Abstract

Exact-image theory is applied to the problem of electromagnetic wave scattering from a small dielectric object above an interface separating two isotropic and homogeneous media. The object is assumed to be electrically small and far enough from the interface so that its internal field can be assumed to be uniform. The approach is applicable to any scatter that can be represented by an electric dipole.

© 1991 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  4. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  5. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  6. A. W. Glisson, “An integral equation for electromagnetic scattering from homogeneous dielectric bodies,”IEEE Trans. Antennas Propag. AP-32, 173–175 (1984).
    [CrossRef]
  7. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 309 (1987).
  8. T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
    [CrossRef]
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  10. Y. Rahmat-Samii, R. Mittra, P. Parhami, “Evaluation of Sommerfeld integrals for lossy half-space problems,” Electromagnetics 1, 1–28 (1981).
    [CrossRef]
  11. I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part I: vertical magnetic dipole,”IEEE Trans. Antennas Propag. AP-32, 126–133 (1984).
    [CrossRef]
  12. I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part II: vertical electric dipole,”IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
    [CrossRef]
  13. I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. AP-32, 1027–1032(1984).
    [CrossRef]
  14. G. J. Burke, E. K. Miller, “A comparison of several methods for evaluating the field of a source near an interface,” in Proceedings of the URSI Radio Science Meeting (URSI, Blacksburg, Va., 1987), p. 102.
  15. I. V. Lindell, E. Alanen, K. Mannersalo, “Exact image method for impedance computation of antennas above the ground,”IEEE Trans. Antennas Propag. AP-33, 937–945 (1985).
    [CrossRef]
  16. I. V. Lindell, E. Alanen, A. T. Hujanen, “Exact image theory for the analysis of microstrip structures,”J. Electromag. Waves Appl. 1, 95–108 (1987).
  17. I. V. Lindell, K. I. Nikoskinen, E. Alanen, A. T. Hujanen, “Scalar Green function method for microstrip antenna analysis based on the exact image theory,” Ann. Telecommun. 44, 533–542 (1989).
  18. K. O. Muinonen, A. H. Sihvola, I. V. Lindell, K. A. Lumme, “Scattering by a small object close to an interface. II. Study of backscattering,” J. Opt. Soc. Am. A 8, 477–482 (1991).
    [CrossRef]
  19. I. V. Lindell, “Exact-image method for Gaussian-beam problems involving a planar interface,” J. Opt. Soc. Am. A 4, 2185–2190 (1987).
    [CrossRef]
  20. I. V. Lindell, “On the integration of image sources in exact image method of field analysis,”J. Electron. Waves. Appl. 2, 607–619 (1988).

1991 (1)

1989 (1)

I. V. Lindell, K. I. Nikoskinen, E. Alanen, A. T. Hujanen, “Scalar Green function method for microstrip antenna analysis based on the exact image theory,” Ann. Telecommun. 44, 533–542 (1989).

1988 (1)

I. V. Lindell, “On the integration of image sources in exact image method of field analysis,”J. Electron. Waves. Appl. 2, 607–619 (1988).

1987 (4)

I. V. Lindell, E. Alanen, A. T. Hujanen, “Exact image theory for the analysis of microstrip structures,”J. Electromag. Waves Appl. 1, 95–108 (1987).

I. V. Lindell, “Exact-image method for Gaussian-beam problems involving a planar interface,” J. Opt. Soc. Am. A 4, 2185–2190 (1987).
[CrossRef]

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 309 (1987).

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

1985 (1)

I. V. Lindell, E. Alanen, K. Mannersalo, “Exact image method for impedance computation of antennas above the ground,”IEEE Trans. Antennas Propag. AP-33, 937–945 (1985).
[CrossRef]

1984 (4)

A. W. Glisson, “An integral equation for electromagnetic scattering from homogeneous dielectric bodies,”IEEE Trans. Antennas Propag. AP-32, 173–175 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part I: vertical magnetic dipole,”IEEE Trans. Antennas Propag. AP-32, 126–133 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part II: vertical electric dipole,”IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. AP-32, 1027–1032(1984).
[CrossRef]

1981 (1)

Y. Rahmat-Samii, R. Mittra, P. Parhami, “Evaluation of Sommerfeld integrals for lossy half-space problems,” Electromagnetics 1, 1–28 (1981).
[CrossRef]

1975 (1)

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Alanen, E.

I. V. Lindell, K. I. Nikoskinen, E. Alanen, A. T. Hujanen, “Scalar Green function method for microstrip antenna analysis based on the exact image theory,” Ann. Telecommun. 44, 533–542 (1989).

I. V. Lindell, E. Alanen, A. T. Hujanen, “Exact image theory for the analysis of microstrip structures,”J. Electromag. Waves Appl. 1, 95–108 (1987).

I. V. Lindell, E. Alanen, K. Mannersalo, “Exact image method for impedance computation of antennas above the ground,”IEEE Trans. Antennas Propag. AP-33, 937–945 (1985).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part I: vertical magnetic dipole,”IEEE Trans. Antennas Propag. AP-32, 126–133 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part II: vertical electric dipole,”IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. AP-32, 1027–1032(1984).
[CrossRef]

Asano, S.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 309 (1987).

Bohren, C. F.

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

Burke, G. J.

G. J. Burke, E. K. Miller, “A comparison of several methods for evaluating the field of a source near an interface,” in Proceedings of the URSI Radio Science Meeting (URSI, Blacksburg, Va., 1987), p. 102.

Glisson, A. W.

A. W. Glisson, “An integral equation for electromagnetic scattering from homogeneous dielectric bodies,”IEEE Trans. Antennas Propag. AP-32, 173–175 (1984).
[CrossRef]

Huffman, D. R.

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

Hujanen, A. T.

I. V. Lindell, K. I. Nikoskinen, E. Alanen, A. T. Hujanen, “Scalar Green function method for microstrip antenna analysis based on the exact image theory,” Ann. Telecommun. 44, 533–542 (1989).

I. V. Lindell, E. Alanen, A. T. Hujanen, “Exact image theory for the analysis of microstrip structures,”J. Electromag. Waves Appl. 1, 95–108 (1987).

Inoue, M.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Kerker, M.

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

Lindell, I. V.

K. O. Muinonen, A. H. Sihvola, I. V. Lindell, K. A. Lumme, “Scattering by a small object close to an interface. II. Study of backscattering,” J. Opt. Soc. Am. A 8, 477–482 (1991).
[CrossRef]

I. V. Lindell, K. I. Nikoskinen, E. Alanen, A. T. Hujanen, “Scalar Green function method for microstrip antenna analysis based on the exact image theory,” Ann. Telecommun. 44, 533–542 (1989).

I. V. Lindell, “On the integration of image sources in exact image method of field analysis,”J. Electron. Waves. Appl. 2, 607–619 (1988).

I. V. Lindell, “Exact-image method for Gaussian-beam problems involving a planar interface,” J. Opt. Soc. Am. A 4, 2185–2190 (1987).
[CrossRef]

I. V. Lindell, E. Alanen, A. T. Hujanen, “Exact image theory for the analysis of microstrip structures,”J. Electromag. Waves Appl. 1, 95–108 (1987).

I. V. Lindell, E. Alanen, K. Mannersalo, “Exact image method for impedance computation of antennas above the ground,”IEEE Trans. Antennas Propag. AP-33, 937–945 (1985).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part I: vertical magnetic dipole,”IEEE Trans. Antennas Propag. AP-32, 126–133 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. AP-32, 1027–1032(1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part II: vertical electric dipole,”IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
[CrossRef]

Lumme, K. A.

Mannersalo, K.

I. V. Lindell, E. Alanen, K. Mannersalo, “Exact image method for impedance computation of antennas above the ground,”IEEE Trans. Antennas Propag. AP-33, 937–945 (1985).
[CrossRef]

Miller, E. K.

G. J. Burke, E. K. Miller, “A comparison of several methods for evaluating the field of a source near an interface,” in Proceedings of the URSI Radio Science Meeting (URSI, Blacksburg, Va., 1987), p. 102.

Mittra, R.

Y. Rahmat-Samii, R. Mittra, P. Parhami, “Evaluation of Sommerfeld integrals for lossy half-space problems,” Electromagnetics 1, 1–28 (1981).
[CrossRef]

Muinonen, K. O.

Nikoskinen, K. I.

I. V. Lindell, K. I. Nikoskinen, E. Alanen, A. T. Hujanen, “Scalar Green function method for microstrip antenna analysis based on the exact image theory,” Ann. Telecommun. 44, 533–542 (1989).

Ohtaka, K.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Parhami, P.

Y. Rahmat-Samii, R. Mittra, P. Parhami, “Evaluation of Sommerfeld integrals for lossy half-space problems,” Electromagnetics 1, 1–28 (1981).
[CrossRef]

Rahmat-Samii, Y.

Y. Rahmat-Samii, R. Mittra, P. Parhami, “Evaluation of Sommerfeld integrals for lossy half-space problems,” Electromagnetics 1, 1–28 (1981).
[CrossRef]

Sihvola, A. H.

Stratton, J. A.

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

Takemori, T.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

van de Hulst, H. C.

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

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 309 (1987).

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Yamamoto, G.

Ann. Telecommun. (1)

I. V. Lindell, K. I. Nikoskinen, E. Alanen, A. T. Hujanen, “Scalar Green function method for microstrip antenna analysis based on the exact image theory,” Ann. Telecommun. 44, 533–542 (1989).

Appl. Opt. (1)

Electromagnetics (1)

Y. Rahmat-Samii, R. Mittra, P. Parhami, “Evaluation of Sommerfeld integrals for lossy half-space problems,” Electromagnetics 1, 1–28 (1981).
[CrossRef]

IEEE Trans. Antennas Propag. (5)

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part I: vertical magnetic dipole,”IEEE Trans. Antennas Propag. AP-32, 126–133 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part II: vertical electric dipole,”IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for the Sommerfeld half-space problem, part III: general formulation,”IEEE Trans. Antennas Propag. AP-32, 1027–1032(1984).
[CrossRef]

I. V. Lindell, E. Alanen, K. Mannersalo, “Exact image method for impedance computation of antennas above the ground,”IEEE Trans. Antennas Propag. AP-33, 937–945 (1985).
[CrossRef]

A. W. Glisson, “An integral equation for electromagnetic scattering from homogeneous dielectric bodies,”IEEE Trans. Antennas Propag. AP-32, 173–175 (1984).
[CrossRef]

J. Electromag. Waves Appl. (1)

I. V. Lindell, E. Alanen, A. T. Hujanen, “Exact image theory for the analysis of microstrip structures,”J. Electromag. Waves Appl. 1, 95–108 (1987).

J. Electron. Waves. Appl. (1)

I. V. Lindell, “On the integration of image sources in exact image method of field analysis,”J. Electron. Waves. Appl. 2, 607–619 (1988).

J. Opt. Soc. Am. A (2)

J. Phys. Soc. Jpn. (1)

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Physica (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A, 309 (1987).

Other (5)

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

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

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

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

G. J. Burke, E. K. Miller, “A comparison of several methods for evaluating the field of a source near an interface,” in Proceedings of the URSI Radio Science Meeting (URSI, Blacksburg, Va., 1987), p. 102.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem: a scatterer above an interface separating two half-spaces of different media with the definition of TE and TM field vectors.

Fig. 2
Fig. 2

Real parts of the Green dyadic K(zuz) components of Eq. (34) as functions of the normalized scatterer height kh for ground parameters μ = 1 and = 1.7. Note that for large kh values Kz decays faster than Kt. Also, the asymptotic dependence of the Green dyadic is according to Eq. (36).

Fig. 3
Fig. 3

Same as Fig. 2 for the imaginary part.

Fig. 4
Fig. 4

Same as Fig. 2 for =2.4.

Fig. 5
Fig. 5

Same as Fig. 3 for = 2.4.

Equations (43)

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E i ( r ) = E 0 exp ( - j k · r ) ,
E r ( r ) = R · E 0 exp ( - j k c r ) .
C = J - 2 u z u z ,
k c = C · k = k - 2 u z ( u z · k ) .
R = C · [ R TE u z u z × × kk ( u z × k ) 2 + R TE kk × × ( u z u z × × kk ) k 2 ( u z × k ) 2 ] ,
R TE = μ β - β 1 μ β + β 1 ,
R TM = - β - β 1 β + β 1 ,
β = [ k 2 - ( u z × k ) 2 ] 1 / 2 , β 1 = [ μ k 2 - ( u z × k ) 2 ] 1 / 2 ,
k 2 = ω 2 μ 0 0 ,
p = α · E t ,
J ( r ) = j ω p δ ( r - u z h ) .
J i ( r , ζ ) = [ f TM ( ζ ) J + 1 k 2 g ( ζ ) u z u z × × ] · J c ( r ) ,
f TE ( ζ ) = j B f ( μ , p ) + μ - 1 μ + 1 δ + ( p ) ,
f TM ( ζ ) = - j B f ( , p ) - - 1 + 1 δ + ( p ) ,
g ( ζ ) = - μ 2 - 1 μ ( μ - ) j B f ( μ , p ) - 2 - 1 ( , μ ) j B f ( , p ) ,
f ( γ , p ) = - 8 γ γ 2 - 1 n = 1 n ( γ - 1 γ + 1 ) n J 2 n ( p ) p U + ( p ) ,
p = j B ζ
δ + ( 0 ) = 0 ,
J c ( r ) = C · J ( C · r ) = ( J - 2 u z u z ) · j ω p δ ( r + u z h ) .
E t ( r ) = E i ( r ) + E r ( r ) + E s ( r ) ,
E s ( r ) = - j ω μ 0 V C G ( r - r + u z ζ ) · J i ( r , ζ ) d V d ζ ,
G ( r ) = ( J + 1 k 2 ) G ( r ) ,
G ( r ) = exp [ - j k D ( r ) ] 4 π D ( r ) ,             D ( r ) = r · r .
K TM ( r ) = C G ( r + u z ζ ) f TM ( ζ ) d ζ ,
L ( r ) = C G ( r + u z ζ ) g ( ζ ) d ζ
K ( r ) = ( J + 1 k 2 ) K TM ( r ) + 1 k 2 ( u z u z × × ) L ( r ) .
E s ( r ) = - j ω μ 0 V K ( r - r ) · J c ( r ) d V .
E s ( r ) = ω 2 μ 0 K ( r + u z h ) · ( J - 2 u z u z ) · p ,
p = α · E t ( u z h ) = α · [ E i ( u z h ) + E r ( u z h ) ] + ω 2 μ 0 α · K ( 2 u z h ) · C · p .
p = [ J - ω 2 μ 0 α · K ( 2 u z h ) · C ] - 1 · α · [ E i ( u z h ) + E r ( u z h ) ] .
arg { ζ } = - π / 2 - arg { μ - 1 } .
f ( r ) = [ - k 2 u r u r - ( J - 3 u r u r ) 1 + j k r r 2 ] f ( r ) ,
K ( z u z ) = J t ( K TM ( z ) + C G ( z + ζ ) { 1 [ j k ( z + ζ ) ] 2 + 1 j k ( z + ζ ) } [ f TM ( ζ ) + g ( ζ ) ] d ζ ) - 2 u z u z ( C G ( z + ζ ) { 1 [ j k ( z + ζ ) ] 2 + 1 j k ( z + ζ ) } f TM ( ζ ) d ζ ) ,
K ( z u z ) = J t ( - - 1 + 1 G ( z ) [ 1 + 1 + j k z ( j k z ) 2 ] + 8 2 - 1 0 G ( z - j p / B ) { 1 + ( 2 + 1 ) · [ 1 + j k ( z - j p / B ) ] [ j k ( z - j p / B ) ] 2 } n = 0 n ( - 1 + 1 ) n J 2 n ( p ) p d p ) - 2 u z u z { - - 1 + 1 G ( z ) 1 + j k z ( j k z ) 2 + 8 2 - 1 0 G ( z - j p / B ) × 1 + j k ( z - j p / B ) [ j k ( z - j p / B ) ] 2 n = 0 n ( - 1 + 1 ) n J 2 n ( p ) p d p } .
K ( z u z ) = J t K t ( z ) + u z u z K z ( z ) .
K TM ( 2 h u z ) exp ( - j k 2 h ) 4 π 2 h C exp ( - j k ζ ) f TM ( ζ ) d ζ = G ( 2 h ) R TM ( k ) ,
K ( 2 h u z ) G ( 2 h ) R TM ( k ) J t ,
R TM ( k ) = R TE ( k ) = 1 - 1 + .
p = α [ 1 - ω 2 μ 0 α K TM ( 2 h u z ) ] - 1 ( E i + E r ) α [ 1 + ω 2 μ 0 α R TM G ( 2 h ) [ E i + E r ) = α ( E i + E r ) + ω 2 μ 0 α R TM G ( 2 h ) α ( E i + E r ) ,
f TM ( ζ ) = lim 1 [ - j B f ( , j B ζ ) - - 1 + 1 δ + ( ζ ) ] = lim 1 - B 2 ( + 1 ) 2 ζ = 0 ,
g ( ζ ) = lim 1 [ - + 1 j B f ( , j B ζ ) ] = lim 1 - B 2 + 1 ζ = 0.
K TM ( q h u z ) = exp ( - j k 2 h ) 4 π 2 h C [ exp ( - j k ζ ) 1 + ( ζ / 2 h ) ] f TM ( ζ ) d ζ .
C exp ( - j k ζ ) 1 + ( ζ / 2 h ) f TM ( ζ ) d ζ = C f TM ( ζ ) d ζ = R TM ( 0 ) = - 1.

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