Abstract

An accurate model for the silicon refractive index including its temperature and wavelength dependence is critically important for many disciplines of science and technology. Currently, such a model for temperatures above 22°C in the optical communication bands is not available. The temperature dependence in the spectral response of integrated echelle grating filters made in silicon-on-insulator is solely determined by the optical properties of the slab waveguide, making it largely immune to dimensional uncertainties. This feature renders the echelle filters a reliable tool to evaluate the thermo-optic properties of silicon. Here we investigate the temperature dependence of silicon echelle filters for the wavelength range of both O and C bands, measured between 22°C to 80°C. We show that if a constant thermo-optic coefficient of silicon is assumed for each band, as is common in the literature, the predictions show an underestimate of up to 10% in the temperature-induced channel wavelength shift. We propose and assess a model of silicon refractive index that encompasses both the wavelength and temperature dependence of its thermo-optic coefficients. We start from literature data for bulk silicon and further refine the model using the echelle filter measurement results. This model is validated through accurate predictions of device channel wavelengths and their temperature dependence, including the quadratic term, over a wide wavelength and temperature range. This work also demonstrates a new high-precision method for characterizing the optical properties of a variety of materials.

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2018 (4)

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12(9), 534–539 (2018).
[Crossref]

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[Crossref]

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[Crossref]

D. Melati, P. G. Verly, A. Delâge, P. Cheben, J. H. Schmid, S. Janz, and D.-X. Xu, “Athermal echelle grating filter in silicon-on-insulator using a temperature-synchronized input,” Opt. Express 26(22), 28651–28659 (2018).
[Crossref]

2017 (2)

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Janz, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. Lett. 42(11), 2239–2242 (2017).
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[Crossref]

2016 (2)

D. Melati, A. Waqas, A. Alippi, and A. Melloni, “Wavelength and composition dependence of the thermo-optic coefficient for InGaAsP-based integrated waveguides,” J. Appl. Phys. 120(21), 213102 (2016).
[Crossref]

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, “Roadmap on silicon photonics,” J. Opt. 18(7), 073003 (2016).
[Crossref]

2015 (5)

2014 (1)

D.-X. Xu, J. H. Schmid, G. T. Reed, G. Z. Mashanovich, D. J. Thomson, M. Nedeljkovic, X. Chen, V. Thourhout, S. Keyvaninia, and S. K. Selvaraja, “Silicon Photonic Integration Platform-Have We Found the Sweet Spot?” IEEE J. Sel. Top. Quantum Electron. 20(4), 189–205 (2014).
[Crossref]

2013 (1)

2012 (2)

2011 (1)

B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,” Phys. Rev. Lett. 106(3), 030801 (2011).
[Crossref]

2010 (2)

2009 (2)

M. Uenuma and T. Motooka, “Temperature-independent silicon waveguide optical filter,” Opt. Lett. 34(5), 599–601 (2009).
[Crossref]

X. Wang, S. Xiao, W. Zheng, F. Wang, Y. Li, Y. Hao, X. Jiang, M. Wang, and J. Yang, “Athermal silicon arrayed waveguide grating with polymer-filled slot structure,” Opt. Commun. 282(14), 2841–2844 (2009).
[Crossref]

2008 (2)

2007 (1)

2005 (2)

2000 (1)

F. G. Della Corte, M. Esposito Montefusco, L. Moretti, I. Rendina, and G. Cocorullo, “Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models,” J. Appl. Phys. 88(12), 7115–7119 (2000).
[Crossref]

1999 (1)

G. Cocorullo, F. G. Della Corte, and I. Rendina, “Temperature dependence of the thermo-optic coefficient in crystalline silicon between room temperature and 550 K at the wavelength of 1523 nm,” Appl. Phys. Lett. 74(22), 3338–3340 (1999).
[Crossref]

1997 (1)

Y. Inoue, A. Kaneko, F. Hanawa, H. Takahashi, K. Hattori, and S. Sumida, “Athermal silica-based arrayed-waveguide grating multiplexer,” Electron. Lett. 33(23), 1945–1947 (1997).
[Crossref]

1994 (1)

J. A. Mccaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon, gallium arsenide, and indium phosphide,” Phys. Rev. B 49(11), 7408–7417 (1994).
[Crossref]

1980 (1)

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9(3), 561–658 (1980).
[Crossref]

1959 (1)

F. Lukes, “The temperature-dependence of the refractive index of silicon,” J. Phys. Chem. Solids 11(3–4), 342–344 (1959).
[Crossref]

Ahmed, Z.

Ahn, H.

Alippi, A.

D. Melati, A. Waqas, A. Alippi, and A. Melloni, “Wavelength and composition dependence of the thermo-optic coefficient for InGaAsP-based integrated waveguides,” J. Appl. Phys. 120(21), 213102 (2016).
[Crossref]

Andreas, B.

B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,” Phys. Rev. Lett. 106(3), 030801 (2011).
[Crossref]

Arai, S.

Arbabi, A.

Atsumi, Y.

Azuma, Y.

B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,” Phys. Rev. Lett. 106(3), 030801 (2011).
[Crossref]

Bae, H. K.

Baehr-Jones, T.

Bartl, G.

B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,” Phys. Rev. Lett. 106(3), 030801 (2011).
[Crossref]

Becker, P.

B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,” Phys. Rev. Lett. 106(3), 030801 (2011).
[Crossref]

Berger, M.

Bettin, H.

B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,” Phys. Rev. Lett. 106(3), 030801 (2011).
[Crossref]

Bienstman, P.

Bland-Hawthorn, J.

S. G. Leon-Saval, J. Bland-Hawthorn, and S. Ellis, “Astrophotonics: a promising arena for silicon photonics,” in Silicon Photonics XIV, G. T. Reed and A. P. Knights, eds. (SPIE, 2019), 109230M.

Boeuf, F.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, “Roadmap on silicon photonics,” J. Opt. 18(7), 073003 (2016).
[Crossref]

Bogaerts, W.

Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
[Crossref]

S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing Fabrication and Thermal Tolerances of All-Silicon FIR Wavelength Filters,” IEEE Photonics Technol. Lett. 27(8), 871–874 (2015).
[Crossref]

S. Dwivedi, A. Ruocco, M. Vanslembrouck, T. Spuesens, P. Bienstman, P. Dumon, T. Van Vaerenbergh, and W. Bogaerts, “Experimental Extraction of Effective Refractive Index and Thermo-Optic Coefficients of Silicon-on-Insulator Waveguides Using Interferometers,” J. Lightwave Technol. 33(21), 4471–4477 (2015).
[Crossref]

Borys, M.

B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,” Phys. Rev. Lett. 106(3), 030801 (2011).
[Crossref]

Bowers, J. E.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, “Roadmap on silicon photonics,” J. Opt. 18(7), 073003 (2016).
[Crossref]

Busch, I.

B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,” Phys. Rev. Lett. 106(3), 030801 (2011).
[Crossref]

Calvo, M. L.

Cassan, E.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, “Roadmap on silicon photonics,” J. Opt. 18(7), 073003 (2016).
[Crossref]

Cengel, Y.

Y. Cengel and A. Ghajar, Heat and Mass Transfer: Fundamentals and Applications, 5th edition (McGraw-Hill, 2015).

Chan, E.

Cheben, P.

D. Melati, P. G. Verly, A. Delâge, P. Cheben, J. H. Schmid, S. Janz, and D.-X. Xu, “Athermal echelle grating filter in silicon-on-insulator using a temperature-synchronized input,” Opt. Express 26(22), 28651–28659 (2018).
[Crossref]

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D. Melati, P. G. Verly, A. Delâge, P. Cheben, J. H. Schmid, S. Janz, and D.-X. Xu, “Athermal echelle grating filter in silicon-on-insulator using a temperature-synchronized input,” Opt. Express 26(22), 28651–28659 (2018).
[Crossref]

A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Janz, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,” Opt. Lett. 42(11), 2239–2242 (2017).
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D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, “Roadmap on silicon photonics,” J. Opt. 18(7), 073003 (2016).
[Crossref]

P. Cheben, J. H. Schmid, S. Wang, D.-X. Xu, M. Vachon, S. Janz, J. Lapointe, Y. Painchaud, and M.-J. Picard, “Broadband polarization independent nanophotonic coupler for silicon waveguides with ultra-high efficiency,” Opt. Express 23(17), 22553–22563 (2015).
[Crossref]

D.-X. Xu, J. H. Schmid, G. T. Reed, G. Z. Mashanovich, D. J. Thomson, M. Nedeljkovic, X. Chen, V. Thourhout, S. Keyvaninia, and S. K. Selvaraja, “Silicon Photonic Integration Platform-Have We Found the Sweet Spot?” IEEE J. Sel. Top. Quantum Electron. 20(4), 189–205 (2014).
[Crossref]

Y. Atsumi, D.-X. Xu, A. Delâge, J. H. Schmid, M. Vachon, P. Cheben, S. Janz, N. Nishiyama, and S. Arai, “Simultaneous retrieval of fluidic refractive index and surface adsorbed molecular film thickness using silicon wire waveguide biosensors,” Opt. Express 20(24), 26969–26977 (2012).
[Crossref]

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(22), 22867–79 (2010).
[Crossref]

W. N. Ye, D.-X. Xu, S. Janz, P. Cheben, M.-J. Picard, B. Lamontagne, and N. G. Tarr, “Birefringence Control Using Stress Engineering in Silicon-on-Insulator (SOI) Waveguides,” J. Lightwave Technol. 23(3), 1308–1318 (2005).
[Crossref]

D.-X. Xu, “Polarization Control in Silicon Photonic Waveguide Components Using Cladding Stress Engineering,” in Silicon Photonics II. Topics in Applied Physics, Vol 119., D. J. Lockwood and L. Pavesi, eds. (Springer, 2011), pp. 31–70.

P. Cheben, A. Delâge, S. Janz, and D.-X. Xu, “Echelle and Arrayed Waveguide Gratings for WDM and Spectral Analysis,” in Advances in Information Optics and Photonics (SPIE, 2008), pp. 599–632.

Yang, J.

X. Wang, S. Xiao, W. Zheng, F. Wang, Y. Li, Y. Hao, X. Jiang, M. Wang, and J. Yang, “Athermal silicon arrayed waveguide grating with polymer-filled slot structure,” Opt. Commun. 282(14), 2841–2844 (2009).
[Crossref]

Ye, W. N.

Zheng, W.

X. Wang, S. Xiao, W. Zheng, F. Wang, Y. Li, Y. Hao, X. Jiang, M. Wang, and J. Yang, “Athermal silicon arrayed waveguide grating with polymer-filled slot structure,” Opt. Commun. 282(14), 2841–2844 (2009).
[Crossref]

Zhou, X.

X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12(9), 534–539 (2018).
[Crossref]

Zilkie, A.

D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, “Roadmap on silicon photonics,” J. Opt. 18(7), 073003 (2016).
[Crossref]

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[Crossref]

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S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing Fabrication and Thermal Tolerances of All-Silicon FIR Wavelength Filters,” IEEE Photonics Technol. Lett. 27(8), 871–874 (2015).
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D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, “Roadmap on silicon photonics,” J. Opt. 18(7), 073003 (2016).
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Nat. Commun. (1)

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X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12(9), 534–539 (2018).
[Crossref]

Opt. Commun. (1)

X. Wang, S. Xiao, W. Zheng, F. Wang, Y. Li, Y. Hao, X. Jiang, M. Wang, and J. Yang, “Athermal silicon arrayed waveguide grating with polymer-filled slot structure,” Opt. Commun. 282(14), 2841–2844 (2009).
[Crossref]

Opt. Express (6)

P. Cheben, J. H. Schmid, S. Wang, D.-X. Xu, M. Vachon, S. Janz, J. Lapointe, Y. Painchaud, and M.-J. Picard, “Broadband polarization independent nanophotonic coupler for silicon waveguides with ultra-high efficiency,” Opt. Express 23(17), 22553–22563 (2015).
[Crossref]

G.-D. Kim, H.-S. Lee, C.-H. Park, S.-S. Lee, B. T. Lim, H. K. Bae, and W.-G. Lee, “Silicon photonic temperature sensor employing a ring resonator manufactured using a standard CMOS process,” Opt. Express 18(21), 22215–22221 (2010).
[Crossref]

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(19), 15137–15148 (2008).
[Crossref]

D. Melati, P. G. Verly, A. Delâge, P. Cheben, J. H. Schmid, S. Janz, and D.-X. Xu, “Athermal echelle grating filter in silicon-on-insulator using a temperature-synchronized input,” Opt. Express 26(22), 28651–28659 (2018).
[Crossref]

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(22), 22867–79 (2010).
[Crossref]

Y. Atsumi, D.-X. Xu, A. Delâge, J. H. Schmid, M. Vachon, P. Cheben, S. Janz, N. Nishiyama, and S. Arai, “Simultaneous retrieval of fluidic refractive index and surface adsorbed molecular film thickness using silicon wire waveguide biosensors,” Opt. Express 20(24), 26969–26977 (2012).
[Crossref]

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Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,” Photonics Res. 6(11), 1008–1020 (2018).
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D.-X. Xu, “Polarization Control in Silicon Photonic Waveguide Components Using Cladding Stress Engineering,” in Silicon Photonics II. Topics in Applied Physics, Vol 119., D. J. Lockwood and L. Pavesi, eds. (Springer, 2011), pp. 31–70.

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P. Cheben, A. Delâge, S. Janz, and D.-X. Xu, “Echelle and Arrayed Waveguide Gratings for WDM and Spectral Analysis,” in Advances in Information Optics and Photonics (SPIE, 2008), pp. 599–632.

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S. G. Leon-Saval, J. Bland-Hawthorn, and S. Ellis, “Astrophotonics: a promising arena for silicon photonics,” in Silicon Photonics XIV, G. T. Reed and A. P. Knights, eds. (SPIE, 2019), 109230M.

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Figures (7)

Fig. 1.
Fig. 1. (a) Schematic of an echelle grating filter; (b) Optical image of a fabricated device.
Fig. 2.
Fig. 2. (a) Measured transmission spectra of device R250A(O). System setup loss and waveguide coupling loss are included. a) Transmission spectra at room temperature; (b) Transmission spectra as a function of temperature for one of the channels.
Fig. 3.
Fig. 3. (a) Central wavelength as a function of temperature for two output channels of sample R250(O). The residue, i.e. the difference between the predicted and measured channel wavelength as a function of temperature, for (b) the linear fit and (c) the quadratic fit.
Fig. 4.
Fig. 4. Wavelength shift Δλk/ΔT extracted assuming a linear dependence on temperature for all the devices as listed in Table 1. The channel wavelengths at 25°C are used for the x-axis. Symbols: experiments; solid lines: modeled values using Eq. (6) assuming ηSi = 1.93 × 10−4/K for the O band and ηSi = 1.85 × 10−4/K for the C band. (a) Data for the 1300 nm wavelength range. The experimental scatter is within ± 0.7%. The discrepancy between the experimental and the modeled values is ∼ 6%. (b) Data for the 1550 nm wavelength range. The experimental scatter is within ± 1.5%. The discrepancy between the experimental and the modeled values is ∼ 10%.
Fig. 5.
Fig. 5. Measured (blue diamond) and predicted (red square) channel wavelength (all 10 channels for diffraction order 46) as a function of temperature for sample R400B(O).
Fig. 6.
Fig. 6. Comparison of the measured and calculated Δλk/ΔT (using a linear fit) as a function of wavelength for both the O and C bands using the proposed models for the silicon refractive index and a silicon thickness of 220 nm. Results using a constant ηSi for each wavelength band, as already presented in Fig. 4, are also included for comparison. The calculations were performed for the wavelength range of 1260–1370 nm and 1470–1580 nm respectively, as indicated by the blue and green symbols. The solid blue and green lines are for guiding the eye.
Fig. 7.
Fig. 7. Comparison of measured and calculated (a) D1 and (b) D2 using a 2nd order polynomial fit to the channel wavelengths, as a function of wavelength for both the O and C bands using the new models for the silicon refractive index nSi(λ, T). The calculations were performed for the wavelength range of 1260–1370 nm and 1470–1580 nm respectively, as indicated by the blue and green symbols. The solid green and blue lines are for guiding the eye.

Tables (4)

Tables Icon

Table 1. Design parameters for the echelle grating filters used in the measurements. R: Rowland circle radius; m: diffraction order; FSR: free spectral range.

Tables Icon

Table 2. Sellmeier coefficients for the refractive index of silicon at the reference temperature of T0=295 K (22 °C) as in Frey et al. [31].

Tables Icon

Table 3. Coefficients for the silicon refractive index model as described in Eq. 9.

Tables Icon

Table 4. The modeled silicon refractive index and its thermo-optic coefficient compared with other published sources.

Equations (11)

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

m λ = n e f f L ,
m λ k = n e f f Λ ( sin θ + s i n φ k ) .
σ k = Λ ( s i n θ + s i n φ k ) m = λ k n e f f ( λ k ) .
n e f f ( λ k , T ) = n r e f ( T ) + ( λ k λ r e f ) α ( T ) + ( λ k λ r e f ) 2 β ( T ) .
λ k = λ r e f ( α 2 β 1 2 β σ k ) ( α 2 β 1 2 β σ k ) 2 n r e f λ r e f σ k β .
Δ λ k Δ T λ k ( T ) n g ( T ) [ n e f f T + 2 n e f f T 2 Δ T + ( λ k λ r e f ) 2 n e f f T λ ] .
d λ k d T λ k ( T ) n g ( λ k , T ) n e f f ( λ k , T ) T .
n 0 2 ( λ , T 0 ) 1 = 1 3 S i ( T 0 ) λ 2 λ 2 λ i 2 ( T 0 ) .
n T ( λ , T ) = C 1 ( λ ) + C 2 ( T T 0 ) + C 3 ( T T 0 ) 2 .
n S i ( λ , T ) = n 0 ( λ , T 0 ) + C 1 ( λ ) ( T T 0 ) + C 2 ( T T 0 ) 2 / 2 + C 3 ( T T 0 ) 3 / 3 ,
C 1 ( λ ) = R 0 + R 1 λ + R 2 λ 2 .

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