Abstract

A unitary analytical approach making use of duality properties of Maxwell’s equations is developed for determining the propagation modes in slab and circular waveguides that comprise a gyrotropic material of dielectric or ferrite type, at which either the permittivity or the permeability tensor is altered by a longitudinally applied quasistatic magnetic field. Both types of electric and magnetic walls are considered. Closed form dimensionless relations are obtained for completely or partially filled and open-wall configurations, with the limit case of unmagnetized gyrotropic material being included. Examples are given for both the slab and the circular gyrotropic waveguides at arbitrary values of material parameters.

© 2010 Optical Society of America

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References

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  1. P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys. 38, 1482–1490 (1967).
    [CrossRef]
  2. I. V. Lindell, “Field decomposition in special gyrotropic media,” Microwave Opt. Technol. Lett. 12, 29–31 (1996).
    [CrossRef]
  3. A. Eroglu and J. K. Lee, “Wave propagation and dispersion characteristics for a nonreciprocal electrically gyrotropic medium,” PIER 62, 237–260 (2006).
    [CrossRef]
  4. H. Dötsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
    [CrossRef]
  5. R. A. Waldron, Ferrites: An Introduction for Microwave Engineers (Van Nostrand, 1961).
  6. B. M. Dillon, A. A. P. Gibson, and J. P. Webb, “Cut-off and phase constants of partially filled axially magnetized, gyromagnetic waveguides using finite elements,” IEEE Trans. Microwave Theory Tech. 41, 803–808 (1993).
    [CrossRef]
  7. D. Yevick, “A guide to electrical field propagation techniques for guided-wave optics,” Opt. Quantum Electron. 26, S185–S197 (1994).
    [CrossRef]
  8. L. Zhang and S. Xu, “Edge-element analysis of anisotropic waveguides with full permittivity and permeability matrices,” Int. J. Infrared Millim. Waves 16, 1351–1360 (1995).
    [CrossRef]
  9. E. Cojocaru, “Third-order triangular finite elements for waveguiding problems,” arXiv:1003.5609.
  10. P. Lüsse, P. Stuwe, J. Schüle, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
    [CrossRef]
  11. A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26, 1423–1431 (2008).
    [CrossRef]
  12. P. J. B. Clarricoats and D. E. Chambers, “Properties of cylindrical waveguides containing isotropic and anisotropic media,” Proc. IEE 110, 2163–2173 (1963).
  13. W. E. Salmond and C. Yeh, “Ferrite-filled elliptical waveguides. I. Propagation characteristics,” J. Appl. Phys. 41, 3210–3220 (1970).
    [CrossRef]
  14. J. Helszajn and A. A. P. Gibson, “Mode nomenclature of circular gyromagnetic and anisotropic waveguides with magnetic and open walls,” IEE Proc., Part H 134, 488–496 (1987).
  15. M. L. Kales, “Modes in wave guides containing ferrites,” J. Appl. Phys. 24, 604–608 (1953).
    [CrossRef]
  16. I. V. Lindell and A. H. Sihvola, “Electromagnetic boundary and its realization with anisotropic metamaterial,” Phys. Rev. E 79, 026604 (2009).
    [CrossRef]
  17. J. R. Gillies and P. Hlawiczka, “Elliptically polarized modes in gyrotropic waveguides: II. An alternative treatment of the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 10, 1891–1904 (1977).
    [CrossRef]
  18. P. Hlawiczka, “A gyrotropic waveguide with dielectric boundaries: the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 11, 1157–1166 (1978).
    [CrossRef]
  19. R. E. Colin, Field Theory of Guided Waves (IEEE, 1991).
  20. C. T. Tai, “Evanescent modes in a partially filled gyromagnetic rectangular wave guide,” J. Appl. Phys. 31, 220–221 (1960).
    [CrossRef]
  21. T. E. Murphy, “Optical modesolver,” http://www.photonics.umd.edu/.

2009 (1)

I. V. Lindell and A. H. Sihvola, “Electromagnetic boundary and its realization with anisotropic metamaterial,” Phys. Rev. E 79, 026604 (2009).
[CrossRef]

2008 (1)

2006 (1)

A. Eroglu and J. K. Lee, “Wave propagation and dispersion characteristics for a nonreciprocal electrically gyrotropic medium,” PIER 62, 237–260 (2006).
[CrossRef]

2005 (1)

1996 (1)

I. V. Lindell, “Field decomposition in special gyrotropic media,” Microwave Opt. Technol. Lett. 12, 29–31 (1996).
[CrossRef]

1995 (1)

L. Zhang and S. Xu, “Edge-element analysis of anisotropic waveguides with full permittivity and permeability matrices,” Int. J. Infrared Millim. Waves 16, 1351–1360 (1995).
[CrossRef]

1994 (2)

P. Lüsse, P. Stuwe, J. Schüle, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[CrossRef]

D. Yevick, “A guide to electrical field propagation techniques for guided-wave optics,” Opt. Quantum Electron. 26, S185–S197 (1994).
[CrossRef]

1993 (1)

B. M. Dillon, A. A. P. Gibson, and J. P. Webb, “Cut-off and phase constants of partially filled axially magnetized, gyromagnetic waveguides using finite elements,” IEEE Trans. Microwave Theory Tech. 41, 803–808 (1993).
[CrossRef]

1987 (1)

J. Helszajn and A. A. P. Gibson, “Mode nomenclature of circular gyromagnetic and anisotropic waveguides with magnetic and open walls,” IEE Proc., Part H 134, 488–496 (1987).

1978 (1)

P. Hlawiczka, “A gyrotropic waveguide with dielectric boundaries: the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 11, 1157–1166 (1978).
[CrossRef]

1977 (1)

J. R. Gillies and P. Hlawiczka, “Elliptically polarized modes in gyrotropic waveguides: II. An alternative treatment of the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 10, 1891–1904 (1977).
[CrossRef]

1970 (1)

W. E. Salmond and C. Yeh, “Ferrite-filled elliptical waveguides. I. Propagation characteristics,” J. Appl. Phys. 41, 3210–3220 (1970).
[CrossRef]

1967 (1)

P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

1963 (1)

P. J. B. Clarricoats and D. E. Chambers, “Properties of cylindrical waveguides containing isotropic and anisotropic media,” Proc. IEE 110, 2163–2173 (1963).

1960 (1)

C. T. Tai, “Evanescent modes in a partially filled gyromagnetic rectangular wave guide,” J. Appl. Phys. 31, 220–221 (1960).
[CrossRef]

1953 (1)

M. L. Kales, “Modes in wave guides containing ferrites,” J. Appl. Phys. 24, 604–608 (1953).
[CrossRef]

Bahlmann, N.

Chambers, D. E.

P. J. B. Clarricoats and D. E. Chambers, “Properties of cylindrical waveguides containing isotropic and anisotropic media,” Proc. IEE 110, 2163–2173 (1963).

Clarricoats, P. J. B.

P. J. B. Clarricoats and D. E. Chambers, “Properties of cylindrical waveguides containing isotropic and anisotropic media,” Proc. IEE 110, 2163–2173 (1963).

Cojocaru, E.

E. Cojocaru, “Third-order triangular finite elements for waveguiding problems,” arXiv:1003.5609.

Colin, R. E.

R. E. Colin, Field Theory of Guided Waves (IEEE, 1991).

Dillon, B. M.

B. M. Dillon, A. A. P. Gibson, and J. P. Webb, “Cut-off and phase constants of partially filled axially magnetized, gyromagnetic waveguides using finite elements,” IEEE Trans. Microwave Theory Tech. 41, 803–808 (1993).
[CrossRef]

Dötsch, H.

Eroglu, A.

A. Eroglu and J. K. Lee, “Wave propagation and dispersion characteristics for a nonreciprocal electrically gyrotropic medium,” PIER 62, 237–260 (2006).
[CrossRef]

Fallahkhair, A. B.

Gerhardt, R.

Gibson, A. A. P.

B. M. Dillon, A. A. P. Gibson, and J. P. Webb, “Cut-off and phase constants of partially filled axially magnetized, gyromagnetic waveguides using finite elements,” IEEE Trans. Microwave Theory Tech. 41, 803–808 (1993).
[CrossRef]

J. Helszajn and A. A. P. Gibson, “Mode nomenclature of circular gyromagnetic and anisotropic waveguides with magnetic and open walls,” IEE Proc., Part H 134, 488–496 (1987).

Gillies, J. R.

J. R. Gillies and P. Hlawiczka, “Elliptically polarized modes in gyrotropic waveguides: II. An alternative treatment of the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 10, 1891–1904 (1977).
[CrossRef]

Hammer, M.

Helszajn, J.

J. Helszajn and A. A. P. Gibson, “Mode nomenclature of circular gyromagnetic and anisotropic waveguides with magnetic and open walls,” IEE Proc., Part H 134, 488–496 (1987).

Hertel, P.

Hlawiczka, P.

P. Hlawiczka, “A gyrotropic waveguide with dielectric boundaries: the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 11, 1157–1166 (1978).
[CrossRef]

J. R. Gillies and P. Hlawiczka, “Elliptically polarized modes in gyrotropic waveguides: II. An alternative treatment of the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 10, 1891–1904 (1977).
[CrossRef]

Kales, M. L.

M. L. Kales, “Modes in wave guides containing ferrites,” J. Appl. Phys. 24, 604–608 (1953).
[CrossRef]

Lee, J. K.

A. Eroglu and J. K. Lee, “Wave propagation and dispersion characteristics for a nonreciprocal electrically gyrotropic medium,” PIER 62, 237–260 (2006).
[CrossRef]

Li, K. S.

Lindell, I. V.

I. V. Lindell and A. H. Sihvola, “Electromagnetic boundary and its realization with anisotropic metamaterial,” Phys. Rev. E 79, 026604 (2009).
[CrossRef]

I. V. Lindell, “Field decomposition in special gyrotropic media,” Microwave Opt. Technol. Lett. 12, 29–31 (1996).
[CrossRef]

Lüsse, P.

P. Lüsse, P. Stuwe, J. Schüle, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[CrossRef]

Murphy, T. E.

Pershan, P. S.

P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

Popkov, A. F.

Salmond, W. E.

W. E. Salmond and C. Yeh, “Ferrite-filled elliptical waveguides. I. Propagation characteristics,” J. Appl. Phys. 41, 3210–3220 (1970).
[CrossRef]

Schüle, J.

P. Lüsse, P. Stuwe, J. Schüle, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[CrossRef]

Sihvola, A. H.

I. V. Lindell and A. H. Sihvola, “Electromagnetic boundary and its realization with anisotropic metamaterial,” Phys. Rev. E 79, 026604 (2009).
[CrossRef]

Stuwe, P.

P. Lüsse, P. Stuwe, J. Schüle, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[CrossRef]

Tai, C. T.

C. T. Tai, “Evanescent modes in a partially filled gyromagnetic rectangular wave guide,” J. Appl. Phys. 31, 220–221 (1960).
[CrossRef]

Unger, H. -G.

P. Lüsse, P. Stuwe, J. Schüle, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[CrossRef]

Waldron, R. A.

R. A. Waldron, Ferrites: An Introduction for Microwave Engineers (Van Nostrand, 1961).

Webb, J. P.

B. M. Dillon, A. A. P. Gibson, and J. P. Webb, “Cut-off and phase constants of partially filled axially magnetized, gyromagnetic waveguides using finite elements,” IEEE Trans. Microwave Theory Tech. 41, 803–808 (1993).
[CrossRef]

Wilkens, L.

Xu, S.

L. Zhang and S. Xu, “Edge-element analysis of anisotropic waveguides with full permittivity and permeability matrices,” Int. J. Infrared Millim. Waves 16, 1351–1360 (1995).
[CrossRef]

Yeh, C.

W. E. Salmond and C. Yeh, “Ferrite-filled elliptical waveguides. I. Propagation characteristics,” J. Appl. Phys. 41, 3210–3220 (1970).
[CrossRef]

Yevick, D.

D. Yevick, “A guide to electrical field propagation techniques for guided-wave optics,” Opt. Quantum Electron. 26, S185–S197 (1994).
[CrossRef]

Zhang, L.

L. Zhang and S. Xu, “Edge-element analysis of anisotropic waveguides with full permittivity and permeability matrices,” Int. J. Infrared Millim. Waves 16, 1351–1360 (1995).
[CrossRef]

Zhuromskyy, O.

IEE Proc., Part H (1)

J. Helszajn and A. A. P. Gibson, “Mode nomenclature of circular gyromagnetic and anisotropic waveguides with magnetic and open walls,” IEE Proc., Part H 134, 488–496 (1987).

IEEE Trans. Microwave Theory Tech. (1)

B. M. Dillon, A. A. P. Gibson, and J. P. Webb, “Cut-off and phase constants of partially filled axially magnetized, gyromagnetic waveguides using finite elements,” IEEE Trans. Microwave Theory Tech. 41, 803–808 (1993).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

L. Zhang and S. Xu, “Edge-element analysis of anisotropic waveguides with full permittivity and permeability matrices,” Int. J. Infrared Millim. Waves 16, 1351–1360 (1995).
[CrossRef]

J. Appl. Phys. (4)

P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

M. L. Kales, “Modes in wave guides containing ferrites,” J. Appl. Phys. 24, 604–608 (1953).
[CrossRef]

C. T. Tai, “Evanescent modes in a partially filled gyromagnetic rectangular wave guide,” J. Appl. Phys. 31, 220–221 (1960).
[CrossRef]

W. E. Salmond and C. Yeh, “Ferrite-filled elliptical waveguides. I. Propagation characteristics,” J. Appl. Phys. 41, 3210–3220 (1970).
[CrossRef]

J. Lightwave Technol. (2)

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26, 1423–1431 (2008).
[CrossRef]

P. Lüsse, P. Stuwe, J. Schüle, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[CrossRef]

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

J. Phys. D: Appl. Phys. (2)

J. R. Gillies and P. Hlawiczka, “Elliptically polarized modes in gyrotropic waveguides: II. An alternative treatment of the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 10, 1891–1904 (1977).
[CrossRef]

P. Hlawiczka, “A gyrotropic waveguide with dielectric boundaries: the longitudinally magnetized case,” J. Phys. D: Appl. Phys. 11, 1157–1166 (1978).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

I. V. Lindell, “Field decomposition in special gyrotropic media,” Microwave Opt. Technol. Lett. 12, 29–31 (1996).
[CrossRef]

Opt. Quantum Electron. (1)

D. Yevick, “A guide to electrical field propagation techniques for guided-wave optics,” Opt. Quantum Electron. 26, S185–S197 (1994).
[CrossRef]

Phys. Rev. E (1)

I. V. Lindell and A. H. Sihvola, “Electromagnetic boundary and its realization with anisotropic metamaterial,” Phys. Rev. E 79, 026604 (2009).
[CrossRef]

PIER (1)

A. Eroglu and J. K. Lee, “Wave propagation and dispersion characteristics for a nonreciprocal electrically gyrotropic medium,” PIER 62, 237–260 (2006).
[CrossRef]

Proc. IEE (1)

P. J. B. Clarricoats and D. E. Chambers, “Properties of cylindrical waveguides containing isotropic and anisotropic media,” Proc. IEE 110, 2163–2173 (1963).

Other (4)

R. E. Colin, Field Theory of Guided Waves (IEEE, 1991).

T. E. Murphy, “Optical modesolver,” http://www.photonics.umd.edu/.

R. A. Waldron, Ferrites: An Introduction for Microwave Engineers (Van Nostrand, 1961).

E. Cojocaru, “Third-order triangular finite elements for waveguiding problems,” arXiv:1003.5609.

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Figures (6)

Fig. 1
Fig. 1

Schematic of different slab waveguides comprising a dielectric gyrotropic material of thickness h with material parameters ϵ ¯ and μ, infinitely extending in x and z directions. (a) Completely filled slab waveguide bounded by two plane-parallel walls; (b) partially filled slab waveguide bounded by two plane-parallel walls separated by a distance h 0 , with the gyrotropic material at 0 < y < h , and an isotropic dielectric of material parameters ( ϵ 1 , μ 1 ) at h < y < h 0 ; (c) slab waveguide backed by a PEC or PMC plane at y = 0 and embedded at y > h into the isotropic dielectric of parameters ( ϵ 1 , μ 1 ) ; (d) open-wall slab waveguide with the gyrotropic material at 0 < y < h embedded into two isotropic dielectrics of parameters ( ϵ 1 , μ 1 ) at y > h and ( ϵ 2 , μ 2 ) at y < 0 .

Fig. 2
Fig. 2

Schematic of different circular waveguides comprising a dielectric gyrotropic material of radius ρ with material parameters ϵ ¯ and μ. (a) Completely filled circular waveguide bounded by a wall; (b) partially filled circular waveguide with a wall at ρ 0 > ρ , the gyrotropic material at r < ρ , and an isotropic dielectric of material parameters ( ϵ 1 , μ 1 ) at ρ < r < ρ 0 ; (c) open-wall circular waveguide with the gyrotropic material embedded into the isotropic dielectric of parameters ( ϵ 1 , μ 1 ) .

Fig. 3
Fig. 3

Diagrams of the normalized propagation constant β / k 0 dependence on ϵ for propagating modes in (a) completely filled, (b) partially filled with air when h 0 / h = 1.25 , (c) backed by a PEC or PMC plane in air, and (d) open-wall in air slab waveguide comprising a dielectric gyrotropic material when k 0 h = 2.1 . In (a)–(c), with i = 1 , 2 : solid curve, HE i e mode; dashed curve, HE i m mode; dotted-dashed curve, EH i e mode; dotted curve, EH i m mode.

Fig. 4
Fig. 4

Diagrams of the normalized propagation constant β / k 0 dependence on ϵ for propagating modes in (a) completely filled, (b) partially filled with air when ρ 0 / ρ = 1.25 , and (c) open-wall in air circular waveguide comprising a dielectric gyrotropic material when k 0 ρ = 2.1 . In (a) and (b), with n = 0 , ± 1 : solid curve, HE n , 1 e mode; dashed curve, HE n , 1 m mode; dotted-dashed curve, EH n , 1 e mode; dotted curve, EH n , 1 m mode.

Fig. 5
Fig. 5

Diagrams of the normalized propagation constant β / k 0 dependence on μ for propagating modes in a completely filled circular waveguide comprising a ferrite gyrotropic material when k 0 ρ = 2.1 . With n = 0 , ± 1 : solid curve, HE n , 1 e mode; dashed curve, HE n , 1 m mode; dotted-dashed curve, EH n , 1 e mode; dotted curve, EH n , 1 m mode.

Fig. 6
Fig. 6

Representations of the unsuppressed normalized TM fields for the first two modes of an open-wall circular dielectric gyrotropic waveguide calculated numerically (by the vectorial finite difference method) in (a) and (b), and numerically (by the same method) and analytically in (c) and (d), at the values of β / k 0 specified on the graphs. The contours in (a) and (b) represent the magnitude of the normalized unsuppressed field specified on either graph in decibels starting from 0 dB in the center to −15 dB toward the boundary at radius ρ = 1 μ m in steps of −3 dB. In (c) and (d), the analytical (solid curve) and numerical (dotted curve) results are almost overlapped.

Tables (1)

Tables Icon

Table 1 Relevant Definitions for Dielectric and Ferrite Gyrotropic Waveguides

Equations (136)

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ϵ ¯ = ϵ 0 [ ϵ j ϵ 0 j ϵ ϵ 0 0 0 ϵ z ] ,
μ ¯ = μ 0 [ μ j μ 0 j μ μ 0 0 0 μ z ] ,
ϵ μ ,     E H ,     H E .
B = μ H ,
D = ϵ E t + j ϵ z ̂ × E t + ϵ z E z z ̂ ,
t × E + j β z ̂ × E = j ω μ H ,
t × H + j β z ̂ × H = j ω D .
t × E t = j k 0 η 0 μ H z z ̂ ,
t × H t = j k 0 ϵ z η 0 E z z ̂ ,
j β z ̂ × E t j k 0 η 0 μ H t = z ̂ × t E z ,
j β z ̂ × H t + j k 0 ϵ η 0 E t k 0 ϵ η 0 z ̂ × E t = z ̂ × t H z ,
κ 2 = k 0 2 ϵ μ β 2 ,     k 2 = k 0 2 μ ϵ ,
κ 2 E t + j k 2 z ̂ × E t = j β t E z j k 0 η 0 μ z ̂ × t H z ,
k 2 E t + j κ 2 z ̂ × E t = k 0 η 0 μ t H z β z ̂ × t E z ,
κ 2 H t + j k 2 z ̂ × H t = k 0 ϵ η 0 t E z + j β t H z + j k 0 ϵ η 0 z ̂ × t E z ,
k 2 H t + j κ 2 z ̂ × H t = k 0 ϵ η 0 t E z β z ̂ × t H z + j k 0 ϵ η 0 z ̂ × t E z .
( κ 4 k 4 ) E t = t ( j β κ 2 E z + k 2 k 0 η 0 μ H z ) j z ̂ t ( j β k 2 E z + k 0 κ 2 η 0 μ H z ) ,
( κ 4 k 4 ) H t = t ( j β κ 2 H z β 2 k 0 ϵ η 0 E z ) + j z ̂ t [ j β k 2 H z + k 0 η 0 ( κ 2 ϵ k 2 ϵ ) E z ] .
t 2 H z + a H z + b E z = 0 ,
t 2 E z + c E z + d H z = 0 ,
a = κ 2 k 2 ϵ ϵ ,     b = j β k 0 η 0 ϵ z ϵ ϵ ,
c = κ 2 ϵ z ϵ ,     d = j β k 0 η 0 ϵ μ ϵ .
H z = s 1 u 1 + s 2 u 2 ,     E z = q 1 u 1 + q 2 u 2 ,
| s 1 s 2 q 1 q 2 | 0.
t 2 u i + k 0 2 s i u i = 0 ,     i = 1 , 2.
| a k 0 2 s b d c k 0 2 s | = 0 ,
q i = s i ( k 0 2 s i a ) / b .
κ 4 k 4 = ϵ ϵ z k 0 4 s 1 s 2 ,
j β κ 2 E z + k 2 k 0 η 0 μ H z = j β k 0 2 s 1 s 2 i = 1 2 k 0 2 s i κ 2 b u i ,
j β k 2 E z + k 0 κ 2 η 0 μ H z = k 0 3 η 0 ϵ μ ϵ z s 1 s 2 ( u 1 + u 2 ) ,
j β κ 2 H z β 2 k 0 ϵ η 0 E z = j β k 0 2 ϵ ϵ z s 1 s 2 ( u 1 + u 2 ) ,
j β k 2 H z + k 0 η 0 ( κ 2 ϵ k 2 ϵ ) E z = k 0 3 ϵ η 0 s 1 s 2 i = 1 2 k 0 2 s i a b u i .
E z = i = 1 2 k 0 2 s i a b s i u i ,
H z = s 1 u 1 + s 2 u 2 ,
E t = η 0 k 0 ϵ t [ i = 1 2 ( s i κ 2 k 0 2 ) u i ] j η 0 μ k 0 z ̂ × t ( u 1 + u 2 ) ,
H t = j β k 0 2 t ( u 1 + u 2 ) ϵ β ϵ z ̂ × t [ i = 1 2 ( s i a k 0 2 ) u i ] .
D = ϵ E ,
B = μ H t + j μ z ̂ × H t + μ z H z z ̂ .
κ 2 = k 0 2 ϵ μ β 2 ,     k 2 = k 0 2 ϵ μ ,
a = κ 2 k 2 μ μ ,     b = j β k 0 η 0 μ μ z μ ,
c = κ 2 μ z μ ,     d = j β k 0 η 0 ϵ μ μ ,
t 2 E z + a E z + b H z = 0 ,
t 2 H z + c H z + d E z = 0 ,
E z = s 1 u 1 + s 2 u 2 ,     H z = q 1 u 1 + q 2 u 2 ,
E z = s 1 u 1 + s 2 u 2 ,
H z = i = 1 2 k 0 2 s i a b s i u i ,
E t = j β k 0 2 t ( u 1 + u 2 ) μ β μ z ̂ × t [ i = 1 2 ( s i a k 0 2 ) u i ] ,
H t = 1 k 0 η 0 μ t [ i = 1 2 ( s i κ 2 k 0 2 ) u i ] + j ϵ k 0 η 0 z ̂ × t ( u 1 + u 2 ) .
t 2 H z + a H z = 0     for   TE   modes ,
t 2 E z + c E z = 0     for   TM   modes ,
k 0 2 s 1 c = κ 2 ϵ z ϵ ,     k 0 2 s 2 a = κ 2 k 0 2 ϵ μ ϵ 2 ϵ 2 ,
s 1 cutoff = ϵ z μ ,     s 2 cutoff = ϵ μ ( 1 ϵ 2 ϵ 2 ) ,
t 2 E z + a E z = 0     for   TM   modes ,
t 2 H z + c H z = 0     for   TE   modes ,
k 0 2 s 1 c = κ 2 μ z μ ,     k 0 2 s 2 a = κ 2 k 0 2 ϵ μ μ 2 μ 2 ,
s 1 cutoff = ϵ μ z ,     s 2 cutoff = ϵ μ ( 1 μ 2 μ 2 ) ,
u 1 = A 1   cos ( k 0 y s 1 ) + A 2   sin ( k 0 y s 1 ) ,
u 2 = B 1   cos ( k 0 y s 2 ) + B 2   sin ( k 0 y s 2 ) ,
α i v = k 0 v s i ,
E x = j η 0 μ [ s 1 ( A 1   sin   α 1 y + A 2   cos   α 1 y ) + s 2 ( B 1   sin   α 2 y + B 2   cos   α 2 y ) ] ,
H x = k 0 ϵ β ϵ [ p 1 s 1 ( A 1   sin   α 1 y + A 2   cos   α 1 y ) + p 2 s 2 ( B 1   sin   α 2 y + B 2   cos   α 2 y ) ] ,
p i = s i a / k 0 2 ,     i = 1 , 2.
2 p 1 p 2 s 1 s 2 ( 1 cos   α 1 h   cos   α 2 h )
= ( s 1 p 1 2 + s 2 p 2 2 ) sin   α 1 h   sin   α 2 h ,
2 p 1 p 2 s 1 s 2 ( 1 cos   α 1 h   cos   α 2 h )
= ( s 1 p 2 2 + s 2 p 1 2 ) sin   α 1 h   sin   α 2 h ,
[ ( R + S ) sin α 1 h α 2 h 2 + ( R S ) sin α 1 h + α 2 h 2 ] [ ( R + S ) sin α 1 h α 2 h 2 ( R S ) sin α 1 h + α 2 h 2 ] = 0 ,
sin   α 1 h   sin   α 2 h = 0 ,
E z = j η 0 [ D 1   cos ( k 1 y ) + D 2   sin ( k 1 y ) ] ,
H z = C 1   cos ( k 1 y ) + C 2   sin ( k 1 y ) ,
E x = j η 0 k 0 μ 1 k 1 [ C 1   sin ( k 1 y ) + C 2   cos ( k 1 y ) ] ,
H x = k 0 ϵ 1 k 1 [ D 1   sin ( k 1 y ) + D 2   cos ( k 1 y ) ] ,
k 1 2 = k 0 2 ϵ 1 μ 1 β 2 .
P 2 { s 1 s 2 [ 2 p 1 p 2 ( p 1 2 + p 2 2 ) cos   α 1 h   cos   α 2 h ] p 1 p 2 ( s 1 + s 2 ) sin   α 1 h   sin   α 2 h } + Q 2 ϵ 1 μ ϵ z μ 1 [ 2 p 1 p 2 s 1 s 2 ( 1 cos   α 1 h   cos   α 2 h ) ( s 1 p 1 2 + s 2 p 2 2 ) sin   α 1 h   sin   α 2 h ] + P Q ( s 1 s 2 ) [ s 2 ( k 1 μ k 0 μ 1 p 2 k 0 ϵ 1 k 1 ϵ z p 1 s 1 ) sin   α 1 h   cos   α 2 h s 1 ( k 1 μ k 0 μ 1 p 1 k 0 ϵ 1 k 1 ϵ z p 2 s 2 ) sin   α 2 h   cos   α 1 h ] = 0 ,
P 2 ϵ 1 μ ϵ z μ 1 { s 1 s 2 [ 2 p 1 p 2 ( p 1 2 + p 2 2 ) cos   α 1 h   cos   α 2 h ] p 1 p 2 ( s 1 + s 2 ) sin   α 1 h   sin   α 2 h } + Q 2 [ 2 p 1 p 2 s 1 s 2 ( 1 cos   α 1 h   cos   α 2 h ) ( s 1 p 2 2 + s 2 p 1 2 ) sin   α 1 h   sin   α 2 h ] P Q ( s 1 s 2 ) [ s 2 ( k 1 μ k 0 μ 1 p 1 k 0 ϵ 1 k 1 ϵ z p 2 s 1 ) sin   α 1 h   cos   α 2 h s 1 ( k 1 μ k 0 μ 1 p 2 k 0 ϵ 1 k 1 ϵ z p 1 s 2 ) sin   α 2 h   cos   α 1 h ] = 0 ,
P = sin [ k 1 ( h 0 h ) ] ,     Q = cos [ k 1 ( h 0 h ) ] .
( Q s 1 k 0 ϵ 1 k 1 ϵ z sin   α 1 h + P   cos   α 1 h ) ( Q k 1 μ k 0 μ 1 sin   α 2 h + P s 2   cos   α 2 h ) = 0 ,
( Q   sin   α 1 h + P s 1 k 0 ϵ 1 k 1 ϵ z cos   α 1 h ) ( Q s 2   sin   α 2 h + P k 1 μ k 0 μ 1 cos   α 2 h ) = 0 ,
E z = j η 0 C 1   exp [ γ 1 ( y h ) ] ,
H z = D 1   exp [ γ 1 ( y h ) ] ,
E x = j η 0 k 0 μ 1 γ 1 D 1   exp [ γ 1 ( y h ) ] ,
H x = k 0 ϵ 1 γ 1 C 1   exp [ γ 1 ( y h ) ] ,
γ 1 2 = β 2 k 0 2 ϵ 1 μ 1 .
2 p 1 p 2 s 1 s 2 ( ϵ z μ 1 ϵ 1 μ ) + s 1 s 2 [ 2 ϵ 1 μ p 1 p 2 ϵ z μ 1 ( p 1 2 + p 2 2 ) ] cos   α 1 h   cos   α 2 h + [ ϵ 1 μ ( s 1 p 1 2 + s 2 p 2 2 ) ϵ z μ 1 p 1 p 2 ( s 1 + s 2 ) ] sin   α 1 h   sin   α 2 h + ( s 1 s 2 ) s 2 ( γ 1 k 0 ϵ z μ p 2 + k 0 γ 1 ϵ 1 μ 1 p 1 s 1 ) sin   α 1 h   cos   α 2 h ( s 1 s 2 ) s 1 ( γ 1 k 0 ϵ z μ p 1 + k 0 γ 1 ϵ 1 μ 1 p 2 s 2 ) sin   α 2 h   cos   α 1 h = 0 ,
2 p 1 p 2 s 1 s 2 ( ϵ z μ 1 ϵ 1 μ ) + s 1 s 2 [ 2 ϵ z μ 1 p 1 p 2 + ϵ 1 μ ( p 1 2 + p 2 2 ) ] cos   α 1 h   cos   α 2 h + [ ϵ 1 μ p 1 p 2 ( s 1 + s 2 ) ϵ z μ 1 ( s 1 p 2 2 + s 2 p 1 2 ) ] sin   α 1 h   sin   α 2 h + ( s 1 s 2 ) s 2 ( γ 1 k 0 ϵ z μ p 1 + k 0 γ 1 ϵ 1 μ 1 p 2 s 1 ) sin   α 1 h   cos   α 2 h ( s 1 s 2 ) s 1 ( γ 1 k 0 ϵ z μ p 2 + k 0 γ 1 ϵ 1 μ 1 p 1 s 2 ) sin   α 2 h   cos   α 1 h = 0 ,
( s 1 k 0 ϵ 1 γ 1 ϵ z sin   α 1 h cos   α 1 h ) ( γ 1 μ k 0 μ 1 sin   α 2 h + s 2   cos   α 2 h ) = 0 ,
( sin   α 1 h + s 1 k 0 ϵ 1 γ 1 ϵ z cos   α 1 h ) ( s 2   sin   α 2 h γ 1 μ k 0 μ 1 cos   α 2 h ) = 0 ,
E z = j η 0 C 2   exp ( γ 2 y ) ,     H z = D 2   exp ( γ 2 y ) ,
E x = j η 0 k 0 μ 2 γ 2 D 2   exp ( γ 2 y ) ,
H x = k 0 ϵ 2 γ 2 C 2   exp ( γ 2 y ) ,
2 p 1 p 2 s 1 s 2 ( ϵ z μ 1 ϵ 1 μ ) ( ϵ 2 μ ϵ z μ 2 ) + [ ξ 1 ζ 2 + ξ 2 ζ 1 ϵ z μ s 1 s 2 ( s 1 s 2 ) 2 ( γ 2 γ 1 ϵ 1 μ 1 + γ 1 γ 2 ϵ 2 μ 2 ) ] cos   α 1 h   cos   α 2 h + [ ξ 1 ζ 1 + ξ 2 ζ 2 ( s 1 s 2 ) 2 ( γ 1 γ 2 k 0 2 ϵ z 2 μ 2 + k 0 2 γ 1 γ 2 ϵ 1 μ 1 ϵ 2 μ 2 s 1 s 2 ) ] sin   α 1 h   sin   α 2 h + ( s 1 s 2 ) [ ϵ z μ ( γ 1 k 0 ξ 2 + γ 2 k 0 ζ 2 ) + s 1 s 2 ( k 0 γ 1 ϵ 1 μ 1 ξ 1 + k 0 γ 2 ϵ 2 μ 2 ζ 1 ) ] sin   α 1 h   cos   α 2 h ( s 1 s 2 ) [ ϵ z μ ( γ 1 k 0 ξ 1 + γ 2 k 0 ζ 1 ) + s 1 s 2 ( k 0 γ 1 ϵ 1 μ 1 ξ 2 + k 0 γ 2 ϵ 2 μ 2 ζ 2 ) ] sin   α 2 h   cos   α 1 h = 0 ,
ξ 1 = s 1 ( ϵ 2 μ p 1 ϵ z μ 2 p 2 ) ,
ξ 2 = s 2 ( ϵ 2 μ p 2 ϵ z μ 2 p 1 ) ,
ζ 1 = s 1 ( ϵ 1 μ p 1 ϵ z μ 1 p 2 ) ,
ζ 2 = s 2 ( ϵ 1 μ p 2 ϵ z μ 1 p 1 ) .
[ ( ϵ 1 κ ϵ ϵ z γ 1 ϵ ϵ z γ 2 ϵ 2 κ ) sin   α 1 h ( 1 + ϵ 1 γ 2 ϵ 2 γ 1 ) cos   α 1 h ] [ ( μ γ 1 μ 2 κ μ 1 κ μ γ 2 ) sin   α 2 h + ( μ 1 μ 2 + γ 1 γ 2 ) cos   α 2 h ] = 0.
tan   α 1 h = κ ( ϵ 1 γ 2 + ϵ 2 γ 1 ) ϵ ϵ z ϵ 1 ϵ 2 κ 2 ϵ ϵ z γ 1 γ 2 ,
tan   α 2 h = κ μ ( μ 1 γ 2 + μ 2 γ 1 ) μ 1 μ 2 κ 2 μ 2 γ 1 γ 2 .
u 1 = A 1 J n ( k 0 r s 1 ) e j n θ ,
u 2 = A 2 J n ( k 0 r s 2 ) e j n θ ,
E z = j η 0 k 0 ϵ β ϵ z ϵ e j n θ i = 1 2 A i p i s i J n ( α i r ) ,
H z = e j n θ i = 1 2 A i s i J n ( α i r ) ,
E θ = j η 0 e j n θ i = 1 2 A i [ n k 0 r ϵ ( s i κ 2 k 0 2 ) J n ( α i r ) + μ s i J n ( α i r ) ] ,
H θ = e j n θ i = 1 2 A i [ n β k 0 2 r J n ( α i r ) + k 0 ϵ β ϵ p i s i J n ( α i r ) ] ,
f 1 g 2 f 2 g 1 = 0 ,
f i = p i s i ,
g i = n k 0 ρ ϵ ( s i κ 2 k 0 2 ) + μ s i J n ( α i ρ ) ,
f i = s i ,
g i = n β k 0 2 ρ + k 0 ϵ β ϵ p i s i J n ( α i ρ ) ,
J n ( α i ρ ) = J n ( α i ρ ) / J n ( α i ρ ) .
( k 0 2 ϵ z μ ϵ z ϵ β 2 k 0 2 s 1 s 1 ) J n ( α 1 ρ ) ( k 0 2 ϵ z μ ϵ z ϵ β 2 k 0 2 s 2 s 2 ) J n ( α 2 ρ ) + ϵ n ϵ k 0 ρ ϵ z μ ( 1 s 2 1 s 1 ) = 0 ,
( k 0 2 ϵ eff μ β 2 k 0 2 s 1 s 1 ) J n ( α 1 ρ ) ( k 0 2 ϵ eff μ β 2 k 0 2 s 2 s 2 ) J n ( α 2 ρ ) ϵ ϵ n ρ β 2 k 0 3 ( 1 s 1 1 s 2 ) = 0 ,    
ϵ eff = ( ϵ 2 ϵ 2 ) / ϵ ,
J n ( α 1 ρ ) [ J n ( α 2 ρ ) n ϵ ϵ α 2 ρ J n ( α 2 ρ ) ] = 0 ,
J n ( α 1 ρ ) J n ( α 2 ρ ) = 0 ,
E z = j η 0 A 3 F ( k 1 r ) e j n θ ,
H z = A 4 G ( k 1 r ) e j n θ ,
E θ = j η 0 [ A 3 n β k 1 2 r F ( k 1 r ) + A 4 k 0 k 1 μ 1 G ( k 1 r ) ] e j n θ ,
H θ = [ A 4 n β k 1 2 r G ( k 1 r ) + A 3 k 0 k 1 ϵ 1 F ( k 1 r ) ] e j n θ ,
F ( k 1 r ) = J n ( k 1 r ) J n ( k 1 ρ 0 ) Y n ( k 1 r ) Y n ( k 1 ρ 0 ) ,
G ( k 1 r ) = J n ( k 1 r ) J n ( k 1 ρ 0 ) Y n ( k 1 r ) Y n ( k 1 ρ 0 ) ,
f i = n β 2 ϵ k 0 3 ρ ϵ ( 1 k 0 2 k 1 2 s i ) + p i s i [ J n ( α i ρ ) k 0 ϵ 1 k 1 ϵ z s i F ( k 1 ρ ) ] ,
g i = n k 0 ρ ϵ μ ( s i κ 2 k 0 2 k 0 2 ϵ k 1 2 ϵ z p i s i ) + s i [ J n ( α i ρ ) k 0 μ 1 k 1 μ s i G ( k 1 ρ ) ] ,
F ( k 1 ρ ) = F ( k 1 ρ ) / F ( k 1 ρ ) ,
G ( k 1 ρ ) = G ( k 1 ρ ) / G ( k 1 ρ ) .
[ s 1 J n ( α 1 ρ ) κ 2 ϵ 1 k 0 k 1 ϵ F ( k 1 ρ ) ] [ s 2 J n ( α 2 ρ ) κ 2 μ 1 k 0 k 1 μ G ] = n 2 β 2 k 0 4 ρ 2 ϵ μ ( 1 κ 2 k 1 2 ) 2 ,
[ ϵ 1 μ ϵ z μ 1 F ( k 1 ρ ) J n ( α 1 ρ ) ] [ μ 1 ϵ 1 ϵ eff G ( k 1 ρ ) ϵ eff μ J n ( α 2 ρ ) + n ϵ k 0 ρ ϵ ] = 0 ,
E z = j η 0 A 3 K n ( γ 1 r ) e j n θ ,
H z = A 4 K n ( γ 1 r ) e j n θ ,
E θ = j η 0 [ A 3 n β γ 1 2 r K n ( γ 1 r ) + A 4 k 0 μ 1 γ 1 K n ( γ 1 r ) ] e j n θ ,
H θ = [ A 4 n β γ 1 2 r K n ( γ 1 r ) + A 3 k 0 ϵ 1 γ 1 K n ( γ 1 r ) ] e j n θ ,
f i = n β 2 ϵ k 0 3 ρ ϵ ( 1 + k 0 2 γ 1 2 s i ) + p i s i [ J n ( α i ρ ) + k 0 ϵ 1 γ 1 ϵ z s i K n ( γ 1 ρ ) ] ,
g i = n p i k 0 ρ ϵ μ ( 1 + k 0 2 ϵ γ 1 2 ϵ z s i ) + s i [ J n ( α i ρ ) + k 0 μ 1 γ 1 μ s i K n ( γ 1 ρ ) ] .
[ γ 1 ϵ ϵ z J n ( α 1 ρ ) + κ ϵ 1 K n ( γ 1 ρ ) ] [ γ 1 μ J n ( α 2 ρ ) + κ μ 1 K n ( γ 1 ρ ) ] = n 2 β 2 ( κ 2 + γ 1 2 ) 2 k 0 2 κ 2 γ 1 2 ρ 2 .
k 0 h ϵ z ϵ ( ϵ μ β 2 k 0 2 ) = n π     for   TM   modes ,
k 0 h ϵ μ β 2 k 0 2 = n π     for   TE   modes ,

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