Abstract

Cylindrical electromagnetic bandgap structures excited by a dipole source are analyzed by a rigorous semi-analytical method using the T-matrix approach and reflection and transmission matrices of cylindrical harmonic wave expansions as the basis. Resonance and stopband characteristics in the transmission spectra of the cylindrical bandgap structures for the excited cylindrical harmonic waves are studied. The relationship between the transmission spectra of the cylindrical harmonic waves and the radiation patterns of the dipole source in both H-plane and E-plane is numerically investigated for three different configurations of cylindrical bandgap structures and different locations of the dipole source with respect to the cylindrical bandgap structures.

© 2012 Optical Society of America

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References

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  1. R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).
  2. K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC, 2005).
  3. G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14, 3323–3332 (1997).
    [CrossRef]
  4. G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
    [CrossRef]
  5. K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
    [CrossRef]
  6. G. Guida, D. Maystre, G. Tayeb, and P. Vincent, “Mean-field theory of two-dimensional metallic photonic crystals,” J. Opt. Soc. Am. B 15, 2308–2315 (1998).
    [CrossRef]
  7. K. Watanabe, R. Petit, and M. Neviere, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
    [CrossRef]
  8. V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals,” Int. J. Infrared Millimeter Terahertz Waves 30, 1102–1112(2009).
    [CrossRef]
  9. P. St. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
    [CrossRef]
  10. H. Boutayeb and T. Denidni, “Metallic cylindrical EBG structures with defects: Directivity analysis and design optimization,” IEEE Trans. Antennas Propag. 55, 3356–3361(2007).
    [CrossRef]
  11. H. Boutayeb, A.-C. Tarot, and K. Mahdjoubi, “Focusing characteristics of a metallic cylindrical electromagnetic band gap structure with defects,” Progr. Electromagn. Res. 66, 89–103 (2006).
    [CrossRef]
  12. C. Biancotto and P. Record, “Design of a beam forming dielectric cylindrical EBG antenna,” Progr. Electromagn. Res. B 18, 327–346 (2009).
    [CrossRef]
  13. G. Palikaras, A. Feresidis, and J. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Antennas Wireless Propagat. Lett. 3, 87–89 (2004).
    [CrossRef]
  14. V. Jandieri and K. Yasumoto, “Electromagnetic scattering by layered cylindrical arrays of circular rods,” IEEE Trans. Antennas Propag. 59, 2437–2441 (2011).
    [CrossRef]
  15. K. Yasumoto, V. Jandieri, and B. Gupta, “Guidance and scattering of electromagnetic waves by layered cylindrical arrays of circular rods,” in Proceedings of IEEE Applied Electromagnetics Conference (IEEE, 2009), pp. 1–4.
  16. V. Jandieri and K. Yasumoto, “Stopband and resonance characteristics of cylindrical electromagnetic bandgap structures,” PIERS Online 7, 761–765 (2011).
  17. V. Jandieri and K. Yasumoto, “Analysis of scattering from a finite array of circular cylinders using a model of layered cylindrical arrays,” Optics Commun. 284, 4109–4113 (2011).
    [CrossRef]
  18. V. Jandieri, K. Yasumoto, A. Sharma, and H. Chauhan, “Modal analysis of specific microstructured optical fibers using a model of layered cylindrical arrays of circular rods,” IEICE Trans. Electron. E93-C, 17–23 (2010).
    [CrossRef]
  19. T. Laroche and C. Girard, “Near field optical properties of single plasmonic nanowires,” Appl. Phys. Lett. 89, 233119 (2006).
    [CrossRef]
  20. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  21. W. C. Chew, Waves and Fields in Inhomogeneous Media(Van Nostrand Reinhold, 1990).

2011 (3)

V. Jandieri and K. Yasumoto, “Electromagnetic scattering by layered cylindrical arrays of circular rods,” IEEE Trans. Antennas Propag. 59, 2437–2441 (2011).
[CrossRef]

V. Jandieri and K. Yasumoto, “Stopband and resonance characteristics of cylindrical electromagnetic bandgap structures,” PIERS Online 7, 761–765 (2011).

V. Jandieri and K. Yasumoto, “Analysis of scattering from a finite array of circular cylinders using a model of layered cylindrical arrays,” Optics Commun. 284, 4109–4113 (2011).
[CrossRef]

2010 (1)

V. Jandieri, K. Yasumoto, A. Sharma, and H. Chauhan, “Modal analysis of specific microstructured optical fibers using a model of layered cylindrical arrays of circular rods,” IEICE Trans. Electron. E93-C, 17–23 (2010).
[CrossRef]

2009 (2)

C. Biancotto and P. Record, “Design of a beam forming dielectric cylindrical EBG antenna,” Progr. Electromagn. Res. B 18, 327–346 (2009).
[CrossRef]

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals,” Int. J. Infrared Millimeter Terahertz Waves 30, 1102–1112(2009).
[CrossRef]

2007 (1)

H. Boutayeb and T. Denidni, “Metallic cylindrical EBG structures with defects: Directivity analysis and design optimization,” IEEE Trans. Antennas Propag. 55, 3356–3361(2007).
[CrossRef]

2006 (3)

H. Boutayeb, A.-C. Tarot, and K. Mahdjoubi, “Focusing characteristics of a metallic cylindrical electromagnetic band gap structure with defects,” Progr. Electromagn. Res. 66, 89–103 (2006).
[CrossRef]

T. Laroche and C. Girard, “Near field optical properties of single plasmonic nanowires,” Appl. Phys. Lett. 89, 233119 (2006).
[CrossRef]

P. St. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
[CrossRef]

2004 (2)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

G. Palikaras, A. Feresidis, and J. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Antennas Wireless Propagat. Lett. 3, 87–89 (2004).
[CrossRef]

2002 (1)

2000 (1)

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

1998 (1)

1997 (1)

Biancotto, C.

C. Biancotto and P. Record, “Design of a beam forming dielectric cylindrical EBG antenna,” Progr. Electromagn. Res. B 18, 327–346 (2009).
[CrossRef]

Boutayeb, H.

H. Boutayeb and T. Denidni, “Metallic cylindrical EBG structures with defects: Directivity analysis and design optimization,” IEEE Trans. Antennas Propag. 55, 3356–3361(2007).
[CrossRef]

H. Boutayeb, A.-C. Tarot, and K. Mahdjoubi, “Focusing characteristics of a metallic cylindrical electromagnetic band gap structure with defects,” Progr. Electromagn. Res. 66, 89–103 (2006).
[CrossRef]

Chauhan, H.

V. Jandieri, K. Yasumoto, A. Sharma, and H. Chauhan, “Modal analysis of specific microstructured optical fibers using a model of layered cylindrical arrays of circular rods,” IEICE Trans. Electron. E93-C, 17–23 (2010).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media(Van Nostrand Reinhold, 1990).

Cocchi, A.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

Denidni, T.

H. Boutayeb and T. Denidni, “Metallic cylindrical EBG structures with defects: Directivity analysis and design optimization,” IEEE Trans. Antennas Propag. 55, 3356–3361(2007).
[CrossRef]

Feresidis, A.

G. Palikaras, A. Feresidis, and J. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Antennas Wireless Propagat. Lett. 3, 87–89 (2004).
[CrossRef]

Girard, C.

T. Laroche and C. Girard, “Near field optical properties of single plasmonic nanowires,” Appl. Phys. Lett. 89, 233119 (2006).
[CrossRef]

Guida, G.

Gupta, B.

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals,” Int. J. Infrared Millimeter Terahertz Waves 30, 1102–1112(2009).
[CrossRef]

K. Yasumoto, V. Jandieri, and B. Gupta, “Guidance and scattering of electromagnetic waves by layered cylindrical arrays of circular rods,” in Proceedings of IEEE Applied Electromagnetics Conference (IEEE, 2009), pp. 1–4.

Jandieri, V.

V. Jandieri and K. Yasumoto, “Stopband and resonance characteristics of cylindrical electromagnetic bandgap structures,” PIERS Online 7, 761–765 (2011).

V. Jandieri and K. Yasumoto, “Electromagnetic scattering by layered cylindrical arrays of circular rods,” IEEE Trans. Antennas Propag. 59, 2437–2441 (2011).
[CrossRef]

V. Jandieri and K. Yasumoto, “Analysis of scattering from a finite array of circular cylinders using a model of layered cylindrical arrays,” Optics Commun. 284, 4109–4113 (2011).
[CrossRef]

V. Jandieri, K. Yasumoto, A. Sharma, and H. Chauhan, “Modal analysis of specific microstructured optical fibers using a model of layered cylindrical arrays of circular rods,” IEICE Trans. Electron. E93-C, 17–23 (2010).
[CrossRef]

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals,” Int. J. Infrared Millimeter Terahertz Waves 30, 1102–1112(2009).
[CrossRef]

K. Yasumoto, V. Jandieri, and B. Gupta, “Guidance and scattering of electromagnetic waves by layered cylindrical arrays of circular rods,” in Proceedings of IEEE Applied Electromagnetics Conference (IEEE, 2009), pp. 1–4.

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Laroche, T.

T. Laroche and C. Girard, “Near field optical properties of single plasmonic nanowires,” Appl. Phys. Lett. 89, 233119 (2006).
[CrossRef]

Mahdjoubi, K.

H. Boutayeb, A.-C. Tarot, and K. Mahdjoubi, “Focusing characteristics of a metallic cylindrical electromagnetic band gap structure with defects,” Progr. Electromagn. Res. 66, 89–103 (2006).
[CrossRef]

Maystre, D.

Monorchio, A.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

Neviere, M.

Palikaras, G.

G. Palikaras, A. Feresidis, and J. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Antennas Wireless Propagat. Lett. 3, 87–89 (2004).
[CrossRef]

Pelosi, G.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

Petit, R.

Record, P.

C. Biancotto and P. Record, “Design of a beam forming dielectric cylindrical EBG antenna,” Progr. Electromagn. Res. B 18, 327–346 (2009).
[CrossRef]

Russell, P. St.

Sharma, A.

V. Jandieri, K. Yasumoto, A. Sharma, and H. Chauhan, “Modal analysis of specific microstructured optical fibers using a model of layered cylindrical arrays of circular rods,” IEICE Trans. Electron. E93-C, 17–23 (2010).
[CrossRef]

Tarot, A.-C.

H. Boutayeb, A.-C. Tarot, and K. Mahdjoubi, “Focusing characteristics of a metallic cylindrical electromagnetic band gap structure with defects,” Progr. Electromagn. Res. 66, 89–103 (2006).
[CrossRef]

Tayeb, G.

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Vardaxoglou, J.

G. Palikaras, A. Feresidis, and J. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Antennas Wireless Propagat. Lett. 3, 87–89 (2004).
[CrossRef]

Vincent, P.

Watanabe, K.

Yasumoto, K.

V. Jandieri and K. Yasumoto, “Stopband and resonance characteristics of cylindrical electromagnetic bandgap structures,” PIERS Online 7, 761–765 (2011).

V. Jandieri and K. Yasumoto, “Analysis of scattering from a finite array of circular cylinders using a model of layered cylindrical arrays,” Optics Commun. 284, 4109–4113 (2011).
[CrossRef]

V. Jandieri and K. Yasumoto, “Electromagnetic scattering by layered cylindrical arrays of circular rods,” IEEE Trans. Antennas Propag. 59, 2437–2441 (2011).
[CrossRef]

V. Jandieri, K. Yasumoto, A. Sharma, and H. Chauhan, “Modal analysis of specific microstructured optical fibers using a model of layered cylindrical arrays of circular rods,” IEICE Trans. Electron. E93-C, 17–23 (2010).
[CrossRef]

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals,” Int. J. Infrared Millimeter Terahertz Waves 30, 1102–1112(2009).
[CrossRef]

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

K. Yasumoto, V. Jandieri, and B. Gupta, “Guidance and scattering of electromagnetic waves by layered cylindrical arrays of circular rods,” in Proceedings of IEEE Applied Electromagnetics Conference (IEEE, 2009), pp. 1–4.

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC, 2005).

Appl. Phys. Lett. (1)

T. Laroche and C. Girard, “Near field optical properties of single plasmonic nanowires,” Appl. Phys. Lett. 89, 233119 (2006).
[CrossRef]

IEEE Antennas Wireless Propagat. Lett. (1)

G. Palikaras, A. Feresidis, and J. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Antennas Wireless Propagat. Lett. 3, 87–89 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

V. Jandieri and K. Yasumoto, “Electromagnetic scattering by layered cylindrical arrays of circular rods,” IEEE Trans. Antennas Propag. 59, 2437–2441 (2011).
[CrossRef]

H. Boutayeb and T. Denidni, “Metallic cylindrical EBG structures with defects: Directivity analysis and design optimization,” IEEE Trans. Antennas Propag. 55, 3356–3361(2007).
[CrossRef]

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973–980 (2000).
[CrossRef]

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

IEICE Trans. Electron. (1)

V. Jandieri, K. Yasumoto, A. Sharma, and H. Chauhan, “Modal analysis of specific microstructured optical fibers using a model of layered cylindrical arrays of circular rods,” IEICE Trans. Electron. E93-C, 17–23 (2010).
[CrossRef]

Int. J. Infrared Millimeter Terahertz Waves (1)

V. Jandieri, K. Yasumoto, and B. Gupta, “Directivity of radiation from a localized source coupled to electromagnetic crystals,” Int. J. Infrared Millimeter Terahertz Waves 30, 1102–1112(2009).
[CrossRef]

J. Lightwave Technol. (1)

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

J. Opt. Soc. Am. B (1)

Optics Commun. (1)

V. Jandieri and K. Yasumoto, “Analysis of scattering from a finite array of circular cylinders using a model of layered cylindrical arrays,” Optics Commun. 284, 4109–4113 (2011).
[CrossRef]

PIERS Online (1)

V. Jandieri and K. Yasumoto, “Stopband and resonance characteristics of cylindrical electromagnetic bandgap structures,” PIERS Online 7, 761–765 (2011).

Progr. Electromagn. Res. (1)

H. Boutayeb, A.-C. Tarot, and K. Mahdjoubi, “Focusing characteristics of a metallic cylindrical electromagnetic band gap structure with defects,” Progr. Electromagn. Res. 66, 89–103 (2006).
[CrossRef]

Progr. Electromagn. Res. B (1)

C. Biancotto and P. Record, “Design of a beam forming dielectric cylindrical EBG antenna,” Progr. Electromagn. Res. B 18, 327–346 (2009).
[CrossRef]

Other (5)

K. Yasumoto, V. Jandieri, and B. Gupta, “Guidance and scattering of electromagnetic waves by layered cylindrical arrays of circular rods,” in Proceedings of IEEE Applied Electromagnetics Conference (IEEE, 2009), pp. 1–4.

R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC, 2005).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1965).

W. C. Chew, Waves and Fields in Inhomogeneous Media(Van Nostrand Reinhold, 1990).

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Figures (11)

Fig. 1.
Fig. 1.

Cross-sectional view of N-layered cylindrical structure formed by M circular rods with radius rv periodically distributed on each of N layered arrays. Radii of the N-layered cylindrical surfaces are Rv(v=1.2.3,,N). Excitation by a dipole source placed in the innermost region (0) is considered.

Fig. 2.
Fig. 2.

Schematic view of scattering process through the v-th layer of the cylindrical arrays and local coordinate systems attached to each of M circular rods. b(v) and c¯(v)are the amplitude vectors of incoming and outgoing cylindrical waves.

Fig. 3.
Fig. 3.

Cross-sectional view of three different configurations of three-layered cylindrical EBG structure of metallic circular rods with r1=r2=r3=0.05R1 periodically located on the concentric circular layers. Radii of the first, second and third circular layers are R1, R2=2R1 and R3=3R1, respectively. (a) Twelve circular rods are symmetrically distributed along the first layer, second layer, and third layer, and the source is located at the global origin. (b) One additional metallic circular rod having a radius rorig=0.3R1 is additionally placed at the origin of cylindrical structure (a) and the source is located at a distance ds from the origin. (c) Twelve circular rods are symmetrically distributed along the first layer, 24 circular rods along the second layer, and 36 circular rods along the third layer. The dipole source is located at the global origin.

Fig. 4.
Fig. 4.

Frequency response of |F1,0m,0(ω,θ)| for one- layered cylindrical EBG structure composed of 12 metallic circular rods with r1=0.05R1. Red line -θ=26°, blue line -θ=60°, black line -θ=90°.

Fig. 5.
Fig. 5.

Radiation patterns of dipole source in H-plane (blue line) and E-plane (red line) at two resonance frequencies (Fig. 4) of one-layered cylindrical EBG structure: R1ω/c=0.37 [(a), (b)] and R1ω/c=0.845 [(c), (d)]. Dipole source is located at the origin ds=0.

Fig. 6.
Fig. 6.

Frequency response of |F¯˜3,0m,0(ω,θ)| for three-layered cylindrical EBG structure shown in Fig. 3(a). Red curve,θ=26°; blue curve, θ=60°; black curve, θ=90°.

Fig. 7.
Fig. 7.

Radiation patterns of dipole source in H-plane (blue line) and E-plane (red line) at three resonance frequencies of three-layered cylindrical EBG structure: R1ω/c=0.37 [(a), (b)], R1ω/c=0.845 [(c), (d)] and R1ω/c=1.305 [(e), (f)]. Dipole source is located at the origin ds=0.

Fig. 8.
Fig. 8.

Frequency response of three components of generalized transmission matrix |F¯˜3,0m,n(ω,θ)| for three-layered cylindrical EBG structure shown in Fig. 3(b). Red curve, θ=26°; blue curve, θ=60°; black curve, θ=90°.

Fig. 9.
Fig. 9.

Radiation patterns of dipole source in H-plane (blue line) and E-plane (red line) at two resonance frequencies of three-layered cylindrical EBG structure shown in Fig. 3(b), where the dipole source is located at ds/R1=0.35 and one additional perfect conductor circular rod having radius rorig/R1=0.3 is placed at the origin: R1ω/c=0.37 [(a), (b)] and R1ω/c=0.845 [(c), (d)].

Fig. 10.
Fig. 10.

Frequency response of the seven components of the generalized transmission matrix |F¯˜3,0m,0(ω,θ=90°)| for three-layered cylindrical EBG structure shown in Fig. 3(c).

Fig. 11.
Fig. 11.

Radiation patterns of dipole in H-plane (blue line) and E-plane (red line) at two resonance frequencies of three-layered cylindrical EBG structure [Fig. 3(c)], where the dipole source is located at the global origin O: R1ω/c=0.825 [(a), (b)] and R1ω/c=1.34 [(c), (d)].

Equations (34)

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

ψ¯s(ν)=exp(iξz)Ψ¯T·j=1Mβ¯jν·a¯jνforρ>Rν
ψ¯s(ν1)=exp(iξz)Φ¯T·j=1Mη¯jν·a¯jνforρ<Rν
ψ¯s(ν)=[Ezs(ν)H^zs(ν)]
a¯jν=[aje,νajh,ν],aje,ν=[aj,me,ν],ajh,ν=[aj,mh,ν]
Ψ¯T=[ΨT00ΨT],Ψ=[Hm(1)(κρ)exp(imφ)]
Φ¯T=[ΦT00ΦT],Φ=[Jm(κρ)exp(imφ)]
β¯jν=[βjν00βjν],βjν=[βj,mnν]={Jmn(κRν)exp[im(j1)θM]}
η¯jν=[ηjν00ηjν],ηjν=[ηj,mnν]={Hmn(1)(κRν)exp[im(j1)θM]},
ψ¯r(ν)=Ψ¯T·R¯ν,ν1·b¯(ν)
ψ¯t(ν1)=Φ¯T·F¯ν1,ν·b¯(ν)
R¯ν,ν1=j=1Mβ¯jν·T¯¯+,jν
F¯ν1,ν=I¯+j=1Mη¯jν·T¯¯+,jν
T¯¯+,jν=q=1MσqνT¯να¯qνΘ¯j1(j=1,2,,M)
σqν==1Mexp[i(1)(p1)θM](Λlν)1
Λlν=I+p=2Mexp[i(1)(p1)θM]Apν
Θ¯=[Θ00Θ],Θ=[Θmn]=[exp(imθM)δmn]
Apν=T¯νK¯pν(p=2,3,,M)
K¯pν=[Kpν00Kpν],Kpν=[Kp,mnν]={Hmn(1)(κdpν)(1)mexp[i(m+n)ζp]}
ζp=π2(p1)θM2(p=2,3,,M)
dpν=2Rνsin[(p1)θM/2]
α¯qν=[αqν00αqν],αqν=[αq,mnν]={(1)mnJmn(κRν)exp[in(q1)θM]},
ψ¯t(ν)=Ψ¯T·F¯ν,ν1·c¯(ν1),
ψ¯r(ν1)=Φ¯T·R¯ν1,ν·c¯(ν1),
F¯ν,ν1=I¯+j=1Mβ¯jν·T¯¯,jν,
R¯ν1,ν=j=1Mη¯jν·T¯¯,jν,
T¯¯,jν=q=1MσqνT¯νγ¯qνΘj1(j=1,2,3,,M),
γ¯qν=[γqν00γqν],γqν=[γq,mnν]={(1)mnHmn(1)(κRν)exp[in(q1)θM]}.
E˜zi(ρ,φ,ξ)=ωμ0I4κ2k02ΨT·se,
se=[Jm(κds)exp(imϕs)],
[Ezt(N)H^zt(N)]=ωμ0I4exp(ik0r)πrsin2θm=exp(imφ)exp[i(2m+1)π4][cm(ξ)e(N)cm(ξ)h(N)]ξ=k0cosθ,
[cm(ξ)e(N)cm(ξ)h(N)]=F¯˜N,0(ξ)[sme(ξ)0],
F¯˜N,0(ξ)=F¯N,N1(ξ)·Γ¯N1,N2(ξ)Γ¯ν,ν1(ξ)Γ¯1,0(ξ),
Γ¯ν,ν1(ξ)=[I¯R¯ν,ν1(ξ)·R¯˜ν,ν+1(ξ)]1·F¯ν,ν1(ξ),
R¯˜v,v+1(ξ)=R¯v,v+1(ξ)+F¯v,v+1(ξ)·R¯˜ν+1,ν+2(ξ)·[I¯R¯ν+1,ν(ξ)·R¯˜ν+1,ν+2(ξ)]1·F¯v+1,ν(ξ).

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