## Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications |

Optics Express, Vol. 19, Issue 1, pp. 206-217 (2011)

http://dx.doi.org/10.1364/OE.19.000206

Acrobat PDF (942 KB)

### Abstract

Silicon-based plasmonic waveguides can be used to simultaneously transmit electrical signals and guide optical energy with deep subwavelength localization, thus providing us with a well needed connecting link between contemporary nanoelectronics and silicon photonics. In this paper, we examine the possibility of employing the large third-order nonlinearity of silicon to create active and passive photonic devices with silicon-based plasmonic waveguides. We unambiguously demonstrate that the relatively weak dependance of the Kerr effect, two-photon absorption (TPA), and stimulated Raman scattering on optical intensity, prevents them from being useful in *μ*m-long plasmonic waveguides. On the other hand, the TPA-initiated free-carrier effects of absorption and dispersion are much more vigorous, and have strong potential for a variety of practical applications. Our work aims to guide research efforts towards the most promising nonlinear optical phenomena in the thriving new field of silicon-based plasmonics.

© 2011 Optical Society of America

## 1. Introduction

1. G. T. Reed and A. P. Knights, *Silicon Photonics: An Introduction* (Wiley, Hoboken, 2004). [CrossRef]

9. J. Y. Lee, L. Yin, G. P. Agrawal, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express **18**, 11514–11523 (2010). [CrossRef] [PubMed]

10. L. Tang, S. Latif, and D. A. B. Miller, “Plasmonic device in silicon CMOS,” Electron. Lett. **45**, 706 (2009). [CrossRef]

13. J. N. Caspers, N. Rotenberg, and H. M. van Driel, “Ultrafast silicon-based active plasmonics at telecom wavelengths,” Opt. Express **18**, 19761–19769 (2010). [CrossRef] [PubMed]

1. G. T. Reed and A. P. Knights, *Silicon Photonics: An Introduction* (Wiley, Hoboken, 2004). [CrossRef]

6. B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. **12**, 412–421 (2006). [CrossRef]

7. R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. **12**, 1678–1687 (2006). [CrossRef]

14. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nature Mater. **9**, 193–204 (2010). [CrossRef]

18. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

19. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. **70**, 1–87 (2007). [CrossRef]

20. U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B **64**, 125420(1–10) (2001). [CrossRef]

11. J. A. Dionne, L. A. Sweatlock, M. T. Sheldon, A. P. Alivisatos, and H. A. Atwater, “Silicon-based plasmonics for on-chip photonics,” IEEE J. Sel. Top. Quantum Electron. **16**, 295–306 (2010). [CrossRef]

13. J. N. Caspers, N. Rotenberg, and H. M. van Driel, “Ultrafast silicon-based active plasmonics at telecom wavelengths,” Opt. Express **18**, 19761–19769 (2010). [CrossRef] [PubMed]

21. M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express **16**, 1385–1392 (2008). [CrossRef] [PubMed]

23. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express **12**, 4072–4079 (2004). [CrossRef] [PubMed]

24. M. Makarova, Y. Gong, S.-L. Cheng, Y. Nishi, S. Yerci, R. Li, L. D. Negro, and J. Vuc̆ović, “Photonic crystal and plasmonic silicon-based light sources,” IEEE J. Sel. Top. Quantum Electron. **16**, 132–140 (2010). [CrossRef]

26. R. J. Walters, R. V. A. van Loon, I. Brunets, J. Schmitz, and A. Polman, “A silicon-based electrical source of surface plasmon polaritons,” Nature Mater. **9**, 21–25 (2010). [CrossRef]

27. A. V. Krasavin and A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express **18**, 11791–11799 (2010). [CrossRef] [PubMed]

## 2. Nonlinear propagation equation

*β*(

*ω*)—from its much more slow variation due to the nonlinear effects in silicon. The electric and magnetic field vectors

**E**(

*x,z, t*) and

**H**(

*x,z, t*) of the SPP mode are thus represented in the form [28

28. B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B **27**, 956–965 (2010). [CrossRef]

30. X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. **42**, 160–170 (2006). [CrossRef]

*A*(

*z, t*) is the temporal envelope,

**E**

_{0}(

*x*,

*ω*) and

**H**

_{0}(

*x*,

*ω*) describe the lateral profile of the SPP mode (unperturbed by the nonlinear effects) at the carrier frequency

*ω*, and 𝒩 is a normalization constant. From here onwards, we consider a slot MSM waveguide that is infinite in the

*y*direction and assume that SPPs propagate along its

*z*axis (see Fig. 1). Nevertheless, as will be evident from the following discussion, the conclusions we reach are also valid for MSM nanowires.

**e**

*is the unit vector in the*

_{z}*z*direction and an asterisk denotes complex conjugation, then the SPP-mode power per unit width of the plasmonic waveguide (in the

*y*direction) has the form where

*L*

_{SPP}= (2Im

*β*)

^{−1}is the propagation length of SPPs. Even though, formally, there is ambiguity in the corresponding intensity,

*I*(

*z, t*) =

*P*(

*z,t*)/𝒟, due to an extension of the mode field to infinity in the

*x*direction, the tight confinement of SPPs within the silicon layer makes the effective mode thickness 𝒟 approximately equal to the layer thickness

*d*. Mathematically, 𝒟 is given by a form naturally arising during derivation of the propagation equation. With this definition, we can apply the same propagation equation for both the MSM and SOI waveguides.

**E**

_{0}= {

*E*, 0,

_{x}*E*} and

_{z}**H**

_{0}= {0,

*H*, 0}. The spatial profile of these modes is given by [31–33] and

_{y}*H*(

_{y}*x*,

*ω*) =

*Y*(

_{j}*ω*)

*E*(

_{x}*x*,

*ω*). Here,

*Y*(

_{j}*ω*) =

*ɛ*

_{0}

*ɛ*/

_{j}ω*β*(

*ω*),

*k*= (

_{j}*ɛ*

_{j}k^{2}–

*β*

^{2})

^{1/2}for (

*j*= 1, 2),

*k*=

*ω*/

*c*, and

*c*is the speed of light in vacuum.

*β*is found from the SPPs’ dispersion relation [16] where we assume

*ɛ*

_{1}(

*ω*) to be real and

*ɛ*

_{2}(

*ω*) to be complex. In the case of ultrathin MSM waveguides with

*d*≪ |

*k*

_{1}|

^{−1}, Eq. (5) leads to the following explicit form of

*β*: As an example, the relative error in |

*β*(

*ω*,

*d*)| calculated using this formula at the wavelength of 1.55

*μ*m is below 1.5% for a silver–silicon–silver (Ag/Si/Ag) waveguide with

*d*≤ 50 nm. A simple analysis of the function

*β*(

*ω*,

*d*) in the limit

*d*→ 0 reveals that, in contrast to SOI waveguides where

*β*tends to

*β*∼ −(2/

*d*)(

*ɛ*

_{1}/

*ɛ*

_{2}). As a consequence of this limiting behavior,

*L*

_{SPP}∼

*d*|

*ɛ*

_{2}|

^{2}/(4

*ɛ*

_{1}Im

*ɛ*

_{2}), i.e., the propagation length of SPPs tends to zero in ultrathin waveguides. This feature clearly forces a compromise between strong localization of the field energy and its guiding over reasonable distances.

*A*(

*z, t*) closely follows that detailed in Ref. [28

28. B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B **27**, 956–965 (2010). [CrossRef]

28. B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B **27**, 956–965 (2010). [CrossRef]

*N*of TPA-generated free carriers are found to be: where

*β*= d

_{n}*/d*

^{n}β*ω*is the

^{n}*n*th order dispersion parameter.

*τ*

_{eff}is the effective free-carrier lifetime,

*n*

_{2}= 6 × 10

^{−5}cm

^{2}/GW is the nonlinear Kerr parameter,

*n*

_{0}= 3.484,

*β*

_{TPA}= 0.5 cm/GW,

*σ*= 1.45 × 10

_{α}^{−21}m

^{2}, and

*σ*= 5.3 × 10

_{n}^{−27}m

^{3}[34

34. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. **16**, 200–215 (2010). [CrossRef]

36. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. **82**, 2954–2956 (2003). [CrossRef]

*η*< 1 are, respectively, the longitudinal enhancement factor (LEF) and the nonlinear overlap factor (NOF) introduced in Ref. [28

**27**, 956–965 (2010). [CrossRef]

*appearing in*

_{κλμν}*η*characterizes the anisotropy of the Kerr effect and TPA, which depends on the orientation of the principal axes in silicon [37

37. C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express **18**, 21427–21448 (2010). [CrossRef] [PubMed]

34. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. **16**, 200–215 (2010). [CrossRef]

38. I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach,” J. Lightwave Technol. **27**, 3241–3248 (2009). [CrossRef]

39. 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]

*η*, Γ, and ℬ) that affect the SPP propagation (see Appendix B for details). Figure 2(a) shows how these parameters vary with the thickness of the MSM waveguide (Ag/Si/Ag) at the telecommunication wavelength

*λ*= 1.55

*μ*m. We assume that the waveguide is fabricated as shown in Fig. 1 and use the dielectric function of silver from Ref. [40

40. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express **18**, 6191–6204 (2010). [CrossRef] [PubMed]

_{MSM}and Γ

_{MSM}diverge as

*d*approaches zero, while

*η*

_{MSM}weakly depends on

*d*and tends to unity in this limit; this behavior reflects the increasing confinement of SPPs within the silicon layer. On the other hand, the inability of SOI waveguides to tightly confine optical mode when

*d*is below 200 nm, results in ℬ

_{SOI}→ 0,

*η*

_{SOI}→ 0 and Γ

_{SOI}→ 1. Hence, as was expected, the nonlinear effects in an MSM plasmonic waveguide become much more pronounced compared with a SOI waveguide as the waveguides’ cross sections decrease. Because the same parameters affect different third-order effects, this conclusion covers all of them, including stimulated Raman scattering (SRS), even though it is neglected in our model.

*d*and becomes proportional to Γ when

*d*tends to zero. For

*λ*= 1.55

*μ*m, PAF exceed 0.1 and starts affecting the phase of SPPs in waveguides thinner than 20 nm. Meanwhile, the intensity of SPPs is affected in thicker waveguides with ϖ ≳ 0.01, as will be seen later.

## 3. Efficiency of nonlinear effects in MSM plasmonic waveguides

### 3.1. Simplified propagation equation

*μ*m [see Fig. 2(b)], dispersive effects of second and higher orders have little impact on the SPP [27

27. A. V. Krasavin and A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express **18**, 11791–11799 (2010). [CrossRef] [PubMed]

*β*

_{2}, which determines the dispersion length

*L*of optical pulses. As can be found using these data, the dispersion length of a 100-fs pulse in a 25-nm-long MSM waveguide exceeds 1 mm, i.e., it comprises of more than 1000 propagation lengths. A further examination shows that this ratio between

_{D}*L*and

_{D}*L*

_{SPP}holds approximately for shorter waveguides as well.

38. I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach,” J. Lightwave Technol. **27**, 3241–3248 (2009). [CrossRef]

*τ*=

*t*–

*β*

_{1}

*z*in Eq. (8) is not purely real due to the small imaginary part of

*β*

_{1}; this part is, however, insignificant for even 100-fs pulses and can be safely neglected. Interestingly,

*β*

_{eff}and

*ζ*can change their sign and become negative in thin MSM waveguides. If this were to take place, TPA will result in amplification of SPPs, and FCD will increase the refractive index of silicon (just as the Kerr effect does). Unfortunately, as will become evident from the following discussion, these unusual properties are unlikely to be utilized in practice.

_{i}*ζ*and

_{r}*ζ*are proportional to the effective free-carrier lifetime. By varying

_{i}*τ*

_{eff}, we can shift the relative importance of the third-order nonlinear effects and free-carrier effects. The effective free-carrier lifetime can be substantially reduced by removing carriers from the optical mode region through the application of a static electric field. If

*V*is the external voltage applied to the metallic parts of the MSM waveguide, then

*τ*

_{eff}is given by the relation where

*μ*≈ 1000 cm

^{2}/(V · s) [39

39. 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]

*τ*is the free-carrier lifetime in the absence of rejection. For example, by applying a voltage of 10 mV to a 100-nm-thick MSM waveguide, we reduce

_{c}*τ*

_{eff}from ns-range to 5 ps, while staying well below the breakdown field of ≈ 3 × 10

^{5}V/cm for silicon.

### 3.2. Propagation of SPPs in the quasi-CW regime

*T*

_{0}≫

*τ*

_{eff}(quasi-CW pulse), so that we may set

*A*(

*z*,

*τ*–

*τ*

_{eff}

*q*) ≈

*A*(

*z,*

*τ*) in Eq. (8). The resulting equation has a well known analytical solution [41

41. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. **35**, 55–57 (2010). [CrossRef] [PubMed]

*I*

_{0}(

*τ*) is the temporal profile of the input pulse used to excite the SPP inside the MSM waveguide. It is easy to see that solution (9) is applicable when while the solution (10) is valid as long as

*τ*

_{eff}= 5 ns (

*V*= 0), free-carrier effects are by far much stronger than the Kerr effect and TPA for all intensities of practical interest. As

*τ*

_{eff}is reduced to 50 ps (

*V*= 1 mV), TPA and the Kerr effect start to dominate for

*I*

_{K}∼ 1 GW/cm

^{2}in MSM waveguides thinner than 15 nm. To make FCA and FCD negligible as compared to the third-order effects in thicker waveguides,

*τ*

_{eff}= 5 ps (

*V*= 10 mV) is required.

*L*

_{SPP}through MSM waveguides of different thicknesses. It was assumed that

*τ*

_{eff}= 5 ns, and the free-carrier effects dominate. One can see that the FCD can produce nonlinear phase shifts ∼

*π*within a length of only 1–2

*μ*m for a 30-nm-thick MSM waveguide. At the same time, FCA may be rather moderate, if the pulse intensity is not very high. These results imply that the free-carrier effects can be successfully employed to realize all FCA- and FCD-based device functionalities on a nanoscale that have been previously demonstrated on a centimeter scale with the SOI technology. These applications include all-optical switching [41

41. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. **35**, 55–57 (2010). [CrossRef] [PubMed]

45. M. W. Geis, S. J. Spector, R. C. Williamson, and T. M. Lyszczarz, “Submicrosecond, submilliwatt, silicon-on-insulator thermooptic switch,” IEEE Photon. Technol. Lett. **16**, 2514–2516 (2004). [CrossRef]

46. S. Abdollahi and M. K. Moravvej-Farshi, “Effects of heat induced by two-photon absorption and free-carrier absorption in silicon-on-insulator nanowaveguides operating as all-optical wavelength converters,” Appl. Opt. **48**, 2505–2514 (2009). [CrossRef] [PubMed]

47. D. F. Logan, P. E. Jessop, A. P. Knights, G. Wojcik, and A. Goebel, “Optical modulation in silicon waveguides via charge state control of deep levels,” Opt. Express **17**, 18571–18580 (2009). [CrossRef]

49. E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. **91**, 201115(1–3) (2007). [CrossRef]

*φ*

_{K}(

*L*

_{SPP})/

*π*≲ 0.005, and the relative decrease in intensity is below 0.5%, irrespective of the specific values of

*I*

_{0}< 10 GW/cm

^{2}and

*d*< 230 nm. Thus, for making use of the third-order nonlinear effects in silicon for the SPPs, the prediction is rather unfavorable: they are too weak to develop on the length scale of

*L*

_{SPP}and unlikely to be employed in practice. It is easy to ascertain the reason behind such a result. In the limit

*d*→ 0, the coefficients

*β*

_{eff}and

*γ*

_{eff}behave like ∝ Γ

^{2}, whereas

*ζ*and

_{r}*ζ*grow much faster—as a product ℬΓ

_{i}^{2}[see Fig. 3(b)]. Physically, this discrepancy is due to different dependance of the free-carrier and third-order nonlinear effects on optical intensity. Since SRS also stems from the third-order susceptibility, it is clear that neither Raman amplification nor lasing are likely to be possible with the silicon-based plasmonic waveguides (despite the fact that SRS coefficient in silicon is much bigger than the TPA coefficient). Of course, one can always introduce gain into silicon layer to compensate for ohmic losses and allow SPPs to build up sufficient gain or phase shift [21

21. M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express **16**, 1385–1392 (2008). [CrossRef] [PubMed]

23. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express **12**, 4072–4079 (2004). [CrossRef] [PubMed]

27. A. V. Krasavin and A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express **18**, 11791–11799 (2010). [CrossRef] [PubMed]

*d*∼ 1 nm), in which the field intensity is extremely high. This situation seems unusual for SOI waveguides, where nonlinear effects produced by strong optical fields significantly perturb the lateral mode profile. In MSM waveguides, the intensity of the electric field becomes more and more uniform inside the silicon layer, as its thickness decreases. As a consequence, the intensity-dependant nonlinear effects cannot modify the lateral profile of the SPP mode, and the accuracy of the approximations used to derive Eq. (7) improves.

## 4. Conclusions

*λ*= 1.55

*μ*m, our conclusions are valid for other waveguide geometries and wavelengths, and are not specific to silver. Our estimates show that, even though all nonlinear effects are substantially stronger in a metal-silicon-metal (MSM) plasmonic waveguide than in a SOI waveguide of the same thickness, the reduced length of the MSM waveguide makes the Kerr effect, two-photon absorption, and stimulated Raman scattering too inefficient for device applications. Fortunately, the free-carrier effects remain strong enough even in a 10-nm-thick MSM waveguide that is only 500-nm long, and they can dramatically affect the propagation of SPPs. Therefore, it is free-carrier absorption and free-carrier dispersion that should be used to control light in nano-sized, silicon-based plasmonic waveguides.

## Appendix A

*iωμ*

_{0}

**H̃**= ∇ ×

**Ẽ**, Employing the vector-triple-product identity and decomposing the electric field into transverse and longitudinal components, we obtain

## Appendix B

*k*as

_{j}*k′*and

_{j}*k″*for (

_{j}*j*= 1, 2) and using Eq. (4), we obtain and

*E*= 0 for the TM SPP mode (see Fig. 1), only the following eight components of the anisotropy tensor contribute to the material polarization [37

_{y}37. C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express **18**, 21427–21448 (2010). [CrossRef] [PubMed]

*ρ*≈ 1.27 in the 1.55-

*μ*m region. With these components, we find that The same expressions are applicable to SOI waveguides, after the replacement sinh(

*νk*″

_{1}

*d*)/

*k*″

_{1}→

*νd*.

## Acknowledgments

## References and links

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27. | A. V. Krasavin and A. V. Zayats, “Silicon-based plasmonic waveguides,” Opt. Express |

28. | B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B |

29. | S. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express |

30. | X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. |

31. | J. Bures, |

32. | J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B |

33. | A. W. Snyder and J. D. Love, |

34. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. |

35. | H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. |

36. | M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. |

37. | C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express |

38. | I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach,” J. Lightwave Technol. |

39. | Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express |

40. | A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express |

41. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. |

42. | Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. |

43. | Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express |

44. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express |

45. | M. W. Geis, S. J. Spector, R. C. Williamson, and T. M. Lyszczarz, “Submicrosecond, submilliwatt, silicon-on-insulator thermooptic switch,” IEEE Photon. Technol. Lett. |

46. | S. Abdollahi and M. K. Moravvej-Farshi, “Effects of heat induced by two-photon absorption and free-carrier absorption in silicon-on-insulator nanowaveguides operating as all-optical wavelength converters,” Appl. Opt. |

47. | D. F. Logan, P. E. Jessop, A. P. Knights, G. Wojcik, and A. Goebel, “Optical modulation in silicon waveguides via charge state control of deep levels,” Opt. Express |

48. | J. Basak, L. Liao, A. Liu, D. Rubin, Y. Chetrit, H. Nguyen, D. Samara-Rubio, R. Cohen, N. Izhaky, and M. Paniccia, “Developments in gigascale silicon optical modulators using free carrier dispersion mechanisms,” Adv. Opt. Technol. |

49. | E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett. |

**OCIS Codes**

(040.6040) Detectors : Silicon

(130.0250) Integrated optics : Optoelectronics

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 21, 2010

Revised Manuscript: December 13, 2010

Manuscript Accepted: December 17, 2010

Published: December 22, 2010

**Citation**

Ivan D. Rukhlenko, Malin Premaratne, and Govind P. Agrawal, "Nonlinear propagation in silicon-based plasmonic waveguides from the
standpoint of applications," Opt. Express **19**, 206-217 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-206

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