Ge-rich graded-index Si<sub>1-</sub>xGex waveguides with broadband tight mode confinement and flat anomalous dispersion for nonlinear mid-infrared photonics

J. M. Ramirez; V. Vakarin; J. Frigerio; P. Chaisakul; D. Chrastina; X. Le Roux; A. Ballabio; L. Vivien; G. Isella; D. Marris-Morini

doi:10.1364/OE.25.006561

Ge-rich graded-index Si_1- _x Ge _x waveguides with broadband tight mode confinement and flat anomalous dispersion for nonlinear mid-infrared photonics

J. M. Ramirez, V. Vakarin, J. Frigerio, P. Chaisakul, D. Chrastina, X. Le Roux, A. Ballabio, L. Vivien, G. Isella, and D. Marris-Morini

Optics Express
Vol. 25,
Issue 6,
pp. 6561-6567
(2017)
•https://doi.org/10.1364/OE.25.006561

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J. M. Ramirez, V. Vakarin, J. Frigerio, P. Chaisakul, D. Chrastina, X. Le Roux, A. Ballabio, L. Vivien, G. Isella, and D. Marris-Morini, "Ge-rich graded-index Si_1-xGex waveguides with broadband tight mode confinement and flat anomalous dispersion for nonlinear mid-infrared photonics," Opt. Express 25, 6561-6567 (2017)

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Related Topics
Table of Contents Category
- Integrated Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Dispersion compensation devices (130.2035)
- Kerr effect (190.3270)
- Nonlinear optics, integrated optics (190.4390)

About this Article
History
- Original Manuscript: January 20, 2017
- Revised Manuscript: March 8, 2017
- Manuscript Accepted: March 8, 2017
- Published: March 13, 2017

Accessible

Open Access

Abstract

This work explores the use of Ge-rich graded-index Si_1-_xGe_x rib waveguides as building blocks to develop integrated nonlinear optical devices for broadband operation in the mid-IR. The vertical Ge gradient concentration in the waveguide core renders unique properties to the guided optical mode, providing tight mode confinement over a broadband mid-IR wavelength range from λ = 3 µm to 8 µm. Additionally, the gradual vertical confinement pulls the optical mode upwards in the waveguide core, overlapping with the Ge-rich area where the nonlinear refractive index is larger. Moreover, the Ge-rich graded-index Si_1-_xGe_x waveguides allow efficient tailoring of the chromatic dispersion curves, achieving flat anomalous dispersion for the quasi-TM optical mode with D ≤ 14 ps/nm/km over a ~1.4 octave span while retaining an optimum third-order nonlinear parameter, γ_eff. These results confirm the potential of Ge-rich graded-index Si_1-_xGe_x waveguides as an attractive platform to develop mid-IR nonlinear approaches requiring broadband dispersion engineering.

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References

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|
Author
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Publication

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[Crossref] [PubMed]
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[Crossref]
P. Barritault, M. Brun, P. Labeye, O. Lartigue, J. M. Hartmann, and S. Nicoletti, “Mlines characterization of the refractive index profile of SiGe gradient waveguides at 2.15 µm,” Opt. Express 21(9), 11506–11515 (2013).
[Crossref] [PubMed]
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[Crossref] [PubMed]
J. M. Ramirez, V. Vakarin, C. Gilles, J. Frigerio, A. Ballabio, P. Chaisakul, X. L. Roux, C. Alonso-Ramos, G. Maisons, L. Vivien, M. Carras, G. Isella, and D. Marris-Morini, “Low-loss Ge-rich Si0.2Ge0.8 waveguides for mid-infrared photonics,” Opt. Lett. 42(1), 105–108 (2017).
[Crossref] [PubMed]
F. De Leonardis, B. Troia, R. A. Soref, and V. M. N. Passaro, “Dispersion of nonresonant third-order nonlinearities in GeSiSn ternary alloys,” Sci. Rep. 6(1), 32622 (2016).
[Crossref] [PubMed]
P. Moontragoon, R. A. Soref, and Z. Ikonic, “The direct and indirect bandgaps of unstrained SixGe1-x-ySny and their photonic device applications,” J. Appl. Phys. 112(7), 073106 (2012).
[Crossref]
L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear Group IV photonics based on silicon and germanium: from near-infrared to mid-infrared,” Nanophotonics 3(4–5), 247–268 (2014).
B. Tatian, “Fitting refractive-index data with the Sellmeier dispersion formula,” Appl. Opt. 23(24), 4477–4485 (1984).
[Crossref] [PubMed]
N. Singh, D. D. Hudson, and B. J. Eggleton, “Silicon-on-sapphire pillar waveguides for Mid-IR supercontinuum generation,” Opt. Express 23(13), 17345–17354 (2015).
[Crossref] [PubMed]
L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20(2), 1685–1690 (2012).
[Crossref] [PubMed]
N. Singh, D. D. Hudson, Y. Yu, C. Grillet, S. D. Jackson, A. Casas-Bedoya, A. Read, P. Atanackovic, S. G. Duvall, S. Palomba, B. Luther-Davies, S. Madden, D. J. Moss, and B. J. Eggleton, “Midinfrared supercontinuum generation from 2 to 6 μm in a silicon nanowire,” Optica 2(9), 797–802 (2015).
[Crossref]
A. Kordts, M. H. P. Pfeiffer, H. Guo, V. Brasch, and T. J. Kippenberg, “Higher order mode suppression in high-Q anomalous dispersion SiN microresonators for temporal dissipative Kerr soliton formation,” Opt. Lett. 41(3), 452–455 (2016).
[Crossref] [PubMed]
N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]
M. Foster, K. Moll, and A. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12(13), 2880–2887 (2004).
[Crossref] [PubMed]
L. Carletti, M. Sinobad, P. Ma, Y. Yu, D. Allioux, R. Orobtchouk, M. Brun, S. Ortiz, P. Labeye, J. M. Hartmann, S. Nicoletti, S. Madden, B. Luther-Davies, D. J. Moss, C. Monat, and C. Grillet, “Mid-Infrared nonlinear optical response of Si-Ge waveguides with ultra-short optical pulses,” Opt. Express 23(25), 32202–32214 (2015).
[Crossref] [PubMed]

2017 (1)

J. M. Ramirez, V. Vakarin, C. Gilles, J. Frigerio, A. Ballabio, P. Chaisakul, X. L. Roux, C. Alonso-Ramos, G. Maisons, L. Vivien, M. Carras, G. Isella, and D. Marris-Morini, “Low-loss Ge-rich Si0.2Ge0.8 waveguides for mid-infrared photonics,” Opt. Lett. 42(1), 105–108 (2017).
[Crossref] [PubMed]

2016 (2)

A. Kordts, M. H. P. Pfeiffer, H. Guo, V. Brasch, and T. J. Kippenberg, “Higher order mode suppression in high-Q anomalous dispersion SiN microresonators for temporal dissipative Kerr soliton formation,” Opt. Lett. 41(3), 452–455 (2016).
[Crossref] [PubMed]

F. De Leonardis, B. Troia, R. A. Soref, and V. M. N. Passaro, “Dispersion of nonresonant third-order nonlinearities in GeSiSn ternary alloys,” Sci. Rep. 6(1), 32622 (2016).
[Crossref] [PubMed]

2015 (3)

N. Singh, D. D. Hudson, and B. J. Eggleton, “Silicon-on-sapphire pillar waveguides for Mid-IR supercontinuum generation,” Opt. Express 23(13), 17345–17354 (2015).
[Crossref] [PubMed]

N. Singh, D. D. Hudson, Y. Yu, C. Grillet, S. D. Jackson, A. Casas-Bedoya, A. Read, P. Atanackovic, S. G. Duvall, S. Palomba, B. Luther-Davies, S. Madden, D. J. Moss, and B. J. Eggleton, “Midinfrared supercontinuum generation from 2 to 6 μm in a silicon nanowire,” Optica 2(9), 797–802 (2015).
[Crossref]

L. Carletti, M. Sinobad, P. Ma, Y. Yu, D. Allioux, R. Orobtchouk, M. Brun, S. Ortiz, P. Labeye, J. M. Hartmann, S. Nicoletti, S. Madden, B. Luther-Davies, D. J. Moss, C. Monat, and C. Grillet, “Mid-Infrared nonlinear optical response of Si-Ge waveguides with ultra-short optical pulses,” Opt. Express 23(25), 32202–32214 (2015).
[Crossref] [PubMed]

2014 (5)

L. Zhang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Nonlinear Group IV photonics based on silicon and germanium: from near-infrared to mid-infrared,” Nanophotonics 3(4–5), 247–268 (2014).

M. Brun, P. Labeye, G. Grand, J. M. Hartmann, F. Boulila, M. Carras, and S. Nicoletti, “Low loss SiGe graded index waveguides for mid-IR applications,” Opt. Express 22(1), 508–518 (2014).
[Crossref] [PubMed]

J. S. Penadés, C. Alonso-Ramos, A. Z. Khokhar, M. Nedeljkovic, L. A. Boodhoo, A. Ortega-Moñux, I. Molina-Fernández, P. Cheben, and G. Z. Mashanovich, “Suspended SOI waveguide with sub-wavelength grating cladding for mid-infrared,” Opt. Lett. 39(19), 5661–5664 (2014).
[Crossref] [PubMed]

F. De Leonardis, B. Troia, and V. M. Passaro, “Mid-IR optical and nonlinear properties of germanium on silicon optical waveguides,” J. Lightwave Technol. 32(22), 3747–3757 (2014).
[Crossref]

A. Malik, S. Dwivedi, L. Van Landschoot, M. Muneeb, Y. Shimura, G. Lepage, J. Van Campenhout, W. Vanherle, T. Van Opstal, R. Loo, and G. Roelkens, “Ge-on-Si and Ge-on-SOI thermo-optic phase shifters for the mid-infrared,” Opt. Express 22(23), 28479–28488 (2014).
[Crossref] [PubMed]

2013 (4)

A. Malik, M. Muneeb, S. Pathak, Y. Shimura, J. Van Campenhout, R. Loo, and G. Roelkens, “Germanium-on-silicon mid-infrared arrayed waveguide grating multiplexers,” IEEE Photonics Technol. Lett. 25(18), 1805–1808 (2013).
[Crossref]

S. Khan, J. Chiles, J. Ma, and S. Fathpour, “Silicon-on-nitride waveguides for mid-and near-infrared integrated photonics,” Appl. Phys. Lett. 102(12), 121104 (2013).
[Crossref]

H. Lin, L. Li, Y. Zou, S. Danto, J. D. Musgraves, K. Richardson, S. Kozacik, M. Murakowski, D. Prather, P. T. Lin, V. Singh, A. Agarwal, L. C. Kimerling, and J. Hu, “Demonstration of high-Q mid-infrared chalcogenide glass-on-silicon resonators,” Opt. Lett. 38(9), 1470–1472 (2013).
[Crossref] [PubMed]

P. Barritault, M. Brun, P. Labeye, O. Lartigue, J. M. Hartmann, and S. Nicoletti, “Mlines characterization of the refractive index profile of SiGe gradient waveguides at 2.15 µm,” Opt. Express 21(9), 11506–11515 (2013).
[Crossref] [PubMed]

2012 (3)

L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20(2), 1685–1690 (2012).
[Crossref] [PubMed]

Y. C. Chang, P. Wägli, V. Paeder, A. Homsy, L. Hvozdara, P. van der Wal, J. Di Francesco, N. F. de Rooij, and H. P. Herzig, “Cocaine detection by a mid-infrared waveguide integrated with a microfluidic chip,” Lab Chip 12(17), 3020–3023 (2012).
[Crossref] [PubMed]

P. Moontragoon, R. A. Soref, and Z. Ikonic, “The direct and indirect bandgaps of unstrained SixGe1-x-ySny and their photonic device applications,” J. Appl. Phys. 112(7), 073106 (2012).
[Crossref]

2011 (2)

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]

B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
[Crossref] [PubMed]

2010 (2)

A. Spott, Y. Liu, T. Baehr-Jones, R. R. Ilic, and M. Hochberg, “Silicon Waveguides and Ring Resonators at 5.5 µm,” Appl. Phys. Lett. 97, 213501 (2010).
[Crossref]

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
[Crossref]

2004 (1)

M. Foster, K. Moll, and A. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12(13), 2880–2887 (2004).
[Crossref] [PubMed]

1994 (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

1984 (1)

B. Tatian, “Fitting refractive-index data with the Sellmeier dispersion formula,” Appl. Opt. 23(24), 4477–4485 (1984).
[Crossref] [PubMed]

Agarwal, A.

Agarwal, A. M.

Allioux, D.

Alonso-Ramos, C.

Atanackovic, P.

Baehr-Jones, T.

A. Spott, Y. Liu, T. Baehr-Jones, R. R. Ilic, and M. Hochberg, “Silicon Waveguides and Ring Resonators at 5.5 µm,” Appl. Phys. Lett. 97, 213501 (2010).
[Crossref]

Baets, R.

Ballabio, A.

Barritault, P.

Beausoleil, R. G.

Boodhoo, L. A.

Boulila, F.

Brasch, V.

Brun, M.

Capasso, F.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Carletti, L.

Carras, M.

Casas-Bedoya, A.

Chaisakul, P.

Chang, Y. C.

Cheben, P.

Chiles, J.

S. Khan, J. Chiles, J. Ma, and S. Fathpour, “Silicon-on-nitride waveguides for mid-and near-infrared integrated photonics,” Appl. Phys. Lett. 102(12), 121104 (2013).
[Crossref]

Cho, A. Y.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Danto, S.

De Leonardis, F.

F. De Leonardis, B. Troia, R. A. Soref, and V. M. N. Passaro, “Dispersion of nonresonant third-order nonlinearities in GeSiSn ternary alloys,” Sci. Rep. 6(1), 32622 (2016).
[Crossref] [PubMed]

F. De Leonardis, B. Troia, and V. M. Passaro, “Mid-IR optical and nonlinear properties of germanium on silicon optical waveguides,” J. Lightwave Technol. 32(22), 3747–3757 (2014).
[Crossref]

de Rooij, N. F.

Di Francesco, J.

Duvall, S. G.

Dwivedi, S.

Eggleton, B. J.

N. Singh, D. D. Hudson, and B. J. Eggleton, “Silicon-on-sapphire pillar waveguides for Mid-IR supercontinuum generation,” Opt. Express 23(13), 17345–17354 (2015).
[Crossref] [PubMed]

Faist, J.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Fathpour, S.

S. Khan, J. Chiles, J. Ma, and S. Fathpour, “Silicon-on-nitride waveguides for mid-and near-infrared integrated photonics,” Appl. Phys. Lett. 102(12), 121104 (2013).
[Crossref]

Foster, M.

M. Foster, K. Moll, and A. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12(13), 2880–2887 (2004).
[Crossref] [PubMed]

Frigerio, J.

Gaeta, A.

M. Foster, K. Moll, and A. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12(13), 2880–2887 (2004).
[Crossref] [PubMed]

Gilles, C.

Grand, G.

Green, W. M.

Grillet, C.

Guo, H.

Hartmann, J. M.

Herzig, H. P.

Hochberg, M.

A. Spott, Y. Liu, T. Baehr-Jones, R. R. Ilic, and M. Hochberg, “Silicon Waveguides and Ring Resonators at 5.5 µm,” Appl. Phys. Lett. 97, 213501 (2010).
[Crossref]

Homsy, A.

Hon, N. K.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]

Hu, J.

Hudson, D. D.

N. Singh, D. D. Hudson, and B. J. Eggleton, “Silicon-on-sapphire pillar waveguides for Mid-IR supercontinuum generation,” Opt. Express 23(13), 17345–17354 (2015).
[Crossref] [PubMed]

Hutchinson, A. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994).
[Crossref] [PubMed]

Hvozdara, L.

Ikonic, Z.

P. Moontragoon, R. A. Soref, and Z. Ikonic, “The direct and indirect bandgaps of unstrained SixGe1-x-ySny and their photonic device applications,” J. Appl. Phys. 112(7), 073106 (2012).
[Crossref]

Ilic, R. R.

A. Spott, Y. Liu, T. Baehr-Jones, R. R. Ilic, and M. Hochberg, “Silicon Waveguides and Ring Resonators at 5.5 µm,” Appl. Phys. Lett. 97, 213501 (2010).
[Crossref]

Isella, G.

Jackson, S. D.

Jalali, B.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]

Khan, S.

S. Khan, J. Chiles, J. Ma, and S. Fathpour, “Silicon-on-nitride waveguides for mid-and near-infrared integrated photonics,” Appl. Phys. Lett. 102(12), 121104 (2013).
[Crossref]

Khokhar, A. Z.

Kimerling, L. C.

Kippenberg, T. J.

Kordts, A.

Kozacik, S.

Kuyken, B.

Labeye, P.

Lartigue, O.

Lepage, G.

Li, L.

Lin, H.

Lin, P. T.

Lin, Q.

Liu, X.

Liu, Y.

A. Spott, Y. Liu, T. Baehr-Jones, R. R. Ilic, and M. Hochberg, “Silicon Waveguides and Ring Resonators at 5.5 µm,” Appl. Phys. Lett. 97, 213501 (2010).
[Crossref]

Loo, R.

Luther-Davies, B.

Ma, J.

S. Khan, J. Chiles, J. Ma, and S. Fathpour, “Silicon-on-nitride waveguides for mid-and near-infrared integrated photonics,” Appl. Phys. Lett. 102(12), 121104 (2013).
[Crossref]

Ma, P.

Madden, S.

Maisons, G.

Malik, A.

Marris-Morini, D.

Mashanovich, G. Z.

Michel, J.

Molina-Fernández, I.

Moll, K.

M. Foster, K. Moll, and A. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12(13), 2880–2887 (2004).
[Crossref] [PubMed]

Monat, C.

Moontragoon, P.

P. Moontragoon, R. A. Soref, and Z. Ikonic, “The direct and indirect bandgaps of unstrained SixGe1-x-ySny and their photonic device applications,” J. Appl. Phys. 112(7), 073106 (2012).
[Crossref]

Moss, D. J.

Muneeb, M.

Murakowski, M.

Musgraves, J. D.

Nedeljkovic, M.

Nicoletti, S.

Orobtchouk, R.

Ortega-Moñux, A.

Ortiz, S.

Osgood, R. M.

Paeder, V.

Palomba, S.

Passaro, V. M.

F. De Leonardis, B. Troia, and V. M. Passaro, “Mid-IR optical and nonlinear properties of germanium on silicon optical waveguides,” J. Lightwave Technol. 32(22), 3747–3757 (2014).
[Crossref]

Passaro, V. M. N.

F. De Leonardis, B. Troia, R. A. Soref, and V. M. N. Passaro, “Dispersion of nonresonant third-order nonlinearities in GeSiSn ternary alloys,” Sci. Rep. 6(1), 32622 (2016).
[Crossref] [PubMed]

Pathak, S.

Penadés, J. S.

Pfeiffer, M. H. P.

Prather, D.

Ramirez, J. M.

Read, A.

Richardson, K.

Roelkens, G.

Roux, X. L.

Shimura, Y.

Singh, N.

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F. De Leonardis, B. Troia, and V. M. Passaro, “Mid-IR optical and nonlinear properties of germanium on silicon optical waveguides,” J. Lightwave Technol. 32(22), 3747–3757 (2014).
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Lab Chip (1)

Nanophotonics (1)

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Optica (1)

Sci. Rep. (1)

F. De Leonardis, B. Troia, R. A. Soref, and V. M. N. Passaro, “Dispersion of nonresonant third-order nonlinearities in GeSiSn ternary alloys,” Sci. Rep. 6(1), 32622 (2016).
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Editorial “Extending opportunities,” Nat. Photonics6(7), 407 (2012).

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Fig. 1 a) Representative scheme of the graded-index Si_1-xGe_x waveguide cross-section. b) Calculated vertical refractive index profile of a graded-index Si_1- _x Ge _x waveguide (h_core = 6 µm and λ = 3 µm). y (µm) corresponds to the vertical position in the waveguide, being its origin located at the interface between the Si substrate and the graded-index Si_1-xGe_x waveguide, as marked with a horizontal dashed line in figure (a)).

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Fig. 2 Top panels in (a), (b) and (c) show the dispersion parameter (D) of the quasi-TE polarization mode (in multicolor scale) of a waveguide with h_core = 6 µm as a function of the etching depth and the waveguide width for λ = 3 µm / 5.5 µm / 8 µm. Multicolored areas correspond to positive (anomalous) dispersion. The grey dot indicates the selected waveguide parameters, with width = 4 µm and etching depth = 4 µm. Bottom panels in (a), (b) and (c) display the simulated optical propagation mode (quasi-TE) of the selected waveguide geometry for increasing λ, according to top panels. (d) Spectral evolution of the dispersion parameter of the quasi-TE (black squares) and quasi-TM (red dots) polarizations for the optimized waveguide design (h_core = 6 µm, width = 4 µm, etch depth = 4 µm).

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Fig. 3 (a) Color map of the calculated Kerr coefficient, n₂, extracted from [21], as a function of the Ge concentration and wavelength, λ. A constant value of n₂ ≈ 1 × 10⁻¹⁷ m²/W was obtained for the Ge concentrations not displayed (i. e. for Ge conc. < 65%). (b) Spatial distribution of n₂ in the waveguide, with the simulated modal intensity superimposed for λ = 3 µm (inner lines denote higher intensity). (c) Calculated quasi-TE mode profile (|E_|||, black curves, left y-axis) and n₂ (red curves, right y-axis) along the waveguide vertical direction for λ = 3 µm (solid lines) and λ = 8 µm (dashed dotted lines). (d) Calculated nonlinear parameter (γ_eff, filled data, left y-axis) and nonlinear modal effective area (A_eff ^NL, empty data, right y-axis) of the quasi-TE (squares) and quasi-TM (circles) optical modes as a function of λ.

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Equations (1)

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γ_{e f f} (λ) = 2 π \iint_{W g} n_{2} (x, y) \cdot S_{z}^{2} d x d y / λ {(\int \int_{- \infty}^{\infty} S_{z} d x d y)}^{2}