## Vectorial modal analysis of dielectric waveguides based on a coupled transverse-mode integral equation. I. Mathematical formulation

JOSA A, Vol. 23, Issue 6, pp. 1468-1477 (2006)

http://dx.doi.org/10.1364/JOSAA.23.001468

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### Abstract

We propose a rigorous full-vector integral-equation formulation for analyzing modal characteristics of the complex, two-dimensional, rectangular-like dielectric waveguide that is divisible into vertical slices of one-dimensional layered structures. The entire electromagnetic mode field is completely determined by the *y*-component electric and magnetic field functions on the interfaces between slices. These interfacial functions are governed by a system of vector-coupled transverse-mode integral equations (VCTMIE) whose kernels are made of orthonormal sets of both TE-to-*y* and TM-to-*y* modes from each slice. To solve for the unknown functions, we construct sets of suitable expansion functions and turn VCTMIE into a nonlinear matrix equation via orthogonal projection. The eigenvectors of the matrix provide the mode field solutions of the complex dielectric waveguide.

© 2006 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.4430) General : Numerical approximation and analysis

(130.0130) Integrated optics : Integrated optics

(230.4170) Optical devices : Multilayers

(230.7370) Optical devices : Waveguides

**ToC Category:**

Optical Devices

**History**

Original Manuscript: August 26, 2005

Revised Manuscript: December 4, 2005

Manuscript Accepted: December 8, 2005

**Citation**

Hung-Wen Chang, Tso-Lun Wu, and Meng-Huei Sheng, "Vectorial modal analysis of dielectric waveguides based on a coupled transverse-mode integral equation. I. Mathematical formulation," J. Opt. Soc. Am. A **23**, 1468-1477 (2006)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-23-6-1468

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### References

- E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).
- Y. H. Cheng and W. G. Lin, "Investigation of rectangular dielectric waveguides: an iteratively equivalent index method," IEE Proc.: Optoelectron. 137, 323-329 (1990). [CrossRef]
- J. S. Lee and S. Y. Shin, "On the validity of the effective-index method for rectangular dielectric waveguides," J. Lightwave Technol. 11, 1320-1324 (1993). [CrossRef]
- Y. Cai, T. Mizumoto, and Y. Naito, "Improved perturbation feedback method for the analysis of rectangular dielectric waveguides," J. Lightwave Technol. 9, 1231-1237 (1991). [CrossRef]
- J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).
- R. Mittra, Y. L. Hou, and V. Jannejad, "Analysis of open dielectric waveguides using mode-matching technique and variational methods," IEEE Trans. Microwave Theory Tech. 28, 36-43 (1980). [CrossRef]
- S. T. Peng and A. A. Oliner, "Guidance and leakage properties of a class of open dielectric waveguides: part I—mathematical formulations," IEEE Trans. Microwave Theory Tech. 29, 843-855 (1981). [CrossRef]
- E. W. Kolk, N. H. G. Baken, and H. Blok, "Domain integral equation analysis of integrated optical channel and ridged waveguides in stratified media," J. Lightwave Technol. 38, 78-85 (1990).
- H. Y. Yang, J. A. Castaneda, and N. G. Alexopoulos, "An integral equation analysis of an infinite array of rectangular dielectric waveguides," IEEE Trans. Microwave Theory Tech. 38, 873-880 (1990). [CrossRef]
- K. Sabetfakhri and L. P. B. Katehi, "An integral transform technique for analysis of planar dielectric structures," IEEE Trans. Microwave Theory Tech. 42, 1052-1062 (1994). [CrossRef]
- G. Athanasoulias and N. K. Uzunoglu, "An accurate and efficient entire-domain basis Galerkin's method for the integral equation analysis of integrated rectangular dielectric waveguides," IEEE Trans. Microwave Theory Tech. 43, 2794-2804 (1995). [CrossRef]
- V. A. Kalinin and B. K. J. C. Nauwelaers, "Free space dyadic Green's function applied to the full-wave numerical analysis of planar transmission lines and dielectric waveguides," IEE Proc., Part H: Microwaves, Antennas Propag. 143, 328-334 (1996). [CrossRef]
- T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, "Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide," J. Lightwave Technol. 11, 429-433 (1993). [CrossRef]
- U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993). [CrossRef]
- R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000). [CrossRef]
- A. S. Sudbo, "Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides," Pure Appl. Opt. 2, 211-233 (1993). [CrossRef]
- A. S. Sudbo, "Improved formulation of film mode matching method for mode field calculations in dielectric waveguides," Pure Appl. Opt. 3, 381-388 (1994). [CrossRef]
- H. W. Chang and T. L. Wu, "Vectorial modal analysis of dielectric waveguides based on coupled transverse-mode integral equation. II. Numerical analysis," J. Opt. Soc. Am. A 23, 1478-1487 (2006). [CrossRef]
- R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
- A. Ishimaru, Electromagnetic Propagation, Radiation, and Scattering (Prentice Hall, 1991).
- Weng Cho Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).
- T. L. Wu and H. W. Chang, "Analysis of TE to x and TM to x modes for dielectric slab waveguides," in Proceedings of Optics and Photonics Taiwan 2001 (National Sun Yat-sen University, 2001), pp. 146-148.
- T. L. Wu and H. W. Chang, "Guiding mode expansion of a TE and TM transverse-mode integral equation for dielectric slab waveguides with an abrupt termination," J. Opt. Soc. Am. A 18, 2823-2832 (2001). [CrossRef]
- A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992). [CrossRef]
- R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000).
- T. E. Rozzi, "Rigorous analysis of the step discontinuity in a planar dielectric waveguide," IEEE Trans. Microwave Theory Tech. 26, 738-746 (1978). [CrossRef]

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