OSA's Digital Library

Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Vol. 20, Iss. 4 — Apr. 1, 2003
  • pp: 633–647

Homogenization expansion for resonances of microstructured photonic waveguides

Steven E. Golowich and Michael I. Weinstein  »View Author Affiliations


JOSA B, Vol. 20, Issue 4, pp. 633-647 (2003)
http://dx.doi.org/10.1364/JOSAB.20.000633


View Full Text Article

Enhanced HTML    Acrobat PDF (597 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We develop a homogenization expansion approach to photonic waveguides whose transverse structures are N-fold rotationally symmetric. Examples include microstructured or holey optical fibers with air holes arranged in one or more concentric rings. We carry out a homogenization expansion for large N about the N= limit. Our multiple scale analysis applies to the scalar approximation of structures in which the microfeatures have arbitrary geometry and large index contrasts and lead to a natural efficient computational algorithm for the waveguide modes and spectral characteristics. In this paper we focus on structures that possess leaky modes. The leading order (N=) equations describe the modes of an averaged structure. We derive an expansion in powers of 1/N of corrections to the leading order behavior and show that the leading order nontrivial contribution arises at order 1/N2. We numerically calculate this leading order correction to the complex effective indices (scattering resonances) for the leaky modes of various microstructured photonic waveguides whose imaginary parts give the leakage rates. We observe that in many instances a two-term truncation of the homogenization expansion gives good agreement with full simulations, even for fairly small values of N, whereas the leading order (averaged) theory yields a substantial underestimate of the leakage rates.

© 2003 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2400) Fiber optics and optical communications : Fiber properties

Citation
Steven E. Golowich and Michael I. Weinstein, "Homogenization expansion for resonances of microstructured photonic waveguides," J. Opt. Soc. Am. B 20, 633-647 (2003)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-20-4-633


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
  2. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25-27 (2000). [CrossRef]
  3. J. Jasapara, R. Bise, and R. Windeler, “Chromatic dispersion measurements in a photonic bandgap fiber,” in Optical Fiber Communication, Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002).
  4. T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 1660-1662 (2001). [CrossRef]
  5. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50-56 (2000). [CrossRef]
  6. L. Poladian, N. A. Issa, and T. M. Monro, “Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,” Opt. Express 10, 449-454 (2002). [CrossRef] [PubMed]
  7. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding-mode-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084-1100 (2000). [CrossRef]
  8. M. J. Steel and R. M. Osgood, Jr., “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229-231 (2001). [CrossRef]
  9. A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, M. C. J. Large, J. Zagari, N. A. P. Nicorovici, R. C. McPhedran, and C. M. de Sterke, “Ring structures in microstructured polymer optical fibers,” Opt. Express 9, 813-820 (2001). [CrossRef] [PubMed]
  10. A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, Vol. 5 of Studies in Mathematics and Its Applications (North-Holland, Amsterdam, 1978).
  11. G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics (Cambridge University, Cambridge, England, 2002).
  12. F. Santosa and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 53, 1636-1668 (1993). [CrossRef]
  13. S. Moskow and M. Vogelius, “First-order corrections to the homogenized eigenvalues of a periodic composite medium: a convergence proof,” Proc. R. Soc. Edinburgh Sect. A Math. 127, 1263-1299 (1997). [CrossRef]
  14. S. E. Golowich and M. I. Weinstein, “Scattering resonances and homogenization theory,” Bell Labs preprint (Bell Laboratories, Murray Hill, N.J., 2002).
  15. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  16. P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory (Springer-Verlag, New York, 1996).
  17. M. Abramowitz and I. E. Stegun, eds., Handbook of Mathematical Functions (National Institute of Standards and Technology, Gaithersburg, MD., 1972).
  18. The values for the LP01 mode calculated by the Fourier expansion were taken from Fig. 3 in Ref. 6.
  19. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, Boston, 1991).
  20. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).
  21. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978). [CrossRef]

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.


Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited