Abstract

We propose a rigorous transverse-mode integral equation formulation for analyzing TE and TM electromagnetic radiation fields on the facet of dielectric slab waveguides with an abrupt termination in free space. Both exact waveguide guiding modes and discretized radiation modes are included in the kernels of the integral equation. To reduce the size of the matrix that approximates the exact integral equation, we expand the unknown field at the junctions in terms of guiding modes of a selected waveguide with sufficiently large normalized frequency and core thickness. By direct matrix inversion, we obtain numerical solutions of the scattered fields at the junctions. Our method can be used to study the field distribution as well as the energy reflection and transmission coefficients of dielectric waveguides with multiple step discontinuities.

© 2001 Optical Society of America

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References

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  1. R. Mittra, S. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).
  2. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  3. T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, Singapore, 1989).
  4. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
  5. Pochi Yeh, Optical Waves in Layered Media (Wiley, Singapore, 1991).
  6. J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  7. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–289 (1970).
    [CrossRef]
  8. A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [CrossRef]
  9. A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  10. W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
  11. D. Marcuse, “Radiation losses of step-tapered waveguides,” Appl. Opt. 19, 3676–3681 (1981).
    [CrossRef]
  12. P. G. Suchoski, V. Ramaswamy, “Exact numerical technique for the analysis of step discontinuities and tapers in optical dielectric waveguides,” J. Opt. Soc. Am. A 3, 194–203 (1986).
    [CrossRef]
  13. P. Clarricoats, A. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
    [CrossRef]
  14. G. A. Hockham, A. B. Sharpe, “Dielectric waveguide discontinuities,” Electron. Lett. 8, 230–231 (1972).
    [CrossRef]
  15. T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–745 (1978).
    [CrossRef]
  16. G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
    [CrossRef]
  17. C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
    [CrossRef]
  18. P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).
  19. I. G. Tigelis, T. G. Theodoropoulos, “Radiation properties of an abruptly terminated five-layer symmetric slab waveguide,” J. Opt. Soc. Am. A 14, 1260–1267 (1997).
    [CrossRef]
  20. H. Derudder, D. De Zutter, F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
    [CrossRef]
  21. M. Öz, R. R. Krchnavek, “Power loss analysis at a step discontinuity of a multimode optical waveguide,” J. Lightwave Technol. 16, 2451–2457 (1998).
    [CrossRef]
  22. I. G. Tigelis, “Abrupt transition coupling between two-layered symmetric slab waveguides,” J. Opt. Soc. Am. A 15, 84–91 (1998).
    [CrossRef]
  23. B. M. Azizur Rahman, J. B. Davies, “Analysis of optical waveguide discontinuities,” J. Lightwave Technol. 6, 52–57 (1988).
    [CrossRef]
  24. S. J. 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]
  25. Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
    [CrossRef]
  26. L. R. Gomaa, “Beam propagation method applied to a step discontinuity in dielectric planar waveguide,” IEE Proc. J 135, 205–206 (1988).
  27. Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
    [CrossRef]
  28. J. S. Chuang, T. L. Wu, H. W. Chang, “Field analysis of dielectric waveguide junctions using spectral-domain integral equation with exact bases,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 632–634.
  29. T. L. Wu, M. H. Sheng, H. W. Chang, “The design of AR-coatings for dielectric layered waveguides using 2D numerical model,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 349–351.

1998 (3)

1997 (2)

I. G. Tigelis, T. G. Theodoropoulos, “Radiation properties of an abruptly terminated five-layer symmetric slab waveguide,” J. Opt. Soc. Am. A 14, 1260–1267 (1997).
[CrossRef]

Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
[CrossRef]

1993 (1)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

1991 (1)

Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

1989 (1)

S. J. 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]

1988 (3)

L. R. Gomaa, “Beam propagation method applied to a step discontinuity in dielectric planar waveguide,” IEE Proc. J 135, 205–206 (1988).

C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
[CrossRef]

B. M. Azizur Rahman, J. B. Davies, “Analysis of optical waveguide discontinuities,” J. Lightwave Technol. 6, 52–57 (1988).
[CrossRef]

1986 (1)

1982 (1)

G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

1981 (1)

1978 (1)

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–745 (1978).
[CrossRef]

1977 (1)

W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).

1973 (1)

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

1972 (3)

A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
[CrossRef]

P. Clarricoats, A. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
[CrossRef]

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

1970 (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–289 (1970).
[CrossRef]

Adams, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Azizur Rahman, B. M.

B. M. Azizur Rahman, J. B. Davies, “Analysis of optical waveguide discontinuities,” J. Lightwave Technol. 6, 52–57 (1988).
[CrossRef]

Brooke, G. H.

G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

Burns, W. K.

W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).

Chang, H. C.

Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
[CrossRef]

Chang, H. W.

J. S. Chuang, T. L. Wu, H. W. Chang, “Field analysis of dielectric waveguide junctions using spectral-domain integral equation with exact bases,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 632–634.

T. L. Wu, M. H. Sheng, H. W. Chang, “The design of AR-coatings for dielectric layered waveguides using 2D numerical model,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 349–351.

Chen, C. H.

S. J. 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]

Chew, W. C.

Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

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

Chiou, Y. P.

Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
[CrossRef]

Chuang, J. S.

J. S. Chuang, T. L. Wu, H. W. Chang, “Field analysis of dielectric waveguide junctions using spectral-domain integral equation with exact bases,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 632–634.

Chung, S. J.

S. J. 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]

Clarricoats, P.

P. Clarricoats, A. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
[CrossRef]

Davies, J. B.

B. M. Azizur Rahman, J. B. Davies, “Analysis of optical waveguide discontinuities,” J. Lightwave Technol. 6, 52–57 (1988).
[CrossRef]

De Zutter, D.

H. Derudder, D. De Zutter, F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

Derudder, H.

H. Derudder, D. De Zutter, F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

Gomaa, L. R.

L. R. Gomaa, “Beam propagation method applied to a step discontinuity in dielectric planar waveguide,” IEE Proc. J 135, 205–206 (1988).

Hockham, G. A.

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

Itoh, T.

T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, Singapore, 1989).

Kendall, P. C.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Kharadly, M. M. Z.

G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

Krchnavek, R. R.

Lee, S.

R. Mittra, S. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).

Liu, Q. H.

Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

Love, J. D.

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

Marcuse, D.

D. Marcuse, “Radiation losses of step-tapered waveguides,” Appl. Opt. 19, 3676–3681 (1981).
[CrossRef]

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–289 (1970).
[CrossRef]

Mittra, R.

R. Mittra, S. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).

Olyslager, F.

H. Derudder, D. De Zutter, F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

Öz, M.

Ramaswamy, V.

Roberson, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Roberts, D. A.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Robson, P. N.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

Rozzi, T. E.

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–745 (1978).
[CrossRef]

Sharpe, A.

P. Clarricoats, A. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
[CrossRef]

Sharpe, A. B.

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

Sheng, M. H.

T. L. Wu, M. H. Sheng, H. W. Chang, “The design of AR-coatings for dielectric layered waveguides using 2D numerical model,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 349–351.

Snyder, A. W.

A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
[CrossRef]

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

Suchoski, P. G.

Theodoropoulos, T. G.

Tigelis, I. G.

Vassallo, C.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Wu, T. L.

T. L. Wu, M. H. Sheng, H. W. Chang, “The design of AR-coatings for dielectric layered waveguides using 2D numerical model,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 349–351.

J. S. Chuang, T. L. Wu, H. W. Chang, “Field analysis of dielectric waveguide junctions using spectral-domain integral equation with exact bases,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 632–634.

Yariv, A.

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Yeh, Pochi

Pochi Yeh, Optical Waves in Layered Media (Wiley, Singapore, 1991).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–289 (1970).
[CrossRef]

Electron. Lett. (3)

P. Clarricoats, A. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett. 8, 28–29 (1972).
[CrossRef]

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

H. Derudder, D. De Zutter, F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

IEE Proc. J (2)

L. R. Gomaa, “Beam propagation method applied to a step discontinuity in dielectric planar waveguide,” IEE Proc. J 135, 205–206 (1988).

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proc. J 140, 49–55 (1993).

IEEE J. Quantum Electron. (2)

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).

IEEE Photon. Technol. Lett. (1)

Y. P. Chiou, H. C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

S. J. 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]

Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

T. E. Rozzi, “Rigorous analysis of the step discontinuity in a planar dielectric waveguide,” IEEE Trans. Microwave Theory Tech. MTT-26, 738–745 (1978).
[CrossRef]

G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

J. Lightwave Technol. (2)

M. Öz, R. R. Krchnavek, “Power loss analysis at a step discontinuity of a multimode optical waveguide,” J. Lightwave Technol. 16, 2451–2457 (1998).
[CrossRef]

B. M. Azizur Rahman, J. B. Davies, “Analysis of optical waveguide discontinuities,” J. Lightwave Technol. 6, 52–57 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Other (8)

J. S. Chuang, T. L. Wu, H. W. Chang, “Field analysis of dielectric waveguide junctions using spectral-domain integral equation with exact bases,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 632–634.

T. L. Wu, M. H. Sheng, H. W. Chang, “The design of AR-coatings for dielectric layered waveguides using 2D numerical model,” in 1998 International Photonics Conference Proceedings (College of Electrical Engineering, National Taiwan University, Taiwan, 1998), pp. 349–351.

R. Mittra, S. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, New York, 1971).

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

T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, Singapore, 1989).

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

Pochi Yeh, Optical Waves in Layered Media (Wiley, Singapore, 1991).

J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995).

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Figures (14)

Fig. 1
Fig. 1

(a) Symmetric slab dielectric waveguide and the junction and (b) the symmetric model of (a).

Fig. 2
Fig. 2

Index profiles for the slab waveguide and the expansion basis.

Fig. 3
Fig. 3

Power reflectivity of the fundamental mode (TE case): our results (solid curves), Vassallo17 (crosses), and Kendall et al.18 (solid dots).

Fig. 4
Fig. 4

Same as Fig. 3 but for the TM case.

Fig. 5
Fig. 5

Power reflectivities R1 and RE (TE case).

Fig. 6
Fig. 6

Power reflectivities R1 and RE (TM case).

Fig. 7
Fig. 7

Convergence of the y-component electric field on the junction by using all guiding modes of a basis waveguide with varying core thickness (TE case) with parameters nh=3.6, nq=3.42, λ=0.86 μm, and 2t=0.25 μm. nhB=7.2, and nqB=3.42.

Fig. 8
Fig. 8

Convergence of the y-component electric field on the junction by using a subset of lower-order guiding modes of the same basis waveguide as that producing the top curve in Fig. 7.

Fig. 9
Fig. 9

Convergence of the y-component magnetic field on the junction (TM case). Parameters are the same as those in Fig. 7.

Fig. 10
Fig. 10

Convergence of the y-component magnetic field on the junction (TM case). Parameters are the same as those in Fig. 8.

Fig. 11
Fig. 11

Effect of waveguide index contrast on the TE mode, y-component electric fields on the junction with parameters nh=3.6, nq=nh(1-Δ), λ=0.86 μm, and 2t=0.25 μm.

Fig. 12
Fig. 12

Effect of waveguide index contrast on the TM mode, y-component magnetic fields on the junction with the same parameters as those in Fig. 11.

Fig. 13
Fig. 13

Magnetic field distribution for the TE case. Parameters: nh=3.6, nq=3.42, λ=0.86 μm, and 2t=0.25 μm.

Fig. 14
Fig. 14

Electric field distribution for the TM case. Parameters are the same as those in Fig. 13.

Tables (4)

Tables Icon

Table 1 TE Power Reflectivity of the Fundamental Mode by Using Mode Functions of Region I As the Expansion Basis with Parameters nh=3.6, nq=3.42, λ=0.86 μm, and 2t=0.25 μm

Tables Icon

Table 2 TM Power Reflectivity of the Fundamental Mode by Using Mode Functions of Region I As the Expansion Basis with the Same Parameters As Those in Table 1

Tables Icon

Table 3 TE Power Reflectivity of the Fundamental Mode by Using a Guiding Mode Expansion of the Basis Waveguide with Parameters nh=3.6, nq=3.42, λ=0.86 μm, 2t=0.25 μm, nhB=7.2, and nqB=3.42

Tables Icon

Table 4 TM Power Reflectivity of the Fundamental Mode by Using a Guiding Mode Expansion of the Basis Waveguide with the Same Parameters As Those in Table 3

Equations (57)

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

EyI(x, z, t)=n=1ξnI(x)exp[j(ωt-βnIz)],
ξnI(x)=C sinh[q(L-x)]sinh(qLd),LxtC cos(hx)cos(ht),tx0,
q=β2-nqωc21/2,h=nhωc2-β21/2.
h tan(ht)=q coth(qLd).
X tan(X)=V2-X2 coth(rV2-X2),
ξnI(x)=C sin[qc(L-x)]sinh(qcLd),LxtC cos(hx)cos(ht),tx0,
qc=nqωc2-β21/2.
X tan(X)=X2-V2 cot(rX2-V2).
ξm(x)|ξn*(x)=δmn,
HyI(x, z, t)=m=1HmI(x)exp[j(ωt-βmIz)],
HmI(x)=C cosh[q(L-x)]cosh(qLd),LxtC cos(hx)cos(ht),tx0,
X tan(X)=εhεq V2-X2 tanh(rV2-X2).
HmI(x)=C cos[qc(L-x)]cos(qcLd),LxtC cos(hx)cos(ht),tx0,
X tan(X)=εhεq X2-V2 tan(rX2-V2).
Hm(x)| 1εr(x)|Hn*(x)=δmn,
ϕnII(x)=2L1/2 cos(2n-1)πx2L,n=1,2,3 ,.
EyI(x, z)=n=1rnϕnI(x)exp(+jβnIz)+ϕiI(x)exp(-jβiIz),
EyII(x, z)=n=1tnϕnII(x)exp(-jβnIIz),
EyI(x, z)=n=1anϕnI(x)exp(+jβnIz)-2j sin(βiIz)ϕiI(x),
rn=anifinan-1ifi=n.
HxI(x, z)=n=1anynIϕnI(x)exp(+jβnIz)-2yiI cos(βiIz)ϕiI(x),
HxII(x, z)=-n=1tnynIIϕnII(x)exp(-jβnIIz).
ynl=βnlωμl,l=I,II,
n=1N1anϕnI(x)=n=1N2tnϕnII(x),
n=1N1anynIϕnI(x)-2yiIϕiI(x)=-n=1N2tnynIIϕnII(x).
an=0LE(x)ϕnI(x)dx,tn=0LE(x)ϕnII(x)dx.
0L[GEI(x, x)-GEII(x, x)]E(x)dx=2Hi(x),
GEI(x, x)=n=1ynIϕnI(x)ϕnI(x),
GEII(x, x)=-n=1ynIIϕnII(x)ϕnII(x),
Hi(x)=yiIϕiI(x).
HyI(x, z)=n=1anψnI(x)exp(+jβnIz)-2j sin(βiIz)ψiI(x),
HyII(x, z)=n=1tnψnII(x)exp(-jβnIIz),
ψnII(x)=1L1/2,n=02L1/2 cosnπxL,n=1,2,3 ,.
0L[GHI(x, x)-GHII(x, x)]H(x)dx=2Ei(x),
GHI(x, x)=-n=1ηnI ψnI(x)εrI(x) ψnI(x)εrI(x),
GHII(x, x)=n=1ηxII ψnII(x)εrII(x) ψnII(x)εrII(x),
Ei(x)=-ηiI ψiI(x)εrI(x).
ηnl=βnlωε0,l=I,II.
E(x)=n=1NcnφnB(x)=cˆφ¯B,
cˆ(WB,IYIϕ¯I-WB,IIYIIϕ¯II)=hˆϕ¯I,
[WB,l]nm=φnB(x)|ϕml(x),l=I,II.
cˆG=bˆ,
G=WB,IYIWI,B-WB,IIYIIWII,B,
bn=2yiIϕiI(x)|φnB(x).
an=k=1NckφkB(x)|ϕnI(x),
tn=k=1NckφkB(x)|ϕnII(x).
G=WB,IZIWI,B-WB,IIZIIWII,B,
bn=2ηiIψiI(x)| 1εrI(x)|φnB(x).
[WB,l]nm=φnB(x)| 1εrl(x)|ψml(x),l=I,II.
Einc(x, z)=ϕiI(x)exp(-jβiIz)yˆ,
Er(x, z)=n=1rnϕnI(x)exp(+jβnIz)yˆ.
Et(x, z)=n=1tnϕnII(x)exp(-jβnIIz)yˆ.
Pinc=Re-zˆ·[Einc(x, 0)×Hinc*(x, 0)]dx=Re(yiI),
Pr=Re-zˆ·[Er(x, 0)×Hr*(x, 0)]dx=n=1|rn|2 Re(ynI),
Pt=Re-zˆ·[Et(x, 0)×Ht*(x, 0)]dx=n=1|tn|2 Re(ynII).
RE=PrPinc,TE=PtPinc.
R1=Pr1Pinc1=|r1|2.

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