Abstract

The boundary integral equations that are called guided-mode extracted integral equations are applied to the investigation of the power-coupling-properties between two arbitrarily ended dielectric slab waveguides. The integral equations derived in this paper can be solved by the conventional boundary-element method. The reflection and coupling coefficients of the guided wave, as well as the scattering power, are calculated numerically for the case of incident TE guided-mode waves. The results presented are checked by the energy conservation law and the reciprocity theorem. Numerical results are presented for several geometries of coupling, including systems with three-layered symmetrical and asymmetrical slab waveguides.

© 2003 Optical Society of America

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  1. G. A. Hockham, A. B. Sharpe, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
    [CrossRef]
  2. K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
    [CrossRef]
  3. P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
    [CrossRef]
  4. T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
    [CrossRef]
  5. 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]
  6. C. S. Rocha, “Scattering of surface waves at dielectric slab waveguide with axial ascending discontinuity,” IEEE Trans. Magn. 34, 2720–2723 (1998).
    [CrossRef]
  7. G. Kweon, I. Park, “Splicing losses between dissimilar optical waveguides,” J. Lightwave Technol. 17, 690–703 (1999).
    [CrossRef]
  8. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  9. C. N. Capsalis, N. K. Uzunoglu, I. G. Tigelis, “Coupling between two abruptly terminated single-mode optical fibers,” J. Opt. Soc. Am. B 5, 1624–1630 (1988).
    [CrossRef]
  10. C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
    [CrossRef]
  11. S. Chung, C. H. Chen, “A partial variational approach for arbitrary discontinuities in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 208–214 (1989).
    [CrossRef]
  12. K. Hirayama, M. Koshiba, “Analysis of discontinuities in an open dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 37, 761–767 (1989).
    [CrossRef]
  13. K. Hirayama, M. Koshiba, “Numerical analysis of arbitrarily shaped discontinuities between planar dielectric waveguides with different thicknesses,” IEEE Trans. Microwave Theory Tech. 38, 260–264 (1990).
    [CrossRef]
  14. T. P. Shen, R. F. Wallis, A. A. Maradudin, “Fresnel-like behavior of guided waves,” J. Opt. Soc. Am. A 4, 2120–2132 (1987).
    [CrossRef]
  15. K. Tanaka, M. Kojima, “New boundary integral equations for computer-aided design of dielectric waveguide circuits,” J. Opt. Soc. Am. A 6, 667–674 (1989).
    [CrossRef]
  16. K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
    [CrossRef]
  17. K. Tanaka, M. Tanaka, “Computer-aided design of dielectric optical waveguide bends by the boundary-element method based on guided-mode, extracted integral equations,” J. Opt. Soc. Am. A 13, 1362–1368 (1996).
    [CrossRef]
  18. D. N. Chien, M. Tanaka, K. Tanaka, “Numerical simulation of an arbitrarily ended, asymmetrical slab waveguide by guided-mode, extracted integral equations,” J. Opt. Soc. Am. A 19, 1649–1657 (2002).
    [CrossRef]
  19. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 1.

2002

1999

1998

C. S. Rocha, “Scattering of surface waves at dielectric slab waveguide with axial ascending discontinuity,” IEEE Trans. Magn. 34, 2720–2723 (1998).
[CrossRef]

1996

1993

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[CrossRef]

1990

K. Hirayama, M. Koshiba, “Numerical analysis of arbitrarily shaped discontinuities between planar dielectric waveguides with different thicknesses,” IEEE Trans. Microwave Theory Tech. 38, 260–264 (1990).
[CrossRef]

1989

K. Tanaka, M. Kojima, “New boundary integral equations for computer-aided design of dielectric waveguide circuits,” J. Opt. Soc. Am. A 6, 667–674 (1989).
[CrossRef]

S. Chung, C. H. Chen, “A partial variational approach for arbitrary discontinuities in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 208–214 (1989).
[CrossRef]

K. Hirayama, M. Koshiba, “Analysis of discontinuities in an open dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 37, 761–767 (1989).
[CrossRef]

1988

1987

C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
[CrossRef]

T. P. Shen, R. F. Wallis, A. A. Maradudin, “Fresnel-like behavior of guided waves,” J. Opt. Soc. Am. A 4, 2120–2132 (1987).
[CrossRef]

1986

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

1984

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

T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
[CrossRef]

1979

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

1977

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

1972

G. A. Hockham, A. B. Sharpe, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
[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]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 1.

Capsalis, C. N.

C. N. Capsalis, N. K. Uzunoglu, I. G. Tigelis, “Coupling between two abruptly terminated single-mode optical fibers,” J. Opt. Soc. Am. B 5, 1624–1630 (1988).
[CrossRef]

C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
[CrossRef]

Chen, C. H.

S. Chung, C. H. Chen, “A partial variational approach for arbitrary discontinuities in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 208–214 (1989).
[CrossRef]

Chien, D. N.

Chung, S.

S. Chung, C. H. Chen, “A partial variational approach for arbitrary discontinuities in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 208–214 (1989).
[CrossRef]

Citerne, J.

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

Fukumitsu, O.

T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
[CrossRef]

Gelin, P.

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

Hirayama, K.

K. Hirayama, M. Koshiba, “Numerical analysis of arbitrarily shaped discontinuities between planar dielectric waveguides with different thicknesses,” IEEE Trans. Microwave Theory Tech. 38, 260–264 (1990).
[CrossRef]

K. Hirayama, M. Koshiba, “Analysis of discontinuities in an open dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 37, 761–767 (1989).
[CrossRef]

Hockham, G. A.

G. A. Hockham, A. B. Sharpe, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
[CrossRef]

Kawano, K.

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

Kojima, M.

Koshiba, M.

K. Hirayama, M. Koshiba, “Numerical analysis of arbitrarily shaped discontinuities between planar dielectric waveguides with different thicknesses,” IEEE Trans. Microwave Theory Tech. 38, 260–264 (1990).
[CrossRef]

K. Hirayama, M. Koshiba, “Analysis of discontinuities in an open dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 37, 761–767 (1989).
[CrossRef]

Kweon, G.

Maradudin, A. A.

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Mitomi, O.

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

Miyazawa, H.

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

Ootera, H.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[CrossRef]

Park, I.

Petenzi, M.

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

Rocha, C. S.

C. S. Rocha, “Scattering of surface waves at dielectric slab waveguide with axial ascending discontinuity,” IEEE Trans. Magn. 34, 2720–2723 (1998).
[CrossRef]

Sharpe, A. B.

G. A. Hockham, A. B. Sharpe, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
[CrossRef]

Shen, T. P.

Takenaka, T.

T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
[CrossRef]

Tanaka, K.

Tanaka, M.

Tashima, H.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[CrossRef]

Tigelis, I. G.

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.

C. N. Capsalis, N. K. Uzunoglu, I. G. Tigelis, “Coupling between two abruptly terminated single-mode optical fibers,” J. Opt. Soc. Am. B 5, 1624–1630 (1988).
[CrossRef]

C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
[CrossRef]

Wallis, R. F.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 1.

Yoshino, Y.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Electron. Lett.

G. A. Hockham, A. B. Sharpe, “Dielectric-waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
[CrossRef]

P. Gelin, M. Petenzi, J. Citerne, “New rigorous analysis of the step discontinuity in a slab dielectric waveguide,” Electron. Lett. 15, 355–356 (1979).
[CrossRef]

T. Takenaka, O. Fukumitsu, “Accurate analysis of the abrupt discontinuity in a dielectric waveguide,” Electron. Lett. 19, 806–807 (1983).
[CrossRef]

IEEE Trans. Magn.

C. S. Rocha, “Scattering of surface waves at dielectric slab waveguide with axial ascending discontinuity,” IEEE Trans. Magn. 34, 2720–2723 (1998).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

C. N. Capsalis, N. K. Uzunoglu, “Coupling between an abruptly terminated optical fiber and a dielectric planar waveguide,” IEEE Trans. Microwave Theory Tech. MTT-35, 1043–1051 (1987).
[CrossRef]

S. Chung, C. H. Chen, “A partial variational approach for arbitrary discontinuities in planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 37, 208–214 (1989).
[CrossRef]

K. Hirayama, M. Koshiba, “Analysis of discontinuities in an open dielectric slab waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Tech. 37, 761–767 (1989).
[CrossRef]

K. Hirayama, M. Koshiba, “Numerical analysis of arbitrarily shaped discontinuities between planar dielectric waveguides with different thicknesses,” IEEE Trans. Microwave Theory Tech. 38, 260–264 (1990).
[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]

J. Lightwave Technol.

K. Kawano, H. Miyazawa, O. Mitomi, “New calculations for coupling laser diode to multimode fiber,” J. Lightwave Technol. LT-4, 368–374 (1986).
[CrossRef]

G. Kweon, I. Park, “Splicing losses between dissimilar optical waveguides,” J. Lightwave Technol. 17, 690–703 (1999).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Radio Sci.

K. Tanaka, M. Tanaka, H. Tashima, H. Ootera, Y. Yoshino, “New integral equation method for CAD of open waveguide bends,” Radio Sci. 28, 1219–1227 (1993).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 1.

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Figures (10)

Fig. 1
Fig. 1

Geometry of the problem under consideration. (a) Matching-plate coupling, (b) tilted-ends coupling, (c) locations of boundaries.

Fig. 2
Fig. 2

Variation of ΓT with tilt angle ϕ for the case of incidence from waveguide A (solid curve) and from waveguide B (dotted curve); k0d1=k0d2=1, n1=n4=1.5, Δij=10% (i=1, j=2, 3 or i=4, j=5, 6), k0l=0.01, n7=1.0 (air gap).

Fig. 3
Fig. 3

Variation of ΓT and ΓS with the ratio d1/d2 for the geometry with k0d1=1.1, n1=1.6, n4=n1(1-0.1)=1.44, n2=n3=n5=n6=1.0. Solid curve, present analysis; dotted curve, Chung and Chen.11

Fig. 4
Fig. 4

Dependences of ΓR, ΓT, ΓS, and ΓTOTAL on location of virtual boundaries; k0d1=k0d2=1, n1=3.6, n4=1.5, Δij=10% (i=1,j=2, 3 or i=4,j=5, 6), ϕ=0°.

Fig. 5
Fig. 5

Variation with the ratio d2/d1 of ΓR and ΓT for the geometry with k0d1=1, n1=3.6, n4=1.5. (a) Symmetrical structure with Δij=10% (i=1,j=2, 3 or i=4,j=5, 6); (b) asymmetrical structure with Δ13=60%, Δ46=30%, Δ12=Δ45=10%, ϕ=0°.

Fig. 6
Fig. 6

Variation of ΓR and ΓT with the refractive index n7 of the matching plate for the geometry with k0d1=1, k0l=π/2, n1=3.6, n4=1.5. (a) Symmetrical structure with k0d2=1.6, Δij=10% (i=1, j=2, 3 or i=4, j=5, 6); (b) asymmetrical structure with k0d2=1.8, Δ13=60%, Δ46=30%, Δ12=Δ45=10%.

Fig. 7
Fig. 7

Variation of ΓR and ΓT with matching plate thickness k0l for the geometry with k0d1=1, n1=3.6, n4=1.5. (a) Symmetrical structure with k0d2=1.6, n7=(n1n4)1/2, Δij=10% (i=1, j=2, 3 or i=4, j=5, 6); (b) asymmetrical structure with k0d2=1.8, n7=1.4, Δ13=60%, Δ46=30%, Δ12=Δ45=10%.

Fig. 8
Fig. 8

Variation of ΓR and ΓT with the tilt angle ϕ for the geometry with k0d1=1, n1=3.6, n4=1.5. (a) Symmetrical structure with k0d2=1.6, Δij=10% (i=1,j=2, 3 or i=4, j=5, 6); (b) asymmetrical structure with k0d2=1.8, Δ13=60%, Δ46=30%, Δ12=Δ45=10%.

Fig. 9
Fig. 9

Total electric field distribution |E|2 of the power coupling (a) without matching plate and (b) with matching plate, with k0l=π/2, n7=(n1n4)1/2; the parameters of the waveguides are k0d1=1.0, k0d2=1.6, n1=3.6, n4=1.5, Δij=10% (i=1, j=2, 3 or i=4, j=5, 6).

Fig. 10
Fig. 10

Total electric field distribution |E|2 of (a) vertical-end coupling and (b) tilted-end coupling (φ=+5°); the parameters of the waveguides are k0d1=1.0, k0d2=1.8, n1=3.6, n4=1.5, Δ13=60%, Δ46=30%, Δ12=Δ45=10%.

Tables (1)

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Table 1 Power Coefficients for Incidence from Slab Waveguide A a

Equations (27)

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E(x)=E(z, x)=E(r, θ),
12 E(x)=CG1(x|x) E(x)n-E(x) G1(x|x)ndl,
G1(x|x)=-j4 H0(2)(n1k0|x-x|),
E(x)=E-(A)(x)+RE+(A)(x)+EC(x) ,
x onC12+C13,
E(x)=EC(x),x onC12+C17+C13.
12 EC(x)=CG1(x|x) EC(x)n-EC(x) G1(x|x)ndl-RU1+(A)(x)-U1-(A)(x),
U1(A)(x)=C01G1(x|x) E(A)(x)n-E(A)×(x) G1(x|x)ndl.
G1(x|x)=A1(r)g1(θ|x),
A1(r)=-j42jπn1k0r1/2exp(-jn1k0r),
g1(θ|x)=exp[jn1k0(zcos θ+xsin θ)].
R=Cg1(θ|x) EC(x)n-EC(x) g1(θ|x)ndl-u1-(A)(θ)/u1+(A)(θ),
u1(A)(θ)=C01g1(θ|x) E(A)(x)n-E(A)(x) g1(θ|x)ndl.
12 EC(x)=CP1(x|x) EC(x)n-EC(x) P1(x|x)ndl-S1(x),
P1(x|x)=G1(x|x)-g1(θ|x) U1+(A)(x)u1+(A)(θ),
S1(x)=U1-(A)(x)-u1-(A)(θ) U1+(A)(x)u1+(A)(θ).
E(x)=TE-(B)(x)+EC(x),x onC45+C46,
E(x)=EC(x),x onC45+C47+C46.
T=Cg4(θ|x) EC(x)n-EC(x) g4(θ|x)ndl/u4-(B)(θ),
u4-(B)(θ)=C04g4(θ|x) E-(B)(x)n-E-(B)(x) g4(θ|x)ndl.
12 EC(x)=CP4(x|x) EC(x)n-EC(x) P4(x|x)ndl,
P4(x|x)=G4(x|x)-g4(θ|x) U4-(B)(x)u4-(B)(θ),
U4-(B)(x)=C04G4(x|x) E-(B)(x)n-E-(B)×(x) G4(x|x)ndl.
12 EC(x)=-C46+C46+C67G6(x|x) EC(x)n-EC(x) G6(x|x)ndl-TU6-(B)(x),
U6-(B)(x)=C06G6(x|x) E-(B)(x)n-E-(B)(x) G6(x|x)ndl.
E6S(r, θ)=-j42jπn6k0r1/2exp(-jn6k0r)B6(θ),
B6(θ)=-C46+C46+C67g6(θ|x) EC(x)n-EC(x) g6(θ|x)ndl-Tu6-(B)(θ),

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