Abstract

We derive approximate analytic expressions for the effective susceptibility tensor of a nonlinear composite, consisting of silicon nanocrystals embedded in fused silica. Two types of composites are considered: by assuming that (i) the crystallographic axes of different crystallites are the same, or (ii) crystallites are oriented randomly. In the first case, the tensor properties of the effective third-order susceptibility are shown to coincide with those of the bulk silicon. In the second case, however, the tensor properties of the susceptibility of the composite material are found to be quite different due to drastic modification of light interaction with optical phonons inside the composite. The newly derived expressions should be useful for modeling nonlinear optical phenomena in silica fibers and waveguides doped with silicon nanocrystals.

© 2012 OSA

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  1. R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for λ = 1.3 and 1.6 μm,” IEEE J. Quantum Electron.22, 873–879 (1986).
    [CrossRef]
  2. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
    [CrossRef]
  3. L. Pavesi and D. Lockwood, eds., Silicon Photonics, vol. 94 of Topics in Applied Physics (Springer-Verlag, Berlin, 2004).
  4. D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics4, 511–517 (2010).
    [CrossRef]
  5. I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express17, 5807–5814 (2009).
    [CrossRef] [PubMed]
  6. A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
    [CrossRef] [PubMed]
  7. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express17, 22124–22137 (2009).
    [CrossRef] [PubMed]
  8. M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4, 498–499 (2010).
    [CrossRef]
  9. L. Pavesi and R. Turan, eds., Silicon Nanocrystals: Fundamentals, Synthesis and Applications (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2010).
  10. L. Khriachtchev, ed., Silicon Nanophotonics: Basic Principles, Present Status and Perspectives (Pan Stanford, Singapore, 2009).
  11. I. D. Rukhlenko and M. Premaratne, “Optimization of nonlinear performance of silicon-nanocrystal cylindrical nanowires,” IEEE Photon. J.4, 952–959 (2012).
    [CrossRef]
  12. F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron.2011, 751498 (2011).
  13. P. Sanchis, J. Blasco, A. Martinez, and J. Marti, “Design of silicon-based slot waveguide configurations for optimum nonlinear performance,” J. Lightwave Technol.25, 1298–1305 (2007).
    [CrossRef]
  14. V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol.2008, 279502 (2008).
  15. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett.37, 2295–2297 (2012).
    [CrossRef] [PubMed]
  16. D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B37, 8719–8724 (1988).
    [CrossRef]
  17. X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B38, 10970–10973 (1988).
    [CrossRef]
  18. D. J. Bergman, “The dielectric constant of a composite material – a problem in classical physics,” Phys. Rep.43, 377–407 (1978).
    [CrossRef]
  19. S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt.59, 954–961 (2012).
    [CrossRef]
  20. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).
  21. R. W. Boyd, R. J. Gehr, G. L. Fischer, and J. E. Sipe, “Nonlinear optical properties of nanocomposite materials,” Pure Appl. Opt.5, 505–512 (1996).
    [CrossRef]
  22. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).
  23. P. M. Hui, P. Cheung, and D. Stroud, “Theory of third harmonic generation in random composites of nonlinear dielectrics,” J. Appl. Phys.84, 3451–3458 (1998).
    [CrossRef]
  24. D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B12, 3368–3373 (1975).
    [CrossRef]
  25. J. Sipe and R. Boyd, “Nanocomposite materials for nonlinear optics based on local field effects,” in “Optical Properties of Nanostructured Random Media,”, vol. 82 of Topics Appl. Phys., V. M. Shalaev, ed. (Springer-Verlag, BerlinHeidelberg, 2002), pp. 1–19.
    [CrossRef]
  26. G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
    [CrossRef] [PubMed]
  27. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).
  28. J. Wei, A. Wirth, M. C. Downer, and B. S. Mendoza, “Second-harmonic and linear optical spectroscopic study of silicon nanocrystals embedded in SiO2,” Phys. Rev. B84, 165316 (2011).
    [CrossRef]
  29. Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett.78, 766 (2001).
    [CrossRef]
  30. W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68, 085318 (2003).
    [CrossRef]
  31. J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett.83, 4045–4048 (1999).
    [CrossRef]
  32. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express15, 16604–16644 (2007).
    [CrossRef] [PubMed]
  33. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media (Cambridge Univ. Press, Cambridge, 2011).
  34. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).
  35. R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, FL, 1994).
  36. W. Grieshaber, E. Belorizky, and M. L. Berre, “A general method for tensor averaging and an application to polycrystalline materials,” Solid State Commun.93, 805–809 (1995).
    [CrossRef]
  37. I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Continuous-wave Raman amplification in silicon waveguides: Beyond the undepleted pump approximation,” Opt. Lett.34, 536–538 (2009).
    [CrossRef] [PubMed]
  38. L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett.34, 476–478 (2009).
    [CrossRef] [PubMed]
  39. I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photon. J.2, 423–435 (2010).
    [CrossRef]
  40. C. Torres-Torres, A. López-Suárez, R. Torres-Martínez, A. Rodriguez, J. A. Reyes-Esqueda, L. Castaneda, J. C. Alonso, and A. Oliver, “Modulation of the propagation speed of mechanical waves in silicon quantum dots embedded in a silicon-nitride film,” Opt. Express20, 4784–4789 (2012).
    [CrossRef] [PubMed]

2012

2011

F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron.2011, 751498 (2011).

J. Wei, A. Wirth, M. C. Downer, and B. S. Mendoza, “Second-harmonic and linear optical spectroscopic study of silicon nanocrystals embedded in SiO2,” Phys. Rev. B84, 165316 (2011).
[CrossRef]

2010

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics4, 511–517 (2010).
[CrossRef]

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4, 498–499 (2010).
[CrossRef]

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photon. J.2, 423–435 (2010).
[CrossRef]

2009

2008

V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol.2008, 279502 (2008).

2007

2003

W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68, 085318 (2003).
[CrossRef]

2001

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett.78, 766 (2001).
[CrossRef]

1999

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett.83, 4045–4048 (1999).
[CrossRef]

1998

P. M. Hui, P. Cheung, and D. Stroud, “Theory of third harmonic generation in random composites of nonlinear dielectrics,” J. Appl. Phys.84, 3451–3458 (1998).
[CrossRef]

1996

R. W. Boyd, R. J. Gehr, G. L. Fischer, and J. E. Sipe, “Nonlinear optical properties of nanocomposite materials,” Pure Appl. Opt.5, 505–512 (1996).
[CrossRef]

1995

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

W. Grieshaber, E. Belorizky, and M. L. Berre, “A general method for tensor averaging and an application to polycrystalline materials,” Solid State Commun.93, 805–809 (1995).
[CrossRef]

1988

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B37, 8719–8724 (1988).
[CrossRef]

X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B38, 10970–10973 (1988).
[CrossRef]

1986

R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for λ = 1.3 and 1.6 μm,” IEEE J. Quantum Electron.22, 873–879 (1986).
[CrossRef]

1978

D. J. Bergman, “The dielectric constant of a composite material – a problem in classical physics,” Phys. Rep.43, 377–407 (1978).
[CrossRef]

1975

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B12, 3368–3373 (1975).
[CrossRef]

Agrawal, G. P.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett.37, 2295–2297 (2012).
[CrossRef] [PubMed]

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photon. J.2, 423–435 (2010).
[CrossRef]

L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett.34, 476–478 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Continuous-wave Raman amplification in silicon waveguides: Beyond the undepleted pump approximation,” Opt. Lett.34, 536–538 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express17, 5807–5814 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express17, 22124–22137 (2009).
[CrossRef] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express15, 16604–16644 (2007).
[CrossRef] [PubMed]

M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media (Cambridge Univ. Press, Cambridge, 2011).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).

Alonso, J. C.

Belorizky, E.

W. Grieshaber, E. Belorizky, and M. L. Berre, “A general method for tensor averaging and an application to polycrystalline materials,” Solid State Commun.93, 805–809 (1995).
[CrossRef]

Belyakov, V. A.

V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol.2008, 279502 (2008).

Bergman, D. J.

X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B38, 10970–10973 (1988).
[CrossRef]

D. J. Bergman, “The dielectric constant of a composite material – a problem in classical physics,” Phys. Rep.43, 377–407 (1978).
[CrossRef]

Berre, M. L.

W. Grieshaber, E. Belorizky, and M. L. Berre, “A general method for tensor averaging and an application to polycrystalline materials,” Solid State Commun.93, 805–809 (1995).
[CrossRef]

Blasco, J.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

P. Sanchis, J. Blasco, A. Martinez, and J. Marti, “Design of silicon-based slot waveguide configurations for optimum nonlinear performance,” J. Lightwave Technol.25, 1298–1305 (2007).
[CrossRef]

Bowers, J. E.

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics4, 511–517 (2010).
[CrossRef]

Boyd, R.

J. Sipe and R. Boyd, “Nanocomposite materials for nonlinear optics based on local field effects,” in “Optical Properties of Nanostructured Random Media,”, vol. 82 of Topics Appl. Phys., V. M. Shalaev, ed. (Springer-Verlag, BerlinHeidelberg, 2002), pp. 1–19.
[CrossRef]

Boyd, R. W.

R. W. Boyd, R. J. Gehr, G. L. Fischer, and J. E. Sipe, “Nonlinear optical properties of nanocomposite materials,” Pure Appl. Opt.5, 505–512 (1996).
[CrossRef]

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).

Brudny, V. L.

W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68, 085318 (2003).
[CrossRef]

Burdov, V. A.

V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol.2008, 279502 (2008).

Cai, W.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).

Castaneda, L.

Cheung, P.

P. M. Hui, P. Cheung, and D. Stroud, “Theory of third harmonic generation in random composites of nonlinear dielectrics,” J. Appl. Phys.84, 3451–3458 (1998).
[CrossRef]

Dadap, J. I.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett.83, 4045–4048 (1999).
[CrossRef]

Daldosso, N.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Dissanayake, C.

Downer, M. C.

J. Wei, A. Wirth, M. C. Downer, and B. S. Mendoza, “Second-harmonic and linear optical spectroscopic study of silicon nanocrystals embedded in SiO2,” Phys. Rev. B84, 165316 (2011).
[CrossRef]

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett.78, 766 (2001).
[CrossRef]

Eisenthal, K. B.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett.83, 4045–4048 (1999).
[CrossRef]

Fauchet, P. M.

Fedeli, J. M.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Fischer, G. L.

R. W. Boyd, R. J. Gehr, G. L. Fischer, and J. E. Sipe, “Nonlinear optical properties of nanocomposite materials,” Pure Appl. Opt.5, 505–512 (1996).
[CrossRef]

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

Freude, W.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

Galan, J. V.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Garanovich, I. L.

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photon. J.2, 423–435 (2010).
[CrossRef]

Garcia-Ruperez, J.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Garrido, B.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Gautier, P.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Gehr, R. J.

R. W. Boyd, R. J. Gehr, G. L. Fischer, and J. E. Sipe, “Nonlinear optical properties of nanocomposite materials,” Pure Appl. Opt.5, 505–512 (1996).
[CrossRef]

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

Grieshaber, W.

W. Grieshaber, E. Belorizky, and M. L. Berre, “A general method for tensor averaging and an application to polycrystalline materials,” Solid State Commun.93, 805–809 (1995).
[CrossRef]

Guider, R.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Heinz, T. F.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett.83, 4045–4048 (1999).
[CrossRef]

Hernandez, S.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Hui, P. M.

P. M. Hui, P. Cheung, and D. Stroud, “Theory of third harmonic generation in random composites of nonlinear dielectrics,” J. Appl. Phys.84, 3451–3458 (1998).
[CrossRef]

X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B38, 10970–10973 (1988).
[CrossRef]

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B37, 8719–8724 (1988).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

Jenekhe, S. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

Jiang, Y.

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett.78, 766 (2001).
[CrossRef]

Jordana, E.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Kivshar, Y. S.

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photon. J.2, 423–435 (2010).
[CrossRef]

Koos, C.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

Lebour, Y.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Leonardis, F. D.

F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron.2011, 751498 (2011).

Leuthold, J.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

Li, Z.

R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, FL, 1994).

Liang, D.

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics4, 511–517 (2010).
[CrossRef]

Lin, Q.

Lockwood, R.

V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol.2008, 279502 (2008).

López-Suárez, A.

Lorenzo, J.

R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for λ = 1.3 and 1.6 μm,” IEEE J. Quantum Electron.22, 873–879 (1986).
[CrossRef]

Marti, J.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

P. Sanchis, J. Blasco, A. Martinez, and J. Marti, “Design of silicon-based slot waveguide configurations for optimum nonlinear performance,” J. Lightwave Technol.25, 1298–1305 (2007).
[CrossRef]

Martinez, A.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

P. Sanchis, J. Blasco, A. Martinez, and J. Marti, “Design of silicon-based slot waveguide configurations for optimum nonlinear performance,” J. Lightwave Technol.25, 1298–1305 (2007).
[CrossRef]

Maytorena, J. A.

W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68, 085318 (2003).
[CrossRef]

Meldrum, A.

V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol.2008, 279502 (2008).

Mendoza, B. S.

J. Wei, A. Wirth, M. C. Downer, and B. S. Mendoza, “Second-harmonic and linear optical spectroscopic study of silicon nanocrystals embedded in SiO2,” Phys. Rev. B84, 165316 (2011).
[CrossRef]

W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68, 085318 (2003).
[CrossRef]

Mochan, W. L.

W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68, 085318 (2003).
[CrossRef]

Murray, R. M.

R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, FL, 1994).

Oliver, A.

Osaheni, J. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

Painter, O. J.

Paniccia, M.

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4, 498–499 (2010).
[CrossRef]

Passaro, V. M. N.

F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron.2011, 751498 (2011).

Pavesi, L.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Premaratne, M.

Reyes-Esqueda, J. A.

Rodriguez, A.

Rukhlenko, I. D.

Saarinen, J. J.

S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt.59, 954–961 (2012).
[CrossRef]

Sanchis, P.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

P. Sanchis, J. Blasco, A. Martinez, and J. Marti, “Design of silicon-based slot waveguide configurations for optimum nonlinear performance,” J. Lightwave Technol.25, 1298–1305 (2007).
[CrossRef]

Sastry, S. S.

R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, FL, 1994).

Shalaev, V.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).

Shan, J.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett.83, 4045–4048 (1999).
[CrossRef]

Sipe, J.

J. Sipe and R. Boyd, “Nanocomposite materials for nonlinear optics based on local field effects,” in “Optical Properties of Nanostructured Random Media,”, vol. 82 of Topics Appl. Phys., V. M. Shalaev, ed. (Springer-Verlag, BerlinHeidelberg, 2002), pp. 1–19.
[CrossRef]

Sipe, J. E.

S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt.59, 954–961 (2012).
[CrossRef]

R. W. Boyd, R. J. Gehr, G. L. Fischer, and J. E. Sipe, “Nonlinear optical properties of nanocomposite materials,” Pure Appl. Opt.5, 505–512 (1996).
[CrossRef]

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

Soref, R.

R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for λ = 1.3 and 1.6 μm,” IEEE J. Quantum Electron.22, 873–879 (1986).
[CrossRef]

Spano, R.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Stroud, D.

P. M. Hui, P. Cheung, and D. Stroud, “Theory of third harmonic generation in random composites of nonlinear dielectrics,” J. Appl. Phys.84, 3451–3458 (1998).
[CrossRef]

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B37, 8719–8724 (1988).
[CrossRef]

X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B38, 10970–10973 (1988).
[CrossRef]

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B12, 3368–3373 (1975).
[CrossRef]

Sukhorukov, A. A.

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photon. J.2, 423–435 (2010).
[CrossRef]

Torres-Martínez, R.

Torres-Torres, C.

Volkov, S. N.

S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt.59, 954–961 (2012).
[CrossRef]

Wei, J.

J. Wei, A. Wirth, M. C. Downer, and B. S. Mendoza, “Second-harmonic and linear optical spectroscopic study of silicon nanocrystals embedded in SiO2,” Phys. Rev. B84, 165316 (2011).
[CrossRef]

Weller-Brophy, L. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

White, C. W.

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett.78, 766 (2001).
[CrossRef]

Wilson, P. T.

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett.78, 766 (2001).
[CrossRef]

Wirth, A.

J. Wei, A. Wirth, M. C. Downer, and B. S. Mendoza, “Second-harmonic and linear optical spectroscopic study of silicon nanocrystals embedded in SiO2,” Phys. Rev. B84, 165316 (2011).
[CrossRef]

Withrow, S. P.

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett.78, 766 (2001).
[CrossRef]

Yin, L.

Zeng, X. C.

X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B38, 10970–10973 (1988).
[CrossRef]

Zhang, J.

Adv. Opt. Technol.

V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol.2008, 279502 (2008).

Adv. OptoElectron.

F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron.2011, 751498 (2011).

Appl. Phys. Lett.

Y. Jiang, P. T. Wilson, M. C. Downer, C. W. White, and S. P. Withrow, “Second-harmonic generation from silicon nanocrystals embedded in SiO2,” Appl. Phys. Lett.78, 766 (2001).
[CrossRef]

IEEE J. Quantum Electron.

R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for λ = 1.3 and 1.6 μm,” IEEE J. Quantum Electron.22, 873–879 (1986).
[CrossRef]

IEEE Photon. J.

I. D. Rukhlenko and M. Premaratne, “Optimization of nonlinear performance of silicon-nanocrystal cylindrical nanowires,” IEEE Photon. J.4, 952–959 (2012).
[CrossRef]

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photon. J.2, 423–435 (2010).
[CrossRef]

J. Appl. Phys.

P. M. Hui, P. Cheung, and D. Stroud, “Theory of third harmonic generation in random composites of nonlinear dielectrics,” J. Appl. Phys.84, 3451–3458 (1998).
[CrossRef]

J. Lightwave Technol.

J. Mod. Opt.

S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt.59, 954–961 (2012).
[CrossRef]

Nano Lett.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Nat. Photonics

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics4, 511–517 (2010).
[CrossRef]

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4, 498–499 (2010).
[CrossRef]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

D. J. Bergman, “The dielectric constant of a composite material – a problem in classical physics,” Phys. Rep.43, 377–407 (1978).
[CrossRef]

Phys. Rev. B

J. Wei, A. Wirth, M. C. Downer, and B. S. Mendoza, “Second-harmonic and linear optical spectroscopic study of silicon nanocrystals embedded in SiO2,” Phys. Rev. B84, 165316 (2011).
[CrossRef]

W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68, 085318 (2003).
[CrossRef]

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B37, 8719–8724 (1988).
[CrossRef]

X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B38, 10970–10973 (1988).
[CrossRef]

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B12, 3368–3373 (1975).
[CrossRef]

Phys. Rev. Lett.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett.83, 4045–4048 (1999).
[CrossRef]

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, “Enhanced nonlinear optical response of composite materials,” Phys. Rev. Lett.74, 1871–1874 (1995).
[CrossRef] [PubMed]

Pure Appl. Opt.

R. W. Boyd, R. J. Gehr, G. L. Fischer, and J. E. Sipe, “Nonlinear optical properties of nanocomposite materials,” Pure Appl. Opt.5, 505–512 (1996).
[CrossRef]

Solid State Commun.

W. Grieshaber, E. Belorizky, and M. L. Berre, “A general method for tensor averaging and an application to polycrystalline materials,” Solid State Commun.93, 805–809 (1995).
[CrossRef]

Other

L. Pavesi and R. Turan, eds., Silicon Nanocrystals: Fundamentals, Synthesis and Applications (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2010).

L. Khriachtchev, ed., Silicon Nanophotonics: Basic Principles, Present Status and Perspectives (Pan Stanford, Singapore, 2009).

M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media (Cambridge Univ. Press, Cambridge, 2011).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).

R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, FL, 1994).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

J. Sipe and R. Boyd, “Nanocomposite materials for nonlinear optics based on local field effects,” in “Optical Properties of Nanostructured Random Media,”, vol. 82 of Topics Appl. Phys., V. M. Shalaev, ed. (Springer-Verlag, BerlinHeidelberg, 2002), pp. 1–19.
[CrossRef]

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).

L. Pavesi and D. Lockwood, eds., Silicon Photonics, vol. 94 of Topics in Applied Physics (Springer-Verlag, Berlin, 2004).

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Figures (3)

Fig. 1
Fig. 1

(a) Identically oriented Si NCs embedded in a SiO2 matrix of permittivity ε2. Nanocrystals are characterized by permittivity ε1, nonlinear susceptibility tensor χ k l m n ( 3 ), and volume filling factor f; electric field (E1x, E1y, E1z) inside Si NCs is assumed to be uniform. (b) Homogeneous Si-NCs/SiO2 composite and the space-averaged electric field (x, y, z) inside it; the composite is characterized by the effective parameters εeff and χ k l m n eff.

Fig. 2
Fig. 2

Ratios εeff/ε1 and ξ are plotted as a function of filling factor f for Si-NCs/SiO2 (solid curves) and Si-NCs/Si3N4 (dashed curves) composites using ε1 = 12 with ε2 = 2.1 for SiO2 and ε2 = 4.1 for Si3N4.

Fig. 3
Fig. 3

(a) Randomly oriented Si NCs embedded in SiO2 matrix. Orientation of each nanocrystal, with respect to the Cartesian axes α, β, and γ, is characterized by the respective directions of its crystallographic axes x, y, and z. (b) Rotation by an angle ψ ∈ [0, 2π) around a unit vector u (set by angles ϑ and φ) brings crystallographic axes of Si NC into coincidence with the axes α, β, and γ.

Equations (28)

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

k = 1 V E k ( r ) d V
ε eff = 1 V ε ( r ) ( E ( r ) ) 2 d V ,
ϑ j ( r ) = { 1 when r is inside the j th medium ; 0 otherwise .
ε eff ( ε 1 , ε 2 , f ) = 1 4 [ u + ( u 2 + 8 ε 1 ε 2 ) 1 / 2 ] ,
χ k l m n eff = 1 V 1 χ k l m n ( 3 ) ( r ) E k ( r ) E l ( r ) E m * ( r ) E n ( r ) 4 d V 1 ,
𝒟 k = ε eff k + l m n χ k l m n eff l m * n ,
D 1 k = ε 1 E 1 k + l m n χ k l m n ( 3 ) E 1 l E 1 m * E 1 n ε ^ 1 E 1 k ,
𝒟 k ε eff ( ε ^ 1 , ε 2 , f ) k ,
𝒟 k ε eff ( ε 1 , ε 2 , f ) k + ε eff ( ε ^ 1 , ε 2 , f ) ε ^ 1 ( ε ^ 1 ε 1 ) k = ε eff k + ε eff ε 1 k E 1 k l m n χ k l m n ( 3 ) E 1 l E 1 m * E 1 n .
ε aux E k ( r ) d V = ε ( r ) E k ( r ) d V .
k ε aux ε 1 = 1 V ϑ 1 ( r ) E k ( r ) d V = f E 1 k ,
ε eff ε 1 = f E 1 2 2 f E 1 2 2 = 1 f ( ε aux ε 1 ) 2 .
𝒟 k ε eff k + 1 f ε eff ε 1 | ε eff ε 1 | l m n χ k l m n ( 3 ) l m * n .
χ k l m n eff = 1 f ε eff ε 1 | ε eff ε 1 | χ k l m n ( 3 ) .
ξ = [ ( 3 f 1 ) ε eff + ε 2 ] 2 f ( u 2 + 8 ε 1 ε 2 ) .
χ k l m n ( 3 ) ( ω ; ω 1 , ω 2 , ω 3 ) = χ x x x x e ( ω ) 𝒦 k l m n + 1 2 [ H ( ω 1 + ω 2 ) k l m n + H ( ω 2 + ω 3 ) k n m l ] ,
𝒦 k l m n = ( ρ / 3 ) ( δ k l δ m n + δ k m δ l n + δ k n δ l m ) + ( 1 ρ ) δ k l δ l m δ m n ,
k l m n = δ k m δ ln + δ k n δ l m 2 δ k l δ l m δ m n ,
χ κ λ μ ν ( 3 ) = k l m n R κ k R λ l R μ m R ν n χ k l m n ( 3 ) ,
R ( ϑ , φ , ψ ) = ( cos ψ cos ϑ sin ψ sin φ sin ϑ sin ψ cos ϑ sin ψ cos ψ cos φ sin ϑ sin ψ sin φ sin ϑ sin ψ cos φ sin ϑ sin ψ cos ψ ) + ( 1 cos ψ ) ( cos 2 φ sin 2 ϑ cos φ sin φ sin 2 ϑ cos φ cos ϑ sin ϑ cos φ sin φ sin 2 ϑ sin 2 φ sin 2 ϑ sin φ cos ϑ sin ϑ cos φ cos ϑ sin ϑ sin φ cos ϑ sin ϑ cos 2 ϑ ) .
χ κ λ μ ν eff = 1 f ε eff ε 1 | ε eff ε 1 | χ κ λ μ ν ( 3 ) .
χ κ λ μ ν eff = 1 8 π 2 0 π sin ϑ d ϑ 0 2 π d φ 0 2 π d ψ χ κ λ μ ν eff ( ϑ , φ , ψ ) .
δ k l δ l m δ m n = 8 45 ( δ k l δ m n + δ k m δ ln + δ k n δ l m ) + 1 9 δ k l δ l m δ m n .
𝒦 κ λ μ ν = 8 + 7 ρ 45 ( δ κ λ δ μ ν + δ κ μ δ λ ν + δ κ ν δ λ μ ) + 1 ρ 9 δ κ λ δ λ μ δ μ ν .
α α α α = β β β β = γ γ γ γ = 29 + 16 ρ 45 1.1 , α α β β = α β α β = α β β α = = 8 + 7 ρ 45 0.375 ,
κ λ μ ν = 29 45 ( δ κ μ δ λ ν + δ κ ν δ λ μ ) 16 45 δ κ λ δ μ ν 2 9 δ κ λ δ λ μ δ μ ν .
α β α β = α β β α = β γ β γ = = 29 45 .
α α α α = β β β β = γ γ γ γ = 32 45 , α α β β = α α γ γ = γ γ β β = = 16 45 .

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