Abstract

A novel approach is proposed for obtaining the analytical solutions of the coupled-mode equations (CMEs); the method is applicable for an arbitrary number of coupled waveguides. The mathematical aspects of the CMEs and their solution by use of Chebyshev polynomials are discussed. When mode coupling between only adjacent waveguides is considered (denoted weak coupling), the first and second kinds of the usual Chebyshev polynomials are appropriate for evaluating the CMEs for linearly distributed and circularly distributed multiwaveguide systems, respectively. However, when one is considering the coupling effects between nonadjacent waveguides also (denoted strong coupling), it is necessary to use redefined generalized Chebyshev polynomials to express general solutions in a form similar to those for the weak-coupling case. As concrete examples, analytical solutions for 2×2, 3×3, and 4×4 linearly distributed directional couplers are obtained by the proposed approach, which treats the calculation as a nondegenerate eigenvalue problem. In addition, for the 3×3 circularly distributed directional coupler, which gives rise to a degenerate eigenvalue problem, an analytical solution is obtained in an improved way. Also, for comparison and without loss of generality, to clarify the difference between the two coupling cases, analytical solutions for a 5×5 circularly distributed directional coupler are obtained by use of the usual and the redefined generalized Chebyshev polynomials.

© 2004 Optical Society of America

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References

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  1. J. W. Arkright, “Novel structure for monolithic fused-fiber 1×4 couplers,” Electron. Lett. 27, 1767–1768 (1991).
    [CrossRef]
  2. Y. Huang, Q. Zeng, “A novel structure single-mode optical fiber splitter,” Acta Opt. Sin. 15, 248–251 (1995) (in Chinese).
  3. A. Biswas, “Theory of optical couplers,” Opt. Quantum Electron. 35, 221–235 (2003).
    [CrossRef]
  4. P. A. Buah, B. M. A. Rahman, K. T. V. Grattan, “Numerical study of soliton switching in active three-core nonlinear fiber couplers,” IEEE J. Quantum Electron. 33, 874–878 (1997).
    [CrossRef]
  5. S.-Q. Yao, Z.-H. Wang, “A dense-wavelength-division-multiplexer by using a three arm Mach-Zehnder interferometer,” Acta Opt. Sin. 20, 952–956 (2000) (in Chinese).
  6. Q. Wang, Y. Zhang, Y. C. Soh, “All-fiber 3×3 interleaver design with flat-top passband,” IEEE Photonics Technol. Lett. 16, 168–170 (2004).
    [CrossRef]
  7. M. Wrage, P. Glas, D. Fischer, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” J. Opt. Commun. 205, 367–375 (2002).
    [CrossRef]
  8. Y. H. Chew, T. T. Tjhung, F. V. C. Mendis, “Performance of single- and double-ring resonators using 3×3 optical fiber coupler,” J. Lightwave Technol. 11, 1998–2008 (1993).
    [CrossRef]
  9. R. W. C. Vance, J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc.: Optoelectron. 141, 231–241 (1994).
  10. A. W. Synder, “Coupled-mode theory for optical fiber,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [CrossRef]
  11. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  12. A. Yariv, “Coupled-mode theory for guided-wave optics,” J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  13. H. A. Haus, L. Molter-Ore, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. 19, 840–844 (1983).
  14. A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
  15. S.-L. Chuang, “A coupled mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5, 174–183 (1987).
  16. N. Kishi, E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36, 1861–1868 (1988).
  17. C.-S. Chang, H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).
  18. G.-D. Peng, A. Ankiewicz, “Modified Gaussian approach for the design of optical fiber couplers of arbitrary core shapes,” Appl. Opt. 30, 2533–2545 (1991).
    [PubMed]
  19. S.-X. She, L. Qiao, “Analysis of three channel waveguide directional couplers by a variational method and weighted residual method,” Opt. Commun. 87, 271–276 (1988).
  20. A. Hardy, W. Streifer, M. Osinski, “Weak coupling of parallel waveguides,” Opt. Lett. 13, 161–163 (1988).
    [PubMed]
  21. H. Kubo, K. Yasumoto, “Numerical analysis of three-parallel embedded optical waveguides,” J. Lightwave Technol. 7, 1924–1931 (1989).
    [CrossRef]
  22. A. Ankiewicz, A. W. Synder, X. H. Zheng, “Coupling between parallel optical fiber cores–critical examination,” J. Lightwave Technol. LT-4 317, 1317–1323 (1986).
    [CrossRef]
  23. R. Falcial, A. M. Scheggi, A. Schena, “Approximate calculation method for predicting selective properties of fused monomode biconical couplers,” Int. J. Optoelectron. 5, 41–46 (1990).
  24. L. Sun, P. Ye, “General analysis of 3×3 optical fiber directional couplers,” Microwave Opt. Technol. Lett. 2, 52–54 (1989).
    [CrossRef]
  25. Y. Chen, “Asymmetric triple-core couplers,” Opt. Quantum Electron. 24, 539–553 (1991).
    [CrossRef]
  26. D. B. Mortimore, “Theory and fabrication of 4×4 single-mode fused optical fiber couplers,” Appl. Opt. 29, 371–374 (1990).
    [CrossRef] [PubMed]
  27. A. Kowalski, “On the analysis of optical fibers described in terms of Chebyshev polynomials,” J. Lightwave Technol. 8, 164–167 (1990).
    [CrossRef]
  28. K. Mehrany, B. Rashidian, “Polynomial expansion for extraction of electromagnetic eigenmodes in layered structures,” J. Opt. Soc. Am. B 20, 2434–2441 (2003).
    [CrossRef]
  29. Z. Wang, D. Guo, Introduction to Special Functions (Peking U. Press, Beijing, 2000), pp. 168, 645 (in Chinese).
  30. S. Gradsheyn, I. M. Ryshik, Table of Integrals (Academic, New York, 1990), p. 30.

2004 (1)

Q. Wang, Y. Zhang, Y. C. Soh, “All-fiber 3×3 interleaver design with flat-top passband,” IEEE Photonics Technol. Lett. 16, 168–170 (2004).
[CrossRef]

2003 (2)

2002 (1)

M. Wrage, P. Glas, D. Fischer, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” J. Opt. Commun. 205, 367–375 (2002).
[CrossRef]

2000 (1)

S.-Q. Yao, Z.-H. Wang, “A dense-wavelength-division-multiplexer by using a three arm Mach-Zehnder interferometer,” Acta Opt. Sin. 20, 952–956 (2000) (in Chinese).

1997 (1)

P. A. Buah, B. M. A. Rahman, K. T. V. Grattan, “Numerical study of soliton switching in active three-core nonlinear fiber couplers,” IEEE J. Quantum Electron. 33, 874–878 (1997).
[CrossRef]

1995 (1)

Y. Huang, Q. Zeng, “A novel structure single-mode optical fiber splitter,” Acta Opt. Sin. 15, 248–251 (1995) (in Chinese).

1994 (3)

R. W. C. Vance, J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc.: Optoelectron. 141, 231–241 (1994).

C.-S. Chang, H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[CrossRef]

1993 (1)

Y. H. Chew, T. T. Tjhung, F. V. C. Mendis, “Performance of single- and double-ring resonators using 3×3 optical fiber coupler,” J. Lightwave Technol. 11, 1998–2008 (1993).
[CrossRef]

1991 (3)

J. W. Arkright, “Novel structure for monolithic fused-fiber 1×4 couplers,” Electron. Lett. 27, 1767–1768 (1991).
[CrossRef]

G.-D. Peng, A. Ankiewicz, “Modified Gaussian approach for the design of optical fiber couplers of arbitrary core shapes,” Appl. Opt. 30, 2533–2545 (1991).
[PubMed]

Y. Chen, “Asymmetric triple-core couplers,” Opt. Quantum Electron. 24, 539–553 (1991).
[CrossRef]

1990 (3)

A. Kowalski, “On the analysis of optical fibers described in terms of Chebyshev polynomials,” J. Lightwave Technol. 8, 164–167 (1990).
[CrossRef]

D. B. Mortimore, “Theory and fabrication of 4×4 single-mode fused optical fiber couplers,” Appl. Opt. 29, 371–374 (1990).
[CrossRef] [PubMed]

R. Falcial, A. M. Scheggi, A. Schena, “Approximate calculation method for predicting selective properties of fused monomode biconical couplers,” Int. J. Optoelectron. 5, 41–46 (1990).

1989 (2)

L. Sun, P. Ye, “General analysis of 3×3 optical fiber directional couplers,” Microwave Opt. Technol. Lett. 2, 52–54 (1989).
[CrossRef]

H. Kubo, K. Yasumoto, “Numerical analysis of three-parallel embedded optical waveguides,” J. Lightwave Technol. 7, 1924–1931 (1989).
[CrossRef]

1988 (3)

N. Kishi, E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36, 1861–1868 (1988).

S.-X. She, L. Qiao, “Analysis of three channel waveguide directional couplers by a variational method and weighted residual method,” Opt. Commun. 87, 271–276 (1988).

A. Hardy, W. Streifer, M. Osinski, “Weak coupling of parallel waveguides,” Opt. Lett. 13, 161–163 (1988).
[PubMed]

1987 (1)

S.-L. Chuang, “A coupled mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5, 174–183 (1987).

1986 (2)

A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).

A. Ankiewicz, A. W. Synder, X. H. Zheng, “Coupling between parallel optical fiber cores–critical examination,” J. Lightwave Technol. LT-4 317, 1317–1323 (1986).
[CrossRef]

1983 (1)

H. A. Haus, L. Molter-Ore, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. 19, 840–844 (1983).

1973 (1)

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

1972 (1)

Ankiewicz, A.

G.-D. Peng, A. Ankiewicz, “Modified Gaussian approach for the design of optical fiber couplers of arbitrary core shapes,” Appl. Opt. 30, 2533–2545 (1991).
[PubMed]

A. Ankiewicz, A. W. Synder, X. H. Zheng, “Coupling between parallel optical fiber cores–critical examination,” J. Lightwave Technol. LT-4 317, 1317–1323 (1986).
[CrossRef]

Arkright, J. W.

J. W. Arkright, “Novel structure for monolithic fused-fiber 1×4 couplers,” Electron. Lett. 27, 1767–1768 (1991).
[CrossRef]

Biswas, A.

A. Biswas, “Theory of optical couplers,” Opt. Quantum Electron. 35, 221–235 (2003).
[CrossRef]

Buah, P. A.

P. A. Buah, B. M. A. Rahman, K. T. V. Grattan, “Numerical study of soliton switching in active three-core nonlinear fiber couplers,” IEEE J. Quantum Electron. 33, 874–878 (1997).
[CrossRef]

Chang, C.-S.

C.-S. Chang, H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).

Chang, H.-C.

C.-S. Chang, H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).

Chen, Y.

Y. Chen, “Asymmetric triple-core couplers,” Opt. Quantum Electron. 24, 539–553 (1991).
[CrossRef]

Chew, Y. H.

Y. H. Chew, T. T. Tjhung, F. V. C. Mendis, “Performance of single- and double-ring resonators using 3×3 optical fiber coupler,” J. Lightwave Technol. 11, 1998–2008 (1993).
[CrossRef]

Chuang, S.-L.

S.-L. Chuang, “A coupled mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5, 174–183 (1987).

Falcial, R.

R. Falcial, A. M. Scheggi, A. Schena, “Approximate calculation method for predicting selective properties of fused monomode biconical couplers,” Int. J. Optoelectron. 5, 41–46 (1990).

Fischer, D.

M. Wrage, P. Glas, D. Fischer, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” J. Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Glas, P.

M. Wrage, P. Glas, D. Fischer, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” J. Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Gradsheyn, S.

S. Gradsheyn, I. M. Ryshik, Table of Integrals (Academic, New York, 1990), p. 30.

Grattan, K. T. V.

P. A. Buah, B. M. A. Rahman, K. T. V. Grattan, “Numerical study of soliton switching in active three-core nonlinear fiber couplers,” IEEE J. Quantum Electron. 33, 874–878 (1997).
[CrossRef]

Guo, D.

Z. Wang, D. Guo, Introduction to Special Functions (Peking U. Press, Beijing, 2000), pp. 168, 645 (in Chinese).

Hardy, A.

A. Hardy, W. Streifer, M. Osinski, “Weak coupling of parallel waveguides,” Opt. Lett. 13, 161–163 (1988).
[PubMed]

A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).

Haus, H. A.

H. A. Haus, L. Molter-Ore, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. 19, 840–844 (1983).

Huang, W.-P.

Huang, Y.

Y. Huang, Q. Zeng, “A novel structure single-mode optical fiber splitter,” Acta Opt. Sin. 15, 248–251 (1995) (in Chinese).

Kishi, N.

N. Kishi, E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36, 1861–1868 (1988).

Kowalski, A.

A. Kowalski, “On the analysis of optical fibers described in terms of Chebyshev polynomials,” J. Lightwave Technol. 8, 164–167 (1990).
[CrossRef]

Kubo, H.

H. Kubo, K. Yasumoto, “Numerical analysis of three-parallel embedded optical waveguides,” J. Lightwave Technol. 7, 1924–1931 (1989).
[CrossRef]

Love, J. D.

R. W. C. Vance, J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc.: Optoelectron. 141, 231–241 (1994).

Mehrany, K.

Mendis, F. V. C.

Y. H. Chew, T. T. Tjhung, F. V. C. Mendis, “Performance of single- and double-ring resonators using 3×3 optical fiber coupler,” J. Lightwave Technol. 11, 1998–2008 (1993).
[CrossRef]

Molter-Ore, L.

H. A. Haus, L. Molter-Ore, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. 19, 840–844 (1983).

Mortimore, D. B.

Osinski, M.

Peng, G.-D.

Qiao, L.

S.-X. She, L. Qiao, “Analysis of three channel waveguide directional couplers by a variational method and weighted residual method,” Opt. Commun. 87, 271–276 (1988).

Rahman, B. M. A.

P. A. Buah, B. M. A. Rahman, K. T. V. Grattan, “Numerical study of soliton switching in active three-core nonlinear fiber couplers,” IEEE J. Quantum Electron. 33, 874–878 (1997).
[CrossRef]

Rashidian, B.

Ryshik, I. M.

S. Gradsheyn, I. M. Ryshik, Table of Integrals (Academic, New York, 1990), p. 30.

Scheggi, A. M.

R. Falcial, A. M. Scheggi, A. Schena, “Approximate calculation method for predicting selective properties of fused monomode biconical couplers,” Int. J. Optoelectron. 5, 41–46 (1990).

Schena, A.

R. Falcial, A. M. Scheggi, A. Schena, “Approximate calculation method for predicting selective properties of fused monomode biconical couplers,” Int. J. Optoelectron. 5, 41–46 (1990).

She, S.-X.

S.-X. She, L. Qiao, “Analysis of three channel waveguide directional couplers by a variational method and weighted residual method,” Opt. Commun. 87, 271–276 (1988).

Soh, Y. C.

Q. Wang, Y. Zhang, Y. C. Soh, “All-fiber 3×3 interleaver design with flat-top passband,” IEEE Photonics Technol. Lett. 16, 168–170 (2004).
[CrossRef]

Streifer, W.

A. Hardy, W. Streifer, M. Osinski, “Weak coupling of parallel waveguides,” Opt. Lett. 13, 161–163 (1988).
[PubMed]

A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).

Sun, L.

L. Sun, P. Ye, “General analysis of 3×3 optical fiber directional couplers,” Microwave Opt. Technol. Lett. 2, 52–54 (1989).
[CrossRef]

Synder, A. W.

A. Ankiewicz, A. W. Synder, X. H. Zheng, “Coupling between parallel optical fiber cores–critical examination,” J. Lightwave Technol. LT-4 317, 1317–1323 (1986).
[CrossRef]

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

Tjhung, T. T.

Y. H. Chew, T. T. Tjhung, F. V. C. Mendis, “Performance of single- and double-ring resonators using 3×3 optical fiber coupler,” J. Lightwave Technol. 11, 1998–2008 (1993).
[CrossRef]

Vance, R. W. C.

R. W. C. Vance, J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc.: Optoelectron. 141, 231–241 (1994).

Wang, Q.

Q. Wang, Y. Zhang, Y. C. Soh, “All-fiber 3×3 interleaver design with flat-top passband,” IEEE Photonics Technol. Lett. 16, 168–170 (2004).
[CrossRef]

Wang, Z.

Z. Wang, D. Guo, Introduction to Special Functions (Peking U. Press, Beijing, 2000), pp. 168, 645 (in Chinese).

Wang, Z.-H.

S.-Q. Yao, Z.-H. Wang, “A dense-wavelength-division-multiplexer by using a three arm Mach-Zehnder interferometer,” Acta Opt. Sin. 20, 952–956 (2000) (in Chinese).

Wrage, M.

M. Wrage, P. Glas, D. Fischer, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” J. Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Yamashita, E.

N. Kishi, E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36, 1861–1868 (1988).

Yao, S.-Q.

S.-Q. Yao, Z.-H. Wang, “A dense-wavelength-division-multiplexer by using a three arm Mach-Zehnder interferometer,” Acta Opt. Sin. 20, 952–956 (2000) (in Chinese).

Yariv, A.

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

Yasumoto, K.

H. Kubo, K. Yasumoto, “Numerical analysis of three-parallel embedded optical waveguides,” J. Lightwave Technol. 7, 1924–1931 (1989).
[CrossRef]

Ye, P.

L. Sun, P. Ye, “General analysis of 3×3 optical fiber directional couplers,” Microwave Opt. Technol. Lett. 2, 52–54 (1989).
[CrossRef]

Zeng, Q.

Y. Huang, Q. Zeng, “A novel structure single-mode optical fiber splitter,” Acta Opt. Sin. 15, 248–251 (1995) (in Chinese).

Zhang, Y.

Q. Wang, Y. Zhang, Y. C. Soh, “All-fiber 3×3 interleaver design with flat-top passband,” IEEE Photonics Technol. Lett. 16, 168–170 (2004).
[CrossRef]

Zheng, X. H.

A. Ankiewicz, A. W. Synder, X. H. Zheng, “Coupling between parallel optical fiber cores–critical examination,” J. Lightwave Technol. LT-4 317, 1317–1323 (1986).
[CrossRef]

Acta Opt. Sin. (2)

Y. Huang, Q. Zeng, “A novel structure single-mode optical fiber splitter,” Acta Opt. Sin. 15, 248–251 (1995) (in Chinese).

S.-Q. Yao, Z.-H. Wang, “A dense-wavelength-division-multiplexer by using a three arm Mach-Zehnder interferometer,” Acta Opt. Sin. 20, 952–956 (2000) (in Chinese).

Appl. Opt. (2)

Electron. Lett. (1)

J. W. Arkright, “Novel structure for monolithic fused-fiber 1×4 couplers,” Electron. Lett. 27, 1767–1768 (1991).
[CrossRef]

IEE Proc.: Optoelectron. (1)

R. W. C. Vance, J. D. Love, “Design procedures for passive planar coupled waveguide devices,” IEE Proc.: Optoelectron. 141, 231–241 (1994).

IEEE J. Quantum Electron. (3)

P. A. Buah, B. M. A. Rahman, K. T. V. Grattan, “Numerical study of soliton switching in active three-core nonlinear fiber couplers,” IEEE J. Quantum Electron. 33, 874–878 (1997).
[CrossRef]

H. A. Haus, L. Molter-Ore, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. 19, 840–844 (1983).

A. Hardy, W. Streifer, “Coupled mode solutions of multi-waveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).

IEEE Photonics Technol. Lett. (1)

Q. Wang, Y. Zhang, Y. C. Soh, “All-fiber 3×3 interleaver design with flat-top passband,” IEEE Photonics Technol. Lett. 16, 168–170 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

N. Kishi, E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36, 1861–1868 (1988).

Int. J. Optoelectron. (1)

R. Falcial, A. M. Scheggi, A. Schena, “Approximate calculation method for predicting selective properties of fused monomode biconical couplers,” Int. J. Optoelectron. 5, 41–46 (1990).

J. Lightwave Technol. (6)

H. Kubo, K. Yasumoto, “Numerical analysis of three-parallel embedded optical waveguides,” J. Lightwave Technol. 7, 1924–1931 (1989).
[CrossRef]

A. Ankiewicz, A. W. Synder, X. H. Zheng, “Coupling between parallel optical fiber cores–critical examination,” J. Lightwave Technol. LT-4 317, 1317–1323 (1986).
[CrossRef]

A. Kowalski, “On the analysis of optical fibers described in terms of Chebyshev polynomials,” J. Lightwave Technol. 8, 164–167 (1990).
[CrossRef]

Y. H. Chew, T. T. Tjhung, F. V. C. Mendis, “Performance of single- and double-ring resonators using 3×3 optical fiber coupler,” J. Lightwave Technol. 11, 1998–2008 (1993).
[CrossRef]

C.-S. Chang, H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).

S.-L. Chuang, “A coupled mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5, 174–183 (1987).

J. Opt. Commun. (1)

M. Wrage, P. Glas, D. Fischer, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” J. Opt. Commun. 205, 367–375 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

J. Quantum Electron. (1)

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

Microwave Opt. Technol. Lett. (1)

L. Sun, P. Ye, “General analysis of 3×3 optical fiber directional couplers,” Microwave Opt. Technol. Lett. 2, 52–54 (1989).
[CrossRef]

Opt. Commun. (1)

S.-X. She, L. Qiao, “Analysis of three channel waveguide directional couplers by a variational method and weighted residual method,” Opt. Commun. 87, 271–276 (1988).

Opt. Lett. (1)

Opt. Quantum Electron. (2)

A. Biswas, “Theory of optical couplers,” Opt. Quantum Electron. 35, 221–235 (2003).
[CrossRef]

Y. Chen, “Asymmetric triple-core couplers,” Opt. Quantum Electron. 24, 539–553 (1991).
[CrossRef]

Other (2)

Z. Wang, D. Guo, Introduction to Special Functions (Peking U. Press, Beijing, 2000), pp. 168, 645 (in Chinese).

S. Gradsheyn, I. M. Ryshik, Table of Integrals (Academic, New York, 1990), p. 30.

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Figures (4)

Fig. 1
Fig. 1

Schematic of linearly distributed directional coupler.

Fig. 2
Fig. 2

Schematic of circularly distributed directional coupler.

Fig. 3
Fig. 3

Eigenfunction Φn(x) versus normalized propagation constant x for a circularly distributed directional coupler; (a) weak-coupling case γ=0; (b) strong-coupling case γ=0.1; solid curves, n=4; dashed curves, n=6; dotted curves n=8.

Fig. 4
Fig. 4

Eigenfunction Φn(x) versus normalized propagation constant x for a strong-coupling circularly distributed directional coupler for n=6 and various relative coupling coefficients γ; solid curve, γ=0; dashed curve, γ=0.05; dotted curve, γ=0.1; dashed-dotted curve, γ=0.2.

Equations (184)

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

uiz=j K2 μi-1+j K2 μi+1,
μ0z=j K2 μ1,μNz=j K2 μN-1,0<i<N.
2xΦi(x)=Φi-1(x)+Φi+1(x),i=1,, N-1,
2xΦ0(x)=Φ1(x),
2xΦN(x)=ΦN-1(x).
Φn(xm)=2N+21/2sin[(n+1)ϕm],
ϕm=mπN+2,xm=βm/K=cos ϕm,
n=0,, N,m=1,, N+1.
2xmΦN(xm)=2N+21/2sin(Nϕm)=ΦN-1(xm),
m=1N+1Φk(xm)Φl(xm)=δk,l,
n=0NΦn(xm)Φn(xl)=δm,l.
X(τ)
=Φ0(x1)exp(jx1τ)Φ0(xN+1)exp(jxN+1τ)ΦN(x1)exp(jx1τ)ΦN(xN+1)exp(jxN+1τ),
 
μ(τ)=X(τ)l=0NclΦl(x1)l=0NclΦl(x2)l=0NclΦl(xN+1)=m=1N+1l=0NclΦl(xm)Φ0(xm)exp(jxmτ)m=1N+1l=0NclΦl(xm)Φ1(xm)exp(jxmτ)m=1N+1l=0NclΦl(xm)ΦN(xm)exp(jxmτ).
μ(0)=[c0, c1, , cN]T,
X-1(0)μ(0)=Φ0(x1)Φ1(x1)ΦN(x1)Φ0(x2)Φ1(x2)ΦN(x2)Φ0(xN+1)Φ1(xN+1)ΦN(xN+1)×c0c1cN=l=0NclΦl(x1)l=0NclΦl(x2)l=0NclΦl(xN+1),
R(τ)=mΦ02(xm)exp(jxmτ)mΦ0(xm)Φ1(xm)exp(jxmτ)mΦ0(xm)ΦN(xm)exp(jxmτ)mΦ1(xm)Φ0(xm)exp(jxmτ)mΦ12(xm)exp(jxmτ)mΦ1(xm)ΦN(xm)exp(jxmτ)mΦN(xm)Φ0(xm)exp(jxmτ)mΦN(xm)Φ1(xm)exp(jxmτ)mΦN2(xm)exp(jxmτ).
μiz=j K2 μi-1+j K2 μi+1,
μ0z=j K2 μ1+j K2 μN,
μNz=j K2 μ0+j K2 μN-1.0<i<N
2xΦi(x)=Φi-1(x)+Φi+1(x),
2xΦ0(x)=Φ1(x)+ΦN(x),
2xΦN(x)=Φ0(x)+ΦN-1(x).0<i<N.
Φn(xm)=2N+11/2cos(nϕm),
m0,0nN,
Φn(x0)=1N+11/2
xm=βm/K=cos ϕm,
ϕm=2π mN+1,m=0,, N.
2xmΦn(xm)=22N+11/2cos ϕmcos(nϕm)=Φn+1(xm)+Φn-1(xm),0<n<N,
2xmΦ0(xm)=22N+11/2cos ϕm=2N+1 [cos ϕm+cos(Nϕm)]=Φ1(xm)+ΦN(xm),
2xmΦN(xm)=22N+11/2cos ϕmcos(Nϕm)=ΦN-1(xm)+Φ0(xm).
μ˜iz=j K2 (μ˜i-1+μ˜i+1)+j K2 (μ˜i-2+μ˜i+2),2iN-2
μ˜0z=j K2 (μ˜N+μ˜1)+j K2 (μ˜N-1+μ˜2),
μ˜1z=j K2 (μ˜0+μ˜2)+j K2 (μ˜N+μ˜3),
μ˜Nz=j K2 (μ˜N-1+μ˜0)+j K2 (μ˜N-2+μ˜1),
μ˜N-1z=j K2 (μ˜N-2+μ˜N)+j K2 (μ˜N-3+μ˜0),
2xΦ˜i(x)=Φ˜i-1(x)+Φ˜i+1(x)+γ[Φ˜i-2(x)+Φ˜i+2(x)],2iN-2,
2xΦ˜0(x)=Φ˜N(x)+Φ˜1(x)+γ[Φ˜N-1(x)+Φ˜2(x)],
2xΦ˜1(x)=Φ˜0(x)+Φ˜2(x)+γ[Φ˜N(x)+Φ˜3(x)],
2xΦ˜N(x)=Φ˜N-1(x)+Φ˜0(x)+γ[Φ˜N-2(x)+Φ˜1(x)],
2xΦ˜N-1(x)=Φ˜N-2(x)+Φ˜N(x)+γ[Φ˜N-3(x)+Φ˜0(x)].
Φ˜n(xm)=2N+11/2cos(nϕm),0nN,
xm=cos ϕm+γ cos(2ϕm),
ϕm=2πm/(N+1),m=0, 1, 2,, N.
x=cos ϕ,Φn(x)=cos(nϕ)=cos(n cos-1 x).
x=cos ϕ+γ cos(2ϕ)=cos ϕ+γ(2 cos2 ϕ-1),
cos ϕ=-1/4γ+[(1/4γ)2+(γ+x)/2γ]1/2=f(x),
Φ˜n(x)=cos(nϕ)=cos[n cos-1f(x)].
xm=cos ϕm=cos(mπ/3),m=1, 2,
x1=cos(π/3)=(1/2),x2=cos(2π/3)=-12,
Φ0(x1)=1/2,Φ0(x2)=1/2,
Φ1(x1)=1/2,Φ1(x2)=-1/2.
R(τ)=12exp(jτ/2)12exp(-jτ/2)12exp(jτ/2)-12exp(-jτ/2) 121212-12=cosτ2j sinτ2j sinτ2cosτ2,
xm=cos ϕm=cos(mπ/4),m=1, 2, 3,
x1=1/2,x2=0,x3=-1/2,
R(τ)
=12expjτ21212exp-jτ212expjτ20-12exp-jτ212expjτ2-1212exp-jτ2×121212120-1212-1212,
R(τ)
=12+12cosτ2j2sinτ2-12+12cosτ2j2sinτ2cosτ2j2sinτ2-12+12cosτ2j2sinτ212+12cosτ2.
xm=cos ϕm=cos(mπ/5),m=1, 2, 3, 4
x1=cos(π/5),x2=cos(2π/5),
x3=-x2,x4=-x1,
Rn,l=4rn,l/5,
r1,1=r4,4=(1-x12)cos(x1τ)+(1-x22)cos(x2τ),
r2,2=r3,3=(1-x22)cos(x1τ)+(1-x12)cos(x2τ),
r1,2=r2,1=r3,4=r4,3=j[(1-x12)(1-x22)]1/2×[sin(x1τ)+sin(x2τ)],
r1,3=r2,4=r3,1=r4,2=[(1-x12)(1-x22)]1/2×[cos(x1τ)-cos(x2τ)],
r1,4=r4,1=j(1-x12)sin(x1τ)-j(1-x22)sin(x2τ),
r2,3=r3,2=j(1-x22)sin(x1τ)-j(1-x12)sin(x2τ).
ϕm=2π m1+2,m=0, 1, 2.
xm=cos ϕm,
x0=1,x1=x2=-1/2,
μ1=α11exp(jx0τ)+(α12+α13τ)exp(jx1τ),
μ2=α21exp(jx0τ)+(α22+α23τ)exp(jx1τ),
μ3=α31exp(jx0τ)+(α32+α33τ)exp(jx1τ).
α11=α21=α31=A/3.
α12=2B,
α22=C-B,
α32=-C-B,
α13=α23=α33=0.
μ1=A3exp(jx0τ)+2B exp(jx1τ),
μ2=A3exp(jx0τ)+(C-B)exp(jx1τ),
μ3=A3exp(jx0τ)-(C+B)exp(jx1τ).
a1=A/3+2B,a2=A/3+C-B,
a3=A/3-(C+B),
A=a1+a2+a3,B=16(2a1-a2-a3),
C=13(a1+a2-2a3).
μ1μ2μ3=γ1γ2γ2γ2γ1γ2γ2γ2γ1 a1a2a3=R(τ)a1a2a3,
γ1=13exp(jτ)+23exp(-jτ/2),
γ2=13exp(jτ)-13exp(-jτ/2).
μ0z=j K2 (μ4+μ1),μ1z=j K2 (μ0+μ2),
μ2z=j K2 (μ1+μ3),μ3z=j K2 (μ2+μ4),
μ4z=j K2 (μ3+μ0).
2xΦ0(x)=Φ4(x)+Φ1(x),
2xΦ1(x)=Φ0(x)+Φ2(x),
2xΦ2(x)=Φ1(x)+Φ3(x),
2xΦ3(x)=Φ2(x)+Φ4(x),
2xΦ4(x)=Φ3(x)+Φ0(x).
Φn(xm)=(2/5)1/2cos(nϕm),0n4,
xm=cos ϕm,
ϕm=(2/5)mπ,m=0, 1, 2, 3, 4.
μ0=α00exp(jx0τ)+(α01+α04τ)exp(jx1τ)+(α02+α03τ)exp(jx2τ),
μ1=α10exp(jx0τ)+(α11+α14τ)exp(jx1τ)+(α12+α13τ)exp(jx2τ),
μ2=α20exp(jx0τ)+(α21+α24τ)exp(jx1τ)+(α22+α23τ)exp(jx2τ),
μ3=α30exp(jx0τ)+(α31+α34τ)exp(jx1τ)+(α32+α33τ)exp(jx2τ),
μ4=α40exp(jx0τ)+(α41+α44τ)exp(jx1τ)+(α42+α43τ)exp(jx2τ).
2xiα0i-(α4i+α1i)=0,
2xiα1i-(α0i+α2i)=0,
2xiα2i-(α1i+α3i)=0,
2xiα3i-(α2i+α4i)=0,
2xiα4i-(α3i+α0i)=0,
αi3=αi4=0,
μ0=b0exp(jx0τ)+b1exp(jx1τ)+b2exp(jx2τ),
μ1=b0exp(jx0τ)+b3exp(jx1τ)+b4exp(jx2τ),
μ2=b0exp(jx0τ)+(2x1b3-b1)exp(jx1τ)+(2x2b4-b2)exp(jx2τ),
μ3=b0exp(jx0τ)-2x1(b1+b3)exp(jx1τ)-2x2(b2+b4)exp(jx2τ),
μ4=b0exp(jx0τ)+(2x1b1-b3)exp(jx1τ)+(2x2b2-b4)exp(jx2τ).
b0=(a0+a1+a2+a3+a4)/5,
b1=(A+Bx2)/E,
b2=-(A+Bx1)/E,b3=(C+Dx2)/E,
b4=-(C+Dx1)/E,
A=2a0-3a1+2a2+2a3-3a4,
B=8a0-2a1-2a2-2a3-2a4,
C=-3a0+2a1-3a2+2a3+2a4,
D=-2a0+8a1-2a2-2a3-2a4,
E=10(x2-x1).
μ0μ1μ2μ3μ4=c1c2c3c3c2c2c1c2c3c3c3c2c1c2c3c3c3c2c1c2c2c3c3c2c1 a0a1a2a3a4,
c1=[exp(jx0τ)+2 exp(jx1τ)+2 exp(jx2τ)]/5,
c2=[2 exp(jx0τ)+(5-1)exp(jx1τ)-(5+1)exp(jx2τ)]/10,
c3=[2 exp(jx0τ)-(5+1)exp(jx1τ)+(5-1)exp(jx2τ)]/10.
μ˜0z=j K2 (μ˜4+μ˜1)+j K2 (μ˜3+μ˜2),
μ˜1z=j K2 (μ˜0+μ˜2)+j K2 (μ˜4+μ˜3),
μ˜2z=j K2 (μ˜1+μ˜3)+j K2 (μ˜0+μ˜4),
μ˜3z=j K2 (μ˜2+μ˜4)+j K2 (μ˜1+μ˜0),
μ˜4z=j K2 (μ˜3+μ˜0)+j K2 (μ˜2+μ˜1).
2xΦ˜0(x)=Φ˜4(x)+Φ˜1(x)+γ[Φ˜3(x)+Φ˜2(x)],
2xΦ˜1(x)=Φ˜0(x)+Φ˜2(x)+γ[Φ˜4(x)+Φ˜3(x)],
2xΦ˜2(x)=Φ˜1(x)+Φ˜3(x)+γ[Φ˜0(x)+Φ˜4(x)],
2xΦ˜3(x)=Φ˜2(x)+Φ˜4(x)+γ[Φ˜1(x)+Φ˜0(x)],
2xΦ˜4(x)=Φ˜3(x)+Φ˜0(x)+γ[Φ˜2(x)+Φ˜1(x)].
x0=1+γ,
x1=cos(2π/5)+γ cos(4π/5),
x2=cos(4π/5)+γ cos(8π/5),
x3=cos(6π/5)+γ cos(12π/5)=cos(4π/5)+γ cos(8π/5)=x2,
x4=cos(8π/5)+γ cos(16π/5)=cos(2π/5)+γ cos(4π/5)=x1,
μ˜0=α00exp(jx0τ)+(α01+α04τ)exp(jx1τ)+(α02+α03τ)exp(jx2τ),
μ˜1=α10exp(jx0τ)+(α11+α14τ)exp(jx1τ)+(α12+α13τ)exp(jx2τ),
μ˜2=α20exp(jx0τ)+(α21+α24τ)exp(jx1τ)+(α22+α23τ)exp(jx2τ),
μ˜3=α30exp(jx0τ)+(α31+α34τ)exp(jx1τ)+(α32+α33τ)exp(jx2τ),
μ˜4=α40exp(jx0τ)+(α41+α44τ)exp(jx1τ)+(α42+α43τ)exp(jx2τ).
μ˜0μ˜1μ˜2μ˜3μ˜4=c1c2c3c3c2c2c1c2c3c3c3c2c1c2c3c3c3c2c1c2c2c3c3c2c1 a0a1a2a3a4,
c1=[exp(jx0τ)+2 exp(jx1τ)+2 exp(jx2τ)]/5,
c2=[2 exp(jx0τ)+(5-1)exp(jx1τ)-(5+1)exp(jx2τ)]/10,
c3=[2 exp(jx0τ)-(5+1)exp(jx1τ)+(5-1)exp(jx2τ)]/10.
ei(z)=1nexp(-jβ0z)m=1ncos2πn miexp{-jΔβ(d)×[cos(2πm/n)]z},
ei(z)=2n+1exp(-jβ0z)m=1nsinmlπn+1sinmiπn+1×exp{-jΔβ(d)cos[mπ/(n+1)]z},
m=0N+1cos m JN+2 π
=cosN+12JN+2 πsinN+22JN+2 πsinJN+2π2  =cosN+22JN+2 πsinN+12JN+2 πsinJN+2π2  +1.
 
m=1N+1sin(k+1)ϕmsin(l+1)ϕm
=12m=1N+1[cos(k-l)ϕm-cos(k+l+2)ϕm]=12m=0N+1[cos(k-l)ϕm-cos(k+l+2)ϕm]=N+22 δk,l,
n=0Nsin(n+1)ϕmsin(n+1)ϕl
=12n=0Ncos(n+1) m-lN+2 π-cos(n+1) m+lN+2 π=N+22 δm,l,
m=1N+1Φk(xm)Φl(xm)=δk,l,
n=0NΦn(xm)Φn(xl)=δm,l.
n=0Ncos2πn mN+1cos2πn lN+1
=12n=0Ncos2πn m+lN+1+cos2πn m-lN+1,
n=0Ncosn 2π(m±l)N+1
=cosN22π(m±l)N+1sinN+122π(m±l)N+1sin2π(m±l)2(N+1)  .
n=0Ncos2πn mN+1cos2πn lN+1=N+12 δm,l.
n=0Ncos2πn mN+1cos2πn lN+1=N+1;
n=0NΦn(xm)Φn(xl)=δm,l.
m=0NΦn(xm)Φl(xm)=δn,l.
γ1=13exp(jϕ2),γ2=13exp(jϕ1),
μ1μ2μ3=13exp(jΔϕ)111exp(jΔϕ)111exp(jΔϕ)×a1a2a3exp-j π18,
(a1, a2, a3)=0,j2expj-δ2+π18,-j2expjδ2+π18,
μ1μ2μ3=231/2sin(δ/2)sin[(δ+2π/3)/2]exp(-jπ/3)sin[(δ-2π/3)/2]exp(-jπ/3),
P1=|μ1|2=23sin2δ2=13 (1-cos δ),
P2=|μ2|2=23sin212δ+2π3=131-cosδ+2π3,
P3=|μ3|2=23sin212δ-2π3=131-cosδ-2π3,
(a1, a2, a3)=expj2π3+δ+π18, expj π18,expj-δ+π18
μ1μ2μ3=13EFG,G=sin[3(δ+2π/3)/2]sin[(δ+2π/3)/2],
E=sin(3δ/2)sin(δ/2),F=sin[3(δ+4π/3)/2]sin[(δ+4π/3)/2]exp-j 2π3.

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