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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).
  2. 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]
  3. 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]
  4. 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]
  5. J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).
  6. 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]
  7. 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]
  8. 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).
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).
    [CrossRef]
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
  20. A. Ishimaru, Electromagnetic Propagation, Radiation, and Scattering (Prentice Hall, 1991).
  21. Weng Cho Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).
  22. 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.
  23. 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]
  24. A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
    [CrossRef]
  25. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000).
  26. T. E. Rozzi, "Rigorous analysis of the step discontinuity in a planar dielectric waveguide," IEEE Trans. Microwave Theory Tech. 26, 738-746 (1978).
    [CrossRef]

2006

2001

2000

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]

1996

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]

1995

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]

1994

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]

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]

1993

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]

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]

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]

1992

A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
[CrossRef]

1991

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]

1990

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]

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]

1981

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]

1980

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]

1978

T. E. Rozzi, "Rigorous analysis of the step discontinuity in a planar dielectric waveguide," IEEE Trans. Microwave Theory Tech. 26, 738-746 (1978).
[CrossRef]

1969

E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).

Alexopoulos, N. G.

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]

Athanasoulias, G.

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]

Baken, N. H. G.

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).

Bjarklev, A.

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]

Blok, H.

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).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000).

Cai, Y.

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]

Castaneda, J. A.

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]

Chang, H. W.

Cheng, Y. H.

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]

Cho Chew, Weng

Weng Cho Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

Goell, J. E.

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).

Gopinath, A.

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]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

Helfert, S.

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]

Hou, Y. L.

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]

Ishimaru, A.

A. Ishimaru, Electromagnetic Propagation, Radiation, and Scattering (Prentice Hall, 1991).

Jannejad, V.

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]

Kalinin, V. A.

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]

Katehi, L. P. B.

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]

Kolk, E. W.

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).

Lee, J. S.

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]

Lin, W. G.

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]

Lumholt, O.

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]

Marcatili, E. A.

E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).

Mittra, R.

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]

Mizumoto, T.

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]

Naito, Y.

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]

Nauwelaers, B. K. J. C.

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]

Oliner, A. A.

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]

Pedersen, B.

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]

Peng, S. T.

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]

Povlsen, J. H.

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]

Pregla, R.

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]

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).
[CrossRef]

Rasmussen, T.

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]

Rogge, U.

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).
[CrossRef]

Rottwitt, K.

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]

Rozzi, T. E.

T. E. Rozzi, "Rigorous analysis of the step discontinuity in a planar dielectric waveguide," IEEE Trans. Microwave Theory Tech. 26, 738-746 (1978).
[CrossRef]

Sabetfakhri, K.

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]

Scarmozzino, R.

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]

Shin, S. Y.

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]

Sudbo, A. S.

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]

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, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
[CrossRef]

Uzunoglu, N. K.

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]

Wu, T. L.

Yang, H. Y.

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]

Bell Syst. Tech. J.

E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).

IEE Proc., Part H: Microwaves, Antennas Propag.

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]

IEE Proc.: Optoelectron.

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]

IEEE J. Sel. Top. Quantum Electron.

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]

IEEE Trans. Microwave Theory Tech.

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]

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]

T. E. Rozzi, "Rigorous analysis of the step discontinuity in a planar dielectric waveguide," IEEE Trans. Microwave Theory Tech. 26, 738-746 (1978).
[CrossRef]

J. Lightwave Technol.

A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
[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).

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]

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]

J. Opt. Soc. Am. A

Pure Appl. Opt.

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]

Other

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.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

General RWG divided into ( M + 1 ) regions (slices) made of layered dielectric waveguide.

Fig. 2
Fig. 2

Definitions of P , Q , R , S generalized impedance matrices.

Fig. 3
Fig. 3

Fundamental mode field components of TE-to-y and TM-to-y modes for a slab waveguide. The x coordinate is in micrometers, while the y coordinate represents relative field intensity. Due to symmetry, only half the figures (positive x) are shown. λ = 1.5 μm , n 1 = 1.5 , n 0 = 1.0 , d = 1 .

Tables (1)

Tables Icon

Table 1 Notation and Symbols

Equations (76)

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

ϕ B ( y ) + k 0 2 ϵ r ( y ) ϕ B ( y ) = λ B ϕ B ( y ) ,
[ ϕ D ( y ) ϵ r ( y ) ] + k 0 2 ϕ D ( y ) = λ D ϕ D ( y ) ϵ r ( y ) ,
D y ( 1 ) ( x , y , z ) = n c D , n ( 1 ) ϕ D , n ( 1 ) ( y ) ψ D , n ( 1 , r ) ( x ) exp ( j β z ) ,
D y ( m ) ( x , y , z ) = n ϕ D , n ( m ) ( y ) [ c D , n ( m , l ) ψ D , n ( m , l ) ( x ) + c D , n ( m , r ) ψ D , n ( m , r ) ( x ) ] exp ( j β z ) ,
D y ( M + 1 ) ( x , y , z ) = n c D , n ( M + 1 ) ϕ D , n ( M + 1 ) ( y ) ψ D , n ( M + 1 , l ) ( x ) exp ( j β z ) ,
B y ( 1 ) ( x , y , z ) = n c B , n ( 1 ) ϕ B , n ( 1 ) ( y ) ψ B , n ( 1 , r ) ( x ) exp ( j β z ) ,
B y ( m ) ( x , y , z ) = n ϕ B , n ( m ) ( y ) [ c B , n ( m , l ) ψ B , n ( m , l ) ( x ) + c B , n ( m , r ) ψ B , n ( m , r ) ( x ) ] exp ( j β z ) ,
B y M + 1 ( x , y , z ) = n c B , n ( M + 1 ) ϕ B , n ( M + 1 ) ( y ) ψ B , n ( M + 1 , l ) ( x ) exp ( j β z ) .
ψ F , n ( 1 , r ) ( x ) = exp ( k x F , n ( 1 ) x ) exp ( k x F , n ( 1 ) x 1 ) ,
ψ F , n ( 1 , r ) ( x ) = sin [ k x F , n ( 1 ) ( x x 0 ) ] sin ( k x F , n ( 1 ) Δ x 1 ) .
ψ F , n ( 1 , r ) ( x ) = cos [ k x F , n ( 1 ) ( x x 0 ) ] cos ( k x F , n ( 1 ) Δ x 1 ) .
ψ F , n ( m , l ) ( x ) = sin [ k x F , n ( m ) ( x m x ) ] sin ( k x F , n ( m ) Δ x m ) ,
ψ F , n ( m , r ) ( x ) = sin [ k x F , n ( m ) ( x x m 1 ) ] sin ( k x F , n ( m ) Δ x m ) ,
c D , n ( m , l ) = E m 1 ( y ) [ ϕ D , n ( m ) ( y ) ] * d y ϕ D , n ( m ) ( y ) 1 ϵ r ( m ) ( y ) [ ϕ D , n ( m ) ( y ) ] * d y ,
c B , n ( m , l ) = H m 1 ( y ) [ ϕ B , n ( m ) ( y ) ] * d y ϕ B , n ( m ) ( y ) [ ϕ B , n ( m ) ( y ) ] * d y ,
c D , n ( m , r ) = E m ( y ) [ ϕ D , n ( m ) ( y ) ] * d y ϕ D , n ( m ) ( y ) 1 ϵ r ( m ) ( y ) [ ϕ D , n ( m ) ( y ) ] * d y ,
c B , n ( m , r ) = H m ( y ) [ ϕ B , n ( m ) ( y ) ] * d y ϕ B , n ( m ) ( y ) [ ϕ B , n ( m ) ( y ) ] * d y .
D y ( m ) ( x , y ) = n ϕ D , n ( m ) ( y ) { E ( m 1 ) ( y ) [ ϕ D , n ( m ) ( y ) ] * d y ψ D , n ( m , l ) ( x ) + E m ( y ) [ ϕ D , n ( m ) ( y ) ] * d y ψ D , n ( m , r ) ( x ) } ,
B y ( m ) ( x , y ) = n ϕ B , n ( m ) ( y ) { H m 1 ( y ) [ ϕ B , n ( m ) ( y ) ] * d y ψ D , n ( m , l ) ( x ) + H m ( y ) [ ϕ B , n ( m ) ( y ) ] * d y ψ D , n ( m , r ) } .
Φ D ( y ) = 1 ϵ r ( y ) d [ ϕ D ( y ) ] d y , Φ B ( y ) = 1 μ r d [ ϕ B ( y ) ] d y .
Ψ F , n ( m , l ) ( x ) = d d x [ ψ F , n ( m , l ) ( x ) ] = k x F , n ( m ) cos [ k x F , n ( m ) ( x m x ) ] sin [ k x F , n ( m ) Δ x m ] ,
Ψ F , n ( m , r ) ( x ) = d d x [ ψ F , n ( m , r ) ( x ) ] = k x F , n ( m ) cos [ k x F , n ( m ) ( x x m 1 ) ] sin [ k x F , n ( m ) Δ x m ] .
E z ( 1 ) ( x , y ) = j β [ k c e , n ( 1 ) ] 2 ϵ ( 1 ) ( y ) D y ( 1 ) y j ω [ k c h , n ( 1 ) ] 2 B y ( 1 ) x = G e e ( 1 , r ) ( x , y ; y ) E 1 ( y ) d y + G e , h ( 1 , r ) ( x , y ; y ) H 1 ( y ) d y ,
G e e ( 1 , r ) ( x , y ; y ) = n j β [ k c e , n ( 1 ) ] 2 ψ D , n ( 1 , r ) ( x ) Φ D , n ( 1 ) ( y ) [ ϕ D , n ( 1 ) ( y ) ] * ,
G e h ( 1 , r ) ( x , y ; y ) = n j ω [ k c h , n ( 1 ) ] 2 Ψ B , n ( 1 , r ) ( x ) ϕ B , n ( 1 ) ( y ) [ ϕ B , n ( 1 ) ( y ) ] * .
H z ( 1 ) ( x , y ) = j ω [ k c e , n ( 1 ) ] 2 D y ( 1 ) x + j β [ k c h , n ( 1 ) ] 2 μ ( 1 ) ( y ) B y ( 1 ) y = G h e ( 1 , r ) ( x , y ; y ) E 1 ( y ) d y + G h h ( 1 , r ) ( x , y ; y ) H 1 ( y ) d y ,
G h e ( 1 , r ) ( x , y ; y ) = n j ω [ k c e , n ( 1 ) ] 2 Ψ D , n ( 1 , r ) ( x ) ϕ D , n ( 1 ) ( y ) [ ϕ D , n ( 1 ) ( y ) ] * ,
G h h ( 1 , r ) ( x , y ; y ) = n j β [ k c h , n ( 1 ) ] 2 ψ B , n ( 1 , r ) ( x ) Φ B , n ( 1 ) ( y ) [ ϕ B , n ( 1 ) ( y ) ] * .
[ H z ( 1 ) ( x , y ) E z ( 1 ) ( x , y ) ] = d y [ G h e ( 1 , r ) ( x , y ; y ) G h h ( 1 , r ) ( x , y ; y ) G e e ( 1 , r ) ( x , y ; y ) G e h ( 1 , r ) ( x , y ; y ) ] [ E 1 ( y ) H 1 ( y ) ] .
[ H z ( 1 ) ( x 1 δ , y ) E z ( 1 ) ( x 1 δ , y ) ] = d y [ R h e ( 1 ) ( y , y ) R h h ( 1 ) ( y , y ) R e e ( 1 ) ( y , y ) R e h ( 1 ) ( y , y ) ] [ E 1 ( y ) H 1 ( y ) ] ,
R h e ( 1 ) ( y , y ) = h j ω [ k c e , n ( 1 ) ] 2 Ψ D , n ( 1 , r ) ( x 1 ) ϕ D , n ( 1 ) ( y ) [ ϕ D , n ( 1 ) ( y ) ] * ,
R h h ( 1 ) ( y , y ) = n j β [ k c h , n ( 1 ) ] 2 ψ B , n ( 1 , r ) ( x 1 ) Φ B , n ( 1 ) ( y ) [ ϕ B , n ( 1 ) ( y ) ] * ,
R e e ( 1 ) ( y , y ) = n j β [ k c e , n ( 1 ) ] 2 ψ D , n ( 1 , r ) ( x 1 ) Φ D , n ( 1 ) ( y ) [ ϕ D , n ( 1 ) ( y ) ] * ,
R e h ( 1 ) ( y , y ) = n j ω [ k c h , n ( 1 ) ] 2 Ψ B , n ( 1 , r ) ( x 1 ) ϕ B , n ( 1 ) ( y ) [ ϕ B , n ( 1 ) ( y ) ] * .
E z ( m ) ( x , y ) = G e e ( m , l ) ( x , y ; y ) E m 1 ( y ) d y + G e h ( m , l ) ( x , y ; y ) H m 1 ( y ) d y + G e e ( m , r ) ( x , y ; y ) E m ( y ) d y + G e h ( m , r ) ( x , y ; y ) H m ( y ) d y ,
G e e ( m , l ) ( x , y ; y ) = n j β [ k c e , n ( m ) ] 2 ψ D , n ( m , l ) ( x ) Φ D , n ( m ) ( y ) [ ϕ D , n ( m ) ( y ) ] * ,
G e h ( m , l ) ( x , y ; y ) = n j ω [ k c h , n ( m ) ] 2 Ψ B , n ( m , l ) ( x ) ϕ B , n ( m ) ( y ) [ ϕ B , n ( m ) ( y ) ] * ,
G e e ( m , r ) ( x , y ; y ) = n j β [ k c e , n ( m ) ] 2 ψ D , n ( m , r ) ( x ) Φ D , n ( m ) ( y ) [ ϕ D , n ( m ) ( y ) ] * ,
G e h ( m , r ) ( x , y ; y ) = n j ω [ k c h , n ( m ) ] 2 Ψ B , n ( m , r ) ( x ) ϕ B , n ( m ) ( y ) [ ϕ B , n ( m ) ( y ) ] * .
H z ( m ) ( x , y ) = G h e ( m , l ) ( x , y ; y ) E m 1 ( y ) d y + G h h ( m , l ) ( x , y ; y ) H m 1 ( y ) d y + G h e ( m , r ) ( x , y ; y ) E m ( y ) d y + G h h ( m , r ) ( x , y ; y ) H m ( y ) d y ,
G h e ( m , l ) ( x , y ; y ) = n j ω [ k c e , n ( m ) ] 2 Ψ D , n ( m , l ) ( x ) ϕ D , n ( m ) ( y ) [ ϕ D , n ( m ) ( y ) ] * ,
G h h ( m , l ) ( x , y ; y ) = n j β [ k c h , n ( m ) ] 2 ψ B , n ( m , l ) ( x ) Φ B , n ( m ) ( y ) [ ϕ B , n ( m ) ( y ) ] * ,
G h e ( m , r ) ( x , y ; y ) = n j ω [ k c e , n ( m ) ] 2 Ψ D , n ( m , r ) ( x ) ϕ D , n ( m ) ( y ) [ ϕ D , n ( m ) ( y ) ] * ,
G h h ( m , r ) ( x , y ; y ) = n j β [ k c h , n ( m ) ] 2 ψ B , n ( m , r ) ( x ) Φ B , n ( m ) ( y ) [ ϕ B , n ( m ) ( y ) ] * .
[ H z ( m ) ( x m 1 + δ , y ) E z ( m ) ( x m 1 + δ , y ) ] = d y [ Q h e ( m ) ( y , y ) Q h h ( m ) ( y , y ) Q e e ( m ) ( y , y ) Q e h ( m ) ( y , y ) ] [ E m 1 ( y ) H m 1 ( y ) ] + d y [ S h e ( m ) ( y , y ) 0 0 S e h ( m ) ( y , y ) ] [ E m ( y ) H m ( y ) ] ,
[ H z ( m ) ( x m δ , y ) E z ( m ) ( x m δ , y ) ] = d y [ P h e ( m ) ( y , y ) 0 0 P e h ( m ) ( y , y ) ] [ E m 1 ( y ) H m 1 ( y ) ] + d y [ R h e ( m ) ( y , y ) R h h ( m ) ( y , y ) R e e ( m ) ( y , y ) R e h ( m ) ( y , y ) ] [ E m ( y ) H m ( y ) ] ,
P u v ( m ) = G u v ( m , l ) x = x m , Q u v ( m ) = G u v ( m , l ) x = x m 1 ,
R h e ( m ) = G h e ( m , l ) x = x m , S u v ( m ) = G u v ( m , r ) x = x m 1 ,
d y [ R h e ( 1 ) ( y , y ) R h h ( 1 ) ( y , y ) R e e ( 1 ) ( y , y ) R e h ( 1 ) ( y , y ) ] [ E 1 ( y ) H 1 ( y ) ] = d y [ Q h e ( 2 ) ( y , y ) Q h h ( 2 ) ( y , y ) Q e e ( 2 ) ( y , y ) Q e h ( 2 ) ( y , y ) ] [ E 1 ( y ) H 1 ( y ) ] .
d y [ R ( 1 ) Q ( 2 ) ] F 1 = 0 ,
F i = [ E i ( y ) H i ( y ) ] , i = 1 , 2 , , M .
A ( m ) = [ A h e ( m ) A h h ( m ) A e e ( m ) A e h ( m ) ] .
d y [ [ R ( 1 ) Q ( 2 ) ] S ( 2 ) 0 0 P ( 2 ) [ R ( 2 ) Q ( 3 ) ] S ( 3 ) 0 0 0 P ( M 1 ) [ R ( M 1 ) Q ( M ) ] S ( M ) 0 0 p ( M ) [ R ( M ) Q ( M + 1 ) ] ] [ F 1 F 2 F M 1 F M ] = 0 .
1 μ 0 × [ 1 j ω ϵ 0 ϵ r ( y ) × B ¯ ] = j ω B ¯ .
× ( ϕ A ¯ ) = ϕ × A ¯ + ϕ × A ¯
B = 0 ,
2 B ¯ + k 0 2 ϵ r ( y ) B ¯ ϵ r ( y ) 1 ϵ r ( y ) × ( × B ¯ ) = 0 .
2 B y + k 0 2 ϵ r ( y ) B y = 0 .
B y ( x , y , z ) = ϕ B ( y ) exp ( j k x B x ) exp ( j β z ) .
[ ϕ B ( y ) ] + k 0 2 ϵ r ( y ) ϕ B ( y ) = ( β 2 + k x B 2 ) ϕ B ( y ) .
μ 0 × [ 1 μ 0 × E ¯ ] = k 0 2 ϵ r ( y ) E ¯ .
( E ¯ ) 2 E ¯ = k 0 2 ϵ r ( y ) E ¯ .
2 D ¯ ϵ r ( y ) [ ϵ r ( y ) E r 2 ( y ) D y ] + k 0 2 D ¯ = 0 .
2 D y ϵ r ( y ) + [ ϵ r ( y ) ϵ y 2 ( y ) D y ] + k 0 2 D y = 0 .
D y ( x , y , z ) = ϕ D ( y ) exp ( j k x D x ) exp ( j β z ) .
[ ϕ D ( y ) ϵ r ( y ) ] + k 0 2 ϕ D ( y ) = ( β 2 + k x D 2 ) ϕ D ( y ) ϵ r ( y ) .
L B [ ϕ B ( y ) ] = ϕ B ( y ) + k 0 2 ϵ r ( y ) ϕ B ( y ) = λ B ϕ B ( y ) ,
L D [ ϕ D ( y ) ] = [ ϕ D ( y ) ϵ r ( y ) ] + k 0 2 ϕ D ( y ) = λ D ϕ D ( y ) ϵ r ( y ) ,
λ B = k c B 2 = β 2 + k x B 2 = ϵ r ( y ) k 0 2 k y B 2 ( y ) ,
λ D = k c D 2 = β 2 + k x D 2 = ϵ r ( y ) k 0 2 k y D 2 ( y ) .
ϕ B , m ( y ) ϕ B , n ( y ) d y = δ m , n ,
ϕ D , m ( y ) ϕ D , n ( y ) ϵ ( y ) d y = δ m , n ,
E x = 1 k c B 2 ϵ r ( y ) 2 D y x y + j ω k c D 2 B y z ,
E z = 1 k c B 2 ϵ r ( y ) 2 D y z y j ω k c D 2 B y x ,
H x = j ω k c B 2 D y z + 1 k c D 2 μ r 2 B y x y ,
H z = j ω k c B 2 D y x + 1 k c D 2 μ r 2 B y z y .

Metrics