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

Ivan D. Rukhlenko; Malin Premaratne; Govind P. Agrawal

doi:10.1364/OE.19.000206

1. Introduction

Silicon is beginning to be considered for on-chip integration of recently demonstrated photonic devices [1–9] with mainstream electronic components [10–13]. Such an integration will eliminate the basic drawbacks of electronic circuitry and become a significant milestone in the development of power-efficient, high-speed optical supercomputers. One of the major obstacles we face when trying to bring photonics functionalities to the scale of nanoelectronics is the diffraction of light. Specifically, the diffraction phenomenon places a fundamental limit on the miniaturization of photonic devices fabricated with silicon-on-insulator (SOI) technology. Even the smallest SOI waveguides have dimensions of ∼ 300 nm [1, 6, 7], which are about ten times what is required for successful integration of electronics and photonics for novel breakthrough applications.

Assistance in the controlling of light on the nanoscale comes from the paradigm of plasmonics [14–18]. In certain situations, light can interact with collective oscillation of electron-hole plasma and form coupled waves known as surface plasmon polaritons (SPPs) [19, 20]. The metal–semiconductor nanowires that guide SPPs are not restricted by the diffraction limit of light and are suitable for carrying information at bit rates comparable to that of high-bandwidth fiber-optic channels. A marriage of silicon photonics with plasmonics will merge the advantages of optical data transfer with the strongest points of modern nanotechnology, facilitating the development of all-optical chips for future supercomputers.

Before such supercomputers can be realized on a silicon platform, a number of technical issues need to be solved and fundamental questions answered. Many of these issues have recently been investigated, and continue to receive much attention [11–13]. Among them is the problem of strong dissipation of SPPs which normally decay within a distance of a few tens of micrometers [21–23]. Another challenge is to create a CMOS-compatible source of SPPs that would allow their generation through electrical pumping [24–26]. Of primary importance are also the questions of the coupling efficiency of closely spaced nanowires, and their interaction with the environment [27]. In this paper, we address a further important question that naturally arises in connection with the adoption of silicon-based functionalities by plasmonic nanowires, which is whether or not the nonlinear effects in silicon are strong enough to control the propagation of SPPs along silicon–metal nanowaveguides. This is a nontrivial issue, as deep subwavelength localization of an optical field makes the nonlinear effects more pronounced, but the short lifetime of SPPs severely limits the interaction length.

We start our analysis by deriving a general integro-differential equation for the slowly varying envelope of the SPP mode in a slot metal–silicon–metal (MSM) plasmonic waveguide. The planar geometry of the waveguide was selected intentionally to allow for an analytic approach. General considerations indicate that the conclusions obtained with our model will hold true for other geometries, specifically for nanowires. Our analysis takes fully into account the vectorial nature of the SPP mode and incorporates all linear and nonlinear effects that may possibly affect its propagation. We examine the impact of different nonlinear effects on the intensity and phase of SPPs, using two approximate analytical solutions of our integro-differential equation in the quasi-continuous-wave (quasi-CW) regime. The results of our analysis are unequivocal: third-order nonlinear effects themselves are too weak to distinctly affect SPPs, but the free-carrier effects initiated by two-photon absorption (TPA) can be used to manipulate light squeezed tightly into the SPP mode.

2. Nonlinear propagation equation

It is convenient to describe the propagation of SPPs along MSM plasmonic waveguides within the frame of the slowly varying envelope approximation (SVEA). As applied to MSM waveguides, the idea of SVEA is to separate the rapid spatial oscillations and steep spatial decay of the SPP field—described by the complex-valued propagation constant 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 [28–30]

E (x, z, t) \approx \sqrt{\frac{2}{𝒩}} A (z, t) E_{0} (x, ω) exp [i (β z - ω t)] + c . c .,

H (x, z, t) \approx \sqrt{\frac{2}{𝒩}} A (z, t) H_{0} (x, ω) exp [i (β z - ω t)] + c . c .,

where A(z, t) is the temporal envelope, E₀(x,ω) and H₀(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.

Fig. 1 Geometry of a metal–silicon–metal plasmonic waveguide and electromagnetic field components of the TM SPP mode. Two metallic regions of permittivity ɛ₂ are separated by a silicon layer of thickness d and permittivity ɛ₁.

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It can be readily verified using Eq. (1) that, if we set

𝒩 = \frac{1}{2} \int_{- \infty}^{+ \infty} e_{z} [E_{0} (x, ω) \times H_{0}^{*} (x, ω)] d x + c . c .,

where e_z is the unit vector in the 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

P (z, t) = {| A (z, t) |}^{2} exp (- z / L_{SPP}),

where L_SPP = (2Imβ)⁻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

𝒟 = \frac{1}{Q} {(\int_{- \infty}^{+ \infty} {| E_{0} |}^{2} d x)}^{2}, Q = \int_{- \infty}^{+ \infty} {| E_{0} |}^{4} d x,

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.

The slot MSM waveguide in Fig. 1 supports only antisymmetric, transverse-magnetic (TM) SPP modes of the form E₀ = {E_x, 0, E_z} and H₀ = {0, H_y, 0}. The spatial profile of these modes is given by [31–33]

E_{x} (x, ω) = {\begin{array}{l} cos (k_{1} x), & | x | < d / 2, \\ cos (k_{1} d / 2) (ɛ_{1} / ɛ_{2}) exp [i k_{2} (| x | - d / 2)], & | x | > d / 2, \end{array}

E_{z} (x, ω) = \frac{k_{1}}{i β} \times {\begin{array}{l} sin (k_{1} x), & | x | \leq d / 2, \\ sin (k_{1} d / 2) sign (x) exp [i k_{2} (| x | - d / 2)], & | x | \geq d / 2, \end{array}

and H_y(x,ω) = Y_j(ω)E_x(x,ω). Here, Y_j(ω) = ɛ₀ɛ_jω/β(ω), k_j = (ɛ_jk² – β²)^1/2 for (j = 1, 2), k = ω/c, and c is the speed of light in vacuum.

The propagation constant β is found from the SPPs’ dispersion relation [16]

tanh [\frac{i k_{1} (ω) d}{2}] = \frac{ɛ_{1} (ω)}{ɛ_{2} (ω)} \frac{k_{2} (ω)}{k_{1} (ω)},

where we assume ɛ₁(ω) to be real and ɛ₂(ω) to be complex. In the case of ultrathin MSM waveguides with d ≪ |k₁|⁻¹, Eq. (5) leads to the following explicit form of β:

β (ω, d) \approx \frac{1}{d} {ɛ_{2} {(k d)}^{2} + {[ɛ_{1} / ɛ_{2} - \sqrt{{(ɛ_{1} / ɛ_{2})}^{2} + (ɛ_{1} - ɛ_{2}) {(k d)}^{2}}]}^{2}}^{1 / 2} .

As an example, the relative error in |β (ω, d)| calculated using this formula at the wavelength of 1.55 μm 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

\sqrt{ɛ_{2}} k

, in MSM waveguides it diverges like β ∼ −(2/d)(ɛ₁/ɛ₂). As a consequence of this limiting behavior, L_SPP ∼ d|ɛ₂|²/(4ɛ₁Imɛ₂), 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.

Our derivation of the propagation equation for A(z, t) closely follows that detailed in Ref. [28] and is not reproduced here. The major difference of the plasmonic problem from optical propagation through SOI waveguides is associated with the decay of SPPs. This decay is taken into account by employing the equality (see Appendix A)

i (β + β *) {| E_{x} |}^{2} + E_{x} \frac{d E_{z}^{*}}{d x} - E_{x}^{*} \frac{d E z}{d x} = i ω μ_{0} (E_{0} \times H_{0}^{*} + E_{0}^{*} + H_{0}) e_{z},

instead of Eq. (A6) in Ref. [28]. The resulting equations for the slowly varying envelope of the fundamental SPP mode and the density N of TPA-generated free carriers are found to be:

\frac{\partial A}{\partial z} + \sum_{n = 1}^{\infty} i^{n - 1} \frac{β_{b}}{n!} \frac{\partial^{n} A}{\partial t^{n}} = i γ \frac{1 + ir}{1 + i ϖ} {| A |}^{2} A - \frac{ℬ}{1 + i ϖ} (\frac{σ_{α}}{2} + i k σ_{n}) N A,

\frac{\partial N}{\partial t} = - \frac{N}{τ_{eff}} + \frac{{\hat{β}}_{TPA}}{2 \bar{h} ω 𝒟 d} {| A |}^{4},

where β_n = dⁿβ/dωⁿ is the nth order dispersion parameter.

Other parameters appearing in the preceding two equations are defined as

γ = \frac{k n_{2} η}{𝒟} {(\frac{n_{0} k Γ}{Re β})}^{2}, r = \frac{β_{TPA}}{2 k n_{2}}, ℬ = \frac{ɛ_{0} n_{0} c}{𝒩} \int_{- d / 2}^{+ d / 2} {| E_{0} |}^{2} d x, Γ = \frac{Re β}{μ_{0} ω 𝒩} \int_{- \infty}^{+ \infty} {| E_{0} |}^{2} d x,

η = \frac{1}{Q} \sum_{κ λ μ v} \int_{- d / 2}^{+ d / 2} ℰ_{κ λ μ v} E_{κ}^{*} E_{λ}^{*} E_{μ}^{*} E_{ν} d x, {\hat{β}}_{TPA} = β_{TPA} \frac{η}{1 + ϖ^{2}} {(\frac{n_{0} k Γ}{Re β})}^{2} .

Here, τ_eff is the effective free-carrier lifetime, n₂ = 6 × 10⁻⁵ cm²/GW is the nonlinear Kerr parameter, n₀ = 3.484, β_TPA = 0.5 cm/GW, σ_α = 1.45 × 10⁻²¹ m², and σ_n = 5.3 × 10⁻²⁷ m³ [34–36]. The parameters Γ ≥ 1 and 0 < η < 1 are, respectively, the longitudinal enhancement factor (LEF) and the nonlinear overlap factor (NOF) introduced in Ref. [28]. They show how the nonlinear effects in silicon are enhanced due to the presence of the longitudinal electric field and weakened because SPPs reside partially inside the metal. Using Eq. (6) it is easy to show that, in order for Γ to be close to unity, both the longitudinal electric field and its transverse variation should be small. The fourth-rank tensor ℰ_κλμν appearing in η characterizes the anisotropy of the Kerr effect and TPA, which depends on the orientation of the principal axes in silicon [37]. The factor ℬ > 0 is responsible for enhancement of the free-carrier effects due to confinement of the electric field within the semiconductor layer.

Notice that we do not include in Eq. (7a) the term accounting for linear absorption, since this type of loss is automatically included by the imaginary part of the propagation constant. Except for this term, the fundamental difference between Eqs. (7a) and (7b) and the analogous equations for SOI waveguides [34, 38, 39] lies in the presence of the new parameter

ϖ = \frac{Im β}{μ_{0} ω 𝒩} \int_{- \infty}^{+ \infty} {| E_{x} |}^{2} d x,

which owes its existence to the decay of SPPs along the MSM waveguide. We refer to this parameter as the plasmonic attenuation factor (PAF). The PAF is seen not only to reduce the efficiency of different nonlinear effects, but also to affect their phase. As a result of this influence, for instance, FCA can change the refractive index of the silicon layer, while FCD can affect the intensity of the SPP mode. It should be also noted that various dispersion parameters in Eq. (7a) are complex-valued.

Using the spatial mode profiles given in Eq. (4), we analytically calculate the effective mode thickness 𝒟 and other factors (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 μm. We assume that the waveguide is fabricated as shown in Fig. 1 and use the dielectric function of silver from Ref. [40]. For comparison, dashed curves show the same parameters calculated for a single-mode SOI waveguide. It is seen that the factors ℬ_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.

Fig. 2 (a) Parameters Γ, η, ℬ, and 𝒟 for the fundamental mode as functions of silicon layer thickness d in MSM (Ag/Si/Ag) and SOI waveguides; circles correspond to 𝒟 = d. (b) Plasmonic attenuation factor ϖ and SPP propagation length L_SPP for the same Ag/Si/Ag waveguide.

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In Fig. 2(b), we plot the PAF for the same Ag/Si/Ag plasmonic waveguide as a function of its thickness. This factor grows monotonously with decreasing d and becomes proportional to Γ when d tends to zero. For λ = 1.55 μm, 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

In this section we analyze the evolution of a SPP by solving Eq. (7), and estimate quantitatively the magnitude of nonlinear changes in its intensity and phase expected for realistic MSM waveguides.

3.1. Simplified propagation equation

Equation (7) holds for broad range of experimental conditions and can be used to simulate propagation of the TM SPPs over arbitrary distances along an MSM waveguide. In practice, however, it is inefficient to use waveguides that are vastly longer than the propagation length of SPPs. Since this length is typically less than 10 μm [see Fig. 2(b)], dispersive effects of second and higher orders have little impact on the SPP [27]. To illustrate this point, we plot in Fig. 3(a) the real part of group-velocity dispersion parameter β₂, which determines the dispersion length L_D 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 L_D and L_SPP holds approximately for shorter waveguides as well.

Fig. 3 (a) Group velocity v_g = (Reβ₁)⁻¹ and dispersion parameter Re(β₂) as a function of the silicon-layer thickness in an MSM plasmonic waveguide; dashed curve shows the group velocity in an SOI waveguide. (b) Ratios R₁ = |β_eff|/(ζ_r𝒟) and R₂ = γ_eff/(|ζ_i|𝒟) as functions of waveguide thickness d for τ_eff = 5 ns, 50 ps, and 5 ps. Other parameters are the same as in Fig. 2.

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By neglecting the dispersion terms in Eq. (7), we can write it in the following compact integro-differential form [38]:

\frac{1}{A} \frac{\partial A}{\partial z} = - (\frac{β_{eff}}{2} - i γ_{eff}) {| A |}^{2} - (\frac{ζ_{r}}{2} + i ζ_{i}) \int_{0}^{\infty} {| A (z, τ - τ_{eff} q) |}^{4} exp (- q) d q,

where the four parameters defined as

β_{eff} = 2 γ \frac{r - ϖ}{1 + ϖ^{2}}, γ_{eff} = γ \frac{1 + ϖ r}{1 + ϖ^{2}}, ζ_{r} = \frac{ℬ {\hat{β}}_{TPA} τ_{eff}}{2 \bar{h} ω 𝒟 d} \frac{σ_{α} + 2 k σ_{n} ϖ}{1 + ϖ^{2}}, ζ_{i} = \frac{2 k σ_{n} - σ_{α} ϖ}{σ_{α} + 2 k σ_{n} ϖ} \frac{ζ_{r}}{2},

are responsible for the TPA, Kerr effect, FCA, and FCD, respectively. The retarded time, τ = t – β₁z in Eq. (8) is not purely real due to the small imaginary part of β₁; this part is, however, insignificant for even 100-fs pulses and can be safely neglected. Interestingly, β_eff and ζ_i 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.

Similar to the situation in SOI waveguides, the parameters ζ_r and ζ_i are proportional to the effective free-carrier lifetime. By varying 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

τ_{eff}^{- 1} = τ_{c}^{- 1} + 2 μ V / d^{2},

where μ ≈ 1000 cm²/(V · s) [39] is the carrier mobility and τ_c 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 τ_eff from ns-range to 5 ps, while staying well below the breakdown field of ≈ 3 × 10⁵ V/cm for silicon.

3.2. Propagation of SPPs in the quasi-CW regime

Consider an optical pulse of width T₀ ≫ τ_eff (quasi-CW pulse), so that we may set A(z, τ – τ_effq) ≈ A(z, τ) in Eq. (8). The resulting equation has a well known analytical solution [41] of the form $A (z, τ) = \sqrt{I (z, τ)} exp [i φ (z, τ)]$ . In order to separately estimate the efficiency of different nonlinear effects in MSM plasmonic waveguides, we use explicit solutions of this equation in two specific cases; (i) when TPA and the Kerr effect dominate over FCA and FCD, the solution is given by

I_{K} (z, τ) = \frac{I_{0} (τ)}{1 + β_{eff} I_{0} (τ) 𝒟 z}, φ_{K} (z, τ) = \frac{γ_{eff}}{β_{eff}} ln | 1 + β_{eff} I_{0} (τ) 𝒟 z |;

(ii) in the opposite limit, where FCA and FCD dominate over TPA and the Kerr effect, the solution has the form

I_{FC} (z, τ) = \frac{I_{0} (τ)}{\sqrt{1 + 2 ζ_{r} I_{0}^{2}} (τ) 𝒟^{2} z}, φ_{FC} (z, τ) = - \frac{ζ_{i}}{2 ζ_{r}} ln | 1 + 2 ζ_{r} I_{0}^{2} (τ) 𝒟^{2} z | .

Here, I₀(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

max [I_{K} (z, τ)] ≪ min (\frac{| β_{eff} |}{ζ_{r} 𝒟}, \frac{γ_{eff}}{| ζ_{i} | 𝒟}),

while the solution (10) is valid as long as

I_{FC} (z, τ) ≫ max (\frac{| β_{eff} |}{ζ_{r} 𝒟}, \frac{γ_{eff}}{| ζ_{i} | 𝒟}) .

The two ratios involved in the above inequalities are plotted in Fig. 3(b) for three values of the dc voltage. It is seen that, if 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/cm² 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.

Of primary importance for practical applications is the extent of changes induced by different nonlinear effects in the SPP mode. As we mentioned earlier, the natural characteristic length that should be used to estimate these changes is the propagation length of SPPs. Figure 4 shows the intensity and phase of the quasi-CW SPP, once it covers a distance of 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 μm 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 [41–45], wavelength conversion [46], optical modulation [47, 48], and pulse compression [49].

Fig. 4 (a) Intensity and (b) nonlinear phase shift as functions of d for a quasi-CW SPP excited inside a plasmonic waveguide (Ag/Si/Ag) when free-carrier effects dominate over TPA and the Kerr effect; dashed curve shows propagation length of SPPs. Simulation parameters are the same as in Fig. 2.

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Similar estimations for the Kerr effect and TPA performed with Eq. (9) show that φ_K(L_SPP)/π ≲ 0.005, and the relative decrease in intensity is below 0.5%, irrespective of the specific values of I₀ < 10 GW/cm² 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 ∝ Γ², whereas ζ_r and ζ_i grow much faster—as a product ℬΓ² [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 [21–23, 27]. However, in this case, the advantage of strong localization of optical power, conferred by MSM waveguides, will be completely offset by their increased length.

As a closing remark, it is worth noting that the derived equations are quite applicable to MSM waveguides with ultrathin silicon layers (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

We have examined the possibility of building photonic devices by employing the nonlinear optical phenomena in silicon-based plasmonic waveguides. Although, for simplicity and concreteness, we analyzed a slot silver–silicon–silver plasmonic waveguide at λ = 1.55 μm, 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

Equation (6) is a special case of a more general relation

i (β + β *) {| {\tilde{E}}_{T} |}^{2} + {\tilde{E}}_{T} \nabla_{T} {\tilde{E}}_{z}^{*} - {\tilde{E}}_{T}^{*} \nabla_{T} {\tilde{E}}_{z} = i ω μ_{0} (\tilde{E} \times \tilde{H} * + \tilde{E} \times \tilde{H}) e_{z},

which holds for any solution of the Maxwell’s equations of the form

\begin{matrix} \tilde{E} (r) = E_{0} (x, y) exp (i β z), & \tilde{H} (r) = H_{0} (x, y) exp (i β z) \end{matrix} .

To prove this relation, we rewrite its right side using the law of induction, iωμ₀H̃ = ∇ × Ẽ,

i ω μ_{0} (\tilde{E} \times {\tilde{H}}^{*} + {\tilde{E}}^{*} \times \tilde{H}) e_{z} = e_{z} [{\tilde{E}}^{*} \times [\nabla \times \tilde{E}]] - c . c .

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

\begin{array}{l} e_{z} [{\tilde{E}}^{*} \times [\nabla \times \tilde{E}]] = e_{z} [\nabla ({\tilde{E}}^{*} \tilde{E}) - ({\tilde{E}}^{*} \nabla) \tilde{E}] = {\tilde{E}}^{*} \frac{\partial \tilde{E}}{\partial z} - ({\tilde{E}}^{*} \nabla) E_{z} \\ = i β {| \tilde{E} |}^{2} - ({\tilde{E}}_{T}^{*} \nabla_{T}) E_{z} - i β {\tilde{E}}_{z}^{2} = i β {| {\tilde{E}}_{T} |}^{2} - ({\tilde{E}}_{T}^{*} \nabla_{T}) E_{z} . \end{array}

Substituting this result in Eq. (12), we arrive at Eq. (11).

Appendix B

Denoting the real and imaginary parts of k_j as k′_j and k″_j for (j = 1, 2) and using Eq. (4), we obtain

\begin{matrix} ℐ_{1} \equiv \int_{- d / 2}^{+ d / 2} {| E_{0} |}^{2} d x = Π_{+}^{(1)} - {| \frac{k_{1}}{β} |}^{2} Π_{-}^{(1)}, Π_{\pm}^{(ν)} = \frac{1}{2} [\frac{sin (ν {k'}_{1} d)}{{k'}_{1}} \pm \frac{sinh (ν {k ″}_{1} d)}{{k ″}_{1}}], \\ \int_{- \infty}^{+ \infty} | E_{0} |^{2} d x = ℐ_{1} + \frac{1}{{k ″}_{2}} {{| \frac{ɛ_{1}}{ɛ_{2}} cos (\frac{k_{1} d}{2}) |}^{2} + {| \frac{k_{1}}{β} sin (\frac{k_{1} d}{2}) |}^{2}}, \\ \int_{- \infty}^{+ \infty} {| E_{x} |}^{2} d x = Π_{+}^{(1)} + \frac{1}{{k ″}_{2}} {| \frac{ɛ_{1}}{ɛ_{2}} cos (\frac{k_{1} d}{2}) |}^{2}, \end{matrix}

\begin{matrix} Q = Λ_{+} + {| \frac{k_{1}}{β} |}^{4} Λ_{-} - \frac{1}{4} {| \frac{k_{1}}{β} |}^{2} Π_{-}^{(2)} + \frac{1}{2 {k ″}_{2}} {{| \frac{ɛ_{1}}{ɛ_{2}} cos (\frac{k_{1} d}{2}) |}^{2} + {| \frac{k_{1}}{β} sin (\frac{k_{1} d}{2}) |}^{2}}^{2}, \\ Λ_{\pm} = \frac{d}{4} + \frac{Π_{+}^{(2)}}{8} \pm \frac{1}{2} Re [\frac{sin (k_{1} d)}{k_{1}}], \end{matrix}

and

𝒩 = ɛ_{0} ω {Re (\frac{ɛ_{1}}{β}) Π_{+}^{(1)} + Re (\frac{ɛ_{2}}{β}) \frac{1}{{k ″}_{2}} {| \frac{ɛ_{1}}{ɛ_{2}} cos (\frac{k_{1} d}{2}) |}^{2}} .

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

ℰ_{x x x x} = 1, ℰ_{z z z z} = \frac{1 + ρ}{2}, ℰ_{x x z z} = ℰ_{z z x x} = ℰ_{x z z x} = ℰ_{z x x z} = ℰ_{x z x z} = ℰ_{z x z x} = \frac{ρ}{3},

where ρ ≈ 1.27 in the 1.55-μm region. With these components, we find that

\begin{array}{l} η Q = Λ_{+} + \frac{1 + ρ}{2} {| \frac{k_{1}}{β} |}^{4} Λ_{-} - \frac{ρ}{6} {| \frac{k_{1}}{β} |}^{2} Π_{-}^{(2)} \\ - \frac{2}{3} ρ {Re (\frac{k_{1}^{2}}{β^{2}}) [\frac{d}{4} - \frac{Π_{+}^{(2)}}{8}] + \frac{1}{2} Im (\frac{k_{1}^{2}}{β^{2}}) Im [\frac{sin (k_{1} d)}{k_{1}}]} \end{array}

The same expressions are applicable to SOI waveguides, after the replacement sinh(νk″₁d)/k″₁ → νd.

Acknowledgments

The work of I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal was sponsored by the Australian Research Council (ARC) through its Discovery Grant scheme under grants DP0877232 and DP110100713. The work of G. P. Agrawal was also supported by the NSF Award ECCS-0801772.

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

Abstract

1. Introduction

2. Nonlinear propagation equation

3. Efficiency of nonlinear effects in MSM plasmonic waveguides

3.1. Simplified propagation equation

3.2. Propagation of SPPs in the quasi-CW regime

4. Conclusions

Appendix A

Appendix B

Acknowledgments

References and links

Cited By

Figures (4)

Equations (31)

Optics Express