Abstract

Birefringence of optical waveguides is closely related to the anisotropy of thermal stresses in the core. Closed-form solutions for estimating the thermal stresses in the cladding layers and in the core of an embedded channel waveguide are presented. The solutions are verified by finite-element simulation. It is found that the thermal stress in the core in the direction perpendicular to the wafer cannot be ignored, and one can tune the core stress anisotropy in the plane normal to the light-propagation direction by varying the thermal-expansion mismatch between the upper cladding layer and the substrate.

© 2003 Optical Society of America

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References

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  1. M. Huang, Int. J. Solids Structures 40, 1615 (2003).
    [CrossRef]
  2. A. Kilian, J. Kirchhof, B. Kuhlow, G. Przyrembel, and W. Wischmann, J. Lightwave Technol. 18, 193 (2000).
    [CrossRef]
  3. S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
    [CrossRef]
  4. X. Zhao, Y. Z. Xu, and C. Li, IEEE Photon. Technol. Lett. 15, 398 (2003).
    [CrossRef]
  5. X. Zhao, C. Li, and Y. Z. Xu, Opt. Lett. 28, 564 (2003).
    [CrossRef] [PubMed]
  6. J. Canning, Opt. Commun. 191, 225 (2001).
    [CrossRef]
  7. S. Timoshenko, J. Opt. Soc. Am. Rev. Sci. Instrum. 11, 233 (1925).
    [CrossRef]
  8. X. Yan, “Underfill selection and its impact on the reliability of flip chip assembles,” presented at the Delphi Automotive Systems Analytical Design Forum, Kokomo, Ind., March, 1999.
  9. G. N. Savin, Stress Concentration Around Holes (Pergamon, New York, 1961) translated by E. Gros.

2003 (3)

X. Zhao, Y. Z. Xu, and C. Li, IEEE Photon. Technol. Lett. 15, 398 (2003).
[CrossRef]

X. Zhao, C. Li, and Y. Z. Xu, Opt. Lett. 28, 564 (2003).
[CrossRef] [PubMed]

M. Huang, Int. J. Solids Structures 40, 1615 (2003).
[CrossRef]

2001 (1)

J. Canning, Opt. Commun. 191, 225 (2001).
[CrossRef]

2000 (1)

1998 (1)

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

1925 (1)

S. Timoshenko, J. Opt. Soc. Am. Rev. Sci. Instrum. 11, 233 (1925).
[CrossRef]

Bell, A. J.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

Bricheno, T.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

Canning, J.

J. Canning, Opt. Commun. 191, 225 (2001).
[CrossRef]

Cureton, C.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

Day, S.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

Huang, M.

M. Huang, Int. J. Solids Structures 40, 1615 (2003).
[CrossRef]

Kilian, A.

Kirchhof, J.

Kuhlow, B.

Li, C.

X. Zhao, C. Li, and Y. Z. Xu, Opt. Lett. 28, 564 (2003).
[CrossRef] [PubMed]

X. Zhao, Y. Z. Xu, and C. Li, IEEE Photon. Technol. Lett. 15, 398 (2003).
[CrossRef]

Moule, D.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

Ojha, S. M.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

Przyrembel, G.

Savin, G. N.

G. N. Savin, Stress Concentration Around Holes (Pergamon, New York, 1961) translated by E. Gros.

Taylor, J.

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

Timoshenko, S.

S. Timoshenko, J. Opt. Soc. Am. Rev. Sci. Instrum. 11, 233 (1925).
[CrossRef]

Wischmann, W.

Xu, Y. Z.

X. Zhao, C. Li, and Y. Z. Xu, Opt. Lett. 28, 564 (2003).
[CrossRef] [PubMed]

X. Zhao, Y. Z. Xu, and C. Li, IEEE Photon. Technol. Lett. 15, 398 (2003).
[CrossRef]

Yan, X.

X. Yan, “Underfill selection and its impact on the reliability of flip chip assembles,” presented at the Delphi Automotive Systems Analytical Design Forum, Kokomo, Ind., March, 1999.

Zhao, X.

X. Zhao, Y. Z. Xu, and C. Li, IEEE Photon. Technol. Lett. 15, 398 (2003).
[CrossRef]

X. Zhao, C. Li, and Y. Z. Xu, Opt. Lett. 28, 564 (2003).
[CrossRef] [PubMed]

Electron. Lett. (1)

S. M. Ojha, C. Cureton, T. Bricheno, S. Day, D. Moule, A. J. Bell, and J. Taylor, Electron. Lett. 34, 78 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

X. Zhao, Y. Z. Xu, and C. Li, IEEE Photon. Technol. Lett. 15, 398 (2003).
[CrossRef]

Int. J. Solids Structures (1)

M. Huang, Int. J. Solids Structures 40, 1615 (2003).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. Rev. Sci. Instrum. (1)

S. Timoshenko, J. Opt. Soc. Am. Rev. Sci. Instrum. 11, 233 (1925).
[CrossRef]

Opt. Commun. (1)

J. Canning, Opt. Commun. 191, 225 (2001).
[CrossRef]

Opt. Lett. (1)

Other (2)

X. Yan, “Underfill selection and its impact on the reliability of flip chip assembles,” presented at the Delphi Automotive Systems Analytical Design Forum, Kokomo, Ind., March, 1999.

G. N. Savin, Stress Concentration Around Holes (Pergamon, New York, 1961) translated by E. Gros.

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Figures (3)

Fig. 1
Fig. 1

(a) Waveguide structure. (b) Embedded waveguide. (c) Stress diagram.

Fig. 2
Fig. 2

Biaxial stress in a 30µm silica lower cladding layer as a function of thickness of the upper cladding layers with various compositions. The bottom set of lines represents Eq. (2), and the filled circles are the accurate solution for a trilayer structure.8 The curves at the upper part are the results of Ref. 6.

Fig. 3
Fig. 3

Thermal stresses in the core as functions of the ratio of the modulus of the core to that of the upper cladding.

Equations (6)

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αwducdt=dlcdtαs+4Euc*tucEs*tsαuc-αs+4Elc*tlcEs*tsαlc-αs,
σlc=Elc*αw-αlcΔT,
σxxcore=1/Ecore+κ1-1/Eucαuc-αcore+νcore/Ecore-νuc/Eucκ1αw-αuc+αuc-αcore1/Ecore+κ1-1/Euc1/Ecore+κ2-1/Euc-νcore/Ecore-νuc/Euc2ΔT,
σyycore=1/Ecore+κ2-1/Eucκ1αw-αuc+αuc-αcore+νcore/Ecore-νuc/Eucαuc-αcore1/Ecore+κ1-1/Euc1/Ecore+κ2-1/Euc-νcore/Ecore-νuc/Euc2ΔT,
σzzcore=Ecoreαw-αcoreΔT+νcoreσxxcore+σyycore.
σyycore-σxxcore=κ1-κ2αcore-αucΔT/Euc1/Ecore+κ1-1/Euc1/Ecore+κ2-1/Euc-νcore/Ecore-νuc/Euc2+κ1αw-αucΔT1-νcore/Ecore-1-νuc/Euc+κ2/Euc1/Ecore+κ1-1/Euc1/Ecore+κ2-1/Euc-νcore/Ecore-νuc/Euc2.

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