticles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104–111 (1996).
[CrossRef]

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussianbeams: II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light scatteringcalculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef] [PubMed]

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

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

G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, includinga discussion of Gaussian beams,” Part. Part. System Charact. 11, 299–308 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, “Interaction between a Gaussian beam and an infinite cylinder usingnon Σ-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
[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 finiteseries,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory forarbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (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. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical resultsusing a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

H. Mignon, F. Onofri, G. Gréhan, G. Gouesbet, C. Tropea, “Mesure de particules non sphériques par phase-Doppler: cas des cylindres et des particules irrégulières,” in Recueil des actes du 4eme congrès francophone de vélocimètrie laser (Poitiers-Futuroscope, A. V. L., Poitiers, France, 1994), pp. 1–4.

K. F. Ren, G. Gouesbet, G. Gréhan, “Scattering of a Gaussian beam by an infinite cylinder: numerical result in GLMT-framework,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Técnico, Lisbon, Portugal, 1996), pp. 6.5.1–6.5.8.

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “Phase-Doppler measurements of non-spherical particles: cylindrical and multilayer particles,” in Proceedings of the 4th International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nuremberg, Germany, 1995), pp. 275–284.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” Chap. 10 of Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), pp. 339–384.

Kozaki, S.

S. Kozaki, “Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder,” J. Opt. Soc. Am. 72, 1470–1474 (1982).
[CrossRef]

S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conductingcylinder,” IEEE Trans. Antennas Propag. AP-30, 881–887 (1982).
[CrossRef]

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[CrossRef]

Krattiger, B.

Lenglart, E.

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussionof electromagnetic scattering theory,” J. Math. Phys. 9, 4705–4710 (1996).
[CrossRef]

Letellier, C.

Lock, J. A.

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 finiteseries,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory forarbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (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. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical resultsusing a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” Chap. 10 of Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), pp. 339–384.

Martinot-Lagarde, G.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussianbeams: II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Mignon, H.

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

H. Mignon, F. Onofri, G. Gréhan, G. Gouesbet, C. Tropea, “Mesure de particules non sphériques par phase-Doppler: cas des cylindres et des particules irrégulières,” in Recueil des actes du 4eme congrès francophone de vélocimètrie laser (Poitiers-Futuroscope, A. V. L., Poitiers, France, 1994), pp. 1–4.

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “Phase-Doppler measurements of non-spherical particles: cylindrical and multilayer particles,” in Proceedings of the 4th International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nuremberg, Germany, 1995), pp. 275–284.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Onofri, F.

F. Onofri, D. Blondel, G. Gréhan, G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayeredparticles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104–111 (1996).
[CrossRef]

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

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “Phase-Doppler measurements of non-spherical particles: cylindrical and multilayer particles,” in Proceedings of the 4th International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nuremberg, Germany, 1995), pp. 275–284.

H. Mignon, F. Onofri, G. Gréhan, G. Gouesbet, C. Tropea, “Mesure de particules non sphériques par phase-Doppler: cas des cylindres et des particules irrégulières,” in Recueil des actes du 4eme congrès francophone de vélocimètrie laser (Poitiers-Futuroscope, A. V. L., Poitiers, France, 1994), pp. 1–4.

Park, P. K.

N. G. Alexopoulos, P. K. Park, “Scattering of waves with normal amplitude distribution from cylinders,” IEEE Trans. Antennas Propag. AP-20, 216–217 (1972).
[CrossRef]

Pouligny, B.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussianbeams: II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Rao, T. C. K.

T. C. K. Rao, “Scattering by a radially inhomogeneous cylindrical dielectric shelldue to an incident Gaussian beam,” Can. J. Phys. 67, 471–475 (1989).
[CrossRef]

Ren, K. F.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

G. Gouesbet, C. Letellier, K. F. Ren, G. Gréhan, “Discussion of two quadrature methods to evaluate beam shape coefficientsin generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, G. Gouesbet, “On prediction of reverse radiation pressure by generalized Lorenz–Mietheory,” Appl. Opt. 35, 2702–2710 (1996).
[CrossRef] [PubMed]

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

K. F. Ren, G. Gouesbet, G. Gréhan, “Scattering of a Gaussian beam by an infinite cylinder: numerical result in GLMT-framework,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Técnico, Lisbon, Portugal, 1996), pp. 6.5.1–6.5.8.

Roddier, F.

F. Roddier, Distributions et Transformation de Fourier (McGraw-Hill, New York, 1982).

Schwartz, L.

L. Schwartz, Méthodes mathématiques pour les sciences physiques (Hermann, Paris, 1965).

Slimani, F.

Souli, N.

Tropea, C.

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

H. Mignon, F. Onofri, G. Gréhan, G. Gouesbet, C. Tropea, “Mesure de particules non sphériques par phase-Doppler: cas des cylindres et des particules irrégulières,” in Recueil des actes du 4eme congrès francophone de vélocimètrie laser (Poitiers-Futuroscope, A. V. L., Poitiers, France, 1994), pp. 1–4.

van de Hulst, H. C.

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

Xu, T. H.

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

Zimmermann, E.

Appl. Opt. (8)

Can. J. Phys. (1)

T. C. K. Rao, “Scattering by a radially inhomogeneous cylindrical dielectric shelldue to an incident Gaussian beam,” Can. J. Phys. 67, 471–475 (1989).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conductingcylinder,” IEEE Trans. Antennas Propag. AP-30, 881–887 (1982).
[CrossRef]

N. G. Alexopoulos, P. K. Park, “Scattering of waves with normal amplitude distribution from cylinders,” IEEE Trans. Antennas Propag. AP-20, 216–217 (1972).
[CrossRef]

J. Appl. Phys. (2)

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamentalGaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. Math. Phys. (1)

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussionof electromagnetic scattering theory,” J. Math. Phys. 9, 4705–4710 (1996).
[CrossRef]

J. Opt. (Paris) (7)

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by usingthe theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[CrossRef]

G. Gouesbet, “The separability theorem revisited with application to light scatteringtheory,” J. Opt. (Paris) 26, 123–135 (1995).
[CrossRef]

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. (Paris) 28, 45–65 (1997).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory forarbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. (Paris) 27, 35–50 (1996).
[CrossRef]

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

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

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, “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]

E. Zimmermann, R. Dändliker, N. Souli, B. Krattiger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder comparedwith a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398–403 (1995).
[CrossRef]

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

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

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

G. Gouesbet, G. Gréhan, “Interaction between a Gaussian beam and an infinite cylinder usingnon Σ-separable potentials,” J. Opt. Soc. Am. A 11, 3261–3273 (1994).
[CrossRef]

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitelong homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
[CrossRef]

J. A. Lock, “Morphology-dependent resonances of an infinitely long circular cylinderilluminated by a diagonally incident plane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A 14, 653–651 (1997).
[CrossRef]

Opt. Diag. Eng. (1)

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

Part. Part. Syst. Charact. (3)

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
[CrossRef]

F. Onofri, D. Blondel, G. Gréhan, G. Gouesbet, “On the optical diagnosis and sizing of spherical coated and multilayeredparticles with phase-Doppler anemometry,” Part. Part. Syst. Charact. 13, 104–111 (1996).
[CrossRef]

G. Gouesbet, “Scattering of a first-order Gaussian beam by an infinite cylinder witharbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

Part. Part. System Charact. (1)

G. Gouesbet, G. Gréhan, “Interaction between shaped beams and an infinite cylinder, includinga discussion of Gaussian beams,” Part. Part. System Charact. 11, 299–308 (1994).
[CrossRef]

Phys. Rev. A (1)

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

Pure Appl. Opt. (1)

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussianbeams: II. GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Other (10)

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” Chap. 10 of Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), pp. 339–384.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

H. Mignon, F. Onofri, G. Gréhan, G. Gouesbet, C. Tropea, “Mesure de particules non sphériques par phase-Doppler: cas des cylindres et des particules irrégulières,” in Recueil des actes du 4eme congrès francophone de vélocimètrie laser (Poitiers-Futuroscope, A. V. L., Poitiers, France, 1994), pp. 1–4.

F. Onofri, H. Mignon, G. Gouesbet, G. Gréhan, “Phase-Doppler measurements of non-spherical particles: cylindrical and multilayer particles,” in Proceedings of the 4th International Congress on Optical Particle Sizing (Nürnberg Messe GmbH, Nuremberg, Germany, 1995), pp. 275–284.

L. Schwartz, Méthodes mathématiques pour les sciences physiques (Hermann, Paris, 1965).

F. Roddier, Distributions et Transformation de Fourier (McGraw-Hill, New York, 1982).

E. Butkov, Mathematical Physics (Addison-Wesley, Reading, Mass.) 1968).

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

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1976).

K. F. Ren, G. Gouesbet, G. Gréhan, “Scattering of a Gaussian beam by an infinite cylinder: numerical result in GLMT-framework,” in Proceedings of the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Técnico, Lisbon, Portugal, 1996), pp. 6.5.1–6.5.8.

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Figures (8)

Fig. 1
Fig. 1

The geometry under study.

Fig. 2
Fig. 2

Intensity evolution along the x axis for a quasi-Gaussian beam.

Fig. 3
Fig. 3

Intensity evolution along the y axis for a quasi-Gaussian beam.

Fig. 4
Fig. 4

Intensity evolution along the z axis for a quasi-Gaussian beam.

Fig. 5
Fig. 5

Comparisons between Maxwellian beams at finite order [ O ( s 2 ) , O ( s 4 ) , and O ( s 6 )] and a quasi-Gaussian beam.

Fig. 6
Fig. 6

Comparison between the quasi-Gaussian beam and the localized approximation.

Fig. 7
Fig. 7

Evolution of the scattered intensity with z (in micrometers) as the parameter for a quasi-Gaussian beam.

Fig. 8
Fig. 8

Evolution of the scattered intensity with the rescaled distance R as the parameter for a quasi-Gaussian beam.

Equations (78)

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

U TM i = E 0 k 2   m = - + ( - i ) m   exp ( im φ )
× I m , TM ( γ ) , J m ( R 1 - γ 2 ) exp ( i γ Z ) ,
U TE i = H 0 k 2   m = - + ( - i ) m   exp ( im φ ) × I m , TE ( γ ) , J m ( R 1 - γ 2 ) exp ( i γ Z ) ,
Z = kz , R = k ρ ,
G ( Z ,   R ,   φ ) = J m ( R 1 - γ 2 ) exp ( im φ ) exp ( i γ Z ) .
I m , TM ( γ ) I m , TE ( γ ) = I m , TM I m , TE δ ( γ ) .
δ ( γ ) ,   f ( γ ) = f ( 0 ) .
δ ( γ ) f ( γ ) d γ = f ( 0 ) .
U TM i = E 0 k 2   m = - + ( - i ) m   exp ( im φ ) I m , TM J m ( R ) ,
U TE i = H 0 k 2   m = - + ( - i ) m   exp ( im φ ) I m , TE J m ( R ) ,
I m , TM = 0 ,
I m , TE = ( - 1 ) m + 1 .
U TM i = E 0 k 2   m = - + ( - i ) m   exp ( im φ ) × I m , TM ( γ ) J m ( R 1 - γ 2 ) exp ( i γ Z ) d γ ,
U TE i = H 0 k 2   m = - + ( - i ) m   exp ( im φ ) × I m , TE ( γ ) J m ( R 1 - γ 2 ) exp ( i γ Z ) d γ .
I m , TM ( γ ) , ( 1 - γ 2 ) J m ( R 1 - γ 2 ) exp ( i γ Z )
= 1 2 π ( - i ) m   0 2 π   E z i E 0   exp ( - im φ ) d φ ,
I m , TE ( γ ) , ( 1 - γ 2 ) J m ( R 1 - γ 2 ) exp ( i γ Z )
= 1 2 π ( - i ) m   0 2 π   H z i H 0   exp ( - im φ ) d φ ,
s = 1 / kw 0 ,
E z i = 0 ,
H z i = H 0 [ - 1 + s 2 ( R 2   sin 2   φ + 2 iR   cos   φ + Z 2 ) ] × exp ( iR   cos   φ ) .
I m , TM ( γ ) = 0 ,
I m , TE ( γ ) = ( - 1 ) m [ - δ ( γ ) + s 2 ( m 2 - 2 ) δ ( γ ) - s 2 δ ( γ ) ] ,
E z i = k 2 U TM i 2 Z + U TM i ,
H z i = k 2 U TE i 2 Z + U TE i
δ ( n ) ( γ ) ,   f ( γ ) = ( - 1 ) n [ f ( γ ) ] γ = 0 ( n ) ,
I m , TM ( γ ) = - 1 + 1 I m , TM ( γ ) δ ( γ - γ ) d γ ,
U TM i = E 0 k 2   m = - + ( - i ) m   exp ( im φ ) × - 1 + 1 I m , TM ( γ ) δ ( γ - γ ) d γ ,   J m ( R 1 - γ 2 ) × exp ( i γ Z ) .
U TM i = E 0 k 2   m = - + ( - i ) m   exp ( im φ ) × - 1 + 1 I m , TM ( γ ) δ ( γ - γ ) ,   J m ( R 1 - γ 2 ) × exp ( i γ Z ) d γ .
δ ( γ - γ ) ,   f ( γ ) = f ( γ ) .
U TM i = E 0 k 2   m = - + ( - i ) m   exp ( im φ ) × - 1 + 1 I m , TM ( γ ) J m ( R 1 - γ 2 ) exp ( i γ Z ) d γ .
U TE i = H 0 k 2   m = - + ( - i ) m   exp ( im φ ) × - 1 + 1 I m , TE ( γ ) J m ( R 1 - γ 2 ) exp ( i γ Z ) d γ .
- 1 + 1 I m , TM ( γ ) ( 1 - γ 2 ) J m ( R 1 - γ 2 ) exp ( i γ Z ) d γ
= 1 2 π ( - i ) m   0 2 π   E z i E 0   exp ( - im φ ) d φ .
δ ( γ - γ ) = 1 2 π   - +   exp [ i ( γ - γ ) Z ] d Z ,
- 1 + 1 I m , TM ( γ ) ( 1 - γ 2 ) J m ( R 1 - γ 2 ) δ ( γ - γ ) d γ
= i m 4 π 2   0 2 π   exp ( - im φ ) - +   E z i E 0   exp ( - i γ Z ) d Z d φ .
I m , TM ( γ ) = i m 4 π 2 ( 1 - γ 2 ) J m ( R 1 - γ 2 ) × 0 2 π   exp ( - im φ ) - +   E z i E 0 × exp ( - i γ Z ) d Z d φ ,
I m , TE ( γ ) = i m 4 π 2 ( 1 - γ 2 ) J m ( R 1 - γ 2 ) × 0 2 π   exp ( - im φ ) - +   H z i H 0 × exp ( - i γ Z ) d Z d φ .
I m , TE ( γ ) = i m 4 π 2 ( 1 - γ 2 ) J m ( R 1 - γ 2 ) × ( - Q 1 + s 2 R 2 Q 2 + 2 iRs 2 Q 3 + s 2 Q 4 ) ,
Q 1 = 0 2 π   exp ( - im φ ) exp ( iR   cos   φ ) d φ × - +   exp ( - i γ Z ) d Z ,
Q 2 = 0 2 π   exp ( - im φ ) exp ( iR   cos   φ ) sin 2   φ d φ × - +   exp ( - i γ Z ) d Z ,
Q 3 = 0 2 π   exp ( - im φ ) exp ( iR   cos   φ ) cos   φ d φ × - +   exp ( - i γ Z ) d Z ,
Q 4 = 0 2 π   exp ( - im φ ) exp ( iR   cos   φ ) d φ × - + Z 2   exp ( - i γ Z ) d Z .
- +   exp ( i γ Z ) d Z = 2 π δ ( γ ) = 2 π δ ( - γ ) = - +   exp ( - i γ Z ) d Z ,
- + Z 2   exp ( - i γ Z ) d Z = - 2 π δ ( γ ) .
0 2 π   exp ( - im φ ) exp ( iR   cos   φ ) d φ
= 2 π ( - i ) m   J m ( R ) ,
0 2 π   exp ( - im φ ) exp ( iR   cos   φ ) sin 2   φ   d φ
= 2 π ( - i ) m   [ J m ( R ) + J m ( R ) ] ,
0 2 π   exp ( - im φ ) exp ( iR   cos   φ ) cos   φ   d φ
= - i   2 π ( - i ) m   J m ( R ) .
I m , TE ( γ ) = ( - 1 ) m ( 1 - γ 2 ) J m ( R 1 - γ 2 )   ( - δ ( γ ) J m ( R ) + s 2 δ ( γ ) { R 2 [ J m ( R ) + J m ( R ) ] + 2 RJ m ( R ) } - s 2 δ ( γ ) J m ( R ) ) .
J m ( R ) = 1 2 [ J m - 1 ( R ) - J m + 1 ( R ) ] ,
J m - 1 ( R ) + J m + 1 ( R ) = 2 m R   J m ( R ) ,
I m , TE ( γ ) = ( - 1 ) m ( 1 - γ 2 ) J m ( R 1 - γ 2 ) ×   [ - δ ( γ ) J m ( R ) + s 2 m 2 δ ( γ ) J m ( R ) + s 2 R δ ( γ ) J m ( R ) - s 2 δ ( γ ) J m ( R ) ] .
δ ( γ ) Q ( γ ) ,   F ( γ ) = δ ( γ ) Q ( γ ) F ( γ ) d γ = Q ( 0 ) F ( 0 ) = δ ( γ ) Q ( 0 ) ,   F ( γ ) .
δ ( γ ) ( 1 - γ 2 ) J m ( R 1 - γ 2 ) = δ ( γ ) J m ( R ) .
δ ( γ ) ( 1 - γ 2 ) J m ( R 1 - γ 2 ) ,   F ( γ ) = δ ( γ ) ,   F ( γ ) ( 1 - γ 2 ) J m ( R 1 - γ 2 ) = 2 γ 2   F ( γ ) ( 1 - γ 2 ) J m ( R 1 - γ 2 ) γ = 0 = F ( 0 ) J m ( R ) + 2 F ( 0 ) J m ( R ) + RF ( 0 ) J m ( R ) J m 2 ( R ) .
δ ( γ ) ( 1 - γ 2 ) J m ( R 1 - γ 2 )
= δ ( γ ) J m ( R ) + 2 δ ( γ ) J m ( R ) + R δ ( γ ) J m ( R ) J m 2 ( R ) .
I m , TE ( γ ) = ( - 1 ) m [ - δ ( γ ) + s 2 ( m 2 - 2 ) δ ( γ ) - s 2 δ ( γ ) ] ,
E z = 0 ,
E p = E 0 Ψ 0 ( sin   φ ) 1 + 2 Q l   ρ   cos   φ exp ( ik ρ   cos   φ ) ,
E φ = E 0 Ψ 0 cos   φ - 2 Q l   ρ   sin 2   φ exp ( ik ρ   cos   φ ) ,
H z = - H 0 Ψ 0   exp ( ik ρ   cos   φ ) ,
H ρ = - H 0 Ψ 0   2 Q l   z ( cos   φ ) exp ( ik ρ   cos   φ ) ,
H φ = - H 0 Ψ 0   2 Q l   z ( sin   φ ) exp ( ik ρ   cos   φ ) ,
Ψ 0 = iQ   exp - iQ w 0 2   ( ρ 2   sin 2   φ + z 2 ) ,
Q = 1 i - ( 2 / l ) ρ   cos   φ ,
l = kw 0 2 .
I m , TM ( γ ) = i m 4 π 2 ( 1 - γ 2 ) J m ( R p 1 - γ 2 ) ×   0 2 π   exp ( - im φ ) d φ - + E z i E 0 R = R p ×   exp ( - i γ Z ) d Z ,
F ˆ : kr ( n + 1 / 2 ) , θ = π / 2 ,
F ˆ : kr [ ( n - 1 ) ( n + 2 ) ] 1 / 2 , θ = π / 2 .
G ˆ : R m , φ = π / 2 ,
I m , TM ( γ ) = 0 ,
I m , TE ( γ ) = ( - 1 ) m 2 π ( 1 - γ 2 ) s   exp - m 2 s 2 - γ 2 4 s 2 ,
H z i = H 0   exp ( iR   cos   φ ) [ - 1 + s 2 ( Z 2 + R 2   sin 2   φ + 2 iR   cos   φ ) ] .

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