Abstract

An analytical-numerical technique for the solution of the plane-wave scattering problem by a set of dielectric cylinders embedded in a dielectric slab is presented. Scattered fields are expressed by means of expansions into cylindrical functions, and the concept of plane-wave spectrum of a cylindrical function is employed to define reflection and transmission through the planar interfaces. Multiple reflection phenomena due to the presence of a layered geometry are taken into account. Solutions can be obtained for both TM and TE polarizations and for near- and far-field regions. The numerical approach is described and the method is validated by comparison with examples given in the literature, with very good agreement. Results are presented for the scattering by a finite grid of three cylinders embedded in a slab.

© 2010 Optical Society of America

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References

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  1. D. J. Daniels, Surface Penetrating Radar, 2nd ed. (IEE, 2004).
    [CrossRef]
  2. M. A. Taubenblatt, “Light scattering from cylindrical structures on surfaces,” Opt. Lett. 15, 255-257 (1990).
    [CrossRef] [PubMed]
  3. R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A 14, 1500-1504 (1997).
    [CrossRef]
  4. D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368-1376 (2002).
    [CrossRef]
  5. M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” Radio Sci. 40, 11 (2005).
  6. F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Electromagnetic scattering by perfectly-conducting cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208-1217 (2009).
    [CrossRef]
  7. N. P. Zhuck and A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16-21 (1994).
    [CrossRef]
  8. S. C. Lee, “Light scattering by closely spaced parallel cylinders embedded in a finite dielectric slab,” J. Opt. Soc. Am. A 16, 1350-1361 (1999).
    [CrossRef]
  9. Q. A. Naqvi and A. A. Rizvi, “Low contrast circular cylinder buried in a grounded dielectric layer,” J. Electromagn. Waves Appl. 12, 1527-1536 (1998).
    [CrossRef]
  10. C. H. Kuo and M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54, 2392-2401 (2006).
    [CrossRef]
  11. M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by buried dielectric cylindrical structures,” Radio Sci. 40, RS6S18 (2005).
    [CrossRef]
  12. G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
    [CrossRef]
  13. R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “Numerical study of the reflection of cylindrical waves of arbitrary order by a generic planar interface,” J. Electromagn. Waves Appl. 13, 27-50 (1999).
    [CrossRef]
  14. R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “A quadrature algorithm for the evaluation of a 2D radiation integral with highly oscillating kernel,” J. Electromagn. Waves Appl. 14, 1353-1370 (2000).
    [CrossRef]
  15. I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory (North-Holland, 1966).
  16. A. Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Sci. 29, 1023-1033 (1994).
    [CrossRef]
  17. W. H. Press, S. A. Teukolski, V. T. Vetterling, B. P. Flannery, and M. Metcalf, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge Univ. Press, 1992).
  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 2nd ed. (Dover, 1972).

2009

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Electromagnetic scattering by perfectly-conducting cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208-1217 (2009).
[CrossRef]

2006

C. H. Kuo and M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54, 2392-2401 (2006).
[CrossRef]

2005

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by buried dielectric cylindrical structures,” Radio Sci. 40, RS6S18 (2005).
[CrossRef]

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” Radio Sci. 40, 11 (2005).

2002

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368-1376 (2002).
[CrossRef]

2000

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “A quadrature algorithm for the evaluation of a 2D radiation integral with highly oscillating kernel,” J. Electromagn. Waves Appl. 14, 1353-1370 (2000).
[CrossRef]

1999

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “Numerical study of the reflection of cylindrical waves of arbitrary order by a generic planar interface,” J. Electromagn. Waves Appl. 13, 27-50 (1999).
[CrossRef]

S. C. Lee, “Light scattering by closely spaced parallel cylinders embedded in a finite dielectric slab,” J. Opt. Soc. Am. A 16, 1350-1361 (1999).
[CrossRef]

1998

Q. A. Naqvi and A. A. Rizvi, “Low contrast circular cylinder buried in a grounded dielectric layer,” J. Electromagn. Waves Appl. 12, 1527-1536 (1998).
[CrossRef]

1997

1994

N. P. Zhuck and A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16-21 (1994).
[CrossRef]

A. Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Sci. 29, 1023-1033 (1994).
[CrossRef]

1993

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
[CrossRef]

1990

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 2nd ed. (Dover, 1972).

Borghi, R.

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “A quadrature algorithm for the evaluation of a 2D radiation integral with highly oscillating kernel,” J. Electromagn. Waves Appl. 14, 1353-1370 (2000).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “Numerical study of the reflection of cylindrical waves of arbitrary order by a generic planar interface,” J. Electromagn. Waves Appl. 13, 27-50 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A 14, 1500-1504 (1997).
[CrossRef]

Cincotti, G.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
[CrossRef]

Daniels, D. J.

D. J. Daniels, Surface Penetrating Radar, 2nd ed. (IEE, 2004).
[CrossRef]

Di Vico, M.

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” Radio Sci. 40, 11 (2005).

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by buried dielectric cylindrical structures,” Radio Sci. 40, RS6S18 (2005).
[CrossRef]

Elsherbeni, A. Z.

A. Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Sci. 29, 1023-1033 (1994).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolski, V. T. Vetterling, B. P. Flannery, and M. Metcalf, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge Univ. Press, 1992).

Frezza, F.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Electromagnetic scattering by perfectly-conducting cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208-1217 (2009).
[CrossRef]

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” Radio Sci. 40, 11 (2005).

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by buried dielectric cylindrical structures,” Radio Sci. 40, RS6S18 (2005).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “A quadrature algorithm for the evaluation of a 2D radiation integral with highly oscillating kernel,” J. Electromagn. Waves Appl. 14, 1353-1370 (2000).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “Numerical study of the reflection of cylindrical waves of arbitrary order by a generic planar interface,” J. Electromagn. Waves Appl. 13, 27-50 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A 14, 1500-1504 (1997).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
[CrossRef]

Furnò, F.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
[CrossRef]

Gori, F.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
[CrossRef]

Kuo, C. H.

C. H. Kuo and M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54, 2392-2401 (2006).
[CrossRef]

Lawrence, D. E.

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368-1376 (2002).
[CrossRef]

Lee, S. C.

Metcalf, M.

W. H. Press, S. A. Teukolski, V. T. Vetterling, B. P. Flannery, and M. Metcalf, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge Univ. Press, 1992).

Moghaddam, M.

C. H. Kuo and M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54, 2392-2401 (2006).
[CrossRef]

Naqvi, Q. A.

Q. A. Naqvi and A. A. Rizvi, “Low contrast circular cylinder buried in a grounded dielectric layer,” J. Electromagn. Waves Appl. 12, 1527-1536 (1998).
[CrossRef]

Pajewski, L.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Electromagnetic scattering by perfectly-conducting cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208-1217 (2009).
[CrossRef]

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” Radio Sci. 40, 11 (2005).

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by buried dielectric cylindrical structures,” Radio Sci. 40, RS6S18 (2005).
[CrossRef]

Ponti, C.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Electromagnetic scattering by perfectly-conducting cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208-1217 (2009).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolski, V. T. Vetterling, B. P. Flannery, and M. Metcalf, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge Univ. Press, 1992).

Rizvi, A. A.

Q. A. Naqvi and A. A. Rizvi, “Low contrast circular cylinder buried in a grounded dielectric layer,” J. Electromagn. Waves Appl. 12, 1527-1536 (1998).
[CrossRef]

Santarsiero, M.

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “A quadrature algorithm for the evaluation of a 2D radiation integral with highly oscillating kernel,” J. Electromagn. Waves Appl. 14, 1353-1370 (2000).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “Numerical study of the reflection of cylindrical waves of arbitrary order by a generic planar interface,” J. Electromagn. Waves Appl. 13, 27-50 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A 14, 1500-1504 (1997).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
[CrossRef]

Santini, C.

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “A quadrature algorithm for the evaluation of a 2D radiation integral with highly oscillating kernel,” J. Electromagn. Waves Appl. 14, 1353-1370 (2000).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “Numerical study of the reflection of cylindrical waves of arbitrary order by a generic planar interface,” J. Electromagn. Waves Appl. 13, 27-50 (1999).
[CrossRef]

Sarabandi, K.

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368-1376 (2002).
[CrossRef]

Schettini, G.

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Electromagnetic scattering by perfectly-conducting cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208-1217 (2009).
[CrossRef]

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” Radio Sci. 40, 11 (2005).

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by buried dielectric cylindrical structures,” Radio Sci. 40, RS6S18 (2005).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “A quadrature algorithm for the evaluation of a 2D radiation integral with highly oscillating kernel,” J. Electromagn. Waves Appl. 14, 1353-1370 (2000).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “Numerical study of the reflection of cylindrical waves of arbitrary order by a generic planar interface,” J. Electromagn. Waves Appl. 13, 27-50 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A 14, 1500-1504 (1997).
[CrossRef]

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
[CrossRef]

Sneddon, I. N.

I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory (North-Holland, 1966).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 2nd ed. (Dover, 1972).

Taubenblatt, M. A.

Teukolski, S. A.

W. H. Press, S. A. Teukolski, V. T. Vetterling, B. P. Flannery, and M. Metcalf, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge Univ. Press, 1992).

Vetterling, V. T.

W. H. Press, S. A. Teukolski, V. T. Vetterling, B. P. Flannery, and M. Metcalf, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge Univ. Press, 1992).

Yarovoy, A. G.

N. P. Zhuck and A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16-21 (1994).
[CrossRef]

Zhuck, N. P.

N. P. Zhuck and A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16-21 (1994).
[CrossRef]

IEEE Trans. Antennas Propag.

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368-1376 (2002).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, and G. Schettini, “Electromagnetic scattering by perfectly-conducting cylinders buried in a dielectric slab through the cylindrical wave approach,” IEEE Trans. Antennas Propag. 57, 1208-1217 (2009).
[CrossRef]

N. P. Zhuck and A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16-21 (1994).
[CrossRef]

C. H. Kuo and M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54, 2392-2401 (2006).
[CrossRef]

J. Electromagn. Waves Appl.

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “Numerical study of the reflection of cylindrical waves of arbitrary order by a generic planar interface,” J. Electromagn. Waves Appl. 13, 27-50 (1999).
[CrossRef]

R. Borghi, F. Frezza, M. Santarsiero, C. Santini, and G. Schettini, “A quadrature algorithm for the evaluation of a 2D radiation integral with highly oscillating kernel,” J. Electromagn. Waves Appl. 14, 1353-1370 (2000).
[CrossRef]

Q. A. Naqvi and A. A. Rizvi, “Low contrast circular cylinder buried in a grounded dielectric layer,” J. Electromagn. Waves Appl. 12, 1527-1536 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192-198 (1993).
[CrossRef]

Opt. Lett.

Radio Sci.

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by a finite set of perfectly conducting cylinders buried in a dielectric half-space: a spectral-domain solution,” Radio Sci. 40, 11 (2005).

A. Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Sci. 29, 1023-1033 (1994).
[CrossRef]

M. Di Vico, F. Frezza, L. Pajewski, and G. Schettini, “Scattering by buried dielectric cylindrical structures,” Radio Sci. 40, RS6S18 (2005).
[CrossRef]

Other

I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory (North-Holland, 1966).

W. H. Press, S. A. Teukolski, V. T. Vetterling, B. P. Flannery, and M. Metcalf, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge Univ. Press, 1992).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 2nd ed. (Dover, 1972).

D. J. Daniels, Surface Penetrating Radar, 2nd ed. (IEE, 2004).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of the scattering problem.

Fig. 2
Fig. 2

Decomposition of the total field.

Fig. 3
Fig. 3

(a) Geometrical layout for a dielectric cylinder buried in a grounded dielectric slab; (b) Comparison between our results (curves) and results in Fig. 3 of [9] (circles).

Fig. 4
Fig. 4

Behavior of the magnitude of the expansion coefficients c m for different numbers of reflections R for the geometrical layout of Fig. 3a when Λ = 12 π .

Fig. 5
Fig. 5

(a) Geometrical layout for a dielectric cylinder on a planar interface; (b) Far-field scattered intensity in arbitrary units as a function of the scattering angle θ = θ 90 ° , with a = 0.35 μ m , n c = 1.46 , n 2 = 3.8 , φ i = 30 ° , and TM polarization. Comparison between our results (curves) and Fig. 2 of [2] (circles).

Fig. 6
Fig. 6

(a) Geometrical layout for a dielectric cylinder below a planar interface; (b) Far-field radar cross section σ as a function of the scattering angle θ = θ 90 ° for a half-space of permittivity ϵ 1 = ϵ 2 = 1.2 , with α = 0.32 π , χ = 2.6 π , n c = 1.5 , φ i = 30 ° , and TE polarization. Comparison between our results (curve) and Fig. 2 of [4] (circles).

Fig. 7
Fig. 7

(a) Geometrical layout for a grid of three equally spaced silicon cylinders ( n c = 3.4211 ) with radius of 2.5 μ m and spacing d = λ embedded in a layer 50 μ m thick at a depth of 20 μ m , with refraction index n 1 = 1.333 . Scattered near-field is evaluated along a line at x = λ 2 for (b) different refraction indexes n 2 , (c) cylinder radius a, (d) cylinder–cylinder distance d.

Equations (39)

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

V i ( ξ , ζ ) = V 0 e i ( n i ξ + n i ζ ) ,
V r ( ξ , ζ ) = V 0 Γ 01 ( n i ) e i ( n i ξ + n i ζ ) ,
V t 1 ( ξ , ζ ) = V 0 T 01 ( n i ) e i n 1 [ n t 1 ( ξ Λ ) + n t 1 ζ ] = V 0 T 01 ( n i ) e i n 1 [ n t 1 ( χ p Λ ) + n t 1 η p ] l = + i l J l ( n 1 ρ p ) e i l θ p e i l φ t 1 ,
V r 1 ( ξ , ζ ) = V 0 T 01 ( n i ) Γ 12 ( n i ) e i n 1 [ n t 1 ( ξ Λ ) + n t 1 ζ ] = V 0 T 01 ( n i ) Γ 12 ( n i ) e i n 1 [ n t 1 ( χ p Λ ) + n t 1 η p ] l = + i l J l ( n 1 ρ p ) e i l θ p e i l φ r 1 ,
V t 2 ( ξ , ζ ) = V 0 T 01 ( n i ) T 12 ( n i ) e i n 2 [ n t 2 ( ξ Λ ) + n t 2 ζ ] .
V s ( ξ , ζ ) = V 0 p = 1 N m = + c pm C W m ( n 1 ξ p , n 1 ζ p ) .
H m ( 1 ) ( n 1 ρ q ) e i m θ q = e i m θ qp l = + i l H m + l ( 1 ) ( n 1 ρ qp ) e i l θ qp J l ( n 1 ρ p ) e i l θ p ,
V s ( ξ , ζ ) = V 0 l = + J l ( n 1 ρ p ) e i l θ p q = 1 N m = + c qm [ C W m l ( n 1 ξ qp , n 1 ζ qp ) ( 1 δ qp ) + H l ( 1 ) ( n 1 ρ p ) J l ( n 1 ρ p ) δ qp δ l m ] ,
C W m ( ξ , ζ ) = 1 2 π + F m ( ξ , n s ) e i n s ζ d n s ,
F m ( ξ , n s ) = 2 1 ( n s ) 2 e i | ξ | 1 ( n s ) 2 { e i m arccos n s , ξ 0 e i m arccos n s , ξ 0 } .
R W m ( 1 , ) ( j ) ( ξ , ζ ) = 1 2 π + [ Γ 10 ( n s ) ] g [ Γ 12 ( n s ) ] r F m ( ξ , n s ) e i n s ζ d n s ,
V sr ( j ) 1 , 0 ( ξ , ζ ) = V 0 l = + J l ( n 1 ρ p ) e i l θ p q = 1 N m = + c qm R W m + l 1 , 0 ( j ) { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n 1 ( η p η q ) } ,
{ h = j , if j = 2 , 4 , 6 , h = j 1 , if j = 1 , 3 , 5 , } ,
R W m 1 , 0 ( j ) { n 1 [ ( h Λ χ q + ( 1 ) j χ p ) ] , n 1 ( η p η q ) } = 1 2 π + [ Γ 10 ( n s ) ] g [ Γ 12 ( n s ) ] r F m { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n s } e in s n 1 ( η p η q ) dn s .
V sr ( j ) 1 , 2 ( ξ , ζ ) = V 0 l = + J l ( n 1 ρ p ) e i l θ p q = 1 N m = + c qm R W m + l 1 , 2 ( j ) { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n 1 ( η p η q ) } ,
{ h = j , if j = 2 , 4 , 6 , h = j + 1 , if j = 1 , 3 , 5 , } ,
R W m 1 , 2 ( j ) { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n 1 ( η p η q ) } = 1 2 π + [ Γ 12 ( n s ) ] h 2 [ Γ 10 ( n s ) ] h 2 F m { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n s } e in s n 1 ( η p η q ) dn s .
T W m ( ξ , ζ ; χ q , n 1 ) = 1 2 π + T 10 ( n s ) F m ( n ξ , n s ) e i n st ( ξ + χ q ) e i n st ζ d n s ,
V st 1 , 0 ( ξ , ζ ) = V 0 q = 1 N m = + c qm T W m ( ξ q , ζ q ; χ q , n 1 ) .
{ n st = 1 ( n 1 n s ) 2 n st = n 1 n s } .
{ n srt = 1 ( n 1 n sr ) 2 n srt = n 1 n sr } ,
R T W m ( j ) ( ξ , ζ ; χ q , Λ , n 1 , n 2 ) = 1 2 π + T 10 ( n s ) [ Γ 10 ( n s ) ] g [ Γ 12 ( n s ) ] r F m ( χ q + Λ , n s ) e i n st ( ξ + χ ) e i n s ζ d n s ,
V srt ( j ) 1 , 0 ( ξ , ζ ) = V 0 q = 1 N m = + c qm R T W m ( j ) [ ξ , ζ ; χ q , + ( 1 ) j + 1 h Λ , n 1 , n 2 ] ,
{ h = j , if j = 2 , 4 , 6 , h = j + 1 , if j = 1 , 3 , 5 , } .
V cp ( ξ , ζ ) = V 0 l = + d p l J l ( n cp ρ p ) e i l θ p .
[ V t 1 + V r 1 + V s + j = 1 + V sr ( j ) 1 , 0 + j = 1 + V sr ( j ) 1 , 2 ] ρ p = k 0 α p = [ V cp ] ρ p = k 0 α p
[ ρ p ( V t 1 + V r 1 + V s + j = 1 + V sr ( j ) 1 , 0 + j = 1 + V sr ( j ) 1 , 2 ) ] ρ p = k 0 α p = t q [ V cp ] ρ p = k 0 α p
t = { 1 , for TM polarization ( n 1 n cp ) 2 , for TE polarization } .
q = 1 N m = + A l m qp ( 1 ) c qm B l p ( 1 ) = d p l G l p ( 1 ) ,
q = 1 N m = + A l m qp ( 2 ) c qm B l p ( 2 ) = d p l G m l ( 2 ) ,
l = 0 , ± 1 , , ±
p = 1 , , N ,
A l m qp ( 1 , 2 ) = i l T l ( 1 , 2 ) { C W m l ( n 1 ξ qp , n 1 ζ qp ) ( 1 δ qp ) + δ qp δ l m T l ( 1 , 2 ) ( n 1 ρ p ) + j = 1 + R W m + l 1 , 0 ( j ) { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n 1 ( η p η q ) } + j = 1 + R W m + l 1 , 2 ( j ) { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n 1 ( η p η q ) } } ,
B l p ( 1 , 2 ) = T l ( 1 , 2 ) { T 01 ( n i ) e i n 1 [ n t 1 ( χ p Λ ) + n t 1 η p ] e i l φ t 1 + Γ 12 ( n i ) T 01 ( n i ) e i n 1 [ n t 1 ( χ p Λ ) + n t 1 η p ] e i l φ r 1 } ,
G l p ( 1 ) = J l ( n cp α p ) H l ( 1 ) ( n 1 α p ) ,
G l p ( 2 ) = g p J l ( n cp α p ) H l ( 1 ) ( n 1 α p ) ,
q = 1 N m = + D m l qp c qm = M l p ,
d p l = q = 1 N m = + A l m qp ( 1 ) c qm B l p ( 1 ) G l p ( 1 ) = q = 1 N m = + A l m qp ( 2 ) c qm B l p ( 2 ) G l p ( 2 ) .
d p l = n 1 J l ( n 1 α p ) H l ( 1 ) ( n 1 α p ) J l ( n 1 α p ) H l ( 1 ) ( n 1 α p ) n 1 J l ( n 1 α p ) H l ( 1 ) ( n 1 α p ) n cp J l ( n 1 α p ) H l ( 1 ) ( n 1 α p ) × { q = 1 N m = + i l c qm { C W m l ( n 1 ξ qp , n 1 ζ qp ) ( 1 δ qp ) + j = 1 + R W m + l 1 , 0 ( j ) { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n 1 ( η p η q ) } + j = 1 + R W m + l 1 , 2 ( j ) { n 1 [ h Λ χ q + ( 1 ) j χ p ] , n 1 ( η p η q ) } } + T l ( 1 , 2 ) { T 01 ( n i ) e i n 1 [ n t 1 ( χ p Λ ) + n t 1 η p ] e i l φ t 1 + Γ 12 ( n i ) T 01 ( n i ) e i n 1 [ n t 1 ( χ p Λ ) + n t 1 η p ] e i l φ r 1 } } .

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