OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 21, Iss. 8 — Aug. 30, 2004
  • pp: 1518–1528

Analytical solutions of coupled-mode equations for multiwaveguide systems, obtained by use of Chebyshev and generalized Chebyshev polynomials

Yi-Chao Meng, Qi-Zhi Guo, Wei-Han Tan, and Zhao-Ming Huang  »View Author Affiliations

JOSA A, Vol. 21, Issue 8, pp. 1518-1528 (2004)

View Full Text Article

Enhanced HTML    Acrobat PDF (236 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(000.3860) General : Mathematical methods in physics
(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers
(130.2790) Integrated optics : Guided waves
(230.7370) Optical devices : Waveguides

Original Manuscript: November 3, 2003
Revised Manuscript: February 19, 2004
Manuscript Accepted: February 19, 2004
Published: August 1, 2004

Yi-Chao Meng, Qi-Zhi Guo, Wei-Han Tan, and Zhao-Ming Huang, "Analytical solutions of coupled-mode equations for multiwaveguide systems, obtained by use of Chebyshev and generalized Chebyshev polynomials," J. Opt. Soc. Am. A 21, 1518-1528 (2004)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited