Abstract

A simple numerical method based on generalized telegraphist’s equations as a full vector–wave analysis tool for dielectric waveguide problems is presented. The method is applied to various guiding structures for single-mode and multimode computation. The generalized telegraphist’s equation formulates the problem as a matrix eigenvalue equation whose solution spectrum of eigenvalues directly gives the modal propagation constants. Accuracies of better than 0.08% are possible for calculating the propagation constants.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. A. J. Marcatilli, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2132 (1969).
  2. W. V. McLevige, T. Itoh, R. Mittra, “New waveguide structures for millimeter-wave and optical integrated circuits,” IEEE Trans. Microwave Theory Tech. MTT-29, 788–794 (1975).
    [CrossRef]
  3. M. D. Feit, J. A. Fleck, “Calculation of dispersion in graded-index multimode fibers by propagating-beam method,” Appl. Opt. 18, 2843–2851 (1979).
    [CrossRef] [PubMed]
  4. C. Yeh, W. P. Brown, R. Szejin, “Multimode inhomogeneous fiber couplers,” Appl. Opt. 18, 489–485 (1979).
    [CrossRef] [PubMed]
  5. J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. 48, 2133–2160 (1969).
  6. R. Pregla, “A method for the analysis of coupled rectangular dielectric waveguides,” Arch. Elektron. Übertragung 28, 349–357 (1974).
  7. E. Schweig, W. B. Bridges, “Computer analysis of dielectric waveguides: a finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531–541 (1984).
    [CrossRef]
  8. K. Bierwirth, N. Schulz, F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-34, 1104–1114 (1984).
  9. D. G. Corr, J. B. Davies, “Computer analysis of the fundamental and higher order modes in single and coupled microstrip,” IEEE Trans. Microwave Theory Tech. MT-20, 669–678 (1972).
    [CrossRef]
  10. S. Ahmed, P. Daly, “Finite-element methods for inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 116, 1661–1664 (1969).
    [CrossRef]
  11. C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
    [CrossRef]
  12. C. Yeh, K. Ha, S. B. Dong, W. P. Brown, “Single mode optical waveguides,” Appl. Opt. 18, 1490–1504 (1979).
    [CrossRef] [PubMed]
  13. L. Manía, T. Corzani, E. Valentinuzzi, “The finite element method in the analysis of optical waveguides,” in Proceedings of NATO Advance Study Institute on Integrated Optics: Physics and Applications (NATO, Brussels, Belgium, 1981), pp. 335–359.
  14. N. Mabaya, P. E. Lagasse, P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
    [CrossRef]
  15. K. Hayata, M. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic field component,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
    [CrossRef]
  16. S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).
  17. S. A. Schelkunoff, “Generalized telegraphist’s equations for waveguides,” Bell Syst. Tech. J. 31, 784–801 (1952).
  18. K. Ogusu, “Numerical analysis of the rectangular dielectric waveguide and its modifications,” IEEE Trans. Microwave Theory Tech. MTT-25, 874–885 (1977).
    [CrossRef]
  19. H. Shinonaga, S. Kurazono, “Y dielectric waveguide for millimeter- and submillimeter-wave,” IEEE Trans. Microwave Theory Tech. MTT-29, 542–546 (1981).
    [CrossRef]
  20. S. M. Saad, “Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 894–899 (1985).
    [CrossRef]
  21. F. L. Ng, “Tabulation of methods for the numerical solution of hollow waveguide problem,” IEEE Trans. Microwave Theory Tech. MTT-22, 322–329 (1974).
  22. J. B. Davies, “Review of methods for numerical solution of hollow-waveguide problem,” Proc. Inst. Electr. Eng. 119, 33–37 (1972).
    [CrossRef]
  23. C. Hafner, R. Ballisti, “Electromagnetic waves on cylindrical structures calculated by the method of moments and by the point matching technique,” in IEEE International Convention Digest on the AP-S Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 331–333.
  24. N. Marcuvitz, Waveguide Handbook (McGraw-Hill, New York, 1986), Chap. 2.
    [CrossRef]
  25. L.-C. So, “Numerical analysis of optical dielectric waveguides and modulators, Ph.D. dissertation (Cornell University, Ithaca, N.Y., 1991), Chap. 3.

1986 (1)

K. Hayata, M. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic field component,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[CrossRef]

1985 (1)

S. M. Saad, “Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 894–899 (1985).
[CrossRef]

1984 (2)

E. Schweig, W. B. Bridges, “Computer analysis of dielectric waveguides: a finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531–541 (1984).
[CrossRef]

K. Bierwirth, N. Schulz, F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-34, 1104–1114 (1984).

1981 (2)

H. Shinonaga, S. Kurazono, “Y dielectric waveguide for millimeter- and submillimeter-wave,” IEEE Trans. Microwave Theory Tech. MTT-29, 542–546 (1981).
[CrossRef]

N. Mabaya, P. E. Lagasse, P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

1979 (3)

1977 (1)

K. Ogusu, “Numerical analysis of the rectangular dielectric waveguide and its modifications,” IEEE Trans. Microwave Theory Tech. MTT-25, 874–885 (1977).
[CrossRef]

1975 (2)

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

W. V. McLevige, T. Itoh, R. Mittra, “New waveguide structures for millimeter-wave and optical integrated circuits,” IEEE Trans. Microwave Theory Tech. MTT-29, 788–794 (1975).
[CrossRef]

1974 (2)

R. Pregla, “A method for the analysis of coupled rectangular dielectric waveguides,” Arch. Elektron. Übertragung 28, 349–357 (1974).

F. L. Ng, “Tabulation of methods for the numerical solution of hollow waveguide problem,” IEEE Trans. Microwave Theory Tech. MTT-22, 322–329 (1974).

1972 (2)

J. B. Davies, “Review of methods for numerical solution of hollow-waveguide problem,” Proc. Inst. Electr. Eng. 119, 33–37 (1972).
[CrossRef]

D. G. Corr, J. B. Davies, “Computer analysis of the fundamental and higher order modes in single and coupled microstrip,” IEEE Trans. Microwave Theory Tech. MT-20, 669–678 (1972).
[CrossRef]

1969 (3)

S. Ahmed, P. Daly, “Finite-element methods for inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 116, 1661–1664 (1969).
[CrossRef]

E. A. J. Marcatilli, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2132 (1969).

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

1955 (1)

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).

1952 (1)

S. A. Schelkunoff, “Generalized telegraphist’s equations for waveguides,” Bell Syst. Tech. J. 31, 784–801 (1952).

Ahmed, S.

S. Ahmed, P. Daly, “Finite-element methods for inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 116, 1661–1664 (1969).
[CrossRef]

Arndt, F.

K. Bierwirth, N. Schulz, F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-34, 1104–1114 (1984).

Ballisti, R.

C. Hafner, R. Ballisti, “Electromagnetic waves on cylindrical structures calculated by the method of moments and by the point matching technique,” in IEEE International Convention Digest on the AP-S Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 331–333.

Bierwirth, K.

K. Bierwirth, N. Schulz, F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-34, 1104–1114 (1984).

Bridges, W. B.

E. Schweig, W. B. Bridges, “Computer analysis of dielectric waveguides: a finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531–541 (1984).
[CrossRef]

Brown, W. P.

Corr, D. G.

D. G. Corr, J. B. Davies, “Computer analysis of the fundamental and higher order modes in single and coupled microstrip,” IEEE Trans. Microwave Theory Tech. MT-20, 669–678 (1972).
[CrossRef]

Corzani, T.

L. Manía, T. Corzani, E. Valentinuzzi, “The finite element method in the analysis of optical waveguides,” in Proceedings of NATO Advance Study Institute on Integrated Optics: Physics and Applications (NATO, Brussels, Belgium, 1981), pp. 335–359.

Daly, P.

S. Ahmed, P. Daly, “Finite-element methods for inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 116, 1661–1664 (1969).
[CrossRef]

Davies, J. B.

D. G. Corr, J. B. Davies, “Computer analysis of the fundamental and higher order modes in single and coupled microstrip,” IEEE Trans. Microwave Theory Tech. MT-20, 669–678 (1972).
[CrossRef]

J. B. Davies, “Review of methods for numerical solution of hollow-waveguide problem,” Proc. Inst. Electr. Eng. 119, 33–37 (1972).
[CrossRef]

Dong, S. B.

C. Yeh, K. Ha, S. B. Dong, W. P. Brown, “Single mode optical waveguides,” Appl. Opt. 18, 1490–1504 (1979).
[CrossRef] [PubMed]

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

Eguchi, M.

K. Hayata, M. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic field component,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Goell, J. E.

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

Ha, K.

Hafner, C.

C. Hafner, R. Ballisti, “Electromagnetic waves on cylindrical structures calculated by the method of moments and by the point matching technique,” in IEEE International Convention Digest on the AP-S Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 331–333.

Hayata, K.

K. Hayata, M. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic field component,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[CrossRef]

Itoh, T.

W. V. McLevige, T. Itoh, R. Mittra, “New waveguide structures for millimeter-wave and optical integrated circuits,” IEEE Trans. Microwave Theory Tech. MTT-29, 788–794 (1975).
[CrossRef]

Koshiba, M.

K. Hayata, M. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic field component,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[CrossRef]

Kurazono, S.

H. Shinonaga, S. Kurazono, “Y dielectric waveguide for millimeter- and submillimeter-wave,” IEEE Trans. Microwave Theory Tech. MTT-29, 542–546 (1981).
[CrossRef]

Lagasse, P. E.

N. Mabaya, P. E. Lagasse, P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

Mabaya, N.

N. Mabaya, P. E. Lagasse, P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

Manía, L.

L. Manía, T. Corzani, E. Valentinuzzi, “The finite element method in the analysis of optical waveguides,” in Proceedings of NATO Advance Study Institute on Integrated Optics: Physics and Applications (NATO, Brussels, Belgium, 1981), pp. 335–359.

Marcatilli, E. A. J.

E. A. J. Marcatilli, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2132 (1969).

Marcuvitz, N.

N. Marcuvitz, Waveguide Handbook (McGraw-Hill, New York, 1986), Chap. 2.
[CrossRef]

McLevige, W. V.

W. V. McLevige, T. Itoh, R. Mittra, “New waveguide structures for millimeter-wave and optical integrated circuits,” IEEE Trans. Microwave Theory Tech. MTT-29, 788–794 (1975).
[CrossRef]

Mittra, R.

W. V. McLevige, T. Itoh, R. Mittra, “New waveguide structures for millimeter-wave and optical integrated circuits,” IEEE Trans. Microwave Theory Tech. MTT-29, 788–794 (1975).
[CrossRef]

Ng, F. L.

F. L. Ng, “Tabulation of methods for the numerical solution of hollow waveguide problem,” IEEE Trans. Microwave Theory Tech. MTT-22, 322–329 (1974).

Ogusu, K.

K. Ogusu, “Numerical analysis of the rectangular dielectric waveguide and its modifications,” IEEE Trans. Microwave Theory Tech. MTT-25, 874–885 (1977).
[CrossRef]

Oliver, W.

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

Pregla, R.

R. Pregla, “A method for the analysis of coupled rectangular dielectric waveguides,” Arch. Elektron. Übertragung 28, 349–357 (1974).

Saad, S. M.

S. M. Saad, “Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 894–899 (1985).
[CrossRef]

Schelkunoff, S. A.

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).

S. A. Schelkunoff, “Generalized telegraphist’s equations for waveguides,” Bell Syst. Tech. J. 31, 784–801 (1952).

Schulz, N.

K. Bierwirth, N. Schulz, F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-34, 1104–1114 (1984).

Schweig, E.

E. Schweig, W. B. Bridges, “Computer analysis of dielectric waveguides: a finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531–541 (1984).
[CrossRef]

Shinonaga, H.

H. Shinonaga, S. Kurazono, “Y dielectric waveguide for millimeter- and submillimeter-wave,” IEEE Trans. Microwave Theory Tech. MTT-29, 542–546 (1981).
[CrossRef]

So, L.-C.

L.-C. So, “Numerical analysis of optical dielectric waveguides and modulators, Ph.D. dissertation (Cornell University, Ithaca, N.Y., 1991), Chap. 3.

Suzuki, M.

K. Hayata, M. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic field component,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[CrossRef]

Szejin, R.

Valentinuzzi, E.

L. Manía, T. Corzani, E. Valentinuzzi, “The finite element method in the analysis of optical waveguides,” in Proceedings of NATO Advance Study Institute on Integrated Optics: Physics and Applications (NATO, Brussels, Belgium, 1981), pp. 335–359.

Vandenbulcke, P.

N. Mabaya, P. E. Lagasse, P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

Yeh, C.

Appl. Opt. (3)

Arch. Elektron. Übertragung (1)

R. Pregla, “A method for the analysis of coupled rectangular dielectric waveguides,” Arch. Elektron. Übertragung 28, 349–357 (1974).

Bell Syst. Tech. J. (4)

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

E. A. J. Marcatilli, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2132 (1969).

S. A. Schelkunoff, “Conversion of Maxwell’s equations into generalized telegraphist’s equations,” Bell Syst. Tech. J. 34, 995–1043 (1955).

S. A. Schelkunoff, “Generalized telegraphist’s equations for waveguides,” Bell Syst. Tech. J. 31, 784–801 (1952).

IEEE Trans. Microwave Theory Tech. (10)

K. Ogusu, “Numerical analysis of the rectangular dielectric waveguide and its modifications,” IEEE Trans. Microwave Theory Tech. MTT-25, 874–885 (1977).
[CrossRef]

H. Shinonaga, S. Kurazono, “Y dielectric waveguide for millimeter- and submillimeter-wave,” IEEE Trans. Microwave Theory Tech. MTT-29, 542–546 (1981).
[CrossRef]

S. M. Saad, “Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 894–899 (1985).
[CrossRef]

F. L. Ng, “Tabulation of methods for the numerical solution of hollow waveguide problem,” IEEE Trans. Microwave Theory Tech. MTT-22, 322–329 (1974).

W. V. McLevige, T. Itoh, R. Mittra, “New waveguide structures for millimeter-wave and optical integrated circuits,” IEEE Trans. Microwave Theory Tech. MTT-29, 788–794 (1975).
[CrossRef]

E. Schweig, W. B. Bridges, “Computer analysis of dielectric waveguides: a finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531–541 (1984).
[CrossRef]

K. Bierwirth, N. Schulz, F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-34, 1104–1114 (1984).

D. G. Corr, J. B. Davies, “Computer analysis of the fundamental and higher order modes in single and coupled microstrip,” IEEE Trans. Microwave Theory Tech. MT-20, 669–678 (1972).
[CrossRef]

N. Mabaya, P. E. Lagasse, P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

K. Hayata, M. Koshiba, M. Eguchi, M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic field component,” IEEE Trans. Microwave Theory Tech. MTT-34, 1120–1124 (1986).
[CrossRef]

J. Appl. Phys. (1)

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

Proc. Inst. Electr. Eng. (2)

S. Ahmed, P. Daly, “Finite-element methods for inhomogeneous waveguides,” Proc. Inst. Electr. Eng. 116, 1661–1664 (1969).
[CrossRef]

J. B. Davies, “Review of methods for numerical solution of hollow-waveguide problem,” Proc. Inst. Electr. Eng. 119, 33–37 (1972).
[CrossRef]

Other (4)

C. Hafner, R. Ballisti, “Electromagnetic waves on cylindrical structures calculated by the method of moments and by the point matching technique,” in IEEE International Convention Digest on the AP-S Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 331–333.

N. Marcuvitz, Waveguide Handbook (McGraw-Hill, New York, 1986), Chap. 2.
[CrossRef]

L.-C. So, “Numerical analysis of optical dielectric waveguides and modulators, Ph.D. dissertation (Cornell University, Ithaca, N.Y., 1991), Chap. 3.

L. Manía, T. Corzani, E. Valentinuzzi, “The finite element method in the analysis of optical waveguides,” in Proceedings of NATO Advance Study Institute on Integrated Optics: Physics and Applications (NATO, Brussels, Belgium, 1981), pp. 335–359.

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 (7)

Fig. 1
Fig. 1

Schematic diagram of an arbitrarily shaped inhomogeneous waveguide of permittivity variation ∊(x, y) enclosed by a hypothetical metal box.

Fig. 2
Fig. 2

Electric field vector diagrams for four symmetry families categorized by the Ex symmetry condition and their relations to fundamental modes in step-index circular and rectangular waveguides.

Fig. 3
Fig. 3

Normalized propagation constants versus increasing width x0 of the square metallic enclosure for different basis orders. Step-index circular waveguide profile: nco = 2.0, ncl = 1.0, r0 = 1.0 μm. (a) λ = 1.25664 μm, (b) λ = 2.5 μm.

Fig. 4
Fig. 4

ω-β diagram of a step-index circular waveguide of an index ratio: (a) nco/ncl = 1.01, (b) nco/ncl = 2.0. —, exact, x x x, GETE with 16 TE–21 TM.

Fig. 5
Fig. 5

Modal power distributions of a step-index circular waveguide profile: nco = 2.0, ncl = 1.0, r0 = 1.0 μm, λ = 1.25644 μm: —, exact, - · - · - · -, GETE with 16 TE–21 TM for (a) HE11, (b) TE01, (c) TM01 modes. G

Fig. 6
Fig. 6

Dispersion relations of step-index rectangular waveguide profiles: (a) n1/n2 = 1.05, a = b; (b) n1/n2 = 1.5, a = b; (c) n1/n2 = 1.05, a = 2b; (d) n1/n2 = 1.5, a = 2b. - · - · - · -, Goell’s,5 x x x, GETE with 49 TE–49 TM.

Fig. 7
Fig. 7

Dispersion relations of dielectric coupler: - · - · - · -, Marcatilli’s approximation, x x x, GETE with 25 TE–30 TM.

Tables (2)

Tables Icon

Table 1 Modal Indices m and n for the Four Symmetry Families

Tables Icon

Table 2 Comparison of Normalized Propagation Constants βn with the Exact Solution for Fundamental Modes in a Step-Index Fiber with Increasing Basis Functionsa

Equations (10)

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

E t z = 1 j ω t [ 1 t · ( H × a ^ z ) ] - j ω μ ( H t × a ^ z ) , H t z = 1 j ω μ t t · ( a ^ z × E t ) - j ω ( a ^ z × E t ) ,
E t ( x , y ) i = 1 N V ( i ) ( z ) e ( i ) ( x , y ) + j = 1 M V [ j ] ( z ) e [ j ] ( x , y ) , H t ( x , y ) i = 1 N I ( i ) ( z ) h ( i ) ( x , y ) + j = 1 M I [ j ] ( z ) h [ j ] ( x , y ) ,
Ω R Ω E · e ( i ) d Ω = 0 , Ω R Ω H · e ( i ) d Ω = 0 , Ω R Ω E · e ( j ) d Ω = 0 , Ω R Ω H · e ( j ) d Ω = 0 ,
R Ω E ( x , y ) = E t z - 1 j ω t [ 1 t · ( H t × a ^ z ) ] + j ω μ ( H t × a ^ z ) , R Ω H ( x , y ) = H t z - 1 j ω μ t t · ( a ^ z × E t ) + j ω μ ( a ^ z × E t ) ;
d V ( n ) ( z ) d z = - 1 j ω 0 i = 1 M Z 1 ( n , i ) I ( i ) ( z ) - j ω μ I ( n ) ( z ) , d V [ m ] ( z ) d z = - j ω μ I [ m ] ( z ) , d I ( n ) ( z ) d z = - j ω 0 [ i = 1 M Y 1 ( n , i ) V ( i ) ( z ) + j = 1 N Y 2 ( n , i ) V [ j ] ( z ) ] , d I [ m ] ( z ) d z = - j ω 0 [ i = 1 M Y 3 ( m , i ) V ( i ) ( z ) + j = 1 N Y 4 ( m , j ) V [ j ] ( z ) ] - k c [ m ] 2 j ω μ V [ m ] ( z ) ,
Z 1 ( n , i ) = k c ( i ) 2 k c ( n ) 2 1 r ( x , y ) ϕ ( i ) ( x , y ) ϕ ( n ) ( x , y ) d Ω , Y 1 ( n , i ) = r ( x , y ) h ( i ) ( x , y ) · h ( n ) ( x , y ) d Ω , Y 2 ( n , j ) = r ( x , y ) h [ j ] ( x , y ) · h ( n ) ( x , y ) d Ω , Y 3 ( m , i ) = r ( x , y ) h ( i ) ( x , y ) · h [ m ] ( x , y ) d Ω , Y 4 ( m , j ) = r ( x , y ) h [ j ] ( x , y ) · h [ m ] ( x , y ) d Ω ,
A i j ( V ( i ) V [ j ] ) = β 2 ( V ( i ) V [ j ] ) ,
A ˜ i j ( I ( i ) I [ j ] ) = β 2 ( I ( i ) I [ j ] ) ,
[ ρ λ / n co ( 1 - n cl 2 n co 2 ) 1 / 2 ] ,
F ρ / [ ρ λ / n co ( 1 - n cl 2 n co 2 ) 1 / 2 ] ,

Metrics