Abstract

An inversion method is presented for recovering size information from the optical transform pattern for spherical particles. Comparisons are made with the Shifrin inversion method for two different types of particle-size distribution. For a monodispersed size distribution optical transform experiments reveal that the new method shows better capabilities in recovering information about the size and number of scatterers. Using a noise-free continuous distribution in a computer simulation, we found that the two methods are equally capable of recovering the form of the original distribution function. Then shot noise is simulated in the transform pattern, and it is shown that the new inversion method is less susceptible to error.

© 1991 Optical Society of America

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References

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  1. P. J. Wyatt, “Differential light scattering: a physical method for identifying living bacterial cells,” Appl. Opt. 7, 1879–1896 (1968).
    [CrossRef] [PubMed]
  2. A. H. Lefebvre, Gas Turbine Combustion (McGraw-Hill, New York, 1983).
  3. G. E. Bromage, “Predictions of exceptionally strong ‘4430’ in the backscattered light from reflection nebulae,” Astrophys. Space Sci. 18, 449–461 (1972).
    [CrossRef]
  4. W. L. Anderson, R. E. Beissner, “Counting and classifying small objects by far field light scattering,” Appl. Opt. 10, 1503–1508 (1971).
    [CrossRef] [PubMed]
  5. G. Mie, “Beiträge zur optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  6. K. S. Shifrin, A. Y. Perleman, “Inversion of light scattering data for the determination of spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).
  7. 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).
  8. J. R. Hodkinson, “Particle sizing by means of the forward scattering lobe,” Appl. Opt. 5, 839–844 (1966).
    [CrossRef] [PubMed]
  9. E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
    [CrossRef]
  10. R. Santer, M. Herman, “Particle size distributions from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22, 2294–2301 (1983).
    [CrossRef] [PubMed]
  11. A. Ben-David, B. M. Herman, “Method for determining particle size distribution by nonlinear inversion of backscattered radiation,” Appl. Opt. 24, 1037–1042 (1985).
    [CrossRef] [PubMed]
  12. D. Holve, S. A. Self, “Optical particle sizing for in situ measurements. Part 1,” Appl. Opt. 18, 1632–1645 (1979).
    [CrossRef] [PubMed]
  13. D. Holve, S. A. Self, “Optical particle sizing for in situ measurements. Part 2,” Appl. Opt. 18, 1646–1652 (1979).
    [CrossRef] [PubMed]
  14. A. L. Fymat, “Remote monitoring of environmental particulate pollution: a problem in inversion of first-kind integral equations,” Appl. Math. Comp. 1, 131–185 (1975).
    [CrossRef]
  15. D. Kouzelis, S. M. Candel, E. Esposito, S. Zikiout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).
  16. L. P. Bayvel, J. C. Knight, G. N. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).
  17. A. L. Fymat, K. D. Mease, “Reconstructing 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), pp. 195–231.
  18. 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 Akad. Nauk SSSR Fiz. Atmos. Okeana, 3, 749–753 (1967).
  19. N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).
  20. W. L. Anderson, “Particle identification and counting by Fourier-optical pattern recognition,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), Chap. 3.
  21. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 110.
  22. J. W. Nicholson, “Notes on Bessel functions,” Q. J. Math. 42, 216–224 (1911).
  23. E. D. Hirleman, “On-line calibration technique for laser diffraction droplet sizing instruments,” in Proceedings of the Twentieth International Gas Turbine Conference, (American Society of Mechanical Engineers, New York, 1983), paper 83-GT-232.
  24. W. J. Oliver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds., Appl. Math. Ser. 55 (U. S. GPO, Washington, D.C., 1964), Chap. 9.
  25. J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci. 28, 1400–1426 (1972).
    [CrossRef]
  26. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1927).
  27. R. N. Bracewell, The Fourier Transform and its Application (McGraw-Hill, New York, 1965), Chap. 5.

1987 (1)

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

1985 (1)

1983 (1)

1979 (2)

1975 (1)

A. L. Fymat, “Remote monitoring of environmental particulate pollution: a problem in inversion of first-kind integral equations,” Appl. Math. Comp. 1, 131–185 (1975).
[CrossRef]

1972 (2)

G. E. Bromage, “Predictions of exceptionally strong ‘4430’ in the backscattered light from reflection nebulae,” Astrophys. Space Sci. 18, 449–461 (1972).
[CrossRef]

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci. 28, 1400–1426 (1972).
[CrossRef]

1971 (1)

1968 (1)

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 Akad. Nauk SSSR Fiz. Atmos. Okeana, 3, 749–753 (1967).

1966 (1)

1911 (1)

J. W. Nicholson, “Notes on Bessel functions,” Q. J. Math. 42, 216–224 (1911).

1908 (1)

G. Mie, “Beiträge zur optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Anderson, W. L.

W. L. Anderson, R. E. Beissner, “Counting and classifying small objects by far field light scattering,” Appl. Opt. 10, 1503–1508 (1971).
[CrossRef] [PubMed]

W. L. Anderson, “Particle identification and counting by Fourier-optical pattern recognition,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), Chap. 3.

Bayvel, L. P.

L. P. Bayvel, J. C. Knight, G. N. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Beissner, R. E.

Ben-David, A.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Application (McGraw-Hill, New York, 1965), Chap. 5.

Bromage, G. E.

G. E. Bromage, “Predictions of exceptionally strong ‘4430’ in the backscattered light from reflection nebulae,” Astrophys. Space Sci. 18, 449–461 (1972).
[CrossRef]

Candel, S. M.

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikiout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Chin, J. H.

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).

Esposito, E.

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikiout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Fymat, A. L.

A. L. Fymat, “Remote monitoring of environmental particulate pollution: a problem in inversion of first-kind integral equations,” Appl. Math. Comp. 1, 131–185 (1975).
[CrossRef]

A. L. Fymat, K. D. Mease, “Reconstructing 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), pp. 195–231.

George, N.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).

Hansen, J. E.

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci. 28, 1400–1426 (1972).
[CrossRef]

Herman, B. M.

Herman, M.

Hirleman, E. D.

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

E. D. Hirleman, “On-line calibration technique for laser diffraction droplet sizing instruments,” in Proceedings of the Twentieth International Gas Turbine Conference, (American Society of Mechanical Engineers, New York, 1983), paper 83-GT-232.

Hodkinson, J. R.

Holve, D.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 110.

Knight, J. C.

L. P. Bayvel, J. C. Knight, G. N. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

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 Akad. Nauk SSSR Fiz. Atmos. Okeana, 3, 749–753 (1967).

Kouzelis, D.

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikiout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Lefebvre, A. H.

A. H. Lefebvre, Gas Turbine Combustion (McGraw-Hill, New York, 1983).

Mease, K. D.

A. L. Fymat, K. D. Mease, “Reconstructing 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), pp. 195–231.

Mie, G.

G. Mie, “Beiträge zur optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Nicholson, J. W.

J. W. Nicholson, “Notes on Bessel functions,” Q. J. Math. 42, 216–224 (1911).

Oliver, W. J.

W. J. Oliver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds., Appl. Math. Ser. 55 (U. S. GPO, Washington, D.C., 1964), Chap. 9.

Perleman, A. Y.

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

Robertson, G. N.

L. P. Bayvel, J. C. Knight, G. N. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Santer, R.

Self, S. A.

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 Akad. Nauk SSSR Fiz. Atmos. Okeana, 3, 749–753 (1967).

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

Spindel, A.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).

Thomasson, J. T.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1927).

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1927).

Wyatt, P. J.

Zikiout, S.

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikiout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

Ann. Phys. (1)

G. Mie, “Beiträge zur optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Math. Comp. (1)

A. L. Fymat, “Remote monitoring of environmental particulate pollution: a problem in inversion of first-kind integral equations,” Appl. Math. Comp. 1, 131–185 (1975).
[CrossRef]

Appl. Opt. (7)

Astrophys. Space Sci. (1)

G. E. Bromage, “Predictions of exceptionally strong ‘4430’ in the backscattered light from reflection nebulae,” Astrophys. Space Sci. 18, 449–461 (1972).
[CrossRef]

Izv. USSR Akad. Nauk SSSR Fiz. Atmos. Okeana (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 Akad. Nauk SSSR Fiz. Atmos. Okeana, 3, 749–753 (1967).

J. Atmos. Sci. (1)

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci. 28, 1400–1426 (1972).
[CrossRef]

Part. Charact. (1)

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

Q. J. Math. (1)

J. W. Nicholson, “Notes on Bessel functions,” Q. J. Math. 42, 216–224 (1911).

Other (13)

E. D. Hirleman, “On-line calibration technique for laser diffraction droplet sizing instruments,” in Proceedings of the Twentieth International Gas Turbine Conference, (American Society of Mechanical Engineers, New York, 1983), paper 83-GT-232.

W. J. Oliver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds., Appl. Math. Ser. 55 (U. S. GPO, Washington, D.C., 1964), Chap. 9.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1927).

R. N. Bracewell, The Fourier Transform and its Application (McGraw-Hill, New York, 1965), Chap. 5.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).

W. L. Anderson, “Particle identification and counting by Fourier-optical pattern recognition,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), Chap. 3.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 110.

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

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).

A. H. Lefebvre, Gas Turbine Combustion (McGraw-Hill, New York, 1983).

D. Kouzelis, S. M. Candel, E. Esposito, S. Zikiout, “Particle sizing by laser light diffraction: improvements in optics and algorithms,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

L. P. Bayvel, J. C. Knight, G. N. Robertson, “Application of the Shifrin inversion to the Malvern particle sizer,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988).

A. L. Fymat, K. D. Mease, “Reconstructing 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), pp. 195–231.

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Figures (11)

Fig. 1
Fig. 1

Geometry for calculating the optical transform pattern for a collection of spherical particles, P. At the back focal plane F of the lens L is the transform pattern recorded by the photodetector PD.19 The observation angle θ is measured radially from the lens axis to a point on the detector.

Fig. 2
Fig. 2

Quadrant representing the particles in the experiments. The quadrant represents 4000 11.0-μm radius clear apertures on a chrome background. The quadrant size measures 2.5 mm square.

Fig. 3
Fig. 3

Normalized intensity, as given by Eq. (1), weighted by the cube of the observation angle θ versus λ for 4000 particles of 11.0-μm radius.

Fig. 4
Fig. 4

Theoretical and experimental results of Shifrin's inversion formula, Eq. (A10), versus the particle radius. We used the mask shown in Fig. 2 for f (a) with θmax = 0.1875 rad.

Fig. 5
Fig. 5

Theoretical and experimental results of Fm, Eq. (22), versus the particle radius. The mask shown in Fig. 2 for f (a) with θmax = 0.1875 rad was used.

Fig. 6
Fig. 6

Expanded view of the experimental plot shown in Fig. 5.

Fig. 7
Fig. 7

Normalized intensity, as given by Eq. (1), weighted by the cube of the observation angle θ versus the observation angle. We used the Γ size distribution of Eq. (24) for f (a).

Fig. 8
Fig. 8

Results of Eq. (32) and Shifrin's inversion formula versus the particle size. The weighted intensity values shown in Fig. 7 for θ3I (θ)/I0 with θmax = 0.04 rad are used. The Γ size distribution is also plotted for reference. Results are read by using the scale on the right-hand side of the plot. In addition the differences between results of the two inversion formulas and the Γ distribution versus particle size are plotted. The values of the differences are read by using the scale on the left-hand side of the plot.

Fig. 9
Fig. 9

Plot of the log of the normalized intensity, given by Eq. (1), versus the observation angle θ. The Γ size distribution of Eq. (24) for f (a) is used.

Fig. 10
Fig. 10

Plot of the log of the normalized intensity, as shown in Fig. 9, plus 100% shot noise versus the observation angle θ.

Fig. 11
Fig. 11

Results of Eq. (32) and Shifrin's inversion formula versus the particle size. The intensity values shown in Fig. 10 for I (θ) with θmax = 0.04 rad are used. The Γ size distribution is also plotted for reference.

Tables (1)

Tables Icon

Table I Results of Fm and Shifrin's Inversion Formula, Eq. (A10), to Find N and a′ by Using Two Masks Consisting of Different Sizes and Numbers of Particles for f (a)

Equations (58)

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

I ( θ ) = I 0 0 a 2 f ( a ) J 1 2 ( k a θ ) F 2 θ 2 d a .
0 θ J n ( x θ ) J n ( y θ ) d θ = 1 x δ ( x y ) ,
J 1 2 ( k a θ ) = 2 π 0 1 J 2 ( 2 k a θ t ) ( 1 t 2 ) 1 / 2 d t .
F 2 θ 3 I ( θ ) I 0 = 2 θ π 0 a 2 f ( a ) 0 1 J 2 ( 2 k a θ t ) ( 1 t 2 ) 1 / 2 d t d a .
F 2 0 θ Σ 3 I ( θ ) I 0 J 2 ( 2 k a θ w ) d θ = 2 π 0 a 2 f ( a ) 0 1 1 ( 1 t 2 ) 1 / 2 0 θ J 2 ( 2 k a θ t ) J 2 ( 2 k a θ w ) d θ d t d a ,
F 2 0 θ 3 I ( θ ) I 0 J 2 ( 2 k a θ w ) d θ = 1 2 π k 2 a w 0 a f ( a ) 0 1 1 ( 1 t 2 ) 1 / 2 δ ( a w a t ) d t d a .
0 1 1 ( 1 t 2 ) 1 / 2 δ ( a w a t ) d t = { 0 a w > a , a ( a 2 a 2 w 2 ) 1 / 2 a w a .
F 2 0 θ 3 I ( θ ) I 0 J 2 ( 2 k a θ w ) d θ = 1 4 π k 2 a w a w a f ( a ) 2 a ( a 2 a 2 w 2 ) 1 / 2 d a .
A [ g ( z ) ] = r 2 z g ( z ) ( z 2 r 2 ) 1 / 2 d z ,
g ( z ) = 1 π z d d r A [ g ( z ) ] ( r 2 z 2 ) 1 / 2 d r .
F 2 d d a 0 θ 3 I ( θ ) I 0 a J 2 ( 2 k a θ w ) d θ = 1 4 π k 2 d d a w A [ z f ( z ) ] .
F 2 d d a 0 θ 3 I ( θ ) I 0 a 1 J 2 ( 2 k a θ w ) ( w 2 1 ) 1 / 2 d w d θ = 1 4 π k 2 1 d d a w A [ z f ( z ) ] ( w 2 1 ) 1 / 2 d w = a f ( a ) 4 k 2 .
1 J 2 ( 2 k a θ w ) ( w 2 1 ) 1 / 2 d w = π 2 J 1 ( k a θ ) Y 1 ( k a θ ) ,
2 π k 2 F 2 a d d a 0 θ 3 I ( θ ) I 0 a J 1 ( k a θ ) Y 1 ( k a θ ) d θ = f ( a ) .
f ( a ) = 2 π k 2 F 2 a 0 θ 3 I ( θ ) I 0 d d a [ a J 1 ( k a θ ) Y 1 ( k a θ ) ] d θ .
f ( a ) = N δ ( a a ) ,
I ( θ ) = I 0 N 2 ( k a 2 F ) 2 .
J 1 ( z ) ( 2 π z ) 1 / 2 cos ( z 3 π 4 ) ,
θ 3 I ( θ ) I 0 N a π k F 2 N a π k F 2 sin ( 2 k a θ ) .
2 π k 2 F 2 a d d θ [ θ 3 I ( θ ) I 0 ] θ J 1 ( k a θ ) Y 1 ( k a θ ) d θ 2 a 2 N π a 2 [ sin ( 2 k a θ 2 k a θ ) a a + sin ( 2 k a θ + 2 k a θ ) a + a ] .
2 π k 2 F 2 a d d θ [ θ 3 I ( θ ) I 0 ] θ J 1 ( k a θ ) Y 1 ( k a θ ) d θ = 2 k N π θ + N π a sin ( 4 k a θ ) .
F m ( a ) = 2 π k 2 F 2 0 θ 3 I ( θ ) I 0 a J 1 ( k a θ ) Y 1 ( k a θ ) d θ .
f ( a ) = 1 a d d a F m ( a ) = 1 a d d a a a f ( a ) d a = 1 a d d a a N a δ ( a a ) d a .
f ( a ) = { C ( a R 0 ) 1 / b 3 exp [ ( a R 0 ) b t ] a R 0 0 a < R 0 ,
r k = 0 a k f ( a ) d a , k = 0 , 1 , 2 , ,
C = 1 Γ ( 1 b 2 ) ( t b ) 1 / b 2 .
θ 3 I ( θ ) I 0 1 π k F 2 0 a f ( a ) d a 1 π k F 2 0 a f ( a ) sin ( 2 k a θ ) d a 3 4 π k 2 F 2 θ 0 f ( a ) cos ( 2 k a θ ) d a .
1 π 2 k 2 F 2 0 a f ( a ) d a [ 2 k sin ( 2 k a θ ) + 3 4 k a 2 θ 2 sin ( 2 k a θ ) 3 2 a θ sin ( 2 k a θ ) ] .
θ max [ 2 k sin ( 2 k a θ ) + 3 4 k a 2 θ 2 sin ( 2 k a θ ) 3 2 a θ sin ( 2 k a θ ) ] d θ = 0 .
θ max = m π 4 k a + Δ 2 k a ,
Δ = m π + ( m 2 π 2 6 ) 1 / 2 .
f ( a ) = 2 π k 2 F 2 a 0 θ max θ 3 I ( θ ) I 0 d d a [ a J 1 ( k a θ ) Y 1 ( k a θ ) ] d θ ,
e n = I n A n η .
N n = n n e n ,
e n T = e n + N n .
N n = N n N m % e m .
N n = n n e n n m e m % e m .
I n T = I n + n n ( A m I n ) 1 / 2 n m ( A n I m ) 1 / 2 % I m .
f ( a ) = 2 π k 2 F 2 a 0 θ 3 I ( θ ) I 0 d d θ [ θ J 1 ( k a θ ) Y 1 ( k a θ ) ] d θ .
f ( a ) = 2 π k 2 F 2 a 0 { θ 3 I ( θ ) I 0 d d θ [ θ J 1 ( k a θ ) Y 1 ( k a θ ) ] + d d θ [ θ 4 I ( θ ) I 0 J 1 ( k a θ ) Y 1 ( k a θ ) ] } d θ .
C = 2 π k 2 F 2 a 0 d d θ [ θ 4 I ( θ ) I 0 J 1 ( k a θ ) Y 1 ( k a θ ) ] d θ .
C = 2 π k 2 a 0 a 2 f ( a ) 0 d d θ [ θ 2 J 1 2 ( k a θ ) J 1 ( k a θ ) Y 1 ( k a θ ) ] d θ d a .
C = 2 π k 2 a 0 a 2 f ( a ) 0 { θ J 1 2 ( k a θ ) J 1 ( k a θ ) Y 1 ( k a θ ) + θ d d θ [ θ J 1 2 ( k a θ ) J 1 ( k a θ ) Y 1 ( k a θ ) ] } d θ d a .
C = 2 π k 2 a 0 a 2 f ( a ) [ 0 θ J 1 2 ( k a θ ) J 1 ( k a θ ) Y 1 ( k a θ ) d θ + d d k k 0 θ J 1 2 ( k a θ ) J 1 ( k a θ ) Y 1 ( k a θ ) d θ ] d a .
0 θ J 1 2 ( k a θ ) J 1 ( k a θ ) Y 1 ( k a θ ) d θ = 1 k 2 0 z J 1 2 ( z a ) J 1 ( z a ) Y 1 ( z a ) dz = 1 k 2 G ( a , a ) ,
C = 2 π a 0 a 2 f ( a ) G ( a , a ) d a + 2 π k 2 a 0 a 2 f ( a ) d d k 1 k G ( a , a ) d a .
D = 2 π k 2 F 2 a lim θ a θ a 3 I ( θ a ) I o 0 θ a d [ θ J 1 ( k a θ ) Y 1 ( k a θ ) ] .
f s ( a ) = 2 π k 2 F 2 a 0 [ θ 3 I ( θ ) I 0 ] d d θ [ θ J 1 ( k a θ ) Y 1 ( k a θ ) ] d θ .
θ max sin ( 2 k a θ ) d θ 3 8 k 2 a 2 θ max sin ( 2 k a θ max ) = 0 .
2 k a θ max m π 2 sin ( θ + m π 2 ) d θ = 0
sin ( m π 2 ) 2 k a θ max m π 2 cos ( θ ) d θ = 0 .
0 cos ( θ ) d θ = 0 ,
θ max = m π 4 k a + Δ 2 k a ,
m π 4 k a + Δ 2 k a m π 4 k a sin ( 2 k a θ ) d θ 3 2 k a ( m π + 2 Δ ) sin ( m π 2 ) cos ( Δ ) = 0 ,
sin ( Δ ) + 3 m π + 2 Δ cos ( Δ ) = 0 .
Δ 2 + 2 m π Δ + 6 = 0
Δ = m π + ( m 2 π 2 6 ) 1 / 2 .
θ max = m π 4 k a + Δ 2 k a

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