Abstract

A method for calculating the propagation constants of allowed guided and leaky modes in multilayer planar waveguides is presented. We develop a two-way graph model to describe the tangential fields propagating in the waveguides. According to the special structure of the graph model, it is convenient to employ a topology scheme to derive analytical and closed-form dispersion equations for TE and TM modes. By comparing the dispersion equations formulated by series-expansion methods, approximation methods, and transfer-matrix methods, we find that the use of these equations for finding the eigenmodes has some benefits. First, this method can be easily employed to solve eigenmodes accurately in numerical computation without using series truncation. Second, the dispersion equations are exact. Moreover, all the eigenmodes can be determined according to the formulas without losing roots or causing numerical instability even for a waveguide with thick layers.

© 2007 Optical Society of America

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  1. S. M. Saad, "Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides," IEEE Trans. Microwave Theory Tech. 33, 894-899 (1985).
    [CrossRef]
  2. E. Anemogiannis and E. N. Glytsis, "Multilayer waveguides: efficient numerical analysis of general structures," J. Lightwave Technol. 10, 1344-1351 (1992).
    [CrossRef]
  3. J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).
  4. K. Mehrany and B. Rashidian, "Polynomial expansion for extraction of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
    [CrossRef]
  5. J. Ctyroky, S. Helfert, and R. Pregla, "Analysis of a deep waveguide Bragg grating," Opt. Quantum Electron. 30, 343-358 (1998).
    [CrossRef]
  6. J. Chiwell, "Thin films field transfer matrix theory of planar multiplayer waveguides and reflection from prism-loaded waveguides," J. Opt. Soc. Am. A 1, 742-753 (1984).
    [CrossRef]
  7. L. M. Walpita, "Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix," J. Opt. Soc. Am. A 2, 595-602 (1985).
    [CrossRef]
  8. A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, "Numerical analysis of planar optical waveguides using matrix approach," J. Lightwave Technol. LT-5, 660-667 (1987).
    [CrossRef]
  9. C. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Express 7, 260-272 (2000).
    [CrossRef] [PubMed]
  10. S. B. Gaal, H. J. W. M. Hoekstra, and P. V. Lambeck, "Determining PML modes in 2-D stratified media," J. Lightwave Technol. 21, 293-298 (2003).
    [CrossRef]
  11. K. H. Schlereth, "The complex propagation constant of multilayer waveguides: an algorithm for a personal computer," IEEE J. Quantum Electron. 26, 627-630 (1990).
    [CrossRef]
  12. L. Sun and M. E. Marhic, "Numerical study of attenuation in multilayer infrared waveguides by the circle-chain convergence method," J. Opt. Soc. Am. B 8, 478-483 (1991).
    [CrossRef]
  13. C. A. Hulse and A. Knoesen, "Iterative calculation of complex propagation constants of modes in multilayer planar waveguides," IEEE J. Quantum Electron. 28, 2682-2684 (1992).
    [CrossRef]
  14. R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, "Numerical determination of planar waveguide modes using the analyticity of the dispersion relation," Opt. Lett. 16, 1316-1318 (1991).
    [CrossRef] [PubMed]
  15. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Efficient solution of eigenvalue equations of optical waveguiding structures," J. Lightwave Technol. 12, 2080-2084 (1994).
    [CrossRef]
  16. A. Bakhtazad, H. Abiri, and R. Ghayour, "A general transform for regularizing planar open waveguide dispersion relation," J. Lightwave Technol. 15, 383-390 (1997).
    [CrossRef]
  17. M. S. Kwon and S. Y. Shin, "Simple and fast numerical analysis of multilayer waveguide modes," Opt. Commun. 233, 119-126 (2004).
    [CrossRef]
  18. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method," J. Lightwave Technol. 17, 929-941 (1999).
    [CrossRef]
  19. J. Petracek and K. Singh, "Determination of leaky modes in planar multilayer waveguides," IEEE Photonics Technol. Lett. 14, 810-812 (2002).
    [CrossRef]
  20. K. Mehrany, S. Khorasani, and B. Rashidian, "Variational approach for extraction of eigenmodes in layered waveguides," J. Opt. Soc. Am. B 19, 1978-1981 (2002).
    [CrossRef]
  21. T. D. Visser, H. Blok, and D. Lenstra, "Modal analysis of a planar waveguide with gain and losses," IEEE J. Quantum Electron. 31, 1803-1810 (1995).
    [CrossRef]
  22. S. J. Mason, "Feedback theory--further properties of signal flow graphs," Proc. IRE 44, 920-926 (1956).
    [CrossRef]
  23. W. Mayeda, Graph Theory (Wiley, 1972).
  24. M. N. S. Swamy and K. Thulasiraman, Graphs, Networks, and Algorithms (Wiley, 1981).

2004 (1)

M. S. Kwon and S. Y. Shin, "Simple and fast numerical analysis of multilayer waveguide modes," Opt. Commun. 233, 119-126 (2004).
[CrossRef]

2003 (2)

2002 (2)

K. Mehrany, S. Khorasani, and B. Rashidian, "Variational approach for extraction of eigenmodes in layered waveguides," J. Opt. Soc. Am. B 19, 1978-1981 (2002).
[CrossRef]

J. Petracek and K. Singh, "Determination of leaky modes in planar multilayer waveguides," IEEE Photonics Technol. Lett. 14, 810-812 (2002).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

J. Ctyroky, S. Helfert, and R. Pregla, "Analysis of a deep waveguide Bragg grating," Opt. Quantum Electron. 30, 343-358 (1998).
[CrossRef]

1997 (1)

A. Bakhtazad, H. Abiri, and R. Ghayour, "A general transform for regularizing planar open waveguide dispersion relation," J. Lightwave Technol. 15, 383-390 (1997).
[CrossRef]

1995 (1)

T. D. Visser, H. Blok, and D. Lenstra, "Modal analysis of a planar waveguide with gain and losses," IEEE J. Quantum Electron. 31, 1803-1810 (1995).
[CrossRef]

1994 (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Efficient solution of eigenvalue equations of optical waveguiding structures," J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

1992 (2)

C. A. Hulse and A. Knoesen, "Iterative calculation of complex propagation constants of modes in multilayer planar waveguides," IEEE J. Quantum Electron. 28, 2682-2684 (1992).
[CrossRef]

E. Anemogiannis and E. N. Glytsis, "Multilayer waveguides: efficient numerical analysis of general structures," J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

1991 (2)

1990 (1)

K. H. Schlereth, "The complex propagation constant of multilayer waveguides: an algorithm for a personal computer," IEEE J. Quantum Electron. 26, 627-630 (1990).
[CrossRef]

1987 (1)

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, "Numerical analysis of planar optical waveguides using matrix approach," J. Lightwave Technol. LT-5, 660-667 (1987).
[CrossRef]

1985 (2)

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

L. M. Walpita, "Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix," J. Opt. Soc. Am. A 2, 595-602 (1985).
[CrossRef]

1984 (1)

1969 (1)

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

1956 (1)

S. J. Mason, "Feedback theory--further properties of signal flow graphs," Proc. IRE 44, 920-926 (1956).
[CrossRef]

Abiri, H.

A. Bakhtazad, H. Abiri, and R. Ghayour, "A general transform for regularizing planar open waveguide dispersion relation," J. Lightwave Technol. 15, 383-390 (1997).
[CrossRef]

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method," J. Lightwave Technol. 17, 929-941 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Efficient solution of eigenvalue equations of optical waveguiding structures," J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

E. Anemogiannis and E. N. Glytsis, "Multilayer waveguides: efficient numerical analysis of general structures," J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

Bakhtazad, A.

A. Bakhtazad, H. Abiri, and R. Ghayour, "A general transform for regularizing planar open waveguide dispersion relation," J. Lightwave Technol. 15, 383-390 (1997).
[CrossRef]

Berini, P.

Blok, H.

T. D. Visser, H. Blok, and D. Lenstra, "Modal analysis of a planar waveguide with gain and losses," IEEE J. Quantum Electron. 31, 1803-1810 (1995).
[CrossRef]

Chen, C.

Chiwell, J.

Ctyroky, J.

J. Ctyroky, S. Helfert, and R. Pregla, "Analysis of a deep waveguide Bragg grating," Opt. Quantum Electron. 30, 343-358 (1998).
[CrossRef]

Feng, D.

Forbes, G. W.

Gaal, S. B.

Gaylord, T. K.

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, "Numerical analysis of planar optical waveguides using matrix approach," J. Lightwave Technol. LT-5, 660-667 (1987).
[CrossRef]

Ghayour, R.

A. Bakhtazad, H. Abiri, and R. Ghayour, "A general transform for regularizing planar open waveguide dispersion relation," J. Lightwave Technol. 15, 383-390 (1997).
[CrossRef]

Glytsis, E. N.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method," J. Lightwave Technol. 17, 929-941 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Efficient solution of eigenvalue equations of optical waveguiding structures," J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

E. Anemogiannis and E. N. Glytsis, "Multilayer waveguides: efficient numerical analysis of general structures," J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

Goell, J. E.

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

Helfert, S.

J. Ctyroky, S. Helfert, and R. Pregla, "Analysis of a deep waveguide Bragg grating," Opt. Quantum Electron. 30, 343-358 (1998).
[CrossRef]

Hoekstra, H. J. W. M.

Houde-Walter, S. N.

Hulse, C. A.

C. A. Hulse and A. Knoesen, "Iterative calculation of complex propagation constants of modes in multilayer planar waveguides," IEEE J. Quantum Electron. 28, 2682-2684 (1992).
[CrossRef]

Khorasani, S.

Knoesen, A.

C. A. Hulse and A. Knoesen, "Iterative calculation of complex propagation constants of modes in multilayer planar waveguides," IEEE J. Quantum Electron. 28, 2682-2684 (1992).
[CrossRef]

Kwon, M. S.

M. S. Kwon and S. Y. Shin, "Simple and fast numerical analysis of multilayer waveguide modes," Opt. Commun. 233, 119-126 (2004).
[CrossRef]

Lambeck, P. V.

Lenstra, D.

T. D. Visser, H. Blok, and D. Lenstra, "Modal analysis of a planar waveguide with gain and losses," IEEE J. Quantum Electron. 31, 1803-1810 (1995).
[CrossRef]

Marhic, M. E.

Mason, S. J.

S. J. Mason, "Feedback theory--further properties of signal flow graphs," Proc. IRE 44, 920-926 (1956).
[CrossRef]

Mayeda, W.

W. Mayeda, Graph Theory (Wiley, 1972).

Mehrany, K.

Petracek, J.

J. Petracek and K. Singh, "Determination of leaky modes in planar multilayer waveguides," IEEE Photonics Technol. Lett. 14, 810-812 (2002).
[CrossRef]

Pregla, R.

J. Ctyroky, S. Helfert, and R. Pregla, "Analysis of a deep waveguide Bragg grating," Opt. Quantum Electron. 30, 343-358 (1998).
[CrossRef]

Rashidian, B.

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. 33, 894-899 (1985).
[CrossRef]

Schlereth, K. H.

K. H. Schlereth, "The complex propagation constant of multilayer waveguides: an algorithm for a personal computer," IEEE J. Quantum Electron. 26, 627-630 (1990).
[CrossRef]

Shenoy, M. R.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, "Numerical analysis of planar optical waveguides using matrix approach," J. Lightwave Technol. LT-5, 660-667 (1987).
[CrossRef]

Shin, S. Y.

M. S. Kwon and S. Y. Shin, "Simple and fast numerical analysis of multilayer waveguide modes," Opt. Commun. 233, 119-126 (2004).
[CrossRef]

Singh, K.

J. Petracek and K. Singh, "Determination of leaky modes in planar multilayer waveguides," IEEE Photonics Technol. Lett. 14, 810-812 (2002).
[CrossRef]

Smith, R. E.

Sun, L.

Swamy, M. N. S.

M. N. S. Swamy and K. Thulasiraman, Graphs, Networks, and Algorithms (Wiley, 1981).

Tanev, S.

Thulasiraman, K.

M. N. S. Swamy and K. Thulasiraman, Graphs, Networks, and Algorithms (Wiley, 1981).

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, "Numerical analysis of planar optical waveguides using matrix approach," J. Lightwave Technol. LT-5, 660-667 (1987).
[CrossRef]

Tzolov, V. P.

Visser, T. D.

T. D. Visser, H. Blok, and D. Lenstra, "Modal analysis of a planar waveguide with gain and losses," IEEE J. Quantum Electron. 31, 1803-1810 (1995).
[CrossRef]

Walpita, L. M.

Bell Syst. Tech. J. (1)

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

IEEE J. Quantum Electron. (3)

K. H. Schlereth, "The complex propagation constant of multilayer waveguides: an algorithm for a personal computer," IEEE J. Quantum Electron. 26, 627-630 (1990).
[CrossRef]

C. A. Hulse and A. Knoesen, "Iterative calculation of complex propagation constants of modes in multilayer planar waveguides," IEEE J. Quantum Electron. 28, 2682-2684 (1992).
[CrossRef]

T. D. Visser, H. Blok, and D. Lenstra, "Modal analysis of a planar waveguide with gain and losses," IEEE J. Quantum Electron. 31, 1803-1810 (1995).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

J. Petracek and K. Singh, "Determination of leaky modes in planar multilayer waveguides," IEEE Photonics Technol. Lett. 14, 810-812 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

J. Lightwave Technol. (6)

E. Anemogiannis and E. N. Glytsis, "Multilayer waveguides: efficient numerical analysis of general structures," J. Lightwave Technol. 10, 1344-1351 (1992).
[CrossRef]

A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, "Numerical analysis of planar optical waveguides using matrix approach," J. Lightwave Technol. LT-5, 660-667 (1987).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Efficient solution of eigenvalue equations of optical waveguiding structures," J. Lightwave Technol. 12, 2080-2084 (1994).
[CrossRef]

A. Bakhtazad, H. Abiri, and R. Ghayour, "A general transform for regularizing planar open waveguide dispersion relation," J. Lightwave Technol. 15, 383-390 (1997).
[CrossRef]

S. B. Gaal, H. J. W. M. Hoekstra, and P. V. Lambeck, "Determining PML modes in 2-D stratified media," J. Lightwave Technol. 21, 293-298 (2003).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method," J. Lightwave Technol. 17, 929-941 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

M. S. Kwon and S. Y. Shin, "Simple and fast numerical analysis of multilayer waveguide modes," Opt. Commun. 233, 119-126 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

J. Ctyroky, S. Helfert, and R. Pregla, "Analysis of a deep waveguide Bragg grating," Opt. Quantum Electron. 30, 343-358 (1998).
[CrossRef]

Proc. IRE (1)

S. J. Mason, "Feedback theory--further properties of signal flow graphs," Proc. IRE 44, 920-926 (1956).
[CrossRef]

Other (2)

W. Mayeda, Graph Theory (Wiley, 1972).

M. N. S. Swamy and K. Thulasiraman, Graphs, Networks, and Algorithms (Wiley, 1981).

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Figures (5)

Fig. 1
Fig. 1

Geometry of a multilayer waveguide.

Fig. 2
Fig. 2

Graph model for the relationships between tangential fields at the upper and lower boundaries of layer i for the TE modes.

Fig. 3
Fig. 3

Combined graph model for the TE modes of the N-layer waveguide.

Fig. 4
Fig. 4

Graph model for the relationships between tangential fields at the upper and lower boundaries of layer i for the TM modes.

Fig. 5
Fig. 5

F ( γ ) max and D ( γ ) max as functions of the thickness d 2 of the waveguide: F ( γ ) max is the maximum of the absolute value of the characteristic function by the transfer-matrix method, as given in Refs. [2, 6, 9, 15], for the range of 1.3 < β k 0 < 2.4 and α = 0 ; D ( γ ) max is calculated by our method for the same range.

Tables (2)

Tables Icon

Table 1 Calculated Effective Indices for TE and TM Guided Modes of a Three-Layer Waveguide

Tables Icon

Table 2 Calculated Effective Indices for Eigenmodes of an ARROW Waveguide

Equations (22)

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d 2 E y i ( x ) d x 2 + k i 2 E y i ( x ) = 0 ,
E y i ( x ) = E i 1 sin k i ( x i x ) sin ( k i d i ) + E i sin k i ( x x i 1 ) sin ( k i d i ) ,
j ω μ 0 H z i ( x ) = E i 1 k i cos k i ( x i x ) sin ( k i d i ) E i k i cos k i ( x x i 1 ) sin ( k i d i ) .
E i = E i 1 1 cos k i d i + H ¯ i tan k i d i k i ,
H ¯ i 1 = H ¯ i 1 cos k i d i + E i 1 k i tan k i d i ,
E y , 0 ( x ) = E 0 exp [ j k 0 , x ] , for x < x 0 ,
E y , N + 1 ( x ) = E N exp [ j k N + 1 ( x x N ) ] , for x > x N .
H ¯ 0 = j k 0 E 0 ,
H ¯ N = j k N + 1 E N .
L p , q = ( k q k p ) tan k q d q tan k p d p i = p q cos 2 k i d i ,
S p , q = s = 0 q p i 2 s = p + s q i 2 s 1 = p + s 1 i 2 s 1 i 2 s 1 = p + s 1 i 2 s 1 i 2 s 2 = p + s 2 i 2 s 1 1 i 2 = p + 1 i 3 i 1 = p i 2 1 u = 1 s ( L i 2 u 1 , j 2 u ) .
D ( γ ) = S 0 , N + 1 ( γ ) = 0 .
d 2 H y i ( x ) d x 2 + k i 2 H y i ( x ) = 0 ,
H i = H i 1 1 cos k i d i + E ¯ i n z z , i 2 tan k i d i k i ,
E ¯ i 1 = E ¯ i 1 cos k i d i + H i 1 k i n z z , i 2 tan k i d i ,
E ¯ 0 = ( j k 0 n z z , 0 2 ) H 0 ,
E ¯ N = ( j k N + 1 n z z , N + 1 2 ) H N .
L p , q = k q n z z , p 2 k p n z z , q 2 tan k q d q tan k p d p i = p q cos 2 k i d i ,
A 2 = n 4 = 2 3 n 3 = 1 n 4 1 n 2 = 1 n 3 n 1 = 0 n 2 1 ( L n 1 , n 2 L n 3 , n 4 ) for s = 2
A 3 = n 6 = 3 3 n 5 = 2 n 6 1 n 4 = 2 n 5 n 3 = 1 n 4 1 n 2 = 1 n 3 n 1 = 0 n 2 1 ( L n 1 , n 2 L n 3 , n 4 L n 5 , n 6 )
for s = 3 .
D ( γ ) = 1 ( L 0 , 1 + L 0 , 2 + L 0 , 3 + L 1 , 2 + L 1 , 3 + L 2 , 3 ) + L 0 , 1 ( L 1 , 2 + L 1 , 3 + L 2 , 3 ) + L 2 , 3 ( L 0 , 2 + L 1 , 2 ) L 0 , 1 L 1 , 2 L 2 , 3 = 0 .

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