Abstract

The propagation characteristics of the leaky modes in planar anisotropic waveguides with a multilayer structure have been investigated by means of a compact rigorous formalism. The leakage losses and leaky transition angle have been studied for the fundamental and first hybrid modes. An inhomogeneous waveguide and buffered step index type structure have been discussed. Particular attention has been devoted to the variation of the loss coefficient of the leaky modes as a function of buffer thickness and buffer refractive index. A notably different behavior has been obtained for various configurations.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Thylen, “Integrated Optics in LiNbO3: Recent Developments in Devices for Telecommunications,” IEEE/OSA J. Lightwave Technol. LT-6, 847–861 (1988).
    [CrossRef]
  2. E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
    [CrossRef]
  3. T. K. Gaylord, A. Knoesen, “Passive Integrated Optical Anisotropy-Based Devices,” J. Mod. Opt. 35, 925–946 (1988).
    [CrossRef]
  4. D. P. Gia Russo, J. H. Harris, “Wave Propagation in Anisotropic Thin-Film Optical Waveguides,” J. Opt. Soc. Am. 63, 138–145 (1973).
    [CrossRef]
  5. W. K. Burns, J. Warner, “Mode Dispersion in Uniaxial Optical Waveguides,” J. Opt. Soc. Am. 64, 441–446 (1974).
    [CrossRef]
  6. D. Marcuse, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part I: Optical Axis not in Plane of Slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
    [CrossRef]
  7. D. Marcuse, I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
    [CrossRef]
  8. W. K. Burns, S. K. Sheem, A. F. Milton, “Approximate Calculation of Leaky-Mode Loss Coefficients for Ti-Diffused LiNbO3 Waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
    [CrossRef]
  9. J. Ctyroky, M. Cada, “Generalized WKB Method for the Analysis of Light Propagation in Inhomogeneous Anisotropic Optical Waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
    [CrossRef]
  10. J. Ctyroky, M. Cada, “Guided and Semileaky Modes in Anisotropic Waveguides of the LiNbO3 Type,” Opt. Commun. 27, 353–357 (1978).
    [CrossRef]
  11. E. A. Kolovsky, D. V. Petrov, A. V. Tsarev, I. B. Yakovkin, “An Exact Method for Analysing Light Propagation in Anisotropic Inhomogeneous Optical Waveguides,” Opt. Commun. 43, 21–25 (1982).
    [CrossRef]
  12. M. Koshiba, H. Kumagami, M. Suzuki, “Finite-Element Solution of Planar Arbitrarily Anisotropic Diffused Optical Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 773–778 (1985).
    [CrossRef]
  13. K. Yamanouchi, T. Kamiya, K. Shibayama, “New Leaky Surface Waves in Anisotropic Metal-Diffused Optical Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–304 (1978).
    [CrossRef]
  14. S. K. Sheem, W. K. Burns, A. F. Milton, “Leaky-Mode Propagation in Ti-Diffused LiNbO3 and LiTaO3 Waveguides,” Opt. Lett. 3, 76–78 (1978).
    [CrossRef] [PubMed]
  15. R. A. Stienberg, T. G. Giallorenzi, “Modal Fields of Anisotropic Channel Waveguides,” J. Opt. Soc. Am. 67, 523–533 (1977).
    [CrossRef]
  16. M. N. Armenise, M. De Sario, “Optical Rectangular Waveguide in Titanium-Diffused Lithium Niobate Having its Optical Axis in the Transverse Plane,” J. Opt. Soc. Am. 72, 1514–1521 (1982).
    [CrossRef]
  17. J. T. Chilwell, I. J. 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]
  18. D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4 × 4-Matrix Formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  19. M. O. Vassell, “Structure of Optical Guided Modes in Planar Multilayers of Optically Anisotropic Materials,” J. Opt. Soc. Am. 64, 166–173 (1974).
    [CrossRef]
  20. P. Yeh, “Electromagnetic Propagation in Birefringent Layered Media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  21. A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
    [CrossRef]
  22. 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]
  23. T. Tamir, “Leaky-Wave Antennas,” in Antenna Theory, Part 2, R. E. Collin, F. J. Zucker, Eds. (McGraw-Hill, New York, 1969).
  24. T. Tamir, F. Y. Kou, “Varieties of Leaky Waves and Their Excitation Along Multilayered Structures,” IEEE J. Quantum Electron. QE-22, 544–551 (1986) and references therein.
    [CrossRef]
  25. L. Torner, F. Canal, “Guided-to-Leaky Mode Transition in Planar Dielectric Uniaxal Waveguides,” submitted for publication.
  26. L. Torner, Ph.D. Thesis, U. Politécnica de Catalunya, Barcelona (1989).

1988 (3)

T. K. Gaylord, A. Knoesen, “Passive Integrated Optical Anisotropy-Based Devices,” J. Mod. Opt. 35, 925–946 (1988).
[CrossRef]

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

L. Thylen, “Integrated Optics in LiNbO3: Recent Developments in Devices for Telecommunications,” IEEE/OSA J. Lightwave Technol. LT-6, 847–861 (1988).
[CrossRef]

1987 (1)

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
[CrossRef]

1986 (1)

T. Tamir, F. Y. Kou, “Varieties of Leaky Waves and Their Excitation Along Multilayered Structures,” IEEE J. Quantum Electron. QE-22, 544–551 (1986) and references therein.
[CrossRef]

1985 (2)

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]

M. Koshiba, H. Kumagami, M. Suzuki, “Finite-Element Solution of Planar Arbitrarily Anisotropic Diffused Optical Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 773–778 (1985).
[CrossRef]

1984 (1)

1982 (2)

E. A. Kolovsky, D. V. Petrov, A. V. Tsarev, I. B. Yakovkin, “An Exact Method for Analysing Light Propagation in Anisotropic Inhomogeneous Optical Waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

M. N. Armenise, M. De Sario, “Optical Rectangular Waveguide in Titanium-Diffused Lithium Niobate Having its Optical Axis in the Transverse Plane,” J. Opt. Soc. Am. 72, 1514–1521 (1982).
[CrossRef]

1981 (1)

J. Ctyroky, M. Cada, “Generalized WKB Method for the Analysis of Light Propagation in Inhomogeneous Anisotropic Optical Waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
[CrossRef]

1979 (3)

D. Marcuse, I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

W. K. Burns, S. K. Sheem, A. F. Milton, “Approximate Calculation of Leaky-Mode Loss Coefficients for Ti-Diffused LiNbO3 Waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
[CrossRef]

P. Yeh, “Electromagnetic Propagation in Birefringent Layered Media,” J. Opt. Soc. Am. 69, 742–756 (1979).
[CrossRef]

1978 (4)

J. Ctyroky, M. Cada, “Guided and Semileaky Modes in Anisotropic Waveguides of the LiNbO3 Type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

D. Marcuse, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part I: Optical Axis not in Plane of Slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

K. Yamanouchi, T. Kamiya, K. Shibayama, “New Leaky Surface Waves in Anisotropic Metal-Diffused Optical Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–304 (1978).
[CrossRef]

S. K. Sheem, W. K. Burns, A. F. Milton, “Leaky-Mode Propagation in Ti-Diffused LiNbO3 and LiTaO3 Waveguides,” Opt. Lett. 3, 76–78 (1978).
[CrossRef] [PubMed]

1977 (1)

1974 (2)

1973 (1)

1972 (1)

Armenise, M. N.

Berreman, D. W.

Burns, W. K.

Cada, M.

J. Ctyroky, M. Cada, “Generalized WKB Method for the Analysis of Light Propagation in Inhomogeneous Anisotropic Optical Waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
[CrossRef]

J. Ctyroky, M. Cada, “Guided and Semileaky Modes in Anisotropic Waveguides of the LiNbO3 Type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

Canal, F.

L. Torner, F. Canal, “Guided-to-Leaky Mode Transition in Planar Dielectric Uniaxal Waveguides,” submitted for publication.

Chilwell, J. T.

Ctyroky, J.

J. Ctyroky, M. Cada, “Generalized WKB Method for the Analysis of Light Propagation in Inhomogeneous Anisotropic Optical Waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
[CrossRef]

J. Ctyroky, M. Cada, “Guided and Semileaky Modes in Anisotropic Waveguides of the LiNbO3 Type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

De Sario, M.

Gaylord, T. K.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

T. K. Gaylord, A. Knoesen, “Passive Integrated Optical Anisotropy-Based Devices,” J. Mod. Opt. 35, 925–946 (1988).
[CrossRef]

Gia Russo, D. P.

Giallorenzi, T. G.

Harris, J. H.

Hodgkinson, I. J.

Kaminow, I. P.

D. Marcuse, I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

Kamiya, T.

K. Yamanouchi, T. Kamiya, K. Shibayama, “New Leaky Surface Waves in Anisotropic Metal-Diffused Optical Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–304 (1978).
[CrossRef]

Knoesen, A.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

T. K. Gaylord, A. Knoesen, “Passive Integrated Optical Anisotropy-Based Devices,” J. Mod. Opt. 35, 925–946 (1988).
[CrossRef]

Kolovsky, E. A.

E. A. Kolovsky, D. V. Petrov, A. V. Tsarev, I. B. Yakovkin, “An Exact Method for Analysing Light Propagation in Anisotropic Inhomogeneous Optical Waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Koshiba, M.

M. Koshiba, H. Kumagami, M. Suzuki, “Finite-Element Solution of Planar Arbitrarily Anisotropic Diffused Optical Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 773–778 (1985).
[CrossRef]

Kou, F. Y.

T. Tamir, F. Y. Kou, “Varieties of Leaky Waves and Their Excitation Along Multilayered Structures,” IEEE J. Quantum Electron. QE-22, 544–551 (1986) and references therein.
[CrossRef]

Kumagami, H.

M. Koshiba, H. Kumagami, M. Suzuki, “Finite-Element Solution of Planar Arbitrarily Anisotropic Diffused Optical Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 773–778 (1985).
[CrossRef]

Marcuse, D.

D. Marcuse, I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

D. Marcuse, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part I: Optical Axis not in Plane of Slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

Milton, A. F.

W. K. Burns, S. K. Sheem, A. F. Milton, “Approximate Calculation of Leaky-Mode Loss Coefficients for Ti-Diffused LiNbO3 Waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
[CrossRef]

S. K. Sheem, W. K. Burns, A. F. Milton, “Leaky-Mode Propagation in Ti-Diffused LiNbO3 and LiTaO3 Waveguides,” Opt. Lett. 3, 76–78 (1978).
[CrossRef] [PubMed]

Moharam, M. G.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

Neyer, A.

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
[CrossRef]

Petrov, D. V.

E. A. Kolovsky, D. V. Petrov, A. V. Tsarev, I. B. Yakovkin, “An Exact Method for Analysing Light Propagation in Anisotropic Inhomogeneous Optical Waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Sheem, S. K.

W. K. Burns, S. K. Sheem, A. F. Milton, “Approximate Calculation of Leaky-Mode Loss Coefficients for Ti-Diffused LiNbO3 Waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
[CrossRef]

S. K. Sheem, W. K. Burns, A. F. Milton, “Leaky-Mode Propagation in Ti-Diffused LiNbO3 and LiTaO3 Waveguides,” Opt. Lett. 3, 76–78 (1978).
[CrossRef] [PubMed]

Shibayama, K.

K. Yamanouchi, T. Kamiya, K. Shibayama, “New Leaky Surface Waves in Anisotropic Metal-Diffused Optical Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–304 (1978).
[CrossRef]

Stienberg, R. A.

Suzuki, M.

M. Koshiba, H. Kumagami, M. Suzuki, “Finite-Element Solution of Planar Arbitrarily Anisotropic Diffused Optical Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 773–778 (1985).
[CrossRef]

Tamir, T.

T. Tamir, F. Y. Kou, “Varieties of Leaky Waves and Their Excitation Along Multilayered Structures,” IEEE J. Quantum Electron. QE-22, 544–551 (1986) and references therein.
[CrossRef]

T. Tamir, “Leaky-Wave Antennas,” in Antenna Theory, Part 2, R. E. Collin, F. J. Zucker, Eds. (McGraw-Hill, New York, 1969).

Thylen, L.

L. Thylen, “Integrated Optics in LiNbO3: Recent Developments in Devices for Telecommunications,” IEEE/OSA J. Lightwave Technol. LT-6, 847–861 (1988).
[CrossRef]

Torner, L.

L. Torner, F. Canal, “Guided-to-Leaky Mode Transition in Planar Dielectric Uniaxal Waveguides,” submitted for publication.

L. Torner, Ph.D. Thesis, U. Politécnica de Catalunya, Barcelona (1989).

Tsarev, A. V.

E. A. Kolovsky, D. V. Petrov, A. V. Tsarev, I. B. Yakovkin, “An Exact Method for Analysing Light Propagation in Anisotropic Inhomogeneous Optical Waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Vassell, M. O.

Voges, E.

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
[CrossRef]

Walpita, L. M.

Warner, J.

Yakovkin, I. B.

E. A. Kolovsky, D. V. Petrov, A. V. Tsarev, I. B. Yakovkin, “An Exact Method for Analysing Light Propagation in Anisotropic Inhomogeneous Optical Waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Yamanouchi, K.

K. Yamanouchi, T. Kamiya, K. Shibayama, “New Leaky Surface Waves in Anisotropic Metal-Diffused Optical Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–304 (1978).
[CrossRef]

Yeh, P.

IEEE J. Quantum Electron. (5)

D. Marcuse, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part I: Optical Axis not in Plane of Slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

D. Marcuse, I. P. Kaminow, “Modes of a Symmetric Slab Optical Waveguide in Birefringent Media. Part II: Slab with Coplanar Optical Axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

W. K. Burns, S. K. Sheem, A. F. Milton, “Approximate Calculation of Leaky-Mode Loss Coefficients for Ti-Diffused LiNbO3 Waveguides,” IEEE J. Quantum Electron. QE-15, 1282–1289 (1979).
[CrossRef]

J. Ctyroky, M. Cada, “Generalized WKB Method for the Analysis of Light Propagation in Inhomogeneous Anisotropic Optical Waveguides,” IEEE J. Quantum Electron. QE-17, 1064–1070 (1981).
[CrossRef]

T. Tamir, F. Y. Kou, “Varieties of Leaky Waves and Their Excitation Along Multilayered Structures,” IEEE J. Quantum Electron. QE-22, 544–551 (1986) and references therein.
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Yamanouchi, T. Kamiya, K. Shibayama, “New Leaky Surface Waves in Anisotropic Metal-Diffused Optical Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-26, 298–304 (1978).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (4)

L. Thylen, “Integrated Optics in LiNbO3: Recent Developments in Devices for Telecommunications,” IEEE/OSA J. Lightwave Technol. LT-6, 847–861 (1988).
[CrossRef]

E. Voges, A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” IEEE/OSA J. Lightwave Technol. LT-5, 1229–1238 (1987).
[CrossRef]

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid Guided Modes in Uniaxial Dielectric Planar Waveguides,” IEEE/OSA J. Lightwave Technol. LT-6, 1083–1104 (1988).
[CrossRef]

M. Koshiba, H. Kumagami, M. Suzuki, “Finite-Element Solution of Planar Arbitrarily Anisotropic Diffused Optical Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 773–778 (1985).
[CrossRef]

J. Mod. Opt. (1)

T. K. Gaylord, A. Knoesen, “Passive Integrated Optical Anisotropy-Based Devices,” J. Mod. Opt. 35, 925–946 (1988).
[CrossRef]

J. Opt. Soc. Am. (7)

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

Opt. Commun. (2)

J. Ctyroky, M. Cada, “Guided and Semileaky Modes in Anisotropic Waveguides of the LiNbO3 Type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

E. A. Kolovsky, D. V. Petrov, A. V. Tsarev, I. B. Yakovkin, “An Exact Method for Analysing Light Propagation in Anisotropic Inhomogeneous Optical Waveguides,” Opt. Commun. 43, 21–25 (1982).
[CrossRef]

Opt. Lett. (1)

Other (3)

T. Tamir, “Leaky-Wave Antennas,” in Antenna Theory, Part 2, R. E. Collin, F. J. Zucker, Eds. (McGraw-Hill, New York, 1969).

L. Torner, F. Canal, “Guided-to-Leaky Mode Transition in Planar Dielectric Uniaxal Waveguides,” submitted for publication.

L. Torner, Ph.D. Thesis, U. Politécnica de Catalunya, Barcelona (1989).

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 (11)

Fig. 1
Fig. 1

Schematic waveguiding multilayer structure. Propagation is along x.

Fig. 2
Fig. 2

Loss coefficient of the [TM0,TE0][l] mode supported by G1 as a function of the optical axis orientation. φ l is the guided-to-leaky mode transition angle.

Fig. 3
Fig. 3

Detail of Fig. 2 showing the secondary maximum of the loss coefficient as a function of φ for the homogeneous waveguide and the monotonous decrease which occurs for the Gaussian profile.

Fig. 4
Fig. 4

Loss coefficient for the leaky mode in Fig. 2 as a function of the cover refractive index for φ = 40°. To emphasize the effects due to the Gaussian profile the curves have been referred to the value of the loss coefficient for n c = 1.

Fig. 5
Fig. 5

Same as in Fig. 4 but for φ = 30°.

Fig. 6
Fig. 6

Effective indices of the hybrid modes supported by G2 as a function of φ. At φ = 0,90° the modes are pure TE and TM. The [TM0,TE0][g] mode is a predominantly extraordinary guided hybrid mode, whereas the [TE0,TM0][l] mode is a predominantly ordinary leaky hybrid mode. Solid line: pure guided mode. Dashed line: leaky mode.

Fig. 7
Fig. 7

Loss coefficient as a function of the buffer refractive index. Mode [TE0,TM0][l]. Buffer thickness: D b = 0.5 μm.

Fig. 8
Fig. 8

Loss coefficient as a function of the decimal logarithm of the λ-scaled buffer thickness. Mode [TE0,TM0][l]. Buffer refractive index: n b = 2.

Fig. 9
Fig. 9

Same as in Fig. 7 for the [TE1,TM1][l] mode supported by the multimode version (D = 6 μm) of G2.

Fig. 10
Fig. 10

Same as in Fig. 8 for the [TE1,TM1][l] mode supported by the multimode version (D = 6 μm) of G2.

Fig. 11
Fig. 11

Guided-to-leaky mode transition angle for the [TE0,TM0][l] mode supported by G2 as a function of the decimal logarithm of the λ-scaled buffer thickness. Dashed line: transition angle in the absence of the buffer layer.

Tables (1)

Tables Icon

Table I Accurate Numerical Values Obtained for the Waveguide with a Gaussian Profile (G1) and Comparison with the Finite Element Results Reported In Ref. 12

Equations (44)

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

[ ] D = ( o o e ) ,
x x = o ( sin 2 φ + cos 2 φ cos 2 θ ) + e cos 2 φ sin 2 θ , y y = o ( cos 2 φ + sin 2 φ cos 2 θ ) + e sin 2 φ sin 2 θ , z z = o sin 2 θ + e cos 2 θ , x y = ( e - o ) sin φ cos φ sin 2 θ , x z = ( e - o ) cos φ sin θ cos θ , y z = ( e - o ) sin φ sin θ cos θ .
E c ( z ) = E c exp ( γ c z ) , z < 0 ,
E s ( z ) = E s exp [ γ s ( D - z ) ] , z > D ,
2 E + μ ω 2 [ ] E = ( · E ) ,
γ o = ± β 2 - μ ω 2 o ,
γ e = ± 1 z z β 2 ( x x z z - x z 2 ) - μ ω 2 o e z z ( ± ) i x z z z β .
[ E x H y E y H x ] z 2 = T [ E x H y E y H x ] z 1 .
E c ( z ) = [ Λ x o 1 Λ z o ] E y o exp ( γ o z ) + [ 1 Λ y e Λ z e ] E x e exp ( γ e z ) ,
E c ( z ) = [ Λ x o 1 1 Λ y e Λ z o Λ z e ] { E y o exp ( γ o z ) E x e exp ( γ e z ) } .
H c ( z ) = - i μ ω [ - γ o - γ e Λ y e γ o Λ x o - i β Λ z o γ e - i β Λ z e i β i β Λ y e ] { E y o exp ( γ o z ) E x e exp ( γ e z ) } .
Λ x o = x y γ o 2 - i y z γ o β ( o - x x ) γ o 2 + β 2 ( o - z z ) + 2 i x z γ o β ,
Λ z o = x y ( μ ω 2 x z - i γ o β ) - y z ( γ o 2 + μ ω 2 x x ) ( o - x x ) γ o 2 + β 2 ( o - z z ) + 2 i x z γ o β ,
Λ y e = μ ω 2 x y ( β 2 - μ ω 2 z z ) + y z ( μ ω 2 x z - i γ e β ) ( γ e 2 - β 2 + μ ω 2 y y ) ( μ ω 2 z z - β 2 ) - ( μ ω 2 y z ) 2 ,
Λ z e = ( γ e 2 - β 2 + μ ω 2 y y ) ( i γ e β - μ ω 2 x z ) + ( μ ω 2 ) 2 x y y z ( γ e 2 - β 2 + μ ω 2 y y ) ( μ ω 2 z z - β 2 ) - ( μ ω 2 y z ) 2 .
E s ( z ) = [ Λ x o * 1 1 Λ y e * Λ z o * Λ z e * ] { E y o * exp [ γ o * ( D - z ) ] E x e * exp [ γ e * ( D - z ) ] } ,
H s ( z ) = - i μ ω [ γ o * γ e * Λ y e * - ( γ o * Λ x o * + i β Λ z o * ) - ( γ e * + i β Λ z e * ) i β i β Λ y e * ] × { E y o * exp [ γ o * ( D - z ) ] E x e * exp [ γ e * ( D - z ) ] } .
[ Λ x o * 1 ν 1 o ν 1 e τ o * τ e * ν 2 o ν 2 e 1 Λ y e * ν 3 o ν 3 e - γ o * - γ e * Λ y e * ν 4 o ν 4 e ] ( E y o * E x e * E y o E x e ) = 0 ,
τ o * γ o * Λ x o * + i β Λ z o * ,
τ e * γ e * + i β Λ z e * ,
ν j { - Ω j , j = 1 , 3 , i μ ω Ω j , j = 2 , 4 ,
Ω j o T j 1 Λ x o + T j 3 - i μ ω ( T j 2 τ o - γ o T j 4 ) ,
Ω j e T j 1 + T j 3 Λ y e - i μ ω ( T j 2 τ e - γ e Λ y e T j 4 ) .
Λ x o * [ τ e * N 34 - Λ y e * N 24 - γ e * Λ y e * N 23 ] - τ o * [ N 34 - Λ y e * N 14 - γ e * Λ y e * N 13 ] + [ N 24 - τ e * N 14 - γ e * Λ y e * N 12 ] + γ o * [ N 23 - τ e * N 13 + Λ y e * N 12 ] = 0 ,
N j l ν j o ν l e - ν j e ν l o .
Λ z e = μ ω 2 x z - i γ e β β 2 - μ ω 2 z z .
˜ o e z z ,
β ˜ β ˜ z z ,
γ ˜ e β ˜ 2 - μ ω 2 ˜ = γ e + i x z z z β ,
γ c = ± β 2 - k 0 2 n c 2 ,
Ω j o = T j 3 + i γ c μ ω T j 4 ,
Ω j e = T j 1 + i k 0 2 n c 2 μ ω γ c T j 2 ,
G 1 : { n o f = 2.2946 n e f = 2.2108 n o s = 2.2866 n e s = 2.2028 n c = 1 D = 2 μ m } .
G 2 : { n o f = 2.1856 n e f = 2.190 n o s = 2.1834 n e s = 2.1878 D = 2 μ m } .
{ T 21 + i ω ˜ γ ˜ e T 22 + i ω ˜ * γ ˜ e * T 11 - ω 2 ˜ ˜ * γ ˜ e γ ˜ e * T 12 } × { T 43 + i γ o μ ω T 44 + i γ o * μ ω T 33 - γ o γ o * ( μ ω ) 2 T 34 } - { T 23 + i γ o μ ω T 24 + i ω ˜ * γ ˜ e * T 13 - ˜ * γ o μ γ ˜ e * T 14 } × { T 41 + i ω ˜ γ ˜ e T 42 + i γ o * μ ω T 31 - ˜ γ o * μ γ ˜ e T 32 } = 0.
U l m = ( λ o 2 α 1 2 + μ o ) U ¯ l m ,
U ¯ 11 = μ o cos ( ω D λ o ) + λ o 2 α 1 2 cos ( ω D λ e ) , U ¯ 12 = i λ o o [ μ o sin ( ω D λ o ) + λ o 3 λ e α 1 2 sin ( ω D λ e ) ] , U ¯ 13 = α 1 λ o 2 [ cos ( ω D λ o ) - cos ( ω D λ e ) ] , U ¯ 14 = - i μ α 1 λ o [ sin ( ω D λ o ) - λ o λ e sin ( ω D λ e ) ] , U ¯ 21 = i o λ o [ μ o sin ( ω D λ o ) + α 1 2 λ o λ e sin ( ω D λ e ) ] , U ¯ 23 = i o α 1 λ o [ sin ( ω D λ o ) - λ e λ o sin ( ω D λ e ) ] , U ¯ 24 = - μ o α 1 [ cos ( ω D λ o ) - cos ( ω D λ e ) ] , U ¯ 33 = α 1 2 λ o 2 cos ( ω D λ o ) + μ o cos ( ω D λ e ) , U ¯ 34 = - i μ λ e [ λ o λ e α 1 2 sin ( ω D λ o ) + μ o sin ( ω D λ e ) ] , U ¯ 43 = - i λ o μ [ λ o 2 α 1 2 sin ( ω D λ o ) + λ e λ o μ o sin ( ω D λ e ) ] , }
U ¯ 22 = U ¯ 11 U ¯ 44 = U ¯ 33 , U ¯ 31 = - U ¯ 24 U ¯ 32 = - U ¯ 14 , U ¯ 41 = - U ¯ 23 U ¯ 42 = - U ¯ 13 , }
λ o = μ o - α 2 ,
λ e = μ e - α 2 x x / o ,
α 1 = o - x x x y .
U b = ( u TM 0 0 u TE ) ,
u = [ cos ( k 0 D b β b ) i ( η / a ) sin ( k 0 D b β b ) i ( a / η ) sin ( k 0 D b β b ) cos ( k 0 D b β b ) ] .
β b n b 2 - N 2 ,

Metrics