Abstract

The exact solution, Born approximation, and its variational improvement are obtained for the scattering of electromagnetic waves from two randomly separated Rayleigh dielectric cylinders. This special model is used to test a recently developed vector stochastic variational principle. The variational results are shown to account accurately for the geometric polarizability of the cylinders as well as for multiple scattering and interference.

© 1983 Optical Society of America

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  1. L. Cairo and T. Kahan, Variational Techniques in Electromagnetism (Gordon and Breach, New York, 1965).
  2. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Parts I and II (McGraw-Hill, New York, 1953).
  3. D. S. Jones, "A critique of the variational methods in scattering problems," IRE Trans. Antennas Propag. AP-4, 297–301 (1956).
  4. R. W. Hart and R. A. Farrell, "A variational principle for scattering from rough surfaces," IEEE Trans. Antennas Propag. AP-25, 708–710 (1977).
  5. E. P. Gray, R. W. Hart, and R. A. Farrell, "An application of a variational principle for scattering by random rough surfaces," Radio Sci. 13, 333–348 (1978).
  6. V. Twersky, "Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders and by two parallel cylinders," J. Appl. Phys. 23, 407–414 (1952).
  7. J. A. Krill and R. A. Farrell, "Comparisons between variational, perturbational, and exact solutions for scattering from a random rough surface model," J. Opt. Soc. Am. 68, 768–774 (1978).
  8. J. A. Krill and R. H. Andreo, "Vector stochastic variational principles for electromagnetic wave scattering," IEEE Trans. Antennas Propag. AP-28, 770–776 (1980).
  9. R. H. Andreo and J. A. Krill, "Vector stochastic variational expressions for scatterers with dielectric, conductive, and magnetic properties," J. Opt. Soc. Am. 71, 978–982 (1981).
  10. The variationally invariant approximation for the small packing density limit of the many-Rayleigh-cylinder model is of the same form as the two-cylinder case (i.e., the N = 2 case), except that factors of 1/N go to zero instead of to 0.5 (as was also found7 for the hemicylinder case). Our studies of trial function effects5,18 suggest that the plane-wave Born trial field should give accurate results for ka ≲ 1. We do not present the many-cylinder results because the purpose of this study was to demonstrate that variational calculations with simple trial field can account for polarization effects, and we do not have an exact solution for the many-cylinder model.
  11. J. A. Krill, R. H. Andreo, and R. A. Farrell, "Calculational procedures for variational, Born, and exact solutions for electromagnetic scattering from two randomly separated dielectric Rayleigh cylinders," JHU/APL Tech. Rep. (Johns Hopkins University, Laurel, Md., 1983).
  12. G. Olaofe, "Scattering by two cylinders," Radio Sci. 5, 1351–1360 (1970).
  13. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1966).
  14. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  15. This result can also be obtained using the theory of distribution or generalized functions as discussed inJ. J. H. Wang, "A unified and consistent view on the singularities of the electric dyadic Green's function in the source region," IEEE Trans. Antennas Propag. AP-30, 463–468 (1982).
  16. A. D. Yaghjian, "Electric dyadic Green's functions in the source region," Proc. IEEE 68, 248–263 (1980).
  17. J. A. Krill, R. H. Andreo, and R. A. Farrell, "A computational alternative for variational expressions that involve dyadic Green functions," IEEE Trans. Antennas Propag. AP-30, 1003–1005 (1982).
  18. M. L. Feinstein and R. A. Farrell, "Trial functions in variational approximations to long wavelength scattering," J. Opt. Soc. Am. 72, 223–231 (1982).

1982

This result can also be obtained using the theory of distribution or generalized functions as discussed inJ. J. H. Wang, "A unified and consistent view on the singularities of the electric dyadic Green's function in the source region," IEEE Trans. Antennas Propag. AP-30, 463–468 (1982).

J. A. Krill, R. H. Andreo, and R. A. Farrell, "A computational alternative for variational expressions that involve dyadic Green functions," IEEE Trans. Antennas Propag. AP-30, 1003–1005 (1982).

M. L. Feinstein and R. A. Farrell, "Trial functions in variational approximations to long wavelength scattering," J. Opt. Soc. Am. 72, 223–231 (1982).

1981

1980

J. A. Krill and R. H. Andreo, "Vector stochastic variational principles for electromagnetic wave scattering," IEEE Trans. Antennas Propag. AP-28, 770–776 (1980).

1978

J. A. Krill and R. A. Farrell, "Comparisons between variational, perturbational, and exact solutions for scattering from a random rough surface model," J. Opt. Soc. Am. 68, 768–774 (1978).

E. P. Gray, R. W. Hart, and R. A. Farrell, "An application of a variational principle for scattering by random rough surfaces," Radio Sci. 13, 333–348 (1978).

1977

R. W. Hart and R. A. Farrell, "A variational principle for scattering from rough surfaces," IEEE Trans. Antennas Propag. AP-25, 708–710 (1977).

1970

G. Olaofe, "Scattering by two cylinders," Radio Sci. 5, 1351–1360 (1970).

1956

D. S. Jones, "A critique of the variational methods in scattering problems," IRE Trans. Antennas Propag. AP-4, 297–301 (1956).

1952

V. Twersky, "Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders and by two parallel cylinders," J. Appl. Phys. 23, 407–414 (1952).

Andreo, R. H.

J. A. Krill, R. H. Andreo, and R. A. Farrell, "A computational alternative for variational expressions that involve dyadic Green functions," IEEE Trans. Antennas Propag. AP-30, 1003–1005 (1982).

R. H. Andreo and J. A. Krill, "Vector stochastic variational expressions for scatterers with dielectric, conductive, and magnetic properties," J. Opt. Soc. Am. 71, 978–982 (1981).

J. A. Krill and R. H. Andreo, "Vector stochastic variational principles for electromagnetic wave scattering," IEEE Trans. Antennas Propag. AP-28, 770–776 (1980).

J. A. Krill, R. H. Andreo, and R. A. Farrell, "Calculational procedures for variational, Born, and exact solutions for electromagnetic scattering from two randomly separated dielectric Rayleigh cylinders," JHU/APL Tech. Rep. (Johns Hopkins University, Laurel, Md., 1983).

Cairo, L.

L. Cairo and T. Kahan, Variational Techniques in Electromagnetism (Gordon and Breach, New York, 1965).

Farrell, R. A.

M. L. Feinstein and R. A. Farrell, "Trial functions in variational approximations to long wavelength scattering," J. Opt. Soc. Am. 72, 223–231 (1982).

J. A. Krill, R. H. Andreo, and R. A. Farrell, "A computational alternative for variational expressions that involve dyadic Green functions," IEEE Trans. Antennas Propag. AP-30, 1003–1005 (1982).

E. P. Gray, R. W. Hart, and R. A. Farrell, "An application of a variational principle for scattering by random rough surfaces," Radio Sci. 13, 333–348 (1978).

J. A. Krill and R. A. Farrell, "Comparisons between variational, perturbational, and exact solutions for scattering from a random rough surface model," J. Opt. Soc. Am. 68, 768–774 (1978).

R. W. Hart and R. A. Farrell, "A variational principle for scattering from rough surfaces," IEEE Trans. Antennas Propag. AP-25, 708–710 (1977).

J. A. Krill, R. H. Andreo, and R. A. Farrell, "Calculational procedures for variational, Born, and exact solutions for electromagnetic scattering from two randomly separated dielectric Rayleigh cylinders," JHU/APL Tech. Rep. (Johns Hopkins University, Laurel, Md., 1983).

Feinstein, M. L.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Parts I and II (McGraw-Hill, New York, 1953).

Gray, E. P.

E. P. Gray, R. W. Hart, and R. A. Farrell, "An application of a variational principle for scattering by random rough surfaces," Radio Sci. 13, 333–348 (1978).

Hart, R. W.

E. P. Gray, R. W. Hart, and R. A. Farrell, "An application of a variational principle for scattering by random rough surfaces," Radio Sci. 13, 333–348 (1978).

R. W. Hart and R. A. Farrell, "A variational principle for scattering from rough surfaces," IEEE Trans. Antennas Propag. AP-25, 708–710 (1977).

Jones, D. S.

D. S. Jones, "A critique of the variational methods in scattering problems," IRE Trans. Antennas Propag. AP-4, 297–301 (1956).

Kahan, T.

L. Cairo and T. Kahan, Variational Techniques in Electromagnetism (Gordon and Breach, New York, 1965).

Krill, J. A.

J. A. Krill, R. H. Andreo, and R. A. Farrell, "A computational alternative for variational expressions that involve dyadic Green functions," IEEE Trans. Antennas Propag. AP-30, 1003–1005 (1982).

R. H. Andreo and J. A. Krill, "Vector stochastic variational expressions for scatterers with dielectric, conductive, and magnetic properties," J. Opt. Soc. Am. 71, 978–982 (1981).

J. A. Krill and R. H. Andreo, "Vector stochastic variational principles for electromagnetic wave scattering," IEEE Trans. Antennas Propag. AP-28, 770–776 (1980).

J. A. Krill and R. A. Farrell, "Comparisons between variational, perturbational, and exact solutions for scattering from a random rough surface model," J. Opt. Soc. Am. 68, 768–774 (1978).

J. A. Krill, R. H. Andreo, and R. A. Farrell, "Calculational procedures for variational, Born, and exact solutions for electromagnetic scattering from two randomly separated dielectric Rayleigh cylinders," JHU/APL Tech. Rep. (Johns Hopkins University, Laurel, Md., 1983).

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Parts I and II (McGraw-Hill, New York, 1953).

Olaofe, G.

G. Olaofe, "Scattering by two cylinders," Radio Sci. 5, 1351–1360 (1970).

Twersky, V.

V. Twersky, "Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders and by two parallel cylinders," J. Appl. Phys. 23, 407–414 (1952).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wang, J. J. H.

This result can also be obtained using the theory of distribution or generalized functions as discussed inJ. J. H. Wang, "A unified and consistent view on the singularities of the electric dyadic Green's function in the source region," IEEE Trans. Antennas Propag. AP-30, 463–468 (1982).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1966).

Yaghjian, A. D.

A. D. Yaghjian, "Electric dyadic Green's functions in the source region," Proc. IEEE 68, 248–263 (1980).

IEEE Trans. Antennas Propag.

This result can also be obtained using the theory of distribution or generalized functions as discussed inJ. J. H. Wang, "A unified and consistent view on the singularities of the electric dyadic Green's function in the source region," IEEE Trans. Antennas Propag. AP-30, 463–468 (1982).

J. A. Krill, R. H. Andreo, and R. A. Farrell, "A computational alternative for variational expressions that involve dyadic Green functions," IEEE Trans. Antennas Propag. AP-30, 1003–1005 (1982).

R. W. Hart and R. A. Farrell, "A variational principle for scattering from rough surfaces," IEEE Trans. Antennas Propag. AP-25, 708–710 (1977).

J. A. Krill and R. H. Andreo, "Vector stochastic variational principles for electromagnetic wave scattering," IEEE Trans. Antennas Propag. AP-28, 770–776 (1980).

IRE Trans. Antennas Propag.

D. S. Jones, "A critique of the variational methods in scattering problems," IRE Trans. Antennas Propag. AP-4, 297–301 (1956).

J. Appl. Phys.

V. Twersky, "Multiple scattering of radiation by an arbitrary planar configuration of parallel cylinders and by two parallel cylinders," J. Appl. Phys. 23, 407–414 (1952).

J. Opt. Soc. Am.

Radio Sci.

E. P. Gray, R. W. Hart, and R. A. Farrell, "An application of a variational principle for scattering by random rough surfaces," Radio Sci. 13, 333–348 (1978).

G. Olaofe, "Scattering by two cylinders," Radio Sci. 5, 1351–1360 (1970).

Other

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1966).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

The variationally invariant approximation for the small packing density limit of the many-Rayleigh-cylinder model is of the same form as the two-cylinder case (i.e., the N = 2 case), except that factors of 1/N go to zero instead of to 0.5 (as was also found7 for the hemicylinder case). Our studies of trial function effects5,18 suggest that the plane-wave Born trial field should give accurate results for ka ≲ 1. We do not present the many-cylinder results because the purpose of this study was to demonstrate that variational calculations with simple trial field can account for polarization effects, and we do not have an exact solution for the many-cylinder model.

J. A. Krill, R. H. Andreo, and R. A. Farrell, "Calculational procedures for variational, Born, and exact solutions for electromagnetic scattering from two randomly separated dielectric Rayleigh cylinders," JHU/APL Tech. Rep. (Johns Hopkins University, Laurel, Md., 1983).

L. Cairo and T. Kahan, Variational Techniques in Electromagnetism (Gordon and Breach, New York, 1965).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Parts I and II (McGraw-Hill, New York, 1953).

A. D. Yaghjian, "Electric dyadic Green's functions in the source region," Proc. IEEE 68, 248–263 (1980).

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