Abstract

Employing a semi-analytic approach, we study the influence of key structural and optical parameters on the thermo-optic characteristics of photonic crystal waveguide (PCW) structures on a silicon-on-insulator (SOI) platform. The power consumption and spatial temperature profile of such structures are given as explicit functions of various structural, thermal and optical parameters, offering physical insight not available in finite-element simulations. Agreement with finite-element simulations and experiments is demonstrated. Thermal enhancement of the air-bridge structure is analyzed. The practical limit of thermo-optic switching power in slow light PCWs is discussed, and the scaling with key parameters is analyzed. Optical switching with sub-milliwatt power is shown viable.

© 2012 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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  20. C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. 97(18), 183302 (2010).
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    [CrossRef]

2011

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

2010

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: An analytic approach,” Phys. Rev. B 82(23), 235306 (2010).
[CrossRef]

C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. 97(18), 183302 (2010).
[CrossRef]

Y. Cui, K. Liu, D. L. MacFarlane, and J.-B. Lee, “Thermo-optically tunable silicon photonic crystal light modulator,” Opt. Lett. 35(21), 3613–3615 (2010).
[CrossRef] [PubMed]

2008

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]

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]

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

2007

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]

2006

2005

V. M. N. Passaro, F. Magno, and A. V. Tsarev, “Investigation of thermo-optic effect and multi-reflector tunable filter/multiplexer in SOI waveguides,” Opt. Express 13(9), 3429–3437 (2005).
[CrossRef] [PubMed]

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. Express 13(18), 7174–7188 (2005).
[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,” Nature 438(7064), 65–69 (2005).
[CrossRef] [PubMed]

E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides,” Phys. Rev. B 72(11), 115102 (2005).
[CrossRef]

2004

2003

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

1994

L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. 41(1), 69–75 (1994).
[CrossRef]

Antoniadis, D. A.

L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. 41(1), 69–75 (1994).
[CrossRef]

Beggs, D. M.

Camargo, E. A.

Celler, G. K.

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

Chakravarty, S.

C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. 97(18), 183302 (2010).
[CrossRef]

Chen, R. T.

C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. 97(18), 183302 (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]

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]

Chong, H. M. H.

Chung, J. E.

L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. 41(1), 69–75 (1994).
[CrossRef]

Cristoloveanu, S.

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

Cui, Y.

Davis, B. L.

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

De La Rue, R. M.

Dulkeith, E.

E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides,” Phys. Rev. B 72(11), 115102 (2005).
[CrossRef]

El-Kady, I.

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

Fan, S. H.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Fathpour, S.

Fejer, M. M.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Flik, M. I.

L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. 41(1), 69–75 (1994).
[CrossRef]

Goodson, K. E.

L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. 41(1), 69–75 (1994).
[CrossRef]

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,” Nature 438(7064), 65–69 (2005).
[CrossRef] [PubMed]

Harris, J. S.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Huo, Y.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Hussein, M. I.

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

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. B 82(23), 235306 (2010).
[CrossRef]

Iodice, M.

Jalali, B.

Jiang, W.

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: An analytic approach,” Phys. Rev. B 82(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]

Joannopoulos, J. D.

M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[CrossRef] [PubMed]

Kim, B.

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

Krauss, T. F.

Lai, W.-C.

C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. 97(18), 183302 (2010).
[CrossRef]

Lee, B. S.

C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. 97(18), 183302 (2010).
[CrossRef]

Lee, J.-B.

Leseman, Z. C.

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

Lin, C.-Y.

C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. 97(18), 183302 (2010).
[CrossRef]

Liu, K.

MacFarlane, D. L.

Magno, F.

Mazzi, G.

McNab, S. J.

E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides,” Phys. Rev. B 72(11), 115102 (2005).
[CrossRef]

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,” Nature 438(7064), 65–69 (2005).
[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,” Nature 438(7064), 65–69 (2005).
[CrossRef] [PubMed]

O’Faolain, L.

Olsson-III, R. H.

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

Pan, J.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Passaro, V. M. N.

Povinelli, M. L.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Reinke, C. M.

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

Sandhu, S.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Scaccabarozzi, L.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Sirleto, L.

Soljacic, M.

M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004).
[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. B 82(23), 235306 (2010).
[CrossRef]

Soref, R.

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

Su, L. T.

L. T. Su, J. E. Chung, D. A. Antoniadis, K. E. Goodson, and M. I. Flik, “Measurement and modeling of self-heating in SOI nMOSFETS,” IEEE Trans. Electron. Dev. 41(1), 69–75 (1994).
[CrossRef]

Su, M. F.

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

Timp, R.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Tinker, M. T.

Tsarev, A. V.

Vlasov, Y. A.

E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides,” Phys. Rev. B 72(11), 115102 (2005).
[CrossRef]

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,” Nature 438(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]

Wang, X.

C.-Y. Lin, X. Wang, S. Chakravarty, B. S. Lee, W.-C. Lai, and R. T. Chen, “Wideband group velocity independent coupling into slow light silicon photonic crystal waveguide,” Appl. Phys. Lett. 97(18), 183302 (2010).
[CrossRef]

White, T. P.

Yamanaka, K.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. H. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

AIP Advances

C. M. Reinke, M. F. Su, B. L. Davis, B. Kim, M. I. Hussein, Z. C. Leseman, R. H. Olsson-III, and I. El-Kady, “Thermal conductivity prediction of nanoscale phononic crystal slabs using a hybrid lattice dynamics-continuum mechanics technique,” AIP Advances 1(4), 041403 (2011).
[CrossRef]

Appl. Phys. Lett.

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Figures (4)

Fig. 1
Fig. 1

Configurations of Si active PCW structures. (a) SOI; (b) Air-bridge (membrane).

Fig. 2
Fig. 2

Temperature profiles in the top Si layer of a PCW (center: x = 0) for various hole radii and in a homogenized slab with κeff(r). Inset: 3D temperature profile in a PCW with r/a = 0.25. One period of the PCW along the y axis is shown.

Fig. 3
Fig. 3

ΔTox/Q vs. waveguide length L (for tox=2μm). Inset: 3D temperature distribution in a chip obtained from finite element simulation for a PCW structure having tSi=250nm, W=400nm, tox=2μm, κeff(r=0.25a) on a 200μm×200μm substrate with a thickness of 100μm.

Fig. 4
Fig. 4

Qπ vs. group index (ng) for r/a = 0.25 for an SOI PCW structure (for various tox), and the estimated 3dB-propagation-loss length for a membrane PCW structure. Inset: The thermal enhancement for a membrane PCW with tox = 2μm.

Tables (1)

Tables Icon

Table 1 Values of Xspr(r) and κeff(r) for Various Hole Sizes

Equations (9)

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T(x)exp[ ( | x |W/2 )/ X spr ( r ) ],for| x |>W/2,
X spr = X Si = [ t Si t ox κ Si / κ ox ] 1/2 ,
κ eff (r)= κ ox X spr 2 (r)/( t Si t ox ).
Q= κ ox L[W+2 X spr (r)](Δ T ox / t ox ),
Δϕ=2πΔnσL n g /(nλ),
Q π =nλ κ ox [W+2 X spr (r)]/[2σ t ox n g (dn/dT)].
Q membrane /2= κ eff L t Si (ΔT) membrane (ΔT) edge W membrane /2 = κ ox L X Si (ΔT) edge t ox ,
Q membrane = κ ox L(2 X Si ) (ΔT) membrane t ox κ eff (r) X Si κ eff (r) X Si + κ Si W membrane /2 ,
(ΔT) membrane (ΔT) SOI X spr (r) X Si κ eff (r) X Si + κ Si W membrane /2 κ eff (r) X Si .

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