Abstract

The conventional integral equation governing the electric field inside dielectric particles is reformed to bridge and to provide mathematical foundations for analytic techniques widely used to estimate such a field. The solution of the reformed equation inside a dielectric slab explained how inner-field formulations based on the Rayleigh, the Rayleigh–Gans, the quasi-static, and the Shifrin approximations can be supported by the particles. It also confirmed the approach employed to reform the integral equation. The analysis performed uncovered the differences between the depolarization tensor characterizing electrostatic fields inside the particles and the source dyadic resulting from the extraction of the singularity of the integral equation kernel.

© 1997 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 63–84.
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 131–150.
  3. M. A. Karam, D. M. LeVine, Y. M. M. Antar, A. Stogryn, “Improvement of the Rayleigh approximation for scattering from a small scatterer,” IEEE Trans. Antennas Propag. 43, 681–687 (1995).
    [CrossRef]
  4. C. Acquista, “Light scattering by tenuous particles: a generalization of the Rayleigh-Gans-Record approach,” Appl. Opt. 15, 2932–2936 (1976).
    [CrossRef] [PubMed]
  5. R. Schiffer, K. O. Thielheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2476–2483 (1979).
    [CrossRef]
  6. K. S. Shifrin, “Scattering of light in a turbid medium (Moscow, 1951),” NASA Tech. Transl. TT F-47 (1968).
  7. L. D. Cohen, R. D. Haracz, A. Cohen, C. Acquista, “Scattering of light from arbitrarily oriented finite cylinders,” Appl. Opt. 22, 742–748 (1983).
    [CrossRef] [PubMed]
  8. R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
    [CrossRef]
  9. R. D. Haracz, L. D. Cohen, A. Cohen, “Perturbation theory for scattering from dielectric spheroids and short cylinders,” Appl. Opt. 23, 436–441 (1984).
    [CrossRef] [PubMed]
  10. M. A. Karam, A. K. Fung, “Electromagnetic scattering from a layer of finite randomly oriented circular cylinders over a rough interface with application to vegetation,” Int. J. Remote Sensing 9, 1109–1134 (1989).
    [CrossRef]
  11. M. A. Karam, A. K. Fung, Y. M. M. Antar, “Electromagnetic wave scattering from some vegetation samples,” IEEE Trans. Geosci. Remote Sensing 26, 799–808 (1988).
    [CrossRef]
  12. D. M. LeVine, “The radar cross section of dielectric disks,” IEEE Trans. Antennas Propag. AP-32, 6–12 (1984).
    [CrossRef]
  13. S.-W. Lee, J. Boersma, C.-L. Law, G. A. Deschamps, “Singularity in Green’s function and its numerical evaluation,” IEEE Trans. Antennas Propag. AP-28, 311–317 (1980).
  14. J. Van Bladel, Singular Electromagnetic Fields and Sources (IEEE, New York, 1995).
  15. A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
    [CrossRef]
  16. W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesly, Reading, Mass., 1956), pp. 73–85.
  17. J. A. Straton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 3.
  18. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.
  19. M. A. Karam, “A molecular optics approach to electromagnetic wave interactions with stratified media,” J. Opt. Soc. Am. A 13, 2208–2218 (1996).
    [CrossRef]

1996

1995

M. A. Karam, D. M. LeVine, Y. M. M. Antar, A. Stogryn, “Improvement of the Rayleigh approximation for scattering from a small scatterer,” IEEE Trans. Antennas Propag. 43, 681–687 (1995).
[CrossRef]

1989

M. A. Karam, A. K. Fung, “Electromagnetic scattering from a layer of finite randomly oriented circular cylinders over a rough interface with application to vegetation,” Int. J. Remote Sensing 9, 1109–1134 (1989).
[CrossRef]

1988

M. A. Karam, A. K. Fung, Y. M. M. Antar, “Electromagnetic wave scattering from some vegetation samples,” IEEE Trans. Geosci. Remote Sensing 26, 799–808 (1988).
[CrossRef]

1985

R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
[CrossRef]

1984

1983

1980

S.-W. Lee, J. Boersma, C.-L. Law, G. A. Deschamps, “Singularity in Green’s function and its numerical evaluation,” IEEE Trans. Antennas Propag. AP-28, 311–317 (1980).

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

1979

R. Schiffer, K. O. Thielheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2476–2483 (1979).
[CrossRef]

1976

1968

K. S. Shifrin, “Scattering of light in a turbid medium (Moscow, 1951),” NASA Tech. Transl. TT F-47 (1968).

Acquista, C.

Antar, Y. M. M.

M. A. Karam, D. M. LeVine, Y. M. M. Antar, A. Stogryn, “Improvement of the Rayleigh approximation for scattering from a small scatterer,” IEEE Trans. Antennas Propag. 43, 681–687 (1995).
[CrossRef]

M. A. Karam, A. K. Fung, Y. M. M. Antar, “Electromagnetic wave scattering from some vegetation samples,” IEEE Trans. Geosci. Remote Sensing 26, 799–808 (1988).
[CrossRef]

Boersma, J.

S.-W. Lee, J. Boersma, C.-L. Law, G. A. Deschamps, “Singularity in Green’s function and its numerical evaluation,” IEEE Trans. Antennas Propag. AP-28, 311–317 (1980).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 131–150.

Cohen, A.

Cohen, L. D.

Deschamps, G. A.

S.-W. Lee, J. Boersma, C.-L. Law, G. A. Deschamps, “Singularity in Green’s function and its numerical evaluation,” IEEE Trans. Antennas Propag. AP-28, 311–317 (1980).

Fung, A. K.

M. A. Karam, A. K. Fung, “Electromagnetic scattering from a layer of finite randomly oriented circular cylinders over a rough interface with application to vegetation,” Int. J. Remote Sensing 9, 1109–1134 (1989).
[CrossRef]

M. A. Karam, A. K. Fung, Y. M. M. Antar, “Electromagnetic wave scattering from some vegetation samples,” IEEE Trans. Geosci. Remote Sensing 26, 799–808 (1988).
[CrossRef]

Haracz, R. D.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 131–150.

Karam, M. A.

M. A. Karam, “A molecular optics approach to electromagnetic wave interactions with stratified media,” J. Opt. Soc. Am. A 13, 2208–2218 (1996).
[CrossRef]

M. A. Karam, D. M. LeVine, Y. M. M. Antar, A. Stogryn, “Improvement of the Rayleigh approximation for scattering from a small scatterer,” IEEE Trans. Antennas Propag. 43, 681–687 (1995).
[CrossRef]

M. A. Karam, A. K. Fung, “Electromagnetic scattering from a layer of finite randomly oriented circular cylinders over a rough interface with application to vegetation,” Int. J. Remote Sensing 9, 1109–1134 (1989).
[CrossRef]

M. A. Karam, A. K. Fung, Y. M. M. Antar, “Electromagnetic wave scattering from some vegetation samples,” IEEE Trans. Geosci. Remote Sensing 26, 799–808 (1988).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.

Law, C.-L.

S.-W. Lee, J. Boersma, C.-L. Law, G. A. Deschamps, “Singularity in Green’s function and its numerical evaluation,” IEEE Trans. Antennas Propag. AP-28, 311–317 (1980).

Lee, S.-W.

S.-W. Lee, J. Boersma, C.-L. Law, G. A. Deschamps, “Singularity in Green’s function and its numerical evaluation,” IEEE Trans. Antennas Propag. AP-28, 311–317 (1980).

LeVine, D. M.

M. A. Karam, D. M. LeVine, Y. M. M. Antar, A. Stogryn, “Improvement of the Rayleigh approximation for scattering from a small scatterer,” IEEE Trans. Antennas Propag. 43, 681–687 (1995).
[CrossRef]

D. M. LeVine, “The radar cross section of dielectric disks,” IEEE Trans. Antennas Propag. AP-32, 6–12 (1984).
[CrossRef]

Panofsky, W. K. H.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesly, Reading, Mass., 1956), pp. 73–85.

Phillips, M.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesly, Reading, Mass., 1956), pp. 73–85.

Schiffer, R.

R. Schiffer, K. O. Thielheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2476–2483 (1979).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, “Scattering of light in a turbid medium (Moscow, 1951),” NASA Tech. Transl. TT F-47 (1968).

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.

Stogryn, A.

M. A. Karam, D. M. LeVine, Y. M. M. Antar, A. Stogryn, “Improvement of the Rayleigh approximation for scattering from a small scatterer,” IEEE Trans. Antennas Propag. 43, 681–687 (1995).
[CrossRef]

Straton, J. A.

J. A. Straton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 3.

Thielheim, K. O.

R. Schiffer, K. O. Thielheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2476–2483 (1979).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.

Van Bladel, J.

J. Van Bladel, Singular Electromagnetic Fields and Sources (IEEE, New York, 1995).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 63–84.

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Appl. Opt.

IEEE Trans. Antennas Propag.

M. A. Karam, D. M. LeVine, Y. M. M. Antar, A. Stogryn, “Improvement of the Rayleigh approximation for scattering from a small scatterer,” IEEE Trans. Antennas Propag. 43, 681–687 (1995).
[CrossRef]

D. M. LeVine, “The radar cross section of dielectric disks,” IEEE Trans. Antennas Propag. AP-32, 6–12 (1984).
[CrossRef]

S.-W. Lee, J. Boersma, C.-L. Law, G. A. Deschamps, “Singularity in Green’s function and its numerical evaluation,” IEEE Trans. Antennas Propag. AP-28, 311–317 (1980).

IEEE Trans. Geosci. Remote Sensing

M. A. Karam, A. K. Fung, Y. M. M. Antar, “Electromagnetic wave scattering from some vegetation samples,” IEEE Trans. Geosci. Remote Sensing 26, 799–808 (1988).
[CrossRef]

Int. J. Remote Sensing

M. A. Karam, A. K. Fung, “Electromagnetic scattering from a layer of finite randomly oriented circular cylinders over a rough interface with application to vegetation,” Int. J. Remote Sensing 9, 1109–1134 (1989).
[CrossRef]

J. Appl. Phys.

R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
[CrossRef]

R. Schiffer, K. O. Thielheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2476–2483 (1979).
[CrossRef]

J. Opt. Soc. Am. A

NASA Tech. Transl.

K. S. Shifrin, “Scattering of light in a turbid medium (Moscow, 1951),” NASA Tech. Transl. TT F-47 (1968).

Proc. IEEE

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Other

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesly, Reading, Mass., 1956), pp. 73–85.

J. A. Straton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 3.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 63–84.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 131–150.

J. Van Bladel, Singular Electromagnetic Fields and Sources (IEEE, New York, 1995).

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Figures (1)

Fig. 1
Fig. 1

Principal volume V δ containing the field point at .

Equations (56)

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Ēr¯=Ēir¯+εr-1VG¯¯r¯, r¯Ēr¯dr¯.
G¯¯r¯, r¯=k2I¯¯+¯k2exp-jkr¯-r¯4πr¯-r¯,
Ēr¯=Ēir¯+εr-1V-VδG¯¯r¯, r¯·Ēr¯dr¯+εr-1×VδG¯¯r¯, r¯·Ēr¯-G¯¯sr¯, r¯·Ēr¯dr¯-εr-1sδRˆds¯4πR2Ēr¯,
G¯¯sr¯, r¯=14π¯1r¯-r¯.
Ēsr¯=Ēsir¯+εr-1V-VδG¯¯sr¯, r¯·E¯sr¯dr¯+εr-1VδG¯¯sr¯, r¯Ēsr¯-Ēsr¯dr¯-εr-1SδRˆds¯4πR2·Ēsr¯.
J¯r¯=-εr-14π¯V-Vδ¯1r¯-r¯·Ēsr¯dr¯,
J¯sr¯=-εr-14π¯S+SδĒsr¯r¯-r¯·ds¯,
J¯r¯=-εr-14πS+SδRˆĒsrR2·ds¯.
Ēsr¯=E¯sir¯-εr-1SRˆds¯4πR2·Ēsr¯+εr-1VδG¯¯sr¯, r¯Ēsr¯-Ēsr¯dr¯+εr-1SδRˆds¯4πR2·Ēsr¯-Ēsr¯.
Ēsr=Esi-εr-1SRˆds¯4πr¯-r¯2·Ēsr¯.
Ēs=Ēsi-εr-1L¯¯·Ēs,
L¯¯=SRˆds¯4πR2.
L¯¯=t=13 gttxˆtxˆt,
t=13gtt=1.
Ēs=α¯¯·Ēsi=t=13αttxˆtxˆt·Ēsi=t=131εr-1gtt+1xˆtxˆt·Eˆsi.
Ēr¯=Ēir¯+εr-1r¯r¯G¯¯r¯, r¯·Ēr¯dr¯-εr-1t=13 gttxˆtxˆt,
I¯¯+εr-1t=13gttxˆtxˆt·Ēr¯=Ēir¯+εr-1r¯r¯G¯¯r¯, r¯·Ēr¯dr¯.
Ēr¯=α¯¯·Ēir¯+εr-1α¯¯·r¯r¯G¯¯r¯, r¯·Ēr¯dr¯,
Ēir¯=Ēi exp-jk¯i·r¯=Eνiνˆi+Ehihˆiexp-jk¯i·r¯,
k¯i=kxixˆ+kyiŷ-kzizˆ=ksin θicos ϕixˆ+sin ϕiŷ-cos θizˆ, hˆi=zˆ×k¯izˆ×k¯i=-kyixˆ+kxiŷkρ=-sin ϕixˆ+cos ϕiŷ, νˆi=hˆi×k¯ihˆi×k¯i=-kzicos ϕixˆ+sin ϕiŷ-kρzˆk,  kρ=k sin θi.
Ēr¯ Ēi exp-jk¯i·r¯.
Ēr α¯¯eq·Ēi,
G¯¯r¯, r¯=k28jπ2-dqx-dqyI¯¯+¯k2×exp-jqxx-x+qyy-yλ×exp-jλz-z,
α¯¯eq=t=13αtt1+jk36πν0εr-1αttxˆtxˆt,
Ēr¯α¯¯·Ēir¯+εr-1δα¯¯·r¯r¯G¯¯r¯, r¯·Ēr¯dxdy,
E¯r¯=n=0 δnE¯nr¯.
Ēnr¯=εr-1α¯¯·r¯r¯G¯¯r¯, r¯·Ēn-1r¯dxdy,  n>0,
E¯0r¯=α¯¯·Ēir¯.
Er¯=E¯- expjγz+Ē+×exp-jγzexp-jkxx+kyy,
12π-dx expjkx-qxx=δkx-qx, 12π-dy expjky-qyy=δky-qy
Ē- expjγz+Ē+ exp-jγzexp-jkxx+kyy=α¯¯·Ēi exp-jkxix+kyiy-kziz+εr-1k22jkzα¯¯·-d0dz I¯¯+¯k2exp-jkxx+kyy+λz-zĒ- expjγz+Ē+ exp-jγz.
Ē- expjγz+Ē+ exp-jγz=α¯¯·Ēi expjkziz+k2εr-12jkzi exp-jkzizα¯¯·I¯¯-k¯rk¯rk2·-dzdz expjkzizĒ- expjγz+Ē+ exp-jγz+k2εr-12jkziα¯¯·I¯¯-k¯ik¯ik2×expjkzizz0dz exp-jkzizĒ- expjγz+Ē+ exp-jγz,
k¯r=kxixˆ+kyiŷ+kzizˆ.
C¯i expjkziz+C¯r exp-jkziz+C¯¯-·Ē- expjγz+C¯¯+·Ē+ exp-jγz=0,
C¯i=α¯¯·Ēi+k2εr-12kziα¯¯·I¯¯-k¯ik¯ik2·Ē-kzi-γ+Ē+kzi+γ,
C¯r=k2ε-12kziα¯¯·I¯¯-k¯rk¯rk2·Ē-kzi+γ exp-jkzi+γd+Ē+kzi-γ exp-jkzi-γd,
C¯¯±=-I¯¯+k2εr-12kziα¯¯·I¯¯-k¯rk¯r/k2kziγ+I¯¯-k¯ik¯i/k2kzi±γ.
C¯i=C¯r=0,
C¯¯±·Ē±=0.
γ2-kzi2I¯¯+εr-1α¯¯2·γkzik¯ik¯i-k¯rk¯r+k¯ik¯i+k¯rk¯r-2k2I¯¯·Ē-=0.
k¯ik¯i=k¯-k¯-+γ-kzi2zˆzˆ+γ-kzizˆk¯-+k¯-zˆ,  k¯rk¯r=k¯-k¯-+γ+kzi2zˆzˆ+γ+kzizˆk¯-+k¯-zˆ,
k-=kxixˆ+kyiŷ-γzˆ.
k-2-k2I¯¯+εr-1α¯¯·k¯-k¯--k-2-k2zˆzˆ-k2I¯¯·Ē-=0.
Ē-=Ehĥ-+Eνvˆ-,
k-2-k2-εr-1k2ĥ-·α¯¯·ĥ-Eh=0,  k-2-k2-εr-1k-2-k2vˆ-·α¯¯·zˆzˆ·vˆ-+k2vˆ-·α¯¯·vˆ- Eν=0.
α¯¯=αρxˆxˆ+ŷŷ+αzzˆzˆ=αρI¯¯+αz-αρzˆzˆ,
k-2-k2-εr-1k2αρEh=0,
k-2-k2-εr-1k2αρ-εr-1νˆ-·zˆ2×αzk-2-αρk2Eν=0.
αzk-2-αρk2=0,
k-2-k2-εr-1k2αρ=0,
αz1+αρεr-1=αρ,
gρ-gz=-1.
2gρ+gz=1.
gρ=0, gz=1.
k-2=k2εr.
gρ=14π  xˆ·Rˆzˆ·xˆ/R2ds=14π  ŷ·Rˆzˆ·ŷ/R2ds=0, gz=14π zˆ·Rˆ/R2ds=14πdΩ=1,

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