Abstract

A reformulated integral equation is solved inside an elliptic disk particle for an electromagnetic field formulation bridging the quasi-static and the physical optics approximations. The scattering amplitude tensor elements associated with such a field formulation are derived and then used to formulate the extinction cross sections. It is shown that the extinction cross sections have a frequency dependence and an incidence angle dependence similar to those associated with the physical optics approximation, and they have a particle shape dependence similar to that associated with the quasi-static approximation. Furthermore, at the high-frequency limits, it is shown that those cross sections could reach the value known in the literature by the extinction paradox, namely, twice the particle geometric shadow area.

© 1998 Optical Society of America

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References

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  1. M. A. Karam, “Electromagnetic wave interactions with dielectric particles. I. Integral equation reformation,” Appl. Opt. 36, 5238–5245 (1997).
    [CrossRef] [PubMed]
  2. 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]
  3. D. M. LeVine, “The radar cross section of dielectric disks,” IEEE Trans. Antennas Propag. AP-32, 6–12 (1984).
    [CrossRef]
  4. D. M. LeVine, A. Schneider, R. H. Lang, H. G. Carter, “Scattering from thin dielectric disks,” IEEE Trans. Antennas Propag. AP-33, 1410–1413 (1985).
    [CrossRef]
  5. R. Schiffer, K. O. Thielheim, “Light scattering by dielectric needles and disks,” J. Appl. Phys. 50, 2476–2483 (1979).
    [CrossRef]
  6. M. A. Karam, A. K. Fung, “Leaf-shape effects in electromagnetic wave scattering from vegetation,” IEEE Trans. Geosci. Remote Sensing 27, 687–697 (1989).
    [CrossRef]
  7. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  8. M. A. Karam, “Molecular optics approach to electromagnetic wave interactions with stratified media,” J. Opt. Soc. Am. A 13, 2208–2218 (1996).
    [CrossRef]
  9. M. A. Karam, A. K. Fung, “Vector forward scattering theorem,” Radio Sci. 17, 752–756 (1982).
    [CrossRef]
  10. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  11. D. M. LeVine, M. A. Karam, “Dependence of attenuation in a vegetation canopy on frequency and plant water content,” IEEE Trans. Geosci. Remote Sensing 34, 1090–1096 (1996).
    [CrossRef]
  12. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [CrossRef] [PubMed]

1997

1996

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

D. M. LeVine, M. A. Karam, “Dependence of attenuation in a vegetation canopy on frequency and plant water content,” IEEE Trans. Geosci. Remote Sensing 34, 1090–1096 (1996).
[CrossRef]

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, “Leaf-shape effects in electromagnetic wave scattering from vegetation,” IEEE Trans. Geosci. Remote Sensing 27, 687–697 (1989).
[CrossRef]

1985

D. M. LeVine, A. Schneider, R. H. Lang, H. G. Carter, “Scattering from thin dielectric disks,” IEEE Trans. Antennas Propag. AP-33, 1410–1413 (1985).
[CrossRef]

1984

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

1982

M. A. Karam, A. K. Fung, “Vector forward scattering theorem,” Radio Sci. 17, 752–756 (1982).
[CrossRef]

1979

S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
[CrossRef] [PubMed]

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

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]

Asano, S.

Carter, H. G.

D. M. LeVine, A. Schneider, R. H. Lang, H. G. Carter, “Scattering from thin dielectric disks,” IEEE Trans. Antennas Propag. AP-33, 1410–1413 (1985).
[CrossRef]

Fung, A. K.

M. A. Karam, A. K. Fung, “Leaf-shape effects in electromagnetic wave scattering from vegetation,” IEEE Trans. Geosci. Remote Sensing 27, 687–697 (1989).
[CrossRef]

M. A. Karam, A. K. Fung, “Vector forward scattering theorem,” Radio Sci. 17, 752–756 (1982).
[CrossRef]

Karam, M. A.

M. A. Karam, “Electromagnetic wave interactions with dielectric particles. I. Integral equation reformation,” Appl. Opt. 36, 5238–5245 (1997).
[CrossRef] [PubMed]

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

D. M. LeVine, M. A. Karam, “Dependence of attenuation in a vegetation canopy on frequency and plant water content,” IEEE Trans. Geosci. Remote Sensing 34, 1090–1096 (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, “Leaf-shape effects in electromagnetic wave scattering from vegetation,” IEEE Trans. Geosci. Remote Sensing 27, 687–697 (1989).
[CrossRef]

M. A. Karam, A. K. Fung, “Vector forward scattering theorem,” Radio Sci. 17, 752–756 (1982).
[CrossRef]

Kong, J. A.

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

Lang, R. H.

D. M. LeVine, A. Schneider, R. H. Lang, H. G. Carter, “Scattering from thin dielectric disks,” IEEE Trans. Antennas Propag. AP-33, 1410–1413 (1985).
[CrossRef]

LeVine, D. M.

D. M. LeVine, M. A. Karam, “Dependence of attenuation in a vegetation canopy on frequency and plant water content,” IEEE Trans. Geosci. Remote Sensing 34, 1090–1096 (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]

D. M. LeVine, A. Schneider, R. H. Lang, H. G. Carter, “Scattering from thin dielectric disks,” IEEE Trans. Antennas Propag. AP-33, 1410–1413 (1985).
[CrossRef]

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

Schiffer, R.

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

Schneider, A.

D. M. LeVine, A. Schneider, R. H. Lang, H. G. Carter, “Scattering from thin dielectric disks,” IEEE Trans. Antennas Propag. AP-33, 1410–1413 (1985).
[CrossRef]

Shin, R. T.

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

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]

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).

van de Hulst, H. C.

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

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]

D. M. LeVine, A. Schneider, R. H. Lang, H. G. Carter, “Scattering from thin dielectric disks,” IEEE Trans. Antennas Propag. AP-33, 1410–1413 (1985).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing

M. A. Karam, A. K. Fung, “Leaf-shape effects in electromagnetic wave scattering from vegetation,” IEEE Trans. Geosci. Remote Sensing 27, 687–697 (1989).
[CrossRef]

D. M. LeVine, M. A. Karam, “Dependence of attenuation in a vegetation canopy on frequency and plant water content,” IEEE Trans. Geosci. Remote Sensing 34, 1090–1096 (1996).
[CrossRef]

J. Appl. Phys.

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

Radio Sci.

M. A. Karam, A. K. Fung, “Vector forward scattering theorem,” Radio Sci. 17, 752–756 (1982).
[CrossRef]

Other

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

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

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Figures (6)

Fig. 1
Fig. 1

Problem configuration.

Fig. 2
Fig. 2

Extinction efficiency of an oblate spheroid (circular disk) as a function of size parameter (size parameter = ka 1, a 1 = 2.5d, ε r = 1.7689, θ i = 0°).

Fig. 3
Fig. 3

Vertically polarized extinction efficiency of an elliptic disk as a function of size parameter (a 1 = 6.5 cm, a 2 = 3.5 cm, d = 4 mm, ε r = 9.6 - j3.03, size parameter = kd/2, θ i = 0°, ϕ i = 90°).

Fig. 4
Fig. 4

Vertically polarized extinction efficiency of an elliptic disk as a function of size parameter (a 1 = 6.5 cm, a 2 = 3.5 cm, d = 4 mm, ε r = 9.6 - j3.03, size parameter = kd/2, θ i = 0°, ϕ i = 90°).

Fig. 5
Fig. 5

Vertically polarized extinction efficiency of an elliptic disk as a function of the azimuth angle (a 1 = 6.5 cm, a 2 = 3.5 cm, d = 4 mm, ε r = 9.6 - j3.03, kd = 0.4, θ i = 0°).

Fig. 6
Fig. 6

Horizontally polarized extinction efficiency of an elliptic disk as a function of the angle of incidence (a 1 = 6.5 cm, a 2 = 3.5 cm, d = 4 mm, ε r = 9.6 - j3.03, kd = 0.4, ϕ i = 0°).

Equations (52)

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E ¯ r ¯ = α ¯ ¯ · E ¯ i r ¯ + α ¯ ¯ · r ¯ r ¯ ε r - 1 G ¯ ¯ r ¯ ,   r ¯ · E ¯ r ¯ d r ¯ .
α ¯ ¯ = t = 1 3   α tt x ˆ t x ˆ t = t = 1 3 1 ε r - 1 g tt + 1   x ˆ t x ˆ t ,
G ¯ ¯ r ¯ ,   r ¯ = k 2 I ¯ ¯ + ¯ k 2 exp - jk | r ¯ - r ¯ | 4 π | r ¯ - r ¯ | ,
E ¯ i r ¯ = E ¯ i exp - j k ¯ i · r ¯ = q = ν , h   E qi q ˆ i exp - j k ¯ i · r ¯ ,
k ¯ i = k xi x ˆ + k yi y ˆ - k zi z ˆ = k sin   θ i cos   ϕ i x ˆ + sin   ϕ i y ˆ - cos   θ i z ˆ , h ˆ i = z ˆ × k ¯ i | z ˆ × k ¯ i | = - sin   ϕ i x ˆ + cos   ϕ i y ˆ , ν ˆ i = h ˆ i × k ¯ i | h ˆ i × k ¯ i | = - cos   θ i cos   ϕ i x ˆ + sin   ϕ i y ˆ - sin   θ i z ˆ .
E ¯ r ¯ = E ¯ - exp - j k ¯ - · r ¯ + E ¯ + exp - j k ¯ + · r ¯ .
k ¯ ± = k xi x ˆ + k yi y ˆ ± k 1 z z ˆ , k xi 2 + k yi 2 + k 1 z 2 = k 2 ε r = k 1 2 .
E ¯ ± = E h ± h ˆ ± + E ν ± ν ˆ ± ,
h ˆ ± = z ˆ × k ¯ ± | z ˆ × k ¯ ± | = - sin   ϕ i x ˆ + cos   ϕ i y ˆ , ν ˆ ± = h ˆ ± × k ˆ ± | h ˆ ± × k ˆ ± | = ± k 1 z cos   ϕ i x ˆ + sin   ϕ i y ˆ - k   sin   θ i z ˆ / k 1 .
G ¯ ¯ r ¯ ,   r ¯ = k 2 8 j π 2 - d k x - d k y I ¯ ¯ + ¯ k 2 × exp - j k x x - x + k y y - y λ × exp - j λ | z - z | ,
E ¯ r ¯ = α ¯ ¯ · E ¯ i r ¯ + k 2 ε r - 1 8 j π 2   α ¯ ¯ · - - dk x dk y λ × - a 1 a 1 d x   - a 2 1 - x / a 1 2 a 2 1 - x / a 1 2 d y × exp - j k x x - x + k y y - y × - d z d z I ¯ ¯ - k ̄ k ̄ / k 2 exp - j λ z - z × E ¯ - exp jk 1 z z + E ¯ + exp - jk 1 z z + z 0 d z I ¯ ¯ - K ¯ K ¯ / k 2 exp - j λ z - z × E ¯ - exp jk 1 z z + E ¯ + exp - jk 1 z z × exp - j k xi x + k yi y .
K ¯ = k x x ˆ + k y y ˆ - λ z ˆ , k ¯ = k x x ˆ + k y y ˆ + λ z ˆ .
E ¯ r ¯ = α ¯ ¯ · E ¯ i r ¯ + α ¯ ¯ 4 π 2 · - -   dk x dk y - a 1 a 1 d x × - α 2 1 - x / α 1 2 α 2 1 - x / α 1 2 d y exp - j k x x - x + k y y - y · C ¯ r λ exp - j λ z + C ¯ t λ exp j λ z + C ¯ - λ exp jk 1 z z + C ¯ + λ exp - jk 1 z z × exp - j k x x + k y y ,
C ¯ r λ = 1 2 I ¯ ¯ - k ¯ k ¯ / k 2 · k 1 z / λ - 1 E ¯ - exp - jk 1 z d - k 1 z / λ + 1 E ¯ + exp jk 1 z d exp - j λ d , C ¯ t λ = 1 2 I ¯ ¯ - KK ¯ / k 2 · k 1 z / λ - 1 E ¯ + + k 1 z / λ + 1 E ¯ - , C ¯ ± λ = I ¯ ¯   ±   k 1 z 2 k 2 λ k ¯ k ¯ - KK ¯ ± 1 2 k 2 k ¯ k ¯ + KK ¯ · E ¯ ± .
- a 1 a 1 d x   - a 2 1 - x / a 1 2 a 2 1 - x / a 1 2 d y   exp - j k x - k xi x + k y - k yi y - a 1 a 1 d x   exp - j k x - k xi x - a 2 a 2 d y × exp - j k y - k yi y , 4 π 2 δ k x - k xi δ k y - k yi ,
E r = α ¯ ¯ · C ¯ r k zi exp - jk zi z + E ¯ i - C ¯ t k zi exp jk zi z + C ¯ - k zi exp jk 1 z z + C ¯ + k zi exp - jk 1 z z × exp - j k xi x + k yi y .
E ν + = r ν E ν - exp - 2 jk 1 z d , r ν = k 1 z - ε r k zi k 1 z + ε r k zi ,
E h + = r h E h - exp - 2 jk 1 z d , r h = k 1 z - k zi k 1 z + k zi .
E ν - = 2 k zi ε r k zi ε r + k 1 z 1 - r ν 2 exp - 2 jk 1 z d   E ν i ,
E h - = 2 k zi k zi + k 1 z 1 - r h 2 exp - 2 jk 1 z d   E hi ,
E ¯ r = α ¯ ¯ · C ¯ - k zi exp jk 1 z z + C ¯ + k zi exp - jk 1 z z exp - j k xi x + k yi y ,
C ¯ ± k zi = I ¯ ¯ ± k 1 z 2 k 2 k zi k ¯ r k ¯ r - k ¯ i k ¯ i ± 1 2 k 2 k ¯ r k ¯ r + k ¯ i k ¯ i · E ¯ ± ,
k ¯ r = k xi x ˆ + k yi y ˆ + k zi z ˆ .
k ¯ r = k ¯ - + k 1 z + k zi z ˆ , k ¯ i = k ¯ - + k 1 z - k zi z ˆ .
C ¯ - k zi = α ¯ ¯ · I ¯ ¯ - k ¯ - k ¯ - / k 2 + ε r - 1 z ˆ z ˆ · E ¯ - .
k ¯ r = k ¯ + - k 1 z - k zi z ˆ , k ¯ i = k ¯ + - k 1 z + k zi z ˆ .
C ¯ + k zi = α ¯ ¯ · I ¯ ¯ - k ¯ + k ¯ + / k 2 + ε r - 1 z ˆ z ˆ · E ¯ + .
E ¯ r ¯ = α ¯ ¯ m · E ¯ - exp - j k ¯ - · r ¯ + E ¯ + exp - j k ¯ + · r ¯ ,
α ¯ ¯ m = α 11 x ˆ x ˆ + α 22 y ˆ y ˆ + ε r α 33 z ˆ z ˆ .
f ¯ ¯ k ¯ s ,   k ¯ i · E ¯ i = k 2 4 π q = ν , h   q ˆ s q ˆ s × ν 0 d r ¯ ε r - 1 E ¯ r ¯ exp j k ¯ s · r ¯ = p = ν , h q = ν , h   f pq k ¯ s ,   k ¯ i p ˆ s q ˆ i .
f ¯ ¯ k ¯ s ,   k ¯ i · E ¯ i = k 2 ν 0 ε r - 1 4 π q = ν , h   q ˆ s q ˆ s · α ¯ ¯ m · E ¯ - - k zs ,   k 1 z + E ¯ + + k zs ,   k 1 z μ k ¯ s ,   k ¯ i ,
± k zs ,   k 1 z = 1 - exp - jd k zs k 1 z jd k zs k 1 z , μ k ¯ s ,   k ¯ i = 2 J 1 Q Q , Q 2 = k xs - k xi a 1 2 + k ys - k yi a 2 2 ,
f ν ν k ¯ s ,   k ¯ i = k 2 ν 0 ε r - 1 4 π ν ˆ s · x ˆ x ˆ · ν ˆ i α 11 + ν ˆ s · y ˆ × y ˆ · ν ˆ i α 22 A 1 ν k zs ,   k 1 z + ν ˆ s · z ˆ z ˆ · ν ˆ i α 33 A 2 ν k zs ,   k 1 z μ k ¯ s ,   k ¯ i , f h ν k ¯ s ,   k ¯ i = k 2 ν 0 ε r - 1 4 π h ˆ s · x ˆ x ˆ · ν ˆ i α 11 + h ˆ s · y ˆ × y ˆ · ν ˆ i α 22 A 1 ν k zs ,   k 1 z + h ˆ s · z ˆ z ˆ · ν ˆ i α 33 A 2 ν k zs ,   k 1 z μ k ¯ s ,   k ¯ i , f ν h k ¯ s ,   k ¯ i = k 2 ν 0 ε r - 1 4 π ν ˆ s · x ˆ x ˆ · h ˆ i α 11 + ν ˆ s · y ˆ y ˆ · h ˆ i α 22 + ν ˆ s · z ˆ z ˆ · h ˆ i α 33 A 1 h k zs ,   k 1 z μ k ¯ s ,   k ¯ i , f hh k ¯ s ,   k ¯ i = k 2 ν 0 ε r - 1 4 π h ˆ s · x ˆ x ˆ · h ˆ i α 11 + h ˆ s · y ˆ y ˆ · h ˆ i α 22 + h ˆ s · z ˆ z ˆ · h ˆ i α 33 A 1 h k zs ,   k 1 z μ k ¯ s ,   k ¯ i ,
A 1 ν k zs ,   k 1 z = - k zs ,   k 1 z - r ν + k zs ,   k 1 z exp - 2 jk z 1 d k 1 z E ν - k zi ε r   E ν i , A 2 ν k zs ,   k 1 z = - k zs ,   k 1 z + r ν + k zs ,   k 1 z exp - 2 jk z 1 d ε r   E ν - / E ν i , A 1 h k zs ,   k 1 z = - k zs ,   k 1 z + r h + k zs ,   k 1 z exp - 2 jk z 1 d E h - / E hi .
ν ˆ - · x ˆ = - ν ˆ + · x ˆ = k 1 z k zi ε r ν ˆ i · x ˆ , ν ˆ - · y ˆ = - ν ˆ + · y ˆ = k 1 z k zi ε r ν ˆ i · y ˆ , ν ˆ - · z ˆ = ν ˆ + · z ˆ = 1 ε r ν ˆ i · z ˆ .
κ ep k ¯ i = - 4 π k Im f pp k ¯ i ,   k ¯ i .
κ e ν k ¯ i = - k ν 0 Im ε r - 1 cos 2   θ i α 11 cos 2   ϕ i + α 22 sin 2   ϕ i A 1 ν - k zi ,   k 1 z + α 33 sin 2   θ i A 2 ν - k zi ,   k 1 z , κ eh k ¯ i = - k ν 0 Im ε r - 1 α 11 sin 2   ϕ i + α 22 cos 2   ϕ i A 1 h - k zi ,   k 1 z .
κ e ν k ¯ i 2 S g Re α 11 cos 2   ϕ i + α 22 sin 2   ϕ i + α 33 ε r sin   θ i tan   θ i ε r - sin 2   θ i , κ eh k ¯ i 2 S g Re α 11 sin 2   ϕ i + α 22 cos 2   ϕ i ,
κ e ν k ¯ i 2 S g Re 1 + sin   θ i tan   θ i ε r - sin 2   θ i , κ eh k ¯ i 2 S g .
κ e ν k ¯ i κ eh k ¯ i 2 S g .
κ e ν k ¯ i = - k ν 0 Im ε r - 1 cos 2   θ i α 11 cos 2   ϕ i + α 22 sin 2   ϕ i + α 33 sin 2   θ i , κ eh k ¯ i = - k ν 0 Im ε r - 1 α 11 sin 2   ϕ i + α 22 cos 2   ϕ i .
κ e ν k ¯ i = k ν 0 ε r cos 2   θ i | α 11 | 2 cos 2   ϕ i + | α 22 | 2 sin 2   ϕ i + | α 33 | 2 sin 2   θ i , κ eh k ¯ i = k ν 0 ε r | α 11 | 2 sin 2   ϕ i + | α 22 | 2 cos 2   ϕ i .
κ e ν k ¯ i = - k ν 0 Im ε r - 1 A 1 ν - k zi ,   k 1 z cos 2   θ i + 1 / ε r A 2 ν - k zi ,   k 1 z sin 2 θ i , κ eh k ¯ i = - k ν 0 Im ε r - 1 A 1 h - k zi ,   k 1 z .
- k zs ,   k 1 z + k zs ,   k 1 z 1 .
E ν - 2 k zi ε r k zi ε r + k 1 z 1 - r ν 2   E ν i , E h - 2 k zi k zi + k 1 z 1 - r h 2   E hi .
A 1 ν k zs ,   k 1 z 2 k zi ε r ε r k zi + k 1 z 1 + r ν , A 2 ν k zs ,   k 1 z 2 ε r k zi ε r k zi + k 1 z 1 - r ν , A 1 h k zs ,   k 1 z 2 k zi k zi + k 1 z 1 + r h .
A 1 ν k zs ,   k 1 z A 2 ν k zs ,   k 1 z A 1 h k zs ,   k 1 z 1 .
A 1 ν - k zi ,   k 1 z - - k zi ,   k 1 z k 1 z E ν - k zi ε r   E ν i , A 2 ν - k zi ,   k 1 z - - k zi ,   k 1 z ε r   E ν - E ν i , A 1 h - k zi ,   k 1 z - - k zi ,   k 1 z E h - E hi .
E ν - 2 k zi k zi ε r + k 1 z   E ν i , E h - 2 k zi k zi + k 1 z   E hi .
- - k zi ,   k 1 z 1 jd k 1 z - k zi .
A 1 ν - k zi ,   k 1 z 2 k zi jd k z 1 - k zi k zi ε r + k 1 z , A 2 ν - k zi ,   k 1 z 2 k zi ε r jd k 1 z - k zi k iz + k zi ε r , A 1 h - k zi ,   k 1 z 2 k zi jd k z 1 - k zi k zi + k z 1 .
A 1 ν - k zi ,   k 1 z 2 jkd ε r - 1 cos   θ i , A 2 ν - k zi ,   k 1 z 2 ε r jkd ε r - 1 ε r - sin 2 θ i , A 1 h - k zi ,   k 1 z 2   cos   θ i jkd ε r - 1 .

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