Abstract

We report a method for compensation of errors caused by temperature fluctuations in refractive index measurements using Silicon photonic microring sensors. The method involves determination of resonance wavelength shifts caused by thermal fluctuations using real-time measurement of on-chip temperature variations and thermo-optic coefficient (TOC) of analyte liquids. Resistive metal lines patterned around Silicon microrings are used to track temperature variations and TOC of analyte is calculated by measuring wavelength shifts caused by controlled increments in device temperature. The TOC of de-ionized water is determined to be −1.12 × 10−4/°C, with an accuracy of ±8.26 × 10−6/°C. In our system, chip-surface temperature variations were measured with an instrument limited precision of 0.004 °C yielding a factor of 16 enhancement in tracking accuracy compared to conventional, bottom-of-chip temperature measurement. We show that refractive index detection limit of the microring sensor is also improved by the same factor.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  7. J. Koo and C. Kleinstreuer, “Viscous dissipation effects in microtubes and microchannels,” Int. J. Heat Mass Transfer 47, 3159–3169 (2004).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  20. C.-B. Kim and C. B. Su, “Measurement of the refractive index of liquids at 1.3 and 1.5 micron using a fibre optic Fresnel ratio meter,” Meas. Sci. Technol. 15, 1683–1686 (2004).
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2017 (1)

2016 (1)

2015 (2)

C.-L. Lee, H.-Y. Ho, J.-H. Gu, T.-Y. Yeh, and C.-H. Tseng, “Dual hollow core fiber-based Fabry-Perot interferometer for measuring the thermo-optic coefficients of liquids,” Opt. Lett. 40, 459–462 (2015).
[Crossref] [PubMed]

C. V. Rumens, M. A. Ziai, K. E. Belsey, J. C. Batchelor, and S. J. Holder, “Swelling of pdms networks in solvent vapours; applications for passive rfid wireless sensors,” J. Mater. Chem. C 3, 10091–10098 (2015).
[Crossref]

2014 (1)

W. Bogaerts, M. Fiers, and P. Dumon, “Design Challenges in Silicon Photonics,” IEEE J. Sel. Topics Quantum Electron. 20, 1–8 (2014).
[Crossref]

2012 (3)

2010 (3)

2009 (1)

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

2008 (2)

2007 (1)

2006 (1)

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

2004 (3)

C.-B. Kim and C. B. Su, “Measurement of the refractive index of liquids at 1.3 and 1.5 micron using a fibre optic Fresnel ratio meter,” Meas. Sci. Technol. 15, 1683–1686 (2004).
[Crossref]

J. Koo and C. Kleinstreuer, “Viscous dissipation effects in microtubes and microchannels,” Int. J. Heat Mass Transfer 47, 3159–3169 (2004).
[Crossref]

J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. 29, 2861–2863 (2004).
[Crossref]

1996 (1)

Alvarez, M.

M. Estevez, M. Alvarez, and L. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photonics Rev. 6, 463–487 (2012).
[Crossref]

Baehr-Jones, T.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Baets, R.

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15, 7610–7615 (2007).
[Crossref] [PubMed]

Bailey, R. C.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Bartolozzi, I.

Batchelor, J. C.

C. V. Rumens, M. A. Ziai, K. E. Belsey, J. C. Batchelor, and S. J. Holder, “Swelling of pdms networks in solvent vapours; applications for passive rfid wireless sensors,” J. Mater. Chem. C 3, 10091–10098 (2015).
[Crossref]

Belsey, K. E.

C. V. Rumens, M. A. Ziai, K. E. Belsey, J. C. Batchelor, and S. J. Holder, “Swelling of pdms networks in solvent vapours; applications for passive rfid wireless sensors,” J. Mater. Chem. C 3, 10091–10098 (2015).
[Crossref]

Bienstman, P.

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15, 7610–7615 (2007).
[Crossref] [PubMed]

Bogaerts, W.

W. Bogaerts, M. Fiers, and P. Dumon, “Design Challenges in Silicon Photonics,” IEEE J. Sel. Topics Quantum Electron. 20, 1–8 (2014).
[Crossref]

Bolivar, P. H.

Cheben, P.

Chen, H.

Chen, Y.

Ciddor, P. E.

Claes, T.

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

De Koninck, Y.

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

De Vos, K.

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15, 7610–7615 (2007).
[Crossref] [PubMed]

Delâge, A.

Densmore, A.

Dumon, P.

W. Bogaerts, M. Fiers, and P. Dumon, “Design Challenges in Silicon Photonics,” IEEE J. Sel. Topics Quantum Electron. 20, 1–8 (2014).
[Crossref]

Estevez, M.

M. Estevez, M. Alvarez, and L. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photonics Rev. 6, 463–487 (2012).
[Crossref]

Fédéli, J.-M.

Fiers, M.

W. Bogaerts, M. Fiers, and P. Dumon, “Design Challenges in Silicon Photonics,” IEEE J. Sel. Topics Quantum Electron. 20, 1–8 (2014).
[Crossref]

Frey, B. J.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

D. B. Leviton and B. J. Frey, “Temperature-dependent absolute refractive index measurements of synthetic fused silica,” arXiv:0805.0091 (2008).

Girones, J.

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

Gleeson, M. A.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Grajower, M.

Gu, J.-H.

Gunn, L. C.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Gunn, W. G.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Han, W.-T.

Henschel, W.

Ho, H.-Y.

Hochberg, M.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Holder, S. J.

C. V. Rumens, M. A. Ziai, K. E. Belsey, J. C. Batchelor, and S. J. Holder, “Swelling of pdms networks in solvent vapours; applications for passive rfid wireless sensors,” J. Mater. Chem. C 3, 10091–10098 (2015).
[Crossref]

Hu, J.

Iqbal, M.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Izuhara, T.

Janz, S.

Jeon, S.-W.

Ju, S.

Keinan, G.

Kim, C.-B.

C.-B. Kim and C. B. Su, “Measurement of the refractive index of liquids at 1.3 and 1.5 micron using a fibre optic Fresnel ratio meter,” Meas. Sci. Technol. 15, 1683–1686 (2004).
[Crossref]

Kim, Y. H.

Kimerling, L.

Kleinstreuer, C.

J. Koo and C. Kleinstreuer, “Viscous dissipation effects in microtubes and microchannels,” Int. J. Heat Mass Transfer 47, 3159–3169 (2004).
[Crossref]

Koo, J.

J. Koo and C. Kleinstreuer, “Viscous dissipation effects in microtubes and microchannels,” Int. J. Heat Mass Transfer 47, 3159–3169 (2004).
[Crossref]

Kurz, H.

Lapointe, J.

Lechuga, L.

M. Estevez, M. Alvarez, and L. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photonics Rev. 6, 463–487 (2012).
[Crossref]

Lee, B. H.

Lee, C.-L.

Leviton, D. B.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

D. B. Leviton and B. J. Frey, “Temperature-dependent absolute refractive index measurements of synthetic fused silica,” arXiv:0805.0091 (2008).

Levy, U.

Lipson, M.

Loock, H.-P.

Lopinski, G.

Lu, Y.-Q.

Ma, R.

Madison, T. J.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

Mazurski, N.

McKinnon, R.

Messaoudène, S.

Michel, J.

Mischki, T.

Naiman, A.

Niehusmann, J.

Painter, O.

Park, C.-S.

Park, S. J.

Popelka, S.

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

Post, E.

Preston, K.

Qiu, S.-J.

Raghunathan, V.

Robinson, J. T.

Rumens, C. V.

C. V. Rumens, M. A. Ziai, K. E. Belsey, J. C. Batchelor, and S. J. Holder, “Swelling of pdms networks in solvent vapours; applications for passive rfid wireless sensors,” J. Mater. Chem. C 3, 10091–10098 (2015).
[Crossref]

Sanders, C.

Saunders, J. E.

Schacht, E.

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15, 7610–7615 (2007).
[Crossref] [PubMed]

Schmid, J. H.

Spaugh, B.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Stern, L.

Su, C. B.

C.-B. Kim and C. B. Su, “Measurement of the refractive index of liquids at 1.3 and 1.5 micron using a fibre optic Fresnel ratio meter,” Meas. Sci. Technol. 15, 1683–1686 (2004).
[Crossref]

Tseng, C.-H.

Tybor, F.

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

Vachon, M.

Vörckel, A.

Wahlbrink, T.

Waldron, P.

Xu, D. X.

Xu, D.-X.

Xu, F.

Ye, W. N.

Yeh, T.-Y.

Ziai, M. A.

C. V. Rumens, M. A. Ziai, K. E. Belsey, J. C. Batchelor, and S. J. Holder, “Swelling of pdms networks in solvent vapours; applications for passive rfid wireless sensors,” J. Mater. Chem. C 3, 10091–10098 (2015).
[Crossref]

Appl. Opt. (2)

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

M. Iqbal, M. A. Gleeson, B. Spaugh, F. Tybor, W. G. Gunn, M. Hochberg, T. Baehr-Jones, R. C. Bailey, and L. C. Gunn, “Label-Free Biosensor Arrays Based on Silicon Ring Resonators and High-Speed Optical Scanning Instrumentation,” IEEE J. Sel. Topics Quantum Electron. 16, 654–661 (2010).
[Crossref]

W. Bogaerts, M. Fiers, and P. Dumon, “Design Challenges in Silicon Photonics,” IEEE J. Sel. Topics Quantum Electron. 20, 1–8 (2014).
[Crossref]

IEEE Photon. J. (1)

K. De Vos, J. Girones, T. Claes, Y. De Koninck, S. Popelka, E. Schacht, R. Baets, and P. Bienstman, “Multiplexed Antibody Detection With an Array of Silicon-on-Insulator Microring Resonators,” IEEE Photon. J. 1, 225–235 (2009).
[Crossref]

Int. J. Heat Mass Transfer (1)

J. Koo and C. Kleinstreuer, “Viscous dissipation effects in microtubes and microchannels,” Int. J. Heat Mass Transfer 47, 3159–3169 (2004).
[Crossref]

J. Mater. Chem. C (1)

C. V. Rumens, M. A. Ziai, K. E. Belsey, J. C. Batchelor, and S. J. Holder, “Swelling of pdms networks in solvent vapours; applications for passive rfid wireless sensors,” J. Mater. Chem. C 3, 10091–10098 (2015).
[Crossref]

Laser Photonics Rev. (1)

M. Estevez, M. Alvarez, and L. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photonics Rev. 6, 463–487 (2012).
[Crossref]

Meas. Sci. Technol. (1)

C.-B. Kim and C. B. Su, “Measurement of the refractive index of liquids at 1.3 and 1.5 micron using a fibre optic Fresnel ratio meter,” Meas. Sci. Technol. 15, 1683–1686 (2004).
[Crossref]

Opt. Express (6)

V. Raghunathan, W. N. Ye, J. Hu, T. Izuhara, J. Michel, and L. Kimerling, “Athermal operation of Silicon waveguides: spectral, second order and footprint dependencies,” Opt. Express 18, 17631–17639 (2010).
[Crossref] [PubMed]

J. T. Robinson, K. Preston, O. Painter, and M. Lipson, “First-principle derivation of gain in high-index-contrast waveguides,” Opt. Express 16, 16659–16669 (2008).
[Crossref] [PubMed]

D. X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,” Opt. Express 16, 15137–15148 (2008).
[Crossref] [PubMed]

K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15, 7610–7615 (2007).
[Crossref] [PubMed]

D.-X. Xu, M. Vachon, A. Densmore, R. Ma, S. Janz, A. Delâge, J. Lapointe, P. Cheben, J. H. Schmid, E. Post, S. Messaoudène, and J.-M. Fédéli, “Real-time cancellation of temperature induced resonance shifts in SOI wire waveguide ring resonator label-free biosensor arrays,” Opt. Express 18, 22867–22879 (2010).
[Crossref] [PubMed]

Y. H. Kim, S. J. Park, S.-W. Jeon, S. Ju, C.-S. Park, W.-T. Han, and B. H. Lee, “Thermo-optic coefficient measurement of liquids based on simultaneous temperature and refractive index sensing capability of a two-mode fiber interferometric probe,” Opt. Express 20, 23744–23754 (2012).
[Crossref] [PubMed]

Opt. Lett. (3)

Optica (1)

Proc. SPIE (1)

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[Crossref]

Other (1)

D. B. Leviton and B. J. Frey, “Temperature-dependent absolute refractive index measurements of synthetic fused silica,” arXiv:0805.0091 (2008).

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Figures (8)

Fig. 1
Fig. 1 Design and simulation of Silicon microring TOC sensor (a) A schematic showing the photonic sensor chip. (b) Simulation result showing the distribution of electric field (TE-like mode) along the cross section of a photonic wire waveguide.
Fig. 2
Fig. 2 Images of fabricated microrings and metal rings for thermo-optic coefficient measurement. (a) Scanning Electron Microscope (SEM) image of Silicon microring surrounded by platinum metal ring (b) Magnified view of the wire waveguide and metal ring (c) Microscope image showing gold contact lines for electrical probing of metal rings
Fig. 3
Fig. 3 Measurements of metal ring resistance versus temperature (a) With air covered microrings. The reference data was obtained using a calibrated thermal chuck. Three trials were performed using our characterization assembly (b) With DI-water covered microrings. The R2 value exceeded 0.99 for all linear fits.
Fig. 4
Fig. 4 Measurements for determination of TOC: Silicon (a) Positions of a transmission resonance for different temperatures. (b) Shift in resonance position as a function of temperature (3 trials). R2 value for all curve fits exceeded 0.99.
Fig. 5
Fig. 5 Measurements for determination of TOC: De-ionized water (a) Positions of a transmission resonance for different temperatures. FWHM of resonances is about 0.3 nm. relative to that of air cladding (b) Shift in resonance position as a function of temperature. R2 value for all curve fits exceeded 0.99.
Fig. 6
Fig. 6 Measurements for determination of TOC: Organic liquids (a) Resonance wavelength shifts for ethanol cladding. (b) Shifts in resonance wavelength for IPA cladding. In this case, the slope is negative owing to very strong (negative) TOC of the analyte.
Fig. 7
Fig. 7 Determination of error limits of TOC of water. Seven iterations of wavelength shift slope measurements were performed.
Fig. 8
Fig. 8 Measurement of ambient thermal fluctuations. (a) Wavelength shift and resistance measurements on Silicon microring and a metal ring spaced apart by about 500 µm (centre to centre), showing uncorrelated variations of quantities. (b) Measurements for concentric microring and metal ring as shown in Fig. 2(b). (c) Cross-correlation plots of wavelength shift and resistance variation signals. A good correlation is observed between the signals for small separation between microring and metal ring at zero delay position. In contrast, no clear trend is visible in the correlation output for large separation between metal and silicon rings. (d) Temperature variation as calculated by resistance measurement, compared with the read-out of the temperature controller.

Tables (2)

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Table 1 Simulation of confinement factors for different claddings of the photonic wire (387 nm × 220 nm) at working wavelength of 1510 nm.

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Table 2 Comparison of thermo-optic coefficients determined by our study with values reported in literature near 1550 nm wavelength band.

Equations (12)

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Δ λ o b s = Δ λ s e n + ( δ λ δ T ) c l Δ T
δ n e f f δ T = Γ c o n c o T + Γ b o x n b o x T + Γ c l n c l T
Γ s = n g n s ( s ϵ | E | 2 dxdy t o t ϵ | E | 2 dxdy ) = n g n s γ s
n c l T = ( δ n e f f δ T ) c l Γ c o n c o T Γ b o x n b o x T Γ c l
n e f f T = n g λ r e s Δ λ Δ T
n s i T = ( n e f f T ) a i r Γ b o x n b o x T Γ c o
Δ λ o b s = Δ λ s e n + λ r e s n g ( Γ c o ( δ n c o δ T ) + Γ o x ( δ n o x δ T ) + Γ c l ( δ n c l δ T ) ) Δ T
Δ λ o b s Δ λ s e n + ( ( Δ λ Δ T ) a i r + λ r e s n g Γ c l ( δ n c l δ T ) ) Δ T
Δ λ s e n = Δ λ o b s ( K 1 + K 2 ( δ n c l δ T ) ) Δ T
Δ n c l = Δ λ s e n ( Δ λ Δ n c l )
e Δ λ / Δ T = n c l γ c l σ Δ λ / Δ T λ r e s
e w = | n c l γ c l ( 1 λ r e s Δ λ Δ T 1 n c o ( n s i T ) γ c o w ) ( n c l γ c l 2 γ c l w ) ( 1 λ r e s Δ λ Δ T γ c o n c o n s i T ) | σ w

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