Abstract

The full near-field Lorenz-Mie theory is described and applied to Gabor microholography to predict the intensities that are recorded on a microholographic plate. The influence of light polarization and the complex refractive index are discussed. Then, the full near-field Lorenz-Mie theory is compared with the Fraunhofer, the Fresnel, and the far-field Lorenz-Mie theories. Criteria of validity of the simplified theories are deduced.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gabor, “Microscopy by Reconstructed Wave-Fronts,” Proc. R. Soc. London Ser. A 197, 454 (1949).
    [Crossref]
  2. D. Gabor, “Microscopy by Reconstructed Wave-Fronts: II,” Proc. Phys. Soc. London Sect. B 64, 449 (1951).
    [Crossref]
  3. H. Royer, “L’utilisation de la microholographie dans les chambres à bulles,” J. Opt. 12, 347 (1981).
    [Crossref]
  4. H. Royer, “Une application de la microholographie ultra-rapide: la métrologie des brouillards,” Nouv. Rev. Opt. 5, 87 (1974).
    [Crossref]
  5. G. Haussmann, W. Lauterborn, “Determination of Size and Position of Fast Moving Gas Bubbles in Liquids by Digital 3-D Image Processing of Hologram Reconstructions,” Appl. Opt. 19, 3529 (1980).
    [Crossref] [PubMed]
  6. G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
    [Crossref]
  7. R. Menzel, F. M. Shofner, “An Investigation of Fraunhofer Holography for Velocimetry Applications,” Appl. Opt. 9, 2073 (1970).
    [Crossref] [PubMed]
  8. C. Ozkul, “Traitement optique des figures de diffraction de Fraunhofer pour une analyse avec une ligne de microphotodiode,” Opt. Acta 28, 1543 (1981).
    [Crossref]
  9. G. B. Parrent, B. J. Thompson, “On the Fraunhofer (Far-Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
    [Crossref]
  10. R. Hickling, “Holography of Liquid Droplets,” J. Opt. Soc. Am. 59, 1334 (1969).
    [Crossref]
  11. L. Lorenz, Oeuvres scientifiques de L. Lorenz, revues et annotées par H. Valentiner, Librairie Lehman et Stage, Copenhague (1898).
  12. G. Mie, “Beiträge fur Optik truber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377 (1908).
    [Crossref]
  13. P. Debye, “Der Licktdruck auf Kugeln von beliebigen Material,” Ann. Phys. IV 30, 57 (1909).
    [Crossref]
  14. H. van de Hulst, Light Scattering by Small Particles (Wiley, London, 1957).
  15. M. Kerker, The Scattering of Light and Other Electromagnetic Radiations (Academic, New York, 1969).
  16. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  17. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. 13, 97 (1982).
    [Crossref]
  18. G. Grehan, F. Slimani, G. Gouesbet, Le programme holomidi pour application de la théorie de Lorenz-Mie à la microholographie et à la détermination d’indices: rapport interne TTI/GSG/91/3/III.
  19. F. Slimani, Application des théories de Lorenz-Mie, Fresnel et Fraunhofer à l’étude des figures de diffraction de gouttelettes de brouillard enregistrées en microholographie, Thèse de 3e cycle, Le Havre (15Juin1982).
  20. G. Grehan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
    [Crossref] [PubMed]
  21. B. J. Thompson, “A New Method of Measuring Particle Size by Diffractioin Techniques,” Jpn. J. Appl. Phys. 4, 302 (1965).
  22. G. Mozer, L. S. Janicijevic, R. Beztulahu, “Fresnel Diffraction of a Circular Gaussian Wave due to a Wood Zone Plate,” J. Opt. (Paris) 12, 309 (1981).
    [Crossref]
  23. J. D. Trolinger, “Particle Field Holography,” Opt. Eng. 14, 383 (1975).
    [Crossref]

1982 (1)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. 13, 97 (1982).
[Crossref]

1981 (3)

G. Mozer, L. S. Janicijevic, R. Beztulahu, “Fresnel Diffraction of a Circular Gaussian Wave due to a Wood Zone Plate,” J. Opt. (Paris) 12, 309 (1981).
[Crossref]

H. Royer, “L’utilisation de la microholographie dans les chambres à bulles,” J. Opt. 12, 347 (1981).
[Crossref]

C. Ozkul, “Traitement optique des figures de diffraction de Fraunhofer pour une analyse avec une ligne de microphotodiode,” Opt. Acta 28, 1543 (1981).
[Crossref]

1980 (1)

1979 (1)

1976 (1)

G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
[Crossref]

1975 (1)

J. D. Trolinger, “Particle Field Holography,” Opt. Eng. 14, 383 (1975).
[Crossref]

1974 (1)

H. Royer, “Une application de la microholographie ultra-rapide: la métrologie des brouillards,” Nouv. Rev. Opt. 5, 87 (1974).
[Crossref]

1970 (1)

1969 (1)

1965 (1)

B. J. Thompson, “A New Method of Measuring Particle Size by Diffractioin Techniques,” Jpn. J. Appl. Phys. 4, 302 (1965).

1964 (1)

G. B. Parrent, B. J. Thompson, “On the Fraunhofer (Far-Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
[Crossref]

1951 (1)

D. Gabor, “Microscopy by Reconstructed Wave-Fronts: II,” Proc. Phys. Soc. London Sect. B 64, 449 (1951).
[Crossref]

1949 (1)

D. Gabor, “Microscopy by Reconstructed Wave-Fronts,” Proc. R. Soc. London Ser. A 197, 454 (1949).
[Crossref]

1909 (1)

P. Debye, “Der Licktdruck auf Kugeln von beliebigen Material,” Ann. Phys. IV 30, 57 (1909).
[Crossref]

1908 (1)

G. Mie, “Beiträge fur Optik truber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377 (1908).
[Crossref]

Beztulahu, R.

G. Mozer, L. S. Janicijevic, R. Beztulahu, “Fresnel Diffraction of a Circular Gaussian Wave due to a Wood Zone Plate,” J. Opt. (Paris) 12, 309 (1981).
[Crossref]

Debye, P.

P. Debye, “Der Licktdruck auf Kugeln von beliebigen Material,” Ann. Phys. IV 30, 57 (1909).
[Crossref]

Gabor, D.

D. Gabor, “Microscopy by Reconstructed Wave-Fronts: II,” Proc. Phys. Soc. London Sect. B 64, 449 (1951).
[Crossref]

D. Gabor, “Microscopy by Reconstructed Wave-Fronts,” Proc. R. Soc. London Ser. A 197, 454 (1949).
[Crossref]

Gouesbet, G.

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. 13, 97 (1982).
[Crossref]

G. Grehan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
[Crossref] [PubMed]

G. Grehan, F. Slimani, G. Gouesbet, Le programme holomidi pour application de la théorie de Lorenz-Mie à la microholographie et à la détermination d’indices: rapport interne TTI/GSG/91/3/III.

Grehan, G.

G. Grehan, G. Gouesbet, “Mie Theory Calculations: New Progress, with Emphasis on Particle Sizing,” Appl. Opt. 18, 3489 (1979).
[Crossref] [PubMed]

G. Grehan, F. Slimani, G. Gouesbet, Le programme holomidi pour application de la théorie de Lorenz-Mie à la microholographie et à la détermination d’indices: rapport interne TTI/GSG/91/3/III.

Gréhan, G.

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. 13, 97 (1982).
[Crossref]

Haussmann, G.

Hickling, R.

Janicijevic, L. S.

G. Mozer, L. S. Janicijevic, R. Beztulahu, “Fresnel Diffraction of a Circular Gaussian Wave due to a Wood Zone Plate,” J. Opt. (Paris) 12, 309 (1981).
[Crossref]

Kerker, M.

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

Lauterborn, W.

Lorenz, L.

L. Lorenz, Oeuvres scientifiques de L. Lorenz, revues et annotées par H. Valentiner, Librairie Lehman et Stage, Copenhague (1898).

Menzel, R.

Mie, G.

G. Mie, “Beiträge fur Optik truber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377 (1908).
[Crossref]

Mozer, G.

G. Mozer, L. S. Janicijevic, R. Beztulahu, “Fresnel Diffraction of a Circular Gaussian Wave due to a Wood Zone Plate,” J. Opt. (Paris) 12, 309 (1981).
[Crossref]

Ozkul, C.

C. Ozkul, “Traitement optique des figures de diffraction de Fraunhofer pour une analyse avec une ligne de microphotodiode,” Opt. Acta 28, 1543 (1981).
[Crossref]

Parrent, G. B.

G. B. Parrent, B. J. Thompson, “On the Fraunhofer (Far-Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
[Crossref]

Royer, H.

H. Royer, “L’utilisation de la microholographie dans les chambres à bulles,” J. Opt. 12, 347 (1981).
[Crossref]

H. Royer, “Une application de la microholographie ultra-rapide: la métrologie des brouillards,” Nouv. Rev. Opt. 5, 87 (1974).
[Crossref]

Shofner, F. M.

Slimani, F.

F. Slimani, Application des théories de Lorenz-Mie, Fresnel et Fraunhofer à l’étude des figures de diffraction de gouttelettes de brouillard enregistrées en microholographie, Thèse de 3e cycle, Le Havre (15Juin1982).

G. Grehan, F. Slimani, G. Gouesbet, Le programme holomidi pour application de la théorie de Lorenz-Mie à la microholographie et à la détermination d’indices: rapport interne TTI/GSG/91/3/III.

Stratton, J. A.

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

Thompson, B. J.

G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
[Crossref]

B. J. Thompson, “A New Method of Measuring Particle Size by Diffractioin Techniques,” Jpn. J. Appl. Phys. 4, 302 (1965).

G. B. Parrent, B. J. Thompson, “On the Fraunhofer (Far-Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
[Crossref]

Trolinger, J. D.

J. D. Trolinger, “Particle Field Holography,” Opt. Eng. 14, 383 (1975).
[Crossref]

Tyler, G. A.

G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
[Crossref]

van de Hulst, H.

H. van de Hulst, Light Scattering by Small Particles (Wiley, London, 1957).

Ann. Phys. (1)

G. Mie, “Beiträge fur Optik truber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377 (1908).
[Crossref]

Ann. Phys. IV (1)

P. Debye, “Der Licktdruck auf Kugeln von beliebigen Material,” Ann. Phys. IV 30, 57 (1909).
[Crossref]

Appl. Opt. (3)

J. Opt. (2)

H. Royer, “L’utilisation de la microholographie dans les chambres à bulles,” J. Opt. 12, 347 (1981).
[Crossref]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. 13, 97 (1982).
[Crossref]

J. Opt. (Paris) (1)

G. Mozer, L. S. Janicijevic, R. Beztulahu, “Fresnel Diffraction of a Circular Gaussian Wave due to a Wood Zone Plate,” J. Opt. (Paris) 12, 309 (1981).
[Crossref]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

B. J. Thompson, “A New Method of Measuring Particle Size by Diffractioin Techniques,” Jpn. J. Appl. Phys. 4, 302 (1965).

Nouv. Rev. Opt. (1)

H. Royer, “Une application de la microholographie ultra-rapide: la métrologie des brouillards,” Nouv. Rev. Opt. 5, 87 (1974).
[Crossref]

Opt. Acta (3)

C. Ozkul, “Traitement optique des figures de diffraction de Fraunhofer pour une analyse avec une ligne de microphotodiode,” Opt. Acta 28, 1543 (1981).
[Crossref]

G. B. Parrent, B. J. Thompson, “On the Fraunhofer (Far-Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
[Crossref]

G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
[Crossref]

Opt. Eng. (1)

J. D. Trolinger, “Particle Field Holography,” Opt. Eng. 14, 383 (1975).
[Crossref]

Proc. Phys. Soc. London Sect. B (1)

D. Gabor, “Microscopy by Reconstructed Wave-Fronts: II,” Proc. Phys. Soc. London Sect. B 64, 449 (1951).
[Crossref]

Proc. R. Soc. London Ser. A (1)

D. Gabor, “Microscopy by Reconstructed Wave-Fronts,” Proc. R. Soc. London Ser. A 197, 454 (1949).
[Crossref]

Other (6)

L. Lorenz, Oeuvres scientifiques de L. Lorenz, revues et annotées par H. Valentiner, Librairie Lehman et Stage, Copenhague (1898).

G. Grehan, F. Slimani, G. Gouesbet, Le programme holomidi pour application de la théorie de Lorenz-Mie à la microholographie et à la détermination d’indices: rapport interne TTI/GSG/91/3/III.

F. Slimani, Application des théories de Lorenz-Mie, Fresnel et Fraunhofer à l’étude des figures de diffraction de gouttelettes de brouillard enregistrées en microholographie, Thèse de 3e cycle, Le Havre (15Juin1982).

H. van de Hulst, Light Scattering by Small Particles (Wiley, London, 1957).

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

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (21)

Fig. 1
Fig. 1

Coordinate system.

Fig. 2
Fig. 2

Isophote close behind a 20-μm opaque particle.

Fig. 3
Fig. 3

Isophote close behind a 20-μm transparent particle.

Fig. 4
Fig. 4

Isophote 20 μm behind a 20-μm opaque particle.

Fig. 5
Fig. 5

Isophote 20 μm behind a 20-μm transparent particle.

Fig. 6
Fig. 6

Isophote 4.5 mm behind a 20-μm opaque particle.

Fig. 7
Fig. 7

Isophote 4.5 mm behind a 20-μm transparent particle.

Fig. 8
Fig. 8

Isophote diagram behind a 20-μm opaque particle: evolution with the particle–hologram separation between 11 and 601 μm.

Fig. 9
Fig. 9

Isophote diagram behind a 20-μm transparent particle: evolution with the particle–hologram separation between 11 and 601 μm.

Fig. 10
Fig. 10

NFLM: comparison with Fraunhofer’s theory for a 50-μm particle and a Fraunhofer parameter of 0.14.

Fig. 11
Fig. 11

NFLM: comparison with Fraunhofer’s theory for a 50-μm particle and a Fraunhofer parameter of 0.28.

Fig. 12
Fig. 12

NFLM: comparison with Fraunhofer’s theory for a 200-μm particle and a Fraunhofer parameter of 0.30.

Fig. 13
Fig. 13

NFLM: comparison with Fresnel’s theory for a 50-μm opaque particle and a Fraunhofer parameter of 3.7.

Fig. 14
Fig. 14

NFLM: comparison with Fresnel’s theory for a 50-μm opaque particle and a Fraunhofer parameter of 2.8.

Fig. 15
Fig. 15

NFLM: comparison with Fresnel’s theory for a 50-μm opaque particle and a Fraunhofer parameter of 1.4.

Fig. 16
Fig. 16

NFLM: comparison with Fresnel’s theory for a 50-μm transparent particle and a Fraunhofer parameter of 1.4.

Fig. 17
Fig. 17

NFLM: comparison with Fresnel’s theory for a 50-μm transparent particle and a Fraunhofer parameter of 0.56.

Fig. 18
Fig. 18

NFLM: comparison with Fresnel’s theory for a 50-μm transparent particle and a Fraunhofer parameter of 0.28.

Fig. 19
Fig. 19

NFLM: comparison with the far-field Lorenz-Mie (FFLM) theory for a 50-μm particle and a Fraunhofer parameter of 0.30.

Fig. 20
Fig. 20

NFLM: comparison with the FFLM theory for a 50-μm particle and a Fraunhofer parameter of 0.20.

Fig. 21
Fig. 21

NFLM: comparison with the FFLM theory for a 50-μm particle and a Fraunhofer parameter of 0.10.

Equations (56)

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

π d 2 4 λ z = F < C F 1 ,
C min < π d 2 4 λ z .
z < π d 2 4 λ C min = z max ,
z > π d 2 4 λ C F = z min ,
E 0 / H 0 = - ( μ / ɛ ) 1 / 2 ,
U T M s = + E 0 k 2 cos ϕ n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) a n ξ n ( k r ) P n 1 ( cos θ ) ,
U T E s = - H 0 k 2 sin ϕ n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) b n ξ n ( k r ) P n 1 ( cos θ ) ,
P n 1 ( cos θ ) = - sin θ d P n ( cos θ ) d cos θ ,
ξ n ( k r ) = ψ n ( k r ) + i χ n ( k r ) ,
ψ n ( k r ) = k r ψ n ( 1 ) ( k r ) = ( π k r 2 ) 1 / 2 J n + 1 / 2 ( k r ) ,
χ n ( k r ) = ( - 1 ) n ( π k r 2 ) 1 / 2 J - n - 1 / 2 ( k r ) ,
a n = ψ n ( α ) ψ n ( β ) - m ψ n ( α ) ψ n ( β ) ξ n ( α ) ψ n ( β ) - m ξ n ( α ) ψ n ( β ) ,
b n = m ψ n ( α ) ψ n ( β ) - ψ n ( α ) ψ n ( β ) m ξ n ( α ) ψ n ( β ) - ξ n ( α ) ψ n ( β ) ,
E r s = E 0 cos ϕ n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) a n [ ξ n ( k r ) + ξ n ( k r ) ] P n 1 ( cos θ ) ,
E θ s = E 0 k r cos ϕ n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) [ a n ξ n ( k r ) τ ( cos θ ) - i b n ξ n ( k r ) π n ( cos θ ) ] ,
E ϕ s = - E 0 k r sin ϕ n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) [ a n ξ n ( k r ) + π n ( cos θ ) - i b n ξ n ( k r ) τ n ( cos θ ) ] ,
H r s = E 0 ( ɛ μ ) 1 / 2 sin ϕ n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) b n [ ξ n ( k r ) + ξ n ( k r ) ] P n 1 ( cos θ ) ,
H θ s = E 0 k r ( ɛ μ ) 1 / 2 sin ϕ n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) [ - i a n ξ n ( k r ) π n ( cos θ ) + b n ξ n ( k r ) τ n ( cos θ ) ] ,
H ϕ s = E 0 k r ( ɛ μ ) 1 / 2 cos ϕ n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) [ - i a n ξ n ( k r ) τ n ( cos θ ) + b n ξ n ( k r ) π n ( cos θ ) ] ,
π n ( cos θ ) = P n 1 ( cos θ ) / sin θ ,
τ n ( cos θ ) = d P n 1 ( cos θ ) / d θ ,
d ξ n ( k r ) d r = k ξ n ( k r ) ,
d 2 ξ n ( k r ) d r 2 = k 2 ξ n ( k r ) ,
H 0 i ω μ k = - i E 0 ,
k = ω ( ɛ μ ) 1 / 2 .
E ϕ , TE = + i ω μ r U TE θ .
E r i = E 0 cos ϕ sin θ exp ( - i k r cos θ ) ,
E θ i = E 0 cos ϕ cos θ exp ( - i k r cos θ ) ,
E ϕ i = - E 0 sin ϕ exp ( - i k r cos θ ) ,
H r i = E 0 ɛ μ sin ϕ sin θ exp ( - i k r cos θ ) ,
H θ i = E 0 ɛ μ sin ϕ cos θ exp ( - i k r cos θ ) ,
H ϕ i = E 0 ɛ μ cos ϕ exp ( - i k r cos θ ) ,
E r t = E 0 cos ϕ { sin θ exp ( - i k r cos θ ) + n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) × a n [ ξ ( k r ) + ξ n ( k r ) ] P n 1 ( cos θ ) } ,
E θ t = E 0 cos ϕ k r { k r cos θ exp ( - i k r cos θ ) + n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) × [ a n ξ n ( k r ) τ n ( cos θ ) - i b n ξ n ( k r ) π n ( cos θ ) ] } ,
E ϕ t = - E 0 sin ϕ k r { k r exp ( - i k r cos θ ) + n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) × [ a n ξ n ( k r ) π n ( cos θ ) - i b n ξ n ( k r ) τ n ( cos θ ) ] } ,
H r t = E 0 ( ɛ μ ) 1 / 2 sin ϕ { sin θ exp ( - i k r cos θ ) + n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) × b n [ ξ n ( k r ) + ξ n ( k r ) ] P n 1 ( cos θ ) } ,
H θ t = E 0 k r ( ɛ μ ) 1 / 2 sin ϕ { k r cos θ exp ( i k r cos θ ) + n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) × [ - i a n ξ n ( k r ) π n ( cos θ ) + b n ξ n ( k r ) τ n ( cos θ ) ] } ,
H ϕ t = E 0 k r ( ɛ μ ) 1 / 2 cos ϕ { k r exp ( - i k r cos θ + n = 1 i n + 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) × [ - i a n ξ n ( k r ) τ n ( cos θ ) + b n ξ n ( k r ) π n ( cos θ ) ] } .
S = 1 2 Re [ E × H * ] ,
S = 1 2 Re [ cos θ ( E θ t H ϕ t * - E ϕ t H θ t * ) - sin θ ( E ϕ t H r t * - E r t H ϕ t * ) ]
P = 20. / 60 , Z = 11 , D O 2 I = 1 , 60 , Z = Z + ( I - 1 ) * P . 2 CONTINUE ,
S = 1 - π d 2 2 λ z sin ( π ρ 2 λ z ) · [ 2 J 1 ( π d ρ λ z ) π d ρ λ z ] + [ π d 2 4 λ z 2 J 1 ( π d ρ λ z ) 2 π d ρ λ z ] ,
ρ = x 2 + y 2 .
F < C F = 0.2 ,
S = A A * ,
A = 2 π exp ( i 2 π z λ ) i λ z exp ( i π ρ 2 λ z ) d / 2 exp ( i π t 2 λ z ) J 0 ( 2 π τ t λ z ) t d t .
F < C F o = 2.
F < C F p = 0.5.
E r s = 0 , E θ s = i E 0 exp ( - i k r ) k r cos ϕ n = 1 2 n + 1 n ( n + 1 )
× [ a n τ n ( cos θ ) + b n π n ( cos θ ) ] ,
E ϕ s = - i E 0 exp ( - i k r ) k r sin ϕ n = 1 2 n + 1 n ( n + 1 ) × [ a n π n ( cos θ ) + b n τ n ( cos θ ) ] ,
H r s = 0 ,
H θ s = - ( ɛ μ ) 1 / 2 E ϕ ,
H ϕ s = ( ɛ μ ) 1 / 2 E θ ,
F < C FFLM = 0.2
R = 10 d 2 ,

Metrics