Abstract

We investigate the influence of the substrate on a photonic crystal thermo-optic device on a silicon-on-insulator (SOI) platform. The substrate-induced thermo-optic tuning is obtained as a function of key physical parameters, based on a semi-analytic theory that agrees well with numeric simulations. It is shown that for some devices, the substrate’s contribution to the thermo-optic tuning can exceed 10% for a heater located in the waveguide core and much higher for some other configurations. The slow response of the substrate may also significantly slow down the overall response time of the device. Strategies of minimizing the substrate’s influence are discussed.

© 2013 OSA

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References

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  1. R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
    [CrossRef]
  2. G. T. Reed, ed., Silicon Photonics: the State of the Art (Wiley, 2008).
  3. G. K. Celler and S. Cristoloveanu, “Frontiers of silicon-on-insulator,” J. Appl. Phys.93(9), 4955–4978 (2003).
    [CrossRef]
  4. L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron.14(4), 1132–1139 (2008).
    [CrossRef]
  5. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438(7064), 65–69 (2005).
    [CrossRef] [PubMed]
  6. M. T. Tinker and J.-B. Lee, “Thermal and optical simulation of a photonic crystal light modulator based on the thermo-optic shift of the cut-off frequency,” Opt. Express13(18), 7174–7188 (2005).
    [CrossRef] [PubMed]
  7. T. Chu, H. Yamada, S. Ishida, and Y. Arakawa, “Thermooptic switch based on photonic-crystal line-defect waveguides,” IEEE Photon. Technol. Lett.17(10), 2083–2085 (2005).
    [CrossRef]
  8. L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett.19(5), 342–344 (2007).
    [CrossRef]
  9. D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett.33(2), 147–149 (2008).
    [CrossRef] [PubMed]
  10. N. Ishikura, T. Baba, E. Kuramochi, and M. Notomi, “Large tunable fractional delay of slow light pulse and its application to fast optical correlator,” Opt. Express19(24), 24102–24108 (2011).
    [CrossRef] [PubMed]
  11. L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007).
    [CrossRef]
  12. M. Chahal, G. K. Celler, Y. Jaluria, and W. Jiang, “Thermo-optic characteristics and switching power limit of slow-light photonic crystal structures on a silicon-on-insulator platform,” Opt. Express20(4), 4225–4231 (2012).
    [CrossRef] [PubMed]
  13. Y. Jaluria, Design and Optimization of Thermal Systems, 2nd ed. (CRC Press, 2008).
  14. Y. Jaluria, Natural Convection Heat and Mass Transfer. (Pergamon Press, 1980).
  15. R. F. David, “Computerized thermal analysis of hybrid circuits,” IEEE Trans. Parts Hybrids Packag.13(3), 283–290 (1977).
    [CrossRef]
  16. F. N. Masana, “A closed form solution of junction to substrate thermal resistance in semiconductor chips,” IEEE Trans. Comp. Packag, Manufact. Technol.19, 539–545 (1996).
  17. W. H. Hayt and J. A. Buck, Engineering Electromagnetics, 7th ed. (McGraw-Hill, 2006).
  18. F. P. Incropera, D. P. Dewitt, T. L. Bergman, and A. S. Lavine, Fundamentals of Heat and Mass Transfer, 6th ed. (Wiley, 2007).
  19. W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B82(23), 235306 (2010).
    [CrossRef]
  20. M. Y. Chen, H. Subbaraman, and R. T. Chen, “Photonic crystal fiber beamformer for multiple X band phased-array antenna transmissions,” IEEE Photon. Technol. Lett.20(5), 375–377 (2008).
    [CrossRef]
  21. E. Brookner, Practical Phased Array Antenna Systems. (Artech House, 1991), Chap. 1.
  22. J. A. Rogers, M. G. Lagally, and R. G. Nuzzo, “Synthesis assembly and applications of semiconductor nanomembranes,” Nature477(7362), 45–53 (2011).
    [CrossRef] [PubMed]
  23. S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
    [CrossRef] [PubMed]

2012 (2)

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

M. Chahal, G. K. Celler, Y. Jaluria, and W. Jiang, “Thermo-optic characteristics and switching power limit of slow-light photonic crystal structures on a silicon-on-insulator platform,” Opt. Express20(4), 4225–4231 (2012).
[CrossRef] [PubMed]

2011 (2)

2010 (1)

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B82(23), 235306 (2010).
[CrossRef]

2008 (3)

M. Y. Chen, H. Subbaraman, and R. T. Chen, “Photonic crystal fiber beamformer for multiple X band phased-array antenna transmissions,” IEEE Photon. Technol. Lett.20(5), 375–377 (2008).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron.14(4), 1132–1139 (2008).
[CrossRef]

D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett.33(2), 147–149 (2008).
[CrossRef] [PubMed]

2007 (2)

L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett.19(5), 342–344 (2007).
[CrossRef]

2006 (1)

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
[CrossRef]

2005 (3)

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438(7064), 65–69 (2005).
[CrossRef] [PubMed]

T. Chu, H. Yamada, S. Ishida, and Y. Arakawa, “Thermooptic switch based on photonic-crystal line-defect waveguides,” IEEE Photon. Technol. Lett.17(10), 2083–2085 (2005).
[CrossRef]

M. T. Tinker and J.-B. Lee, “Thermal and optical simulation of a photonic crystal light modulator based on the thermo-optic shift of the cut-off frequency,” Opt. Express13(18), 7174–7188 (2005).
[CrossRef] [PubMed]

2003 (1)

G. K. Celler and S. Cristoloveanu, “Frontiers of silicon-on-insulator,” J. Appl. Phys.93(9), 4955–4978 (2003).
[CrossRef]

1996 (1)

F. N. Masana, “A closed form solution of junction to substrate thermal resistance in semiconductor chips,” IEEE Trans. Comp. Packag, Manufact. Technol.19, 539–545 (1996).

1977 (1)

R. F. David, “Computerized thermal analysis of hybrid circuits,” IEEE Trans. Parts Hybrids Packag.13(3), 283–290 (1977).
[CrossRef]

Arakawa, Y.

T. Chu, H. Yamada, S. Ishida, and Y. Arakawa, “Thermooptic switch based on photonic-crystal line-defect waveguides,” IEEE Photon. Technol. Lett.17(10), 2083–2085 (2005).
[CrossRef]

Ayazi, A.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Baba, T.

Baehr-Jones, T.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Beggs, D. M.

Celler, G. K.

Chahal, M.

Chen, A.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Chen, M. Y.

M. Y. Chen, H. Subbaraman, and R. T. Chen, “Photonic crystal fiber beamformer for multiple X band phased-array antenna transmissions,” IEEE Photon. Technol. Lett.20(5), 375–377 (2008).
[CrossRef]

Chen, R. T.

M. Y. Chen, H. Subbaraman, and R. T. Chen, “Photonic crystal fiber beamformer for multiple X band phased-array antenna transmissions,” IEEE Photon. Technol. Lett.20(5), 375–377 (2008).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron.14(4), 1132–1139 (2008).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett.19(5), 342–344 (2007).
[CrossRef]

L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007).
[CrossRef]

Chen, X.

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron.14(4), 1132–1139 (2008).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett.19(5), 342–344 (2007).
[CrossRef]

L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007).
[CrossRef]

Chu, T.

T. Chu, H. Yamada, S. Ishida, and Y. Arakawa, “Thermooptic switch based on photonic-crystal line-defect waveguides,” IEEE Photon. Technol. Lett.17(10), 2083–2085 (2005).
[CrossRef]

Cristoloveanu, S.

G. K. Celler and S. Cristoloveanu, “Frontiers of silicon-on-insulator,” J. Appl. Phys.93(9), 4955–4978 (2003).
[CrossRef]

David, R. F.

R. F. David, “Computerized thermal analysis of hybrid circuits,” IEEE Trans. Parts Hybrids Packag.13(3), 283–290 (1977).
[CrossRef]

Gould, M.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Gu, L.

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron.14(4), 1132–1139 (2008).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett.19(5), 342–344 (2007).
[CrossRef]

L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007).
[CrossRef]

Hamann, H. F.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438(7064), 65–69 (2005).
[CrossRef] [PubMed]

Hochberg, M.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Huang, S.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Integlia, R. A.

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B82(23), 235306 (2010).
[CrossRef]

Ishida, S.

T. Chu, H. Yamada, S. Ishida, and Y. Arakawa, “Thermooptic switch based on photonic-crystal line-defect waveguides,” IEEE Photon. Technol. Lett.17(10), 2083–2085 (2005).
[CrossRef]

Ishikura, N.

Jaluria, Y.

Jen, A. K. Y.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Jiang, W.

M. Chahal, G. K. Celler, Y. Jaluria, and W. Jiang, “Thermo-optic characteristics and switching power limit of slow-light photonic crystal structures on a silicon-on-insulator platform,” Opt. Express20(4), 4225–4231 (2012).
[CrossRef] [PubMed]

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B82(23), 235306 (2010).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron.14(4), 1132–1139 (2008).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett.19(5), 342–344 (2007).
[CrossRef]

L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007).
[CrossRef]

Krauss, T. F.

Kuramochi, E.

Lagally, M. G.

J. A. Rogers, M. G. Lagally, and R. G. Nuzzo, “Synthesis assembly and applications of semiconductor nanomembranes,” Nature477(7362), 45–53 (2011).
[CrossRef] [PubMed]

Lee, J.-B.

Luo, J.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Masana, F. N.

F. N. Masana, “A closed form solution of junction to substrate thermal resistance in semiconductor chips,” IEEE Trans. Comp. Packag, Manufact. Technol.19, 539–545 (1996).

McNab, S. J.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438(7064), 65–69 (2005).
[CrossRef] [PubMed]

Notomi, M.

Nuzzo, R. G.

J. A. Rogers, M. G. Lagally, and R. G. Nuzzo, “Synthesis assembly and applications of semiconductor nanomembranes,” Nature477(7362), 45–53 (2011).
[CrossRef] [PubMed]

O’Boyle, M.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438(7064), 65–69 (2005).
[CrossRef] [PubMed]

O’Faolain, L.

Rogers, J. A.

J. A. Rogers, M. G. Lagally, and R. G. Nuzzo, “Synthesis assembly and applications of semiconductor nanomembranes,” Nature477(7362), 45–53 (2011).
[CrossRef] [PubMed]

Song, W.

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B82(23), 235306 (2010).
[CrossRef]

Soref, R. A.

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
[CrossRef]

Subbaraman, H.

M. Y. Chen, H. Subbaraman, and R. T. Chen, “Photonic crystal fiber beamformer for multiple X band phased-array antenna transmissions,” IEEE Photon. Technol. Lett.20(5), 375–377 (2008).
[CrossRef]

Tinker, M. T.

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438(7064), 65–69 (2005).
[CrossRef] [PubMed]

Wang, L.

L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007).
[CrossRef]

White, T. P.

Yamada, H.

T. Chu, H. Yamada, S. Ishida, and Y. Arakawa, “Thermooptic switch based on photonic-crystal line-defect waveguides,” IEEE Photon. Technol. Lett.17(10), 2083–2085 (2005).
[CrossRef]

Yip, H. L.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Zhou, X. H.

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Adv. Mater. (Deerfield Beach Fla.) (1)

S. Huang, J. Luo, H. L. Yip, A. Ayazi, X. H. Zhou, M. Gould, A. Chen, T. Baehr-Jones, M. Hochberg, and A. K. Y. Jen, “Efficient poling of electro-optic polymers in thin films and silicon slot waveguides by detachable pyroelectric crystals,” Adv. Mater. (Deerfield Beach Fla.)24(10), OP42–OP47 (2012).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, “High speed silicon photonic crystal waveguide modulator for low voltage operation,” Appl. Phys. Lett.90(7), 071105 (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12(6), 1678–1687 (2006).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Physical mechanism of p-i-n diode based photonic crystal silicon electrooptic modulators for gigahertz operation,” IEEE J. Sel. Top. Quantum Electron.14(4), 1132–1139 (2008).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

T. Chu, H. Yamada, S. Ishida, and Y. Arakawa, “Thermooptic switch based on photonic-crystal line-defect waveguides,” IEEE Photon. Technol. Lett.17(10), 2083–2085 (2005).
[CrossRef]

L. Gu, W. Jiang, X. Chen, and R. T. Chen, “Thermooptically tuned photonic crystal waveguide silicon-on-insulator Mach-Zehnder interferometers,” IEEE Photon. Technol. Lett.19(5), 342–344 (2007).
[CrossRef]

M. Y. Chen, H. Subbaraman, and R. T. Chen, “Photonic crystal fiber beamformer for multiple X band phased-array antenna transmissions,” IEEE Photon. Technol. Lett.20(5), 375–377 (2008).
[CrossRef]

IEEE Trans. Comp. Packag, Manufact. Technol. (1)

F. N. Masana, “A closed form solution of junction to substrate thermal resistance in semiconductor chips,” IEEE Trans. Comp. Packag, Manufact. Technol.19, 539–545 (1996).

IEEE Trans. Parts Hybrids Packag. (1)

R. F. David, “Computerized thermal analysis of hybrid circuits,” IEEE Trans. Parts Hybrids Packag.13(3), 283–290 (1977).
[CrossRef]

J. Appl. Phys. (1)

G. K. Celler and S. Cristoloveanu, “Frontiers of silicon-on-insulator,” J. Appl. Phys.93(9), 4955–4978 (2003).
[CrossRef]

Nature (2)

J. A. Rogers, M. G. Lagally, and R. G. Nuzzo, “Synthesis assembly and applications of semiconductor nanomembranes,” Nature477(7362), 45–53 (2011).
[CrossRef] [PubMed]

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438(7064), 65–69 (2005).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. B (1)

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B82(23), 235306 (2010).
[CrossRef]

Other (6)

E. Brookner, Practical Phased Array Antenna Systems. (Artech House, 1991), Chap. 1.

W. H. Hayt and J. A. Buck, Engineering Electromagnetics, 7th ed. (McGraw-Hill, 2006).

F. P. Incropera, D. P. Dewitt, T. L. Bergman, and A. S. Lavine, Fundamentals of Heat and Mass Transfer, 6th ed. (Wiley, 2007).

G. T. Reed, ed., Silicon Photonics: the State of the Art (Wiley, 2008).

Y. Jaluria, Design and Optimization of Thermal Systems, 2nd ed. (CRC Press, 2008).

Y. Jaluria, Natural Convection Heat and Mass Transfer. (Pergamon Press, 1980).

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Figures (7)

Fig. 1
Fig. 1

Schematic configuration of an active Si photonic crystal waveguide structure on SOI (not drawn to scale).

Fig. 2
Fig. 2

Temperature profile in the substrate along z-axis with the temperature at the bottom surface of the substrate as the reference, T0; for a heating power of 5mW. (a) Results for a chip with tox = 2μm, tsub = 500μm and L as a parameter; the inset shows a typical structure and steady state 3-D FEM simulation result for a simulation chip area of 800μm × 800μm with tSi = 250nm, tox = 2μm, tsub = 500μm, L = 150μm, and W = 400nm. The corresponding photonic crystal structure has a hole radius r = 0.25a, where a is the lattice constant. (b) Results for a chip with L = 100μm, tsub = 500μm and various tox; the inset shows the schematic of heat spreading in the cross section of the device.

Fig. 3
Fig. 3

Fractional delay perturbation due to the substrate as a function of L; (a) For tox = 2μm with tsub as a parameter; (b) For tsub = 500μm with tox as a parameter.

Fig. 4
Fig. 4

The trend of fractional delay perturbation: Semi-analytic results for finite L with different oxide layer thicknesses for tsub = 500μm.

Fig. 5
Fig. 5

Transient temperature response to a square-wave heating power with a period of 400μs, a duty cycle of 50%. L = 150μm, tsub = 250μm, and tox = 1μm. All temperature values are normalized by the steady-state temperature ΔTtot,max. The ΔTsub trace is magnified 5 times for ease of view. For each trace, the 90% point of the rising edge is marked with a star. (10% points of the rising edges of these curves are all close to t = 0, they are not marked to avoid cluttered view).

Fig. 6
Fig. 6

Trend for fsub(L→∞) with tox as the variable for different substrate thicknesses. The inset shows the influence of the hole radius (with tox = 2μm).

Fig. 7
Fig. 7

Simulation result for a non-optimal case where the heater is located at the PCW lateral edge, 7μm from the PCW core (other parameters of this structure are same as in Fig. 5). All temperatures are evaluated in the vertical plane at x = xcore. Note ΔTcoreΔTtot(xcore) = ΔTox(xcore) + ΔTsub(xcore)

Equations (7)

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W'=(W+2 X spr +2 t ox ) and L'=(L+2 t ox ).
V(z)V(0)= 0 z ρ s πε arctan[ (L/2)(W/2) z' (L/2) 2 + (W/2) 2 +z ' 2 ] dz'.
T(z)T(0)= 2Q/W'L' πκ 0 z arctan[ (L'/2)(W'/2) z' (L'/2) 2 + (W'/2) 2 +z ' 2 ] dz'.
{zarctan[ (L'/2)(W'/2)/zμ(z) ](L'/2)arccoth[μ(z)/(W'/2)] (W'/2)arccoth[μ(z)/(L'/2)] }| 0 z ,
Δ T ox =Q t ox /( κ ox L[W+2 X spr ]).
Δ T sub,L = Q Den π κ Si [ t sub W'/2 arctan( W'/2 t sub )+ 1 2 ln( 1+ ( t sub W'/2 ) 2 ) ].
Δ T sub,L Q Den π κ Si ( 1+ln 2 t sub W+2 X spr +2 t ox ).

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