Abstract

A new variational approach to the extraction of guided and leaky modes in layered waveguides is proposed. To verify the method we compare the results of analysis of a typical test case and those from other references and find them in agreement. The efficiency of the proposed approach is compared with that of other reported methods.

© 2002 Optical Society of America

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References

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  1. K.-H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
    [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. M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J-134, 333–338 (1987).
  4. 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]
  5. L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
    [CrossRef]
  6. 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]
  7. L. F. Abd-ellal, L. M. Delves, and J. K. Reid, “A numerical method for locating the zeros and poles of a mermomorphic function,” in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon & Breach, London, 1970), pp. 47–59.
  8. 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]
  9. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  10. S. Khorasani and B. Rashidian, “Modified TMM for layered waveguides with conducting interfaces,” in Optoelectronics, Materials, and Devices, Tien Pei Lee and Qiming Wang, eds., Proc. SPIE 4580, 167–169 (2001).
    [CrossRef]
  11. H. Bach, “On the down-hill method,” Commun. ACM 12, 675–677 (1969).
    [CrossRef]
  12. K. Mehrany, S. Khorasani, and B. Rashidian, “Differential transfer matrix method for nonhomogeneous anisotropic media,” presented at the 2000 OSA Annual Meeting, Providence, R.I., October 22–26, 2000.
  13. P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound wave and leaky wave ranges for a partially dielectric-loaded open guiding structures,” Intensive Care Med. 38, 1831–1836 (1990).
  14. A. A. Oliner, “Leaky waves: basic properties and applications,” presented at the Asia Pacific Microwave Conference, Hong Kong, China, December 2–5, 1997.
  15. J. Chiwell and I. Hodgkinson, “Thin-films field transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]
  16. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Optimization 9, 112–147 (1999).
    [CrossRef]
  17. S. L. Ray and A. F. Peterson, “Error and convergence in numerical implementations of the conjugate gradient method,” IEEE Trans. Antennas Propag. AP-36, 1824–1827 (1988).
    [CrossRef]

2001 (1)

S. Khorasani and B. Rashidian, “Modified TMM for layered waveguides with conducting interfaces,” in Optoelectronics, Materials, and Devices, Tien Pei Lee and Qiming Wang, eds., Proc. SPIE 4580, 167–169 (2001).
[CrossRef]

1999 (2)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Optimization 9, 112–147 (1999).
[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]

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 (1)

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

1990 (2)

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

P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound wave and leaky wave ranges for a partially dielectric-loaded open guiding structures,” Intensive Care Med. 38, 1831–1836 (1990).

1988 (1)

S. L. Ray and A. F. Peterson, “Error and convergence in numerical implementations of the conjugate gradient method,” IEEE Trans. Antennas Propag. AP-36, 1824–1827 (1988).
[CrossRef]

1987 (2)

M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J-134, 333–338 (1987).

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]

1984 (1)

1969 (1)

H. Bach, “On the down-hill method,” Commun. ACM 12, 675–677 (1969).
[CrossRef]

1967 (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[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]

Bach, H.

H. Bach, “On the down-hill method,” Commun. ACM 12, 675–677 (1969).
[CrossRef]

Chiwell, J.

Delves, L. M.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Frezza, F.

P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound wave and leaky wave ranges for a partially dielectric-loaded open guiding structures,” Intensive Care Med. 38, 1831–1836 (1990).

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]

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]

Hodgkinson, I.

Khorasani, S.

S. Khorasani and B. Rashidian, “Modified TMM for layered waveguides with conducting interfaces,” in Optoelectronics, Materials, and Devices, Tien Pei Lee and Qiming Wang, eds., Proc. SPIE 4580, 167–169 (2001).
[CrossRef]

Koshiba, M.

M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J-134, 333–338 (1987).

Kumagami, H.

M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J-134, 333–338 (1987).

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Optimization 9, 112–147 (1999).
[CrossRef]

Lampariello, P.

P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound wave and leaky wave ranges for a partially dielectric-loaded open guiding structures,” Intensive Care Med. 38, 1831–1836 (1990).

Lyness, J. N.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Oliner, A. A.

P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound wave and leaky wave ranges for a partially dielectric-loaded open guiding structures,” Intensive Care Med. 38, 1831–1836 (1990).

Peterson, A. F.

S. L. Ray and A. F. Peterson, “Error and convergence in numerical implementations of the conjugate gradient method,” IEEE Trans. Antennas Propag. AP-36, 1824–1827 (1988).
[CrossRef]

Rashidian, B.

S. Khorasani and B. Rashidian, “Modified TMM for layered waveguides with conducting interfaces,” in Optoelectronics, Materials, and Devices, Tien Pei Lee and Qiming Wang, eds., Proc. SPIE 4580, 167–169 (2001).
[CrossRef]

Ray, S. L.

S. L. Ray and A. F. Peterson, “Error and convergence in numerical implementations of the conjugate gradient method,” IEEE Trans. Antennas Propag. AP-36, 1824–1827 (1988).
[CrossRef]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Optimization 9, 112–147 (1999).
[CrossRef]

Schlereth, K.-H.

K.-H. Schlereth and M. Tacke, “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]

Tacke, M.

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

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]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Optimization 9, 112–147 (1999).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Optimization 9, 112–147 (1999).
[CrossRef]

Commun. ACM (1)

H. Bach, “On the down-hill method,” Commun. ACM 12, 675–677 (1969).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

IEEE Trans. Antennas Propag. (1)

S. L. Ray and A. F. Peterson, “Error and convergence in numerical implementations of the conjugate gradient method,” IEEE Trans. Antennas Propag. AP-36, 1824–1827 (1988).
[CrossRef]

Intensive Care Med. (1)

P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound wave and leaky wave ranges for a partially dielectric-loaded open guiding structures,” Intensive Care Med. 38, 1831–1836 (1990).

J. Lightwave Technol. (4)

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]

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 (1)

Math. Comput. (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

M. Koshiba and H. Kumagami, “Theoretical study of silicon-clad planar diffused optical waveguides,” Proc. Inst. Electr. Eng. J-134, 333–338 (1987).

Proc. SPIE (1)

S. Khorasani and B. Rashidian, “Modified TMM for layered waveguides with conducting interfaces,” in Optoelectronics, Materials, and Devices, Tien Pei Lee and Qiming Wang, eds., Proc. SPIE 4580, 167–169 (2001).
[CrossRef]

SIAM J. Optimization (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Optimization 9, 112–147 (1999).
[CrossRef]

Other (4)

A. A. Oliner, “Leaky waves: basic properties and applications,” presented at the Asia Pacific Microwave Conference, Hong Kong, China, December 2–5, 1997.

K. Mehrany, S. Khorasani, and B. Rashidian, “Differential transfer matrix method for nonhomogeneous anisotropic media,” presented at the 2000 OSA Annual Meeting, Providence, R.I., October 22–26, 2000.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

L. F. Abd-ellal, L. M. Delves, and J. K. Reid, “A numerical method for locating the zeros and poles of a mermomorphic function,” in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed. (Gordon & Breach, London, 1970), pp. 47–59.

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Figures (3)

Fig. 1
Fig. 1

Layered waveguide.

Fig. 2
Fig. 2

Illustration of the maxima of Jm as given by Eqs. (5) for the TE mode with m=5 (solid curve), m=3 (dotted–dashed curve), and m=2 (dashed curve).

Fig. 3
Fig. 3

Illustration of the maxima of Jm as given by Eqs. (5) for the TM mode with m=5 (solid curve), m=3 (dotted–dashed curve), and m=2 (dashed curve).

Tables (2)

Tables Icon

Table 1 Effective Indices for TE and TM Guided Propagating Modesa

Tables Icon

Table 2 Effective Indices for TE and TM Leaky Propagating Modesa

Equations (15)

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

2kmdm-Φm,m+1-Φm,m-1=2νπ,νN
Um+1+Um+1-=Qmm+1Um+Um-,
exp(jΦm,m+1)=-q21m/+2q22m/+2exp(j2kmXm)Rm,m+1 exp(j2kmXm),
exp(jΦm,m-1)=-q12m1q11m1exp(-j2kmXm-1)Rm,m-1 exp(-j2kmXm-1),
Jm-12+12 Re(Rm,m+1Rm,m-1),1<m<l+2,
Um+Um-=Q1mU1+U1-,
Ul+2+Ul+2-=Qm/+2Um+Um-.
Ul+2+0=Q1l+20U1-,
q2211/+2=q21ml+2q121m+q22ml+2q221m=0
-q21ml+2q22ml+2×q121mq221m=q21ml+2q22ml+2×q12m1q11m1=1.
Jm-12+12Rm,m+1Rm,m-1.
q211l+2=q21ml+2q111m+q22ml+2q211m=0
q211l+2=q11ml+2q121m+q12ml+2q221m=0.
-q21ml+2q22ml+2×q111mq211m=1
-q11ml+2q12ml+2×q121mq221m=1.

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