Abstract

Ring resonators with TiO2 core confinement factors from 0.07 to 0.42 are fabricated and measured for thermal sensitivity achieving −2.9 pm/K thermal drift in the best case. Materials used are CMOS compatible (TiO2, SiO2 and Si3N4) on a Si substrate. The under discussed role of stress in thermo-optic behavior is clearly observed when contrasting waveguides buried in SiO2 to those with etched sidewalls revealed to air. Multiphysics simulations are conducted to provide a theoretical explanation of this phenomenon in contrast to the more widely reported theories on thermo-optic behavior dominated by confinement factor.

© 2014 Optical Society of America

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References

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  1. T. H. Kim, B. G. You, H. J. Lee, and T. H. Rhee, “Athermal AWG multiplexer/demultiplexer for E/C-band WDM-PON application,” in 2007 Asia Opt. Fiber Commun. Optoelectron. Conf. 2 (IEEE, 2007), pp. 330–332.
  2. N. Ooba, Y. Hibino, Y. Inoue, A. Sugita, “Athermal silica-based arrayed-waveguide grating multiplexer using bimetal plate temperature compensator,” Electron. Lett. 36(21), 1800–1801 (2000).
    [CrossRef]
  3. Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. P. Alipour, A. Atabaki, A. Eftekhar, and A. Adibi, “Athermal performance in titania-clad microring Resonators on SOI,” in Integrated Photonics Research, Silicon and Nanophotonics and Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IMC6.
    [CrossRef]
  9. F. Qiu, A. M. Spring, F. Yu, S. Yokoyama, “Complementary metal–oxide–semiconductor compatible athermal silicon nitride/titanium dioxide hybrid micro-ring resonators,” Appl. Phys. Lett. 102(5), 051106 (2013).
    [CrossRef]
  10. S. S. Djordjevic, K. Shang, B. Guan, S. T. S. Cheung, L. Liao, J. Basak, H.-F. Liu, S. J. B. Yoo, “CMOS-compatible, athermal silicon ring modulators clad with titanium dioxide,” Opt. Express 21(12), 13958–13968 (2013).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. M. Huang, X. Yan, “Thermal-stress effects on the temperature sensitivity of optical waveguides,” J. Opt. Soc. Am. B 20(6), 1326–1333 (2003).
    [CrossRef]
  14. A. Arbabi, L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of Si3N4 and SiOx using microring resonances,” Opt. Lett. 38(19), 3878–3881 (2013).
    [CrossRef] [PubMed]
  15. C. Ottermann, K. Bange, “Correlation between the density of TiO2 films and their properties,” Thin Solid Films 286(1–2), 32–34 (1996).
    [CrossRef]
  16. M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40(7), 1615–1632 (2003).
    [CrossRef]
  17. C. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, K. Bange, “Young’s modulus and density of thin TiO2 films produced by different methods,” Mater. Res. Soc. 436, 251–256 (1996).
    [CrossRef]
  18. “Stress-optical effects with generalized plane strain.” COMSOL Multiphysics 4.3a (2012).
  19. A. Khan, J. Philip, P. Hess, “Young’s modulus of silicon nitride used in scanning force microscope cantilevers,” J. Appl. Phys. 95(4), 1667–1672 (2004).
    [CrossRef]
  20. National Physical Laboratory Kaye and Laby Table of Physical and Chemical Constants, Version 1.1,(2010), http://www.kayelaby.npl.co.uk/general_physics/2_3/2_3_5.html . Accessed Sept. 4, 2013.
  21. C.-C. Lee, C.-L. Tien, W.-S. Sheu, C.-C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
    [CrossRef]
  22. P. Temple-Boyer, C. Rossi, E. Scheid, “Residual stress in low pressure chemical vapor deposition SiNx films deposited from silane and ammonia,” J. Vac. Sci. Technol. A 16(4), 2003–2007 (1998).
    [CrossRef]

2013 (4)

2010 (2)

2009 (1)

2004 (1)

A. Khan, J. Philip, P. Hess, “Young’s modulus of silicon nitride used in scanning force microscope cantilevers,” J. Appl. Phys. 95(4), 1667–1672 (2004).
[CrossRef]

2003 (2)

M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40(7), 1615–1632 (2003).
[CrossRef]

M. Huang, X. Yan, “Thermal-stress effects on the temperature sensitivity of optical waveguides,” J. Opt. Soc. Am. B 20(6), 1326–1333 (2003).
[CrossRef]

2001 (1)

C.-C. Lee, C.-L. Tien, W.-S. Sheu, C.-C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

2000 (1)

N. Ooba, Y. Hibino, Y. Inoue, A. Sugita, “Athermal silica-based arrayed-waveguide grating multiplexer using bimetal plate temperature compensator,” Electron. Lett. 36(21), 1800–1801 (2000).
[CrossRef]

1999 (1)

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

1998 (2)

H. Tanobe, Y. Kondo, Y. Kadota, K. Okamoto, Y. Yoshikuni, “Temperature insensitive arrayed waveguide gratings on InP substrates,” IEEE Photonics Technol. Lett. 10(2), 235–237 (1998).
[CrossRef]

P. Temple-Boyer, C. Rossi, E. Scheid, “Residual stress in low pressure chemical vapor deposition SiNx films deposited from silane and ammonia,” J. Vac. Sci. Technol. A 16(4), 2003–2007 (1998).
[CrossRef]

1997 (1)

T. D. Visser, H. Blok, B. Demeulenaere, D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33(10), 1763–1766 (1997).
[CrossRef]

1996 (2)

C. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, K. Bange, “Young’s modulus and density of thin TiO2 films produced by different methods,” Mater. Res. Soc. 436, 251–256 (1996).
[CrossRef]

C. Ottermann, K. Bange, “Correlation between the density of TiO2 films and their properties,” Thin Solid Films 286(1–2), 32–34 (1996).
[CrossRef]

Abe, M.

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

Albert, J.

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

Anderson, O.

C. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, K. Bange, “Young’s modulus and density of thin TiO2 films produced by different methods,” Mater. Res. Soc. 436, 251–256 (1996).
[CrossRef]

Arbabi, A.

Bange, K.

C. Ottermann, K. Bange, “Correlation between the density of TiO2 films and their properties,” Thin Solid Films 286(1–2), 32–34 (1996).
[CrossRef]

C. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, K. Bange, “Young’s modulus and density of thin TiO2 films produced by different methods,” Mater. Res. Soc. 436, 251–256 (1996).
[CrossRef]

Basak, J.

Blok, H.

T. D. Visser, H. Blok, B. Demeulenaere, D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33(10), 1763–1766 (1997).
[CrossRef]

Cardenas, J.

Cheung, S. T. S.

Demeulenaere, B.

T. D. Visser, H. Blok, B. Demeulenaere, D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33(10), 1763–1766 (1997).
[CrossRef]

Djordjevic, S. S.

Goddard, L. L.

Guan, B.

Guha, B.

Hess, P.

A. Khan, J. Philip, P. Hess, “Young’s modulus of silicon nitride used in scanning force microscope cantilevers,” J. Appl. Phys. 95(4), 1667–1672 (2004).
[CrossRef]

C. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, K. Bange, “Young’s modulus and density of thin TiO2 films produced by different methods,” Mater. Res. Soc. 436, 251–256 (1996).
[CrossRef]

Hibino, Y.

N. Ooba, Y. Hibino, Y. Inoue, A. Sugita, “Athermal silica-based arrayed-waveguide grating multiplexer using bimetal plate temperature compensator,” Electron. Lett. 36(21), 1800–1801 (2000).
[CrossRef]

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

Hill, K. O.

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

Himeno, A.

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

Ho, S.-T.

Hu, J.

Huang, D.

Huang, H.

Huang, M.

M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40(7), 1615–1632 (2003).
[CrossRef]

M. Huang, X. Yan, “Thermal-stress effects on the temperature sensitivity of optical waveguides,” J. Opt. Soc. Am. B 20(6), 1326–1333 (2003).
[CrossRef]

Inoue, Y.

N. Ooba, Y. Hibino, Y. Inoue, A. Sugita, “Athermal silica-based arrayed-waveguide grating multiplexer using bimetal plate temperature compensator,” Electron. Lett. 36(21), 1800–1801 (2000).
[CrossRef]

Izuhara, T.

Jaing, C.-C.

C.-C. Lee, C.-L. Tien, W.-S. Sheu, C.-C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Johnson, D. C.

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

Kadota, Y.

H. Tanobe, Y. Kondo, Y. Kadota, K. Okamoto, Y. Yoshikuni, “Temperature insensitive arrayed waveguide gratings on InP substrates,” IEEE Photonics Technol. Lett. 10(2), 235–237 (1998).
[CrossRef]

Khan, A.

A. Khan, J. Philip, P. Hess, “Young’s modulus of silicon nitride used in scanning force microscope cantilevers,” J. Appl. Phys. 95(4), 1667–1672 (2004).
[CrossRef]

Kimerling, L.

Kondo, Y.

H. Tanobe, Y. Kondo, Y. Kadota, K. Okamoto, Y. Yoshikuni, “Temperature insensitive arrayed waveguide gratings on InP substrates,” IEEE Photonics Technol. Lett. 10(2), 235–237 (1998).
[CrossRef]

Kuschnereit, R.

C. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, K. Bange, “Young’s modulus and density of thin TiO2 films produced by different methods,” Mater. Res. Soc. 436, 251–256 (1996).
[CrossRef]

Lee, C.-C.

C.-C. Lee, C.-L. Tien, W.-S. Sheu, C.-C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Lenstra, D.

T. D. Visser, H. Blok, B. Demeulenaere, D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33(10), 1763–1766 (1997).
[CrossRef]

Liao, L.

Lipson, M.

Liu, H.-F.

Liu, W.

Michel, J.

Moooka, T.

Okamoto, K.

H. Tanobe, Y. Kondo, Y. Kadota, K. Okamoto, Y. Yoshikuni, “Temperature insensitive arrayed waveguide gratings on InP substrates,” IEEE Photonics Technol. Lett. 10(2), 235–237 (1998).
[CrossRef]

Ooba, N.

N. Ooba, Y. Hibino, Y. Inoue, A. Sugita, “Athermal silica-based arrayed-waveguide grating multiplexer using bimetal plate temperature compensator,” Electron. Lett. 36(21), 1800–1801 (2000).
[CrossRef]

Ottermann, C.

C. Ottermann, K. Bange, “Correlation between the density of TiO2 films and their properties,” Thin Solid Films 286(1–2), 32–34 (1996).
[CrossRef]

C. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, K. Bange, “Young’s modulus and density of thin TiO2 films produced by different methods,” Mater. Res. Soc. 436, 251–256 (1996).
[CrossRef]

Philip, J.

A. Khan, J. Philip, P. Hess, “Young’s modulus of silicon nitride used in scanning force microscope cantilevers,” J. Appl. Phys. 95(4), 1667–1672 (2004).
[CrossRef]

Qiu, F.

F. Qiu, A. M. Spring, F. Yu, S. Yokoyama, “Complementary metal–oxide–semiconductor compatible athermal silicon nitride/titanium dioxide hybrid micro-ring resonators,” Appl. Phys. Lett. 102(5), 051106 (2013).
[CrossRef]

Raghunathan, V.

Rossi, C.

P. Temple-Boyer, C. Rossi, E. Scheid, “Residual stress in low pressure chemical vapor deposition SiNx films deposited from silane and ammonia,” J. Vac. Sci. Technol. A 16(4), 2003–2007 (1998).
[CrossRef]

Scheid, E.

P. Temple-Boyer, C. Rossi, E. Scheid, “Residual stress in low pressure chemical vapor deposition SiNx films deposited from silane and ammonia,” J. Vac. Sci. Technol. A 16(4), 2003–2007 (1998).
[CrossRef]

Shang, K.

Sheu, W.-S.

C.-C. Lee, C.-L. Tien, W.-S. Sheu, C.-C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Spring, A. M.

F. Qiu, A. M. Spring, F. Yu, S. Yokoyama, “Complementary metal–oxide–semiconductor compatible athermal silicon nitride/titanium dioxide hybrid micro-ring resonators,” Appl. Phys. Lett. 102(5), 051106 (2013).
[CrossRef]

Sugita, A.

N. Ooba, Y. Hibino, Y. Inoue, A. Sugita, “Athermal silica-based arrayed-waveguide grating multiplexer using bimetal plate temperature compensator,” Electron. Lett. 36(21), 1800–1801 (2000).
[CrossRef]

Tanaka, T.

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

Tanobe, H.

H. Tanobe, Y. Kondo, Y. Kadota, K. Okamoto, Y. Yoshikuni, “Temperature insensitive arrayed waveguide gratings on InP substrates,” IEEE Photonics Technol. Lett. 10(2), 235–237 (1998).
[CrossRef]

Temple-Boyer, P.

P. Temple-Boyer, C. Rossi, E. Scheid, “Residual stress in low pressure chemical vapor deposition SiNx films deposited from silane and ammonia,” J. Vac. Sci. Technol. A 16(4), 2003–2007 (1998).
[CrossRef]

Tien, C.-L.

C.-C. Lee, C.-L. Tien, W.-S. Sheu, C.-C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Tu, Y.

Uenuma, M.

Visser, T. D.

T. D. Visser, H. Blok, B. Demeulenaere, D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33(10), 1763–1766 (1997).
[CrossRef]

Yan, X.

Ye, W. N.

Yokoyama, S.

F. Qiu, A. M. Spring, F. Yu, S. Yokoyama, “Complementary metal–oxide–semiconductor compatible athermal silicon nitride/titanium dioxide hybrid micro-ring resonators,” Appl. Phys. Lett. 102(5), 051106 (2013).
[CrossRef]

Yoo, S. J. B.

Yoshikuni, Y.

H. Tanobe, Y. Kondo, Y. Kadota, K. Okamoto, Y. Yoshikuni, “Temperature insensitive arrayed waveguide gratings on InP substrates,” IEEE Photonics Technol. Lett. 10(2), 235–237 (1998).
[CrossRef]

Yu, F.

F. Qiu, A. M. Spring, F. Yu, S. Yokoyama, “Complementary metal–oxide–semiconductor compatible athermal silicon nitride/titanium dioxide hybrid micro-ring resonators,” Appl. Phys. Lett. 102(5), 051106 (2013).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

F. Qiu, A. M. Spring, F. Yu, S. Yokoyama, “Complementary metal–oxide–semiconductor compatible athermal silicon nitride/titanium dioxide hybrid micro-ring resonators,” Appl. Phys. Lett. 102(5), 051106 (2013).
[CrossRef]

Electron. Lett. (2)

N. Ooba, Y. Hibino, Y. Inoue, A. Sugita, “Athermal silica-based arrayed-waveguide grating multiplexer using bimetal plate temperature compensator,” Electron. Lett. 36(21), 1800–1801 (2000).
[CrossRef]

Y. Hibino, M. Abe, T. Tanaka, A. Himeno, J. Albert, D. C. Johnson, K. O. Hill, “Temperature-insensitive UV-induced Bragg gratings in silica-based planar lightwave circuits on Si,” Electron. Lett. 35(21), 1844–1845 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. D. Visser, H. Blok, B. Demeulenaere, D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33(10), 1763–1766 (1997).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

H. Tanobe, Y. Kondo, Y. Kadota, K. Okamoto, Y. Yoshikuni, “Temperature insensitive arrayed waveguide gratings on InP substrates,” IEEE Photonics Technol. Lett. 10(2), 235–237 (1998).
[CrossRef]

Int. J. Solids Struct. (1)

M. Huang, “Stress effects on the performance of optical waveguides,” Int. J. Solids Struct. 40(7), 1615–1632 (2003).
[CrossRef]

J. Appl. Phys. (1)

A. Khan, J. Philip, P. Hess, “Young’s modulus of silicon nitride used in scanning force microscope cantilevers,” J. Appl. Phys. 95(4), 1667–1672 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Vac. Sci. Technol. A (1)

P. Temple-Boyer, C. Rossi, E. Scheid, “Residual stress in low pressure chemical vapor deposition SiNx films deposited from silane and ammonia,” J. Vac. Sci. Technol. A 16(4), 2003–2007 (1998).
[CrossRef]

Mater. Res. Soc. (1)

C. Ottermann, R. Kuschnereit, O. Anderson, P. Hess, K. Bange, “Young’s modulus and density of thin TiO2 films produced by different methods,” Mater. Res. Soc. 436, 251–256 (1996).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Rev. Sci. Instrum. (1)

C.-C. Lee, C.-L. Tien, W.-S. Sheu, C.-C. Jaing, “An apparatus for the measurement of internal stress and thermal expansion coefficient of metal oxide films,” Rev. Sci. Instrum. 72(4), 2128–2133 (2001).
[CrossRef]

Thin Solid Films (1)

C. Ottermann, K. Bange, “Correlation between the density of TiO2 films and their properties,” Thin Solid Films 286(1–2), 32–34 (1996).
[CrossRef]

Other (4)

“Stress-optical effects with generalized plane strain.” COMSOL Multiphysics 4.3a (2012).

National Physical Laboratory Kaye and Laby Table of Physical and Chemical Constants, Version 1.1,(2010), http://www.kayelaby.npl.co.uk/general_physics/2_3/2_3_5.html . Accessed Sept. 4, 2013.

P. Alipour, A. Atabaki, A. Eftekhar, and A. Adibi, “Athermal performance in titania-clad microring Resonators on SOI,” in Integrated Photonics Research, Silicon and Nanophotonics and Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IMC6.
[CrossRef]

T. H. Kim, B. G. You, H. J. Lee, and T. H. Rhee, “Athermal AWG multiplexer/demultiplexer for E/C-band WDM-PON application,” in 2007 Asia Opt. Fiber Commun. Optoelectron. Conf. 2 (IEEE, 2007), pp. 330–332.

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Figures (4)

Fig. 1
Fig. 1

Cross section of waveguide designs with TiO2 or hybrid cores.

Fig. 2
Fig. 2

(a) Measured thermal drift with separate confinement models simulated for buried and revealed rings. (b) Measurments compared to other literature with TiO2 cladding demonstrates a surpression of TOC in buried TiO2.

Fig. 3
Fig. 3

Comparison of the (a) confinement, (b) stress, and (c) combined stress/confinement models.

Fig. 4
Fig. 4

Thermally induced stress profiles for Buried and revealed type waveguides. Figures are von Mises Stress profiles in Pa with a 10K temperature difference. A 10x displacement enhancement is applied for better visualization.

Tables (2)

Tables Icon

Table 1 Comparison of thermo-optic models and measured data

Tables Icon

Table 2 Material Properties for Simulation (λ0 = 1550nm)

Equations (5)

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

m λ r =2π n eff ( λ r ,T)R(T) d λ r dT = n eff λ r n g R R T + λ r n g n eff T λ r n g ( n eff α sub + n eff T )
where, n eff T k Γ k n k T .
m λ r =2π n eff ( λ r ,σ,T)R(σ,T) d λ r dT = λ r n g ( n eff σ dσ dT + n eff T )+ n eff λ r n g R R σ dσ dT + n eff λ r n g R R T d λ r dT λ r n g ( n eff α sub + d n eff dT +β dσ dT )
where, d n eff dT n eff σ dσ dT + n eff T .
[ d n xx d n yy d n zz d n yz d n xz d n xy ]=[ n xx n 0 n yy n 0 n zz n 0 n yz 0 n xz 0 n xy 0 ]=[ B 1 B 2 B 2 0 0 0 B 2 B 1 B 2 0 0 0 B 2 B 2 B 1 0 0 0 0 0 0 B 3 0 0 0 0 0 0 B 3 0 0 0 0 0 0 B 3 ][ σ xx σ yy σ zz σ yz σ xz σ xy ]

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