Abstract

Absorption cross-sectional measurements and model predictions of 25 disks with moderate electrical conductivity were made. The disk radii span part of the resonance region, and their thicknesses are an order of magnitude less than the skin depth. A photoacoustic apparatus, with a 35-GHz microwave source, was used to measure the absorption cross sections. The disks tend to be more absorbing when they are edge-on rather than broadside to the incident microwaves. Also they absorb most efficiently near a size parameter of 1. Model predictions generally agree with the measured magnitudes and trends in the data.

© 1998 Optical Society of America

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References

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  1. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Wiley, New York, 1969).
  2. R. Devore, D. B. Hodge, R. G. Kouyoumjian, “Backscattering cross sections of circular disks for arbitrary incidence,” J. Appl. Phys. 42, 3075–3083 (1971).
    [CrossRef]
  3. H. Weil, C. M. Chu, “Scattering and absorption of electromagnetic radiation by thin dielectric disks,” Appl. Opt. 15, 1832–1836 (1976).
    [CrossRef] [PubMed]
  4. L. E. Allan, G. C. McCormick, “Measurements of the backscatter matrix of dielectric bodies,” IEEE Trans. Antennas Propag. AP-28, 166–169 (1980).
    [CrossRef]
  5. H. Weil, C. M. Chu, “Scattering and absorption by thin flat aerosols,” Appl. Opt. 19, 2066–2071 (1980).
    [CrossRef] [PubMed]
  6. D. B. Hodge, “Scattering by circular metallic disks,” IEEE Trans. Antennas Propag. AP-28, 707–712 (1980).
    [CrossRef]
  7. D. M. LeVine, R. Meneghini, R. H. Lang, S. S. Seker, “Scattering from arbitrarily oriented dielectric disks in the physical optics regime,” J. Opt. Soc. Am. 73, 1255–1262 (1983).
    [CrossRef]
  8. J. W. Sheperd, A. R. Holt, “The scattering of electromagnetic radiation from finite dielectric circular cylinders,” J. Phys. A: Math. Nucl. Gen. 16, 651–662 (1983).
    [CrossRef]
  9. T. M. Willis, H. Weil, “Disk scattering and absorption by an improved computational method,” Appl. Opt. 26, 3987–3995 (1987).
    [CrossRef] [PubMed]
  10. Cabot Corporation, Carbon Blacks for Specialty Applications, North American Tech. Rep. S-136 (Cabot Corporation Special Blacks Division, Billerica, Mass., 1995).
  11. M. J. Venner, “The absorption and scattering cross-section of moderately conducting disks at 35 GHz,” Ph.D. dissertation (New Mexico State University, Las Cruces, N.M., 1997), pp. 10–12.
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1980), pp. 611–613.
  13. D. J. Sommers, Conductive Carbon Black in Plastics, Tech. Rep. S-39 (Cabot Corporation Special Blacks Division, Billerica, Mass., 1985), p. 2.
  14. A. R. Von Hipple, Dielectric Materials and Applications (Wiley, New York, 1954), pp. 330–331.
  15. Ref. 12, pp. 624–625.
  16. K. Gurton, C. W. Bruce, “Parametric study of the absorption cross section for a moderately conducting thin cylinder,” Appl. Opt. 34, 2822–2828 (1995).
    [CrossRef] [PubMed]
  17. Ref. 11, pp. 22–26.
  18. T. M. Willis, H. Weil, “Internal induced fields, scattering and absorption of electromagnetic radiation by disc shaped aerosols; an improved computational formulation and computer code,” Radiation Laboratory Rep. RL023618-1-T (Electrical Engineering and Computer Sciences Department, University of Michigan, Ann Arbor, Mich., 1986).
  19. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).
  20. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 365–369.
  21. Ref. 19, pp. 394–407.
  22. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 336–338.

1995

1987

1983

D. M. LeVine, R. Meneghini, R. H. Lang, S. S. Seker, “Scattering from arbitrarily oriented dielectric disks in the physical optics regime,” J. Opt. Soc. Am. 73, 1255–1262 (1983).
[CrossRef]

J. W. Sheperd, A. R. Holt, “The scattering of electromagnetic radiation from finite dielectric circular cylinders,” J. Phys. A: Math. Nucl. Gen. 16, 651–662 (1983).
[CrossRef]

1980

L. E. Allan, G. C. McCormick, “Measurements of the backscatter matrix of dielectric bodies,” IEEE Trans. Antennas Propag. AP-28, 166–169 (1980).
[CrossRef]

D. B. Hodge, “Scattering by circular metallic disks,” IEEE Trans. Antennas Propag. AP-28, 707–712 (1980).
[CrossRef]

H. Weil, C. M. Chu, “Scattering and absorption by thin flat aerosols,” Appl. Opt. 19, 2066–2071 (1980).
[CrossRef] [PubMed]

1976

1971

R. Devore, D. B. Hodge, R. G. Kouyoumjian, “Backscattering cross sections of circular disks for arbitrary incidence,” J. Appl. Phys. 42, 3075–3083 (1971).
[CrossRef]

Allan, L. E.

L. E. Allan, G. C. McCormick, “Measurements of the backscatter matrix of dielectric bodies,” IEEE Trans. Antennas Propag. AP-28, 166–169 (1980).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1980), pp. 611–613.

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Wiley, New York, 1969).

Bruce, C. W.

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).

Chu, C. M.

Devore, R.

R. Devore, D. B. Hodge, R. G. Kouyoumjian, “Backscattering cross sections of circular disks for arbitrary incidence,” J. Appl. Phys. 42, 3075–3083 (1971).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 365–369.

Gurton, K.

Hodge, D. B.

D. B. Hodge, “Scattering by circular metallic disks,” IEEE Trans. Antennas Propag. AP-28, 707–712 (1980).
[CrossRef]

R. Devore, D. B. Hodge, R. G. Kouyoumjian, “Backscattering cross sections of circular disks for arbitrary incidence,” J. Appl. Phys. 42, 3075–3083 (1971).
[CrossRef]

Holt, A. R.

J. W. Sheperd, A. R. Holt, “The scattering of electromagnetic radiation from finite dielectric circular cylinders,” J. Phys. A: Math. Nucl. Gen. 16, 651–662 (1983).
[CrossRef]

Kouyoumjian, R. G.

R. Devore, D. B. Hodge, R. G. Kouyoumjian, “Backscattering cross sections of circular disks for arbitrary incidence,” J. Appl. Phys. 42, 3075–3083 (1971).
[CrossRef]

Lang, R. H.

LeVine, D. M.

McCormick, G. C.

L. E. Allan, G. C. McCormick, “Measurements of the backscatter matrix of dielectric bodies,” IEEE Trans. Antennas Propag. AP-28, 166–169 (1980).
[CrossRef]

Meneghini, R.

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).

Seker, S. S.

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Wiley, New York, 1969).

Sheperd, J. W.

J. W. Sheperd, A. R. Holt, “The scattering of electromagnetic radiation from finite dielectric circular cylinders,” J. Phys. A: Math. Nucl. Gen. 16, 651–662 (1983).
[CrossRef]

Sommers, D. J.

D. J. Sommers, Conductive Carbon Black in Plastics, Tech. Rep. S-39 (Cabot Corporation Special Blacks Division, Billerica, Mass., 1985), p. 2.

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Wiley, New York, 1969).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 336–338.

Venner, M. J.

M. J. Venner, “The absorption and scattering cross-section of moderately conducting disks at 35 GHz,” Ph.D. dissertation (New Mexico State University, Las Cruces, N.M., 1997), pp. 10–12.

Von Hipple, A. R.

A. R. Von Hipple, Dielectric Materials and Applications (Wiley, New York, 1954), pp. 330–331.

Weil, H.

T. M. Willis, H. Weil, “Disk scattering and absorption by an improved computational method,” Appl. Opt. 26, 3987–3995 (1987).
[CrossRef] [PubMed]

H. Weil, C. M. Chu, “Scattering and absorption by thin flat aerosols,” Appl. Opt. 19, 2066–2071 (1980).
[CrossRef] [PubMed]

H. Weil, C. M. Chu, “Scattering and absorption of electromagnetic radiation by thin dielectric disks,” Appl. Opt. 15, 1832–1836 (1976).
[CrossRef] [PubMed]

T. M. Willis, H. Weil, “Internal induced fields, scattering and absorption of electromagnetic radiation by disc shaped aerosols; an improved computational formulation and computer code,” Radiation Laboratory Rep. RL023618-1-T (Electrical Engineering and Computer Sciences Department, University of Michigan, Ann Arbor, Mich., 1986).

Willis, T. M.

T. M. Willis, H. Weil, “Disk scattering and absorption by an improved computational method,” Appl. Opt. 26, 3987–3995 (1987).
[CrossRef] [PubMed]

T. M. Willis, H. Weil, “Internal induced fields, scattering and absorption of electromagnetic radiation by disc shaped aerosols; an improved computational formulation and computer code,” Radiation Laboratory Rep. RL023618-1-T (Electrical Engineering and Computer Sciences Department, University of Michigan, Ann Arbor, Mich., 1986).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1980), pp. 611–613.

Appl. Opt.

IEEE Trans. Antennas Propag.

L. E. Allan, G. C. McCormick, “Measurements of the backscatter matrix of dielectric bodies,” IEEE Trans. Antennas Propag. AP-28, 166–169 (1980).
[CrossRef]

D. B. Hodge, “Scattering by circular metallic disks,” IEEE Trans. Antennas Propag. AP-28, 707–712 (1980).
[CrossRef]

J. Appl. Phys.

R. Devore, D. B. Hodge, R. G. Kouyoumjian, “Backscattering cross sections of circular disks for arbitrary incidence,” J. Appl. Phys. 42, 3075–3083 (1971).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. A: Math. Nucl. Gen.

J. W. Sheperd, A. R. Holt, “The scattering of electromagnetic radiation from finite dielectric circular cylinders,” J. Phys. A: Math. Nucl. Gen. 16, 651–662 (1983).
[CrossRef]

Other

Cabot Corporation, Carbon Blacks for Specialty Applications, North American Tech. Rep. S-136 (Cabot Corporation Special Blacks Division, Billerica, Mass., 1995).

M. J. Venner, “The absorption and scattering cross-section of moderately conducting disks at 35 GHz,” Ph.D. dissertation (New Mexico State University, Las Cruces, N.M., 1997), pp. 10–12.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1980), pp. 611–613.

D. J. Sommers, Conductive Carbon Black in Plastics, Tech. Rep. S-39 (Cabot Corporation Special Blacks Division, Billerica, Mass., 1985), p. 2.

A. R. Von Hipple, Dielectric Materials and Applications (Wiley, New York, 1954), pp. 330–331.

Ref. 12, pp. 624–625.

Ref. 11, pp. 22–26.

T. M. Willis, H. Weil, “Internal induced fields, scattering and absorption of electromagnetic radiation by disc shaped aerosols; an improved computational formulation and computer code,” Radiation Laboratory Rep. RL023618-1-T (Electrical Engineering and Computer Sciences Department, University of Michigan, Ann Arbor, Mich., 1986).

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 365–369.

Ref. 19, pp. 394–407.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 336–338.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Wiley, New York, 1969).

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Figures (9)

Fig. 1
Fig. 1

Disk coordinate system.

Fig. 2
Fig. 2

Possible disk orientations relative to the incident wave: (a) E-axis rotation, (b) H-axis rotation, and (c) k-axis rotation.

Fig. 3
Fig. 3

Data and model comparison for all three orientations. The dashed curves are model predictions. ●, E-axis rotation; ■, H-axis rotation; ◆, k-axis rotation.

Fig. 4
Fig. 4

Total field amplitude (arbitrary units) on the disk surface: (a) broadside incidence and (b) edge-on incidence.

Fig. 5
Fig. 5

Comparison of disk absorption data to a one-dimensional thin-film model and a disk model prediction versus angle of incidence: —, film model; ┄, disk model; ■, data.

Fig. 6
Fig. 6

Diameter dependence of the absorption cross section: (a) broadside incidence and (b) edge-on incidence. Curves depict the calculations and symbols depict the data: ■, <12; ▲, 12–17; ▼, 17–22; ◆, 22–27; ●, >27 μm.

Fig. 7
Fig. 7

Calculated absorption, total scattering, and extinction for disks with a thickness of 15 μm.

Fig. 8
Fig. 8

Thickness dependence of the cross section for selected disks at 0-deg incidence. Curves depict the calculations and symbols depict the data: ka = ■, 0.348; ▲, 0.910; ▼, 1.889; ◆, 3.264.

Fig. 9
Fig. 9

Thickness dependence on the radiative properties of thin films with differing conductivity: (1) 200, (2) 400, (3) 500, and (4) 600 Ω-1 m-1.

Equations (6)

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

ε ˆ / ε 0 = ε / ε 0 + i σ ˆ ω ε 0 = n 2 - κ 2 + i 2 n κ ,
n 2 = 1 2 + 1 2 1 + σ ω ε 0 2 1 / 2 ,
κ = σ 2 ω ε 0 n .
σ a = 1 I rad d Q d t ,
σ a = Vc v RI rad d Δ p d t ,
r = r 12 + r 23   exp i β 1 + r 12 r 23   exp i β ,

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