Abstract

The radiation characteristics of an abruptly terminated five-layer symmetric monomode slab waveguide are treated analytically by means of an integral equation, which has been appropriately chosen for the corresponding boundary-value problem of the even TE modes. To obtain the solutions of this equation, representing the electric-field distribution on the terminal plane, a Neumann-series iterative procedure is applied. Then the reflection coefficient of the surface wave, the power of the reflected radiation modes, and the far-field radiation pattern are computed. Numerical results are presented for several cases of abruptly terminated waveguides.

© 1997 Optical Society of America

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  1. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell. Syst. Tech. J. 49, 273–290 (1970).
    [CrossRef]
  2. G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
    [CrossRef]
  3. T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
    [CrossRef]
  4. K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
    [CrossRef]
  5. A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
    [CrossRef]
  6. H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
    [CrossRef]
  7. K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
    [CrossRef]
  8. M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
    [CrossRef]
  9. M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
    [CrossRef]
  10. R. Baets, P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. 21, 1972–1978 (1982).
    [CrossRef] [PubMed]
  11. K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. LT-6, 590–600 (1988).
    [CrossRef]
  12. C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. LT-3, 408–415 (1985).
    [CrossRef]
  13. H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. 30, 2827–2835 (1994).
    [CrossRef]
  14. Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
    [CrossRef]
  15. S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low loss applications,” J. Lightwave Technol. 11, 567–575 (1993).
    [CrossRef]
  16. D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electro-optic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
    [CrossRef]
  17. T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1811–1824 (1995).
    [CrossRef]
  18. D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, 1964), Chap. 8.
  19. N. K. Uzunoglu, C. N. Capsalis, I. G. Tigelis, “Scattering from an abruptly terminated single-mode fiber termination,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
    [CrossRef]

1995 (1)

T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1811–1824 (1995).
[CrossRef]

1994 (1)

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. 30, 2827–2835 (1994).
[CrossRef]

1993 (1)

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low loss applications,” J. Lightwave Technol. 11, 567–575 (1993).
[CrossRef]

1992 (2)

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electro-optic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
[CrossRef]

1989 (1)

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

1988 (1)

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. LT-6, 590–600 (1988).
[CrossRef]

1987 (1)

1985 (1)

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. LT-3, 408–415 (1985).
[CrossRef]

1984 (1)

K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
[CrossRef]

1983 (1)

M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
[CrossRef]

1982 (1)

1979 (2)

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
[CrossRef]

1978 (2)

H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[CrossRef]

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

1976 (1)

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

1970 (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell. Syst. Tech. J. 49, 273–290 (1970).
[CrossRef]

Aoki, K.

K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
[CrossRef]

Baets, R.

Benech, P.

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electro-optic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

Brooke, G. H.

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

Capsalis, C. N.

N. K. Uzunoglu, C. N. Capsalis, I. G. Tigelis, “Scattering from an abruptly terminated single-mode fiber termination,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
[CrossRef]

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. LT-3, 408–415 (1985).
[CrossRef]

Chaudhuri, S. K.

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low loss applications,” J. Lightwave Technol. 11, 567–575 (1993).
[CrossRef]

Chow, Y. L.

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low loss applications,” J. Lightwave Technol. 11, 567–575 (1993).
[CrossRef]

Fikioris, J. G.

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. LT-3, 408–415 (1985).
[CrossRef]

Goss, A.

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low loss applications,” J. Lightwave Technol. 11, 567–575 (1993).
[CrossRef]

Hamid, M.

A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
[CrossRef]

Hardy, A.

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

Haus, H. A.

M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
[CrossRef]

Hirai, H.

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. LT-6, 590–600 (1988).
[CrossRef]

Imada, Y.

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. LT-6, 590–600 (1988).
[CrossRef]

Inagaki, S.

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Ittipiboon, A.

A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
[CrossRef]

Jones, D. S.

D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, 1964), Chap. 8.

Khalil, D.

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electro-optic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

Kharadly, M. M. Z.

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

Kumagai, N.

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Kuznetsov, M.

M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
[CrossRef]

Lagasse, P. E.

Lin, H.-B.

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. 30, 2827–2835 (1994).
[CrossRef]

Marcuse, D.

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell. Syst. Tech. J. 49, 273–290 (1970).
[CrossRef]

Marom, E.

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

Morishita, K.

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Munowitz, M.

M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
[CrossRef]

Rozzi, T. E.

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

Safavi-Naeini, S.

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low loss applications,” J. Lightwave Technol. 11, 567–575 (1993).
[CrossRef]

Su, J.-Y.

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. 30, 2827–2835 (1994).
[CrossRef]

Tedjini, S.

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electro-optic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

Theodoropoulos, T. G.

T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1811–1824 (1995).
[CrossRef]

Tigelis, I. G.

T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1811–1824 (1995).
[CrossRef]

N. K. Uzunoglu, C. N. Capsalis, I. G. Tigelis, “Scattering from an abruptly terminated single-mode fiber termination,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
[CrossRef]

Tsutsumi, K.

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. LT-6, 590–600 (1988).
[CrossRef]

Uchida, K.

K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
[CrossRef]

Uzunoglu, N. K.

N. K. Uzunoglu, C. N. Capsalis, I. G. Tigelis, “Scattering from an abruptly terminated single-mode fiber termination,” J. Opt. Soc. Am. A 4, 2150–2157 (1987).
[CrossRef]

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. LT-3, 408–415 (1985).
[CrossRef]

Vezzetti, D. J.

M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
[CrossRef]

Wang, W.-S.

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. 30, 2827–2835 (1994).
[CrossRef]

Wei, P.-K.

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. 30, 2827–2835 (1994).
[CrossRef]

Weissman, Z.

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

Yajima, H.

H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[CrossRef]

Yuba, Y.

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. LT-6, 590–600 (1988).
[CrossRef]

Appl. Opt. (1)

Bell. Syst. Tech. J. (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell. Syst. Tech. J. 49, 273–290 (1970).
[CrossRef]

Electron. Lett. (1)

G. H. Brooke, M. M. Z. Kharadly, “Step discontinuities on dielectric waveguides,” Electron. Lett. 12, 473–475 (1976).
[CrossRef]

IEEE J. Quantum Electron. (4)

H. Yajima, “Coupled mode analysis of dielectric planar branching waveguides,” IEEE J. Quantum Electron. QE-14, 749–755 (1978).
[CrossRef]

M. Kuznetsov, H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1515 (1983).
[CrossRef]

H.-B. Lin, J.-Y. Su, P.-K. Wei, W.-S. Wang, “Design and application of very low-loss abrupt bends in optical waveguides,” IEEE J. Quantum Electron. 30, 2827–2835 (1994).
[CrossRef]

Z. Weissman, A. Hardy, E. Marom, “Mode-dependent radiation loss in Y junctions and directional couplers,” IEEE J. Quantum Electron. 25, 1200–1208 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

D. Khalil, S. Tedjini, P. Benech, “Asymmetric excitement of symmetric monomode Y-junction: the radiation mode effects of radiation modes in Mach–Zehnder electro-optic modulators,” IEEE Trans. Microwave Theory Tech. 40, 2235–2242 (1992).
[CrossRef]

K. Uchida, K. Aoki, “Scattering of surface waves on transverse discontinuities in symmetrical three-layer dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-32, 11–19 (1984).
[CrossRef]

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–746 (1978).
[CrossRef]

K. Morishita, S. Inagaki, N. Kumagai, “Analysis of discontinuities in dielectric waveguides by means of the least squares boundary residual method,” IEEE Trans. Microwave Theory Tech. MTT-27, 310–315 (1979).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

T. G. Theodoropoulos, I. G. Tigelis, “Radiation modes of a five-layer symmetric slab waveguide,” Int. J. Infrared Millim. Waves 16, 1811–1824 (1995).
[CrossRef]

J. Lightwave Technol. (4)

S. Safavi-Naeini, Y. L. Chow, S. K. Chaudhuri, A. Goss, “Wide angle phase-corrected Y-junction of dielectric waveguides for low loss applications,” J. Lightwave Technol. 11, 567–575 (1993).
[CrossRef]

K. Tsutsumi, Y. Imada, H. Hirai, Y. Yuba, “Analysis of single-mode optical Y-junctions by the bounded step and bend approximation,” J. Lightwave Technol. LT-6, 590–600 (1988).
[CrossRef]

C. N. Capsalis, J. G. Fikioris, N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” J. Lightwave Technol. LT-3, 408–415 (1985).
[CrossRef]

M. Munowitz, D. J. Vezzetti, “Numerical modelling of coherent coupling and radiation fields in planar Y-branch interferometers,” J. Lightwave Technol. 10, 1570–1573 (1992).
[CrossRef]

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

Proc. Inst. Electr. Eng. (1)

A. Ittipiboon, M. Hamid, “Scattering of surface waves at a slab waveguide discontinuity,” Proc. Inst. Electr. Eng. 126, 798–804 (1979).
[CrossRef]

Other (1)

D. S. Jones, Theory of Electromagnetism (Pergamon, Oxford, 1964), Chap. 8.

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Figures (6)

Fig. 1
Fig. 1

Geometry of an abruptly terminated five-layer symmetric slab waveguide.

Fig. 2
Fig. 2

Normalized propagation constant β versus normalized parameter V for the first three even-TE modes for relative differences Δ13=0%, 5%, and 50%.

Fig. 3
Fig. 3

Variation of the electric-field magnitude |E(x)| at the terminal plane z=0 with x, with n0=1, d=0.5 μm, and Δ13=0.

Fig. 4
Fig. 4

(a) Variation of the square magnitude of the reflection coefficient R0 of the guided mode with the core width D-d, with n0=1, d=0.5 μm, and Δ13=0. (b) Variation of the power of the reflected radiation modes, Prad, with the core width D-d, with n0=1, d=0.5 μm, and Δ13=0.

Fig. 5
Fig. 5

(a) Variation of the square magnitude of the reflection coefficient R0 of the guided mode with the core width D-d, with n0=1, d=0.5 μm, and Δ32=0.5%. (b) Same as (a), but for Δ32=5%.

Fig. 6
Fig. 6

(a) Normalized radiation pattern U(θ, ϕ=0°) with n0 =1, d=0.5 μm, D=1 μm, and Δ13=0%. (b) Same as (a), but for D=0.9 μm and Δ32=1%.

Tables (1)

Tables Icon

Table 1 Convergence of the Iterative Solution for the Electric-Field Distribution and the Reflection Coefficient of the Guided Mode at z=0 for an Abruptly Terminated Five-Layer Symmetric Slab Waveguide

Equations (46)

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

U0(x)
=β02ωμ0 A exp[-h3(x-D)],B cos(h2x)+Γ sin(h2x),Δ cosh(h1x),B cos(h2x)-Γ sin(h2x),A exp[+h3(x+D)],D<x<+d<x<D-d<x<d-D<x<-d-<x<-D,
 
h1 tanh(h1d)=-h2{tan[h2(D-d)+tan-1(h2/h3)]}-1.
Ψ(x, q)
=1π cos[q(x-D)+b],B(q)cos[σ(x-d)+a],C(q)cos(ρx),B(q)cos[σ(x+d)-a],cos[q(x+D)-b],D<x<+d<x<D-d<x<d-D<x<-d-<x<-D,
 
-+U02(x)dx=1,
-+Ψ(x, q)Ψ(x, q)dx=δ(q-q),
U0(x)U0(x)+0+Ψ(x, q)Ψ(x, q)dq=δ(x-x).
Φ1(x, z)=U0(x)[exp(-jβ0z)+R0 exp(+jβ0z)]+0+R(q)Ψ(x, q)exp[+jγ(q)z]dq.
ΦII(x, z)=0+T(λ)ϕ0(x, λ)exp[-jδ(λ)z]dλ,
-+ϕ0(x, λ)ϕ0(x, λ)dx=δ(λ-λ),
-+ϕ0(x, λ)ϕ0(x, λ)dλ=δ(x-x).
R0=-1+-+E(x)U0(x)dx,
R(q)=-+E(x)Ψ(x, q)dx,
T(λ)=-+E(x)ϕ(x, λ)dx.
2β0U0(x)=-+E(x)Ξ(x, x)dx,
Ξ(x, x)=β0U0(x)U0(x)+0+γ(q)Ψ(x, q)Ψ(x, q)dq+0+δ(λ)ϕ0(x, λ)ϕ0(x, λ)dλ.
E(x)=E0(x)+-+E(x)K(x, x)dx,
E0(x)=2β0k0(n0+n3) U0(x),
K(x, x)=-1k0(n0+n3) (β0-k0n3)U0(x)U0(x)+0+[γ(q)-k0n3]Ψ(x, q)Ψ(x, q)dq+0+[δ(λ)-k0n3]ϕ0(x, λ)ϕ0(x, λ)dλ.
EK(x)=E0(x)+n=1KCn(x),K=1, 2, ,
Cn(x)=-+dx1-+dx2-+dxn K(x, x1)×K(x1, x2)K(xn-1, xn)E(xn).
E1(x)=E0(x)-2β0k02(n0+n3)2 (β0-k0n3)U0(x)-2β0k02(n0+n3)2 0+dλ[δ(λ)-k0n0]×Uϕ0(λ)ϕ0(x, λ),
R0(0)=-1+2β0k0(n0+n3),
R(0)(q)=0.
R0(1)=R0(0)-2β0k02(n0+n3)2 (β0-k0n3)+0+ dλ[δ(λ)-k0n0][Uϕ0(λ)]2,
R(1)(q)=R(0)(q)-2β0k02(n0+n3)2×0+dλ[δ(λ)-k0n0]×Ψϕ0(q, λ)Uϕ0(λ),
Prad=0+|R(q)|2 dq.
limr ΦΠ(r, θ)=(2/k0n0r)1/2exp(-jk0n0r+jπ/4)×k0n0(cos θ)T(λ=k0n0 sin θ)
Eθ(r, θ, ϕ)=(sin θ cos ϕ)ΦΠ(r, θ),
Eϕ(r, θ, ϕ)=(cos ϕ)Φn(r, θ)
(r+andθ0°).
VC=k0(D-d)n22-n32=mπ+tan-1×n32-n12n22-n32 tanh(k0dn32-n12),
B=Acos(h2D)+sin(h2D) h3h2,
Γ=Asin(h2D)-cos(h2D) h3h2,
Δ=A 1cosh(h1d) cos[h2(D-d)]+h3h2 sin[h2(D-d)],
A=β0ωμ0 1cosh2(h1d) cos[h2(D-d)]+h3h2 sin[h2(D-d)]2sinh(2h1d)4h1+0.5d+12 (D-d)1+h32h22+h32h2+sin[2h2(D-d)]4h2-h32h2 cos[2h2(D-d)]-h324h23 sin[2h2(D-d)]+12h3-1/2,
tan a=(ρ/σ)tan(ρd),
tan b=(σ/q)tan[σ(D-d)+a],
B(q)=cos bcos[σ(D-d)+a],
C(q)=B(q)cos a/cos(ρd).
E2(x)=E1(x)+2β0k03(n0+n3)3 (β0-k0n3)2U0(x)+2β0(β0-k0n3)k03(n0+n3)3 0+dλ[δ(λ)-k0n0]×Uϕ0(λ)ϕ0(x, λ)+2β0(β0-k0n3)k03(n0+n3)3 U0(x)0+dλ[δ(λ)-k0n0][Uϕ0(λ)]2+2β0k03(n0+n3)3×0+dλ[δ(λ)-k0n0]Uϕ0(λ)×0+dq[γ(q)-k0n3]Ψ(x, q)Ψϕ0(q, λ)+2β0k03(n0+n3)3 0+dλ[δ(λ)-k0n0]2×Uϕ0(λ)ϕ0(x, λ),
R0(2)=R0(1)+2β0(β0-k0n3)2k03(n0+n3)3+2 2β0(β0-k0n3)k03(n0+n3)3 0+dλ[δ(λ)-k0n0][×Uϕ0(λ)]2+2β0k03(n0+n3)3×0+dλ[δ(λ)-k0n0]2[Uϕ0(λ)]2,
R(2)(q)=R(1)(q)+2β0(β0-k0n3)k03(n0+n3)3×0+dλ[δ(λ)-k0n0]Ψϕ0(q, λ)Uϕ0(λ)+2β0[γ(q)-k0n3]k03(n0+n3)3×0+dλ[δ(λ)-k0n0]Ψϕ0(q, λ)×Uϕ0(λ)+2β0k03(n0+n3)3×0+dλ[δ(λ)-k0n0]2Ψϕ0(q, λ)Uϕ0(λ).

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