Abstract

We report an experimental and theoretical study of whispering-gallery-mode propagation of subpicosecond terahertz pulses in a dielectric cylinder coupled by means of a dielectric slab waveguide. We observed repetitive cavity pulses from this structure for which the output pulse shapes were determined by the multi-whispering-gallery-mode coupling into the cylinder. A coupled-mode theory derived for this cylindrical system and coupling structure gives reasonably good agreement with the experiment in both the frequency and time domains.

© 2003 Optical Society of America

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References

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  1. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, Cambridge, UK, 1992).
  2. C. Vedrenne and J. Arnaud, “Whispering-gallery modes of dielectric resonators,” IEE Proc. 129, 183–187 (1982).
  3. D. Cros and P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1674 (1990).
    [CrossRef]
  4. L. Collot, V. Lefèvr-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
    [CrossRef]
  5. S. Chiller and R. L. Byer, “High-resolution spectroscopy of whispering gallery modes in large dielectrics,” Opt. Lett. 16, 1138–1140 (1991).
    [CrossRef]
  6. A. T. Rosenberger, “Nonlinear optical effects in the whispering-gallery modes of microspheres,” in Operational Characteristics and Crystal Growth of Nonlinear Optical Materials, R. B. Lal D.O. Frazier, eds., Proc. SPIE 3793, 179–186 (1999).
    [CrossRef]
  7. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
    [CrossRef] [PubMed]
  8. V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
    [CrossRef]
  9. P. Rabiei, W. H. Steier, C. Chang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. 20, 1968–1975 (2002).
    [CrossRef]
  10. V. S. Ilchenko, X. S. Yao, and L. Maleki, “Pigtailing the high-Q microsphere cavity: a simple fiber coupler for optical whispering-gallery modes,” Opt. Lett. 24, 723–725 (1999).
    [CrossRef]
  11. P. Barthia and I. J. Bahl, Millimeter Wave Engineering and Applications (Wiley, New York, 1984).
  12. G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
    [CrossRef]
  13. D. Kajfez and P. Guillon, Dielectric Resonators (Artech House, Norwood, Mass., 1986).
  14. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “THz waveguides,” J. Opt. Soc. Am. B 17, 851–863 (2000).
    [CrossRef]
  15. R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
    [CrossRef]
  16. R. Mendis and D. Grischkowsky, “Undistorted guided wave propagation of sub-picosecond THz pulses,” Opt. Lett. 26, 846–848 (2001).
    [CrossRef]
  17. R. Mendis and D. Grischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wire. Compon. Lett. 11, 444–446 (2001).
    [CrossRef]
  18. D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [CrossRef]
  19. R. W. Shaw, W. B. Whitten, M. D. Barnes, and J. M. Ramsey, “Time-domain observation of optical pulse propagation in whispering-gallery modes of glass spheres,” Opt. Lett. 23, 1301–1303 (1998).
    [CrossRef]
  20. J. Zhang and D. Grischkowsky, “Whispering-gallery mode terahertz pulses,” Opt. Lett. 27, 661–663 (2002).
    [CrossRef]
  21. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [CrossRef]
  22. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  23. S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
    [CrossRef]
  24. Q. Han, Y. Kogami, Y. Tomabechi, and K. Matsumura, “Coupling characteristics of eccentric arranged dielectric disk and ring,” IEEE Trans. Microwave Theory Tech. 44, 2017–2024 (1996).
    [CrossRef]
  25. J. Zhang, Ph.D. thesis, “A cylindrical dielectric whispering-gallery mode terahertz cavity coupled with a dielectric slab waveguide” (Oklahoma State University, Stillwater, Oklahoma, 2002).
  26. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
    [CrossRef]
  27. J. D. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

2002 (3)

2001 (2)

R. Mendis and D. Grischkowsky, “Undistorted guided wave propagation of sub-picosecond THz pulses,” Opt. Lett. 26, 846–848 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wire. Compon. Lett. 11, 444–446 (2001).
[CrossRef]

2000 (3)

G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
[CrossRef]

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “THz waveguides,” J. Opt. Soc. Am. B 17, 851–863 (2000).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
[CrossRef]

1999 (3)

1998 (2)

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

R. W. Shaw, W. B. Whitten, M. D. Barnes, and J. M. Ramsey, “Time-domain observation of optical pulse propagation in whispering-gallery modes of glass spheres,” Opt. Lett. 23, 1301–1303 (1998).
[CrossRef]

1996 (1)

Q. Han, Y. Kogami, Y. Tomabechi, and K. Matsumura, “Coupling characteristics of eccentric arranged dielectric disk and ring,” IEEE Trans. Microwave Theory Tech. 44, 2017–2024 (1996).
[CrossRef]

1993 (1)

L. Collot, V. Lefèvr-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

1991 (1)

1990 (2)

D. Cros and P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1674 (1990).
[CrossRef]

D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

1987 (1)

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
[CrossRef]

1982 (1)

C. Vedrenne and J. Arnaud, “Whispering-gallery modes of dielectric resonators,” IEE Proc. 129, 183–187 (1982).

1972 (1)

Annino, G.

G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
[CrossRef]

Arnaud, J.

C. Vedrenne and J. Arnaud, “Whispering-gallery modes of dielectric resonators,” IEE Proc. 129, 183–187 (1982).

Barnes, M. D.

Bertolini, D.

G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
[CrossRef]

Brune, M.

L. Collot, V. Lefèvr-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Byer, R. L.

Cassettari, M.

G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
[CrossRef]

Chang, C.

Chiller, S.

Chuang, S.-L.

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
[CrossRef]

Collot, L.

L. Collot, V. Lefèvr-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Cros, D.

D. Cros and P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1674 (1990).
[CrossRef]

Dalton, L. R.

Fattinger, Ch.

Fittipaldi, M.

G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
[CrossRef]

Gallot, G.

Gorodetsky, M. L.

Grischkowsky, D.

Guillon, P.

D. Cros and P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1674 (1990).
[CrossRef]

Han, Q.

Q. Han, Y. Kogami, Y. Tomabechi, and K. Matsumura, “Coupling characteristics of eccentric arranged dielectric disk and ring,” IEEE Trans. Microwave Theory Tech. 44, 2017–2024 (1996).
[CrossRef]

Haroche, S.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

L. Collot, V. Lefèvr-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Ilchenko, V. S.

Jamison, S. P.

Keiding, S.

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Kogami, Y.

Q. Han, Y. Kogami, Y. Tomabechi, and K. Matsumura, “Coupling characteristics of eccentric arranged dielectric disk and ring,” IEEE Trans. Microwave Theory Tech. 44, 2017–2024 (1996).
[CrossRef]

Lefèvre-Seguin, V.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Lefèvr-Seguin, V.

L. Collot, V. Lefèvr-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Longo, I.

G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
[CrossRef]

Maleki, L.

Martinelli, M.

G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
[CrossRef]

Matsumura, K.

Q. Han, Y. Kogami, Y. Tomabechi, and K. Matsumura, “Coupling characteristics of eccentric arranged dielectric disk and ring,” IEEE Trans. Microwave Theory Tech. 44, 2017–2024 (1996).
[CrossRef]

McGowan, R. W.

Mendis, R.

R. Mendis and D. Grischkowsky, “Undistorted guided wave propagation of sub-picosecond THz pulses,” Opt. Lett. 26, 846–848 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wire. Compon. Lett. 11, 444–446 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
[CrossRef]

Rabiei, P.

Raimond, J. M.

L. Collot, V. Lefèvr-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Raimond, J.-M.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Ramsey, J. M.

Rosenberger, A. T.

A. T. Rosenberger, “Nonlinear optical effects in the whispering-gallery modes of microspheres,” in Operational Characteristics and Crystal Growth of Nonlinear Optical Materials, R. B. Lal D.O. Frazier, eds., Proc. SPIE 3793, 179–186 (1999).
[CrossRef]

Shaw, R. W.

Snyder, A. W.

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Steier, W. H.

Tomabechi, Y.

Q. Han, Y. Kogami, Y. Tomabechi, and K. Matsumura, “Coupling characteristics of eccentric arranged dielectric disk and ring,” IEEE Trans. Microwave Theory Tech. 44, 2017–2024 (1996).
[CrossRef]

Treussart, F.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

van Exter, M.

Vedrenne, C.

C. Vedrenne and J. Arnaud, “Whispering-gallery modes of dielectric resonators,” IEE Proc. 129, 183–187 (1982).

Velichansky, V. L.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Volikov, P. S.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Whitten, W. B.

Yao, X. S.

Zhang, J.

Europhys. Lett. (1)

L. Collot, V. Lefèvr-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed in fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

IEE Proc. (1)

C. Vedrenne and J. Arnaud, “Whispering-gallery modes of dielectric resonators,” IEE Proc. 129, 183–187 (1982).

IEEE Microw. Wire. Compon. Lett. (1)

R. Mendis and D. Grischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wire. Compon. Lett. 11, 444–446 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

D. Cros and P. Guillon, “Whispering gallery dielectric resonator modes for W-band devices,” IEEE Trans. Microwave Theory Tech. 38, 1667–1674 (1990).
[CrossRef]

Q. Han, Y. Kogami, Y. Tomabechi, and K. Matsumura, “Coupling characteristics of eccentric arranged dielectric disk and ring,” IEEE Trans. Microwave Theory Tech. 44, 2017–2024 (1996).
[CrossRef]

J. Appl. Phys. (1)

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000).
[CrossRef]

J. Chem. Phys. (1)

G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, “Dielectric properties of materials using whispering-gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization,” J. Chem. Phys. 112, 2308–2314 (2000).
[CrossRef]

J. Lightwave Technol. (2)

P. Rabiei, W. H. Steier, C. Chang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. 20, 1968–1975 (2002).
[CrossRef]

S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. LT-5, 5–15 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (3)

Nature (1)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Opt. Commun. (1)

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Opt. Lett. (5)

Proc. SPIE (1)

A. T. Rosenberger, “Nonlinear optical effects in the whispering-gallery modes of microspheres,” in Operational Characteristics and Crystal Growth of Nonlinear Optical Materials, R. B. Lal D.O. Frazier, eds., Proc. SPIE 3793, 179–186 (1999).
[CrossRef]

Other (6)

P. Barthia and I. J. Bahl, Millimeter Wave Engineering and Applications (Wiley, New York, 1984).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, Cambridge, UK, 1992).

D. Kajfez and P. Guillon, Dielectric Resonators (Artech House, Norwood, Mass., 1986).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

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

J. Zhang, Ph.D. thesis, “A cylindrical dielectric whispering-gallery mode terahertz cavity coupled with a dielectric slab waveguide” (Oklahoma State University, Stillwater, Oklahoma, 2002).

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Figures (11)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Cross section of the slab–cylinder coupling structure.

Fig. 3
Fig. 3

Measured time domain pulses: (a) reference scan without cylinder, (b) sample scan with the cylinder in contact with the slab waveguide.

Fig. 4
Fig. 4

(a) Main transmitted pulses of Fig. 3 and, (b) their corresponding spectra. The dashed curves represent the reference scan, the open circles represent the sample scan, and the solid curves are the calculation results for the sample scan.

Fig. 5
Fig. 5

Output pulses for (a) the first and, (b) the second cavity pulses (4b and 5b, respectively). The solid curves are calculation results and the open circles represent the experiment.

Fig. 6
Fig. 6

Spectra of (a) the first and (b) the second cavity pulses (4b and 5b, respectively). The heavy curves are calculation results and the light curves with open circles represent the experiment.

Fig. 7
Fig. 7

Coordinate system of the slab–cylinder coupling structure.

Fig. 8
Fig. 8

Coupling coefficients for the slab TM0 mode and WG1 mode at 1.0 and 0.5 THz. The solid curves are real parts and the dashed curves represent imaginary parts. (a) C01 at 1 THz. (b) C01 at 0.5 THz. (c) P01 at 1 THz. (d) P01 at 0.5 THz.

Fig. 9
Fig. 9

Effective refractive indices of the slab TM modes and the cylindrical WGMs. The straight dashed line indicates the essentially constant refractive index of silicon nSi=3.417.

Fig. 10
Fig. 10

Amplitude evolution during the coupling process at 1.0 THz when the slab TM0 mode is initially excited.

Fig. 11
Fig. 11

Amplitude transfer functions of the main transmitted pulse |H0(ω)| and the cavity pulses |H1(ω)| and |H2(ω)|.

Equations (62)

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

(E1×H2-E2×H1)
=ik0μ0 (ε2-ε1)E1E2,
ddθ0(E1×H2-E2×H1)θˆdr
=ik0μ00(ε2-ε1)E1E2rdr,
εc(r, θ)=cinsidecylinder,1outsidecylinder.
εs(r, θ)=sinsideslab,1outsideslab.
εT(r, θ)=cinsidecylinder,sinsideslab,1inair.
ET(r, θ)=as(θ)Est(r, θ)+εsεT θˆEsθ(r, θ)+ac(θ)Ect(r, θ)+εcεT θˆEcθ(r, θ),
HT(r, θ)=as(θ)[Hst(r, θ)+θˆHsθ(r, θ)]+ac(θ)[Hct(r, θ)+θˆHcθ(r, θ)],
ddθ0(ET×Hs--Es-×HT)θˆdr
=ik0μ00(εs-εT)ETEs-rdr,
ddθ0(ET×Hc--Ec-×HT)θˆdr
=ik0μ00(εc-εT)ETEc-rdr,
ddθ [Pss(θ)as(θ)+Psc(θ)ac(θ)]=iCssas(θ)+iCscac(θ),
ddθ [Pcs(θ)as(θ)+Pcc(θ)ac(θ)]=iCcsas(θ)+iCccac(θ),
Ppq(θ)=140exp(iΔΦpq)(ept×hqt+eqt × hpt)θˆdr,
Cpq(θ)=k40μ00Δεpexp(iΔΦpq)×epteqt-εqεT epθeqθrdr,
PssPcc=1.
ddθn=0NPmn(θ)an(θ)=in=0NCmn(θ)an(θ),
ddθ [P(θ)A(θ)]=iC(θ)A(θ),
P(θ)=1P01P02P0NP10100P20010PN0001,
C(θ)=C00C01C02C0NC10C1100C200C220CN000CNN.
H1(ω)=m=1Namfs(ω)a0fcm(ω)exp[i2πlm(ω)].
H2(ω)=mamfs(ω)amfcm(ω)a0fcm(ω)exp[i4πlm(ω)].
hsmx(y)=-1Nsmkns2βm0μ0cos UmYcos Um,
esmy(y)=1Nsmcos UmYcos Um,
esmz(y)=-1Nsmins2bWmβmsin UmYsin Um;
hsmx(y)=-1Nsmkns2βm0μ0exp(-Wm|Y|)exp(-Wm),
esmy(y)=ns2Nsmexp(-Wm|Y|)exp(-Wm),
esmz(y)=-1Nsmins2bWmβmY|Y|exp(-Wm|Y|)exp(-Wm);
Um=bk2ns2-βm2,Wm=bβm2-k2,
Y=y/b.
½-esm×hsmzˆdx=1.
ns2Wm=Umtan Um.
ns2Wm=-Umcot Um.
hcmx(r)=1Ncm0μ0Jl(kncr)Jl(knca),
ecmr(r)=1Ncmlknc2rJl(kncr)Jl(knca),
ecmθ(r)=1NcmincJl(kncr)Jl(knca);
hcmx(r)=1Ncm0μ0Hl(1)(kr)Hl(1)(ka),
ecmr(r)=1NcmlkrHl(1)(kr)Hl(1)(ka),
ecmθ(r)=iNcm[Hl(1)(kr)]Hl(1)(ka);
½0ecm×hcmθˆdr=1.
Jl(knca)ncJl(knca)=[Hl(1)](ka)Hl(1)(ka).
nsm(f )=cνsm(f)=cβm(f )2πf
ncm(f )=cνcm(f)=clm(f )2πaf,
Es(r, θ)=Est(r, θ)+θˆEsθ(r, θ)=[est(r, θ)+θˆesθ(r, θ)]exp(iβz),
Hs(r, θ)=Hst(r, θ)+θˆHsθ(r, θ)=[hst(r, θ)+θˆhsθ(r, θ)]exp(iβz).
Ec(r, θ)=Ect(r, θ)+θˆEcθ(r, θ)=[ect(r, θ)+θˆecθ(r, θ)]exp(ilθ),
Hc(r, θ)=Hct(r, θ)+θˆHcθ(r, θ)=[hct(r, θ)+θˆhcθ(r, θ)]exp(ilθ).
est(r, θ)=rˆ[esy(y)cos θ+esz(y)sin θ],
hst(r, θ)=xˆhsx(y),
esθ(r, θ)=esz(y)cos θ-esx(y)sin θ,
hsθ(r, θ)=0,
ect(r, θ)=rˆecr(r),ecθ(r, θ)=ecθ(r),
hct(r, θ)=xˆhcx(r),hcθ(r, θ)=0,
y=r cos θ-(a+b),z=r sin θ.
Ec-(r, θ)=[ect(r, θ)-θˆecθ(r, θ)]exp(-ilθ),
Hc-(r, θ)=[-hct(r, θ)+θˆhcθ(r, θ)]exp(-ilθ),
Es-(r, θ)=[est(r, θ)-θˆesθ(r, θ)]exp(-iβz),
Hs-(r, θ)=[-hst(r, θ)+θˆhsθ(r, θ)]exp(-iβz).
ET(r, θ)=as(z)Est(r, θ)+εsεT θˆEsθ(r, θ)+ac(θ)Ect(r, θ)+εcεT θˆEcθ(r, θ)=as(z)est(r, θ)+εsεT θˆesθ(r, θ)exp(iβz)+ac(θ)ect(r, θ)+εcεT θˆecθ(r, θ)exp(ilθ),
HT(r, θ)=as(z)[Hst(r, θ)+θˆHsθ(r, θ)]+ac(θ)[Hct(r, θ)+θˆHcθ(r, θ)]=as(z)[hst(r, θ)+θˆhsθ(r, θ)]exp(iβz)+ac(θ)[hct(r, θ)+θˆhcθ(r, θ)]exp(ilθ).

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