Abstract

A four-port device, consisting of a codirectional coupler with a grating overlay, is analyzed using four-wave coupled-wave theory. Both fully synchronous and asynchronous cases are treated, and the results from the coupled-wave analysis are then related to modal analysis. The implications of the results for both active and passive devices based round the same geometry are then discussed.

© 1985 Optical Society of America

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References

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  1. S. E. Miller, “Coupled Wave Theory and Waveguide Applications,” Bell Syst. Tech. J. 33, 661 (1954).
  2. E. A. J. Marcatili, “Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071 (1969).
  3. H. Kogelnik, R. V. Schmidt, “Switched Directional Couplers with Alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396 (1976).
    [Crossref]
  4. R. C. Alferness, “Polarization Independent Optical Directional Coupler Switch Using Weighted Coupling,” Appl. Phys. Lett. 35, 748 (1979).
    [Crossref]
  5. H. F. Taylor, “Frequency Selective Coupling in Parallel Dielectric Waveguides,” Opt. Commun. 8, 421 (1973).
    [Crossref]
  6. R. C. Alferness, P. S. Cross, “Filter Characteristics of Co-directionally Coupled Waveguides with Weighted Coupling,” IEEE J. Quantum Electron. QE-14, 843 (1978).
    [Crossref]
  7. J. S. Wilkinson, M. G. F. Wilson, “Directional Coupler Wavelength Filter,” presented at IERE Colloquium on Integrated Optics, London, 12 June 1984.
  8. N. Tsukada, “Modification of the Coupling Coefficient by Periodic Modulation of the Propagation Constants,” Opt. Commun. 22, 113 (1977).
    [Crossref]
  9. M. Kondo, Y. Ohta, M. Fujiwara, M. Sakaguchi, “Integrated Optical Switch Matrix for Single-Mode Fibre Networks,” IEEE J. Quantum Electron. QE-18, 1759 (1982).
    [Crossref]
  10. V. Ramaswamy, M. D. Divino, R. D. Standley, “Balanced Bridge Modulator Switch Using Ti-diffused LiNbO3 Strip Waveguides,” Appl. Phys. Lett. 32, 644 (1978).
    [Crossref]
  11. K. H. Tietgen, R. Th. Kersten, “180° Turns in Integrated Optics,” Opt. Commun. 36, 281 (1981).
    [Crossref]
  12. K. Sukada, A. Yariv, “Analysis of Optical Propagation in a Corrugated Dielectric Waveguide,” Opt. Commun. 8, 1 (1973).
    [Crossref]
  13. H. Stoll, A. Yariv, “Coupled-Mode Analysis of Periodic Dielectric Waveguides,” Opt. Commun. 8, 5 (1973).
    [Crossref]
  14. C. Elachi, C. Yeh, “Periodic Structures in Integrated Optics,” J. Appl. Phys. 44, 3146 (1973).
    [Crossref]
  15. R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band Grating Filters for Thin-Film Optical Waveguides,” Appl. Phys. Lett. 25, 651 (1974).
    [Crossref]
  16. J. E. Bjorkholm, C. V. Shank, “Distributed Feedback Lasers in Thin Film Optical Waveguides,” IEEE J. Quantum Electron. QE-8, 833 (1972).
    [Crossref]
  17. H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
    [Crossref]
  18. A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
    [Crossref]
  19. C. Elachi, C. Yeh, “Frequency Selective Coupler for Integrated Optics Systems,” Opt. Commun. 7, 201 (1973).
    [Crossref]
  20. P. Yeh, H. F. Taylor, “Contradirectional Frequency-Selective Couplers for Guided-Wave Optics,” Appl. Opt. 19, 2848 (1980).
    [Crossref] [PubMed]
  21. A. Yariv, “Coupled-Wave Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
    [Crossref]
  22. A. Yi-Yan, I. Andonovic, E. Y. B. Pun, B. Bjortop, “Fabrication of Periodic Ti:LiNbO3 Waveguides by Single and Double Diffusion,” Appl. Phys. Lett. 43, 19 (1983).
    [Crossref]
  23. J. M. Hammer, “Metal Diffused Stripe Waveguides: Approximate Closed Form Solution for Lower Order Modes,” Appl. Opt. 15, 319 (1976).
    [Crossref] [PubMed]
  24. P. Vandenbulke, P. E. Lagasse, “Static Field Analysis of Thin Film Electro-optic Light Modulators and Switches,” Wave Electron. 1, 295 (1975).

1983 (1)

A. Yi-Yan, I. Andonovic, E. Y. B. Pun, B. Bjortop, “Fabrication of Periodic Ti:LiNbO3 Waveguides by Single and Double Diffusion,” Appl. Phys. Lett. 43, 19 (1983).
[Crossref]

1982 (1)

M. Kondo, Y. Ohta, M. Fujiwara, M. Sakaguchi, “Integrated Optical Switch Matrix for Single-Mode Fibre Networks,” IEEE J. Quantum Electron. QE-18, 1759 (1982).
[Crossref]

1981 (1)

K. H. Tietgen, R. Th. Kersten, “180° Turns in Integrated Optics,” Opt. Commun. 36, 281 (1981).
[Crossref]

1980 (1)

1979 (1)

R. C. Alferness, “Polarization Independent Optical Directional Coupler Switch Using Weighted Coupling,” Appl. Phys. Lett. 35, 748 (1979).
[Crossref]

1978 (2)

V. Ramaswamy, M. D. Divino, R. D. Standley, “Balanced Bridge Modulator Switch Using Ti-diffused LiNbO3 Strip Waveguides,” Appl. Phys. Lett. 32, 644 (1978).
[Crossref]

R. C. Alferness, P. S. Cross, “Filter Characteristics of Co-directionally Coupled Waveguides with Weighted Coupling,” IEEE J. Quantum Electron. QE-14, 843 (1978).
[Crossref]

1977 (2)

N. Tsukada, “Modification of the Coupling Coefficient by Periodic Modulation of the Propagation Constants,” Opt. Commun. 22, 113 (1977).
[Crossref]

A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
[Crossref]

1976 (2)

J. M. Hammer, “Metal Diffused Stripe Waveguides: Approximate Closed Form Solution for Lower Order Modes,” Appl. Opt. 15, 319 (1976).
[Crossref] [PubMed]

H. Kogelnik, R. V. Schmidt, “Switched Directional Couplers with Alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396 (1976).
[Crossref]

1975 (1)

P. Vandenbulke, P. E. Lagasse, “Static Field Analysis of Thin Film Electro-optic Light Modulators and Switches,” Wave Electron. 1, 295 (1975).

1974 (1)

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band Grating Filters for Thin-Film Optical Waveguides,” Appl. Phys. Lett. 25, 651 (1974).
[Crossref]

1973 (6)

C. Elachi, C. Yeh, “Frequency Selective Coupler for Integrated Optics Systems,” Opt. Commun. 7, 201 (1973).
[Crossref]

K. Sukada, A. Yariv, “Analysis of Optical Propagation in a Corrugated Dielectric Waveguide,” Opt. Commun. 8, 1 (1973).
[Crossref]

H. Stoll, A. Yariv, “Coupled-Mode Analysis of Periodic Dielectric Waveguides,” Opt. Commun. 8, 5 (1973).
[Crossref]

C. Elachi, C. Yeh, “Periodic Structures in Integrated Optics,” J. Appl. Phys. 44, 3146 (1973).
[Crossref]

H. F. Taylor, “Frequency Selective Coupling in Parallel Dielectric Waveguides,” Opt. Commun. 8, 421 (1973).
[Crossref]

A. Yariv, “Coupled-Wave Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[Crossref]

1972 (2)

J. E. Bjorkholm, C. V. Shank, “Distributed Feedback Lasers in Thin Film Optical Waveguides,” IEEE J. Quantum Electron. QE-8, 833 (1972).
[Crossref]

H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
[Crossref]

1969 (1)

E. A. J. Marcatili, “Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071 (1969).

1954 (1)

S. E. Miller, “Coupled Wave Theory and Waveguide Applications,” Bell Syst. Tech. J. 33, 661 (1954).

Alferness, R. C.

R. C. Alferness, “Polarization Independent Optical Directional Coupler Switch Using Weighted Coupling,” Appl. Phys. Lett. 35, 748 (1979).
[Crossref]

R. C. Alferness, P. S. Cross, “Filter Characteristics of Co-directionally Coupled Waveguides with Weighted Coupling,” IEEE J. Quantum Electron. QE-14, 843 (1978).
[Crossref]

Andonovic, I.

A. Yi-Yan, I. Andonovic, E. Y. B. Pun, B. Bjortop, “Fabrication of Periodic Ti:LiNbO3 Waveguides by Single and Double Diffusion,” Appl. Phys. Lett. 43, 19 (1983).
[Crossref]

Bjorkholm, J. E.

J. E. Bjorkholm, C. V. Shank, “Distributed Feedback Lasers in Thin Film Optical Waveguides,” IEEE J. Quantum Electron. QE-8, 833 (1972).
[Crossref]

Bjortop, B.

A. Yi-Yan, I. Andonovic, E. Y. B. Pun, B. Bjortop, “Fabrication of Periodic Ti:LiNbO3 Waveguides by Single and Double Diffusion,” Appl. Phys. Lett. 43, 19 (1983).
[Crossref]

Cross, P. S.

R. C. Alferness, P. S. Cross, “Filter Characteristics of Co-directionally Coupled Waveguides with Weighted Coupling,” IEEE J. Quantum Electron. QE-14, 843 (1978).
[Crossref]

Divino, M. D.

V. Ramaswamy, M. D. Divino, R. D. Standley, “Balanced Bridge Modulator Switch Using Ti-diffused LiNbO3 Strip Waveguides,” Appl. Phys. Lett. 32, 644 (1978).
[Crossref]

Elachi, C.

C. Elachi, C. Yeh, “Periodic Structures in Integrated Optics,” J. Appl. Phys. 44, 3146 (1973).
[Crossref]

C. Elachi, C. Yeh, “Frequency Selective Coupler for Integrated Optics Systems,” Opt. Commun. 7, 201 (1973).
[Crossref]

Flanders, D. C.

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band Grating Filters for Thin-Film Optical Waveguides,” Appl. Phys. Lett. 25, 651 (1974).
[Crossref]

Fujiwara, M.

M. Kondo, Y. Ohta, M. Fujiwara, M. Sakaguchi, “Integrated Optical Switch Matrix for Single-Mode Fibre Networks,” IEEE J. Quantum Electron. QE-18, 1759 (1982).
[Crossref]

Hammer, J. M.

Kersten, R. Th.

K. H. Tietgen, R. Th. Kersten, “180° Turns in Integrated Optics,” Opt. Commun. 36, 281 (1981).
[Crossref]

Kogelnik, H.

H. Kogelnik, R. V. Schmidt, “Switched Directional Couplers with Alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396 (1976).
[Crossref]

H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
[Crossref]

Kondo, M.

M. Kondo, Y. Ohta, M. Fujiwara, M. Sakaguchi, “Integrated Optical Switch Matrix for Single-Mode Fibre Networks,” IEEE J. Quantum Electron. QE-18, 1759 (1982).
[Crossref]

Lagasse, P. E.

P. Vandenbulke, P. E. Lagasse, “Static Field Analysis of Thin Film Electro-optic Light Modulators and Switches,” Wave Electron. 1, 295 (1975).

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071 (1969).

Miller, S. E.

S. E. Miller, “Coupled Wave Theory and Waveguide Applications,” Bell Syst. Tech. J. 33, 661 (1954).

Nakamura, M.

A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
[Crossref]

Ohta, Y.

M. Kondo, Y. Ohta, M. Fujiwara, M. Sakaguchi, “Integrated Optical Switch Matrix for Single-Mode Fibre Networks,” IEEE J. Quantum Electron. QE-18, 1759 (1982).
[Crossref]

Pun, E. Y. B.

A. Yi-Yan, I. Andonovic, E. Y. B. Pun, B. Bjortop, “Fabrication of Periodic Ti:LiNbO3 Waveguides by Single and Double Diffusion,” Appl. Phys. Lett. 43, 19 (1983).
[Crossref]

Ramaswamy, V.

V. Ramaswamy, M. D. Divino, R. D. Standley, “Balanced Bridge Modulator Switch Using Ti-diffused LiNbO3 Strip Waveguides,” Appl. Phys. Lett. 32, 644 (1978).
[Crossref]

Sakaguchi, M.

M. Kondo, Y. Ohta, M. Fujiwara, M. Sakaguchi, “Integrated Optical Switch Matrix for Single-Mode Fibre Networks,” IEEE J. Quantum Electron. QE-18, 1759 (1982).
[Crossref]

Schmidt, R. V.

H. Kogelnik, R. V. Schmidt, “Switched Directional Couplers with Alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396 (1976).
[Crossref]

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band Grating Filters for Thin-Film Optical Waveguides,” Appl. Phys. Lett. 25, 651 (1974).
[Crossref]

Shank, C. V.

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band Grating Filters for Thin-Film Optical Waveguides,” Appl. Phys. Lett. 25, 651 (1974).
[Crossref]

J. E. Bjorkholm, C. V. Shank, “Distributed Feedback Lasers in Thin Film Optical Waveguides,” IEEE J. Quantum Electron. QE-8, 833 (1972).
[Crossref]

H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
[Crossref]

Standley, R. D.

V. Ramaswamy, M. D. Divino, R. D. Standley, “Balanced Bridge Modulator Switch Using Ti-diffused LiNbO3 Strip Waveguides,” Appl. Phys. Lett. 32, 644 (1978).
[Crossref]

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band Grating Filters for Thin-Film Optical Waveguides,” Appl. Phys. Lett. 25, 651 (1974).
[Crossref]

Stoll, H.

H. Stoll, A. Yariv, “Coupled-Mode Analysis of Periodic Dielectric Waveguides,” Opt. Commun. 8, 5 (1973).
[Crossref]

Sukada, K.

K. Sukada, A. Yariv, “Analysis of Optical Propagation in a Corrugated Dielectric Waveguide,” Opt. Commun. 8, 1 (1973).
[Crossref]

Taylor, H. F.

P. Yeh, H. F. Taylor, “Contradirectional Frequency-Selective Couplers for Guided-Wave Optics,” Appl. Opt. 19, 2848 (1980).
[Crossref] [PubMed]

H. F. Taylor, “Frequency Selective Coupling in Parallel Dielectric Waveguides,” Opt. Commun. 8, 421 (1973).
[Crossref]

Tietgen, K. H.

K. H. Tietgen, R. Th. Kersten, “180° Turns in Integrated Optics,” Opt. Commun. 36, 281 (1981).
[Crossref]

Tsukada, N.

N. Tsukada, “Modification of the Coupling Coefficient by Periodic Modulation of the Propagation Constants,” Opt. Commun. 22, 113 (1977).
[Crossref]

Vandenbulke, P.

P. Vandenbulke, P. E. Lagasse, “Static Field Analysis of Thin Film Electro-optic Light Modulators and Switches,” Wave Electron. 1, 295 (1975).

Wilkinson, J. S.

J. S. Wilkinson, M. G. F. Wilson, “Directional Coupler Wavelength Filter,” presented at IERE Colloquium on Integrated Optics, London, 12 June 1984.

Wilson, M. G. F.

J. S. Wilkinson, M. G. F. Wilson, “Directional Coupler Wavelength Filter,” presented at IERE Colloquium on Integrated Optics, London, 12 June 1984.

Yariv, A.

A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
[Crossref]

K. Sukada, A. Yariv, “Analysis of Optical Propagation in a Corrugated Dielectric Waveguide,” Opt. Commun. 8, 1 (1973).
[Crossref]

H. Stoll, A. Yariv, “Coupled-Mode Analysis of Periodic Dielectric Waveguides,” Opt. Commun. 8, 5 (1973).
[Crossref]

A. Yariv, “Coupled-Wave Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[Crossref]

Yeh, C.

C. Elachi, C. Yeh, “Frequency Selective Coupler for Integrated Optics Systems,” Opt. Commun. 7, 201 (1973).
[Crossref]

C. Elachi, C. Yeh, “Periodic Structures in Integrated Optics,” J. Appl. Phys. 44, 3146 (1973).
[Crossref]

Yeh, P.

Yi-Yan, A.

A. Yi-Yan, I. Andonovic, E. Y. B. Pun, B. Bjortop, “Fabrication of Periodic Ti:LiNbO3 Waveguides by Single and Double Diffusion,” Appl. Phys. Lett. 43, 19 (1983).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

A. Yi-Yan, I. Andonovic, E. Y. B. Pun, B. Bjortop, “Fabrication of Periodic Ti:LiNbO3 Waveguides by Single and Double Diffusion,” Appl. Phys. Lett. 43, 19 (1983).
[Crossref]

R. C. Alferness, “Polarization Independent Optical Directional Coupler Switch Using Weighted Coupling,” Appl. Phys. Lett. 35, 748 (1979).
[Crossref]

V. Ramaswamy, M. D. Divino, R. D. Standley, “Balanced Bridge Modulator Switch Using Ti-diffused LiNbO3 Strip Waveguides,” Appl. Phys. Lett. 32, 644 (1978).
[Crossref]

R. V. Schmidt, D. C. Flanders, C. V. Shank, R. D. Standley, “Narrow-band Grating Filters for Thin-Film Optical Waveguides,” Appl. Phys. Lett. 25, 651 (1974).
[Crossref]

Bell Syst. Tech. J. (2)

S. E. Miller, “Coupled Wave Theory and Waveguide Applications,” Bell Syst. Tech. J. 33, 661 (1954).

E. A. J. Marcatili, “Dielectric Rectangular Waveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071 (1969).

IEEE J. Quantum Electron. (6)

H. Kogelnik, R. V. Schmidt, “Switched Directional Couplers with Alternating Δβ,” IEEE J. Quantum Electron. QE-12, 396 (1976).
[Crossref]

R. C. Alferness, P. S. Cross, “Filter Characteristics of Co-directionally Coupled Waveguides with Weighted Coupling,” IEEE J. Quantum Electron. QE-14, 843 (1978).
[Crossref]

M. Kondo, Y. Ohta, M. Fujiwara, M. Sakaguchi, “Integrated Optical Switch Matrix for Single-Mode Fibre Networks,” IEEE J. Quantum Electron. QE-18, 1759 (1982).
[Crossref]

J. E. Bjorkholm, C. V. Shank, “Distributed Feedback Lasers in Thin Film Optical Waveguides,” IEEE J. Quantum Electron. QE-8, 833 (1972).
[Crossref]

A. Yariv, M. Nakamura, “Periodic Structures for Integrated Optics,” IEEE J. Quantum Electron. QE-13, 233 (1977).
[Crossref]

A. Yariv, “Coupled-Wave Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[Crossref]

J. Appl. Phys. (2)

C. Elachi, C. Yeh, “Periodic Structures in Integrated Optics,” J. Appl. Phys. 44, 3146 (1973).
[Crossref]

H. Kogelnik, C. V. Shank, “Coupled-Wave Theory of Distributed Feedback Lasers,” J. Appl. Phys. 43, 2327 (1972).
[Crossref]

Opt. Commun. (6)

C. Elachi, C. Yeh, “Frequency Selective Coupler for Integrated Optics Systems,” Opt. Commun. 7, 201 (1973).
[Crossref]

K. H. Tietgen, R. Th. Kersten, “180° Turns in Integrated Optics,” Opt. Commun. 36, 281 (1981).
[Crossref]

K. Sukada, A. Yariv, “Analysis of Optical Propagation in a Corrugated Dielectric Waveguide,” Opt. Commun. 8, 1 (1973).
[Crossref]

H. Stoll, A. Yariv, “Coupled-Mode Analysis of Periodic Dielectric Waveguides,” Opt. Commun. 8, 5 (1973).
[Crossref]

N. Tsukada, “Modification of the Coupling Coefficient by Periodic Modulation of the Propagation Constants,” Opt. Commun. 22, 113 (1977).
[Crossref]

H. F. Taylor, “Frequency Selective Coupling in Parallel Dielectric Waveguides,” Opt. Commun. 8, 421 (1973).
[Crossref]

Wave Electron. (1)

P. Vandenbulke, P. E. Lagasse, “Static Field Analysis of Thin Film Electro-optic Light Modulators and Switches,” Wave Electron. 1, 295 (1975).

Other (1)

J. S. Wilkinson, M. G. F. Wilson, “Directional Coupler Wavelength Filter,” presented at IERE Colloquium on Integrated Optics, London, 12 June 1984.

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Figures (15)

Fig. 1
Fig. 1

Four-port device geometry.

Fig. 2
Fig. 2

Variation of the distribution of power among the output ports with κbd for κad = 1.5 and κcd = 0. Device operating in full synchronism.

Fig. 3
Fig. 3

Variation of the distribution of power among the output ports with κbd for κad = 1.5 and κcd = κbd/2. Device operating in full synchronism.

Fig. 4
Fig. 4

Variation of the distribution of power among the output ports with κbd for κad = 1.5 and κcd = κbd. Device operating in full synchronism.

Fig. 5
Fig. 5

Δβλd – Δβed plane showing operating points and trajectories for which numerical results are presented in Figs. 612.

Fig. 6
Fig. 6

Variation of the distribution of power among the output ports with Δβλd for κad = 1.5, κbd = 1.5, κcd = 1.3, and Δβed = 0. Trajectory 1.

Fig. 7
Fig. 7

Variation of the distribution of power among the output ports with Δβed for κad = 1.5, κbd = 1.5, κcd = 1.3, and Δβλd = 0. Trajectory 2.

Fig. 8
Fig. 8

Variation of the distribution of power among the output ports with Δβλd for κad = 1.5, κbd = 1.5, κcd = 1.3, and Δβed = 20. Trajectory 3.

Fig. 9
Fig. 9

Variation of the distribution of power among the output ports with Δβλd for κad = 1.5, κbd = 1.5, κcd = 1.3, and Δβed = −20. Trajectory 4.

Fig. 10
Fig. 10

Variation of the distribution of power among the output ports with Δβed for κad = 1.5, κbd = 1.5, kcd = 1.3, and Δβλd = 20. Trajectory 5.

Fig. 11
Fig. 11

Variation of the distribution of power among the output ports with Δβed for κad = 1.5, κbd = 1.5, κcd = 1.3, and Δβλd = −20. Trajectory 6.

Fig. 12
Fig. 12

Variation of the distribution of power among the output ports with Δβλd as predicted by the approximate two-wave solutions, for the same parameters as Fig. 8.

Fig. 13
Fig. 13

(a) Symmetric and antisymmetric modes; (b), and (c) schematic of the reflection process for symmetric and antisymmetric modes.

Fig. 14
Fig. 14

Variation of the distribution of power between the forward and backward characteristic modes for (a) a symmetric mode launch (full line) and (b) an antisymmetric mode launch (dashed line) with Δβλd for κad = 1.5, κbd = 1.5, and κcd = 0.

Fig. 15
Fig. 15

Four-port device used in conjunction with Y-junctions.

Equations (30)

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| K | = 2 π Λ = 2 β 0 .
d A 1 d x = j { κ 12 A 2 + κ 13 A 3 + κ 14 A 4 } .
κ 12 = κ 21 = κ 34 = κ 43 = κ a , κ 13 = κ 31 = κ 24 = κ 42 = κ b , κ 14 = κ 41 = κ 23 = κ 32 = κ c .
a = j κ a , b = j κ b , c = j κ c , and D = d d x
( D a b c a D c b b c D a c b a D ) ( A 1 A 2 A 3 A 4 ) = ( 0 0 0 0 ) .
d d x { | A 1 | 2 + | A 2 | 2 | A 3 | 2 | A 4 | 2 } = 0 .
A = j = 1 4 C j v j exp ( λ j x ) .
λ 4 2 [ a 2 b 2 c 2 ] λ 2 + { [ a 4 + b 4 + c 4 ] 2 [ a 2 b 2 + b 2 c 2 + a 2 c 2 ] } = 0 .
λ 1 , 2 = ± a 2 ( b + c ) 2 ; λ 3 , 4 = ± a 2 ( b c ) 2 .
A 1 = 1 , A 2 = 0 on x = 0 ; A 3 = 0 , A 4 = 0 on x = d .
A 1 ( x = d ) = γ 1 2 [ γ 1 cosh γ 1 d + γ 2 sinh γ 1 d ] + γ 4 2 [ γ 4 cosh γ 4 d γ s sinh γ 4 d ] , A 2 ( x = d ) = γ 1 2 [ γ 1 cosh γ 1 d + γ 2 sinh γ 1 d ] γ 4 2 [ γ 4 cosh γ 4 d γ 5 sinh γ 4 d ] , A 3 ( x = 0 ) = γ 3 sinh γ 1 d 2 [ γ 1 cosh γ 1 d + γ 2 sinh γ 1 d ] γ 6 sinh γ 4 d 2 [ γ 4 cosh γ 4 d γ 5 sinh γ 4 d ] , A 4 ( x = 0 ) = γ 3 sinh γ 1 d 2 [ γ 1 cosh γ 1 d + γ 2 sinh γ 1 d ] + γ 6 sinh γ 4 d 2 [ γ 4 cosh γ 4 d γ 5 sinh γ 4 d ] ,
γ 1 = a 2 ( b + c ) 2 , γ 4 = a 2 ( b c ) 2 , γ 2 = a , γ 5 = a , γ 3 = b + c , γ 6 = b c .
A 1 ( x = d ) = cosh γ 2 d = cos ( κ a d ) , A 2 ( x = d ) = sinh γ 2 d = j sin ( κ a d ) , A 3 ( x = 0 ) = A 4 ( x = 0 ) = 0 .
A 1 ( x = d ) = 1 / cosh γ 1 d = 1 / cosh ( κ b d ) , A 3 ( x = 0 ) = ( γ 4 / γ 1 ) tanh γ 1 d = j tanh ( κ b d ) , A 2 ( x = d ) = A 4 ( x = 0 ) = 0 .
A 1 ( x = d ) = 1 / 2 exp ( j κ a d ) ; A 2 ( x = d ) = 1 / 2 exp ( j κ a d ) , A 3 ( x = 0 ) = j / 2 ; A 4 ( x = 0 ) = j / 2 .
( D + δ 1 a b c a D + δ 2 c b b c D δ 1 a c b a D δ 2 ) ( A 1 A 2 A 3 A 4 ) = ( 0 0 0 0 ) ,
δ 1 = j ( Δ β λ + Δ β e ) , δ 2 = j ( Δ β λ Δ β e ) .
λ 4 { 2 ( a 2 b 2 c 2 ) + ( δ 1 2 + δ 2 2 ) } λ 2 + { ( a 4 + b 4 + c 4 ) 2 ( a 2 b 2 + b 2 c 2 + a 2 c 2 ) 2 δ 1 δ 2 ( a 2 + c 2 ) ( δ 1 2 + δ 2 2 ) b 2 + δ 1 2 δ 2 2 + 4 a b c ( δ 1 + δ 2 ) } = 0 ,
λ 2 = { a 2 b 2 c 2 } + 1 / 2 { δ 1 2 + δ 2 2 } ± 16 b 2 c 2 + 4 a 2 ( δ 1 + δ 2 ) 2 4 c 2 ( δ 1 δ 2 ) 2 16 abc ( δ 1 + δ 2 ) + ( δ 1 + δ 2 ) 2 ( δ 1 δ 2 ) 2 .
γ 1 = ( a + δ λ ) 2 ( b + c ) 2 , γ 4 = ( a δ λ ) 2 ( b c ) 2 , γ 2 = a + δ λ , γ 5 = a δ λ , γ 3 = b + c , γ 6 = b c . }
( D + δ 1 c c D δ 2 ) ( A 1 A 4 ) = ( 0 0 ) .
A 1 = exp ( δ e d ) δ λ 2 c 2 δ λ 2 c 2 cosh ( d δ λ 2 c 2 ) + δ λ sinh ( d δ λ 2 c 2 ) on x = d , A 4 = c sinh ( d δ λ 2 c 2 ) δ λ 2 c 2 cosh ( d δ λ 2 c 2 ) + δ λ sinh ( d δ λ 2 c 2 ) on x = 0 ,
( D + δ 1 b b D δ 1 ) ( A 1 A 3 ) = ( 0 0 ) .
A 1 = ( δ λ + δ e ) 2 b 2 ( δ λ + δ e ) 2 b 2 cosh [ d ( δ λ + δ e ) 2 b 2 ] + ( δ λ + δ e ) sinh [ d ( δ λ + δ e ) 2 b 2 ] ( on x = d ) , A 3 = b sinh [ d ( δ λ + δ e ) 2 b 2 ] ( δ λ + δ e ) 2 b 2 cosh [ d ( δ λ + δ e ) 2 b 2 ] + ( δ λ + δ e ) sinh [ d ( δ λ + δ e ) 2 b 2 ] ( on x = 0 ) .
M s f = A 1 + A 2 ; M a f = A 1 A 2 ; M s b = A 3 + A 4 ; M a b = A 3 A 4 .
( ( D + a ) 0 ( b + c ) 0 0 ( D a ) 0 ( b c ) ( b + c ) 0 ( D a ) 0 0 ( b c ) 0 ( D + a ) ) ( M s f M a f M s b M a b ) = ( 0 0 0 0 ) .
A 1 = A 2 = cos ( κ a d ) j sin ( κ a d ) = exp ( j κ a d ) .
M s f ( x = d ) = γ 1 γ 1 cosh γ 1 d + γ 2 sinh γ 1 d , [ γ given by Eq . ( 16 ) ] M s b ( x = 0 ) = γ 3 sinh γ 1 d γ 1 cosh γ 1 d + γ 2 sinh γ 1 d ,
M a f ( x = d ) = γ 4 γ 4 cosh γ 4 d + γ 5 sinh γ 4 d , [ γ given by Eq . ( 16 ) ] M a b ( x = 0 ) = γ sinh γ d 4 γ 4 cosh γ 4 d + γ 5 sinh γ 4 d .
[ ( D + a + δ λ ) 0 ( b + c ) 0 0 ( D a + δ λ ) 0 ( b c ) ( b + c ) 0 ( D a δ λ ) 0 0 ( b c ) 0 ( D + a δ λ ) ] ( M s f M a f M s b M a b ) = ( 0 0 0 0 ) .

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