Abstract

Five integral transform solutions to the Fredholm integral equation that describes the forward-scattering properties of the distributions of spherical particles in the Fraunhofer diffraction regime have been studied. We have systematically reformulated the family of solutions, including four derivations by three other research groups and one developed as part of this work, using a standardized notation. This synthesis elucidates the mathematical interrelationships and fosters an understanding of the inversion performance of the five solutions. Finally a series of numerical experiments was carried out to demonstrate the relative performance of the techniques when applied to identical sets of simulated forward-scattering signatures.

© 1992 Optical Society of America

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  2. J. R. Hodkinson, “Particle sizing by means of the forward scattering lobe,” Appl. Opt. 5, 839–844 (1966).
    [CrossRef] [PubMed]
  3. E. D. Hirleman, “Optimal scaling for Fraunhofer diffraction particle sizing instruments,” Part. Charact. 4, 128–133 (1988).
  4. E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,” Opt. Eng. 23, 610–619 (1984).
  5. S. Boron, B. Waldie, “Particle sizing by forward lobe scattered intensity ratio technique: errors introduced by applying diffraction in the Mie regime,” Appl. Opt. 17, 1644–1648 (1978).
    [CrossRef] [PubMed]
  6. J. H. Koo, “Particle size analysis using integral transform techniques on Fraunhofer diffraction patterns,” Ph.D. dissertation (Arizona State University, Tempe, Ariz.; George Washington University, Washington, D.C., 1987).
  7. A. N. Tikhonov, V. Y. Arsenin, Solution on Ill-Posed Problems (Winston, Washington, D.C., 1977).
  8. J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angle,” J. Phys. Chem. 59, 841–844 (1955).
    [CrossRef]
  9. J. H. Chin, C. M. Sliepcevich, M. Tribus, “Determination of particle size distribution in polydisperse systems by means of measurement of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
    [CrossRef]
  10. J. H. Chin, “Particle size distributions from angular variation of intensity of forward-scattered light,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1955).
  11. K. S. Shifrin, “Calculation of a certain class of definite integrals containing the square of a first order Bessel function,” Tr. Vses. Zoachn. Lesotekhnich. Inst. 2 (1956).
  12. K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 3, 749–753 (1967) (English translation).
  13. L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium powder,” Combust. Explosion Shock Waves 12, 258–262 (1976).
    [CrossRef]
  14. J. B. Abbiss, “Theoretical aspects of the determination of particle size distributions from measurements of scattered light intensity,” Tech. Rep. 70151 (Royal Aircraft Establishment, Ministry of Technology, Farnborough, England, 1970).
  15. A. L. Fymat, K. D. Mease, “Reconstruction of the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. Fymat, V. E. Zuev, eds. (Elsevier, New York, 1978).
  16. L. V. Ruscello, E. D. Hirleman, “Determining droplet size distributions with a photodiode array,” presented at the Fall Meeting of the Western States Section of the Combustion Institute, Tempe, Ariz., 1981.
  17. J. H. Koo, “A new family of intergral transform techniques for laser light diffraction particle sizing,” presented at the Spring Meeting of the Western States Section of the Combustion Institute, Salt Lake City, Utah, 1988.
  18. J. H. Koo, E. D. Hirleman, “Comparative study of laser diffraction droplet size analysis using integral transform techniques: factors affecting the reconstruction of droplet size distributions,” presented at the Joint Spring Meeting of the Canadian and Western States Sections of the Combustion Institute, Banff, Alberta, Canada, 1986.
  19. Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).
    [CrossRef]
  20. K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 559–561 (1966) (English translation).
  21. K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 514–518 (1966) (English translation).
  22. E. C. Titchmarsh, “Extension of Fourier’s integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Series 2 23, xxii–xxiv (1924).
  23. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
    [CrossRef]
  24. K. S. Shifrin, A. Y. Perelman, “Inversion of light scattering data for the determination of the spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967, pp. 131–167.
  25. G. D. Petrov, R. N. Sokolov, V. A. Vasilev, A. M. Kapkov, “Measurement of distribution by the method of small angles from the dimensions of particles suspended in a flow,” J. Eng. Phys. USSR 16, 300–303 (1969) (English translation).
    [CrossRef]
  26. G. D. Petrov, R. N. Sokolov, V. A. Vasilev, “Particle size distribution in various zones of a spray jet,” J. Eng. Phys. USSR 18, 77–80 (1970) (English translation).
    [CrossRef]
  27. R. N. Sokolov, G. D. Petrov, V. N. Funtakov, F. A. Kudryavitskii, “Measuring the range of particle sizes in a flame,” Meas. Tech. 10, 1534–1536 (1973) (English translation).
    [CrossRef]
  28. V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Determination of the maximum scattered radiation collection angle in particle size measurement” Meas. Tech. USSR 2, 268–270 (1974) (English translation).
    [CrossRef]
  29. V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Instrumentation error in measuring the spectrum of hydrosuspensions by means of the small-angle method,” Meas. Tech. USSR 7, 1125–1127 (1977) (English translation).
  30. A. L. Fymat, “Analytical inversions in remote sensing of particle size distribution 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
    [CrossRef] [PubMed]
  31. K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), pp. 232–236.
  32. C. Fox, “A generalization of the Fourier–Bessel integral transform,” Proc. London Math. Soc. Ser. 2 29, 401–452 (1929).
    [CrossRef]
  33. H. Bateman, “On the inversion of a definite integral” Proc. London Math. Soc. Ser. 4, 461–498 (1906).
    [CrossRef]
  34. C. R. Wylie, Advanced Engineering Mathematics (McGraw-Hill, New York, 1966), p. 228.
  35. L. N. Voitova, F. A. Kudrevitskii, G. D. Petrov, V. L. Rovinskii, R. N. Sokolov, “Particle Size Distribution for the Flame from a Mixture Containing Magnesium Powder,” Fiz. Goreniya Vzryva 12, 292–296 (1976) (in Russian).
  36. J. H. Koo, E. D. Hirleman, “Investigation of integral transform techniques for laser diffraction particle size analysis,” in Technical Digest of the Fall Technical Meeting of the Eastern Section of the Combustion Institute (The Combustion Institute, Pittsburgh, Pa., 1985).
  37. Lord Rayleigh, “On images formed without reflection or refraction,” Philos. Mag. 11, 214–218 (1881). Also available in Scientific Papers by Lord Rayleigh, Vol. 1: 1869–1881 (Dover, New York, 1964), pp. 513–517.
    [CrossRef]

1988 (1)

E. D. Hirleman, “Optimal scaling for Fraunhofer diffraction particle sizing instruments,” Part. Charact. 4, 128–133 (1988).

1984 (1)

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,” Opt. Eng. 23, 610–619 (1984).

1978 (2)

S. Boron, B. Waldie, “Particle sizing by forward lobe scattered intensity ratio technique: errors introduced by applying diffraction in the Mie regime,” Appl. Opt. 17, 1644–1648 (1978).
[CrossRef] [PubMed]

A. L. Fymat, “Analytical inversions in remote sensing of particle size distribution 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
[CrossRef] [PubMed]

1977 (1)

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Instrumentation error in measuring the spectrum of hydrosuspensions by means of the small-angle method,” Meas. Tech. USSR 7, 1125–1127 (1977) (English translation).

1976 (2)

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium powder,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

L. N. Voitova, F. A. Kudrevitskii, G. D. Petrov, V. L. Rovinskii, R. N. Sokolov, “Particle Size Distribution for the Flame from a Mixture Containing Magnesium Powder,” Fiz. Goreniya Vzryva 12, 292–296 (1976) (in Russian).

1974 (1)

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Determination of the maximum scattered radiation collection angle in particle size measurement” Meas. Tech. USSR 2, 268–270 (1974) (English translation).
[CrossRef]

1973 (1)

R. N. Sokolov, G. D. Petrov, V. N. Funtakov, F. A. Kudryavitskii, “Measuring the range of particle sizes in a flame,” Meas. Tech. 10, 1534–1536 (1973) (English translation).
[CrossRef]

1970 (1)

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, “Particle size distribution in various zones of a spray jet,” J. Eng. Phys. USSR 18, 77–80 (1970) (English translation).
[CrossRef]

1969 (1)

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, A. M. Kapkov, “Measurement of distribution by the method of small angles from the dimensions of particles suspended in a flow,” J. Eng. Phys. USSR 16, 300–303 (1969) (English translation).
[CrossRef]

1967 (1)

K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 3, 749–753 (1967) (English translation).

1966 (3)

J. R. Hodkinson, “Particle sizing by means of the forward scattering lobe,” Appl. Opt. 5, 839–844 (1966).
[CrossRef] [PubMed]

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 559–561 (1966) (English translation).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 514–518 (1966) (English translation).

1956 (1)

K. S. Shifrin, “Calculation of a certain class of definite integrals containing the square of a first order Bessel function,” Tr. Vses. Zoachn. Lesotekhnich. Inst. 2 (1956).

1955 (2)

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angle,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Determination of particle size distribution in polydisperse systems by means of measurement of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

1929 (1)

C. Fox, “A generalization of the Fourier–Bessel integral transform,” Proc. London Math. Soc. Ser. 2 29, 401–452 (1929).
[CrossRef]

1924 (1)

E. C. Titchmarsh, “Extension of Fourier’s integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Series 2 23, xxii–xxiv (1924).

1906 (1)

H. Bateman, “On the inversion of a definite integral” Proc. London Math. Soc. Ser. 4, 461–498 (1906).
[CrossRef]

1881 (1)

Lord Rayleigh, “On images formed without reflection or refraction,” Philos. Mag. 11, 214–218 (1881). Also available in Scientific Papers by Lord Rayleigh, Vol. 1: 1869–1881 (Dover, New York, 1964), pp. 513–517.
[CrossRef]

Abbiss, J. B.

J. B. Abbiss, “Theoretical aspects of the determination of particle size distributions from measurements of scattered light intensity,” Tech. Rep. 70151 (Royal Aircraft Establishment, Ministry of Technology, Farnborough, England, 1970).

Agrawal, Y. C.

Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).
[CrossRef]

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solution on Ill-Posed Problems (Winston, Washington, D.C., 1977).

Bateman, H.

H. Bateman, “On the inversion of a definite integral” Proc. London Math. Soc. Ser. 4, 461–498 (1906).
[CrossRef]

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

Boitova, L. N.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium powder,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

Boron, S.

S. Boron, B. Waldie, “Particle sizing by forward lobe scattered intensity ratio technique: errors introduced by applying diffraction in the Mie regime,” Appl. Opt. 17, 1644–1648 (1978).
[CrossRef] [PubMed]

Chigier, N. A.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,” Opt. Eng. 23, 610–619 (1984).

Chin, J. H.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angle,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Determination of particle size distribution in polydisperse systems by means of measurement of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

J. H. Chin, “Particle size distributions from angular variation of intensity of forward-scattered light,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1955).

Fox, C.

C. Fox, “A generalization of the Fourier–Bessel integral transform,” Proc. London Math. Soc. Ser. 2 29, 401–452 (1929).
[CrossRef]

Funtakov, V. N.

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Instrumentation error in measuring the spectrum of hydrosuspensions by means of the small-angle method,” Meas. Tech. USSR 7, 1125–1127 (1977) (English translation).

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Determination of the maximum scattered radiation collection angle in particle size measurement” Meas. Tech. USSR 2, 268–270 (1974) (English translation).
[CrossRef]

R. N. Sokolov, G. D. Petrov, V. N. Funtakov, F. A. Kudryavitskii, “Measuring the range of particle sizes in a flame,” Meas. Tech. 10, 1534–1536 (1973) (English translation).
[CrossRef]

Fymat, A. L.

A. L. Fymat, “Analytical inversions in remote sensing of particle size distribution 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
[CrossRef] [PubMed]

A. L. Fymat, K. D. Mease, “Reconstruction of the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. Fymat, V. E. Zuev, eds. (Elsevier, New York, 1978).

Hirleman, E. D.

E. D. Hirleman, “Optimal scaling for Fraunhofer diffraction particle sizing instruments,” Part. Charact. 4, 128–133 (1988).

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,” Opt. Eng. 23, 610–619 (1984).

L. V. Ruscello, E. D. Hirleman, “Determining droplet size distributions with a photodiode array,” presented at the Fall Meeting of the Western States Section of the Combustion Institute, Tempe, Ariz., 1981.

J. H. Koo, E. D. Hirleman, “Comparative study of laser diffraction droplet size analysis using integral transform techniques: factors affecting the reconstruction of droplet size distributions,” presented at the Joint Spring Meeting of the Canadian and Western States Sections of the Combustion Institute, Banff, Alberta, Canada, 1986.

J. H. Koo, E. D. Hirleman, “Investigation of integral transform techniques for laser diffraction particle size analysis,” in Technical Digest of the Fall Technical Meeting of the Eastern Section of the Combustion Institute (The Combustion Institute, Pittsburgh, Pa., 1985).

Hodkinson, J. R.

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

Kapkov, A. M.

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, A. M. Kapkov, “Measurement of distribution by the method of small angles from the dimensions of particles suspended in a flow,” J. Eng. Phys. USSR 16, 300–303 (1969) (English translation).
[CrossRef]

Kolmakov, I. B.

K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 3, 749–753 (1967) (English translation).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 514–518 (1966) (English translation).

Koo, J. H.

J. H. Koo, “Particle size analysis using integral transform techniques on Fraunhofer diffraction patterns,” Ph.D. dissertation (Arizona State University, Tempe, Ariz.; George Washington University, Washington, D.C., 1987).

J. H. Koo, “A new family of intergral transform techniques for laser light diffraction particle sizing,” presented at the Spring Meeting of the Western States Section of the Combustion Institute, Salt Lake City, Utah, 1988.

J. H. Koo, E. D. Hirleman, “Comparative study of laser diffraction droplet size analysis using integral transform techniques: factors affecting the reconstruction of droplet size distributions,” presented at the Joint Spring Meeting of the Canadian and Western States Sections of the Combustion Institute, Banff, Alberta, Canada, 1986.

J. H. Koo, E. D. Hirleman, “Investigation of integral transform techniques for laser diffraction particle size analysis,” in Technical Digest of the Fall Technical Meeting of the Eastern Section of the Combustion Institute (The Combustion Institute, Pittsburgh, Pa., 1985).

Kudrevitskii, F. A.

L. N. Voitova, F. A. Kudrevitskii, G. D. Petrov, V. L. Rovinskii, R. N. Sokolov, “Particle Size Distribution for the Flame from a Mixture Containing Magnesium Powder,” Fiz. Goreniya Vzryva 12, 292–296 (1976) (in Russian).

Kudryavitskii, F. A.

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Instrumentation error in measuring the spectrum of hydrosuspensions by means of the small-angle method,” Meas. Tech. USSR 7, 1125–1127 (1977) (English translation).

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium powder,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Determination of the maximum scattered radiation collection angle in particle size measurement” Meas. Tech. USSR 2, 268–270 (1974) (English translation).
[CrossRef]

R. N. Sokolov, G. D. Petrov, V. N. Funtakov, F. A. Kudryavitskii, “Measuring the range of particle sizes in a flame,” Meas. Tech. 10, 1534–1536 (1973) (English translation).
[CrossRef]

Mease, K. D.

A. L. Fymat, K. D. Mease, “Reconstruction of the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. Fymat, V. E. Zuev, eds. (Elsevier, New York, 1978).

Oechsle, V.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,” Opt. Eng. 23, 610–619 (1984).

Perelman, A. Y.

K. S. Shifrin, A. Y. Perelman, “Inversion of light scattering data for the determination of the spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967, pp. 131–167.

Petrov, G. D.

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Instrumentation error in measuring the spectrum of hydrosuspensions by means of the small-angle method,” Meas. Tech. USSR 7, 1125–1127 (1977) (English translation).

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium powder,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

L. N. Voitova, F. A. Kudrevitskii, G. D. Petrov, V. L. Rovinskii, R. N. Sokolov, “Particle Size Distribution for the Flame from a Mixture Containing Magnesium Powder,” Fiz. Goreniya Vzryva 12, 292–296 (1976) (in Russian).

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Determination of the maximum scattered radiation collection angle in particle size measurement” Meas. Tech. USSR 2, 268–270 (1974) (English translation).
[CrossRef]

R. N. Sokolov, G. D. Petrov, V. N. Funtakov, F. A. Kudryavitskii, “Measuring the range of particle sizes in a flame,” Meas. Tech. 10, 1534–1536 (1973) (English translation).
[CrossRef]

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, “Particle size distribution in various zones of a spray jet,” J. Eng. Phys. USSR 18, 77–80 (1970) (English translation).
[CrossRef]

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, A. M. Kapkov, “Measurement of distribution by the method of small angles from the dimensions of particles suspended in a flow,” J. Eng. Phys. USSR 16, 300–303 (1969) (English translation).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On images formed without reflection or refraction,” Philos. Mag. 11, 214–218 (1881). Also available in Scientific Papers by Lord Rayleigh, Vol. 1: 1869–1881 (Dover, New York, 1964), pp. 513–517.
[CrossRef]

Riley, J. B.

Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).
[CrossRef]

Robinskii, V. L.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium powder,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

Rovinskii, V. L.

L. N. Voitova, F. A. Kudrevitskii, G. D. Petrov, V. L. Rovinskii, R. N. Sokolov, “Particle Size Distribution for the Flame from a Mixture Containing Magnesium Powder,” Fiz. Goreniya Vzryva 12, 292–296 (1976) (in Russian).

Ruscello, L. V.

L. V. Ruscello, E. D. Hirleman, “Determining droplet size distributions with a photodiode array,” presented at the Fall Meeting of the Western States Section of the Combustion Institute, Tempe, Ariz., 1981.

Shifrin, K. S.

K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 3, 749–753 (1967) (English translation).

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 559–561 (1966) (English translation).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 514–518 (1966) (English translation).

K. S. Shifrin, “Calculation of a certain class of definite integrals containing the square of a first order Bessel function,” Tr. Vses. Zoachn. Lesotekhnich. Inst. 2 (1956).

K. S. Shifrin, A. Y. Perelman, “Inversion of light scattering data for the determination of the spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967, pp. 131–167.

K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), pp. 232–236.

Sliepcevich, C. M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Determination of particle size distribution in polydisperse systems by means of measurement of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angle,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

Sokolov, R. N.

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Instrumentation error in measuring the spectrum of hydrosuspensions by means of the small-angle method,” Meas. Tech. USSR 7, 1125–1127 (1977) (English translation).

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium powder,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

L. N. Voitova, F. A. Kudrevitskii, G. D. Petrov, V. L. Rovinskii, R. N. Sokolov, “Particle Size Distribution for the Flame from a Mixture Containing Magnesium Powder,” Fiz. Goreniya Vzryva 12, 292–296 (1976) (in Russian).

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Determination of the maximum scattered radiation collection angle in particle size measurement” Meas. Tech. USSR 2, 268–270 (1974) (English translation).
[CrossRef]

R. N. Sokolov, G. D. Petrov, V. N. Funtakov, F. A. Kudryavitskii, “Measuring the range of particle sizes in a flame,” Meas. Tech. 10, 1534–1536 (1973) (English translation).
[CrossRef]

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, “Particle size distribution in various zones of a spray jet,” J. Eng. Phys. USSR 18, 77–80 (1970) (English translation).
[CrossRef]

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, A. M. Kapkov, “Measurement of distribution by the method of small angles from the dimensions of particles suspended in a flow,” J. Eng. Phys. USSR 16, 300–303 (1969) (English translation).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solution on Ill-Posed Problems (Winston, Washington, D.C., 1977).

Titchmarsh, E. C.

E. C. Titchmarsh, “Extension of Fourier’s integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Series 2 23, xxii–xxiv (1924).

Tribus, M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angle,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Determination of particle size distribution in polydisperse systems by means of measurement of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

van de Hulst, H.C.

H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Vasilev, V. A.

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, “Particle size distribution in various zones of a spray jet,” J. Eng. Phys. USSR 18, 77–80 (1970) (English translation).
[CrossRef]

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, A. M. Kapkov, “Measurement of distribution by the method of small angles from the dimensions of particles suspended in a flow,” J. Eng. Phys. USSR 16, 300–303 (1969) (English translation).
[CrossRef]

Voitova, L. N.

L. N. Voitova, F. A. Kudrevitskii, G. D. Petrov, V. L. Rovinskii, R. N. Sokolov, “Particle Size Distribution for the Flame from a Mixture Containing Magnesium Powder,” Fiz. Goreniya Vzryva 12, 292–296 (1976) (in Russian).

Waldie, B.

S. Boron, B. Waldie, “Particle sizing by forward lobe scattered intensity ratio technique: errors introduced by applying diffraction in the Mie regime,” Appl. Opt. 17, 1644–1648 (1978).
[CrossRef] [PubMed]

Wylie, C. R.

C. R. Wylie, Advanced Engineering Mathematics (McGraw-Hill, New York, 1966), p. 228.

Appl. Opt. (2)

S. Boron, B. Waldie, “Particle sizing by forward lobe scattered intensity ratio technique: errors introduced by applying diffraction in the Mie regime,” Appl. Opt. 17, 1644–1648 (1978).
[CrossRef] [PubMed]

A. L. Fymat, “Analytical inversions in remote sensing of particle size distribution 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
[CrossRef] [PubMed]

Appl. Opt. (1)

Combust. Explosion Shock Waves (1)

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium powder,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

Fiz. Goreniya Vzryva (1)

L. N. Voitova, F. A. Kudrevitskii, G. D. Petrov, V. L. Rovinskii, R. N. Sokolov, “Particle Size Distribution for the Flame from a Mixture Containing Magnesium Powder,” Fiz. Goreniya Vzryva 12, 292–296 (1976) (in Russian).

Izv. USSR Acad. Sci. Atmos. Oceanic Phys. (3)

K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 3, 749–753 (1967) (English translation).

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 559–561 (1966) (English translation).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. USSR Acad. Sci. Atmos. Oceanic Phys. 2, 514–518 (1966) (English translation).

J. Eng. Phys. USSR (1)

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, “Particle size distribution in various zones of a spray jet,” J. Eng. Phys. USSR 18, 77–80 (1970) (English translation).
[CrossRef]

J. Eng. Phys. USSR (1)

G. D. Petrov, R. N. Sokolov, V. A. Vasilev, A. M. Kapkov, “Measurement of distribution by the method of small angles from the dimensions of particles suspended in a flow,” J. Eng. Phys. USSR 16, 300–303 (1969) (English translation).
[CrossRef]

J. Phys. Chem. (2)

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angle,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Determination of particle size distribution in polydisperse systems by means of measurement of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Meas. Tech. (1)

R. N. Sokolov, G. D. Petrov, V. N. Funtakov, F. A. Kudryavitskii, “Measuring the range of particle sizes in a flame,” Meas. Tech. 10, 1534–1536 (1973) (English translation).
[CrossRef]

Meas. Tech. USSR (2)

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Determination of the maximum scattered radiation collection angle in particle size measurement” Meas. Tech. USSR 2, 268–270 (1974) (English translation).
[CrossRef]

V. N. Funtakov, R. N. Sokolov, G. D. Petrov, F. A. Kudryavitskii, “Instrumentation error in measuring the spectrum of hydrosuspensions by means of the small-angle method,” Meas. Tech. USSR 7, 1125–1127 (1977) (English translation).

Opt. Eng. (1)

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,” Opt. Eng. 23, 610–619 (1984).

Part. Charact. (1)

E. D. Hirleman, “Optimal scaling for Fraunhofer diffraction particle sizing instruments,” Part. Charact. 4, 128–133 (1988).

Philos. Mag. (1)

Lord Rayleigh, “On images formed without reflection or refraction,” Philos. Mag. 11, 214–218 (1881). Also available in Scientific Papers by Lord Rayleigh, Vol. 1: 1869–1881 (Dover, New York, 1964), pp. 513–517.
[CrossRef]

Proc. London Math. Soc. Ser. (1)

H. Bateman, “On the inversion of a definite integral” Proc. London Math. Soc. Ser. 4, 461–498 (1906).
[CrossRef]

Proc. London Math. Soc. Ser. 2 (1)

C. Fox, “A generalization of the Fourier–Bessel integral transform,” Proc. London Math. Soc. Ser. 2 29, 401–452 (1929).
[CrossRef]

Proc. London Math. Soc. Series 2 (1)

E. C. Titchmarsh, “Extension of Fourier’s integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Series 2 23, xxii–xxiv (1924).

Tr. Vses. Zoachn. Lesotekhnich. Inst. (1)

K. S. Shifrin, “Calculation of a certain class of definite integrals containing the square of a first order Bessel function,” Tr. Vses. Zoachn. Lesotekhnich. Inst. 2 (1956).

Other (15)

K. S. Shifrin, Physical Optics of Ocean Water (American Institute of Physics, New York, 1988), pp. 232–236.

C. R. Wylie, Advanced Engineering Mathematics (McGraw-Hill, New York, 1966), p. 228.

J. H. Koo, E. D. Hirleman, “Investigation of integral transform techniques for laser diffraction particle size analysis,” in Technical Digest of the Fall Technical Meeting of the Eastern Section of the Combustion Institute (The Combustion Institute, Pittsburgh, Pa., 1985).

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

K. S. Shifrin, A. Y. Perelman, “Inversion of light scattering data for the determination of the spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967, pp. 131–167.

J. B. Abbiss, “Theoretical aspects of the determination of particle size distributions from measurements of scattered light intensity,” Tech. Rep. 70151 (Royal Aircraft Establishment, Ministry of Technology, Farnborough, England, 1970).

A. L. Fymat, K. D. Mease, “Reconstruction of the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. Fymat, V. E. Zuev, eds. (Elsevier, New York, 1978).

L. V. Ruscello, E. D. Hirleman, “Determining droplet size distributions with a photodiode array,” presented at the Fall Meeting of the Western States Section of the Combustion Institute, Tempe, Ariz., 1981.

J. H. Koo, “A new family of intergral transform techniques for laser light diffraction particle sizing,” presented at the Spring Meeting of the Western States Section of the Combustion Institute, Salt Lake City, Utah, 1988.

J. H. Koo, E. D. Hirleman, “Comparative study of laser diffraction droplet size analysis using integral transform techniques: factors affecting the reconstruction of droplet size distributions,” presented at the Joint Spring Meeting of the Canadian and Western States Sections of the Combustion Institute, Banff, Alberta, Canada, 1986.

Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).
[CrossRef]

H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

J. H. Chin, “Particle size distributions from angular variation of intensity of forward-scattered light,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1955).

J. H. Koo, “Particle size analysis using integral transform techniques on Fraunhofer diffraction patterns,” Ph.D. dissertation (Arizona State University, Tempe, Ariz.; George Washington University, Washington, D.C., 1987).

A. N. Tikhonov, V. Y. Arsenin, Solution on Ill-Posed Problems (Winston, Washington, D.C., 1977).

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Figures (2)

Fig. 1
Fig. 1

Results of numerical experiments that involve the reconstruction of particle-size distributions from simulated forward-scattering data for five integral transform solutions. The assumed particle-size distributions were lognormal on an area basis, with geometric standard deviation σ = 1.2 and mean diameters Dm of (a) 7, (b) 9, (c) 12, (d) 17, (e) 27, (f) 47, (g) 83, (h) 150, (i) 275, and (j) 506 μm. The diffraction instrument configuration assumed 64 log-scaled3 detector elements that spanned radii from rmin = 0. 1 mm to rmax = 20 mm and gave a detector–bandwidth ratio of 200, a transform lens focal length f = 300 mm, and λ = 0.6328 μm.

Fig. 2
Fig. 2

Plots of five quantitative measurements of the reconstruction performance of the five integral transform solutions for the cases shown in Fig. 1. The criteria, which are calculated from the assumed and reconstructed particle-size distributions, are (a) chi-squared, (b) the correlation coefficient, (c) D32, (d) DA0.5, and (e) the ARS. The parameter Xm is defined by Eq. (64).

Equations (74)

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i a ( θ ) = I inc / k 2 0 J 1 2 ( α θ ) / ( θ 2 - a α b - 2 ) n b ( α ) d α .
i a ( θ ) = I inc / k 2 0 k a b ( α , θ ) n b ( α ) d α ,
k a b ( α , θ ) = J 1 2 ( α θ ) / ( θ 2 - a α b - 2 ) .
ϕ ( t ) = 2 π 0 d / d t [ t J β 2 ( r t ) ] r f ( r ) d r ,
f ( s ) = - 0 J β ( s t ) Y β ( s t ) t ϕ ( t ) d t .
( 2 π k 2 / I inc ) i 3 ( θ ) = 2 π 0 [ θ J 1 2 ( α θ ) ] α n 1 ( α ) d a .
d / d θ [ ( 2 π k 2 / I inc ) i 3 ( θ ) ] = 2 π 0 d / d θ [ θ J 1 2 ( α θ ) ] α n 1 ( α ) d α ,
n 1 ( α ) = - 0 [ J 1 ( α θ ) Y 1 ( α θ ) ] θ d / d θ [ ( 2 π k 2 / I inc ) i 3 ( θ ) ] d θ ,
n 2 ( α ) = - ( k 2 / I inc ) 0 [ 2 π J 1 ( α θ ) Y 1 ( α θ ) ] d / d θ [ i 3 ( θ ) ] d θ ,
n b ( α ) = - k 2 / I inc 0 h i ( α θ ) d / d θ [ e i ( θ ) ] d θ ,
h CS ( x ) = 2 π J 1 ( x ) Y 1 ( x ) x ,
d / d θ [ e CS ( θ ) ] = d / d θ [ i 3 ( θ ) ] ,
n 2 ( α ) = - k 2 / I inc 0 h CS ( α θ ) d / d θ [ e CS ( θ ) ] d θ .
0 h i ( α θ ) d / d θ [ e i ( θ ) ] d θ = h i ( α θ ) e i ( θ ) 0 - 0 d / d θ [ h i ( α θ ) ] e i ( θ ) d θ ,
h i ( α θ ) e i ( θ ) 0 = 0 ,
n b ( α ) = - k 2 / I inc 0 - d / d θ [ h i ( α θ ) ] e i ( θ ) d θ .
h i + 1 ( α θ ) = - d / d θ [ h i ( α θ ) ] ,
h i + 1 ( x ) = - d / d x [ h i ( x ) ] = - 1 / α d / d θ [ h i ( α θ ) ] ,
d / d θ [ e i + 1 ( θ ) ] = d / d θ [ e i ( θ ) ] d θ = e i ( θ ) + C
e i + 1 ( θ ) = [ e i ( θ ) + C ] d θ ,
n 2 ( α ) = - k 2 / I inc [ h CS ( α θ ) e CS ( θ ) ] 0 - K 2 / I inc 0 - d / d θ [ h CS ( α θ ) ] e CS ( θ ) d θ ,
n 2 ( α ) = - k 2 / I inc h CS ( α θ ) d / d θ [ e SKI ( θ ) ] 0 - 0 α h SK I ( α θ ) d / d θ [ e SKI ( θ ) ] d θ ,
h SK I ( x ) - d / d x [ h CS ( x ) ] = - 2 π Y 1 ( x ) [ 2 x J 0 ( x ) - J 1 ( x ) ] - 4 ,
h SK I ( x ) = { 2.0 when x 1 4 sin ( 2 x ) when x 1 .
d / d θ [ e SK I ( θ ) ] = d / d θ [ e CS ( θ ) ] d θ = d / d θ [ i 3 ( θ ) ] d θ = i 3 ( θ ) + C ,
d / d θ [ e SK I ( θ ) ] i 3 ( θ ) - i 3 ( ) .
h CS ( 0 ) = { 0 when θ 0 2 = cos ( 2 α θ ) when θ , d / d θ [ e SK I ( θ ) = { - i 3 ( ) θ 0 0 θ .
n 1 ( α ) = - k 2 / I inc 0 h SK I ( α θ ) d / d θ [ e SK I ( θ ) ] d θ ,
i 3 ( θ ) = I inc / k 2 0 [ α θ J 1 2 ( α θ ) ] n 1 ( α ) d α .
i 3 ( θ ) = I inc / k 2 0 ( 2 / π ) cos 2 ( α θ - 3 π / 4 ) n 1 ( α ) d α             ( valid for large θ ) .
i 3 ( θ ) = F 1 ( θ ) - F 2 ( θ )             ( valid for large θ ) ,
F 1 ( θ ) = I inc / π k 2 0 n 1 ( α ) d α = constant             ( not a function of θ ) ,
F 2 ( θ ) = I inc / π k 2 0 n 1 ( α ) sin ( 2 α θ ) d α .
F 2 ( θ ) = I inc / π k 2 sin ( 2 α 0 θ ) ,
F 1 ( θ ) = I inc / π k 2 N tot D 10 .
i 3 ( ) = I inc / k 2 0 1 / π n 1 ( α ) d α ,
d / d θ [ e SK I ( θ ) ] i 3 ( θ ) - i 3 ( ) = I inc / k 2 0 α θ J 1 2 ( α θ ) n 1 ( α ) d α - 0 1 / π n 1 ( α ) d α = I inc / k 2 0 [ α θ J 1 2 ( α θ ) - 1 / π ] n 1 ( α ) d α .
n 1 ( α ) = - k 2 / I inc h SK I ( α θ ) e SK I ( θ ) 0 - k 2 / I inc 0 - d / d θ [ h SK I ( α θ ) ] e SK I ( θ ) d θ .
n 1 ( α ) = - k 2 / I inc h SK I ( α θ ) d / d θ [ e SK II ( θ ) ] 0 - k 2 / I inc 0 α h SK II ( α θ ) d / d θ [ e SK II ( θ ) ] d θ ,
h SK II ( x ) - d / d x [ h SK I ( x ) ] = 4 π { ( 1 / x - x ) J 1 ( x ) Y 1 ( x ) + J 0 ( x ) [ x Y 0 ( x ) - Y 1 ( x ) ] } - 4 / x .
h SK II ( x ) = { 0 when x 1 - 8 cos ( 2 x ) when x 1 .
d / d θ [ e SK II ( θ ) ] = d / d θ [ e SK I ( θ ) ] d θ ,
d / d θ [ e SK II ( θ ) ] = 0 θ [ i 3 ( θ ) - i 3 ( ) ] d θ .
d / d θ [ e SK II ( θ ) ] = I inc / k 2 0 { α θ 2 / 2 [ J 1 2 ( α θ ) + J 0 2 ( α θ ) ] - α θ [ J 0 ( α θ ) J 1 ( α θ ) + 1 / π ] } n 0 ( α ) d α .
d / d θ [ e SK II ( θ ) ] = I inc / π k 2 0 cos ( 2 α θ ) n 0 ( α ) d α .
n 0 ( α ) = - k 2 / I inc 0 h S K I I ( α θ ) d / d θ [ e S K I I ( θ ) ] d θ ,
E ( x ) = J 0 2 ( x ) + J 1 2 ( x ) .
E ( x ) = 1 - E ( x ) = 1 - J 0 2 ( x ) - J 1 2 ( x ) .
F c ( θ ) = 0 θ i 0 ( ζ ) 2 π ζ d ζ .
F c ( θ ) = π I inc 0 a 2 n ( a ) d a - π I inc 0 J 0 2 ( α θ ) a 2 n ( a ) d a - π I inc 0 J 1 2 ( α θ ) a 2 n ( a ) d a ,
I inc 0 J 0 2 ( α θ ) a 2 n ( a ) d a = I inc 0 a 2 n ( a ) d a - 2 0 i 1 ( θ ) d θ - i 2 ( θ ) .
0 J 0 2 ( α θ ) a 2 n ( a ) d a = λ / 2 π 2 θ 0 a n ( a ) d a + λ / 2 π 2 θ 0 a n ( a ) sin ( 2 α θ ) d a ,
0 J 1 2 ( α θ ) a 2 n ( a ) d a = λ / 2 π 2 θ 0 a n ( a ) d a - λ / 2 π 2 θ 0 a n ( a ) sin ( 2 α θ ) d a
0 J 1 2 ( α θ ) a 2 n ( a ) d a 0 J 0 2 ( α θ ) a 2 n ( α ) d a .
i 2 ( θ ) I inc / k 2 0 J 0 2 ( α θ ) n 2 ( α ) d α
I inc 0 a 2 n ( a ) d a = 2 0 θ lim i 1 ( θ ) d θ + 2 i 2 ( θ lim ) ,
I inc 0 J 0 2 ( α 0 ) a 2 n ( a ) d a = 2 0 θ lim i 1 ( θ ) d θ + 2 i 2 ( θ lim ) - i 2 ( θ ) - 2 0 i 1 ( θ ) d θ A ( θ ) .
A ( θ ) = I inc / k 2 0 J 0 2 ( α θ ) n 2 ( α ) d α .
F 0 ( θ ) ( 2 π k 2 θ ) A 0 = 2 π 0 [ θ J 0 2 ( α θ ) ] α n 1 ( α ) d α ,
d / d θ [ F 0 ( θ ) ] = 2 π 0 d / d θ [ θ J 0 2 ( α θ ) ] α n 1 ( α ) d α ,
n 1 ( α ) = - 0 J 0 ( α θ ) Y 0 ( α θ ) θ d / d θ [ F 0 ( θ ) ] d θ .
n 2 ( α ) = - 2 π k 2 0 J 0 ( α θ ) Y 0 ( α θ ) α θ d / d θ [ A θ θ ] d θ .
n 2 ( α ) = - 2 π k 2 0 d / d θ [ A θ θ ] J 0 2 ( α θ ) Y 0 ( α θ ) α θ d θ .             .
n 2 ( α ) = - k 2 / I inc 0 h MP ( α θ ) d / d θ [ e MP ( θ ) ] d θ ,
h MP ( x ) = 2 π J 0 ( x ) Y 0 ( x ) x ,
d / d θ [ e MP ( θ ) ] = d / d θ [ A ( θ ) θ ] .
A ( θ ) θ = I inc θ / k 2 0 J 0 2 ( α θ ) n 2 ( α ) d α ,
i 3 ( θ ) = I inc θ / k 2 0 J 1 2 ( α θ ) n 2 ( α ) d α .
n 2 ( α ) = - k 2 / I inc h MP ( α θ ) d / d θ [ e NEW ( θ ) ] | 0 - k 2 / I inc 0 h NEW ( α θ ) d / d θ [ e NEW ( θ ) ] .
d / d θ [ e NEW ( θ ) ] = d / d θ [ e MP ( θ ) ] d θ = d / d θ [ A ( θ ) ] d θ = A ( θ ) θ + C .
d / d θ [ e NEW ( θ ) ] A ( θ ) θ - A ( ) .
h NEW ( x ) = - d / d x [ h MP ( x ) ] = - 2 π Y 0 ( x ) [ J 0 ( x ) - 2 x J 1 ( x ) ] - 4 ,
n 1 ( α ) = - k 2 / I inc 0 h NEW ( α θ ) d / d θ [ e NEW ( θ ) ] d θ ,
x m α m θ m ,

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