1. Introduction

Metal nanosystems play a central role in the emerging field of nanooptics and plasmonics and have been the subject of extensive research, both fundamental and with view to applications. In particular, the need for compact and high-performance optical devices has driven the interest in plasmonic waveguides which can be used to carry optical signals to different parts of subwavelength components of plasmonic optoelectronic circuids (see e.g. [1]). In metal nanowaveguides the nonlinear susceptibility is significantly enhanced (see e.g. [2–4]) and are related with peculiarities of intraband and interband nonlinear electronic processes in metal under the influence of electromagnetic fields [5]. In the last decades, different types of waveguides has been theoretically investigted, such as nonlinear plasmonic planar waveguides [6–9], metal nanowires [10, 11], plasmonic slot waveguides [12] and periodic plasmonic waveguides [13]. In such sub-wavelength waveguides TM modes exhibits a significant longitudinal component of the electric field, which plays an important role in nonlinear propagation (see e.g. [14]).

Among the different types of plasmonic waveguides metal nanowires have the most simplest waveguide structure and are of great interest due to their potential in several fields. Subwavelength waveguiding of plasmons in such structures has been observed in a number of experiments [15–18]. The nonlinear optical behaviour of crystalline metal nanowires has been experimetally studied in several papers (see e.g. [19–21]). The experimetal results in these papers indicate that depending on parameters crystalline metal nanowires exhibit both the property of saturable absorption and of optical limiting. Similar nonlinear effects has also been found in metal nanoparticle composites (see e.g. [22,23] and references therein). In a theoretical study of this phenomenon in [24] it was shown that the physical origin of the saturation effect is related to the intrinsic Kerr nonlinearity of the metal nanoparticles and the linear metal loss leading to a shift of the plasmon resonance and therefore to a reduction of loss with increasing intensity. The saturable loss in metal nanocomposites can be used for modelocking and Q-switching of lasers as proposed and theoretically studied in [25] and experimentally realized by several groups (see e.g. [26] and refereces therein). Although the experimental observations for metal nanowires [19–21] were done in samples of crystaline structures, one can expect that analogous phenomena occure also in single metal nanowires. Note that in previous theoretical studies of the nolinear propagation in single metal nanowires [10, 11] only positive values of the real and imaginary part of the nonlinear susceptibility were predicted.

In this paper we reexamine the problem of nonlinear propagation in plasmonic waveguides and use an alternative method for the derivation of the nonlinear propagation equation for the slowly varying field amplitude of plasmon-polaritons in metal nanowaveguides. In our approach we use the Lorentz reciprocity theorem (see e.g. [13, 27, 28] and references therein) for the derivation of the nonlinear coefficient and study its dependence on the parameters of the incident field and of the waveguide for the case of metal-air interfaces and metal nanowires. The main result of our work is the prediction that the sign of the real and imaginary part of the effective nonlinear coefficient of gold nanowires at visible wavelengths can be shifted from positive to negative values by changing wire radii and wavelengths. This means in dependence on the parameters our predictions describe both the property of saturable absorption and optical limiting as well a positive or negative nonlinear phaseshifts. A physical explanation of this phenomenon is given.

2. Derivation of the nonlinear coefficient using the reciprocity theorem

We use the mode expansion for the electric and the magnetic fields $\overrightarrow{F}=(\overrightarrow{E};\overrightarrow{H})$ in the plasmonic waveguide which can be expressed as $\overrightarrow{\mathrm{F}}(\overrightarrow{r},t)=(1/2)[\overrightarrow{F}(\overrightarrow{r})\exp (-i\omega t)+c.c]$, where $\overrightarrow{F}(\overrightarrow{r})$ is a time-independent amplitude for the mode m. In the following we restrict ourself to the fundamental mode m = 0 which can be presented by

Here z and ẑ are the coordinate and the unit vector in the direction of propagation, ${\overrightarrow{r}}_{\perp }$ is the position vector in the transverse plane, s₀ is defined as $s_0=(1/2){\displaystyle \int \mathrm{Re}\left({\overrightarrow{e}}_0\times {\overrightarrow{h}}_0^{\ast }\right)}\cdot \widehat{z}d\sigma$, where the integral is performed in the transverse plane, β = κ + iα/2 is the SPP propagation constant, Ψ is normalized so that |Ψ|² is equal to the power flow along the z direction $P(z)=(1/2){\displaystyle \int \mathrm{Re}\left(\overrightarrow{E}\times {\overrightarrow{H}}^{\ast }\right)\cdot \widehat{z}d\sigma \approx |\mathrm{\Psi }{|}^2}$, and ${\overrightarrow{e}}_0(r_{\perp })$ and ${\overrightarrow{h}}_0(r_{\perp })$ describe spatial transverse distribution of the fundamental mode.

The Lorentz reciprocity theorem states the following relation for two electromagnetic fields $\left({\overrightarrow{E}}_1,{\overrightarrow{H}}_1\right)$ and $\left({\overrightarrow{E}}_2,{\overrightarrow{H}}_2\right)$ which are solutions of the Maxwell equations corresponding to the relative permittivity ${\epsilon }_1(\overrightarrow{r})$ and ${\epsilon }_2(\overrightarrow{r})$, respectively, [13, 27, 28]:

We substitute for $\left({\overrightarrow{E}}_1,{\overrightarrow{H}}_1\right)$ and $\left({\overrightarrow{E}}_2,{\overrightarrow{H}}_2\right)$ the unperturbed backward propagating field $\left({\overrightarrow{e}}^{-},{\overrightarrow{h}}^{-}\right)$ and the perturbed forward propagating field $\left(\overrightarrow{E},\overrightarrow{H}\right)$, correspondingly. In the first order of perturbation, Eq. (3) leads to the following equation describing the amplitude Ψ(z)

Here we use the relations $e_{0x}^{-}=e_{0x}$, $e_{0y}^{-}=e_{0y}$, $e_{0z}^{-}=-e_{0z}$, $h_{0x}^{-}=-h_{0x}$, $h_{0y}^{-}=-h_{0y}$, $h_{0z}^{-}=h_{0z}$.

3. Nonlinear metal-air interface

To benchmark the derived formula (5), we apply it to the simple analytically solvable case of a single gold-air interface, for which (ε = ε_gold ·θ (−x) + ε_air · θ (x)) for δε = δε₁ · θ (−x) + δε₂ · θ (x), where θ (x) is the Heaviside step function. For the calculations we used the·measured wavelength-dependent permittivity of gold [29]. Using the formula of effective refractive index $n_{\text{eff}}=\sqrt{{\epsilon }_1{\epsilon }_2/\left({\epsilon }_1+{\epsilon }_2\right)}$ for a single interface and substracting from the perturbed refractive index the unperturbed one, we can obtain the following analytical formula of δn_eff:

On the other hand the expression for δn_eff derived in [6, 14] is

 

Fig. 1 Response of effective refractive index in a single gold-air interface to a perturbation of the permittivity in gold δε₁ = 0.01, δε₂ = 0. The solid curve, blue circles and red crosses represent the results by Eq. (5), the analytical formula Eq. (6), and of Eq. (7), respectively.

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In Fig. 1(a) we have studied the influence of a perturbation of the dielectric function on the effective refraction index without specifying the physical nature of this perturbation. To investigate an intrinsic nonlinear response of SPPs in plasmonic waveguides, we now consider a Kerr-type nonlinear contribution to the permittivity ${\epsilon }_{\text{nl}}=(3/4){\chi }^{(3)}(\omega ;\omega ,-\omega ,\omega ){\left|\overrightarrow{E}\right|}^2$ [30, 31]. Equation (5) then leads to the nonlinear propagation equation

We apply the Eqs. (8) and (9) to a single gold-air interface. Since the intrinsic nonlinearity of gold is much larger than that of the dielectric, we neglect the nonlinear susceptibility of the dielectric and use the wavelength-depending values of the complex nonlinear susceptibility of gold ${\chi }_{\text{gold}}^{(3)}$ from [11]. In Fig. 2(a) the nolinear phaseshift is presented at the wavelength of 535 nm. We can see that a negative real part of γ leads to a negative nonlinear phase shift [see Fig. 2(a)] in spite of a positive real part of ${\chi }_{\text{gold}}^{(3)}$. In Fig. 2(b) the nonlinear contribution to the effctive loss of the interface is presented. As can be seen the negative imaginary part of γ for the wavelength of 502 nm leads to a negative nonlinear nonlinear absorption cefficient or to saturable loss in spite of a positive imaginary part of ${\chi }_{\text{gold}}^{(3)}$. Our results are in good agreement with results of a full-dimensional simulation which were obtained by numerically solving Maxwell equations in the frequency domain [32] (blue circles in Fig. 2).

 

Fig. 2 Power-dependence of the nonlinear phase shift for the wavelength of 535 nm and a propagation distance of 1000 nm (a) and nonlinear absorption for the wavelength of 502 nm and a propagation distance of 500 nm (b) for a single gold-air interface. The black line is the results obtained by Eq. (5), the red crosses by Eq. (7) and the blue circles correspond to a full-dimensional simulation.

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Let us compare the prediction provided by (9) with a result of [7] using a Green-function formalism. We obtained γ = (1.16 + i1.00) × 10⁻⁷ W⁻¹m⁻¹ for the single gold-air interface at λ = 796 nm using the measured value of χ^(3)gold = (4.67 + i3.03) × 10⁻¹⁹ m²/V² [33], which is in a good agreement with the result γ = (1.03 + i0.98) × 10⁻⁷ W⁻¹m⁻¹ in [7].

4. Nonlinear metal nanowires

Let us now study the effective nonlinear response of gold nanowires surrounded by silica glass. We consider a silica glass host, since this material was used in previous investigations of nanowires (e.g. [11]). The standard Bessel-function formalism for linear modes in cylindrically symmetric structures is used, with wavenumber found numerically. Here we focus on the TM mode (m = 0) because the fundamental TM mode shows the strongest nonlinear response and a well-confined spatial distribution [11]. We used the Sellmeier expansion for the permittivity of silica glass [31], which is included in the abovementioned numerical calculation. The non-linearity of silica glass is disregarded because it is several orders of magnitude smaller than that of gold. As shown in Fig. 3 for two different wire radii, in the visible region a large nonlinear coefficient |γ| > 105 W⁻¹m⁻¹ is attained. This is attributed to the large nonlinear susceptibility of gold in the visible region and comparably strong confinement and enhancement of the field. Note that both Reγ and Imγ changes the sign in the visible region, while in the infrared region it has only the positive sign. This means depending on frequency gold nanowires can be used for positive or negative nonlinear self-phase modulation.

 

Fig. 3 Effective nonlinear coefficient of a gold nanowires with a wire radii 200 nm (a) and 50 nm (b) surrounded by silica glass for the fundamental TM mode (m=0). The red solid and blue dashed line shows the real and imaginary part of the effective nonlinear coefficient, respectively.

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In Fig. 4 the dependence of Re γ on the wire radii is plotted. As can be seen Re γ has a negative sign in the range of wire radii beetween about 30 nm and 50 nm. This gives the possibility to design the required properties of nonlinear device by simply choosing the radius of the plasmonic waveguides, and possibly to applications such as for switching or quasi-phase-matching. Fig. 5 shows that Im γ also change the sign at two values of wire radii at a fixed wavelength. This implies that gold nanowires in the visible region would exhibit saturated absorption or reverse-saturated absorption for different wire radii. Note that the dependence of the sign of the effective nonlinear coefficients on the waveguide radius is a rather surprising phenomenon because the linear and nonlinear properties of the component materials do not depend on the wire radius.

 

Fig. 4 Real part of effective nonlinear coefficient of gold nanowires surrounded by silica glass for the fundamental TM mode (m=0) at the wavelength of 720 nm versus the wire radius. The third-order nonlinear susceptibility of gold ${\chi }_{\text{gold}}^{(3)}=(2+i15)\times {10}^{-17}{\mathrm{m}}^2/{\mathrm{V}}^2$ from [11] is used.

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Fig. 5 Imaginary part of effective nonlinear coefficient of gold nanowires surrounded by silica glass for the fundamental TM mode (m=0) at the wavelength of 630 nm versus the wire radius. The third-order nonlinear susceptibility of gold ${\chi }_{\text{gold}}^{(3)}=(20-i5)\times {10}^{-17}{\mathrm{m}}^2/{\mathrm{V}}^2$ from [11] is used.

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The high intrinsic nonlinearity of SPP is naturally accompanied with a high loss of the SPP. In order to define the performance of a lossy nonlinear waveguide, we introduce a figure of merit F = |γ|L_spp, where L_spp = 1/α is the SPP propagation length. The input power needed for a certain nonlinear effect is inversely proportional to this figure of merit. In Fig. 6 one can see that F exhibits a maximum which varies in position from 610 nm for 30-nm nanowires to 750 nm for 200-nm nanowires and can be positioned at either side of the resonance. The lower limit of all curves is determined by the mode cutoff. The maximum of the nonlinearity as well as its sign change is attributed to an additional phase of $({\overrightarrow{E}}^2-2E_z^2)/|{\overrightarrow{E}}^2|$, which is related to the gold loss. The accumulated additional phase is influenced by two factors, local additional phase distribution and field enhancement distribution, as one can see in Eq. (9). With increasing the radius, the local additional phase in the central part increases [Fig. 7(a)], but the local field magnitude in the central part decreases [Fig. 7(b)]. Thus the accumulated additional phase is maximum for a certain radius.

 

Fig. 6 SPP propagation length L_spp (a) and figure of merit F = |γ| L_spp (b) of gold nanowires with wire radii 30nm (black solid), 50 nm (red dashed) and 200 nm (blue crosses) surrounded by silica glass for the fundamental TM mode (m=0).

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Fig. 7 Additional phase ϕ and field enhancement |E|⁴/|E₀4 due to metal loss versus r in the gold nanowire with different radii R = 30, 50, 80 nm. The wavelength is 630 nm and ${\chi }_{\text{gold}}^{(3)}=(20-i5)\times {10}^{-17}{\mathrm{m}}^2/{\mathrm{V}}^2$ from [11] is used.

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Let us discuss the physical reason for the above decribed sign change of both the real and imaginary part of the nonlinear susceptibility of the metal nanowire. We remind a similar phenomenon of saturated absorption in metal nanocomposites where the sign change is the result of the fact that near the plasmon resonance the local field enhancement factor x = E_in/E₀ has an imaginary component that arises by the linear loss of the metal nanoparticles and leads to a phase difference between the field inside the nanoparticles E_in and the external field E₀ [24, 34]. A simple explanation of the sign change of the nonlinearity and of saturable loss can be provided by the dependence of the field enhancement factor for spherical nanoparticles on wavelengths, radius and intensity. This means that the sign change of the nonlinear susceptibility is a result of both the high metal nonlinearity and the high metal loss. For the nonlinear metal nanowires a qualitatively analogous effect arises. The sensitive dependence of the sign of the effective nonlinear susceptipility of the nanowire is related with the dependence of the plasmon resonance on the waveguide radius.

5. Conclusion

In summary, by using the Lorentz reciprocity theorem we reconsidered the problem to model the nonlinear response of SPPs in plasmonic waveguides and derived a formula for the effective complex SPP nonlinear susceptibility in these waveguides. For the case of gold nanowires surrounded by silica glass we calculated the nonlinear coeficient in dependence on the wavelength and the wire radius. We predicted a large nonlinear coefficient (|γ| > 10⁵ W⁻¹m⁻¹) and have shown that the sign of the real and imaginary part of the nonlinear susceptibility depend on the wavelength and can also be changed for a fixed wavelength by changing the wire radius. This opens the possibility to design the nonlinear properties of metal nanowires by the choice of the nanowire radius which can realize both positive and negative phase shifts by self-phase modulation as well as saturable loss or optical limiting which could be used in future ultracompact photonic devices. We discussed the physical origin of these effects.