Abstract

A procedure based on the T-matrix method is devised to study the electromagnetic response of chiral, (lossy) dielectric, nonspherical objects exposed to an arbitrary incident field. Reductions in the method for axisymmetric objects are discussed. Using the technique thus developed, the plane wave scattering and absorption characteristics of lossy dielectric, axisymmetric scatterers (spheres as well as prolate and oblate spheroids), with and without chiral properties, are examined at frequencies above 50 GHz. The relative permittivity of the objects is assumed to be frequency dependent, whereas the chiral parameters are set to be constant in the numerical study. From the computed results, it appears that chiral spheres are the most effective objects in retarding the progress of an incident plane wave regardless of its polarization.

© 1985 Optical Society of America

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References

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  1. C. F. Bohren, “Light Scattering by an Optically Active Sphere,” Chem. Phys. Lett. 29, 458 (1974).
    [CrossRef]
  2. D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On Electromagnetic Waves in Chiral Media,” Appl. Phys. 18, 211 (1979).
    [CrossRef]
  3. N. Engheta, A. R. Mickelson, “Transition Radiation Caused by a Chiral Plate,” IEEE Trans. Antennas Propag. AP-30, 1213 (1982).
    [CrossRef]
  4. I. Tinoco, M. P. Freeman, “The Optical Activity of Oriented Helices,” J. Phys. Chem. 61, 1196 (1957).
    [CrossRef]
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 8.
  6. P. C. Waterman, “Scattering by Dielectric Obstacles,” Alta Freq. 38 (Speciale), 348 (1969).
  7. P. W. Barber, C. Yeh, “Scattering of Electromagnetic Waves by Arbitrarily Shaped Bodies,” Appl. Opt. 14, 2864 (1975).
    [CrossRef] [PubMed]
  8. D. J. N. Wall, “Methods of Overcoming Numerical Instabilities Associated with the T-matrix Method,” in Acoustic, Electromagnetic and Elastic Wave Scattering, V. K. Varadan, V. V. Varadan, Eds. (Pergamon, New York, 1980).
  9. A. Lakhtakia, M. F. Iskander, C. H. Durney, “Absorption Characteristics of Lossy Dielectric Objects using an Iterative Extended Boundary Condition Method of Solution,” IEEE Trans. Microwave Theory Tech. MTT-31, 640 (1983).
    [CrossRef]
  10. A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Iterative Extended Boundary Condition Method for Scattering by Objects of High Aspect Ratios,” J. Acoust. Soc. Am. 76, 906 (1984).
    [CrossRef]
  11. C. Yeh, “Theoretical and Numerical Methods of Quantitation of Absorption Patterns in Man and Animal Bodies,” in Proceedings, NATO ASI on Theoretical Methods for Determining the Interaction of Electromagnetic Waves with Structures (Sijth-off and Noordhoff, The Netherlands, 1981).
    [CrossRef]
  12. A. Lakhtakia, “Near-field Scattering and Absorption by Lossy Dielectrics at Resonance Frequencies,” Ph.D. Dissertation, U. Utah, Salt Lake City (1983), pp. 3–10.
  13. R. A. Satten, “Time Reversal Symmetry and Electromagnetic Polarizaton Fields,” J. Chem. Phys. 28, 742 (1958).
    [CrossRef]
  14. R. F. Harrington, Introduction to Electromagnetic Engineering (McGraw-Hill, New York, 1958).
  15. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peter Peregrinus, Stevenage, 1976).
  16. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 2 (McGraw-Hill, New York, 1953).
  17. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  18. C. H. Durney et al., Radio Frequency Radiation Dosimetry Handbook (Departments of Electrical Engineering and Bioengineering, U. Utah, Salt Lake City, 1978), pp. 33–36.

1984

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Iterative Extended Boundary Condition Method for Scattering by Objects of High Aspect Ratios,” J. Acoust. Soc. Am. 76, 906 (1984).
[CrossRef]

1983

A. Lakhtakia, M. F. Iskander, C. H. Durney, “Absorption Characteristics of Lossy Dielectric Objects using an Iterative Extended Boundary Condition Method of Solution,” IEEE Trans. Microwave Theory Tech. MTT-31, 640 (1983).
[CrossRef]

1982

N. Engheta, A. R. Mickelson, “Transition Radiation Caused by a Chiral Plate,” IEEE Trans. Antennas Propag. AP-30, 1213 (1982).
[CrossRef]

1979

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On Electromagnetic Waves in Chiral Media,” Appl. Phys. 18, 211 (1979).
[CrossRef]

1975

1974

C. F. Bohren, “Light Scattering by an Optically Active Sphere,” Chem. Phys. Lett. 29, 458 (1974).
[CrossRef]

1969

P. C. Waterman, “Scattering by Dielectric Obstacles,” Alta Freq. 38 (Speciale), 348 (1969).

1958

R. A. Satten, “Time Reversal Symmetry and Electromagnetic Polarizaton Fields,” J. Chem. Phys. 28, 742 (1958).
[CrossRef]

1957

I. Tinoco, M. P. Freeman, “The Optical Activity of Oriented Helices,” J. Phys. Chem. 61, 1196 (1957).
[CrossRef]

Barber, P. W.

Bohren, C. F.

C. F. Bohren, “Light Scattering by an Optically Active Sphere,” Chem. Phys. Lett. 29, 458 (1974).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 8.

Durney, C. H.

A. Lakhtakia, M. F. Iskander, C. H. Durney, “Absorption Characteristics of Lossy Dielectric Objects using an Iterative Extended Boundary Condition Method of Solution,” IEEE Trans. Microwave Theory Tech. MTT-31, 640 (1983).
[CrossRef]

C. H. Durney et al., Radio Frequency Radiation Dosimetry Handbook (Departments of Electrical Engineering and Bioengineering, U. Utah, Salt Lake City, 1978), pp. 33–36.

Engheta, N.

N. Engheta, A. R. Mickelson, “Transition Radiation Caused by a Chiral Plate,” IEEE Trans. Antennas Propag. AP-30, 1213 (1982).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 2 (McGraw-Hill, New York, 1953).

Freeman, M. P.

I. Tinoco, M. P. Freeman, “The Optical Activity of Oriented Helices,” J. Phys. Chem. 61, 1196 (1957).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Introduction to Electromagnetic Engineering (McGraw-Hill, New York, 1958).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 8.

Iskander, M. F.

A. Lakhtakia, M. F. Iskander, C. H. Durney, “Absorption Characteristics of Lossy Dielectric Objects using an Iterative Extended Boundary Condition Method of Solution,” IEEE Trans. Microwave Theory Tech. MTT-31, 640 (1983).
[CrossRef]

Jaggard, D. L.

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On Electromagnetic Waves in Chiral Media,” Appl. Phys. 18, 211 (1979).
[CrossRef]

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peter Peregrinus, Stevenage, 1976).

Lakhtakia, A.

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Iterative Extended Boundary Condition Method for Scattering by Objects of High Aspect Ratios,” J. Acoust. Soc. Am. 76, 906 (1984).
[CrossRef]

A. Lakhtakia, M. F. Iskander, C. H. Durney, “Absorption Characteristics of Lossy Dielectric Objects using an Iterative Extended Boundary Condition Method of Solution,” IEEE Trans. Microwave Theory Tech. MTT-31, 640 (1983).
[CrossRef]

A. Lakhtakia, “Near-field Scattering and Absorption by Lossy Dielectrics at Resonance Frequencies,” Ph.D. Dissertation, U. Utah, Salt Lake City (1983), pp. 3–10.

Mickelson, A. R.

N. Engheta, A. R. Mickelson, “Transition Radiation Caused by a Chiral Plate,” IEEE Trans. Antennas Propag. AP-30, 1213 (1982).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On Electromagnetic Waves in Chiral Media,” Appl. Phys. 18, 211 (1979).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 2 (McGraw-Hill, New York, 1953).

Papas, C. H.

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On Electromagnetic Waves in Chiral Media,” Appl. Phys. 18, 211 (1979).
[CrossRef]

Satten, R. A.

R. A. Satten, “Time Reversal Symmetry and Electromagnetic Polarizaton Fields,” J. Chem. Phys. 28, 742 (1958).
[CrossRef]

Stratton, J. A.

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

Tinoco, I.

I. Tinoco, M. P. Freeman, “The Optical Activity of Oriented Helices,” J. Phys. Chem. 61, 1196 (1957).
[CrossRef]

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Iterative Extended Boundary Condition Method for Scattering by Objects of High Aspect Ratios,” J. Acoust. Soc. Am. 76, 906 (1984).
[CrossRef]

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Iterative Extended Boundary Condition Method for Scattering by Objects of High Aspect Ratios,” J. Acoust. Soc. Am. 76, 906 (1984).
[CrossRef]

Wall, D. J. N.

D. J. N. Wall, “Methods of Overcoming Numerical Instabilities Associated with the T-matrix Method,” in Acoustic, Electromagnetic and Elastic Wave Scattering, V. K. Varadan, V. V. Varadan, Eds. (Pergamon, New York, 1980).

Waterman, P. C.

P. C. Waterman, “Scattering by Dielectric Obstacles,” Alta Freq. 38 (Speciale), 348 (1969).

Yeh, C.

P. W. Barber, C. Yeh, “Scattering of Electromagnetic Waves by Arbitrarily Shaped Bodies,” Appl. Opt. 14, 2864 (1975).
[CrossRef] [PubMed]

C. Yeh, “Theoretical and Numerical Methods of Quantitation of Absorption Patterns in Man and Animal Bodies,” in Proceedings, NATO ASI on Theoretical Methods for Determining the Interaction of Electromagnetic Waves with Structures (Sijth-off and Noordhoff, The Netherlands, 1981).
[CrossRef]

Alta Freq.

P. C. Waterman, “Scattering by Dielectric Obstacles,” Alta Freq. 38 (Speciale), 348 (1969).

Appl. Opt.

Appl. Phys.

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On Electromagnetic Waves in Chiral Media,” Appl. Phys. 18, 211 (1979).
[CrossRef]

Chem. Phys. Lett.

C. F. Bohren, “Light Scattering by an Optically Active Sphere,” Chem. Phys. Lett. 29, 458 (1974).
[CrossRef]

IEEE Trans. Antennas Propag.

N. Engheta, A. R. Mickelson, “Transition Radiation Caused by a Chiral Plate,” IEEE Trans. Antennas Propag. AP-30, 1213 (1982).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

A. Lakhtakia, M. F. Iskander, C. H. Durney, “Absorption Characteristics of Lossy Dielectric Objects using an Iterative Extended Boundary Condition Method of Solution,” IEEE Trans. Microwave Theory Tech. MTT-31, 640 (1983).
[CrossRef]

J. Acoust. Soc. Am.

A. Lakhtakia, V. K. Varadan, V. V. Varadan, “Iterative Extended Boundary Condition Method for Scattering by Objects of High Aspect Ratios,” J. Acoust. Soc. Am. 76, 906 (1984).
[CrossRef]

J. Chem. Phys.

R. A. Satten, “Time Reversal Symmetry and Electromagnetic Polarizaton Fields,” J. Chem. Phys. 28, 742 (1958).
[CrossRef]

J. Phys. Chem.

I. Tinoco, M. P. Freeman, “The Optical Activity of Oriented Helices,” J. Phys. Chem. 61, 1196 (1957).
[CrossRef]

Other

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Chap. 8.

D. J. N. Wall, “Methods of Overcoming Numerical Instabilities Associated with the T-matrix Method,” in Acoustic, Electromagnetic and Elastic Wave Scattering, V. K. Varadan, V. V. Varadan, Eds. (Pergamon, New York, 1980).

R. F. Harrington, Introduction to Electromagnetic Engineering (McGraw-Hill, New York, 1958).

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peter Peregrinus, Stevenage, 1976).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 2 (McGraw-Hill, New York, 1953).

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

C. H. Durney et al., Radio Frequency Radiation Dosimetry Handbook (Departments of Electrical Engineering and Bioengineering, U. Utah, Salt Lake City, 1978), pp. 33–36.

C. Yeh, “Theoretical and Numerical Methods of Quantitation of Absorption Patterns in Man and Animal Bodies,” in Proceedings, NATO ASI on Theoretical Methods for Determining the Interaction of Electromagnetic Waves with Structures (Sijth-off and Noordhoff, The Netherlands, 1981).
[CrossRef]

A. Lakhtakia, “Near-field Scattering and Absorption by Lossy Dielectrics at Resonance Frequencies,” Ph.D. Dissertation, U. Utah, Salt Lake City (1983), pp. 3–10.

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Figures (9)

Fig. 1
Fig. 1

Normalized scattering and absorption cross-sections, 2Csca/πa2 (----) and 2Cabs/πa2 (…….), for a 0.4-mm diam lossy dielectric sphere exposed to a linearly polarized plane wave as a function of frequency. A, β = 10−4 m. B, β = 0 m. When β = 0 m, the sphere is nonchiral. The complex relative permittivity of the sphere is shown in Table I as a function of frequency.

Fig. 2
Fig. 2

Same as Fig. 1 but for a prolate spheroid (a = 0.2 mm, a/b = 3/2) on which an E-polarized plane wave is incident.

Fig. 3
Fig. 3

Same as Fig. 2 but for H-polarized plane wave incidence.

Fig. 4
Fig. 4

Same as Fig. 2 but for K-polarized plane wave incidence.

Fig. 5
Fig. 5

Same as Fig. 1 but for an oblate spheroid (a = 0.2 mm, a/b = 2/3) on which a K-polarized plane wave is incident.

Fig. 6
Fig. 6

Comparison of 4Cext normalized by its high-frequency limiting value given in Sec. VI. Results are given for chiral scatterers: the sphere of Fig. 1 and the prolate spheroids of Figs. 2 and 3.

Fig. 7
Fig. 7

Comparison of 4Cext normalized by its high-frequency limiting value given in Sec. VI. Results are given for chiral scatterers: the prolate spheroid of Fig. 4 and the oblate spheroid of Figs. 5.

Fig. 8
Fig. 8

Normalized absorption cross section, 2Cabs/πa2, for a 0.4-mm diam chiral, lossy dielectric sphere exposed to a linearly polarized plane wave as a function of frequency. The relative permittivity /1 = 5.0 + j0.1 is constant throughout the frequency range of calculations.

Fig. 9
Fig. 9

Normalized scattering cross section, 2Csca/πa2, for a 0.4-mm diam chiral, lossy dielectric sphere exposed to a linearly polarized plane wave as a function of frequency. The relative permittivity /1 = 5.0 + j0.1 is constant throughout the frequency range of calculations.

Tables (1)

Tables Icon

Table I Complex Relative Permittivity of the Chiral Scatterers and the Wave Numbers kR and kL as Functions of Frequency used for the Computations of Figs. 17

Equations (55)

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

D = E + α × E , B = μ H + β μ × H
2 [ E H ] = [ K ] 2 [ E H ] , × [ E H ] = [ K ] [ E H ] , · [ E H ] = [ 0 0 ]
[ K ] = [ 1 ( k β ) 2 ] 1 [ k 2 β j ω μ j ω k 2 β ]
k = ω { μ } 1 / 2 ,
[ E H ] = [ A ] [ Q L Q R ]
{ 2 + ( k ) L 2 } Q L = 0 ; { 2 + ( k ) R 2 } Q R = 0 ,
× Q L = ( k ) L Q L ; · Q L = 0 ,
× Q R = ( k ) R Q R ; · Q R = 0 .
[ A ] = [ 1 a R a L 1 ] ,
( k ) L = ( k ) / { 1 ( k ) β } ; a L = j ( / μ ) 1 / 2 ,
( k ) R = ( k ) / { 1 + k β } ; a R = j ( / μ ) 1 / 2 .
E = Q L + a R Q R ; H = Q R + a L Q L .
E ( r ) = E i ( r ) + E s ( r ) ,
E s ( r ) = × S K + ( r o ) · G ( k 1 r | k 1 r o ) d s o + × × S ( 1 / j ω 1 ) J + ( r o ) · G ( k 1 r | k 1 r o ) d s o ,
k 1 = ω { μ 1 1 } 1 / 2 ;
E i ( r ) = × S K + ( r o ) · G ( k 1 r | k 1 r o ) d s o + × × S ( 1 / j ω 1 ) J + ( r o ) · G ( k 1 r | k 1 r o ) d s o , r V ,
E s ( r ) = × S K + ( r o ) · G ( k 1 r | k 1 r o ) d s o + × × S ( 1 / j ω 1 ) J + ( r o ) · G ( k 1 r | k 1 r o ) d s o , r V 1 ,
J ( r ) + J + ( r ) = 0 , or , n × H + ( r ) = n × H ( r ) , r S ,
K ( r ) + K + ( r ) = 0 , or , n × E + ( r ) = n × E ( r ) , r S .
E i ( r ) = × S n × E int · G ( k 1 r | k 1 r o ) d s o × × S ( 1 / j ω 1 ) n × H int · G ( k 1 r | k 1 r o ) d s o , r V ,
E s ( r ) = × S n × E int · G ( k 1 r | k 1 r o ) d s o × × S ( 1 / j ω 1 ) n × H int · G ( k 1 r | k 1 r o ) d s o , r V 1 ;
E i ( r ) = ν = σ m n D n m [ a ν M ν ( 1 ) ( k 1 r ) + b ν N ν ( 1 ) ( k 1 r ) ] ; H i ( r ) = ( 1 / j ω μ 1 ) × E i ( r ) ,
D n m = ( 2 δ m 0 ) { ( 2 n + 1 ) / [ 4 n ( n + 1 ) ] } { ( n m ) ! / ( n + m ) ! } ,
E s ( r ) = ν = σ m n D n m [ c ν M ν ( 3 ) ( k 1 r ) + d ν N ν ( 3 ) ( k 1 r ) ] ; H s ( r ) = ( 1 / j ω μ 1 ) × E s ( r ) ,
G ( k 1 r | k 1 r o ) = ( j k 1 / π ) ν = σ m n D n m { M ν ( 3 ) ( k 1 r > ) M ν ( 1 ) ( k 1 r < ) + N ν ( 3 ) ( k 1 r > ) N ν ( 1 ) ( k 1 r > ) } ,
a ν = ( j k 1 2 / π ) { S d s o [ n × E int ( r o ) · N ν ( 3 ) ( k 1 r o ) ( k 1 / j ω 1 ) n × H int ( r o ) · M ν ( 3 ) ( k 1 r o ) ] } ,
b ν = ( j k 1 2 / π ) { S d s o [ n × E int ( r o ) · M ν ( 3 ) ( k 1 r o ) ( k 1 / j ω 1 ) n × H int ( r o ) · N ν ( 3 ) ( k 1 r o ) ] } ,
c ν = ( j k 1 2 / π ) { S d s o [ n × E int ( r o ) · N ν ( 1 ) ( k 1 r o ) ( k 1 / j ω 1 ) n × H int ( r o ) · M ν ( 1 ) ( k 1 r o ) ] } ,
d ν = ( j k 1 2 / π ) { S d s o [ n × E int ( r o ) · M ν ( 1 ) ( k 1 r o ) ( k 1 / j ω 1 ) n × H int ( r o ) · N ν ( 1 ) ( k 1 r o ) ] } .
Q L ( r ) = ν = σ m n f ν [ M ν ( 1 ) ( k L r ) + N ν ( 1 ) ( k L r ) ] ,
Q R ( r ) = ν = σ m n g ν [ M ν ( 1 ) ( k R r ) + N ν ( 1 ) ( k R r ) ] ,
[ a ν ; b ν ] T = [ Y 3 ] [ f ν ; g ν ] T ,
[ c ν ; d ν ] T = [ Y 1 ] [ f ν ; g ν ] T ,
[ Y 3 ] = [ I 3 ν ν K 3 v ν J 3 v ν L 3 v ν ] ; [ Y 1 ] = [ I 1 ν ν K 1 v ν J 1 v ν L 1 v ν ] .
I 3 ν ν = ( j k 1 2 / π ) { S d s o n × [ M ν ( 1 ) ( k L r o ) + N ν ( 1 ) ( k L r o ) ] · [ N ν ( 3 ) ( k 1 r o ) + η M ν ( 3 ) ( k 1 r o ) ] } ,
J 3 ν ν = ( j k 1 2 / π ) { S d s o n × [ M ν ( 1 ) ( k R r o ) N ν ( 1 ) ( k R r o ) ] · [ N ν ( 3 ) ( k 1 r o ) η M ν ( 3 ) ( k 1 r o ) ] } a R ,
K 3 ν ν = ( j k 1 2 / π ) { S d s o n × [ M ν ( 1 ) ( k L r o ) + N ν ( 1 ) ( k L r o ) ] · [ M ν ( 3 ) ( k 1 r o ) + η N ν ( 3 ) ( k 1 r o ) ] } ,
J 3 ν ν = ( j k 1 2 / π ) { S d s o n × [ M ν ( 1 ) ( k R r o ) N ν ( 1 ) ( k R r o ) ] · [ M ν ( 3 ) ( k 1 r o ) η N ν ( 3 ) ( k 1 r o ) ] } a R ,
η = [ ( μ 1 / 1 ) ( / μ ) ] 1 / 2 .
[ c ν ; d ν ] T = [ Y 1 ] [ Y 3 ] 1 [ a ν ; b ν ] T = [ T ] [ a ν ; b ν ] T ,
[ a e m n a o m n b e m n b o m n ] = [ A 3 m n n ( k L ) B 3 m n n ( k L ) A 3 m n n ( k R ) B 3 m n n ( k R ) B 3 m n n ( k L ) A 3 m n n ( k L ) B 3 m n n ( k R ) A 3 m n n ( k R ) C 3 m n n ( k L ) D 3 m n n ( k L ) C 3 m n n ( k R ) D 3 m n n ( k R ) D 3 m n n ( k L ) C 3 m n n ( k L ) D 3 m n n ( k R ) C 3 m n n ( k R ) ] [ f e m n f o m n g e m n a R g o m n a R ] ,
A 3 m n n ( p ) = ( j k 1 2 / π ) { S d s o n · ( [ M e m n ( 1 ) ( p r o ) × N e m n ( 3 ) ( k 1 r o ) ] + [ η N e m n ( 1 ) ( p r o ) × M e m n ( 3 ) ( k 1 r o ) ] ) } ,
B 3 m n n ( p ) = ( j k 1 2 / π ) { S d s o n · ( [ N e m n ( 1 ) ( p r o ) × N o m n ( 3 ) ( k 1 r o ) ] + [ η M e m n ( 1 ) ( p r o ) × M o m n ( 3 ) ( k 1 r o ) ] ) } ,
C 3 m n n ( p ) = ( j k 1 2 / π ) { S d s o n · ( [ N e m n ( 1 ) ( p r o ) × M e m n ( 3 ) ( k 1 r o ) ] + [ η M e m n ( 1 ) ( p r o ) × N e m n ( 3 ) ( k 1 r o ) ] ) } ,
D 3 m n n ( p ) = ( j k 1 2 / π ) { S d s o n · ( [ M e m n ( 1 ) ( p r o ) × M o m n ( 3 ) ( k 1 r o ) ] + [ η N e m n ( 1 ) ( p r o ) × N o m n ( 3 ) ( k 1 r o ) ] ) } ,
r ( θ ) = a { cos 2 θ + ( a / b ) 2 sin 2 θ } 1 / 2 ,
C sca = ( π / k 1 2 ) ν D n m [ | c ν | 2 + | d ν | 2 ]
f = | r exp ( j k 1 r ) E s ( r , π θ 0 , π ϕ 0 ) | , k 1 r
C abs = ( 120 π ) Re { S d s o n ( r o ) · E int ( r o ) × H int * ( r o ) } ;
C ext = C abs + C sca .
ξ i ( r ) = exp { j k 1 ( x sin θ 0 cos ϕ 0 + y sin θ 0 sin ϕ 0 + z cos θ 0 ) · r } .
( a ) E polarization : θ 0 = 90 ° , ϕ 0 = 0 ° , E i = z ξ i ; ( b ) H polarization : θ 0 = 90 ° , ϕ 0 = 0 ° , E i = y ξ i ; and ( c ) K polarization : θ 0 = 0 ° , ϕ 0 = 0 ° , E i = x ξ i or E i = y ξ i .
k R / k = [ ( 1 + β k ) 2 + ( β k ) 2 ] 1 ; k L / k = [ ( 1 β k ) 2 + ( β k ) 2 ] 1 ,
{ / β } [ k R / k ] = 2 [ k ( 1 + β k ) + β k 2 ] [ ( 1 + β k ) 2 + ( β k ) 2 ] 1 , { / β } [ k L / k ] = 2 [ k ( 1 β k ) β k 2 ] [ ( 1 β k ) 2 + ( β k ) 2 ] 1 .
{ / β } [ k L / k ] = { / β } [ k R / k ] = 2 k ,

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