Abstract

We theoretically consider the existence of multiple nonzero components of the second-order nonlinear susceptibility tensor, χ(2), generated via strain-induced symmetry breaking in crystalline silicon. We determine that, in addition to the previously reported χxxy(2) component, the χyyy(2) component also becomes nonzero based on the remaining symmetry present in the strained material. In order to characterize these two nonlinearities, we fabricate Fabry–Perot waveguide resonators on 250 nm thick silicon-on-insulator wafers clad with 180 nm of compressively stressed (1.275GPa) silicon nitride. We measure the shifts in these devices’ modal effective indices in response to several bias electric fields and calculate the χeff,xxy(2) and χeff,yyy(2) nonlinear susceptibility tensor elements induced by the breaking of the guiding material’s inversion symmetry. Through the incorporation of finite element simulations encompassing the theoretical distribution of strain, the applied bias field, and the optical modes supported by the waveguide geometry, we extract two phenomenological scaling coefficients which relate the induced optical nonlinearities to the local strain gradient.

© 2014 Optical Society of America

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2013

2011

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrin, M. Nagel, F. Merget, and H. Kurz, Opt. Express 19, 17212 (2011).
[CrossRef]

I. Avrutsky and R. Soref, Opt. Express 19, 21707 (2011).
[CrossRef]

2010

2009

2006

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

2003

M. Huang, Int. J. Solids Struct. 40, 1615 (2003).
[CrossRef]

G. Li and P. Yu, J. Lightwave Technol. 21, 2010 (2003).
[CrossRef]

Amberg, P.

Andersen, K.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Asghari, M.

Avrutsky, I.

Bianco, F.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Bjarklev, A.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Bolten, J.

Borel, P.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Borga, E.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Cazzanelli, M.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Chmielak, B.

Cunningham, J.

Degoli, E.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Dong, P.

Fage-Pedersen, J.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Fainman, Y.

A. Grieco, B. Slutsky, and Y. Fainman, “Characterization of waveguide loss using distributed Bragg reflectors,” Appl. Phys. B, doi: 10.1007/s00340-013-5543-x (2013).
[CrossRef]

Feng, D.

Frandsen, L.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Gardes, F.

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, Nat. Photonics 4, 518 (2010).
[CrossRef]

Ghulinyan, M.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Grieco, A.

A. Grieco, B. Slutsky, and Y. Fainman, “Characterization of waveguide loss using distributed Bragg reflectors,” Appl. Phys. B, doi: 10.1007/s00340-013-5543-x (2013).
[CrossRef]

Hansen, O.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Ho, R.

Hon, N.

N. Hon, K. Tsia, D. Solli, B. Jalali, and J. Khurgin, Proceedings of the 6th IEEE International Conference on Group IV Photonics (IEEE, 2009).

Huang, M.

M. Huang, Int. J. Solids Struct. 40, 1615 (2003).
[CrossRef]

Jacobsen, R.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Jalali, B.

N. Hon, K. Tsia, D. Solli, B. Jalali, and J. Khurgin, Proceedings of the 6th IEEE International Conference on Group IV Photonics (IEEE, 2009).

Khurgin, J.

N. Hon, K. Tsia, D. Solli, B. Jalali, and J. Khurgin, Proceedings of the 6th IEEE International Conference on Group IV Photonics (IEEE, 2009).

Krishnamoorthy, A.

Kristensen, M.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Kung, C.

Kurz, H.

Lavrinenko, A.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Lexau, J.

Li, G.

Liang, H.

Liao, S.

Luo, Y.

Luppi, E.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Mashanovich, G.

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, Nat. Photonics 4, 518 (2010).
[CrossRef]

Matheisen, C.

Mekis, A.

Merget, F.

Modotto, D.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Moulin, G.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Nagel, M.

Ossicini, S.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Ou, H.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Pavesi, L.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Peucheret, C.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Pierobon, R.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Pinckney, N.

Pinguet, T.

Pucker, G.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Qian, W.

Raj, K.

Reed, G.

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, Nat. Photonics 4, 518 (2010).
[CrossRef]

Ripperda, C.

Shafiiha, R.

Sharipov, R.

R. Sharipov, “Quick introduction to tensor analysis,” arXiv preprint math/0403252 (2004).

Shi, J.

Slutsky, B.

A. Grieco, B. Slutsky, and Y. Fainman, “Characterization of waveguide loss using distributed Bragg reflectors,” Appl. Phys. B, doi: 10.1007/s00340-013-5543-x (2013).
[CrossRef]

Solli, D.

N. Hon, K. Tsia, D. Solli, B. Jalali, and J. Khurgin, Proceedings of the 6th IEEE International Conference on Group IV Photonics (IEEE, 2009).

Soref, R.

Thacker, H.

Thomson, D.

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, Nat. Photonics 4, 518 (2010).
[CrossRef]

Tsia, K.

N. Hon, K. Tsia, D. Solli, B. Jalali, and J. Khurgin, Proceedings of the 6th IEEE International Conference on Group IV Photonics (IEEE, 2009).

Véniard, V.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Wabnitz, S.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Wahlbrin, T.

Wahlbrink, T.

Waldow, M.

Yu, P.

Zheng, D.

Zheng, X.

Zsigri, B.

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Int. J. Solids Struct.

M. Huang, Int. J. Solids Struct. 40, 1615 (2003).
[CrossRef]

J. Lightwave Technol.

Nat. Mater.

M. Cazzanelli, F. Bianco, E. Borga, G. Pucker, M. Ghulinyan, E. Degoli, E. Luppi, V. Véniard, S. Ossicini, D. Modotto, S. Wabnitz, R. Pierobon, and L. Pavesi, Nat. Mater. 11, 148 (2011).
[CrossRef]

Nat. Photonics

G. Reed, G. Mashanovich, F. Gardes, and D. Thomson, Nat. Photonics 4, 518 (2010).
[CrossRef]

Nature

R. Jacobsen, K. Andersen, P. Borel, J. Fage-Pedersen, L. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[CrossRef]

Opt. Express

Other

N. Hon, K. Tsia, D. Solli, B. Jalali, and J. Khurgin, Proceedings of the 6th IEEE International Conference on Group IV Photonics (IEEE, 2009).

A. Grieco, B. Slutsky, and Y. Fainman, “Characterization of waveguide loss using distributed Bragg reflectors,” Appl. Phys. B, doi: 10.1007/s00340-013-5543-x (2013).
[CrossRef]

R. Sharipov, “Quick introduction to tensor analysis,” arXiv preprint math/0403252 (2004).

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Figures (5)

Fig. 1.
Fig. 1.

(a) A single unit cell of silicon in the absence of strain, showing a few bonds to illustrate the relative locations of atoms within the cell. (b) A single unit cell of silicon in the presence of a y-oriented gradient of x-oriented strain, demonstrating the loss of centrosymmetry and the existence of rotational symmetry of 180° about the dashed line.

Fig. 2.
Fig. 2.

ESEM side view of the edge of a 250 nm tall waveguide clad with 180 nm of silicon nitride through PECVD. Two distinct regions have formed. One comes in contact with the top surface of the waveguide, whereas the other grows on the substrate.

Fig. 3.
Fig. 3.

(a) Waveguide cross section resulting from the outlined fabrication process. (b) The device outline showing, from left to right, the input facet, a 90° bend, the cavity located between two Bragg mirrors, and the output facet.

Fig. 4.
Fig. 4.

Transmission spectra of the (a) TE- and (b) TM-like modes, in the absence of an applied voltage.

Fig. 5.
Fig. 5.

Observed index changes of the TE- and TM-like modes’ resonances in response to applied voltages ranging from 50V to +50V.

Equations (9)

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

Msymm=[100;010;001].
Masymm=[100;010;001].
Δn=nλΔλ.
Δneff,TE=χeff,xxy(2)Eavgneff,TE+3χeff,xxyy(3)Eavg22neff,TE,
Δneff,TM=χeff,yyy(2)Eavg2neff,TM+χeff,yyyy(2)Eavg22neff,TM.
Δnx,Si=C1sxxC2(syy+szz)+Cxxy(dexxdy)Eynx,Si,
Δny,Si=C1syyC2(sxx+szz)+Cyyy(dexxdy)Ey2nx,Si,
χxxy(2)=Cxxy(dexxdy),
χyyy(2)=Cyyy(dexxdy).

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