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 Optical Society of America

1. Introduction

Even though silicon was recognized as an important material for photonics technology more than 25 years ago [1], the relevant theoretical concepts have begun to be put into practice only recently [2, 3]. A number of breakthroughs in the field of silicon photonics have made possible the development of a variety of functional nonlinear devices, with both active and passive silicon elements. These devices not only can generate and amplify optical signals [4,5] but can also modulate and switch them—either all-optically or electro-optically—at speeds approaching hundreds of gigabits per second [6, 7]. The realization of such ultrafast silicon photonic devices brings us closer to the moment when they will be combined with the microelectronics technology to build high-performance, low-cost, photonic integrated circuits [8].

A crucial step towards the development of silicon photonics was the discovery of unique optical properties of low-dimensional silicon [9,10]. In particular, it was found that the ultrafast Kerr effect in silicon nanocrystals (Si NCs) may be, respectively, 10000 and 100 times stronger than that in fused silica (SiO2) and bulk silicon. Much like silicon-on-insulator waveguides, silica glass doped with Si NCs (Si-NCs/SiO2 composite) enables tight confinement of optical fields, while its refractive index may be tuned to any value between 1.45 and 2.2 by simply changing the density of the Si NCs [1113]. Furthermore, the optical response of Si NCs has a pronounced dependence on their size, thus providing a flexibility in the engineering of their nonlinear properties. These and other features guarantee that Si NCs can serve silicon photonics by improving the performance of optical memories, wavelength converters, and modulators, as well as by enabling power-efficient amplifiers and light sources [14].

Numerical modeling of light propagation through Si-NC-doped silica waveguides requires a knowledge of the effective optical parameters of the Si-NCs/SiO2 composite [11, 15]. The reason is that it is impracticable to study the nonlinear effects in Si NCs by solving Maxwell equations for the whole composite while treating each NC individually. To do so, one would have to consider more than 100000 nanocrystals per micrometer of waveguide with a 0.01-μm2 cross section, in which Si NCs of 2.5 nm diameter have a volume fraction as small as 10%. It is evident that it would be extremely challenging to solve numerically such a nonlinear problem.

To enable theoretical studies of the nonlinear optical phenomena in a Si-NCs/SiO2 composite, we relate its effective susceptibility to the third-order susceptibility tensor of silicon and linear permittivities of the composite’s constituents. We begin by considering the situation in which all nanocrystals have the same orientation with respect to the macroscopic sample of the composite. In our derivation we closely follow the effective-medium approach to the calculation of the nonlinear susceptibilities of granular matter [16]. It was initially applied by Zeng et al. [17] to random composites featuring a weakly nonlinear relation between electric field and electric displacement of the form D = (ε + χ|E|2)E, where χ is a scalar. Our derivation shows that the values of all components of the effective susceptibility tensor are reduced (with respect to those in silicon) by the same factor that depends on the volume fraction of Si NCs. We then extend our analysis and calculate the effective susceptibility for a situation in which NCs are randomly orientated in space with a uniform distribution. In this case, the anisotropy of the nonlinear optical response of the whole composite is different from that of bulk silicon. In particular, the Raman response resulting from the vibrational subsystem of the NCs is modified the most.

2. Identically oriented nanocrystals

2.1. Linear effective permittivity of Si-NCs/SiO2 composite

The response of both Si NCs and fused silica to a weak optical field is essentially linear and isotropic. In this instance, the space-averaged electric displacement 𝒟k (k = x, y, z) and space-averaged electric field

k=1VEk(r)dV
inside the Si-NCs/SiO2 composite are related simply through the linear effective permittivity εeff as 𝒟k = εeffk. If the local electric field E(r) = (Ex, Ey, Ez) is known everywhere inside the composite, then the linear effective permittivity may be calculated using the definition [18]
εeff=1Vε(r)(E(r))2dV,
where 2=x2+y2+z2 and the integration is evaluated over the entire volume V of the composite. The space-dependent permittivity ε(r) = ε1ϑ1(r) + ε2ϑ2(r) is expressed here through the permittivity ε1 of silicon and permittivity ε2 of silica, as well as through ϑ functions defined as
ϑj(r)={1whenrisinsidethejthmedium;0otherwise.

Since no analytical expression generally exists for the local field, one often resorts to the mean-field approch in order to calculate εeff [19, 20]. Specifically, for the Si-NCs/SiO2 composite with the volume fraction f of the NCs, the effective-medium theory gives

εeff(ε1,ε2,f)=14[u+(u2+8ε1ε2)1/2],
where u = (3f − 1)ε1 + (2 − 3f)ε2. This equation assumes that Si NCs are spherical in shape and their mean size is much smaller than the optical wavelength; this assumption is valid for most practical situations of interest [6,10].

2.2. Third-order effective susceptibility of Si-NCs/SiO2 composite

To simplify the following calculation, we neglect the third-order susceptibility χ(3) of silica completely in the following calculation. This assumption is justified in practice because χ(3) for silicon NCs is much larger than that for silica and is valid as long as the filling factor of Si NCs exceeds, respectively, 0.001% and 0.1% when Raman scattering and Kerr effect are considered. We also assume the average NC diameter to be larger than the exciton Bohr radius in bulk silicon (which is about 5 nm) and neglect the effect of quantum confinement on the third-order susceptibility of silicon. With these simplifications, we can define the effective susceptibility tensor of a Si-NCs/SiO2 composite in a way similar to Eq. (2) [16, 17]:

χklmneff=1V1χklmn(3)(r)Ek(r)El(r)Em*(r)En(r)4dV1,
where χklmn(3)(r) is the third-order susceptibility of silicon (which may be a function of coordinates if the crystallographic axes of different nanocrystals do not coincide) and the integration is over the nanocrystals’ volume V1 = fV. Without knowing the exact field distribution inside Si NCs, however, it is more practical to introduce this susceptibility using averaged fields and displacements,
𝒟k=εeffk+lmnχklmnefflm*n,
and resort to the effective-medium theory [21]. If we assume that the crystallographic axes x, y, and z of all Si NCs have the same spatial orientation, as in Fig. 1(a), then the components of the electric displacement inside the nanocrystals is given by the expression
D1k=ε1E1k+lmnχklmn(3)E1lE1m*E1nε^1E1k,
where the nonlinear permittivity ε̂1 of Si NCs is implicitly defined. Now χklmneff may be expressed through χklmn(3) by invoking an approximate relation
𝒟kεeff(ε^1,ε2,f)k,
which is valid provided the nonlinear terms in Eqs. (5) and (6) are small compared to the linear ones [17].

 

Fig. 1 (a) Identically oriented Si NCs embedded in a SiO2 matrix of permittivity ε2. Nanocrystals are characterized by permittivity ε1, nonlinear susceptibility tensor χklmn(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 χklmneff.

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Expanding the function εeff(ε̂1, ε2, f) in Taylor series about the linear permittivity of silicon, yields

𝒟kεeff(ε1,ε2,f)k+εeff(ε^1,ε2,f)ε^1(ε^1ε1)k=εeffk+εeffε1kE1klmnχklmn(3)E1lE1m*E1n.
The local field E1k may be related to the averaged field k using an auxiliary effective permittivity εaux, which satisfies the relation
εauxEk(r)dV=ε(r)Ek(r)dV.
By differentiating both sides of this relation with respect to ε1, we obtain
kεauxε1=1Vϑ1(r)Ek(r)dV=fE1k,
where the angle brackets denote averaging over the volume of Si NCs. In a similar fashion, Eq. (2) gives
εeffε1=fE122fE122=1f(εauxε1)2.

In deriving this result, we have assumed that electric field is almost uniform inside the nanocrystals, i.e., E1k ≈ 〈E1k〉. With this assumption and Eqs. (8) to (10), we obtain

𝒟kεeffk+1fεeffε1|εeffε1|lmnχklmn(3)lm*n.
The comparison of this equality with Eq. (5) shows that
χklmneff=1fεeffε1|εeffε1|χklmn(3).
The effective susceptibility tensor is thus obtained via a multiplication of χklmn(3) by the scalar attenuation factor, which according to Eq. (4) is given by
ξ=[(3f1)εeff+ε2]2f(u2+8ε1ε2).
This result is valid to the first order in χklmn(3). Since electric field is constant inside a dielectric sphere placed in an initially uniform electric field [22], Eq. (11) becomes more accurate for weakly interacting Si NCs, i.e., for a sample with a smaller filling factor. In the case of a larger filling factor, an accurate relation between the third-order susceptibility of silicon and that of Si-NCs/SiO2 composite may be obtained numerically within the framework of the generalized effective-medium approach developed by Stroud [23,24].

The effective permittivity and attenuation factor of the Si-NCs/SiO2 composite are plotted in Fig. 2 as solid curves. It is seen that the components of the effective susceptibility tensor are about 200 times smaller for a Si-NCs/SiO2 composite than those of Si NCs for moderate filling factors of about 10%. However, thanks to the strong optical nonlinearities of Si NCs, these may still be comparable to, and even exceed, the nonlinear coefficients in bulk silicon. For example, the Kerr coefficient n2 at a wavelength of 1.55 μm is approximately equal to 4×10−14 cm2/W for bulk silicon and to 2.5 × 10−14 cm2/W for the Si-NCs/SiO2 composite in which Si NCs with n2 = 2 × 10−12 cm2/W occupy 17% of the volume [6]. The value of n2eff may be increased by using a host medium with higher permittivity [25, 26]. For instance, if Si3N4 with ε2 = 4.1 is used instead of SiO2 in the above example, we obtain n2eff8.1×1014cm2/W. For f ≪ 1, the attenuation factor for Si-NCs/SiO2 may be approximated as ξ ≈ 81 f/(2+ε1/ε2)4 ≈ 0.22 f [20,27].

 

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.

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It is interesting to note that Si-NCs/SiO2 composite may exhibit dipolar second-harmonic (SH) due to the effect of NC interfaces, despite the fact that bulk silicon is a centrosymmetric medium and can generate SH only through the quadrupolar nonlinearity [28, 29]. The theories of SH generation by a single centrosymmetric NC and a disordered composite of such NCs were developed in Refs. [30,31].

It is also worth noting that Eq. (11) naturally generalizes the result of Zeng et al. [17] to the case of identically oriented crystallites possessing a tensorial third-order susceptibility.

3. Randomly oriented nanocrystals

Consider now the situation where the crystallographic axes of different Si NCs have all possible orientations in space, as is shown schematically in Fig. 3(a). The effective nonlinear susceptibility in this case is still defined by Eq. (5), but the constitutive relation in Eq. (6) is no longer valid for an arbitrary nanocrystal. As a result of this, each component of the effective susceptibility tensor becomes dependent on several components of χklmn(3), and its calculation requires a knowledge of the exact tensorial form of the nonlinear optical susceptibility of silicon.

 

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.

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3.1. Nonlinear optical susceptibility of silicon

The third-order susceptibility of silicon may be represented as a sum of contributions from bound electrons and optical phonons [32]

χklmn(3)(ω;ω1,ω2,ω3)=χxxxxe(ω)𝒦klmn+12[H(ω1+ω2)klmn+H(ω2+ω3)knml],
where χxxxxe(ω) is a complex constant, H(ω) is the Raman gain profile,
𝒦klmn=(ρ/3)(δklδmn+δkmδln+δknδlm)+(1ρ)δklδlmδmn,
klmn=δkmδln+δknδlm2δklδlmδmn,
ρ=3χxxyye/χxxxxe is the anisotropy factor, and δij is the Kronecker delta. The first term in Eq. (12) leads to the Kerr effect and two-photon absorption, and the remaining terms lead to stimulated Raman scattering [33, 34]. It is important to keep in mind that these effects not only exhibit different tensorial properties but are also characterized by significantly different response times.

3.2. Susceptibility tensor averaging

Suppose that the crystallographic axes {x,y,z} of a certain sub-ensemble of the entire Si-NC ensemble may be brought in to coincide with the reference frame {α, β, γ} of the macroscopic sample via their rotation by angles from ψ to ψ + dψ around radius vectors lying within the infinitesimal solid angle dΩ = sinϑ dϑ dφ about the unit vector u, whose position in the reference frame is determined by polar angle ϑ and azimuth φ [see Fig. 3(b)]. Then the transformed susceptibility tensor characterizing the sub-ensemble can be written as

χκλμν(3)=klmnRκkRλlRμmRνnχklmn(3),
where the rotation matrix is given by the Rodrigues’ formula [35]
R(ϑ,φ,ψ)=(cosψcosϑsinψsinφsinϑsinψcosϑsinψcosψcosφsinϑsinψsinφsinϑsinψcosφsinϑsinψcosψ)+(1cosψ)(cos2φsin2ϑcosφsinφsin2ϑcosφcosϑsinϑcosφsinφsin2ϑsin2φsin2ϑsinφcosϑsinϑcosφcosϑsinϑsinφcosϑsinϑcos2ϑ).

As discussed in Section 2.2, the contribution of the sub-ensemble of almost identically oriented Si NCs to the effective third-order susceptibility of the Si-NCs/SiO2 composite is given by the expression

χκλμνeff=1fεeffε1|εeffε1|χκλμν(3).
Its averaging over a uniform distribution of nanocrystal orientations in space yields the effective susceptibility tensor of the entire composite,
χκλμνeff=18π20πsinϑdϑ02πdφ02πdψχκλμνeff(ϑ,φ,ψ).
This expression may be evaluated either directly [using Eqs. (14) and (15)] or using a general method based on finding the rotationally invariant part of the tensor in spherical coordinates [36].

Equations (14)(16) show that the effective susceptibility tensor of the Si-NCs/SiO2 composite that consists of randomly oriented nanocrystals may be conveniently split into electronic and Raman parts, as is done in Eq. (12). The tensor properties of each part can be found by averaging the respective tensors over the possible spacial orientations of the crystallites. This procedure may be simplified by noticing that tensors δklδmn, δkmδln, and δknδlm are rotationally invariant. Applying the averaging procedure defined in Eq. (16) to the triple product in Eq. (13), we obtain

δklδlmδmn=845(δklδmn+δkmδln+δknδlm)+19δklδlmδmn.

Using the preceding result in Eq. (16), the averaged value of 𝒦κλμν is found to be

𝒦κλμν=8+7ρ45(δκλδμν+δκμδλν+δκνδλμ)+1ρ9δκλδλμδμν.
Since the susceptibility tensor of an individual Si NC has only 21 nonzero components, this tensor also has only 21 nonzero components. The numerical factors for these components from Eq. (17) are found to be
αααα=ββββ=γγγγ=29+16ρ451.1,ααββ=αβαβ=αββα==8+7ρ450.375,
where we used ρ ≈ 1.27 near the 1.55-μm wavelength [32]. It is seen that the averaging of the electronic part of the susceptibility tensor leads to about 10% increase in the values of diagonal components and 13% increase in the values of off-diagonal components.

A similar calculation can be performed for the Raman part of the third-order susceptibility. We find that 〈κλμν〉 is given by

κλμν=2945(δκμδλν+δκνδλμ)1645δκλδμν29δκλδλμδμν.
We see that the averaging over NC orientations drastically modifies the interaction of an optical field with phonons in the Si-NCs/SiO2 composite from that in individual nanocrystals. Even though the tensor κλμν has only 12 nonzero components (equal to unity), the averaged tensor 〈κλμν〉 is characterized by 21 nonzero components. The components analogous to those of tensor klmn have a value that is reduced considerably from 1:
αβαβ=αββα=βγβγ==2945.
However, this reduction leads to additional 9 components of the averaged tensor to become nonzero with values
αααα=ββββ=γγγγ=3245,ααββ=ααγγ=γγββ==1645.

These results imply that the dynamics of Raman amplification in a Si-NCs/SiO2 composite will be different from that in a silicon-on-insulator waveguide [37]. It simply follows from the fact that, unlike the crystallographic axes of individual nanocrystals, the reference axes associated with the composite as a whole can be chosen arbitrarily. One consequence of this feature is that stimulated Raman scattering occurs in Si-NCs/SiO2 composites regardless of the polarizations of pump and signal waves.

From a practical viewpoint, it is important to note that the averaging of the third-order susceptibility over the orientations of Si NCs does not make the nonlinear response of the Si-NCs/SiO2 composite isotropic. This feature makes fused silica doped with randomly oriented Si NCs an attractive medium for polarization-sensitive applications, such as optical switching [38] or power equalization [39], that utilize nonlinear polarization rotation through cross-phase modulation. This statement does not apply to such nonlinear phenomena as thermo-optic effect [7] and electrostriction [40], which are essentially isotropic in individual Si NCs. Since such effects are often related to free carriers generated through two-photon absorption, they develop on nanosecond time scales and may cause the nonlinear response of the composite to vary with pulse width.

We emphasize that our theory is only applicable to those nonlinearities of silicon that can be described by the third-order susceptibility tensor; it does not include such effects as free-carrier absorption, which is essentially a fifth-order effect [32].

4. Conclusions

We have derived approximate analytic expressions for the effective third-order susceptibility of Si-NCs/SiO2 composites consisting of nanocrystals with either identically or randomly oriented crystallographic axes. We showed that when the orientations of the crystallographic axes in different crystallites are the same, the effective susceptibility of the composite may be simply obtained via a multiplication of the susceptibility tensor of silicon by a constant factor. On the other hand, if the crystallites are oriented in space randomly, then the effective susceptibility has tensor properties which are significantly different from those of silicon susceptibility. In the case of the Kerr nonlinearity, the values are enhanced by 10% or more depending on the susceptibility component.

Much more dramatic changes occur in the case of Raman susceptibility due to a modification of the interaction between the optical field and phonons inside the composite. In particular, some components that vanish in bulk silicon or planar silicon waveguides acquire a finite value in the case of Si-NCs/SiO2 composites. The new form of Raman susceptibility has practical consequences if Si NCs are used to make Raman amplifiers and lasers in place of silicon planar waveguides. Results obtained in this paper should be useful for modeling nonlinear propagation through Si-NCs/SiO2 fibers, which are promising candidates for realization of all-optical functions on a photonic chip.

Acknowledgments

This work is sponsored by the Australian Research Council, through its Discovery Early Career Researcher Award DE120100055. The work of W. Zhu and M. Premaratne is supported by the Australian Research Council through its Discovery Grant scheme under grant DP110100713.

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References

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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. Photonics 4, 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. Photonics 4, 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. Express 17, 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. Express 17, 22124–22137 (2009).
    [Crossref] [PubMed]
  8. M. Paniccia, “Integrating silicon photonics,” Nat. Photonics 4, 498–499 (2010).
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  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. B 37, 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. B 38, 10970–10973 (1988).
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  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. B 12, 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. B 84, 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. B 68, 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. Express 15, 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. Express 20, 4784–4789 (2012).
    [Crossref] [PubMed]

2012 (4)

2011 (2)

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. B 84, 165316 (2011).
[Crossref]

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

2010 (5)

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics 4, 498–499 (2010).
[Crossref]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[Crossref]

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4, 511–517 (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 (4)

2008 (1)

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 (2)

2003 (1)

W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
[Crossref]

2001 (1)

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 (1)

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 (1)

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 (1)

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 (2)

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]

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]

1988 (2)

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B 37, 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. B 38, 10970–10973 (1988).
[Crossref]

1986 (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]

1978 (1)

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

1975 (1)

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 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]

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]

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, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express 17, 5807–5814 (2009).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009).
[Crossref] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 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. B 38, 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. Photonics 4, 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. B 68, 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. B 84, 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. Photonics 4, 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. B 38, 10970–10973 (1988).
[Crossref]

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B 37, 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. Photonics 4, 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. Photonics 4, 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. Photonics 4, 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. B 68, 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. B 84, 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. B 68, 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. B 68, 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. Photonics 4, 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. B 37, 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. B 38, 10970–10973 (1988).
[Crossref]

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 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. B 84, 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. B 84, 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. B 38, 10970–10973 (1988).
[Crossref]

Zhang, J.

Adv. Opt. Technol. (1)

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. (1)

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. (1)

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. (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]

IEEE Photon. J. (2)

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. (1)

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. (1)

J. Mod. Opt. (1)

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. (1)

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 (3)

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics 4, 498–499 (2010).
[Crossref]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[Crossref]

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4, 511–517 (2010).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rep. (1)

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 (5)

D. Stroud and P. M. Hui, “Nonlinear susceptibilities of granular matter,” Phys. Rev. B 37, 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. B 38, 10970–10973 (1988).
[Crossref]

W. L. Mochan, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68, 085318 (2003).
[Crossref]

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. B 84, 165316 (2011).
[Crossref]

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
[Crossref]

Phys. Rev. Lett. (2)

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]

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]

Pure Appl. Opt. (1)

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. (1)

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 (10)

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).

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).

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]

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).

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)

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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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